## Abstract

The objective of this study is to investigate the impact of different porous metal samples on the hydro-thermal characteristics of a single cylinder with porous fins using computational fluid dynamics. Commercially used porous samples with pore densities of 10, 20, and 40 PPI were used in this study for heat recovery from exhaust flue gas. The three-dimensional computational domain with porous aluminum fins attached to a tube over which high-temperature exhaust gas flows in a crossflow arrangement mimics a waste heat recovery system. Computations were performed at Reynolds number of 6000–9000, using the realizable κ-ϵ turbulence model. Three fin diameter-to-tube diameter ratios (D_{f} /D = 2, 2.5, and 3) were considered. The local thermal nonequilibrium model is implemented for energy transfer, as it is more accurate for a high-temperature gradient scenario in a waste heat recovery system. The foam sample with the highest pore density was observed to have the highest pressure drop due to low permeability. A maximum heat transfer and Nusselt number were achieved for a 40 PPI foam sample due to a reduced flowrate inside the porous zone. The overall performance of metal foam samples at varying fin diameters was evaluated based on the area goodness factor (j/f) and a heat transfer coefficient ratio to pumping power per unit heat transfer surface (*Z*/*E*). The analysis of these two parameters suggests using 20 PPI foam at D_{f} /D = 2.

## 1 Introduction

Energy is an essential commodity for humankind. In developing countries like India, the demand for energy is increasing rapidly. India is the third largest consumer of energy, and energy demands have doubled since 2000. The majority of this demand is still fulfilled by coal and oil. As of 2022, the country's power sector was strongly dependent on fossil fuels (57.7%), with coal having the highest share at 49.3%. Burning fossil fuels leads to CO_{2} and NO* _{x}* emissions, thereby making several environmental impacts like global warming and air pollution. The dependence on fossil fuels and their environmental effects can be reduced by using renewable energy sources [1,2] and waste heat recovery techniques [3,4]. The exhaust gas of an internal combustion engine dissipates nearly 60–70% of fuel energy into the atmosphere as waste [5]. This discarded energy can reduce fuel requirements if appropriately utilized. Hence, waste heat recovery is beneficial rather than discharging it into the atmosphere.

Waste heat from fossil fuel combustion can be used for heat exchangers, heat pipes, air preheaters, and many other thermal transport technologies [6] and in aircraft applications [7,8]. However, in the case of most exhaust gas heat exchangers, the heat transfer coefficient on the exhaust gas side tends to be lower than that on the cooling medium side. This is primarily due to the impurities present in flue gas [9,10]. These impurities include CO_{2}, water vapor, N_{2}, O_{2}, SO_{X}, NO_{X,} and particulate matter (ash). The exact composition of these impurities depends on the type of fuel used for combustion [11] and will alter the heat transfer coefficient. A low heat transfer coefficient significantly reduces the efficiency of a heat exchanger. Therefore, using different techniques for heat transfer augmentation in a waste heat recovery system is vital. Prominent passive techniques for heat transfer augmentation include fins [12,13], microchannels [14], porous media [15,16], and nanofluids [17]. Among these, fins are the most common passive heat transfer enhancement mode and have been used for a long time. However, the weight of a heat exchanger is significantly increased by implication of fins. A recent study from our research group focused on using trimmed annular fins for material saving and weight reduction in crossflow heat exchangers [18,19]. Conventional solid fins are not feasible in waste heat recovery systems of aircraft and automobiles due to a significant increase in weight, noncompact structure, and thermal wearing at high temperatures. Utilizing fins made from high-porosity metal foams can overcome this issue and provide substantial advantages over solid fins. Utilizing porous media is proven to be a prominent passive heat transfer enhancement technique. Researchers have used porous media as fin, wrapping, and inserts for enhanced heat transfer applications. Some key characteristics of porous metal foams include (i) low weight, (ii) high surface-to-volume ratio, (iii) resistance to high-temperature wear and humidity, and (iv) noise attenuation [20]. The tortuous structure of an open-cell metal foam aids fluid mixing, thereby raising heat exchanger performance [21].

In recent decades, porous media has attracted many researchers. In addition to an increment in heat transfer, porous media is also used as a passive technique to control flow oscillations, flow-induced vibrations, and vortex shedding from a bluff body [22,23]. An extensive review of thermal management on heat exchangers using porous materials was carried out by Rashidi et al. [24] and Habibishandiz and Saghir [25]. Mahjoob and Vafai [26] investigated various fluid and thermal transport models within metal foam heat exchangers. Their analysis indicated that employing the local thermal nonequilibrium (LTNE) model for energy transfer in porous materials would be advantageous. The forced convection heat transfer using high-porosity metal foams was experimentally and numerically studied by Calmidi and Mahajan [27]. The main objective of this study was to determine thermal dispersion and thermal nonequilibrium effects in metal foams. Their results suggest that the thermal dispersion is low for a considerable difference in fluid and solid medium thermal conductivity. An experimental study to investigate the impact of porous material in a plate-fin heat exchanger was carried out by Kim et al. [28]. Their study involved six different aluminum-alloy porous materials with different porosity and permeability. This study also gave correlations to predict friction factor and heat transfer based on the results of experimental data. Feng et al. [29] did a numerical and experimental analysis on finned metal foam heat sinks for jet impingement. In this study, aluminum foams were used, and the Nusselt number and pressure drop were observed to depend on the height of the foam sample.

Due to their dual advantage of flow control and heat transfer enhancement, researchers used porous wrapping over solid surfaces [30–32]. Experimental studies by Chumpia and Hooman [33,34] on a single tube wrapped with porous material showed that the foam-wrapped heat exchanger can aid heat transfer while keeping pressure drop at the same magnitude as a finned tube. Boules et al. [35] did an experimental study investigating heat transfer enhancement from a cylinder with complete and segmented porous wrapping. Their results concluded that applying metal foams significantly enhances heat transfer compared to bare cylinders at the expense of pressure drop. A numerical study on forced convection from a metal foam-wrapped cylinder in a waste heat recovery system using high-temperature exhaust gas was carried out by Wang et al. [36]. Their study used the Darcy–Brinkman–Forchheimer (DBF) momentum equations to model the flow inside porous media. This study advises the use of porous wrapping in a waste heat recovery system to get higher heat transfer than a bare cylinder. The presence of porous material changes the flow structure downstream of a cylinder and influences the pressure field. Chen et al. [37] did an experimental analysis to study the thermal performance of staggered tube bundles wrapped with metallic foams. On the basis of experimental results, they proposed correlations to determine the friction factor and Nusselt number for an array of porous-wrapped tubes. Mao et al. [38] gave correlations to predict permeability, friction factor, form drag coefficient, and overall heat transfer coefficient for different metal foams. They also did a numerical analysis to check the consistency of predicted correlations. Odabaee and Hooman [39] conducted a numerical study on a four-row staggered tube bundle with metal foam wrapping. They concluded that, for porous-wrapped tubes, the air-side thermal resistance could be significantly reduced than conventional solid finned tubes. However, a rise in pressure drop is a crucial drawback in the case of porous wrappings. To overcome this disadvantage, Alvandifar et al. [40] used partial metal foam wrapping in a five-row tube bundle. They conclude that partial wrapping of porous media gives nearly the same Nusselt number, while the pressure drop is significantly reduced.

In addition to porous wrappings, researchers have also utilized porous fins to investigate the thermal behavior of heat exchangers [41,42]. A numerical investigation on forced convective heat transfer from a cylinder fitted with longitudinal permeable fins was conducted by Abu-Hijleh [43]. Their results concluded that permeable fins gave a higher Nusselt number than solid fins. Due to the wakes formed, the fins downstream are observed to be less effective than their upstream counterpart. A three-dimensional comparative study on the performance of solid and porous fins for exhaust gas heat exchanger was done by Zhao et al. [10]. In their research, the LTNE model was implemented to study the thermal characteristics of a porous foam sample. It was concluded that porous fins can enhance heat transfer performance by nearly 77% compared to their solid counterpart. It was also stated that the porous samples give a higher pressure drop (almost 18%) than solid fins. An experimental analysis of natural convection from a cylinder having porous fins was done by Kiwan et al. [44]. This study investigated two porous samples by varying cylinder diameter and the number of porous fins.

According to Jamin and Mohamad [45], the thermal resistance is the highest for low Prandtl number fluids (*Pr* ≈ 1) like exhaust gas, air, etc. This makes heat recovery difficult and requires a large surface area. The porous media reduces thermal resistance at the interface, thereby enhancing heat transfer. Also, the tortuous nature of porous media makes fluid mixing easier. Hence, enhanced heat transfer from exhaust flue gas is carried out using porous metal fins. In the prevailing literature, the majority of studies use porous-wrapped tubes for heat transfer enhancement. However, they are associated with higher pressure drops due to added flow resistance. A porous fin, on the other hand, overcomes this issue by creating a free-flow channel between two consecutive fins. For modeling the fluid flow and heat transfer inside the porous media, we have implemented the DBF model and LTNE, respectively. The LTNE model is recommended for high-temperature scenarios, which are common in waste heat recovery systems. The objective of this research article is to assess the effectiveness of various porous samples in facilitating heat transfer within a single-finned tube. The fins are considered to be made of porous aluminum foam, which is used in industry. High-temperature exhaust gas moves over the finned tube arrangement in crossflow to mimic the scenario in a waste heat recovery system. All computations were performed on a 3D domain in Reynolds number range 6000 ≤ *Re* ≤ 9000. The fin diameter is varied such that the fin diameter-to-tube diameter ratio falls in a range 2 ≤ *D _{f}*/

*D*≤ 3.

## 2 Model Formulation and Numerical Methodology

### 2.1 Flow Domain.

A single cylinder having an annular porous metal fin attached to its surface is placed in the flow of a high-temperature exhaust gas. The porous finned cylinder is oriented such that a crossflow arrangement is obtained. Three values of fin diameter-to-tube diameter ratio (*D _{f}*/

*D*) were considered. The cylinder having diameter

*D*is placed at the midplane between two plates (channel height 18

*D*) at an upstream and downstream distance of 60

*D*and 125

*D*, respectively. The schematic of the flow domain is shown in Fig. 1. Three commercially available porous samples were used [27,39]. The physical characteristics of the used samples are listed in Table 1. Additionally, the following assumptions were made for simplification: (i) porous material is homogenous, isotropic, and saturated with exhaust gas [46]; (ii) the physical properties of exhaust gas are constant and are obtained from the literature [10]; and (iii) flow inside the porous region is laminar.

Sample no. | Pore density (ϕ) | Porosity (ε) | Pore diameter (d) in m_{p} | Permeability (K) in m^{2} |
---|---|---|---|---|

1 | 10 | 0.9486 | 0.00313 | 1.2 × 10^{−7} |

2 | 20 | 0.9005 | 0.00258 | 0.9 × 10^{−7} |

3 | 40 | 0.9132 | 0.00180 | 0.53 × 10^{−7} |

Sample no. | Pore density (ϕ) | Porosity (ε) | Pore diameter (d) in m_{p} | Permeability (K) in m^{2} |
---|---|---|---|---|

1 | 10 | 0.9486 | 0.00313 | 1.2 × 10^{−7} |

2 | 20 | 0.9005 | 0.00258 | 0.9 × 10^{−7} |

3 | 40 | 0.9132 | 0.00180 | 0.53 × 10^{−7} |

### 2.2 Governing Equations.

As the flow domain has a porous and nonporous region, two different sets of governing equations were used. The Reynolds-averaged Navier–Stokes equations and the Darcy–Brinkman–Forchheimer model were used as momentum equations in nonporous and porous regions, respectively. The LTNE model is used for energy equations in the porous region. As per the analysis of Torabi et al. [47], the LTNE model is suggested when a significant difference in thermal conductivities of fluid and solid material exists.

For a nonporous zone, the following set of Reynolds-averaged Navier–Stokes equations is valid:

- Continuity equation:(1)$\u2207\u22c5V=0$
- Momentum equation:(2)$\rho (DVDt)=\u2212\u2207P+\mu \u22072V+\u2207\u22c5(\u2212\rho V\u2032V\u2032\xaf)$

In Eqs. (1)–(3), the nonprimed quantities represent mean component, and primed quantities are fluctuating velocity components. Also, the Kronecker delta (*δ*) is 1 when $\u2207V=(\u2207V)T$ and 0 otherwise. The Reynolds stresses are modeled using the realizable *κ-ϵ* turbulence model. This model is widely used in the literature for flow interactions between porous and nonporous zones [48,49]. Details of this model can be found in the said articles.

- (3)Energy equation:(4)$\u2202(\rho e)\u2202t+\u2207\u22c5(V(\rho e+P))=keff\u22072T$

In Eq. (4), *e* is total energy and *k*_{eff} is effective thermal conductivity (*k _{f}* +

*k*).

_{t}The equations applied for a porous zone are written as follows:

- (4)Continuity equation:(5)$\u2207\u22c5Vp=0$
- (5)Momentum equation:(6)$\rho \epsilon 2(\epsilon \u2202Vp\u2202t+Vp\u22c5\u2207Vp)=\u2212\u2207P+\mu \epsilon \u22072Vp\u2212\mu KVp\u2212\rho CFK|Vp|Vp$
- (6)
Energy equation:

*h*) and interfacial area density (

_{sf}*a*). The surface area density for metal foams is calculated using an expression given by Calmidi [52].

_{sf}*hsf*.

### 2.3 Boundary Conditions.

*u = U*

_{in}) and temperature (

*T = T*

_{in}). The right edge denotes the outlet boundary applied with the pressure outlet condition (

*P = P*

_{atm}). The cylinder surface highlighted in grey in Fig. 1(b) has a no-slip condition and constant surface temperature (

*T = T*). At the top and bottom walls of the domain (Fig. 1(a)), no-slip and adiabatic wall conditions are employed. Normal velocity components and heat flux are zero at symmetry planes (Fig. 1(b)). At the porous/nonporous interface, the continuity of velocity, stresses, temperature, and heat flux is ensured. A mathematical representation of the interface conditions is given as follows.

_{w}### 2.4 Numerical Scheme.

A meshing tool divides the computational domain into a finite number of control volumes. Fine meshing is done in the porous zone. The final mesh is shown in Fig. 2. Previously stated governing equations and boundary conditions are solved in the computational domain using ansys fluent, a finite volume-based numerical solver. A SIMPLE algorithm is utilized for pressure–velocity coupling. For the discretization of convective terms in momentum and energy equations, a QUICK scheme is used. Additionally, a second-order implicit method is implemented for transient conditions. The convergence criteria are set to 10^{−10} for the energy equation and 10^{−09} for the other remaining equations.

### 2.5 Grid and Domain Independence.

The grid and domain independence studies are performed at maximum and minimum pore density (10 PPI and 40 PPI). Four meshes with 572,000, 774,000, 1,184,050, and 1,476,000 control volumes were generated to confirm that mesh size does not influence the results. The results of the grid independence study are represented in Table 2. The errors of respective grid sizes are compared with the finest mesh, and the variations are mentioned in parentheses. As per Table 2, the deviation for a mesh of 1,184,050 is negligible (<1%) for all cases. Hence, considering computational cost and accuracy, the mesh of 1,184,050 elements is used for this study. Extensive tests were performed to finalize the upstream and downstream lengths of the computational domain. These results are summarized in Tables 3 and 4, respectively. Based on these findings, the distances upstream and downstream were adjusted to 60*D* and 125*D*, respectively. For modeling the unsteady flow regime in a porous medium, attention must be given to time-step sensitivity to get accurate results [46]. Table 5 studies different time-steps and shows that minimum deviations are observed for a time-step of Δ*t* = 0.0075 s. Accordingly, this time-step is selected for the current problem.

No. of control volumes | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

572,000 | 242.29 (4.33%) | 32.29 (−3.53%) | 349.24 (0.27%) | 33.35 (−2.19%) |

774,000 | 237.88 (2.43%) | 32.36 (−3.32%) | 349.11 (0.24%) | 33.71 (−1.14%) |

1,184,050 | 234.49 (0.97%) | 33.23 (−0.72%) | 349.03 (0.22%) | 33.83 (−0.79%) |

1,476,000 | 232.24 | 33.47 | 348.27 | 34.10 |

No. of control volumes | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

572,000 | 242.29 (4.33%) | 32.29 (−3.53%) | 349.24 (0.27%) | 33.35 (−2.19%) |

774,000 | 237.88 (2.43%) | 32.36 (−3.32%) | 349.11 (0.24%) | 33.71 (−1.14%) |

1,184,050 | 234.49 (0.97%) | 33.23 (−0.72%) | 349.03 (0.22%) | 33.83 (−0.79%) |

1,476,000 | 232.24 | 33.47 | 348.27 | 34.10 |

X_{u} | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

40D | 236.32 (−0.67%) | 31.19 (−4.56%) | 348.89 (−0.14%) | 33.16 (−2.61%) |

60D | 237.88 (−0.02%) | 32.36 (−0.99%) | 349.11 (−0.07%) | 33.71 (−0.99%) |

80D | 237.93 | 32.68 | 349.37 | 34.05 |

X_{u} | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

40D | 236.32 (−0.67%) | 31.19 (−4.56%) | 348.89 (−0.14%) | 33.16 (−2.61%) |

60D | 237.88 (−0.02%) | 32.36 (−0.99%) | 349.11 (−0.07%) | 33.71 (−0.99%) |

80D | 237.93 | 32.68 | 349.37 | 34.05 |

X_{d} | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

100D | 237.65 (−0.09%) | 32.98 (1.94%) | 348.71 (−0.11%) | 32.88 (−2.46%) |

125D | 237.75 (−0.05%) | 32.51 (0.46%) | 348.82 (−0.08%) | 33.58 (−0.38%) |

150D | 237.88 | 32.36 | 349.11 | 33.71 |

X_{d} | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

100D | 237.65 (−0.09%) | 32.98 (1.94%) | 348.71 (−0.11%) | 32.88 (−2.46%) |

125D | 237.75 (−0.05%) | 32.51 (0.46%) | 348.82 (−0.08%) | 33.58 (−0.38%) |

150D | 237.88 | 32.36 | 349.11 | 33.71 |

Δt (s) | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

0.1 | 236.89 (−0.42%) | 33.58 (3.77%) | 348.19 (−0.24%) | 34.86 (3.04%) |

0.01 | 237.53 (−0.14%) | 33.11 (2.31%) | 348.84 (−0.05%) | 34.84 (2.98%) |

0.0075 | 237.65 (−0.09%) | 32.67 (0.96%) | 348.85 (−0.05%) | 34.12 (0.85%) |

0.005 | 237.88 | 32.36 | 349.03 | 33.83 |

Δt (s) | 10 PPI | 40 PPI | ||
---|---|---|---|---|

Nu | ΔP | Nu | ΔP | |

0.1 | 236.89 (−0.42%) | 33.58 (3.77%) | 348.19 (−0.24%) | 34.86 (3.04%) |

0.01 | 237.53 (−0.14%) | 33.11 (2.31%) | 348.84 (−0.05%) | 34.84 (2.98%) |

0.0075 | 237.65 (−0.09%) | 32.67 (0.96%) | 348.85 (−0.05%) | 34.12 (0.85%) |

0.005 | 237.88 | 32.36 | 349.03 | 33.83 |

### 2.6 Validation.

To ensure that the current numerical approach gives accurate results, it is matched with the available experimental and numerical results. In the current scientific literature, finding a study that uses porous annular fins in crossflow is difficult. Hence, we have used the experimental data of Hamadouche et al. [55], which studies the convective heat transfer occurring within a rectangular channel containing open-cell metal foam. As shown in Fig. 3(a), the maximum absolute deviation in Nusselt number is 5.8% for a 40 PPI aluminum foam having a 20 mm height. The deviations can arise due to uncertainties in experimental measurements and approximations involved in modeling the turbulent flow. The model is also compared with analytical data of nondimensional flow velocity distribution in a tube filled with porous metal foam [53]. Figure 3(b) represents a reasonable agreement between the results of the current model and analytical expression. Figures 3(c) and 3(d) illustrate that the highest absolute deviation is 2.6% for the Nusselt number and 1.4% for the pressure drop compared to the numerical findings of Zhao et al. [10]. The model successfully reproduces the results from the literature with reasonable accuracy. Hence, it can be used to evaluate the current problem in detail.

## 3 Results and Discussion

This section deals with analyzing flow interactions and thermal functioning of a porous annular fin. The flow dynamics are studied with the help of streamlines passing through the porous region and pressure drop across the domain. Along with these, pumping power and friction factor are determined to get additional insight into hydrodynamics. On the other hand, thermal parameters like the Nusselt number and heat transfer rate are evaluated as a function of Reynolds number. Finally, all metal foam samples were compared for overall hydro-thermal achievements based on performance evaluation criteria.

### 3.1 Flow Dynamics.

The presence of metal foam over a cylinder can significantly affect the wake formation. Hence, studying wake formation is crucial in decoding flow dynamics. Figure 4 shows the streamlines with an *x*-velocity scale on a plane passing through the middle of a porous fin for a 40 PPI metal foam at *Re =* 6000. It has been observed that the vortices formed downstream are attenuated at higher fin diameter. The recirculation zone infiltrates into the porous zone at a lower fin diameter. However, no such infiltration is observed at higher fin diameters. For *D _{f}* /

*D*= 2.5, a smaller size recirculation is observed, which is damped out at higher values of fin diameter. The velocity inside the porous zone is reduced due to the hindrance created by metal foam fibers.

The pressure drop (Δ*P*) across the inlet and outlet of the flow domain is represented in Fig. 5 for different diameter ratios and pore density. The tortuous structure of a metal foam impedes the flow and acts as an additional barrier that the fluid must overcome. This tortuousness makes the fluid to follow a longer and convoluted path inside the porous region. At a particular fin height, a marginal increase in pressure drop is observed for all the metal foam samples studied (Figs. 5(a)–5(c)). The 40 PPI metal foam gave the highest pressure drop among the porous samples due to its lowest permeability. Higher pore density and lower permeability are the two properties of a porous sample that make fluid flow difficult. Higher pore density is associated with lower pore size and reduced flow area, whereas permeability refers to the ability of a material to allow fluid to pass through the porous medium. Thus, low permeability restrains the flow and creates a higher pressure gradient. Due to higher flow velocity, the pressure drop increases by more than twice as the Reynolds number increases from 6000 to 9000. At *D _{f}*/

*D*= 2, the pressure drop is nearly 1.5–4% higher at 40 PPI than at 10 PPI foam. Consequently, at the highest fin diameter (

*D*/

_{f}*D*= 3), a maximum rise of 5% is observed in pressure drop. However, at a particular pore density, as the

*D*/

_{f}*D*ratio increases, the flow restrictions cause pressure drop to rise (Figs. 5(d)–5(f)). The rise in fin diameter causes the pressure drop to increase by nearly 50% at all Reynolds numbers and pore density. Pressure contours on a plane passing through the mid-section of the fin show an analogous pattern to the pressure drop results (Fig. 6). The contours indicate the formation of a local negative pressure zone at the fin's rear. The span and intensity of this region are extended with a rise in Reynolds number and fin diameter. This negative pressure zone is due to the flow separation and wake formation behind the body. At the stagnation point, the fluid experiences a sudden reduction in velocity. This stagnation process converts the entire kinetic energy head into a pressure head, giving a high-pressure zone in the front of the body.

In the aforementioned equation, *A*_{in} is an area of inlet section. Figure 7 represents the variation of pumping power along the Reynolds number for various foam samples at different fin diameters and pore densities. The pumping power and pressure drop show a similar trend by virtue of their interdependent relation. According to Figs. 7(a)–7(c), the maximum pumping power is required for the 40 PPI foam at all values of *D _{f}*/

*D*. This augmentation in pumping requirements is observed due to a higher pressure drop arising from significant flow restrictions for a 40 PPI metal foam sample. Figures 7(d)–7(f) indicate that the rise in fin height (

*D*/

_{f}*D*ratio) escalates the pumping power of the overall flow domain. In relation to pressure drop, the pumping power is 50% higher across all Reynolds numbers and pore density. Further, it is beneficial to obtain the results in a nondimensional manner. For this, friction factor is obtained from Eq. (16).

The variation of friction factor against Reynolds number is plotted in Fig. 8 for all ratios of fin diameter to tube diameter and pore density. The friction factor is an essential parameter in fluid dynamics as it measures the resistance to flow caused by surface roughness. In contrast to pressure drop, friction factor reduces at higher flow velocity. This is because a significant increase in inlet velocity at a higher Reynolds number dominates the overall value of the friction factor. With an increase in Reynolds number, the turbulence intensifies, causing efficient and smoother fluid flow with less energy loss. At higher pore density (*ϕ*), a reduction in pore diameter (*d _{p}*) reduces the flow passage through porous media. Hence, the highest and lowest friction factors are observed at all fin diameters for 40 PPI and 20 PPI foam, respectively (Figs. 8(a)–8(c)). A comparison of Figs. 8(d)–8(f) indicates that regardless of pore density, the highest friction factor is observed at maximum fin height (

*D*/

_{f}*D =*3).

### 3.2 Thermal Investigations.

Along with other thermal parameters like Nusselt number and heat transfer rate, studying the temperature distribution in the flow domain is helpful. Therefore, the temperature contours on a plane passing through the midsection of the fin are shown in Fig. 9 for *Re* = 6000. For all cases, it is observed that the low-temperature zone is found on the fin downstream. This is apparent as the hot fluid approaching from upstream will start losing heat as soon as it flows through the porous region. Careful observation of Fig. 9 shows an evident difference in the temperature distribution at a variety of pore densities and fin diameters. The temperature distribution varies significantly with fin diameter at all pore densities (10, 20, and 40 PPI). Contours reveal that at all pore densities, the temperature in the wake increases at higher fin diameters. With an increase in fin height, the residence time of fluid inside the porous fin is substantially higher. The reduced thermal resistance at the interface contributes to higher heat extraction from hot flue gases. However, at a particular fin diameter, the temperature in the wake is the highest for 10 PPI sample and the lowest for 40 PPI sample. This results from high permeability at lower pore density, which makes it easier for a fluid to move through a porous zone, thereby increasing the thermal interactions. Among all cases, the lowest temperature region occurs at the cylinder's rear stagnation point.

In the aforementioned equation, *Ḣ*_{out} and *Ḣ*_{in} are enthalpy flowrates at the outlet and inlet, respectively. As depicted in Figs. 10(a)–10(c), the maximum heat transfer rate is observed for 40 PPI foam sample. A decrease in pore size (*d _{p}*) and fiber diameter (

*d*) will increase the area density (

_{f}*a*), according to Eq. (9). This will result in higher heat transfer for a metal foam having a greater pore density. At the point where pore density is maximized and permeability is minimized, fluids navigate through complex, tortuous pathways, mixing with fluid at different temperatures. This intermixing enhances heat extraction within porous fins. At all fin diameters (

_{sf}*D*/

_{f}*D =*2, 2.5, and 3), the heat transfer rate augments by nearly 48% as the pore density increases. The heat transfer rate increases slightly by 2.5% or less when 40 PPI foam is used instead of 20 PPI foam. However, when a specific foam sample is used with different fin diameters, a maximum heat transfer rate is observed for

*D*/

_{f}*D =*3 (Figs. 10(d)–10(f)). At higher values of

*D*/

_{f}*D*ratio, the residence time of hot fluid inside the porous media is substantially larger than at lower values. Consequently, better thermal interactions occur between metal foam and fluid at higher fin diameter, resulting in an augmentation in heat transfer rate. With a 10 PPI sample, increasing the fin diameter results in a 4–6% increase in the heat transfer rate. Similarly, for a 40 PPI foam, a 6–8% rise in heat transfer rate is observed.

In Eq. (18), *A* is the heat transfer surface area. The variation of Nusselt number against Reynolds number is shown in Fig. 11. The Nusselt number shows an identical nature to that of the heat transfer rate. Figures 11(a)–11(c) indicate that the foam having a minimum pore size gives the maximum Nusselt number. Due to low permeability at higher pore density, the hot fluid moves slowly through the porous region. Consequently, heat dissipation increases, and a rise in Nusselt number is observed. At *D _{f}*/

*D =*2, an augmentation of nearly 45% is observed for a 40 PPI foam compared to 10 PPI. On the other hand, for

*D*/

_{f}*D =*2.5 and 3, the Nusselt number augments by nearly 48% with pore density. When the fin diameter is increased for a constant pore density, there is an observed increase in the Nusselt number (Figs. 11(d)–11(f)). At a particular Reynolds number and pore density, the hot fluid spends more time in the porous zone at a higher fin diameter, and a higher surface area for heat transfer results in prominent heat transfer; thereby, augmentation in Nusselt number is achieved. An augmentation of nearly 4–6% is achieved for a 10 PPI sample, and for a 40 PPI sample, the Nusselt number increases by almost 6–8% with fin diameter.

*j*) is determined using the following expression:

Figure 12 indicates that the Colburn factor reduces with Reynolds number. This observation is not new, as the Colburn analogy [56] describes that the friction factor and Colburn factor are directly proportional, thus showing a similar behavior as that of the friction factor. Furthermore, the quantity (*RePr*^{1/3}) varies linearly with Reynolds number, justifying the reason for the low Colburn factor at higher *Re*. Due to the higher Nusselt number, a higher Colburn factor is observed for foam samples having more pore density. Following its relationship with Nusselt number, the Colburn factor shows a 45% and 48% rise when pore density is changed from 10 PPI to 40 PPI at *D _{f}*/

*D =*2 and 3, respectively. Similarly, an increase of 4–6% is observed for 10 PPI foam with fin diameter; for a 40 PPI foam, the Colburn factor rises by 6–8% at the highest value of fin diameter.

### 3.3 Overall Performance Evaluation.

*j*/

*f*) and (ii) a ratio of heat transfer coefficient to pumping power per unit heat transfer surface (

*Z*/

*E*). According to the literature, these parameters are recommended for performance evaluation [57,58]. The expression for

*Z*/

*E*is given in Eq. (21), where

*Z*represents heat transfer per unit heat exchanger volume per unit temperature difference and

*E*signifies power provided per unit heat exchanger volume.

Figure 13 shows the area goodness factor for different metal foam samples and fin diameters. A higher *j*/*f* value indicates a lower frontal area of the heat exchanger for a specified amount of heat transfer. This parameter is vital in determining the heat exchanger's compactness. Irrespective of the fin diameter, it has been observed that the foam with a pore density of 20 PPI gave the highest area goodness factor. However, for a specific foam sample, the maximum area goodness factor is obtained for *D _{f}*/

*D =*2. The

*Z*/

*E*factor shows a similar nature (Fig. 14). A higher value of

*Z*/

*E*indicates better heat transfer at a specified pumping power. Based on these results, it is recommended to use sample 2 (20 PPI) at a diameter ratio of

*D*/

_{f}*D =*2 to achieve better heat transfer at the expense of minimum pumping power.

## 4 Conclusions

A 3D computational fluid dynamics was used to investigate the hydrodynamic and thermal behavior for a crossflow of high-temperature exhaust gas over a cylinder fitted with porous fins. The resulting arrangement mimics a scenario in a waste heat recovery system. Porous samples with varying pore densities were studied using the LTNE model. The streamlines reveal that the wake size reduces at a higher fin diameter. Because of the larger pore size, the 10 PPI metal foam exhibits the smallest pressure drop across all fin diameters.

The pressure drop increased marginally (nearly 5%) as pore density increased. A study of pumping power showed that higher pump work is needed at higher fin diameter and pore density. The result of the heat transfer rate revealed that a 40 PPI foam gives the highest heat transfer due to a slower flowrate inside the porous zone. Conversely, a larger fin diameter increases heat transfer for a particular pore density. The Nusselt number showed an analogous behavior with the heat transfer rate. Overall performance evaluation parameters (*j*/*f* and *Z*/E) suggested using 20 PPI samples at *D _{f}*/

*D =*2 for better heat transfer at a minimum expense of pumping power. This analysis conveys that the porous metal fins can efficiently remove heat from exhaust flue gases by overcoming the disadvantage of significantly higher pressure drop observed in the case of porous wrappings. Additionally, implementing the porous fins for waste heat recovery will result in a considerable reduction in fuel consumption and emission of pollutants, as this extracted heat can be used to preheat the feed streams of a process. Based on this study, it is recommended that porous fins can be a promising alternative for heat transfer augmentation from exhaust flue gases.

## Acknowledgment

Amit Kumar Dhiman would like to acknowledge the Science and Engineering Research Board (SERB), Department of Science and Technology (DST), Government of India (GoI) for the providence of the MATRICS grant (File No: MTR/2021/000867).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## Nomenclature

*e*=total energy (J)

*f*=friction factor

*h*=heat transfer coefficient (W/m

^{2}K)*j*=Colburn factor

*k*=thermal conductivity (W/m K)

- $d\xaf$ =
specific diameter (m)

*A*=heat transfer surface area (m

^{2})*D*=tube diameter (m)

*E*=power provided per unit heat exchanger volume (W/m

^{3})*K*=permeability (m

^{2})*Q*=heat transfer rate (W)

*T*=temperature (K)

*Z*=heat transfer per unit heat exchanger volume per unit temperature difference (W/m

^{3}K)- $V$ =
velocity vector (m/s)

*Ḣ*=enthalpy flowrate (W)

*a*=_{sf}interfacial area density (m

^{2}/m^{3})*d*=_{f}fiber diameter (m)

*d*=_{p}pore diameter (m)

*h*=_{sf}interfacial heat transfer coefficient (W/m

^{2}K)*C*=_{F}Forchheimer coefficient

*D*=_{f}fin diameter (m)

- Nu =
Nusselt number

- Pr =
Prandtl number

- Re =
Reynolds number

*Z*/*E*=ratio of heat transfer for unit temperature difference to power provided (1/K)

- Δ
*P*= pressure drop (Pa)

- Δ
*T*_{lm}= log mean temperature difference (K)

*κ*=turbulent kinetic energy (m

^{2}/s^{2})

### Abbreviations

### Subscripts

### Greek Symbols

### Superscripts

## References

_{2}Conversion to Syngas With Desired H

_{2}/CO Ratios Using Flue Gas of Power Plants Without CO

_{2}Separation