A localized meshless approach using radial basis functions for conjugate heat transfer problems in a heat exchanger

International Journal of Refrigeration 110 (2020) 38–46

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A localized meshless approach using radial basis functions for conjugate heat transfer problems in a heat exchanger Agung Tri Wijayanta a,b,∗, Pranowo b,c,∗ a

Department of Mechanical Engineering, Faculty of Engineering, Universitas Sebelas Maret, Jl. Ir. Sutami 36A Kentingan, Surakarta 57126, Indonesia Research Group of Sustainable Thermofluids, Universitas Sebelas Maret, Jl. Ir. Sutami 36A Kentingan, Surakarta 57126, Indonesia c Department of Informatics, Universitas Atma Jaya Yogyakarta, Jl. Babarsari 44 Yogyakarta 55281, Indonesia b

a r t i c l e

i n f o

Article history: Received 29 June 2019 Revised 21 September 2019 Accepted 29 October 2019 Available online 5 November 2019 Keywords: Conjugate heat transfer Natural convection Heat exchanger Meshless method Radial basis function

a b s t r a c t An understanding of conjugate heat transfer is important for enhancing the thermal performance of heat transfer devices. This study discusses the numerical solutions of the conjugate natural convective heat transfer problem using a local radial basis function meshless method. The temperature difference is assumed to be small enough to ensure the validity of the Boussinesq approximation. An artificial compressibility method is used to couple pressure to the continuity equation. The meshless method requires only one set of nodes, which are distributed in the spatial domain. The semi-algebraic equations are integrated using the second-order Runge–Kutta method. Three benchmark problems of conjugate natural convection with one vertical wall, conjugate natural convection with centrally vertical partition, and natural convection with a conducting solid in the middle of the square cavity are analyzed. The numerical results are compared with three benchmark solutions and experimental results; it was observed that the experimental and predicted results showed good agreement with each other. This numerical study is to contribute to understanding the knowledge of heat and fluid flow mechanisms for conjugate heat transfer in a heat exchanger acquired through experiments. © 2019 Elsevier Ltd and IIR. All rights reserved.

Approche localisée sans maillage en utilisant des fonctions radiales de base pour les problèmes de transfert de chaleur conjugués dans un échangeur de chaleur Mots clés: Transfert de chaleur conjugué; Convection naturelle; Échangeur de chaleur; Méthode sans maillage; Fonction radiale de base

1. Introduction Advanced heat exchangers, equipped with an enhanced surface area over which heat transfer can occur while maintaining a compact size, have become an integral part of refrigeration systems. Heat transfer and flow phenomena in the refrigeration system are complicated. The influence of conjugate heat transfer is often not adequately addressed. Conjugate heat transfer process occurs due

∗ Corresponding authors at: Research Group of Sustainable Thermofluids, Universitas Sebelas Maret, Jl. Ir. Sutami 36A Kentingan, Surakarta 57126, Indonesia. E-mail addresses: [email protected], [email protected] (A.T. Wijayanta), [email protected] (Pranowo).

https://doi.org/10.1016/j.ijrefrig.2019.10.025 0140-7007/© 2019 Elsevier Ltd and IIR. All rights reserved.

to the interaction between the buoyancy forced flow of fluid and the conduction along a solid wall. In the practical systems, such as evaporator of a refrigerator, the convection around the surrounding fluid significantly affects the conduction in a tube wall. Consequently, the convection in the fluid and the conduction in the solid body should be simultaneously accounted. Conjugate heat transfer must be determined to provide local heat transfer coefficient correlations for the comprehensive design of future thermal management in the modern refrigeration system. Studies on natural convection, especially those based on numerical investigations, have gained much research attention of late. Conjugate heat transfer with natural convection is an important consideration in real problems because the effect of wall conduction on convection heat transfer cannot be ignored. Validation of numerical computations by available experimental data

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Nomenclature a kr p u, v t x, y H W Pr Ra Nuo Gr r n N B Hx Dx , Dy Dxx , Dyy

ratio of thermal diffusivity ratio of thermal conductivity dimensionless pressure dimensionless velocity components pseudo-time Cartesian co-ordinates system Height of enclosure Width of enclosure Prandtl number Rayleigh number average Nusselt number Grashoff number Euclidian distance normal direction, number of stencil nodes in a local domain total number of nodes, number of partitions interpolation matrix matrix containing first x-derivative of RBF first-order differentiation matrix second-order differentiation matrix

Greek symbols α vector of expansion coefficients β artificial compressibility parameter θ dimensionless temperature θh hot wall dimensionless temperature θc cold wall dimensionless temperature φ Radial basis function ɛ shape parameter x position of the node vector Subscripts c cold f fluid h hot s solid

have provided reasonable results; rendering the validated numerical methods powerful tools (Anderson et al., 1984). Zhang and Lian (2014) studied numerically conjugate heat transfer analysis using a simplified household refrigerator model at two Rayleigh numbers Ra = 106 and 108 . Ga et al. (2017) investigated natural convection inside the refrigerator compartment for analyzing heat leakage at the gasket region. Using a commercial computational software, You et al. (2017) studied the physical mechanism of unsteady conjugate heat transfer in magnetic refrigerator by analyzing the temperature contours. Until now, numerical computations of conjugate natural convection have been dominated by mesh-based methods, such as the finite difference, finite volume, finite element, and spectral methods. The semi-implicit method for pressure-linked equations (SIMPLE) Algorithm described by Patankar (1980) was employed for this purpose. They examined three models to account for wall conduction, i.e., fully two-dimensional conjugate analysis, onedimensional wall conduction, and the lumped parameter approach. All three methods predicted nearly the same values for the Nusselt numbers. Misra and Sarkar (1997) studied the effect of vertical wall conduction on natural convective heat transfer using the finite element method (FEM). The solid conducting wall was treated as a fluid with a high viscosity. It was found that the Nusselt number decreased as the thickness of the conducting wall increased. When the wall conductivity was increased gradually, it was observed that

39

the Nusselt number increased and approached the standard square cavity results. House et al. (1990) examined the effect of a centered conducting body on natural convection using the finite difference method and the SIMPLE algorithm (Patankar, 1980). The entire domain was assumed to be fluid and velocity components in the solid domain were forced to be zero by setting the Prandtl number to infinity. Their crucial conclusion is that the heat transfer rate is enhanced in a body with a low thermal conductivity and vice versa for small body sizes. Das and Reddy (2006) extended the work of House et al. (1990) to study the effect of the inclination angle. Dong and Li (2004) conducted a numerical study on conjugate natural convection in a complicated enclosure using a vorticitystream function formulation with an unstructured quadrilateral mesh. They investigated the influence of the conductivity ratio, geometrical shape, and Rayleigh number. Oztop et al. (2011) investigated conjugate natural convection in air-filled tubes inserted in a square cavity. The numerical method was employed using the finite volume method with an unstructured triangular mesh. The tube has a finite thickness and the same fluid is filled inside and outside the tube. Higher-order methods, such as the spectral multi-domain method, were employed by many researchers to solve the conjugate natural heat transfer process. Ha et al. (2002) obtained twodimensional solutions of unsteady natural convection in a horizontal enclosure with a square body using the Chebyshev spectral multi-domain method. Lee and Ha (2005) studied natural convection in a horizontal enclosure heated below, and cold above with a solid body placed at the center of the enclosure. They examined the effect of boundary conditions of the solid body, i.e., conducting, isothermal, and adiabatic bodies. Zhang et al. (2011) investigated the effect of inclination of conjugate conduction-natural convection in an enclosure, which is bounded by four finite thicknesses and conducting walls with a time-periodic sidewall temperature and inclination. One sidewall was exposed to a timeperiodic sinusoidal temperature, while the temperature at the opposite wall was kept constant. The rest of the walls were kept adiabatic. The meshless method is an alternative in this context; its calculations do not require a mesh, i.e., it only needs the distribution points in the domain space. Recently, our previous works applied a meshless method called the direct meshless local PetrovGalerkin (DMLPG) method for solving natural convection in a porous medium (Pranowo and Wijayanta, 2018a) and in a square cavity (Pranowo and Wijayanta, 2018b, 2019). Divo and Kassab (20 05, 20 06, 20 07, 20 08) developed a new localized radial basis function (RBF) meshless method for natural convection and conjugate forced convective heat transfer problems. They used a domain decomposition technique to reduce the computational load. Using this approach, the nearly singular matrix can be avoided and the conditioning numbers of semi-algebraic equations can be reduced significantly. A review of the available literature shows that at present, studies on the mathematical modeling of conjugate natural convection using meshless methods are limited and poorly described. In this study, we propose a local RBF meshless method based on Sarra’s approach (Sarra, 2012), which is simple and easy to implement for solving conjugated natural convective heat transfer problems. The novelty of this work is its numerical method, which is unique to solve the problems of conjugate heat transfer in a heat exchanger. The numerical method was also used to analyze three benchmark problems and the obtained numerical results were compared with the experimental data. Therefore, this study is to contribute to understanding the numerical approach to estimate the performance based on the knowledge of heat and fluid flow mechanisms for conjugate heat transfer acquired through experiments.

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2. Governing equations The governing partial differential equations for the fluid are the coupled mass, momentum, and energy conservation equations, applicable for two dimensions. The dimensionless equations are as follows (Quere, 1991):

∂ u ∂v + =0 ∂x ∂y  2  ∂u ∂u ∂u ∂p Pr ∂ u ∂ 2u +u +v =− + + ∂t ∂x ∂y ∂ x Ra0.5 ∂ x2 ∂ y2  2  ∂v ∂v ∂v ∂p Pr ∂ v ∂ 2v +u +v =− + + + Pr θ f ∂t ∂x ∂y ∂ y Ra0.5 ∂ x2 ∂ y2  2  ∂θf ∂θf ∂θf ∂ θ f ∂ 2θ f 1 +u +v = + ∂t ∂x ∂y Ra0.5 ∂ x2 ∂ y2

(1) (2) (3) (4)

The energy equation in the solid is:

 2  ∂ θs a ∂ θs ∂ 2 θs = + ∂t Ra0.5 ∂ x2 ∂ y2

where Pr is the Prandtl number, Ra is the Rayleigh number, kr is the ratio of thermal conductivity, (u, v) are velocity components, p is the pressure, θ f is the fluid temperature, and θ s is the solid temperature. The correlation between thermal conductivity ratio and temperature at the interfaces can be written as

kr

Fig. 1. Stencils with a center node.

(5)

  ∂ θ f  ∂ θs  = ∂ n s ∂n f

(6)

In this investigation, we employ the local RBF method, which was developed by Sarra (2012), for solving conjugate natural convection. The spatial domain is discretized with a set of N nonoverlapping nodes x = {x1 , ..., xN } in a two-dimensional space. At each of the nodes, the spatial partial derivatives are approximated using multiquadric (MQ) RBF as the basis function (Sarra and Kansa, 2009).

where n is the normal to the interface of the solid and fluid.

ϕ (r, ε ) =

3. Numerical technique

where ɛ is the shape parameter and r is the Euclidian distance. The local RBF interpolant for any smooth function f(x) can be constructed as

An artificial compressibility (Chorin, 1967) method has been proposed to solve the governing equations of the fluid. This method adds a pseudo-time derivative to the continuity equation to couple the pressure with velocity.

  ∂p ∂ u ∂v +β + =0 ∂t ∂x ∂y

(7)

In this approach, t does not represent the actual time but a pseudo-time and β denotes an artificial compressibility parameter. By using artificial compressibility, the governing equations are rearranged and these equations become

∂q = L (q ) ∂t 

q= p u

⎡

(8a)

v θ f θs 

−β

T

∂ u ∂v + ∂x ∂y

(8b)



⎤

Ra0.5

∂

+

∂

n  j=1

r = x2 =

   α j ϕ xi − x j 2 , ε

(9)

(10)

x21 + x21

where α is a vector of expansion coefficients. The surrounding nodes of each node xj are called stencils (Fig. 1) and n is the number of nodes in a stencil. If Eq. (11) is written in a vectormatrix notation, we have N × N linear systems to be solved for the expansion coefficients. The matrix B is called the interpolation matrix.

Bα = f

(11)

The partial derivative of the function can be obtained after solving Eq. (11). Supposing that Dx is the first derivative operator at a stencil corresponding to the center node xj , one obtains

Dx = B−1 Hx

⎢ ⎥ ⎢ ⎥  2  ⎢ ∂u ⎥ 2 ⎢−u − v ∂ u − ∂ p + Pr ∂ u + ∂ u ⎥ ⎢ ∂x ⎥ 0 . 5 2 2 ∂ y ∂ x Ra ∂x ∂y ⎢ ⎥ ⎢ ⎥  2  2 ⎢ ∂v ⎥ ∂v ∂ p Pr ∂ v ∂ v ⎥ L (q ) = ⎢ −u − v − + + + Pr θ f⎥ 0 . 5 2 2 ⎢ ∂x ∂ y ∂ y Ra ∂x ∂y ⎢ ⎥  2  ⎢ ⎥ ∂θf ∂ θ f ∂ 2θ f ⎢ ∂θf ⎥ 1 ⎢−u ⎥ −v + + ∂ y Ra0.5 ∂ x2 ⎢ ∂x ⎥ ∂ y2 ⎢ ⎥  ⎣ a  ∂ 2θ ⎦ 2 ∂ θ s x2

f ( xi ) =

1 + ε2 r2

(12)

Dx is also called the first-order differentiation matrix. Matrix Hx is a N × N linear system with entries:

hi j =

 ∂  ϕ x i − x j 2 ∂x

(13)

The rest of the partial derivative operators (Dy , Dxx , Dyy ) are obtained in the same manner. The discretizing equation of Eq. (2) in space using the local RBF is

∂u Pr = −uDx u − vDy u − Dx p + (Dxx u + Dyy u ) ∂t Ra0.5

s y2

(8c)

(14)

The semi-algebraic equations are advanced explicitly in pseudotime using the second-order Runge–Kutta method. The numerical

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41

Fig. 2. (a) The problem geometry and (b) node distribution of conjugate natural convection with left vertical solid conduction.

solution has no physical meaning before a steady-state solution is reached. Implementation of the algorithm for the problem of incompressible fluid flow is as follows

q

n+1

n

=q +

t 2

( K1 + K2 )

(15)

K1 = L ( qn ) K2 = L(q + t K1 ) n

(16)

Table 1 Comparison of the average Nusselt number Nu0 between our method and previously reported methods for Ra = 103 . kr

Reference

Numerical method

Nu0

1

Present study Misra and Sarkar (1997)

Local RBF FEM

0.88 0.90

5

Present study Misra and Sarkar (1997)

Local RBF FEM

1.05 1.06

25

Present study Misra and Sarkar (1997)

Local RBF FEM

1.09 1.10

200

Present study Misra and Sarkar (1997)

Local RBF FEM

1.10 1.12

4. Results and discussion The local RBF method in the framework of artificial compressibility is used to analyze the following three benchmark problems.

Table 2 Comparison of the average Nusselt number Nu0 between our method and previously reported methods for Ra = 104 . kr

Reference

Numerical method

Nu0

1

Present study Misra and Sarkar (1997)

Local RBF FEM

1.42 1.45

4.1. Conjugate natural convection with left vertical solid conduction

5

Present study Misra and Sarkar (1997)

Local RBF FEM

1.96 2.02

The first case chosen for testing the accuracy of the present method is the conjugate natural convection with left vertical solid conduction. Fig. 2(a) shows the geometry of the domain and the boundary conditions. No-slip velocity boundary conditions are imposed on the walls. The right side wall is heated, the left side wall is cooled, and the top and bottom walls are insulated. The cavityfilled fluid consists of 41 × 41 nodes and 41 × 9 nodes lie in the solid wall (see Fig. 2(b)). The temperature along the interface of the solid and fluid is obtained by imposing the boundary condition described below at the interface (see Eq. (6)):

25

Present study Misra and Sarkar (1997)

Local RBF FEM

2.13 2.19

200

Present study Misra and Sarkar (1997)

Local RBF FEM

2.17 2.23

  ∂ θ f  ∂ θs  kr = ∂ x s ∂x f

(17)

Most of the available conjugate heat transfer studies solved the governing equations for the fluid across the entire domain, which consists of a solid wall and cavity-filled fluid (House et al., 1990; Misra and Sarkar, 1997; Das and Reddy, 2006). The solid is assumed to be a fluid with a very high viscosity and the velocity components are neglected. In the present method, numerical solutions are obtained separately for both domains.

The numerical results are presented in terms of temperature fields and average Nusselt numbers of the vertical wall (Nuo ).

 N uo = −

1 0

∂θ dy ∂x

(18)

The Nusselt numbers obtained in the present study are compared with the Nusselt numbers obtained by Misra and Sarkar (1997) in Tables 1, 2, and 3 for Ra = 103 , 104 , and 105 , respectively. The comparison indicates good agreement. Fig. 3 shows the temperature contours (isotherms) for Ra = 105 at kr = 5.0. The temperature gradient in the solid part and around the interface is flatter, indicating that the flux flow is low. If the value of kr increases, the temperature gradient will also increase in the area around the interface (for more comparison, see Supplementary material S-1). The phenomenon of fluid flow and

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Table 3 Comparison of the average Nusselt number Nu0 between our method and previously reported methods for Ra = 105 . kr

Reference

Numerical method

Nu0

1

Present study Misra and Sarkar (1997)

Local RBF FEM

2.26 2.20

5

Present study Misra and Sarkar (1997)

Local RBF FEM

3.80 3.72

25

Present study Misra and Sarkar (1997)

Local RBF FEM

4.37 4.33

200

Present study Misra and Sarkar (1997)

Local RBF FEM

4.56 4.49

Fig. 4. The problem geometry and boundary conditions of conjugate natural convection in a partitioned enclosure. Table 4 Comparison of the average Nusselt numbers Nuo between our method and previously reported methods for Pr = 0.71.

Fig. 3. Isotherm of conjugate natural convection with left vertical solid conduction at Ra = 105 for kr = 5.0.

heat transfer is closer to pure natural convection (Davis, 1983; Najafi and Enjilela, 2013). The increase of flux due to an increase in kr is also seen at the condition for the increase of Nusselt number (see Tables 1–3). 4.2. Conjugate natural convection in a partitioned enclosure The interaction of natural convection heat transfer with a centrally positioned conducting partition is used as the second case. The vertical partition reduces the heat transfer rate. The geometry of the domain and boundary conditions for the second case are shown in Fig. 4. The solid walls and partition are subjected to no-slip velocity boundary conditions. The left wall is heated, the right wall is cooled, and the top and bottom walls are insulated. The boundary condition at the interface of the fluid region and the solid partition region is described using Eq. (17). The numerical results of the present method are compared with other numerical methods and experimental correlations. Khatamifar et al. (2017, 2018) studied this problem numerically using the finite volume method (FVM) on a staggered grid and solved semi-discretized equations using the SIMPLE algorithm (Patankar, 1980). A grid of 200 × 200 points is selected to perform the numerical calculation with high accuracy. Kahveci (2007) investigated this problem numerically using the polynomial-based differential quadrature (PDQ) method over the Rayleigh number range of 104 –106 . The PDQ method is a high-order method and hence it needs fewer grid points than low-order conventional methods, such as FDM, FVM, and FEM, to ensure reliable accuracy. Based on a grid independence study, Kahveci (2007) showed that

Ra

Reference

Numerical method

Nu0 kr = 1

kr = 100

104

Present study Kahveci (2007) Khatamifar et al. (2017)

Local RBF PDQ FVM

1.07 1.06 1.09

1.17 1.19 1.17

105

Present study Kahveci (2007) Khatamifar et al. (2017)

Local RBF PDQ FVM

1.77 1.79 1.76

2.08 2.14 2.11

106

Present study Kahveci (2007) Khatamifar et al. (2017)

Local RBF PDQ FVM

2.94 2.80 2.82

3.89 3.93 4.03

a non-uniform grid of 41 × 41 points can be used to achieve excellent accuracy. The values of the average Nusselt numbers in the present study are compared with those obtained in Kahveci (2007) and Khatamifar et al. (2017) for a partition thickness of 0.1 and Pr = 0.71. The calculations are carried out using 101 × 101 nodes, which are distributed uniformly. Table 4 shows the results of comparison for Ra up to 106 ; the numerical results of the present method are in excellent agreement with those of the PDQ and FV methods. Figs. 5 and 6 depict the temperature contours (isotherms) for Ra = 105 at kr = 1 and kr = 100, respectively. In the left enclosure, buoyancy forces cause the fluid to rise near the hot wall until it reaches the top wall and then moves toward the partition. The fluid undergoes cooling and thus, it flows down and turns toward the hot left wall. On the other hand, in the right enclosure, the cold right wall causes the fluid to flow down and turn toward a hotter partition; the fluid warms up so that it flows up until it reaches the upper wall and turns right toward the cold right wall (see S-2 and S-3 completely). The effect of increasing the Ra value is presented here for air medium (Pr = 0.71). For Ra = 104 , the influence of the viscous force is dominant compared to the buoyancy force and the fluid flow is weak. The isotherm lines are almost vertical lines and heat transfer in the fluid is dominated by the effect of thermal diffusion. If the Ra increases, the buoyancy force becomes more dominant and the circulation flow strengthens; the isotherm lines bend like the letter "s" according to the fluid circulation flow. The Nu number also increased (see Table 4).

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Fig. 5. Isotherm of conjugate natural convection in a partitioned enclosure at kr = 1.0 for Ra = 105 .

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Fig. 7. Isotherm of conjugate natural convection in a partitioned enclosure at for Ra = 258,0 0 0 and Pr = 3300.

this indicates that a high kr value causes the partition that was originally conductive to changes in the isotherm. The present numerical model is also verified by the experimental results of Anderson and Bejan (1981) and CuckovicDzodzo et al. (1999). Anderson and Bejan (1981) investigated heat transfer through a number of vertical partitions using water as the working fluid in the Rayleigh number range of 109 to 1010 . They derived the empirical correlation for water as follows.

N uo = 0.167Ra0.25 (H/W )−0.25 (N + 1 )−0.61

Fig. 6. Isotherm of a conjugate natural convection in a partitioned enclosure at kr = 100 for Ra = 105 .

The effect of thermal conductivity ratio (kr ) is explained in this section (see S-2). A low kr value indicates high thermal resistance and the partition is difficult to pass on heat. If kr = 1, the thermal conductivity fluid and solid partition are the same; the capability of the partition to conduct heat is equal to that of the fluid. Therefore, the consequence is that the temperature gradient in the enclosure is close to the temperature gradient in the partition. A high kr value causes the gradient in the partition to be close to zero and the temperature distribution in the partition is almost constant. This physical mechanism is appropriately described by the following equation (see also Eq. (17)).

  ∂ θs  1 ∂ θ f  = ∂ x s kr ∂ x  f

(19)

For this physical mechanism, it indicates an absence of isotherm lines in the partition (see S-3). The temperature in the partition is almost uniform, which was also observed by Khatamifar (2018);

(20)

where H/W is the aspect ratio (in this case, H/W = 1) and N is the number of partitions (= 1). Cuckovic-Dzodzo et al. (1999) conducted an experimental investigation on natural convection heat transfer in an enclosure with and without a vertical partition. Glycerol was used as the working fluid and the vertical partition was made of plexiglass. The enclosure was 38 × 38 × 38 mm3 in size. The thicknesses of the walls and partition were 8 mm and 2 mm, respectively. It is assumed that the properties of glycerol, i.e. density, thermal expansion coefficient, thermal conductivity, and dynamic viscosity, depend on the temperature; this dependence is described in more detail in Cuckovic-Dzodzo et al. (1999), and based on their experimental results, they developed the following empirical correlation.

N uo = 0.201Ra0.276 (N + 1 )−1.4

(21)

This correlation is valid in the range of 38,0 0 0 < Ra < 369,0 0 0 and 2700 < Pr < 70 0 0. Fig. 7 shows the isotherm distributions for Pr = 3300 at Ra = 258,0 0 0. Numerical calculations are performed using 3 types of Ra and Pr property values as described by Cuckovic-Dzodzo et al. (1999) – (a) Ra = 96,0 0 0 and Pr = 5100, (b) Ra = 182,0 0 0 and Pr = 3900, and (c) Ra = 258,0 0 0 and Pr = 3300 (completely see S-4). A comparison between the shapes of the fluid flow pattern of the calculation of the local RBF method and the experimental results are shown in Fig. 8. Fig. 8(a) shows a photograph of the experimental result reported in Cuckovic-Dzodzo et al. (1999) (figure courtesy of the Publisher of Ref. in Cuckovic-Dzodzo et al., 1999). Fig. 8(a) provides the numerical result of the present study using local RBF. The comparison indicates a good match between the experimental and numerical results. Fig. 9 compares the average Nusselt numbers calculated using the local RBF method and

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Fig. 8. Flow pattern of conjugate natural convection in a partitioned enclosure at Ra = 105 and Pr = 5100. (a) Experimental result (courtesy of the Publisher of Ref. in Cuckovic-Dzodzo et al., 1999) and (b) local RBF result (present study).

Table 5 Comparison of the average Nusselt numbers for Ra = 105 . kr

0.2

1

0.5

Present study House et al. (1990)

4.5607 4.6257

4.4641 4.5061

4.2528 4.3221

- at y = 0 and y = 1, 0 ≤ x ≤ 1, ∂θ /∂ y = 0 - at the interfaces

  ∂ θs  ∂ θs  kr = ∂ n s ∂n f

Fig. 9. Comparison of the average Nusselt numbers for conjugate natural convection in a partitioned enclosure.

the experimental correlation Eqs. (20) and (21). The present result using the local RBF method still offers a good agreement when compared to the empirical correlation provided by CuckovicDzodzo et al. (1999). When compared with the empirical correlation proposed by Anderson and Bejan (1981), the result shows a similar tendency although there exists a significant difference. This is because the empirical correlation described in Eq. (20) is obtained using water as the working fluid, which has a large Pr value compared to glycerol. However, as shown in Fig. 9, all the results indicate that an increase in Ra increases the Nuo . 4.3. Conjugate natural convection in a square cavity containing a conducting block In the third case, conjugate natural convection in a square cavity containing a conducting block is considered to extend our previous work (Pranowo and Wijayanta, 2018b). The effect of the vertical position of the conducting block on the heat transfer characteristics is investigated. The geometry problem is illustrated in Fig. 10 and the boundary conditions are: - at x = 0, 0 ≤ y ≤ 1, θ = 0.5 - at x = 1, 0 ≤ y ≤ 1, θ = −0.5

where n is normal to the interface of the solid and fluid. No-slip velocity boundary conditions are imposed on the cavity walls and on the outer sides of the conducting block. The vertical position of the conducting block can be changed by varying the value of Yh from 0.15 to 0.3 with increments of 0.05. The numerical calculations are performed using 61 × 61 nodes, of which 31 × 31 nodes lay on the conducting solid and the rest of the nodes are in the fluid domain. The various parameters considered here are Pr = 0.71 (air), Ra = 105 , and kr = 0.2, 1.0, and 5.0. Fig. 11 depicts the isotherms for Ra = 105 , kr = 5, and at vertical positions of the conducting block for Yh = 0.20 (see S-5 for various Yh). The average Nusselt numbers at Ra = 105 for various kr values are listed in Table 5. The isotherms of the temperature field of conjugate natural convection for various kr values at Ra = 105 is provided in the S-6. The numerical results of the problem with a centered block (Yh = 0.25) here are validated with the results of House et al. (1990). One can observe that the heat transfer is enhanced by a smaller ratio of thermal conductivity. The results obtained in the present study agree with House et al. (1990). The rate of heat transfer from the vertical wall to the fluid increases as the kr decreases compared to the Yh (see again S-5 and S-6 for comparison). Fig. 12 shows the average Nusselt numbers at different vertical positions of block at various kr values. It can be observed from Fig. 12 that the heat transfer rate from the vertical wall to the fluid increases if the kr decreases. This occurs because at a low kr , the block becomes adiabatic (see again S-6 (a) and (b)). The heat flux entering the block decreases and hence the flux in the fluid increases. The maximum heat transfer is achieved when the conducting block is in the center of the cavity.

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Fig. 10. (a) Problem geometry and boundary conditions and (b) distribution of nodes in a square cavity containing a conducting block for Yh = 0.25.

5. Conclusion Improving numerical methods is the best approach to estimate the performance based on the knowledge of heat and fluid flow mechanisms acquired through experiments. In the case of conjugate natural convection with a conducting solid in the center of the square cavity, the position of the vertical conducting block was varied, and the numerical results show that the maximum heat transfer rate could be achieved when the conducting block is positioned precisely at the center of the cavity. Acknowledgments

Fig. 11. Isotherm of conjugate natural convection in a square cavity containing a conducting block for kr = 5.0 at Yh = 0.20.

This research work was funded by the Institute of Research and Community Services, Universitas Sebelas Maret, Indonesia (No.: 516/UN27.21/PP/2019). Support provided by the Indonesian Ministry of Research, Technology and Higher Education under World Class University (WCU) Program managed by Institut Teknologi Bandung was acknowledged. Agung Tri Wijayanta (first author) wishes to thank the Engineering Faculty, Universitas Sebelas Maret for the partial financial assistance to visit Essex University, Colchester, United Kingdom during the revision of this manuscript. The first author also acknowledges the support from the Education and Culture Attaché, the Embassy of the Republic of Indonesia in London. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.ijrefrig.2019.10.025. References

Fig. 12. The average Nusselt number vs. vertical positions of the block at various kr values.

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