1 Introduction

Open-cell metallic foams have been widely used in a various range of technical applications, such as combustion, purification processes, solar collector, electronics cooler, fuel cell, and even biomedical engineering due to their superior thermal capabilities [1,2,3]. Metallic foams have many advantages over solid materials, considering the fluid flow characteristics and heat transfer. The material of metallic foams can be copper, steel, titanium, nickel and aluminum, while many attractive compounds of thermal, physical, and mechanical properties, namely high thermal conductivity, high permeability, ultra-light, high porosity, uniform pore size, and high stiffness with a low specific weight, can be attributed to the metallic foams [4]. Although the heat transfer performance can be improved via the metallic foams compared with the no-foam case, microstructures of the metal foam will result in a massive pressure drop. Therefore, the partially filled metallic foam is proposed to minimize the undesirable pressure drop, while the heat transfer rate of this configuration will be higher than that of the no-foam case [5].

Numerous studies have been done to analyze open-cell metal foams in terms of pressure drop, equivalent fluid-dynamics properties, and forced convection configuration to optimize a compact heat exchange unit. The advantages of applying open-cell metal foams for heat transfer augmentation have been examined experimentally [6,7,8,9], analytically [10,11,12], and numerically [13,14,15].

Qu et al. [16] introduced an analytical solution to study thermal and fluid flow characteristics in an annulus partly coated with metallic foams. The flow passed through the metallic foam region was modeled by the Brinkman–Darcy equation, while the heat transfer equations were developed using the thermal non-equilibrium model. They investigated the influence of various parameters on Nusselt number (Nu), friction factor, and j/f1/3. They concluded that porosity did not have any effects on the coefficient of flow heterogeneity. Although a comprehensive study was accomplished to study the impacts of determinant parameters on the heat transfer characteristics as well as fluid flow characteristics, heat transfer performance of an annulus partly coated with metallic foam on two wall surfaces subject to asymmetrical heat fluxes was not investigated. The influences of endothermicity and exothermicity upon the temperature distribution of a partly filled porous channel were investigated using the LTNE conditions and the Darcy–Brinkman model of momentum transport [17]. The fluid and solid phases were assumed to feature exothermicity (generation) and endothermicity (internal heat consumption). They deduced that the heat sources in both solid and fluid phases had a determinant role on the thermal characteristics of a channel partly coated with metallic foam. An analytical study was developed to determine the thermal features of the forced convection flow in a channel partly coated with porous media [18]. They applied the non-equilibrium heat transfer model and the Brinkman-extended Darcy momentum methodology for the region occupied by the foam. They reported that the effects of porosity and pore density on the flow and heat transfer features could be influenced by the height of the foam.

Xu [19] applied the LTE and LTNE models to study heat transfer characteristics and fluid flow features in a fully filled micro-annulus. It was reported that the velocity became uniform as the Kn number and a modified Darcy number inverse were increased. Although increasing a modified Darcy number inverse resulted in improving the heat transfer performance, homogenizing the flow field could limit the heat transfer enhancement. In this research, the annulus was fully filled with porous medium, while the heat fluxes on the interior and exterior boundaries of the annulus were supposed to be different. In other words, the impacts of heat flux ratio on the thermal characteristics of a partly filled annulus were not considered in the study. Adding nanofluid can improve the thermal performance of a channel. Therefore, the efficacy of using nanofluid and porous media in enhancing the thermal characteristics of a channel subjected to a specified heat flux was evaluated by Nohooji et al. [20]. Various arrangements were considered for the porous material to evaluate the thermal characteristics of the channel in their research. The numerical results revealed that the effects of porous media in enhancing the heat transfer rate were greater than those of adding nanofluids. The influence of applying aluminum oxide nanoparticles in an aqueous solution of carboxymethyl cellulose in a microtube was analyzed in a research conducted by Goodarzi et al. [21]. The numerical results showed that heat transfer rate was improved via raising the coefficient of slip as well as adding nanoparticles to the base fluid.

Chen et al. [22] accomplished a numerical analysis to evaluate the characteristics of high-temperature heat transfer in a foam-filled heat exchange unit. The flow behavior and thermal transport inside the foam regions were simulated using the LTNE model as well as the equation of Forchheimer-extended Darcy. The influences of various parameters, namely size of a heat exchanger, foam structural specifications, and thermal radiation on the overall efficiency of the heat exchange unit, were investigated. The numerical results revealed that increasing the pore density or decreasing the porosity would improve the effectiveness of the heat exchanger. Thermal transport in microchannels partly included with a metallic foam was studied by Xu [23]. The author supposed the thermal asymmetry boundary conditions for the two walls sandwiching the channel. The effects of various parameters, namely the velocity jumps at the fluid–porous interfaces, the flow inertia in the porous medium region, and the flow and thermal slips at the fluid–solid interfaces, were considered in the simulation. The author also reported the relationship between the thermal performance and flow heterogeneity coefficient in the analysis.

The forced convection heat transfer in a pipe included with metallic foam as well as asymmetric inlet temperature under LTNE conditions was evaluated by Yue et al. [24] using COMSOL Multiphysics. They calculated some key parameters, including the solid and fluid temperature distribution, the fluid velocity profile, and Nu. They reported that the entrance temperature function strongly affected the distribution feature of Nu. Yerramalle et al. [25] developed a numerical analysis to study mixed convection in a horizontal channel and in the presence of a single heat source. The channel was supposed to be partially filled with the porous material. Temperature profiles of solid as well as fluid phases were obtained using the LTNE model. They concluded that heat transfer rate was dependent on porosity, thermal conductivity, and diameter of particle. Various configurations were proposed by Jadhav et al. [26] to improve the heat transfer rate of a horizontal conduit with partially filled high porosity metallic foams, while the pressure drop was reasonable. The pore density varied from 10 to 45 PPI, while the porosity was supposed to be between 0.90 and 0.95. They employed Darcy-extended Forchheimer methodology for the metallic foam zone, while the fluid region was simulated using two-equation turbulence k-ω model. They concluded that increasing the air velocity would enhance the velocity distribution at the exit of tube in the porous and clear region. Petracci and Gori [27] conducted an experimental study to evaluate the influences of a layer of aluminum foam covering a smooth circular cylinder on improving the heat transfer rate. They deduced that the heat transfer of foam-covered cylinder increased due to an increase in the exchange surface. Wang et al. [28] applied the two-velocity two-temperature model to study the convective heat transfer of an annular duct analytically. The annular duct was totally filled with a bidisperse porous medium. Asymmetric heat fluxes were applied to heat the annulus from interior and exterior walls. They reported that Nu was maximized when the heat flux ratio was assumed to be 0.175.

Based on the literature, metallic foam partially or fully filled tube [29], channel [30], and even annulus [26] have been studied by many researchers. However, the effect of metallic foam on both surfaces of an annulus has not been investigated yet. In the current study, a fully developed forced convection through a metallic foam partially filled annulus from the inner and outer surfaces is numerically examined comprehensively. Both surfaces of the annulus are exposed to asymmetric heat flux ratio. Ultimately, the impact of some critical factors on heat transfer and fluid flow inside foam and open regions is thoroughly explored, and the system performance is compared with a smooth annulus.

2 Mathematical Modeling

2.1 Problem Statement

The details of configuration for an annulus partly filled with porous material on two wall surfaces subject to asymmetrical heat fluxes are illustrated in Fig. 1. The metallic foam is attached to both the interior and exterior walls of an annulus with the inner radius of \(r_{1}\) and outer radius of \(r_{2}\). The outer- and inner-edge radii of the metallic foam of the interior and exterior boundaries are denoted by \(r_{i,o}\) and \(r_{i,i}\), respectively. The interior and exterior surfaces of the annulus are subjected to constant heat fluxes of \(q_{{{\text{w}}1}}\) and \(q_{{{\text{w}}2}}\), respectively. The heat fluxes subject to two wall surfaces of the annulus are removed via a single-phase fluid flow with an average velocity of \(u_{{\text{m}}}\).

Fig. 1
figure 1

Configuration details for an annulus partly coated with porous material on two wall surfaces subject to asymmetrical heat fluxes

Full size image

In this research, the fully developed laminar flow with a specified thermal–physical properties through an annulus partly coated with metallic foam on two wall surfaces subject to asymmetrical heat fluxes is studied, while natural convective heat transfer and thermal radiation are neglected in mathematical modeling. Furthermore, the metallic foam sintered on the interior and exterior surfaces of the annulus is supposed to be homogenous and isotropic.

2.2 Governing Equations

The momentum and energy equations for an annulus including with porous material on two wall surfaces are written as follows for fully developed forced convection in the foam and no-foam (fluid) zones:

Foam region on the inner wall \((r_{1} < r < r_{i,o} )\):

$$- \frac{{{\text{d}}p}}{{{\text{d}}x}} + \frac{{\mu_{{\text{f}}} }}{\varepsilon }\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{{\partial r}}\right) - \frac{{\mu_{{\text{f}}} u}}{K} = 0$$
(1)
$$\frac{{k_{{{\text{se}}}} }}{r}\frac{\partial }{\partial r}\left(r\frac{{\partial T_{{\text{s}}} }}{\partial r}\right) - h_{{{\text{sf}}}} a_{{{\text{sf}}}} (T_{{\text{s}}} - T_{{\text{f}}} ) = 0$$
(2)
$$\frac{{(k_{{{\text{fe}}}} + k_{{\text{d}}} )}}{r}\frac{\partial }{\partial r}(r\frac{{\partial T_{{\text{f}}} }}{\partial r}) + h_{{{\text{sf}}}} a_{{{\text{sf}}}} (T_{{\text{s}}} - T_{{\text{f}}} ) = \rho_{{\text{f}}} c_{{\text{f}}} u\frac{{\partial T_{{\text{f}}} }}{\partial x}$$
(3)

where \(\rho_{{\text{f}}}\), \(c_{{\text{f}}}\), K and \(\varepsilon\) stand for density of fluid, specific heat of fluid, permeability, and porosity of porous material, respectively.

Fluid region \((r_{i,o} < r < r_{i,i} )\):

$$- \frac{{d{\text{p}}}}{{d{\text{x}}}} + \frac{{\mu_{{\text{f}}} }}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{{\partial r}}\right) = 0$$
(4)
$$\frac{{k_{{\text{f}}} }}{r}\frac{\partial }{\partial r}\left(r\frac{{\partial T_{{\text{f}}} }}{\partial r}\right) = \rho_{{\text{f}}} c_{{\text{f}}} u\frac{{\partial T_{{\text{f}}} }}{\partial x}$$
(5)

Foam region on the outer wall \((r_{i,i} < r < r_{2} )\):

$$- \frac{{{\text{d}}p}}{{{\text{d}}x}} + \frac{{\mu_{{\text{f}}} }}{\varepsilon }\frac{1}{r}\frac{\partial }{\partial r}\left(r\frac{\partial u}{{\partial r}}\right) - \frac{{\mu_{{\text{f}}} u}}{K} = 0$$
(6)
$$\frac{{k_{{{\text{se}}}} }}{r}\frac{\partial }{\partial r}\left(r\frac{{\partial T_{{\text{s}}} }}{\partial r}\right) - h_{{{\text{sf}}}} a_{{{\text{sf}}}} (T_{{\text{s}}} - T_{{\text{f}}} ) = 0$$
(7)
$$\frac{{(k_{{{\text{fe}}}} + k_{{\text{d}}} )}}{r}\frac{\partial }{\partial r}\left(r\frac{{\partial T_{{\text{f}}} }}{\partial r}\right) + h_{{{\text{sf}}}} a_{{{\text{sf}}}} (T_{{\text{s}}} - T_{{\text{f}}} ) = \rho_{{\text{f}}} c_{{\text{f}}} u\frac{{\partial T_{{\text{f}}} }}{\partial x}$$
(8)

where kd stands for the dispersion conductivity and kd is calculated as follows [6]:

$$k_{{\text{d}}} (u) = C_{{\text{d}}} \rho_{{\text{f}}} c_{{\text{f}}} \sqrt K u$$
(9)

The thermal dispersion coefficient is denoted by Cd, while fluid specific heat, fluid density, and porous medium permeability are shown by cf, ρf, and K, respectively. The dispersion conductivity (kd) can be ignored under the following condition:

$$k_{{\text{d}}} < < k_{{{\text{fe}}}}$$
(10)

After substituting Eq. (9) into Eq. (10), Eq. (11) can be obtained as follows:

$$\frac{{C_{{\text{d}}} \rho_{{\text{f}}} c_{{\text{f}}} \sqrt K u}}{{k_{{{\text{fe}}}} }} < < 1$$
(11)

In Eqs. (3) and (8), the dispersion conductivity (kd) should be neglected since the permeability in the porous region of a duct partly coated with metallic foam is very low. Thus, the velocity in the clear region is three orders of magnitude more than that in the foam region for a fluid flow with a big frontal velocity. Lower permeability would result in a lower dispersion conductivity (kd). Further details can be found in references [6, 16].

Based on the boundary condition of uniform heat flux on the interior and exterior surfaces of the annulus, the energy balance can be written as follows:

$$\rho_{{\text{f}}} u_{{\text{m}}} \pi (r_{2}^{2} - r_{1}^{2} )c_{{\text{f}}} {\text{d}}T_{{\text{f,b}}} = (q_{{{\text{w}}1}} 2\pi r_{1} + q_{{{\text{w}}2}} 2\pi r_{2} ){\text{d}}x$$
(12)

where the heat flux on the interior and exterior surfaces are denoted by qw1 and qw2, respectively. The heat flux ratio is defined as below:

$$\zeta = \frac{{q_{{{\text{w}}2}} }}{{q_{{{\text{w}}1}} }}$$
(13)

After substituting Eq. (13) into Eq. (12), one can write:

$$\rho_{{\text{f}}} u_{{\text{m}}} c_{{\text{f}}} \frac{{{\text{d}}T_{{\text{f,b}}} }}{{{\text{d}}x}} = \frac{{4q_{{{\text{w}}1}} (r_{1} + \zeta r_{2} )}}{{2(r_{2} - r_{1} )(r_{1} + r_{2} )}}$$
(14)

After combining Eq. (14) with Eqs. (3), (5) and (8), the energy equations for the foam and fluid zones can be simplified as follows:for the foam region:

$$\frac{{k_{{{\text{fe}}}} }}{r}\frac{\partial }{\partial r}(r\frac{{\partial T_{{\text{f}}} }}{\partial r}) + h_{{{\text{sf}}}} a_{{{\text{sf}}}} (T_{{\text{s}}} - T_{{\text{f}}} ) = \frac{u}{{u_{{\text{m}}} }} \times \frac{{4q_{{{\text{w}}1}} }}{{2(r_{2} - r_{1} )}} \times \frac{{(r_{1} + \zeta r_{2} )}}{{(r_{1} + r_{2} )}}$$
(15)

for the fluid region:

$$\frac{{k_{{\text{f}}} }}{r}\frac{\partial }{\partial r}(r\frac{{\partial T_{{\text{f}}} }}{\partial r}) = \frac{u}{{u_{{\text{m}}} }} \times \frac{{4q_{{{\text{w}}1}} }}{{2(r_{2} - r_{1} )}} \times \frac{{(r_{1} + \zeta r_{2} )}}{{(r_{1} + r_{2} )}}$$
(16)

2.3 Boundary Conditions

Two types of closure conditions are needed to solve Eqs. (1)–(8): boundary conditions and interfacial coupling conditions. The following equations can be imposed at the interior and exterior surfaces of the annulus as the boundary conditions:

For the inner wall (r = r1):

$$u = 0$$
(17a)
$$T_{{\text{f}}} = T_{{\text{s}}} = T_{{{\text{w}}1}}$$
(17b)

For the outer wall (r = r2):

$$u = 0$$
(18a)
$$T_{{\text{f}}} = T_{{\text{s}}}$$
(18b)
$$q_{{{\text{w}}2}}^{\prime \prime } = k_{{{\text{se}}}} \frac{{\partial T_{{\text{s}}} }}{\partial r} + k_{{{\text{fe}}}} \frac{{\partial T_{{\text{f}}} }}{\partial r}$$
(18c)

At the fluid–porous boundary, the interfacial coupling conditions must be applied. The continuity of velocity and shear stress is applied for the fluid flow to close the governing equations.at the fluid–porous boundary of the metallic foam on the inner wall:

$$\left. u \right|_{{r_{i,o}^{ - } }} = \left. u \right|_{{r_{i,o}^{ + } }}$$
(19)
$$\left. {\frac{{\mu_{{\text{f}}} }}{\varepsilon }\frac{\partial u}{{\partial r}}} \right|_{{r_{i,o}^{ - } }} = \left. {\mu_{{\text{f}}} \frac{\partial u}{{\partial r}}} \right|_{{r_{i,o}^{ + } }}$$
(20)

at the fluid–porous boundary of the metallic foam on the outer wall:

$$\left. u \right|_{{r_{i,i}^{ - } }} = \left. u \right|_{{r_{i,i}^{ + } }}$$
(21)
$$\left. {\mu_{{\text{f}}} \frac{\partial u}{{\partial r}}} \right|_{{r_{i,i}^{ - } }} = \left. {\frac{{\mu_{{\text{f}}} }}{\varepsilon }\frac{\partial u}{{\partial r}}} \right|_{{r_{i,i}^{ + } }}$$
(22)

Furthermore, the following equations are applied for the continuity of heat flux and fluid temperature at the interface of porous–fluid metallic foams:for the metallic foam on the inner wall:

$$\left. {T_{{\text{f}}} } \right|_{{r_{i,o}^{ - } }} = \left. {T_{{\text{f}}} } \right|_{{r_{i,o}^{ + } }}$$
(23)
$$\left. {(k_{{{\text{se}}}} \frac{{\partial T_{{\text{s}}} }}{\partial r} + k_{{{\text{fe}}}} \frac{{\partial T_{{\text{f}}} }}{\partial r})} \right|_{{r_{i,o}^{ - } }} = \left. {k_{{\text{f}}} \frac{{\partial T_{{\text{f}}} }}{\partial r}} \right|_{{r_{i,o}^{ + } }}$$
(24)

for the metallic foam on the outer wall:

$$\left. {T_{{\text{f}}} } \right|_{{r_{i,i}^{ - } }} = \left. {T_{{\text{f}}} } \right|_{{r_{i,i}^{ + } }}$$
(25)
$$\left. {k_{{\text{f}}} \frac{{\partial T_{{\text{f}}} }}{\partial r}} \right|_{{r_{i,i}^{ - } }} = \left. {(k_{{{\text{se}}}} \frac{{\partial T_{{\text{s}}} }}{\partial r} + k_{{{\text{fe}}}} \frac{{\partial T_{{\text{f}}} }}{\partial r})} \right|_{{r_{i,i}^{ + } }}$$
(26)

Temperature field cannot be obtained using the interfacial boundary conditions presented in Eqs. (23)-(26) since the energy equations are not closed. Energy equations must be closed using a novel interface condition. Ochoa-Tapia and Whitakar [31] proposed an interface condition for non-equilibrium heat transfer at interface of the porous–fluid:

$$\vec{n}_{{\text{p,f}}} (\left. {k_{{{\text{se}}}} \nabla T_{{\text{s}}} } \right|_{{r_{i,o}^{ - } }} ) = h_{{{\text{sf}}}} (\left. {T_{{\text{f}}} } \right|_{{r_{i,o}^{ + } }} - \left. {T_{{\text{s}}} } \right|_{{r_{i,o}^{ - } }} )$$
(27a)

After axial heat conduction is ignored, Eq. (27a) is simplified to:for the metallic foam on the inner wall:

$$- k_{{{\text{se}}}} \left. {\frac{{\partial T_{{\text{s}}} }}{\partial r}} \right|_{{r_{i,o}^{ - } }} = h_{{{\text{sf}}}} (\left. {T_{{\text{s}}} } \right|_{{_{{r_{i,o}^{ - } }} }} - \left. {T_{{\text{f}}} } \right|_{{_{{r_{i,o}^{ + } }} }} )$$
(27b)

for the metallic foam on the outer wall:

$$- k_{{{\text{se}}}} \left. {\frac{{\partial T_{{\text{s}}} }}{\partial r}} \right|_{{r_{i,i}^{ + } }} = h_{{{\text{sf}}}} (\left. {T_{f} } \right|_{{_{{r_{i,i}^{ - } }} }} - \left. {T_{{\text{s}}} } \right|_{{_{{r_{i,i}^{ + } }} }} )$$
(27c)

Permeability, local heat transfer coefficient between fluid and solid, effective thermal conductivity, and specific surface area are shown by K, hsf, ke, and asf, respectively. The correlations presented in Table 1 can be applied to calculate the values of these parameters.

Table 1 Correlations of parameters for metallic foams
Full size table

2.4 Normalizing the Closure Conditions and Governing Equations

The following dimensionless parameters are defined to normalize closure conditions and the governing equations presented in the previous sections:

$$R = \frac{2r}{{D_{{\text{h}}} }},\,R_{2} = \frac{{r_{2} }}{{r_{1} }},\,U = \frac{u}{{u_{{\text{m}}} }},\,A = \frac{{h_{{{\text{sf}}}} D_{{\text{h}}} }}{{k_{{{\text{se}}}} }},\,B = \frac{{k_{{\text{f}}} }}{{k_{{{\text{se}}}} }},C = \frac{{k_{{{\text{fe}}}} }}{{k_{{{\text{se}}}} }},\,D = \frac{{h_{{{\text{sf}}}} a_{{{\text{sf}}}} D_{{\text{h}}}^{2} }}{{k_{{{\text{se}}}} }}$$
$$P = \frac{K}{{\mu_{{\text{f}}} u_{{\text{m}}} }}\frac{{{\text{d}}p}}{{{\text{d}}x}},Da = \frac{K}{{D_{{\text{h}}}^{2} }},s = \sqrt {\frac{\varepsilon }{Da}} ,\zeta = \frac{{q_{{{\text{w}}2}} }}{{q_{{{\text{w}}1}} }},\theta_{f} = \frac{{T_{{\text{f}}} - T_{{{\text{w}}1}} }}{{\left( {q_{{{\text{w}}1}} + q_{{{\text{w}}2}} } \right)\frac{{D_{{\text{h}}} }}{{k_{{{\text{se}}}} }}}}$$
(28)

After normalizing momentum and energy equations for thermally and hydraulically fully developed forced convection, these equations can be reformulated as follows:for the metallic foam on the inner wall \(\left( {\frac{{2r_{1} }}{{D_{h} }} < R < R_{i,o} } \right)\):

$$R^{2} \frac{{\partial^{2} U}}{{\partial R^{2} }} + R\frac{\partial U}{{\partial R}} - \frac{{R^{2} s^{2} }}{4}(U + P) = 0$$
(29)
$$\frac{{\partial^{2} \theta_{s} }}{{\partial R^{2} }} + \frac{1}{R}\frac{{\partial \theta_{s} }}{\partial R} - \frac{D}{4}(\theta_{s} - \theta_{f} ) = 0$$
(30)
$$C\left( {\frac{{\partial^{2} \theta_{f} }}{{\partial R^{2} }} + \frac{1}{R}\frac{{\partial \theta_{f} }}{\partial R}} \right) + \frac{D}{4}\left( {\theta_{s} - \theta_{f} } \right) = \frac{{U\left( {1 + \zeta R_{2} } \right)}}{{\left( {1 + R_{2} } \right)\left( {1 + \zeta } \right)}}$$
(31)

for the fluid region \(\left( {R_{i,o} < R < R_{i,i} } \right)\):

$$\frac{1}{R}\frac{\partial }{\partial R}\left( {R\frac{\partial U}{{\partial R}}} \right) = \frac{P}{4Da}$$
(32)
$$B\left( {\frac{{\partial^{2} \theta_{f} }}{{\partial R^{2} }} + \frac{1}{R}\frac{{\partial \theta_{f} }}{\partial R}} \right) = \frac{{U(1 + \zeta R_{2} )}}{{(1 + R_{2} )(1 + \zeta )}}$$
(33)

For the metallic foam on the outer wall \(\left( {R_{i,i} < R < \frac{{2r_{2} }}{{D_{h} }}} \right)\):

$$R^{2} \frac{{\partial^{2} U}}{{\partial R^{2} }} + R\frac{\partial U}{{\partial R}} - \frac{{R^{2} s^{2} }}{4}(U + P) = 0$$
(34)
$$\frac{{\partial^{2} \theta_{s} }}{{\partial R^{2} }} + \frac{1}{R}\frac{{\partial \theta_{s} }}{\partial R} - \frac{D}{4}(\theta_{s} - \theta_{f} ) = 0$$
(35)
$$C\left( {\frac{{\partial^{2} \theta_{f} }}{{\partial R^{2} }} + \frac{1}{R}\frac{{\partial \theta_{f} }}{\partial R}} \right) + \frac{D}{4}\left( {\theta_{s} - \theta_{f} } \right) = \frac{{U\left( {1 + \zeta R_{2} } \right)}}{{\left( {1 + R_{2} } \right)\left( {1 + \zeta } \right)}}$$
(36)

The corresponding boundary conditions are normalized as follows:

$$R = \frac{{2r_{1} }}{{D_{h} }}:\,U = 0;\,\theta_{f} = \theta_{s} = 0$$
(37)
$$\begin{gathered} R = R_{i,o} :\,\left. U \right|_{{R_{i,o}^{ - } }} = \left. U \right|_{{R_{i,o}^{ + } }} ;\,\frac{1}{\varepsilon }\left. {\frac{\partial U}{{\partial R}}} \right|_{{R_{i,o}^{ - } }} = \left. {\frac{\partial U}{{\partial R}}} \right|_{{R_{i,o}^{ + } }} ;\,\hfill \\ \left. {\theta_{f} } \right|_{{R_{i,o}^{ - } }} = \left. {\theta_{f} } \right|_{{R_{i,o}^{ + } }} ;\left. {\left( {\frac{{\partial \theta_{s} }}{\partial R} + C\frac{{\partial \theta_{f} }}{\partial R}} \right)} \right|_{{R_{i,o}^{ - } }} = B\left. {\frac{{\partial \theta_{f} }}{\partial R}} \right|_{{R_{i,o}^{ + } }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left. {\frac{{\partial \theta_{s} }}{\partial R}} \right|_{{R_{i,o}^{ - } }} = A_{i} (\left. {\theta_{s} } \right|_{{R_{i,o}^{ - } }} - \left. {\theta_{f} } \right|{}_{{R_{i,o}^{ + } }}) \hfill \\ \end{gathered}$$
(38)
$$\begin{gathered} R = R_{i,i} :\,\left. U \right|_{{R_{i,i}^{ - } }} = \left. U \right|_{{R_{i,i}^{ + } }} ;\,\left. {\frac{\partial U}{{\partial R}}} \right|_{{R_{i,i}^{ - } }} = \left. {\frac{1}{\varepsilon }\frac{\partial U}{{\partial R}}} \right|_{{R_{i,i}^{ + } }} ;\,\hfill\\ \left. {\theta_{f} } \right|_{{R_{i,i}^{ - } }} = \left. {\theta_{f} } \right|_{{R_{i,i}^{ + } }} ;\,\,B\left. {\frac{{\partial \theta_{f} }}{\partial R}} \right|_{{R_{i,i}^{ - } }} = \left. {\left( {\frac{{\partial \theta_{s} }}{\partial R} + C\frac{{\partial \theta_{f} }}{\partial R}} \right)} \right|_{{R_{i,i}^{ + } }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \left. {\frac{{\partial \theta_{s} }}{\partial R}} \right|_{{R_{i,i}^{ + } }} = A_{i} (\left. {\theta_{f} } \right|_{{R_{i,i}^{ - } }} - \left. {\theta_{s} } \right|_{{R_{i,i}^{ + } }} ) \hfill \\ \end{gathered}$$
(39)
$$R = \frac{{2r_{2} }}{{D_{h} }}:\,U = 0;\,\,\theta_{f} = \theta_{s} ;\,\,C\frac{{\partial \theta_{f} }}{\partial R} + \frac{{\partial \theta_{s} }}{\partial R} = \frac{\zeta }{{2\left( {1 + \zeta } \right)}}$$
(40)

2.5 Definition of Key Parameters

The following equation can be applied to compute the Reynolds number of forced convection in the annulus:

$${\text{Re}} = \frac{{\rho_{f} u_{m} 2(r_{2} - r_{1} )}}{{\mu_{f} }}$$
(41)

The friction factor is obtained as follows:

$$fr = \frac{{ - \frac{dp}{{dx}} \times 2 \times (r_{2} - r_{1} )}}{{\rho_{f} \frac{{u_{m}^{2} }}{2}}} = \frac{ - 2P}{{Da.{\text{Re}} }}$$
(42)

The following equation is integrated to calculate the dimensionless temperature of bulk-mean fluid (\(\theta_{f,b}\)):

$$\theta_{f,b} = \frac{{\frac{1}{A}\int\limits_{A} {U\theta_{f} dA} }}{{\frac{1}{A}\int\limits_{A} {UdA} }} = \frac{1}{A}\int\limits_{A} {U\theta_{f} dA} = \frac{{D_{h}^{2} }}{4A}\left\{ {\int\limits_{0}^{2\pi } {\int\limits_{{\frac{{2r_{1} }}{{D_{h} }}}}^{{\frac{{2r_{2} }}{{D_{h} }}}} {U\theta_{f} RdRd\varphi } } } \right\} = \frac{2}{{\left( {\left( {\frac{{2r_{2} }}{{D_{h} }}} \right)^{2} - \left( {\frac{{2r_{1} }}{{D_{h} }}} \right)^{2} } \right)}}\left\{ {\int\limits_{{\frac{{2r_{1} }}{{D_{h} }}}}^{{\frac{{2r_{2} }}{{D_{h} }}}} {U\theta_{f} RdR} } \right\}$$
(43)

Then, Nu at the interior and exterior surfaces can be obtained as follows:

$$Nu_{1} = \frac{{q_{w1} }}{{T_{w1} - T_{f,b} }} \times \frac{{2(r_{2} - r_{1} )}}{{k_{f} }} = \frac{ - 1}{{\theta_{f,b} \times (1 + \zeta ) \times B}}$$
(44)
$$Nu_{2} = \frac{{q_{w2} }}{{T_{w2} - T_{f,b} }} \times \frac{{2(r_{2} - r_{1} )}}{{k_{f} }} = \frac{1}{{B \times (\theta_{w2} - \theta_{f,b} )}} \times \frac{\zeta }{{\left( {1 + \zeta } \right)}}$$
(45)

The effective Nu can be defined as:

$$Nu_{eff} = \frac{{Nu_{1} r_{1} + Nu_{2} r_{2} }}{{r_{1} + r_{2} }} = \frac{{Nu_{1} + Nu_{2} R_{2} }}{{\left( {1 + R_{2} } \right)}}$$
(46)

Applying metallic foam on the inner and outer walls of the annulus will increase the heat transfer rate and flow resistance. The following equation is used for the comprehensive performance evaluation criteria \(\left( {j/f^{1/3} } \right)\,\) to consider these two effects simultaneously:

$$j/f^{1/3} = \frac{{Nu_{eff} }}{{{\text{Re}} \times (\Pr \times fr)^{1/3} }}$$
(47)

The ratio of the mass flow rate in the foam region to the total mass flow rate (coefficient of the flow heterogeneity) is defined as follows:

$$\begin{gathered} \xi = \frac{{q_{m,p} }}{{q_{m} }} = \frac{{\int\limits_{{{\text{inner - wall}}\,{\text{foam}}}} {u2\pi rdr} + \int\limits_{{{\text{outer - wall}}\,\,{\text{foam}}}} {u2\pi rdr} }}{{\int\limits_{{{\text{inner - wall}}\,{\text{foam}}}} {u2\pi rdr} + \int\limits_{{{\text{fluid}}\,{\text{region}}}} {u2\pi rdr} + \int\limits_{{{\text{outer - wall}}\,\,{\text{foam}}}} {u2\pi rdr} }} \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\ \end{gathered}$$
(48)

After dividing the numerator and denominator of Eq. (48) by \(u_{m}\), Eq. (48) can be reformulated using the non-dimensional parameters defined in Eq. (28):

$$\begin{aligned}{}& \xi \\ &= \frac{{\int\limits_{{{\text{inner - wall}}\,{\text{foam}}}} {URdR} + \int\limits_{{{\text{outer - wall}}\,\,{\text{foam}}}} {URdR} }}{{\int\limits_{{{\text{inner - wall}}\,{\text{foam}}}} {URdR} + \int\limits_{{{\text{fluid}}\,{\text{region}}}} {URdR} + \int\limits_{{{\text{outer - wall}}\,\,{\text{foam}}}} {URdR} }}\end{aligned}$$
(49)

A dimensionless radial position \(\left( \eta \right)\) is defined to appraise the influence of foam thickness on the heat transfer rate and flow resistance:

$$\eta = \frac{{r - r_{1} }}{{r_{2} - r_{1} }} = \frac{{\frac{{RD_{h} }}{{2r_{1} }} - 1}}{{R_{2} - 1}}$$
(50)

It should be mentioned that the dimensionless foam thickness on the inner (\(\eta_{i,o}\)) and outer walls (\(\eta_{i,i}\)) can be calculated using the following equations:

$$\eta_{i,o} = \frac{{r_{i,o} - r_{1} }}{{r_{2} - r_{1} }} = \frac{{\frac{{R_{i,o} D_{h} }}{{2r_{1} }} - 1}}{{R_{2} - 1}}$$
(51)
$$\eta_{i,i} = 1 - \frac{{r_{i,i} - r_{1} }}{{r_{2} - r_{1} }} = 1 - \frac{{\frac{{R_{i,i} D_{h} }}{{2r_{1} }} - 1}}{{R_{2} - 1}}$$
(52)

3 Numerical Method and Accuracy

Thermally and hydrodynamically fully developed laminar flow through a horizontal annulus partially covered with metallic foam in the presence of asymmetric heat flux ratio at inner and outer walls has been considered. To study velocity and temperature profiles inside the annulus, the non-dimensional parameters presented in Eq. (28) (except the non-dimensional pressure gradient (P)) are calculated using the input variables. Then, the non-dimensional governing equations, Eqs. (29)–(36), are solved in conjunction with the boundary conditions of Eqs. (37)–(40) by using MATLAB built-in bvp4c method. The solutions are acquired by initial guess for, P; consequently, a reciprocal algorithm was required to gain the appropriate P to satisfy the continuity equation. The equations of the foam and fluid regions are simultaneously computed, and the continuity equation is applied for the whole cross section, including foam and open regions. The convergence criterion is deemed to be 10–6 for the relative error between assumed value of P with the computed one. If the stopping criterion is not satisfied, the value of non-dimensional pressure gradient is modified, and the numerical process is repeated to satisfy the continuity equation on the whole domain. To avoid the grid dependency, the developed code is run on three different integration intervals (dR) of 10–5, 5 × 10–6, and 10–6 to confirm the grid independency of the results. It should be noted that all obtained results are conducted using the integration interval dR = 5 × 10–6. After the numerical solution is converged, the desired outputs are computed using the modified value of the non-dimensional pressure gradient (P). The flowchart of the numerical code developed in the present study is illustrated in Fig. 2.

Fig. 2
figure 2

The flowchart of the numerical code developed in the present study

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The precision of the developed numerical code is indicated in Fig. 3 in which flow heterogeneity coefficient (ξ) and Nu for the case of the metallic foam on the interior wall and no foam on the exterior wall (\(\eta_{i,i} \to 0\)) are compared to the reported results of Que et al. [16] These parameters are computed with the variations of foam thickness on the inner wall from zero (no foam on cross section) to 1 (fully filled with the foam). To evaluate the numerical setup, ξ and Nu are calculated for different pore densities and kr, respectively. The obtained results demonstrate the numerical results perfectly matches the reported analytical data.

Fig. 3
figure 3

Comparison of the numerical results of the present method with those of the analytical approach developed by Qu et al. [16]. a Effects of dimensionless foam thickness on ξ for various pore densities b Effects of dimensionless thickness of foam on the Nu for various values of kr

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4 Results and Discussion

The effects of different values of critical factors on the velocity profile, flow heterogeneity, thermal distribution, friction factor, Nu, and the system performance are studied for different sizes of foam thickness on both walls. For some critical parameters, the results are compared with different cases of fully filled annulus and smooth one.

4.1 Velocity Profile

As is commonly understood, higher porosity in the foam indicates larger empty spaces within the material. As depicted in Fig. 4a,b, increasing porosity leads to greater permeability of the foam media and thus higher fluid velocity. Due to the principle of mass conservation, this results in a decrease in the velocity peak observed in the open region as porosity increases.

Fig. 4
figure 4

Porosity and pore density effects on the velocity profile for various foam thicknesses

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After examining the velocity profiles, it was determined that the velocity within the foam media is significantly lower than in the open region. This can be explained by the Brinkman–Darcy flow model for porous media. The velocity within the foam is relatively uniform due to the homogenous metallic foam, but at the interface between the foam and the fluid, the flow velocity experiences a sudden increase, resulting in a peak velocity in the open region. This is due to an increase in permeability from the foam region to the open region, and because of the principle of total mass conservation, the velocity reaches its maximum in the open region. Increasing the foam thickness on both walls results in more fluid flowing through the foam region, as the porous media becomes wider. As a result, the velocity peak in the open region is doubled when the foam thickness is tripled, as depicted in Fig. 4a-d.

Figure 4c,d shows how changes in pore density affect the velocity profile. A higher pore density reduces the flow velocity within the foam region, which impacts the velocity peak in the open region and increases it. This is because increasing pore density directly affects the permeability of the foam region and causes it to decrease. These effects are most pronounced in the annulus with a higher foam thickness because the fluid flow encounters a larger foam area. Therefore, the fluid flows rapidly through the open region, and the fluid velocity reaches a very high peak due to the principle of total mass flow conservation. This effect has been observed in other studies of porous media, and it underscores the importance of considering the overall geometry and structure of a porous medium when studying fluid flow through it.

4.2 Flow Heterogeneity

The flow heterogeneity coefficient measures the ratio of mass flow through the porous zone to the total mass flow, and it is influenced by critical parameters such as foam thickness, porosity, and pore density, as demonstrated in Fig. 5. The results in Fig. 5a and b indicate that an increase in foam thickness leads to a higher flow ratio within the foam region. Additionally, the data reveal that the flow heterogeneity coefficient is less sensitive to changes in porosity compared to pore density. When porosity increases, permeability also increases, allowing more space for fluid to flow through the foam region. Consequently, an increase in porosity leads to higher ξ, as illustrated in Fig. 5a.

Fig. 5
figure 5

Porosity and pore density effects on the coefficient of flow heterogeneity for various foam thicknesses

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Figure 5b illustrates the effects of pore density and foam thickness on the flow heterogeneity coefficient. As pore density increases in the porous zones, the resistance to fluid flow through the foam region also increases, resulting in less fluid flow passing through the high flow resistance area and more fluid flow traveling through the low flow resistance open region. The data in Fig. 5a,b demonstrate that the porous zone flow ratio depends more on pore density than porosity. These results suggest that controlling pore density and foam thickness can lead to significant changes in the flow heterogeneity coefficient, which can have important implications for a range of industrial and engineering applications.

4.3 Temperature Distribution

The dimensionless temperature, θ = (T−Tw1)/(qw1 (1 + ζ) Dh/kse), is defined to represent the temperature difference between the fluid and the inner wall, with kse being the effective thermal conductivity of the solid. This thermal conductivity is influenced by the porosity and thermal conductivity ratio. To isolate the effect of these parameters, -θ/kse is replaced with θ. To study the effect of different heat flux ratios on temperature profiles, the term (1 + ζ) in θ is removed, and −θ(1 + ζ) is used instead of θ. The temperature distributions are investigated to study the influence of various control parameters, such as pore density, conductivity ratio, porosity, heat flux ratio, and Re number, as shown in Figs. 6,7. In thermal equilibrium condition, the fluid flow is hydrodynamically and thermally fully developed as it passes through the foam region. Hence, the temperature difference between the metal foam and fluid flow inside the foam region is relatively small.

Fig. 6
figure 6

The impacts of porosity and pore density on the temperature distribution for various foam thicknesses

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Fig. 7
figure 7

The impacts of thermal conductivity and heat flux ratios on the temperature distribution for various foam thicknesses

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In Fig. 6, the temperature profiles are displayed for two different foam thicknesses on the walls while varying the porosity and pore density. The temperature difference increases in both the foam and open regions as the porosity increases, as shown in Fig. 6a,b. This is because increasing porosity decreases the specific surface area of the foam and increases its permeability, which leads to a higher temperature difference between the fluid and wall due to thermal aggregation in the annulus where heat source is applied. Additionally, heat transfers from the walls to the foam region and then radially to the center of the open region. Increasing the foam thickness leads to a rise in both fluid and solid temperatures due to an increase in flow heterogeneity coefficient (Fig. 5a), resulting in heat dissipation. Therefore, the temperature difference peak in the open region is higher for the case of lower foam thickness compared to higher foam thickness, as demonstrated in Fig. 6a,b.

Figure 6c,d shows the impact of pore density on the temperature difference. An increase in pore density results in a decrease in the specific surface area and permeability, which significantly reduces the coefficient of flow heterogeneity, as indicated in Fig. 5b. As a result, the temperature difference in the porous zone decreases with an increase in pore density, while it increases in the clear region. In the lower pore density, the local convective heat transfer coefficient is higher in the open region due to the higher foam region flow ratio, causing a lower temperature difference.

In Fig. 7a,b, the temperature distribution is shown for different thermal conductivity ratios (kr). An increase in kr means that the ratio of thermal conductivities between the fluid and solid is larger. At low kr values, the thermal resistance to conduction through the foam region decreases, which leads to a reduction in temperature difference inside the foam region. As the thickness of the foam increases, the temperature difference decreases further, as shown in Fig. 4b,d, where there is a decrease in the velocity peak at the middle of the annulus.

Figure 7c,d shows the effects of heat flux ratio variations on the temperature. The heat flux ratio represents the ratio of the applied heat flux at the exterior wall to the interior wall. A value of ζ = 0.25 means that the heat flux at the interior wall is four times greater than that at the exterior wall. When ζ = 0, the entire applied heat flux to the system is qw1, resulting in the maximum temperature difference between the two wall surfaces. An increase in ζ from 0 to 1 causes the applied heat flux to vary from qw1 to 2qw1, leading to an increase in the system’s total energy and, therefore, an increase in the temperature of the walls and fluid inside the open and foam regions. As the heat flux ratio increases, the temperature difference between the outer and inner walls decreases, as shown in Fig. 7b,c. As expected, in higher foam thickness on either wall, the maximum difference in temperature decreases due to heat dissipation occurring in the foam region.

4.4 Friction Factor

The very low-velocity profile in the foam region is attributed to the frictional drag at the fluid–solid interface in this region, as shown in Fig. 4. The friction factor, defined in Eq. (42), represents the flow resistance due to existing obstacles in the foam region and can be controlled by various factors such as porosity, pore density, Reynolds number, and foam thickness, as depicted in Fig. 8. A higher permeability leads to a lower friction factor, as shown in Fig. 8, indicating that the annulus without metal foam has the lowest friction factor, while the fully filled annulus with metal foam has the highest. Increasing foam thickness significantly increases the friction factor, but increasing the porosity of the metal foam reduces it. In cases of higher foam thickness, increasing porosity results in a more substantial decline in the friction factor, as shown in Fig. 8a.

Fig. 8
figure 8

The impacts of porosity, pore density, and Re number on the friction factor for various foam thicknesses

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Pore density is another determinant of the friction factor value. Higher pore density gives lower permeability; consequently, the frictional drag at the fluid–solid interface in the foam region increases. Figure 8b indicates the highest increase in the friction factor belongs to the case of a fully filled annulus. In addition, in this case, there is a remarkable increase in the friction factor over increasing pore density. This can be attributed to the high flow resistance from the metal foam region.

Reynolds number is another controlling parameter that directly affects friction factor in such a way that in the case of the empty annulus, as one knows, the friction factor is inversely proportional to Re number. One of the reasons for increasing Re number is the increase in the mean flow velocity, leading to augment pressure drop. However, the rate of increase in flow velocity is greater than the increase in pressure drop, as stated in Ref. [18]. Therefore, based on the definition of friction factor, Eq. (38), friction factor has a downward trend toward an increase in Re number.

4.5 Thermal Performance

This section examines the thermal behavior of a partially filled annulus with metal foam of varying thicknesses compared to that of an empty annulus and a filled one. The heat flux is applied to both sides of the annulus, and the heat transfer occurs through conduction (via the metal foam) or convection (via fluid flow inside the foam region) heat transfer. Thus, the heat transfer encounters three thermal resistances as it moves toward the open region. The first thermal resistance is the convective resistance between the laminar fluid flow and the heated wall, and the second is the conduction resistance of the metal foam. Once the heat penetrates the foam region, the local convective thermal resistance between the fluid flow and the metal foam occurs. Therefore, the thermal performance of the system is affected by the combined effect of these three thermal resistances. Furthermore, these thermal resistances are influenced by changes in porosity, pore density, and thermal conductivity ratio.

Figure 9a illustrates the effect of porosity on the effective Nu for various foam thicknesses, as well as empty and fully filled annuli. Nu remains constant for an empty annulus but decreases for a fully filled one. In the presence of metal foam in the annulus, Nu decreases as porosity increases. This is because an increase in porosity leads to greater permeability and flow heterogeneity, resulting in a decrease in the local convective thermal resistance but an increase in the heat conduction thermal resistance due to the reduced foam diameter or kse. Because of the problem’s governing conditions, such as asymmetric heat flux on the walls and the presence of metal foam on both walls, the resulting changes in these two resistances ultimately lead to a deterioration in heat transfer. The increase in the latter resistance is higher than the decrease in the former one. As porosity and foam thickness increase, the ratio of these resistances also increases, leading to a significant reduction in Nu.

Fig. 9
figure 9

The impacts of porosity, pore density, and conductivity ratio on the effective Nu for various foam thicknesses

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Figure 9b demonstrates the effect of pore density on the effective Nu as foam thickness increases. An increase in pore density results in a reduction in flow heterogeneity, as shown in Fig. 5b, thereby increasing the local convective thermal resistance. However, the heat conduction thermal resistance tends to remain unchanged with an increase in pore density. As a result, Nu decreases at low pore densities, then levels off, and reaches a constant value. This trend is more noticeable in thicker foams; however, the trend will be opposite for a fully filled annulus due to the sharp increase in flow heterogeneity and consequently an increase in the local convective thermal resistance.

Figure 9c displays the influence of the fluid–solid thermal conductivity ratio (kr) on the effective Nu for different foam thicknesses. The results show that the effective Nu for the partially filled annulus is higher than that for the smooth annulus, and with an increase in kr, the heat transfer deteriorates sharply. At the optimum kr value (approximately kr ~ 0.035), the effective Nu of the partially filled annulus becomes lower than that of the smooth annulus. In lower values of kr, the conduction thermal resistance dominates, while in higher values of kr, the heat conduction and fluid convection in the porous medium become worse than that of the smooth annulus.

4.6 System Performance

In order to evaluate the system performance, a critical parameter, j/f1/3 has been formulated by heat transfer augmentation with the penalty of pressure drop, Eq. (47). The impact of porosity and foam thickness on j/f1/3 is demonstrated in Fig. 10a. The results show that when ε > 0.6, the partially filled annulus performs worse than the smooth annulus, but for the fully filled annulus and 0.6 < ε < 0.83, j/f1/3 is significantly higher than that of the smooth annulus, but then drops sharply. Figure 10b illustrates the impact of pore density on j/f1/3 for different foam thicknesses. In the range of pore density studied, all cases, even the fully filled annulus, perform worse than the smooth annulus due to low permeability. Furthermore, increasing Re number decreases the performance in each foam thickness and smooth annulus, as shown in Fig. 10c. The performance of the system deteriorates with increasing foam thickness on both walls compared to the smooth annulus because the friction factor decreases as Re number increases (Fig. 8c).

Fig. 10
figure 10

The impacts of porosity, pore density, Re number, and conductivity ratio on j/f1/3 for various foam thicknesses

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Figure 10d shows how j/f1/3 changes over a wide range of kr values. The results reveal that j/f1/3 is higher than that of a smooth annulus only for low fluid-to-solid thermal conductivity ratios (kr < 0.002). Therefore, the system with partially filled metal foam of kr > 0.002 is less effective than a smooth annulus. The critical value of kr for the fully filled annulus case is about 0.006. Overall, the findings suggest that the metallic foam partially filled annulus offers better performance than an empty annulus for a particular range of kr when considering heat transfer enhancement with the penalty of pressure drop loss.

5 Conclusion

This study aimed to investigate the effect of metallic foam on both walls of an annulus with asymmetric heat flux ratio under fully developed laminar forced convection. The numerical method used was validated with previous works for various thicknesses of partially filled metallic foam annulus on the inner wall. The results showed that foam thickness, porosity, pore density, and thermal conductivity ratio had significant impacts on flow heterogeneity coefficient, friction factor, and heat transfer.

Flow heterogeneity or foam region flow ratio was found to depend directly on permeability. Increasing porosity and decreasing pore density increased flow heterogeneity. The friction factor indicated the flow resistance in the foam region, and increasing permeability reduced the friction factor. Additionally, increasing the foam thickness increased flow heterogeneity and friction factor by increasing the area of the foam region.

Thermal performance was found to be controlled by three thermal resistances along the radial direction of the cross section. These three thermal resistances were affected by porosity, pore density, foam thickness, and thermal conductivity ratio. To evaluate the system performance, a critical parameter, j/f1/3, was defined based on heat transfer augmentation and friction factor increase. The results showed that for a fixed conductivity ratio of 0.01, all partially filled cases had lower performance than the empty annulus for a wide range of porosity, pore density, and Re number. However, the results also demonstrated that the performance was highly dependent on the conductivity ratio. For kr values below 0.002, the partially filled annulus performed better than the empty one, while for the case of the fully filled annulus, this critical kr increased to 0.006.

Overall, this study provides valuable insights into the impact of metallic foam on forced convection heat transfer and identifies the critical factors affecting the performance of metallic foam-filled annuli. In addition to the main outcomes listed above, the novelty of this work lies in the thorough investigation of the impact of metallic foam on both walls of an annulus under the influence of asymmetric heat flux ratio. The presented numerical method was accurately validated with previous works for different thicknesses of partially filled metallic foam annulus on the inner wall. The results provide valuable insights into the complex interplay between foam thickness, porosity, pore density, and thermal conductivity ratio on flow heterogeneity, friction factor, and heat transfer. This research also introduces a critical parameter, j/f1/3, for evaluating system performance based on heat transfer augmentation and friction factor increase. The findings of this study can help inform the design of heat exchangers, such as those used in power generation, chemical processing, and refrigeration systems.