web.uettaxila.edu.pkweb.uettaxila.edu.pk/MED/PG_downloads/PhD_thesis...ii OPTIMAL PERFORMANCE...

PhD Thesis

OPTIMAL PERFORMANCE ANALYSIS OF A SOLAR THERMAL ENERGY STORAGE PLANT BASED ON

LIQUID AMMONIA

Submitted by

Engr. Sadaf Siddiq (08F-UET/PhD-ME-47)

Supervised by

Prof. Dr. Shahab Khushnood

Department of Mechanical Engineering Faculty of Mechanical and Aeronautical Engineering

University of Engineering and Technology Taxila, Pakistan

July 2013

ii

OPTIMAL PERFORMANCE ANALYSIS OF A SOLAR THERMAL ENERGY STORAGE PLANT BASED ON

LIQUID AMMONIA

by


A proposal submitted for research leading to the degree of Doctor of Philosophy

in MECHANICAL ENGINEERING

Approved by

External Examiners

________________________________

(Engr. Dr. M. Javed Hyder) Dean of Engineering,

Pakistan Institute of Engineering & Applied Sciences Nilore, Islamabad.

________________________________ (Engr. Dr. Ejaz M. Shahid)

Associate Professor, Department of Mechanical Engineering,

University of Engineering & Technology, Lahore.

Internal Examiner (Research Supervisor)

________________________________ (Engr. Dr. Shahab Khushnood)

Professor, Department of Mechanical Engineering,

University of Engineering & Technology, Taxila.

Department of Mechanical Engineering Faculty of Mechanical and Aeronautical Engineering

University of Engineering & Technology Taxila, Pakistan.

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DECLARATION

I declare that all material in this thesis is my own work and that which is not, has been

identified and appropriately referenced. No material in this work has been submitted or

approved for the award of a degree by this or any other university.

Signature: _____________________________

Author’s Name: ________________________

It is certified that the work in this thesis is carried out and completed under my supervision. Supervisor: Prof. Dr. Shahab Khushnood Department of Mechanical Engineering Faculty of Mechanical and Aeronautical Engineering University of Engineering and Technology Taxila, Pakistan.

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ABSTRACT

This work focuses on extending the use of a solar thermal energy plant from an intermittent

energy source to a base load power plant by incorporating an efficient thermal storage feature.

A reference 10 MWe solar thermal plant design is considered with liquid ammonia as a

working fluid for energy production, in a Rankine Cycle, as well as a thermal storage

medium. During periods of no solar insolence, the recovery system, based on an industrial

ammonia synthesis system, is used to drive the power conversion unit and enable continuous

operation.

A thermofluid model, based on the continuity, momentum and energy conservation equations,

is used to carry out a numerical simulation of the plant, to determine the process variables

and subsequently carry out an integrated plant energy recovery analysis. The objective of this

work is to maximize the efficiency of the plant by a detailed consideration of the most critical

process in the plant: the energy recovery unit. This is carried out by (i) estimating the

sensitivity of non-uniform catalyst concentration in a synthesis reactor, and (ii) obtaining an

optimal configuration from a variational Lagrangian cost functional and applying

Pontryagin’s Maximum Principle. The optimal configuration is used to recommend a re-

design of the synthesis reactor and to quantify the energy recovery benefits emanating from

such a recommendation. Industrial optimal configurations are achieved by carrying out the

analysis with the simulation code, Aspen Plus™, to design a heat removal system

surrounding the catalyst beds, and incorporating the effect of standard industrial processes

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such as purge gas removal, quench gas recycling, and recycle ratio to achieve the optimal

temperature profile obtained for the synthesis reactor considered in this work.

This work quantifies the maximum energy recovery in a base-load solar thermal plant

utilizing the existing environment of chemical process industry. It is concluded that a one-

dimensional model, with mass and energy conservation equations using the Temkin-Pyzhev

activity and pressure-based kinetics rate expressions, predicted an optimal ammonia

conversion of 0.2137 with a thermal energy availability of 20 MWth. A comprehensive

process simulation using Aspen Plus™ predicts an optimal ammonia conversion of 0.2762

mole fraction at exit, with two inter-bed heat exchangers having optimal temperature drops of

205K and 95K respectively, and yielding a thermal availability of 45.6 MWth. The thermal

energy availability of a base-load solar thermal plant can be increased by 15% in the

ammonia conversion and over 25% in thermal energy availability for energy recovery.

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To my family . . .

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ACKNOWLEDGEMENTS

During the development of my PhD studies at University of Engineering & Technology

Taxila, several persons and institutions collaborated directly and indirectly with my research.

Without their support it would be impossible for me to finish my work. That is why I wish to

dedicate this section to recognize their support.

I want to start expressing a sincere acknowledgement to my advisor, Prof. Dr. Shahab

Khushnood because he gave me the opportunity to research under his kind guidance and

supervision. I received motivation; encouragement and support from him during all my

studies. I owe Special thanks to Dr. Zafar Ullah Koreshi for the his support, guidance, and

transmitted knowledge for the completion of my work. With him, I have learned writing

papers for conferences and journals and sharing my ideas with the scientific community. I

also want to thank the example, motivation, inspiration and support I received from Dr.

Tasneem M. Shah, Dr. Arshad H. Qureshi and Dr. M. Bilal Khan.

The Grant from University of Engineering & Technology Taxila provided the funding and

resources for the development of this research and validation of my work. At last, but the

most important I would like to thank my family, for their unconditional support, inspiration,

love and prayers.

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NOMENCLATURE

A Cross-sectional area (m2)

ANU Austrailian National University

pC Specific heat at constant pressure (kJ kmol-1 K-1)

rC Compression Ratio

CSP Concentrating Solar Power

E Activation energy (kJ kmol-1)

sfF → Force (external, fluid to solid)

0NF Initial nitrogen molar flow rate (kmol h-1)

Gt Giga-ton (109 ton)

Η Hamiltonian

Η Enthalpy per unit mass

J Functional

*, ii JJ Molar Fluxes

K Kinetic Energy

aK Equilibrium constant

KBR Kellogg Brown and Root™

L Length of synthesis reactor (m)

MTD Metric tonnes per day

Mtoe Million ton of oil equivalent

MWe Megawatt electric

MWth Megawatt thermal

OEM One Equation Model

P Pressure (MPa)

Ρ Linear Momentum

PMP Pontryagin’s Maximum Principle

PV PhotoVoltaic

Q Heat

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R Universal gas constant 8.3144 kJ kmol-1 K-1

AR Reaction rate (kmol NH3 h-1 m-3 catalyst)

RK-4 4th order Runge-Kutta

S Surface Area (m2)

T Temperature (K)

TEM Two Equation Model

TSP Thermal Storage Plant

TWh Terawatt-hours (1012 Watt-hrs)

U Internal Energy (kJ)

U Internal Energy per unit mass

W Watts

ia Activity for speciei

c Total Molar Concentration

ic Molar Concentration of Specie i

pd Particle Diameter

g Gravitational acceleration (9.81 ms-2)

*, ii jj Mass Fluxes

kWe kilowatt Electric

kWchem. kilowatt chemical

kWth kilowatt thermal

m Mass (kg)

0in Initial mole flow rate of speciei (kmol h-1)

ppm Parts per million

r Molar Production

t Time

u Control variable

v Velocity (m s-1)

w Work

x Distance along catalyst bed (m)

x

x′ Normalized Distance along catalyst bed (m)

iy Mole fraction for specie i

o

iy Initial Mole fraction for specie i

Nzz, Fractional conversion of Nitrogen

Greek

rH∆ Heat of reaction (kJ kmol-1 NH3)

ε Extent of reaction

Φ Potential Energy

iφ Fugacity coefficient for speciei

ω Mass Flow Rate (kghr-1)

τ Shear Stress

η Catalyst effectiveness factor

ψ The void space of the bed

λ Lagrange multiplier

)(xξ Catalyst spatial factor

)(x′θ Optimal Temperature

ρ Density (kg m-3)

σr Vector containing state variables

Subscripts

eqm Equilibrium

i Species in a multi component system, Ni ,....4,3,2,1=

opt Optimal

s Isentropic

tot Total amount of entity in a macroscopic system

0 Evaluated at a surface

2,1 Evaluated at cross sections 1 and 2

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Table of Contents

ABSTRACT ....................................................................................................................................................... IV

ACKNOWLEDGEMENTS ............................................................................................................................ VII

NOMENCLATURE ...................................................................................................................................... VIII

TABLE OF CONTENTS .................................................................................................................................. XI

TABLE LIST ................................................................................................................................................... XIII

FIGURE LIST ................................................................................................................................................ XIV

1 INTRODUCTION ..................................................................................................................................... 1

1.1 SOLAR ENERGY: POTENTIAL AS A RENEWABLE ENERGY SOURCE ............................................................ 2 1.2 SOLAR POWER PLANTS IN OPERATION ...................................................................................................... 5

1.2.1 PV Plants ........................................................................................................................................ 5 1.2.2 Solar Thermal Plants ...................................................................................................................... 7

1.3 THERMAL ENERGY STORAGE REQUIREMENT ............................................................................................ 9 1.4 THERMAL STORAGE MATERIALS .............................................................................................................. 9 1.5 USE OF L IQUID AMMONIA AS STORAGE MATERIAL ................................................................................ 11

1.5.1 Poperties of Liquid Ammonia ....................................................................................................... 12 1.5.2 Dissociation and Synthesis of Ammonia ....................................................................................... 12 1.5.3 Commercial uses of Ammonia ...................................................................................................... 13 1.5.4 Industrial proprietary processes for Ammonia Production .......................................................... 14

1.5.4.1 Haldor Topsoe Ammonia Synthesis Process ........................................................................................... 15 1.5.4.2 Kellog Brown & Roots (KBR) Advanced Ammonia Process (KAPP).................................................... 15 1.5.4.3 Krupp Uhde GmbH Ammonia Process ................................................................................................... 16 1.5.4.4 ICI-Leading Concept Ammonia (LCA) Process...................................................................................... 17 1.5.4.5 The Linde Ammonia Concept (LAC) Ammonia (LCA) Process ............................................................ 17

1.6 THERMODYNAMIC CYCLES FOR SOLAR THERMAL POWER PLANTS ........................................................ 18 1.7 LITERATURE REVIEW .............................................................................................................................. 19 1.8 THESIS MOTIVATION ............................................................................................................................... 22 1.9 OBJECTIVES ............................................................................................................................................ 23 1.10 SUMMARY OF FOLLOWING CHAPTERS ............................................................................................... 24

2 DESCRIPTION OF THE THERMAL STORAGE PLANT .............................................................. 25

2.1 PLANT FEATURES .................................................................................................................................... 25 2.1.1 Process Design ............................................................................................................................. 25 2.1.2 Opertational Parameters .............................................................................................................. 27

2.2 OVERALL PLANT LAYOUT AND DESCRIPTION ......................................................................................... 28 2.2.1 Ammonia Dissociation .................................................................................................................. 29 2.2.2 Ammonia Synthesis ....................................................................................................................... 30 2.2.3 Syn Gas and Ammonia Storage .................................................................................................... 31 2.2.4 Heat Exchangers and Transport Piping ....................................................................................... 31 2.2.5 Compressors and Pumps .............................................................................................................. 31

2.3 THERMAL STORAGE PLANT PROCESS FLOW DIAGRAM ........................................................................... 32

3 MODELLING & SIMULATION OF THERMAL STORAGE PLANT .......................................... 34

3.1 MATHEMATICAL MODELLING .............................................................................................................. 34 3.1.1 Review of Mathematical Models of TSP ....................................................................................... 34 3.1.2 Mathematical Models for TSP ...................................................................................................... 41

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3.1.2.1 TEM Model using Activity based Temkin-Pyzhev Form (TP-A): .......................................................... 43 3.1.2.2 TEM Model using Partial Pressure based Temkin-Pyzhev Form (TP-B): ............................................... 45

3.2 MODELING UNIT OPERATIONS ................................................................................................................ 48 3.2.1 Dissociation Reactor .................................................................................................................... 48 3.2.2 Synthesis Reactor-Aspen Plus Model............................................................................................ 51 3.2.3 Synthesis Reactor- HYSYS Model ................................................................................................. 53 3.2.4 Flash Tank .................................................................................................................................... 55 3.2.5 Purge Gas & Recycle .................................................................................................................... 56 3.2.6 Heat Exchangers and Waste Heat Recovery................................................................................. 57

3.2.6.1 Counter Flow Heat Exchanger (CF-HX): ................................................................................................ 57 3.2.6.2 Thermal Heat Exchanger (SRIN-HX): .................................................................................................... 58 3.2.6.3 Thermal Heat Exchanger (SROUT-HX): ................................................................................................ 58

3.3 MODELING THE INTEGRATED PLANT ...................................................................................................... 59 3.4 TWO EQUATION MODEL (TEM) VALIDATION : ....................................................................................... 59

4 PLANT OPTIMIZATION...................................................................................................................... 62

4.1 REVIEW OF OPTIMIZATION TECHNIQUES ............................................................................................. 62 4.2 OPTIMAL ANALYSIS USING VARIATIONAL CALCULUS ........................................................................ 72 4.3 PARAMETRIC SENSITIVITY ANALYSIS .................................................................................................... 78

4.3.1 Effect of Temperature on Dissociation ......................................................................................... 78 4.3.2 Effect of Flow Rate on Dissociation ............................................................................................. 79 4.3.3 Effect of Pressure on Synthesis ..................................................................................................... 80 4.3.4 Effect of Temperature on Synthesis .............................................................................................. 80 4.3.5 Effect of Flash Temperature on Liquid Ammonia Separation ...................................................... 82 4.3.6 Effect of Purge Fraction on Ammonia Liquification .................................................................... 83 4.3.7 Effect of Recycle Stream on Synthesis .......................................................................................... 84

5 AN OPTIMAL STORAGE PLANT ...................................................................................................... 85

5.1 PROCESS MODIFICATIONS ..................................................................................................................... 85 5.1.1 Optimal Analysis Problem Formulation- Process Modifications ................................................. 85 5.1.2 OEM using Activity based Temkin-Pehzev form (OEM-TPA) ...................................................... 86 5.1.3 OEM using Partial Pressure based Temkin-Pehzev form (OEM-TPB) ........................................ 89 5.1.4 Process Modifications Validation: ............................................................................................... 94

5.2 DESIGN MODIFICATIONS ....................................................................................................................... 97 5.2.1 The Proposed Design .................................................................................................................... 99 5.2.2 Design Modifications Validation: ............................................................................................... 101

6 CONCLUSIONS AND FUTURE WORK ......................................................................................... 103

REFERENCES .................................................................................................................................................. 106

APPENDIX A. AMMONIA 3D PHASE DIAGRAMS ...................................................................... 121

APPENDIX B MATLAB™ PROGRAMS FOR AMMONIA SIMULATION ............................... 122

APPENDIX B1: MATLAB ™ PROGRAM FOR OUTPUT OF STEADY STATE SYNTHESIS REACTOR ....................... 123 APPENDIX B2: MATLAB ™ PROGRAM FOR FINDING EQUILIBRIUM CONCENTRATIONS ................................... 149 APPENDIX B3: MATLAB ™ PROGRAM FOR FINDING OUTPUT OF COUNTER-FLOW SYNTHESIS REACTOR ....... 153 APPENDIX B4: MATLAB ™ PROGRAM FOR FINDING OUTPUT OF STEADY STATE SYNTHESIS REACTOR WITH 3

CATALYST ZONES .......................................................................................................................................... 157

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Table List

Tables Page TABLE 1.1: World’s Largest (25MWe or above) PV Plants in Operation .............................. 6 TABLE 1.2: Solar Themal Plants in Operation ........................................................................ 7 TABLE 1.3: Haldor Topsoe Ammonia Converter Features ................................................... 15

TABLE 2.1: Overall Plant Design for a 10 MW(e) Baseload Plant ....................................... 26 TABLE 3.1: Equations of change of Multi-component Mixtures in terms of the Molecular

Fluxes .............................................................................................................................. 35 TABLE 3.2: Coefficients of the correction factor polynomial in terms of pressure .............. 38 TABLE 3.3: Input Data for Dissociation Reactor .................................................................. 49 TABLE 3.4: Input Data for Synthesis Reactor ....................................................................... 51 TABLE 3.5: Reaction Input for Temkin-Pyzhev Power-Law Expression in Aspen Plus™ .. 52 TABLE 3.6: Flash Tank Output .............................................................................................. 56 TABLE 3.7: Molar Flow Rates of Components in and out of Splitter ................................... 56 TABLE 3.8: Percentage errors in 1-D Models compared with HYSYS™ and Aspen Plus™61 TABLE 4.1: Optimal solution for the exit conditions ............................................................ 67

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Figure List

Figures Page

Figure 1.1 Volume Reduction with Phase Change Materials ................................................. 10 Figure 1.2 Materials for medium and high heat storage ......................................................... 10 Figure 1.3 Energy densities for different energy carriers ....................................................... 12 Figure 2.1: Thermal Storage Plant Schematic ........................................................................ 29 Figure 2.2: Array of 400 m2 Paraboloidal Solar Collectors [3] .............................................. 30 Figure 2.3: TSP Process Flow Diagram .................................................................................. 32 Figure 3.1 : Conversion of Nitrogen along a single-bed catalyst ........................................... 39 Figure 3.2 : Syngas temperature in converter ......................................................................... 39 Figure 3.3 : Molar flow rate in converter ................................................................................ 40 Figure 3.4 : Syngas compression requirement ........................................................................ 40 Figure 3.5 : 3-Bed Homogeneous Reactor with TP-A Kinetics ............................................. 45 Figure 3.6 : 3-Bed Homogeneous Reactor with TP-B Kinetics .............................................. 47

Figure 3.7 : PFR Dissociation Reactor in Aspen Plus™ ........................................................ 49 Figure 3.8 : Dissociation Reactor Exit Composition .............................................................. 50 Figure 3.9 : Dissociation Reactor Temperature Profile .......................................................... 50 Figure 3.10 : PFR Synthesis Reactor in Aspen Plus™ ........................................................... 51 Figure 3.11 : Synthesis Reactor Exit Composition (Aspen Plus™) ....................................... 52

Figure 3.12 : Synthesis Reactor Temperature Profile (Aspen Plus™) ................................... 53 Figure 3.13 : Plug Flow Reactor in HYSYS™ ....................................................................... 53 Figure 3.14 : Synthesis Reactor Temperature Profile (HYSYS™) ........................................ 54 Figure 3.15 : Synthesis Reactor Exit Composition (HYSYS™) ............................................ 55 Figure 3.16 : Flash Tank in Aspen Plus™ .............................................................................. 55 Figure 3.17 : Splitter in Aspen Plus™ .................................................................................... 56 Figure 3.18 : Mixer in Aspen Plus™ ...................................................................................... 57 Figure 3.19 : Counter Flow Heat Exchanger (CF-HX)........................................................... 57 Figure 3.20 : Thermal Heat Exchanger (SRIN-HX) ............................................................... 58 Figure 3.21 : Thermal Heat Exchanger (SROUT-HX) ........................................................... 58 Figure 3.22 : Integrated Plant .................................................................................................. 59 Figure 3.23 : Comparison of 1-D (TP-B) model, HYSYS™, and Aspen Plus™ results ....... 60 Figure 4.1 : Optimization Process ........................................................................................... 63 Figure 4.2 : Mathematical Methodology to solve governing equations ................................. 64 Figure 4.3 : Counter-Flow Ammonia Synthesis Reactor ........................................................ 66 Figure 4.4 : Temperature & Concentration Profiles along Converter Length ........................ 66

Figure 4.5 : GA Search Algorithm .......................................................................................... 67 Figure 4.6 : Four-Bed Synthesis Reactor ................................................................................ 68 Figure 4.7 : Effect of Quench gas on conversion efficiency ................................................... 69 Figure 4.8 : GA Algorithm for obtaining optimal temperature distribution ........................... 70

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Figure 4.9 : Optimal and Normal Ammonia Production Rates .............................................. 71 Figure 4.10 : Optimal and Normal Nitrogen Conversion and Reaction rates ......................... 71 Figure 4.11 : Effect of Temperature on Dissociation ............................................................. 79 Figure 4.12 : Effect of Flow Rate on Dissociation ................................................................. 79 Figure 4.13 : Effect of Pressure on Synthesis ......................................................................... 80 Figure 4.14 : Effect of Temperature on Synthesis .................................................................. 81 Figure 4.15 :Temperature & Pressure Parametric Sensitivity for Synthesis .......................... 81 Figure 4.16 :Effect of Flash Temperature on Ammonia Flow Rate ....................................... 82

Figure 4.17 :Effect of Flash Temperature on Ammonia Mole Fraction ................................. 83 Figure 4.18 :Effect of Purge Fraction on Ammonia Liquification ......................................... 83 Figure 5.1 : Homogeneous reactor with 1-D Model (TEM-TPA) showing gas temperatureT ,

equilibrium temperatureeqmT , and optimal temepratureoptT ............................................ 87

Figure 5.2 : Temperature in homogeneous reactor compared with one-equation optimal temperatureoptT and equilibrium temperatureeqmT . ......................................................... 91

Figure 5.3 : Homogeneous reactor: (a) ammonia mole fraction, (b) temperature profile, and (c) hydrogen/nitrogen/ammonia mole fractions. .................................................................. 92

Figure 5.4 : Homogeneous reactor with OEM-TPA, showing gas temperatureT , equilibrium temperatureeqmT , and optimal temperatureoptT . .............................................................. 93

Figure 5.5 : The Proposed Energy Recovery Plant with Process Modifications ................... 93

Figure 5.6 : PFR reactor beds with cooling between beds 1 and 2 ......................................... 94 Figure 5.7 : PFR reactor beds with cooling between beds ...................................................... 95 Figure 5.8 : Effect of temperature drop in the inter-bed heat exchanger, after the first bed, on

the ammonia mole fraction at reactor outlet. .................................................................. 95 Figure 5.9 : Effect of temperature drop in the inter-bed heat exchangers, after the first and

second beds, on the ammonia mole fraction at reactor outlet. ........................................ 96 Figure 5.10 : Homogeneous reactor: Nitrogen conversion in catalyst bed. ............................ 98

Figure 5.11 : Effect of varying spatial composition in reactor beds on the mole fraction of ammonia in the reactor compared with the reference (homogeneous) design with spatial concentration [1.00, 1.00, 1.00] ...................................................................................... 99

Figure 5.12 : Effect of varying spatial composition in reactor beds (1.50, 1.25, 1.00); a) nitrogen conversion, b) actual, optimal and equilibrium temperatures, c) hydrogen, nitrogen and ammonia mole fractions. .......................................................................... 100

Figure 5.13 : Bed1: Temperature Profile with different Catalyst Distribution ..................... 101 Figure 5.14 : Bed2: Temperature Profile with different Catalyst Distribution ..................... 102 Figure A.1 Ammonia 3D Phase Diagram ............................................................................. 121


1

1 INTRODUCTION In the coming centuries of the decline of the world’s fossil energy stocks, an electricity

production mix will establish which will be inevitably dominated increasingly by alternate &

renewable energies.

The alternate energy sources are available in form of solar energy, wind energy,

hydroelectric, geothermal, wave and tidal power etc. The current global energy consumption

is 15TWe per year while the solar energy potential is estimated to be 86000 TWe per year

[46].

Solar energy can be utilized either as a direct photovoltaic (PV) source, where the light is

converted directly into electrical energy or as concentrated solar power where a fluid is

heated by concentrating the solar thermal energy to produce electricity in a thermal power

plant. Solar thermal energy is concentrated using different techniques, such as, Parabolic

Trough, dish System and power tower etc.

The success of solar thermal systems for electricity production hinges very crucially on the

selection, mechanical design and optimal operation of an energy storage system which can

enable the continuous operation of a power plant. The energy storage systems being

investigated include solid graphite, encapsulated Phase Change Materials (PCMs) in a

graphite matrix, and liquid ammonia [72].

This work focuses on extending the use of a solar thermal energy plant from an intermittent

energy source to a base load power plant by incorporating an efficient thermal storage feature.

A reference 10 MWe solar thermal plant design is considered with liquid ammonia as a

working fluid for energy production in a Rankine Cycle as well as a thermal storage medium.


2

During periods of no solar insolence, the recovery system, based on an industrial ammonia

synthesis system, is used to drive the power conversion unit and enable continuous operation.

The objective of this work is to increase the efficiency of ammonia synthesis process for

maximum heat recovery and hence to improve the performance of Solar Thermal Storage

plant.

1.1 Solar Energy: Potential as a Renewable Energy Source Solar energy currently accounts for an installed capacity of about 23 GWe, compared with

geothermal (installed capacity 10.7 GW), and wind (160 GW) [8]. This is insignificant in the

global scenario where in 2010, the total primary energy consumption was 12002.4 Mtoe [8]

consisting of oil (33.8%), coal (29.6%), natural gas (23.8%), hydroelectric (6.5%) and

nuclear (5.6%). Even though renewable sources such as solar, geothermal and wind are not

presently significant, they offer the promise of providing clean and sustainable energy by

mitigating the effect of the carbon release from fossil fuels, in the form of greenhouse gases

[8], [14]. Such reductions are necessary for the environment and are binding on states

signatory to the Kyoto Protocol [117]. Emission of greenhouse gases (carbon dioxide,

methane, nitrous oxide, hydrofluorocarbons, perfluorocarbons and sulphur hexafluoride) as

well as toxic and pollutant gases, also have a harmful effect on people.

The Kyoto Protocol of 1997 [117] came into force on 16th February 2005 and establishes

quantified limitations on greenhouse gases, to promote sustainable development and calls for

member states to develop new forms of renewable energy and innovative environmentally

sound technologies.


3

According to a study by the World Energy Outlook [14] a Reference Scenario studies the

period 2006-2030 and estimates an increase in world primary energy demand of 45% from an

annual of 11730 Mtoe to over 17010 Mtoe at 1.6% increase per year. While oil remains the

dominant fuel, its share decreases from 34% to 30% over this period. In the same period, gas

rises at 1.8% per year to 22% while coal, at an annual increase of 2% rises from 26% to 29%.

Thus the fossil fuels contribute to 81% of the total primary energy demand by 2030.

Notwithstanding the impact of a nuclear renaissance, the contribution of nuclear power to

primary energy drops from 6% to 5%; this is an electricity generation share from 15% to 10%

by 2030.

In this period, renewable energy sources take second place after coal for electricity

generation. The contribution of hydropower drops from 16% to 14% while non-hydro

renewables, growing at an average annual rate of 7.2% increase from less than 1% to 4%.

The absolute magnitude of the non-hydro renewables increases from 66 Mtoe in 2006 to 350

Mtoe by 2030.

The power outlook has coal contribution to electricity generation increasing from 41% in

2006 to 44% in 2030, while the share of renewable grows from 18% to 23% in the same

period. The world’s final electricity consumption grows from 15665 TWh to 28141 TWh at

an average annual growth of 2.5%. This corresponds, in the Reference Scenario, to an

electricity generation of 18921 TWh in 2006 to 33265 TWh in 2030.

The factors accelerating the share of renewables are climate change, to attain the CO2 ppm

goal, the higher cost of oil and gas and energy security. Among the renewables, hydropower


4

will continue to be the dominant while others will include wind, solar, biomass and

geothermal energy.

The sun, as the primary source of energy for the solar system, supplies over 30,000 TWyr/yr

which, compared with the global energy requirement of the order of 20 TWyr/yr over the

next generation, may be considered to be a virtually inexhaustible source [46]. Solar energy

is useable as thermal energy, bioenergy from photosynthesis, and as a source for photovoltaic

conversion. Solar energy is truly renewable and sustainable as it is non-depletable, carbon

emission free, scalable, readily accessible, robust and flexible. The issues which will ensure

its place in the future energy scenario is its economic competitiveness in comparison with

existing technologies. A key technological issue that lies at the core of economic

competitiveness of solar energy -- thermal energy storage, is the focus of this thesis.

For electricity generation, the solar energy options available are photovoltaic (PV)

technology and concentrated solar power (CSP) technology. PV technology is based on the

direct conversion of photon energy from the Sun to electricity. Since the energy from the sun

is spread over a large range of wavelengths, a PV collector is designed to utilize as much of

the available spectrum as possible. The primary limitation is the detection window of the

sensor material forming the collector. The efficiency of a PV collector has remained low

(about 20%) and thus its application has been generally limited to mini-power requirements

such as off-grid homes [168][175]. However, larger PV plants have been built and the total

PV technology had a global installed capacity of 6 GWe in 2006, growing by 2009 to

15GWe and by the end of 2009 to 23 GWe, but had the disadvantage of having the highest

generating cost (US$ 5500-9000 per kWh in 2007) compared to all renewable technologies.


5

The Reference Scenario estimates the cost to reduce to US$ 2600 per kWh by 2030. CSP

technology uses optics to focus sunlight to a small receiver where the energy can be utilized

to convert water to superheated steam for electricity generation in a turbine. The total

installed capacity of CSP was 354 MWe in 2006. This technology is expected to be

comparable in cost (US$ 2-3 per kW: 2007) with gas-fired, but generally more expensive

than coal-fired generation, wind and nuclear.

1.2 Solar Power Plants in Operation 1.2.1 PV Plants Though PV technology is considered to be of use for small off-grid locations, large plants

have been built and are currently in operation [46]. The PV power generation technology saw

a 70% increase in 2008 alone, to 13GWe. Two notable areas of growth witnessed in 2008

were the Building Integrated PV Plants (BIPV) in Europe, and the utility-scaled PV plants (>

200 kWe), By the end of 2008, over 1800 such plants were in operation worldwide. Several

of these plants can be considered to be large, with the 200 MWe Huanghe Hydropower

Golmud Solar Park plant, completed in China in 2011, to be the largest PV plant in the world.

Plants of this magnitude are currently under development in Europe, China, India, Japan, the

United States of America and other countries. Table 1.1 presents World’s largest PV plants in

operation while 38 more plants with a cumulative nominal power of about 13000 MWe are

planned or under construction and are expected to complete by 2019.


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TABLE 1.1: World’s Largest (25MWe or above) PV Plants in Operation S.No. Name Country Nominal

Power(MWe) 1 Huanghe Hydropower Golmud Solar Park China 200 2 Perovo Solar Park Ukraine 100 3 Sarnia Photovoltaic Power Plan Canada 97 4 Montalto di Castro Photovoltaic Power

Station Italy 84.2

5 Solarpark Senftenberg Germany 82 6 Finsterwalde Solar Park Germany 80.7 7 Okhotnykovo Solar Park Ukraine 80 8 Lopburi Solar Farm Thailand 73 9 Lieberose Photovoltaic Park Germany 71.8 10 San Bellino Photovoltaic Power Plant Italy 70 11 Le Gabardan Solar Park France 67.2 12 Olmedilla Photovoltaic Park Spain 60 13 Sault Ste Marie Solar Park Canada 60 14 Strasskirchen Solar Park Germany 54 15 Tutow Solar Park Germany 52 16 Waldpolenz Solar Park Germany 50 17 Longyuan Golmud Solar Park China 50 18 Hongsibao Solar Park China 48 19 Serenissima Solar Park Italy 48 20 Copper Mountain Solar Facility USA 47.6

21 Puertollano Photovoltaic Park Spain 46 22 Moura photovoltaic power station Portugal 45 23 Kothen Solar Park Germany 45 24 Avenal Solar Facility USA 42.7 25 Cellino San Marco Solar Park Italy 40 26 Bitta Solar Park India 39.5 27 Fürstenwalde Solar Park Germany 38.3 28 Ralsko Solar Park Ra 1 Czech Republic 38 29 Reckahn Solar Park Germany 36.2 30 Alfonsine Solar Park Italy 35.1 31 Vepřek Solar Park Czech Republic 35 32 San Luis Valley Solar Ranch USA 34.4 33 Sant'Alberto Solar Park Spain 34 34 Planta Solar La Magascona & La Magasquila Italy 33 35 Ernsthof Solar Park Germany 32 36 Arnedo Solar Plant Spain 31.8 37 Parc Solaire Curbans USA 30.2


7

38 Long Island Solar Farm USA 30 39 Planta Solar Dulcinea Spain 30 40 Cottbus Drewitz Solar Park Germany 30 41 Agua Caliente Solar Project USA 30 42 Gunthawad Solar Farm India 30 43 Cimarron Solar Farm USA 30 44 Merida/Don Alvaro Solar Park Spain 29.9 45 Planta Solar Ose de la Vega Spain 27.5 46 Webberville Solar Park USA 26.4 47 Ševětín Solar Park Czech Republic 26 48 Solarpark Heideblick Germany 25.7 49 Solarpark Eiche Germany 25

1.2.2 Solar Thermal Plants The CSP technology showed a small generation increase by 0.06GWe to 0.5GWe by the end

of 2008. The world’s largest solar site is in California, owned by NextEra Energy Resources

[45] The power produced is 354 MWe, which is purchased by Southern California Edison

and provides to more than 230,000 homes at peak power during the day. It is thus as large as

a nuclear reactor such as CHASNUPP, and would be sufficient for a city of the size of

Islamabad. The site is spread over 1500 acres, and has more than 900,000 mirrors.

Other large CSP plants in the range of 30-150 MWe are also located in the United States and

Spain [21],[47]. Several other countries including Abu Dhabi, Algeria, Egypt, Israel,

Portugal and Morocco have projects underway [45]. One of the plants, a 20MWe CSP is

integrated with a 450MWe natural-gas combined-cycle plant in Morocco. Table 1.2 lists

solar thermal power plants in operation in different parts of the world with total capacity

amounting to 1702.65 MWe. The total capacity of under construction (to be completed by

2014) solar thermal plants is 2106.9 MWe.

TABLE 1.2: Solar Themal Plants in Operation

Serial #

Name Country Capacity (MWe)

Technology

1 Solar Energy Generating Systems

USA 354 parabolic trough

2 Solnova Solar Power Station Spain 150 parabolic trough 3 Andasol solar power station Spain 150 parabolic trough


8

4 Extresol Solar Power Station Spain 100 parabolic trough 5 Palma del Rio Solar Power

Station Spain 100 parabolic trough

6 Manchasol Power Station Spain 100 parabolic trough 7 Valle Solar Power Station Spain 100 parabolic trough 8 Martin Next Generation Solar

Energy Center USA 75 ISCC

9 Nevada Solar One USA 64 parabolic trough 10 Ibersol Ciudad Real Spain 50 parabolic trough 11 Alvarado I Spain 50 parabolic trough 12 La Florida Spain 50 parabolic trough 13 Majadas de Tiétar Spain 50 parabolic trough 14 La Dehesa Spain 50 parabolic trough 15 Helioenergy 1 Spain 50 parabolic trough 16 Lebrija-1 Spain 50 parabolic trough 17 Solacor 1 Spain 50 parabolic trough 18 Puerto Errado 1+2 Spain 31.4 fresnel reflector 19 Hassi R'mel integrated solar

combined cycle power station Algeria 25 ISCC

20 PS20 solar power tower Spain 20 solar power tower

21 Kuraymat Plant Egypt 20 ISCC 22 Beni Mathar Plant Morocco 20 ISCC 23 Yazd integrated solar combined

cycle power station Iran 17 parabolic trough

24 Gemasolar Spain 17 solar power tower 25 PS10 solar power tower Spain 11 solar power tower 26 Kimberlina Solar Thermal

Energy Plant USA 5 fresnel reflector

27 Sierra SunTower USA 5 solar power tower 28 Archimede solar power plant Italy 5 parabolic trough 29 Thai Solar Energy (TSE) 1 Thailand 5 parabolic trough 30 Liddell Power Station Solar

Steam Generator Australia 2 fresnel reflector

31 Keahole Solar Power USA 2 parabolic trough 32 Maricopa Solar USA 1.5 dish stirling 33 Jülich Solar Tower Germany 1.5 solar power tower 34 Saguaro Solar Power Station USA 1 parabolic trough 35 Shiraz solar power plant Iran 0.25 parabolic trough Overall Capacity 1702.65


9

1.3 Thermal Energy Storage Requirement The major drawback of CSPs at the moment is the lack of thermal storage due to which

operation is only possible when daylight is available. Only two plants have storage viz the

Andasol-1 [48] plant in Spain which has more than seven hours of full-load thermal storage

capability, and a 280 MWe plant planned in Arizona which will also have a six-hour storage

capacity.

1.4 Thermal Storage Materials The thermal energy storage technologies can be classified [72] by the mechanism of heat viz

(i) sensible, (ii) latent, (iii) sorptive, and (iv) chemical. In the sensible heat storage systems,

there is the possibility of liquid (water tank, aquifier, thermal oil) and solid systems (building

mass, concrete, and ground etc.) [5]. In the latent heat storage systems, both organic

(parrafins) and inorganic (hydrate salts) compounds can be used. In the sorptive, both

absorption and adsorption systems can be used. Finally, in the chemical storage, energy can

be stored in chemical bonds which can be broken endothermically and recovered in a

synthesis exothermically.

When single-phase heat transfer fluids such as thermal oil or pressurized water are used, a

sensible heat storage system using concrete has been developed and experimentally tested

[51] in the temperature range 300-400 oC and found to be an attractive options for CSPs.

Storage materials and technology will also depend on the temperatures in the plant [66]. For

domestic hot water and space heating, the temperatures will be less than 100 oC; for process

heat, 100-250 oC; for electricity generation 250-1000 oC, while for hydrogen production they

Engr. Sadaf Siddiq (08F-UET/PhD-ME

will be in excess of 1000 oC. The storage capacity of some pha

below [66]. It can be seen that the highest storage capacity is for salts.

Figure 1.1 Volume Reduction with Phase Change Materials

Figure 1.2 Materials for medium and high heat storage

ME-47)

10

C. The storage capacity of some phase change materials is shown

. It can be seen that the highest storage capacity is for salts.

Volume Reduction with Phase Change Materials

Materials for medium and high heat storage

se change materials is shown


11

Salts freeze 350-500 oC and boil at ~1000 oC. They have a high volumetric heat capacity and

may be used, even in graphite blocks. Liquid fluoride salts are also widely available, as they

are used in aluminium metal extraction Hall electrolysis process in which aluminium oxide

is dissolved in cryolite which is a sodium-aluminium fluoride salt. Fluoride salts are

compatible with graphite upto 1400 oC [176].

1.5 Use of Liquid Ammonia as Storage Material Liquid ammonia is a candidate for large solar-thermal systems due to the storage of thermal

energy in its chemical bonds during, for example, solar insolation and recovery from

subsequent exothermic synthesis. To compare different storage opportunities, the energy

storage density is a value which is useful to determine the required size of storage for a

required amount of energy. With the kind of energy carrier, the amount of stored energy

varies strongly. A comparison between different energy carriers is presented in Figure 1.3

[79]. It is clear that thermo-chemical energy carriers offer the suitable most energy densities

i.e. of the order of 10 MJ/kg.

Ammonia is an abundantly produced chemical, globally and in Pakistan. It has an important

use as a fertilizer to boost agricultural production. Thus it is used in a synthesis process of

natural gas with carbon dioxide resulting in the formation of urea fertilizer, or carbamide

(NH2)2CO. In Pakistan, there are eight large urea fertilizer plants based on the reforming and

synthesis of natural gas mainly from the Sui and Marri gas fields. At an international price of

US$ 300/tonne, this represents an annual sales value of US$1,500 million. This amounts to

an average production of about 1600 tonnes per day (TPD) per plant [25].


12

Figure 1.3 Energy densities for different energy carriers 1.5.1 Poperties of Liquid Ammonia Ammonia (NH3) stays in the liquid form at temperatures higher than its melting point

73.77− oC and has a density of 681.9 kg/m3 at its boiling point -33.34 oC ; it must thus be

kept at very low temperature or stored at very high pressure [165]. Liquid ammonia was first

produced on an industrial scale in Germany, during the First World War, by the Haber -

Bosch process [110].

1.5.2 Dissociation and Synthesis of Ammonia The dissociation of ammonia

223 32 HNNH +⇔ (∆H = 66.9 kJ/mol)


13

is an endothermic reaction that can be carried out by thermo-catalytic decomposition using

catalysts: ruthenium, indium, nickel, Fe-Al-L, Fe-Cr. Typical temperatures are in the range of

850 – 1000 oC. Approximately 1.4 kW power per cubic meter of hydrogen is typically used.

Conversely, the synthesis of ammonia from nitrogen and hydrogen reactant gases

322 23 NHHN ⇔+ (∆H = -92.22 kJ/mol)

is an exothermic reaction for which the pressure required is in the range 130 – 250 bar, and

the temperature required is in the range 250 – 600 oC. High temperature gives higher reaction

rate, but as reaction is exothermic, higher temperature according to Le Chatelier’s principle

causes the reaction to move in the reverse direction hence a reduction in product. Similarly,

higher Temperature reduces the equilibrium constant and hence the amount of product

decreases; this is the Van’t Hoff equation

R

S

RT

HK

oo ∆+∆−=ln

An increase in pressure, however, causes a forward reaction and is thus favorable. Synthesis

is achieved by using catalysts such as osmium, ruthenium, and iron-based catalysts [110].

1.5.3 Commercial uses of Ammonia Ammonia is one of the most widely produced chemicals, amounting to over 15 million tones

in 2009. Its major uses are as fertilizer and for production of nitrogen containing compounds

such as nitric acid. It is used as a refrigerant and in textile processing. A very important

emerging use of ammonia is Hydrogen production, by its decomposition, to be used in

Hydrogen Fuel cells.


14

1.5.4 Industrial proprietary processes for Ammonia Production

Global fertilizer industry produces about 170 million tones of fertilizer nutrients every year

[42] for boosting agricultural output. Fertilizers are based on nitrogen, phosphorus or

potassium. Nitrogen accounts for 78% of the earth’s atmosphere. Since plants can not breathe

nitrogen, it must be converted to a suitable form such as ammonia. The Haber-Bosch process,

first demonstrated by Fritz Haber in 1909 and scaled up to an industrial process by Carl

Bosch in 1913. Both Haber and Bosch were awarded Nobel Prizes for their inventions, and

ammonia was used in Germany in the First World War for the manufacture of explosives. A

greater use of the Haber-Bosch process was in the manufacture of fertilizers such as urea and

ammonium nitrate. About 70% of the ammonia produced is from natural gas as feedstock

and the rest is mainly from coal. The Haber-Bosch process, involving the steam reforming of

methane to produce hydrogen is used with nitrogen taken from the air, to produce ammonia.

The typical size of urea plants is 1000 MeT per day with a capital cost of US$ 150 million.

The total production of ammonia was 130 million tones in 2000, produced in 80 countries

and 85% of which was used for nitrogen fertilizer production. The largest chemical industry

in the world is in the U.S. [19], with ammonia being the most important intermediate

chemical compound produced in 41 plants. The energy intensity for ammonia manufacture in

the U.S. is 39.3 GJ/tonne (including feedstocks HHV). The theoretical minimum for

ammonia production by steam reforming is 21.6 GJ/tonne which represents the ideal goal.

The technology is now mature, with the market dominated by five licensers-Haldor Topsøe,

M.W. Kellogg, Uhde, ICI, and Brown & Root, of which Haldor Topsøe has a 50 per cent

world market share as supplier of the technology [42].


15

1.5.4.1 Haldor Topsoe Ammonia Synthesis Process

The conventional sequence of process steps are optimized by the introduction of improved

catalysts (KM high strength, versatile, stable and poison-resistant catalyst, mainly magnetite

Fe3O4 with promoters mainly oxides of calcium, aluminum and potassium, operating

temperatures 340-550 oC [21]), new equipment design (such as improved synthesis

converters), and process optimization studies. The carbon monoxide concentrations have

been minimized at the exit of the shift converters, and a low-energy carbon dioxide removal

process (such as selexol) has been used. New syn converters S-250 and S-300 are improved

versions of the previous single bed S-50 and two-bed S-200 radial flow converters. Topsoe

recommends S-300, developed in 1999, for all new plants [[21], [24].

TABLE 1.3: Haldor Topsoe Ammonia Converter Features

Type Basic Design Comments S-50 One catalyst bed Simplest and cheapest S-200 Two catalyst beds and one interbed

heat exchanger Commissioned in 1979; 130 units installed

S-250 Combination of the S-200 followed by the S-50

S-300 Three catalyst beds with two interbed heat exchangers

Higher conversion for same catalyst volume of S-250; installed first in 1991.

1.5.4.2 Kellog Brown & Roots (KBR) Advanced Ammonia Process (KAPP)

KAAP uses a traditional high-pressure heat exchange based steam reforming process

integrated with a low-pressure advanced ammonia synthesis process. The steam reforming of

hydrocarbon based on Kellogg Brown and Root Reforming Exchange System (KRES) is


16

carried out which reduces energy consumption and capital cost besides reduced emissions

and enhanced reliability.

After reforming, carbon monoxide is removed from the shift converter, and carbon dioxide is

removed from the process gas using hot potassium carbonate solution, methyl diethanol

amine (MDEA) etc.

KAAP uses a high activity graphite supported ruthenium catalyst, typically three stages, after

one stage of traditional iron catalyst. This is claimed to increase the activity 10 to 20 times

enabling very high conversion at a lower pressure of 90 bar [10].

KBR is a large player in the ammonia and urea industry. It has been involved in the licensing,

design, engineering and/or construction of more than 200 ammonia plants and 62 urea

projects in the range of 600 to 3500 MTPD worldwide, representing approximately half of

current global ammonia production [23].

1.5.4.3 Krupp Uhde GmbH Ammonia Process

The Krupp Uhde Gmbh process uses the traditional reforming process followed by a

medium-pressure ammonia synthesis loop[86].

The primary reforming is carried out at a pressure of 40 bar and temperature range of 800-

850 oC. Enhanced reliability is attained by using a top-fired steam reformer with high alloy

steel tubes. Process air is added in the secondary reformer through nozzles installed in the

wall of vessel thus providing proper mixing of the air and reformer gas. This also provides

high energy efficiency in high pressure steam generation and superheating. As in other

processes, carbon monoxide is converted to carbon dioxide in HT and LT shift converters,

and the MDEA or Benfield system is used for carbon dioxide removal.


17

The ammonia synthesis loop uses two radial flow ammonia converters with three catalyst

beds, containing iron catalyst, and waste heat boiler located downstream of each reactor. The

converters have small grain iron catalyst.

Since 1994, Uhde has built 15 new ammonia and 13 new urea plants with annual production

capacities of more than 8 million tonnes of ammonia and 10 million tonnes of urea with

individual capacities ranging from 600 to 3,300 mtpd of ammonia and from 1,050 to 3,500

mtpd of urea [36]. Uhde has also been awarded a contract to build a 3300 MTPD “Uhde

Dual-Pressure Process” ammonia plant for Saudi Arabian Fertilizer Company (SAFCO) in

Al Jubail, Saudi Arabia [78].

1.5.4.4 ICI-Leading Concept Ammonia (LCA) Process

In this process, ammonia synthesis takes place at low pressure of below 100 kg/cm2g

(approximately 100 bar) using ICI’s highly active cobalt promoted catalyst. This process has

an energy consumption of approximately 7.2 Gcal/ MeT (30.1 GJ/MeT) ammonia for a 450

MeT per day plant [19].

1.5.4.5 The Linde Ammonia Concept (LAC) Ammonia (LCA) Process

The LAC process consists essentially of a modern hydrogen plant and a standard nitrogen

unit with a third-party license from Casale for a high efficiency ammonia synthesis loop [34].

Ammonia Casale [16] is one of the oldest companies in the business of synthetic ammonia

production, having been founded in Switzerland in 1921. To date it has been active in the

design of over 150 ammonia synthesis reactors and in the constructionof several new plants.


18

The CO shift conversion is carried out in a single stage in the tube cooled isothermal shift

converter and gas is sent to pressure swing absorption (PSA) unit wherein the process gas is

purified to 99.99 mole % hydrogen . A low temperature air separation in cold box is used to

produce pure nitrogen. BASF’s MDEA process is also eliminated in this process used for

CO2 removal.

The ammonia synthesis loop is based on Casale axial-radial three-bed converter with internal

heat exchanger giving a high conversion. The energy consumption for ammonia production

is about 29.3 GJ/ MeT [16].

Thus far, four plants based on the relatively new Linde Ammonia Concept have been

constructed with capacities of between 230 to 1,350 MTPD of ammonia.

1.6 Thermodynamic Cycles for Solar Thermal Power Plants

The two commonly used thermodynamic cycles for solar plants are the Brayton and Rankine

Cycles depending on the temperatures of the working fluid. Power towers employing PCM

salts are able to achieve very high temperatures, typically in excess of 1000 oC which transfer

heat to inert gases such as helium, and at a lower temperature, water is converted to

superheated steam. Such plants draw heavily from the experience and resources available

with high temperature gas reactors in the nuclear industry. While the thermodynamic

efficiency of such systems is high, special materials and high safety features are required for

this technology [140].

Solar power plants based on the concentrating parabolic systems, ordinarily use water as a

working fluid and are thus based on the Rankine Cycle [130].


19

1.7 Literature Review The literature survey covered a wide range of areas, this section reviews the potential of solar

energy as a renewable source for a sustainable and clean energy future, Solar thermal power

and its components, thermodynamic aspects of candidate solar plants, thermal storage

materials, energy inputs and outputs from various thermal storage materials, energy recovery

industrial process and energy efficiency analyses for plant performance and design

parameters for realized Solar thermal power plants. This design data from realized Solar

thermal power plants has been used as a starting point for component and overall simulation,

as well as optimization formulations for carrying out sensitivity analyses leading to an

optimal pant design. Modeling and Simulation techniques for component and integrated plant

design are discussed in section 3.1.1 while review of optimization techniques is presented in

section 4.1.

Concentrating solar power is a method of increasing solar power density. CSP has been

theorized and contemplated by inventors for thousands of years. The first documented use of

concentrated power comes from the great Greek scientist Archimedes (287-212 B.C.) in 212

B.C. [175]. The modern solar concentration is believed to begin by the experiments of

Athanasius Kircher (1601-1680) in seventeenth century [175]. Solar concentrators then

began being used as furnaces in chemical and metallurgical experiments [161]. In eighteenth

and nineteenth centuries CSP applications were restricted to low pressure steam generation

and solar pumps etc.

CSP systems can provide energy storage fully integrated within the electricity-generating

plant [2][4][5]. Solar thermal radiation can be concentrated using parabolic mirrors in the


20

form of dishes, power towers, troughs and linear Fresnel etc. in commercial CSP systems.

The efficiency of these mechanisms can be evaluated on the basis of geometric concentration

ratio. The geometric concentration ratio for parabolic troughs and linear Fresnel systems can

be up to 100 and in excess of 1000 for power towers and dishes. This thermal energy can be

used to produce steam for immediate electricity generation, or alternatively it can be stored

prior to electricity generation using sensible heat storage in solids [27][49][66], molten salts

[88], phase change materials [9][39][145][153], or thermochemical storage cycles [15].

Thermochemical energy storage for CSP is less mature than molten salt and other thermal

storage methods, but it has the potential to achieve higher storage densities and hence smaller

storage size. Reactions involving ammonia, hydroxides, carbonates, hydrides, and sulfates

are the important candidates for thermochemical energy storage [15][67]. At first,

thermochemical storage loops based on methane reforming received considerable attention

[58][115][138][141]. Methane reforming is still under research for solar enhancement of

natural gas [30] and hydrogen production [31]. A lot of research is being conducted on solar

fuel production by making use of thermochemical processes [17][38].

The concept of ammonia-based energy storage for concentrating solar power systems was

first proposed by Carden in 1974 at the Australian National University [174][177] followed

by the researchers at Colorado State University in early 1980’s [163].

Researchers at ANU [132][167] and Colorado State University [163] have concluded after

theoretical analysis and experimental results [109] that dish concentrators are the most

suitable solar receiver designs for ammonia dissociation because they provide a

circumferentially homogenous solar flux profile [136] which can facilitate thermochemical


21

reactor design inferring that only simple control systems are necessary, thus the mobile

receiver can be maintained at a light weight, and solar transients are easy to handle

[132][137]. Feasilbility of parabolic trough systems have also been investigated for use with

CSP employing ammonia based energy storage systems [97].

Prototype solar ammonia receiver/reactors, Mark I and Mark II were tested in 1994 and 1998

respectively both employing a 200-mm long cavity type reactor mounted on a 20-m2 faceted

paraboloidal dish. Haldor-Topsøe DNK-2R iron-cobalt catalyst was used in the annular

catalyst beds [108]. These reactors were rated for 1.0– 2.2-kWchem conversion. Recent work

is being conducted on paraboloibal dishes of area 400-m2, 489-m2 and newly constructed

500-m2 for a base load plant size of upto 10 MWe [11].

For solar collector/receiver design improvements, investigations into convection losses from

cavity receivers have been undertaken [12][81] as these improvements can amount to solar-

to-chemical efficiency gains of up to 7% absolute [106].

The kinetic mechanisms for the synthesis and decomposition of ammonia have been

described by various authors for ironbased catalysts [120][186][189][195] and for ruthenium-

based catalysts [111][121][126].

Comprehensive studies for solar energy heat [104],[106]] recovery have been carried out on

an experimental 1-kWchem. synthesis reactor by Kreetz and Lovegrove [106] in a laboratory-

scale high-pressure closed-loop system with a feed-gas mass flow rate of 0.3 g s-1 at

pressures ranging from 9.3 to 19 MPa. With external pre-heating of the feed gas, average

external wall temperature varying between 250-480°C and peak internal reactor temperatures

varying between 253-534°C, the maximum reaction was reported by Kreetz and Lovegrove


22

[106] to have been achieved at approximately 450°C. In their ‘optimal’ system, a net heat

recovery rate of 391 W was reported. The study by Kreetz and Lovegrove [106] was extended

to a 10 kW system with an ammonia synthesis tubular reactor at a pressure of 20 MPa and a

flow rate of 0.9 g s-1 [79]. The larger system, with a controlled linear temperature profile in

the reactor wall, and the gas inlet temperature kept to 50 °C lower than the wall temperature

at the inlet, resulted in a maximum thermal output achieved at an average wall temperature of

475°C produced with an inlet temperature of 500 °C and a slope of -50 °C m-1. Such studies

have attempted to achieve optimal heat recovery by varying the inlet temperature arbitrarily

instead of attaining the optimal temperature suggested by theoretical models, such as

variational methods.

1.8 Thesis Motivation

Thermal Storage plants using ammonia as storage medium can take advantage of the well-

understood and extensively deployed ammonia dissociation and synthesis technologies. Their

efficiency, however, will depend on the optimization of the process parameters typical of the

system pressure and temperatures in the dissociation and synthesis reactors taken together

with those at the solar receiver.

A lot of research has been carried out on solar collector design and dissociation efficiencies

of more than 90% have been practically achieved using cavity type dissociation reactors in

conjunction with paraboloidal dish type solar receivers[105][106]. The motivation of plant

optimization is to maximize the efficiency of the plant by maximizing the heat recovery from

the most critical process in the plant: the synthesis reactor. This research is of great value to

industry as well because the same optimization techniques can be used for improving


23

ammonia production rates at pressures lower than the industry standard pressures, hence

cutting the costs.

1.9 Objectives

The use of liquid ammonia, as a thermo-chemical energy storage medium, for endothermic

dissociation by solar energy during insolence and subsequent energy recovery by exothermic

synthesis is considered to be a strong candidate for the design of a base-load solar thermal

power plant.

The technology of ammonia production is well established as is the modeling and simulation

of ammonia synthesis. However, optimization of the process is an on-going challenge as

technological innovations enable better designs resulting in improved efficiency. As part of

this optimization challenge, this thesis considers the possible improvement in the recovery of

exothermic thermal energy by optimization of the ammonia synthesis process. While

ammonia production has remained almost the same for decades, the energy consumption has

reduced as technology improvements have been incorporated especially for the fertilizer

industry where over 90% of the energy utilization is for ammonia synthesis [76].

The objective of the study will be achieved by:

Parametric Sensitivity studies leading to an optimized design of a TSP

i. Numerical Simulation of Conservation Equations,

ii. Optimize physical dimensions of Synthesis Reactor,

iii. Optimal distribution of Catalyst (Optimal Control analysis),

iv. Overall Thermal Energy Recovery Analysis.


24

1.10 Summary of Following Chapters The necessary background information is given in Chapter 1. Chapter 2 describes the thermal

storage plant features such as process design, operational parameters and process flow

diagram etc. Chapter 3 deals with the modeling and simulation of the components of thermal

storage plant while optimization of thermal storage plant has been done in chapter four both

by variational calculus and process engineering codes. The fifth chapter presents the optimal

TSP model, designed in the light of sensitivity and parmetric analyses in chapter four.

Conclusions and future recommendations are presented in Chapter 6.

25

2 DESCRIPTION OF THE THERMAL STORAGE

PLANT

2.1 Plant Features A thermal storage plant may be used as a baseload plant, when it operates on a continuous

basis just like a coal-fired, nuclear or hydroelectric power plant, or as a traditional PV

intermittent solar plant. The baseload operation is only possible if the plant has an integrated

thermal storage feature.

The major components of a baseload plant are the receiver system, a storage system, an

energy recovery system, a power conversion unit, and associated plant systems such as

compressors, pumps and heat exchangers.

The objective of the Thermal Storage Plant (TSP) considered here is to maximize the overall

efficiency of the plant, which is essentially the optimization of the ammonia synthesis

process.

2.1.1 Process Design

This section considers some basic aspects of the overall plant design with the objective of

getting orders of magnitude. Table 2.1 shows such overall conditions for a conceptual MS-

Excel calculation for a baseload plant of 10 MWe. It is assumed that a solar insolation of 1

kW/m2 is available for 8 hours in a day. With 400 parabolic dishes, each of area 400 m2 of

the type available to the ANU group [101][105], the thermal power intercepted by the plant is

26

4.608 TJ in a day. These assumptions are optimistic even for the high solar insolence of

about 20 MJ/m2 for Pakistan [150].

TABLE 2.1: Overall Plant Design for a 10 MW(e) Baseload Plant

BASIC DATA Dissociation Energy kJ/mol 66 Synthesis Energy kJ/mol 46.6 Power Density Watts / m^2 1000 Insolation Hours per Day Hr 8 Dish Area m^2 400 No. of Dishes 400 extent of dissociation reaction 0.9 Electrical Power Reqd (24hrs) MW(e) 10 Synthesis Conversion 0.2 Rankine Cycle Effciency 0.4 POWER INPUT

Thermal Power Available/day kW-hr/m^2 per day 8

Thermal Power Available/day MJ/m^2 per day 28.8

ThPower/day on one dish MJ per dish per day 11520

ThPower/day on all dishes MJ per day 4608000

Flow of NH3 per dish mol per dish during insolation 193939.39

Flow of NH3 per dish kg NH3 per dish during insolation 3296.97

Flow of NH3 MTD NH3 during insolation 1318.79

Flow rate of NH3 kg/s NH3 45.79 Flow rate of NH3 per dish kg/s NH3 0.1145 Electrical Energy Needed MJ(e) per day 288000 Thermal Energy Needed MJ(th) per day 720000 POWER OUTPUT Recoverd Synthesis Energy MJ per day 921600 Converted Synthesis Energy MJ per day 368640 Overall Efficiency % 8

27

Key design parameters are the thermal power intercepted by the plant during insolence (4.6

TJ), the thermal energy needed (2.1 TJ) for a baseload of 10 MW(e), the recovered synthesis

energy (0.922 TJ) and the final converted energy (0.368 TJ). In this scenario, three

efficiencies are assumed viz (i) the extent of dissociation (0.9) [47], (ii) the synthesis

conversion (0.2), and (iii) the Rankine efficiency (0.4) [[15],[79],[130].

The purpose of the present research is to estimate the best possible synthesis conversion, by

optimizing the catalyst distribution, to investigate the feasibility of such baseload operation.

2.1.2 Opertational Parameters

The operational parameters of TSP have to be chosen carefully as the use of a reversible

reaction to store energy is governed by the dependency of the thermodynamic equilibrium

composition on temperature and pressure. Conceptually, if a sample of ammonia were heated

slowly (quasi-statically), it would begin to decompose at temperatures of several hundred

degrees, around 700 K at 200 atmospheres (20MPa). Complete dissociation would only be

approached asymptotically at very high temperatures. The amount of energy absorbed at each

step would be proportional to the fraction of ammonia split. Reversing the process and

withdrawing heat would see ammonia resynthesize, with heat released progressively [196].

To implement this on an industrial scale, the limitations of reaction kinetics must also be

taken into account. Reaction rates are zero at equilibrium by definition; they increase by the

degree of departure from equilibrium (and in the direction needed to return the system to

equilibrium) and also increase rapidly with temperature in proportion to the well-known

28

Arrhenius factor. Thus, a real system absorbs heat at temperatures higher than the

equilibrium curves suggest and then releases it at lower temperatures.

The input temperature for the power cycle is an extremely important issue for all thermal-

based energy storage systems, not just thermochemical ones. Electric power generation via a

thermal cycle is limited by the second law of thermodynamics – lower temperature thermal

inputs reduce the efficiency of power generation. Thus, in designing and examining thermal

energy storage systems, it is necessary to consider both “thermal efficiencies” (energy

out/energy in) and “second law efficiencies” (potential for work out/potential for work in).

A TSP will have operational parameters, pressures, temperatures and flow rates, similar to

those in the ammonia units of urea fertilizer plants in the chemical process industry. These

require pressures in the range of 130-250 bar and temperatures in the range 250-600 oC for

flow rates typically of the order of 50 kg s-1 for a 1500 MTD ammonia plant. Such high

pressures require compression which is expensive in terms of equipment cost as well as

energy utilization

2.2 Overall Plant Layout and Description The schematic diagram of TSP is shown in figure 2.1[2]. In this closed loop system, a fixed

inventory of ammonia passes alternately between energy-storing (solar dissociation) and

energy-releasing (synthesis) reactors, both of which contain a catalyst bed. Coupled with a

Rankine power cycle, the energy-releasing reaction could be used to produce baseload power

29

for the grid. At 20 MPa and 300 K, the enthalpy of reaction is 66.8 kJ/mol, equivalent to 1.09

kWh/kg of ammonia, or 2.43 MJ/L, with the corresponding density of 0.6195 kg/L [165].

Figure 2.1: Thermal Storage Plant Schematic 2.2.1 Ammonia Dissociation Having the advantage of solar concentration of 3000 suns [105], a mirrored paraboloidal dish

focuses solar radiation onto a dissociation reactor (cavity type) through which anhydrous

ammonia is pumped. The reactor contains an annular catalyst bed which facilitates the

dissociation of ammonia at requisite temperature and pressure into gaseous nitrogen and

hydrogen termed “syngas”. The fact that the ammonia dissociation reaction has no possible

side reactions makes solar dissociation reactors easy to control and implement [2][160].

Typically, 400 such reactors mounted on paraboloidal dishes, of area 400 m2 each, are used

in an array patteren to feed the ammonia synthesis reactor.

30

Figure 2.2: Array of 400 m2 Paraboloidal Solar Collectors [3]

2.2.2 Ammonia Synthesis A reactor is used for energy recovery from the exothermic synthesis reaction in which syngas

is synthesized to produce ammonia in the presence of an annular catalyst bed. Since

ammonia synthesis is a developed technology for more than 100 years, synthesis reactors

used for TSP are based on standard and proprietary industrial technologies from companies

that include Haldor-Topsoe, Kellogg Brown & Root (KBR), AkzoNobel (formerly Imperial

Chemical Industries (ICI)), and Cassal [16],[21],[23],[24].

For the reference TSP in this work, KBR Advanced Ammonia Process (KAAP™) synthesis

convertor is chosen. In the KBR Advanced Ammonia Process (KAAP™)[23], the synthesis

converter uses a combination of catalysts to maximize the conversion and heat recovery, such

as one stage of traditional magnetite catalyst, followed by three stages of a proprietary

KAAP™ catalyst consisting of ruthenium on a stable, high-surface-area graphite carbon base

(KBR). This KAAP™ catalyst has an intrinsic activity ten to twenty times higher than

31

conventional magnetite catalyst and is used to lower the synthesis operating pressure to 90

bar which is one-half to two-thirds the operating pressure of a conventional magnetite

ammonia synthesis loop and hence cutting plant costs.

2.2.3 Syn Gas and Ammonia Storage The closed-loop TSP operates at a pressure (150 bar) above ambient temperature saturation

pressure of ammonia and the ammonia fraction in storage is present largely as a liquid which

causes automatic phase separation of ammonia. Thus, a common storage tank can be used to

store both syngas and liquid ammonia.

2.2.4 Heat Exchangers and Transport Piping

The heat exchangers shown in Fig. 2.1 serve to transfer heat from exiting reaction products to

the cold incoming reactants. In this way, the transport piping and energy storage volume are

all operated at close to ambient temperature, reducing thermal losses from the system, as well

as eliminating the need for costly specialized equipment.

2.2.5 Compressors and Pumps

Compressors are used for the pressure management of high pressure storage vessel and

synthesis loop. In the dissociation part of the system, a liquid ammonia feed pump is

incorporated with each paraboloidal dish. These pumps are used to control the actual process

conditions within the ammonia dissociation reactor. Mass flow control aims for 80% of the

ammonia feed being dissociated on average.

32

2.3 Thermal Storage Plant Process Flow Diagram Figure 2.3 presents a simplified process flow diagram of thermal storage plant. The input

stream (DIS-IN) to the solar driven dissociation reactor (DISRCTR), consisting of liquid

ammonia, is pumped from the high pressure storage tank (S-TANK).

The stream DIS-IN is pre-heated by passing it through the counter flow heat exchanger (CF-

HX) in order to increase its temperature. The output syngas stream (DIS-OUT), consisting of

nitrogen, hydrogen and small amounts of other gases, looses heat in heat exchanger (CF-HX)

and is fed into storage tank (S-TANK). The feed-stream (FEED1) from storage tank is

compressed to the pressure required for synthesis, 150 bar. Due to the unfavourable reaction

equilibrium, only part of the Syngas is converted to ammonia on a single pass through the

Synthesis Reactor (SYNRCTR). Since the unconverted Syngas is valuable, the majority of it

is recycled back to the SYNRCTR. A Mixer is used to combine the Recycle Stream (FEED2)

and fresh stream FEED1.

Figure 2.3: TSP Process Flow Diagram

33

This mixed stream (MIX-OUT), heated in SRIN-HX to a temperature of 370 oC and is fed

into the catalyst-containing synthesis reactor (SYNRCTR) where the synthesis reaction, in

the forward direction, converts nitrogen and hydrogen into ammonia and hence producing

energy.

The effluent stream passes through the recovery heat exchanger (SROUT-HX) into the

Knock-Out drum FLASH, where the liquid ammonia is sent back to the storage tank through

stream PRODNH3 and stream VAPOR is carried to the purging system. The VAPOR stream

from Flash tank (FLASHT) contain traces of undesirable gasses such as Argon, Carbon

Monoxide and Carbon Dioxide. Argon has high partial pressure while Carbon Monoxide and

Carbon Dioxide are poisons for the Catalyst. Some of the cycle gas must be purged from the

Synthesis Loop. Otherwise, the argon that enters the loop in the Syngas has no way to leave

and will build up in concentration. This will reduce the rate of the ammonia synthesis

reaction to an unacceptable level. To prevent this from happening, a small amount of the

cycle gas must be purged, the amount being determined by the amount of argon in the feed

and its acceptable level in the Synthesis Converter feed (generally about 10 mol %). A

splitter is used to divide the VAPOR stream into PURGE and RECYCLE streams.

Another re-cycle compressor (RCOMP) is required at this stage to restore the pressure to the

required level till the stream (FEED2) is mixed with the feed stream (FEED1) and enters as

stream MIXER-OUT.

34

3 MODELLING & SIMULATION OF THERMAL

STORAGE PLANT

3.1 Mathematical Modelling 3.1.1 Review of Mathematical Models of TSP

The synthesis of ammonia can be modeled using the laws of conservation of mass,

momentum and energy for non-isothermal multi-component systems undergoing chemical

reactions and mass transfer [85]. In the case of unsteady flow the governing equations are:

Mass:

0222021 ωρωωω +⟩⟨Σ=+Σ−Σ= Svmdt

dtot

3.1

Mass of Species i:

Nirmdt

dtotiiiitoti ,......3,2,1,021, =++Σ−Σ= ωωω

3.2

Momentum:

sftottot FFgmSpv

vSp

v

v

dt

d→−+++

⟩⟨

⟩⟨Σ−+⟩⟨

⟩⟨Σ=Ρ 022222

22

11111

21 )()( ωωωω

3.3

(Total) energy:

QQwHghv

vHgh

v

vUK

dt

dtottottot −++++

⟩⟨

⟩⟨Σ−++⟩⟨

⟩⟨Σ=+Φ+∧∧

02222

32

1111

31 )

21

()21

()( ωω

3.4

35

In terms of molar quantities, the continuity equation is expressed in terms of the molar

concentration c, and the mole fractions yi as

∑=

=−+⋅∇−=

∇⋅+∂∂ N

iiiii NiRyRJyvt

yc

1

** ,.....3,2,1)()(β

β 3.5

TABLE 3.1: Equations of change of Multi-component Mixtures in terms of the

Molecular Fluxes Total mass:

)( vDt

D ⋅∇−= ρρ

Species mass: (i=1,2,3,…..N) ii

i rjDt

D +⋅∇−= )(ωρ

Momentum: g

Dt

Dv ρτρρ +⋅∇−−∇= ][

Energy: )(])[()()()

21

( 2 gvvpvqvUDt

D ⋅−⋅⋅∇−⋅∇−⋅∇−=+∧

ρτρ

The above have been expressed by Dashti [64] as

( )

×−−×−−=∇−=

=∆−+

=

pp

NHrp

oN

NH

d

v

d

vv

dx

dP

RHdx

dTvC

AF

R

dx

dz

2

323

22 )1(

75.11

150

0)(

/2

3

2

3

ρψ

ψµψ

ψµ

ηρ

η

3.6

A simpler analysis ignores the pressure drop in flow reducing to the conservation equations

for mass and energy with reaction kinetics, used by Yuguo [152] and Dashti [64]

AF

R

dx

dzo

N

NH

/23

η=

3.7

0)(3

=∆−+ NHrp RHdx

dTvC ηρ

3.8

36

For the reaction kinetics, the Temkin-Pyzhev [64],[152] form for the synthesis reaction rate

as a function of the pressure, temperature, and activities is used

−=≡

5.1

5.12

2

3

3

2

232

H

NH

NH

HNaNHA a

a

a

aaKkRR

3.9

where the activities are defined as Pya iii φ= . The individual activities are:

[ ]{ }

+−++=+−++=

+−=

−−−−

−−−−

−−−−−+−

262532

262633

300/)941.5011901.0(2)98.151263.0()541.08402.3(

102761216.0101142945.0104487672.0102028538.01438996.0

104775207.010270727.010295896.0102028538.093431737.0

300exp

3

2

5.0129.0

2

PXTXPXTX

PXTXPXTX

eePePe

NH

N

PTTTH

φφ

φ

3.10

The Arrhenius rate form is given as:

−=RT

Ekk o exp

3.11

and the specific heat capacities of hydrogen, nitrogen, methane and argon of the syngas (T in

Kelvin, Cp in J/mol-K) are expressed as:

=−++=

−+−=−+−=

−−−

−−−

−−−

9675.4*184.4

)1063.210303.0102.175.4(184.4

)106861.01001930.01003753.0903.6(184.4

)102079.01009563.01004567.0952.6(184.4

392524

392522

332222

ArC

TXTXTXCHC

TXTXTXNC

TXTXTXHC

P

p

p

p

3.12

For ammonia, the Shomate equations [28] are given as:

232

3 ***t

EtDtCtBANHCP ++++=

3.13

with T in the range 298-1400 K, and .1000

Tt =

37

18917.0;921168.1;37599.15;77119.49;99563.19 ==−=== EDCBA

have been used for the temperature range of interest (500-800K) and compared with

simplified expressions [96]:

+=+=+=

−

−

−

TXC

TXC

TXC

NHP

NP

HP

3,

3,

3,

1011.2575.29

1093.427.27

1051.301.27

3

2

2

3.14

In this model, the compressor power requirements [96], [130] for isentropic compression, are

obtained with the ideal gas assumption, as

−

−=

− γγ

γγ

/)1(

1

21 1

1 P

PRTws

3.15

The actual work will of course be larger than sw and the ratio actuals ww /

will depend on the

compressor efficiency. In terms of the initial and final temperatures )&( 21 TT , the work can

also be found from:

)( 21 TTCHw ps −=∆−=

3.16

More generally, for a multi-stage compressor with n units, with the compression ratios (Cr) in

all the stages equal, the total work can be estimated from

( )[ ]γγ

γγ /)1(

1 11

−−−

= rs CRTn

w

3.17

where, for an ideal gas 111 RTVP = . The above expressions are based on the assumption that

specific heats remain constant in the pressure and temperature range.

38

Similar models are also used to determine the compression requirement for recycle in the

converter, refrigeration duty, vaporizer and purge systems. Additional objectives will be to

quantify process-variable trade-offs with an aim to progress towards an “optimal design”.

The above model can be used to carry out an energy balance of the Process Flow Diagram

(PFD). The ‘energy input’ components are thus the compressors (FCOMP, RCOMP), while

the ‘energy output’ system is the heat recovery system.

The equilibrium constant is obtained from

689.26.2001

10848863.11051925.5log691122.2log 275 +++−−= −−

TTXTXTKa

3.18

The overall synthesis rate (kmol of ammonia produced per hr per unit volume of catalyst) is

ARη where η is the catalyst effect factor [64]. It is defined as:

36

35

24

2321 ZbTbZbTbZbTbbo ++++++=η

3.19

and may have a significant effect on the overall efficiency. Equation 3.19 is in terms of T and

conversion percentage Z. The coefficients of this equation for three different pressures have

been depicted in table 3.2.

TABLE 3.2: Coefficients of the correction factor polynomial in terms of pressure

Pressure (bar)

0b 1b 2b 3b 4b

5b 6b

150 -17.539096 0.07697849

6.900548 -1.08279e-4 -26.42469

4.927648e-8

38.937

225 -8.2125534 0.03774149

6.190112 -5.354571e-5 -20.86963 2.379142e-8 27.88

300 -4.6757259 0.02354872

4.687353 -3.463308e-5 -11.28031 1.540881e-8 10.46

39

The resulting equations are coupled non-linear partial differential equations, which are

converted to ordinary differential equations and solved by numerical integration [135].

The above model has been used by Siddiq et al [32], [33] to obtain the molar flow rates and

temperature of the syngas in a single-bed catalyst convertor, and the net power produced by

the TSP. For a plant of magnitude similar to that described in Dashti [64] the results for the

nitrogen conversion, syngas temperature and molar flow rates are shown in Figs. 3.1-3.4.

Figure 3.1 : Conversion of Nitrogen along a single-bed catalyst

Figure 3.2 : Syngas temperature in converter

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2

0.25

Distance (m)

Con

vers

ion

of N

2 (Z

)

Conversion of N2 in the synthesis convertor

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5650

655

660

665

670

675

Distance (m)

Tem

pera

ture

(K

)

Temperature in synthesis convertor

40

Figure 3.3 : Molar flow rate in converter

Figure 3.4 : Syngas compression requirement It was found that the overall energy availability from the exothermic synthesis of ammonia is

of the order of 32.4 MW(th), of which about 50% is available in the top 30% of the

convertor. This energy availability was assessed in comparison with the syngas compression

requirement shown in Figure 3.4. As shown, the compressor power is dependent on the

syngas flow-rate, the compression ratio and the initial temperature. As a rough guide a

compression ratio of 3, typical of industrial multistage compressors, would require of the

order of 5 MW(e). The two dominating energy factors of a thermal storage plant based on the

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

2000

4000

6000

8000

10000

12000

Distance (m)

Mol

ar F

low

Rat

e (k

mol

s/hr

)

Molar flow rate of H2, N

2 and NH

3 in convertor

H2

N2

NH3

1 1.5 2 2.5 3 3.5 4 4.5 50

1000

2000

3000

4000

5000

6000

7000

8000

Compression Ratio

Com

pres

sor

Pow

er

(kW

)

41

endothermic dissociation and exothermic synthesis of ammonia are quantified. It is

determined that the syngas compression may greatly exceed the useful work realizable from

such a plant. Factors to investigate in greater detail include the hydrogen/nitrogen ratio, the

purge gas ratio, the ammonia content of the convertor feed-stream and the catalyst effect. The

computer program developed, using the model described above, can investigate all these

effects and quantify the sensitivity of each independent parameter on the overall system

efficiency, restricted, for the moment to only two important and dominating energy

components. Ammonia concentration in the convertor feed, for example, is an important

parameter as it determines the ammonia production, recirculation rate, and refrigeration

requirement. Yuguo and Changing [152] found that, when inerts and the H2/N2 ratio

remained constant, for ammonia content increase from 2 to 2.5%, power consumption

increased from 6 to 6.1 kW-hr/kmol and ammonia production increased 3% from 985 to

1015 MTD.

3.1.2 Mathematical Models for TSP Synthesis reactor is considered a fixed bed catalytic reactor and Pseudo homogenous models

are the simplest to use in Catalytic fixed bed reactor modeling. This model assumes that all

the catalyst surface is totally exposed to the bulk conditions and having the same conditions

as the bulk fluid and can thus be fully described by bulk variables (temperature,

concentration, pressure). The Intrinsic reaction rates are used in the reaction model without

any modification.

42

The simplest Pseudo homogenous model is the one dimensional plug flow model, in which

the fluid is assumed to move as a plug through the reactor tube and the reaction rate which

depends on local species concentration, and temperature, is described as rate of specie

generation or consumption per unit reaction volume [148].

A flat velocity profile and uniform temperature and concentration at the radial cross-section

are assumed. The model ignores axial and radial dispersion and heterogeneous effects due to

solid-gas inter-phase gradients [82],[93],[113][134],[141]. This results in coupled non-linear

first-order ordinary differential equations of the initial value type which can be numerically

solved to obtain concentration and temperature profiles as a function of the axial position.

These equations are considerably simpler than a two-dimensional model which accounts for

axial and radial dispersion but results in coupled partial differential equations requiring

advanced numerical techniques and more computational effort.

For simulation of the synthesis process, mass and energy conservation equations are used,

henceforth referred to as the Two-Equation Model (TEM), for chemically reacting species

similar to the ones used by Dashti et. al. [64] and Yuguo and Changying [152]. A catalyst

spatial factor )(xξ is introduced which may be chosen to be a free parameter or specified

subject to a constraint of total available catalyst quantity. In the former case, a three-zone

single catalyst bed, for example could have discrete values ],,[ 321 ξξξ for )(xξ . The one-

dimensional steady-state mass and energy conservation equations in a pseudo-homogeneous

model [64], for the fractional conversion of nitrogen ,z , along the synthesis reactor length x

and the temperature of the process gas, in a volume element of an adiabatic bed, are given as:

43

AF

Rx

dx

dzo

N

A

/2)(ηξ=

3.20

and

Ap

r RxvC

H

dx

dT)(ηξ

ρ

∆−−=

3.21

Where η is the catalyst activity factor, it is defined to investigate the effects of temperature

and density of the catalyst interior and the difference between these parameters with those of

the catalyst surface, AF oN / is the initial molar flow rate of nitrogen, ρ is the density, rH∆−

is the heat of reaction and AR is the reaction rate.

In the above, the momentum conservation equation is ignored, which has been used, after

numerical solution of the above two equations, by Dashti et. al. [64] for estimating the

pressure drop along the catalyst beds. This introduces a small error as the pressure drop, in

the KBR reactor considered in this work, is less than 2% of the system pressure [64].

3.1.2.1 TEM Model using Activity based Temkin-Pyzhev Form (TP-A):

For the reaction kinetics, the Activity based Temkin-Pyzhev [64] form is used for the

synthesis reaction rate

−= 5.1

5.12

2

3

3

2

22

H

NH

NH

HNaA a

a

a

aaKkR

3.22

The equilibrium constant aK is given by

689.2/6.20017848863.1551925.5log691122.2log 2 ++−+−−−= TTeTeTKa

44

and the reaction constant is given by the Arrhenius rate form [93] as

−=RT

EXk exp10849.8 14

where E=170.56 kJ/mol and respective activities (3

,, NHHN aaa ) and fugacity coefficients are

defined by equation 3.10.

The governing conservation, non-linear differential, equations are numerically solved using

the Runge-Kutta fourth order method in MATLAB™.

The three-bed KBR reactor data used is [10],[23] : Pressure=15 MPa, Inlet Temperature =

643 K, mass flow rate = 183600 kg h-1, mole fractions: 6567006.02

=Hy , 2363680.02

=Ny ,

0269300.03

=NHy , 0202874.0=Ary , 0597140.04

=CHy . Figure 3.5 shows the homogeneous

reactor with the TP-A model. The temperature increases from 643 K to 780 K with a

maximum ammonia mole fraction of 0.1162, corresponding to a molar flow increasing from

486 kmol h-1 to 1930 kmol h-1. It can be seen that the reaction is almost complete at 1.5 m

from inlet.

45

Figure 3.5 : 3-Bed Homogeneous Reactor with TP-A Kinetics

3.1.2.2 TEM Model using Partial Pressure based Temkin-Pyzhev Form (TP-B):

For the reaction kinetics, the Partial Pressure (Power Law) based Temkin-Pyzhev [142] (TP-

B) form is used for the synthesis reaction rate:

αα −

⋅−

⋅=

1

3

2

22

3

1

2

3

3

2

2

H

NH

NH

HNA p

pk

p

ppkR

3.23

for which the rate constants for synthesis and decomposition are

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5600

650

700

750

800

Distance (m)

Tem

pera

ture

(K)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

Distance (m)

Mol

e F

ract

ion

H2

N2

NH3

46

−=RT

kk o

20800exp11 and

−=RT

kk o

47400exp22 , 4

1 1078954.1 ×=ok kmol/m3-atm1.5

catalyst bed-hr, and 162 105714.2 ×=ok kmol-atm0.5/m3 catalyst bed-hr.

The parameter α in the kinetic expressions depends on the catalyst characteristics as well as

the range of operating variables [141] and although Temkin and Pyzhev [195] assigned the

value 0.5 in the power-law model for iron catalysts, Dyson and Simon [183] showed that both

values 0.5 and 0.75 satisfied experimental data. Another point that has been observed for this

expression is that the decomposition rate coefficient 2k , assumed constant, decreases with

pressure [194]. However different values of α can be used for the same catalyst. A limitation

of this expression, clearly, is that it is not valid for very low ammonia composition due to the

divergence of the first term. The value α=0.6 has been used in this work since Dashti et al

[64] have validated the results with industrial data, based on the Kellogg Brown and Root™

horizontal ammonia synthesis reactor.

The governing conservation, non-linear differential, equations are again numerically solved

using the Runge-Kutta fourth order method in MATLAB™. The three-bed KBR reactor data

is used to simulate the behavior of the reactor shown in figure 3.6.

47

Figure 3.6 : 3-Bed Homogeneous Reactor with TP-B Kinetics

With the TP-B model, Figure 3.6 shows a temperature increases from 643 K at inlet to 788 K

with a maximum ammonia mole fraction of 0.1221, and a mole flow increase from 486 kmol

h-1 to 2018 kmol h-1. The error of the power-law model, relative to the activity-based model,

is thus ~6% in ∆T and ~5% in the ammonia molar flow rate. A better model has been found

to be the activity based Temkin-Pyzhev model (TP-A) which does not have a singularity for

zero ammonia composition, which can be the case for a syngas inlet without recycle, and for

which 2k is constant.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5600

650

700

750

800

Distance (m)

Tem

pera

ture

(K)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

Distance (m)

Mol

e F

ract

ion

H2

N2

NH3

48

3.2 Modeling Unit Operations

The TSP model provides a useful description of the process. The simulations have been

developed using Aspen Plus™ and HYSYS™ modules of the AspenTech™ Process

Engineering Simulation Suit. These simulations make use of many of the capabilities of

AspenTech™ Process Engineering Simulation Suit including unit operation models, physical

property methods, models and data, and flowsheeting capabilities like convergence design

specs etc. The model provides rigorous mass and energy balance information for dissociation

process, ammonia production process, Flash tank and other components in storage plant [7].

This model can be used to support the conceptual process design. The TSP model is meant to

be used as a guide for process optimization for maximum energy recovery. It can also be

used as a starting point for more sophisticated models.

Both the Dissociation and Synthesis reactors are considered as fixed bed catalytic reactors.

In AspenTech™ Process Engineering Simulation Suit, the PFR is augmented by including

axial and radial dispersions, for both mass and heat transport [7] resulting in coupled partial

differential equations requiring advanced numerical techniques and more computational

effort.

3.2.1 Dissociation Reactor The dissociation reactor, considered as a fixed bed catalytic reactor, is modeled using a Plug

Flow Reactor (PFR) with a specified uniform external heat flux profile in Aspen Plus™ as

49

shown in figure 3.7. If the length of Plug Flow Reactor is less than its diameter, the axial

dispersion of mass and heat are not significant [7],[148].

Figure 3.7 : PFR Dissociation Reactor in Aspen Plus™

The dissociation is carried out in a temperature range of 520-580 oC and a pressure of 150

bar using Ni-Pt catalyst [128]. The data presented in table 3.3 is used as input to stream

DIS-IN.

TABLE 3.3: Input Data for Dissociation Reactor

Parameter Value Process Pressure P=15MPa (150 bar) Inlet Temperature T=793 K (520 oC) Mass Flow Rate m& =0.125 kgs-1 Reactor Length L=1m Reactor Diameter D=1.5m

The Temkin-Pyzhev reaction rate is specified in Aspen Plus™ through the stoichiometric

coefficients for ammonia (-1), nitrogen (0.5), and hydrogen (1.5) and with basis specified to

be partial pressures, for a liquid-vapor reaction phase. The reaction is specified as a Power-

law class for a kinetic expression of the form:

33exp NHoNH p

RT

Ekr

−=

50

where 3NHr is the reaction rate for decomposition of ammonia, ok is the frequency factor and

its value is 1.33x108 kmol m-3 s-1 Pa-1 while activation energy E equals 190000 kJ kmol-1 [6].

Figure 3.8 : Dissociation Reactor Exit Composition

Figure 3.9 : Dissociation Reactor Temperature Profile

Simulation results of dissociation reactor are presented in figures 3.8 and 3.9. Ammonia

conversion completes at reactor length of 1m while the maximum possible temperature at

reactor exit is around 535 oC.

51

3.2.2 Synthesis Reactor-Aspen Plus Model

The Synthesis reactor is modeled using an Adiabatic Plug Flow Reactor (PFR) as shown in

figure 3.10. Since Synthesis reactor is adiabatic, radial dispersion in negligible because there

is no driving force for long range gradients to exist in the radial direction hence resulting in

two dimensional model [7],[148].

Figure 3.10 : PFR Synthesis Reactor in Aspen Plus™ The three-bed KBR reactor data presented in table 3.4 is used as input to stream SYN-IN.

TABLE 3.4: Input Data for Synthesis Reactor

Parameter Value Process Pressure P=15MPa (150 bar) Inlet Temperature T=643 K (370 oC) Mass Flow Rate

totalm& =183600 kgh-1

Hydrogen Mole Fraction 6567006.0=Hy Nitrogen Mole Fraction 2363680.0=Ny

Ammonia Mole Fraction 0269300.03

=NHy

Argon Mole Fraction 0202874.0=Ary Methane Mole Fraction 0597140.0

4=CHy

Reactor Length L=4.5m Reactor Diameter D=5.046m

52

The Temkin-Pyzhev reaction rate is specified in Aspen Plus™ through the stoichiometric

coefficients for nitrogen (-1), hydrogen (-3) and ammonia (+2). The basis is specified to be

partial pressures, for a vapor reaction phase. The reaction is specified as a Power-law class

for a kinetic expression form of partial pressure based Temkin Pehzev (TP-B):

][][ ceDrivingFortorKineticFacr ∗=

The components are specified in Table 3.5.

TABLE 3.5: Reaction Input for Temkin-Pyzhev Power-Law Expression in Aspen Plus™

Forward Reverse Nitrogen 1 0 Hydrogen 1.8 -1.2 Ammonia -1.2 0.8 A 9.79227 37.785812 B -10467 -23852.69 C - - D - -

The parameter α in the kinetic expression is taken to be equal to 0.6.

Figure 3.11 : Synthesis Reactor Exit Composition (Aspen Plus™)

53

Figures 3.11 and 3.12 show a temperature increases from 370 oC at inlet to 513.82 oC with a

maximum ammonia mole fraction of 0.1231 at reactor length 1.5 m.

Figure 3.12 : Synthesis Reactor Temperature Profile (Aspen Plus™)

3.2.3 Synthesis Reactor- HYSYS Model

Synthesis Reactor is modeled as Plug Flow Reactor PFR-100 shown in figure 3.13 The

Temkin Pyzhev kinetic rate expression, with 6.0=α is used in HYSYS™.

Figure 3.13 : Plug Flow Reactor in HYSYS™

The Temkin-Pyzhev Model B is specified in HYSYS™ through the stoichiometric

coefficients for nitrogen (-1), hydrogen (-3) and ammonia (+2) and the forward order of the

rate information (nitrogen 1.00, hydrogen 1.80, and ammonia -1.20) and the backward order

(nitrogen 0, hydrogen -1.20, and ammonia 0.80). The base component is specified to be

nitrogen, the basis is specified to be pa

minimum temperature -273.1

reaction rate to be specified in the form

bTRTEAk ∗−∗= )/exp(

'

)/exp( ''' bTRTEAk ∗−∗=

Figure 3.14 and Fig.3.15 show that for an inlet temperature of 370

no pressure drop is assumed, and saturation is observed at 1.5m for an ammonia

fraction of 0.123.

Figure 3.14 : Synthesis Reactor Temperature Profile

54

nitrogen, the basis is specified to be partial pressures, for a vapor reaction phase with

273.1oC and maximum temperature 3000oC. HYSYS™ requires the

reaction rate to be specified in the form )(* ' fkBasisfkr ∗−=

, with 17895=A and 87027= kJE

with 16' 105714.2 ×=A , kJE 109832.1 5' ×=

show that for an inlet temperature of 370oC, the outlet is 513.7

no pressure drop is assumed, and saturation is observed at 1.5m for an ammonia

: Synthesis Reactor Temperature Profile (HYSYS™)

rtial pressures, for a vapor reaction phase with

C. HYSYS™ requires the

)(' Basisf where

0,/ =bkmolkJ ,

kmol/ , 0' =b .

C, the outlet is 513.7oC,

no pressure drop is assumed, and saturation is observed at 1.5m for an ammonia mole

Figure 3.15 : Synthesis Reactor Exit Composition (HYSYS™) 3.2.4 Flash Tank

The Flash Tank is modeled using

feed into two outlet streams, using rigor

Figure 3.16 : Flash Tank in Aspen Plus™

55

: Synthesis Reactor Exit Composition (HYSYS™)

is modeled using a FLASH2 seperator (figure 3.16). It is

feed into two outlet streams, using rigorous vapor-liquid equilibrium.

: Flash Tank in Aspen Plus™

used to separate

56

TABLE 3.6: Flash Tank Output

Molar Composition

Equilibrium Liquid Mole

Fraction

Equilibrium Vapor Mole

Fraction Hydrogen 0.5776749 0.00638187 0.61373434 Nitrogen 0.21166301 0.03253954 0.2229691 Ammonia 0.12316376 0.93099292 0.07217439 Argon 0.02218853 0.00406039 0.02333276

3.2.5 Purge Gas & Recycle A splitter is used to divide the VAPOR stream into PURGE and RECYCLE streams with a

purge split fraction of 0.095.

Figure 3.17 : Splitter in Aspen Plus™

Stream data for VAPOR, PURGE and RECYCLE streams is presented in table 3.7.

TABLE 3.7: Molar Flow Rates of Components in and out of Splitter

Component VAPOR PURGE RECYCLE Molar Flow

Rate (kmol/hr)

HYDROGEN 9526.794 905.04543 8621.74857 NITROGEN 3461.075 328.802125 3132.272875 AMMONIA 1120.339 106.432205 1013.9068 ARGON 362.1868 34.407746 327.779

57

A mixer (MIXER) is used to combine the Feed stream (FEED1) from storage tank and

Recycle stream (RECYCLE) as shown in figure 3.18

Figure 3.18 : Mixer in Aspen Plus™

3.2.6 Heat Exchangers and Waste Heat Recovery

3.2.6.1 Counter Flow Heat Exchanger (CF-HX):

The stream DIS-IN is pre-heated by passing it through the counter flow heat exchanger (CF-

HX) in order to increase its temperature from 30 oC to 370 oC.

Heat duty of CF-HX= + 6.898 kWth

Figure 3.19 : Counter Flow Heat Exchanger (CF-HX)

58

3.2.6.2 Thermal Heat Exchanger (SRIN-HX):

The temperature of the output from mixer (MIX-OUT) is 30 oC so it is heated in SRIN-HX to

a temperature of 370 oC prior to enterance in SYNRCTR.

Figure 3.20 : Thermal Heat Exchanger (SRIN-HX)

Heat Duty of SRIN-HX = +51828.236 kWth

3.2.6.3 Thermal Heat Exchanger (SROUT-HX):

Heat generated in SYNRCTR is extracted in thermal heat exchanger SROUT-HX prior to

entry in FLASHT.

Figure 3.21 : Thermal Heat Exchanger (SROUT-HX)

Heat Duty of SROUT-HX: -76532.016 kWth

59

3.3 Modeling the Integrated Plant

Figure 3.22 : Integrated Plant Net Heat Recovery = + 6.898 kWth + 51828.236 kWth -76532.016 kWth = -24696.882 kWth

Net Heat Recovery = - 24.7 MWth

3.4 Two Equation Model (TEM) Validation: The one-dimensional pseudo-homogeneous model has been validated with industrial data to

quantify its accuracy for modeling and simulation of the ammonia conversion process.

Elnashaie and Elshishini [142] have compared results of the outlet temperatures from a

three-bed industrial ammonia reactor with inter-stage cooling and found a maximum

deviation of less than 2%. Industrial data from the Kellogg type ammonia synthesis reactor,

used in this work, is also considered for model validation [54][64][152] and optimization

with one- and two-dimensional models. Yuguo and Changying [152], using a one-

dimensional model for an ammonia plant incorporating unit operations for compression,

60

ammonia synthesis, vapor-liquid separation, and refrigeration, calculate 11.58% ammonia in

converter effluent compared with industrial data of 12%, i.e., a relative error of -3.5%.

Sadeghi and Kavianiboroujeni [54], validating the models for comparison of the bed outlet

temperatures with data from a four-bed Kellogg ammonia reactor, compute the first-bed

outlet temperature as 505.58°C from a one-dimensional steady-state heterogeneous model,

and 502.85°C from a two-dimensional rigorous cylindrical model, compared with the

industrial data temperature of 501.5°C giving a relative error of 0.81% and 0.27%

respectively. The relative errors reported in the outlet temperatures for all beds are in the

range 0.8% to 1.5% for the one-dimensional model compared with 0.2% to 0.4% with the

two-dimensional model.

Figure 3.23 : Comparison of 1-D (TP-B) model, HYSYS™, and Aspen Plus™ results

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5640

660

680

700

720

740

760

780

800

Distance (m)

Tem

pera

ture

(K)

RK-4HYSYSAspenPlus

1D (TP-B)

HYSYS

Aspen Plus

61

Figure 3.23 illustrates the comparison between the 1-D model (TP-B), HYSYS™ and Aspen

Plus™ results. Though the 1-D model varies significantly from the HYSYS™ and Aspen

Plus™ results, there is little difference in the saturation values. Thus all models give

saturation, and hence the physical dimensions of the first bed of the reactor as between 1.5m

and 1.75m.

Percentage errors in 1-D, TEM-TPA and TEM-TPB models compared with 2-D HYSYS™

and Aspen Plus™ models are presented in table 3.8.

TABLE 3.8: Percentage errors in 1-D Models compared with HYSYS™ and Aspen Plus™

Model 0T∆ % error 0

3NHy % error

1-D RK-4 TP-A 137 1.7 0.1162 -5.5 1-D RK-4 TP-B 145 7.6 0.1221 -0.7 HYSYS™ 134.7 0 0.1230 0 Aspen Plus™ 134.7 - 0.1230 -

It can be concluded that a one-dimensional model is adequate, and justified, to simulate the

ammonia conversion process and to carry out an optimization analysis, and uses considerably

less computational effort than a two-dimensional rigorous model.

62

4 PLANT OPTIMIZATION

4.1 Review of Optimization Techniques Optimization is a widely used engineering tool for enhancing the energy efficiency, and

hence the economic competitiveness, in chemical industry. It is aimed at finding some

‘optimal’ set of design and operating parameters which enables industry to function in a ‘best

possible’ way.

Design parameters, such as physical dimensions and materials of plant components, are the

basis of fabrication and subsequent plant erection. Operating parameters, on the other hand,

involve process variables, such as flow rates, pressures and temperatures, which are

integrated into the overall plant units. As an example, a urea fertilizer chemical process plant

has four large integrated units: reforming, ammonia, urea, and utilities, all of which involve

inter-dependent process variables. The overall system is thus not only large in magnitude, but

its interdependence in the form of re-cycle streams, for example, renders the determination of

optimality as a complex non-linear problem. The process of optimization is thus both

theoretical and practical as it is based on mathematical models which are non-linear and

utilize material properties at actual plant conditions. These can be obtained from on-line

chemical analysis during plant operation. Once formulated, the system of equations needs to

be solved by analytical or numerical methods often requiring powerful computer hardware

and sophisticated simulation software. Optimization analysis for a solar thermal plant

requires a mathematical framework to model the underlying processes, the physical and

63

chemical data of the plant and its materials, and numerical techniques to carry out a

simulation to achieve the objective of determining optimality.

Figure 4.1 : Optimization Process The objective of optimization studies is to determine optimality which will result in

maximum efficiency of the plant, by which it will be taken to mean optimality of the

synthesis reactor in the integrated plant. Several other objectives, as diverse as plant and

equipment sizing, minimizing inventory charges, and allocating resources among several

processes, may be defined [65][100]. Other terms that are analogously used for objective

function, process model, and constraints are economic cost (or profitability) function and

feasible solution. An optimization problem, formulated as an objective function ),( 21 xxf

with two independent variables 21 , xx , subject to an equality constraint 0),( 21 =xxg and

possibly N inequality constraints of the form Nixxhi L2,1,0,0),( 21 =≥ will then have a

feasible region illustrated by Figure 4.1 [100]. An optimal solution will then be determined as

falling in the feasible region defined as satisfying all the constraints and having an extremum.

For such a problem, the dashed lines in the figure would show the infeasible regions for the

64

inequality constraints while the solid line would show the equality constraint. Of all the

values of the variables 21 , xx lying in the feasible region, one would need to determine the

particular one set, or a number of sets, which would correspond to the extremum.

Possible obstacles in the optimization process are non-linearities, discontinuities,

insensitivity of the objective function to independent variables, and local extrema confused

with a global extremum.

Figure 4.2 : Mathematical Methodology to solve governing equations

The mathematical methodology is then identified to solve the governing equations and carry

out a simulation to obtain the optimal set of parameters. These methodologies may be

65

broadly classified as deterministic or stochastic. The former are illustrated in Figure 4.2 [100].

Thus, numerical techniques such as the Runge-Kutta method for solving a non-linear set of

coupled first-order ordinary differential equations may be used, while the latter is based on

either an analogous stochastic simulation of the phenomena or on stochastic methods to solve

a deterministic set of governing equations. A typical example of a full stochastic simulation

is analog, or biased, Monte Carlo simulation, while a typical example of a stochastic

technique for a deterministic set of governing equations are random search methods such as

Genetic Algorithms (GA).

One way of classifying modeling problems [100] is from the degrees of freedom

evf NNN −= , in terms of the number of variables vN , and the number of independent

equations eN . When 0=fN the problem is exactly determined and for a linear set of

independent equations, there is a unique solution, while for non-linear equations there may

be multiple solutions or no real solution. Such problems do not constitute an optimization

problem. When 0>fN , the problem is underdetermined and at least one process variable

can be optimized. Conversely 0<fN , constitutes an over-determined problem and the set

of equations has no solutions and methods, such as the least squares method, can be used to

determine the unknowns.

Widely used optimization methods include the following areas [100]: one-dimensional search,

unconstrained multivariable optimization, linear programming, nonlinear programming,

optimization involving discrete variables, and global optimization (operations research,

66

including Monte Carlo, heuristic methods, GA and evolutionary methods). Some widely used

techniques for obtaining optimality are (i) the deterministic variational functional optimality

based on Pontryagin’s Maximum Principle, and (ii) the Genetic Algorithm optimization

search [80][100]. The GA search method has been used [65],[100] for the optimization of an

ammonia synthesis reactor [65] shown in Figure 4.3.

Figure 4.3 : Counter-Flow Ammonia Synthesis Reactor

First, the governing equations are numerically solved to obtain the temperature and

concentration profiles shown in Figure 4.4.

Figure 4.4 : Temperature & Concentration Profiles along Converter Length

67

Figure 4.4 uses the same scale for Temperature and Concentration, it is not clear in reference

65 , whether these values of temperature and concentration have been normalized to show

them on the same scale or not.

The numerical solution is followed by an optimization exercise in which an objective

function ),,,(2 gfN TTNxf is defined along with the three governing equations, taken as

constraints. Thus, the problem has one variable (x; the reactor length) one degree of freedom

)1( =fN and, being underdetermined, can be solved for optimality. The GA search is carried

out as shown in Figure 4.5.

Figure 4.5 : GA Search Algorithm The optimal solution found for the exit conditions shown in Table 4.1.

TABLE 4.1: Optimal solution for the exit conditions

68

The optimization thus concludes that a reactor with initial guesses of Nitrogen concentration,

Mole fraction, feed gas and reacting gas temperatures, reactor length and cost result in an

optimal solution with values given in table 4.1.

Figure 4.6 : Four-Bed Synthesis Reactor

Sadeghi and Kavianiboroujeni [54] have used the Genetic Algorithm for a 1-D and 2-D

optimization of a Kellog-type ammonia plant, located at Khorasan (Iran). The axial reactor

(Figure 4.6) has four promoted-Fe catalytic fixed beds with a heat exchanger at the top.

Syngas flows vertically upwards in the spaces between the two walls of the reactor, where it

is pre-heated, then turns down through the beds and from the bottom fourth bed again turns

upwards, exiting from the top of the reactor. The independent parameters investigated are the

quench flow and quench temperature. Between the beds, the hot syngas is mixed with quench

streams (shown in three streams between the beds) to control the temperature.

69

Figure 4.7 : Effect of Quench gas on conversion efficiency The purpose of injecting the quench streams is, clearly, to increase the ammonia production

from the reactor; and since the forward synthesis chemical reaction is favoured by high

pressure and low temperature, the challenge is to reduce the temperature at every bed exit.

The ammonia and nitrogen conversion obtained by Sadeghi and Kavianiboroujeni [54] by a

numerical solution of the mass and energy balance equations is shown in the Figure 4.7. It

can be seen that most of the ammonia conversion takes place in the first bed even though it is

the shortest. The effect of the quench gas is to reduce the ammonia conversion after every

bed exit but this picks up as flow proceeds.

The paper uses GA for obtaining optimal temperature distribution, in this nonlinear

optimization problem, resulting from a given quench flow, and subsequently optimal quench

flow given quench temperature. In the optimization problem, the objective function is the

70

ammonia outlet flow-rate, while the constraints are (i) KT 800< , in the reactor for avoiding

hotspots, (ii) an ascending nitrogen conversion during optimal flow: xxx ZZ ∆+< || , and (iii)

for the syngas: outin TT < .

Figure 4.8 : GA Algorithm for obtaining optimal temperature distribution

The flow-chart, Figure 4.8, for this optimization is reproduced from the paper [54] and their

results, in Figure 4.9, give maximum ammonia conversion at a quench temperature of 650 K

and a maximum conversion flow-rate of 47%

71

Figure 4.9 : Optimal and Normal Ammonia Production Rates The important changes between normal and optimal operations for nitrogen conversion and

reaction rate are shown in Figure 4.10.

Figure 4.10 : Optimal and Normal Nitrogen Conversion and Reaction rates GA used for obtaining optimal temperature distribution has increased the ammonia

production of the Khorasan plant by 3.3% (8,470 tons per year).

When the reactor is isothermal, different catalysts can be loaded to enhance productivity and

when it is non-isothermal i.e. it has temperature gradients, then the catalyst may be non-

72

uniformly loaded or different catalysts may be used for different temperature regions, to

enhance productivity. Finding the optimal (spatial) distribution of catalyst is crucial to

optimizing the performance. Numerical search can be carried out by dividing the reactor in

zones and assuming uniform values of catalyst material in each zone; this will mostly result

in a sub-optimal solution.

4.2 Optimal Analysis using Variational Calculus

Variational methods have also been used to find optimal configurations for process variables

in a synthesis convertor. These methods originate from the works of Leibniz (1646-1716) and

Newton (1643-1727) credited for inventing and formalizing Calculus; followed by

“Variational Calculus” attributed to Leonhard Euler (1707-1783) through his published work

of 1733. Among the several contributors to variational calculus were Lagrange (1736-1813),

Legendre (1752-1833), Gauss (1777-1855), Cauchy (1789-1857), and Poisson (1781-1840).

The field of Optimal Control [180] is an area of optimization in which the “best possible”

strategy is chosen using the calculus of variations. While calculus can be used for

optimization of a function of variables, calculus of variations is used to obtain the extremum,

or stationary condition, of a functional (function of a function) by finding the function which

extremizes the functional. The first variational calculus optimal problem was the

Brachistochrone (shortest time) Problem solved by Bernoulli in 1696 [185]. The formulations

used are by Lev Semenovich Pontryagin (1908-1988), who developed the Maximum

Principle, and the terminology of “Bang-Bang control” to steer a system with maximum or

minimum control parameters, and of Bellman (1920-1984) who extended works of Hamilton

73

(1805-1865) and Jacobi (1804-1851) to the now well-known Hamilton-Jacobi-Bellman

(HJB) equations in Dynamic Programming.

Variational calculus [181],[182],[185] is used in areas that include optimal control, particle

transport, mechanics, optics and chemical plant design [158]. There is a vast range of

problems that determine complexity, such as whether the functional involves one or several

functions, derivatives of functions, and one or more than one independent variable. Another

class of variational calculus problems involves constrained problems with algebraic, integral

or differential equation constraints.

The optimality conditions for Pontryagin’s Maximum Priciple (PMP) can be derived from

the first principle of conservation of matter. Consider the reversible Ammonia synthesis

process

322 23 NHHN ⇔+

The mass balance equation for this reaction can be written in dimensionless form as:

],,,[3 aNHHN

N Kyyyfxd

dy =′

4.1

Where Hy ,3NHy and Ny represent the mole fractions of hydrogen, ammonia and nitrogen

respectively while x′ is the normalized distance along catalyst bed (m).

Mole fractions of hydrogen (Hy ) and ammonia (3NHy ) are related to nitrogen mole fraction

( Ny ) as:

)(233 N

oN

oNHNH yyyy −+=

4.2

74

)(3 NoN

oHH yyyy −−=

4.3

Where oNHy

3, o

Ny and oHy are initial mole fractions of ammonia, nitrogen and hydrogen

respectively.

In equation 4.2 and 4.3, the term NNoN zyy =− )( represents the converted moles of nitrogen

and fractional conversion of nitrogen can be presented as:

oN

NoN

N y

yyz

−=

4.4

So equations 4.2 and 4.3 in terms of fractional conversion of nitrogen are

NoN

oNHNH zyyy 2

33+=

4.5

NoN

oHH zyyy 3−=

4.6

Therefore mole fractions of hydrogen and ammonia can be removed from the reactor model

equation and making it only a function of Ny

))(),(())(,( xxyfKyfxd

dyNaN

N ′′≡−=′

θθ

4.7

Where ),( θNyf is the rate of reaction equation in terms of nitrogen mole fraction Ny and

temperature θ at any point along the length of reactor such that

at )1,0(;,0 ∈′== xyyt oNN

75

)(x′θ is the control variable for which the optimal variation along the length of the reactor is

sought, that maximizes the exit conversion.

Equation 4.4 can be written as:

)1( NoNN

oN

oNNN zyyyyzy −=⇒−=−

4.8

Differentiating Equation 4.8 w.r.t x′ yields

),(1 θN

NoN

NNoN

N yGxd

dy

yxd

dz

xd

dzy

xd

dy −=′

−=′

⇒

′−=

′ 4.9

The objective is to maximize the nitrogen conversion xd

dzN

′, so we can write

∫∫ ′−=′′

−=1

0

1

0

),(1

xdyGxdxd

dy

yM N

NoN

θ

4.10

Now if we consider that the optimal temperature profile that gives the maximum conversion

( M ) is )(x′θ and consider an infinitesimal variation in )(x′θ to make it δθθ + then )(xyN ′

will change to NN yy δ+ and M to MM δ+ . Thus,

∫ ′∂∂+

∂∂+−=+

1

0

]),([ xdG

yy

GyGMM N

NN δθ

θδθδ

4.11

and since, ∫ ′−=1

0

),( xdyGM N θ

thus,

∫ ′∂∂+

∂∂−=

1

0

] xdG

yy

GM N

N

δθθ

δδ

4.12

Comparing equation 4.7 and 4.9 yields

76

))(),(( xxyfxd

dzy

xd

dyN

NoN

N ′′=′

−=′

θ

4.13

Consider the change in f

δθθ

δδ∂∂+

∂∂= f

yy

ff N

N 4.14

δθθ

δδδ∂∂+

∂∂=

′=

′−⇒

fy

y

f

xd

dy

xd

dzy N

N

NNoN

δθθ

δδ∂∂+

∂∂=

′⇒

fy

y

f

xd

yd N

N

N

4.15

Multiplying equation 4.15 by the Lagrange Multiplier (sensitivity coefficient/ adjoint

variable), )(x′λ integrating from 0=′x to 1=′x and adding the result to equation 4.12 gives:

∫ ′

∂∂−

∂∂−

∂∂+

∂∂+

′=

1

0

xdG

yy

Gfy

y

f

xd

ydM N

NN

N

N δθθ

δδθθ

λδλδλδ

4.16

Rearranging equation 4.16 yields

∫∫ ′′

+′

∂∂−

∂∂−

∂∂+

∂∂=

1

0

1

0

xdyxd

dxd

Gy

y

Gfy

y

fM NN

NN

N

δλδθθ

δδθθ

λδλδ

4.17

The last term on the right hand side of equation 4.17 is integrated by parts to give:

∫∫ ′′

−−=′′

1

0

1

0

)0().0()1().1()( xdxd

dyyyxdy

xd

dNNNN

λδλδλδδλ

4.18

If we consider the feed concentration to have a fixed value, then, 0)0( =Nyδ and we impose

77

the boundary condition on the Lagrange multiplier as 0)1( =λ and define the Hamiltonian (

H ) of the system as,

GfH −= .λ

4.19

Then equation 4.16 can be written as,

∫∫ ′+′∂

′−

∂∂=

1

0

1

0

xdH

xdyxd

d

y

HM N

N

δθδθδλδ

4.20

For )(x′θ to be the optimal temperature profile, we must have, 0=Mδ .Thus the optimality

conditions are:

0=′

−∂∂

xd

d

y

H

N

λ and 0=∂

δθH

for all values of x′ 4.21

The adjoint equation is therefore:

Ny

H

xd

d

∂∂=

′λ

and 0)1( =λ 4.22

and the optimality condition according to Pontryagin’s maximum principle is:

0=∂δθH

4.23

Which gives the global maxima for the objective function [155],[180]

From the optimality condition (4.23), it can be shown that the problem reduces simply to that

of finding the temperature profile )(x′θ that maximizes the net rate of reaction at each point

along the length of the reactor. Since N

Ny

yfG ),(θ−= , the Hamiltonian can be written in

terms of ),( Nyf θ as follows:

78

xd

dzf

xd

dy

yff

yfGfH NN

oN

oN ′

+=′

−=−=−= .1

.1

.. λλλλ

Or

fxd

dzH N .λ+

′=

4.24

4.3 Parametric Sensitivity Analysis This section considers the optimal design of TSP by calculating the process variable

sensitivities for different components.

Sensitivity analysis is an optimization tool for determining how a process reacts to varying

key operating and design variables. It can be used to vary one or more flowsheet variables

and study the effect of that variation on other flowsheet variables. It is a valuable tool for

performing “what if” studies. Sensitivity analysis can be used to verify if the solution to a

design specification lies within the range of the manipulated variable [55].

4.3.1 Effect of Temperature on Dissociation

Objective: Minimize the molar composition of Ammonia in stream DIS-OUT

Manipulated Variable: Process Temperature

The results of Temperature Sensitivity of dissociation reaction at a pressure of 150 bar are

shown in figure 4.11. The favorable range of values for process temperature are from 450-

850 oC after which catalyst burns out and there is no conversion of ammonia.

79

Figure 4.11 : Effect of Temperature on Dissociation

4.3.2 Effect of Flow Rate on Dissociation

Objective: Minimize the molar composition of Ammonia in stream DIS-OUT

Manipulated Variable: Mass flow rate of ammonia in stream DIS-IN

Figure 4.12 : Effect of Flow Rate on Dissociation

The results of Flow rate Sensitivity of dissociation reaction at a pressure of 150 bar and

temperature of 520 oC are shown in figure 4.12. If the mass flow rate of ammonia in stream

80

DIS-IN increases 550 kg/hr, conversion process is not complete and mole fraction of

ammonia in stream DIS-OUT increases.

4.3.3 Effect of Pressure on Synthesis

Objective: Maximize the molar composition of Ammonia in stream SYN-OUT

Manipulated Variable: Process Pressure

Figure 4.13 : Effect of Pressure on Synthesis

The results of Pressure Sensitivity of Synthesis reaction at a temperature of 370 oC are

shown in figure 4.13. There exist nearly a direct proportionality in between increase in

pressure and mole fraction of ammonia at reactor exit from 100 bar to 500 bar. The limiting

case are the pressures below 100 bar, where ammonia production decreases.

4.3.4 Effect of Temperature on Synthesis

Objective: Maximize the molar composition of Ammonia in stream SYN-OUT.

Manipulated Variable: Process Temperature

81

Figure 4.14 : Effect of Temperature on Synthesis The results of Temperature Sensitivity of Synthesis reaction at a pressure of 150 bar are

shown in figure 4.14. As temperature increases, there is a decrease in ammonia production in

the reactor. It is clear from figure 4.14 that a temperature range of 280-400 oC is suitable for

a process pressure of 150 bar.

Figure 4.15 presents the results of parametric sensitivity for synthesis reactor. For higher

pressures, Ammonia production is possible at even lower process temperatures.

Figure 4.15 :Temperature & Pressure Parametric Sensitivity for Synthesis

82

4.3.5 Effect of Flash Temperature on Liquid Ammonia Separation For choosing flash temperature, there are two constraints, maximum liquification of

ammonia is desirable and the mole fraction of ammonia should be maximum in the stream

NH3PROD.

To achieve the desired results, two sensitivities are designed that yield exactly opposite

results to each other. Figure 4.16 depicts a decrease in molar flow of ammonia in stream

PRODNH3 with increase in temperature while an increase in flash temperature favors

increase in mole fraction of ammonia in product stream (figure 4.17).

Figure 4.16 :Effect of Flash Temperature on Ammonia Flow Rate Figure 4.16 depicts a decrease in molar flow of ammonia in stream PRODNH3 with increase

in temperature while an increase in flash temperature favors increase in mole fraction of

ammonia in product stream (figure 4.17).

83

Figure 4.17 :Effect of Flash Temperature on Ammonia Mole Fraction 4.3.6 Effect of Purge Fraction on Ammonia Liquification

The effect of purge fraction in Splitter on Ammonia liquification in stream PRODNH3 is

presented in figure 4.18 which predicts a direct proportionality between the two factors but

the maximum purge fraction is limited by the fact that Syngas is precious and higher purge

fractions will result in decrease in mass flow rate of the closed loop system.

Figure 4.18 :Effect of Purge Fraction on Ammonia Liquification

84

4.3.7 Effect of Recycle Stream on Synthesis

By using Recycle Stream, ammonia mole fraction in SYN-OUT stream increases from

0.1231 to 0.1355.

85

5 AN OPTIMAL STORAGE PLANT

5.1 Process Modifications This section is aimed at finding the optimal process parameters specially optimal temperature

profile for synthesis reactor using the principles of variational calculus described in section

4.2.

5.1.1 Optimal Analysis Problem Formulation- Process Modifications

The optimization problem for the activity based Two Equation Model (TEM-TPA) can be

formulated as: maximize ∫=−=ΕL

dxdx

dzzLz

0

)0()( subject to the constraints given by the

governing equations 3.20 and 3.21. The Hamiltonian is written as

2211 ffdx

dz λλ ++=Η

5.1

Where 21 λλ and are Lagrange multipliers and 21 fandf are constraint equations referring to

mass and energy conservation equations.

Equation 5.1 yields the Lagrange multipliers (Co-state equations) as:

zdx

d

∂Η∂−=1λ

5.2

and

Tdx

d

∂Η∂−=2λ

5.3

86

with the boundary conditions 0)()( 21 == LL λλ . For a stationary Hamiltonian 0/ =∂Η∂ u ,

where u is the control variable that represents the optimal temperature.

5.1.2 OEM using Activity based Temkin-Pehzev form (OEM-TPA)

Considering the case 02 →f , i.e. only the mass conservation equation, henceforth referred

to as the One-Equation Model (OEM-TPA), as considered by Mansson and Andresen [157],

an optimal condition can be found for the temperature T as the control variable. This reduces

to the case of a single Lagrange multiplier λλ ≡1 . The Hamiltonian is thus 1)1( fλ+=Η ,

and the optimality condition is

∂∂==

∂Η∂≡

∂Η∂

AF

R

TTu N

A

/20

0

ηξ

5.4

The optimal temperature distribution can be obtained by solving the equation

000

=∂∂+

∂∂

T

R

TR A

A ηη

5.5

Figure 5.1 shows the process gas temperature, the optimal temperature obtained from

equation 5.5, and the equilibrium temperature [1]. The gas temperatureT increases rapidly in

the reactor saturating to 780 K while the predicted optimal temperatureoptT starts at a high

value, decreases rapidly and saturates to 738 K. The equilibrium temperatureeqmT , following

the same trend as optT saturates to the higher value of 786 K. Best ammonia conversion is

thus achieved by realizing this optimal profile.

87

Figure 5.1 : Homogeneous reactor with 1-D Model (TEM-TPA) showing gas temperatureT , equilibrium temperature eqmT , and optimal temeprature optT

The equilibrium temperature,eqmT , is defined as the temperature at which the optimal

concentration would be at equilibrium. Thus, for the synthesis reaction 322 23 NHHN →+ ,

the optimal molar fractions oiX and system pressureP are used to obtain the equilibrium

constant at zero pressure0aK from which eqmT is found.

)()(

)(1

22

3

,3

,

2,

20NoHo

NHoa XX

X

PK = .

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5600

650

700

750

800

850

900

950

1000

Topt

Teqm

T

Distance (m)

Tem

pera

ture

(K)

88

For the equilibrium composition [96], we determine the extentε of the reaction, with the

optimal composition taken as the initial composition, from a numerical solution of the

equilibrium relationship

23

22

)()()3(

)2()2(3 PTK

nn

nnoptaoo

NoH

oT

oNH =

−−−+

εεεε

.

The optimal and equilibrium temperatures show a trend, especially at reactor inlet, which

represent a theoretical ‘goal’ to achieve optimality. It can be seen from Figure 5.1 that there

is a substantial difference between the inlet temperatures, actual and optimal, but this

decreases as the reaction proceeds. Ideally, the inlet gas should be heated to the optimal

temperature and a heat exchanger should gradually remove the heat of the (exothermic)

reaction so that the resulting temperature decreases, rather than increasing, as gas flows in the

reactor. The optimal profile of figure 5.1 is thus in line with the requirement of a high

temperature to favor a high reaction rate at inlet, but as the exothermic ammonia synthesis

reaction proceeds, heat is generated and, according to Le Chatelier’s principle, the reaction is

driven in the backward direction to favor reactants. Thus optimality requires a gradual

decrease in temperature, to increase the equilibrium constant given by the Van’t Hoff

equation, and subsequently increase product formation [93],[96]. The actual temperature, as

shown in figure 5.1 is in sharp contrast to the optimal temperature since the inlet temperature

is limited by process parameters and the limits of the catalyst supplied by the manufacturer.

This work evaluates the approach-to-optimal by the use of inter-bed heat exchangers which

succesively reduce inlet temperatures and permit saturation to the optimal profile, achieving

89

higher conversion, in line with the S-300 Haldor Topsøe™ [21] which is a feature of all new

ammonia plants.

5.1.3 OEM using Partial Pressure based Temkin-Pehzev form (OEM-TPB)

A simpler formulation [141] is here considered for the determination of the optimal heat

removal strategy with the objective of maximizing ammonia conversion. In this case, the

energy equation (3.21) can be written, with a control variable u~ , as

uxTxzfuRxHdx

dTvC Arp

~)](),([~)()( 2 −≡−∆= ηξρ

5.6

We thus seek heat removal max

~0 uu ≤≤ which can be found by expressing the Hamiltonian as

)( 2211 uffdx

dz −++=Η λλ

5.7

where )/(~pvCuu ρ≡ . The Costate equations are then

])1[( 22

11

1

z

f

z

f

zdx

d

∂∂+

∂∂+−=

∂Η∂−= λλλ

5.8

and

])1[( 22

11

2

T

f

T

f

Tdx

d

∂∂+

∂∂+−=

∂Η∂−= λλλ

5.9

With boundary conditions )(0)( 21 LL λλ == .

The Stationarity condition is defined as:

0)1(0 22

21

1 =−∂∂+

∂∂+==

∂∂ λλλ

u

f

u

f

u

H

5.10

90

20 λ−==∂∂

u

H

5.11

Substituting value 2λ of in equation 5.9 yields

0])1[( 22

11

2 =∂∂+

∂∂+−=

T

f

T

f

dx

d λλλ

5.12

This formulation yields the optimal temperature from a solution of

0

1

3

2

222

2

3

121

0 2

3

3

2

2=

−

=

∂∂

−αα

H

NH

NH

HN

A

P

Pk

RT

E

P

PPk

RT

E

T

R

5.13

Solving Eqn. 5.13 is thus analogous to maximizing the ammonia conversion in the reactor.

For equation 5.13, the optimal temperature is found in a compact form:

βln12

R

EETopt

−=

5.14

where22

3

3

2

110

220

NH

NH

PP

P

Ek

Ek≡β .

The effect of heat removal at the exit of the first bed is shown in figure 5.2. The drop in inter-

bed temperature, taken here as 106 K, is arbitrary and can be adjusted according to the power

requirement, but this will have an effect on the entrance temperature for the second bed, and

hence on the saturation length.

91

Figure 5.2 : Temperature in homogeneous reactor compared with one-equation optimal temperature optT and equilibrium temperature eqmT . The mole fractions of hydrogen, nitrogen and ammonia at bed exits are: 0.5894, 0.2153,

0.1089; 0.5534, 0.2041, 0.1526; and 0.5419, 0.2005, 0.1667 (2648 kmol h-1) respectively.

Figure 5.3 shows the results of the homogeneous one-equation optimal model (OEM-TPB).

The optimal nitrogen mole fraction maintains an almost-constant gap with the equilibrium

mole fraction, which is the driving force of the reaction. A similar difference is observed

between the optimal and equilibrium temperatures. When the optimal temperature profile is

used for the reacting gas, the nitrogen conversion is 0.3003 compared with 0.2543 and the

subsequent ammonia composition at the reactor exit is 1246-MTD i.e. a 15% increase over

0.5 1 1.5 2 2.5 3 3.5 4 4.5650

700

750

800

850

900

950

1000

Topt T

eqm

Distance (m)

Tem

pera

ture

(K)

92

the Reference Design. Clearly, such a temperature profile is difficult to achieve but can serve

as a reference ideal goal.

Figure 5.3 : Homogeneous reactor: (a) ammonia mole fraction, (b) temperature profile, and (c) hydrogen/nitrogen/ammonia mole fractions.

With two inter-bed heat exchangers the resulting temperature profile, shown in figure 5.4,

yields an enhanced ammonia mole fraction at exit of 0.1817 (2850 kmol h-1) and an energy

availability of 109.02 kJ h-1 (30.28 MWth). The process flow diagram of the energy recovery

plant with propsed process modifications is presented in figure 5.5.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2

Distance (m)

Am

mon

ia M

ole

Fr

0 1 2 3 4600

650

700

750

800

Distance (m)

Tem

pera

ture

(K)

0 1 2 3 40

0.2

0.4

0.6

0.8

Distance (m)

Mol

e F

ract

ion H2

N2

NH3

(a)

(b) (c)

93

Figure 5.4 : Homogeneous reactor with OEM-TPA, showing gas temperatureT , equilibrium temperature eqmT , and optimal temperature optT .

Figure 5.5 : The Proposed Energy Recovery Plant with Process Modifications

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5600

650

700

750

800

850

900

950

1000

Topt

Teqm

T

Distance (m)

Tem

pera

ture

(K)

94

5.1.4 Process Modifications Validation:

This section uses HYSYS™ and AspenPlusTM to estimate the sensitivity of bed temperatures

on the overall thermal energy recovery. It assumes a homogenous catalyst concentration in

each of the three beds. The objective is to arrive at a set of optimal exit temperatures specific

to the heat exchangers rather than the reactor.

The effect of heat exchangers between reactors is simulated with the configuration shown in

Fig. 5.6. The Plug Flow Reactors PFR-100 and PFR-102 represent the first reactor for which

saturation is achieved at the length of the first bed and the second reactor with a lower inlet

temperature.

Figure 5.6 : PFR reactor beds with cooling between beds 1 and 2 The sensitivity of the temperature drop, in the inter-bed heat exchanger E-100, to the

ammonia mole fraction at reactor exit is shown in Fig. 5.8. The value of 1T∆ , arbitrarily set

to 106 K in the previous sections, is seen in Fig. 5.8 to have an optimal value of 205K, for

which the optimal exit mole fraction is 0.223, and the thermal energy availability is 31.5

MWth.

Figure 5.7 shows the Aspen Plus™ three-bed (reactor) configuration with two inter-bed

(reactor) heat exchangers.

95

Figure 5.7 : PFR reactor beds with cooling between beds The optimality now must differentiate between the bed configurations and the reactor

configurations. For the bed configuration, we consider the first bed of length 1.5m and beds 2

and 3 of 4.5 m each to simulate the full effect of reactors.

Delta T1

NH

_3 M

olar

Fra

ctio

n

-260.0 -240.0 -220.0 -200.0 -180.0 -160.0 -140.0 -120.0 -100.0 -80.0 -60.0 -40.0 -20.0

0.13

0.14

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0.22

0.23

0.24

Figure 5.8 : Effect of temperature drop in the inter-bed heat exchanger, after the first bed, on the ammonia mole fraction at reactor outlet.

96

With Aspen Plus™ the results indicate, with inter-bed temperature drops of 106 oC and 86 oC

respectively, an exit ammonia mole fraction of 0.2137. If the first temperature drop is

constant, an optimal is found for the second temperature drop at a value 90 oC, for which the

subsequent exit ammonia mole fraction increases to 0.2143. We now consider a theoretical

maximum by allowing infinite reactor length. For the same temperature drops, we obtain an

exit ammonia mole fraction of 0.2165, which is markedly better than the finite case

previously considered.

Delta T2 (C)

NH

_3 M

ole

Fra

ctio

n

-260.0 -240.0 -220.0 -200.0 -180.0 -160.0 -140.0 -120.0 -100.0 -80.0 -60.0 -40.0 -20.0

0.23

0.24

0.25

0.26

0.27

0.28

Figure 5.9 : Effect of temperature drop in the inter-bed heat exchangers, after the first and second beds, on the ammonia mole fraction at reactor outlet.

97

In order to obtain the maximum achievable ammonia conversion, and hence a maximum

available thermal energy availability, we consider a sensitivity analysis of two parameters viz

the inter-reactor temperature drops. These are shown in Fig. 5.8 and Fig. 5.9. Figure 5.8

shows an optimal temperature drop of 205oC, as mentioned above, for which the exit

ammonia mole fraction is 0.2230. Similarly, Fig.5.9 shows an optimal at a temperature drop

of 95oC, for which the final exit ammonia mole fraction is 0.2762, and a thermal energy

availability of 45.6 MWth.

5.2 Design Modifications

Best ammonia conversion is achieved by realizing the optimal temperature profile as

described in section 5.1. For a solar thermal power plant, this inlet temperature will be

constrained by the maximum temperature achievable at the dissociation side of the plant.

Thus, it is practically difficult to have an inlet gas temperature high enough as 870 K as

predicted by theory. We thus seek the industrially viable option so that the resulting

temperature difference, between actual and optimal, is minimized.

Two technology options that can make this possible include pre-heating the inlet gas and

progressively removing the heat of reaction from the reactor acordingly, and with the given

inlet tempeature and increasing profile, removing process gas heat from the ‘first catalyst bed’

at first saturation and achieving optimal temperature in the second bed, and repeating the

procedure in the third bed. The first option has been extensively investigated [65][77][100]

using a counter-flow arrangement in which heat at reactor exit is used for pre-heating the

98

feed gas. Another technological option [1] consisting of varying the catalyst concentration

taking advantage of the ‘importance’ of the beds, has been shown to enable optimal

configuration.

Figure 5.10 shows a steep 65% conversion in the first bed, a slower but increasing 35%

conversion in the second bed, followed by a slight 10% improvement in the third bed. All

beds are taken to be of height 1.5 m. A significant feature of figure 5.10 is the saturation of

the conversionz to a value 0.2534 at the reactor exit. Figure 5.3a shows the increase of

ammonia in the three beds at approximately linear rates with decreasing gradients in

successive beds. The homogeneous catalyst loading indicates that the third bed has a very

small contribution.

Figure 5.10 : Homogeneous reactor: Nitrogen conversion in catalyst bed.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.05

0.1

0.15

0.2

0.25

0.3

Distance (m)

Con

vers

ion

of N

2 (

Z)

99

5.2.1 The Proposed Design

Figure 5.11 shows the comparison of the reference (homogeneous) design with the proposed

process modifications. The three cases shown, between the reference (1,1,1) and the optimal,

are a 10% and a 20% increase in catalyst concentration in the first bed only, followed by a

simultaneous 50% increase in the first bed and a 25% increase in the second bed

(1.50,1.25,1.00). The first two cases are shown to marginally improve the conversion since

the design change is small, while the last case achieves the optimal conversion before the exit

from the first bed. It is seen that there is further scope for improvement in subsequent catalyst

beds.

Figure 5.11 : Effect of varying spatial composition in reactor beds on the mole fraction of ammonia in the reactor compared with the reference (homogeneous) design with spatial concentration [1.00, 1.00, 1.00]

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Distance (m)

Mol

e F

ract

ion

NH 3

optimal1.00 1.00 1.001.10 1.00 1.001.20 1.00 1.001.50 1.25 1.00

Reference Design

Proposed Design

100

In Figure 5.12, detailed results are illustrated for a 50% and 25% concentration increase in

the first and second beds respectively. The nitrogen conversion increases from 0.2534 to

0.2832 while the temperature profile moves closer to the OEM equilibrium for the first and

second beds.

Figure 5.12 : Effect of varying spatial composition in reactor beds (1.50, 1.25, 1.00); a) nitrogen conversion, b) actual, optimal and equilibrium temperatures, c) hydrogen, nitrogen and ammonia mole fractions. Saturation in the first bed appears in the last 0.30 m of the first bed. The engineering

implications are that, with fixed bed physical dimensions, this space could be utilized for

some other scheme such as pre-heating for the second bed inlet to bring inlet temperature

closer to the optimal. The highest temperature in the first bed increases to 780 K (from 775 K

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

Distance (m)

Con

vers

ion

of N

2 (

Z)

1 2 3 4600

700

800

900

1000

Topt

Teqm

Distance (m)

Tem

pera

ture

(K)

0 1 2 3 40

0.2

0.4

0.6

0.8

Distance (m)

Mol

e F

ract

ion H2

N2NH3

(a)

(b) (c)

101

in the homogeneous case) which brings it closer to the maximum permissible catalyst

temperature, and hence a reduction in the reactor safety margin. The ammonia mole fraction

at reactor exit has now increased to 0.1856; equivalent to 1186-MTD. Apart from an increase

in the ammonia production, the heat recovery potential from the first inter-bed heat

exchanger increases to 18.395 MW(th) and 14.277 MW(th) from the second inter-bed heat

exchanger. When the spatial concentration is also increased in the third bed so that the

concentrations are [1.50,1.25,1.25], the nitrogen conversion increases to 0.2901, and the

ammonia production increases to 0.1902 (1210-MTD); the heat recovery is 18.395 MW(th)

and 13.880 MW(th) (total: 32.275 MW(th)).

5.2.2 Design Modifications Validation: This section uses AspenPlusTM to estimate the effect of heterogeneous catalyst distribution

on the temperature profile in bed 1 and bed 2 of the KBR Synthesis reactor .

Figure 5.13 : Bed1: Temperature Profile with different Catalyst Distribution

102

Figure 5.13 presents the temperature profiles in the reactor bed1 with catalyst spatial factors

of 1 and 1.5. It is clear that in case of catalyst spatial facor of 1.5, the temperature value

saturates at reactor length of 0.7m, thereby approaching the optimal temperature profile faster

than in the uniform case; the equilibrium temperature can be achieved before the exit of the

first bed and hence a wider span is available for heat/energy extraction to be used by Solar

Thermal power plant.

Figure 5.14 : Bed2: Temperature Profile with different Catalyst Distribution

The same results can be observed in figure 5.14 for temperature profile of bed2, where a

catalyst spatial factor of 1.25 is used.

103

6 CONCLUSIONS AND FUTURE WORK

This work is aimed at maximizing the heat recovery and hence improving the overall

efficiency of a thermal storage plant, by estimating optimal geometric and process variables

for system components including ammonia dissociation reactor, heat exchangers, Flash tank,

purge gas removal, recycle streams and most importantly ammonia synthesis reactor.

Kellogg KBR industrial ammonia synthesis reactor has been used for computing the

ammonia production using mass and energy conservation equations with homogeneous as

well as non-uniform catalyst distributions. The momentum conservation equation has been

ignored since the pressure drop in this reactor has been shown [64] to be not more than 2% of

the system pressure. The optimal and equilibrium temperature profiles are then computed in

the reactor and compared with the temperature profile in the homogeneous reactor.

The optimal and equilibrium temperature profiles have been used as the desired profiles by

incorporating process, rather than design, changes in the heat recovery system. This led to a

study of a heterogeneous, or non-uniform, catalyst distribution in the beds.

The results indicated that optimal ammonia conversion requires a high inlet temperature to

favor a high reaction rate as opposed to the comparatively lower inlet temperatures which are

found in industrial reactors. The optimal design, based on the optimal temperature profile,

gives a 15% increase in ammonia yield, from 1082 MTD in the homogeneous configuration

104

to 1246 MTD in the optimal configuration. A non-uniform catalyst distribution can be used

to take advantage of the importance of the first bed, followed by successive beds, thereby

approaching the optimal temperature profile faster than in the uniform case; as a result of the

previous argument, the equilibrium temperature can be achieved before the exit of the first

bed; heterogeneous catalyst distribution, is a strategy which increases the concentration by 50%

in the first bed and 25% in the second bed, brings the temperature profile close to equilibrium

results with a 10% increase in ammonia conversion. A larger number of beds could be used,

in principle, to search for a catalyst distribution which would yield the optimal temperature

profile leading to a 15% increase in ammonia yield.

Finally it is concluded that a one-dimensional model, with mass and energy conservation

equations using the Temkin-Pyzhev activity and pressure-based kinetics rate expressions,

predicted an optimal ammonia conversion of 0.2137 with a thermal energy availability of 20

MWth. A comprehensive process simulation using Aspen Plus™ predicts an optimal

ammonia conversion of 0.2762 mole fraction at exit, with two inter-bed heat exchangers

having optimal temperature drops of 205K and 95K respectively, and yielding a thermal

availability of 45.6 MWth. The thermal energy availability of a base-load solar thermal plant

can be increased by 15% in the ammonia conversion and over 25% in thermal energy

availability for energy recovery.

105

Construction of experimental setup is recommended for validation of results presented in this

thesis and further research in this area. It is being considered at University of Engineering &

Technology Taxila through undergraduate students and development of initial funding

proposal is in progress.

Further work is suggested to quantify the performance increase in the considered system by

conducting an exergy analysis of the system components for better quantification of the

efficiency improvements as the major determinant of achievable performance for such a

system is the degree of thermodynamic irreversibility associated with the heat recovery

process.

The major irreversibilities occur within the exothermic reactor and the counterflow heat

exchanger between ingoing and outgoing reactants. In the suggested study, optimum reactor

control will yield exergetic efficiencies, which should translate to overall solar to electric

conversion efficiencies.

106

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121

APPENDIX A. AMMONIA 3D PHASE DIAGRAMS

Figure A.1 Ammonia 3D Phase Diagram Critical Point*

Substance cT cP cρ Boiling Point

K MPa 3/ mkg K

Nitrogen 126.192 3.3958 31.3 77.355 Hydrogen 33.145 1.2964 31.263 20.369 Ammonia 405.4 11.333 225.0 239.823

122

APPENDIX B MATLAB™ PROGRAMS FOR AMMONIA

SIMULATION

S. No.

Name Functions Called Description Remarks

1

SSATS

ACTVT(P(IS),T(IS),x(1),x(2),x(4)) EQNCN(T(IS)) ETA(P,T,Z) RTCNT(T(IS),R_un) RRATE(kArh,Ka,aH2,aN2,aNH3) SPCHT(T(IS),P(IS))

Solves 2 equations in one zone; RK4 method

2

SSEqmProg

None

Computes eqm mol fractions, extent and K

3

RunSSoptim

SSoptim

Solves 3 equations from Babu; counter flow feed/reacting Gas and temperature

4

SSATS_NonUnif

ACTVT(P(IS),T(IS),x(1),x(2),x(4)) EQNCN(T(IS)) ETA(P,T,Z) RTCNT(T(IS),R_un) RRATE(kArh,Ka,aH2,aN2,aNH3) SPCHT(T(IS),P(IS))

Same as SSATS- except it has 3 zones allowing user variable catalyst activity

5

MansonOneEqn

OptimalT

Solves one equation with optimal temperature found from numerical solution of Hamiltonian

123

APPENDIX B1: MATLAB ™ PROGRAM FOR OUTPUT OF STEADY STATE SYNTHESIS REACTOR

This section lists the Program SSATS and its functions: ACTVT, COMPR, DeltaH, EQNCN, ETA, RCTHT, RRATE, RTCNT, SPCHT.

Program SSATS computes output for a synthesis react or

Data: inlet conditions

Output: outlet conditions

% Program Name: SSATS

% Steady State Ammonia Thermal Storage

% C:\MATLAB7\work\Ammonia\SSATS

% Author: Engr. Sadaf Siddiq ([email protected])

% First Written: JULY 2010

% Last Update: APRIL 2011

% open output file

resl=fopen('out1.txt','w'); % INPUT and STREAM OUT PUT

tic

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% INPUT

% System Pressure (atm)

% Temp (K)

% Total Mass Flow Rate (kg/hr) =

% Mole fractions (H,N,Ar,Amm,Met) [x] =

% Area = 20.0 ; % cross-section area of catalyst m^ 2

% Length of converter Lmax (m)

% OUTPUT

% graph of mol fractions of H, N, Amm vs length

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% program parameters

NS = 10; % number of streams

NC = 5; % number of components, H2,N2,NH3,Ar,CH4

fprintf (resl,'\n No. of Streams = %3.0f \n',NS) ;

fprintf (resl,'\n No. of Components = %3.0f \n',NC) ;

124

%%%%%%%%%%%%%%%%%%%%%%%%% DATA **********************

R_un = 8.314472; % universal gas const J/(mol-K)

%R_un = 1545.349 ; % lbf.ft/lbmol.R

%R_un = 1545.349;

fprintf (resl,'\n Universal Gas Const = %12.6f kJ/ (mol-K)\n',R_un);

% chemical data from http://www-jmg.ch.cam.ac.uk/tools/magnus/PeriodicTable.html

% component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane

MW1=2.0*1.0079759; MW2=2.0*14.0067231; MW3= 39.9476 613;

MW4=(14.0067231+3.0*1.0079759); MW5 = (12.0110369+4 .0*1.0079759);

MW = [ MW1;MW2;MW3;MW4;MW5 ];

fprintf (resl,'\n ID Name Mol Wt (kg/kmol)\n') ;

fprintf (resl,'\n 1 Hydrogen %8.4f ',MW(1));

fprintf (resl,'\n 2 Nitrogen %8.4f ',MW(2));

fprintf (resl,'\n 3 Argon %8.4f ',MW(3));

fprintf (resl,'\n 4 Ammonia %8.4f ',MW(4));

fprintf (resl,'\n 5 Methane %8.4f ',MW(5));

fprintf (resl,'\n ----------------------------\n');

% STREAM 1: SYN GAS ENTERING C-1 ******************************************

IS=1;

fprintf (resl,'\n Stream No: %3.0f \n',IS);

% STREAM INPUT

P(1) = 100.0; % atm

P_SIunits(1) = P(1)*1.0e5; %Pascals

T(1) = 300.0; % Kelvin

TotMassFlRate(IS) = 183600; % kg/hr Dashti p.20

%TotMassFlRate(IS) =1000; % kg/hr Dashti p.20


x= [ 0.6567006; 0.2363680; 0.0202874; 0.026930; 0.0 59714]; % mole fractions

Area = 20.0 ; % cross-section area of catalyst m^2

% Radius = 1.0; % try a radius of 1 m

% Area = 3.142 * Radius^2 ;

fprintf (resl,'\n Area = %6.2f m^2',Area);

HghtReactor = 5.6; % meters

125

VolCatalyst = Area * HghtReactor;

fprintf (resl,'\n Vol of catalyst = %6.3f m^3 \n',V olCatalyst);

%x= [ 0.7426; 0.2475; 0.0099; 0.0; 0.0]; % mole fra ctions

%mfm = [ 750.0 ; 250.0 ; 10.0; 0.0; 0.0]; % kg-mo ls/hr

%

% stream computation............................... ........

MWt(IS) = 0.0; % average molecular weight of stream

xT = 0.0; % total mol fr. this is should add up to 1.0

for i = 1:NC

MWt(IS)=MWt(IS)+x(i)*MW(i);

xT = xT +x(i);

end

RR(IS) = R_un/MWt(IS); % gas constant for this stre am (gas)

%-------------------------------

% mfrT = 0.0; % total mass flow rate/hr

% for j=1:NC

% mfr(j)=mfm(j)*MW(j); % kg/hr

% mfrT = mfrT + mfr(j);

% end

% MolarFlRateMix(IS) = mfrT/MWt(IS); % moles per hr

%--------------------------------

MolarFlRateMix(IS) = TotMassFlRate(IS)/MWt(IS); % i n kmols/hr

mfrT=0.0;

for jk=1:NC

mfr(jk) = x(jk)*MolarFlRateMix(IS);

mfrT = mfrT + mfr(jk);

mfm(jk) = mfr(jk)/MW(jk);

end

% end of stream computation........................ .......

%

126

% STREAM OUTPUT

fprintf (resl,'\n --------------------------------- --------------------\n');

fprintf (resl,'\n STREAM No: %3.0f \n',IS);

fprintf (resl,'\n NC Mole Fr Molar flow rate Mass flow rate');

fprintf (resl,'\n kg-mols/hr kg/hr\n');

for i=1:NC

fprintf (resl,'\n %2.0f %8.4f %8.4f %8.4f ',i,x(i),mfm(i),mfr(i))

end

fprintf (resl,'\n \n Total %6.4f %12.4f\n',xT,mfrT);

fprintf (resl,'\n Mol Wt of mixture MWt( %2.0f ) = %8.4f kg/kmol \n',IS,MWt(IS));

fprintf (resl,'\n Gas Const of mixture RR ( %2.0f ) = %8.4f kJ/(kmol-K) \n',IS,RR(IS));

fprintf (resl,'\n Molar Flow Rate of Stream %3.0f i s %8.2f kmoles / hr \n',IS,MolarFlRateMix(IS));

fprintf (resl,'\n --------------------------------- --------------------\n');

% COMPRESSOR WORK FOR STREAM 1

% make the mixture Cp, Cv and gamma

CpMix = 0.0;

RMix = 0.0;

% find the Cp values at this temp T1 and press P(1)

[CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T(1),P(1));

Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4;

jj=1;

for j= NC+1:2*NC

Cp(j) = Cp(jj)/MW(jj);

jj=jj+1;

end

fprintf (resl,'\n --------------------------------- --------\n');

fprintf (resl,'\n Cp values from function SPCHT at T = %8.2f K and P = %6.2f atm\n',T(1),P(1));

fprintf (resl,'\n Component Cp kJ/(kmol-K) kJ/( kg-K)\n');

fprintf (resl,'\n 1 Hydrogen %12.4e %12.4e ',Cp(1 ),Cp(6));

fprintf (resl,'\n 1 Nitrogen %12.4e %12.4e ',Cp(2 ),Cp(7));

fprintf (resl,'\n 1 Argon %12.4e %12.4e ',Cp(3 ),Cp(8));

fprintf (resl,'\n 1 Ammonia %12.4e %12.4e ',Cp(4 ),Cp(9));

127

fprintf (resl,'\n 1 Methane %12.4e %12.4e ',Cp(5 ),Cp(10));

fprintf (resl,'\n --------------------------------- ---------\n');

for i = 1:NC

% individual gas constants

Rg(i) = R_un/MW(i);

RMix = RMix + Rg(i)*mfr(i)/mfrT;

CpMix = CpMix + Cp(i)*mfr(i)/mfrT;

end

CvMix = CpMix - RMix;

GammaMix = CpMix/CvMix;

fprintf (resl,'\n Mixture properties (based on mass ratios)\n');

fprintf (resl,'\n CpMix CvMix RMix GammaMix\n');

fprintf (resl,'\n %12.4e %12.4e %12.4e % 8.2f \n',CpMix,CvMix,RMix,GammaMix);

% compressor work

n=1;

P(2) = 1.5 * P(1); % discharge pressure

PressRatio = P(2)/P(1);

[Ws]=COMPR(n,T(1),RMix,P(1),P(2),GammaMix); % gives Work in kJ/kmol of gas

PowerComp = Ws * MolarFlRateMix(1); % kJ/hr

PowerComp = (1./(60.*60.))*PowerComp; % kW

fprintf (resl,'\n Compressor Power \n');

fprintf (resl,'\n Reciprocating Compressor\n')

fprintf (resl,'\n No of stages = %2.0f ',n);

fprintf (resl,'\n Suction Pressure = %12.4e atm \ n',P(1));

fprintf (resl,'\n Discharge Pressure = %12.4e atm \ n',P(2));

fprintf (resl,'\n Pressure Ratio = %6.2f \ n\n',PressRatio);

fprintf (resl,'\n Compressor Work = %12.4e kJ/km ol \n',Ws);

fprintf (resl,'\n Compressor Work = %12.4e kW \ n',PowerComp);

% example from K.V.Narayanan p.134

% the R used in this is the universal Gas Const; th is prog uses RgasMix

% (check again)

128

%[Ws]=COMPR(1,300,8.314,1,10,1.3);

%PowerComp = Ws * 1.114e-3;

%fprintf (resl,'\n Compressor Work = %12.4e kW \n',PowerComp);

%------------------------------ SYNTHESIS --------- -----------------

% STREAM 3

% give input data for stream

% molar flow rate; mole fractions, pressure and te mperature

% output will be mole fractions, temperature and pr essure

% solve the equations using RK-4 method

% STREAM 3: SYN GAS ENTERING R-1 ******************************************

IS=3;

fprintf (resl,'\n Stream No: %3.0f \n',IS);

% STREAM INPUT

P(IS) = 150 ; % bar assume 1 bar = 1 atm

P_SIunits(IS) = P(IS)*1.0e5; %Pascals

T(IS) = 371+273; % Kelvin

TotMassFlRate(IS) = 183600; % kg/hr Dashti p.20


x= [ 0.6567006; 0.2363680; 0.0202874; 0.026930; 0.0 59714]; % mole fractions

% mfm = [ 750.0 ; 250.0 ; 10.0; 0.0; 0.0]; % kg-m ols/hr

%

% stream computation............................... ........

MWt(IS) = 0.0; % average molecular weight of stream

xT = 0.0; % total mol fr. this is should add up to 1.0

for i = 1:NC

MWt(IS)=MWt(IS)+x(i)*MW(i);

xT = xT +x(i);

end

129

RR(IS) = R_un/MWt(IS); % gas constant for this stre am (gas)

TotMolarFlRate(IS) = TotMassFlRate(IS)/MWt(IS); % k moles per hr

sumMol = 0.0;

sumMas = 0.0;

for j=1:NC

MolFlRate(j) = x(j) * TotMolarFlRate(IS);

sumMol = sumMol + MolFlRate(j);

MassFlRate(j)= x(j) * TotMassFlRate(IS);

sumMas = sumMas + MassFlRate(j);

end

% end of stream computation........................ .......

%

% STREAM OUTPUT

fprintf (resl,'\n --------------------------------- --------------------\n');

fprintf (resl,'\n STREAM No: %3.0f \n\n',IS);

fprintf (resl,'\n\n Pressure = %8.2f atm Temperat ure = %8.2f \n\n',P(IS),T(IS));

fprintf (resl,'\n NC Mole Fr Molar flow rate Mass flow rate');

fprintf (resl,'\n kmoles/hr kg/hr\n');

for i=1:NC

fprintf (resl,'\n %2.0f %8.4f %8.2f %8.2f ',i,x(i),MolFlRate(i),MassFlRate(i))

end

fprintf (resl,'\n \n Total %6.4f %12.4f %12.4f\n',xT,sumMol,su mMas);

fprintf (resl,'\n Mol Wt of mixture MWt(%2.0f) = %8.2f kg/kmol \n',IS,MWt(IS));

fprintf (resl,'\n Gas Const of mixture RR (%2.0f) = %8.2f kJ/(kmol-K) \n',IS,RR(IS));

fprintf (resl,'\n Molar Flow Rate of Stream %3.0f i s %8.2f kmoles / hr \n',IS,TotMolarFlRate(IS));

fprintf (resl,'\n Mass Flow Rate of Stream %3.0f i s %8.2f kg /hr \n',IS,TotMassFlRate(IS));

fprintf (resl,'\n --------------------------------- --------------------\n');

130

% SECTION BEGINS ================================= ====================

% do this section if you want to check manually the orders of magnitude

% otherwise remove this section

% activities: a_i = y_i * phi_i * P

% y_i is the mole fraction which is available above , as x(i)

% obtain phi_i from the function

% function[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P ,T,yH2,yN2,yNH3)

[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P(IS),T(IS) ,x(1),x(2),x(4));

% now compute the equilibrium constant K

% function[Ka] = EQNCN(T)

[Ka] = EQNCN(T(IS));

% now find Arrhenius Rate Constant

[kArh]= RTCNT(T(IS),R_un);

% now find reaction rate

[RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3);

[CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T(IS),P(IS));


fprintf (resl,'\n check this part carefully; return ed from SPCHT\n');

fprintf (resl,'\n temp = %6.2f pressure = %6.2f \ n',T(IS),P(IS));

fprintf (resl,'\n CpH2 = %8.2f kJ/kmol-K',CpH2);

fprintf (resl,'\n CpN2 = %8.2f kJ/kmol-K',CpN2);

fprintf (resl,'\n CpAr = %8.2f kJ/kmol-K',CpAr);

fprintf (resl,'\n CpNH3= %8.2f kJ/kmol-K',CpNH3);

fprintf (resl,'\n CpCH4= %8.2f kJ/kmol-K',CpCH4);

fprintf (resl,'\n now convert to kJ/kg-K');

jj=1;

for j= NC+1:2*NC

Cp(j) = Cp(jj)/MW(jj); % kJ/(kg-K)

fprintf (resl,'\n Cp( %2.0f )= %6.2f kJ/kg-K ', j,Cp(j));

jj=jj+1;

end

CpMix = 0.0;

for ii = 1:NC

CpMix = CpMix + Cp(ii+5)*MassFlRate(ii)/TotMassFlRa te(IS);

131

end

fprintf (resl,'\n\n Mixture Cp kJ/kg-K');

fprintf (resl,'\n CpMix = %6.2f KJ/kg-K',CpMix);

[HtReact]= DeltaH(P(IS),T(IS));

% X is the height from the top of the catalyst (m) and StepX is the

% increment in X

fprintf (resl,'\n Test Section output begins ...\n' );

fprintf (resl,'\n Pressure = %6.2f atm Temp = % 6.2f',P(IS),T(IS));

X = 0.0; StepX=0.5;

fprintf (resl,'\n X = %6.2f m StepX = %6.2f',X,St epX);

Zold = 0; % no N2 has been converted as yet

fprintf (resl,'\n Zold = %6.2f',Zold);

Told = T(IS); % this is temp at the inlet of this v olume

N20 = MolFlRate(2); % kmols/hr of N2 entering this volume element

fprintf (resl,'\n N20 = %6.2f kmol/hr ',N20);

fprintf (resl,'\n Mdot = %12.4e kg/hr',TotMassFlRat e(IS));

fprintf (resl,'\n Cp = %12.4e kJ/(kg-K) ',CpMix);

fprintf (resl,'\n x(1) = %8.4f x(2)=%8.4f x(4 )=%8.4f ',x(1),x(2),x(4));

fprintf (resl,'\n RNH3 = %12.4e kmol/(hr-m^3) ',RNH 3);

fprintf (resl,'\n HtReact = %12.4e kJ/kmol ',HtReac t);

dZdX = Area*RNH3 /(2.0*N20);

dTdX = abs(HtReact)*RNH3*Area/(TotMassFlRate(IS)*Cp Mix);

fprintf (resl,'\n\n dZ/dx = %12.4e conversion p er meter',dZdX);

fprintf (resl,'\n dT/dx = %12.4e K per meter \n',dTdX);

% now use the above gradients to find new values fo r moles of H, N, Ammonia

132

% the volume of this box is now Area * StepX

DeltaVolume = Area * StepX;

Znew = Zold + dZdX * StepX;

Tnew = Told + dTdX * StepX;

fprintf (resl,'\n New values Z = %6.2f T = %6. 2f K \n',Znew,Tnew);

% now compute new moles and mole fractions

% converted moles of N2

delta = Znew * MolFlRate(2);

% new mole rates (kmols/hr) and mass flow rates (kg /hr)

for ik =1:NC

MolFlRateOld(ik) = MolFlRate(ik);

MassFlRateOld(ik) = MassFlRate(ik);

MolFlRateNew(ik) = 0.0; % need to compute this now

MassFlRateNew(ik)= 0.0; % need to compute this now

xold(ik) = x(ik); % old mole fractions

xnew(ik) = 0.0; % new mole fractions

end

% new totals

TotMolarFlRateOld(IS)=TotMolarFlRate(IS);

TotMolarFlRateNew(IS)=0.0;

TotMoleFrOld=0.0;

TotMoleFrNew=0.0;

TotMassFlRateOld(IS)=TotMassFlRate(IS);

TotMassFlRateNew(IS)=0.0;

MolFlRateNew(1) = MolFlRateOld(1) - 3.0* delta; % H 2

MolFlRateNew(2) = MolFlRateOld(2) - 1.0* delta; % N 2

MolFlRateNew(3) = MolFlRateOld(3); % A rgon

MolFlRateNew(4) = MolFlRateOld(4) + 2.0* delta; % N H3

MolFlRateNew(5) = MolFlRateOld(5); % C H4

%

133

for ik1 = 1:NC

MassFlRateNew(ik1) = MW(ik1) * MolFlRateNew(ik1 );

end

for ik2 = 1:NC

TotMolarFlRateNew(IS) = TotMolarFlRateNew(IS) + MolFlRateNew(ik2);

TotMassFlRateNew(IS) = TotMassFlRateNew(IS) + MassFlRateNew(ik2);

end

for ik11 = 1:NC

xnew(ik11) = MolFlRateNew(ik11)/TotMolarFl RateNew(IS);

end

for ik12 = 1:NC

TotMoleFrOld = TotMoleFrOld + xold(ik12);

TotMoleFrNew = TotMoleFrNew + xnew(ik12);

end

% now write a summary of the change

fprintf (resl,'\n\n SUMMARY AFTER CONVERSION in th is volume box Area*StepX \n');

fprintf (resl,'\n BEFORE CONVERSI ON AFTER CONVERSION\n');

fprintf (resl,'\n i MolFlRate mol fr MassFlRate MolFlRate mol fr MassFlRate');

fprintf (resl,'\n kmol/hr kg/hr kmol/hr kg/hr');

for ik3=1:NC

fprintf (resl,'\n %3.0f %15.4e %8.4f %12.4e %20. 4e %8.4f %12.4e ',ik3,MolFlRateOld(ik3),xold(ik3),MassFlRateOld(ik3 ),MolFlRateNew(ik3),xnew(ik3),MassFlRateNew(ik3));

end

fprintf (resl,'\n %21.4e %8.4f %12.4e %21.4e %9.4f %13.4e', TotMolarFlRateOld(IS),TotMoleFrOld,TotMassFlRateOld(IS),TotMolarFlRateNe w(IS),TotMoleFrNew,TotMassFlRateNew(IS));

% how much energy is given off in this volume?

% mass flow rate is the same before or after so can use either

Power = TotMassFlRateOld(IS)*CpMix*(Tnew-Told); % ( kg/hr)*(kJ/kg-K)*(K) = kJ/hr

134

Power = Power/(60*60); % kW

fprintf (resl,'\n Power given off in synthesis of a mmonia in this vol element = %12.4e kW\n',Power);

fprintf (resl,'\n Test Section ends.............\n\ n\n');

%SECTION ENDS ==================================== ===

Z=0.0; % N2 conversion percentage start with ) sin ce no N2 is converted at t=0

[Eta1]=ETA(P(IS),T(IS),Z);%

fprintf (resl,'\n\n Pressure = %8.2f atm Temperat ure = %8.2f \n',P(IS),T(IS));

%fprintf (resl,'\n Temp phiH2 phiN2 p hiNH3 aH2 aN2 aNH3 Ka kArh RNH3\n ');

%fprintf (resl,'\n %6.1f %10.2e %10.2e %10.2e %10.2e %10.2e %10.2e %10.2e %10.2e %10.2e\n',T(IS),phiH2,phiN2,phiNH3,aH2,aN2,aNH3,Ka, kArh,RNH3);

%-------------------------------------------------- --------

% begin RK 4th order for 2 1st-order coupled ODEs

fprintf (resl,'\n\n R-K method \n\n');

%Area is defined above

FoverA = MolFlRate(2)/Area; % molar flow rate for N itrogen at inlet

fprintf (resl,'\n FoverA = %8.3f kmols per hr per m ^2\n',FoverA);

Z = 0.0;

T = T(IS);

P = P(IS);

%LL = [0 ; 0.50; 1.00; 1.50; 2.00; 2.50; 3.00; 3.50 ; 4.00; 4.50; 5.00];

ZZ(1)=0.0; % initial value of conversion percentage

TT(1)=T; % initial value of temperature (K)

Lmin =0.0;

Lmax= HghtReactor; % height of the convtr in meters

NPTS = 20;

h = (Lmax-Lmin)/NPTS;

LL(1)=Lmin;

135

for i = 2: NPTS

LL(i) = LL(i-1)+h;

end

LL

L = LL(1);

i = 1 ;


Hmoles(i) = MolFlRate(1);

Nmoles(i) = MolFlRate(2);

Rmoles(i) = MolFlRate(3);

Amoles(i) = MolFlRate(4);

Mmoles(i) = MolFlRate(5);

TotalMoles(i) = Hmoles(i) + Nmoles(i) + Rmoles(i) + Amoles(i) + Mmoles(i);

AmolesPC(i) = Amoles(i)/TotalMoles(i);

HmoleFr(i)=x(1);

NmoleFr(i)=x(2);

AmoleFr(i)=x(4);

%

fprintf (resl,'\n i x(1) x(2) x(4) Total Mole Fl Rate');

%

Toptimal(i) = 800.0;

NPTS1=NPTS-1;

while (i <= NPTS1) %------------------------------- --- RK4 LOOP BEGINS

i

fprintf (resl,'\n %3.0f %8.4f %8.4f %8.4f %15.4e',i,x(1),x( 2),x(4),TotalMoles(i));

136

%%%%%%%%%%%%%%%%%%%%%%%%%%% k1_z k1_T %%%%%%%%%%%%%%%%%%%%

% first function

% evaluate RHS of mass conservation equation FUNC1

[Eta1]=ETA(P,T,Z);

[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4));

[Ka] = EQNCN(T);

[kArh]= RTCNT(T,R_un);


FUNC1 = Eta1*RNH3/(2.0*FoverA);

k1_z = FUNC1;

% second function

% evaluate RHS of energy conservation equation FUNC 2

[CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T,P);


jj=1;

for j= NC+1:2*NC


jj=jj+1;

end

CpMix = 0.0;

for ii = 1:NC


end

[HtReact]= DeltaH(P,T);

FUNC2 = ( HtReact * Eta1 * RNH3 ) / ((TotMassFlRate (IS)/Area) * CpMix );

k1_T = -FUNC2;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% k2_z k2_P

L = LL(i) + h/2. ;

Z = ZZ(i) + k1_z * h/2.0;

T = TT(i) + k1_T * h/2.0;

% first function

[Eta1]=ETA(P,T,Z);

137


[Ka] = EQNCN(T);




k2_z = FUNC1;

% second function



jj=1;

for j= NC+1:2*NC


jj=jj+1;

end

CpMix = 0.0;

for ii = 1:NC


end



k2_T = -FUNC2;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%

k3_z k3_P

Z = ZZ(i) + k2_z * h/2.0;

T = TT(i) + k2_T * h/2.0;

% first function

[Eta1]=ETA(P,T,Z);


[Ka] = EQNCN(T);


138



k3_z = FUNC1;

% second function



jj=1;

for j= NC+1:2*NC


jj=jj+1;

end

CpMix = 0.0;

for ii = 1:NC


end



k3_T = -FUNC2;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% k4_z k4_P

L = LL(i) + h;

Z = ZZ(i) + k3_z * h;

T = TT(i) + k3_T * h;

% first function

[Eta1]=ETA(P,T,Z);


[Ka] = EQNCN(T);



139


k4_z = FUNC1;

% second function



jj=1;

for j= NC+1:2*NC


jj=jj+1;

end

CpMix = 0.0;

for ii = 1:NC


end



k4_T = -FUNC2;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% get new values for Z, T

i = i + 1;

ZZ(i) = ZZ(i-1) + (h/6.)*(k1_z + 2.0*k2_z + 2.0*k3_ z + k4_z);

TT(i) = TT(i-1) + (h/6.)*(k1_T + 2.0*k2_T + 2.0*k3_ T + k4_T);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% update mole fractions


ConvertedMolesOfN = ZZ(i)*MolFlRate(2);

Hmoles(i) = MolFlRate(1) - 3.0*( ConvertedMolesOfN ) ;

Nmoles(i) = (1.0-ZZ(i)) *MolFlRate(2);

140

Rmoles(i) = MolFlRate(3);

Amoles(i) = MolFlRate(4) + 2.0*( ConvertedMolesOfN ) ;

Mmoles(i) = MolFlRate(5);

%

TotalMoles(i) = Hmoles(i) + Nmoles(i) + Rmoles(i) + Amoles(i) + Mmoles(i);

AmolesPC(i) = Amoles(i)/TotalMoles(i);

% mole fractions

HmoleFr(i) = Hmoles(i)/TotalMoles(i);

NmoleFr(i) = Nmoles(i)/TotalMoles(i);

AmoleFr(i) = Amoles(i)/TotalMoles(i);

Power(i-1) = TotMassFlRate(IS)*CpMix*(TT(i)-TT(i-1) ); % (kg/hr)*(kJ/kg.K)*K=kJ/hr

Power(i-1) = Power(i-1)/(60*60); % kW

% now to go back to the beginning of the loop, set new local parameters

% P,T,Z. and also need P, T, x(1), x(2), x(4) for n ext iteration step

x(1) = Hmoles(i)/TotalMoles(i);

x(2) = Nmoles(i)/TotalMoles(i);

x(4) = Amoles(i)/TotalMoles(i);

MolFlowRate(2) = Nmoles(i); % this is the N2 inlet for the next volume element

FoverA = MolFlRate(2)/Area; % molar flow rate for N itrogen at inlet

% Pressure remains same

P=P;

T=TT(i);

Z=ZZ(i);

% now go to find optimal temp at this point ------- -------------- 10 may 2011

% scan between 500 and 800

Tchk=495;

fprintf (resl,'\n Temp (K) diff=fLHS-fRHS');

141

for iTchk = 1:60

Tchk = Tchk+5;

[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,Tchk,x(1) ,x(2),x(4));

[Ka] = EQNCN(Tchk);

FatT = Ka^2*aN2*aH2^1.5/aNH3 - aNH3/aH2^1.5;

fRHS = -( (20.523e3)/Tchk^2 ) * FatT;

% now find deriv at this point

T1=Tchk-1; T2=Tchk+1;

[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T1,x(1),x (2),x(4));

[Ka] = EQNCN(T1);

FatT1 = Ka^2*aN2*aH2^1.5/aNH3 - aNH3/aH2^1.5;

[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T2,x(1),x (2),x(4));

[Ka] = EQNCN(T2);

FatT2 = Ka^2*aN2*aH2^1.5/aNH3 - aNH3/aH2^1.5;

fLHS = 0.5*(FatT2-FatT1);

diff=fLHS-fRHS;

rootT(iTchk) = Tchk;

rootY(iTchk) = diff;

fprintf(resl,'\n %3.0f %6.2f %12.4e',iTchk,rootT(iT chk),rootY(iTchk));

end

Topt=0;

for jTchk = 1:59

jT1=jTchk+1;

if ((rootY(jTchk)>0)&(rootY(jT1)<0))

Topt = rootT(jTchk+1);

end

end

for jTchk = 1:59

jT1=jTchk+1;

if ((rootY(jTchk)<0)&(rootY(jT1)>0))

Topt = rootT(jTchk+1);

end

end

Toptimal(i) = Topt;

% what if Topt is not found? set it to max

142

if (Topt==0)

Toptimal(i)=800.0;

end

%-------------------------------------------------- --------------

% now go back

end % ------------------------------------- RK4 LOOP ENDS

fprintf (resl,'\n %3.0f %8.4f %8.4f %8.4f %15.4e',i,x(1),x( 2),x(4),TotalMoles(i));

fprintf (resl,'\n\n\n Summary of RK numerical solut ion \n \n \n');

fprintf (resl,'\n Moles Conversion % of N \n');

fprintf (resl,'\n i L H2 N2 Ar NH3 CH4 Total Z T Toptimal\n' );

for i = 1: NPTS

fprintf (resl,'\n %3.0f %6.4f %8.2f %8.2f %8.2f %8.2f %8.2f %10.2f %8.4f %8.2f %8.2f',i,LL(i),Hmoles(i),Nmoles(i),Rmoles( i),Amoles(i),Mmoles(i),TotalMoles(i),ZZ(i),TT(i),Toptimal(i));

end

% Power available from exothermic reactions

fprintf (resl,'\n\n\n\n i Power (kW)\n');

sumPower =0.0;

for i = 1: NPTS1

fprintf (resl,'\n %3.0f %12.4e ',i,Power(i));

sumPower=sumPower+Power(i);

end

fprintf (resl,'\n total exothermic energy avaiable is %12.4e kW ',sumPower);

ThisPlot=4;

143

if (ThisPlot==1)

plot (LL,ZZ,'-k')

xlabel ('Distance (m)')

ylabel ('Conversion of N_2 (Z)')

title 'Conversion of N_2 in the synthesis con vertor'

end

if (ThisPlot==2)

plot (LL,TT,'-k')


ylabel ('Temperature (K)')

title 'Temperature in synthesis convertor'

end

if (ThisPlot==3)

plot (LL,Hmoles,'-k')

hold on

plot (LL,Nmoles,'-.k')

hold on

plot (LL,Amoles,'--k')


ylabel ('Molar Flow Rate (kmols/hr)')

h = legend('H_2','N_2','NH_3',2);

title 'Molar flow rate of H_2, N_2 and NH_3 i n convertor'

end

if (ThisPlot==4)

subplot(2,2,1)

plot (LL,AmolesPC,'-k')


144

ylabel ('Ammonia Mole Fr')

grid on

% title 'Mole % NH_3'

subplot(2,2,2)

plot (LL,ZZ,'-k')


ylabel ('Conversion of N_2 (Z)')

% title 'Conversion of N_2'

grid on

subplot(2,2,3)

plot (LL,TT,'-k')

hold on

% now plot the optimal temp on this

plot (LL,Toptimal,'--k');


ylabel ('Temperature (K)')

h = legend('Temp','Topt',2);

grid on

% title 'Temperature'

subplot(2,2,4)

%plot (LL,Hmoles,'-k')

plot (LL,HmoleFr,'-k')

hold on

%plot (LL,Nmoles,'-.k')

plot (LL,NmoleFr,'-.k')

hold on

%plot (LL,Amoles,'--k')

plot (LL,AmoleFr,'--k')


%ylabel ('Molar Flow Rate (kmols/hr)')

ylabel ('Mole Fractions')

grid on

h = legend('H_2','N_2','NH_3',2);

145

% title 'Mol fl rate of H_2, N_2 and NH_3'

end

toc

fclose(resl);

function[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T ,yH2,yN2,yNH3)

%% Program Name: ACTVT

% ACTIVITY CALCULATION OF H2, N2 & N H3

%

% C:\MATLAB7\work\Ammonia\ACTVT.m

% Author: Engr. Sadaf Siddiq ([email protected])

% Written: JULY 2010

% Pressure in atm

% Temp in K

A=(-3.8402*T^0.125)+0.541;

B=(-0.1263*T^0.5)-15.98;

C=(-0.011901*T)-5.941;

% Fugacity Coefficients

tt1 = exp(A)*P; tt2 = exp(B)*P^2; tt3 = exp(C)* exp(-P/300); % check last term sign again

phiH2=exp(tt1-tt2 +300*tt3);

phiN2=0.93431737+((0.2028538E-3)*T)+((0.295896E -3)*P)-((0.270727E-6)*T^2)+(0.4775207E-6*P^2);

phiNH3=0.1438996+((0.2028538E-2)*T)-((0.4487672 E-3)*P)-((0.1142945E-5)*T^2)+((0.2761216E-6)*P^2);

% Activities

aH2=phiH2*yH2*P;

aN2=phiN2*yN2*P;

aNH3=phiNH3*yNH3*P;

function[Ws]=COMPR(n,T,RMix,P1,P2,gamma)

146

% Ws = Work Done On Compressor

% r = Pressure ratio for each stage in compress ion

% n = Number of compressor stages

% T = inlet temperature (K)

% RMix = Gas constant for this gas mixture

% P1 = inlet pressure (Pa)

% P2 = discharge pressure (Pa)

tt1 = gamma/(gamma-1.0);

tt2 = 1./tt1;

ratio = (P2/P1)^(1./n);

tt3 = tt1 * RMix * T;

tt4 = (1.0 - ratio^tt2);

Ws= n* tt3 * tt4 ;

function[HtReact]= DeltaH(P,T)

% heat of reaction in kJ/kmol

% DASHTI

tt1 = 0.54526 + 846.609/T + 459.734e6/T^3;

tt2 = -0.2525e-3*T^2 + 1.69197e-6*T^3-9157.09;

HtReact = 4.184*( -tt1*P - 5.34685*T + tt2 );

function[Ka] = EQNCN(T)

% equilibrium constant

% T in K

Ka=10^(-(2.691122*log10(T))-(5.519265E-5*T)+(1.8488 63E-7*(T^2))+(2001.6/T)+2.689);

function[Eta1]= ETA(P,T,Z)

% eta is the catalyst activity factor

% P units of atm

if (P <= 150)

b0 = -17.539096; b1=0.07697849; b2=6.900548; b3 = -1.08279e-4;

b4 = -26.42469; b5=4.927648e-8; b6=38.937;

end

if ((P > 150)&(P<=250))

b0 = -8.2125534; b1=0.03774149; b2=6.190112; b3 = -5.354571e-5;

b4 = -20.86963; b5=2.379142e-8; b6=27.88;

147

end

if (P > 250)

b0 = -4.6757259; b1=0.2354872; b2=4.687353; b3= -3.463308e-5;

b4 = -11.28031; b5=1.540881e-8; b6=10.46;

end

Eta1 = b0 + b1*T + b2*Z + b3*T^2 + b4*Z^2 + b5*T^3 + b6*Z^3;

function[delHr]= RCTHT(T,P)

delHr=(-(0.54526+(846.609/T)+(459.734E6/T^3))*P-5.3 4685*T-(0.2525E-3*T^2)+(1.069197E-6*T^3)-9157.09);

function[RNH3]= RRATE(kArh,Ka,aH2,aN2,aNH3)

% reaction rate; units: kmol of NH3/(hr-m^3 of cat alyst)

tt1 = aH2^1.5;

RNH3 = 2.0 * kArh * ( Ka^2* aN2 * tt1/aNH3 - aNH3 /tt1 );

function[kArh]= RTCNT(T,R)

% rate constant

k0 = 8.849e14;

% 1 J = 2.389x10-4 kcal

% E = 40765 kcal/kmol

E = 40765 / 2.389e-4; % J/kmol

E = 1.0e-3*E ; % kJ/kmol

kArh = k0*exp(-E/(R*T));

function[CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T,P)

% specific heat capacities

% cp in kJ/(kmol-K);

% T in K

% P in atm

CpH2=6.952-(0.04567E-2*T)+(0.09563E-5*T^2)-(0.2079E -9*T^3);

CpH2=4.184*CpH2;

CpN2=6.903-(0.03753E-2*T)+(0.1930E-5*T^2)-(0.6861E- 9*T^3);

CpN2=4.184*CpN2;

% this expression from Dashti does not give results matching with nist

% a simple model from VKN

%CpNH3a=6.5846-(0.61251E-2*T)+(0.23663E-5*T^2)-(1.5 981E-9*T^3);

148

%CpNH3b=(96.1678-0.067571*P+(0.2225+1.6847E-4*P)*T+ (1.289E-4-1.0095E-7*P)*T^2);

%CpNH3 = 4.184*(CpNH3a+CpNH3b);

% use Shomate equations from NIST 298-1400 K

A=19.99563; B=49.77119; C=-15.37599; D=1.921168; E= 0.189174;

t = T/1000;

CpNH3=A + B*t + C*t^2 + D*t^3 + E/t^2;

CpCH4=4.75+(1.2E-2*T)+(0.303E-5*T^2)-(2.63E-9*T^3);

CpCH4=4.184*CpCH4;

CpAr=4.9675;

CpAr=4.184*CpAr;

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APPENDIX B2: MATLAB ™ PROGRAM FOR FINDING EQUILIBRIUM CONCENTRATIONS This section lists the Program SSEqmProg Program SSEqmProg computes equilibrium mole fractio n, extent & Equilibrium Constant K

% Program Name: SSEqmProg % To find the eqm concn % % C:\MATLAB7\work\Ammonia\SSEqmProg.m % Author: Engr. Sadaf Siddiq ([email protected]) % % Jan 2011 % % open output file resl=fopen('out1.txt','w'); % INPUT and STREAM OUT PUT tic %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % step 1 specify an initial conc N2 mole = 1 H2= 3 NH3 = 0 % step 2 specify T, P % step 3 evaluate K(T) % step 4 calc eps (extent of reaction) % step 5 plot eps vs T for fixed P % STATIC DATA fprintf (resl,'\n physical data'); fprintf (resl,'\n universal data'); R_un = 8.314; % universal gas const J/(mol-K) fprintf (resl,'\n Universal Gas Const = %12.6f kJ/ (mol-K)\n',R_un); % chemical data from http://www-jmg.ch.cam.ac.uk/tools/magnus/PeriodicTable.html % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane MW1=2.0*1.0079759; MW2=2.0*14.0067231; MW4= 39.9476 613; MW3=(14.0067231+3.0*1.0079759); MW5 = (12.0110369+4 .0*1.0079759); MW = [ MW1;MW2;MW3;MW4;MW5 ]; %----------------- % STEP 1 initial composition n(2) = 1.0; % moles of nitrogen n(1) = 3*n(2); % moles of hydrogen n(3) = 0.0; % ammonia mole fraction nI = 0.0; % inerts mole fraction kappa= 0; % ratio of Ar/Methane n(4) = (kappa/(1+kappa))*nI; n(5) = nI - n(4); fprintf (resl,'\n ID Name Mol Wt (kg/kmol) Moles\n'); fprintf (resl,'\n 1 Hydrogen %8.4f %21.4f',MW(1), n(1)); fprintf (resl,'\n 2 Nitrogen %8.4f %21.4f',MW(2), n(2)); fprintf (resl,'\n 3 Ammonia %8.4f %21.4f',MW(4), n(3)); fprintf (resl,'\n 4 Argon %8.4f %21.4f',MW(3), n(4)); fprintf (resl,'\n 5 Methane %8.4f %21.4f',MW(5), n(5));

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fprintf (resl,'\n ----------------------------\n'); %------------------ NP = 6; NT = 20; % STEP 2 specify T, P Pressure = 0.0; % atm for j = 1:NP Pressure = Pressure+50 ; % atm ppp(j) = Pressure; % pressure of reaction fprintf (resl,'\n\n System pressure is %8.2 f atm',Pressure); temp = -50.0; % deg C for i = 1:NT temp = temp + 50.0; Temp = temp +273.15 % K %temp = 675-273.15; Temp=temp+273.15; % STEP 3 evaluate K(T,P) %[Ka] = EQNCN(Temp); % Dashti expression % Ka from Narayanan p.417 Eq. 9.58 lnKa = 79201/(R_un*Temp) - (48.8/R_un)*log(Temp) + (17.38e-3/R_un)*Temp + 14.169; Ka = (exp(1))^lnKa; % STEP 4 calc eps alpha = sqrt( 27.0 * Ka * Pressure^2 ); beta = alpha + 4.0; tt1 = 1.0 - alpha/beta; eps = 1.0e23; if (tt1>=0) tt2 = sqrt(tt1); eps1 = 1.0 + tt2; eps2 = 1.0 - tt2; end %fprintf (resl,'\n eps1 = %12.4f eps2 = %12.4e \n ',eps1,eps2); % now select the value less than 1 if (eps1 <= 1 ) eps = eps1; end if (eps2 <= 1 ) eps = eps2; end % what if both epsilons are +ve ? ttt(j,i) = temp; % deg C TTT(j,i) = Temp; % Kelvin eee(j,i) = eps; % extent of reaction kkk(j,i) = log10(Ka); % log10 of eqm const aaa(j,i) = 2*eps/(4 - 2*eps); end % end of temperature loop

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% write the results fprintf (resl,'\n No. temp (C) Temp (K) Ka eps'); for iP = 1: NT fprintf (resl,'\n %4.0f %9.2f %9.2f %12.4e %12.4f',iP,ttt(j ,iP),TTT(j,iP),kkk(j,iP),eee(j,iP)); end end % end of pressure loop % STEP 5 plot the results PlotExtent = 0; PlotNH3eqm = 1; PlotEqmConst = 0; if (PlotExtent == 1) for plotP = 1:NP for k = 1:NT X(k) = ttt(plotP,k); % temp deg C Y(k) = eee(plotP,k); % extent of reaction % Y(k) = kkk(plotP,k); % eqm const end plot(X,Y,'-k') hold on end xlabel (' Temp (ôC)','Fontsize',16) ylabel ('Extent (\epsilon ) ','Fontsize',16) grid off text(400,0.15,'50 ','Fontsize',12) text(560,0.35,'300','Fontsize',12) text(530,0.90,'\bf Pressure','Fontsize',12) text(450,0.85,'50,100,150,200,250,300 atm','Fontsiz e',12) end % end of plot of extent reaction if (PlotNH3eqm == 1) for plotP = 1:NP for k = 1:NT X(k) = ttt(plotP,k); % temp deg C Y(k) = aaa(plotP,k); % mol fraction of ammonia in eqm mixture % Y(k) = kkk(plotP,k); % eqm const end plot(X,Y,'-k') hold on end xlabel (' Temp (ôC)','Fontsize',16) ylabel ('NH_3 in Eqm. Mix. (mol. fr.) ','Fontsize', 16) grid on text(320,0.15,'50 ','Fontsize',12) text(480,0.35,'300','Fontsize',12) text(530,0.90,'\bf Pressure','Fontsize',12) text(450,0.85,'50,100,150,200,250,300 atm','Fontsiz e',12) end % end of plot of ammonia in eqm mixture if (PlotEqmConst == 1) for plotP = 1:NP for k = 1:NT X(k) = ttt(plotP,k); % temp deg C

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Y(k) = kkk(plotP,k); % eqm const end plot(X,Y,'-k') hold on end xlabel (' Temp (ôC)','Fontsize',16) ylabel ('Eqm. Const. (log_{10}K_a) ','Fontsize',16) grid on end % end of plot of Equilibrium Constant toc fclose(resl);

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APPENDIX B3: MATLAB ™ PROGRAM FOR FINDING OUTPUT OF COUNTER-FLOW SYNTHESIS REACTOR This section lists the Program RunSSoptim and its function SSoptim Program SSoptim computes output for counter flow sy nthesis reactor by solving three equations as described by B.V.Babu: c ounterflow feed/reacting Gas and temperature Data: inlet conditions Output: outlet conditions % Program Name: RunSSoptim % % C:\MATLAB7\work\Ammonia\RunSSoptim.m % Author: Engr. Sadaf Siddiq ([email protected]) % Last modified May 28 2011 % % % open output file resl=fopen('out1.txt','w'); % INPUT and STREAM OUT PUT tic options = odeset('RelTol',1e-6,'AbsTol',[1e-8 1e-8 1e-8]); [t,Y] = ode15s(@SSoptimKREETZ,[0 0.8],[663 663 0.02 51],options); % ammonia moles AmmMolesInit = 0.0029; NMolesInit = 0.0251; Amm = AmmMolesInit + 2*( NMolesInit - Y(:,3)); PlotTemp = 1; PlotComp = 0; PlotBoth = 0; % for plot showing temperatures only if (PlotTemp ==1) plot (t,Y(:,1),'-ok') hold on plot (t,Y(:,2),'-sk') grid on xlabel ('Distance (m)') ylabel ('Temperature (K)') h = legend('Feed Gas','Reacting Gas',1); % title 'Feed Gas Temperature and Reacting Gas Temperature' end

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% for plot showing molar compositions only if (PlotComp ==1) plot (t,Y(:,3),'-ok') hold on plot (t,Amm,'-sk') grid on xlabel ('Distance (m)') ylabel ('Molar Flow Rate (kmols/hr)') h = legend('N_2','NH_3',1); title 'Nitrogen and Ammonia Molar Flow Rates' end if (PlotBoth ==1) subplot(2,1,1) plot (t,Y(:,1),'-ok') hold on plot (t,Y(:,2),'-sk') grid on xlabel ('Distance (m)') ylabel ('Temperature (K)') h = legend('Feed Gas','Reacting Gas',1); % title 'Feed Gas Temperature and Reacting Gas Temperature' subplot (2,1,2) plot (t,Y(:,3),'-ok') hold on plot (t,Amm,'-sk') grid on xlabel ('Distance (m)') ylabel ('Molar Flow Rate (kmols/hr)') h = legend('N_2','NH_3',1); %title 'Nitrogen and Ammonia Molar Flow Rates ' end toc fclose(resl); function dy = SSoptim(t,y) dy = zeros(3,1); % a column vector % % last modified 28 May 2011 % % MATLAB commands to run this program % options = odeset('RelTol',1e-6,'AbsTol',[1e-8 1 e-8 1e-8]); % [t,Y] = ode15s(@SSoptim,[0 8],[700 700 701.2],o ptions); % plot(t,Y(:,1),'-',t,Y(:,2),'-.',t,Y(:,3),'.') Cpf=3.0 ; % kJ/(kg-K) cold feed gas Cpg=3.1 ; % kJ/(kg-K) hot reacting gas flowing do wnwards U=140.0; % heat transfer coefficient kW/(m^2-K)

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s1=2.*3.14159*7.9/1000; % surface area of cooling tubes per length of reactor W=1.08; % mass flow rate kg/s f=1.0; % catalyst effectiveness or concentratio n H=-100000.0;% kJ/(kmol N_2) R= 8.314472; % universal gas constant kJ/(kmol-K) Press=150; % system pressure atm s2=3.14159*7.9*7.9*1.0e-6; % cross-section area of catalyst zone % Moles kmol/h-m^2 at Feed % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane xFeed= [ 0.6567006; 0.2363680; 0.0202874; 0.026930; 0.059714]; % mole fractions TotalFeed = 0.1062; % kmols / hr H2F = xFeed(1) * TotalFeed; % kmol/h N2F = xFeed(2) * TotalFeed ; % kmol/h NH3F = xFeed(4) * TotalFeed; % kmol/h InertsF = (xFeed(3) + xFeed(5)) * TotalFeed; % kmol/h %fprintf (resl,'\n Total Feed = %12.4f moles/h-m^2 \n',TotalFeed); % dy(1) is dTf dy(2) is dTg dy(3 ) is dN % feed gas temp eqn dy(1)=((-U*s1)/(W*Cpf))*(y(2)-y(1)); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % hot gas temp %Tf = TfLast; T=y(2); %this is the temp of the gas flowing downwa rds convertedN = N2F - y(3); H2Now = H2F - 3*convertedN ; N2Now = y(3); NH3Now= NH3F + 2*convertedN; InertsNow = InertsF; TotalMolesNow = TotalFeed - 2.0*convertedN ; % new mole fractions xH = H2Now/TotalMolesNow; xN = N2Now/TotalMolesNow; xA = NH3Now/TotalMolesNow; [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(Press,T,xH, xN,xA); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); dndx = -f* RNH3/2.0 ; dy(2) =(-U*s1)/(W*Cpg)*(y(2)-y(1))+ ( (-H*s2)/(W*Cp g) ) * ( -dndx ); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % moles of N2 % parameter T represents moles of N2 (normalized) %Tg=TgLast; Tg = y(2); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(Press,T,xH, xN,xA);

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[Ka] = EQNCN(T); [kArh]= RTCNT(T,R); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); dy(3) = -f* s2* RNH3/2.0 ;

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APPENDIX B4: MATLAB ™ PROGRAM FOR FINDING OUTPUT OF STEADY STATE SYNTHESIS REACTOR WITH 3 CATALYST ZONES % Program Name: SSATSNonUnif % Steady State Ammonia Thermal Stora ge % % C:\MATLAB7\work\Ammonia\SSATS_NonUnif_New % % First Written: JULY 2010 % Last Update: 24 May 2011 % open output file resl=fopen('outSSATSnu.txt','w'); % INPUT and STRE AM OUTPUT tic %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % INPUT % System Pressure (atm) % Temp (K) % Total Mass Flow Rate (kg/hr) = % Mole fractions (H,N,Ar,Amm,Met) [x] = % Area = 20.0 ; % cross-section area of catalyst m^ 2 % Length of converter Lmax (m) % OUTPUT % graph of mol fractions of H, N, Amm vs length %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % program parameters NS = 10; % number of streams NC = 5; % number of components, H2,N2,NH3,Ar,CH4 fprintf (resl,'\n No. of Streams = %3.0f \n',NS) ; fprintf (resl,'\n No. of Components = %3.0f \n',NC) ; %%%%%%%%%%%%%%%%%%%%%%%%% DATA ********************** R_un = 8.314472; % universal gas const J/(mol-K) %R_un = 1545.349 ; % lbf.ft/lbmol.R %R_un = 1545.349; fprintf (resl,'\n Universal Gas Const = %12.6f kJ/ (mol-K)\n',R_un); % chemical data from http://www-jmg.ch.cam.ac.uk/tools/magnus/PeriodicTable.html % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane MW1=2.0*1.0079759; MW2=2.0*14.0067231; MW3= 39.9476 613; MW4=(14.0067231+3.0*1.0079759); MW5 = (12.0110369+4 .0*1.0079759); MW = [ MW1;MW2;MW3;MW4;MW5 ]; fprintf (resl,'\n ID Name Mol Wt (kg/kmol)\n') ; fprintf (resl,'\n 1 Hydrogen %8.4f ',MW(1)); fprintf (resl,'\n 2 Nitrogen %8.4f ',MW(2)); fprintf (resl,'\n 3 Argon %8.4f ',MW(3)); fprintf (resl,'\n 4 Ammonia %8.4f ',MW(4)); fprintf (resl,'\n 5 Methane %8.4f ',MW(5)); fprintf (resl,'\n ----------------------------\n'); % STREAM 1: SYN GAS ENTERING C-1 ****************** ************************ IS=1; fprintf (resl,'\n Stream No: %3.0f \n',IS); % STREAM INPUT

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P(1) = 100.0; % atm P_SIunits(1) = P(1)*1.0e5; %Pascals T(1) = 300.0; % Kelvin TotMassFlRate(IS) = 0.3*60*60/1000; % kg/hr this for comp; for stream 3, line 207 % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane x= [ 0.6567006; 0.2363680; 0.0202874; 0.026930; 0. 059714]; % mole fractions % x= [ 0.78; 0.20 ; 0.0 ; 0.02; 0.0] Area = 3.14159 * (1.58e-2)^2 ; % cross-section area of catalyst m^2 fprintf (resl,'\n Area = %12.4e m^2',Area); HghtReactor = 0.8; % meters VolCatalyst = Area * HghtReactor; fprintf (resl,'\n Vol of catalyst = %12.4e m^3 \n', VolCatalyst); %x= [ 0.7426; 0.2475; 0.0099; 0.0; 0.0]; % mole fra ctions %mfm = [ 750.0 ; 250.0 ; 10.0; 0.0; 0.0]; % kg-mo ls/hr % % stream computation............................... ........ MWT(IS) = 0.0; % average molecular weight of stream xT = 0.0; % total mol fr. this is should add up to 1.0 for i = 1:NC MWT(IS)=MWT(IS)+x(i)*MW(i); xT = xT +x(i); end RR(IS) = R_un/MWT(IS); % gas constant for this stre am (gas) %------------------------------- % mfrT = 0.0; % total mass flow rate/hr % for j=1:NC % mfr(j)=mfm(j)*MW(j); % kg/hr % mfrT = mfrT + mfr(j); % end % MolarFlRateMix(IS) = mfrT/MWT(IS); % moles per hr %-------------------------------- MolarFlRateMix(IS) = TotMassFlRate(IS)/MWT(IS); % i n kmols/hr mfrT=0.0; mfmT=0.0; for jk=1:NC mfr(jk) = x(jk)*MolarFlRateMix(IS); % units kmo l/hr mfrT = mfrT + mfr(jk); mfm(jk) = mfr(jk)*MW(jk); % units should be kg /hr mfmT = mfmT + mfm(jk); end % end of stream computation........................ .......

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% % STREAM OUTPUT fprintf (resl,'\n --------------------------------- --------------------\n'); fprintf (resl,'\n STREAM No: %3.0f \n',IS); fprintf (resl,'\n NC Mole Fr Molar flow rate Mass flow rate'); fprintf (resl,'\n kg-mols/hr kg/hr\n'); for i=1:NC fprintf (resl,'\n %2.0f %8.4f %8.4f %8.4f ',i,x(i),mfr(i),mfm(i)) end fprintf (resl,'\n \n Total %6.4f %8.4f %8.4f\n',xT,mfrT,mfmT); fprintf (resl,'\n Mol Wt of mixture MWt( %2.0f ) = %8.4f kg/kmol ',IS,MWT(IS)); fprintf (resl,'\n Gas Const of mixture RR ( %2.0f ) = %8.4f kJ/(kmol-K) ',IS,RR(IS)); fprintf (resl,'\n Molar Flow Rate of Stream %3.0f i s %8.2f kmoles / hr ',IS,MolarFlRateMix(IS)); fprintf (resl,'\n --------------------------------- --------------------\n'); % COMPRESSOR WORK FOR STREAM 1 % make the mixture Cp, Cv and gamma CpMix = 0.0; RMix = 0.0; % find the Cp values at this temp T1 and press P(1) [CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T(1),P(1)); % kJ/(kmol-K) Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4; jj=1; for j= NC+1:2*NC Cp(j) = Cp(jj)/MW(jj); jj=jj+1; end fprintf (resl,'\n --------------------------------- --------\n'); fprintf (resl,'\n Cp values from function SPCHT at T = %8.2f K and P = %6.2f atm\n',T(1),P(1)); fprintf (resl,'\n Component Cp kJ/(kmol-K) kJ/( kg-K)\n'); fprintf (resl,'\n 1 Hydrogen %12.4e %12.4e ',Cp(1 ),Cp(6)); fprintf (resl,'\n 1 Nitrogen %12.4e %12.4e ',Cp(2 ),Cp(7)); fprintf (resl,'\n 1 Argon %12.4e %12.4e ',Cp(3 ),Cp(8)); fprintf (resl,'\n 1 Ammonia %12.4e %12.4e ',Cp(4 ),Cp(9)); fprintf (resl,'\n 1 Methane %12.4e %12.4e ',Cp(5 ),Cp(10)); fprintf (resl,'\n --------------------------------- ---------\n'); for i = 1:NC % individual gas constants Rg(i) = R_un/MW(i); % kJ/(kmol-K) * kmol/kg = kJ/(k g-K) %RMix = RMix + Rg(i)*mfm(i)/mfmT; % based on mass fl rates RMix = RMix + R_un *mfr(i)/mfrT; % based on molar fl rates CpMix = CpMix + Cp(i)*mfr(i)/mfrT; % based on molar flow rates end CvMix = CpMix - RMix; % molar basis GammaMix = CpMix/CvMix; fprintf (resl,'\n Mixture properties (based on mola r ratios)\n'); fprintf (resl,'\n CpMix (kJ/(kmol-K) CvMix RMix GammaMix\n'); fprintf (resl,'\n %12.4e %12.4e %12.4e % 8.2f \n',CpMix,CvMix,RMix,GammaMix);

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% compressor work n=1; P(2) = 1.5 * P(1); % discharge pressure PressRatio = P(2)/P(1); [Ws]=COMPR(n,T(1),RMix,P(1),P(2),GammaMix); % gives Work in kJ/kmol of gas PowerComp = Ws * MolarFlRateMix(1); % kJ/hr PowerComp = (1./(60.*60.))*PowerComp; % kW fprintf (resl,'\n Compressor Power \n'); fprintf (resl,'\n Reciprocating Compressor\n') fprintf (resl,'\n No of stages = %2.0f ',n); fprintf (resl,'\n Suction Pressure = %12.4e atm \ n',P(1)); fprintf (resl,'\n Discharge Pressure = %12.4e atm \ n',P(2)); fprintf (resl,'\n Pressure Ratio = %6.2f \ n\n',PressRatio); fprintf (resl,'\n Compressor Work = %12.4e kJ/km ol \n',Ws); fprintf (resl,'\n Compressor Work = %12.4e kW \ n',PowerComp); % example from K.V.Narayanan p.134 % the R used in this is the universal Gas Const; th is prog uses RgasMix % (check again) %[Ws]=COMPR(1,300,8.314,1,10,1.3); %PowerComp = Ws * 1.114e-3; %fprintf (resl,'\n Compressor Work = %12.4e kW \n',PowerComp); %------------------------------ SYNTHESIS --------- ----------------- % STREAM 3 % give input data for stream % molar flow rate; mole fractions, pressure and te mperature % output will be mole fractions, temperature and pr essure % solve the equations using RK-4 method % STREAM 3: SYN GAS ENTERING R-1 ****************** ************************ IS=3; fprintf (resl,'\n Stream No: %3.0f \n',IS); % STREAM INPUT P(IS) =150.0 ; % atm P_SIunits(IS) = P(IS)*1.0e5; %Pascals %T(IS) = 371+273; % Kelvin Dashti paper inlet T %T(IS) = 291 + 273 - 10 ; % Kelvin with Manson Set 2 T(IS) = 390 +273; % Kreetz and Lovegrove 1999 InletTemp = T(IS); TotMassFlRate(IS) = 0.3*60*60/1000; % kg/hr Das hti p.20 % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane %x= [ 0.6567006; 0.2363680; 0.0202874; 0.026930; 0. 059714]; % mole fractions %x= [ 0.6567006; 0.2363680; 0.0202874; 0.026930; 0. 059714]; % mole fractions % mfm = [ 750.0 ; 250.0 ; 10.0; 0.0; 0.0]; % kg-m ols/hr % % stream computation............................... ........ MWT(IS) = 0.0; % average molecular weight of stream

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xT = 0.0; % total mol fr. this is should add up to 1.0 for i = 1:NC MWT(IS)=MWT(IS)+x(i)*MW(i); xT = xT +x(i); end RR(IS) = R_un/MWT(IS); % gas constant for this stre am (gas) TotMolarFlRate(IS) = TotMassFlRate(IS)/MWT(IS); % k moles per hr sumMol = 0.0; sumMas = 0.0; for j=1:NC MolFlRate(j) = x(j) * TotMolarFlRate(IS); % kmol/hr sumMol = sumMol + MolFlRate(j); MassFlRate(j)= MolFlRate(j) * MW(j); % kmol/hr * k g/kmol = kg/hr sumMas = sumMas + MassFlRate(j); end % end of stream computation........................ ....... % % STREAM OUTPUT fprintf (resl,'\n --------------------------------- --------------------\n'); fprintf (resl,'\n STREAM No: %3.0f \n\n',IS); fprintf (resl,'\n\n Pressure = %8.2f atm Temperat ure = %8.2f \n\n',P(IS),T(IS)); fprintf (resl,'\n NC Mole Fr Molar flow rate Mass flow rate'); fprintf (resl,'\n kmoles/hr kg/hr\n'); for i=1:NC fprintf (resl,'\n %2.0f %8.4f %8.4f %8.4f ',i,x(i),MolFlRate(i),MassFlRate(i)) end fprintf (resl,'\n \n Total %6.4f %12.4f %12.4f\n',xT,sumMol,sumMas); fprintf (resl,'\n Mol Wt of mixture MWt(%2.0f) = %8.2f kg/kmol \n',IS,MWT(IS)); fprintf (resl,'\n Gas Const of mixture RR (%2.0f) = %8.2f kJ/(kmol-K) ',IS,RR(IS)); fprintf (resl,'\n Molar Flow Rate of Stream %3.0f i s %8.4f kmoles / hr ',IS,TotMolarFlRate(IS)); fprintf (resl,'\n Mass Flow Rate of Stream %3.0f i s %8.4f kg /hr ',IS,TotMassFlRate(IS)); fprintf (resl,'\n --------------------------------- --------------------\n'); % SECTION BEGINS ================================= ==================== % do this section if you want to check manually the orders of magnitude % otherwise remove this section % activities: a_i = y_i * phi_i * P % y_i is the mole fraction which is available above , as x(i) % obtain phi_i from the function % function[phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P ,T,yH2,yN2,yNH3) [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P(IS),T(IS) ,x(1),x(2),x(4)); % now compute the equilibrium constant K % function[Ka] = EQNCN(T) [Ka] = EQNCN(T(IS)); % now find Arrhenius Rate Constant [kArh]= RTCNT(T(IS),R_un);

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% now find reaction rate [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); [CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T(IS),P(IS)); Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4; jj=1; for j= NC+1:2*NC Cp(j) = Cp(jj)/MW(jj); % kJ/(kg-K) jj=jj+1; end CpMix = 0.0; for ii = 1:NC CpMix = CpMix + Cp(ii+5)*MassFlRate(ii)/TotMassFlRa te(IS); end [HtReact]= DeltaH(P(IS),T(IS)); % X is the height from the top of the catalyst (m) and StepX is the % increment in X fprintf (resl,'\n Test Section output begins ...\n' ); fprintf (resl,'\n Pressure = %6.2f atm Temp = % 6.2f',P(IS),T(IS)); X = 0.0; StepX=0.5; fprintf (resl,'\n X = %6.2f m StepX = %6.2f',X,St epX); Zold = 0; % no N2 has been converted as yet fprintf (resl,'\n Zold = %6.2f',Zold); Told = T(IS); % this is temp at the inlet of this v olume N20 = MolFlRate(2); % kmols/hr of N2 entering this volume element fprintf (resl,'\n N20 = %6.2f kmol/hr ',N20); fprintf (resl,'\n Mdot = %12.4e kg/hr',TotMassFlRat e(IS)); fprintf (resl,'\n Cp = %12.4e kJ/(kg-K) ',CpMix); fprintf (resl,'\n x(1) = %8.4f x(2)=%8.4f x(4 )=%8.4f ',x(1),x(2),x(4)); fprintf (resl,'\n RNH3 = %12.4e kmol/(hr-m^3) ',RNH 3); fprintf (resl,'\n HtReact = %12.4e kJ/kmol ',HtReac t); dZdX = Area*RNH3 /(2.0*N20); dTdX = abs(HtReact)*RNH3*Area/(TotMassFlRate(IS)*Cp Mix); fprintf (resl,'\n\n dZ/dx = %12.4e conversion p er meter',dZdX); fprintf (resl,'\n dT/dx = %12.4e K per meter \n',dTdX); % now use the above gradients to find new values fo r moles of H, N, Ammonia % the volume of this box is now Area * StepX DeltaVolume = Area * StepX; Znew = Zold + dZdX * StepX; Tnew = Told + dTdX * StepX; fprintf (resl,'\n New values Z = %6.2f T = %6. 2f K \n',Znew,Tnew); % now compute new moles and mole fractions % converted moles of N2 delta = Znew * MolFlRate(2); % new mole rates (kmols/hr) and mass flow rates (kg /hr)

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for ik =1:NC MolFlRateOld(ik) = MolFlRate(ik); MassFlRateOld(ik) = MassFlRate(ik); MolFlRateNew(ik) = 0.0; % need to compute this now MassFlRateNew(ik)= 0.0; % need to compute this now xold(ik) = x(ik); % old mole fractions xnew(ik) = 0.0; % new mole fractions end % new totals TotMolarFlRateOld(IS)=TotMolarFlRate(IS); TotMolarFlRateNew(IS)=0.0; TotMoleFrOld=0.0; TotMoleFrNew=0.0; TotMassFlRateOld(IS)=TotMassFlRate(IS); TotMassFlRateNew(IS)=0.0; MolFlRateNew(1) = MolFlRateOld(1) - 3.0* delta; % H 2 MolFlRateNew(2) = MolFlRateOld(2) - 1.0* delta; % N 2 MolFlRateNew(3) = MolFlRateOld(3); % A rgon MolFlRateNew(4) = MolFlRateOld(4) + 2.0* delta; % N H3 MolFlRateNew(5) = MolFlRateOld(5); % C H4 % for ik1 = 1:NC MassFlRateNew(ik1) = MW(ik1) * MolFlRateNew(ik1 ); end for ik2 = 1:NC TotMolarFlRateNew(IS) = TotMolarFlRateNew(IS) + MolFlRateNew(ik2); TotMassFlRateNew(IS) = TotMassFlRateNew(IS) + MassFlRateNew(ik2); end for ik11 = 1:NC xnew(ik11) = MolFlRateNew(ik11)/TotMolarFl RateNew(IS); end for ik12 = 1:NC TotMoleFrOld = TotMoleFrOld + xold(ik12); TotMoleFrNew = TotMoleFrNew + xnew(ik12); end % now write a summary of the change fprintf (resl,'\n\n SUMMARY AFTER CONVERSION in th is volume box Area*StepX \n'); fprintf (resl,'\n BEFORE CONVERSI ON AFTER CONVERSION\n'); fprintf (resl,'\n i MolFlRate mol fr MassFlRate MolFlRate mol fr MassFlRate'); fprintf (resl,'\n kmol/hr kg/hr kmol/hr kg/hr'); for ik3=1:NC

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fprintf (resl,'\n %3.0f %15.4e %8.4f %12.4e %20. 4e %8.4f %12.4e ',ik3,MolFlRateOld(ik3),xold(ik3),MassFlRateOld(ik3 ),MolFlRateNew(ik3),xnew(ik3),MassFlRateNew(ik3)); end fprintf (resl,'\n %21.4e %8.4f %12.4e %21.4e %9.4f %13.4e',TotMolarFlRateOld(IS),TotMoleFrOld,TotMassF lRateOld(IS),TotMolarFlRateNew(IS),TotMoleFrNew,TotMassFlRateNew(IS)); % how much energy is given off in this volume? % mass flow rate is the same before or after so can use either Power = TotMassFlRateOld(IS)*CpMix*(Tnew-Told); % ( kg/hr)*(kJ/kg-K)*(K) = kJ/hr Power = Power/(60*60); % kW fprintf (resl,'\n Power given off in synthesis of a mmonia in this vol element = %12.4e kW\n',Power); fprintf (resl,'\n Test Section ends.............\n\ n\n'); %SECTION ENDS ==================================== === Z=0.0; % N2 conversion percentage start with ) sin ce no N2 is converted at t=0 [Eta1]=ETA(P(IS),T(IS),Z);% fprintf (resl,'\n\n Pressure = %8.2f atm Temperat ure = %8.2f \n',P(IS),T(IS)); %fprintf (resl,'\n Temp phiH2 phiN2 p hiNH3 aH2 aN2 aNH3 Ka kArh RNH3\n '); %fprintf (resl,'\n %6.1f %10.2e %10.2e %10.2e %10.2 e %10.2e %10.2e %10.2e %10.2e %10.2e\n',T(IS),phiH2,phiN2,phiNH3,aH2,aN2,a NH3,Ka,kArh,RNH3); %-------------------------------------------------- -------- % begin RK 4th order for 2 1st-order coupled ODEs fprintf (resl,'\n\n R-K method \n\n'); %Area is defined above FoverA = MolFlRate(2)/Area; % molar flow rate for N itrogen at inlet Z = 0.0; T = T(IS); P = P(IS); ZZ(1)=0.0; % initial value of conversion percentage TT(1)=T; % initial value of temperature (K) Lmin =0.0; Lmax= HghtReactor; % height of the convtr in meters NZONES = 3; NPTZ=10; % 10 meshes in each zone NPTS = NPTZ*NZONES; % 30 meshes total h = (Lmax-Lmin)/NPTS;

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LL(1)=Lmin; for i = 2: NPTS+1 LL(i) = LL(i-1)+h; end LL L = LL(1); i = 1 ; % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane Hmoles(i) = MolFlRate(1); Nmoles(i) = MolFlRate(2); Rmoles(i) = MolFlRate(3); Amoles(i) = MolFlRate(4); Mmoles(i) = MolFlRate(5); TotalMoles(i) = Hmoles(i) + Nmoles(i) + Rmoles(i) + Amoles(i) + Mmoles(i); HmolesFr(i) = Hmoles(i)/TotalMoles(i); % for plotti ng, instead of mole fl rates NmolesFr(i) = Nmoles(i)/TotalMoles(i); AmolesFr(i) = Amoles(i)/TotalMoles(i); AmolesPC(i) = Amoles(i)/TotalMoles(i); % %%%%%%%%%%%%%%% patch added 18 May 2011 %%%%%%% for printout only Tsend = T; TotMassSend = TotMassFlRate(IS); [MWtMixture,CpMixKg,CpMixMo,TotMolarFlRateMix,Moles ,masses]=CpSummary(Tsend,P,x,MW,TotMassSend); CpKg(i) = CpMixKg; % kJ/(kg-K) CpMo(i) = CpMixMo; % kJ/(kmol-K) MWmix(i)= MWtMixture; % kg/kmol MdotCpKT(i) = ((TotMassFlRate(IS))/3600)*CpKg(i)*Ts end; % kg/s * kJ/(kg-K) * K = kW TMFMix(i)=TotMolarFlRateMix; hhh(i)=Moles(1); nnn(i)=Moles(2); rrr(i)=Moles(3); amm(i)=Moles(4); ch4(i)=Moles(5); massh(i)=masses(1); massn(i)=masses(2); massr(i)=ma sses(3); massa(i)=masses(4); massc(i)=masses(5); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of patch 18 May 2011 %%%%%%%% %%%%%%%%%%%%%% patch added 15 May 2011 % now find, at this point as an initial condition, the equilibrium position Epsilon=0.0; i=1; [Epsilon] = EquilibriumC (P,T,Hmoles(i),Nmoles(i),A moles(i),TotalMoles(i)); HmolesEq(i) = Hmoles(i) - 3.0*Epsilon ; NmolesEq(i) = Nmoles(i) - Epsilon ; RmolesEq(i) = Rmoles(i) ; AmolesEq(i) = Amoles(i) + 2.0*Epsilon ; MmolesEq(i) = Mmoles(i) ; % TotalMolesEq(i) = HmolesEq(i) + NmolesEq(i) + Rmole sEq(i) + AmolesEq(i) + MmolesEq(i);

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AmolesPCEq(i) = AmolesEq(i)/TotalMolesEq(i); TTeps(1) = Epsilon; % now find equilibrium temp [Teqm] = EquilibriumT (x,P,T); TTeqm(i)=Teqm; % what if Teqm is not found? set it to max if (Teqm==0) TTeqm(i)=800.0; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% EachZoneHgt = Lmax/NZONES; LBDRY(1) = 0.0; for ibdry = 2:NZONES+1 LBDRY(ibdry)=LBDRY(ibdry-1)+EachZoneHgt; end LBDRY %%%%%%%%%%%%%%%%%%%%%%%%% CATALYST LOADING %%%%%%%%%%%%%%%%%% fprintf (resl,'\n No of catalyst zones = %2.0f \n', NZONES); frac = [1.5;1.25;1.0]; fprintf (resl,'\n Zone LBDRY Fraction of Catalys t'); for icat = 1:NZONES fprintf (resl,'\n %2.0f %6.2f %6.2f ',icat,LBD RY(icat+1),frac(icat)); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CatFrac = frac(1); % start with the fraction of zon e 1 fprintf (resl,'\n i x(1) x(2) x(3) x(4) x(5) Total Mole Fl Rate'); fprintf (resl,'\n %3.0f %8.4f %8.4f %8.4f %8.4f %8 .4f %15.4e',i,x(1),x(2),x(3),x(4),x(5),TotalMoles(i)); Toptimal(i) = 896.0; NPTS1=NPTS-1; PZ1 = 0; PZ2 =0; PZ3 = 0; % limits of i in zones % in zone 1 i goes from 1 to NPTZ+1 % 2 i NPTZ+1 to 2*NPTZ+1 % 3 i 2*NPTZ+1 to 3*NPTZ+1 iLIM1 = NPTZ+1; iLIM2 = 2*NPTZ+1; iLIM3 = 3*NPTZ+1 ; while (i <= NPTS) %-------------------------------- -- RK4 LOOP BEGINS i; % zone is defined by height from top L if (i<=iLIM1) thisZone = 1;

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LastZone = thisZone; PZ1 = PZ1 +1; % this counter runs from 1 to las t value of PZ1=NPTZ % fprintf (resl,'\n PZ1 = %3.0f ',PZ1); TMB(1,PZ1)=TT(i); LMB(1,PZ1)=LL(i); end if ((i>=iLIM1)&(i<=iLIM2)) thisZone = 2; if (LastZone==1) TT(i)=TT(3); LastZone = thisZone; end PZ2 = PZ2 +1; % this counter runs from 1 to las t value of PZ2 (20) % fprintf (resl,'\n PZ2 = %3.0f ',PZ2); TMB(2,PZ2)=TT(i); LMB(2,PZ2)=LL(i); end if ((i>=iLIM2)&(i<=iLIM3)) thisZone = 3; if (LastZone==2) % TT(i)=TT(2); LastZone = thisZone; end PZ3 = PZ3 +1; % this counter runs from 1 to las t value of PZ3 % fprintf (resl,'\n PZ3 = %3.0f ',PZ3); TMB(3,PZ3)=TT(i); LMB(3,PZ3)=LL(i); end CatFrac = frac(thisZone); %%%%%%%%%%%%%%%%%%%%%%%%%%% k1_z k1_T %%%%%%%%%%%%%%%%%%%% % first function % evaluate RHS of mass conservation equation FUNC1 [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4)); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); FUNC1 = Eta1*CatFrac*RNH3/(2.0*FoverA); k1_z = FUNC1; % second function % evaluate RHS of energy conservation equation FUNC 2 [CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T,P);

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Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4; jj=1; for j= NC+1:2*NC Cp(j) = Cp(jj)/MW(jj); % kJ/(kg-K) jj=jj+1; end CpMix = 0.0; for ii = 1:NC CpMix = CpMix + Cp(ii+5)*MassFlRate(ii)/TotMassFlRa te(IS); end [HtReact]= DeltaH(P,T); FUNC2 = ( HtReact * Eta1 * CatFrac*RNH3 ) / ((TotMa ssFlRate(IS)/Area) * CpMix ); k1_T = -FUNC2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k2_z k2_P L = LL(i) + h/2. ; Z = ZZ(i) + k1_z * h/2.0; T = TT(i) + k1_T * h/2.0; % first function [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4)); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); FUNC1 = Eta1*CatFrac*RNH3/(2.0*FoverA); k2_z = FUNC1; % second function [CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T,P); Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4; jj=1; for j= NC+1:2*NC Cp(j) = Cp(jj)/MW(jj); % kJ/(kg-K) jj=jj+1; end CpMix = 0.0; for ii = 1:NC CpMix = CpMix + Cp(ii+5)*MassFlRate(ii)/TotMassFlRa te(IS); end [HtReact]= DeltaH(P,T); FUNC2 = ( HtReact * Eta1 * CatFrac* RNH3 ) / ((TotM assFlRate(IS)/Area) * CpMix ); k2_T = -FUNC2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k3_z k3_P Z = ZZ(i) + k2_z * h/2.0; T = TT(i) + k2_T * h/2.0; % first function [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4));

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[Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); FUNC1 = Eta1*CatFrac* RNH3/(2.0*FoverA); k3_z = FUNC1; % second function [CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T,P); Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4; jj=1; for j= NC+1:2*NC Cp(j) = Cp(jj)/MW(jj); % kJ/(kg-K) jj=jj+1; end CpMix = 0.0; for ii = 1:NC CpMix = CpMix + Cp(ii+5)*MassFlRate(ii)/TotMassFlRa te(IS); end [HtReact]= DeltaH(P,T); FUNC2 = ( HtReact * Eta1 * CatFrac* RNH3 ) / ((TotM assFlRate(IS)/Area) * CpMix ); k3_T = -FUNC2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k4_z k4_P L = LL(i) + h; Z = ZZ(i) + k3_z * h; T = TT(i) + k3_T * h; % first function [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4)); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); FUNC1 = Eta1*RNH3/(2.0*FoverA); k4_z = FUNC1; % second function [CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T,P); Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4; jj=1; for j= NC+1:2*NC Cp(j) = Cp(jj)/MW(jj); % kJ/(kg-K) jj=jj+1; end CpMix = 0.0; for ii = 1:NC CpMix = CpMix + Cp(ii+5)*MassFlRate(ii)/TotMassFlRa te(IS); end [HtReact]= DeltaH(P,T);

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FUNC2 = ( HtReact * Eta1 * RNH3 ) / ((TotMassFlRate (IS)/Area) * CpMix ); k4_T = -FUNC2; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % get new values for Z, T i = i + 1; ZZ(i) = ZZ(i-1) + (h/6.)*(k1_z + 2.0*k2_z + 2.0*k3_ z + k4_z); TT(i) = TT(i-1) + (h/6.)*(k1_T + 2.0*k2_T + 2.0*k3_ T + k4_T); % if this is the last point, it will not go back, s o store the last point if (i==(NPTS+1)) TMB(3,PZ3+1)=TT(i); LMB(3,PZ3+1)=LL(i); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % update mole fractions % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane ConvertedMolesOfN = ZZ(i)*MolFlRate(2); Hmoles(i) = MolFlRate(1) - 3.0*( ConvertedMolesOfN ) ; Nmoles(i) = (1.0-ZZ(i)) *MolFlRate(2); Rmoles(i) = MolFlRate(3); Amoles(i) = MolFlRate(4) + 2.0*( ConvertedMolesOfN ) ; Mmoles(i) = MolFlRate(5); % TotalMoles(i) = Hmoles(i) + Nmoles(i) + Rmoles(i) + Amoles(i) + Mmoles(i); AmolesPC(i) = Amoles(i)/TotalMoles(i); Power(i-1) = TotMassFlRate(IS)*CpMix*(TT(i)-TT(i-1) ); % (kg/hr)*(kJ/kg.K)*K=kJ/hr Power(i-1) = Power(i-1)/(60*60); % kW % now to go back to the beginning of the loop, set new local parameters % P,T,Z. and also need P, T, x(1), x(2), x(4) for n ext iteration step x(1) = Hmoles(i)/TotalMoles(i); x(2) = Nmoles(i)/TotalMoles(i); x(3) = Rmoles(i)/TotalMoles(i); x(4) = Amoles(i)/TotalMoles(i); x(5) = Mmoles(i)/TotalMoles(i); %%%%%%%%%%%%%%% patch added 18 May 2011 %%%%%%% for printout only Tsend = TT(i); TotMassSend = TotMassFlRate(IS); [MWtMixture,CpMixKg,CpMixMo,TotMolarFlRateMix,Moles ,Masses]=CpSummary(Tsend,P,x,MW,TotMassSend); CpKg(i) = CpMixKg; % kJ/(kg-K) CpMo(i) = CpMixMo; % kJ/(kmol-K) MWmix(i)= MWtMixture; % kg/kmol MdotCpKT(i) = ((TotMassFlRate(IS))/3600)*CpKg(i)*Ts end; % kg/s * kJ/(kg-K) * K = kW TMFMix(i)=TotMolarFlRateMix; hhh(i)=Moles(1); nnn(i)=Moles(2); rrr(i)=Moles(3); amm(i)=Moles(4); ch4(i)=Moles(5); massh(i)=masses(1); massn(i)=masses(2); massr(i)=ma sses(3); massa(i)=masses(4); massc(i)=masses(5); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of patch 18 May 2011 %%%%%%%%

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MolFlowRate(2) = Nmoles(i); % this is the N2 inlet for the next volume element %FoverA = MolFlRate(2)/Area; % molar flow rate for Nitrogen at inlet % Pressure remains same P=P; T=TT(i); Z=ZZ(i); fprintf (resl,'\n %3.0f %8.4f %8.4f %8.4f %8.4f %8. 4f %15.4e',i,x(1),x(2),x(3),x(4),x(5),TotalMoles(i)); % now go to find optimal temp at this point ------- -------------- 10 may 2011 [Topt] = OptimalT (x,P); Toptimal(i) = Topt; % what if Topt is not found? set it to max if (Topt==0) Toptimal(i)=850.0; end % now go back %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% patch added May 15 2011 % now find equilibrium temp [Teqm] = EquilibriumT (x,P,T); TTeqm(i)=Teqm; % what if Teqm is not found? set it to max if (Teqm==0) TTeqm(i)=800.0; end % now find, at this point as an initial condition, the equilibrium position %HmolesInit=Hmoles(i) %NmolesInit=Nmoles(i) %AmmMolesInit=Amoles(i) %TotMoles=TotalMoles(i) Epsilon=0.0; [Epsilon] = EquilibriumC (P,T,Hmoles(i),Nmoles(i),A moles(i),TotalMoles(i)); TTeps(i) = Epsilon; HmolesEq(i) = Hmoles(i) - 3.0*Epsilon ; NmolesEq(i) = Nmoles(i) - Epsilon ; RmolesEq(i) = Rmoles(i) ; AmolesEq(i) = Amoles(i) + 2.0*Epsilon ; MmolesEq(i) = Mmoles(i) ; % TotalMolesEq(i) = HmolesEq(i) + NmolesEq(i) + Rmole sEq(i) + AmolesEq(i) + MmolesEq(i); HmolesFr(i) = Hmoles(i)/TotalMoles(i); % for plotti ng, instead of mole fl rates NmolesFr(i) = Nmoles(i)/TotalMoles(i); AmolesFr(i) = Amoles(i)/TotalMoles(i); AmolesPCEq(i) = AmolesEq(i)/TotalMolesEq(i); % now to go back to the beginning of the loop, set new local parameters % P,T,Z. and also need P, T, x(1), x(2), x(4) for n ext iteration step

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%xEq(1) = HmolesEq(i)/TotalMolesEq(i); %xEq(2) = NmolesEq(i)/TotalMolesEq(i); %xEq(4) = AmolesEq(i)/TotalMolesEq(i); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of patch added on May 13, 2011 %%%%%%%%%%%%% patch entered on 17 may 2011 % compute new mass fractions since Cp changes % stream computation............................... ........ %%%%%%%%%%%%% end of patch 17 may 2011 %%%%%%%%%%%%%%%%%% end % ------------------------------------- RK4 LOOP ENDS fprintf (resl,'\n\n i Toptimal Tequilibrium Epsilon'); for i2 = 1:NPTS+1 fprintf (resl,'\n %3.0f %8.2f %8.2f %12.4e',i2,Toptimal(i2),TTeqm(i2),TTeps(i2)); end fprintf (resl,'\n\n\n Summary of RK numerical solut ion \n \n \n'); fprintf (resl,'\n Moles Conversion % of N \n'); fprintf (resl,'\n i L H2 N2 Ar NH3 CH4 Total Z T Topt\n'); for i = 1: NPTS+1 fprintf (resl,'\n %3.0f %6.4f %8.4f %8.4f % 8.4f %8.4f %8.4f %10.4f %6.4f %6.2f %6.2f',i,LL(i),Hmoles(i),Nmoles(i),Rmoles(i),Amoles (i),Mmoles(i),TotalMoles(i),ZZ(i),TT(i),Toptimal(i)); end fprintf (resl,'\n\n\n Summary of Molar Quantities computed from CpSummary '); fprintf (resl,'\n i L H2 N2 Ar NH3 CH4 Total Z T CpKg CpMo MWtMix Power k W\n'); for i = 1: NPTS+1 fprintf (resl,'\n %2.0f %4.3f %8.4f %8.4f %8.4 f %8.4f %8.4f %10.4f %6.4f %6.2f %6.2f %6.2f %6.2f %6.2f',i,LL(i),hhh(i),nnn(i),rrr(i),amm(i),ch4(i),T MFMix(i),ZZ(i),TT(i),CpKg(i),CpMo(i),MWmix(i),MdotCpKT(i)); end % temps in each catalyst bed % straight line between x = LMB(1,20), y =TMB(1,20) % and x = LMB(2,1) , y =TMB(2,1) xSt12(1) = LMB(1,iLIM1); ySt12(1) = TMB(1,iLIM1); xSt12(2) = LMB(2,1) ; ySt12(2) = TMB(2,1); % straight line between x = LMB(2,20), y =TMB(2,20) % and x = LMB(3,1) , y =TMB(3,1) xSt23(1) = LMB(2,NPTZ+1); ySt23(1) = TMB(2,NPTZ+1); xSt23(2) = LMB(3,1) ; ySt23(2) = TMB(3,1);

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LMB; TMB; for ibed = 1: 3 fprintf (resl,'\n Bed no: %2.0f ',ibed); fprintf (resl,'\n j Length Temp '); for ip1 = 1:NPTZ+1 fprintf (resl,'\n %3.0f %9.3f %9.3f',ip1,LMB(ibed,ip1),TMB(ibed,ip1)); end end for ip1=1:NPTZ+1 LMB1(ip1) = LMB(1,ip1); LMB2(ip1) = LMB(2,ip1); LMB3(ip1) = LMB(3,ip1); TMB1(ip1) = TMB(1,ip1); TMB2(ip1) = TMB(2,ip1); TMB3(ip1) = TMB(3,ip1); end % Power available from exothermic reactions fprintf (resl,'\n\n\n\n i Power (kW)\n'); sumPower =0.0; for i = 1: NPTS1 fprintf (resl,'\n %3.0f %12.4e ',i,Power(i)); sumPower=sumPower+Power(i); end fprintf (resl,'\n total exothermic energy available is %12.4e kW ',sumPower); Power = TotMassFlRateOld(IS)*CpMix*(Tnew-Told); % ( kg/hr)*(kJ/kg-K)*(K) = kJ/hr Power = Power/(60*60); % kW fprintf (resl,'\n Power given off in synthesis of a mmonia in this vol element = %12.4e kW\n',Power); % Energy Available between 1st and 2nd bed TempBed1 = TMB(1,NPTZ+1); fprintf(resl,'\n Exit Temp from 1st Bed = %8.2f K\n ',TempBed1); fprintf(resl,'\n i Cp kJ/(kg-K) Mass Fl R ate(i) kg/hr'); CpMix1 = 0.0; for ii = 1:NC CpMix1 = CpMix1 + Cp(ii+5)*MassFlRate(ii)/TotMassFl Rate(IS); fprintf(resl,'\n %3.0f %8.2f %8.2f',ii,Cp(ii +5),MassFlRate(ii)); end fprintf(resl,'\n Total Mass Flow Rate = %12.2f kg/h r',TotMassFlRate(IS)); fprintf(resl,'\n Cp Mixture = %8.2f kJ/kg-K',CpMix1 ); HeatValue1 = (1./3600)*TotMassFlRate(IS)*CpMix1*Tem pBed1 ; fprintf (resl,'\n Power Value 1 = %12.4e kW',HeatV alue1); TempBed2 = TMB(2,1); fprintf(resl,'\n\n Exit Temp from 2nd Bed = %8.2f K \n',TempBed2); fprintf(resl,'\n i Cp kJ/(kg-K) Mass Fl R ate(i) kg/hr'); CpMix2 = 0.0; for ii = 1:NC CpMix2 = CpMix2 + Cp(ii+5)*MassFlRate(ii)/TotMassFl Rate(IS);

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fprintf(resl,'\n %3.0f %8.2f %8.2f',ii,Cp(ii +5),MassFlRate(ii)); end fprintf(resl,'\n Total Mass Flow Rate = %12.2f kg/h r',TotMassFlRate(IS)); fprintf(resl,'\n Cp Mixture = %8.2f kJ/kg-K',CpMix2 ); HeatValue2 = (1./3600)*TotMassFlRate(IS)*CpMix2*Tem pBed2 ; fprintf (resl,'\n Power Value 2 = %12.4e kW ',HeatV alue2); PowerExtracted = HeatValue1 - HeatValue2; fprintf (resl,'\n\n Power Extracted = %12.4e kW',Po werExtracted); InletTemp LL ZZ ThisPlot=4; if (ThisPlot==1) plot (LL,ZZ,'-k') grid on xlabel ('Distance (m)') ylabel ('Conversion of N_2 (Z)') xlim([0 HghtReactor]) % title 'Conversion of N_2 in the synthesis co nvertor' end if (ThisPlot==2) % plot (LL,TT,'-k') plot (LMB1,TMB1,'-k') hold on plot(xSt12,ySt12,'-k') hold on plot (LMB2,TMB2,'-k') hold on plot(xSt23,ySt23,'-k') hold on plot (LMB3,TMB3,'-k') hold on % now plot the optimal temp on this % plot (LL,Toptimal,'-k'); % hold on % plot (LL,TTeqm,'--k') grid on xlabel ('Distance (m)') ylabel ('Temperature (K)') title 'Temperature in synthesis convertor' end if (ThisPlot==3) %plot (LL,Hmoles,'-k') plot (LL,HmolesFr,'-k') hold on %plot (LL,Nmoles,'-.k')

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plot (LL,NmolesFr,'-k') hold on %plot (LL,Amoles,'--k') plot (LL,AmolesFr,'-k') xlabel ('Distance (m)') % ylabel ('Molar Flow Rate (kmols/hr)') ylabel ('Molar Fraction)') h = legend('H_2','N_2','NH_3',2); %title 'Molar flow rate of H_2, N_2 and NH_3 in convertor' title 'Molar fractions in convertor' end if (ThisPlot==4) subplot(2,2,1:2) % plot (LL,AmolesPC,'-k') % xlabel ('Distance (m)') % ylabel ('Ammonia Mole Fr') % grid on % title 'Mole % NH_3' plot (LL,ZZ,'-k') xlim([0 HghtReactor]) % ylim([0 0.301]) grid on xlabel ('Distance (m)') ylabel ('Conversion of N_2 (Z)') % title 'Conversion of N_2 in the synthesis c onvertor' subplot(2,2,3) % plot (LL,TT,'-k') plot (LMB1,TMB1,'-k') hold on plot(xSt12,ySt12,'-k') hold on plot (LMB2,TMB2,'-k') hold on plot(xSt23,ySt23,'-k') hold on plot (LMB3,TMB3,'-k') hold on % now plot the optimal temp on this plot (LL,Toptimal,'-k'); hold on plot (LL,TTeqm,'--k') grid on text (0.3,750,'T_{opt}'); text (0.5,850,'T_{eqm}'); xlim([0 HghtReactor]) %ylim([600 1000]) xlabel ('Distance (m)') ylabel ('Temperature (K)') % title 'Temperature' subplot(2,2,4) %plot (LL,Hmoles,'-k') plot (LL,HmolesFr,'-k') hold on

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%plot (LL,Nmoles,'-.k') plot (LL,NmolesFr,'-.k') hold on %plot (LL,Amoles,'--k') plot (LL,AmolesFr,'--k') grid on xlim([0 HghtReactor]) xlabel ('Distance (m)') % ylabel ('Molar Flow Rate (kmols/hr)') ylabel ('Molar Fraction') h = legend('H_2','N_2','NH_3',2); %title 'Molar flow rate of H_2, N_2 and NH_3 in convertor' %title 'Molar fractions in convertor' end if (ThisPlot==5) subplot(2,1,1) % plot (LL,TT,'-k') plot (LMB1,TMB1,'-k') hold on plot(xSt12,ySt12,'-k') hold on plot (LMB2,TMB2,'-k') hold on plot(xSt23,ySt23,'-k') hold on plot (LMB3,TMB3,'-k') hold on % now plot the optimal temp on this plot (LL,Toptimal,'-k'); hold on plot (LL,TTeqm,'--k') grid on text (0.15,810,'T_{opt}'); text (0.62,870,'T_{eqm}'); xlim([0 HghtReactor]) %ylim([600 1000]) xlabel ('Distance (m)') ylabel ('Temperature (K)') % title 'Temperature' subplot(2,1,2) %plot (LL,Hmoles,'-k') plot (LL,HmolesFr,'-k') hold on %plot (LL,Nmoles,'-.k') plot (LL,NmolesFr,'-.k') hold on %plot (LL,Amoles,'--k') plot (LL,AmolesFr,'--k') grid on xlim([0 HghtReactor]) xlabel ('Distance (m)') % ylabel ('Molar Flow Rate (kmols/hr)') ylabel ('Mole Fraction') h = legend('H_2','N_2','NH_3',2);

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%title 'Molar flow rate of H_2, N_2 and NH_3 in convertor' %title 'Molar fractions in convertor' end toc fclose(resl)

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APPENDIX B5: MATLAB ™ PROGRAM FOR FINDING OPTIMAL

TEMPERATURE This section lists the Program MansonOneEqn and its function OptimalT

% Program Name: MansonOneEqn % Steady State Ammonia Thermal Stora ge % % C:\MATLAB7\work\Ammonia\MansonOneEqn.m % % First Written: JULY 2010 % Last Update: Oct 19 2011 % open output file resl=fopen('outm.txt','w'); % INPUT and STREAM OUT PUT tic fprintf (resl,'\n MansonOneEqn.m \n'); fprintf (resl,'\n MansonOneEqn.m \n'); fprintf (resl,'\n MansonOneEqn.m \n'); fprintf (resl,'\n MansonOneEqn.m \n'); fprintf (resl,'\n MansonOneEqn.m \n'); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % program parameters NC = 5; % number of components, H2,N2,NH3,Ar,CH4 fprintf (resl,'\n No. of Components = %3.0f \n',NC) ; R_un = 8.314472; % universal gas const J/(mol-K) fprintf (resl,'\n Universal Gas Const = %12.6f kJ/ (mol-K)\n',R_un); % chemical data from http://www-jmg.ch.cam.ac.uk/tools/magnus/PeriodicTable.html % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane MW1=2.0*1.0079759; MW2=2.0*14.0067231; MW3= 39.9476 613; MW4=(14.0067231+3.0*1.0079759); MW5 = (12.0110369+4 .0*1.0079759); MW = [ MW1;MW2;MW3;MW4;MW5 ]; fprintf (resl,'\n ID Name Mol Wt (kg/kmol)\n') ; fprintf (resl,'\n 1 Hydrogen %8.4f ',MW(1)); fprintf (resl,'\n 2 Nitrogen %8.4f ',MW(2)); fprintf (resl,'\n 3 Argon %8.4f ',MW(3)); fprintf (resl,'\n 4 Ammonia %8.4f ',MW(4)); fprintf (resl,'\n 5 Methane %8.4f ',MW(5)); fprintf (resl,'\n ----------------------------\n'); % STREAM INPUT P = 150.0; % atm P_SIunits = P*1.0e5; %Pascals TotMassFlRate = 183600; % kg/hr Dashti p.20 % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane x= [ 0.6567006; 0.2363680; 0.0202874; 0.026930; 0.0 59714]; % mole fractions % x= [ 0.75; 0.235 ; 0.0 ; 0.015; 0.0]; % mole frac tions %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% [Topt] = OptimalT (x,P); Toptimal(1) = Topt; % what if Topt is not found? set it to max if ((Topt==0)|(Topt>800))

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Toptimal(1)=780.0; end T=Toptimal(1); TT(1)=T; % for plotting Teqm=0; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% patch added May 13 2011 % now find equilibrium temp [Teqm] = EquilibriumT (x,P,T); TTeqm(1)=Teqm; % what if Teqm is not found? set it to max if (Teqm==0) TTeqm(1)=780.0; end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of patch added on May 13, 2011 Area = 20.0 ; % cross-section area of catalyst m^2 fprintf (resl,'\n Area = %6.2f m^2',Area); HghtReactor = 4.50; % meters VolCatalyst = Area * HghtReactor; fprintf (resl,'\n Vol of catalyst = %6.3f m^3 \n',V olCatalyst); % stream computation............................... ........ MWt = 0.0; % average molecular weight of stream xT = 0.0; % total mol fr. this is should add up to 1.0 for i = 1:NC MWt=MWt+x(i)*MW(i); xT = xT +x(i); end RR = R_un/MWt; % gas constant for this stream (gas) MolarFlRateMix = TotMassFlRate/MWt; % in kmols/hr TotMolarFlRate = TotMassFlRate/MWt; % kmoles per hr mfrT=0.0; mfmT=0.0; for jk=1:NC mfr(jk) = x(jk)*MolarFlRateMix; mfrT = mfrT + mfr(jk); mfm(jk) = mfr(jk)*MW(jk); mfmT = mfmT + mfm(jk); end % end of stream computation........................ ....... % % STREAM OUTPUT fprintf (resl,'\n --------------------------------- --------------------\n'); fprintf (resl,'\n NC Mole Fr Molar flow rate Mass flow rate'); fprintf (resl,'\n kg-mols/hr kg/hr\n'); for i=1:NC fprintf (resl,'\n %2.0f %8.4f %8.4f %8.4f ',i,x(i),mfr(i),mfm(i)) end fprintf (resl,'\n \n Total %6.4f %12.4f %12. 4f\n',xT,mfrT,mfmT); fprintf (resl,'\n Mol Wt of mixture MWt = %8.4f kg/kmol \n',MWt); fprintf (resl,'\n Gas Const of mixture RR = %8.4f kJ/(kmol-K) \n',RR); fprintf (resl,'\n Molar Flow Rate of Stream is %8.2 f kmoles / hr \n',MolarFlRateMix);

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fprintf (resl,'\n --------------------------------- --------------------\n'); % make the mixture Cp, Cv and gamma CpMix = 0.0; RMix = 0.0; % find the Cp values at this temp T1 and press P [CpH2,CpN2,CpNH3,CpCH4,CpAr] = SPCHT(T,P); % kJ/(km ol-K) Cp(1) = CpH2; Cp(2) = CpN2; Cp(3) = CpAr; Cp(4)=CpN H3; Cp(5)=CpCH4; jj=1; for j= NC+1:2*NC Cp(j) = Cp(jj)/MW(jj); % kJ/(kg-K) jj=jj+1; end fprintf (resl,'\n --------------------------------- --------\n'); fprintf (resl,'\n Cp values from function SPCHT at T = %8.2f K and P = %6.2f atm\n',T,P); fprintf (resl,'\n Component Cp kJ/(kmol-K) kJ/( kg-K)\n'); fprintf (resl,'\n 1 Hydrogen %12.4e %12.4e ',Cp(1 ),Cp(6)); fprintf (resl,'\n 2 Nitrogen %12.4e %12.4e ',Cp(2 ),Cp(7)); fprintf (resl,'\n 3 Argon %12.4e %12.4e ',Cp(3 ),Cp(8)); fprintf (resl,'\n 4 Ammonia %12.4e %12.4e ',Cp(4 ),Cp(9)); fprintf (resl,'\n 5 Methane %12.4e %12.4e ',Cp(5 ),Cp(10)); fprintf (resl,'\n --------------------------------- ---------\n'); for i = 1:NC % individual gas constants Rg(i) = R_un/MW(i); %RMix = RMix + Rg(i)*mfr(i)/mfrT; RMix = RMix + R_un*mfr(i)/mfrT; % kJ/(kmol-K) CpMix = CpMix + Cp(i)*mfr(i)/mfrT; % kJ/(kmol-K) end CvMix = CpMix - RMix; GammaMix = CpMix/CvMix; fprintf (resl,'\n Mixture properties (based on mass ratios)\n'); fprintf (resl,'\n CpMix CvMix RMix GammaMix\n'); fprintf (resl,'\n %12.4e %12.4e %12.4e % 8.2f \n',CpMix,CvMix,RMix,GammaMix); sumMol = 0.0; sumMas = 0.0; for j=1:NC MolFlRate(j) = x(j) * TotMolarFlRate; % kmol/hr sumMol = sumMol + MolFlRate(j); MassFlRate(j)= MolFlRate(j) * MW(j); % kmol/hr * k g/kmol = kg/hr sumMas = sumMas + MassFlRate(j); end % STREAM OUTPUT fprintf (resl,'\n --------------------------------- --------------------\n'); fprintf (resl,'\n\n Pressure = %8.2f atm Temperat ure = %8.2f \n\n',P,T); fprintf (resl,'\n NC Mole Fr Molar flow rate Mass flow rate'); fprintf (resl,'\n kmoles/hr kg/hr\n'); for i=1:NC fprintf (resl,'\n %2.0f %8.4f %8.2f %8.2f ',i,x(i),MolFlRate(i),MassFlRate(i)) end

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fprintf (resl,'\n \n Total %6.4f %12.4f %12.4f\n',xT,sumMol,sumMas); fprintf (resl,'\n Mol Wt of mixture MWt = %8.2f kg/kmol \n',MWt); fprintf (resl,'\n Gas Const of mixture RR = %8.2f kJ/(kmol-K) ',RR); fprintf (resl,'\n Molar Flow Rate of Stream is %8.2 f kmoles / hr ',TotMolarFlRate); fprintf (resl,'\n Mass Flow Rate of Stream is %8.2 f kg /hr ',TotMassFlRate); fprintf (resl,'\n --------------------------------- --------------------\n'); Z=0.0; % N2 conversion percentage start with ) sin ce no N2 is converted at t=0 [Eta1]=ETA(P,T,Z);% fprintf (resl,'\n\n Pressure = %8.2f atm Temperat ure = %8.2f \n',P,T); %-------------------------------------------------- -------- % begin RK 4th order for 1st-order coupled ODE fprintf (resl,'\n\n R-K method \n\n'); FoverA = MolFlRate(2)/Area; % molar flow rate for N itrogen at inlet Z = 0.0; ZZ(1)=0.0; % initial value of conversion percentage Toptimal(1)=T; TT(1)=T; % for plotting Lmin =0.0; Lmax= HghtReactor; % height of the convtr in meters NPTS = 20 ; % meshes total h = (Lmax-Lmin)/NPTS; LL(1)=Lmin; for i = 2: NPTS+1 LL(i) = LL(i-1)+h; end LL L = LL(1); i = 1 ; % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane Hmoles(i) = MolFlRate(1); Nmoles(i) = MolFlRate(2); Rmoles(i) = MolFlRate(3); Amoles(i) = MolFlRate(4); Mmoles(i) = MolFlRate(5); TotalMoles(i) = Hmoles(i) + Nmoles(i) + Rmoles(i) + Amoles(i) + Mmoles(i); HmolesFr(i) = Hmoles(i)/TotalMoles(i); % for plotti ng, instead of mole fl rates NmolesFr(i) = Nmoles(i)/TotalMoles(i); AmolesFr(i) = Amoles(i)/TotalMoles(i); AmolesPC(i) = Amoles(i)/TotalMoles(i); fprintf (resl,'\n i x(1) x(2) x(4) Total Mole Fl Rate'); fprintf (resl,'\n %3.0f %8.4f %8.4f %8.4f %15.4e',i,x(1),x(2),x(4),TotalMoles(i));

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%%%%%%%%%%%%%% patch added 15 May 2011 % now find, at this point as an initial condition, the equilibrium position Epsilon=0.0; i=1; [Epsilon] = EquilibriumC (P,T,Hmoles(i),Nmoles(i),A moles(i),TotalMoles(i)) HmolesEq(i) = Hmoles(i) - 3.0*Epsilon ; NmolesEq(i) = Nmoles(i) - Epsilon ; RmolesEq(i) = Rmoles(i) ; AmolesEq(i) = Amoles(i) + 2.0*Epsilon ; MmolesEq(i) = Mmoles(i) ; % TotalMolesEq(i) = HmolesEq(i) + NmolesEq(i) + Rmole sEq(i) + AmolesEq(i) + MmolesEq(i); AmolesPCEq(i) = AmolesEq(i)/TotalMolesEq(i); NmolesPCEq(i) = NmolesEq(i)/TotalMolesEq(i); TTeps(1) = Epsilon; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% CatFrac = 1.0; NPTS1=NPTS-1; while (i <= NPTS) %-------------------------------- -- RK4 LOOP BEGINS i %%%%%%%%%%%%%%%%%%%%%%%%%%% k1_z %%%%%%%%%%%%%%%%%%%% % first function % evaluate RHS of mass conservation equation FUNC1 [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4)); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); FUNC1 = Eta1*CatFrac*RNH3/(2.0*FoverA); k1_z = FUNC1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k2_z k2_P L = LL(i) + h/2. ; Z = ZZ(i) + k1_z * h/2.0; % first function [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4)); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); FUNC1 = Eta1*CatFrac*RNH3/(2.0*FoverA); k2_z = FUNC1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k3_z k3_P Z = ZZ(i) + k2_z * h/2.0; % first function [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4)); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3);

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FUNC1 = Eta1*CatFrac* RNH3/(2.0*FoverA); k3_z = FUNC1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % k4_z L = LL(i) + h; Z = ZZ(i) + k3_z * h; % first function [Eta1]=ETA(P,T,Z); [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T,x(1),x( 2),x(4)); [Ka] = EQNCN(T); [kArh]= RTCNT(T,R_un); [RNH3]=RRATE(kArh,Ka,aH2,aN2,aNH3); FUNC1 = Eta1*RNH3/(2.0*FoverA); k4_z = FUNC1; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % get new values for Z i = i + 1; ZZ(i) = ZZ(i-1) + (h/6.)*(k1_z + 2.0*k2_z + 2.0*k3_ z + k4_z); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % update mole fractions % component ID: 1 : H2 2: N2 3: Ar 4: Amm onia 5:Methane ConvertedMolesOfN = ZZ(i)*MolFlRate(2); Hmoles(i) = MolFlRate(1) - 3.0*( ConvertedMolesOfN ) ; Nmoles(i) = (1.0-ZZ(i)) *MolFlRate(2); Rmoles(i) = MolFlRate(3); Amoles(i) = MolFlRate(4) + 2.0*( ConvertedMolesOfN ) ; Mmoles(i) = MolFlRate(5); % TotalMoles(i) = Hmoles(i) + Nmoles(i) + Rmoles(i) + Amoles(i) + Mmoles(i); HmolesFr(i) = Hmoles(i)/TotalMoles(i); % for plotti ng, instead of mole fl rates NmolesFr(i) = Nmoles(i)/TotalMoles(i); AmolesFr(i) = Amoles(i)/TotalMoles(i); AmolesPC(i) = Amoles(i)/TotalMoles(i); NmolesPC(i) = Nmoles(i)/TotalMoles(i); % now to go back to the beginning of the loop, set new local parameters % P,T,Z. and also need P, T, x(1), x(2), x(4) for n ext iteration step x(1) = Hmoles(i)/TotalMoles(i); x(2) = Nmoles(i)/TotalMoles(i); x(4) = Amoles(i)/TotalMoles(i); MolFlowRate(2) = Nmoles(i); % this is the N2 inlet for the next volume element FoverA = MolFlRate(2)/Area; % molar flow rate for N itrogen at inlet Z=ZZ(i); fprintf (resl,'\n %3.0f %8.4f %8.4f %8.4f %15.4e',i,x(1),x(2),x(4),TotalMoles(i)); [Topt] = OptimalT (x,P); Toptimal(i) = Topt; % what if Topt is not found? set it to max if (Topt==0) Toptimal(i)=800.0; end

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% now go back T=Toptimal(i); TT(i)=T; % for plotting T %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% patch added May 13 2011 % now find equilibrium temp [Teqm] = EquilibriumT (x,P,T); TTeqm(i)=Teqm; % what if Teqm is not found? set it to max if (Teqm==0) TTeqm(i)=800.0; end % now find, at this point as an initial condition, the equilibrium position %HmolesInit=Hmoles(i) %NmolesInit=Nmoles(i) %AmmMolesInit=Amoles(i) %TotMoles=TotalMoles(i) Epsilon=0.0; [Epsilon] = EquilibriumC (P,T,Hmoles(i),Nmoles(i),A moles(i),TotalMoles(i)) TTeps(i) = Epsilon; HmolesEq(i) = Hmoles(i) - 3.0*Epsilon ; NmolesEq(i) = Nmoles(i) - Epsilon ; RmolesEq(i) = Rmoles(i) ; AmolesEq(i) = Amoles(i) + 2.0*Epsilon ; MmolesEq(i) = Mmoles(i) ; % TotalMolesEq(i) = HmolesEq(i) + NmolesEq(i) + Rmole sEq(i) + AmolesEq(i) + MmolesEq(i); AmolesPCEq(i) = AmolesEq(i)/TotalMolesEq(i); NmolesPCEq(i) = NmolesEq(i)/TotalMolesEq(i); % now to go back to the beginning of the loop, set new local parameters % P,T,Z. and also need P, T, x(1), x(2), x(4) for n ext iteration step %xEq(1) = HmolesEq(i)/TotalMolesEq(i); %xEq(2) = NmolesEq(i)/TotalMolesEq(i); %xEq(4) = AmolesEq(i)/TotalMolesEq(i); %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% end of patch added on May 13, 2011 end % ------------------------------------- RK4 LOOP ENDS fprintf (resl,'\n\n i Toptimal Tequilibrium Epsilon'); for i2 = 1:NPTS+1 fprintf (resl,'\n %3.0f %8.2f %8.2f %12.4e',i2,Toptimal(i2),TTeqm(i2),TTeps(i2)); end Toptimal TTeqm fprintf (resl,'\n\n\n Summary of RK numerical solut ion'); fprintf (resl,'\n Moles Conversion % of N \n');

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fprintf (resl,'\n i L H2 N2 Ar NH3 CH4 Total Z T \n'); for i = 1: NPTS+1 fprintf (resl,'\n %3.0f %6.4f %8.2f %8.2f % 8.2f %8.2f %8.2f %10.2f %8.4f %8.2f',i,LL(i),Hmoles(i),Nmoles(i),Rmoles(i),Amoles (i),Mmoles(i),TotalMoles(i),ZZ(i),TT(i)); end %------------------- sept 4 2011 ------------------ ---- LLHom = [0;0.15;0.30;0.45;0.60;0.75;0.90;1.05;1.20;1.35;1.5 0;1.65;1.80;1.95;2.10;2.25;2.40;2.55;2.70;2.85;3.00;3.15;3.30;3.45;3.60;3.7 5;3.90;4.05;4.20;4.35;4.50]; NmolesHom = [0.2364;0.2349;0.2332;0.2312;0.2289;0.2262;0.2233;0 .2202;0.2174;0.2153;0.2142;0.2132;0.2121;0.2109;0.2097;0.2085;0.2073;0.2062 ;0.2051;0.2041;0.2032;0.2025;0.2020;0.2015;0.2012;0.2009;0.2008;0.2007;0.20 06;0.2005;0.2005]; fprintf (resl,'\n Comparison of Optimal, Equilibriu m from this program, and Normal from SSATS_NonUnif'); fprintf (resl,'\n i L Nopt Neq ' ); for izzz = 1:NPTS+1 fprintf (resl,'\n %3.0f %6.4f %8.4f %8.4f',izzz,LL(izzz),NmolesPC(izzz),NmolesPCEq(izzz )); end fprintf (resl,'\n i L Nhom'); for izzz = 1:31 fprintf (resl,'\n %3.0f %6.4f %8.4f ',izzz,LLHom(izzz),NmolesHom(izzz)); end %%%%%%%%%%----------------------------------------- ----- ThisPlot=5; if (ThisPlot==1) plot (LL,ZZ,'-k') xlabel ('Distance (m)') ylabel ('Conversion of N_2 (Z)') title 'Conversion of N_2 in the synthesis con vertor' end if (ThisPlot==2) plot (LL,TT,'-k') xlabel ('Distance (m)') ylabel ('Temperature (K)') title 'Temperature in synthesis convertor' end if (ThisPlot==3) plot (LL,Hmoles,'-k') hold on plot (LL,Nmoles,'-.k') hold on plot (LL,Amoles,'--k')

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xlabel ('Distance (m)') ylabel ('Molar Flow Rate (kmols/hr)') h = legend('H_2','N_2','NH_3',2); title 'Molar flow rate of H_2, N_2 and NH_3 i n convertor' end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (ThisPlot==4) subplot(2,2,1:2) % plot (LL,AmolesPC,'-k') plot (LL,NmolesPC,'-k') xlabel ('Distance (m)') ylabel ('Mole Fraction N_2') % title 'Optimal Conversion of N_2 in the synt hesis convertor' hold on %plot (LL,AmolesPCEq,'--k') plot (LL,NmolesPCEq,'-k') % now get the results from SSATS_NonUnif (hom cas e) and paste the Nmoles % from a simple run, to compare with the optimal and eqm Nmoles obtained % in this prog hold on plot (LLHom,NmolesHom,'-k') xlim([0.25 4.5]) text (3.1,0.24,'{N_2}_{,hom}'); text (2.1,0.21,'{N_2}_{,opt}'); text (1.1,0.19,'{N_2}_{,eqm}'); % h = legend('opt','eqm',2); % xlabel ('Distance (m)') % ylabel ('Ammonia Mole Fr') grid on % title 'Mole % NH_3' subplot(2,2,3) % plot the optimal temp plot (LL,TT,'-k') hold on % plot the eqm temp plot (LL,TTeqm,'-k') %h = legend('opt','eqm',2); text (0.3,730,'T_{opt}'); text (1.5,800,'T_{eqm}'); grid on xlim([0.2 4.5]) xlabel ('Distance (m)') ylabel ('Temperature (K)') % title 'Temperature' subplot(2,2,4) % plot (LL,Hmoles,'-k') plot (LL,HmolesFr,'-k') hold on xlim([0.225 4.51]) %plot (LL,Nmoles,'-.k')

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plot (LL,NmolesFr,'-.k') hold on %plot (LL,Amoles,'--k') plot (LL,AmolesFr,'--k') xlabel ('Distance (m)') %ylabel ('Molar Flow Rate (kmols/hr)') ylabel ('Optimal Mole Fraction') h = legend('H_2','N_2','NH_3',2); grid on % title 'Mol fl rate of H_2, N_2 and NH_3' end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (ThisPlot==5) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % these are the results for runs below %run C:\MATLAB7\work\Ammonia\SSATS_NonUnif_2Sept201 1 % cat distribution 1 1 1 Length = [0;0.1500;0.3000;0.4500;0.6000;0.7500;0.9000;1.0500 ;1.2000;1.3500;1.5000;1.6500;1.8000;1.9500;2.1000;2.2500;2.4000;2.5500;2.70 00;2.8500;3.0000;3.1500;3.3000;3.4500;3.6000;3.7500;3.9000;4.0500;4.2000;4. 3500;4.5000]; NmolesFr1 = [0.2364;0.2349;0.2332;0.2312;0.2289;0.2262;0.2233;0 .2202;0.2174;0.2153;0.2142;0.2132;0.2121;0.2109;0.2097;0.2085;0.2073;0.2062 ;0.2051;0.2041;0.2032;0.2026;0.2019;0.2013;0.2006;0.2000;0.1993;0.1986;0.19 79;0.1973;0.1966]; AmolesFr1 = [0.0269;0.0325;0.0392;0.0470;0.0560;0.0664;0.0780;0 .0900;0.1009;0.1089;0.1133;0.1172;0.1215;0.1261;0.1307;0.1354;0.1401;0.1446 ;0.1488;0.1526;0.1560;0.1585;0.1610;0.1635;0.1661;0.1687;0.1714;0.1740;0.17 66;0.1792;0.1817]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1.10 1.00 1.00 NmolesFr2 =[0.2364;0.2348;0.2329;0.2306;0.2279;0.2249;0.2215; 0.2183;0.2157;0.2143;0.2137;0.2128;0.2116;0.2105;0.2093;0.2081;0.2069;0.205 8;0.2047;0.2038;0.2030;0.2023;0.2017;0.2010;0.2003;0.1997;0.1990;0.1983;0.1 977;0.1970;0.1964]; AmolesFr2 =[0.0269;0.0331;0.0405;0.0494;0.0598;0.0717;0.0848; 0.0974;0.1073;0.1129;0.1152;0.1189;0.1233;0.1278;0.1325;0.1371;0.1417;0.146 1;0.1503;0.1539;0.1571;0.1596;0.1621;0.1647;0.1672;0.1698;0.1725;0.1751;0.1 777;0.1802;0.1827]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1.20 1.00 1.00 NmolesFr3 =[0.2364;0.2346;0.2325;0.2299;0.2269;0.2234;0.2198; 0.2166;0.2146;0.2138;0.2135;0.2126;0.2114;0.2102;0.2090;0.2078;0.2066;0.205 5;0.2045;0.2036;0.2028;0.2022;0.2015;0.2009;0.2002;0.1995;0.1989;0.1982;0.1 975;0.1969;0.1963]; AmolesFr3 =[0.0269;0.0337;0.0420;0.0519;0.0638;0.0773;0.0916; 0.1040;0.1118;0.1150;0.1

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159;0.1196;0.1241;0.1287;0.1334;0.1381;0.1427;0.147 1;0.1511;0.1546;0.1576;0.1601;0.1626;0.1652;0.1678;0.1704;0.1730;0.1756;0.1 782;0.1807;0.1832]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1.50 1.25 1.00 NmolesFr4 =[0.2364;0.2341;0.2313;0.2277;0.2234;0.2188;0.2153; 0.2138;0.2135;0.2134;0.2134;0.2122;0.2107;0.2091;0.2076;0.2061;0.2049;0.203 8;0.2030;0.2024;0.2021;0.2015;0.2008;0.2002;0.1995;0.1988;0.1982;0.1975;0.1 969;0.1962;0.1956]; AmolesFr4 =[0.0269;0.0356;0.0467;0.0606;0.0773;0.0952;0.1090; 0.1148;0.1160;0.1162;0.1162;0.1210;0.1271;0.1332;0.1392;0.1448;0.1497;0.153 8;0.1569;0.1591;0.1606;0.1628;0.1654;0.1679;0.1705;0.1732;0.1758;0.1783;0.1 808;0.1833;0.1856]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % optimal ammonia concentration plot (LL,AmolesPC,'-k') hold on % equilibrium ammonia concentration %plot (LL,AmolesPCEq,'-k') %hold on plot (Length,AmolesFr1,'--k') hold on plot (Length,AmolesFr2,':k') hold on plot (Length,AmolesFr3,'-.k') hold on plot (Length,AmolesFr4,'+k') hold on xlabel ('Distance (m)') ylabel ('Mole Fraction NH_3') % title 'Optimal NH_3 in the synthesis convert or' % text (3.1,0.24,'{N_2}_{,hom}'); % text (2.1,0.21,'{N_2}_{,opt}'); % text (1.1,0.19,'{N_2}_{,eqm}'); h = legend('optimal','1.00 1.00 1.00','1.10 1.00 1. 00','1.20 1.00 1.00','1.50 1.25 1.00',2); % xlabel ('Distance (m)') % ylabel ('Ammonia Mole Fr') grid on % title 'Mole Fraction % NH_3' end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% if (ThisPlot==6) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % these are the results for runs below %run C:\MATLAB7\work\Ammonia\SSATS_NonUnif_2Sept201 1 % cat distribution 1 1 1 Length = [0;0.1500;0.3000;0.4500;0.6000;0.7500;0.9000;1.0500 ;1.2000;1.3500;1.5000;1.

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6500;1.8000;1.9500;2.1000;2.2500;2.4000;2.5500;2.70 00;2.8500;3.0000;3.1500;3.3000;3.4500;3.6000;3.7500;3.9000;4.0500;4.2000;4. 3500;4.5000]; NmolesFr1 = [0.2364;0.2349;0.2332;0.2312;0.2289;0.2262;0.2233;0 .2202;0.2174;0.2153;0.2142;0.2132;0.2121;0.2109;0.2097;0.2085;0.2073;0.2062 ;0.2051;0.2041;0.2032;0.2026;0.2019;0.2013;0.2006;0.2000;0.1993;0.1986;0.19 79;0.1973;0.1966]; AmolesFr1 = [0.0269;0.0325;0.0392;0.0470;0.0560;0.0664;0.0780;0 .0900;0.1009;0.1089;0.1133;0.1172;0.1215;0.1261;0.1307;0.1354;0.1401;0.1446 ;0.1488;0.1526;0.1560;0.1585;0.1610;0.1635;0.1661;0.1687;0.1714;0.1740;0.17 66;0.1792;0.1817]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1.10 1.00 1.00 NmolesFr2 =[0.2364;0.2348;0.2329;0.2306;0.2279;0.2249;0.2215; 0.2183;0.2157;0.2143;0.2137;0.2128;0.2116;0.2105;0.2093;0.2081;0.2069;0.205 8;0.2047;0.2038;0.2030;0.2023;0.2017;0.2010;0.2003;0.1997;0.1990;0.1983;0.1 977;0.1970;0.1964]; AmolesFr2 =[0.0269;0.0331;0.0405;0.0494;0.0598;0.0717;0.0848; 0.0974;0.1073;0.1129;0.1152;0.1189;0.1233;0.1278;0.1325;0.1371;0.1417;0.146 1;0.1503;0.1539;0.1571;0.1596;0.1621;0.1647;0.1672;0.1698;0.1725;0.1751;0.1 777;0.1802;0.1827]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1.20 1.00 1.00 NmolesFr3 =[0.2364;0.2346;0.2325;0.2299;0.2269;0.2234;0.2198; 0.2166;0.2146;0.2138;0.2135;0.2126;0.2114;0.2102;0.2090;0.2078;0.2066;0.205 5;0.2045;0.2036;0.2028;0.2022;0.2015;0.2009;0.2002;0.1995;0.1989;0.1982;0.1 975;0.1969;0.1963]; AmolesFr3 =[0.0269;0.0337;0.0420;0.0519;0.0638;0.0773;0.0916; 0.1040;0.1118;0.1150;0.1159;0.1196;0.1241;0.1287;0.1334;0.1381;0.1427;0.147 1;0.1511;0.1546;0.1576;0.1601;0.1626;0.1652;0.1678;0.1704;0.1730;0.1756;0.1 782;0.1807;0.1832]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 1.50 1.25 1.00 NmolesFr4 =[0.2364;0.2341;0.2313;0.2277;0.2234;0.2188;0.2153; 0.2138;0.2135;0.2134;0.2134;0.2122;0.2107;0.2091;0.2076;0.2061;0.2049;0.203 8;0.2030;0.2024;0.2021;0.2015;0.2008;0.2002;0.1995;0.1988;0.1982;0.1975;0.1 969;0.1962;0.1956]; AmolesFr4 =[0.0269;0.0356;0.0467;0.0606;0.0773;0.0952;0.1090; 0.1148;0.1160;0.1162;0.1162;0.1210;0.1271;0.1332;0.1392;0.1448;0.1497;0.153 8;0.1569;0.1591;0.1606;0.1628;0.1654;0.1679;0.1705;0.1732;0.1758;0.1783;0.1 808;0.1833;0.1856]; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % optimal nitrogen concentration plot (LL,NmolesPC,'-k') hold on % equilibrium nitrogen concentration %plot (LL,NmolesPCEq,'-k') %hold on plot (Length,NmolesFr1,'--k') hold on plot (Length,NmolesFr2,':k')

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hold on plot (Length,NmolesFr3,'-.k') hold on plot (Length,NmolesFr4,'+k') hold on grid on xlabel ('Distance (m)') ylabel ('Mole Fraction N_2') % title 'Optimal N_2 in the synthesis converto r' % text (3.1,0.24,'{N_2}_{,hom}'); % text (2.1,0.21,'{N_2}_{,opt}'); % text (1.1,0.19,'{N_2}_{,eqm}'); h = legend('opt','1','2','3','4',2); end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% toc fclose(resl); function [Topt] = OptimalT (x,P) % DATA % P, x(1), x(2), x(4) % gives Topt % scan between 500 and 1000 scanTlow = 500; scanTup=1000.0; scanDel = 2.0; Tchk = scanTlow-scanDel; scanT = (scanTup-scanTlow)/scanDel; % fprintf (resl,'\n Temp (K) diff=fLHS-fRHS'); for iTchk = 1:scanT Tchk = Tchk+scanDel; [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,Tchk,x(1) ,x(2),x(4)); [Ka] = EQNCN(Tchk); FatT = Ka^2*aN2*aH2^1.5/aNH3 - aNH3/aH2^1.5; fRHS = -( (20.523e3)/Tchk^2 ) * FatT; % now find deriv at this point T1=Tchk-1; T2=Tchk+1; [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T1,x(1),x (2),x(4)); [Ka] = EQNCN(T1); FatT1 = Ka^2*aN2*aH2^1.5/aNH3 - aNH3/aH2^1.5; [phiH2,phiN2,phiNH3,aH2,aN2,aNH3]=ACTVT(P,T2,x(1),x (2),x(4)); [Ka] = EQNCN(T2); FatT2 = Ka^2*aN2*aH2^1.5/aNH3 - aNH3/aH2^1.5; fLHS = 0.5*(FatT2-FatT1); diff=fLHS-fRHS; rootT(iTchk) = Tchk; rootY(iTchk) = diff; % fprintf(resl,'\n %3.0f %6.2f %12.4e',iTchk,rootT( iTchk),rootY(iTchk)); end Topt=0; for jTchk = 1:scanT-1 jT1=jTchk+1;

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if ((rootY(jTchk)>0)&(rootY(jT1)<0)) Topt = rootT(jTchk+1); end end for jTchk = 1:scanT-1 jT1=jTchk+1; if ((rootY(jTchk)<0)&(rootY(jT1)>0)) Topt = rootT(jTchk+1); end end

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