Steady state models for a multi-component crude distillation column have been developed from the MESH equations. The equations developed are capable of predicting the concentrations (mole fractions) and temperature of any component/fraction of interest from the mixture on the trays of the column. The accuracy of the models was ascertained with plant data of the crude distillation unit of the Port-Harcourt Refinery. The models applied to this distillation column produced a set of forty eight coupled algebraic equations for any component/fraction of interest. These equations were transformed into a matrix and solve by matrix inversion using the Mathlab solver. The model results of the concentrations and temperatures for five components compared favorably with output values from the distillation unit with maximum deviations of 8.33% and 6.62% respectively. The developed model also accurately predicted the tray of withdrawal of the five components/fractions in the distillation column. The models were therefore used to simulate the effects of feed flow rate and feed tray position on the efficiency of the distillation column.
Keywords: modeling, multi-component distillation, effect of feed rate and feed tray position. INTRODUCTION
Separation technology plays an important role in the process industry. Typical separation techniques include distillation, absorption, extraction etc, with distillation being the most frequently used in the chemical process industry. Distillation is the separation of a liquid mixture into its component parts or fractions based on the differences in their volatilities or boiling points. It is widely used in petroleum processing, petrochemical production, natural gas processing, coal tar processing, brewing, liquefied air separation, hydrocarbon solvents production and other industries where product separation of liquid mixtures is required; however its widest application is in the petroleum refineries. In the petroleum refineries the crude oil feedstock is a complex multi-component mixture which has to be separated into groups of compounds within relatively small range of boiling points or fractions. On account of the complexities of feeds and products (Douani, et al., 2007a), the development of the proper algorithm for simulating such processes is critically important (Sridhar and Lucia, 1990).Simulation methods for distillation columns are divided into three categories/methods according to the assumptions made in the development of the model as (Jelinet and Hlavecek (1975);Gani, et al., (1986)):
approximate, equilibrium and rate based methods. The equilibrium based methods are the most commonly used method and is based on the equilibrium assumption between leaving vapor and liquid flows for each stage (Ramesh, et al., 2007). In this method the model equations of the column called MESH (Material balance equation, Equilibrium phase equations, Summation equations and Heat balance equations) are developed and solved. No matter the method used, the model equations that describe multi-component separation processes are non-linear and interdependent (Abdullah et al., 2007); with solutions that are iterative (Boston and Sullivan 1974), complex and difficult to converge (Jaroslav, et al., 1973; Ivakpour and Kasiri, (2008)). The use of the matrix method for the solution of the MESH equations had been performed by Rosendo, (2003) and Douani, et al., (2007b). This method has also been reported to be difficult to converge and require considerable simulation CPU time.
The convergence problem of the solution of the model equations has always been reported in various works (Russell, (1983), Rosendo, (2003), LI et al., (2006), Ramesh et al., (2007)).
In this work, Equilibrium based models are developed for the distillation of a multi-component mixture with
an improved computational algorithm. The method presented in this work is derived from that of Rosendo, (2003) with some adjustments. In these models, the non-linear equation sets (the vapor liquid equilibrium relationships and enthalpy characteristics of the streams in the column) which require iterative procedure such as the Newton-Raphson method for the re-estimation of some parameters between iterations have been eliminated by making appropriate simplifying assumptions in the equilibrium phase relationships (constant relative volatility of component on each tray).The resulting model equations are transformed into a matrix which is solved using the MathLab matrix solver. This solution is fast and converges quite easily.
MODEL DEVELOPMENT
The phenomena occurring in the distillation column can be broken down into the transfer of mass and heat. The modeling of the column was therefore based on the equations describing these phenomena. These equations are the continuity equations for mass and heat transfer. A tray by tray model was obtained by applying the continuity equations on a typical column depicted in Figure 1, using a representative tray j of the column as shown in Figure 2 with its inflow and outflow streams.
Fig 1: Diagram of a typical column showing component distribution on each tray
43rd
Feed
35th
2nd
12th
25th
1st
X1,12, X2,12, X3,12, X4,12, X5,12
X13, X23, X33, X43, X53
X1,25, X2,25, X3,25, X4,25, X5,25
X1,35, X2,35, X3,35, X4,35, X5,35
X1,48, X2,48, X3,48, X4,48, X5,48
3rd
Whole Naphtha
(Component i = 1)
(Component i = 5)
(Component i=2)
Straight run
Kerosene (SRK)
Light Diesel oil
(LDO) (i= 3)
Heavy Diesel Oil (HDO)
(Component i = 4
N = 48 Atmospheric (AR)
residue
X12, X22, X32, X42, X52
X11, X21, X31, X41, X51
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Fig 2: General schematic representation of a tray j.
For a typical column (Figure 1), the material, equilibrium, summation and heat (MESH) equations for any stage j and any component i at steady state were developed as:
MATERIAL BALANCE EQUATION (M equation)
( ) (1)
Equation (1) is the general model equation that predicts the flow of mass of component i in and out of a given tray j. Rearranging equation (1) gives:
( ) ( )
( ) (2)
EQUILIBRIUM (PHASE) RELATIONSHIPS (E equation)
( ) (3)
SUMMATIONS OF MOLE FRACTIONS (S equation)
∑ ∑
( ) (4)
HEAT BALANCE (H equation)
( )
( ) (5)
The following assumptions were imposed on the model
i. The residence time on each plate is such that equilibrium is attained between the liquid and the vapor. Hence
(6)
ii. Constant relative volatility on each tray, Hence
(7)
(8)
iii. Since equilibrium is attained between liquid and vapor, then the temperature of the liquid and vapor streams are the same
(9)
iv. The liquid and vapor flow rates are constant for all trays
(10)
(11)
v. Since the liquid and vapor flow rates in and out of each tray are constant then, the liquid and vapor holdup in each tray will also be constant.
(12)
(13)
vi. Column is well lagged hence heat losses are negligible and for an ideal system heat of mixing is zero
vii. For ideal systems the molar heat of vaporization may be taken as constant and independent of the composition, the temperature change from one tray to the next will be small, hence
=
= similarly =
= and
(14)
Introducing these assumptions and equation (3), the model equation simplifies to:
( ( ) )
(15)
Tray j Q
j
U
j
Fj
Vj+
1
Mj
V
j Lj-1
L
j
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Equation (15) contains Lj-1, the liquid flow rate from tray j-1.This can be eliminated from the model
equation by taking a balance on tray j-1 shown in Figure 3.
A material balance on stage j-1 gives:
(16)
Making 1jL the subject
(17)
Equation (17) can be written in the general form as:
V1
Mj
Uj-1
Fj-1
Uj
Uj+1
U1
F1
U2
F2
F3
Stage 1
Stage 2
Stage j-1
Stage j
M1
M2
Mj-1
V2
Vj-1
Vj
Vj+1 Lj
Lj-1
L2
L1
Fig 3: Tray by tray column showing its boundary
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∑ ( )
(18)
Similarly, equation (18) can be extended to obtain the
expression for jL as:
∑ ( )
(19)
Substituting equations (18) and (19) into equation (15) gives:
[ ∑ ( ) ] [
∑ ( ) ( ) ]
(20)
Equation (20) can be written in a compact form as
(21)
Where:
∑ ( )
(22)
∑ ( )
( ) (23)
(24)
(25)
Similarly the Heat balance equation was simplified using the assumptions to obtain:
{ ( ) }
(26)
Equation (26) is the general model that predicts the temperature of each tray hence the temperature progression along the column.
Substituting equations (18) and (19) into equation (26) and ignoring the superscript i gives:
[ ∑ ( ) ]
[ ∑ ( )
( ) ]
(27)
Equation (27) can also be written in a compact form as:
(28)
Where:
[ ∑ ( ) ] (29)
[ ∑ ( )
( ) ] (30)
(31)
MATERIALS AND METHODS
Determination of parameters: The feed properties and operational conditions of the crude distillation column in the Port-Harcourt Refinery Company (Area 1) are given in Table 1.
Table 1: Operating data of the Crude Distillation Unit.
TRAY Flow Rate (m
3/hr)
Component mole Fraction in
Feed
Vapor liquid ratio Kij
Feed 43 993.7
Naphtha 1 300.1 0.260 1.40
Straight run Kerosene (side stream U12) 12 168.1 0.139 0.44
Solution technique of model equations: The model equations developed (equations 21 and 28) written for any component/fraction of interest gives a system of 48 algebraic equations (for the 48 trays of the crude distillation unit in the Port-Harcourt refinery). Plant operating data from the crude distillation unit (Area 1) of the Port-Harcourt refinery given in Table 1 and the flow rates of the various streams as stated above were substituted into the model equations to obtain the unknown parameters in the model equations. A matrix representation of either equation for any component/fraction written in the form: [A] [G] = [D], where [A] is the coefficient matrix, [G] is the tray component/fraction mole fraction or tray Temperature matrix, and [D] is the constant matrix was developed in the form shown in Figure 4.
iNiNiN
iNiNiN
iii
iii
iii
iii
CBA
CBA
CBA
CBA
CBA
CBA
0000
0000
0
0
000
000
00000
0000
000
00000
111
555
444
331
222
1
1
3
2
1
.
.
.
.
iN
iN
iN
i
i
i
X
X
X
X
X
X
iN
iN
i
i
i
D
D
D
D
D
1
4
3
2
.
.
.
.
.
Fig 4: Matrix representation of model equations
The resulting matrix from equation (21) was solved by the method of matrix inversion using the MathLab software to obtain the mole fraction of any component/fraction on the forty eight (48) trays of the crude distillation column. Similarly the resultant matrix from equation (28) was solved using the same method to obtain the temperature of any component/fraction on the forty eight (48) trays of the crude distillation column.
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DISCUSSION OF RESULTS
A. COMPONENT/FRACTION CONCENTRATION (MOLE FRACTION)
The model developed (equation 21) was used to predict the concentrations (mole fractions) of five components/fractions of a crude oil mixture separated in the distillation column. The mole
fractions of the five components/fractions; naphtha, straight run kerosene (SRK), light diesel oil (LDO), heavy diesel oil (HDO) and atmospheric residue (AR) as predicted by the model equation on the forty eight trays of the crude distillation unit (CDU) of the Port-Harcourt Refining Company are shown in Figures 5, 6, 7, 8 and 9 respectively.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 20 25 30 35 40 45 50
Mo
le F
ract
ion
of
Co
mp
on
en
t
Tray Number
Fig. 5: Composition of Naphta on each Tray.
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0 5 10 15 20 25 30 35 40 45 50
Mo
le F
ract
ion
of
Co
mp
on
en
t
Tray Number
Fig. 6: Composition of SRK on each Tray
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0
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 20 25 30 35 40 45 50
Mo
le F
ract
ion
of
Co
mp
on
en
t
Tray Number
Fig. 7: Composition of LDO on each Tray
0
0.01
0.02
0.03
0.04
0.05
0.06
0 5 10 15 20 25 30 35 40 45 50
Mo
le F
ract
ion
of
Co
mp
on
en
t
Tray Number
Fig. 8: Composition of HDO on each Tray
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Any component/fraction of interest is expected to be withdrawn at the tray where its concentration is highest. Therefore from the model results, the five components of interest; naphtha, straight run kerosene (SRK), light diesel oil (LDO), heavy diesel oil (HDO) and atmospheric residue (AR) should be withdrawn on trays 1, 12, 25, 35 and 48.These trays of withdrawal correspond to the trays where these components/fractions are withdrawn in the CDU of the Port-Harcourt Refining Company. Hence the
model developed accurately predicted the tray of withdrawal of the five components/fraction.
Tray component temperature: Similarly, the model equation developed (equation 28) was used to predict the temperatures of any component/fraction on the trays of the distillation column (the temperature progression of each component of the trays). The temperatures of the five components of interest at the trays of withdrawal (tray temperature) as predicted by the model are presented in Table 2
Table 2: Tray Temperature as Predicted by Model.
S/N Component Tray of Withdrawal Temperature
1. Whole Naphtha 1 156.5
2. Straight Run Kerosene 12 223.91
3. Light Diesel Oil (LDO) 25 268.80
4. Heavy Diesel Oil (HDO) 35 320
5. Atmospheric Residue (AR) 48 329
MODEL VALIDATION
The accuracy of the model equations were validated by comparing the concentration of these components and their respective tray temperatures as predicted
by the model with those obtained from the industrial plant (Crude Distillation Column of Area 1 of the Port Harcourt Refining Company). This comparison is shown in Table 3.
0
0.05
0.1
0.15
0.2
0.25
0.3
0 5 10 15 20 25 30 35 40 45 50
Mo
le F
ract
ion
of
Co
mp
on
en
t
Tray Number
Fig. 9: Composition of Atmospheric Residue on each Tray
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Table 3: Comparison of Plant data and Model predictions
S/N Component Model Prediction Plant Data % Deviation
Composition Temperature Composition Temperature Composition Temperature
1. Whole Naphtha
0.2650 156.5 0.2710 150 2.21 -4.33
2. Straight Run Kerosene
0.1690 223.91 0.1560 210 -8.33 -6.62
3. Light Diesel Oil (LDO)
0.2560 268.80 0.2650 280 3.40 4.00
4. Heavy Diesel Oil (HDO)
0.0570 320 0.0530 310 -7.55 -3.2
5. Atmospheric Residue (AR)
0.2530 329 0.2550 320 7.84 -2.81
Table 3 shows that the maximum deviation between the model prediction of the mole fraction of components/fractions on each tray and plant data using the value at the tray of withdrawal is 8.33% for straight run kerosene, while the maximum deviation between model prediction of the tray temperatures and plant data at the tray of withdrawal is 6.62% for the temperature predictions on tray 12. The model equations were also able to predict the point (tray number) of withdrawal of each component/fraction of interest. These show the accuracy of the models developed and that to a very large extent they can be used to predict the composition and temperature on each tray of the crude distillation column; hence can be used for simulation studies of the column.
Process Simulation: The model equations were used to study the effects of process variables such as the Feed Flow rate and Feed entry point (feed tray) on the performance of the column.
Effect of feed flow rate on column: The feed flow rate is the rate at which feed mixture (liquid, vapor or mixture of both) is pumped into the column. The flow of feed into a column can affect the quantity (moles/hr) and quality (concentration - mole fraction of the components/fractions).The effects of variation of the feed rate on the performance of the distillation column are shown in Table 4.
Table 4: Effect of Feed Rate on Column
Feed Flow Rate Component1 (Naphtha)
Component2 (S.R.K)
Component3 (L.D.O)
Component4 (H.D.O)
Component5 (AR)
1200 0.1246 0.1900 0.3500 0.1800 0.1554
993 0.2650 0.1690 0.2560 0.0570 0.2530
750 0.2685 0.1020 0.1830 0.1560 0.3205
500 0.3072 0.0885 0.1230 0.1945 0.3468
Table 4 shows that the higher the feed flow rate (increase in the feed rate) the greater the composition of the lighter ends in the bottom plate and the heavy components in the upper plate; the lower the feed rate (decrease in the feed rate) there is a reduction of the lighter ends in the bottom region and the heavier ends in the upper region. When the feed rate is increased, its velocity increases, its
residence time (contact time of the vapor-liquid phases on each tray) in the column reduces, causing inefficient separation and a reduction in the percentage purity of each component. This inefficient separation could also be as a result of increased liquid or vapor flow rate as feed rates are increased. Increasing liquid level on the trays causes overloading due to increased liquid hydrostatic
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pressure, restriction of vapor rising through liquid and eventual flooding of the column. Similarly, excessive vapor could lead to over-loading of trays, too much vaporization of the bottom liquid, higher vapor flow and the flooding of trays in an upward progression. When the feed rate is decreased, its velocity decreases, its residence time in the column is increased, there is efficient separation and the percentage purity of each component is increased. However, decreases in the feed flow rates result in decreased vapor and liquid flow rates. At low liquid rates, the vapor forces the liquid back from the slots, while low vapor rates results in pulsating flow, both
condition causes weeping on the trays and reduced efficiency of the column (Richardson et al., 2002).These extreme flow rates result in inefficient separation which the model predicts as shown in Table 4. In practice, an optimum feed rate is often determined (Bennett and Kovak, 2010) when the system is being designed with a range of acceptable variation specified; this can also be established from the model predictions as shown in Table 4.
Effect of feed tray position on the performance of column: The effects of feed tray position on the performance of the column are shown in Table 5.
Table 5: Effect of Feed Tray Position on Column Performance
Composition (Mole Fraction)
Feed Entry Point Naphtha S.R.K L.D.O H.D.O AR
10 0.6124 0.0963 0.0850 0.1003 0.1060
20 0.5433 0.1001 0.1005 0.0811 0.1750
30 0.3110 0.2032 0.2008 0.0836 0.2014
43 0.2650 0.1690 0.2560 0.0570 0.2530
45 0.1569 0.1705 0.1860 0.0340 0.4526
Table 5 shows there is an increase in the purity of the more volatile component hence an improvement in separation as the feed tray is lowered down the column.
As the feed tray is lowered down the column, there are more trays up, that is more points of contact where separation can occur. Therefore as the mixture rise up the column, the less volatile compounds condenses from the rising vapor into the liquid, are withdrawn from their various trays of withdrawal, thus decreasing the concentration and increasing the purity of the more volatile compound (MVC) in the vapor. At the same time, the more volatile compound is vaporized from the liquid on the tray and increasing the concentration of the MVC in the liquid. The higher the feed tray, there are less trays up, hence less points of contact where separation can occur. Also the temperatures at such high trays are not high enough to vaporize and condense the heavier components, thus increasing the concentration and decreasing the purity of the more volatile component in the vapor. These trends the model accurately predicts as shown in Table 5. Similar trends had also
been reported in the works of Bandyopadhyay, (2002).
CONCLUSION
Steady state models have been developed from the principles of conservation of mass and energy for a multi-component distillation column. The models developed were algebraic equations. The models were tested for the distillation of a multi-component mixture (crude oil) in a distillation column of the Port-Harcourt Refinery. The model equations developed were written for any component/fraction of interest and yielded a system of 48 algebraic equations each (for the 48 trays of the crude distillation unit in the Port-Harcourt refinery). The equations were transformed into a diagonal matrix which was solved by matrix inversion using the MathLab solver to obtain the mole fraction and temperature of the component on the respective trays of the distillation column. Specifically in this work, the mole fraction and temperature of five components (whole Naphtha, Straight Run Gasoline, Light Diesel Oil, Heavy Diesel oil and Atmospheric residue) were predicted by the model. The model also accurately predicted the tray of withdrawal of these components. The simulation
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result shows the effects of the feed flow rate and feed tray location on the performance of the column.
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