Analysis of the Adsorption Process and of Desiccant Cooling Systems

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SERI/TR-631-1330UC Category: 59c

Analysis of the AdsorptionProcess and of DesiccantCooling Systems -A Pseudo- Steady-State Modelfor Coupled Heat andMass TransferRobert S. Bartow

December 1982

Prepared Under Task No. 1131.00 and l f 2.11WPA Nor 01-256 and 01-315

Solar Energy Research InstituteA Division of Midwest R esearch lnstitute1617 Cole BoulevardGolden, Colorado 80401Prepared for theU.S. Department of EnergyContract No. EG-77-C-01-4042


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Printed in the United States of AmericaAvailable from:National Technical Information ServiceU.S. Department of Commerce

5285 Port Royal RoadSpringfield, VA 22161Price:Microfiche $3.00Printed Copy $8.00

NOTICEThis report was prepared as an account of work sponsored by the United StatesGovernment. Neither the United States nor the United States Department of Energy,nor any of their employees, nor any of their contractors, subcontractors, or theiremployees, makes any warranty, express or implied, or assumes any legal liabilityor responsibility for the accuracy, completeness or usefulness of any information,apparatus, product or process disclosed, or represents that its use would notinfringe privately owned rights.


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This report documents a computer model that simulates the adiabatic adsorp-tionldesorption process. Developed to predict the performance of desiccantcooling systems, the model has been validated through comparison with experi-mental data for single-blow adsorption and desorption. This report also con-tains a literature review on adsorption analysis, detailed discussions of theadsorption process, and an i n i t a l assessment of the potential for performanceimprovement through advanced component developmentThis research was performed under task 1131.00 of the Solar Desiccant CoolingProgram at the Solar Energy Research Institute during fiscal year 1981. Theauthor would like to acknowledge the contributions of Charles F. Kutscher, incharge of SERZts desiccant laboratory during this work; Harry Pohl, thelaboratory technician; and Terry Penney, who provided a detailed and helpfulreview of the report.Details of the experimental work completed at SERI to validate the computermodel are contained in a separate report, SERI/TR-253-1429.

Approved forSOLAR ENERGY RESEARCH INSTITUTE

Frank Kteith, Chief 1 -5Thermal Research Branch

er, ManagerSolar @herma1 and Xaterials Research D i v f sion

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The primary objective of the research leading to this report was to developand validate a computer program for predicting the performance of solar desic-cant cooling systems. Major considerations in the development of the sirnula-tion model were to make the method of analysis as simple and versatile as pos-sible, whfle maintaining the level of detail necessary for accurate predic-tions. The primary objective of this report is to provide complete documenta-tion of this method of analysis and the computer programs. The report servestwo additional functions. The first is to provide a detailed description ofthe adsorption process and the physical behavior of desiccant beds to facili-tate a more complete understanding of the operation of the dehumidifier in adeci sant cooling cycle. The second is to provide information for an initialassessment of the potential for performance improvements through the develop-ment of advanced, high-performance components.

The computer model documented here is called a pseudo-steady-state model, andit uses a new approach to analyzing transient coupled heat and mass transferas it occurs in adiabatic adsorption. Rather than deriving and solvfng a setof differential equations, the method uses simple effectiveness equations fromthe theory of steady-state heat exchangers and mass exchangers within astraight mard finite difference procedure. This simplifies the mathematicsof the adsorption problem, makes the model easy to adapt to investigate avariety of adsorption issues, and makes it easier to keep track of the physicsof the adsorption process. The body of the report includes a literaturereview of adsorption analysts and a review of available data and correlationsof th e properties of regular density silica gel, which appears to be the mostsuitable available desiccant for solar cooling systemsThe computer model was validated through comparison with experimental data forsingle-blow adsorption and desorption in packed beds of silica gel. Threedata sources were used; the primary one was the SERI Desiccant Test Labora-tory. Measured and predicted results for adsorption show agreement wellwithin experimental uncertainty. This demonstrates that the lumped gas-sidetransfer coefficient, rather than separate gas-stde and solid-side resistancesto mass transfer is sufficiently accurate for adsorption cases. Experimentalresults show that a different effective transfer coefficient must be usedduring single-blow desorption cases. However, relatively good agreementbetween measured and predicted results can still be obtained.G major section of the report deals with the physical behavior of desiccantbeds during single-blow adsorption or desorption and during cyclic operation,as in a cooling system. Of particular Importance is the fact that adsorptioncomprises the progression of two heat and mass transfer waves through a desic-cant bed. Also important are the differences between the behavior of thickbeds, typically used for industrial adsorption applications, and thin beds, asused fn desiccant cooling systems. Fredicted performance of thin beds is mch


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more sensitive to errors in transfer coefficients and desiccant property cor-relations, because only a portion of the second wave front is contained withinthe bed. The detailed discussion of the adsorption process is intended tofacilitate a more complete understanding of the operation of the dehumidifierin a desiccant cooling cycleThe pseudo-steady-state model was incorporated into a computer program for thesimulation of a complete desiccant cooling system. Parametric studies wereperformed on two systems, one representative of existing prototypes and theother containing high effectiveness components. These parametric studiescharacterize the effect on performance of operating conditions, such as indoorconditions, outdoor conditions, dehumidifier wheel rotation speed, andregeneration temperature. They also provide information on the influence ofindividual component effectiveness on overall system thermal performance.This permits an initial assessment of the potential gains in performance thatcould be achieved through the development of advanced, high-effectiveness com-ponents.

Major conclusions include the following:A simple computer model for the adiabatic adsorption/desorption processhas been developed and validated.Close agreement between measured and predicted results for single-blowadsorption demonstrates that a lumped gas-side mass transfer coefficientcan be used.Effective mass transfer coefficients must he reduced for cases of single-blow desorption, presumably because of a dynamtc hysteresis effect insilica gel properties.The thermal performance of a desiccant cooling system varies with indoorand outdoor temperature and humidity, and generally decreases as the dif-ference between indoor and outdoor conditions increases.COPs above 1.0 are technically feasible if high-effectiveness dehu-midifiers and heat exchangers are used.With high-performance components, the ventilation mode is clearlysuperior to the recirculation mode. Thermal COPs in the ventilation modebecome relatively insensitive to outdoor conditions when high-effectiveness components are used.


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TABLE OF COHTEKTS

Page1.0 Introduction ....................................................... 1

......................................................1 Background 11.2 Purpose and Scope.............................................. 21.3 Adsorption Process and Desiccant Cooling Cycle................. 3

2.0 Historical Overview.............................m...................3.0 Pseudo-Steady-State Model........................................... 9

3.1 General Description...........................................m 93.2 Mass Transfer Calculation...................................... 103.3 Intermediate Energy Balance...........m...................m....23.4 Heat Transfer Calculation............m....m....................33.5 Transfer Coefficients ...................mm.................m..43.6 Properties of Moist Air........................................ 163.7 Properties of Silica Gel....................................... 18...................................7.1 Equilibrium Properties 193.7.2 Heat of Adsorption ...............e....e.mm...m.........74.0 Comparison of the Model with Experimental Data ..................... 31

..........1 Data Sources for Single-Blow Adsorption and Desorption 314.2 Comparison with SERI Data...................................... 314.3 Comparison with Pesaran Data .....em...........................34.4 Comparison with Koh Data ...................................... 524.5 Results. ....................................................... 53

..................................0 Desiccant Cooling System Simulation 63.............................................1 System Configuration 635.2 Method of Analysis . . . . . . . . . . . . . . . . . . . . . . .+ . . . . . . . . . . . . . . .m.. . 635.3 Component Analysis and Equations .............................. 66

6.0 The Physical Behavior of Desiccant Beds............................. 696.1 Behavior of Thick Beds During Adiabatic Adsorption.... ........ 696.2 Behavior of Thick Beds During Adiabatic Desorption ............ 716.3 Behavior of Thin bed^..............^^.......^.........^........ 756.4 Behavior During Cyclic Operation.............................Om0......................0 Parametric Studies of Desiccant Cooling Systems 857.1 Effect of Simulation Parameters in the Dehumidifier Model...... 857.2 System specifications ......me.............................m...77.3 Effect of Outdoor condition^.....^...................^^........ 887.4 Effect of Indoor Conditions .................................... 89

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sia+ TR-1330TABLE OF CONTENTS (Concluded)

Page7.5 Effect of Regeneration Temperature ............................ 947.6 Effect of Heat Exchanger Effectiveness......................... 947.7 Effect of Evaporative Cooler 1................................. 957.8 Summary of Parametric Studies.................................. 95

8.0 Conclusions and Recommendations .................................... 998.1 Conclusions Regarding the Pseudo-Steady-State Model ............ 998.2 Conclusions Regarding Cooling System Performance............... 1008.3 Recommendations ...............................................100

9.0 References.......................................................... 03Appendix A: Users Guide to Computer Programs............................. 107

A-1 Overview of Computer Programs........................... 107A-2 Job Control Files and Input Data........................ 108A-3 A Partial Listing of FORTRAN Variables.................. 109Appendix B: Program Listings............................................. 115


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LIST OF FIGURTXS

Page1-1 IGT Solar-MEC@ System i n Recirculat ion Mode ......................... 42-1 Linear Approximation t o P r o p e r t i e s o f Silica Gel andMolecular Sieve................................................... 63-1 Conveyor Belt Concept Used i n Computer Elodel ........................ 93-2 C o n cen t r a t i o n P ro f i l e s fo r S t ead y -S t a t e Mass T ran s fe r Process....... 113-3 (a) Deta i led Res i s tance Model f o r Mass Transfer ..................... 14

(b ) Gas-Side Re sis tan ce Model fo r Mass T ran s fe r ..................... 143-4 Comparison of Various Mass Tran sfer C orre la t io ns f o r Packed Beds .... 17...........-5 Typica l Iso h e m f or Regular- and Low-Density Silica Gel 203-6 Isotherms Based on the Bullock and T h re l k e l d C o r re l a t i o n............ 233-7 Comparison of the Close and Banks Complete Correlation w i t h

S i m p l i f i e d C o r r e l a t i o n Using fl(X) = 2.009 ........................ 243-8 Comparison of th e Clo se and Banks Simpl i f ied Expression w i t h..................ur the r S imp l i f ie d Expression Us ing L inear f2 (X) 253-9 Comparison of th e Rojas C o r re l a t i o n wi t h E q o 3-30 ................... 263-10 Comparison of Various Cor re la t i on s and Data f or the Heat ofAd s o rp t i o n fo r S i l i c a G e l ......................................... 2 94-1 Measured and Predicted Outlet Air Condit ions During..........................................ERI Run 85A (ICORR = 1) 344-2 Measured and Pred i c t ed Ou t l e t Air Conditions DuringSER I Run 85A (ICORR = 2 ) .......................................... 354-3 Measured and Pred i c t ed Ou t l e t Air Condit ions DuringSERI Run 8 5A ( ICORR = 3). ......................................... 364-4 Measured and Predicted Outlet Air Condit ions During

S E N Run 8% ( ICORR = 1. Le = 3) .................................. 374-5 Measured and Predicted Outlet Air Condit ions DuringSERI RUN 85R (ICORR = 1. L e = 6).................................e 84-6 Measured and Pred ic ted Out le t Air Condit ions During

SER I Run 85R ( ICORR = 1. Le = 9 ) 0 * e 0 0 * m * 0 + * 0 * 0 * e 0 0 m 0 * 0 0 0 * 0 * 0 * * * * 0 C 39


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TR-1330

LIST OF FIGWES (Continued)

Page4-7 Measured and Predicted Outlet Air Conditions DuringSERI Run 85R (ICORR = 2 , Le * g).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 04-8 Measured and Predicted Outlet Air Conditions DuringSERI Run 85R (ICORR = 3 , Le = g). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414-9 Measured and Predicted Outlet Air Conditions DuringPesaran Run 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424-10 Measured and Predicted Outlet Air Conditions DuringPesaran Run 10.................................................. 43

4-11 Measured and Predicted Outlet Air Conditions DuringPesaran Run 11................................................... 4 44-12 Measured and Predicted Outlet Air Conditions DuringPesaran Run 12...................................... 454-13 Measured and Predicted Outlet Air Conditions DuringPesaran Run 18 (ICORR = I ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464-14 Measured and Predicted Outlet Air Conditions DuringPesaran Run 18 (ICORR = 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474-15 Measured and Predicted Outlet Air Conditions During

Pesaran Run 18 (ICORR = 3 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 84-16 Measured and Predicted Outlet Air Conditions DuringPesaran Run 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ~ 494-17 Measured and Predicted Outlet Air Conditions DuringPesaran Run 24......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504-18 Measured and Predicted Outlet Air Conditions During Koh Run 2 . . . . . . . 544-19 Measured and Predicted Outlet M r Conditions During Koh Run 4 . . . . . . . 5 54-20 Measured and Predicted Outlet Air Conditions During

Koh Run 6 (ICORR = 2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564-21 Measured and Predicted Outlet Air Conditions DuringKoh Run 6 (ICORR = 3 ) ............................................ 574-22 Measured and Predicted Outlet Air Conditions DuringKoh Run 8 (ICORR = 1)........................................... 584-23 Measured and Predicted Outlet Air Conditions During Koh Run 9 . . . 59


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SE?I .LIST O f PIGIJBaS (Continued)

Page4-24 Measured and Predicted Outlet Air Conditions During Koh Run 10 ..... 604-25 Measured and Predicted Outlet Air Conditions During Koh Run 15...... 615-1 System Schematic and Psychrometric Cycle Diagram for theVentilation Mode.................................................. 645-2 System Schematic and Psychrometric Cycle Diagram for theRecirculation ~clode .............................m..........5

...........-3 Small Element Used in Simulations of Rotary Dehumidifiers 676-1 The First Adsorption Wave in a Thick Desiccant Bed .................. 70

(a ) Desiccant Temperature Profiles ................m......m.......m0(b) Desiccant Loading Profiles...................................... 70(c) Outlet Air Temperature................I........................ 0(d) Outlet Air Humidity..... . . . . . . m . . . m m . . m m . . m a . m m . . . . . . . .... 70.............................e ) Psychrometic Path of Outlet States 70

6-2 Two Adsorption Waves in a Thick Desiccant Bed 72(a) Desiccant Temperature profile^....................^........^^.. 72(b) Desiccant Loading profile^..............^...................... 72(c) Outlet Air temperature...................^^...^^^..... 72(d) Outlet Air hmidity . . . . . . . . . . . .m . . . . . .mm. .m . . . . . ~ . . . . . . . . . . .2(e) Psychrometric Path of Outlet States . . . . . . . . . . . . m m ~ ~ e . ~ . m . m . . . . 72

6-3 The First &sorption Wave in a Thick Desiccant Bed .................. 73(a) Desiccant Temperature profile^..................^.............. 73(b) Desiccant Loading profile^............^.^...^^..^....^...^..... 73( c ) Outlet Air Temperature......................................... 73(d) Outlet Air hmidity ............................................ 73( e ) Psychrometric Path of Outlet state^..................^...^..... 73

6-4 Two Desorption Waves in a Thick Desiccant Bed.........ma............4(a) Desiccant Temperature profile^...........^.^.......^^^.^....^.. 74(b) Desiccant Loading Profiles .................................... 74(c) Outlet Air Temperature......................................... 74............................................d) Outlet Air k m i d i t y 74(e) Psychrometric Path of Outlet States . . . . . . . . . . . . . . . . .mmm... . . . .m4

6-5 The First Adsorption Wave in a Thin Desiccant Bed................ .. 76(a) Desiccant Temperature Profiles ................................ 76(b) Desiccant Loading profile^.............^........^.^............ 76(c) Outlet Air Temperature . . . , . . . . . . . . . ~ . ~ .m . . . . ~m . . . . .m . . . . . . . . . .6(d) Outlet Air Humidity......................m..................... 6(e) Psychrometric Path of Outlet state^..............^.^^^......... 76


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LIST OF FIGURES (Continued)

Page6-6 Two Adsorption Waves in a Thin Desiccant 3ed........................ 77(a) Desiccant Temperature Profiles................................. 77(b) Desiccant Loading Profiles ................................. 77(c) Outlet Air Temperature....................................... 77(d) Outlet Air Humidity ............................................ 77(e) Psychrometric Path of Outlet States............................ 776-7 The First Desorption Wave in a Thin Desiccant Bed................... 78(a) Desiccant Temperature Profiles ................................ 78(b) Desiccant Loading Profiles.................................... 78(c) Outlet Air Temperature ........................................ 78(d) Outlet Air Humidity............................................78

(e) Psychrometric Path of Outlet States............................ 786-8 Two Desorption Waves in a Thin Desiccant Bed........................ 79(a) Desiccant Temperature Profiles ................................ 79(b) Desiccant Loading Profiles ................................... 79(c) Outlet Air Temperature ........................................ 79(d) Outlet Air Humidity ............................................ 79(e) Psychrometric Path of Outlet States............................ 796-9 Behavior of a Thin Desiccant Bed During Cyclic Operation ............ 81.........................a) Temperature Profiles During Adsorption 81(b) Loading Profiles During Adsorption............................. 81.......................c) Outlet Air Temperature hrring Adsorption 81(d) Outlet Air Humidity During Adsorption.......................... 81.........................e) Temperature Profiles During Desorption 82.............................f) Loading Profiles During Desorption 82(g) Outlet Air Temperature During Desorption ...................... 82..........................h) Outlet Air Humidity During Desorption 82(i) Psychrometric Paths of Outlet States........................... 82..............................-1 Predicted COP vs. Simulation Time Step 857-2 (a) COP vs. Regeneration Lewis Number.............................. 86(b) Capacity vs. Regeneration Lewis Number ......................... 867-3 (a) COP vs. Half-Cycle Time........................................ 87

(b) Capacity vs. Half-Cycle Time................................... 877-4 Effect of Outdoor Conditions on Performance of the Base System:(a) COP vs. T. ambient in Ventilation Mode ......................... 90

(b) COP vs. T. ambient in Recirculation Mode ....................... 90(c) Capacity vs. T. ambient in Ventilation Node .................... 90(d) Capacity vs. T. ambient in Recirculation M d e ................. 90xii


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LIST OF FIGURES (Continued)

Page7-5 Effect of Outdoor Conditions on Performance of the HighPerformance System............................................... 91

(a) COP vs. T. ambient in Ventilation Mode......................... 91(b) COP vs. T. ambient in Recirculation Mode....................... 91(c) Capacity vs. T. ambient in Ventilation Mode .................... 91(d) Capacity vs. T. ambient in &circulation Mode.......... . . .. 91

7-6 Effect of Indoor Conditions on Performance of the Base System. ...... 92(a) COP vs. T. room in Ventilation Mode............................ 92..........................b) COP vs. T. room in Recirculation Mode 92( c ) Capacity vs. T. room in Ventilation Mode....................... 92(d) Capacity vs. T. room in Recirculation Mode ..................... 92

7-7 Effect of Indoor Conditions on Performance of the HighPerformance System................................................ 93(a) COP vs. T. room in Ventilation Mde............................ 93(b) COP vs. T. room in Recirculation Mode.......................... 93( c ) Capacity vs. T. room in Ventilation Mode....................... 93.....................d) Capacity vs. T. room in Recirculation Mode 93

7-8 (a) Effect of Regeneration Temperature on COP...................... 94(b) Effect of Regeneration Temperature on Capacity ................. 94

7-9 (a) Effect of Heat Exchanger Effectiveness on COP................,. 95(b) Effect of Heat Exchanger Effectiveness on Capacity............. 95

7-10 Effect of the Effectiveness of Evaporative Cooler 1 onPerformance of the Base System.................................... 96(a) COP...................................a........................6(b) Capacity....................................................... 96

LIST OF TABLES

Page3-1 Typical Properties of Silica Gel .................................... 193-2 Particle Diameter and Surface Area.................................. 193-3 Constants for the Bullock and Threlkeld Correlation for

S i l i c a G e l ..............................a.........................1.............-1 Physical Propert ie s and Dimensions fo r SERI Experiments 32

x i i i


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LIST OF TABLES (Concluded)

Page4-2 Summary of Data Runs and Computer Predictions forSERI Experiments on a 3.5-cm Bed..............................*... 334-3 Physical Properties and Dimensions for Pesaran Experiments.......... 514-4 Summary of Data Runs and Predictions for Pesaran Experiments.. . 514-5 Physical Properties for Koh Experinents............................ 524-6 Summary of Data Runs and Predictions for Koh Experiments............ 527-1 System Specifications for parametric Studies.. . 887-2 Nominal Operating Environment for Parametric Studies................ 88


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Su face area for heat or mass transfer within a packed bed section5(mSpecific heat o f desiccant bed (J/~~'c)Specific heat of air ( J / ~ ~ O C )Specific heat of liquid water ( J / ~ ~ O C )Capacity rate of air on the adsorption side of the s y s t em heatexchanger (w/OC)Capacity rate of air (w/'c or kg/s)Capacity rate of desiccant (WIOC or kg/s)

CLarger capacity rate (w/'c)Smaller capacity rate (w/*c)Capacity rate of air on the regeneration side of the system heatexchanger (WIOC)Particle diameter (m)Effectiveness of exchange processEffectiveness of evaporative cooler 1Effectiveness of evaporative cooler 2Effectiveness of system heat exchanger

2Mass transfer coefficient (kg/m s)2Mass velocity (kg/m s)

Enthalpy of moist air (J/kg)Enthalpy of moist air after mass transfer calculation (J/kg)Heat of adsorption of water (J/kg)Enthalpy of moist desiccant (J /kg)Heat of vaporization of water (J/kg)Integral heat of wetting


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H Heat transf r coefficient (w/'cm2)k 2Thermal conductivity of air (w/'c~ )L Depth of desiccant bed (m)Le Lewis numberm Mass fraction of water vapor in air-m Average mass fraction of water vapor in airms Mass fraction of water vapor in air at equilibrium with the desiccantsurf ce

., 2mH20 Mass flux of water (kglm s)Ma Dry mass of air chunk (kg)Mb Dry mass of bed section (kg)% Mass of water absorbed (kg)Ntu Number of transfer unitsNu Nusselt number'atm Atmospheric pressure (Pa)'sat Saturation pressure (Pa)'v Vapor pressure (Pa)'ve Equilibrium vapor pressure at desiccant surface (Pa)Q Energy transferred (J)Q Rate of energy transfer (W)Re Reynolds numberRg Gas-side mass transfer resistance (s/kg)R *g Modified gas-side mass transfer resistanceRH Relative humidity

2Rs Solid-side mass transfer resistance (m s/kg)Sc Schmidt numberA t Time step (s)


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TR-1330SEPITemperature (OC)Temperature of bed section (OC)Intermediate temperature of bed section (OC)Inlet air temperature (OC)Initial temperature of desiccant bed (OC)Wet bulb temperature (OC)Superficial air velocity (m/s)Humidity ratio (kg/kg dry air)Equilibrium humidity ratio at desiccant surface (kglkg dry atr)Inlet humidity ratio (kg/kg dry air)Saturation humidity ratio at wet bulb temperature (kg/kg dry air)Moisture ratio of water in desiccant (kg/kg dry desiccant)Infti 1 moisture content of deslccant (kg/kg dry desiccant)Moisture conteht at desiccant surface (kg/kg dry desiccant)-X Average moisture content of desiccant (kg/kg dry desiccant)

Y Mass fraction of water in desiccant (XI1 + X)Void fraction

P Dynamic viscosity of air (kg/m s)P 3Density of air (kg/m )

3Pb Density of desiccant bed (kg/m )

msmmse Exiti Inlet1 State at beginning of time step2 State at end o f time s t e p

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SECTION 1.0IrnODUCTLON

Desiccants have been used for many years to provide dry air for a variety ofindustrial and commercial processes. Several manufacturers have marketeddesiccant air conditioning systems for special situations requiring very lowrelative humidities ((10%) which cannot be provided by vapor compressionequipment. Because there has been little competition in this market, theenergy efficiency of drying equipment has not been a primary concern. Thethermal efficiency of industrial and commercial drying equipment tends to below, and parasitic power requirements tend to be high. A significant dif-ference between solar cooling systems and commercial drying systems is thatsolar cooling must compete with conventional vapor compression cooling. Thus,the thermal coefficient of performance must be high and parasitic powerrequirements must be low.Although I36f [ I ] proposed an open-cycle , liquid-desiccant solar-cooling sys-tem in 1955 and Dunkle [ 2 ] proposed an open-cycle, solid-desiccant, solar-cooling system in 1965, active research in desiccant cooling did not beginuntil the mid-1970s. Lunde [3] developed preliminary designs for a coolingsystem using silica gel. Nelson [4] investigated the feasibility of solardesiccant cooling using a simple desiccant model in seasonal simulations.Because of an emphasis on the rapid commercialization of solar technologies,the national desiccant cooling research program moved quickly from theseinitial studies toward the development of prototype cooling systems in thecapacity range of 5 to 10 kW (17000 to 34000 Btu/h) for residential and smallcommercial applications. Solid desiccant systems are a promising alternativefor these smaller-scale applications because they are mechanically simple, canbe driven by flat-plate collectors, and use air as the transport fluid andwater as the refrigerant.Three prototype systems were designed and built under U . S . Department ofEnergy (DOE) contracts beginning in 1977. The Institute of Gas Technology(IGT) uses a molecular sieve impregnated wheel in its Solar-MEC@ system [ 5 ] .AfResearch uses a thin rotating drum packed with silica gel particles in itsSODAC system 161. The Illinois Institute of Technology (IIT) uses Teflon@bonded silica gel sheets in fixed, cross-cooled adsorbers [ 7 ] . All. three sys-tems have performed with COPs between 0.5 and 0.6 under ARI standard operatingconditions.A recent report evaluating residential and commercial solar/gas heating andcooling technologies, writ ten by Booz-Allen and Hamil on for the Gas ReserchInstitute [B], has indicated that advanced desiccant cooling s y s t ems with COPsnear 1.2 would be competitive with vapor compression systems in the 1990s andwould be the solar cooling system of choice for residential applications inregions having moderate heating and cooling loads. The current emphasis ofthe desiccant cooling program is to expand on experience gained with first-generation prototypes and to perform basic research and development on newconcepts and advanced components that will lead to advanced desiccant systemswith this prescribed performance. I


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1.2 PURPOSE AHD SCOPEThis report documents a computer model for adsorption/desorption that wasdeveloped as a research tool to investigate a variety of technical areas con-cerning the behavior of desiccant beds and the performance of cooling sys-tems. This model fills a need for a general-purpose desiccant simulation pro-gram to evaluate alternatives for advanced components. Specific issues thatthe model has been or will be used to investigate involve the effects ofdesiccant bed geometry, of modifying desiccant properties,* and of componentperformance on the system coefficient of performance (COP). The model uses anew approach to simulation of transient coupled heat and mass transfer in thatsimple equations for steady-state heat and mass exchangers are used. Hence,it is referred to as a pseudo-steady-state model. This approach simplifiesthe mathematics of the analysis and is intended to make the model easy toadapt to a variety of adsorption problems. It also should make the model moreacceptable and thus more useful to engineers and research scientists who arenot specialists in numerical analysis.The report is divided into eight sections. In the last portion of this intro-duction is a brief description of the physics of the adsorption problem,intended primarily for those with little or no experience with desiccants.Section 2.0 provides a detailed historical overview of adsorption analysisthat summarizes the different mathematical approaches that various investiga-tors have taken. Section 3.0 describes the new pseudo-steady-state adsorptionmodel, presents the equati-ons used, and gives correlations for the propertiesof moist air and silica gel. Section 4.0 compares predictions using thismodel with experimentally measured outlet air conditions for single-blowadsorption and desorption experiments with packed beds of silica gel. Threeindependent data sources are used to provide a thorough assessment of thevalidity of the model. Section 5.0 outlines the simulation of complete desic-cant cooling systems, giving equations for analysis of components other thanthe dehumidifier and describing the program logic. Section 6.0 describes thephysical behavior of desiccant beds, both in single-blow operation and cyclicsystem operation. This description is qualitative but detailed. It isintended to facilitate a better understanding of the way the desiccant coolingcycle works and the factors that affect the performance of the desiccant bed,an understanding essential to making sound research decisions. Section 7 .0presents results of parametric studies on the performance of two cooling sys-tems, one representative of existing prototypes and the other containingadvanced, high-effectiveness components. Section 8.0 presents conclusions andrecommendations. A practical user's guide to the programs for single-blowsimulations, system simulations, and parametric studies is included asAppendix A. Program listings are also included, as Appendix B.As reported here, the computer model only includes property data of regulardensity silica gel in a packed-bed geometry. The capabilities of the modelwill be expanded to include properties of other desiccant materials, such asmolecular sieve, and to simulate heat and mass transfer in parallel-passage,laminar flow desiccant beds in the near future.

*See SERI/TP-631-1157 [Ref. 491 .


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1.3 ADSORPTIOH PBOCESS AND DESICCANT COOLING CYCLEFew people in the solar community are familiar with desiccants or the adsorp-tion process. Therefore, a brief description of both is included here as anintroduction.Desiccants are a class of adsorbents/absorbents that have a high affinity forwater. They can be either liquids or solids. Examples of liquid desiccantsare salt solutions, such as lithium chloride or calcium chloride, and someorganic liquids, such as trfethylene glycol. Examples of solid desi cants aresilica gel, molecular sieves, and natural zeolites. This report is concernedonly with solid desiccants and, specifically, regular density silica gel.However, the methods used here can be applied to a wide variety of coupledheat and mass transfer problems. Generally, solid desiccants are highlyporous materials that adsorb water by mechanisms of chemical adsorption ofwater molecules onto sites on the walls of the pores, physical adsorption ofsuccessive layers of water molecules, and capillary condensation wlthin thepores. The amount of water that a desiccant will hold at equilibrium is afunction of its temperature and the water vapor pressure or humidity of theair that surrounds it. At high temperatures and low humidities, a desiccantwill contain almost no sorbed water. At room temperature in saturated air,silica gel wflL pick up 35% to 40Z of its weight in moisture.The desiccant cooling cycle takes advantage of this moisture cycling capacityto dehumidify air. Figure 1-1 is a diagram of the Solar-MEC@ desiccant cool-ing system. It consists of a dehumidifier, sensible heat exchanger, twoevaporative coolers, a solar heating coil, and an auxiliary gas burner.In essence, the desiccant cycle permits evaporative cooling to be used inhumid climates by turning hot, humid air into hot, dry air before it is sentthrough an evaporator pad. The heart of a desiccant system and the componentthat is most difficult to simulate mathematically is the dehumidifier ordesiccant wheel. This wheel rotates between two counter-flowing air streams,adsorbing moisture from the conditioning stream and desorbing that moisture tothe solar-heated air of the regeneration stream.Consider a dehumidifier wheel constructed of a thin, packed bed of desiccantparticles held between two metal screens (the construction used byAiResearch). On the adsorption side of the system, warm, moist air is exposedto relatively dry desiccant. Water molecules in the air at the surface of theparticles are adsorbed. This creates a humidity gradient in the air streamand causes other water molecules to migrate toward the surface where they, inturn, are adsorbed as the air flows through the bed. This is a convectivemass transfer process. A second mass transport process takes placesimultaneously within the desiccant particles. During dehumidification, thewater concentration in the desiccant near the surface of the desiccantparticle is higher than it is at the center. This concentration gradientcauses water to diffuse inward.Thus, there are two mass transfer resistances, a gas-side resistance and asolid-side resistance, that determine the rate of transfer of water betweenthe air and the desiccant. Both resistances are important in packed beds andmust be accounted for In simulations-

3


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Air mFrom 8R y ,DryingWheel

SolarCoil

Air- .-

Fig- 1-1. IGT Solar-HEW System in the Recirculation Mode [Ref. 51In addition to the mass transfer problem, the adiabatic dehumidification pro-cess involves a heat transfer problem. When water is adsorbed by the desic-cant, energy is released. For silica gel, the amount of energy, called theheat of adsorption, is typically 10% to 15% greater than the heat of condensa-tion of water. This energy release elevates the temperature of the desiccantand causes heat to be transferred to the air stream. Fortunately, the thermalconductivity of most desiccants is high enough so that temperature gradientswithin the particles can be neglected, and only a single resistance to heattransfer, the convective resistance, need be considered. In summary, adsorp-tion comprises simultaneous heat and mass transfer processes that are coupledby the equilibrium properties of the desiccant and during which thermal energyis generated as a consequence of the mass transfer.

- - - - - -When-the desiccant-- wheel passes to the regeneration side, all Processes listedI above are reversed.- ____I_2 Energy is transferred from the solar-heated airstream to the desiccant. As the temperature of the desiccant increases, waterat the surface is desorbed and picked up by the air. Water from the interiorof the particle diffuses toward the surface where it is desorbed.


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Desiccants have been used for many years to dehumidify air for industrial pro-cesses. The analysis of air drying also has a long history, dating back tothe late 1940s. Beyond the drying application, adsorption has many applica-tions in the chemical industry. The governing transport equations are thesame whether we use silica gel to remove water vapor from air or activatedcarbon to remove methane from helium. As a result, researchers have contri-buted to the state of the art of analysis in a variety of fields. Despitethis long history, few modeling attempts have been completely successful, fortwo reasons. First, the mathematical formulation of the adsorption process asdescribed above includes five dif erential equations for the conservation ofmass and energy. These equations are coupled nonlinearly by additional equa-tions describing the properties of the desiccant. This set of equations mustbe solved numerically, and even with today's computers, their solution is com-putationally expensive. Second, differential equations include transportcoefficients that must be determined experimentally. The experiments are difficult to perform, and few data are available for some of these transportcoefficients. This is especially true with regard to diffusion coefficientsfor water within solid desiccants. The accuracy of any model is limited bythe accuracy of the data supplied to it.To make the problem manageable, the first investtgators of the adsorption pro-cess made sets of fairly restrictive assumptions about the adsorption processand desiccant properties. The history of modeling has been a process ofrelaxing these assumptions step by step. The first modeling work in airdrying was conducted by Hougen and Marshall [ 9 ] . They developed an analyticalsolution for isothermal adsorption where the relationship between the vaporpressure in the air and the equilibrium water content of the desiccant islinear. Figure 2-1 compares linear isotherm with typical isotherms for mole-cular sfeve and silica gel. The linear approximation is relatively good forsilica gel below relative humidities of about 50%. However, it is a poorapproximaion for molecular sieves. Hougen and Marshall also proposed graphi-cal methods to deal with nonlinear desiccant properties, and with adiabaticadsorption where the heat generated during adsorption is significant and mustbe included. An underlying assumption in this work was that a single, air-side mass transfer coefficient could be used, and that the effect of diffusionin the solid could be included in this lumped coefficient. Hougen andMarshall used data from Ahlberg [ l o ] to produce empirical correlations f o rheat and mass transfer coefficients in packed beds of silica gel. Sub-sequently, several other authors have used these correlations. However,because the data set on which they are based is relatively limited andincludes considerable scatter, these coefficients should probably be used onlyas a starting point. Additional problems with the correlations are discussedin Sec. 3.5.Rosen [11,12] expanded the isothermal model to include the separate effects ofmass transfer across a fluid boundary layer and diffuston of the adsorbateinto spherical adsorbent particles. This analytical approach leads to aseries solution that can be evaluated numerically. However, it is still


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Relative Humidity.%

Fig. 2-1. Linear Approximation to Properties of Silica Gel and MolecularSievelimited to linear isotherms and applies only to cases in which the desiccantbed is initially uniformly dry. An alternative approach used by Eagleton andBliss [13] allows for the concentration gradient in the absorbent by means ofa hypothetical solid film coefficient. That is, the mass transfer rate in thesolid is assumed to be proportional to the difference between the surface con-centration and the average concentration of the adsorbate in the particle.They maintained the assumption of a linear isotherm, but allowed for a non-zero intercept. Antonson [14] obtained relatively good agreement with experi-mental results for the isothermal case by considering diffusion in the solidas the only resistance to mass transfer in a helium/ethane/molecular sievesystem.After comparing the relative success of the previous approaches and concludingthat a two-resistance model is preferable [15-171, Carter extended Rosen'swork to include the adiabatic case [18,19]. Carter assumed a simple exponen-tial relationship between vapor pressure and the equilibrium loading of theadsorbent. He used finite difference techniques to solve the resulting set ofequations and obtained good agreement between experimental and calculatedresults for adsorption of water vapor by activated alumina. Meyer andWeber [20] increased the complexity and generality of adsorption modeling in astudy of the adsorption of methane from helium by activated carbon. Theyincluded internal and external resistances to heat transEer as well as masstransfer and used a general equation for equilibrium properties containingeight curve-fitting parameters. Although this is a very detailed model, it


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attained only limited success. presumably because of inaccuracies in theexperimentally measured values of the diffusion coefficient of methane inactivated carbon.Noting the difficulty of obtaining accurate values for the intraparticle dif-fusion coefficient, Bullock and Threlkeld (211 used the lumped, effectiveexternal heat and mass transfer coefficients reported by Hougen and Marshallfor the adiabatic drying of air by silica gel. To facilitate computation,they expressed the equilibrium data for silica gel presented by &bard [ 2 2 ] aspolynomials in temperature and moisture content. Chi and Wasan [23] used thesame approach to study air drying by a porous matrix impregnated with lithiumchloride. More recently, Koh [ 2 4 ] investigated the use of solar energy forthe regeneration of silica gel used for grain drying. Following the methodsof Bullock and Chi, but modifying the transfer coefficients of Hougen andMarshall, Koh was moderately successful in matching experimental and predictedresults for the adiabatic desorption process.All the researchers mentioned have investigated single-blow adsorption ordesotption in fixed beds. Concerns about energy conservation and renewableresources over the past decade have stimulated research in coupled heat andmass transfer in total heat regenerators or enthalpy exchangers and in desic-cant dehumidifiers for solar cooling. Maclaine-cross, Banks, andClose [ 2 5 , 2 6 , 2 7 , 2 8 ] used the method of characteristics to reexpress thegoverning equations for adsorption in terms of combined potentials. Theseequations are analogous to those for transfer alone, and existing solutionsfor rotary heat exchangers can be applied. This analogy method is approximatebecause linearizations are made at several points in its derivation. However,f t is very efficient cornputionally. Pla-Barby 129 1, Holmberg [30] Barker andKettleborough 1311, and Hathiprakasam and Lavan [ 3 2 ] have each analyzed theperformance of adiabatic silica gel dehumidifiers but without comparing pre-dictions with experimental data. Mathiprakasam 1331 predicted the performanceof a cross-cooled silica gel dehumidifier.


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SECTION 3-0PSEDDO-STUDY-STATE MODEL,

Most of the analytical approaches described in the previous section involvethe derivation of a set of differential equations for conservation of mass andenergy within the adsorption system, and the solution of that set of equationsby finite difference techniques. The pseudo-steady-state model developed inthis study uses an approach that simplifies t h e mathematics of the adsorptionproblem. As illustrated in Fig. 3-1, the Desiccant Simulation computer model(DESSIM) can be thought of as a conveyor belt that carries one chunk of air ata time through the desiccant bed. The passage of each air chunk through thebed corresponds to a single time step. The bed is divided into equal sec-tions, and as the air chunk is exposed to each section, mass and heat transfercalculations are perf nned in an uncoupled mannerFirst, all temperatures are held constant while the amount of moisture trans-ferred between the air and the bed section during the time step is determined,and the humidity ratio of the air chunk and moisture content of the desiccantsection are revised. Second, an energy balance is performed that accounts forthe amount of energy released when the water is adsorbed and determines anintermediate temperature for the bed section. Finally, a heat-transfer cal-culation is performed using this intermediate bed temperature and the originalair temperature. Temperatures for the air and the bed are updated, and theair chunk is moved to the next bed sectionAlthough the sorption process is a transient one, these mass transfer and heattransfer calculations are done using equations for steady-state, counterflowmass exchangers and he a t exchangers hence, the name pseudo-steady state.Conceptually, the model carries along a counterflow mass exchanger and acounterflow heat exchanger that each have surface areas for transfer equal tothe surface area in each bed section. Final moisture contents and tempera-tures for the air chunk and the bed section at the end o f each time step aretaken to be the same as the outlet moisture contents and temperatures fromsimple counterflow exchangers. These have steady flows of air and desiccantmaterial with inlet conditions equal to the initial conditions of t h e air

OutletBasket

Air

Fig. 3-1. Conveyor Be l t Concept Used in Computer Model9

Desiccant Bed2 3 * o m N


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chunk and the bed section. Applying these exchanger equations during timesteps that are short compared to the duration of the complete process pre-serves the transient character. The fact that the exchanger equations areanalytical solutions describing the approach of an exchange process toward amaximum effectiveness allows relatively large time and space increments to beused without complicated numerical techniques. This permits simulations to becarried out at a very reasonable computational costIn the remainder of this section, we outline the equations used in the com-puter model.

3.2 W S RANS= CALCULATIONThe mass transfer calculation that is performed as each air chunk is exposedto a bed section is adapted from the effectiveness equations for a counterflowgas/liquid mass exchanger 1461 . The configuration from which these equationsare derived is shown in Fig. 3-2.The air at the desiccant surface is assumed to be in equilibrium with thedesiccant. All temperatures are assumed to be constant. The moisture ratioin the desiccant is assumed to be uniform in the direction perpendicular toflow, and the transfer process is assumed to be controlled by a gas filmresistance, which is modified to account approximately for the effect of asolid-side resistance to the diffusion of water. The applicability of theselast two assumptions is discussed in Sec. 3.5.The rate of mass transfer per unit surface area at any point in the exchangeris given by

wherem = bulk vapor mass fraction in airn, = f(X,Tb), equilibrium vapor mass fraction at the desiccant surfaceg = effective gas-side mass transfer coefficientX = moisture ratio in desiccant (kg water/kg dry desiccant)

Tb = temperature of bed sectionDefining the effectiveness E of the exchanger as the actual moisture transferdivided by the maximum possible moisture transfer, we have

1 - exp [-Ntu(1-CC)]E = 1 - CCexp [-Ntu(1-CC) 1where

Ntu = gAs/CmlnA, = surface area for mass transfer within the bed section


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i Ai r )/z 1msL II J( 1Ii Desiccant i xII t

F i g . 3-2. Cancentrattoo Rof i l ea for Steady-State Mass Transfer Process",in and C,, are the smaller and larger of the capacity rates of the twostreams flowing through the hypothet cal exchanger The capacity rate for theair is

'air ' gas, for the steady-state case= Ma (1 + w l ) / A t , for the pseudo-steady-state case.

The capacity rate for the desiccant material is

whereMa = dry mass of air chunkMb = dry mass of bed section iY a water mass fraction in desiccant, X/(l+X)

A t = tine step defined for the simulation.The partial derivative of the mass fraction of water in the desiccant withrespect to the equilibrium vapor mass fraction at the surface replaces theinverse oE the Henry number used in gasjlfquid mass exchanger analysis, and isanalogous to the specific heat in the expression for heat exchanger capacityrates. The calculatfon of this quantity is discussed in Sec. 3.6.With the effectiveness known, the mass fraction for the outlet air is

m2 = ml - E(ml - m,)


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a nd t h e o u t l e t hu mi di ty r a t i o i s

The amount of water t r a n s f e r r e d t o t h e bed s e c t i o n d u r i ng o ne t i m e s t e p i s

s o t h e new m o is tu r e r a t i o i n t h e b ed i s

3.3 J x l mmDL PTE ENERGY 33AmNCEWhen water i s adsorbed by a des icc ant , energy i s r e l e a s e d . T h i s he a t o fa ds o r p t i on i s u s u a l l y g r e a t e r t h a n t h e h e a t o f v a p o r i z a t i o n o f w a t er , a nd i s af u n c t i o n o f t h e water c on t e n t i n t he de s i c c a n t . I n t he p s eudo- s te a dy - st a temodel , an energy bala nce i s p er fo rm ed a f t e r t h e mass t r a n s f e r c a l c u l a t i o n t ode t e rm i ne t he c hange i n t h e bed s e c t i o n t e m pe r at u r e due t o t he a ds o r p t i on o rdeso rp t i on of wa te r. The t empera ture of the a i r i s assumed t o remain con-s t a n t .The ove r a l l s o r p t i o n p r oc e s s i s assumed t o be ad ia ba ti c, and no work i si nvo lve d . T he r ef o r e , t he e ne r gy ba l a nc e on t he a i r chunk a nd t h e bed s e c t i o nc a n be w r i t t e n s i m p l y a s

H er e, a l l e n t h a l p i e s a r e i n t er ms o f e ne rg y p er u n i t d r y mass o f a i r o r b edm a t e r i a l , a nd 1 and * r e f e r t o s t a t e s b e f or e and a f t e r t h e a d so r pt i on o rdes orp t io n of th e amount of water determined by Eq. 3-7. Cor r e la t i ons fo r th ee n t ha l py o f m o i st a i r ( Se c. 3 .6 ) u s e d r y a i r a nd l i qu i d w a te r a t OC as abase. There fore , the hea t o f vapo r iza t ion i s i nc lu de d i m p l i c i t l y i n t h ee n t h a l p i e s . E n t ha l p i e s f o r t he de s i c c a n t bed c a n be de f i ne d u s i ng t he sameba se, OC and l i q u id water. However, water i n t h e bed i s i n t h e so rb ed s t a t e ,no t t h e l i qu i d s t a t e . The e ne r gy d i f f e r e nc e bet we en t h e l i qu i d a nd s o r beds t a t e s c a n b e a cc ou nt ed f o r e i t h e r by i n c l u d i n g a te rm f o r t h e i n t e g r a l h e a to f w e tt in g d i r e c t l y i n t o t h e e x p r es s io n f o r t h e e n t h a lp y o f t h e d e s i c c a n t o rby i nc l ud i ng a n e x t r a t er m i n t h e e ner gy ba l a nc e, w hi ch a c c ount s f o r t he d i f -f e r e nc e bet we en t he he a t o f a d s o r p t i o n and t h e he a t o f va po r i z a t i on . The l a t -t e r a pp roac h i s use d i n t h i s a na l y s i s . Hence, t he e n t ha l py o f t he bed i sd e f i n e d as

where t he t empera ture i s i n de g r ee s Ce l s i u s . E qua t ion 3-9 becomes

*where h = ha ( T ~ q .a n


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Solving for the intermediate bed temperature 6 ,

This is the initial bed temperature for the heat transfer calculation. Corre-lations for the ratio of the heat of adsorption and the heat of vaporizationare given in Sec. 3.7.

After the temperature change of the bed section caused by the heat of adsorp-tion is determined, a heat transfer calculation is performed. As with themass transfer calculation, equations for a counterflow exchanger are appliedand the steady-state outlet temperatures are used as the final averagetemperatures of the air chunk and the bed section at the end of the timestep. Again, the equation for the exchanger effectiveness is

1 - exp[-Ntu(1 - CC)E = 1 - CC exp[-Ntu(1 - CC)]where, for the heat exchange problem,

N ~ U H A,/C,~~CC = Cmax'Cmin

Cair = Ma cp/AtCbed = M b ( c b + X2cw)/At

H = average heat transfer coefficient.Since the effectiveness is the ratio of actual energy transfer to the maximumpossible energy transfer, the total energy transfer from the air to the bedsection during the time increment is

Applying simple energy balances, new temperatures for the air chunk and bedsection are

The conditions of the bed section are s t o r e d and the a i r chunk is sent 1-0 thenext bed section, where this series of calculations is repeated.


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sea TR-1330

In both the mass transfer and heat transfer calculations, the rates of trans-fer are assumed to be controlled by single resistances in the gas filmadjacent to the desiccant surface. This assumption is a good one for the heattransfer calculation because the heat transfer Biot number is small fortypical packed-bed situations, and the temperature gradient within theparticles would be minor. However, there is wide agreement that for silicagel particles of the size used in packed beds, there is a significant resis-tance to mass diffusion within the solid particle. The appropriate resistancemodel for the overall mass transfer process is shown in Fig. 3-3a. Here, Rgand Rs are the gas-side and solid-side resistances, respectively. In thiscase, the rate of mass transfer would be

Solving for a value of the moisture ratio at the desiccant surface that isdifferent from the average value adds significantly to the complexity and costof a numerical solution to the sorption problem. Furthermore, there may belittle benefit in accurate predictions, because diffusion coefficients fordesiccants are difficult to measure accurately or predict theoretically.To simplify the analysis, the mass transfer rate is calculated as if it werestrictly a gas-side-controlled process. However, the gas-side resistance ismodified to account approximately for the resistance to moisture diffusionwithin the desiccant particles. This gas-side resistance model is shown inFig. 3-3b.

m j = f (ST, T)

Fig- 3-3. (a) Detailed Resistance %del for Hass Transfer(b) Gas-Side Resistance Xodel for Mass Transfer14


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The particle is assumed to have a uniform moisture content and the equilibriumvapor mass fraction of the air at the surface is calculated using the averagemoisture content of the desiccant. The convective mass transfer coefffcient gis then reduced by some factor, which can be determined experimentally orapproximated on the basis of theory.Hougen and Marshall [ 9 ] , who were the first researchers to analyze theadiabatic adsorptfon process, used experimental data from Ahlberg [ l o ] todevelop the following correlations for effective mass transfer and heat trans-fer coefficients for adsorption in packed beds of silica gel.

These transfer coefficients have been used by Bullock and Threlkeld 12'11, Chiand Wasan 1231, Pla-Barby et al. (291, Koh [24], and Nienberg [34]. However,the original Ahlberg data contain a considerable amount of scatter, whichleads to significant uncertainty Also , the correlations of Hougen andHarshall modify the heat transfer coefficient, as well as the mass transfercoefficient, to preserve a Lewis number close to unity. The k w i s number,defined as

is close t o unity for an aidwater mixture in a strictly convective problem.However, in our case, there is no reason to modify the heat transfer coef-ficient, and consequently, modification of the mass transfer correlationresults in an effective ZRwis number greater than unity.Since the heat transfer process is gas-side controlled in typical packed-bedsituations, a correlation for heat transfer alone is used in the pseudo-steady-state model. The correlation given by Handley and Heggs [3S] for theNusselt number is

where

c V = void fraction.Manufacturers' data gfve a void fraction of 0.4, so the expression for theheat transfer coefficient used in the model is


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The mass transfer coefficient is then

where Le is an effective Lewis number that depends on the parameters of thesorption situation.Close and Banks [27] have used this same approach for calculating mass trans-fer coefficients in silica gel beds. Recently, van Leersum 1471 has carriedout an analysis to determine appropriate values of the effective Lewis numberfor rotary dehumidifiers. This analysis adapts Hausen's correction for theheat transfer coefficient in thermal regenerators that have a resistance toheat transfer within the solid matrix, and includes considerations of the con-vective transfer coefficient, the diffusivity of moisture in silica gel, theparticle diameter, the equilibrium properties of silica gel, and the typicaloperating conditions of a rotating dehumidifier within a desiccant coolingsystem. Van Leersum's results indicate that a value between Le = 3 and Le = 4would be an appropriate average for rotary system simulations. However, theeffective Lewis number is a function of several parameters and differentvalues may be appropriate for single-blow simulations.Figure 3-4 compares several correlations for mass transfer in packed beds.Clearly, there is a considerable degree of uncertainty in characterizing thistype of transport phenomenon. Mass transfer relationships based on theHandley and Heggs heat transfer correlation are included in Fig. 3-4 for threevalues of the effective Lewis number; 1, 3, and 9. With Le = 1, the Handleyand Heggs correlation is grouped with the majority of the other correla-tions. With Le = 3, it is similar to the Hougen and Marshall correlation,which is based on silica gel adsorption data. With Le = 9 at low Re, it isrelatively close to the Eagleton and Bliss correlation, which is also based onsilica gel adsorption data. Thus, the use of an effective Lewis number aslarge as 9 is not out of line with previously reported mass transfer correla-tions.

3.6 PROPERTIES OF MOIST A IRThe properties of air that are used at various points in the calculation pro-cedure are enthalpy, specific heat, thermal conductivity, and dynamicviscosity. The following an expression for the enthalpy of moist air wasreported by Maclaine-cross [37]:

Here, temperature T is in degrees Celsius and the enthalpy is units of joulesper kilogram. The reported accuracy of this equation is 0.05%, as compared tostandard tables. In the simulation of complete cooling systems, outlet temp-eratures from the sensible heat exchanger must be calculated from known valuesof enthalpy and humidity ratio. Solving Eq. 3-21 for temperature gives us


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s n a TR-1330

where

The specific heat of moist air is also derived from Eq. 3-21:

Specific heat, conductivity, and viscosity appear in the correlation for theheat and mass transfer coefficients. Because of the inherent uncertainty inthis type of correlation, highly accurate expressions for air properties arenot warranted. Properties of dry air at the inlet temperature to the bed are,therefore, used in calculating transfer coefficients. A constant specificheat and linear interpolation are used to calculate conductivity and viscosityfrom data in Holman [38] for the temperature range of interest.

The following standard psychrometric equations are also used:humidity ratio w = .622 pv/(Patm - Pv) (3-27)

relative Humidity RH = Patm w/[(CL622 + w) Psatl (3-28)where Pv is the vapor pressure, Patm is the ambient pressure, and Psat is thesaturation pressure at a given temperature.

2The saturation pressure (N/m ) is calculated using the simple but accuratecorrelation reported by Maclaine-cross [37]. The reported accuracy of thisequation is 0.08% as compared with standard tables.

3.7 PROPERTIES OF SILICA GeLSilica gel is a highly porous, granular, amorphous form of silica that is man-ufactured by reacting sodium silicate with sulfuric acid. The internal struc-ture of silica gel consists of a vast number of small pores. When used as adesiccant, standard grades of gel can hold up to 40% of their weight in waterby mechanisms of physical adsorption and capillary condensation. Many dif-ferent grades of silica gel are manufactured whose physical and equilibriumproperties vary between grades. Regular-density gel, such as Davison PA40 orSyloide 63, is more appropriate than low-density gel for solar coolingapplications because it demonstrates a favorable and relatively steep isothermshape in the range of relative humidities typically found in desiccantcooling. Figure 3-5 shows this qualitative difference in gel properties.The physical properties of regular-density silica gel are summarized inTables 3-1 and 3-2.

18


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Table 3-1. Typical Properties of S i l i c aGel (ManufacturerT Data,Grace & Co - 1 4 3 1 )Specific heat (cb) . 921 J/kg;CApparent bulk density (pb) 720 kg/mVoid fraction ( E ~ ) 0.4

Table 3-2. Particle Mameter and SurfaceArea 1241

Tyler Average Particle External SurfaceDiameter AreaMesh S i z e (m) 2 3(m

3.7.1 Equilibrium PropertiesCritical information for the simulation of the adsorption process is anexpression for the equilibrium properties of the desiccant being modeled;i.e., an expression for the equilibrium vapor pressure at the desiccant sur-face as a function of the desiccant temperature and moisture content. Thereare several correlations available for the equilibrium properties of silicagel. Lunde and Kester [ 3 9 ] used a multilayer adsorption model along withexperimental data For a moderate range of moisture content t o predict completeiso herms. Rojas [ 4 0 ] measured the equilibrium characteris ics of four gradesof silica gel of different porosity, and fit polynomial expressions of theform [ [ [ 'V]i [PvJiX x A O + A 1 P * 2 i + * 3 7 A 4 P (3-30)sat sat sat sato t h i s data. One of the gels investigated by Rojas is Davison grade PA-40,the same silica gel used in current adsorption experiments at SER I .Jury and Edwards 1411 fit the following equation to their data:


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Regular-Density

0 20 40 60 80 100Relative Humidity,%

Fig. 3-5. Typical Isotherms for Regular- and b w - D e n s i t y Silica GelHubard [ 2 2 ] reported equilibrium data for the silica-gel/air-water system fortemperatures between 4C and 93C (40F and 200F), and moisture contentsbetween 1.5% and 35%. Bullock and Threlkeld [21] represented these data aspolynomials of the form

where

've = equilibrium vapor pressure (in Hg)T = silica gel temperature (OF)X = silica gel moisture content (lb water/lb dry gel).

The constants C1 through C8 are given in Table 3-3; m = 2 and n = 1 for X lessthan 0.05; and m = 1 and n = 0 for X greater than 0.05.


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Table 3 3 . Constants for the k l l o c k and Threlkeld Correlation for S i l i c aGelRange of

Moisture Content Cl l o 6 c q l o4 C3 C4(XI

Range of )Moisture Content( X I


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Close and Banks [27] plotted Hubard's data in the form of Othmer charts and,by applying the Clausius-Clapeyron equation, obtained

where fl(X) and f2(X) are polynomials, and f2(X) is the ratio of the heat ofadsorption to the heat of vaporization. They recommended the simplificationof replacing fl(X) with the simple linear function 2.009X.The polynomial expression of Bullock and Threlkeld wasthis study, because of the convenience of calculating . However, itwas later determined that this correlation gavebination of high temperature and high moisture content was encountered.Figure 3-6 shows a series of isotherms based on the equations of Bullock andThrelkeld. This demonstrates that the correlation should not be used when thegel moisture content is greater than 20% and the temperature is greater thanabout 50C.Once this was discovered, other correlations were investigated more closely.Isotherms from the Close and Banks complete correlation and the simplified oneusing f (X) = 2.009X are compared in Fig. 3-7. This correlation was furthersimplifted to facilitate the calculation of ( a ~ / a m ~ ) ~y using linear equa-tions for h /h which are discussed in the next section. Pigure 3-8shows that tadas ? simplification produces almost no change in the iso-therms. The Rojas correlation for grade PA-40 is included for comparison inboth figures. The Rojas data demonstrate an upper limit on the capacity ofthis grade of silica gel of about 37%, which is typical of regular-densitygels. The other correlations do not taper off toward such a limit. However,the Rojas correlation does not include the fanning of isotherms at differenttemperatures, which is displayed in his data and by the other correlations.To combine both characteristics, the Rojas equation was recast to give vaporpressure as a function of moisture content, and an additional temperaturedependence was included.

Isotherms from this equation are shown in Fig. 3-9.The equilibrium vapor mass function at the surface ms used in the mass trans-fer calculation is determined from the equilibrium vapor pressure using thefollowing psychrometric relationships:

Also required in the mass transfer calculation is the partial derivative ofthe water mass fraction in the desiccant with respect to the equilibriun vapormass fraction of the air at the surface. Reexpressing the derivative,


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Here, Y, the water mass fraction in the des iccant , isY = X / ( 1 + X); kg water/(kg water + kg ge l ) ,

Equations 3-35 and 3-36 combine t o give usm, = 0.622 Pve/(Patm + 0.378 Pv,) ,

The partial derivative (BXI~P,,) can be evaluated by differentiating theappropriate equation for P,, with respect to X and taking the inverse of theresult. The reader is referred to the program l i s t i n g s i n Appendix 3 forspecific equations.The value of ( W/3tns) is determined by combining these results in Eq. 3-37.3.7.2 Heat of AdsorptionThe heat of adsorption of water on silica gel is a function of the water con-tent of the gel. Close and Banks [ 2 7 ] plotted Bubardls data in the form of anOthmer chart to determine the ratio of the heat of adsorption to the heat ofvaporization for water contents between 1% and 35%. Results were presented aspolynomial curve fits, which were used in this study in combination with theequilibrf m correlations of Bullock and Th r e l k e l d .

Close and Banks [ 2 7 ] a l s o a p p l i e d the Clausius-Clapeyron equation to the vaporpressure data o f Hougen, Watson, and Ragatz to obtain heat o.E adsorptioninformation. Their conclusion was that these data must have been for a gradeof silica gel other than the regular density type. Rojas 1401 reported heatof adsorption data for four grades of silica gel. Of particular interest is27


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h i s d a ta f o r r e gu la r - de ns i ty Grade PA-40. Low-density grades exh ib i t a lowerhea t of adsorpt ion s i mi la r t o th e da ta of Hougen e t a l . Bul lock andThre lke ld [21] f i t two polynomials t o Ewing's and Bauer' s da ta f o r the in t e -g r a l hea t of wett ing. The polynomials a r e

When the se e qua t ions a r e d i f f e r e n t i a t e d wi th r e spe c t to X a nd th e he a t o fva por i z a t ion i s added, t he hea t of adso rpt io n i s obta ined . A d i s c o n t i n u i t y i nt h e h e a t o f a d so r pt i on r e s u l t s from a d i s c o n t i n u i t y i n s l o p e a t t h e i n t e r -se ct io n of th e above equat ions . Nienberg [ 3 4 ] f i t a l i n e a r e q ua ti on t o d a t areported by Beecher [ 4 2 ] .Figure 3-10 i s a g ra ph of the r a t i o of the he a t of a dso r p t ion t o the he a t ofvapo r iza t ion , h ,ds/\ap versus ge l mois ture con ten t , which compares a l l th esour ces above. Agreement f o r re gu la r de ns it y ge l i s with in 5% fo r va lues of Xbetween 0.1 and 0.25. However, f o r X l e s s tha n 0.1 the r e i s a l a r g eunce r ta in ty . The fo l lowing l i n ea r equa t ions , which a r e shown as dashed l in esi n Fig. 3-10, were considered t o giv e th e bes t combination of accuracy andcomputa t iona l convenience and a r e used i n t h i s s tudy i n combinat ion wi th theequ il i br i um co rr el a t io ns of Close and Ranks and those based on th e Rojas da ta .

In summary, th e computer program inclu des thr ee s e t s of co rr el a t io ns f o re qu i l ib r ium p r op e r t i e s and th e he a t o f a dso r p t ion . The f i r s t s e t c om bine s th eBul lock and Thre lke ld polynomia ls fo r equi l ib r ium vapor pres sure wi th theClose and Banks equ ati on s fo r hads/hva,. The second s e t combines t h e Closeand Banks equatio n fo r equil ib r ium vhpor press ure with th e above l i ne arexpre ssio ns fo r hads/hvap. The t h i r d s e t combines t h e modif ied equation f o rRojas ' equi l ibr ium da ta wi th the l i ne ar express ions fo r hads/hvap. Theseth r e e c o r r e l a t i on s e t s were inc lude d t o a l low c om pa ri son o f t he r e l a t iv es u c ce s s of t h e d i f f e r e n t c o r r e l a t i o n s i n p r e d i c t i n g ex p er im e nt a l r e s u l t s .


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SECTION 4.0COMPARISON OF TBe MODEL ldI!lX EXPF:RZMgNTAL DATA

4.1 DATA SOURCES FOR SINGLe-BLOW BDSQBPTION BND DESORPTIOBIn terms of both analytical and experimental work, it is most convenient todeal with single-blow adsorption or desorption; i . e . , the case where thedesiccant bed is initially at some uniform s t a t e of temperature and moisturecontent, and air of constant temperature, humidity, and flow rate is passedthrough the bed. If an analytical model can predict the outlet conditions ofprocess air during single-blow experiments, then the same model can be used topredict the periodic performance of desiccant beds in cooling systems. Thisreduces the need for complex and costly experiments on rotary desiccant wheelsor complete cooling system prototypes.Simulation of the single-blow case is a straightforward application of thepseudo-s eady-state calculation procedure described in Sec. 3.0. The physicalproperties and dimensions of the desiccant bed are specified, along with theinitial conditions of the bed and the inlet conditions of the air. Outletconditions of the air at each point in time are simply the final conditions ofeach air chunk after being exposed to the last bed section.In this section, predictions of the pseudo-steady-state model are compared toexperimental data for single-blow adsorption and desorption cases. Three datasources are used to provide a thorough assessment of the validity of the modeland the supporting information on silica gel properties and transport coeffi-cients. The first data source is the Desiccant Test Laboratory at SERI . Thesecond is Pesaran's master's thesis on "Air Dehumidification in Packed SilicaGel Beds" / 3 6 ] . The third data source is Koh's doctoral dissertation on theregeneration of silica gel used for grain drying [ 2 4 ] . For each of these datasources, predictions using each of the three sets of property correlationsdescribed in Secs. 3.6 and 3.7 are compared.

4.2 COMPARISON WITH SEBI DATAThe SERI desiccant test laboratory was designed to t e s t the performance ofdesiccants under operating conditions that would exist in desiccant coolingsystems 1481. The data reported here were taken for a thin, packed bed ofDavison Grade PA-40 regular density silica gel, which was held between twometal screens. The materials used, the dinensions of the bed, the flow r a t e s ,and the inlet air conditions were all chosen to be similar to those found inthe AiResearch prototype cooling system. The physical properties of thesilica gel and the dimensions of the bed are given in Table 4-1.


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sea (al TR-1330Table 4-1. Physical Properties and Dimensionsfo r SEEI Experiments

De sic can t type: Davison Grade PA-40 S i l i c aG e1

P a r t i c l e d ia me te r: 0.00193 m (8-10 mesh)Void f rac t ion :Bulk de ns it y: 850 kg/m3*Su r face a rea : 2 31335 m / mSp e c i f i c h ea t : 9 21 J / k g OC*Bed th ic kn es s: 0.035 mFace ar ea: 0.412 m2+From manufactu rer 's data .*Measured a f t e r s e t t l i n g . Note t h a t t h i s v a l u ei s d i f f e r en t f rom the nominal one repor te d i nthe manufac tu rer ' s da ta .

Data from only two experim ental runs , one ads orp t io n run and one des or pt i onrun , a re repor ted here . (See Ref . [48] fo r a complete re po rt on the SERIdesiccant tes t laboratory and a complete comparison of measured and predictedre su l t s . ) For the adsor p t ion run, da ta a re compared wi th p re d ic t io ns us ingeach o f t h e t h re e sets of s i l i c a g e l p r o pe r ty c o r r e l a t i o n s d e s cr i be d i nSecs. 3.6 and 3 .7. For the des orp t io n run, d at a a r e compared with pre di ct io nsusi ng t he Bullock and Threlkeld c or re la t i on s with Lewis numbers of 3 , 6 , an d9. Data ar e a l s o compared wi th p re d ic t io ns us ing the o the r two co rr e l a t io ns(ICORR = 2, 3) and a Lewis number of 9.Table 4-2 pr es en ts a summary of th e parame ter s fo r th es e comparisons, andgr ap hi ca l r e s u l t s a r e shown i n Figs. 4-1 through 4-8. ICORR in di ca te s whichc o r r e l a t i o n s a r e us ed i n t h e s i m u la t io n . ICORR = 1 i s fo r the Bu l lock andThre lkel d equ i l i bri um co rr el at io n with the Close and Banks polynomial equa-t i o n s fo r h d s/ hv a . ICORR = 2 i s for the Close and Banks equi l ibrium cor-r e l a t i o n wit% l i ne & equa t ion s fo r hads/hvap. ICORR = 3 i s f o r t h e c o r r e l a -t i on adap ted f rom the Ro as d a t a w i t h l i n ea r eq u a t i o n s fo r h ad s/ hv ap .Data p res en t ed h e re a r e t y p i ca l of a l l t h e d a t a o b ta i n ed i n t h e SERI d es i ccan tt e s t l a b o r a t o r y . Data o b t a i n ed d ur i n g t h e f i r s t o ne o r two m i nu t es a reunre l iab le ,however , because of the response ch ar ac te r i s t i c s o f the op t i ca l dewp o i n t h yg ro me te rs . Af t e r t h i s i n i t i a l t r a n s i en t , t h e d a t a fol l o w smooth p a t hsw i th v er y l i t t l e s c a t t e r . The r e l a t i o n s h i p betwee n d a t a and p r e d i c t i o n s i sa l s o t y p i c a l i n t h a t a gr ee me nt i s c l o s e r f o r a d s o r p t i on t h a n f o r d e s o rp t i on .Fi gu re s 4-1 through 4-3 show pre d ic t io ns f o r th e adsorp t ion run us ing t het h r e e sets o f p ro p e r t y co r r e l a t i o n s . T hese f i g u r e s d em on s t ra t e t h a t t h e re i sl i t t l e d i f f e r e nc e b etween t h e se c o r r e l a t i o n s i n p r e d i c t in g o u t l e t a i r condi-t i o n s d u r i n g ad s o rp t i o n . An e f f e c t i v e Lewis number of 3.0 was used i n


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Table 4-2. Summary of Data Buns and Computer Predictions for SePI Experiments on a 3. %em &daPrevious Adsorption Simulation Parametersor Regeneration Conditions

DataRun T Ga X b ICORR Le( kg/m2s) $ 1 ( k g ykg )

a~at, = 83.4 kPa. Two-second time step and 10 bed sections used in simulations.b~alc ulat ed from the previous adsorption or regeneration conditions using the indicatede q u i l i b r t ~ orrelations.

each s imu lat ion and provided a b e t t e r f i t t o t h e d a t a than 2.5 o r 4.0. How-e v e r , t h e s e changes i n Lewis number produced only s m a l l changes i n th e pre-d ic ted b reak through curve.To ob tai n reaso nable agreement between pre dic ted and measured o u t l e t c o n d i t i o nd u r ing d eso r p t io n , t h e e f f e c t i v e L ew i s number had t o be i n c r eased s ig n i -f i c a n t l y . F i g u r e s 4-4 t h r u 4-6 compare d a ta t o p r ed i c t i o n s u s ing Lewis num-bers of 3, 6 , and 9 w it h th e f i r s t c o r r e l a t i o n s e t . ie = 9 p r ov ides t h e b e s tf i t to the data , a l though t h i s f i t i s not as good as tha t ob ta ined fo r adsorp-t ion . F igures 4-7 and 4-8 show p r ed i c t i o n u s in g the second and third correla-t i o n se ts . D i f feren ces between the p red ic t i ons us ing the three c o r r e l a t i o n sare l a r g e r t ha n i n t h e adsorption case. However, the d i f f e r e n c e s a r e n o tmajor. These r e s u l t s a r e d i sc us se d i n g re a t e r d e t a i l i n Sec. 4.5. .

Pesaran [36] conducted s ingle-blow adso rpt i on exper iments on regu lar de nsi tys i l i c a gel i n packed beds. He performed exper iments with two p a r t i c le s iz e san d three bed th i ck n es ses . H i s range of inlet air co n d i t i o n s i s somewhatl im i t e d , w i th i n l e t t emp er a tu res n ea r room temperature and r e l a t i v e l y lowi n l e t h u mid i t i e s b e tw een 0.003 and 0.011 kg/kg. Pred ic t ions us ing the pseudo-st ea dy -st at e model a r e compared wit h dat a from seven of Pesaran ' s runs i nFigs. 4-9 thru 4-17. Most of these runs were performed us ing t he Bullock andT h re l ke l d c o r r e l a t i o n b e fo r e t h e l i m i t a t i o n of tha t c o r r e l a t i o n was d i s -covered. Pred ic t i ons us ing the o t h e r two cor re la t ions a r e g iv en f o r Pesa r an ' sRun 13. A ga in , w i th t h ese o p e r a t i n g co n d i t i o n s , t h e d i f f e r en t co r r e l a t i o n sg i v e v e r y s i m i l a r p r e d i c t i o n s .The p h y s i c a l p r o p e r t i e s o f the silica gel and the dimensions o f the bed asr epor ted by Pesaran are summarized i n Table 4-3. A summary of the parametersof each run i s given i n Table 4-4.


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Table 4-3. Physical Roperties and Dimensionsfor Pesaran Experiaeats [ 3 6 ]Sp e c i f i c heat: 921 J/kg O CVoid f ract ion: 0 . 4Bulk dens i ty : 745 kg/m3Face area: 0.01327 m

Tyler mesh: 3-8 6-12Surface area : 2 3919 m /m 2 31690 rn /mParticle diameter: 0.0039 m 0.0021 m

Table 4-4. w r y f Data Pune and Predictions for Pesaran ~x~erioents--Data Gel . Be d 'in =o Xo Ga SimulationParametersRun Yesh Depth ( k g / b ) ( O C ) (kdkg) ( k 8 h 2 S ) ICORR k8 3-8 0.070 24.3 0.00873 2 0.028 0.408 1 3

Patm = 101235 Pa. Two-second time step and LO b e d sections used in simulations.

Re su lt s show r e l a t i v e l y good agreement between data and pred ic t ions , a l thoughagreement i s not as c l o s e a s with the SERI d a ta . A d i sc r epan cy th a t r ep ea t si t s e l f i s t h a t t h e second pred ic ted process l i ne on the psychrometric c ha r t ,the one t h a t is s i m i l a r t o a constant en thalpy line, r u n s p a r a l l e l t o t he databut i s d i sp l aced to a lower humidity i n s e v e ra l runs. T h is i n d i c a t e s t h e pos-s i b i l i t y t h a t r e po rt ed i n l e t c o n di t io n s f o r these runs are inaccura te , becauseth e psychrometr ic p rocess l i n e must terminate a t t h e i n l e t a i r s t a t e once t hebed i s s a t u r a t e d .


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To s tud y th e p o t e n t i a l o f u s in g so l a r ene rg y to r eg en e r a t e s i l i c a g e l u sed f o rg ra in drying , Koh performed single-blow de sor pti on experim ents on packedbeds. The packed beds Koh stu di ed were th ic k compared t o tho se used i n desi c-can t co o l ing , b ut h i s r eg en e r a t i o n t emp era tu res a r e i n a rang e of i n t e r e s t .The phys ica l p rop er t ies o f th e s i l i c a ge l r epor ted by Koh a re g iven i nTable 4-5. A summary of the par ame ters from each run incl ude d here i s giveni n T ab le 4-6.

Table 4-5. P h y s i c a l P r o p e r t i e s for KohExperiments

P a r t i c l e d ia me te r: 0.00176 mSur face area : 2 31440 m /mVoid f ract ion: 0.4Bulk density: 740 kg/m3S p e

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Analysis of the Adsorption Process and of Desiccant Cooling Systems

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