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J. ChenSchool of Energy and Power Engineering,

University of Shanghai for Science and

Technology,

Room 314, Second Building,

Jungong Road, Yangpu District,

Shanghai 200093, China

e-mail: [email protected]

H. T. Xu1

School of Energy and Power Engineering,


Technology,





Z. Y. WangSchool of Energy and Power Engineering,


Technology,





S. P. HanShanghai Institute of Special Equipment

Inspection and Technical Research,

Room 504, Jinshanjiang Road,

Putuo District,



Thermal Performance Study of aWater Tank for a Solar SystemWith a Fresnel LensThe heat transfer characteristics of a rectangular water tank used in a solar water heat-ing system with a Fresnel Len were investigated qualitatively and quantitatively throughthe theoretical and numerical methods. The water tank is 450 mm� 400 mm� 500 mm insize and consists of 15 layers of coil pipe placed at its center. The MIX number andexergy efficiency were studied to quantify the thermal stratification of this water tank. Aflow field analysis was also carried out to understand the heat transfer mechanism insidethe water tank. Results indicate that the Nusselt number of shell side is increased with thegrowth of Reynolds number. The MIX number suggested that the thermal stratification isenhanced and then reduced with increasing flow rate. A correlation is proposed to pre-dict the Nusselt numbers on the shell side. A detailed flow field analysis indicated that thethermal stratification is highly related to the runoff time, buoyancy force, mixing process,and geometry of the water tank. [DOI: 10.1115/1.4039986]

Keywords: water tank, SDHWS, correlation, thermal stratification, flow field

1 Introduction

Environmental and energy-related problems are recentlybecoming major concerns worldwide [1]. Solar energy has theadvantages of renewability, cleanliness, and sustainability. It isconsidered a promising alternative for depleted traditional fossilfuel energy sources [2,3]. Thus, teams of researchers are focusingon utilizing solar energy, such as heating and cooling [4], photo-voltaic [5], and photo-thermal technologies [6,7].

By and large, solar energy is labeled with intermittent and insta-bility characteristics. Hence, using solar energy resource effi-ciently becomes an intractable challenge facing solar engineers.Many researchers adopt water tanks to offset the intermittent fea-ture of solar energy. Water tanks can also improve the efficiencyand performance of a solar water heating system [8,9]. A watertank, which has large storage capacity and good regulatingproperty, can improve the solar energy collection efficiencysignificantly [10].

Generally, the coiled tube has a better heat transfer performancethan the straight tube [11,12]. Thus, most of the heat exchangerwater tanks use a coiled tube, which is regarded as the compacttype. Alimoradi and Vevsi [13] conducted a comprehensive inves-tigation on the key design parameters of a heat exchanger whichconsists of a helical coil and a cylinder shell. The growth of pitchsize enhances the Nusselt number in the shell. Moreover, larger

height and diameter of shell led to a decreased Nusselt number.The proposed shell and coil correlations can predict the Nusseltnumber well for a wide range of operational condition and designparameters. Salimpour [12] investigated a water tank with a heli-cal coiled tube and a cylinder shell experimentally and developedtwo empirical correlations for the tube and shell. Pimenta andCampos [14] reviewed the corrections for the Newtonian fluid aswell as the non-Newtonian fluid for a coil tube. The special atten-tion was paid on the elastic effect of the non-Newtonian fluid. Theresults suggested that the elastic behavior of the fluid tends toreduce the heat transfer performance as well as the mixing processin a coil tube. Moawed [15] studied the outside surface of heli-cally coiled tubes under forced convection. The results suggestedthat the diameter and pitch ratios have a great impact on the heattransfer. Naphon [16] built a heat exchanger with a helical coiland cylinder shell and also studied thermal features of a helicalcoil with fins. Recently, air bubbles [17] have been injected intothe two sides of a heat exchanger to enhance the performance andeffectiveness of the system. Dizaji et al. [18] also studied air bub-bles injected into a heat exchanger by varying air flow rates tofind an optimal injection volume. Their experiments indicated thata certain amount of air bubbles improve the number of thermalunits by as much as 1.5–4.2 times for the shell side. Four and twoinput parameters are imported into the artificial neural networkmodel to predict the dimensionless parameters of heat transfer ina helical coil, respectively [19]. Jayakumar et al. [20] researchedthe heat transfer process of a heat exchanger with a helical coiland a ring shell. A correlation was built to calculate the Nussultnumber in the helical coil. Jamshidi et al. [21] used Wilson plotsand Taguchi method to find the impact weight of each design

1Corresponding author.Contributed by the Solar Energy Division of ASME for publication in the

JOURNAL OF SOLAR ENERGY ENGINEERING: INCLUDING WIND ENERGY AND BUILDING

ENERGY CONSERVATION. Manuscript received September 12, 2017; final manuscriptreceived April 10, 2018; published online May 7, 2018. Assoc. Editor: Ming Qu.

Journal of Solar Energy Engineering OCTOBER 2018, Vol. 140 / 051005-1Copyright VC 2018 by ASME

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parameter for a heat exchanger with a coil tube and a cylindershell. Experimental tests concluded that pitch and diameter of coilaffect the performance greatly as well as the flow rate of tube andshell side. A comprehensive estimation was conducted by Jayaku-mar et al. [22] to study coil parameters of a helical coil positionedvertically. They also researched the flow field and local Nusseltnumber inside the coil. Mirgolbabaei et al. [23] simulated a heatexchanger composed of a copper coil tube and a cylindrical shellto study the mixed convection process in the shell side for variousReynold numbers.

The majority foregoing researches concentrated on coil-side heattransfer process, whereas minimal attention was paid to the shell-side heat transfer, especially the detailed flow and temperature fieldinside the water tank. The shell shape in the previous literature wasmainly cylindrical. A few studies were also found for water tankswith a rectangular shell. Ten shell shapes were compared in Ref.[24], their results concluded that the thermal stratification of a cyl-inder shell is the worst among ten kinds of shell shapes.

Thus, a water tank built by a rectangular shell and a coiled tubeis proposed for a solar heating water system with a linear poly-methyl methacrylate (PMMA) Fresnel lens. We conducted an on-site testing and a numerical investigation on heat transfer charac-teristics of this water tank. A special attention is paid on the flowfield analysis of this water tank. This study can provide theoreticaland practical support for the application and efficiency of the solardomestic hot water systems (SDHWS).

2 Description of the Experimental System

The schematic of the test equipment is depicted in Fig. 1. Theequipment had two water circulating systems, namely, the coil-and shell-side circulating systems. The coil-side circulatingsystem consists of a linear PMMA Fresnel lens, a flowmeter, aradiation sensor, a thermal collector, a self-sucking pump, the hel-ical coil of the water tank, and several connection pipes. Thedescription of these components can be found in Ref. [25]. TwoS-thermocouples were installed in the coil-side circulating systemto obtain the temperature differential between the thermal collec-tor’s inlet and outlet. We placed six S-thermocouples inside thewater tank to measure the water temperature in the shell side. Thedistance among thermocouples was 10 cm. The flow rates of thecoil and shell sides were obtained from two Z3002 flowmeters;the location of the flowmeters is demonstrated in Fig. 1. All themeasuring instruments adopted are listed in Table 1.

The heat exchanger tank depicted in Fig. 2(a) consists of a heli-cal coil and a rectangular shell. The shell of the water tank wasmade of stainless steel. The helical coil presented in Fig. 2(b) wasmade of copper and placed in the center of the water tank. Thecoil diameter is expressed as 2Rc. The total height of the helicalcoil is Hc. A pitch is defined as b which is the distance between

two neighboring turns. The internal and external diameters of thepipe are called di and do, respectively. The di and do of the helicalsection are 10.922 and 12.7 mm, respectively. The pipe diameter(d) was calculated by averaging the inner and outer diameters.The curvature ratio (d) is the ratio between pipe diameter and coildiameter (d/2Rc). The pitch ratio defined as the length of one turn(H/2Rc) is called nondimensional pitch (k). The dimension of thewater tank is 450 mm� 400 mm� 500 mm. A coil tube with 15layers was placed at the center of the water tank. The diameters ofthe entrance and exit for the shell side are 20 mm. A foam boardof 10 mm was wrapped around the outer surfaces of the heatexchanger to reduce the heat loss from the water tank. The dimen-sions and parameters of the coils and the water tank are summar-ized in Table 2. The heat exchanger was positioned verticallyduring experiments.

We switched on the water pump to circulate the hot waterinside the helical coil when the circulation water in the thermalcollector reaches 40 �C. The coil-side hot water exchanged heatwith the cold water on the shell side. The temperature of eachmonitor point was initially unstable and varies with time. Then,the temperature change at each monitor point became stable afterapproximately 20 min, indicating that the water tank was operat-ing steadily. We recorded the values of flow rate and temperatureonce the steady-state condition was reached. The water flow ratesof the coil and shell side were maintained at two values (0.1 and0.05 kg/s). All temperatures were measured thrice with an accu-racy of 0.1 �C maintained for 10 min, and the average values werecalculated to conduct further analysis.

3 Numerical Method

3.1 Simulation Model and Computational Fluid DynamicsStrategies. A numerical simulation of the water tank was con-ducted to investigate the performance and mechanism of heattransfer process in the water tank. The simulated model was thesame as that of the experimental one. The computational domainwas divided into the coil and shell sides. The detailed meshes ofthe coil and shell sides are displayed in Fig. 3. Figures 3(a) and3(c) present the mesh of the shell side, whereas Figs. 3(b) and3(d) exhibit the mesh of the coil side. The unstructured grid domi-nated the shell-side domain, and the helical coil side was meshedwith structured grids. We refined the mesh inside the helical coiland the surfaces of the pipe. Six meshing numbers were built toconduct a mesh-independent investigation. The mesh numbers ofthese models are 504,564; 1,030,167; 1,502,435; 2,074,532;2,545,221; 3,014,567. Six simulated values of area-averaged Nus-selt number (Nuav) over the coil pipe surface were compared inFig. 4. It was observed that the mesh number has a minimal effecton the Nuav value when the mesh number is larger than 2,000,000.Therefore, the mesh number of 2,074,532 was used in our numeri-cal simulation.

The commercial CFD code, FLUENT, was used in this paper tostudy the heat transfer process of the water tank. The standard k–eturbulent model [26] was adopted. The flow and heat transfercharacteristics were governed by the conservation laws of mass,momentum, and energy as

div qU/� C/grad/ð Þ ¼ S/ (1)

where U is the velocity vector, q is the density, /¼ 1 is for masscontinuity, /¼Vj (j¼ 1, 2, 3) is for momentum conservation,/¼T is for energy transportation, /¼K is for turbulent energy,/¼ E is for the dissipation rate of K, C/ is the diffusion coeffi-cient, and S/ is the source term. The standard k–e turbulencemodel and SIMPLEC algorithm were suitable for the water tankanalysis [22]. First-order upwind scheme was used for the discreti-zation of k–e equations [13]. The internal and external surfaces ofhelical pipe were set as the fluid-solid coupling heat transferboundary conditions. Therefore, the temperature or heat flux atFig. 1 Flow diagram of the experimental setup

051005-2 / Vol. 140, OCTOBER 2018 Transactions of the ASME

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the fluid–solid interfaces was determined by the coupling of thefluid and solid temperature distributions, rather than the prede-fined temperature distributions. The inlet boundary conditions ofthe coil and shell sides were set as the mass flow inlet, and the out-let was defined as the outflow. The shell-side outer surfaces wereset to adiabatic. All the boundary conditions applied to the modelcan be found in Table 3.

Table 1 Instruments used in the experiment

No. Instrument name Specification model Range Accuracy

1 Thermocouple Copper constantan 0–100 �C 60.1 �C2 Water pump GP125 30 L/min3 Radiation sensor I-7017R 0–2000 W/m2 61 W/m2

4 Flow meter Z3002 0–10 L/min 60.1 L/min5 Digital-to-analog converter I-7520

Fig. 2 Heat exchanger tank and helical coil

Table 2 Dimensions of coil pipe and water tank

Parameters Unit (mm)

l 450w 400h 500Rc 150Hc 360di 10.92do 12.7B 24

Fig. 3 Detailed mesh of water tank

Fig. 4 Grid-independent verification

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3.2 Data Reduction. We obtained the Nusselt numbers of thecoil and shell sides through the numerical simulation using thefollowing equations:

Qc ¼ mcCp T2 � T1ð Þ (2)

Qs ¼ msCp T4 � T3ð Þ (3)

qc ¼Qc

Ai(4)

qs ¼Qs

Ao(5)

hc ¼qc

Tc � Thi

(6)

hs ¼qs

Tho� Ts

(7)

Rec ¼uidc

�(8)

Res ¼usds

�(9)

Nuc ¼hcdi

kc(10)

Nus ¼hsds

ks(11)

where Qc and Qs are coil-side and shell-side heat transfer rates,respectively; mc and ms are the coil-side and shell-side mass flowrates, respectively; Cp is the water specific heat capacity; qc and qs

are the heat fluxes of the coil and shell sides, respectively; Ai andAo are the helical coil internal and external surface areas, respec-tively; Tc is the hot water average temperature; Thi

is the helicalcoil internal surface average temperature; Tho

is the helical coilexternal surface average temperature; Ts is the cold water averagetemperature; hc and hs are the coil-side and shell-side heat transfercoefficients, respectively; di is the coil-side hydraulic diameter;ds¼ ((2wl)/(wþ l)) 2wl=wþ l is the shell-side hydraulic diameter;ui is the average inlet velocity of the coil side; us is the averagevelocity of the shell side; v is the water viscosity; kc and ks are thethermal conductivities.

4 Results and Discussion

4.1 Validation of Simulation Results. The experimental testwere conducted by maintaining the flow rates of cold water at0.05 kg/s and flow rates of hot water at 0.1 kg/s, respectively. Wemeasured the entrance and exit temperatures of the coil and shellside. Table 4 lists the recorded, average, and simulated tempera-tures. The simulated temperatures are generally consistent withthe tested ones. We calculate the experimental and numerical heattransfer rates, Qc and Qs, based on the equations in Sec. 3.2. Thetested and simulated heat transfer rates at the coil side are 2634.6and 2801.9 W, correspondingly, thereby resulting in a deviationof approximately 5.97%. The deviation between the simulatedand measured heat transfer rates of the shell is approximately9.77%. The numerical results agree well with the measurementresults, thus verifying the accuracy of the numerical strategies.The tested and simulated temperatures inside the water tank at dif-ferent locations are also presented in Table 5. The locations oftesting points are demonstrated in Fig. 1. It is observed that thedeviation is less than 1.36%, and it can be concluded that our sim-ulation methodology is reliable.

4.2 MIX Number. Anderson [27] proposed the MIX numberderived from the first moment of energy to study temperaturestratification inside the water tank quantitatively. The distributionsof the energy inside the tank imply that the energy and momentumof the tested tank are associated with two theoretical cases,namely, a fully stratified tank and a perfectly mixed tank. Theenergy of two theoretical cases is the same as that of the testedtank. The MIX number means the degree of stratification. TheMIX number falls in between 0 and 1, where 0 represents a fullystratified tank, and 1 represents a perfectly mixed tank

Table 3 Simulation conditions of heat exchange water tank

CaseHot water inlet

temperature (�C)Hot water inlet

mass flow rate (kg/s)Cold water inlettemperature (�C)

Cold water inletmass flow rate (kg/s)

a 40 0.1 20 0.03b 40 0.1 20 0.05c 40 0.1 20 0.08d 40 0.1 20 0.1e 40 0.1 20 0.15f 40 0.1 20 0.2

Table 4 Comparison of experimental and numerical simulation results

Part Inlet temperature (�C) Outlet temperature (�C) Average temperature (�C) Heat transfer rate (W) Relative deviation (%)

Test Coil 40.1 33.8 36.9 2634.6 5.97Simulation Coil 40 33.3 36.8 2801.9Test Shell 21.1 33.1 30.7 2509.2 9.77Simulation Shell 20 33.3 30.9 2781

Table 5 The tested and simulated temperatures inside thewater tank at the cold water mass flow rate of 0.05 kg/s and hotwater mass flow rate of 0.1 kg/s

Testing point Experiment (�C) Simulation (�C) Relative deviation/%

T10 33.1 33.3 0.60T9 33.0 33.2 0.60T8 32.4 32.6 0.61T7 31.3 31.5 0.64T6 29.5 29.9 1.36T5 23.7 23.8 0.42


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MIX ¼ Mstr �Mexp

Mstr �Mmix

(12)

Mexp ¼Xn

i¼1

yi � Ei (13)

Ei ¼ q � Vi � Cp � Ti (14)

Mstr ¼Xn

i¼1

yi � Estr;i (15)

Estr;i ¼ Vhot � q � Cp � Thot þ Vcold � q � Cp � Tcold (16)

Mmix ¼Xn

i¼1

yi � Emix (17)

Emix ¼ VT � q � Cp � Tmix (18)

The experimental and numerical variations in MIX number withdifferent mass flow rates are demonstrated in Fig. 5. The numeri-cal values of various flow rates generally agree well with theexperimental values. The simulated MIX number is slightly lowerthan the experimental MIX number. The largest deviationbetween the experimental and numerical MIX numbers is 8.56%.The MIX number rapidly reduces first and then increases with therise in flow rate. In particular, the stratified behavior of the tankfirst improves and then deteriorates. The MIX numbers at 0.03and 0.05 kg/s are higher than the MIX numbers at other flow ratespossibly, because a low flow rate increases the runoff time, whichcauses a well-distributed temperature field and weakens the tem-perature stratification phenomenon. The MIX number reaches theminimum when the flow rates are 0.08 and 0.10 kg/s. Therefore,the water tank has a perfectly stratified behavior under this cir-cumstance. The mixing process is enforced due to the increasingvelocity at m¼ 0.15 kg/s and m¼ 0.2 kg/s, thereby degrading thestratified behavior of the water tank.

4.3 Exergy Efficiency. The exergy quantifies the workloadthat can be extracted from a system, because the perfect stratifiedand the experimental tanks have the same amount of energy.Thus, the exergy analysis is used as a benchmark for assessing theactual theoretical performance of a system. Significant stratifica-tion indicates high exergy. The exergy analysis is suitable forevaluating stratification performance and exergy efficiency. Shah[28] and Rosen [29] proposed a calculation method for exergyefficiency

u ¼ dexp

dstr

(19)

d ¼ Eexp �Xn

i¼1

mi � Cp � Tcold � lnTi

Tcold

� �(20)

Eexp ¼Xn

i¼1

mi � Cp � Ti � Tcoldð Þ (21)

u represents the deviation in ideal stratified condition varyingfrom 0 to 1, where 0 represents a fully mixed tank, and 1 repre-sents a perfectly stratified tank.

Figure 6 illustrates the exergy efficiency versus mass flow rate.In this figure, the exergy efficiency first upgrade and thendescending later with the rise in mass flow rate. The exergy effi-ciency at 0.03–0.08 kg/s is increased rapidly, thereby indicatingan improved stratified behavior of the experimental tank. Theexergy efficiency is reduced in a flow rate range from 0.08 kg/s to0.2 kg/s, denoting the deterioration in the stratified behavior of the

experiment tank. The numerical exergy efficiency is constantlylarger than the experimental exergy efficiency. The largest relativedeviation between the experimental and the simulated exergy effi-ciencies is approximately 8.7%. Therefore, the simulation resultsare reliable.

4.4 Nusselt Number and Shell Side Correlation. Numeri-cally, the Nusselt numbers of the shell side for all the simulation

Fig. 5 Influence of flow rate on MIX number

Fig. 6 Influence of flow rate on exergy efficiency

Fig. 7 Variation of Nusselt number with Reynolds number


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cases can be obtained by the equations in Sec. 3.2. The variationof the shell-side Nusselt number is depicted in Fig. 7. It is notedthat the growth of the Reynolds number led to rise in the Nusseltnumber. This conclusion is similar to the conclusions in certainpublished literature [13,23]. A high Reynolds number means ahigh water flow rate, which enhances the heat transfer coefficientthat results in increasing Nusselt number.

A few correlations are found based on previous literature to pre-dict the heat transfer coefficients in the shell side of a heatexchanger with a helical coil and a rectangular shell. Thus, weproposed a correlation for the shell-side Nusselt number. Litera-tures [2,3] state that the correlation is a function that depends onReynolds and Prandtl numbers. This correlation for the shell sideaccording to the simulated results is

Nus ¼ 625:42 Re0:117Pr�0:292 (22)

Figure 8 demonstrates the comparison of the calculated Nusseltnumber gained from above correlation and tested Nusselt num-bers. It is noticed that the predicted Nusselt numbers are coinci-dent with the experimental ones. Most of the experimental dataare located within the positive (þ) and negative (�) 10% errorcurves.

4.5 Flow Field Analysis of the Water Tank. Figure 9 showsthe internal temperature and the flow field distributions on thecenter plane in the water tank in a flow rate range from 0.03 kg/sto 0.2 kg/s. Figure 9(a) presents the temperature and the flow fielddistributions at m¼ 0.03 kg/s. The water temperature differencebetween the upper and lower parts of the tank is approximately5 �C. Two vortices are formed at two corners of the upper part ofthe water tank. One vortex centered at the bottom of the watertank. The water flows into the water tank slowly, thereby increas-ing runoff time. The flow field is mainly driven by buoyancyforce. These conditions are the possible reasons for the low

Fig. 8 Comparison of experimental and predicted Nusseltnumbers

Fig. 9 Temperature and velocity contours of heat exchange water tank: (a) m 5 0.03 kg/s, (b) m 5 0.05 kg/s, (c) m 5 0.08 kg/s,(d) m 5 0.1 kg/s, (e) m 5 0.15 kg/s, and (f) m 5 0.2 kg/s


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thermal stratification. In Fig. 9(b), the water temperature differ-ence between the upper and lower parts of the tank is increased to8 �C at m¼ 0.05 kg/s. The right-hand vortex is enlarged, whereasthe left-hand vortex is shrunk. The location of the above vorticesis unchanged. However, the vortex at the bottom of the water tankis moved from the middle to the right side of the water tank due tothe horizontal flow owing to the increased flow rate and the posi-tion of the entrance. In Fig. 9(c), the water temperature differencebetween the upper and lower parts of the tank remains at 8 �Cwhen the flow rate rises to 0.08 kg/s. The size of the vortex at thetop right side is reduced significantly. Thus, a tiny vortex isformed at the bottom right of the water tank. The size and locationof the two vortices at the right side of the water tank areunchanged because of the fully developed bottom horizontal flowoccupied half of the water tank. In Fig. 9(d), the water tempera-ture difference between the upper and lower parts of the tank is7 �C when the flow rate is 0.1 kg/m. The flow field at this flow rateis similar to that at m¼ 0.08 kg/s. As shown in Fig. 9(e), the watertemperature difference between the top and bottom of the tank isreduced to 4 �C. The distribution of the vortices is similar to thedistribution presented in Fig. 9(d) in terms of size and location.The streamline direction is altered from vertical to horizontal dueto the increasing velocity of the water flow. Figure 9(f) depicts theminimal water temperature difference of upper-lower water tank,and the flow pattern of this condition is close to the flow patternpresented in Fig. 9(e).

From Figs. 9(a)–9(f), the outlet temperature is reduced rap-idly with the rise in flow rate. The temperature difference firstincreases and then reduces with the rise in flow rate. The varia-tion tendencies of the temperature difference and MIX numberare similar. The flow field implies that a high flow rate causesthe vortices to be small and moves the vortex location upward.The high flow rate also enables the streamlines inside the watertank to gradually change from vertical to horizontal, therebyresulting in an enhanced mixing process. Thus, the key determi-native issues of thermal stratification are runoff time, buoyancyforce, mixing process, and geometry of the water tank at differ-ent flow rates.

5 Conclusions

This work investigated the heat transfer and temperature strati-fication in a rectangular water tank under different flow rates for aSDHWS with a linear PMMA Fresnel lens through numerical andexperimental methods. The simulated results were validated withthe experimental results, and the property of thermal stratificationinside the water tank was studied qualitatively and quantitatively.

The results revealed that the MIX number first reduced rapidlyand then increased with the rise in flow rate. The vary of exergyefficiency was opposite to the vary of the MIX number. It meansthat the stratified behavior of the experimental tank first improvedand then deteriorated. A correlation of the shell-side Nusselt num-ber was also proposed.

The shell-side water temperature difference between the topand bottom first increased and then decreased with the rise in flowrate. The size of the vortices was shrunk, and their location wasmoved from bottom to the top. The streamline direction waschanged from vertical to horizontal with the rise in flow ratecaused by the interaction between runoff time, buoyancy force,mixing process, and geometry of the water tank.

Funding Data

� National Natural Science Foundation of China (51276117).� Shanghai Pujiang Program (15PJ1406200).

Nomenclature

A ¼ area of coiled tube, m2

b ¼ coil pitch, m

Cp ¼ specific heat capacity, J/kg�kd ¼ coil tube diameter, m

dc ¼ hydraulic diameter of coil side, mds ¼ hydraulic diameter of shell side, m

Eexp ¼ energy of experiment tank, JEi ¼ energy of each water layer, J

Emix ¼ energy of fully mixed tank, JEstr ¼ energy of perfectly stratified tank, J

h ¼ heat transfer coefficient, W/m2�kh ¼ water tank height, m

Hc ¼ coil height, ml ¼ water tank length, m

M ¼ flow rate, kg/smi ¼ quality of each water layer, kg

Mexp ¼ energy momentum of experiment tank, J�mMmix ¼ energy momentum of fully mixed tank, J�mMstr ¼ energy momentum of perfectly stratified tank, J�m

MIX ¼ MIX numberNu ¼ Nusselt numberPr ¼ Prandtl numberq ¼ heat flux, W/m2

Q ¼ heat transfer rate, WRc ¼ curvature radius, mRe ¼ Reynolds numberSu ¼ source termT ¼ temperature, K or �C

Tcold ¼ cold water temperature, K or �CThot ¼ hot water temperature, K or �C

Ti ¼ temperature of each water layer, K or �CTmix ¼ temperature in fully mixed tank, K or �C

u ¼ average velocity, m/sU ¼ velocity vector, m/sVi ¼ volume of each water layer, m3

VT ¼ volume of water tank, m3

w ¼ water tank width, myi ¼ vertical distance from the center of gravity layer to the

bottom of the tank, m

Greek Symbols

P ¼ density, kg/m3

V ¼ viscosity, m2/sC/ ¼ diffusion coefficient

dexp ¼ exergy efficiency of experiment tank, Jdstr ¼ exergy efficiency of perfectly stratified tank, JK ¼ thermal conductivity, W/m2�ku ¼ exergy efficiency

Subscripts

c ¼ coil sideh ¼ helical coilI ¼ innero ¼ outers ¼ shell side

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