Effects of Variable Properties on the Convective Flow of Water near Its Density Extremum in an Inclined Enclosure with Entropy Generation

Abstract

The aim of the current numerical simulation was to understand the effects of the temperature-dependent properties of viscosity and thermal conductivity on the buoyant convection of cold water around its density extremum in a tilting chamber. The equations for thermal conductivity and viscosity were derived based on the reference temperature parameter. The entropy generation and cup mixing temperature were also calculated. The governing mathematical model was solved by the finite-volume-based iterative technique. The obtained results were deliberated for several values of inclination angles and for the density reversal parameter for variable viscosity and thermal conductivity cases. It was detected that density inversion has a strong effect on stream and thermal patterns through the development of a bi-cellular arrangement due to density inversion.

1. Introduction

The convective flow of cold water around its density extremum ($3.98 °\mathrm{C}$) happens universally in the environment and in some engineering and technical processes [1,2,3]. Since the convective stream and thermal transfer of water around its density extremum is a complex phenomenon, experimental and theoretical investigations into the buoyant convective stream of water near its extreme density in a rectangular closed domain have been reported by many researchers [4,5,6,7]. Ho and Tu [8] (2001) numerically and experimentally inspected the free convective stream of water near its extreme density in a rectangular closed container. Moraga and Vega [9] (2004) scrutinized the variable physical properties of water in a 3D buoyant convection flow. Hossain and Rees [10] (2005) deliberated the unsteady laminar convective flow of water subject to density reversal in a rectangular chamber in the presence of (internal) heat generation. Sivasankaran and Ho [11,12,13] explored the effects of temperature-based water properties on the free convection of cold water near its density extreme region in a chamber. Varol et al. [14] made a numerical simulation to find an interesting result from the convective motion of (cold) water at 4 °C in a porous container with a thick bottom wall.

Li et al. [15] explored the three-dimensional Rayleigh–Bénard (RB) convection of water around its density extremum in a cylindrical domain with an aspect ratio of two. Hu et al. [16] experimentally and numerically examined the impact of aspect ratios on the Rayleigh–Bénard (RB) convective flow of (cold) water near its density supremum in cylindrical containers. They found that density inversion has a negative effect on heat transfer rates. Mastiani et al. [17] examined the density extremum effects of nano liquids on combined convection in a lid-driven chamber with various vertical wall movements. They used both non-Boussinesq and Boussinesq approximations in the study. Hua et al. [18] explored the effects of density inversion, aspect ratio, and thermal border conditions on the RB convection of cold water. Huang et al. [19] studied the effects of density inversion on Rayleigh–Bénard fluid motion and heat transfer in a cylindrical container. Hu et al. [20] explored the free convective flow of water-based nanofluids near their density inversions in an annulus. They established that Brownian motion must be taken into account when considering the free convection of water-based nanofluids at their density supremum. Bindhu et al. [21] explored the density inversion effect with various combinations of heater and cooler locations in a buoyant convective flow regime inside a square container. Sivasankaran et al. [22] found the impact of temperature-dependent water properties on buoyant flow around the density extreme region in a square chamber. From all these studies, we understand that flow around the vicinity of the density inversion point is more complicated and requires more detailed investigation.

In some situations, we have a thermal and mechanical system with a certain inclination from the ground (horizontal position). Therefore, it is essential that the flow and thermal transports in a tilted chamber are explored. Much research on the inclination of geometry under various conditions has been conducted during past three decades [23,24,25]. However, research exploring the effects of fluid properties in a tilted geometry is very limited. Cheng et al. [26] numerically deliberated the influence of tilting angles and aspect ratios on convective flow in a rectangular geometry with sinusoidal-type heating. Sivakumar and Sivasankaran [27] explored the impacts of nonuniform heating on mixed convection in a lid-driven inclined cavity. The impact of the direction of moving walls in a lid-driven geometry with sinusoidal heating was studied by Sivasankaran et al. [28]. The effects of various thermal border conditions on buoyant convection in an inclined triangular chamber was explored by Sivasankaran et al. [29]. Alshomrani et al. [30] inspected the influence of tiling angle, heater locations, and cooler locations on buoyant convection in a 3D box. Sivasankaran and Janagi [31] explored the effects of inclination and baffle size on mixed convective streams in a channel cavity.

The second law of thermodynamics explains the concept of entropy in a thermodynamic system. The analysis of entropy generation (EG) or irreversibility analysis is a key factor due to the calculation of energy loss, and EG can compute energy transmission and can estimate thermal performance. Since the irreversibility analysis exists on both due to fluid flow and thermal transport [32,33,34], numerous interdisciplinary sectors need to know about it in detail. Moreover, a reduction in EG is essential in order to save energy and improve the performance of the thermal system. Recently, investigations into entropy optimization have dramatically increased in the thermo-fluid field and its related disciplines. Many basic results on EG, with applications, have been derived by Bejan [35,36,37,38].

Alsabery et al. [39,40] explored entropy generation and convective flow hybrid nanofluids in a wavy walled container with a hot-half partition. They found that varying geometrical parameters induced different effects on entropy generation and Bejan number. Ghalambaz et al. [41] explored thermal convective transport and the entropy generation of hybrid nano liquids in an oddly shaped chamber. Khetib et al. [42] numerically explored the free convection and generation of entropy of nanofluids in a square inclined cavity under a constant magnetic field. They used three different fin configurations (straight, curved, and inclined) in order to analyze the results. Cherif et al. [43] explored heat transferal rates and the generation of entropy for a water-based hybrid nanofluid inside a 3D porous triangular-shaped cavity with a spinning cylinder. Alshare et al. [44], having performed numerical simulations, found an interesting result that thermal transport is responsible for entropy production when the magnetic and frictional effects are marginal. Le et al. [45], having examined entropy in the convection of nano-suspensions in a cavity, found that the addition of nanoparticles reduced the strength of heat transfer and diminished entropy generation. Abderrahmane et al. [46] deliberated the mixed convection and entropy generation of hybrid nanofluids in a 3D triangular enclosed area with a spinning circular cylinder in the middle of the domain. Mourad et al. [47] studied the MHD free convection and EG of (Cu–water) nano liquid in a porous annulus.

We have found that the convective flow of cold water with temperature-based physical properties in inclined cavities has not explored in previous research thus far. Hence, the current study inspects the influence of temperature-dependent viscosity and thermal conductivity of the water near its density extremum point on buoyant convection in a tilted cavity. The entropy generation and cup-mixing temperature are also explored here. The present research may be very useful for environmental thermal systems.

2. Mathematical Modelling

We studied a two-dimensional square chamber of size “D” that was filled with cold water, as shown in Figure 1. The isothermal vertical borders of the enclosed space were well preserved at dissimilar (constant) temperatures T_h* and T_c*, with the condition that T_h^* > T_c^*. The horizontal borders of the enclosed space were adiabatic. The gravitational force acted in a downwards direction. In the x and y directions, the velocity components u and v are given. The chamber was tilted to have an acute angle γ with the horizontal axis. The subsequent assumptions were adopted for this study: the flow is two-dimensional, unsteady, laminar, and incompressible. The Boussinesq approximation was valid. The viscous dissipation and radiation were neglected in this study. Water density has a non-linear relation with temperature, as follows [8,11]:

$$\rho ={\rho }_m\left(1-\beta {\left|T\ast -T_m{}^{\ast }\right|}^b\right)$$

(1)

where β = 9.2971730 × 10⁻⁶, T_m = 4.0293250 °C, ρ_m (=999.9720) is the maximum density of water, and b = 1.894816. The thermal conductivity of water and the viscosity of water vary with temperature. The water viscosity varies with temperature, as follows [11,12]:

μ(T*) = a₀ + a₁ T* + a₂ T*² + a₃ T*³

(2)

where a₀ (=1.791084), a₁ (=−6.144 × 10⁻²), a₂ (=1.451 × 10⁻³), and a₃ (=−1.6826 × 10⁻⁵) are the temperature quantities of the viscosity of water. Equation (2) can be written as:

$$\mu ={\mu }_r\left\{1+A1\epsilon T+A2\left[{\left(1+\epsilon T\right)}^2-1\right]+A3\left[{\left(1+\epsilon T\right)}^3-1\right]\right\}$$

(3)

where $\epsilon =\frac{T_h^{\ast }-T_c^{\ast }}{T_r}$, $A1=\frac{a1T_r}{{\mu }_r}$, $A2=\frac{a2 T_r{}^2}{{\mu }_r}$, and $A3=\frac{a3 T_r{}^3}{{\mu }_r}$. The variable T_r denotes the reference temperature. ε denotes a parameter of the reference temperature, which plays an essential role on the water stream when variable properties are considered. The thermal conductivity of water [11,12,13] varies with temperature, as follows:

k(T*) = b₀ + b₁ T* + b₂ T*² + b₃ T*³

(4)

where b₀ (=0.561965), b₁ (=2.15346 × 10⁻³), b₂ (=−1.55141 × 10⁻⁵), and b₃ (=1.01689 × 10⁻⁷) are the temperature coefficients of the thermal conductivity of water. The equation can be written as:

$$k=k_r\left\{1+B1\epsilon T+B2\left[{\left(1+\epsilon T\right)}^2-1\right]+B3\left[{\left(1+\epsilon T\right)}^3-1\right]\right\}$$

(5)

where $B1=\frac{b1T_r}{k_r}$, $B2=\frac{b2 T_r{}^2}{k_r}$, and $B3=\frac{b3 T_r{}^3}{k_r}$.

The governing equations of vorticity, stream function, and temperature with temperature-dependent liquid properties are:

$$\begin{array}{l}{\rho }_r\left(\frac{\partial \omega }{\partial t}+u\frac{\partial \omega }{\partial x}+v\frac{\partial \omega }{\partial y}\right)=\mu \left(\frac{{\partial }^2\omega }{\partial x^2}+\frac{{\partial }^2\omega }{\partial y^2}\right)+2\frac{\partial \omega }{\partial x}\frac{\partial \mu }{\partial x}+4\frac{\partial v}{\partial y}\frac{{\partial }^2\mu }{\partial x\partial y}+2\frac{\partial \omega }{\partial y}\frac{\partial \mu }{\partial y}\\ +\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\left(\frac{{\partial }^2\mu }{\partial x^2}-\frac{{\partial }^2\mu }{\partial y^2}\right)+\rho \beta \left[\frac{\partial {\left|T^{\ast }-T_m{}^{\ast }\right|}^b}{\partial x}Cos(\gamma )-\frac{\partial {\left|T^{\ast }-T_m{}^{\ast }\right|}^b}{\partial y}Sin(\gamma )\right]\end{array}$$

(6)

$$\frac{{\partial }^2\psi }{\partial x^2}+\frac{{\partial }^2\psi }{\partial y^2}+\omega =0$$

(7)

$$\frac{\partial T^{\ast }}{\partial t}+u\frac{\partial T^{\ast }}{\partial x}+v\frac{\partial T^{\ast }}{\partial y}=\frac{\partial }{\partial x}\left(\frac{k}{{\rho }_r{c}_p}\frac{\partial T^{\ast }}{\partial x}\right)+\frac{\partial }{\partial y}\left(\frac{k}{{\rho }_r{c}_p}\frac{\partial T^{\ast }}{\partial y}\right)$$

(8)

where g denotes acceleration due to gravity, c_p—specific heat, t—time, k—thermal conductivity, T—temperature, μ viscosity, and ρ_r—density.

The initial values are taken as $u=v=0$, $T^{\ast }=T_c^{\ast }$ in the whole domain. The boundary conditions are as follows: the no-slip condition ($u=v=0$) for velocities was taken on all the solid walls at $x=0\&D$,$y=0\&D$. The temperature conditions are $\frac{\partial T^{\ast }}{\partial y}=0$ at $y=0\&D$, $T^{\ast }=T_h^{\ast }$ at $x=0$, and $T^{\ast }=T_c^{\ast }$ at $x=D$.

The non-dimensional variables used here are given below.

$$\left(X, Y\right)=\frac{\left(x,y\right)}{D}, \left(U, V\right)=\frac{\left(u, v\right)}{\raisebox{1ex}{${\mu }_r$}\!\left/ \!\raisebox{-1ex}{$D{\rho }_r$}\right.}, t_0=\frac{t {\mu }_r}{D^2{\rho }_r},{\mu }^{\ast }=\frac{\mu }{{\mu }_r}, k^{\ast }=\frac{k}{k_r}, \mathsf{\Psi }=\frac{\psi }{{\mu }_r/{\rho }_r}, \mathsf{\Omega }=\frac{\omega D^2{\rho }_r }{{\mu }_r} \mathrm{and} T=\frac{T^{\ast }-T_r}{T_h^{\ast }-T_c^{\ast }}$$

The governing model equations after non-dimensionalization are:

$$\begin{array}{l}\frac{\partial \mathsf{\Omega }}{\partial t_0}+U\frac{\partial \mathsf{\Omega }}{\partial X}+V\frac{\partial \mathsf{\Omega }}{\partial Y}={\mu }^{\ast }\left(\frac{{\partial }^2\mathsf{\Omega }}{\partial X^2}+\frac{{\partial }^2\mathsf{\Omega }}{\partial Y^2}\right)+2\frac{\partial \mathsf{\Omega }}{\partial X}\frac{\partial {\mu }^{\ast }}{\partial X}+2\frac{\partial \mathsf{\Omega }}{\partial Y}\frac{\partial {\mu }^{\ast }}{\partial Y}+4\frac{\partial V}{\partial Y}\frac{{\partial }^2{\mu }^{\ast }}{\partial X\partial Y}\\ +\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)\left(\frac{{\partial }^2{\mu }^{\ast }}{\partial X^2}-\frac{{\partial }^2{\mu }^{\ast }}{\partial Y^2}\right)+\frac{Ra}{\mathrm{Pr}}\left[\frac{\partial {\left|T-T_m\right|}^b}{\partial X}Cos(\gamma )-\frac{\partial {\left|T-T_m\right|}^b}{\partial Y}Sin(\gamma )\right]\end{array}$$

(9)

$$\frac{{\partial }^2\mathsf{\Psi }}{\partial X^2}+\frac{{\partial }^2\mathsf{\Psi }}{\partial Y^2}+\mathsf{\Omega }=0$$

(10)

$$\frac{\partial T}{\partial t_0}+U\frac{\partial T}{\partial X}+V\frac{\partial T}{\partial Y}=\frac{1}{\mathrm{Pr}}\left[\frac{\partial }{\partial X}\left(k^{\ast }\frac{\partial T}{\partial X}\right)+\frac{\partial }{\partial Y}\left(k^{\ast }\frac{\partial T}{\partial Y}\right)\right]$$

(11)

The relation of velocity to stream function and vorticity are:

$$U=-{\mathsf{\Psi }}_Y, V={\mathsf{\Psi }}_X \mathrm{and} \mathsf{\Omega }=U_Y-V_X$$

(12)

The dimensionless form of the initial values are: $U=V=\mathsf{\Psi }=T=\mathsf{\Omega }=0$. The boundary conditions are:

$$\mathsf{\Psi }=U=V=0; \frac{\partial T}{\partial Y}=0; Y=1\&0$$

(13)

$$\mathsf{\Psi }=U=V=0; T=0; X=1$$

$$\mathsf{\Psi }=U=V=0, T=1; X=0$$

The non-dimensional parameters that appear in the equations are: $T_m=\frac{T_m-T_r}{T_h^{\ast }-T_c^{\ast }}$, the density inversion parameter; $Ra=\frac{g\beta {\left(T_h-T_c\right)}^b{D}^3{\rho }_r}{{\mu }_r{\alpha }_r}$, the Rayleigh number; $\mathrm{Pr}=\frac{{\mu }_r}{{\rho }_r{\alpha }_r}$(=11.6), the Prandtl number. The density extremum parameter assisted us in tracing the maximum density plane inside the enclosed space. This was due to the existence of two dissimilar horizontal layers of water with dissimilar density gradients divided by the extreme density plane. $T_m$ values are between 0 and 1. Suppose $T_m=1(0)$, the hot (cool) barrier is at temperature $T_m{}^{\ast }$ so the density of water rises (declines) with temperature inside the domain. Computing the thermal gradient on a solid wall is vital in convective flow problems. The rate of heat transfer at the hot (left side) wall was measured using the local Nusselt number and is defined by $Nu=-k^{\ast }{\left.\frac{\partial T}{\partial X}\right|}_{X=0}$. Then, the average Nusselt number is computed by: $\hspace{0.17em}\overline{Nu}\hspace{0.17em}\hspace{0.17em}={\displaystyle \underset{0}{\overset{1}{\int }}NudY}$.

3. Entropy Generation

The EG was computed from well-known velocity and thermal fields because the entropy production resulted from the irreversible process of thermal transfer and viscosity effects. We can use the entropy generation values to discern the optimal settings for real world applications. The local EG (per unit area) was derived subject to heat transmission and fluid friction. That is, the total local EG was the sum of the two quantities (S_Gen = S_heat + S_fluid).

$$S_{Gen}=\frac{k}{T_c^2}\left[{\left(\frac{\partial T^{\ast }}{\partial x}\right)}^2+{\left(\frac{\partial T^{\ast }}{\partial y}\right)}^2\right]+\left(\frac{\mu }{T_c}\right)\left\{{\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)}^2+2\left[{\left(\frac{\partial u}{\partial x}\right)}^2+{\left(\frac{\partial v}{\partial y}\right)}^2\right]\right\}$$

(14)

The dimensionless formula for EG is attained in a usual way:

$$S_{total}=S_{heat}^{\ast }+S_{fluid}^{\ast }$$

$$S_{total}={\left(\frac{\partial \theta }{\partial X}\right)}^2+{\left(\frac{\partial \theta }{\partial Y}\right)}^2+{\varphi }_2\left\{{\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)}^2+2\left[{\left(\frac{\partial U}{\partial X}\right)}^2+{\left(\frac{\partial V}{\partial Y}\right)}^2\right]\right\}$$

The whole EG calculation is achieved by integrating the local EG over the enclosed region.

$$S{G}_{total}={\int }_V^{}{S}_{total}\left(X,Y\right)dA$$

The local Bejan number gives the strength of the EG subject to the irreversibility of thermal transference. It is defined as:

$$B{e}_l=\frac{S_{heat}^{\ast }}{S_{total}}$$

(15)

It is interesting to note that heat transfer irreversibility dominates when $B{e}_l>\frac{1}{2}$. On the other hand, fluid friction irreversibility dominates when $B{e}_l<\frac{1}{2}$. The thermal and viscous irreversibilities are equal at $B{e}_l=\frac{1}{2}$. The mean Bejan number provides the relative impact of thermal current energy transfer irreversibility for the whole domain.

$$Be=\frac{{\int }_A^{}B{e}_l\left(X,Y\right)dA}{{\int }_A^{}dA}$$

(16)

The final steady state values of the thermal and flow fields were taken to estimate the EG inside the enclosed area.

4. Thermal Mixing Analysis

The cup-mixing temperature was derived to determine thermal mixing in the enclosed space. The velocity-weighted mean temperature was more appropriate for convective currents than the spatial average temperature. The cup-mixing temperature (T_cup) is given as:

$$T_{Cup}=\frac{\iint \widehat{V} TdYdX}{\iint \widehat{V}dYdX}$$

where $\widehat{V}\left(X,Y\right)=\sqrt{U^2+V^2}$.

The RMSD (Root Mean Square Deviation) was used to quantify the amount of temperature regularity. The RMSD_Tcup is defined based on the cup-mixing temperature values, as follows:

$$RMS{D}_{T_{cup}}=\sqrt{\frac{{\sum }_{i=1}^N{\left({\theta }_i-T_{Cup}\right)}^2}{N}}$$

(17)

The lower values of RMSD specify greater temperature uniformity in the enclosed space and vice versa. The RMSD values cannot exceed unity because the dimensionless temperature diverges between 0 and 1.

5. Numerical Technique

The non-dimensional modelled Equations (9)–(12) with the initial and border conditions (13) are discretized by the control volume method. The diffusion terms were treated with the central difference scheme and the convective terms were treated with the upwind scheme. The physical domain was divided into a number of control volumes based on a non-uniform grid in both X and Y directions (Figure 1b). The grids were gathered near the walls of the chamber. The grid sizes were examined from 41 × 41 to 161 × 161 for Pr = 11.60, Ra = 10⁶, and T_m = 0. It was perceived from the grid independence examination that a grid of size of 141 × 141 was adequate in order to explore the problem. The resulting algebraic equations were solved by the (Gauss-Seidel) iterative method using the successive over relaxation technique. This sequence was repeated until we reached the preferred accuracy for the convergence condition.

$$\left|\frac{{\phi }_{(n+1)}(i,j)-{\phi }_{(n)}(i,j)}{{\phi }_{(n+1)}(i,j)}\right|\le {10}^{-6}$$

where “I” and “j” denote the grid nodes in X and Y directions, and the subscript “n” denotes the time step.

6. Validation

The computation was performed using ForTran in-house coding. The free convection of cold water near its density supremum with constant properties in a (square) cavity was taken for numerical verification. Since the validation of the in-house code is essential in research utilizing numerical simulations, the justifications for the current computational program were tested against the existing available results for free convection in a (square) cavity filled with cold water near its density supremum (Tong [5]; Lin and Nansteel [2]). The various values of Ra and T_m that were found are depicted in

Table 1. Comparison of $\overline{Nu}$ results for the diverse density extremum parameter and Ra.

Ra	Tm	$\overline{\mathit{N}\mathit{u}}$
		Present	Tong [5] (1999)	Lin and Nansteel [2] (1987)
106	1.0 0.5	9.1452 4.0281	9.2742 4.0272	9.195 4.090
105	1.0 0.5	4.7895 2.0725	4.7143 2.0298	4.709 2.080
104	1.0 0.5	2.2441 1.0691	2.2739 1.0655	2.278 1.076
103	1.0 0.5	1.1127 1.0003	1.1186 1.0007	1.1190 1.0009

. A remarkable agreement was acquired between the current and previous values.

7. Results and Discussion

Numerical simulations exploring the buoyant convective flow of cold water around its density inversion point in an inclined chamber are made for a diverse combination of parameters with variable thermal conductivity and variable viscosity. The interaction of two mechanisms (namely, the inclination angle affecting geometry and buoyancy force due to the temperature gradients present in the body of the liquid) were explored in the present study. The effects result from two different phenomena. The goal of the present research was to inspect the interaction of these two effects on water current and heat transfer in the vicinity of $4 °\mathrm{C}$ inside the cavity with variable fluid properties. The range of the significant parameters were wisely scrutinized. The density reversal parameter diverged from 0 to 1, and the Rayleigh number was fixed at ${10}^6$. The inclination angle varied from 0° to 90°. The reference temperature parameter diverged from 0.0 to 0.3.

Figure 2 demonstrates thermal spreading for the different values of the density inversion parameter and the tilting angle for variable thermal conductivity and variable viscosity cases. At $T_m=0$, the density supreme plane was located at the hot wall. That is, there was no density reversal inside the chamber. There was a single clockwise eddy inside the container at $\gamma =0$. When the tilting angle was increased to ${30}^{°}$, a single eddy stretched along the corner of the container. When the tilting angle was increased to ${45}^{°}$ and ${60}^{°}$, we observed two inner eddies. When $\gamma ={90}^{°}$, the hot wall was located at the lower portion of the container. Three multi-cellular flow patterns existed, with two re-circulating eddies at the hot top and cold bottom corners of the container when $T_m=0$. When $T_m=0.5$, density inversion existed inside the container. That is, two dissimilar (horizontal) water layers formed with two dissimilar density gradients, divided by the density maximum plane. The dual cell structure of the stream was detected for all the tilting angles of the container. However, the size and strength of the eddies depended on the tilting angle. When $\gamma =0^{°}$, the cold cell occupied the majority of the container. However, the hot cell occupied the majority of the titled cavity. At $\gamma ={90}^{°}$, two equally strong eddies formed. The energy transfer from hot (left) wall to cold wall slowed down due to the dual cell pattern. This was because of the transfer of heat energy from one cell to the other cell by the conduction mode. After increasing the density inversion parameter value to $T_m=1.0$, the density extremum plane was located at the cold wall. That is, there was no density reversal inside the chamber. A counter-rotating cell formed at this situation. When tilting the cavity, the cell stretched diagonally when $\gamma ={45}^{°}$, and two inner circulating cells formed inside the stretched eddy. When the tilting angle was further increased to $\gamma ={60}^{°}$, two strong counter-acting eddies formed inside the container. When $\gamma ={90}^{°}$, a tri-cellular pattern was observed. When comparing the cases of variable thermal conductivity and variable viscosity and thermal conductivity, the flow field had similar trends in most of the cases studied, except when $\gamma ={90}^{°}$ and $T_m=0$. Drastic changes to the flow field were found when $\gamma ={90}^{°}$ and $T_m=0$. The stream function value was lower for variable thermal conductivity case.

Figure 3 displays the thermal contours inside the container for various inclination angles, density inversion parameters, and both cases of variable thermal conductivity and variable viscosity. At $T_m=0$ and $\gamma =0^{°}$, vertical thermal stratification was observed. When increasing the tilting angle, thermal stratification disappeared. The thermal boundary layers along both isothermal walls were formed for all the inclination angle values at $T_m=0$. A small difference in thermal distribution was found between the cases of variable thermal conductivity and the variable thermal conductivity and viscosity cases. When $T_m=0.5$, the density maximum plane moved inside the cavity, and strong temperature gradients formed at the density maximum plane. The thermal boundary layers were not found along the isothermal walls at $\gamma =0^{°}$. In this situation, the density extremum plane was near the hot (left) wall. When titling the cavity ($\gamma ={30}^{°}\sim {60}^{°}$), the density extremum plane was near the cold (right) wall. When $\gamma ={90}^{°}$, the density extremum plane was in the middle of the cavity. The difference between the thermal distribution of the two cases was clearly visible at $T_m=0.5$ for all the tilting angles. When the $T_m$($T_m=1$) was increased further, the vertical thermal stratification was found for the vertical cavity ($\gamma =0^{°}$). When the tilting angle was changed, the thermal distribution was altered and vertical temperature stratification existed. However, the boundary layers disappeared when the cavity was tilted. At $\gamma ={90}^{°}$, the hot wall went down, which changed much of the flow and thermal fields. Here, the isotherms were almost straight lines and a conduction type of heat transfer was evident.

Figure 4 demonstrates the time history of the averaged Nusselt numbers for the diverse values of $T_m$ and for the variable viscosity and variable thermal conductivity cases. Initially, the mean Nusselt number fell down and then reached a constant value after a specified time. A shorter time was taken to reach the steady state at $T_m=0.5$. However, the time taken to reach the steady state was greater at $T_m=0.6$. The average Nusselt number was initially overshot when $T_m\le 0.6$. Figure 5 demonstrates the influence of local heat transfer for the various values of $T_m$ and $\gamma$ with variable thermal conductivity and variable viscosity. The heat transfer attained a minimum value at $\gamma ={90}^{°}$ for all the density inversion parameters. A higher local heat transfer rate was observed at $T_m=0$. When increasing the values of $T_m$ from 0 to 1, the local heat transport rate initially declined and then increased later at $\gamma =0^{°}$. However, local heat transfer attained the lowest value at $T_m=1.0$ when $\gamma ={90}^{°}$. Local heat transfer declined when the values of $T_m$ were raised at $\gamma ={90}^{°}$. The local heat transfer rate was high at the bottom edge when $T_m\le 0.5$, whereas it was high at the top edge of the cavity when $T_m>0.6$. A similar trend was observed for $\gamma ={45}^{°}$. That is, the heat transport rate declined near the bottom edge of the cavity when the values of $T_m$ increased. However, the maximum value of local heat transfer was attained at the middle of the hot wall when $\gamma ={90}^{°}$.

Figure 6 displays the local heat transfer rate for various tilting angles and the $T_m$ for both cases of varying properties (viscosity and thermal conductivity cases). The local thermal transfer rate declined when the value of the tilting angle of the chamber was raised. The local heat transfer rate declined with the height of the wall at $T_m=0$. That is, a higher local heat transfer was found at the lower portion of the hot wall at $T_m=0 \mathrm{and} 0.5$. However, the opposite trend was shown for the local heat transfer rate at $T_m=1.0$. Furthermore, when inspecting these figures, a greater heat transfer was found to occur at $T_m=1.0$ compared to $T_m=0 \mathrm{and} 0.5$.

Figure 7a,b demonstrates the influence of averaged heat transfer rates for both cases with various values of inclination and different density inversion parameters. The averaged Nusselt number behaves non-linearly with the density reversal parameter. The minimum average heat transfer rate was obtained at $T_m=0.5$ when $\gamma =0^{°}, {15}^{°}$ with changing $T_m$ values. For tilted cavities with $\gamma ={45}^{°}$, the average Nusselt number decreased when the value of $T_m$ was raised. Two different behaviours were observed between vertical and tilted cavities at the average Nusselt number. In the vertical cavity, the average Nusselt number declined at $T_m=0.5$ and then increased at $T_m=1.0$. In comparing the two cases, the variable thermal conductivity case had a smaller high energy transfer rate than both the variable thermal conductivity and viscosity cases. While the vertical cavity gave a higher (lower) heat transfer rate at either $T_m=0$ or $T_m=1.0$ ($T_m=0.5$), the tilted cavity ($\gamma >{60}^{°}$) gave a higher (lower) heat transfer rate at $T_m=0$ ($T_m=1.0$). The cavity with a lower tilting angle ($\gamma <{30}^{°}$) behaved like a vertical cavity ($\gamma =0^{°}$).

Figure 8a–d exposes the cup-mixing temperature and the RMSDTcup for various inclination angles and both variable thermal conductivity and viscosity cases. The cup-mixing temperature was high at $\gamma ={90}^{°}$ and $T_m=0.8$ for the variable thermal conductivity case, whereas it was higher at $\gamma ={90}^{°}$ and $T_m=0.6$ for both variable viscosity and thermal conductivity cases. When$\gamma =0^{°}$ and $T_m=0.5$, the cup-mixing temperature and its RMSDTcup was lower. Lower values of RMSD indicate good thermal (mixing) uniformity inside the enclosed region. RMSD never exceeded unity here. However, the RMSD values that are close to unity will not cope well with thermal mixing. In comparing both cases, only the variable thermal conductivity case provided better thermal mixing. When $T_m<0.5$, the inclined cavity gave a higher temperature uniformity inside the enclosed region. The RMSD values of the vertical cavity $(\gamma =0^{°}$) were similar for all the values of $T_m$. The RMSD values increased when the values of $T_m$ were raised in both cases.

Figure 9 depicts the entropy generation for various inclination angle values and density extremum parameters for the variable thermal conductivity and variable viscosity cases. The entropy generation was limited to a narrow boundary along the thermal walls for $T_m=0$ and all the values of $\gamma$. That is, strong entropy gradients were found along the isothermal walls. The generation of entropy happened near the boundary at $T_m=0$ and $\gamma =0^{°}$. When the inclination was increased, entropy generation was distributed inside the cavity. At $T_m=1$, entropy generation existed throughout the cavity for all the inclination angles. These are interesting results to note. Entropy production was very strong near the vertical walls at $T_m=0$ and 0.5 for all the values of $\gamma .$

Figure 10 demonstrates the Bejan number versus $T_m$ for various inclination angle values with variable $k\left(T\right)$ and both variable $k\left(T\right)$ and $\mu \left(T\right)$ cases. The Bejan number values were always above 0.65 $(Be>0.5)$ for all the considered cases. This clearly indicates that entropy generation is dominant by heat transfer. The Bejan number increased whe the values of $T_m$ increased. The Bejan number was high (low) at $\gamma =0^{°}\left({90}^{°}\right)$ when $T_m<0.5$. However, the Bejan number was higher (a lower value) at $\gamma ={90}^{°}\left({90}^{°}\right)$ when $T_m>0.5$. In comparing both of these cases, the Bejan number was high for both the variable viscosity and variable thermal conductivity cases.

8. Conclusions

The influence of the tilting angle of the cavity, the variable viscosity and the variable thermal conductivity of cold water near its density supremum on free convection in an inclined closed chamber was investigated numerically. Cup-mixing temperature and entropy generation were also explored.

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The temperature-dependent equations for thermal conductivity and viscosity were derived with regards to the reference temperature parameter.

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The development of the dual cell arrangement and the strength of the cells depends on the density supremum parameter and inclination angle. The dual cell arrangement forbids convective thermal transfer across the chamber.

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The heat transfer rate for the variable thermal conductivity case was higher than the heat transfer rate for both of the varying cases.

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The rate of thermal transport performs nonlinearly with the density extremum parameter and the inclination angle. The mean heat transfer rate declines first and then increases when the density extremum parameter is raised when $\gamma \le {45}^{°}$. However, it always declines when the density extremum parameter increases when $\gamma >{60}^{°}$.

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A small impact on the rate of thermal transport was found when the temperature difference parameter varied.

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The cup-mixing temperature behaves non-linearly with the density extremum parameter. RMSDTcup values are enhanced when the density extremum parameter values increase in both cases.

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The Bejan number increases when the density extremum parameter values increase. Entropy generation results can be used for the effective design of convective flow systems.

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The density extremum leaves a strong effect on the stream and the thermal distribution due to development of the bicellular arrangement. The dual cell arrangement directly prohibits the exchange of thermal energy between the hot region and the cold region. This results in a reduction in the average Nusselt number.