Entropy Generation Analysis on Heat Exchanger in Aluminum Foam

A metal foam heat exchanger in forced convection steady laminar flow represents the physical domain. The 2D sketch of the compact heat exchange system under investigation is presented in Figure1. An array of five tubes enclosed inside the porous medium constituted the tubular heat exchanger. The distance between center-to-center of two consecutive tubes, indicated as 2H, is 38 mm and the diameter d of the tubes is 12 mm. The system is characterized by the height H_tot of 190 mm and the length L_mf of 38 mm.

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Figure 1. Physical domain

The local volume averaging process is applied at the microscopic flow equations as written in [23] in order to replicate the thermal and fluid dynamic behaviors inside the domain. Whitaker wrote the governing equations from the Navier-Stokes and energy equation [24].

The Darcy–Forchheimer-Brinkman model and LTNE condition are consider to describe the presence of metal foam.

The variables of the equations are evaluated estimating the average of the local variables over an appropriate volume, called Representative Elementary Volume (REV), applied the local volume averaging technique. The metal foam is assumed homogeneous and isotropic and the thermal and physical properties of the fluid and solid phases are considered constant. In addition, the viscous dissipation, buoyancy force, and the thermal contact resistances between the tube surface and foam are overlooked. Applying the hypothesis above indicated, the governing equations are the following:

-Continuity equation

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$      (1)

-x- momentum equation

$\rho_{f}\left(\frac{u}{\varepsilon^{2}} \frac{\partial u}{\partial x}+\frac{v}{\varepsilon^{2}} \frac{\partial u}{\partial y}\right)=-\frac{\partial p}{\partial x}+\frac{\mu_{f}}{\varepsilon}\left(\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}\right)+$$-\frac{\mu_{f}}{K} u-\frac{C_{F}}{K^{1 / 2}} \rho_{f} \sqrt{u^{2}+v^{2}} u$      (2)

-y-momentum equation

$\rho_{f}\left(\frac{u}{\varepsilon^{2}} \frac{\partial v}{\partial x}+\frac{v}{\varepsilon^{2}} \frac{\partial v}{\partial y}\right)=-\frac{\partial p}{\partial y}+\frac{\mu_{f}}{\varepsilon}\left(\frac{\partial^{2} v}{\partial x^{2}}+\frac{\partial^{2} v}{\partial y^{2}}\right)+$$-\frac{\mu_{f}}{K} v-\frac{C_{F}}{K^{1 / 2}} \rho_{f} \sqrt{u^{2}+v^{2}} v$     (3)

where, ɛ is the porosity, ρ_f and μ_f are fluid density and viscosity, u and v are correspondingly the velocity components in Cartesian coordinates, K and C_F are respectively the permeability and the inertial coefficient of metal foam.

-Fluid phase energy equation

$\left(\rho_{f} c_{p}\right)_{f}\left(u \frac{\partial T_{f}}{\partial x}+v \frac{\partial T_{f}}{\partial y}\right)=\varepsilon k_{f}\left(\frac{\partial^{2} T_{f}}{\partial x^{2}}+\frac{\partial^{2} T_{f}}{\partial y^{2}}\right)$$+h_{s f} \alpha_{s f}\left(T_{s}-T_{f}\right)$     (4)

-Solid phase energy equation

$(1-\varepsilon) k_{s}\left(\frac{\partial^{2} T_{s}}{\partial x^{2}}+\frac{\partial^{2} T_{s}}{\partial y^{2}}\right)-h_{s f} \alpha_{s f}\left(T_{s}-T_{f}\right)=0$     (5)

where, c_p is the specific heat, k_f and k_s are the fluid phase and solid matrix thermal conductivity, T_f and T_s are the temperature of fluid and solid phase of metal foam, respectively. In the last term in both energy equations, α_sf and h_sf are respectively the specific surface area density and the interfacial heat transfer coefficient between the fluid and solid matrix, due to the LTNE hypothesis.

The equations for the evaluation of the entropy generation for local thermal non-equilibrium porous model are the following [25]:

-Fluid phase entropy generation

$\dot{S}_{g e n, f}^{\prime \prime \prime}=\underbrace{\frac{\varepsilon k_{f}}{T_{f}^{2}}\left[\left(\frac{\partial T_{f}}{\partial x}\right)^{2}+\left(\frac{\partial T_{f}}{\partial y}\right)^{2}\right]+\frac{\alpha_{s f} h_{s f}\left(T_{s}-T_{f}\right)^{2}}{T_{s} T_{f}}}_{\dot{s}_{g e n, \Delta T}^{'''}}$$+\underbrace{\frac{\mu_{f}}{T_{f}} \Phi+\frac{\mu_{f} \varepsilon}{K T_{f}}\left(u^{2}+v^{2}\right)+\frac{C_{F} \rho \varepsilon^{2}}{T_{f} \sqrt{K}}\left(u^{2}+v^{2}\right)^{3 / 2}}_{\dot{s}_{g e n, \Delta p}^{'''}}$      (6)

where, Φ in Eq. (6) is the well-known dissipation function; its form for incompressible fluid at a two-dimensional rectangular coordinate is:

$\Phi=2\left[\left(\frac{\partial u}{\partial x}\right)^{2}+\left(\frac{\partial v}{\partial y}\right)^{2}\right]+\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)^{2}$       (7)

The fourth term on the right-hand side of Eq. (6) considers the irreversibility rate of internal energy per unit volume by viscous drag in the porous section, and the last term in Eq. (6) represents the irreversibility rate of internal energy per unit volume by form drag in the porous section.

-Solid phase entropy generation

$\dot{S}_{g e n, s}^{\prime \prime \prime}=\underbrace{\frac{(1-\varepsilon) k_{s}}{T_{s}^{2}}\left[\left(\frac{\partial T_{s}}{\partial x}\right)^{2}+\left(\frac{\partial T_{s}}{\partial y}\right)^{2}\right]+\frac{\alpha_{s f} h_{s f}\left(T_{f}-T_{s}\right)^{2}}{T_{s} T_{f}}}_{s_{g e n, \Delta T}^{'''}}$     (8)

The correlations of Calmidi are used to evaluate K and C_F [26]:

$K=0.00073(1-\varepsilon)^{0.0224}\left(\frac{d_{f}}{d_{p}}\right)^{-1.11} d_{p}^{2}$      (9)

and

$C_{F}=0.00212(1-\varepsilon)^{-0.132}\left(\frac{d_{f}}{d_{p}}\right)^{-1.63}$      (10)

where, d_f and d_p are respectively the fiber and pore diameter of the aluminum foam. These parameters are in relation with the nature of the metal foam by means of the following relationships [27]:

$\frac{d_{f}}{d_{p}}=1.18 \sqrt{\frac{1-\varepsilon}{3 \pi}}\left(\frac{1}{1-e^{-(1-\varepsilon) / 0.04}}\right)$      (11)

and

$d_{p}=\frac{0.0224}{\omega}$      (12)

w is the pores density of the metal foam, i.e., the number of pores across one inch (PPI). The following correlations are adopted to estimate a_sf and h_sf [28]:

$\alpha_{s f}=\frac{3 \pi d_{f}}{\left(0.59 d_{p}\right)^{2}}\left(1-e^{-(1-\varepsilon) / 0.04}\right)$     (13)

and

$h_{s f}=\left\{\begin{array}{c}\left(0.75 R e_{d_{f}}^{0.4} P r_{a i r}^{0.37}\right)\left(\frac{k_{f}}{d_{f}}\right), 1 \leq R e_{d f} \leq 40 \\ \left(0.51 R e_{d_{f}}^{0.5} P r_{a i r}^{0.37}\right)\left(\frac{k_{f}}{d_{f}}\right), 40 \leq R e_{d f} \leq 10^{3} \\ \left(0.26 R e_{d_{f}}^{0.6} P r_{a i r}^{0.37}\right)\left(\frac{k_{f}}{d_{f}}\right), 10^{3} \leq R e_{d f} \leq 2 \times 10^{5}\end{array}\right.$      (14)

where, Re_df is the local Reynolds number referred to ligament diameter:

$R e_{d_{f}}=\frac{\rho_{f} u_{0} d_{f}}{\mu_{f}}$      (15)

Pr_air is the air Prandtl number that is evaluated as:

$P r_{a i r}=\frac{\mu_{f} c_{p}}{k_{f}}$       (16)

The parameters of the metal foams, for different porosity and PPI, are listed in Table 1.

Table 1. Parameters of the aluminum metal foams

The entropy generation method analyses are accomplished for different inlet air velocities and for a fixed temperature on the external surface of tube equal to 323.16 K. The air flow is laminar and the LTNE model is considered to write the energy equations. The total global entropy generation $\dot{S}_{g e n, t o t}$ is evaluated as:

$\dot{S}_{g e n, t o t}=\dot{S}_{g e n, f}+\dot{S}_{g e n, s}$       (18)

where, $\dot{S}_{g e n, f}$ and $\dot{S}_{g e n, s}$ are the fluid phase and solid matrix global entropy generation, respectively. The global total entropy generation results, evaluated in the porous section, for different porosity values are plotted, in the Figure 3 only for 5 PPI and 40 PPI because the behavior of the $\dot{S}_{g e n, t o t}$ is the same for 10 PPI and 20 PPI values, respectively.

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Figure 3. Global entropy generation as a function of inlet velocity for different porosity and: (a) 5 PPI, (b) 40 PPI

The $\dot{S}_{\text {gen,tot }}$ increases with increasing inlet velocity values for all PPI. Furthermore, it is possible to see, in the Figure 3.a, that, for velocity values minor than 0.85 m/s, the global entropy generation increases with the increase of porosity; while, for higher velocity values, the lower $\dot{S}_{\text {gen,tot }}$ is obtained in corresponding to the porosity equal to 0.978. The same trend is also achieved for 20 PPI. In addition, from the Figure 3.b, one can notice the same behavior of the global entropy generation, but with the difference that there is a decrease of $\dot{S}_{\text {gen,tot }}$ at the highest porosity for a greater velocity value, near to 0.97 m/s. Among the various metal foam samples analyzed, the one showing the same trend of 40 PPI is the foam characterized by a pore density equal to 10.

In the Figure 4, the total global entropy generation, calculated in the porous medium, is presented for several PPI values approximately to the same porosity around 0.90 and 0.97, respectively.

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Figure 4. Global entropy generation as a function of inlet velocity for different PPI values and: (a) ε≅0.90, (b) ε≅0.97

It is possible to see that, both in the Figure 4.a and in Figure 4.b, the total global entropy generation increases with increasing PPI values. This behavior is because as the number of pores per inch increases, there is an increment in the pressure drop, which consequently leads to an increase in the entropic generation component linked to friction.

In the Figure 5, the Bejan number Be evaluated in the metal foam region are showed as a function of Reynolds number. Bejan number Be, a typical parameter employed in the entropy generation method, is defined as:

$B e=\frac{\dot{S}_{g e n, \Delta T}}{\dot{S}_{g e n, \Delta T}+\dot{S}_{g e n, \Delta p}}$      (19)

where, in the denominator the first term indicates the entropy generation due to the heat transfer between finite temperature difference $\dot{S}_{g e n, \Delta T}$ and the second term indicates the entropy generation by the pressure drop due to friction $\dot{S}_{g e n, \Delta p}$. The sum of two terms of denominator of Eq. (19) represents the total global entropy generation. The Bejan number is showed in Figure 5, for 10 PPI and 20 PPI.

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Figure 5. Bejan number as a function of Reynolds number for different porosity and: (a) 10 PPI, (b) 20 PPI

The Bejan number Be decreases with increasing inlet velocity values for all numbers of pores per inch, because of the increment of the term $\dot{S}_{g e n, \Delta p}$. Moreover, it is possible to see, in the Figure 5.a, that, for Reynolds number minor than 800, the Bejan number increases with the increase of porosity; while, for higher Re values, the lower Be is achieved in corresponding to the porosity equal to 0.9724. The same trend is also obtained for 40 PPI. In addition, from the Figure 5.b, it is possible to notice it is possible to note once again the increase in the Bejan number as the porosity increases, but in this case a lower Bejan number is recorded for the higher porosity at a lower Reynolds number value than before ($R e \cong 600$). The metal foam that has the same Reynolds value for which the Bejan number decreases with increasing porosity is the one characterized by a number of pores per inch equal to 5. In the Figure 6, the Bejan number Be is illustrated at varying Reynolds number and for several PPI values at roughly the same porosity around 0.90 and 0.97, respectively. One can observe that, in the Figure 6.a, the Bejan number decreases with increasing PPI values except for 10 PPI and 20 PPI for a Re value greater than 800. In the Figure 6.b and 6.c, however, Be increases as the number of pores per inch decreases for any Reynolds value. The lowest Be value, at which the generation due to friction weighs more, is recorded for 40 PPI and porosity equal to 0.91 and is approximately equal to 0.83.

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(a)

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(b)

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(c)

Figure 6. Bejan number as a function of Re for different PPI values and: (a) $\varepsilon \cong 0.90$, (b) $\varepsilon \cong 0.94$, (c) $\varepsilon \cong 0.97$

This research was partially funded by MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca), grant number PRIN-2017F7KZWS and by Università degli Studi della Campania “Luigi Vanvitelli” with the grant number D.R. n. 138 under NanoTES project - V:ALERE program 2020.