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Simulation of intermittent thermal compression processes using adsorption technology
ARTICLE IN PRESS
Journal of the Franklin Institute 344 (2007) 725–740 www.elsevier.com/locate/jfranklin
Simulation of intermittent thermal compression processes using adsorption technology Mohamed A. Gadalla American University of Sharjah, Mechanical Engineering Department, P.O. Box 26666, Sharjah, UAE Received 17 December 2005; accepted 17 December 2005
Abstract This paper presents a dynamic model to simulate the adsorption–desorption processes associated with intermittent heat pump systems. This simulation plays an important role in sizing the adsorption systems for various types of applications in the design stage. A mathematical model that is based on the control volume approach was first developed and then discretized using the finite difference implicit scheme. The equations for the conservation of mass, momentum, and energy in the bed were derived for high-pressure and low-pressure segments, including the adsorbate (refrigerant), the adsorbent (Linde 13X), and the vessel wall. A pseudo-homogeneous model for the compression system was adopted. The numerical results that describe the adsorption–desorption history were obtained. It was found that the amount of the refrigerant recovered in the desorption process at the end of the cyclic operation is smaller than the amount adsorbed during the adsorption process. This indicates that the time for the regeneration process should be longer than the time for the adsorption process in order to raise the sieve temperature. In order to compare the simulated results with experimental data, numerical values for the heat transfer coefficients were theoretically evaluated. To assure the stability of the simulated results, the incremental time of system operation is kept equal or less than the value obtained from the minimum stability requirement. The simulated results of the temperature distribution history during system operation are in good agreement with the conducted experimental results, which led to the conclusion that the model can be used as an effective tool during the design stage and for the system development. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Adsorption–desorption processes; Thermal compression system; Adsorption technology
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E-mail address: [email protected]. 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.12.007
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1. Introduction World-wide efforts to use adsorption–desorption technology for refrigeration/heat pump systems have been intensified since the energy crises in 1973. This technology holds the potential for replacing the compressor with a low-grade energy source such as waste heat. This in turn reduces the power demands on utility grids and enhances the power factor. Yet, to this date solid–vapor refrigeration and heat pump systems are still under theoretical and laboratory testing and validation stages. Many promising developments in USA, Japan and Europe include the use of porous metal hydrides and composite adsorbents [1–6]. The adsorption and regeneration by a molecular sieve, which are important processes in commercial laboratory separation, have been widely used in chemical engineering operations. The development of the molecular sieves two decades ago contributed significantly to the advancement of adsorption and desorption processes. This development has generated a remarkable progress in many engineering applications, such as dehydration of gases, catalytics, separation of gases, and purification processes [7–9]. The most commonly studied adsorption refrigeration/heat pump systems utilize an adsorbent bed which has its temperature valid, causing the adsorbate to alternately adsorb and regenerate [10–13]. The adsorbing and desorbing fluid results in a pressure change capable of evaporating and condensing the working fluid (refrigerant) within the compression system. All physical adsorption processes from the gas phase are exothermic. In the regeneration process, the adsorbate is regenerated from the adsorbent through one of the regeneration systems, namely, thermal-swing cycle, pressure swing cycles, purge-gasstripping cycles, or displacement systems. In order for an adsorptive heat pump system to be competitive with the common CFCbased compressor-driven heat pump cycles, the system needs the heating performance to be high and utilize a waste heat source. This means that the adsorption technology has the additional advantage over vapor compression systems, in that the electric power input to the compressor is substituted by a heat-driven device, an adsorber. Many challenges of this emerging technology have been addressed by many investigators [14–18]. The performance of an adsorption heat pump system is controlled by various parameters, such that the adsorbent bed characteristics, type of adsorbate (refrigerant), system design and operating conditions. But the adsorption system performance is mainly dependent on the heat transfer rate and the mass transfer limitations during the adsorption–desorption processes. The mass transfer resistances during adsorption–desorption are not only due to the interparticle flow but also to intraparticle diffusion due to concentration gradients; molecular diffusion, Knudsen diffusion, surface diffusion, and Poiseuille flow. The interparticle transfer resistance can be reduced by using a packed bed of high permeability. While the intraparticle resistance can be reduced by using small molecular sieves with large pores [19]. Few investigators studied heat transfer characteristics associated with adsorption technology [20–24]. Balakrishnan and Pei [20] developed correlations for the overall heat transfer coefficient. These correlations are applicable to a particular bed under which they were developed. They also investigated the conduction mode of heat transfer in packed bed subjected to flowing gases. They assumed that the bed wall surface heat transfer as well as the radial conduction to be small and thus negligible. Yamamoto in his mathematical model [21] assumed that the effective thermal conductivity under a steady-state condition and infinite cylinder length depends mainly on
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the thermal conductivity of the solid, fluid, and the void fraction. The author neglected also the effect of the convection mode on the overall heat transfer coefficient. But Dixon and Cresswell [19] found that the effective heat transfer parameters used in pseudohomogeneous models were quite different in steady state as compared to transient conditions. In previous works, it has been generally assumed that the bed permeability is high and the pressure distribution across the adsorber is uniform [25–31]. This in turn justifies neglecting the bed resistance to gas flow. Recently, Chahbani et al. [22] presented a study of the impacts of mass and heat transfers on the performance of thermal wave regenerative adsorption heat pumps. They concluded that the effect of using slow diffusing refrigerants would not be significant and could be easily enhanced by increasing the cycle time. From the previous review, one concludes that the heat transfer characteristics in porous molecular sieves are too complicated. It depends mainly on the developed mathematical model, boundary conditions, initial conditions, and the assumptions used to simplify the energy equation. This paper presents a dynamic simulation model that incorporates the heat transfer characteristics of the refrigerant gas flowing in a zeolite porous molecular sieve to predict temperature distribution and refrigerant adsorbed and regenerated through the intermittent compression system. This study can be used as a complementary tool in designing and developing a zeolite-refrigerant adsorption–desorption intermittent heat pump for the simultaneous cooling and heating applications. 2. Adsorption heat pump system The overall system consisting of a blower, a heating duct, a thermal compression vessel, a water heat exchanger, a Refrigerant 12 cylinder, two one-way valves, a weighing scale, pressure gages, and pressure transducers, is shown in Fig. 1. The setup simulates the actual intermittent refrigeration/heat pump adsorption system. As seen in Fig. 2, the refrigerant passes through the intermittent compression vessel filled with a Linde 13X molecular sieve (adsorbent). This vessel is equipped with 12 thermocouple wells to monitor temperatures at different locations. After the refrigerant (adsorbate) leaves the compression unit, the refrigerant passes through a water heat exchanger. The flow direction is indicated by the arrows on the intermittent thermal compression system as shown in Fig. 2. 3. Compression system modeling The modeling of the adsorption–desorption processes involves energy balances for the heat transfer fluid (HTF), the tube and the adsorbed bed as well as mass transfer kinetic equations. Fig. 3 shows the equilibrium isobar adsorption vessel. This vessel was designed and constructed to investigate the adsorption properties of the refrigerant on the molecular sieves. The controlling of the adsorbate pressure was achieved by a constant oil temperature bath, while an electric oven was used to subject the sieve to the desired temperature. Fig. 2 shows a schematic diagram of the intermittent thermal compression system. The vessel has an opening, closed by a brass plug, used to fill the system with Linde 13X molecular sieve. For thermocouple wells, 12 copper nipples were inserted perpendicularly to the axis of the compression system. A 5-cm diameter, 96-cm long copper pipe concentric
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Fig. 1. The overall adsorption compression system.
Fig. 2. Details of the intermittent compression system.
with the 10-cm pipe passes through the center and is hard soldered to the flanges (as illustrated in Fig. 2). The overall system consists of many components as indicated above. The main aim of modeling is to predict the system dynamic behavior during adsorption and regeneration. Therefore, a two-dimensional transient model was developed to predict the temperature history at different locations in the compression vessel. The modeling of the adsorption–desorption is based on mass and momentum balance of the adsorbate and energy balance equations within the adsorbate stream, within the solid material, and within the vessel wall. An intuitive understanding of the model is required to develop the assumptions for the derivation. A two-dimensional transient model was developed to
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Fig. 3. Boundaries and differential elemental volume for mathematical model.
predict the temperature history at different locations in the compression vessel. The following assumptions were made in the analysis:
The fractional void volume e is independent of time and position. Diffusion is negligible in the system. The gas is free to move in the external void space, while the solid (molecular sieve) is restrained from moving. The gas velocity in the radial direction is negligible. The sieve is homogenous and has the same properties everywhere in the system. The solid (sieve) density is independent of time and position. Pressure gradient exists in the z-direction only. Constant gas viscosity over the range of interest. The gas thermal conductivity is independent of time and position. There is no radiation effect between the gas and solid. The heat transfer area per unit volume of sieve is constant. The gas specific heat is independent of temperature and therefore of time and position. The solid specific heat and thermal conductivity are independent of time and position. Heat losses are negligible. The thermal resistance of the pressure vessel is negligible. The heat of adsorption per kg of refrigerant is constant.
3.1. Mass balance of the adsorbate The equation of continuity for the Refrigerant 12 inside the porous molecular sieve (Linde 13X) about a differential elemental volume (see Fig. 3) of the molecular sieve was developed. The mass flow difference between the amount of refrigerant entering and leaving the control volume plus the amount adsorbed are equal to mass of refrigerant stored in the elemental volume. To sum up the input and output terms during the adsorption process: qrg qðUrg Þ ð1 eÞ qm qðUrg Þ þ þ rs ¼ 0, e qt qt qz qz
(1)
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where m is the mass ratio of the adsorbate captured on the surfaces of the molecular sieve to the mass of the molecular sieve (kg/kg). In addition, rg and rs are the adsorbate (Refrigerant12) and solid densities, respectively, U is the gas average velocity in the zdirection, e is the porosity or the void fraction of the sieve, t is the time, and r and z are the radial and axial coordinates. 3.2. Momentum balance within the adsorbate The momentum balance for the adsorbate inside the porous molecular sieve about the differential elemental volume 2prDrDz yields
qðrg UUÞ qP qðrg UÞ bz U ¼ , qz qz qt
(2)
where bz is the pressure drop coefficient due to the friction between the gas and the solid in the z-direction. For porosity less than 0.8, the pressure drop can be described by the Ergun equation [32] Syamlal and Gidaspow [33]: bz ¼
150ð1 eÞ2 mg eðd p fs Þ2
.
(3)
In addition, P is the pressure inside the compression system, dp is the particle diameter, fs is the particle shape factor, and mg is the gas viscosity. 3.3. Energy balance within the refrigeration stream An energy balance within the refrigerant for an elemental volume, 2prDrDz, is taking place. The sum of the energy input and output yields: rkg
qðrg UT g Þ harðT g T s Þ qðrg T g Þ q2 T g q ðrqT g Þ rcpg ¼ rcpg þ kg qr qr e qz2 qz qt
(4)
where kg is the gas conductivity, cpg is the gas specific heat, h is the solid to gas heat transfer coefficient, a is the heat area transfer area per unit volume, and Ts is the solid temperature. 3.4. Energy balance within the solid An energy balance within the solid adsorbent element yields: ks r
ð1 eÞ q2 T s ð1 eÞ q ðrqT s Þ ahrðT s T g Þ ð1 eÞ qm þ DH fg rs r þ ks e e qr qr e qt e qz2 ð1 eÞ qT s ¼ rrs cps . e qt
ð5Þ
Rudy E Rogers [34] found that the differential heat of adsorption, DHfg , of Refrigerant 12 on a sieve having 10 A˚ pore diameter is about 10% higher than the heat of evaporation, hfg.
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As a result of the above, Eq. (5) becomes ks r
ð1 eÞ q2 T s ð1 eÞ q ðrqT s Þ ahrðT s T g Þ ð1 eÞ qm þ Chfg rs r þ ks e e qr qr e qt e qz2 ð1 eÞ qT s ¼ rrs cps , e qt
ð6Þ
where DHfg is the differential heat of evaporation (heat of adsorption during the adsorption process) and equals Chfg, hfg is the enthalpy of evaporation, and C is a constant; ks and cps are the thermal conductivity and specific heat of the solid, respectively. 3.5. Energy balance within the internal wall of the vessel Due to the considerable length of the experimental apparatus, the amount of energy absorbed by the wall with respect to the amount generated is significant. According to Oscar Meyer [35] by considering a thin copper wall vessel and neglecting the radial temperature gradient, the outside and inside temperatures of the wall become equal. The energy balance within the internal results: q2 T p qT p (7) ¼ cpp rp ðR2o R2i Þ qz2 qt where Ri and Ro are the inside and outside radii of the internal pipe, respectively; kp, cpp, rp, Tp are the thermal conductivity, specific heat, mass density, and temperature of the pipe, respectively; hf is the fluid heat transfer coefficient; and Tf is the fluid (air) temperature. 2Ri hf ðT f T p Þ þ kp ðR2o R2i Þ
4. Simplification and additional assumptions The differential equations governing the process have been derived. The Eqs. (1), (2), (4), (6), and (7) were general enough for application to any model. Additional assumptions can be made in order to apply the mathematical model under study. Eqs. (4) and (6) were simplified with the assumption that temperature equilibrium between the gas and the solid exists, or a pseudo-homogenous model for the compression system could be considered. In addition, it was found that the conduction term, kg, in the gas phase energy balance does not have a significant effect and the gas thermal capacity term is small as compared to the solid phase, and the combined equation becomes qðrg UTÞ ð1 eÞ q2 T ð1 eÞ q2 T ð1 eÞ 1 qT cpg þ k þ ks s 2 2 e qz e qr e r qr qz qðrg TÞ ð1 eÞ qm ð1 eÞ qT ¼ cpg . ð8Þ þ rs cps þ Chfg rs e qt e qt qt In addition, the pressure drop inside the compression system and the refrigerant velocity was assumed to be negligible. Therefore, Eq. (2) can be dropped and Eqs. (1) and (8) take the following forms: ks
qrg ð1 eÞ qm ¼ 0, þ rs e qt qt
(9)
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ks
q2 T q2 T ks qT qm qT þ Chfg rs ¼ rs cps þ k þ s qz2 qr2 qt qt r qr
(10)
These developments have now basically defined the model of the system and have yielded two Eqs. (7) and (10), the basic equations for the combined heat transfer and adsorption in a molecular sieve. 5. Mathematical model Eqs. (7) and (10) represent the basic equations of the model in addition to the adsorption rate ðqm=qtÞ, and the boundary and initial conditions subjected to the compression system. The initial and boundary conditions in the adsorption case for this model are: 1:
Tðt; r; zÞ;
Tð0; Ro prpR2 ; zÞ ¼ T i ,
@ r ¼ R2 ;
qT ¼ 0, qr
@ r ¼ Ro ;
T ¼ T p,
@ z ¼ 0;
qT ¼ 0, qz
@ z ¼ L;
qT ¼ 0, qz
2: T p ðt; zÞ;
(11)
T p ð0; 0pzpLÞ ¼ T pi ,
@ z ¼ 0;
qT p ¼ 0, qz
@ z ¼ L;
qT p ¼ 0, qz
3: @ r ¼ Ri ;
_ pf DT f ¼ 2pRi hf DzðT f T p Þ, mc
(12) (13)
4: T f ðzÞ, @ z ¼ 0;
T f ð0Þ ¼ T fi ,
@ z ¼ L;
_ pf DT f Þ=ð2pRi hf LÞ þ T p , T f ðLÞ ¼ ðmc
(14)
where Ti, Tpi, and Tfi are the initial temperatures of the porous sieve, concentric pipe, and _ is the air mass the fluid (air), respectively; R2 is the radius of the compression unit; and m flow rate. The basic equations, (7), (10), and (13), with the initial and boundary conditions constitute the nonlinear differential equations that form a simultaneous nonlinear system to be solved.
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Fig. 4. Discretized system schematic.
5.1. Model discretization The basic governing equations with the boundary conditions were discretized using the finite difference scheme (fully implicit formulation) and the meshing grids as shown in Fig. 4. The finite difference formulation starts with a subdivision of the compression unit into ordered nodal points. Basically, the compression system is divided into 22 grids along the axial direction (i) and four grids in the radial direction (j) (see Fig. 4). The discretized equations at different nodes are based on the principle of energy conservation and fully presented in detail in the Appendix. 5.2. Stability criteria To assure stability of the computational technique, the time increment (Dt) were kept equal to or less than the value obtained from the maximum allowable time increment needed for stability requirement: i .h X Dtpci jð1=Rij Þ , (15) where Dt is the time increment (h), ci is the thermal capacity ðri cpi DV i Þ, Rij is the thermal resistance between i and adjoining nodes, i is the node of interest, j is the adjoined nodes with i, and DV is the volume element. The obtained value for Dt is 0.00075 h. A 0.0005 h incremental time has been chosen for stability purposes. 6. Results and discussion The conservation equations of the model are solved using the finite difference implicit scheme. The aim of the model is to predict the temperature distribution history and the amount of refrigerant stored inside the thermal compression system. Figs. 5–7 show the temperature distribution history at specific locations inside the compression system. Fig. 5 depicts the temperature distribution at an axial distance, z, equals 10.16 cm and a radial distance r, equals 3.175 cm, while Figs. 6 and 7 present the temperature history at axial
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46 Simulation Experimental
TEMPERATURE (°C)
40
34
28
22
16 0
0.05
0.1 0.15 TIME (HOUR)
0.2
0.25
Fig. 5. Experimental and simulation results for temperature history at r ¼ 3:175 cm and z ¼ 10:16 cm.
46 Simulation Experimental
TEMPERATURE (°C)
40
34
28
22
16 0
0.05
0.1
0.15
0.2
0.25
TIME (HOUR) Fig. 6. Experimental and simulation results for temperature history at r ¼ 3:81 cm and z ¼ 30:48 cm.
distances 30.48 and 50.8 cm and radial distances, r, equal 3.81 and 4.45 cm, respectively. In these figures the simulation results are compared with the experimental runs. The predicted temperature evolution is very close to the experimental measurements. This means that the deviations of predicted results from the experimental findings are very small and within the maximum relative experimental error (0.8%). Four reasons might have caused these deviations: 1. The model might not replicate the physical process due to the assumptions used in developing the physical model. This can be indicated as follows: (a) The pressure is assumed to be constant along the compression system.
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TEMPERATURE (°C)
46
40
34
28
22
16 0
0.05
0.1
0.15
0.2
0.25
TIME (HOUR) Fig. 7. Experimental and simulation results for temperature history at r ¼ 4:45 cm and z ¼ 50:8 cm.
(b) The physical properties of the refrigerant, solid sieve, air, and pipe are assumed to be constants. (c) The amount of energy absorbed by the intermittent compression system, which weighs about 11.5 kg, is significant. (d) Radial temperature gradients are assumed negligible since the compression system wall is thin. 2. The heat convection inside the thermocouple wells has some effects on the thermocouple readings. 3. The accuracy of the heat transfer coefficient and the heat adsorption might not be satisfactory. 4. The truncation errors of the numerical solution might be excessive. Fig. 8 shows the thermal compression system behavior for a cyclic time of 15 min for the adsorption–desorption processes. One can see that the amount of refrigerant recovered in the regeneration process, at the end of the cycle, is smaller than the amount adsorbed in the adsorption process. This indicates that the time for the regeneration process should be longer than the time for the adsorption process in order to raise the sieve temperature. In general, if a designer assumes a minimum required air-conditioning capacity of 14 ton refrigeration, a refrigerant 12 flow rate of 27.2 kg/h with 116.3 kJ/kg cooling energy available is required to establish the new thermal compression hest pump system. If four compression units are connected in parallel with the conventional compressor and cycled every 15 min (two units are in the adsorption mode while the other two are in the regeneration mode), 6.8 kg of refrigerant are required every cycle (15 min), or 1.7 kg refrigerant per unit per cycle. If 75% of the refrigerant adsorbed can be recovered, 2.27 kg of refrigerant are required for each thermal compression unit. The Linde 13X molecular sieve weight required for each unit with an adsorption of 30% to adsorb 2.27 kg of refrigerant is about 7.5 kg. This compression system can provide either heating or cooling. When the refrigerant leaves the compression unit and enters the condenser where energy is
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Refrigerant 12 in the compression system (g)
325 300 275 250 225 200 175 150 125 100 75 50 25 0 0.00
0.25
0.50 Time (hour)
0.75
1.00
Fig. 8. Thermal compression system behavior.
removed and the vapor liquefies, then the system at this stage is in the heat pump mode. If the refrigerant liquid leaves the condenser and passes through the throttling valve to the evaporator, then the refrigerant absorbs heat and provides cooling at this location; thus the heat pump is in the regular air-conditioning and refrigeration mode. Since this thermal compression system uses waste heat or low-grade thermal energy source is technically applicable during the adsorption and regeneration processes. 7. Conclusions Solid–vapor adsorption technology still has considerable scope for investigation and improvement to compete with conventional absorption and vapor compression technologies. A dynamic simulation model for adsorption–desorption processes associated with an intermittent thermal compression adsorption system filled with porous molecular sieve was developed. Based on the simulated results, the following conclusions can be drawn:
The dynamic model can be used as an effective complementary tool during the design stage of intermittent heat pump/refrigeration systems. The mass of the compression system must be minimized to shorten the heating and cooling cyclic time and enhance the system performance. The presence of mass and heat transfer limitations can affect significantly the performance of the adsorption heat pump systems. This would be accomplished by reducing the difference between the inner and outer diameter of the thermal compression unit. The temperature distribution history shows that there is a good agreement between the simulated and experimental results.
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Further R&D work is recommended to increase the chances of the adsorption technology to compete with absorption and conventional vapor compression technologies.
Appendix A. Details of model discretization A.1. For the interior nodes in the molecular sieve The finite difference form of the time-dependent energy balance on mesh point (i,j) located in the porous media is T kþ1 ði; jÞ ¼ fK s 2p½Ro þ ð2j 3ÞD=2Dz½T ks ði; j 1Þ T ks ði; jÞDt=D s K s 2p½Ro þ ð2j 1ÞD=2Dz½T ks ði; jÞ T ks ði; j þ 1ÞDt=D þ K s 2p½Ro þ ð2j 3ÞD=2D½T ks ði 1; jÞ T ks ði; jÞDt=Dz K s 2p½Ro þ ð2j 3ÞD=2D½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2D DzDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2DDz þ T ks ði; jÞ
ðA:1Þ
where D is the increment size along the radial direction, Dz is the increment size in the axial direction, Dt is the time increment, ADSORB is the adsorption rate per unit volume of the molecular sieve, Hfg is the heat of adsorption and the exponent k denotes temperatures at the time level. This equation is valid only from i ¼ 2, 22 and j ¼ 2, 4 (see Fig. 4). A.2. For the porous media of all elements located on the first row and the internal pipe The finite difference formulation for these grids can be presented in the following form: T kþ1 ði; 1Þ ¼ ½hf 2pRi Dz½T k f ði; 1Þ T ks ði; 1ÞDt þ kp 2pRi Dr½T ks ði 1; 1Þ s T ks ði; 1ÞDt=Dz kp 2pRi Dr½T ks ði; 1Þ T ks ði þ 1; 1ÞDt=Dz 1 eÞks 2p½Ro þ D=2Dz½T ks ði; 1Þ T ks ði; 2ÞDt=D þ ð1 eÞks 2p½Ro D=2½T ks ði 1; 1Þ T ks ði; 1ÞDt=Dz ð1 eÞks 2p½Ro D=2½T ks ði; 1Þ T ks ði þ 1; 1ÞDt=Dz þ H fg ADSORBði; jÞ½2p½Ro D=2DzDt=½rs cps 2p½Ro D=2Dz þ cpp rp 2p½Ri DzDr þ T ks ði; 1Þ.
ðA:2Þ
The previous equation is valid for i ¼ 2, 22 and j ¼ 5 only. A.3. For the fluid (air) flowing inside the concentric pipe The following equation represents the energy balance between the fluid flowing through the concentric pipe and the conduit itself: _ cpf ½T k f ðiÞ T k f ði þ 1Þ ¼ hf 2pRi Dz½T k f ðiÞ T k pðiÞ. m This equation is valid only for i ¼ 1, 23 only.
(A.3)
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A.4. For the upper boundary of the molecular sieve The finite difference formulation for the top boundary of the porous media can be cast as follows: T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2Dz½T ks ði; j 1Þ T ks ði; jÞDt=D s þ ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T ks ði 1; jÞ T ks ði; jÞDt=Dz ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2ðD=2ÞDzDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2ðD=2ÞDz þ T ks ði; jÞ.
ðA:4Þ
A.5. For the top right corner of the porous media only The finite difference equation for the upper right corner can be formulated as T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D s ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T k ði 1; jÞ T ks ði; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2Þ þ T ks ði; jÞ.
ðA:5Þ
The previous equation is valid only for i ¼ 23 and j ¼ 5. A.6. For the upper left corner of the porous media only The discretized equation for this corner can be presented as T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D s ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2Þ þ T ks ði; jÞ.
ðA:6Þ
This equation is valid for i ¼ 1 and j ¼ 23 only. A.7. For the right side of the porous media only The equation for this portion is for i ¼ 23 and j ¼ 2, 4 and can be cast in the following form: T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D s ks 2p½Ro þ ð2j 1ÞD=2ðDz=2Þ½T ks ði; jÞ T k ði; j þ 1ÞDt=D þ ks 2p½Ro þ ð2j 3ÞD=2D½T ks ði 1; jÞ T ks ði; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2DðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2DðDz=2Þ þ T ks ði; jÞ.
ðA:7Þ
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A.8. For the left side of the porous media only The finite difference formulation for that side (i ¼ 1, j ¼ 2, 4) can be described as ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D T kþ1 s ks 2p½Ro þ ð2j 1ÞD=2ðDz=2Þ½T ks ði; jÞ T ks ði; j þ 1ÞDt=D ks 2p½Ro þ ð2j 3ÞD=2D½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2DðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2DðDz=2Þ þ T ks ði; jÞ.
ðA:8Þ
References [1] D.J. Miles, S.V. Shelton, Design and testing of a solid-sorption heat-pump system, Appl. Thermal Eng. 16 (1996) 389–394. [2] S. Suda, Experimental evaluation of heat pumps performance in connection with metal hybrid properties, J. Less Common Metals 104 (1984) 211–222. [3] A. Yanoma, M. Yoneta, T. Nitta, A. Okuda, The performance of a large scale hybrid heat pump, JSME Int. J. ser. II 31 (1988) 741–747. [4] M. Nagel, Y. Komasaki, M. Uchida, S. Suda, Y. Matsubara, Operating characteristics of a metal hybrid heat pump for generating cool air, J. Less Common Metals 104 (1984) 307–318. [5] W. Charters, S. Brey, L. Aye, Metal hydride heat pumps, refrigeration, climate control, and energy conversion, Proceedings of Meetings of IIR Commissions E2, E1, B1, B2, Melbourne, 1996. [6] N.C. Srivastava, I.W. Eames, A review of adsorbents and adsorbates in soild–vapour adsorption heat pump systems, Appl. Thermal Eng. 18 (1998) 707–714. [7] M. Buzanowski, R.T. Yang, Approximation for intraparticle diffusion rates in cyclic adsorption, Chem. Eng. Sci. 46 (1991) 2589–2598. [8] M.H. Chahbani, D. Tondeur, Mass transfer kinetics in pressure swing adsorption, Separation Purification Technol. 20 (2000) 185–196. [9] M. Pons, D. Laurent, F. Meunier, Experimental temperature fronts for adsorptive heat applications, Appl. Thermal Eng. 16 (1996) 395–404. [10] F. Meunier, Adsorption Heat pumps Technology, in: Possibilities and Limits ISHPC’99, Proceedings of Sorption Heat Pump, 1999. [11] D.M. Ruthven, Principles of Adsorption and Adsorption Processes, Wiley, New York, 1984. [12] R.Z. Wang, J.Y. Wu, Y.X. Xu, W. Wang, Performance researches and improvements on heat regenerative adsorption refrigerator and heat pump, Energy Conversion Manage. 42 (2002) 233–249. [13] M. Pons, F. Poyelle, Adsorption machines with advanced cycles for heat pumping or cooling applications, Int. J. Refrigeration 22 (1999) 27–37. [14] F. Meunier, Solid sorption cycles for cooling and heat pumping applications, Appl. Thermal Eng. 18 (1998) 715–729. [15] G. Cacciola, G. Restuccia, Progress on adsorption heat pumps, Heat Recovery Systems & CHP 14 (1994) 409–420. [16] R.Z. Wang, J.P. Jia, Y.H. Zhu, Y. Teng, J.C. Wu, Q.B. Wang, Study on a new solid adsorption refrigeration pair: active carbon fiber-methanol, ASME J. Solar Energy Eng. 119 (1994) 214–218. [17] R.Z. Wang, Y.X. Xu, J.Y. Wu, M. Li, H.B. Shou, Research on a combined adsorption heating and cooling system, Appl. Thermal Eng. 22 (2002) 603–617. [18] R.Z. Wang, J.Y. Wu, Y.X. Xu, W. Wang, Performance researches and improvements on heat regenerative adsorption refrigerator and heat pump, Energy Conversion Manage. 42 (2001) 233–249. [19] A.G. Dixon, D.L. Cresswell, Effective heats transfer parameters for transient packed-bed models, AICHE J. 5 (1986) 809–819. [20] A.R. Balakrishnan, D.C.T. Pei, Heat transfer in gas–solid packed bed systems: overall heat transfer rates in adiabatic beds, Ind. Eng. Chem. Process Des. Dev. 18 (1979) 47–50.
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[21] Y. Yamamoto, Adsorption and heat transfer characteristics in Krypton-85 storage cylinder filled with adsorbent, J. Nucl. Sci. Technol. 18 (1981) 856–862. [22] M.H. Chahbani, J. Labidi, J. Paris, Modeling of adsorption heat pumps with heat regeneration, Appl. Thermal Eng. 24 (2004) 431–447. [23] W. Zheng, W.M. Worek, G. Nowakowski, Performance of portioned-bed sorption heat pump systems, J. Enhanced Heat Transfer 2 (1995) 283–293. [24] A.G. Dixon, A.G.D.L. Cresswell, Theoretical prediction of effective heat transfer parameters in packed beds, AICHE J. 4 (1979) 663–676. [25] L.M. Sun, N.Y. Feng, N.Y.M. Pons, Numerical investigation of adsorptive heat pump systems with thermal wave heat regeneration under uniform-pressure conditions, Int. J. Heat Mass Transfer 40 (1997) 281–293. [26] B.K. Sward, M.D. Le Van, F. Meunier, Adsorption heat pump modeling: the thermal wave process with local equilibrium, Appl. Thermal Eng. 20 (2000) 759–780. [27] M. Pons, Analysis of the adsorption cycles with thermal regeneration based on the entropic mean temperatures, Appl. Thermal Eng. 17 (1996) 615–627. [28] Z. Taminot-Telto, R.E. Critoph, Adsorption refrigerant using monolithic carbon–ammonia pair, Int. J. Refrigeration 20 (1997) 146–155. [29] W. Zheng, W.M. Worek, Effect of design and operating parameters on the performance of two-bed sorption heat pump systems, J. Energy Resources Technol. 117 (1995) 67–74. [30] W. Zheng, W.M. Worek, G. Nowakowski, Effect of operating conditions on the performance of two-bed closed-cycle solid-sorption heat pump systems, J. Solar Energy 117 (1995) 181–186. [31] W. Zheng, W.M. Worek, G. Nowakowski, Performance of multi-bed sorption heat pump systems, Int. J. Energy Res. 20 (1996) 339–350. [32] L.E. Sisson, D.D.R. Pitts, R. Donald, Elements of Transport Phenomena, McGraw Hill, New York, 1971. [33] M. Syamlal, D. Gidaspaw, Hydrodynamics of fluidization: prediction of wall to bed heat transfer coefficients, AICHE J. 1 (1985) 127–135. [34] R.E. Rogers, Rate mechanism for adsorption on molecular sieves, Ph.D. dissertation, University of Alabama, 1968. [35] O. Meyer, Dynamics of nonisothermal fixed bed adsorption, Ph.D. dissertation, State University of New York at Buffalo, Ann Arbor, MI, Published by University Microfilms, Inc., 1966.
Journal of the Franklin Institute 344 (2007) 725–740 www.elsevier.com/locate/jfranklin
Simulation of intermittent thermal compression processes using adsorption technology Mohamed A. Gadalla American University of Sharjah, Mechanical Engineering Department, P.O. Box 26666, Sharjah, UAE Received 17 December 2005; accepted 17 December 2005
Abstract This paper presents a dynamic model to simulate the adsorption–desorption processes associated with intermittent heat pump systems. This simulation plays an important role in sizing the adsorption systems for various types of applications in the design stage. A mathematical model that is based on the control volume approach was first developed and then discretized using the finite difference implicit scheme. The equations for the conservation of mass, momentum, and energy in the bed were derived for high-pressure and low-pressure segments, including the adsorbate (refrigerant), the adsorbent (Linde 13X), and the vessel wall. A pseudo-homogeneous model for the compression system was adopted. The numerical results that describe the adsorption–desorption history were obtained. It was found that the amount of the refrigerant recovered in the desorption process at the end of the cyclic operation is smaller than the amount adsorbed during the adsorption process. This indicates that the time for the regeneration process should be longer than the time for the adsorption process in order to raise the sieve temperature. In order to compare the simulated results with experimental data, numerical values for the heat transfer coefficients were theoretically evaluated. To assure the stability of the simulated results, the incremental time of system operation is kept equal or less than the value obtained from the minimum stability requirement. The simulated results of the temperature distribution history during system operation are in good agreement with the conducted experimental results, which led to the conclusion that the model can be used as an effective tool during the design stage and for the system development. r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. Keywords: Adsorption–desorption processes; Thermal compression system; Adsorption technology
Tel.: +971 50 5144440; fax: +971 6 5152979.
E-mail address: [email protected]. 0016-0032/$30.00 r 2006 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2005.12.007
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1. Introduction World-wide efforts to use adsorption–desorption technology for refrigeration/heat pump systems have been intensified since the energy crises in 1973. This technology holds the potential for replacing the compressor with a low-grade energy source such as waste heat. This in turn reduces the power demands on utility grids and enhances the power factor. Yet, to this date solid–vapor refrigeration and heat pump systems are still under theoretical and laboratory testing and validation stages. Many promising developments in USA, Japan and Europe include the use of porous metal hydrides and composite adsorbents [1–6]. The adsorption and regeneration by a molecular sieve, which are important processes in commercial laboratory separation, have been widely used in chemical engineering operations. The development of the molecular sieves two decades ago contributed significantly to the advancement of adsorption and desorption processes. This development has generated a remarkable progress in many engineering applications, such as dehydration of gases, catalytics, separation of gases, and purification processes [7–9]. The most commonly studied adsorption refrigeration/heat pump systems utilize an adsorbent bed which has its temperature valid, causing the adsorbate to alternately adsorb and regenerate [10–13]. The adsorbing and desorbing fluid results in a pressure change capable of evaporating and condensing the working fluid (refrigerant) within the compression system. All physical adsorption processes from the gas phase are exothermic. In the regeneration process, the adsorbate is regenerated from the adsorbent through one of the regeneration systems, namely, thermal-swing cycle, pressure swing cycles, purge-gasstripping cycles, or displacement systems. In order for an adsorptive heat pump system to be competitive with the common CFCbased compressor-driven heat pump cycles, the system needs the heating performance to be high and utilize a waste heat source. This means that the adsorption technology has the additional advantage over vapor compression systems, in that the electric power input to the compressor is substituted by a heat-driven device, an adsorber. Many challenges of this emerging technology have been addressed by many investigators [14–18]. The performance of an adsorption heat pump system is controlled by various parameters, such that the adsorbent bed characteristics, type of adsorbate (refrigerant), system design and operating conditions. But the adsorption system performance is mainly dependent on the heat transfer rate and the mass transfer limitations during the adsorption–desorption processes. The mass transfer resistances during adsorption–desorption are not only due to the interparticle flow but also to intraparticle diffusion due to concentration gradients; molecular diffusion, Knudsen diffusion, surface diffusion, and Poiseuille flow. The interparticle transfer resistance can be reduced by using a packed bed of high permeability. While the intraparticle resistance can be reduced by using small molecular sieves with large pores [19]. Few investigators studied heat transfer characteristics associated with adsorption technology [20–24]. Balakrishnan and Pei [20] developed correlations for the overall heat transfer coefficient. These correlations are applicable to a particular bed under which they were developed. They also investigated the conduction mode of heat transfer in packed bed subjected to flowing gases. They assumed that the bed wall surface heat transfer as well as the radial conduction to be small and thus negligible. Yamamoto in his mathematical model [21] assumed that the effective thermal conductivity under a steady-state condition and infinite cylinder length depends mainly on
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the thermal conductivity of the solid, fluid, and the void fraction. The author neglected also the effect of the convection mode on the overall heat transfer coefficient. But Dixon and Cresswell [19] found that the effective heat transfer parameters used in pseudohomogeneous models were quite different in steady state as compared to transient conditions. In previous works, it has been generally assumed that the bed permeability is high and the pressure distribution across the adsorber is uniform [25–31]. This in turn justifies neglecting the bed resistance to gas flow. Recently, Chahbani et al. [22] presented a study of the impacts of mass and heat transfers on the performance of thermal wave regenerative adsorption heat pumps. They concluded that the effect of using slow diffusing refrigerants would not be significant and could be easily enhanced by increasing the cycle time. From the previous review, one concludes that the heat transfer characteristics in porous molecular sieves are too complicated. It depends mainly on the developed mathematical model, boundary conditions, initial conditions, and the assumptions used to simplify the energy equation. This paper presents a dynamic simulation model that incorporates the heat transfer characteristics of the refrigerant gas flowing in a zeolite porous molecular sieve to predict temperature distribution and refrigerant adsorbed and regenerated through the intermittent compression system. This study can be used as a complementary tool in designing and developing a zeolite-refrigerant adsorption–desorption intermittent heat pump for the simultaneous cooling and heating applications. 2. Adsorption heat pump system The overall system consisting of a blower, a heating duct, a thermal compression vessel, a water heat exchanger, a Refrigerant 12 cylinder, two one-way valves, a weighing scale, pressure gages, and pressure transducers, is shown in Fig. 1. The setup simulates the actual intermittent refrigeration/heat pump adsorption system. As seen in Fig. 2, the refrigerant passes through the intermittent compression vessel filled with a Linde 13X molecular sieve (adsorbent). This vessel is equipped with 12 thermocouple wells to monitor temperatures at different locations. After the refrigerant (adsorbate) leaves the compression unit, the refrigerant passes through a water heat exchanger. The flow direction is indicated by the arrows on the intermittent thermal compression system as shown in Fig. 2. 3. Compression system modeling The modeling of the adsorption–desorption processes involves energy balances for the heat transfer fluid (HTF), the tube and the adsorbed bed as well as mass transfer kinetic equations. Fig. 3 shows the equilibrium isobar adsorption vessel. This vessel was designed and constructed to investigate the adsorption properties of the refrigerant on the molecular sieves. The controlling of the adsorbate pressure was achieved by a constant oil temperature bath, while an electric oven was used to subject the sieve to the desired temperature. Fig. 2 shows a schematic diagram of the intermittent thermal compression system. The vessel has an opening, closed by a brass plug, used to fill the system with Linde 13X molecular sieve. For thermocouple wells, 12 copper nipples were inserted perpendicularly to the axis of the compression system. A 5-cm diameter, 96-cm long copper pipe concentric
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Fig. 1. The overall adsorption compression system.
Fig. 2. Details of the intermittent compression system.
with the 10-cm pipe passes through the center and is hard soldered to the flanges (as illustrated in Fig. 2). The overall system consists of many components as indicated above. The main aim of modeling is to predict the system dynamic behavior during adsorption and regeneration. Therefore, a two-dimensional transient model was developed to predict the temperature history at different locations in the compression vessel. The modeling of the adsorption–desorption is based on mass and momentum balance of the adsorbate and energy balance equations within the adsorbate stream, within the solid material, and within the vessel wall. An intuitive understanding of the model is required to develop the assumptions for the derivation. A two-dimensional transient model was developed to
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Fig. 3. Boundaries and differential elemental volume for mathematical model.
predict the temperature history at different locations in the compression vessel. The following assumptions were made in the analysis:
The fractional void volume e is independent of time and position. Diffusion is negligible in the system. The gas is free to move in the external void space, while the solid (molecular sieve) is restrained from moving. The gas velocity in the radial direction is negligible. The sieve is homogenous and has the same properties everywhere in the system. The solid (sieve) density is independent of time and position. Pressure gradient exists in the z-direction only. Constant gas viscosity over the range of interest. The gas thermal conductivity is independent of time and position. There is no radiation effect between the gas and solid. The heat transfer area per unit volume of sieve is constant. The gas specific heat is independent of temperature and therefore of time and position. The solid specific heat and thermal conductivity are independent of time and position. Heat losses are negligible. The thermal resistance of the pressure vessel is negligible. The heat of adsorption per kg of refrigerant is constant.
3.1. Mass balance of the adsorbate The equation of continuity for the Refrigerant 12 inside the porous molecular sieve (Linde 13X) about a differential elemental volume (see Fig. 3) of the molecular sieve was developed. The mass flow difference between the amount of refrigerant entering and leaving the control volume plus the amount adsorbed are equal to mass of refrigerant stored in the elemental volume. To sum up the input and output terms during the adsorption process: qrg qðUrg Þ ð1 eÞ qm qðUrg Þ þ þ rs ¼ 0, e qt qt qz qz
(1)
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where m is the mass ratio of the adsorbate captured on the surfaces of the molecular sieve to the mass of the molecular sieve (kg/kg). In addition, rg and rs are the adsorbate (Refrigerant12) and solid densities, respectively, U is the gas average velocity in the zdirection, e is the porosity or the void fraction of the sieve, t is the time, and r and z are the radial and axial coordinates. 3.2. Momentum balance within the adsorbate The momentum balance for the adsorbate inside the porous molecular sieve about the differential elemental volume 2prDrDz yields
qðrg UUÞ qP qðrg UÞ bz U ¼ , qz qz qt
(2)
where bz is the pressure drop coefficient due to the friction between the gas and the solid in the z-direction. For porosity less than 0.8, the pressure drop can be described by the Ergun equation [32] Syamlal and Gidaspow [33]: bz ¼
150ð1 eÞ2 mg eðd p fs Þ2
.
(3)
In addition, P is the pressure inside the compression system, dp is the particle diameter, fs is the particle shape factor, and mg is the gas viscosity. 3.3. Energy balance within the refrigeration stream An energy balance within the refrigerant for an elemental volume, 2prDrDz, is taking place. The sum of the energy input and output yields: rkg
qðrg UT g Þ harðT g T s Þ qðrg T g Þ q2 T g q ðrqT g Þ rcpg ¼ rcpg þ kg qr qr e qz2 qz qt
(4)
where kg is the gas conductivity, cpg is the gas specific heat, h is the solid to gas heat transfer coefficient, a is the heat area transfer area per unit volume, and Ts is the solid temperature. 3.4. Energy balance within the solid An energy balance within the solid adsorbent element yields: ks r
ð1 eÞ q2 T s ð1 eÞ q ðrqT s Þ ahrðT s T g Þ ð1 eÞ qm þ DH fg rs r þ ks e e qr qr e qt e qz2 ð1 eÞ qT s ¼ rrs cps . e qt
ð5Þ
Rudy E Rogers [34] found that the differential heat of adsorption, DHfg , of Refrigerant 12 on a sieve having 10 A˚ pore diameter is about 10% higher than the heat of evaporation, hfg.
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As a result of the above, Eq. (5) becomes ks r
ð1 eÞ q2 T s ð1 eÞ q ðrqT s Þ ahrðT s T g Þ ð1 eÞ qm þ Chfg rs r þ ks e e qr qr e qt e qz2 ð1 eÞ qT s ¼ rrs cps , e qt
ð6Þ
where DHfg is the differential heat of evaporation (heat of adsorption during the adsorption process) and equals Chfg, hfg is the enthalpy of evaporation, and C is a constant; ks and cps are the thermal conductivity and specific heat of the solid, respectively. 3.5. Energy balance within the internal wall of the vessel Due to the considerable length of the experimental apparatus, the amount of energy absorbed by the wall with respect to the amount generated is significant. According to Oscar Meyer [35] by considering a thin copper wall vessel and neglecting the radial temperature gradient, the outside and inside temperatures of the wall become equal. The energy balance within the internal results: q2 T p qT p (7) ¼ cpp rp ðR2o R2i Þ qz2 qt where Ri and Ro are the inside and outside radii of the internal pipe, respectively; kp, cpp, rp, Tp are the thermal conductivity, specific heat, mass density, and temperature of the pipe, respectively; hf is the fluid heat transfer coefficient; and Tf is the fluid (air) temperature. 2Ri hf ðT f T p Þ þ kp ðR2o R2i Þ
4. Simplification and additional assumptions The differential equations governing the process have been derived. The Eqs. (1), (2), (4), (6), and (7) were general enough for application to any model. Additional assumptions can be made in order to apply the mathematical model under study. Eqs. (4) and (6) were simplified with the assumption that temperature equilibrium between the gas and the solid exists, or a pseudo-homogenous model for the compression system could be considered. In addition, it was found that the conduction term, kg, in the gas phase energy balance does not have a significant effect and the gas thermal capacity term is small as compared to the solid phase, and the combined equation becomes qðrg UTÞ ð1 eÞ q2 T ð1 eÞ q2 T ð1 eÞ 1 qT cpg þ k þ ks s 2 2 e qz e qr e r qr qz qðrg TÞ ð1 eÞ qm ð1 eÞ qT ¼ cpg . ð8Þ þ rs cps þ Chfg rs e qt e qt qt In addition, the pressure drop inside the compression system and the refrigerant velocity was assumed to be negligible. Therefore, Eq. (2) can be dropped and Eqs. (1) and (8) take the following forms: ks
qrg ð1 eÞ qm ¼ 0, þ rs e qt qt
(9)
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ks
q2 T q2 T ks qT qm qT þ Chfg rs ¼ rs cps þ k þ s qz2 qr2 qt qt r qr
(10)
These developments have now basically defined the model of the system and have yielded two Eqs. (7) and (10), the basic equations for the combined heat transfer and adsorption in a molecular sieve. 5. Mathematical model Eqs. (7) and (10) represent the basic equations of the model in addition to the adsorption rate ðqm=qtÞ, and the boundary and initial conditions subjected to the compression system. The initial and boundary conditions in the adsorption case for this model are: 1:
Tðt; r; zÞ;
Tð0; Ro prpR2 ; zÞ ¼ T i ,
@ r ¼ R2 ;
qT ¼ 0, qr
@ r ¼ Ro ;
T ¼ T p,
@ z ¼ 0;
qT ¼ 0, qz
@ z ¼ L;
qT ¼ 0, qz
2: T p ðt; zÞ;
(11)
T p ð0; 0pzpLÞ ¼ T pi ,
@ z ¼ 0;
qT p ¼ 0, qz
@ z ¼ L;
qT p ¼ 0, qz
3: @ r ¼ Ri ;
_ pf DT f ¼ 2pRi hf DzðT f T p Þ, mc
(12) (13)
4: T f ðzÞ, @ z ¼ 0;
T f ð0Þ ¼ T fi ,
@ z ¼ L;
_ pf DT f Þ=ð2pRi hf LÞ þ T p , T f ðLÞ ¼ ðmc
(14)
where Ti, Tpi, and Tfi are the initial temperatures of the porous sieve, concentric pipe, and _ is the air mass the fluid (air), respectively; R2 is the radius of the compression unit; and m flow rate. The basic equations, (7), (10), and (13), with the initial and boundary conditions constitute the nonlinear differential equations that form a simultaneous nonlinear system to be solved.
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Fig. 4. Discretized system schematic.
5.1. Model discretization The basic governing equations with the boundary conditions were discretized using the finite difference scheme (fully implicit formulation) and the meshing grids as shown in Fig. 4. The finite difference formulation starts with a subdivision of the compression unit into ordered nodal points. Basically, the compression system is divided into 22 grids along the axial direction (i) and four grids in the radial direction (j) (see Fig. 4). The discretized equations at different nodes are based on the principle of energy conservation and fully presented in detail in the Appendix. 5.2. Stability criteria To assure stability of the computational technique, the time increment (Dt) were kept equal to or less than the value obtained from the maximum allowable time increment needed for stability requirement: i .h X Dtpci jð1=Rij Þ , (15) where Dt is the time increment (h), ci is the thermal capacity ðri cpi DV i Þ, Rij is the thermal resistance between i and adjoining nodes, i is the node of interest, j is the adjoined nodes with i, and DV is the volume element. The obtained value for Dt is 0.00075 h. A 0.0005 h incremental time has been chosen for stability purposes. 6. Results and discussion The conservation equations of the model are solved using the finite difference implicit scheme. The aim of the model is to predict the temperature distribution history and the amount of refrigerant stored inside the thermal compression system. Figs. 5–7 show the temperature distribution history at specific locations inside the compression system. Fig. 5 depicts the temperature distribution at an axial distance, z, equals 10.16 cm and a radial distance r, equals 3.175 cm, while Figs. 6 and 7 present the temperature history at axial
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46 Simulation Experimental
TEMPERATURE (°C)
40
34
28
22
16 0
0.05
0.1 0.15 TIME (HOUR)
0.2
0.25
Fig. 5. Experimental and simulation results for temperature history at r ¼ 3:175 cm and z ¼ 10:16 cm.
46 Simulation Experimental
TEMPERATURE (°C)
40
34
28
22
16 0
0.05
0.1
0.15
0.2
0.25
TIME (HOUR) Fig. 6. Experimental and simulation results for temperature history at r ¼ 3:81 cm and z ¼ 30:48 cm.
distances 30.48 and 50.8 cm and radial distances, r, equal 3.81 and 4.45 cm, respectively. In these figures the simulation results are compared with the experimental runs. The predicted temperature evolution is very close to the experimental measurements. This means that the deviations of predicted results from the experimental findings are very small and within the maximum relative experimental error (0.8%). Four reasons might have caused these deviations: 1. The model might not replicate the physical process due to the assumptions used in developing the physical model. This can be indicated as follows: (a) The pressure is assumed to be constant along the compression system.
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TEMPERATURE (°C)
46
40
34
28
22
16 0
0.05
0.1
0.15
0.2
0.25
TIME (HOUR) Fig. 7. Experimental and simulation results for temperature history at r ¼ 4:45 cm and z ¼ 50:8 cm.
(b) The physical properties of the refrigerant, solid sieve, air, and pipe are assumed to be constants. (c) The amount of energy absorbed by the intermittent compression system, which weighs about 11.5 kg, is significant. (d) Radial temperature gradients are assumed negligible since the compression system wall is thin. 2. The heat convection inside the thermocouple wells has some effects on the thermocouple readings. 3. The accuracy of the heat transfer coefficient and the heat adsorption might not be satisfactory. 4. The truncation errors of the numerical solution might be excessive. Fig. 8 shows the thermal compression system behavior for a cyclic time of 15 min for the adsorption–desorption processes. One can see that the amount of refrigerant recovered in the regeneration process, at the end of the cycle, is smaller than the amount adsorbed in the adsorption process. This indicates that the time for the regeneration process should be longer than the time for the adsorption process in order to raise the sieve temperature. In general, if a designer assumes a minimum required air-conditioning capacity of 14 ton refrigeration, a refrigerant 12 flow rate of 27.2 kg/h with 116.3 kJ/kg cooling energy available is required to establish the new thermal compression hest pump system. If four compression units are connected in parallel with the conventional compressor and cycled every 15 min (two units are in the adsorption mode while the other two are in the regeneration mode), 6.8 kg of refrigerant are required every cycle (15 min), or 1.7 kg refrigerant per unit per cycle. If 75% of the refrigerant adsorbed can be recovered, 2.27 kg of refrigerant are required for each thermal compression unit. The Linde 13X molecular sieve weight required for each unit with an adsorption of 30% to adsorb 2.27 kg of refrigerant is about 7.5 kg. This compression system can provide either heating or cooling. When the refrigerant leaves the compression unit and enters the condenser where energy is
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Refrigerant 12 in the compression system (g)
325 300 275 250 225 200 175 150 125 100 75 50 25 0 0.00
0.25
0.50 Time (hour)
0.75
1.00
Fig. 8. Thermal compression system behavior.
removed and the vapor liquefies, then the system at this stage is in the heat pump mode. If the refrigerant liquid leaves the condenser and passes through the throttling valve to the evaporator, then the refrigerant absorbs heat and provides cooling at this location; thus the heat pump is in the regular air-conditioning and refrigeration mode. Since this thermal compression system uses waste heat or low-grade thermal energy source is technically applicable during the adsorption and regeneration processes. 7. Conclusions Solid–vapor adsorption technology still has considerable scope for investigation and improvement to compete with conventional absorption and vapor compression technologies. A dynamic simulation model for adsorption–desorption processes associated with an intermittent thermal compression adsorption system filled with porous molecular sieve was developed. Based on the simulated results, the following conclusions can be drawn:
The dynamic model can be used as an effective complementary tool during the design stage of intermittent heat pump/refrigeration systems. The mass of the compression system must be minimized to shorten the heating and cooling cyclic time and enhance the system performance. The presence of mass and heat transfer limitations can affect significantly the performance of the adsorption heat pump systems. This would be accomplished by reducing the difference between the inner and outer diameter of the thermal compression unit. The temperature distribution history shows that there is a good agreement between the simulated and experimental results.
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Further R&D work is recommended to increase the chances of the adsorption technology to compete with absorption and conventional vapor compression technologies.
Appendix A. Details of model discretization A.1. For the interior nodes in the molecular sieve The finite difference form of the time-dependent energy balance on mesh point (i,j) located in the porous media is T kþ1 ði; jÞ ¼ fK s 2p½Ro þ ð2j 3ÞD=2Dz½T ks ði; j 1Þ T ks ði; jÞDt=D s K s 2p½Ro þ ð2j 1ÞD=2Dz½T ks ði; jÞ T ks ði; j þ 1ÞDt=D þ K s 2p½Ro þ ð2j 3ÞD=2D½T ks ði 1; jÞ T ks ði; jÞDt=Dz K s 2p½Ro þ ð2j 3ÞD=2D½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2D DzDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2DDz þ T ks ði; jÞ
ðA:1Þ
where D is the increment size along the radial direction, Dz is the increment size in the axial direction, Dt is the time increment, ADSORB is the adsorption rate per unit volume of the molecular sieve, Hfg is the heat of adsorption and the exponent k denotes temperatures at the time level. This equation is valid only from i ¼ 2, 22 and j ¼ 2, 4 (see Fig. 4). A.2. For the porous media of all elements located on the first row and the internal pipe The finite difference formulation for these grids can be presented in the following form: T kþ1 ði; 1Þ ¼ ½hf 2pRi Dz½T k f ði; 1Þ T ks ði; 1ÞDt þ kp 2pRi Dr½T ks ði 1; 1Þ s T ks ði; 1ÞDt=Dz kp 2pRi Dr½T ks ði; 1Þ T ks ði þ 1; 1ÞDt=Dz 1 eÞks 2p½Ro þ D=2Dz½T ks ði; 1Þ T ks ði; 2ÞDt=D þ ð1 eÞks 2p½Ro D=2½T ks ði 1; 1Þ T ks ði; 1ÞDt=Dz ð1 eÞks 2p½Ro D=2½T ks ði; 1Þ T ks ði þ 1; 1ÞDt=Dz þ H fg ADSORBði; jÞ½2p½Ro D=2DzDt=½rs cps 2p½Ro D=2Dz þ cpp rp 2p½Ri DzDr þ T ks ði; 1Þ.
ðA:2Þ
The previous equation is valid for i ¼ 2, 22 and j ¼ 5 only. A.3. For the fluid (air) flowing inside the concentric pipe The following equation represents the energy balance between the fluid flowing through the concentric pipe and the conduit itself: _ cpf ½T k f ðiÞ T k f ði þ 1Þ ¼ hf 2pRi Dz½T k f ðiÞ T k pðiÞ. m This equation is valid only for i ¼ 1, 23 only.
(A.3)
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A.4. For the upper boundary of the molecular sieve The finite difference formulation for the top boundary of the porous media can be cast as follows: T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2Dz½T ks ði; j 1Þ T ks ði; jÞDt=D s þ ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T ks ði 1; jÞ T ks ði; jÞDt=Dz ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2ðD=2ÞDzDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2ðD=2ÞDz þ T ks ði; jÞ.
ðA:4Þ
A.5. For the top right corner of the porous media only The finite difference equation for the upper right corner can be formulated as T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D s ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T k ði 1; jÞ T ks ði; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2Þ þ T ks ði; jÞ.
ðA:5Þ
The previous equation is valid only for i ¼ 23 and j ¼ 5. A.6. For the upper left corner of the porous media only The discretized equation for this corner can be presented as T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D s ks 2p½Ro þ ð2j 3ÞD=2ðD=2Þ½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2ðD=2ÞðDz=2Þ þ T ks ði; jÞ.
ðA:6Þ
This equation is valid for i ¼ 1 and j ¼ 23 only. A.7. For the right side of the porous media only The equation for this portion is for i ¼ 23 and j ¼ 2, 4 and can be cast in the following form: T kþ1 ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D s ks 2p½Ro þ ð2j 1ÞD=2ðDz=2Þ½T ks ði; jÞ T k ði; j þ 1ÞDt=D þ ks 2p½Ro þ ð2j 3ÞD=2D½T ks ði 1; jÞ T ks ði; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2DðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2DðDz=2Þ þ T ks ði; jÞ.
ðA:7Þ
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A.8. For the left side of the porous media only The finite difference formulation for that side (i ¼ 1, j ¼ 2, 4) can be described as ði; jÞ ¼ fks 2p½Ro þ ð2j 3ÞD=2ðDz=2Þ½T ks ði; j 1Þ T ks ði; jÞDt=D T kþ1 s ks 2p½Ro þ ð2j 1ÞD=2ðDz=2Þ½T ks ði; jÞ T ks ði; j þ 1ÞDt=D ks 2p½Ro þ ð2j 3ÞD=2D½T ks ði; jÞ T ks ði þ 1; jÞDt=Dz þ H fg ADSORBði; jÞ½2p=ð1 eÞ½Ro þ ð2j 3ÞD=2DðDz=2ÞDtg =½rs cps 2p½Ro þ ð2j 3ÞD=2DðDz=2Þ þ T ks ði; jÞ.
ðA:8Þ
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