The heat transfer characteristics of a rectangular water tank used in a solar water heating system with a Fresnel Len were investigated qualitatively and quantitatively through the theoretical and numerical methods. The water tank is 450 mm × 400 mm × 500 mm in size and consists of 15 layers of coil pipe placed at its center. The MIX number and exergy efficiency were studied to quantify the thermal stratification of this water tank. A flow field analysis was also carried out to understand the heat transfer mechanism inside the water tank. Results indicate that the Nusselt number of shell side is increased with the growth of Reynolds number. The MIX number suggested that the thermal stratification is enhanced and then reduced with increasing flow rate. A correlation is proposed to predict the Nusselt numbers on the shell side. A detailed flow field analysis indicated that the thermal stratification is highly related to the runoff time, buoyancy force, mixing process, and geometry of the water tank.

Introduction

Environmental and energy-related problems are recently becoming major concerns worldwide [1]. Solar energy has the advantages of renewability, cleanliness, and sustainability. It is considered a promising alternative for depleted traditional fossil fuel energy sources [2,3]. Thus, teams of researchers are focusing on utilizing solar energy, such as heating and cooling [4], photovoltaic [5], and photo-thermal technologies [6,7].

By and large, solar energy is labeled with intermittent and instability characteristics. Hence, using solar energy resource efficiently becomes an intractable challenge facing solar engineers. Many researchers adopt water tanks to offset the intermittent feature of solar energy. Water tanks can also improve the efficiency and performance of a solar water heating system [8,9]. A water tank, which has large storage capacity and good regulating property, can improve the solar energy collection efficiency significantly [10].

Generally, the coiled tube has a better heat transfer performance than the straight tube [11,12]. Thus, most of the heat exchanger water tanks use a coiled tube, which is regarded as the compact type. Alimoradi and Vevsi [13] conducted a comprehensive investigation on the key design parameters of a heat exchanger which consists of a helical coil and a cylinder shell. The growth of pitch size 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 Nusselt number well for a wide range of operational condition and design parameters. Salimpour [12] investigated a water tank with a helical coiled tube and a cylinder shell experimentally and developed two empirical correlations for the tube and shell. Pimenta and Campos [14] reviewed the corrections for the Newtonian fluid as well as the non-Newtonian fluid for a coil tube. The special attention was paid on the elastic effect of the non-Newtonian fluid. The results suggested that the elastic behavior of the fluid tends to reduce the heat transfer performance as well as the mixing process in a coil tube. Moawed [15] studied the outside surface of helically coiled tubes under forced convection. The results suggested that the diameter and pitch ratios have a great impact on the heat transfer. Naphon [16] built a heat exchanger with a helical coil and cylinder shell and also studied thermal features of a helical coil with fins. Recently, air bubbles [17] have been injected into the two sides of a heat exchanger to enhance the performance and effectiveness of the system. Dizaji et al. [18] also studied air bubbles injected into a heat exchanger by varying air flow rates to find an optimal injection volume. Their experiments indicated that a certain amount of air bubbles improve the number of thermal units by as much as 1.5–4.2 times for the shell side. Four and two input parameters are imported into the artificial neural network model to predict the dimensionless parameters of heat transfer in a helical coil, respectively [19]. Jayakumar et al. [20] researched the heat transfer process of a heat exchanger with a helical coil and a ring shell. A correlation was built to calculate the Nussult number in the helical coil. Jamshidi et al. [21] used Wilson plots and Taguchi method to find the impact weight of each design parameter for a heat exchanger with a coil tube and a cylinder shell. Experimental tests concluded that pitch and diameter of coil affect the performance greatly as well as the flow rate of tube and shell side. A comprehensive estimation was conducted by Jayakumar et al. [22] to study coil parameters of a helical coil positioned vertically. They also researched the flow field and local Nusselt number inside the coil. Mirgolbabaei et al. [23] simulated a heat exchanger composed of a copper coil tube and a cylindrical shell to study the mixed convection process in the shell side for various Reynold numbers.

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

Thus, a water tank built by a rectangular shell and a coiled tube is proposed for a solar heating water system with a linear polymethyl methacrylate (PMMA) Fresnel lens. We conducted an on-site testing and a numerical investigation on heat transfer characteristics of this water tank. A special attention is paid on the flow field analysis of this water tank. This study can provide theoretical and practical support for the application and efficiency of the solar domestic hot water systems (SDHWS).

Description of the Experimental System

The schematic of the test equipment is depicted in Fig. 1. The equipment had two water circulating systems, namely, the coil- and shell-side circulating systems. The coil-side circulating system consists of a linear PMMA Fresnel lens, a flowmeter, a radiation sensor, a thermal collector, a self-sucking pump, the helical coil of the water tank, and several connection pipes. The description of these components can be found in Ref. [25]. Two S-thermocouples were installed in the coil-side circulating system to obtain the temperature differential between the thermal collector's inlet and outlet. We placed six S-thermocouples inside the water tank to measure the water temperature in the shell side. The distance among thermocouples was 10 cm. The flow rates of the coil and shell sides were obtained from two Z3002 flowmeters; the location of the flowmeters is demonstrated in Fig. 1. All the measuring instruments adopted are listed in Table 1.

Fig. 1
Flow diagram of the experimental setup
Fig. 1
Flow diagram of the experimental setup
Close modal
Table 1

Instruments used in the experiment

No.Instrument nameSpecification modelRangeAccuracy
1ThermocoupleCopper constantan0–100 °C±0.1 °C
2Water pumpGP12530 L/min
3Radiation sensorI-7017R0–2000 W/m2±1 W/m2
4Flow meterZ30020–10 L/min±0.1 L/min
5Digital-to-analog converterI-7520
No.Instrument nameSpecification modelRangeAccuracy
1ThermocoupleCopper constantan0–100 °C±0.1 °C
2Water pumpGP12530 L/min
3Radiation sensorI-7017R0–2000 W/m2±1 W/m2
4Flow meterZ30020–10 L/min±0.1 L/min
5Digital-to-analog converterI-7520

The heat exchanger tank depicted in Fig. 2(a) consists of a helical coil and a rectangular shell. The shell of the water tank was made of stainless steel. The helical coil presented in Fig. 2(b) was made of copper and placed in the center of the water tank. The coil diameter is expressed as 2Rc. The total height of the helical coil is Hc. A pitch is defined as b which is the distance between two neighboring turns. The internal and external diameters of the pipe are called di and do, respectively. The di and do of the helical section are 10.922 and 12.7 mm, respectively. The pipe diameter (d) was calculated by averaging the inner and outer diameters. The curvature ratio (δ) is the ratio between pipe diameter and coil diameter (d/2Rc). The pitch ratio defined as the length of one turn (H/2Rc) is called nondimensional pitch (λ). The dimension of the water tank is 450 mm × 400 mm × 500 mm. A coil tube with 15 layers was placed at the center of the water tank. The diameters of the entrance and exit for the shell side are 20 mm. A foam board of 10 mm was wrapped around the outer surfaces of the heat exchanger to reduce the heat loss from the water tank. The dimensions and parameters of the coils and the water tank are summarized in Table 2. The heat exchanger was positioned vertically during experiments.

Fig. 2
Heat exchanger tank and helical coil
Fig. 2
Heat exchanger tank and helical coil
Close modal
Table 2

Dimensions of coil pipe and water tank

ParametersUnit (mm)
l450
w400
h500
Rc150
Hc360
di10.92
do12.7
B24
ParametersUnit (mm)
l450
w400
h500
Rc150
Hc360
di10.92
do12.7
B24

We switched on the water pump to circulate the hot water inside the helical coil when the circulation water in the thermal collector reaches 40 °C. The coil-side hot water exchanged heat with the cold water on the shell side. The temperature of each monitor point was initially unstable and varies with time. Then, the temperature change at each monitor point became stable after approximately 20 min, indicating that the water tank was operating steadily. We recorded the values of flow rate and temperature once the steady-state condition was reached. The water flow rates of the coil and shell side were maintained at two values (0.1 and 0.05 kg/s). All temperatures were measured thrice with an accuracy of 0.1 °C maintained for 10 min, and the average values were calculated to conduct further analysis.

Numerical Method

Simulation Model and Computational Fluid Dynamics Strategies.

A numerical simulation of the water tank was conducted to investigate the performance and mechanism of heat transfer process in the water tank. The simulated model was the same as that of the experimental one. The computational domain was divided into the coil and shell sides. The detailed meshes of the coil and shell sides are displayed in Fig. 3. Figures 3(a) and 3(c) present the mesh of the shell side, whereas Figs. 3(b) and 3(d) exhibit the mesh of the coil side. The unstructured grid dominated the shell-side domain, and the helical coil side was meshed with structured grids. We refined the mesh inside the helical coil and the surfaces of the pipe. Six meshing numbers were built to conduct a mesh-independent investigation. The mesh numbers of these 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 Nusselt number (Nuav) over the coil pipe surface were compared in Fig. 4. It was observed that the mesh number has a minimal effect on 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 numerical simulation.

Fig. 3
Detailed mesh of water tank
Fig. 3
Detailed mesh of water tank
Close modal
Fig. 4
Grid-independent verification
Fig. 4
Grid-independent verification
Close modal
The commercial CFD code, FLUENT, was used in this paper to study the heat transfer process of the water tank. The standard k–ε turbulent model [26] was adopted. The flow and heat transfer characteristics were governed by the conservation laws of mass, momentum, and energy as
divρUϕΓϕgradϕ=Sϕ
(1)

where U is the velocity vector, ρ is the density, ϕ = 1 is for mass continuity, ϕ = Vj (j = 1, 2, 3) is for momentum conservation, ϕ = T is for energy transportation, ϕ = K is for turbulent energy, ϕ = ɛ is for the dissipation rate of K, Γϕ is the diffusion coefficient, and Sϕ is the source term. The standard k–ε turbulence model and SIMPLEC algorithm were suitable for the water tank analysis [22]. First-order upwind scheme was used for the discretization of k–ε equations [13]. The internal and external surfaces of helical pipe were set as the fluid-solid coupling heat transfer boundary conditions. Therefore, the temperature or heat flux at the fluid–solid interfaces was determined by the coupling of the fluid and solid temperature distributions, rather than the predefined temperature distributions. The inlet boundary conditions of the coil and shell sides were set as the mass flow inlet, and the outlet was defined as the outflow. The shell-side outer surfaces were set to adiabatic. All the boundary conditions applied to the model can be found in Table 3.

Table 3

Simulation conditions of heat exchange water tank

CaseHot water inlet temperature (°C)Hot water inlet mass flow rate (kg/s)Cold water inlet temperature (°C)Cold water inlet mass flow rate (kg/s)
a400.1200.03
b400.1200.05
c400.1200.08
d400.1200.1
e400.1200.15
f400.1200.2
CaseHot water inlet temperature (°C)Hot water inlet mass flow rate (kg/s)Cold water inlet temperature (°C)Cold water inlet mass flow rate (kg/s)
a400.1200.03
b400.1200.05
c400.1200.08
d400.1200.1
e400.1200.15
f400.1200.2

Data Reduction.

We obtained the Nusselt numbers of the coil and shell sides through the numerical simulation using the following equations:
Qc=mcCpT2T1
(2)
Qs=msCpT4T3
(3)
qc=QcAi
(4)
qs=QsAo
(5)
hc=qcTcThi
(6)
hs=qsThoTs
(7)
Rec=uidcν
(8)
Res=usdsν
(9)
Nuc=hcdiλc
(10)
Nus=hsdsλs
(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 flow rates, respectively; Cp is the water specific heat capacity; qc and qs are the heat fluxes of the coil and shell sides, respectively; Ai and Ao are the helical coil internal and external surface areas, respectively; Tc is the hot water average temperature; Thi is the helical coil internal surface average temperature; Tho is the helical coil external surface average temperature; Ts is the cold water average temperature; hc and hs are the coil-side and shell-side heat transfer coefficients, respectively; di is the coil-side hydraulic diameter; ds = ((2wl)/(w + l)) 2wl/w+l is the shell-side hydraulic diameter; ui is the average inlet velocity of the coil side; us is the average velocity of the shell side; v is the water viscosity; λc and λs are the thermal conductivities.

Results and Discussion

Validation of Simulation Results.

The experimental test were conducted by maintaining the flow rates of cold water at 0.05 kg/s and flow rates of hot water at 0.1 kg/s, respectively. We measured the entrance and exit temperatures of the coil and shell side. Table 4 lists the recorded, average, and simulated temperatures. The simulated temperatures are generally consistent with the tested ones. We calculate the experimental and numerical heat transfer rates, Qc and Qs, based on the equations in Sec. 3.2. The tested and simulated heat transfer rates at the coil side are 2634.6 and 2801.9 W, correspondingly, thereby resulting in a deviation of approximately 5.97%. The deviation between the simulated and measured heat transfer rates of the shell is approximately 9.77%. The numerical results agree well with the measurement results, thus verifying the accuracy of the numerical strategies. The tested and simulated temperatures inside the water tank at different locations are also presented in Table 5. The locations of testing points are demonstrated in Fig. 1. It is observed that the deviation is less than 1.36%, and it can be concluded that our simulation methodology is reliable.

Table 4

Comparison of experimental and numerical simulation results

PartInlet temperature (°C)Outlet temperature (°C)Average temperature (°C)Heat transfer rate (W)Relative deviation (%)
TestCoil40.133.836.92634.65.97
SimulationCoil4033.336.82801.9
TestShell21.133.130.72509.29.77
SimulationShell2033.330.92781
PartInlet temperature (°C)Outlet temperature (°C)Average temperature (°C)Heat transfer rate (W)Relative deviation (%)
TestCoil40.133.836.92634.65.97
SimulationCoil4033.336.82801.9
TestShell21.133.130.72509.29.77
SimulationShell2033.330.92781
Table 5

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

Testing pointExperiment (°C)Simulation (°C)Relative deviation/%
T1033.133.30.60
T933.033.20.60
T832.432.60.61
T731.331.50.64
T629.529.91.36
T523.723.80.42
Testing pointExperiment (°C)Simulation (°C)Relative deviation/%
T1033.133.30.60
T933.033.20.60
T832.432.60.61
T731.331.50.64
T629.529.91.36
T523.723.80.42

MIX Number.

Anderson [27] proposed the MIX number derived from the first moment of energy to study temperature stratification inside the water tank quantitatively. The distributions of the energy inside the tank imply that the energy and momentum of the tested tank are associated with two theoretical cases, namely, a fully stratified tank and a perfectly mixed tank. The energy of two theoretical cases is the same as that of the tested tank. The MIX number means the degree of stratification. The MIX number falls in between 0 and 1, where 0 represents a fully stratified tank, and 1 represents a perfectly mixed tank
MIX=MstrMexpMstrMmix
(12)
Mexp=i=1nyi·Ei
(13)
Ei=ρ·Vi·Cp·Ti
(14)
Mstr=i=1nyi·Estr,i
(15)
Estr,i=Vhot·ρ·Cp·Thot+Vcold·ρ·Cp·Tcold
(16)
Mmix=i=1nyi·Emix
(17)
Emix=VT·ρ·Cp·Tmix
(18)

The experimental and numerical variations in MIX number with different mass flow rates are demonstrated in Fig. 5. The numerical values of various flow rates generally agree well with the experimental values. The simulated MIX number is slightly lower than the experimental MIX number. The largest deviation between the experimental and numerical MIX numbers is 8.56%. The MIX number rapidly reduces first and then increases with the rise in flow rate. In particular, the stratified behavior of the tank first improves and then deteriorates. The MIX numbers at 0.03 and 0.05 kg/s are higher than the MIX numbers at other flow rates possibly, because a low flow rate increases the runoff time, which causes a well-distributed temperature field and weakens the temperature stratification phenomenon. The MIX number reaches the minimum when the flow rates are 0.08 and 0.10 kg/s. Therefore, the water tank has a perfectly stratified behavior under this circumstance. The mixing process is enforced due to the increasing velocity at m = 0.15 kg/s and m = 0.2 kg/s, thereby degrading the stratified behavior of the water tank.

Fig. 5
Influence of flow rate on MIX number
Fig. 5
Influence of flow rate on MIX number
Close modal

Exergy Efficiency.

The exergy quantifies the workload that can be extracted from a system, because the perfect stratified and the experimental tanks have the same amount of energy. Thus, the exergy analysis is used as a benchmark for assessing the actual theoretical performance of a system. Significant stratification indicates high exergy. The exergy analysis is suitable for evaluating stratification performance and exergy efficiency. Shah [28] and Rosen [29] proposed a calculation method for exergy efficiency
φ=δexpδstr
(19)
δ=Eexpi=1nmi·Cp·Tcold·lnTiTcold
(20)
Eexp=i=1nmi·Cp·TiTcold
(21)

φ represents the deviation in ideal stratified condition varying from 0 to 1, where 0 represents a fully mixed tank, and 1 represents a perfectly stratified tank.

Figure 6 illustrates the exergy efficiency versus mass flow rate. In this figure, the exergy efficiency first upgrade and then descending later with the rise in mass flow rate. The exergy efficiency at 0.03–0.08 kg/s is increased rapidly, thereby indicating an improved stratified behavior of the experimental tank. The exergy efficiency is reduced in a flow rate range from 0.08 kg/s to 0.2 kg/s, denoting the deterioration in the stratified behavior of the experiment tank. The numerical exergy efficiency is constantly larger than the experimental exergy efficiency. The largest relative deviation between the experimental and the simulated exergy efficiencies is approximately 8.7%. Therefore, the simulation results are reliable.

Fig. 6
Influence of flow rate on exergy efficiency
Fig. 6
Influence of flow rate on exergy efficiency
Close modal

Nusselt Number and Shell Side Correlation.

Numerically, the Nusselt numbers of the shell side for all the simulation cases can be obtained by the equations in Sec. 3.2. The variation of the shell-side Nusselt number is depicted in Fig. 7. It is noted that the growth of the Reynolds number led to rise in the Nusselt number. This conclusion is similar to the conclusions in certain published literature [13,23]. A high Reynolds number means a high water flow rate, which enhances the heat transfer coefficient that results in increasing Nusselt number.

Fig. 7
Variation of Nusselt number with Reynolds number
Fig. 7
Variation of Nusselt number with Reynolds number
Close modal
A few correlations are found based on previous literature to predict the heat transfer coefficients in the shell side of a heat exchanger with a helical coil and a rectangular shell. Thus, we proposed a correlation for the shell-side Nusselt number. Literatures [2,3] state that the correlation is a function that depends on Reynolds and Prandtl numbers. This correlation for the shell side according to the simulated results is
Nus=625.42Re0.117Pr0.292
(22)

Figure 8 demonstrates the comparison of the calculated Nusselt number gained from above correlation and tested Nusselt numbers. It is noticed that the predicted Nusselt numbers are coincident with the experimental ones. Most of the experimental data are located within the positive (+) and negative (−) 10% error curves.

Fig. 8
Comparison of experimental and predicted Nusselt numbers
Fig. 8
Comparison of experimental and predicted Nusselt numbers
Close modal

Flow Field Analysis of the Water Tank.

Figure 9 shows the internal temperature and the flow field distributions on the center plane in the water tank in a flow rate range from 0.03 kg/s to 0.2 kg/s. Figure 9(a) presents the temperature and the flow field distributions at m = 0.03 kg/s. The water temperature difference between the upper and lower parts of the tank is approximately 5 °C. Two vortices are formed at two corners of the upper part of the water tank. One vortex centered at the bottom of the water tank. The water flows into the water tank slowly, thereby increasing runoff time. The flow field is mainly driven by buoyancy force. These conditions are the possible reasons for the low thermal stratification. In Fig. 9(b), the water temperature difference between the upper and lower parts of the tank is increased to 8 °C at m = 0.05 kg/s. The right-hand vortex is enlarged, whereas the left-hand vortex is shrunk. The location of the above vortices is unchanged. However, the vortex at the bottom of the water tank is moved from the middle to the right side of the water tank due to the horizontal flow owing to the increased flow rate and the position of the entrance. In Fig. 9(c), the water temperature difference between the upper and lower parts of the tank remains at 8 °C when the flow rate rises to 0.08 kg/s. The size of the vortex at the top right side is reduced significantly. Thus, a tiny vortex is formed at the bottom right of the water tank. The size and location of the two vortices at the right side of the water tank are unchanged because of the fully developed bottom horizontal flow occupied half of the water tank. In Fig. 9(d), the water temperature difference between the upper and lower parts of the tank is 7 °C when the flow rate is 0.1 kg/m. The flow field at this flow rate is similar to that at m = 0.08 kg/s. As shown in Fig. 9(e), the water temperature difference between the top and bottom of the tank is reduced to 4 °C. The distribution of the vortices is similar to the distribution presented in Fig. 9(d) in terms of size and location. The streamline direction is altered from vertical to horizontal due to the increasing velocity of the water flow. Figure 9(f) depicts the minimal water temperature difference of upper-lower water tank, and the flow pattern of this condition is close to the flow pattern presented in Fig. 9(e).

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

From Figs. 9(a)9(f), the outlet temperature is reduced rapidly with the rise in flow rate. The temperature difference first increases and then reduces with the rise in flow rate. The variation tendencies of the temperature difference and MIX number are similar. The flow field implies that a high flow rate causes the vortices to be small and moves the vortex location upward. The high flow rate also enables the streamlines inside the water tank to gradually change from vertical to horizontal, thereby resulting in an enhanced mixing process. Thus, the key determinative issues of thermal stratification are runoff time, buoyancy force, mixing process, and geometry of the water tank at different flow rates.

Conclusions

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

The results revealed that the MIX number first reduced rapidly and then increased with the rise in flow rate. The vary of exergy efficiency was opposite to the vary of the MIX number. It means that the stratified behavior of the experimental tank first improved and then deteriorated. A correlation of the shell-side Nusselt number was also proposed.

The shell-side water temperature difference between the top and bottom first increased and then decreased with the rise in flow rate. The size of the vortices was shrunk, and their location was moved from bottom to the top. The streamline direction was changed from vertical to horizontal with the rise in flow rate caused 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 =

area of coiled tube, m2

b =

coil pitch, m

Cp =

specific heat capacity, J/kg·k

d =

coil tube diameter, m

dc =

hydraulic diameter of coil side, m

ds =

hydraulic diameter of shell side, m

Eexp =

energy of experiment tank, J

Ei =

energy of each water layer, J

Emix =

energy of fully mixed tank, J

Estr =

energy of perfectly stratified tank, J

h =

heat transfer coefficient, W/m2·k

h =

water tank height, m

Hc =

coil height, m

l =

water tank length, m

M =

flow rate, kg/s

mi =

quality of each water layer, kg

Mexp =

energy momentum of experiment tank, J·m

Mmix =

energy momentum of fully mixed tank, J·m

Mstr =

energy momentum of perfectly stratified tank, J·m

MIX =

MIX number

Nu =

Nusselt number

Pr =

Prandtl number

q =

heat flux, W/m2

Q =

heat transfer rate, W

Rc =

curvature radius, m

Re =

Reynolds number

Sφ =

source term

T =

temperature, K or °C

Tcold =

cold water temperature, K or °C

Thot =

hot water temperature, K or °C

Ti =

temperature of each water layer, K or °C

Tmix =

temperature in fully mixed tank, K or °C

u =

average velocity, m/s

U =

velocity vector, m/s

Vi =

volume of each water layer, m3

VT =

volume of water tank, m3

w =

water tank width, m

yi =

vertical distance from the center of gravity layer to the bottom of the tank, m

Greek Symbols
Ρ =

density, kg/m3

V =

viscosity, m2/s

Γϕ =

diffusion coefficient

δexp =

exergy efficiency of experiment tank, J

δstr =

exergy efficiency of perfectly stratified tank, J

Λ =

thermal conductivity, W/m2·k

φ =

exergy efficiency

Subscripts
c =

coil side

h =

helical coil

I =

inner

o =

outer

s =

shell side

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