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Numerical study of flow and heat transfer characteristics in hot water stores
PII: S0038-092X(98)00051-6
Solar Energy Vol. 64, Nos 1–3, pp. 9–18, 1998 © 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X/98 $—see front matter
NUMERICAL STUDY OF FLOW AND HEAT TRANSFER CHARACTERISTICS IN HOT WATER STORES E. HAHNE and Y. CHEN University of Stuttgart, Institut fu¨r Thermodynamik und Wa¨rmetechnik (ITW ), Pfaffenwaldring 6, D-70550 Stuttgart, Germany Received 25 September 1997; revised version accepted 22 April 1998 Communicated by DOUG HITTLE Abstract—The flow and heat transfer characteristics in a cylindrical hot water store during the charging process under adiabatic thermal boundary conditions were studied numerically in the present paper. The charging efficiency was used to evaluate the thermal stratification. The emphasis was put on the effects of charging temperature differences, charging velocities, charging flow rates and length to diameter ratios on the charging efficiency. The results were summarized both in dimensional and dimensionless forms. They indicate that the charging efficiency depends mainly on the modified Richardson number Ri and H,f Peclet number Pe , which present the combined effects of charging temperature difference and charging H,f velocity on the charging efficiency. If Ri is larger than 0.25, the charging efficiency is above 97%. At a H,f given Richardson number the increase of Peclet number leads to a higher charging efficiency. For H/D less than 4, the increase of the height to diameter ratio H/D can improve the charging efficiency as well. The effect of the Fourier number (or charging flow rate) on the charging efficiency, however, is relatively small. A correlation of the numerical results was obtained for the design of effective hot water stores. © 1998 Elsevier Science Ltd. All rights reserved.
charging flow rate and height to diameter ratio H/D on thermal stratification in hot water stores seem to need more investigation. In order to reveal the flow and heat transfer characteristics in cylindrical hot water stores during a charging process and the effects of charging temperature difference and velocity, charging flow rate and height to diameter ratio H/D on the thermal stratification in the store, a numerical investigation of the flow and heat transfer characteristics in a cylindrical hot water store during charging processes under adiabatic thermal boundary conditions is performed in the present paper.
1. INTRODUCTION
The performance of a solar heating system can be improved significantly as the storage medium in the hot water store remains thermally stratified (Hollands and Lightstone, 1989). The thermal stratification in a hot water store depends on the heat conduction and convection in the storage medium, the heat losses via the tank surface and heat conduction along the tank wall as well as on forced and free convection in the store caused by charging and discharging processes. During the last 20 years many investigations, both numerical and theoretical as well as experimental, have focused on the destratification, which is caused by heat conduction and convection in the storage medium and heat conduction along the tank wall (Jaluria and Gupta, 1982; Hesse and Miller, 1982; Shyu et al., 1989; Murthy et al., 1992; Yoo and Pak, 1993). The effects of different parameters on the thermal stratification in a cylindrical hot water store during the discharging process have also been experimentally and numerically investigated by Lavan and Thompson (1977) and van Berkel (1996). The experimental investigation by Sliwinski et al. (1978) has indicated that the thermal stratification in a hot water store during the charging process depends mainly on the modified Richardson number Ri and Peclet number H,f Pe . However, the effects of the Peclet number, H,f
2. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
The hot water store under investigation is shown schematically in Fig. 1. It is a cylindrical steel water tank. We assume that there are no heat losses to the environment and no heat conduction along the tank wall. The water store is initially filled with cold water. Then hot water is filled in through the inlet port located on the top, and cold water, at the same time, flows out from the outlet port located on the bottom of the store. The flow and heat transfer characteristics in a hot water store during the charging process can be described with the continuity equation, 9
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E. Hahne and Y. Chen
v*=
∂w* ∂w* r − z ∂z* ∂r*
(3)
1 ∂y* w* =− r r* ∂z*
(4)
1 ∂y* w* = z r* ∂r*
(5)
All variables in eqns (1)–(5) are expressed in dimensionless form as follows: w w , w* = r , w* = z , z r H H w w in in w t q−q ini t *= in , h= H q −q in ini and the dimensionless numbers in these equations are: r*=
r
, z*=
z
w H n w H Re = in , Pr = , Pe = in , H,f H,f a H,f n a Gr = H,f
Fig. 1. Schematic of the investigated hot water store.
momentum equation and energy equation. For two-dimensional problems, it is convenient to combine the continuity equation and momentum equation into a vorticity transport equation and a stream function equation by introducing the stream function y and vorticity v. Neglecting the dissipation term in the energy equation, one obtains, subject to the Boussinesq approximation, the dimensionless equations for two-dimensional, incompressible laminar flow in cylindrical coordinates as follows: ∂h ∂(w* v*) ∂(w* v*) Gr r z + =− H,f ∂t * ∂r* ∂z* Re2 ∂r* H,f ∂ 1 ∂ 1 ∂2v* + (1) (r*v*) + Re ∂r* r* ∂r* ∂z*2 H,f 1 ∂ ∂(w* h) ∂h z + (r*w* h)+ r ∂t * r* ∂r* ∂z*
∂v*
gb(q −q )H3 in ini n2
The store height H has been chosen here as the characteristic dimension according to Sliwinski et al. (1978) and Yoo and Pak (1993). The mean water temperature q =(q +q )/2 has f in ini been taken as the characteristic temperature for thermal properties in the dimensionless numbers. Boundary and initial conditions for eqns (1)– (5) are: r*=0:
y*=0, v*=0, ∂h/∂r*=0
r*=R /H: y*=−(r /H )2/2, ∂h/∂r*=0 i i z*=0 or z*=1 and r*≤r /H: y*=−r*2/2, ∂h/∂z*=0 i z*=0 or z*=1 and r*>r /H: y*=−(r /H )2/2, ∂h/∂z*=0 i i t *=0: y*=0, v*=0, h=0, w* =0, w* =0 r z
+
G C
=
1 Pe H,f
C
1 ∂
r* ∂r*
D
A
r*
∂h ∂r*
B
+
H
∂2h ∂z*2
D
(2)
3. NUMERICAL METHODS
For the numerical solution of the governing equations, the finite difference method is used. The vorticity and energy equations have the same form, i.e. parabolic in time and elliptic in space, so they are solved with the ADI method (alternative direction implicit) developed by Peaceman and Rachford (1955). The advantage of the ADI method over other fully implicit
Flow and heat transfer characteristics in hot water stores
methods is that with this method only the solution of a tridiagonal system is required, which occurs only for the usual implicit methods in one dimension. Furthermore, it has a second order accuracy of O(Dt2, Dr2, Dz2) and its ‘‘weak’’ stability conditions are easy to satisfy. For more details about the ADI method see Peaceman and Rachford (1955) and Roache (1976). The stream function equation is an elliptic Poisson equation, which can be solved by the SOR method (successive over-relaxation) developed by Frankel (1950) and Young (1954). During the iteration process a constant relaxation factor Q=1.5 was used. The error of this process is controlled by setting:
K
K
(y* )k+1−(y* )k ≤10−8 i,j i,j
The convection terms in the governing equations in r- and z-directions are approximated by the second upwind scheme (Patankar, 1990) and the MLU method [monotonized linear upwind method (Noll, 1993)], respectively, which permit us to achieve a stable numerical method ( Ku¨blbeck et al., 1980; Noll, 1993). In order to satisfy the mass conservation and the circulation theorem, both locally and globally in cylindrical coordinates, the velocity fields are computed based on the concept of a kinematically consistent velocity devised by Parmenter and Torrance (1975). The vorticity values at the solid boundary are calculated according to the method suggested by Roache (1976). In the calculation, a regular mesh system has been used. Five different grid spacings have
11
been tested. The numerical results of the outlet temperatures during a charging process with different node numbers, the experimental result performed at ITW and the one-dimensional theoretical solution of Yoo and Pak (1993) under the corresponding Peclet number are illustrated in Fig. 2. It shows that the outlet temperature difference between the numerical and experimental results decreases with increasing node number N (or decreasing grid spacing) in the axial direction. This improvement will not be evident if N is greater than 41. The increase of the node number M in the radial direction from 35 to 101 for N=61 leads to little improvement of the numerical results. Because of the heat loss to the surroundings during the experiment, a temperature difference of about 1.5 K between the experimental and numerical (N=61, M=35) results has been appeared 1.5 h after the beginning of the charging process (after 5655 s the store volume is already fully replaced). According to the test results a regular mesh system with node number in the axial direction of N=61 and in the radial direction of M=35 to 61 has been used in the present numerical investigation. The inlet and outlet of the store were generally covered by one or two cells according to Cabelli (1977) and Guo and Wu (1985). A cylindrical hot water store with about 410 litres volume was employed as a prototype of the investigated hot water store. It was initially filled with cold water at 20°C. Then hot water at different temperatures, flow rates and inlet velocities was charged into the store. The height to diameter ratios of the store H/D were
Fig. 2. Comparison of the numerical results with the experimental and theoretical results.
12
E. Hahne and Y. Chen
Fig. 3. Flow and temperature fields in hot water store.
also changed in the investigation. The corresponding Grashof numbers Gr and Reynolds H,f numbers Re are varied from 1.22×107 to H,f 2.63×1012 and 1.49×105 to 1.08×106, respectively.
4. RESULTS AND DISCUSSION
4.1. Flow and temperature fields Two typical flow and temperature fields are given in Fig. 3 in terms of streamlines and
Flow and heat transfer characteristics in hot water stores
isotherms for different charging temperatures and velocities at 3360 s after the beginning of the charging process. As charging time t of the c store we consider the time necessary to fully replace the store volume with a known volume and a given mass flow rate, this time can easily be calculated. Here it is 5600 s. The height to diameter ratio H/D is 2.41. It can be seen that the flow and temperature fields inside the store depend strongly on the modified Richardson number Gr gbDqH Ri = H,f = . H,f Re2 w2 H,f in For low Ri the momentum of the inlet flow H,f jet is much greater than the buoyancy. Most of the inlet hot water would be directly discharged from the bottom of the store. A big eddy inside the store would be developed because of the plume entrainment of the inlet flow jet. Corresponding to this flow characteristic, the water temperature in a narrow area around the inlet flow jet is almost identical to the charging temperature. Between this area and the other part of the store a horizontal thermocline is developed due to heat transfer between the hot inlet water and the cold water in the store. Beyond this thermocline zone the temperatures form almost horizontal isotherms. The vertical temperature gradient, representing the thermal stratification inside the store, is relatively small. With a large modified Richardson number Ri , the momentum of the inlet flow jet H,f becomes low in comparison with the buoyancy. In this case, the inlet hot water cannot penetrate into the cold water zone and down to the bottom of the store. Thus the direct discharging of inlet hot water is inhibited. The hot water flows to the upper part of the store due to the buoyancy effect and the cold water is withdrawn from the bottom. The temperatures inside the store are presented by horizontal isotherms. A large temperature gradient within a small zone gives a good thermal stratification here. 4.2. Charging efficiency A good thermal stratification in a hot water store results in a high storage efficiency at a given storage temperature. The thermal stratification varies, however, with time and charging parameters as well as initial conditions. It is difficult to measure it accurately in practice. In order to evaluate the thermal stratification in a hot water store during a charging process, the charging efficiency g will be used. It is defined c as the ratio of the net stored thermal energy in
13
a store at the end of the charging process to the thermodynamic maximum storable energy:
P
tc m ˙ c [q −q (t)] dt in p in o g= 0 (6) c mc (q −q ) p in ini where m is the total mass of water in the store, m ˙ the charging mass flow rate, t the charging in c time of the store, and q , q and q are the in o ini inlet, outlet and initial water temperatures, respectively. The charging efficiency g is the same as the c storage efficiency for the charging process used by Yoo and Pak (1993). A similar definition, termed the figure of merit (FOM ), was used by Tran et al. (1988) and by Wildin and Truman (1989) to evaluate the thermal performance of the store during a charging and discharging circle. Ideally the charging efficiency g should be c one, if there was no heat transfer between the inlet hot water and the initial cold water in the store. In fact g is less than one due to the c existence of heat transfer inside the store, which is affected by many parameters. 4.2.1. Effects of charging temperature and velocity. Figure 4 shows the effects of charging temperature difference and velocity on the charging efficiency when the charging time t is c 5600 s. The initial water temperature in the store is 20°C and the height to diameter ratio is 2.41. It can be seen that the charging efficiency depends strongly on the charging temperature difference as long as this is small (<1 K ). With small charging temperature differences, at large charging velocities (i.e. small Ri ) compared H,f to a weak buoyancy most of the incoming hot water will be directly discharged from the store [e.g. Figure 3(a) left]. This results in the very low charging efficiency. As the charging temperature difference increases, the buoyancy effect becomes stronger in comparison to the momentum of the inlet flow jet. Thus the direct discharging of hot water is inhibited and the charging efficiency increases. When the charging temperature difference is greater than 20 K, the charging efficiency remains almost constant (97–98%). Only 2–3% of the input energy is lost due to the weak heat transfer across the thermocline. The effect of charging velocity on the charging efficiency is also shown in the same figure. For a given charging temperature difference we have the following results: for a large velocity, the charging efficiency is low. This is due to the
14
E. Hahne and Y. Chen
Fig. 4. Effects of charging temperature difference and velocity on the charging efficiency.
direct discharge of the incoming hot water and the intensive mixing by charging. However, this effect decreases as the charging temperature difference increases. When the charging temperature difference is larger than 20 K, the effect of velocity on the charging efficiency can be neglected. 4.2.2. Effect of charging flow rate. The effect of charging flow rate on the charging efficiency is shown in Fig. 5. With high inlet temperature
(q =80°C ), i.e. large charging temperature in difference and large modified Richardson number (Ri =7.12), the charging efficiency H,f increases only slightly as the charging flow rate varies from 2×10−5 to 7.4×10−5 m3/s. With low inlet temperature (q =22°C ), or small in modified Richardson number (Ri =0.114), H,f the charging efficiency decreases with increase of the charging flow rate. This is due to the flow characteristics in the store. With a large
[10-5m3/s]
Fig. 5. Effects of charging flow rate on the charging efficiency.
Flow and heat transfer characteristics in hot water stores
Ri a good thermal stratification will be develH,f oped by the buoyancy effect. The heat transfer from the inlet hot water to the cold water in the store is mainly by the heat convection and conduction across the thermocline. As the charging flow rate increases, the charging time t decreases, and hence the time for the heat c transfer process in the store. Consequently, little input energy will be lost. With a small modified Richardson number, the increase of the charging flow rate intensifies the mixing in the store, and so the charging efficiency is decreased. However, the effect of the charging flow rate on the charging efficiency is much smaller than the effect of charging temperature difference. 4.2.3. Effect of the height to diameter ratio H/D. The effect of the height to diameter ratio H/D on the charging efficiency is shown in Fig. 6. It can be seen that for a small modified
15
Richardson number Ri the charging efficiency H,f increases steeply as H/D varies from 1 to 4. The variation of the charging efficiency for a large Ri is, however, comparatively small as H/D H,f increases. This result can be explained as the reduction of the heat transfer cross-section and the increase of the modified Peclet number. With the increase of H/D the heat transfer crosssection will be decreased, and hence the input energy loss. On the other hand, the increasing of H/D at a given modified Richardson number means the increase of the modified Peclet number, which, as will be seen in Fig. 7, leads to an increasing of the charging efficiency. At large Richardson number the flow field in the store tends to be a piston flow and a good thermal stratification has been developed, as shown in Fig. 3. The heat transfer across the thermocline is now very weak, so the effect of
Fig. 6. Effects of the height to diameter ratio H/D on the charging efficiency.
Fig. 7. Effects of the modified Richardson number and Peclet number on the charging efficiency.
16
E. Hahne and Y. Chen
H/D on the charging efficiency is also not evident. An H/D value between 3 and 4 appears to be a reasonable compromise in practice. 4.2.4. Effect of the modified Richardson number and the Peclet number. Having presented the results in dimensional form, the effects of different dimensionless numbers on the charging efficiency can be illustrated, in order to reveal some general flow and heat transfer characteristics in hot water stores during a charging process. The effects of the modified Richardson number Ri and the Peclet number Pe on H,f H,f the charging efficiency are presented in Fig. 7. It shows that for a given Peclet number the charging efficiency increases steeply as Ri H,f varies from 0.001 to 0.01. If the modified Richardson number is greater than 0.25, the charging efficiency is almost constant. This result reveals an important flow and heat transfer characteristic in cylindrical hot water stores during a charging process. As mentioned above, the flow and temperature fields depend strongly on the modified Richardson number. At a small Richardson number much input energy is lost due to the direct discharging of the input hot water from the bottom of the store or due to the high outlet temperature caused by the intensive mixing between the inlet hot water and the cold water in the store. These result in a low charging efficiency. As the Richardson number increases the direct discharging of the inlet hot water is gradually inhibited and the mixing between hot and cold water in the store is also diminished. Hence the charging efficiency increases steeply. The experimental result of Sliwinski et al. (1978) has shown that if the modified Richardson number is greater than 0.244, a good thermal stratification can be developed in the store. This result is also indirectly revealed in Fig. 7. It can be seen that for modified Richardson number greater than 0.25 the charging efficiency remains between 97% and 98%. Only 2–3% of the input energy is lost due to the weak heat transfer across the thermocline. The one-dimensional theoretical solution of Yoo and Pak (1993) has indicated that at the corresponding modified Peclet number in Fig. 7 the charging efficiency lies between 98.5% and 98.8%. The effects of the modified Peclet number on the charging efficiency can also be seen from the same figure. It shows that at a given modified Richardson number, for Ri ≤0.1, the H,f larger the Peclet number, the higher the charg-
ing efficiency. If the modified Richardson number is greater than 0.25, the effect of the Peclet number on the charging efficiency is no longer evident. It can be seen from the governing Eqns (1) and (2) that the modified Richardson and Peclet numbers are the two important numbers which determine the flow and heat transfer characteristics in a hot water store. For Ri <0.1 the flow H,f field in the store is still two-dimensional. The increase of Peclet number leads to a weak heat transfer between hot and cold water in the store. So the inlet jet will be cooled slowly. This prevents the inlet flow jet from penetrating into the cold water zone to cause more mixing between hot and cold water in the store. The charging efficiency is, therefore, increased. For Ri >0.25 the flow field in the store tends to H,f be an ideal piston flow. The charging efficiency, as mentioned above, is over 97%. So the effect of Peclet number on the charging efficiency is very small. 4.2.5. Effect of the Fourier number. Figure 8 shows the effect of the Fourier number Fo on H,f the charging efficiency. The theoretical results of Yoo and Pak (1993) are also illustrated. It can be seen that for Ri =0.115 the charging H,f efficiency increases with increasing of the Fourier number. For Ri =7.12, however, the H,f charging efficiency decreases as the Fourier number varies from 4.14×10−4 to 1.66×10−3. The Fourier number represents the dimensionless time of a heat transfer process. For a large modified Richardson number (Ri >0.25) the increasing of the Fourier H,f number prolongs the time for the heat transfer process across the thermocline. This leads to a decrease of the charging efficiency. In comparison with the results of Yoo and Pak (1993), the numerical result is 0.3–0.8% lower than the theoretical value under the same conditions. At a relatively small modified Richardson number the increasing of the Fourier number also makes for more heat transfer to the store domain because of the two-dimensional flow and temperature distribution characteristics in the store. So the charging efficiency increases a little. In comparison to the effect of the modified Richardson number and the Peclet number on the charging efficiency, the effect of the Fourier number is much smaller. 4.3. Correlation of the results Having determined the effects of various parameters on the charging efficiency, we attempt to obtain a correlation that would predict these effects.
Flow and heat transfer characteristics in hot water stores
17
Fig. 8. Effects of the Fourier number on the charging efficiency.
A least-squares fit of all the numerical results yields the correlation given in eqn (7). It can be seen from Fig. 9 that the proposed curve fits the calculated data quite well. The maximum deviation of the correlation from the numerical results is ±15%: g =1−0.206Ri−0.57Pe−0.49Fo−0.74 (H/D)−1.10 c H,f H,f H,f (7) This correlation applies only for water and is valid for: 0.0013
Thermal stratification in a hot water store, which is affected by the flow and heat transfer
characteristics in it, is very important for the efficiency of a solar heating system. It can be improved by detailed design and by regulating charging parameters. It can also easily be evaluated with the use of a charging efficiency. The present numerical study shows that the flow and heat transfer characteristics, and hence the thermal stratification, in the store depend mainly on the modified charging Richardson number Ri and the Peclet number Pe , which H,f H,f represent the combined effects of the charging temperature difference and velocity. If Ri is H,f greater than 0.25, the flow field in the store tends to be the piston flow, an almost onedimensional temperature field and a good thermal stratification are developed in it. The charging efficiency is, under this condition, over 97%. Increasing the Peclet number at a given modified Richardson number results in a higher charging
Fig. 9. Correlation of the charging efficiency.
18
E. Hahne and Y. Chen
efficiency. This effect is especially evident for Richardson number smaller than 0.01. The effect of the Fourier number, representing mainly the effect of charging flow rate (or charging time t ), on the charging efficiency is, c however, relatively small. If H/D is less than 4, the increase of the height to diameter ratio H/D can improve the charging efficiency as well. The numerical results are correlated to predict the charging efficiency in terms of the modified Richardson number Ri , Peclet number Pe , H,f H,f Fourier number Fo as well as the height to H,f diameter ratio H/D of a hot water store. The maximum deviation between the numerical results and the correlation is ±15%. Acknowledgements—The Friedrich-Ebert-Stiftung supported this work by financing the co-author. This is gratefully acknowledged.
NOMENCLATURE c p D Fo g Gr H m m ˙ M N Pr r r i R i Re Ri t t c w z
specific heat ( p=const) [J/(kg K )] store diameter (m) Fourier number, Fo=at/H2 (—) gravitational acceleration (m/s2) Grashof number, Gr=gb(q −q )H3/n2 (—) in ini store height (m) total mass (kg) charging mass flow rate (kg/s) nodes in r-direction (—) nodes in z-direction (—) Prandtl number, Pr=n/a (—) radial coordinate (m) inlet port radius (m) store radius (—) modified Reynolds number, Re =w H/n (—) H,f in modified Richardson number, Ri =Gr/Re2=gb(q −q )H/w2 (—) H,f in ini in time (s) charging time of stores (s) velocity (m/s) axial coordinate (m)
Greek letters a b g n r y v
thermal diffusivity (m2/s) coefficient of expansion (1/K ) efficiency (—) kinematic viscosity (m2/s) fluid density (kg/m3) stream function (m3/s) vorticity (1/s)
Superscripts k *
iterations step dimensionless quantity
Subscripts f H i in ini
fluid store height inner inlet or charging initial
o r z
outlet or discharging radial component axial component
REFERENCES Cabelli A. (1977) Storage tanks—A numerical experiment. Solar Energy 19, 45–54. Frankel S. P. (1950) Convergence rates of iterative treatments of partial differential equations. Math Tables and Other Aids to Computation 4, 65–75. Guo K. L. and Wu S. T. (1985) Numerical study of flow and temperature stratifications in a liquid thermal storage tank. Journal of Solar Energy Engineering 107, 15–20. Hesse C. F. and Miller C. W. (1982) An experimental and numerical study on the effect of the wall in thermocline type cylindrical enclosure—I and II. Solar Energy 28, 2, 145–161. Hollands K. G. and Lightstone M. F. (1989) A review of low-flow, stratified-tank solar water heating systems. Solar Energy 43, 2, 97–105. Jaluria Y. and Gupta S. K. (1982) Decay of thermal stratification in a water body for solar energy storage. Solar Energy 28, 2, 137–143. Ku¨blbeck K., Merker G. P. and Straub J. (1980) Advanced numerical computation of two-dimensional time-dependent free convection in cavities. International Journal of Heat and Mass Transfer 23, 203–217. Lavan Z. and Thompson J. (1977) An experimental study of thermal stratified hot water storage tank. Solar Energy 19, 519–524. Murthy S. S., Nelson J. E. B. and Sitharama Rao T. L. (1992) Effect of wall conductivity on thermal stratification. Solar Energy 49, 4, 273–277. Noll, B. (1993) Numerische Stro¨mungsmechanik. Springer, Berlin. Parmenter E. M. and Torrance K. E. (1975) Kinematically consistent velocity fields for hydrodynamic calculation in curvilinear coordinates. Journal of Computational Physics 19, 404–419. Patankar S. V. (1990) Computational Heat Transfer and Fluid Flow. McGraw-Hill, New York. Peaceman D. W. and Rachford H. H. (1955) The numerical solution of parabolic and elliptic differential equations. Journal of the Society of Industrial Applied Mathematics 3, 28–41. Roache P. J. (1976) Computational Fluid Dynamics. Hermosa, Albuquerque, USA. Shyu P. J., Lin J. Y. and Fang L. J. (1989) Thermal analysis of stratified storage tanks. Journal of Solar Energy Engineering 111, 54–61. Sliwinski B. J., Mech A. R. and Shih T. S. (1978) Stratification in thermal storage during charging, 6th International Heat Transfer Conference, Toronto, Vol. 4, pp. 149–154. Tran N., Kreider J. F. and Brother P. (1988) Final report on field measurement of chilled water storage system performance (PR482), Joint Center for Energy Management, University of Colorado, Boulder, CO. van Berkel J. (1996) Mixing in thermally stratified energy stores. Solar Energy 58, 4–6, 203–211. Wildin M. W. and Truman C. R. (1989) Performance of stratified vertical cylindrical thermal storage tanks, Part I: Scale model tank. ASHRAE Transactions 89, 2, 1086–1095. Yoo H. and Pak E. (1993) Theoretical model of the charging process for stratified thermal storage tank. Solar Energy 51, 6, 513–519. Young D. (1954) Iterative methods for solving partial differential equations of elliptic type. Transactions of the American Mathematical Society 76, 92–111.
Solar Energy Vol. 64, Nos 1–3, pp. 9–18, 1998 © 1998 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0038-092X/98 $—see front matter
NUMERICAL STUDY OF FLOW AND HEAT TRANSFER CHARACTERISTICS IN HOT WATER STORES E. HAHNE and Y. CHEN University of Stuttgart, Institut fu¨r Thermodynamik und Wa¨rmetechnik (ITW ), Pfaffenwaldring 6, D-70550 Stuttgart, Germany Received 25 September 1997; revised version accepted 22 April 1998 Communicated by DOUG HITTLE Abstract—The flow and heat transfer characteristics in a cylindrical hot water store during the charging process under adiabatic thermal boundary conditions were studied numerically in the present paper. The charging efficiency was used to evaluate the thermal stratification. The emphasis was put on the effects of charging temperature differences, charging velocities, charging flow rates and length to diameter ratios on the charging efficiency. The results were summarized both in dimensional and dimensionless forms. They indicate that the charging efficiency depends mainly on the modified Richardson number Ri and H,f Peclet number Pe , which present the combined effects of charging temperature difference and charging H,f velocity on the charging efficiency. If Ri is larger than 0.25, the charging efficiency is above 97%. At a H,f given Richardson number the increase of Peclet number leads to a higher charging efficiency. For H/D less than 4, the increase of the height to diameter ratio H/D can improve the charging efficiency as well. The effect of the Fourier number (or charging flow rate) on the charging efficiency, however, is relatively small. A correlation of the numerical results was obtained for the design of effective hot water stores. © 1998 Elsevier Science Ltd. All rights reserved.
charging flow rate and height to diameter ratio H/D on thermal stratification in hot water stores seem to need more investigation. In order to reveal the flow and heat transfer characteristics in cylindrical hot water stores during a charging process and the effects of charging temperature difference and velocity, charging flow rate and height to diameter ratio H/D on the thermal stratification in the store, a numerical investigation of the flow and heat transfer characteristics in a cylindrical hot water store during charging processes under adiabatic thermal boundary conditions is performed in the present paper.
1. INTRODUCTION
The performance of a solar heating system can be improved significantly as the storage medium in the hot water store remains thermally stratified (Hollands and Lightstone, 1989). The thermal stratification in a hot water store depends on the heat conduction and convection in the storage medium, the heat losses via the tank surface and heat conduction along the tank wall as well as on forced and free convection in the store caused by charging and discharging processes. During the last 20 years many investigations, both numerical and theoretical as well as experimental, have focused on the destratification, which is caused by heat conduction and convection in the storage medium and heat conduction along the tank wall (Jaluria and Gupta, 1982; Hesse and Miller, 1982; Shyu et al., 1989; Murthy et al., 1992; Yoo and Pak, 1993). The effects of different parameters on the thermal stratification in a cylindrical hot water store during the discharging process have also been experimentally and numerically investigated by Lavan and Thompson (1977) and van Berkel (1996). The experimental investigation by Sliwinski et al. (1978) has indicated that the thermal stratification in a hot water store during the charging process depends mainly on the modified Richardson number Ri and Peclet number H,f Pe . However, the effects of the Peclet number, H,f
2. GOVERNING EQUATIONS AND BOUNDARY CONDITIONS
The hot water store under investigation is shown schematically in Fig. 1. It is a cylindrical steel water tank. We assume that there are no heat losses to the environment and no heat conduction along the tank wall. The water store is initially filled with cold water. Then hot water is filled in through the inlet port located on the top, and cold water, at the same time, flows out from the outlet port located on the bottom of the store. The flow and heat transfer characteristics in a hot water store during the charging process can be described with the continuity equation, 9
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E. Hahne and Y. Chen
v*=
∂w* ∂w* r − z ∂z* ∂r*
(3)
1 ∂y* w* =− r r* ∂z*
(4)
1 ∂y* w* = z r* ∂r*
(5)
All variables in eqns (1)–(5) are expressed in dimensionless form as follows: w w , w* = r , w* = z , z r H H w w in in w t q−q ini t *= in , h= H q −q in ini and the dimensionless numbers in these equations are: r*=
r
, z*=
z
w H n w H Re = in , Pr = , Pe = in , H,f H,f a H,f n a Gr = H,f
Fig. 1. Schematic of the investigated hot water store.
momentum equation and energy equation. For two-dimensional problems, it is convenient to combine the continuity equation and momentum equation into a vorticity transport equation and a stream function equation by introducing the stream function y and vorticity v. Neglecting the dissipation term in the energy equation, one obtains, subject to the Boussinesq approximation, the dimensionless equations for two-dimensional, incompressible laminar flow in cylindrical coordinates as follows: ∂h ∂(w* v*) ∂(w* v*) Gr r z + =− H,f ∂t * ∂r* ∂z* Re2 ∂r* H,f ∂ 1 ∂ 1 ∂2v* + (1) (r*v*) + Re ∂r* r* ∂r* ∂z*2 H,f 1 ∂ ∂(w* h) ∂h z + (r*w* h)+ r ∂t * r* ∂r* ∂z*
∂v*
gb(q −q )H3 in ini n2
The store height H has been chosen here as the characteristic dimension according to Sliwinski et al. (1978) and Yoo and Pak (1993). The mean water temperature q =(q +q )/2 has f in ini been taken as the characteristic temperature for thermal properties in the dimensionless numbers. Boundary and initial conditions for eqns (1)– (5) are: r*=0:
y*=0, v*=0, ∂h/∂r*=0
r*=R /H: y*=−(r /H )2/2, ∂h/∂r*=0 i i z*=0 or z*=1 and r*≤r /H: y*=−r*2/2, ∂h/∂z*=0 i z*=0 or z*=1 and r*>r /H: y*=−(r /H )2/2, ∂h/∂z*=0 i i t *=0: y*=0, v*=0, h=0, w* =0, w* =0 r z
+
G C
=
1 Pe H,f
C
1 ∂
r* ∂r*
D
A
r*
∂h ∂r*
B
+
H
∂2h ∂z*2
D
(2)
3. NUMERICAL METHODS
For the numerical solution of the governing equations, the finite difference method is used. The vorticity and energy equations have the same form, i.e. parabolic in time and elliptic in space, so they are solved with the ADI method (alternative direction implicit) developed by Peaceman and Rachford (1955). The advantage of the ADI method over other fully implicit
Flow and heat transfer characteristics in hot water stores
methods is that with this method only the solution of a tridiagonal system is required, which occurs only for the usual implicit methods in one dimension. Furthermore, it has a second order accuracy of O(Dt2, Dr2, Dz2) and its ‘‘weak’’ stability conditions are easy to satisfy. For more details about the ADI method see Peaceman and Rachford (1955) and Roache (1976). The stream function equation is an elliptic Poisson equation, which can be solved by the SOR method (successive over-relaxation) developed by Frankel (1950) and Young (1954). During the iteration process a constant relaxation factor Q=1.5 was used. The error of this process is controlled by setting:
K
K
(y* )k+1−(y* )k ≤10−8 i,j i,j
The convection terms in the governing equations in r- and z-directions are approximated by the second upwind scheme (Patankar, 1990) and the MLU method [monotonized linear upwind method (Noll, 1993)], respectively, which permit us to achieve a stable numerical method ( Ku¨blbeck et al., 1980; Noll, 1993). In order to satisfy the mass conservation and the circulation theorem, both locally and globally in cylindrical coordinates, the velocity fields are computed based on the concept of a kinematically consistent velocity devised by Parmenter and Torrance (1975). The vorticity values at the solid boundary are calculated according to the method suggested by Roache (1976). In the calculation, a regular mesh system has been used. Five different grid spacings have
11
been tested. The numerical results of the outlet temperatures during a charging process with different node numbers, the experimental result performed at ITW and the one-dimensional theoretical solution of Yoo and Pak (1993) under the corresponding Peclet number are illustrated in Fig. 2. It shows that the outlet temperature difference between the numerical and experimental results decreases with increasing node number N (or decreasing grid spacing) in the axial direction. This improvement will not be evident if N is greater than 41. The increase of the node number M in the radial direction from 35 to 101 for N=61 leads to little improvement of the numerical results. Because of the heat loss to the surroundings during the experiment, a temperature difference of about 1.5 K between the experimental and numerical (N=61, M=35) results has been appeared 1.5 h after the beginning of the charging process (after 5655 s the store volume is already fully replaced). According to the test results a regular mesh system with node number in the axial direction of N=61 and in the radial direction of M=35 to 61 has been used in the present numerical investigation. The inlet and outlet of the store were generally covered by one or two cells according to Cabelli (1977) and Guo and Wu (1985). A cylindrical hot water store with about 410 litres volume was employed as a prototype of the investigated hot water store. It was initially filled with cold water at 20°C. Then hot water at different temperatures, flow rates and inlet velocities was charged into the store. The height to diameter ratios of the store H/D were
Fig. 2. Comparison of the numerical results with the experimental and theoretical results.
12
E. Hahne and Y. Chen
Fig. 3. Flow and temperature fields in hot water store.
also changed in the investigation. The corresponding Grashof numbers Gr and Reynolds H,f numbers Re are varied from 1.22×107 to H,f 2.63×1012 and 1.49×105 to 1.08×106, respectively.
4. RESULTS AND DISCUSSION
4.1. Flow and temperature fields Two typical flow and temperature fields are given in Fig. 3 in terms of streamlines and
Flow and heat transfer characteristics in hot water stores
isotherms for different charging temperatures and velocities at 3360 s after the beginning of the charging process. As charging time t of the c store we consider the time necessary to fully replace the store volume with a known volume and a given mass flow rate, this time can easily be calculated. Here it is 5600 s. The height to diameter ratio H/D is 2.41. It can be seen that the flow and temperature fields inside the store depend strongly on the modified Richardson number Gr gbDqH Ri = H,f = . H,f Re2 w2 H,f in For low Ri the momentum of the inlet flow H,f jet is much greater than the buoyancy. Most of the inlet hot water would be directly discharged from the bottom of the store. A big eddy inside the store would be developed because of the plume entrainment of the inlet flow jet. Corresponding to this flow characteristic, the water temperature in a narrow area around the inlet flow jet is almost identical to the charging temperature. Between this area and the other part of the store a horizontal thermocline is developed due to heat transfer between the hot inlet water and the cold water in the store. Beyond this thermocline zone the temperatures form almost horizontal isotherms. The vertical temperature gradient, representing the thermal stratification inside the store, is relatively small. With a large modified Richardson number Ri , the momentum of the inlet flow jet H,f becomes low in comparison with the buoyancy. In this case, the inlet hot water cannot penetrate into the cold water zone and down to the bottom of the store. Thus the direct discharging of inlet hot water is inhibited. The hot water flows to the upper part of the store due to the buoyancy effect and the cold water is withdrawn from the bottom. The temperatures inside the store are presented by horizontal isotherms. A large temperature gradient within a small zone gives a good thermal stratification here. 4.2. Charging efficiency A good thermal stratification in a hot water store results in a high storage efficiency at a given storage temperature. The thermal stratification varies, however, with time and charging parameters as well as initial conditions. It is difficult to measure it accurately in practice. In order to evaluate the thermal stratification in a hot water store during a charging process, the charging efficiency g will be used. It is defined c as the ratio of the net stored thermal energy in
13
a store at the end of the charging process to the thermodynamic maximum storable energy:
P
tc m ˙ c [q −q (t)] dt in p in o g= 0 (6) c mc (q −q ) p in ini where m is the total mass of water in the store, m ˙ the charging mass flow rate, t the charging in c time of the store, and q , q and q are the in o ini inlet, outlet and initial water temperatures, respectively. The charging efficiency g is the same as the c storage efficiency for the charging process used by Yoo and Pak (1993). A similar definition, termed the figure of merit (FOM ), was used by Tran et al. (1988) and by Wildin and Truman (1989) to evaluate the thermal performance of the store during a charging and discharging circle. Ideally the charging efficiency g should be c one, if there was no heat transfer between the inlet hot water and the initial cold water in the store. In fact g is less than one due to the c existence of heat transfer inside the store, which is affected by many parameters. 4.2.1. Effects of charging temperature and velocity. Figure 4 shows the effects of charging temperature difference and velocity on the charging efficiency when the charging time t is c 5600 s. The initial water temperature in the store is 20°C and the height to diameter ratio is 2.41. It can be seen that the charging efficiency depends strongly on the charging temperature difference as long as this is small (<1 K ). With small charging temperature differences, at large charging velocities (i.e. small Ri ) compared H,f to a weak buoyancy most of the incoming hot water will be directly discharged from the store [e.g. Figure 3(a) left]. This results in the very low charging efficiency. As the charging temperature difference increases, the buoyancy effect becomes stronger in comparison to the momentum of the inlet flow jet. Thus the direct discharging of hot water is inhibited and the charging efficiency increases. When the charging temperature difference is greater than 20 K, the charging efficiency remains almost constant (97–98%). Only 2–3% of the input energy is lost due to the weak heat transfer across the thermocline. The effect of charging velocity on the charging efficiency is also shown in the same figure. For a given charging temperature difference we have the following results: for a large velocity, the charging efficiency is low. This is due to the
14
E. Hahne and Y. Chen
Fig. 4. Effects of charging temperature difference and velocity on the charging efficiency.
direct discharge of the incoming hot water and the intensive mixing by charging. However, this effect decreases as the charging temperature difference increases. When the charging temperature difference is larger than 20 K, the effect of velocity on the charging efficiency can be neglected. 4.2.2. Effect of charging flow rate. The effect of charging flow rate on the charging efficiency is shown in Fig. 5. With high inlet temperature
(q =80°C ), i.e. large charging temperature in difference and large modified Richardson number (Ri =7.12), the charging efficiency H,f increases only slightly as the charging flow rate varies from 2×10−5 to 7.4×10−5 m3/s. With low inlet temperature (q =22°C ), or small in modified Richardson number (Ri =0.114), H,f the charging efficiency decreases with increase of the charging flow rate. This is due to the flow characteristics in the store. With a large
[10-5m3/s]
Fig. 5. Effects of charging flow rate on the charging efficiency.
Flow and heat transfer characteristics in hot water stores
Ri a good thermal stratification will be develH,f oped by the buoyancy effect. The heat transfer from the inlet hot water to the cold water in the store is mainly by the heat convection and conduction across the thermocline. As the charging flow rate increases, the charging time t decreases, and hence the time for the heat c transfer process in the store. Consequently, little input energy will be lost. With a small modified Richardson number, the increase of the charging flow rate intensifies the mixing in the store, and so the charging efficiency is decreased. However, the effect of the charging flow rate on the charging efficiency is much smaller than the effect of charging temperature difference. 4.2.3. Effect of the height to diameter ratio H/D. The effect of the height to diameter ratio H/D on the charging efficiency is shown in Fig. 6. It can be seen that for a small modified
15
Richardson number Ri the charging efficiency H,f increases steeply as H/D varies from 1 to 4. The variation of the charging efficiency for a large Ri is, however, comparatively small as H/D H,f increases. This result can be explained as the reduction of the heat transfer cross-section and the increase of the modified Peclet number. With the increase of H/D the heat transfer crosssection will be decreased, and hence the input energy loss. On the other hand, the increasing of H/D at a given modified Richardson number means the increase of the modified Peclet number, which, as will be seen in Fig. 7, leads to an increasing of the charging efficiency. At large Richardson number the flow field in the store tends to be a piston flow and a good thermal stratification has been developed, as shown in Fig. 3. The heat transfer across the thermocline is now very weak, so the effect of
Fig. 6. Effects of the height to diameter ratio H/D on the charging efficiency.
Fig. 7. Effects of the modified Richardson number and Peclet number on the charging efficiency.
16
E. Hahne and Y. Chen
H/D on the charging efficiency is also not evident. An H/D value between 3 and 4 appears to be a reasonable compromise in practice. 4.2.4. Effect of the modified Richardson number and the Peclet number. Having presented the results in dimensional form, the effects of different dimensionless numbers on the charging efficiency can be illustrated, in order to reveal some general flow and heat transfer characteristics in hot water stores during a charging process. The effects of the modified Richardson number Ri and the Peclet number Pe on H,f H,f the charging efficiency are presented in Fig. 7. It shows that for a given Peclet number the charging efficiency increases steeply as Ri H,f varies from 0.001 to 0.01. If the modified Richardson number is greater than 0.25, the charging efficiency is almost constant. This result reveals an important flow and heat transfer characteristic in cylindrical hot water stores during a charging process. As mentioned above, the flow and temperature fields depend strongly on the modified Richardson number. At a small Richardson number much input energy is lost due to the direct discharging of the input hot water from the bottom of the store or due to the high outlet temperature caused by the intensive mixing between the inlet hot water and the cold water in the store. These result in a low charging efficiency. As the Richardson number increases the direct discharging of the inlet hot water is gradually inhibited and the mixing between hot and cold water in the store is also diminished. Hence the charging efficiency increases steeply. The experimental result of Sliwinski et al. (1978) has shown that if the modified Richardson number is greater than 0.244, a good thermal stratification can be developed in the store. This result is also indirectly revealed in Fig. 7. It can be seen that for modified Richardson number greater than 0.25 the charging efficiency remains between 97% and 98%. Only 2–3% of the input energy is lost due to the weak heat transfer across the thermocline. The one-dimensional theoretical solution of Yoo and Pak (1993) has indicated that at the corresponding modified Peclet number in Fig. 7 the charging efficiency lies between 98.5% and 98.8%. The effects of the modified Peclet number on the charging efficiency can also be seen from the same figure. It shows that at a given modified Richardson number, for Ri ≤0.1, the H,f larger the Peclet number, the higher the charg-
ing efficiency. If the modified Richardson number is greater than 0.25, the effect of the Peclet number on the charging efficiency is no longer evident. It can be seen from the governing Eqns (1) and (2) that the modified Richardson and Peclet numbers are the two important numbers which determine the flow and heat transfer characteristics in a hot water store. For Ri <0.1 the flow H,f field in the store is still two-dimensional. The increase of Peclet number leads to a weak heat transfer between hot and cold water in the store. So the inlet jet will be cooled slowly. This prevents the inlet flow jet from penetrating into the cold water zone to cause more mixing between hot and cold water in the store. The charging efficiency is, therefore, increased. For Ri >0.25 the flow field in the store tends to H,f be an ideal piston flow. The charging efficiency, as mentioned above, is over 97%. So the effect of Peclet number on the charging efficiency is very small. 4.2.5. Effect of the Fourier number. Figure 8 shows the effect of the Fourier number Fo on H,f the charging efficiency. The theoretical results of Yoo and Pak (1993) are also illustrated. It can be seen that for Ri =0.115 the charging H,f efficiency increases with increasing of the Fourier number. For Ri =7.12, however, the H,f charging efficiency decreases as the Fourier number varies from 4.14×10−4 to 1.66×10−3. The Fourier number represents the dimensionless time of a heat transfer process. For a large modified Richardson number (Ri >0.25) the increasing of the Fourier H,f number prolongs the time for the heat transfer process across the thermocline. This leads to a decrease of the charging efficiency. In comparison with the results of Yoo and Pak (1993), the numerical result is 0.3–0.8% lower than the theoretical value under the same conditions. At a relatively small modified Richardson number the increasing of the Fourier number also makes for more heat transfer to the store domain because of the two-dimensional flow and temperature distribution characteristics in the store. So the charging efficiency increases a little. In comparison to the effect of the modified Richardson number and the Peclet number on the charging efficiency, the effect of the Fourier number is much smaller. 4.3. Correlation of the results Having determined the effects of various parameters on the charging efficiency, we attempt to obtain a correlation that would predict these effects.
Flow and heat transfer characteristics in hot water stores
17
Fig. 8. Effects of the Fourier number on the charging efficiency.
A least-squares fit of all the numerical results yields the correlation given in eqn (7). It can be seen from Fig. 9 that the proposed curve fits the calculated data quite well. The maximum deviation of the correlation from the numerical results is ±15%: g =1−0.206Ri−0.57Pe−0.49Fo−0.74 (H/D)−1.10 c H,f H,f H,f (7) This correlation applies only for water and is valid for: 0.0013
Thermal stratification in a hot water store, which is affected by the flow and heat transfer
characteristics in it, is very important for the efficiency of a solar heating system. It can be improved by detailed design and by regulating charging parameters. It can also easily be evaluated with the use of a charging efficiency. The present numerical study shows that the flow and heat transfer characteristics, and hence the thermal stratification, in the store depend mainly on the modified charging Richardson number Ri and the Peclet number Pe , which H,f H,f represent the combined effects of the charging temperature difference and velocity. If Ri is H,f greater than 0.25, the flow field in the store tends to be the piston flow, an almost onedimensional temperature field and a good thermal stratification are developed in it. The charging efficiency is, under this condition, over 97%. Increasing the Peclet number at a given modified Richardson number results in a higher charging
Fig. 9. Correlation of the charging efficiency.
18
E. Hahne and Y. Chen
efficiency. This effect is especially evident for Richardson number smaller than 0.01. The effect of the Fourier number, representing mainly the effect of charging flow rate (or charging time t ), on the charging efficiency is, c however, relatively small. If H/D is less than 4, the increase of the height to diameter ratio H/D can improve the charging efficiency as well. The numerical results are correlated to predict the charging efficiency in terms of the modified Richardson number Ri , Peclet number Pe , H,f H,f Fourier number Fo as well as the height to H,f diameter ratio H/D of a hot water store. The maximum deviation between the numerical results and the correlation is ±15%. Acknowledgements—The Friedrich-Ebert-Stiftung supported this work by financing the co-author. This is gratefully acknowledged.
NOMENCLATURE c p D Fo g Gr H m m ˙ M N Pr r r i R i Re Ri t t c w z
specific heat ( p=const) [J/(kg K )] store diameter (m) Fourier number, Fo=at/H2 (—) gravitational acceleration (m/s2) Grashof number, Gr=gb(q −q )H3/n2 (—) in ini store height (m) total mass (kg) charging mass flow rate (kg/s) nodes in r-direction (—) nodes in z-direction (—) Prandtl number, Pr=n/a (—) radial coordinate (m) inlet port radius (m) store radius (—) modified Reynolds number, Re =w H/n (—) H,f in modified Richardson number, Ri =Gr/Re2=gb(q −q )H/w2 (—) H,f in ini in time (s) charging time of stores (s) velocity (m/s) axial coordinate (m)
Greek letters a b g n r y v
thermal diffusivity (m2/s) coefficient of expansion (1/K ) efficiency (—) kinematic viscosity (m2/s) fluid density (kg/m3) stream function (m3/s) vorticity (1/s)
Superscripts k *
iterations step dimensionless quantity
Subscripts f H i in ini
fluid store height inner inlet or charging initial
o r z
outlet or discharging radial component axial component
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