1、4756 A Semi-Empirical Model for Residential Electric Hot Water Tanks Nima Atabaki Student Member ASHRAE ABSTRACT An experimental and numerical study offluid flow and heat transfer in residential electric hot water tanks with side- wall inlets ispresented in thispaper. Thesemi-empirical model propose
2、d here divides the tank into two distinct regions: a piston-Qpe flow region, where the axial flow velociQ is uniform, and a fully mixed uniform temperature region at the bottom of the tank near the cold water inlet. The piston-Qpe region is treated numerically using a quasi-one-dimensional model, wh
3、ile the height of the fully mixed region is based on an experimentally derived correlation. A full-size transparent tank is used to visualize fluid motion in the fully mixed region. Results of these flow visual- ization studies, combined with temperature measurements, have led to the development of
4、an empirical correlation relat- ing the rate of growth of the height of the mixed region to the entrance Richardson (Ri) number and a dimensionless time parameter representing the percentage of tank discharge. The results obtained in this study confrm that the efects of mixing cannot be neglected. T
5、he semi-empirical model is validated against experimental data for constant and varying inlet Ri. The results for constant Ri agree favorably with the experimental data except at the beginning of the discharge process, where the presence of a small stratgcation zone cannot be predicted by the model.
6、 The results also show that the thermocline and the outlet temperature predictions have a high level of agreement with experimental results. The proposed model is also checked for varying Ri conditions. In this case, the proposed model was able to capture accurately the successive formation of two d
7、iferent fully mixed regions separated by an intermediate thermocline. Michel Bernier, PhD Member ASHRAE INTRODUCTION There are approximately 40 million residential electric water heaters in operation in the United States and Canada (E- news 1993; CEA 1994). Since hot water needs are fairly constant
8、throughout the year, residential electric water heaters constitute an excellent load for electric utilities. However, electric water heating may also contribute significantly to daily peak demand. According to various studies, the diversi- fied peak demand is in the range of 1 to 1.5 kW (Ton-That an
9、d Laperrire 1990; Couture 1990; Hiller et al. 1994). In certain regions, these peak demands are met using old polluting power plants. One way to reduce peak demand is to shut off one (or both) heating elements during peak network demand (Bernier 1 996). Ideally, this load-shedding scheme displaces p
10、ower demand to another non-peak period without affecting the availability of hot water during the load-shedding period. Another minor advantage of load shedding is that average heat losses from the tank are slightly lower since the average tank temperature is lower. In order to predict hot water ava
11、ilability under various load-shedding scenarios, it is useful to be able to predict the transient thermal behavior of hot water tanks over a full day. Although two-dimensional and three-dimensional numerical models are feasible, they require an excessive amount of computational resources for such a
12、task. This study proposes an alternative by using a quasi-one-dimensional numerical model to predict the thermal behavior of electrical hot water tanks. This model is enhanced using empirically derived correlations to account for two three-dimensional phenomena occurring in the tank. The first one i
13、s associated with the fluid motion of thermal plumes when heating elements are acti- vated. This phenomenon has been documented by Atabaki Nima Atabaki is a graduate student in the Department of Mechanical Engineering, McGill University, Montral, Qubec, Canada. Michel Bernier is a professor at cole
14、Polytechnique de Montral, Montral, Qubec, Canada. 02005 ASHRAE. 159 Hot Water Outlet n Inlet Thermostat - - Cold Water Upper Heater Thermostat Lower Heater Figure 1 Schematic presentation of a residential hot water tank with sidewall water inlet. (2001) and Atabaki and Bernier (2001). The second phe
15、nom- enon, which is the main focus of this investigation, has to do with the so-called “mixed region” located in the bottom near the cold water inlet. PROBLEM STATEMENT A schematic representation of a residential electric water heater is shown in Figure 1. Typically, the volume of residen- tial hot
16、water tanks ranges from 175 liters (45 US gal) to 300 liters (80 US gal.). Two heating elements, with power ratings from 3 to 6 kW, provide the necessary power to heat the water. These elements are activateddeactivated by controlling ther- mostats located near the heating elements. The lower heating
17、 element is positioned at some distance from the actual bottom to avoid contact with sediments that may accumulate over time on the bottom. The heating elements usually operate in flip-flop mode with the highest priority assigned to the top element. However, the lower element accomplishes most of th
18、e heating since it is located near the cold water inlet. To promote stratification, the water inlet is located near the bottom and hot water exits at the top. Furthermore, the water inlet velocity is kept low to avoid unnecessary mixing and destratification. Consider the process of discharging hot w
19、ater from a resi- dential hot water tank as presented in Figure 2. Lets assume, as shown in Figure 2a, that the initial conditions are such that the water temperature is uniform from bottom to top and that heating elements are deactivated during the discharge process. When hot water is consumed, col
20、d water enters at the bottom. Two forces act on the incoming cold jet. First, because ofits momentum, the water will have a tendency to travel horizon- tally until it strikes the opposing wall. However, since the incoming water has a higher mass density than the neighbor- ing hot water, the jet will
21、 drop toward the bottom of the tank. As shown in Figure 2b, the end result is the development of a stratified region at the bottom of the tank with arelatively steep Hot t E% Cold Waterwcl Inlet Tem perature Stratifie Temperature - rn m I Thermocllne .- 1 Hot Zone :_j Mixed Region Temperature Figure
22、 2 Schematic presentation of the discharging process in a residential hot water tank: (a) tank initially filled with hot water at uniform temperature (no pow); (b) development of a stratified region near the cold water entrance; (c) development of thermocline and the mixed region on its bottom. temp
23、erature gradient. As cold water continues to enter, a fully mixed uniform temperature region is established. As shown in Figure 2c, a steep temperature gradient region, also referred to as a thermocline, separates the cold mixed region from the hot zone. As water consumption continues, the thermocli
24、ne rises in the tank and also becomes thicker due to heat diffusion. As shown in this sequence of events, the top portion of a properly sized hot water tank should not be affected by the cold water entry at the bottom if the mixed region remains confined to the bottom of the tank. The prediction of
25、the height of the mixed region is crucial if one wants to accurately predict the outlet temperature. The main objective of this paper is to predict the evolution of the mixed region in the tank as well as the outlet temperature for various water inlet conditions. LITERATURE REVIEW A complete literat
26、ure review on thermally stratified tanks can be found in Atabaki (2001). The following paragraphs present a summary of articles pertinent to the present study. Lavan and Thompson (1977) experimentally studied ther- mal stratification in hot water storage tanks. They showed that Stratification improv
27、es with increasing lengtwdiameter ratio, Grashof number, Gr (proportional to the inlet-outlet tempera- ture difference), and inlet and outlet port diameters. Cabelli (1 977) identified the entrance Reynolds number, Re, as well as the Richardson number, Ri (Gr/Re2), as dominant parameters influencing
28、 stratification in the tank. He also proposed a one- dimensional model without mixing effect, and he solved it analytically. Cole and Bellinger (1 982) modified the analyti- cal solution of Cabelli (1977) by introducing a correction factor that accounts for the mixing effect at the tank entrance. Th
29、ey showed that the mixing becomes important for Ri 0; x = O forRi21 T = Tmix tO; x = (RVtot/ACs) forRi 1, then the entire tank (O s . 0.4 0.2 1 A Experimental data - Plot Regr. - L an electrical hot water tank (Q = 5.1 7 L/min, Ri = 0.0578, Tai = 46.49“C, Tent = 22.53“C). 0.0 0.0 0.2 0.4 0.6 0.8 1.0
30、 e = (T-T,i 1 I (Tenl-T, ) Figure 7 Experimental results for axial t=O 053 0 t*=O 153 7 t=O 252 Q t*=O 352 - t=O 451 0 t=O 554 t“=O653 0 t=O 752 A t“= 851 temperature measurements (Q = 5.1 7 Limin, Ri = 0.0578, Cni = 46.49“C, Te, = 22.53“C). 0.45 1. For this value oft * = 0.45 1, the three different
31、 thermal zones identified earlier in relation to Figure 2c are clearly separable: a mixed region below V/V, 0.55. Finally, it is interesting to observe the thickening of the ther- mocline with time. This is mainly caused by heatdifision from the hot zone to the mixed region. ASHRAE Transactions: Res
32、earch 0.30 0.25 0.20 0.15 0.10 0.05 R a test-IO 0 test-Il * test-12 - test-I 0 test-2 test-3 0 test-4 e test4 a test-6 = test-7 s test-8 = test-9 0.00 F 1 0.0 0.2 0.4 0.6 0.8 t* Figure 8 Plot of R as a function oft* to determine R, and Rk Figure 9 Plot of R, as a function of the entrance Richardson
33、number. Ri. Experimental Determination of Coefficients Rk and ti, The semi-empirical model presented here relies on the experimental determination of Rk and R, (Equation 3). A total of 12 tests, similar to the one presented in Figure 7, were performed to evaluate these coefficients. Results are plot
34、ted in Figures 8 and 9. Figure 8 shows the variation ofR as a function oft*. The curves shown in Figure 8 were obtained by evalu- ating T, as a function oft* and then calculating R using Equation 7. For example, in test #1 i, which corresponds to the data in Figure 7, T, = 26.12“C at t* = 0.153 and
35、the corre- sponding value of R is 0.095. As shown in Figure 8, the slopes ofthe R vs. t* curves, Le., Rk, are almost identical and range from O. 193 to 0.220. In this study an average value of 0.205 is used. Therefore, R = R0+0.205t*. (9) 165 1 .o 0.8 0.6 - “ . 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 e
36、=(T-T,) ven,-,“,) t*=0.053 t*=0.153 t*=0.252 t=0.352 t=0.451 t=0.554 t=0.653 t=0.752 t=0.851 Num erica1 Figure 10 Axial temperature measurements (Q = 5.17 L/ min, Ri = 0.0578). A comparison between the proposed semi-empirical numerical model and the experimental results. As presented in Equation 5,
37、R, varies with Ri according to R, = mRi n. The values ofR, obtained from Figure 8 and the corresponding Ri are then plotted on logarithmic scales, as presented in Figure 9, where it can be shown that R, = 0.023Ri-0.376. (10) Validation of the Semi-Empirical Model The numerical model presented here w
38、as validated against several experimental tests obtained with the experi- mental facility described here. Figures 10 and 11 present results for a constant Ri while Figure 12 presents the results of a varying Ri. Constant Inlet Ri. As shown in Figure 10, the semi- empirical numerical model generally
39、predicts well the evolu- tion of the axial temperature in the tank. However, in the begin- ning, at t* = 0.053, the temperature predicted by the numerical model at the bottom of the tank deviates from the one measured experimentally. This is due to the existence of a small stratified region. This re
40、gion is formed at the beginning of the discharge process and is not modeled by the present semi-empirical model, which assumes the creation of a mixed region from the start. Figure 11 presents the variation of 3 with t* for three different positions in the tank: the exit and two positions near the m
41、iddle of the tank. The results presented in Figure 11 confirm the good performance of the model. The point situated at V/Vtot = 99.96%, i.e., close to the exit ofthe tank, is the most important point as it represents the water temperature reaching the consumer. The model prediction for this point is
42、 in good agreement with experimental measurements, with a maximum difference of less than fl .OC. 0.6 0.6 - 0: 0.4 1 . Exp., VN, =99,96% . Exp VN,=65.91% . Exp VN,., =32.69% Numerical - 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 t Figure II Evolution of the dimensionless temperature with dimensionless time
43、 (Q = 5.17 Lhin, Ri = 0.0578). Results of a comparison between the proposed semi-empirical numerical model and experimental results. Variable inlet Ri. Tests were performed for cases where the inlet flow rate is suddenly decreased by keeping the inlet temperature constant. This resulted in an increa
44、se of the inlet Richardson number, which, in turn, promotes the creation of a new mixed region below the first one. In this case, a new R, is calculated using the new entrance Richardson number. The height and the temperature of this new region are calculated by the method described earlier. The old
45、 mixed region then becomes part of the piston-type flow region. The performance of the numerical model was verified for the variable inlet conditions using the following scenario: QI = (10.64L)/min; Ri, = 0.0099 Q, = (4.61L)/min; Ri, = 0.0121 for t* 0.26 (1 1) where qni = 40.37“C, Tent= 22.44“C, and
46、 Tamb = 23.58OC at t* = 0.0. Both numerical and experimental results are shown in Figure 12. Fort* = 0.25, the first set ofconditions applies, and the model prediction deviates from experimental results at the bottom of the tank. This is because of the relatively high flow rate (10.64 L/min) inlet j
47、et striking the opposing wall and producing a random temperature distribution at the bottom of the tank in the centerline where the measurements are taken. The other two curves correspond to times after the flow rate change. The predictions of the model for t* = 0.67 and 1.0, show, from top to botto
48、m, the first thermocline (which sepa- rated the first mixed region from the hot zone), the first mixed region, a second thermocline, and, finally, a second mixed region. The predictions presented in Figure 12 agree well with the experimental results and indicate that the model can handle flow rate c
49、hanges adequately. CONCLUSIONS An experimental and numerical study of fluid flow and heat transfer in residential electric hot water tanks with side- 166 ASHRAE Transactions: Research 1 .o 0.8 - 0.6 4 . 0.4 0.2 0.0 O0 0.2 04 0.6 0.8 1 .o 8- V-T, ) / ve, -T, ) - Numerical Exp, Is025 Exp., t-O 67 A Exp,t*=I.C Figure I2 Comparison of the numerical and experimental results for variable inlet conditions. QI = 10.64 L/min and Ri, = 0.00994 (t*0.26). qni = 40.37“C, Tent = 22.44“C, Tamb = 23.58OC. wall inlets is presented in this paper. The semi-em
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