Numerical Study of Entropy Generation in Fully Developed Turbulent Circular Tube Flow Using an Elliptic Blending Turbulence Model

As computational fluid dynamics (CFD) advances, entropy generation minimization based on CFD becomes attractive for optimizing complex heat-transfer systems. This optimization depends on the accuracy of CFD results, such that accurate turbulence models, such as elliptic relaxation or elliptic blending turbulence models, become important. The performance of a previously developed elliptic blending turbulence model (the SST k–ω–φ–α model) to predict the rate of entropy generation in the fully developed turbulent circular tube flow with constant heat flux was studied to provide some guidelines for using this class of turbulence model to calculate entropy generation in complex systems. The flow and temperature fields were simulated by using a CFD package, and then the rate of entropy generation was calculated in post-processing. The analytical correlations and results of two popular turbulence models (the realizable k–ε and the shear stress transport (SST) k–ω models) were used as references to demonstrate the accuracy of the SST k–ω–φ–α model. The findings indicate that the turbulent Prandtl number (Prt) influences the entropy generation rate due to heat-transfer irreversibility. Prt = 0.85 produces the best results for the SST k–ω–φ–α model. For the realizable k–ε and SST k–ω models, Prt = 0.85 and Prt = 0.92 produce the best results, respectively. For the realizable k–ε and the SST k–ω models, the two methods used to predict the rate of entropy generation due to friction irreversibility produce the same results. However, for the SST k–ω–φ–α model, the rates of entropy generation due to friction irreversibility predicted by the two methods are different. The difference at a Reynolds number of 100,000 is about 14%. The method that incorporates the effective turbulent viscosity should be used to predict the rate of entropy generation due to friction irreversibility for the SST k–ω–φ–α model. Furthermore, when the temperature in the flow field changes dramatically, the temperature-dependent fluid properties must be considered.


Introduction
Due to the shortage of fossil energy and a desire for sustainable development, the demand for high-efficiency heat-transfer systems is increasing. Hence, much effort has been made to improve the efficiency of heat-transfer systems. However, much of this work is based on the first law of thermodynamics, which can provide a lot of useful information but does not provide optimal conditions. The entropy generation analysis (EGA), which is based on the second law of thermodynamics, is a powerful tool for heattransfer-process optimization.
The EGA has been used in designing and optimizing thermal systems since its introduction by Bejan [1,2]. However, because earlier attempts were primarily theoretical, they could only deal with simple geometry configurations and flow conditions. Furthermore, some empirical correlations had to be introduced, because obtaining local information is the effect of large wall-fluid temperature difference, and the effect of the temperaturedependent fluid properties were investigated. The findings-for example, the Pr t should be chosen as 0.85, the method that incorporates the effective turbulent viscosity should be used to predict the rate of entropy generation due to friction irreversibility, and the temperaturedependent fluid properties should be considered when the temperature difference is largecan guide the application of the elliptic blending turbulence model to complex problems.
The numerical modeling is briefly described in the next section. Then, in the third section, the results are presented and discussed in detail. Finally, the conclusions are presented.

Numerical Modeling
The problem under consideration is fully developed turbulent flow in a straight circular tube (diameter, D, and length, L) with a uniform constant heat flux (q ) on the sidewall. Because of the axial symmetry of the problem under consideration, an axisymmetric computational model was constructed (Figure 1). tional method for entropy generation rate, the effect of the turbulent Prandtl numb the effect of large wall-fluid temperature difference, and the effect of the tempe dependent fluid properties were investigated. The findings-for example, the Prt be chosen as 0.85, the method that incorporates the effective turbulent viscosity sh used to predict the rate of entropy generation due to friction irreversibility, and th perature-dependent fluid properties should be considered when the temperature ence is large-can guide the application of the elliptic blending turbulence model plex problems.
The numerical modeling is briefly described in the next section. Then, in th section, the results are presented and discussed in detail. Finally, the conclusions a sented.

Numerical Modeling
The problem under consideration is fully developed turbulent flow in a strai cular tube (diameter, D, and length, L) with a uniform constant heat flux ( ) on th wall. Because of the axial symmetry of the problem under consideration, an axisym computational model was constructed (Figure 1).

Governing Equations
The flow under consideration in this study is steady, incompressible, and tur Turbulence was handled by using the Reynolds averaging method, yielding the RANS equations given as follows: For linear eddy viscosity turbulence models, the turbulent stresses are linked to th rate by using the following:

Governing Equations
The flow under consideration in this study is steady, incompressible, and turbulent. Turbulence was handled by using the Reynolds averaging method, yielding the steady RANS equations given as follows: For linear eddy viscosity turbulence models, the turbulent stresses are linked to the strain rate by using the following: For incompressible flow, after using the Boussinesq approximation, the governing equation of energy can be simplified to the mean temperature (T) equation, as given by the following: ∂ρT ∂t The turbulence model yields the eddy viscosity, µ t . The authors' previously developed SST-incorporated elliptic-blending turbulence model (denoted as SST k-ω-ϕ-α model) was used. The model comprises four equations for the turbulent kinetic energy, k; specific dissipation rate, ω; wall-normal turbulent anisotropy, ϕ; and elliptic variable, α, respectively.
The following are the model equations.
∂ρk ∂t ∂ρϕ ∂t The eddy viscosity is defined by a blending formulation: The model's details are not provided here for the sake of brevity, and the interested reader is directed to Yang et al. [22]. Another two popular turbulence models, the realizable k-ε model [23] and the SST k-ω model [24], were also used for comparison.

Entropy Generations
In order to facilitate understanding, it is necessary to first declare the meaning of the superscripts on the rate of entropy generation, S, used in the following: (·) means the quantity per unit length, (·) means the quantity per unit area, and (·) means the quantity per unit volume.
The volumetric rate of entropy generation in viscous flow with convective heat transfer can be written as follows [3]: where the first and second terms on the right-hand side are the local volumetric rate of entropy generation due to friction irreversibility and heat transfer irreversibility, respectively. Both terms can be expressed in cylindrical coordinates for circular tube flow [3]: and where T is the local mean temperature of flow, and µ and λ are the molecular dynamic viscosity and the molecular thermal conductivity of the fluid, respectively. The volumetric rate of entropy generation is affected by absolute temperature, viscosity, thermal conductivity, and local spatial gradients of velocity and temperature. These quantities can be easily obtained by using CFD. The two equations above are valid for laminar flow. For turbulent flow, there are two different processing methods when the mean values of the quantities are used to calculate the rate of entropy generation. One can still use Equations (11) and (12), but µ and λ must be replaced with effective viscosity (µ + µ t ) and effective thermal conductivity (λ + λ t ), respectively, where µ t is the eddy viscosity and λ t is the turbulent thermal conductivity [25,26]. Moreover, Equations (11) and (12) become the following: Another method was developed by Kock and Herwig [27]. Using the time-averaging process gives S gen, f as follows: For S gen,h , Equation (14) still holds. Equation (15) appears to have a natural advantage for k-ε models because the turbulence dissipation rate ε is obtained directly [28]. However, it could pose challenges to turbulence models where the accurate ε is difficult to obtain. The following integral can be used to calculate the total rate of entropy generation in a fluid with volume, V: where For fully developed turbulent circular tube flow, Bejan [3] gave the analytical expressions for the rate of entropy generation per unit length as follows: where T ave is the average temperature in the domain, f is the friction factor, . m is the mass flow rate, q is the heat flux per unit length, and Nu is the Nusselt number.
For Nu, the widely used correlations are the Gnielinski's correlation [29]: and the Dittus-Boelter correlation [29]: where Pr is the molecular Prandtl number. In Gnielinski's correlation (Equation (21)), the friction factor is given by the Petukhov's correlation [29]: This correlation can also be used in Equation (19) to calculate S gen, f .

Numerical Procedures
The CFD package Ansys Fluent (17.0) was used for all computations. In addition, the User-Defined Function functionality was used to implement the SST k-ω-ϕ-α model. The four turbulence variables, k, ω, ϕ, and α, were solved as User-Defined Scalar (UDS) quantities. The production and dissipation terms in these equations were added as "sources" to the transport equations, and the diffusivities of these variables were assigned accordingly. The SST k-ω and realizable k-ε models are pre-coded turbulence models available in Ansys Fluent that can be used directly.
The governing equations were solved by using the pressure-based Coupled algorithm. The second-order upwind scheme was used to discretize the convection terms in the momentum, energy, and turbulence equations. The velocity-pressure coupling process was handled by the Coupled algorithm, which solves the momentum and continuity equations in a closely coupled manner and can improve the rate of solution convergence significantly when compared to the segregated algorithm (such as the SIMPLE algorithm). The gradients and derivatives were evaluated by using the least-squares cell-based method. Furthermore, it should be noted that, in the current study, a double-precision solver was required to obtain a stable and accurate solution. Figure 2 shows the solution procedures used in the present paper. For the cases considered in Sections 3.1-3.4 (using the procedure as Figure 2a), the thermal properties of the materials were assumed to be temperature independent, allowing the governing equations for fluid flow and the energy equation to be solved independently. The isothermal fluid flow was solved first, and then the energy equation was solved by using the converged solution of the fluid flow. For the cases considered in Section 3.5 (using the procedure as Figure 2b), the temperature-dependent thermal properties of the fluid were considered, so that the governing equations for fluid flow and the energy equation were solved continuously in each iteration. After obtaining convergent flow and temperature fields, post-processing was performed to obtain the desired quantities, such as the Nusselt number, friction factor, and rate of entropy generation.

Computational Grid and Boundary Conditions
The computational domain (as shown in Figure 1) was meshed by using a Cartesian grid. In the vertical direction of the wall (r-direction), to capture high resolutions of vari ables close to the wall, the successive ratio mesh was used. In the streamwise direction (x direction), the cells were distributed uniformly. Several different grids were designed to examine the mesh convergence results, and then we decided upon the grid used in the present work. In the test cases, the successive ratio and the number of nodes in the x direction (Nx) kept being constants, as 1.05 and 100, respectively. The number of nodes in the r-direction (Nr) was different. Table 1 shows the results computed by using the SST --model with temperature-independent fluid properties. It was clear that when Nr increases from 150 to 200, the change of the calculated parameters, f, Nu , , and , , whose reference values were evaluated by Equations (23), (21), (20), and (19), respectively, was small. Therefore, the mesh resolution of 200 × 100 was selected for the present work.

Computational Grid and Boundary Conditions
The computational domain (as shown in Figure 1) was meshed by using a Cartesian grid. In the vertical direction of the wall (r-direction), to capture high resolutions of variables close to the wall, the successive ratio mesh was used. In the streamwise direction (x-direction), the cells were distributed uniformly. Several different grids were designed to examine the mesh convergence results, and then we decided upon the grid used in the present work. In the test cases, the successive ratio and the number of nodes in the x-direction (N x ) kept being constants, as 1.05 and 100, respectively. The number of nodes in the r-direction (N r ) was different. Table 1 shows the results computed by using the SST k-ω-ϕ-α model with temperature-independent fluid properties. It was clear that, when N r increases from 150 to 200, the change of the calculated parameters, f, Nu, S gen,h , and S gen, f , whose reference values were evaluated by Equations (19)- (21), and (23), respectively, was small. Therefore, the mesh resolution of 200 × 100 was selected for the present work. Figure 1 also depicts a schematic of the problem's boundary conditions. Given that the turbulent flow in a circular tube is fully developed, it is appropriate to construct a periodic boundary condition for the inlet and outlet. The mass flow rate and temperature were specified at the inlet. To match the Re, the mass flow rate was adjusted. Throughout this study, the inlet temperature was 293.15 K. Constant heat flux was specified at the wall for the energy equation, and the no-slip condition was used for fluid flow. The wall treatments varied depending on the turbulence model. Specifically, for the SST k-ω-ϕ-α model, the conditions u i = 0, k = 0, ϕ = 0, α = 0, and ω = 3ν/ β 1 y 2 1 were specified [22]. For the SST k-ω model, an automatic near-wall treatment method was used [30]. The enhanced wall-treatment method was used for the realizable k-ε model [31].

Data Reduction
The results were obtained through post-processing. The Nusselt number, friction factor, and entropy generation rate per unit length were the relevant parameters used in the analysis. They were computed as follows.
The Nusselt number is denoted by the following: where where h(x) is the local convective heat-transfer coefficient; q" is the heat flux; x is the distance from inlet; c p is the specific heat of the fluid; and T in , T w , and T m are the inlet temperature, wall temperature, and bulk temperature, respectively. The average temperature T ave in Equations (19) and (20) is calculated as [14] T The friction factor is represented by the following: where τ w is the wall shear stress, and U b is the bulk velocity. Given that the aspect ratio of the computational region (L/D) is small in this work, the rates of entropy generation per unit length were calculated by using Equations (17) and (18) as follows: S gen,h = S gen,h /L.

Results and Discussion
In present study, the working fluid is water, whose temperature-independent thermal properties at 293.15 K (used in Sections 3.1-3.4) are listed in Table 2. For various purposes, three circular tubes with different cross-sectional areas (A c ) of 0.000005 m 2 , 0.0005 m 2 , and 0.05 m 2 ; five Reynolds numbers, 10,000, 30,000, 50,000, 70,000, and 100,000; and different heat fluxes, ranging from 500 to 30,000 W/m 2 , were used. The main concern in this paper is the rate of entropy generation per unit length, which was obtained by averaging along the flow direction (Equations (29) and (30)). To achieve accurate results, it requires that the physical quantity in the computational region cannot change too much along the flow direction. Consequently, a value of 0.5 for the aspect ratio (L/D) was selected for all tubes considered in this study. Table 3 lists other parameters used in the following sections. For Sections 3.1-3.3, the motivation for using different heat fluxes for different tubes was to ensure that the wall-bulk temperature difference (∆T) in all tubes was small enough when compared to the bulk flow temperature (T m ), and then the results could be compared with the analytical expressions of Bejan [3] (Equations (19) and (20)), in which the ratio of ∆T to T m was small and, thus, ignored. However, we discovered that, when a large heat flux (for example, 50,000 W/m 2 ) is applied to a large tube (for example, A c = 0.05 m 2 ), the computed rate of entropy generation deviates from the analytical expressions at a low Re. Mwesigye and Huan [16] discovered the same phenomenon. They reasoned that the large error was caused by the incorrect correlations used for Nu and f. However, we discovered in this study that ∆T is too large to be ignored in this case, and the condition that should be satisfied in Bejan's analytical expression is violated. The solution convergence history is influenced by the relaxation factors and the initial conditions. In present work, the relaxation factors for the four turbulent quantities of the SST k-ω-ϕ-α model were set to be 0. Based on same computer hardware (24 CPU cores with 3.1 GHz frequency and 128 GB RAM), the computational efficiency was compared by using a case with constant fluid properties and A c = 0.05 m 2 , Re = 100,000, and q" = 500 W/m 2 . The SST k-ω-ϕ-α, SST k-ω, and realizable k-ε models spent about 45, 32 and 28 min CPU time to obtain a convergent solution, respectively. The reason that the SST k-ω-ϕ-α model takes more computation time is that it has two more governing equations and more complex source terms.

Validation of the Friction Factor and Nusselt Number
The friction factor was used to validate the hydrodynamic performance of the turbulence model. Figure 3 compares friction factors calculated by various turbulence models with Petukhov's correlation [29]. The SST k-ω-ϕ-α model achieves an excellent agreement for all Re. The SST k-ω model and the realizable k-ε model both predict good results at high Re but overestimate the friction factor at low Re. When the Re is lower, the deviation is greater. This is not surprising given that both the SST k-ω model and the realizable k-ε model we used in present work are developed for high Re flow and do not include any low Re corrections, whereas the SST k-ω-ϕ-α model is developed for low and high Re flow and includes some low Re corrections.  The Nusselt number was used to validate the heat-transfer pe noted that the Prt in Equation (4) affects the temperature solution other quantities, such as the Nusselt number, effective thermal co entropy generation. Because the influence factors are so comple stood, Prt is difficult to derive from theory. In CFD applications, P using an empirical formula or as a constant determined by the sl curve in the log region [32]. Even in simple wall shear flows, there value for Prt. For example, the Prt ranges between 0.73 and 0.92 fo [32]. Furthermore, different turbulence models may necessitate dif ent temperature profile predictions. Three constant values, 0.73, 0. to evaluate the effect of Prt on heat-transfer performance.
The computed Nu by different Prt was compared with the (Equation (21)) and the Dittus-Boelter correlation (Equation (22)). sults. It is clear that a lower Prt leads to higher Nu for all turbulence results deviate from the Dittus-Boelter correlation to a large ext SST --model (Figure 4a)   The Nusselt number was used to validate the heat-transfer performance. It should be noted that the Pr t in Equation (4) affects the temperature solution, which, in turn, affects other quantities, such as the Nusselt number, effective thermal conductivity, and rate of entropy generation. Because the influence factors are so complex and not fully understood, Pr t is difficult to derive from theory. In CFD applications, Pr t is typically treated by using an empirical formula or as a constant determined by the slope of the temperature curve in the log region [32]. Even in simple wall shear flows, there is no universal constant value for Pr t . For example, the Pr t ranges between 0.73 and 0.92 for airflow with Pr = 0.71 [32]. Furthermore, different turbulence models may necessitate different Pr t , due to different temperature profile predictions. Three constant values, 0.73, 0.85, and 0.92, were used to evaluate the effect of Pr t on heat-transfer performance.
The computed Nu by different Pr t was compared with the Gnielinski's correlation (Equation (21)) and the Dittus-Boelter correlation (Equation (22)). Figure 4 depicts the results. It is clear that a lower Pr t leads to higher Nu for all turbulence models. All computed results deviate from the Dittus-Boelter correlation to a large extent. The results for the SST k-ω-ϕ-α model (Figure 4a) with Pr t = 0.73 and 0.85 are in good agreement with Gnielinski's correlation. The results with Pr t = 0.92 deviate slightly. All results for the SST k-ω model (Figure 4b) deviate from Gnielinski's correlation. The result with Pr t = 0.92 is the best among them. The result with Pr t = 0.85 for the realizable k-ε model (Figure 4c) is in good agreement with Gnielinski's correlation. Pr t = 0.73 overestimates the Nu. Pr t = 0.92; however, underpredicts it.
Overall, the computed results for all three turbulence models agree better with the Gnielinski's correlation than the Dittus-Boelter correlation, so the Gnielinski's correlation was used where the correlation of Nu is required. k-ω model (Figure 4b) deviate from Gnielinski's correlation. The result with Prt = 0.92 is the best among them. The result with Prt = 0.85 for the realizable k-ε model (Figure 4c) is in good agreement with Gnielinski's correlation. Prt = 0.73 overestimates the Nu. Prt = 0.92; however, underpredicts it.  Overall, the computed results for all three turbulence models agree better with the Gnielinski's correlation than the Dittus-Boelter correlation, so the Gnielinski's correlation was used where the correlation of Nu is required.

Rate of Entropy Generation Due to Heat Transfer Irreversibility
According to Equation (20), the rate of entropy generation per unit length due to heat transfer irreversibility, , , is dependent on Tave and Nu, which is, in turn, dependent on Prt. Consequently, , will be affected by Prt. Because the effect of Prt on , is similar for circular tubes of various sizes, only the results for Ac = 0.000005 m 2 were provided. Figure 5 depicts the variation of , with respect to Re computed by various turbulence models with varying Prt. It is self-evident that a smaller Prt results in a smaller , . This is the inverse of the effect of Prt on Nu. It is simple to understand because , is inversely proportional to Nu. When comparing the SST --model (Figure 5a) with the Bejan's correlation (Equation (20)

Rate of Entropy Generation Due to Heat Transfer Irreversibility
According to Equation (20), the rate of entropy generation per unit length due to heat transfer irreversibility, S gen,h , is dependent on T ave and Nu, which is, in turn, dependent on Pr t . Consequently, S gen,h will be affected by Pr t . Because the effect of Pr t on S gen,h is similar for circular tubes of various sizes, only the results for A c = 0.000005 m 2 were provided. Figure 5 depicts the variation of S gen,h with respect to Re computed by various turbulence models with varying Pr t . It is self-evident that a smaller Pr t results in a smaller S gen,h . This is the inverse of the effect of Pr t on Nu. It is simple to understand because S gen,h is inversely proportional to Nu. When comparing the SST k-ω-ϕ-α model (Figure 5a) with the Bejan's correlation (Equation (20)), we see that the result with Pr t = 0.85 agrees very well. Pr t = 0.73 overpredicts but Pr t = 0.92 underpredicts S gen,h . Comparing the SST k-ω model (Figure 5b) with the Bejan's correlation, we see that all values of Pr t underpredict S gen,h , but the degree of deviation decreases as Re increases. The result with Pr t = 0.92 is the best of the three Pr t . The results with Pr t = 0.85 for the realizable k-ε model (Figure 5c) agree with the Bejan's correlation for high Re. However, for low Re (Re = 10,000), the result with Pr t = 0.92 is better. Based on this finding, considering together the effect of Pr t on Nu, the Pr t that yields good Nu and S gen,h was chosen in the subsequent analysis of each turbulence model, i.e., 0.85, 0.92, and 0.85 for the SST k-ω-ϕ-α model, SST k-ω model, and realizable k-ε model, respectively. The local distribution of the rate of entropy generation per unit volume due to heattransfer irreversibility, , , along the radial direction at Re = 10,000 is shown in Figure  6. It can be seen that , mainly occurs in the near-wall region. It is not surprising because, in the near-wall region, the temperature gradient is large. The distributions of , predicted by the SST k-ω and realizable k-ε models are very close. The , predicted by the SST --model is larger than the other two.  3 10 − × The local distribution of the rate of entropy generation per unit volume due to heattransfer irreversibility, S gen,h , along the radial direction at Re = 10,000 is shown in Figure 6.
It can be seen that S gen,h mainly occurs in the near-wall region. It is not surprising because, in the near-wall region, the temperature gradient is large. The distributions of S gen,h predicted by the SST k-ω and realizable k-ε models are very close. The S gen,h predicted by the SST k-ω-ϕ-α model is larger than the other two. , 6. It can be seen that , mainly occurs in the near-wall region. It is cause, in the near-wall region, the temperature gradient is large. T , predicted by the SST k-ω and realizable k-ε models are very clo dicted by the SST --model is larger than the other two.  Figure 6. Comparison of local distribution of S gen,h .

Rate of Entropy Generation Due to Friction Irreversibility
As mentioned in Section 2.2, there are two methods in CFD applications for calculating the entropy generation due to friction irreversibility, S gen, f . The first uses turbulent viscosity (Equation (13), denoted as Method 1), and the second uses the turbulent dissipation rate (Equation (15), denoted as Method 2). For the k-ε model series, Method 2 is natural. However, for other turbulence models in which ε is not easily determined, Method 2 may encounter difficulty. For example, in the SST k-ω-ϕ-α and SST k-ω models, although ε = β * kω was used when the models were developed, it does not hold everywhere due to some simplifications, modifications, and corrections. Figure 7 depicts a comparison of the results obtained by using Methods 1 and 2. Because the observed phenomena are the same for tubes of different diameters, only the results with A c = 0.000005 m 2 are provided. The results show that S gen, f increases with the increasing Re. For the SST k-ω-ϕ-α model, the results predicted by the two methods are significantly different. Compared with the Bejan's correlation (Equation (19)), the result predicted by Method 1 agrees well, but the result predicted by Method 2 deviates. The difference at Re = 100,000 is about 14%. This is because the equation ε = β * kω does not exactly hold in the SST k-ω-ϕ-α model. Consequently, Method 2 cannot be used to determine S gen, f for this model. However, for the SST k-ω and realizable k-ε models, the results predicted by the two methods are almost indistinguishable, and they are both in good agreement with the Bejan's correlation. Figure 8 shows the local distribution of the rate of entropy generation per unit volume due to friction irreversibility, S gen, f , along the radial direction at Re = 10,000. Similar to S gen,h , the S gen, f mainly occurs in the near-wall region. This is because the velocity gradient is large in the near-wall region. The difference among three models occurs in a very thin region near the wall (0.49 < r/D < 0.5), so that, after being integrated, the difference is small.
As we can see from Equation (19), when the difference in average temperature is small, the S gen, f is inversely proportional to the square of tube diameter at the same Re. This phenomenon is well represented by all three turbulence models. The details are omitted here for brevity. The difference at Re = 100,000 is about 14%. This is because the equation = * does not exactly hold in the SST --model. Consequently, Method 2 cannot be used to determine , for this model. However, for the SST k-ω and realizable k-ε models, the results predicted by the two methods are almost indistinguishable, and they are both in good agreement with the Bejan's correlation.  Figure 8 shows the local distribution of the rate of entropy generation per unit volume due to friction irreversibility, , , along the radial direction at Re = 10,000. Similar , , the , mainly occurs in the near-wall region. This is because gradient is large in the near-wall region. The difference among three model very thin region near the wall (0.49 < r/D < 0.5), so that, after being integrate ence is small. As we can see from Equation (19), when the difference in average tem small, the , is inversely proportional to the square of tube diameter at

Rate of Entropy Generation Due to Heat Transfer Irreversibility with Large ∆T
As previously stated, Bejan's correlation of the rate of entropy generation per unit length due to heat transfer irreversibility (S gen,h , Equation (20)) holds only if ∆T/T m is so small that it can be ignored. In fact, when ∆T/T m cannot be ignored, S gen,h should be calculated by using the following [3].
Considering ∆T = T w − T m and q = πDq , using Equations (24) and (25) can yield The Bejan's correlation will deviate significantly in the large tube with a high heat flux and low Re [16]. To investigate the turbulence model's ability to predict S gen,h when the wall-bulk temperature difference is large, the flow in a large tube with A c = 0.05 m 2 and a low Re of 10,000 was considered. Figure 9 compares the predicted S gen,h by different turbulence models with the analytical results. In this case, S gen,h increases with increasing heat flux. All three turbulence models' results agree well with the analytical expression of Equation (31). Among them, the SST k-ω-ϕ-α model predicts a better result. However, the deviation of the Bejan's correlation (Equation (20)) is very noticeable and increases with increasing heat flux. The deviation is close to 50% at q = 30, 000 W/m 2 . When examining this case in detail, we discovered that the large wall-bulk temperature difference (exceeding 100 K) is the reason for the large error. This finding suggests that, when using the Bejan's correlation (Equation (20)) to validate turbulence models, it is critical to ensure that the wall-bulk temperature difference is small enough that ∆T/T m can be ignored. ropy 2022, 24, x FOR PEER REVIEW Figure 9. Comparison of the predicted ,ℎ by different turbulence models und temperature difference.

Effect of the Temperature-Dependent Fluid Properties
It is well-known that the properties of a fluid are generally tempera For example, the viscosity of water is highly temperature-dependent. Wh ture changes significantly, it is inappropriate to consider fluid properti Ideally, the entropy generation rate will change because of temperatureproperties on the flow field. Hence, the effect of temperature-dependent on the entropy generation rate was investigated in this subsection. Abb Amani [33] used the following correlations to calculate the water propert

Effect of the Temperature-Dependent Fluid Properties
It is well-known that the properties of a fluid are generally temperature-dependent. For example, the viscosity of water is highly temperature-dependent. When the temperature changes significantly, it is inappropriate to consider fluid properties as constants. Ideally, the entropy generation rate will change because of temperature-dependent fluid properties on the flow field. Hence, the effect of temperature-dependent fluid properties on the entropy generation rate was investigated in this subsection. Abbasian Arani and Amani [33] used the following correlations to calculate the water properties.
The flow in the tube with A c = 0.05 m 2 at Re = 10,000, which is based on fluid properties at the inlet (T in = 293.15 K), was computed. Five cases were considered, each with a different heat flux of 2000, 5000, 10,000, 15,000, and 20,000 W/m 2 . Figure 10 depicts the effect of temperature-dependent fluid properties on entropy generation rate due to friction irreversibility, S gen, f . The Bejan's correlation of Equation (19) computed by using constant fluid properties at inlet temperature was included as a reference. The result of Bejan's correlation decreases only slightly as heat flux increases. Conversely, as the heat flux increases, all S gen, f computed by using CFD results with three turbulence models decreased significantly. The observed behavior can be explained by using detailed flow and temperature field information. Figure 11 depicts the SST k-ω-ϕ-α model's simulated temperature and fluid molecular viscosity profiles on the lateral section. The difference in average temperature for different heat fluxes is small. Consequently, the S gen, f predicted by the Bejan's correlation with constant fluid property, which is only dependent on the average temperature, decreases slightly with increasing heat flux. With increasing heat flux in the near-wall region, the local temperature rises dramatically, significantly lowering the molecular fluid viscosity. Furthermore, there is a high velocity gradient near the wall. Consequently, the S gen, f computed using CFD results (Equation (11)) are significantly reduced. The greater the heat flux, the lower the viscosity and, consequently, the entropy generation.
py 2022, 24, x FOR PEER REVIEW wall. Consequently, the , computed using CFD results (Equatio cantly reduced. The greater the heat flux, the lower the viscosity and, entropy generation.    The effect of temperature-dependent fluid properties on the rate of entropy generation due to heat-transfer irreversibility, , , is depicted in Figure 12. The CFD predicted , separately, using temperature-dependent fluid properties (denoted by solid points), and constant fluid properties at inlet temperature (denoted by hollow points) were compared with the Bejan's correlation of Equation (20). When considering the variation of fluid properties with temperature, , is significantly reduced. Similar to , , the greater the heat flux applied, the more , decreases. The effect of temperature-dependent fluid properties on the rate of entropy generation due to heat-transfer irreversibility, S gen,h , is depicted in Figure 12. The CFD predicted S gen,h separately, using temperature-dependent fluid properties (denoted by solid points), and constant fluid properties at inlet temperature (denoted by hollow points) were compared with the Bejan's correlation of Equation (20). When considering the variation of fluid properties with temperature, S gen,h is significantly reduced. Similar to S gen, f , the greater the heat flux applied, the more S gen,h decreases. This result demonstrates that temperature-dependent fluid properti sidered when there is a significant temperature change in the flow field. jen's correlations are inapplicable.

Conclusions
The rate of entropy generation in a circular tube subjected to unifor calculated by using an elliptic blending model (SST ---). The an This result demonstrates that temperature-dependent fluid properties must be considered when there is a significant temperature change in the flow field. In this case, Bajen's correlations are inapplicable.

Conclusions
The rate of entropy generation in a circular tube subjected to uniform heat flux was calculated by using an elliptic blending model (SST k-ω-ϕ-α). The analytical correlations and results that were computed by using the realizable k-ε model and the SST k-ω model were provided as references. As a result of the findings, the following conclusions were reached.
(1) The turbulent Prandtl number influences both the Nusselt number and the entropy generation rate due to heat-transfer irreversibility. For the SST k-ω-ϕ-α model, Pr t = 0.85 yields the best results. The SST k-ω-ϕ-α model produces excellent results for all Reynolds numbers; however, both the realizable k-ε and SST k-ω models perform poorly for low Reynolds numbers.
(2) The rate of entropy generation due to friction irreversibility was well predicted by all three turbulence models. For the realizable k-ε and SST k-ω models, the two methods for calculating S gen, f led to nearly indistinguishable results. However, For the SST k-ω-ϕ-α model, the difference between two methods is significant and it is about 14% at Re = 100,000. The method that employs the effective turbulent viscosity, rather than the turbulent dissipation rate, ε, should be used.
(3) When the temperature in the flow field varies significantly, the change in fluid properties with a temperature considerably affects entropy generation rates due to friction and heat transfer.
In conclusion, the SST k-ω-ϕ-α model outperforms the other two turbulence models. Furthermore, although the case studied in this paper is relatively simple, given the inherent advantages of the SST k-ω-ϕ-α model in complex flows, it has great potential for the application of heat-transfer-system optimization. The application of this model in complex heat-transfer systems (such as separated flow and impinging jet flow) requires continued research and verification.
Finally, it should be pointed out that this paper mainly studied the capabilities of the turbulence model, rather than the detailed behavior of entropy generation rate in circular pipe flow, so we did not focus on the contribution of each part of the entropy generation rate to the total entropy generation rate. In fact, it could be clearly seen that, in small-sized pipe (A c = 0.000005 m 2 ), as shown in Sections 3.3 and 3.4, the entropy generation rate due to heat transfer is only on the order of 10 −3 of the entropy generation rate due to friction. Meanwhile, in large-sized pipe (A c = 0.05 m 2 ), as shown in Section 3.5, the entropy generation rate due to friction is only on the order of 10 −7 of the entropy generation rate due to heat transfer. For the total entropy generation rate, the entropy generation rate due to heat transfer in small-sized pipe and that due to friction in large-sized pipe are completely negligible. Of course, for medium-sized pipes, the entropy generation rates of these two parts are of the same order of magnitude.