Modeling and Computation of Heat Transfer through Permeable Hollow-Pin Systems

Abstract

Heat transfer inside permeable hollow-fin systems is analyzed in this work. Two types are considered: (A) the permeable hollow-pin, and (B) the permeable hollow joint-pin. The governing partial differential equations are solved numerically using a well-known implicit, iterative and finite-difference method. The numerical solutions are validated against various analytical solutions derived based on different constraints. It is found that the permeable hollow-pin can transfer more thermal energy than the solid pin when an external suction flow is present at the outer surfaces. Moreover, the maximum reported heat transfer rate due to permeable hollow-pin is 362 percent above that of solid pin at dimensionless suction flow number equals to 3.0. Furthermore, the maximum reported heat transfer rate due to permeable hollow joint-pin is 44 percent above that of the solid joint-pin at dimensionless suction flow number equals to 2.0. In addition, the permeable hollow joint-pin is found to be capable of transferring more thermal energy than the solid joint-pin at a specific joint-pin lengths ratio depending on the values of the various controlling parameters. Finally, this work demonstrates that by using combined heat transfer enhancement approaches, novel heat transfer enhancers can be proposed.

1. Introduction

The most recent reviews of heat transfer literature [1–3] show that a huge number of researches have been conducted to explore heat transfer enhancement (HTE). These researches discussed the following major HTE approaches: (a) flow profiling [4, 5] such as using twisted tapes, spirals, and wire coils, (b) fins such as slotted and louvered fins [6, 7], (c) electrohydrodynamic effects [8], (d) surface coatings [9], (e) heat transfer additives [10], (f) acoustic streaming [2], and (g) turbulators [2]. These approaches are verified to be effective in enhancing heat transfer [1–3] because they produce at least one of the following effects: introducing a secondary heat transfer surface, disrupting the fluid velocity, disrupting of the laminar sublayer in the turbulent boundary layer, introducing secondary flows, promoting flow attachment, delaying the boundary layer development, redistribution of the flow, enhancing effective thermal conductivity, thermal dispersion, and increasing the radiative properties of the surface. The detailed discussions of these mechanisms can be found in the review report given by Siddique et al. [11]. In addition to the afore mentioned HTE approaches, there are other well-known HTE approaches such as having preamble surfaces subject to suction flows [11]. The HTE approach combined between having fins and permeable surfaces subject to suction flows is the major interest of the present work.

Having preamble surfaces subject to suction flows is a well-established HTE approach in literature [12–17]. But combining this approach with fins is not a well-investigated technique in the literature. To the best knowledge of the author, three works found that they are related to permeable fins [18–20]. The first two of them [18, 19] deal with porous fins [21]. For this kind of fins, it is almost impossible to ascertain suction flows at all fin outside surfaces. As such, the HTE approach in these studies [18, 19, 21] cannot be considered as the HTE approach combined between fins and permeable surfaces subject to suction flows. In addition, the one-half of the permeable fin surfaces in the work of Khaled [20] is considered to be insulated. This is in order to assure external suction flow at the fin surface. Although the latter technique proves the possibility of enhancing heat transfer as compared to solid fin, the thermal efficiency is reduced due to elimination of half the fin surface area from exchanging heat. To avoid the last issued constraint, it is proposed to consider permeable hollow-fins confining internal fluid flows. A fin confining an internal fluid flow represents a very recent HTE approach [22, 23]. This recent tactic facilitates attainment of an external suction flow at the outside fin surfaces. This is because the external masses sucked at the fin outside surface can be transferred to the mass sink reservoir connected to fin base surface via the internal fluid flow.

In this work, heat transfer inside permeable hollow-fins subject to lateral mass transfer across their thicknesses is analyzed. Fins with round cross-sections (hollow-pins) are considered to achieve equal lateral mass transfer rate along the perimeter. Two types of these systems are analyzed: (A) typical permeable hollow-pins and (B) permeable hollow joint-pins. Momentum and energy transport equations applicable to the hollow-pin inside volume, outside volume, pores volume, and the hollow-pin solid volumes are solved numerically. Approximate analytical solutions under different constraints are obtained and compared with the numerical solutions. Various HTE factors are defined and computed as functions of the different controlling parameters. As such, HTE characteristics of the permeable hollow-fin systems are explored.

2. Theory and Problem Formulation

2.1. Modeling of Momentum and Heat Transfer inside the Permeable Hollow-Pin

Consider a permeable hollow-pin having an inner and outer diameters d _i and d _o , respectively. The length of the pin is L. Let the hollow-pin be composed of solid parts of thermal conductivity k _f and pores of diameter d _p . Consider that the solid part temperature field is T _f . The permeable hollow-pin is considered to be extended inside a fluidic reservoir having a surrounding temperature T _R and a far stream pressure P _∞ . The external fluid pressure near the outer pin surface is denoted by P _o . The hollow-pin is attached at its base to a solid surface of temperature T _b which is open to another fluidic reservoir. The pin tip temperature is T _L . The x-axis is taken along the pin centerline starting from the tip and directing towards the base surface as shown in Figure 1. The r-axis is taken along the pin radial direction starting from the pin centerline (see Figure 1). The continuity, momentum, and energy equations for the induced flow inside the hollow-pin have the following dimensionless forms:

$\begin{array}{}\frac{\partial \stackrel{-}{u}}{\partial \stackrel{-}{x}}+\frac{\mathrm{1}}{\stackrel{-}{r}}\frac{\partial }{\partial \stackrel{-}{r}}\left(\stackrel{-}{r}\stackrel{-}{v}\right)=\mathrm{0},\end{array}$

(1)

$\begin{array}{}\stackrel{-}{u}\frac{\partial \stackrel{-}{u}}{\partial \stackrel{-}{x}}+\stackrel{-}{v}\frac{\partial \stackrel{-}{u}}{\partial \stackrel{-}{r}}=\frac{\mathrm{16}}{a\mathrm{Re}}\frac{d{\stackrel{-}{P}}_i}{d\stackrel{-}{x}}+\frac{\mathrm{2}}{a\mathrm{Re}}\left(\frac{\mathrm{1}}{\stackrel{-}{r}}\right)\frac{\partial }{\partial \stackrel{-}{r}}\left(\stackrel{-}{r}\frac{\partial \stackrel{-}{u}}{\partial \stackrel{-}{r}}\right),\end{array}$

(2)

$\begin{array}{}\stackrel{-}{u}\frac{\partial \theta }{\partial \stackrel{-}{x}}+\stackrel{-}{v}\frac{\partial \theta }{\partial \stackrel{-}{r}}=\frac{\mathrm{2}}{a\text{Pe}}\left(\frac{\mathrm{1}}{\stackrel{-}{r}}\right)\frac{\partial }{\partial \stackrel{-}{r}}\left(\stackrel{-}{r}\frac{\partial \theta }{\partial \stackrel{-}{r}}\right),\end{array}$

(3)

where

$\begin{array}{}\stackrel{-}{x}=\frac{x}{L},\end{array}$

(4a)

$\begin{array}{}\stackrel{-}{r}=\frac{\mathrm{2}r}{d_i},\end{array}$

(4b)

$\begin{array}{}\stackrel{-}{u}\left(\stackrel{-}{x},\stackrel{-}{r}\right)=\frac{u\left(x,r\right)}{u_o},\end{array}$

(4c)

$\begin{array}{}\stackrel{-}{v}\left(\stackrel{-}{x},\stackrel{-}{r}\right)=\frac{v\left(x,r\right)}{a{u}_o},\end{array}$

(4d)

$\begin{array}{}{\stackrel{-}{P}}_i\left(\stackrel{-}{x}\right)=\frac{P_i\left(x\right)-P_{\infty }}{P_b-P_{\infty }},\end{array}$

(4e)

$\begin{array}{}{u}_o=\left(\frac{\mathrm{1}}{\mathrm{32}}\right)\left(\frac{d_i^{\mathrm{2}}}{\mu L}\right)\left(P_{\infty }-P_b\right),\end{array}$

(4f)

$\begin{array}{}\theta \left(\stackrel{-}{x},\stackrel{-}{r}\right)=\frac{T\left(x,r\right)-T_R}{T_b-T_R},\end{array}$

(4g)

$\begin{array}{}\mathrm{Re}=\frac{\rho u_o{d}_i}{\mu },\end{array}$

(4h)

$\begin{array}{}a=\frac{d_i}{\mathrm{2}L},\end{array}$

(4i)

$\begin{array}{}\text{Pe}=\mathrm{Re}\text{Pr}\end{array}$

(4j)

$P_i(x)$ , P _b , u, u _o , v, T, ρ, μ, and Pr are the fluid pressure inside the hollow-pin, the internal fluid pressure at the pin base, axial velocity component of the internal flow, reference axial velocity, normal velocity component of the internal flow, inner fluid temperature field, fluid density, fluid dynamic viscosity, and the Prandtl number, respectively. The boundary conditions of (1)–(3) are given by

$\begin{array}{}\stackrel{-}{r}=\mathrm{0}:\frac{\partial \stackrel{-}{u}}{\partial \stackrel{-}{r}}=\mathrm{0},\end{array}$

(5a)

$\begin{array}{}\frac{\partial \theta }{\partial \stackrel{-}{r}}=\mathrm{0},\end{array}$

(5b)

$\begin{array}{}\stackrel{-}{v}=\mathrm{0},\end{array}$

(5c)

$\begin{array}{}\stackrel{-}{r}=\mathrm{1}:\stackrel{-}{u}=\mathrm{0},\end{array}$

(5d)

$\begin{array}{}\theta ={\theta }_f\left(\stackrel{-}{x}\right),\end{array}$

(5e)

where θ _f is the dimensionless temperature field of the solid part, ${\theta }_f=(T_f-T_R)/(T_b-T_R)$ . The inlet thermal condition of (3) is given by

$\begin{array}{}\stackrel{-}{x}=\mathrm{0}:\theta ={\theta }_L,\hspace{1em}\text{Pe}>\mathrm{0},\end{array}$

(6a)

$\begin{array}{}\stackrel{-}{x}=\mathrm{1}:\theta =\mathrm{1},\hspace{1em}\text{Pe}<\mathrm{0},\end{array}$

(6b)

where ${\theta }_L=(T_L-T_R)/(T_b-T_R)$ . For situations where $a\mathrm{Re}\ll 1$ as for micropins applications, the left side of (2) can be ignored. As such, the solutions of (1) and (2) are given by

$\begin{array}{}\stackrel{-}{u}\left(\stackrel{-}{x},\stackrel{-}{r}\right)=\mathrm{2}{\stackrel{-}{P}}_i^{\prime }\left(\mathrm{1}-{\stackrel{-}{r}}^{\mathrm{2}}\right),\end{array}$

(7)

$\begin{array}{}\stackrel{-}{v}\left(\stackrel{-}{x},\stackrel{-}{r}\right)=-{\stackrel{-}{P}}_i^{\prime \prime }\stackrel{-}{r}\left(\mathrm{1}-\frac{{\stackrel{-}{r}}^{\mathrm{2}}}{\mathrm{2}}\right),\end{array}$

(8)

where ${\stackrel{-}{P}}_i^{\prime }=d{\stackrel{-}{P}}_i/d\stackrel{-}{x}$ and ${\stackrel{-}{P}}_i^{\prime \prime }=d^{\mathrm{2}}{\stackrel{-}{P}}_i/d{\stackrel{-}{x}}^{\mathrm{2}}$ . By substituting (7) and (8) in (2), it is noticed that (7) and (8) are valid when $a\mathrm{Re}{\stackrel{-}{P}}_i^{\prime \prime }\ll \mathrm{8.0}$ . The inner Nusselt number is equal to

$\begin{array}{}{\text{Nu}}_i\equiv \frac{h_i{d}_i}{k}=\left\{\frac{\mathrm{2}}{{\theta }_f-{\theta }_m}\right\}\left({\frac{\partial \theta }{\partial \stackrel{-}{r}}|}_{\stackrel{-}{r}=\mathrm{1}}\right),\end{array}$

(9)

where h _i and θ _m are the inner convection heat transfer coefficient $(h_i=q_i^{\mathrm{\prime \prime }}/\lfloor T_f-T_m\rfloor )$ and the inner fluid dimensionless mean bulk temperature, respectively. $q_i^{\mathrm{\prime \prime }}$ is the local heat flux between the pin inner surface and the internal flowing fluid. T _m is the mean bulk temperature at a given cross-section. θ _m can be computed from the following equation:

$\begin{array}{}{\theta }_m\left(\stackrel{-}{x}\right)\equiv \frac{T_m\left(x\right)-T_R}{T_b-T_R}=\mathrm{4}{\int }_{\mathrm{0}}^{\mathrm{1}}\left(\mathrm{1}-{\stackrel{-}{r}}^{\mathrm{2}}\right)\stackrel{-}{r}\theta d\stackrel{-}{r}.\end{array}$

(10)

Figure 1:

Schematic diagram for a permeable hollow-pin with suction flow at the outer surface $(\text{Pe}>\mathrm{0})$ and the corresponding coordinates system.

2.1.1. Integral Energy Equation of the Internal Fluid Flow

The integral form of (3) can be expressed in the following form:

$\begin{array}{}\frac{\mathrm{1}}{\mathrm{2}}{\stackrel{-}{P}}_i^{\prime }\frac{d{\theta }_m}{d\stackrel{-}{x}}=\left[\frac{\mathrm{1}}{\mathrm{2}}{\stackrel{-}{P}}_i^{\prime \prime }+\frac{{\text{Nu}}_i}{a\text{Pe}}\right]\left({\theta }_f-{\theta }_m\right).\end{array}$

(11)

The inlet condition of (11) is given by

$\begin{array}{}\stackrel{-}{x}=\mathrm{0}:{\theta }_m={\theta }_L,\hspace{1em}\text{Pe}>\mathrm{0},\end{array}$

(12a)

$\begin{array}{}\stackrel{-}{x}=\mathrm{1}:{\theta }_m=\mathrm{1},\hspace{1em}\text{Pe}<\mathrm{0}.\end{array}$

(12b)

2.1.2. Approximate Expressions of Nu _i

Under Uniform Heat Flux (UHF) condition at the pin inner surface, (3) under fully developed condition can be approximated by

$\begin{array}{}-\left(\frac{\stackrel{-}{u}}{{\theta }_f-{\theta }_m}\right)\frac{d{\theta }_m}{d\stackrel{-}{x}}+\stackrel{-}{v}{G}^{\prime }=\frac{\mathrm{2}}{a\text{Pe}}\left(\frac{\mathrm{1}}{\stackrel{-}{r}}\right)\frac{d}{d\stackrel{-}{r}}\left(\stackrel{-}{r}{G}^{\prime }\right),\end{array}$

(13)

where $G=({\theta }_f-\theta )/({\theta }_f-{\theta }_m)$ . If the initial estimate of $G^{(\mathrm{1})}$ is taken as $G^{(\mathrm{1})}=\mathrm{1}$ , then the fifth iteration resulted from solving (13) produces the following approximation of the Nusselt number:

$\begin{array}{}{\text{Nu}}_i\cong \frac{\mathrm{48}}{\mathrm{11}}-\mathrm{0.20744}\beta +\mathrm{5.4141}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{3}}{\beta }^{\mathrm{2}}-\mathrm{4.5450}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{4}}{\beta }^{\mathrm{3}}-\mathrm{1.4392}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{6}}{\beta }^{\mathrm{4}},\hspace{1em}\left(\left|\beta \right|\le \mathrm{9.0},\text{UHF}\right),\end{array}$

(14)

where $\beta =a\text{Pe}{\stackrel{-}{P}}_i^{\prime \prime }$ . When the pin inner surface has constant surface temperature denoted by CWT-condition, (3) at fully developed condition is approximated by

$\begin{array}{}-\stackrel{-}{u}\frac{d{\theta }_m}{d\stackrel{-}{x}}G+\stackrel{-}{v}{G}^{\prime }=\frac{\mathrm{2}}{a\text{Pe}}\left(\frac{\mathrm{1}}{\stackrel{-}{r}}\right)\frac{d}{d\stackrel{-}{r}}\left(\stackrel{-}{r}{G}^{\prime }\right).\end{array}$

(15)

If the initial estimate of $G^{(\mathrm{1})}$ is taken as $G^{(\mathrm{1})}=\mathrm{1}$ , then the fourth iteration obtained from solving (15) reveals the following Nusselt number correlation:

$\begin{array}{}{\text{Nu}}_i\cong \frac{\mathrm{3.6595}-\mathrm{0.21435}\beta +\mathrm{3.0851}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{3}}{\beta }^{\mathrm{2}}+\mathrm{3.4241}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{5}}{\beta }^{\mathrm{3}}+\mathrm{6.7755}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{7}}{\beta }^{\mathrm{4}}}{\mathrm{1.0}+\mathrm{9.7937}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{3}}\beta +\mathrm{6.2081}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{5}}{\beta }^{\mathrm{2}}+\mathrm{9.9404}\times \mathrm{1}{\mathrm{0}}^{-\mathrm{7}}{\beta }^{\mathrm{3}}},\hspace{1em}\left(\left|\beta \right|\le \mathrm{11},\text{CWT}\right).\end{array}$

(16)

2.2. Modeling of Momentum and Heat Transfer outside the Permeable Hollow-Pin

The continuity, momentum, and energy equations for the induced flow outside the pin have the following dimensionless forms:

$\begin{array}{}\frac{\mathrm{1}}{r^{*}}\frac{d}{d{r}^{*}}\left(r^{*}{v}^{*}\right)=\mathrm{0},\end{array}$

(17)

$\begin{array}{}\mathrm{2}{v}^{*}\frac{d{v}^{*}}{d{r}^{*}}=-\frac{d{P}_o^{*}}{d{r}^{*}},\end{array}$

(18)

$\begin{array}{}-v^{*}\frac{d{\theta }^{*}}{d{r}^{*}}=\frac{\mathrm{2}}{{\text{Pe}}_v}\left(\frac{\mathrm{1}}{r^{*}}\right)\frac{d}{d{r}^{*}}\left(r^{*}\frac{d{\theta }^{*}}{d{r}^{*}}\right),\end{array}$

(19)

where

$\begin{array}{}{r}^{*}=\frac{\mathrm{2}r}{d_o},\end{array}$

(20a)

$\begin{array}{}{v}^{*}\left(\stackrel{-}{x},r^{*}\right)=\frac{v_o\left(x,r\right)}{v_m\left(x\right)},\end{array}$

(20b)

$\begin{array}{}{P}_o^{*}\left(\stackrel{-}{x},r^{*}\right)=\frac{P_o\left(x,r\right)-P_{\infty }}{\rho {\left[v_m\left(x\right)\right]}^{\mathrm{2}}/\mathrm{2}},\end{array}$

(20c)

$\begin{array}{}{\theta }^{*}\left(\stackrel{-}{x},r^{*}\right)=\frac{T_o\left(x,r\right)-T_R}{T_b-T_R},\end{array}$

(20d)

$\begin{array}{}{\text{Pe}}_v=\frac{\rho v_m\left(x\right)d_o}{\mu }\text{Pr},\end{array}$

(20e)

where v _m , v _o , and T _o are the fluid averaged normal velocity at the pin outer surface ( $v_m>\mathrm{0}$ with external suction flow), outer stream normal velocity component, and the outer fluid temperature field, respectively.

The boundary conditions of (17) and (18) are given by

$\begin{array}{}{r}^{*}=\mathrm{1}:v^{*}=\mathrm{1},\end{array}$

(21a)

$\begin{array}{}{r}^{*}\to \infty :v^{*}=\mathrm{0},\end{array}$

(21b)

$\begin{array}{}{P}_o^{*}=\mathrm{0},\end{array}$

(21c)

when ${\text{Pe}}_v>\mathrm{0}$ , the outer fluid is being sucked into the permeable hollow-pin internal fluidic volume. However, the inner fluid is blown to the permeable pin external fluidic medium when ${\text{Pe}}_v<\mathrm{0}$ . The boundary conditions (19) are taken to be

$\begin{array}{}{r}^{*}=\mathrm{1}:{\theta }^{*}={\theta }_f\left(\stackrel{-}{x}\right),\end{array}$

(22a)

$\begin{array}{}{r}^{*}=\frac{D}{d_o}:{\theta }^{*}=\mathrm{0},\end{array}$

(22b)

where D is the average diameter of the surrounding boundary. Equations (21a), (21b), (21c), (22a), and (22b) necessitate that the boundaries of the surroundings are isothermal and permeable. The solutions of (17)–(19) are given by

$\begin{array}{c}{v}_o\left(x,r\right)=v_m\left(x\right)\left(\frac{d_o}{\mathrm{2}r}\right),\hspace{1em}r\ge \frac{d_o}{\mathrm{2}},P_o\left(x,r\right)=P_{\infty }-\rho \frac{{\left[v_m\left(x\right)\right]}^{\mathrm{2}}}{\mathrm{2}}{\left(\frac{d_o}{\mathrm{2}r}\right)}^{\mathrm{2}},\hspace{1em}r\ge \frac{d_o}{\mathrm{2}},\\ \frac{T_o\left(x\right)-T_{\infty }}{T_f\left(x\right)-T_{\infty }}=\frac{{\left(\mathrm{2}r/D\right)}^{-\text{R}{\text{e}}_v\text{Pr}/\mathrm{2}}-\mathrm{1}}{{\left(d_o/D\right)}^{-\text{R}{\text{e}}_v\text{Pr}/\mathrm{2}}-\mathrm{1}},\hspace{1em}r\ge \frac{d_o}{\mathrm{2}}.\end{array}$

(23)

The outer convection heat transfer coefficient h _o is defined as $h_o=q_o^{\mathrm{\prime \prime }}/(T_f-T_{\infty })$ where $q_o^{\mathrm{\prime \prime }}$ is the local heat flux between the pin outer surface and the external fluid flow. It can be obtained using the outer Nusselt number relationship given by

$\begin{array}{}{\text{Nu}}_o\equiv \frac{h_o{d}_o}{k}=-{\mathrm{2}\frac{\partial \left({\theta }^{*}/{\theta }_f\right)}{\partial r^{*}}|}_{r^{*}=\mathrm{1}}=\frac{{\text{Pe}}_v}{\mathrm{1}-{\left(d_o/D\right)}^{{\text{Pe}}_v/\mathrm{2}}}.\end{array}$

(24)

2.3. Modeling of the Internal Flow Pressure Gradient

Assuming Poiseuille's flow [24] inside the pin pores, the mean velocity inside the pores denoted by v _p is given by

$\begin{array}{}{v}_p=\left(\frac{\mathrm{1}}{\mathrm{16}}\right)\left(\frac{d_p^{\mathrm{2}}}{\mu }\right)\left\{\frac{{\left(P_o\right)}_{r^{*}=\mathrm{1}}-P_i}{d_o-d_i}\right\}.\end{array}$

(25)

The mean velocity inside the pores averaged over the outer surface area is equal to

$\begin{array}{}{v}_m\equiv v_p\phi \left(\frac{d_i}{d_o}\right)=u_o\left[\frac{\mathrm{2}\phi d_p^{\mathrm{2}}L}{d_i{d}_o\left(d_o-d_i\right)}\right]\left\{\frac{\rho v_{mo}^{\mathrm{2}}}{\mathrm{4}\left(P_b-P_{\infty }\right)}+{\stackrel{-}{P}}_i\right\},\end{array}$

(26)

where $\phi =A_{C,p}/(\pi d_{i}L)$ is the ratio of the pin pores cross-sectional area to total hollow-pin inner surface area. In a dimensionless form, (26) can be rewritten as

$\begin{array}{}\sqrt{\frac{Z_P}{a\mathrm{Re}}}=a\left(\frac{d_i}{d_o}\right)\frac{{\left(nL\right)}^{\mathrm{2}}}{\mathrm{16}}\left[{\stackrel{-}{P}}_i-Z_P\right],\end{array}$

(27)

where Z _P and n (index number of the induced flow) are equal to

$\begin{array}{}{Z}_P=\frac{\rho v_m^{\mathrm{2}}}{\mathrm{4}\left(P_{\infty }-P_b\right)},\end{array}$

(28a)

$\begin{array}{}n=\mathrm{2}\frac{d_p}{d_i^{\mathrm{2}}}\sqrt{\frac{\mathrm{2}\phi \left(d_i/d_o\right)}{\mathrm{1}-\left(d_i/d_o\right)}}.\end{array}$

(28b)

By solving (27), Z _P can be found. It is equal to

$\begin{array}{}{Z}_P={\stackrel{-}{P}}_i+\frac{\mathrm{128}{\left(d_o/d_i\right)}^{\mathrm{2}}}{{\left(nL\right)}^{\mathrm{4}}{a}^{\mathrm{3}}\mathrm{Re}}-\frac{\mathrm{128}{\left(d_o/d_i\right)}^{\mathrm{2}}}{{\left(nL\right)}^{\mathrm{4}}{a}^{\mathrm{3}}\mathrm{Re}}\sqrt{\mathrm{1}+\frac{{\left(nL\right)}^{\mathrm{4}}{a}^{\mathrm{3}}\mathrm{Re}}{\mathrm{64}{\left(d_o/d_i\right)}^{\mathrm{2}}}{\stackrel{-}{P}}_i}.\end{array}$

(29)

which can be approximated by the following expression:

$\begin{array}{}{Z}_P=\frac{{\left(nL\right)}^{\mathrm{4}}{a}^{\mathrm{3}}\mathrm{Re}}{\mathrm{256}}{\left(\frac{d_i}{d_o}\right)}^{\mathrm{2}}{\stackrel{-}{P}}_i^{\mathrm{2}}.\end{array}$

(30)

Equation (30) is obtained by eliminating the fourth term and above of the MacLaurin series [25] of the square root term, $\sqrt{\mathrm{1}+z}\cong \mathrm{1}+(\mathrm{1}/\mathrm{2})z-(\mathrm{1}/\mathrm{8})z^{\mathrm{2}}$ . By solving (30) for v _m , the following expression can be obtained:

$\begin{array}{}{\text{Pe}}_v\left(\stackrel{-}{x}\right)=a\text{Pe}\frac{{\left(nL\right)}^{\mathrm{2}}}{\mathrm{2}}{\stackrel{-}{P}}_i\left(\stackrel{-}{x}\right).\end{array}$

(31)

The gradient of the mass flow rate inside the pin can be shown to be equal to the following using the conservation of mass principle as follows:

$\begin{array}{}\frac{d{\dot{m}}_i}{dx}=\mathrm{2}\pi \rho \frac{d}{dx}\left({\int }_{\mathrm{0}}^{d_i/\mathrm{2}}rudr\right)=\rho \phi v_p\pi d_i.\end{array}$

(32)

By utilizing (7) and (27) and the definition of the n-index, (32) can be reduced to the following:

$\begin{array}{}{\stackrel{-}{P}}_i^{\mathrm{\prime \prime }}-{\left(nL\right)}^{\mathrm{2}}{\stackrel{-}{P}}_i=\mathrm{0}.\end{array}$

(33)

The following boundary conditions of (33) are considered:

$\begin{array}{}\stackrel{-}{x}=\mathrm{0}:\frac{d{\stackrel{-}{P}}_i}{d\stackrel{-}{x}}=\mathrm{0},\end{array}$

(34a)

$\begin{array}{}\stackrel{-}{x}=\mathrm{1}:{\stackrel{-}{P}}_i=\mathrm{1}.\end{array}$

(34b)

The boundary condition given by (34a) necessitates that the pin tip is isolated against mass flux. As such, the solution of (33) is equal to

$\begin{array}{}{\stackrel{-}{P}}_i\left(\stackrel{-}{x}\right)=\frac{\cosh \left(nL\stackrel{-}{x}\right)}{\cosh \left(nL\right)}.\end{array}$

(35)

The condition for validity of (7) and (8) changes to $a\mathrm{Re}{(nL)}^{\mathrm{2}}\ll \mathrm{8.0}$ . The total mass flow rate at the pin base is calculated from the following equation:

$\begin{array}{}{{\dot{m}}_i|}_{\stackrel{-}{x}=\mathrm{1}}=\mathrm{2}\pi \rho {\left({\int }_{\mathrm{0}}^{d_i/\mathrm{2}}rudr\right)|}_{\stackrel{-}{x}=\mathrm{1}}=\left(\frac{\pi }{\mathrm{4}}\right)\rho u_o{d}_i^{\mathrm{2}}\left(nL\right)\tanh \left(nL\right).\end{array}$

(36)

The effective length, L _∞ , of the pin that causes the mass flow rate at the base to be 99% of the maximum possible mass flow rate, ${{\dot{m}}_i|}_{\stackrel{-}{x}=\mathrm{1},L\to \infty }$ , is obtained by solving the following equation:

$\begin{array}{}\frac{{{\dot{m}}_i|}_{x=L_{\infty }}}{{{\dot{m}}_i|}_{x=L\to \infty }}=\tanh \left(n{L}_{\infty }\right)=\mathrm{0.99}⟶L_{\infty }=\frac{\mathrm{2.65}}{n}.\end{array}$

(37)

2.4. Modeling of Heat Transfer through the Solid Part of the Permeable Hollow-Pin

The one-dimensional energy equation for the solid part of the hollow-pin can be written as

$\begin{array}{}\frac{d{q}_x}{dx}+\frac{d{q}_{\text{conv}.,o}}{dx}+\frac{d{q}_{\text{conv}.,i}}{dx}+\frac{d{q}_{\text{conv}.,p}}{dx}=\mathrm{0}.\end{array}$

(38)

The axial conduction heat transfer rate through the solid part of the hollow-pin can be shown to be equal to

$\begin{array}{}{q}_x=-k_f\frac{\pi }{\mathrm{4}}\left(d_o^{\mathrm{2}}-d_i^{\mathrm{2}}\right)\left[\mathrm{1}-\frac{\mathrm{2}\left(d_i/d_o\right)\phi }{\mathrm{1}+\left(d_i/d_o\right)}\right]\frac{d{T}_f}{dx}.\end{array}$

(39)

Note that the conduction heat transfer rate across the pores is neglected compared to the conduction rate through the solid part of the hollow-pin as $k_f\gg k$ . The second term in the square bracket of (39) represents the ratio of pores volume to the total volume of the permeable hollow-pin. The convection heat transfer rate per unit length between the hollow-pin outer surface and external fluid flow is equal to

$\begin{array}{}\frac{d{q}_{\text{conv}.,o}}{dx}=h_o\left[\mathrm{1}-\phi \frac{d_i}{d_o}\right]\pi d_o\left(T_f-T_R\right).\end{array}$

(40)

The convection heat transfer rate per unit length between the hollow-pin inner surface and internal fluid flow is given by

$\begin{array}{}\frac{d{q}_{\text{conv}.,i}}{dx}=h_i\left(\mathrm{1}-\phi \right)\pi d_i\left(T_f-T_m\right).\end{array}$

(41)

The convection heat transfer rate per unit length between the pores solid surfaces and the fluid flow inside the pores can be shown to be equal to

$\begin{array}{}\frac{d{q}_{\text{conv}.,p}}{dx}=h_o\phi \left(\frac{d_i}{d_o}\right)\pi d_o\left(T_f-T_R\right)+h_i\phi \pi d_i\left(T_f-T_m\right).\end{array}$

(42)

By utilizing (39)–(42), (38) can be expressed in dimensionless form as

$\begin{array}{}\text{(43a)}& \left(\frac{a^{\mathrm{2}}\Pi }{k/k_f}\right){\theta }_f^{\mathrm{\prime \prime }}-{\text{Nu}}_i\left({\theta }_f-{\theta }_m\right)-{\text{Nu}}_o{\theta }_f=\mathrm{0},\end{array}$

(43a)

where $\Pi ={(d_o/d_i)}^{\mathrm{2}}-\mathrm{1}-\mathrm{2}\phi [(d_o/d_i)-\mathrm{1}]$ . By utilizing (11), (43a) changes to:

$\begin{array}{}\text{(43b)}& \left(\frac{a^{\mathrm{2}}\Pi }{k/k_f}\right){\theta }_f^{\mathrm{\prime \prime }}-\left(\frac{a\text{Pe}{\stackrel{-}{P}}_i^{\prime }{\text{Nu}}_i}{a\text{Pe}{\stackrel{-}{P}}_i^{\mathrm{\prime \prime }}+\mathrm{2}{\text{Nu}}_i}\right)\frac{d{\theta }_m}{d\stackrel{-}{x}}-{\text{Nu}}_o{\theta }_f=\mathrm{0}.\end{array}$

(43b)

The boundary conditions for (43a) and (43b) are taken as:

$\begin{array}{c}\stackrel{-}{x}=\mathrm{0}:\frac{d{\theta }_f}{d\stackrel{-}{x}}=\mathrm{0},\stackrel{-}{x}=\mathrm{1}:{\theta }_f=\mathrm{1}.\end{array}$

(44)

2.4.1. Constraint Averaged Energy Equation of the Permeable Hollow-Pin

For slender hollow-pin $(\mathrm{2}{\text{Nu}}_i/a\text{Pe}{\stackrel{-}{P}}_i^{\mathrm{\prime \prime }}\gg \mathrm{1})$ , (43b) with coefficients evaluated at their averaged independent variables is reducible to

$\begin{array}{c}\left(\frac{a^{\mathrm{2}}\Pi }{k/k_f}\right){\theta }_f^{\mathrm{\prime \prime }}-\frac{\mathrm{1}}{\mathrm{2}}a\text{Pe}{\stackrel{-}{P}}_{i,\text{avg}}^{\prime }{\theta }_f^{\prime }-{\stackrel{-}{\text{Nu}}}_o{\theta }_f=\mathrm{0},{\stackrel{-}{P}}_{i,\text{avg}}^{\prime }=\{\begin{array}{cc}\mathrm{1}-\text{sech}\left(nL\right),\hfill & \text{Pe}>\mathrm{0},\hfill \\ \text{sech}\left(nL\right)-\mathrm{1},\hfill & \text{Pe}<\mathrm{0},\hfill \end{array}\\ {\stackrel{-}{\text{Nu}}}_o=\frac{P_{v,\text{avg}}^{}}{\left(\mathrm{1}-{\left[d_o/D\right]}^{{\text{Pe}}_{v,\text{avg}}/\mathrm{2}}\right)},\\ {\text{Pe}}_{v,\text{avg}}=\frac{a\text{Pe}}{\mathrm{2}}\left(nL\right)\tanh \left(nL\right).\end{array}$

(45)

Based on correlations given by (14) and (16), the previous constraint requires that $a\text{Pe}(nL)\ll \mathrm{5.0}$ . The solution of (45) under conditions given by (44) is

$\begin{array}{c}\frac{T_f-T_R}{T_b-T_R}=\exp \left(s\left[\stackrel{-}{x}-\mathrm{1}\right]\right)\left\{\frac{\cosh \left(w\stackrel{-}{x}\right)-\left(s/w\right)\sinh \left(w\stackrel{-}{x}\right)}{\cosh \left(w\right)-\left(s/w\right)\sinh \left(w\right)}\right\},s=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{k}{k_f}\right)\left(\frac{\text{Pe}}{a\Pi }\right){\stackrel{-}{P}}_{i,\text{avg}}^{\prime },\hspace{1em}\hspace{1em}w=\sqrt{s^{\mathrm{2}}+{\stackrel{-}{\text{Nu}}}_o\frac{\left(k/k_f\right)}{\left(a^2\Pi \right)}}.\end{array}$

(46)

2.5. The Permeable Hollow Joint-Pin

Consider that the permeable hollow-pin discussed in Section 2.1 is joining two different reservoirs of same fluid as shown in Figure 2. The left reservoir has an external free stream pressure $p_{\infty \mathrm{1}}$ and surrounding temperature $T_{R\mathrm{1}}$ while the external free stream pressure and the surrounding temperature of the right reservoir are $p_{\infty \mathrm{2}}$ and $T_{R\mathrm{2}}$ , respectively. The energy equations for the resulting permeable hollow joint-pin system are

$\begin{array}{}\stackrel{-}{u}\frac{\partial {\theta }_{\mathrm{1,2}}}{\partial {\stackrel{-}{x}}_{\mathrm{1,2}}}+\stackrel{-}{v}\frac{\partial {\theta }_{\mathrm{1,2}}}{\partial \stackrel{-}{r}}=\frac{\mathrm{2}}{a{\text{Pe}}_{\mathrm{1,2}}}\left(\frac{\mathrm{1}}{\stackrel{-}{r}}\right)\frac{\partial }{\partial \stackrel{-}{r}}\left(\stackrel{-}{r}\frac{\partial {\theta }_{\mathrm{1,2}}}{\partial \stackrel{-}{r}}\right),\end{array}$

(47)

$\begin{array}{}\left(\frac{a_{\mathrm{1}}^{\mathrm{2}}{\Pi }_{\mathrm{1}}}{k/k_f}\right){\theta }_{f\mathrm{1}}^{\mathrm{\prime \prime }}-{\text{Nu}}_i\left({\theta }_{f\mathrm{1}}-{\theta }_{m\mathrm{1}}\right)-{\text{Nu}}_{o\mathrm{1}}{\theta }_{f\mathrm{1}}=\mathrm{0},\end{array}$

(48)

$\begin{array}{}\left(\frac{a_{\mathrm{2}}^{\mathrm{2}}{\Pi }_{\mathrm{1}}}{k/k_f}\right){\theta }_{f\mathrm{2}}^{\mathrm{\prime \prime }}-{\text{Nu}}_i\left({\theta }_{f\mathrm{2}}-{\theta }_{m\mathrm{2}}\right)-{\text{Nu}}_{o\mathrm{2}}{\theta }_{f\mathrm{2}}=\mathrm{0},\end{array}$

(49)

where the dimensionless variables ${\stackrel{-}{x}}_{\mathrm{1}}$ , ${\stackrel{-}{x}}_{\mathrm{2}}$ , ${\theta }_{\mathrm{1}}$ , ${\theta }_{\mathrm{2}}$ , ${\theta }_{f\mathrm{1}}$ , and ${\theta }_{f\mathrm{2}}$ are given by

$\begin{array}{}{\stackrel{-}{x}}_{\mathrm{1}}=\frac{x}{L_{\mathrm{1}}},\end{array}$

(50a)

$\begin{array}{}{\stackrel{-}{x}}_{\mathrm{2}}=\frac{x}{L_{\mathrm{2}}},\end{array}$

(50b)

$\begin{array}{}{\theta }_{\mathrm{1}}=\frac{T_{\mathrm{1}}-T_{R\mathrm{1}}}{T_{R\mathrm{2}}-T_{R\mathrm{1}}},\end{array}$

(50c)

$\begin{array}{}{\theta }_{\mathrm{2}}=\frac{T_{\mathrm{2}}-T_{R\mathrm{2}}}{T_{R\mathrm{1}}-T_{R\mathrm{2}}},\end{array}$

(50d)

$\begin{array}{}{\theta }_{f\mathrm{1}}=\frac{T_{f\mathrm{1}}-T_{R\mathrm{1}}}{T_{R\mathrm{2}}-T_{R\mathrm{1}}},\end{array}$

(50e)

$\begin{array}{}{\theta }_{f\mathrm{2}}=\frac{T_{f\mathrm{2}}-T_{R\mathrm{2}}}{T_{R\mathrm{1}}-T_{R\mathrm{2}}},\end{array}$

(50f)

$L_{\mathrm{1}}$ , $L_{\mathrm{2}}$ , $T_{\mathrm{1}}$ , and $T_{\mathrm{2}}$ are the left side joint-pin length, right side joint-pin length, left side inner fluid temperature, and the right side inner fluid temperature, respectively. $T_{f\mathrm{1}}$ and $T_{f\mathrm{2}}$ are the left side solid part temperature and the right side solid part temperature, respectively. By using the conservation of mass principle and (36), ${\text{Pe}}_{\mathrm{1}}$ , ${\text{Pe}}_{\mathrm{2}}$ , ${({\text{Pe}}_{v\mathrm{1}})}_{\text{avg}}$ , ${({\text{Pe}}_{v\mathrm{2}})}_{\text{avg}}$ , and the total mass flow rate across the permeable hollow joint-pin can be shown to be equal to

$\begin{array}{}\frac{{\text{Pe}}_{\mathrm{1}}}{{\text{Pe}}_t}=\frac{\cosh \left(n{L}_{\mathrm{1}}\right)\sinh \left(n\left[L-L_{\mathrm{1}}\right]\right)}{\left(L_{\mathrm{1}}/L\right)\sinh \left(nL\right)},\end{array}$

(51a)

$\begin{array}{}\frac{{\text{Pe}}_{\mathrm{2}}}{{\text{Pe}}_t}=-\frac{\sinh \left(n{L}_{\mathrm{1}}\right)\cosh \left(n\left[L-L_{\mathrm{1}}\right]\right)}{\left[\mathrm{1}-\left(L_{\mathrm{1}}/L\right)\right]\sinh \left(nL\right)},\end{array}$

(51b)

$\begin{array}{}{\text{Pe}}_{v\mathrm{1},\text{avg}}=\left(nL\right)\frac{\sinh \left(n{L}_{\mathrm{1}}\right)\sinh \left(n\left[L-L_{\mathrm{1}}\right]\right)}{\mathrm{2}\left(L_{\mathrm{1}}/L\right)\sinh \left(nL\right)}a{\text{Pe}}_t,\end{array}$

(51c)

$\begin{array}{}{\text{Pe}}_{v\mathrm{2},\text{avg}}=-\left(nL\right)\frac{\sinh \left(n{L}_{\mathrm{1}}\right)\sinh \left(n\left[L-L_{\mathrm{1}}\right]\right)}{\mathrm{2}\left[\mathrm{1}-\left(L_{\mathrm{1}}/L\right)\right]\sinh \left(nL\right)}a{\text{Pe}}_t,\end{array}$

(51d)

$\begin{array}{}{{\dot{m}}_i|}_{{\stackrel{-}{x}}_{\mathrm{1}}=\mathrm{1}}=\frac{\pi }{\mathrm{4}}\mu d_i\text{R}{\text{e}}_t\left(nL\right)\left\{\coth \left(n{L}_{\mathrm{1}}\right)-\coth \left(nL\right)\right\}\times {\left[\sinh \left(n{L}_{\mathrm{1}}\right)\right]}^{\mathrm{2}},\end{array}$

(51e)

where ${\text{Pe}}_t$ and Re _t are defined as

$\begin{array}{}{\text{Pe}}_t=\text{R}{\text{e}}_t\text{Pr},\end{array}$

(51f)

$\begin{array}{}\text{R}{\text{e}}_t=\frac{\rho u_t{d}_i}{\mu },\end{array}$

(51g)

$\begin{array}{}{u}_t=\left(\frac{\mathrm{1}}{\mathrm{32}}\right)\left(\frac{d_i^{\mathrm{2}}}{\mu L}\right)\left(P_{\infty \mathrm{1}}-P_{\infty \mathrm{2}}\right),\end{array}$

(51h)

${\text{Nu}}_{o\mathrm{1}}$ and ${\text{Nu}}_{o\mathrm{2}}$ are obtained from (24) by replacing $(\text{Pe},L)$ with $({\text{Pe}}_{\mathrm{1}},L_{\mathrm{1}})$ and $({\text{Pe}}_{\mathrm{2}},L_{\mathrm{2}})$ , respectively. Representative values of outer Nusselt numbers ${\overline{\text{Nu}}}_{o\mathrm{1}}$ and ${\overline{\text{Nu}}}_{o\mathrm{2}}$ can be obtained by substituting the average values of ${\text{Pe}}_{v\mathrm{1}}$ and ${\text{Pe}}_{v\mathrm{2}}$ given in (51c) and (51d) in (24). The boundary conditions of (47) are given by

$\begin{array}{}\stackrel{-}{r}=\mathrm{1}:{\theta }_{\mathrm{1,2}}={\theta }_{f\mathrm{1,2}},\end{array}$

(52a)

$\begin{array}{}\stackrel{-}{r}=\mathrm{1}:\frac{\partial {\theta }_{\mathrm{1,2}}}{\partial \stackrel{-}{r}}=\mathrm{0},\end{array}$

(52b)

$\begin{array}{}{\stackrel{-}{x}}_{\mathrm{1,2}}=\mathrm{0}:\theta ={\theta }_{f\mathrm{1,2}}.\end{array}$

(52c)

The boundary conditions of (48) and (49) are given by

$\begin{array}{}{\stackrel{-}{x}}_{\mathrm{1,2}}=\mathrm{0}:\frac{d{\theta }_{f\mathrm{1,2}}}{d{\stackrel{-}{x}}_{\mathrm{1,2}}}=\mathrm{0},\end{array}$

(53a)

$\begin{array}{}{\stackrel{-}{x}}_{\mathrm{1,2}}=\mathrm{1}:{\theta }_{f\mathrm{1}}=\mathrm{1}-{\theta }_{f\mathrm{2}},\end{array}$

(53b)

$\begin{array}{}\frac{d{\theta }_{f\mathrm{2}}}{d{\stackrel{-}{x}}_{\mathrm{2}}}=\frac{d{\theta }_{f\mathrm{1}}}{d{\stackrel{-}{x}}_{\mathrm{1}}}.\end{array}$

(53c)

The maximum value of mass flow rate via the permeable hollow joint-pin is evaluated by differentiating (51e) and equating it to zero. It is equal to

$\begin{array}{}{\left({{\dot{m}}_i|}_{{\stackrel{-}{x}}_{\mathrm{1}}=\mathrm{1}}\right)}_{\max }=\frac{\pi }{\mathrm{8}}\mu d_i\text{R}{\text{e}}_t\left(nL\right)\tanh \left(\frac{nL}{\mathrm{2}}\right).\end{array}$

(54)

The maximum value given by (54) occurs at $L_1/L=0.5$ .

Figure 2:

Schematic diagram for a permeable hollow joint-pin and the corresponding coordinates system.

2.6. The Thermal Efficiency of the Permeable Hollow-Pin Systems

The maximum heat transfer rate by the permeable hollow-pin, q _max , obtained at $k_f\to \infty$ is equal to

$\begin{array}{}{q}_{\max }=\pi kL\left(T_b-T_R\right){\int }_{\mathrm{0}}^{\mathrm{1}}{\text{Nu}}_{o}d\stackrel{-}{x}.\end{array}$

(55)

The total heat transfer rate through the permeable hollow-pin can be obtained from

$\begin{array}{}{q_t=-q_x|}_{x=L}={\int }_{\mathrm{0}}^L\left(\frac{d{q}_{\text{conv}.,o}}{dx}+\frac{d{q}_{\text{conv}.,i}}{dx}+\frac{d{q}_{\text{conv}.,p}}{dx}\right)dx.\end{array}$

(56)

The thermal efficiency of this type of systems, η, is defined as the ratio of the total heat transfer rate by the permeable hollow-pin to the maximum heat transfer rate. It can be expressed as follows:

$\begin{array}{}\eta \equiv \frac{q_t}{q_{\max }}=\frac{{\int }_{\mathrm{0}}^{\mathrm{1}}\left({\text{Nu}}_i\left({\theta }_f-{\theta }_m\right)+{\text{Nu}}_o{\theta }_f\right)d\stackrel{-}{x}}{{\int }_{\mathrm{0}}^{\mathrm{1}}{\text{Nu}}_{o}d\stackrel{-}{x}}.\end{array}$

(57)

In case of the permeable hollow joint-pin, the thermal efficiency is obtained using (57) with Nusselt numbers, dimensionless temperatures, and $\stackrel{-}{x}$ -coordinate computed for one side of the joint-pin. For the constraint solution given by (46), the thermal efficiency is equal to

$\begin{array}{}\eta \cong \frac{\left[\left(w/s\right)-\left(s/w\right)\right]\tanh \left(w\right)a\text{Pe}{\stackrel{-}{P}}_{i,\text{avg}}^{\prime }}{\mathrm{4}\left[\mathrm{1}-\left(s/w\right)\tanh \left(w\right)\right]{\stackrel{-}{\text{Nu}}}_o}.\end{array}$

(58)

2.7. The Second HTE Factor for the Permeable Hollow-Pin Systems

Define the second HTE factor, γ, as the ratio of the system total heat transfer rate to the heat transfer rate from a solid pin of diameter d _o . Mathematically, it is equal to

$\begin{array}{}\gamma \equiv \frac{q_t}{\sqrt{h_{o,\infty }{k}_f{A}_{C,\mathrm{0}}{P}_{\mathrm{0}}}\left(T_b-T_R\right)\tanh \left(L\sqrt{\left(h_{o,\infty }P\right)/\left(k_f{A}_C\right)}\right)},\end{array}$

(59)

where $A_{C,\mathrm{0}}=\pi d_o^{\mathrm{2}}/\mathrm{4}$ and $P_{\mathrm{0}}=\pi d_o$ are the solid pin cross-sectional area and the perimeter of the solid pin, respectively. The outer convection heat transfer coefficient $h_{o,\infty }$ is obtained from (24) at zero Re _v , $h_{o,\infty }=\mathrm{2}k/[d_o\ln (D/d_o)]$ . By substituting (56) in (58), the following expression is obtained:

$\begin{array}{}\gamma =\frac{{\int }_{\mathrm{0}}^{\mathrm{1}}\left({\text{Nu}}_i\left({\theta }_f-{\theta }_m\right)+{\text{Nu}}_o{\theta }_f\right)d\stackrel{-}{x}}{a\left(d_o/d_i\right)\sqrt{\left(\mathrm{2}{k}_f/k\right)/\left(\ln \left(D/d_o\right)\right)}\tanh \left[\left(\mathrm{1}/a\right)\left(d_i/d_o\right)\sqrt{\left(\mathrm{2}k/k_f\right)/\left(\ln \left(D/d_o\right)\right)}\right]}.\end{array}$

(60)

In case of the permeable hollow joint-pin, the γ-factor is obtained using (60) with Nusselt numbers, dimensionless temperatures, $\stackrel{-}{x}$ -coordinate, a-ratio, and D-diameter computed for one side of the joint-pin. When $\gamma >1$ , the performance of the permeable fin is better than that of the solid one. However, the performance of the solid pin is better than that of permeable fin when $\gamma <1$ .

3. Numerical Methodology and Validations

3.1. Numerical Methodology

Equations (3), (43a), (43b), or (47)–(49) are coupled via the boundary conditions and must be solved using an iterative numerical method. The implicit finite-difference method that Blottner [26] discussed is an appropriate and an accurate numerical method for the present problem. This method is based on discretizing the second derivatives terms using three-point central different quotients. Also, the first derivative term with respect to $\stackrel{-}{r}$ -direction is discretized using the three-point central different quotient. Two-point backward differencing quotient is considered for the first derivative with respect to $\stackrel{-}{x}$ -direction. The resulting tridiagonal systems of algebraic equations obtained by the discretization process at a given $\stackrel{-}{x}$ -section were solved using the well-established Thomas algorithm [26]. The same procedure was repeated for the consecutive $\stackrel{-}{x}$ -sections until $\stackrel{-}{x}=\mathrm{1.0}$ , when $\text{Pe}>\mathrm{0}$ . When $\text{Pe}<\mathrm{0}$ , the marching of the numerical method was started at $\stackrel{-}{x}=\mathrm{1.0}$ and the previous procedure was repeated for the consecutive $-\stackrel{-}{x}$ sections until $\stackrel{-}{x}=\mathrm{0}$ . After that, the values of θ _m or ${\theta }_{m\mathrm{1,2}}$ were calculated using (10). Then, the discretized system of equations was solved again using the Thomas algorithm for the second iteration. Next, the previous procedure was periodically repeated by updating the values of θ _f or ${\theta }_{f\mathrm{1,2}}$ until the maximum change in θ _m or ${\theta }_{m\mathrm{1,2}}$ between the current and previous iterations be less than $\mathrm{1}{\mathrm{0}}^{-\mathrm{6}}$ .

3.2. Validations

When $\text{Pe}>\mathrm{0}$ , both θ _f and θ _m increase as $\stackrel{-}{x}$ increases. Thus, ${\text{Nu}}_i$ given by (14) can substituted in (11), (43a), and (43b). On the other hand, ${\text{Nu}}_i$ correlation given by (16) is the most appropriate correlation to be used when $\text{Pe}<\mathrm{0}$ . For both situations, the coupled equations (43a), (43b), and (11) are one-dimensional equations which can be solved using an iterative numerical method based on the Thomas algorithm. The comparisons between the present numerical solutions (approximate solutions) and the two dimensional numerical solutions discussed in Section 3.1 are shown in Figures 3–10. Meanwhile, the comparison between the highly constraint solution given by (58) and the two-dimensional solution under the same constraint is shown in Figure 3. Very good agreements are noticed between all solutions. This led to more confidence in the obtained results.

Figure 3:

Effects of Pe and $k/k_f$ on η when both $\text{Pe}>\mathrm{0}$ and $\text{Pe}<\mathrm{0}$ .

Figure 4:

Effects of Pe and $k/k_f$ on γ when both $\text{Pe}>\mathrm{0}$ and $\text{Pe}<\mathrm{0}$ .

Figure 5:

Effects of nL and Pe on γ when both $\text{Pe}>\mathrm{0}$ and $\text{Pe}<\mathrm{0}$ .

Figure 6:

Effects of $d_i/d_o$ and Pe on γ when both $\text{Pe}>\mathrm{0}$ and $\text{Pe}<\mathrm{0}$ .

Figure 7:

Effects of φ and Pe on γ when both $\text{Pe}>\mathrm{0}$ and $\text{Pe}<\mathrm{0}$ .

Figure 8:

Effects of $L_1/L$ and ${\text{Pe}}_t$ on η and γ for permeable hollow joint-pin.

Figure 9:

Effects of $L_1/L$ and nL on η and γ for permeable hollow joint-pin.

Figure 10:

Effects of $L_1/L$ and φ on η and γ for permeable hollow joint-pin.

4. Results and Discussions

4.1. The Performance of the Hollow-Pin due to Variation in the Peclet Number

Figure 3 shows the variation of the thermal efficiency $(\eta )$ of the permeable hollow-pin with the Peclet number $(\text{Pe})$ for different values of fluid to pin (solid part) thermal conductivities ratio $(k/k_f)$ . When $\text{Pe}>\mathrm{0}$ , the permeable pin temperatures near the tip region are furtherly convected toward the base region as Pe increases. This effect tends to increase the heat transfer rate at the base which increases the thermal efficiency. However, the outer convection coefficient $(h_o)$ increases as Pe increases when $\text{Pe}>\mathrm{0}$ due to increases in the suction effects at the outer surface. The latter effect tends to decrease the thermal efficiency. Due to the previous effects, the thermal efficiency is maximized at a specific Pe value when $\text{Pe}>\mathrm{0}$ . When $\text{Pe}<\mathrm{0}$ , the permeable pin temperatures near the base region are furtherly convected toward the tip region as $-\text{Pe}$ increases. Furthermore, h _o -coefficients decreases as $-\text{Pe}$ increases due to increases in the blowing effects. Both effects decrease the heat transfer rate through the permeable pin. Thus, the thermal efficiency decreases as $-\text{Pe}$ increases when $\text{Pe}<\mathrm{0}$ . Increasing the pin thermal conductivity which decreases $k/k_f$ causes increases in the heat transfer rate, the thermal efficiency and the second HTE factor $(\gamma )$ . These trends are shown in Figures 3 and 4, respectively. It is shown from Figure 4 that γ increases as Pe increases when $\text{Pe}>\mathrm{0}$ while it decreases as $-\text{Pe}$ increases when $\text{Pe}<\mathrm{0}$ . This is because the increase in $|\text{Pe}|$ causes an increase in the heat transfer rate when $\text{Pe}>\mathrm{0}$ due to increases in both h _o -coefficients and temperature gradients near the base. However, the increase in $|\text{Pe}|$ causes a decrease in the heat transfer rate when $\text{Pe}<\mathrm{0}$ due to increases in blowing effects as $-\text{Pe}$ increases. Figure 4 shows that $\gamma >\mathrm{1}$ when $\text{Pe}>\mathrm{0}$ while it is $\gamma <\mathrm{1}$ when $\text{Pe}<\mathrm{0}$ . This indicates the superiority of the permeable hollow-pin with external suction flow over those with external blowing flow.

4.2. The Performance of the Hollow-Pin due to Variation in Flow Index Number

Figure 5 illustrates the variation of the second HTE factor $(\gamma )$ of the permeable hollow-pin with the induced flow index number $(n)$ for different Peclet numbers. As n increases, the total mass flow rate across the permeable pin increases as dictated from (36). Thus, γ increases as n increases when $\text{Pe}>\mathrm{0}$ due to increases in suction effects. However, γ decreases as n increases when $\text{Pe}<\mathrm{0}$ due to increases in blowing effects at the outer surface. These trends are clearly seen in Figure 5. Moreover, Figure 5 shows that for large n numbers, the γ-factor becomes more sensitive to variation in the Peclet numbers. Figure 5 shows that maximum value of γ is γ = 4.62. Figure 5 shows that HTEs due to permeable hollow-pins is only present when $\text{Pe}>\mathrm{0}$ as $\gamma >\mathrm{1}$ for this case.

4.3. The Performance of the Hollow-Pin due to Variation in $d_i/d_o$

Figure 6 displays the effect of permeable hollow-pin inner to outer diameters ratio $(d_i/d_o)$ on the second HTE factor $(\gamma )$ for different Peclet numbers. The change in $d_i/d_o$ is obtained by varying the outer diameter d _o under same inner diameter d _i . An increase in d _o causes a decrease in n-index (see (28b)) thus, convection heat transfer rate is reduced when $\text{Pe}>\mathrm{0}$ . The latter effect can be explained using (24), (31), and (40). As such, γ-factor decreases as d _o increases or as $d_i/d_o$ decreases. On the other hand, the blowing effect at the pin outer surface when $\text{Pe}<\mathrm{0}$ is reduced when n-index decreases. Therefore, γ-factor increases as d _o increases or as $d_i/d_o$ decreases. These trends are clearly shown in Figure 6. Moreover, Figure 6 shows that for large $d_i/d_o$ ratios, the γ-factor becomes more sensitive to variation in the Peclet number. Furthermore, Figure 6 shows that HTEs due to permeable hollow-pins is only present when $\text{Pe}>\mathrm{0}$ .

4.4. The Performance of the Hollow-Pin due to Variation in φ-ratio

Figure 7 shows the effect of φ-ratio on the second HTE factor $(\gamma )$ of the permeable hollow-pin for different Peclet numbers. An increase in φ-ratio produces an increase in n-index (see (28b)), thus, convection heat transfer rate is increased when $\text{Pe}>\mathrm{0}$ as explained earlier. In addition, the increase in φ-Ratio decreases conduction heat transfer through the pin due to reduction of the higher thermal conductivity volume (the solid volume). Due to these reasons, γ is maximized at a specific φ-ratio as can be seen from Figure 7. On the other hand, the blowing effect at the pin outer surface increases as φincreases when $\text{Pe}<\mathrm{0}$ due the associated increase in n-index. Therefore, γ-factor decreases as φ increases when $\text{Pe}<\mathrm{0}$ . This is shown in Figure 7. Moreover, Figure 7 shows that for large φ-ratios, γ-factor becomes more sensitive to variation in the Peclet numbers. Figure 7 shows that HTEs due to permeable hollow-pins is only present when $\text{Pe}>\mathrm{0}$ .

4.5. The Performance of the Permeable Hollow Joint-Pin

Figures 8–10 illustrate the effect of the relative lengths ratio $(L_{\mathrm{1}}/L)$ on the thermal efficiency and γ-factor of the permeable hollow join-pin for different total Peclet ${\text{Pe}}_t$ numbers, n-indices, and φ-ratios, respectively. As $L_{\mathrm{1}}/L$ increases, the average outer Nusselt number ${\overline{\text{Nu}}}_{o\mathrm{2}}$ decreases which in turn is much smaller than ${\overline{\text{Nu}}}_{o\mathrm{1}}$ . This effect requires an increase in the departure of the base temperature T _b from $T_{R\mathrm{2}}$ thus, the heat transfer rate decreases. Meanwhile, an increase in $L_{\mathrm{1}}/L$ decreases the thermal efficiency of the joint-pin left portion due to an increase in its conduction resistance. Due to the previous reasons, the thermal efficiency of the joint system is expected to decrease as $L_{\mathrm{1}}/L$ increases as shown in Figures 8–10. An increase in ${\text{Pe}}_t$ , nL, or φ causes a further reductions ${\overline{\text{Nu}}}_{o\mathrm{2}}$ (e.g., see Figure 11) and the left side conduction resistance relative to its outer convection resistance. As such, it causes further reduction in the thermal efficiency as shown in Figures 8–10. Since both the flow rate through the permeable hollow joint-pin and the overall average outer Nusselt number $\{{\overline{\text{Nu}}}_o=(L_{\mathrm{1}}/L){\overline{\text{Nu}}}_{o\mathrm{1}}+(\mathrm{1}-L_{\mathrm{1}}/L){\overline{\text{Nu}}}_{o\mathrm{2}}\}$ are maximized at $L_{\mathrm{1}}/L=\mathrm{0.5}$ as illustrated in (53a), (53b), (53c), and Figure 11, respectively. It is expected that γ-factor must be maximized at specific $L_{\mathrm{1}}/L$ -value. This is illustrated in Figures 8–10. This specific value is expected not to coincide with $L_{\mathrm{1}}/L=\mathrm{0.5}$ due to the following factors: (a) the increase in conduction resistance in permeable hollow joint-pins due to presence of fluid volumes, (b) the variation of inner convection resistance across the two sides of system, and (c) the presence of entry region effects near the base section on the right side of the system. An increase in ${\text{Pe}}_t$ , nL, or φ causes further increases in ${\overline{\text{Nu}}}_o$ (e.g., see Figure 11). As such, it causes a further increase in γ-factor as seen in Figures 8–10. Figure 9 shows that maximum value of γ is γ = 1.44.

Figure 11:

Effects of $L_1/L$ and nL on ${\overline{\text{Nu}}}_{o\mathrm{1}}$ , ${\overline{\text{Nu}}}_{o\mathrm{2}}$ , and ${\overline{\text{Nu}}}_o$ for permeable hollow joint-pin.

5. Conclusions

In this work, flow and heat transfer through permeable hollow-pin systems were studied. Two different systems are considered: (A) the permeable hollow-pin system, and (B) the permeable hollow joint-pin system. Numerous mathematical techniques were used to model flow and energy transport in the explored systems. The appropriate dimensionless forms of the governing equations were solved using a well-known finite-difference numerical method. Very good agreements were obtained between the results of numerical solutions and the many analytical solutions derived in this work. The following main findings were found.

• Enhancing of heat transfer in a permeable hollow-pin over that of the solid pin is only reachable when an external suction flow is present at the hollow-pin outer surface.

• Enhancing of heat transfer in a permeable hollow joint-pin over that of a solid joint-pin can be attained at a specific hollow joint-pin relative lengths ratio.

• Factors increasing both the internal fluid flow and the conduction through the solid parts of the permeable hollow-pin and permeable hollow joint-pin systems tend to further enhance the heat transfer rate. Examples of these factors are the pores diameter and the thermal conductivity of the solid parts.

• The maximum reported heat transfer rate due to permeable hollow-pin is 362 percent above that of solid pin at dimensionless suction number equal to 3.0.

• The maximum reported heat transfer rate due to permeable hollow joint-pin is 44 percent above that of solid joint-pin at dimensionless suction number equal to 2.0.

Finally, this work demonstrates that using combined heat transfer enhancement approaches is capable of attaining an accumulated enhancement levels.