A simplified model for the analysis of a phase-change material-based, thermal energy storage system

A simplified model for the analysis of a phase-change material-based, thermal energy storage system

Heat RecoverySystems & CHP Vol. 8, No. 3, pp. 247-254, 1988 Printed in Great Britain 08904332/88 $3.00+ .00 Persamon Preu pk A SIMPLIFIED MODEL FOR ...

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Heat RecoverySystems & CHP Vol. 8, No. 3, pp. 247-254, 1988 Printed in Great Britain

08904332/88 $3.00+ .00 Persamon Preu pk


(Received 21 April 1987; /n revised form 5 August 1987) Abstract--The concept of a phase-change material-based, thermal energy storage system is often used for different applications. A theoretical model to determine the thermal and fluid flow characteristics of a thermal energy storage system using a phase-change material has been developed. The model can be used to predict the energy storage behavior of different phase-change materials used with different heat transfer fluids, flow geometries, flow rates and temperatures. Results have been obtained for the case where the phase-change material is Na2SO(. 10H20 (Glauber's salt) and the heat transfer fluid is water. The variation of the dimensionless temperatures of the fluid and the solid, and the molten fraction of the solid during the phase change process, with dimensionless time, for different values of Biot number, Stefan number, and the flow parameter have been determined. A discussion of the results obtained and the condmfions drawn from them are also given.


area of cross section specificheat diameter


convective heat transfer coefficient thermal conductivity mass mass flow rate radius temperature dimensional time volumetric flow rate

Non-dimensional parameters Bi Biot number (=hD2/k,) Fo Fourier number (dimensionless time) G geometry parameter of the storage configuration Q flowparameter of the system Ste Stefan number (ffiC,(T~- T,~)/,~) Greek symbols thermal diffusivity fl fraction of the total mass of PCM in the molten state 2 latent heat of fusion of the PCM p density 0 dimensionless temperature Subscripts 1 cylindrical container 2 PCM capsule f heat transfer fluid s solid m molten Superscript * initial INTRODUCTON T h e c o n c e p t o f storing energy becomes especially i m p o r t a n t w h e n the supply or d e m a n d o f energy varies c o n s i d e r a b l y with time. A n e x a m p l e o f this is solar energy, which is available o n l y d u r i n g 247



the daylight hours and therefore must be stored for use on cloudy days and nights. One method of storing energy is by a latent-heat storage system (LHSS). This method is particularly attractive due to its high energy-storage density and the characteristics of storing energy at a constant temperature. Some of the common phase-change materials (PCMs) used are ice, paraffins or Glauber salt. Several investigators have studied the advantages of various PCMs and different heat exchanger configurations. An extensive experimental investigation on the performance of a latent heat storage system has been conducted by Abhat [1]. This included a study of the melting and freezing behavior of the PCM and an assessment of the overall thermal performance of the LHSS. An analysis of the energy and exergy values of an LHSS was performed by Bjurstrom and Carlsson [2]. This indicated that the transition temperature of the PCM should be low for efficient operation. An analytical study was performed by Viskanta [3] to study the characteristics of an LHSS. Two models, one with a discrete phase-change temperature and the other with phase change occurring over a range of temperatures were proposed and evaluated. I~jong and Hoogendoorn [4] addressed the problem of improvement of heat transport in paraffins. The testing of pebble bed and phase-change, thermal energy storage devices was performed by Jones and Hill [5] and Marshall [6]. Both these studies were with reference to the ASHRAE standard on sensible heat storage devices. Analytical studies on a tube-shell type heat exchanger, wherein the coolant fluid flows through the tube and the PCM is on the shell side, were performed by Shamsundar and Srinivasan [7, 8]. The axial variation of the fluid temperature flowing through the tube was considered and NTU-effectiveness charts for the heat exchanger were presented. A finite difference approach in two dimensions was used by Sparrow and Hsu [9] and the results compared well with the work done by Shamsundar [8]. Based on the literature survey, it is evident that considerable work on the selection of PCMs, determination of the solid-fluid interface in PCM capsules and the heat transfer characteristics of PCMs has be~n done. The objective of the present paper is to simulate the thermal energy storage system as a complete system, including the heat exchanger, but considering the load and the heat transfer fluid moving equipment external to the system, and to identify the governing parameters of the problem. T H E O R E T I C A L MODEL DEVELOPMENT The schematic diagram of the proposed PCM thermal storage model is shown in Fig. 1. The PCM is in a cylindrical shaped capsule (PCM capsule). The PCM capsule, in turn, is placed in a


"7 I

i = PCM H



:= I=CM copsule I --H,- Cylindrical , container I


l Hold-up tank for the heat transfer fluid

~.~ ~ t r o l

volume L




Fig. 1. Schematic of proposed PCM model.

A phase-changematerial-basedthermalenergystorage system


cylindrical container. The heat transfer fluid flo~S ih ihe annular Space between the PCM capsule and the cylindrical container. Circulation of the heat transfer fluid is maintained using a pump. The heat transfer fluid exchanges heat with the PCM capsule and transfers it to the load through a heat exchanger coil (not shown in Fig. 1.). The load could be a space where cooling or heating is desired. The region of interest for this analysis is the control volume (CV) shown by the dashed lines. The control volume consists of the cylindrical container and the PCM capsule and the annular space containing the flowing fluid. It is divided into two smaller control volumes: the PCM capsule itself and the annular space containing the heat transfer fluid. The load is external to CV and is not treated as a parameter in the analysis. Hence, the magnitude of the load is assumed to be independent of time. If the PCM is melting, the heat transfer fluid "loses" heat to the PCM and facilitates cooling of the load. On the other hand, if the PCM is solidifying then the heat transfer fluid "gains" heat from the PCM and facilitates heating of the load. The heat transfer fluid from the load flows into a hold-up tank. Sufficient fluid volume is maintained in the hold-up tank at all times to prevent the pump from running dry. A "lumped-system" model is proposed to predict the performance of the PCM thermal energy storage system. Based on some simplifying assumptions, the governing equations have been developed in dimensionless form and the governing parameters of the problem have been identified. The following simplifying assumptions are made for the model development. 1. The hold-up tank is assumed to be very large and all the components are assumed to be insulated so that the temperature of the heat transfer fluid in the hold-up tank can be assumed to remain constant and independent of time. 2. The pump driving the heat transfer fluid is assumed to be running all the time so that natural convection effects in the control volume can be neglected. 3. The axial and radial temperature variation for the PCM and the heat transfer fluid are approximated by average values. [e.g. I/2 (Tf.~,p+ Tf.bottom)].Note that the temperature of the PCM and that of the heat transfer fluid have to be different, or else no heat transfer occurs. 4. The thermal storage effect of the PCM capsule and the wall of the cylindrical section is neglected. 5. The thermo-physical properties of the PCM and the heat transfer fluid are assumed to remain invariant during the PCM thermal storage system operation. 6. The heat transfer fluid is assumed to be incompressible. 7. The entire system is assumed to be in a steady state condition with the pump running and the only source of transient behavior in the whole operation is the change of phase of the PCM (melting or freezing). 8. The height of the PCM capsule is considered to be significantly larger than the radius or the PCM capsule is slender (i.e. 1>>R2/H2); the heat transfer from the upper and lower flat surfaces (ends) of the PCM capsule is neglected. In practice, during the discharging of the PCM it is difficult to maintain the temperature of the PCM at saturated conditions. Hence, when the charging of the PCM begins, the PCM will be in the subcooled state and so, the heat transfer between the PCM capsule and the heat transfer fluid under both saturated and subcooled conditions of the PCM should be addressed.

A. Governing equations for the PCM Based on the above-mentioned assumptions, the energy balance for the PCM under subcooled conditions can be written as:

hA2[Tf- T,I=m,C,dT, dt


the initial conditions being: at






Note that T* is the temperature at which the transfer fluid enters the control volume at all times. The temperature Tf in equation (1) represents the temperature of the transfer fluid in the annular cylindrical section and this is the temperature at which the transfer fluid leaves the control volume. H.R.$. g/3---E



Let 0 be the dimensionless temperature defined by:



0 = Tr-~'


where T is the variable temperature. Also, the Biot number (Bi) and the Fourier number (Fo) are defined in the following manner: Bi = hD2/ks;


Fo = ey,t/R 2.


In terms of the dimensionless quantities given by equations (3) and (4), the governing equation for the PCM (equation 1) can be written as: d0,


d--~o = Bi [ 0 f - 0s] and the initial conditions are: for






The equation (5) is valid for T, < T~,,,. When the PCM reaches the saturated state, wherein part of the PCM is in the liquid state and part of it in the solid state, the governing energy equation for the PCM will be:


hA2 [Tf - Tin] -- m,2 --~-,


where ~ is the ratio of the mass of the PCM in the molten state to the total mass of the PCM. The initial condition will be: at the time when T, ffi Tin,/~ ffi 0 if the PCM is being charged and -- 1 if the PCM is being discharged. Note that/~ = 0 implies that the PCM is completely in a solid state and ~ ffi 1 implies that the whole PCM is in liquid state. Equation (7) can be recast into a dimensionless form as:


Bi [ o , - o.] = ~ee ~oo ' where Ste is the dimensionless Stefan number and is given by: Ste = C, (T* - Tin) 2


This is the ratio of the sensible heat content in the PCM to the latent heat of fusion of the PCM. B. Governing equations for the fluid The energy balance for the transfer fluid in the annulus is given by:


vhf C f [ T f - T*] + hA2 [Tr - 7",] + H, ~ [R~ - R2,] Pr C f - ~ = 0


the initial condition being: for



Additional dimensionless parameters are identified as the geometrical parameter, G: Hj

G = -~ [ R2 /R 2, -



and the flow parameter, Q: Q ffi ~'f/,~H2 ~f.


A phase-change material-based thermal energy storage system


Hence, equation (9) can be rewritten as: Q [Of-- I] + [k,/kf]Bi [Of-



Jr"G [=,/=f]~ o



and the initialcondition becomes Fo = 0, 0f = 0f*. The above equation (13) for the transfer fluid is valid for all conditions irrespectiveof whether the P C M isin the molten or a subcooled state.Thus, the model for the P C M thermal storage system has been completely developed and the governing equations have been written in terms of dimensionless quantifies. They are: (a) flow parameter, Q, related to the volume flow rate; (b) Biot number, Bh related to the heat transfer properties; (c) geometric parameter, G, related to the dimensions of the capsule and the container; (d) thermal diffusivityratio (=,/=f),related to the solid and fluid thermal diffusivities; (e) Stefan number, Ste, related to the thermal properties of the P C M ; and (f) Fourier number, Fo, related to the time elapsed, is the independent parameter. A generalized numerical algorithm was used to solve the equations for the two unknowns, the solid P C M temperature (in a subcooled state),or the fraction of the solid in the molten state,/~ (in a molten state),and the transfer fluid temperature, 0f, all in dimensionless forms. It was also assumed that the Glauber's salt underwent ideal congruent behaviour, i.e.melting/freezing with no segregation. RESULTS The proposed model was applied to a case where the phase change material was Glauber's salt, Na2SO4" 10H20 and the heat transfer fluid, water. The parameters studied were the Biot number, ai, Stefan number, Ste, and the flow parameter, Q. The variation of the dimensionless temperatures, 0r and 0., and the molten fraction, ~, with dimensionless time, Fo, are presented for different values of the above parameters. Representative values of the parameters describing the fluid flow and heat transfer of the proposed system were used. The exact conditions of each case in terms of temperatures and fluid flow rates are given below in detail.

I. Influence of Bi Figure 2 shows the variation of the dimensionless fluid and solid temperatures, Of and 8,, with the dimensionless time, Fo, for various Biot numbers. 14


! / I


T~: 50°C Bi : 0.1, 1.0,10.0



FLuid :E



~,,: 0oo2 mss"



1". = 32.38°C






/ 06

7" /'



Bi : I0.0 04

/ _



.- : _ . . . <



1.0 i



~ 04

0.I /





...... i






T~' : 20" C

Tf': T': ~, : T,:


50"C 20°C 0.002 0.(~ m= 3Z38 "C





Fourier number (dimensionLess time )


Bi : IO.O









iJ I00

iOI 00

Fourier numNr (dimensionLess time ) Fig. 2. Dimenmoxdeu temperature vs time ('or various Blot nos.

Fig. 3. Molten fraction vs time for various Blot no$.


252 1.2



Ste : 0 2 5


0.50 1.00

Bi = I0


o.e Fluid Bi : T,~ : ~-V, : T. :


3 0

~. O.e

I0 20"C 0.002 m3 s " 32.38"C


T~ : 2 0 " C V, : 0 . 0 0 2 m 3 s "

=E t3 0.

/ !

/II /

T~ : 32.38 *C

"6 o.e I




Ste : t.OO



w 0.4

,/ /





E 0.2 i:5


Ste = 0.25











/05011 0.25 /


/111 /11 I I //


time )

Fourier number ( d ~ n s i o n L e s s



[ /




Fourier number (dimensiontess t i m e )

Fig. 4. Dimensionless temperature vs time for various Stefan nos.

Fig. 5. Molten fraction vs time for various Stefan nos.

As the Blot number increases, the time required for melting decreases and the temperature reaches a constant plateau while the phase-change mechanism is being completed. Since the Biot number represents the non-dimensional heat transfer coefficient, the time required for melting decreases with an increase in the heat transfer coefficient.However, the fluid temperature, Or, is seen to approach a value of l, independent of the Biot number. This occurs since the firstterm in equation (13) is the more dominant term, causing a sharp fallin the fluid temperature initially, before approaching unity. Figure 3 shows the variation of the molten fraction ~, with the dimensionless time, Fo, for various Biot numbers. It is seen from Fig. 3 that, as the Biot number increases, the slope of/~ increases with time. This indicates that the rate of change of the molten fraction with respect to time, (d~/dFo), increases proportionally, since Of is practically a constant as seen from Fig. 2. For a higher Blot number, the melting occurs sooner than for a lower Biot number. II.

Influence of Ste

Figure 4 shows the effect of Stefan number on the fluid and solid temperatures with time. For higher Stefan numbers, the melting process occurs faster and 0, reaches a plateau more rapidly. This is because an increase in the Stefan number translates into an increase in the initial fluid temperature, T*. The fluid temperatures finally converge to unity similar to Fig. 2. The reasoning for this is the same as mentioned earlier. 14 ~


O : 5000


7500 IO000


O : 5000, 7 5 0 0 , I0000


c~- Lo n o




0 8 --



O : 5000, 7500, I0000 /

06 O4






T~': 20"C T~: 32.38" C




T , ' : 50*C T ; : 20°C T~ : 32 38°C



T; : 50"C


Bi = I0 0.6 -

IE 0.2-


Fourmr number (dimensionless t~me)




o ol

o io


Fourier number (dimensionless brae)

Fig. 6. Dimensionless temperatures vs time for various flow rates.

Fi 8. 7. M o l t e n


vs t i m e f o r v a r i o u s

f l o w rates.

A phase-change material-based thermal energy storage system


Figure 5 shows the variation of the molten PCM fraction with time. The curves are qualitatively similar to those in Fig. 3, since the Biot number is constant and the rate of change of melting with time is proportional to the Stefan number. III. Influence of Q Figures 6 and 7 reflect the variation of the flow rates on temperatures and the molten fraction. The fluid temperatures vary with the flow rate, Q, such that for a higher flow rate there is a lower fluid temperature, all other parameters being unchanged. However, all cases approach a value of unity, as seen in the previous cases. The effect of Q on the PCM temperature is found to be minimal. DISCUSSION The theoretical results lead to a variety of conclusions as follows. (1) The Blot number has a considerable effect on the time-dependent behavior of the PCM, since it is a function of the capsule geometry, the heat transfer coefficient, and the thermal conductivity of the PCM. (2) The Biot number has little or no effect on the fluid temperature. This is interesting to note since it indicates that despite an increase in heat transfer, the temperature of the fluid remains constant. This leads to a constant temperature source of fluid supply, which can be useful in many heating/cooling applications involving a constant load. (3) The Stefan number has little or no (real) effect on the fluid temperature, but does have an effect on the melting rate. This is physically realistic since a higher Stefan number indicates a higher initial fluid temperature, which increases the heat transfer rate. (4) A variation in the fluid flow rate has no (real) effect on the system. This indicates that if we have a higher flow rate, there is no real difference in heat transfer. However, a faster flow rate means an increased thermal capacitance of the fluid. The value of 5000 for Q translated into approximately 18 U.S. gal min -I, so that higher values would mean even higher flow rates.

CONCLUSIONS A simplified model of slender PCM containing tubes with heat transfer fluid flowing on the outside in the axial direction and with negligible warming up/cooling down of the fluid, has been proposed and analyzed theoretically. Three independent, dimensionless parameters (Biot number, Stefan number and flow rate, Q) which characterize the heat transfer and fluid flow of the PCM thermal storage system were identified and their effects on the system performance, (viz. 0,, 0f, ~), studied. The results showed that increased flow rates had little/no effect on the melting rate. However, as can be expected, an increase in the Biot and Stefan numbers led to a marked increase in the melting rate. The dimensionless fluid temperature, 0f, remains approximately a constant (~- 1.0). The model has the following limitations: (1) it is based on a "lumped system" approach; (2) the melting and solidification of the PCM is not modelled by the discrete system approach; (3) the load is assumed to be independent of time; and (4) the axial and radial temperature variations for the PCM and the axial temperature variation of the heat transfer fluid are approximated by average values. However, the proposed model has the following advantages: (1) it is extremely simple, hence, it can be suitably modified for s p ~ a l cases: for example, a time dependent flow rate Q(7o), could be very easily adapted to the model equations; (2) simple, easy=to=use design charts could be generated using the model. It must be pointed out that such charts would be limited to cases where the geometry of the system and operating conditions are well defined/known a priori. This will enable an investigator to get a quick, easy-to=use estimate of how useful a particular PCM thermal energy storage system would be



before proceeding to a more complex modelling analysis. In other words, this model can be a useful tool for simple feasibility studies of a PCM based thermal energy storage system. Acknowledgement--The authors would like to acknowledge the constructive suggestions made by the reviewer, which enhance the quality of the paper. REFERENCES 1. A. Abhat, Low temperature latent heat thermal energy storage, in Thermal Energy Storage (Edited by G. Beghi) pp. 33-91. Reidel, Hingham, U.S.A. (1981). 2. H. Bjurstrom and B. Carlsson, An energy analysis of sensibleand latent heat storage, Heat Recovery Systems 5, 233-250

(1985). 3. R. Viskanta, Phase change heat transfer, in Solar Heat Storage: Latent Heat Material (edited by G. A. Lane), Vol. 1 CRC Press, Palm Beach, U.S.A. (1983). 4. A. G. Dejong and G. J. Hoogendoorn, Improvement of heat transport in parafl~n~ for latent heat stora F systems, in Thermal Storage of Solar Energy (edited by G. D. Ouden) pp. 123-133. Martinus Nijhoff, 's-Gravenhagen, The Netherlands (1981). 5. D. E. Jones and J. E. Hill, An evaluation of ASHRAE standard 94-77 for testing pebble-bed and phase change thermal energy storage devices, ASHRAE Trans. DE-79-4, No. 3, 1979. 6. R.H. Marshall, A theoretical study of ASHRAE standard 94-77 for testing thermal storage devices, in Therma/Starage of Solar Energy (ecfited by G. D. Ouden) pp. 111-122. Mattinus Nijhoff, 's-Gravenlmgen, The Nethedan~ (1981). 7. N. Shamsundar and R. Srinivuan, Effectivenes~-NTU charts for heat recovery for latent heat storage uedts, J. Solar Energy IQ2, 263 (1980). 8. N. S ~ , Formula for freezing outside a circular tube with axial variation of coolant temperature, Int. J~ Heat Mass Transfer 25, 1614-1616 (1982). 9. E. M. Sparrow and C. F. Hsu, Analysis of two-dimensional-freezing on a coolant-carrying tube, Int. J. Heat Mass Transfer 24, 1345-1357 (1981).