A transient analysis of a closed loop solar thermosyphon water heater with heat exchangers

Energy Convers. Mgmt Vol. 31, No. 5, pp. 505-508, 1991 Printed in Great Britain. All rights reserved

0196-8904/91 $3.00 + 0.00 Copyright © 1991 Pergamon Press plc

TECHNICAL NOTE A TRANSIENT ANALYSIS OF A CLOSED LOOP SOLAR THERMOSYPHON WATER HEATER WITH HEAT EXCHANGERS G. N. TIWARIt and S. A. LAWRENCE~ Department of Physics, University of Papua New Guinea, P.O. Box 320, Papua New Guinea (Received 26 January 1989: received for publication 1 June 1990)

Abstract--In this Technical Note, a straightforward transient analysis of a closed loop solar water heater with heat exchanger under a thermosyphon mode has been presented. Analytical expressions for fluid temperature in the flat plate collector (Tf) and storage tank water temperature (Tt) have been obtained. Numerical calculations have been carried out by using the experimental data of Webster et al. (T. L. Webster, J. P. Coutier, J. W. Place and M. Tavana, Sol. Energy 38, 219 (1987); Ref. [1]), except the plate efficiency factor. It is observed that there is a good agreement between the theoretical results obtained by the proposed model and the experimental observations of Webster et al. [1] for the mean storage water temperature.

NOMENCLATURE C w ~ Specific heat of water (J/kg ~C) G , C2= Heat exchanger penalty factor F ' = Plate efficiency factor H = Total power input to collector absorber (kW) Mass of water in storage tank (kg) gw= Mass flow rate of cold water through storage tank (kg/s) qc = Useful heat available from flat plate collector to storage tank through heat exchanger (kW) T = Temperature (°C) (UA) = Total overall heat transfer coefficient (kW/°C) ?l¢ = SHW system efficiency

Subscripts

Ambient air Collector fluid f = Fluid in heat exchanger hx= Heat exchanger t = Storage tank W = Water in storage tank a=

c=

FORMULATION O F T H E S H W S Y S T E M Recently, W e b s t e r e t al. [1] have a n a l y s e d the p e r f o r m a n c e o f d o m e s t i c solar h o t w a t e r ( S H W ) systems with a n d w i t h o u t a h e a t e x c h a n g e r p l a c e d inside the s t o r a g e t a n k . T h e y d e v e l o p e d a theoretical m o d e l o f a w a t e r h e a t e r w o r k i n g u n d e r a t h e r m o s y p h o n m o d e with a parallel h e a t exchanger. T h e i r a n a l y t i c a l m o d e l was similar to t h a t p r o p o s e d by Close [2]. T h e y derived an expression for the useful h e a t available f r o m a fiat plate collector to the s t o r a g e t a n k t h r o u g h a h e a t exchanger, a n d it is given by qc = F ' [ C 1 H - C 2 ( U A ) ¢ ( T t - T~)]

where C1 = C2 +

(UA)p (UA)hx + ( U A ) v + F ' ( U A )¢

tPresent address: Centre of Energy Studies, Indian Institute of Technology, Hauz Khas, New Delhi-ll0 016, India. ~To whom all correspondence should be addressed. 505

(1)

506

TIWARI and LAWRENCE: TECHNICALNOTE

and

(UA)hx

G = (UA)~ + (UA)p + F'(UA)c' With the help of equation (1), the energy balance of the complete system, which has not been considered by Webster et al. [1], can be written as follows (Sodha and Tiwari [3]): dTt F'[C,H(t) - C2(UA)c(T t - Ta)] = MwCw ~ + (UA)t(Tt - Ta) + rnwCw(Tt - Tin).

(2)

In order to write the above energy balance, the following assumptions have been made: (i) The insolation H(t) is a constant over shorter intervals of time (At) and is time dependent over a complete cycle. (ii) The mass flow rate (rnw) through a storage tank is zero during the no hot water withdrawal period [1], otherwise, it has some value depending on the demand pattern. (iii) The first initial condition (T~0) is the temperature of the water initially filled in the storage tank. (iv) The inlet temperature (Tin) of the cold water fed to the storage tank is the same as the ambient air temperature (Ta) which is generally true for all practical purposes. The solution of equation (2) can be written as:

F" FtCI H(t~-}- Tal [1 - exp(--at)] + T,0 e x p ( - - a / ) Tt = L( UA )t + rhwCw+ F'C2(UA )c

(3)

where a

-

(UA)t + rhwCw + F'C2(UA)c M~Cw

H(t) is the average insolation during shorter intervals of time (0 ~ t) for the first set of calculations, and it will have a different value for the next interval of time, Tt0 is the initial water temperature in the storage tank for the first set of calculations, and Tt calculated from the first set becomes Tt0 for the 2nd set of calculations and so on. Equation (3) can be used to evaluate the water temperature in the storage tank with (thw ~ 0) and without (n~w-- 0) withdrawal of hot water from the storage tank. Equation (3) has been used to evaluate the water temperature (Tt) for the following parameters as reported by Webster et al. [1]; (UA )c = 12.9 W/°C~ (UA)p

(UA )~x ( UA )c

2.62 W/°CJ Fig. 7 of Webster et

al.

[1]

0.85 {average value from Fig. 8 of Webster et al. [1]

(UA)h~ = l l W/°C (UA)t =

4.6 W/°C

Mw = 310 kg, Cw = 4190 J/kg°C, Ta = 24°C and t = 3600s. Since neither the value of the plate efficiency factor F' nor the detailed configuration of the flat plate collector used was given by Webster et aL [1], arbitrary values of F ' were taken for the calculations. It was observed that F' = 0.3 gives reasonable agreement between the theoretical value obtained by using equation (3) and the experimental observations of Webster et aL (Fig. 1).

TIWARI and LAWRENCE: TECHNICAL NOTE

......

H (KW) TI: (Experimental Tt (Theoretical, Tf (Experimental TJ: (Theoretical,

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Fig. 1. H o u r l y variation of power input (H), fluid temperature (Tf = T¢ = Thx) and water t e m p e r a t u r e in storage tank.

After knowing the value of the water temperature (Tt), equation (6) of Webster

Tf = TaF'(UA)c + (UA)pTa + Tt(UA)ehx + F' H(t) (UA)hx + (UA)p + F'(UA)c

et al.

[1], i.e. (4)

can be used to theoretically evaluate the fluid temperature (Tr) in the flat plate collector. Without the value of Tt, Tr cannot be determined. The corresponding calculated value of Tr has also been drawn in Fig. I. It is clear from the figure that there is no agreement between the theoretical and experimental results, since the experimental observation of Tf = Tc = Thx is about 5°C higher than Tt, which is not justified from a practical point of view because the fluid temperature in the flat plate collector is always much higher (say the maximum value reaches about 100°C) than that of the storage water temperature. As for the assumption that Tc = Thx = Tr and if equation (4) is used to calculate Tf for F ' = 0.3 (which is also not justified in the present case) for experimental values of Tt, the calculated values of Tf will come out much higher than the reported values for T¢ = Thx = Tr. Following Webster et al. [l], the efficiency of the SHW system can be correctly written as

rl~= CF" [1-(UA)c TtHT~ ].

(5)

Without knowing Tt from equation (3), the above equation cannot be used theoretically to evaluate q¢ as claimed by Webster et al. [1]. The analysis will be very useful to design, fabricate and install solar thermal devices in which the collector loop circuit will be required to operate as a thermosyphon, e.g. indoor solar swimming pool heating, particularly in remote cold climate conditions.

508

TIWARI and LAWRENCE: TECHNICAL NOTE

T h e p r o p o s e d m o d e l is valid for all c l i m a t i c a l c o n d i t i o n s , w h e t h e r clear o r c l o u d y .

REFERENCES 1. T. L. Webster, J. P. Coutier, J. W. Place and M. Tavana, Sol. Energy 38, 219 (1987). 2. D. J. Close, Sol. Energy 6, 33 (1962). 3. M. S. Sodha and G. N. Tiwari, Energy Convers. Mgmt 21, 283 (1981).