4. Formulations

Open-cell metal foams are classified as high-porosity materials that consist of irregularly shaped and tortuous flow passages. Pressure drop and heat transfer through such media are significantly affected by their geometrical characteristics, namely the foam porosity ε, fiber size df, pore diameter dp, pore density ω, and cell shape. Therefore, most aspects regarding granular porous media and packed beds need to be adjusted for metal foams [11].

### 4.1. Estimation of fiber and pore diameter

In practice, the ligament size, or in other words fiber diameter, is usually measured using a microscope. Alternatively, the mean pore diameter can be estimated by counting the number of pores that exist in a particular length of foam, which is usually provided by the manufacturer in terms of pore density (PPI), that is, number of pores per inch. Depending on the representative unit cell used (Figure 6), various models were proposed. Among them is the model proposed by Fourie and Du Plessis [49] for pore size estimation as a function of the width of a cubic representative unit cell and tortuosity (Eq. (1)). Another model was developed by Calmidi [50] to estimate the fiber- to pore-diameter ratio as a function of the porosity and shape function, G = 1 � e �(1 � <sup>ε</sup>)/0.04, for both the cubic unit cell (Eq. (2)) and the three-dimensional structure of a dodecahedron unit cell (Eq. (3)).

Fourie and Du Plessis [49]:

$$d\_p = d \frac{2}{\left(\Im - \chi\right)'} , \quad d = d\_p + d\_f \tag{1}$$

Figure 6. Models used to represent microstructural unit cells in metal foams.

Calmidi [50]:

foams are commonly used as energy absorber in aerospace and military applications, air/ oil separators in aircraft engine gearboxes, baffles to prevent sudden surges in liquids while being penetrated by solid frames, and breather plugs in applications requiring fast equalization of pressure changes. Also, they are utilized in electrodes, fuel cells, bone researches, micrometeorite shields, optics/mirrors, windscreens and so on. Manufacturers' data and the open literature can be further dug for much more applications where highporosity metal foams have outperformed and/or achieved considerable savings in the

Open-cell metal foams are classified as high-porosity materials that consist of irregularly shaped and tortuous flow passages. Pressure drop and heat transfer through such media are significantly affected by their geometrical characteristics, namely the foam porosity ε, fiber size df, pore diameter dp, pore density ω, and cell shape. Therefore, most aspects regarding granular

In practice, the ligament size, or in other words fiber diameter, is usually measured using a microscope. Alternatively, the mean pore diameter can be estimated by counting the number of pores that exist in a particular length of foam, which is usually provided by the manufacturer in terms of pore density (PPI), that is, number of pores per inch. Depending on the representative unit cell used (Figure 6), various models were proposed. Among them is the model proposed by Fourie and Du Plessis [49] for pore size estimation as a function of the width of a cubic representative unit cell and tortuosity (Eq. (1)). Another model was developed by Calmidi [50] to estimate the fiber- to pore-diameter ratio as a function of the porosity and shape

�(1 � <sup>ε</sup>)/0.04, for both the cubic unit cell (Eq. (2)) and the three-dimensional

, d ¼ dp þ df (1)

porous media and packed beds need to be adjusted for metal foams [11].

dp <sup>¼</sup> <sup>d</sup> <sup>2</sup>

Figure 6. Models used to represent microstructural unit cells in metal foams.

ð Þ 3 � χ

4.1. Estimation of fiber and pore diameter

structure of a dodecahedron unit cell (Eq. (3)).

expenses required.

188 Porosity - Process, Technologies and Applications

4. Formulations

function, G = 1 � e

Fourie and Du Plessis [49]:

$$\frac{d\_f}{d\_p} = 2\sqrt{\frac{(1-\varepsilon)}{3\pi}}\frac{1}{G} \tag{2}$$

$$\frac{d\_f}{d\_p} = 1.18 \sqrt{\frac{(1 - \varepsilon)}{3\pi}} \frac{1}{G} \tag{3}$$

#### 4.2. Models developed for predicting tortuosity of high-porosity metal foams

The tortuosity, defined as the total tortuous pore length within a linear length scale divided by the linear length scale in the porous medium [49], was modeled by Du Plessis et al. [51] as a function of porosity only (Eq. (4)). However, experiments conducted by Bhattacharya et al. [52] indicated that the accuracy of tortuosity model proposed by Du Plessis et al. [51] is limited for higher levels of pore density; hence, a tortuosity formulation that accurately covers a wider range of porosity and pore densities was established in terms of porosity and shape function G (Eq. (5)). Recently, an analytical model was proposed by Yang et al. [53] (Eq. (6)), as a simple function of both foam porosity and a pore shape factor. The shape factor β is defined as the ratio of the representative pore perimeter to the perimeter of a typical reference circle with an area equal to that of the representative pore.

Du Plessis et al. [51]:

$$\frac{1}{\chi} = \frac{3}{4\varepsilon} + \frac{\sqrt{9 - 8\varepsilon}}{2\varepsilon} \cos\left\{ \frac{4\pi}{3} + \frac{1}{3} \cos^{-1} \left[ \frac{8\varepsilon^2 - 36\varepsilon + 27}{\left(9 - 8\varepsilon\right)^{3/2}} \right] \right\} \tag{4}$$

Bhattacharya et al. [52]:

$$\frac{1}{\chi} = \frac{\pi}{4\varepsilon} \left\{ 1 - \left( 1.18 \sqrt{\frac{(1-\varepsilon)}{3\pi}} \frac{1}{G} \right)^2 \right\} \tag{5}$$

Yang et al. [53]:

$$\chi = \frac{\beta \varepsilon}{1 - (1 - \varepsilon)^{1/3}} \tag{6}$$

#### 4.3. Models developed for estimating pressure drop across open-cell metal foams

With regard to predicting the pressure drop produced in fluid flows across high-porosity metal foams, a variety of models have been developed, which can be classified into two main categories. The first encompasses those investigations interested in estimating the pressure drop by means of the foam friction factor. Among them is the model presented by Paek et al. [54] for the friction factor as a function of pore Reynolds number (Eq. (7)). Also, the empirical correlations established by Liu et al. [55] offer friction factor estimation for airflow via aluminum foams for a wide range of porosity and various flow regimes (Eq. (8)).

Paek et al. [54]:

$$f = \frac{1}{\text{Re}\_{\text{K}}} + 0.105 \quad f = \frac{(\Delta P/L)\sqrt{K}}{\rho u^2}, \text{ and } \text{Re}\kappa = \frac{\rho \,\mu\sqrt{K}}{\mu} \tag{7}$$

Liu et al. [55]:

$$\begin{aligned} \text{30 } &< \text{Re}\_{D\text{p}} < \text{300 } : f = \frac{22(1 - \varepsilon)}{\text{Re}\_{D\text{p}}} + 0.22\\ \text{Re}\_{D\text{p}} &> \text{300 } : \quad f = 0.22 \end{aligned} \tag{8}$$

$$\begin{aligned} f = \frac{(\Delta P/L)D\_{\text{p}}}{\rho u^{2}} \frac{\varepsilon^{3}}{1 - \varepsilon}, \quad \text{and} \quad \text{Re}\_{D\text{p}} = \frac{\rho u \sqrt{K}}{\mu} \end{aligned} \tag{9}$$

The other category of pressure drop models consists of those concerned with estimating the permeability and inertial coefficient according to Darcy-–Forchheimer's equation:

$$\frac{dp}{d\mathbf{x}} = \frac{\mu}{K}\mu + \frac{\rho \mathbf{F}}{\sqrt{K}}\mu^2 \tag{9}$$

Calmidi [50]:

Bhattacharya et al. [52]:

Tadrist et al. [56]:

Dukhan [57]:

Yang et al. [53]:

F ¼ 0:095

CDð Þ <sup>ε</sup>¼0:<sup>85</sup> <sup>12</sup> <sup>G</sup><sup>0</sup>:<sup>2</sup>

<sup>K</sup> <sup>¼</sup> <sup>0</sup>:00073d<sup>2</sup>

<sup>p</sup>ð Þ <sup>1</sup> � <sup>ε</sup> �0:<sup>224</sup> df

<sup>F</sup> <sup>¼</sup> <sup>0</sup>:00212 1ð Þ � <sup>ε</sup> �0:<sup>132</sup> df

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε 3ð Þ χ � 1

r

<sup>0</sup>:<sup>85</sup> <sup>&</sup>lt; <sup>ε</sup> <sup>&</sup>lt; <sup>0</sup>:<sup>97</sup> : <sup>G</sup> <sup>¼</sup> <sup>1</sup> � <sup>e</sup>�ð Þ <sup>1</sup>�<sup>ε</sup> <sup>=</sup>0:<sup>04</sup> <sup>ε</sup> <sup>≥</sup> <sup>0</sup>:<sup>97</sup> : <sup>G</sup> <sup>¼</sup> <sup>0</sup>:<sup>5831</sup> )

<sup>K</sup> <sup>¼</sup> <sup>ε</sup><sup>3</sup>d<sup>2</sup>

<sup>F</sup> <sup>¼</sup> <sup>β</sup>ð Þ <sup>1</sup> � <sup>ε</sup> ε3

K <sup>d</sup><sup>2</sup> <sup>¼</sup>

f αð Þ 1 � ε

> ffiffiffi K p df

K ¼ a1e

F ¼ ð Þ a2ε þ b<sup>2</sup>

a1, a2, b1, and b<sup>2</sup> are all constants

ε 1 � ð Þ 1 � ε <sup>1</sup>=<sup>3</sup> � �<sup>2</sup>

Many investigations have been conducted to evaluate the effective thermal conductivity ke as a key factor in the thermal analysis of such systems. Among them is the model established by Paek et al. [54] based on one-dimensional heat conduction through a cubic unit cell (Eq. (20)). This model indicates that foam porosity has a direct impact on the overall thermal conductiv-

36β ð Þ 1 � ε

4.4. Effective thermal conductivity models for high-porosity metal foams

ity unlike the pore size, which was found to have a marginal influence.

ffiffiffi K

dp � ��1:<sup>11</sup>

dp � ��1:<sup>63</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 � ε 3π r 1

!�<sup>1</sup>

G

High-Porosity Metal Foams: Potentials, Applications, and Formulations

http://dx.doi.org/10.5772/intechopen.70451

, and, CDð Þ <sup>ε</sup>¼0:<sup>85</sup> ¼ 1:2

<sup>2</sup> , α : 100 � 865 (15)

, β : 0:65 � 2:6 (16)

<sup>b</sup>1<sup>ε</sup> (17)

<sup>1</sup>=<sup>3</sup> � <sup>1</sup> � βε � � h i (19)

<sup>p</sup> (18)

1:18

(12)

191

(13)

(14)

Based on a cubic representative unit cell, a theoretical model for predicting permeability and inertial coefficient was derived by Du Plessis et al. [51] (Eqs. (10) and (11)), as functions of porosity, tortuosity, and the width of cubic representative unit cell. In the study conducted by Calmidi [50], mathematical models were developed for both permeability and inertial coefficient as functions of the fiber and pore diameters as well as the foam porosity (Eqs. (12) and (13)). A correlation was established by Bhattacharya et al. [52] for predicting the inertial coefficient (Eq. (14)), in terms of tortuosity, porosity, shape function, and form drag coefficient CD(ε). Based on Ergun's law <sup>Δ</sup><sup>p</sup> <sup>L</sup> <sup>¼</sup> <sup>α</sup> ð Þ <sup>1</sup>�<sup>ε</sup> <sup>2</sup> <sup>ε</sup>3d<sup>2</sup> <sup>μ</sup><sup>u</sup> <sup>þ</sup> <sup>β</sup> ð Þ <sup>1</sup>�<sup>ε</sup> <sup>ε</sup>3<sup>d</sup> <sup>ρ</sup>u<sup>2</sup> � �, Tadrist et al. [56] developed an empirical model for estimating permeability and inertial coefficient as functions of foam porosity and fiber size (Eqs. (15) and (16)). In the experiments conducted by Dukhan [57], the pressure drop resulting from airflow across aluminum foam was correlated into a model predicting the permeability and inertial coefficient as functions of porosity only (Eqs. (17) and (18)). Recently, an analytical model for estimating the permeability of metal foams (Eq. (19)) was established by Yang et al. [53] according to the cubic representative unit cell. This model offers the capability of estimating the permeability for a wide range of foam porosities ε = 0.55 ~ 0.98 and pore densities ω = 5 ~ 100 PPI.

Du Plessis et al. [51]:

$$K = \frac{\varepsilon^2 d^2}{36\chi(\chi - 1)}\tag{10}$$

$$F = \frac{2.05\chi(\chi - 1)}{\varepsilon^2(3 - \chi)} \frac{\sqrt{K}}{d} \tag{11}$$

Calmidi [50]:

Paek et al. [54]:

Liu et al. [55]:

<sup>f</sup> <sup>¼</sup> <sup>1</sup> Re<sup>K</sup>

190 Porosity - Process, Technologies and Applications

CD(ε). Based on Ergun's law <sup>Δ</sup><sup>p</sup>

Du Plessis et al. [51]:

ε = 0.55 ~ 0.98 and pore densities ω = 5 ~ 100 PPI.

<sup>þ</sup> <sup>0</sup>:<sup>105</sup> <sup>f</sup> <sup>¼</sup> ð Þ <sup>Δ</sup>P=<sup>L</sup> ffiffiffi

<sup>30</sup> <sup>&</sup>lt; ReDP <sup>&</sup>lt; <sup>300</sup> : <sup>f</sup> <sup>¼</sup> 22 1ð Þ � <sup>ε</sup>

ε3 1 � ε

The other category of pressure drop models consists of those concerned with estimating the

<sup>K</sup> <sup>u</sup> <sup>þ</sup> <sup>ρ</sup><sup>F</sup> ffiffiffi K

Based on a cubic representative unit cell, a theoretical model for predicting permeability and inertial coefficient was derived by Du Plessis et al. [51] (Eqs. (10) and (11)), as functions of porosity, tortuosity, and the width of cubic representative unit cell. In the study conducted by Calmidi [50], mathematical models were developed for both permeability and inertial coefficient as functions of the fiber and pore diameters as well as the foam porosity (Eqs. (12) and (13)). A correlation was established by Bhattacharya et al. [52] for predicting the inertial coefficient (Eq. (14)), in terms of tortuosity, porosity, shape function, and form drag coefficient

<sup>ε</sup>3d<sup>2</sup> <sup>μ</sup><sup>u</sup> <sup>þ</sup> <sup>β</sup> ð Þ <sup>1</sup>�<sup>ε</sup>

empirical model for estimating permeability and inertial coefficient as functions of foam porosity and fiber size (Eqs. (15) and (16)). In the experiments conducted by Dukhan [57], the pressure drop resulting from airflow across aluminum foam was correlated into a model predicting the permeability and inertial coefficient as functions of porosity only (Eqs. (17) and (18)). Recently, an analytical model for estimating the permeability of metal foams (Eq. (19)) was established by Yang et al. [53] according to the cubic representative unit cell. This model offers the capability of estimating the permeability for a wide range of foam porosities

� �

<sup>K</sup> <sup>¼</sup> <sup>ε</sup><sup>2</sup>d<sup>2</sup>

<sup>F</sup> <sup>¼</sup> <sup>2</sup>:05χ χð Þ � <sup>1</sup> ε<sup>2</sup>ð Þ 3 � χ

<sup>ε</sup>3<sup>d</sup> <sup>ρ</sup>u<sup>2</sup>

ffiffiffi K p

ReDP >> 300 : f ¼ 0:22

permeability and inertial coefficient according to Darcy-–Forchheimer's equation:

dp dx <sup>¼</sup> <sup>μ</sup>

<sup>L</sup> <sup>¼</sup> <sup>α</sup> ð Þ <sup>1</sup>�<sup>ε</sup> <sup>2</sup>

<sup>f</sup> <sup>¼</sup> ð Þ <sup>Δ</sup>P=<sup>L</sup> DP ρu<sup>2</sup>

K p

ReDP

, and ReDP <sup>¼</sup> <sup>ρ</sup><sup>u</sup> ffiffiffi

<sup>ρ</sup>u<sup>2</sup> , and Re<sup>K</sup> <sup>¼</sup> <sup>ρ</sup><sup>u</sup> ffiffiffi

þ 0:22

9 >=

>;

K p μ

<sup>p</sup> <sup>u</sup><sup>2</sup> (9)

, Tadrist et al. [56] developed an

<sup>36</sup>χ χð Þ � <sup>1</sup> (10)

<sup>d</sup> (11)

K p

<sup>μ</sup> (7)

(8)

$$K = 0.00073d\_p^2(1 - \varepsilon)^{-0.224} \left(\frac{d\_f}{d\_p}\right)^{-1.11} \tag{12}$$

$$F = 0.00212(1 - \varepsilon)^{-0.132} \left(\frac{d\_f}{d\_p}\right)^{-1.63} \tag{13}$$

Bhattacharya et al. [52]:

$$\begin{aligned} F &= 0.095 \frac{\mathbb{C}\_{D(\varepsilon = 0.85)}}{12} G^{0.2} \sqrt{\frac{\varepsilon}{3(\chi - 1)}} \left( 1.18 \sqrt{\frac{(1 - \varepsilon)}{3\pi}} \frac{1}{G} \right)^{-1} \\\ 0.85 &< \varepsilon < 0.97: \quad G = 1 - e^{-(1 - \varepsilon)/0.04} \Bigg\} \quad \text{and} \quad \mathbb{C}\_{D(\varepsilon = 0.85)} = 1.2 \end{aligned} \tag{14}$$

Tadrist et al. [56]:

$$K = \frac{\varepsilon^3 d\_f^2}{a(1-\varepsilon)^2}, \quad \alpha: 100 \sim 865 \tag{15}$$

$$F = \frac{\beta (1 - \varepsilon)}{\varepsilon^3} \frac{\sqrt{K}}{d\_f}, \quad \beta: 0.65 \sim 2.6$$

Dukhan [57]:

$$K = a\_1 e^{b\_1 e} \tag{17}$$

$$F = (a\_2 \varepsilon + b\_2) \sqrt{K} \tag{18}$$

a1, a2, b1, and b<sup>2</sup> are all constants

Yang et al. [53]:

$$\frac{K}{d^2} = \frac{\varepsilon \left(1 - (1 - \varepsilon)^{1/3}\right)^2}{36\beta \left[\left(1 - \varepsilon\right)^{1/3} - \left(1 - \beta\varepsilon\right)\right]}\tag{19}$$

#### 4.4. Effective thermal conductivity models for high-porosity metal foams

Many investigations have been conducted to evaluate the effective thermal conductivity ke as a key factor in the thermal analysis of such systems. Among them is the model established by Paek et al. [54] based on one-dimensional heat conduction through a cubic unit cell (Eq. (20)). This model indicates that foam porosity has a direct impact on the overall thermal conductivity unlike the pore size, which was found to have a marginal influence.

The theoretical model derived by Calmidi and Mahajan [58] for the effective thermal conductivity as a function of the foam porosity (Eq. (21)) was found to match well with the experiments for both air and water as fluid phase. Similarly, Bhattacharya et al. [52] examined a two-dimensional unit cell shaped as a hexagonal honey comb to estimate the effective thermal conductivity, but taking into account circular nodes at each intersection joint rather than the square nodes considered earlier by Calmidi and Mahajan [58]. Although this model showed excellent agreement with the experimental data obtained and its formulations detailed in Eq. (22) look mathematically simpler than that derived by Calmidi and Mahajan [58], it is in fact more complicated because this model is valid only for a limited case when the intersection size R approaches zero, which implies that r ! ∞.

Bhattacharya et al. [52]:

ke <sup>¼</sup> <sup>2</sup>

Boomsma and Poulikakos [59]:

8 < :

ffiffiffi 3 p

ke ¼

RC ¼

d ¼

Dai et al. [60]:

Yang et al. [61]:

Yao et al. [62]:

RD <sup>¼</sup> <sup>2</sup><sup>e</sup>

2

RC ¼

d ¼

2 <sup>p</sup> <sup>π</sup>d<sup>2</sup>

2

2 4

t L kf <sup>þ</sup> ð Þ ks�kf 3

þ ffiffi 3 p <sup>2</sup> � <sup>t</sup> L kf

ffiffiffi 2 p

ð Þ RA þ RB þ RC þ RD

RA <sup>¼</sup> <sup>4</sup><sup>d</sup>

RB <sup>¼</sup> ð Þ <sup>e</sup> � <sup>2</sup><sup>d</sup> <sup>2</sup>

<sup>2</sup>πd<sup>2</sup> <sup>1</sup> � <sup>2</sup><sup>e</sup> ffiffiffi

e<sup>2</sup>ks þ 4 � e<sup>2</sup> ð Þkf

<sup>p</sup> <sup>2</sup> � ð Þ <sup>5</sup>=<sup>8</sup> <sup>e</sup><sup>3</sup> ffiffiffi

<sup>p</sup> � <sup>e</sup> � � <sup>s</sup>

<sup>π</sup> <sup>3</sup> � <sup>4</sup><sup>e</sup> ffiffiffi

3 5

9 = ;

�1 , <sup>t</sup>

2e ½ � <sup>2</sup> þ πdð Þ 1 � e ks þ 4 � 2e ½ � <sup>2</sup> � πdð Þ 1 � e kf

ffiffiffi 2 <sup>p</sup> � <sup>2</sup><sup>e</sup> � �<sup>2</sup>

2

<sup>p</sup> � <sup>2</sup><sup>e</sup> � <sup>π</sup>d<sup>2</sup> <sup>1</sup> � <sup>2</sup><sup>e</sup> ffiffiffi <sup>2</sup> � � � � <sup>p</sup> kf

, and, e ¼ 0:339

, and, e ¼ 0:198

ð Þ e � 2d e<sup>2</sup>ks þ 2e � 4d � ð Þ e � 2d e<sup>2</sup> ½ �kf

2 <sup>p</sup> � <sup>2</sup><sup>ε</sup> � �

2

ffiffiffi 2 <sup>p</sup> � <sup>2</sup><sup>e</sup> ffiffiffi

ks <sup>þ</sup> <sup>2</sup> � ffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi

<sup>p</sup> <sup>2</sup> � ð Þ <sup>3</sup>=<sup>4</sup> <sup>e</sup><sup>3</sup> ffiffiffi

<sup>p</sup> � <sup>e</sup> � � <sup>s</sup>

<sup>π</sup> <sup>3</sup> � <sup>2</sup><sup>e</sup> ffiffiffi

2 <sup>p</sup> <sup>π</sup>d<sup>2</sup> � �kf

2 <sup>p</sup> � <sup>2</sup><sup>ε</sup> � �

2

ke <sup>¼</sup> <sup>1</sup> 3

<sup>2</sup> � � <sup>p</sup> ks <sup>þ</sup> <sup>2</sup> ffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffiffi

<sup>L</sup> <sup>¼</sup> � ffiffiffi 3 <sup>p</sup> � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>3</sup> <sup>þ</sup> ð Þ <sup>1</sup> � <sup>ε</sup> ffiffiffi

http://dx.doi.org/10.5772/intechopen.70451

9

>>>>>>>>>>>>>>>>>>>>>>>>>=

>>>>>>>>>>>>>>>>>>>>>>>>>;

9 >>>>>=

>>>>>;

ð Þ 1 � ε ks (25)

q � �

<sup>1</sup> <sup>þ</sup> <sup>1</sup>ffiffi 3 <sup>p</sup> � <sup>8</sup> 3

High-Porosity Metal Foams: Potentials, Applications, and Formulations

3 <sup>p</sup> � <sup>5</sup>

(22)

193

(23)

(24)

A tetrakaidecahedron unit cell was adopted by Boomsma and Poulikakos [59] to estimate the effective thermal conductivity. Despite the fact that taking tetrakaidecahedron topology into account can provide a better estimation, it leads to more complex formulae, detailed in Eq. (23). Moreover, it was found that this model includes some aspects need to be adjusted as found by Dai et al. [60] (Eq. (24)). Based on the tetrakaidecahedron unit cell as well but with assuming one-dimensional heat conduction along the highly tortuous ligaments, a quite simplistic model (Eq. (25)) was recently derived by Yang et al. [61] for effective thermal conductivity of metal foams saturated with low conducting fluids, for example, air. However, this model is limited to highly conducting foams, where heat conduction is assumed to occur only along the tortuous ligaments ignoring the heat conduction through the fluid phase. More recently, the 3D tetrakaidecahedron unit cell was considered by Yao et al. [62] to establish a more realistic formulation for effective thermal conductivity (Eq. (26)) through taking into account the concavity and orientation of the tri-prism ligaments and for four pyramids nodes. In addition to including no empirical parameters, this model has outperformed, in terms of accuracy, what were reported earlier in the literature [62].

Paek et al. [54]:

$$k\_{\varepsilon} = k\_{\hat{f}}(1-t)^2 + k\_{\ast}t^2 + \frac{2t(1-t)k\_{\hat{f}}k\_{\ast}}{k\_{\hat{f}}t + k\_{\ast}(1-t)}, \quad t = \frac{1}{2} + \cos\left(\frac{1}{3}\cos^{-1}(2\varepsilon - 1) + \frac{4\pi}{3}\right) \tag{20}$$

Calmidi and Mahajan [58]:

$$k\_{\varepsilon} = \left\{ \frac{2}{\sqrt{3}} \left[ \frac{r(\mathbf{\hat{t}})}{k\_{\parallel} + \left(1 + \frac{\mathbf{\hat{t}}}{\mathbf{\hat{t}}}\right)^{\left(\mathbf{\hat{t}} - \frac{\mathbf{\hat{t}}}{\mathbf{\hat{t}}}\right)}} + \frac{(1 - r)(\mathbf{\hat{t}})}{k\_{\parallel} + \frac{\mathbf{\hat{t}}}{\mathbf{\hat{t}}}(\mathbf{\hat{t}})(k\_{\parallel} - k\_{\parallel})} + \frac{\frac{\mathbf{\hat{t}}}{\mathbf{\hat{t}}} - \frac{\mathbf{\hat{t}}}{\mathbf{\hat{t}}}}{k\_{\parallel} + \frac{\mathbf{\hat{t}}}{\mathbf{\hat{t}}}(\mathbf{\hat{t}})(k\_{\parallel} - k\_{\parallel})} \right] \right\}^{-1} $$
 
$$\frac{b}{L} = \frac{-r + \sqrt{r^2 + \frac{2}{\sqrt{3}}(1 - \varepsilon)\left[2 - r\left(1 + \frac{\mathbf{4}}{\sqrt{3}}\right)\right]}}{\frac{2}{3}\left[2 - r\left(1 + \frac{\mathbf{4}}{\sqrt{3}}\right)\right]}, \quad and, r = 0.09$$

Bhattacharya et al. [52]:

The theoretical model derived by Calmidi and Mahajan [58] for the effective thermal conductivity as a function of the foam porosity (Eq. (21)) was found to match well with the experiments for both air and water as fluid phase. Similarly, Bhattacharya et al. [52] examined a two-dimensional unit cell shaped as a hexagonal honey comb to estimate the effective thermal conductivity, but taking into account circular nodes at each intersection joint rather than the square nodes considered earlier by Calmidi and Mahajan [58]. Although this model showed excellent agreement with the experimental data obtained and its formulations detailed in Eq. (22) look mathematically simpler than that derived by Calmidi and Mahajan [58], it is in fact more complicated because this model is valid only for a limited case when the intersection size R approaches zero, which implies

A tetrakaidecahedron unit cell was adopted by Boomsma and Poulikakos [59] to estimate the effective thermal conductivity. Despite the fact that taking tetrakaidecahedron topology into account can provide a better estimation, it leads to more complex formulae, detailed in Eq. (23). Moreover, it was found that this model includes some aspects need to be adjusted as found by Dai et al. [60] (Eq. (24)). Based on the tetrakaidecahedron unit cell as well but with assuming one-dimensional heat conduction along the highly tortuous ligaments, a quite simplistic model (Eq. (25)) was recently derived by Yang et al. [61] for effective thermal conductivity of metal foams saturated with low conducting fluids, for example, air. However, this model is limited to highly conducting foams, where heat conduction is assumed to occur only along the tortuous ligaments ignoring the heat conduction through the fluid phase. More recently, the 3D tetrakaidecahedron unit cell was considered by Yao et al. [62] to establish a more realistic formulation for effective thermal conductivity (Eq. (26)) through taking into account the concavity and orientation of the tri-prism ligaments and for four pyramids nodes. In addition to including no empirical parameters, this model has outperformed, in terms of accuracy, what were

that r ! ∞.

192 Porosity - Process, Technologies and Applications

reported earlier in the literature [62].

ke <sup>¼</sup> kfð Þ <sup>1</sup> � <sup>t</sup> <sup>2</sup> <sup>þ</sup> kst

ke <sup>¼</sup> <sup>2</sup>ffiffi 3

�r þ

b L ¼

Calmidi and Mahajan [58]:

2 þ

<sup>p</sup> <sup>r</sup> <sup>b</sup> ð Þ<sup>L</sup> kf <sup>þ</sup> <sup>1</sup> <sup>þ</sup> <sup>b</sup> ð Þ<sup>L</sup>

> r<sup>2</sup> þ 2 ffiffiffi 3

> > 2

2tð Þ 1 � t kf ks kf <sup>t</sup> <sup>þ</sup> ksð Þ <sup>1</sup> � <sup>t</sup> , t <sup>¼</sup> <sup>1</sup>

> ks � <sup>k</sup> ð Þ<sup>f</sup> 3

<sup>3</sup> <sup>2</sup> � <sup>r</sup> <sup>1</sup> <sup>þ</sup>

<sup>þ</sup> ð Þ <sup>1</sup> � <sup>r</sup> <sup>b</sup> ð Þ<sup>L</sup> kf <sup>þ</sup> <sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

4 ffiffiffi 3 p

p ð Þ 1 � ε 2 � r 1 þ

s � � � �

<sup>3</sup> <sup>b</sup> ð Þ<sup>L</sup> ð Þ ks � kf

( ) " # �<sup>1</sup>

<sup>2</sup> <sup>þ</sup> cos

þ

4 ffiffiffi 3 p

� � � � , and, r <sup>¼</sup> <sup>0</sup>:<sup>09</sup>

kf <sup>þ</sup> <sup>4</sup><sup>r</sup> 3 ffi 3 <sup>p</sup> <sup>b</sup> ð Þ<sup>L</sup> ð Þ ks � kf

1

ffi 3 p <sup>2</sup> � <sup>b</sup> L

<sup>3</sup> cos �<sup>1</sup>ð Þþ <sup>2</sup><sup>ε</sup> � <sup>1</sup> <sup>4</sup><sup>π</sup>

� �

3

(20)

(21)

Paek et al. [54]:

$$k\_{\varepsilon} = \left\{ \frac{2}{\sqrt{3}} \left[ \frac{\frac{t}{L}}{k\_f + \frac{\left(k\_- - k\_f\right)}{3}} + \frac{\frac{\sqrt{3}}{2} - \frac{t}{L}}{k\_f} \right] \right\}^{-1}, \quad \frac{t}{L} = \frac{-\sqrt{3} - \sqrt{3 + (1 - \varepsilon)\left(\sqrt{3} - 5\right)}}{1 + \frac{1}{\sqrt{3}} - \frac{8}{3}} \tag{22}$$

Boomsma and Poulikakos [59]:

$$\begin{aligned} k\_c &= \frac{\sqrt{2}}{(R\_A + R\_B + R\_C + R\_D)} \\ R\_A &= \frac{4d}{[2e^2 + \pi d(1 - e)]k\_s + [4 - 2e^2 - \pi d(1 - e)]k\_f} \\\\ R\_B &= \frac{(e - 2d)^2}{(e - 2d)e^2 k\_s + [2e - 4d - (e - 2d)e^2]k\_f} \\\\ R\_C &= \frac{(\sqrt{2} - 2e)^2}{2\pi d^2 \left(1 - 2e\sqrt{2}\right)k\_s + 2\left[\sqrt{2} - 2e - \pi d^2 \left(1 - 2e\sqrt{2}\right)\right]k\_f} \\\\ R\_D &= \frac{2e}{e^2 k\_s + (4 - e^2)k\_f} \\\\ d &= \sqrt{\frac{\sqrt{2}\left[2 - (5/8)e^3\sqrt{2} - 2e\right]}{\pi\left(3 - 4e\sqrt{2} - e\right)}}, \text{and}, \ c = 0.339 \end{aligned} \tag{23}$$

Dai et al. [60]:

$$\begin{aligned} R\_{\mathbb{C}} &= \frac{\sqrt{2} - 2\varepsilon}{\sqrt{2}\pi d^2 k\_s + \left(2 - \sqrt{2}\pi d^2\right) k\_f} \\\\ d &= \sqrt{\frac{\sqrt{2}\left[2 - (3/4)e^3\sqrt{2} - 2\varepsilon\right]}{\pi\left(3 - 2e\sqrt{2} - \varepsilon\right)}}, \text{and}, \quad e = 0.198 \end{aligned} \tag{24}$$

Yang et al. [61]:

$$k\_{\varepsilon} = \frac{1}{3}(1 - \varepsilon)k\_{s} \tag{25}$$

Yao et al. [62]:

$$\begin{aligned} k\_c &= \frac{1}{(\lambda/k\_A) + ((1-2\lambda)/k\_B) + (\lambda/k\_C)}\\ k\_A &= \frac{\sqrt{2}}{6}\pi\lambda(3-4\lambda)\frac{1+a\_1^2}{a\_1^2}k\_s + \left[1 - \frac{\sqrt{2}}{6}\pi\lambda(3-4\lambda)\frac{1+a\_1^2}{a\_1^2}\right]k\_f\\ k\_B &= \frac{\sqrt{2}}{2}\pi\lambda^2\frac{1+a\_1^2}{a\_1^2}k\_s + \left(1 - \sqrt{2}\pi\lambda^2\frac{1+a\_1^2}{a\_1^2}\right)k\_f\\ k\_C &= \frac{\sqrt{2}}{6}\pi\lambda^2\frac{1+a\_1^2}{a\_1^2}k\_s + \left(1 - \frac{\sqrt{2}}{6}\pi\lambda^2\frac{1+a\_1^2}{a\_1^2}\right)k\_f\\ \varepsilon &= 1 - \frac{\sqrt{2}}{2}\pi\lambda^2(3-5\lambda)\frac{1+a\_1^2}{a\_1^2}, \quad and, \quad a\_1 = 2.01\end{aligned} \tag{26}$$

asf <sup>¼</sup> <sup>3</sup>πdf 0:59dp � �<sup>2</sup> <sup>1</sup> � <sup>e</sup>

> asf <sup>¼</sup> <sup>3</sup> d

Mahajan [63] as it is valid for a limited range of Reynolds numbers (40–1000).

NuDP <sup>¼</sup> hDP kse

> 8 >>>><

> >>>>:

<sup>d</sup> <sup>¼</sup> <sup>1</sup> � <sup>e</sup>�ð Þ <sup>1</sup>�<sup>ε</sup> <sup>=</sup>0:<sup>04</sup> � � df

\*Address all correspondence to: ahmedn.alhusseini@uokufa.edu.iq

<sup>¼</sup> CTRe<sup>0</sup>:<sup>5</sup> df

0:76Re0:<sup>4</sup>

0:52Re0:<sup>5</sup>

0:26Re0:<sup>6</sup>

Ahmed Niameh Mehdy Alhusseny1,2\*, Adel Gharib Nasser2 and Nabeel M J Al-zurf<sup>1</sup>

1 Mechanical Engineering Department, Faculty of Engineering, University of Kufa, Iraq 2 School of Mechanical, Aerospace and Civil Engineering, University of Manchester, UK

<sup>¼</sup> <sup>a</sup>Re<sup>b</sup>

Pr<sup>0</sup>:<sup>37</sup> <sup>¼</sup> <sup>0</sup>:<sup>52</sup>

DP <sup>¼</sup> <sup>a</sup> <sup>ρ</sup>uDP μ � �<sup>b</sup>

<sup>d</sup> Pr<sup>0</sup>:<sup>37</sup>, ð Þ <sup>1</sup> <sup>≤</sup>Re<sup>d</sup> <sup>≤</sup> <sup>40</sup>

<sup>d</sup> Pr<sup>0</sup>:<sup>37</sup>, <sup>40</sup> <sup>≤</sup>Re<sup>d</sup> <sup>≤</sup> 103 � �

<sup>d</sup> Pr<sup>0</sup>:<sup>37</sup>, <sup>10</sup><sup>3</sup> <sup>≤</sup> Re<sup>d</sup> <sup>≤</sup> <sup>2</sup> � 105 � �

ffiffiffiffiffiffiffi udf ευ r

Pr<sup>0</sup>:<sup>37</sup> (29)

9 >>>>=

>>>>;

(30)

(31)

Nusf <sup>¼</sup> hsf df kf

Nusf <sup>¼</sup> hsf <sup>d</sup> kf ¼

With regard to estimating the solid–fluid interfacial heat transfer coefficient in high-porosity metal foams, Calmidi and Mahajan [63] proposed a correlation for interfacial Nusselt number as a function of both the foam porosity and fiber diameter (Eq. (29)). Another model was established by Shih et al. [28] for Nusselt number as a function of the foam porosity and pore diameter (Eq. (30)), where a and b are constants depending on the geometrical characteristics of the foam samples used in the experiments conducted. The correlations developed by Zukauskas [64] for staggered cylinders are widely used as a model to predict the interfacial Nusselt number as a function of the foam porosity, fiber diameter, and the value of Reynolds number (Eq. (31)), which makes it more general than the one proposed by Calmidi and

Fourie and Du Plessis [49]:

Calmidi and Mahajan [63]:

Shih et al. [28]:

Zukauskas [64]:

Author details

�ð Þ <sup>1</sup>�<sup>ε</sup> <sup>=</sup>0:<sup>04</sup> h i (27)

http://dx.doi.org/10.5772/intechopen.70451

195

High-Porosity Metal Foams: Potentials, Applications, and Formulations

ð Þ 3 � χ ð Þ χ � 1 (28)

#### 4.5. Solid–fluid interstitial thermal exchange within open-cell metal foams

The condition of local thermal equilibrium (LTE) often occurs between the fluid and solid phases when a fluid flows across a permeable medium formed of comparably thermally conductive material. This makes the temperature difference between the two phases negligibly small. However, when the solid thermal conductivity is much higher than the corresponding value for the fluid phase, for example, metal foams, the assumption of LTE is no longer valid and usually results in an overestimation of the heat transported between the two phases. Hence, taking into account the local thermal nonequilibrium (LTNE) between the two phases becomes indispensable in metal foams, where two energy equations are coupled together to predict heat transfer in each phase separately.

Three principal heat transfer modes take place when a low conductive fluid flows across the ligaments of highly conductive foam: convection between the solid and fluid phases besides conduction via each one of the two phases. Thus, the three key parameters required for applying the LTNE approach are the effective thermal conductivity of the fluid kfe and solid kse phases in addition to the interstitial specific heat transfer rate between the two phases (asf hsf), which depends on the foam structure and the flow regime across it.

The interstitial heat exchange rate depends on two individual quantities: the interfacial specific surface area asf and the solid-to-fluid interfacial heat transfer coefficient hsf. By utilizing the dodecahedral structure of open-cell foams and taking into account the noncircular fiber cross section, the solid-to-fluid interfacial specific surface area asf was modeled by Calmidi and Mahajan [63] for arrays of cylinders that intersect in three mutually perpendicular directions (Eq. (27)), while Fourie and Du Plessis [49] established another model based on the cubic unitcell representation (Eq. (28)). However, Schampheleire et al. [6] observed that the asf values estimated using Eq. (27) by Calmidi and Mahajan [63] deviates seriously from those obtained experimentally through a µCT scan with differences up to 233%, while the model of Fourie and Du Plessis [49] performs much better with up to 22% deviation from the experimental data of the full µCT scan.

Calmidi and Mahajan [63]:

High-Porosity Metal Foams: Potentials, Applications, and Formulations http://dx.doi.org/10.5772/intechopen.70451 195

$$a\_{sf} = \frac{3\pi d\_f}{\left(0.59d\_p\right)^2} \left[1 - e^{-(1-\epsilon)/0.04}\right] \tag{27}$$

Fourie and Du Plessis [49]:

ke <sup>¼</sup> <sup>1</sup>

ffiffiffi 2 p

ffiffiffi 2 p

ffiffiffi 2 p

<sup>2</sup> πλ<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

<sup>6</sup> πλ<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

ffiffiffi 2 p <sup>2</sup> πλ<sup>2</sup>

kA ¼

194 Porosity - Process, Technologies and Applications

kB ¼

kC ¼

ε ¼ 1 �

predict heat transfer in each phase separately.

the full µCT scan.

Calmidi and Mahajan [63]:

ð Þþ λ=kA ð Þþ ð Þ 1 � 2λ =kB ð Þ λ=kC

1 a2 1

ks <sup>þ</sup> <sup>1</sup> � ffiffiffi

ks þ 1 �

ð Þ <sup>3</sup> � <sup>5</sup><sup>λ</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

4.5. Solid–fluid interstitial thermal exchange within open-cell metal foams

hsf), which depends on the foam structure and the flow regime across it.

ks þ 1 �

2 <sup>p</sup> πλ<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

ffiffiffi 2 p

1 a2 1

The condition of local thermal equilibrium (LTE) often occurs between the fluid and solid phases when a fluid flows across a permeable medium formed of comparably thermally conductive material. This makes the temperature difference between the two phases negligibly small. However, when the solid thermal conductivity is much higher than the corresponding value for the fluid phase, for example, metal foams, the assumption of LTE is no longer valid and usually results in an overestimation of the heat transported between the two phases. Hence, taking into account the local thermal nonequilibrium (LTNE) between the two phases becomes indispensable in metal foams, where two energy equations are coupled together to

Three principal heat transfer modes take place when a low conductive fluid flows across the ligaments of highly conductive foam: convection between the solid and fluid phases besides conduction via each one of the two phases. Thus, the three key parameters required for applying the LTNE approach are the effective thermal conductivity of the fluid kfe and solid kse phases in addition to the interstitial specific heat transfer rate between the two phases (asf

The interstitial heat exchange rate depends on two individual quantities: the interfacial specific surface area asf and the solid-to-fluid interfacial heat transfer coefficient hsf. By utilizing the dodecahedral structure of open-cell foams and taking into account the noncircular fiber cross section, the solid-to-fluid interfacial specific surface area asf was modeled by Calmidi and Mahajan [63] for arrays of cylinders that intersect in three mutually perpendicular directions (Eq. (27)), while Fourie and Du Plessis [49] established another model based on the cubic unitcell representation (Eq. (28)). However, Schampheleire et al. [6] observed that the asf values estimated using Eq. (27) by Calmidi and Mahajan [63] deviates seriously from those obtained experimentally through a µCT scan with differences up to 233%, while the model of Fourie and Du Plessis [49] performs much better with up to 22% deviation from the experimental data of

� �

!

<sup>6</sup> πλ<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

, and, a<sup>1</sup> ¼ 2:01

ffiffiffi 2 p

> 1 a2 1

> > 1 a2 1

<sup>6</sup> πλð Þ <sup>3</sup> � <sup>4</sup><sup>λ</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

" #

kf

kf

1 a2 1

kf

9

>>>>>>>>>>>>>>>>>=

(26)

>>>>>>>>>>>>>>>>>;

<sup>6</sup> πλð Þ <sup>3</sup> � <sup>4</sup><sup>λ</sup> <sup>1</sup> <sup>þ</sup> <sup>a</sup><sup>2</sup>

1 a2 1

1 a2 1

$$a\_{\circ f} = \frac{\mathfrak{Z}}{d} (\mathfrak{Z} - \chi)(\chi - 1) \tag{28}$$

With regard to estimating the solid–fluid interfacial heat transfer coefficient in high-porosity metal foams, Calmidi and Mahajan [63] proposed a correlation for interfacial Nusselt number as a function of both the foam porosity and fiber diameter (Eq. (29)). Another model was established by Shih et al. [28] for Nusselt number as a function of the foam porosity and pore diameter (Eq. (30)), where a and b are constants depending on the geometrical characteristics of the foam samples used in the experiments conducted. The correlations developed by Zukauskas [64] for staggered cylinders are widely used as a model to predict the interfacial Nusselt number as a function of the foam porosity, fiber diameter, and the value of Reynolds number (Eq. (31)), which makes it more general than the one proposed by Calmidi and Mahajan [63] as it is valid for a limited range of Reynolds numbers (40–1000).

Calmidi and Mahajan [63]:

$$\mathrm{Nu}\_{\mathrm{sf}} = \frac{h\_{\mathrm{sf}} d\_{\mathrm{f}}}{k\_{\mathrm{f}}} = \mathrm{C}\_{T} \mathrm{Re}\_{d\_{\mathrm{f}}}^{0.5} \mathrm{Pr}^{0.37} = 0.52 \sqrt{\frac{\mathrm{u} d\_{\mathrm{f}}}{\varepsilon \nu}} \mathrm{Pr}^{0.37} \tag{29}$$

Shih et al. [28]:

$$Nu\_{D\_P} = \frac{hD\_P}{k\_{\rm sc}} = a \text{Re}\_{D\_P}^b = a \left(\frac{\rho \, uD\_P}{\mu}\right)^b \tag{30}$$

Zukauskas [64]:

$$\mathrm{Nu}\_{\mathrm{sf}} = \frac{h\_{\mathrm{sf}}d}{k\_{\mathrm{f}}} = \begin{cases} 0.76 \mathrm{Re}\_{d}^{0.4} \mathrm{Pr}^{0.37}, \ (1 \le \mathrm{Re}\_{d} \le 40) \\\\ 0.52 \mathrm{Re}\_{d}^{0.5} \mathrm{Pr}^{0.37}, \ (40 \le \mathrm{Re}\_{d} \le 10^{3}) \\\\ 0.26 \mathrm{Re}\_{d}^{0.6} \mathrm{Pr}^{0.37}, \ \{10^{3} \le \mathrm{Re}\_{d} \le 2 \times 10^{5}\} \end{cases} \tag{31}$$
  $d = \left(1 - e^{-(1-\varepsilon)/0.04}\right)$   $d\_{f}$ 

### Author details

Ahmed Niameh Mehdy Alhusseny1,2\*, Adel Gharib Nasser2 and Nabeel M J Al-zurf<sup>1</sup> \*Address all correspondence to: ahmedn.alhusseini@uokufa.edu.iq

1 Mechanical Engineering Department, Faculty of Engineering, University of Kufa, Iraq

2 School of Mechanical, Aerospace and Civil Engineering, University of Manchester, UK
