**1. Introduction**

One of the main components of a cooling system of an engine is a radiator. Vehicle radiators are typically fin-and-tube-type compact heat exchangers (HXs) and composed of inlet manifolds, outlet manifolds, tubes and fins as shown in **Figure 1**. Simply, a radiator works with two fluids which are air and anti-freeze water mixture. Hot antifreeze water mixture flows through the tubes, whereas cooling air flows through the fins resulting in heat exchange between both streams.

**Figure 1.** A typical 4-row tractor radiator.

Due to the strong competition in the automotive industry, radiators with better performance (higher cooling capacity, less hydrodynamic loss, less weight, etc.) have been desired. A common tool for the determination of thermal characteristics of vehicle radiators is the experimental testing. However, experimental testing may not be feasible considering the cost and labor-time. Basic understanding of the past experimental data and analytical/computational modeling can significantly enhance the effectiveness of the design and development phase. There are techniques available to analyze HXs such as log mean temperature difference (LMTD) and effectiveness-NTU (ε-NTU). However, these techniques require some parameters known *a priori* such as overall heat transfer coefficients and/or NTU relations for a given HX. There are no general expressions for overall heat transfer coefficients and/or ε-NTU relations valid for any HX. Therefore, these parameters need to be predicted either from analytical expressions [1], experimental data [2, 3] and/or computational models [3–6]. *A priori* knowledge of these parameters is required for the designer. Therefore, implementation of LMTD and/or ε-NTU is not feasible especially for vehicle radiators which may include customdesigned fin configurations. Alternatively, computational fluid dynamics (CFD) analysis can be applied to predict the thermal characteristics of a radiator. However, CFD analysis of a fullsize HX is not feasible due to extremely high number of cells required to resolve the complex nature of the HXs; especially the fin structures. This point is more problematic when the number of fins is high in the case of heavy-duty vehicle radiators. Although fins introduce a significant complexity for the problem, the repetitive and/or regular structure of the fins enables the porous medium based modeling. From computational point of view, this approach offers some unique advantages. The complex fluid flow occurring through fins can be introduced into the model through porous parameters. Although the determination of these porous parameters requires a rigorous, detailed computational model with very fine mesh structure especially within the regions mainly responsible for the fluid friction and heat transfer, this modeling can be performed on a representative unit cell due to the repetitive nature of the fins. Once these effects are included through the porous parameters, the mesh structure simplifies dramatically and considering the whole geometry, the number of degree of freedom of the system drops down to a feasible number (in the order of 10 millions). Besides, the porous modeling does not require any boundary layer meshing since the friction and heat transfer parameters are already included through the porous parameters.

#### **1.1. Porous modeling**

**1. Introduction**

244 Heat Exchangers– Design, Experiment and Simulation

**Figure 1.** A typical 4-row tractor radiator.

streams.

One of the main components of a cooling system of an engine is a radiator. Vehicle radiators are typically fin-and-tube-type compact heat exchangers (HXs) and composed of inlet manifolds, outlet manifolds, tubes and fins as shown in **Figure 1**. Simply, a radiator works with two fluids which are air and anti-freeze water mixture. Hot antifreeze water mixture flows through the tubes, whereas cooling air flows through the fins resulting in heat exchange between both

Due to the strong competition in the automotive industry, radiators with better performance (higher cooling capacity, less hydrodynamic loss, less weight, etc.) have been desired. A common tool for the determination of thermal characteristics of vehicle radiators is the experimental testing. However, experimental testing may not be feasible considering the cost and labor-time. Basic understanding of the past experimental data and analytical/computational modeling can significantly enhance the effectiveness of the design and development phase. There are techniques available to analyze HXs such as log mean temperature difference (LMTD) and effectiveness-NTU (ε-NTU). However, these techniques require some parameters known *a priori* such as overall heat transfer coefficients and/or NTU relations for a given HX. There are no general expressions for overall heat transfer coefficients and/or ε-NTU relations valid for any HX. Therefore, these parameters need to be predicted either from analytical Porous modeling is governed by three models. The simplest model is the Darcy's model which is suggested by Henry Darcy (1856) during his investigations on hydrology of the water supplies of Dijon [7]. Darcy's equation is expressed as:

$$\frac{\Delta p}{l} = -\frac{\mu}{a}V \tag{1}$$

where, Δ*p* is the pressure drop, *l* is the pipe length, *V* is the average velocity, *μ* is the dynamic viscosity and *α* is permeability of porous domain. Permeability depends on the fluid properties and the geometrical properties of the medium. The dependence of the pressure drop on velocity in the Darcy's equation is linear; therefore, Darcy's equation is applicable when the flow is laminar. As the velocity increases, the dependence of the pressure drop on velocity becomes non-linear due to drag caused by solid obstacles. At this point, there are two extended models proposed in the literature namely Forchheimer and Forchheimer-Brinkman model. For moderate Reynolds numbers, including nonlinear effects, pressure drop is defined as Forchheimer's equation [7]:

$$\frac{\Delta p}{l} = -\left(\frac{\mu}{a}V + \frac{C\_r}{\sqrt{a}}\frac{1}{2}\rho V^2\right) \tag{2}$$

where *CF* is the dimensionless form-drag constant and *ρ* is the density of the fluid. The first term denotes the viscous characteristics of porous flow and the second term (also called Forchheimer term) denotes the inertial characteristics. Lastly, Forchheimer-Brinkman model includes additional Laplacian term in addition to Forchheimer's equation. Forchheimer-Brinkman model is expressed as [7]:

$$\frac{\Delta p}{\mu} = -\left(\frac{\mu}{\alpha}V + \frac{c\_F}{\sqrt{\alpha}}\frac{1}{2}\rho V^2 - \bar{\mu}\nabla^2 V\right) \tag{3}$$

where is the effective viscosity. In general, added Laplacian term (also known as Brinkman term) resolves effects of the flow characteristics in a thin boundary layer at the near wall regions. Strictly speaking, the last term becomes important for large porosity (ratio of the fluid volume to the solid volume in a porous medium) values which means the effect is negligible for many practical applications where typically porosity value is relatively small. Eq. (3) without the quadratic term is known as extended Darcy (or Brinkman) model. Therefore, Forchheimer-Brinkman model is the most general model, but the inclusion of the Brinkman and Forchheimer term on the left-hand side can be questionable since the Brinkman term is appropriate for large porosity values, yet there exists uncertainty about the validity of the Forchheimer term at larger porosity values [7].

Velocity definition in porous modeling is specified by using two different descriptions: superficial formulation and physical velocity formulation. Superficial velocity formulation does not take the porosity into account during the evaluation of the continuity, momentum and energy equations. On the other hand, physical velocity formulation includes porosity during the calculation of transport equations [8]. The continuity and momentum transport equation for a porous domain using Forcheimer's model can be written as [2]:

$$
\frac{
\partial
}{
\partial t
}
(\gamma \rho) + \nabla. \left(\gamma \rho \vec{V}\right) = 0
\tag{4}
$$

$$\frac{\partial}{\partial t}\left(\mathcal{\mathcal{p}P}\stackrel{\rightarrow}{V}\right) + \nabla.\Big(\mathcal{\mathcal{p}P}\stackrel{\rightarrow}{V}\Big) = -\mathcal{\mathcal{p}P}p + \nabla.\Big(\mathcal{\mathcal{p}\bar{\tau}}\Big) + \mathcal{\mathcal{p}B}\_f - \left(\frac{\mathcal{\mathcal{p}^2}\mu}{a}\stackrel{\rightarrow}{V} + \mathcal{\mathcal{p}^3}\frac{\mathcal{C}\_2}{2}\rho\middle| \stackrel{\rightarrow}{\mathcal{V}}\Big) \tag{5}$$

where *γ* is the porosity, *C*2 is the inertial coefficient for porous domain and is the body force term.

Besides flow modeling, heat transfer modeling for porous flow is described by using two models which are (i) equilibrium model and (ii) nonequilibrium model. Equilibrium model (one-equation energy model) is used when the porous medium and fluid phase are in thermal equilibrium. However, in most cases, fluid phase and porous medium are not in thermal equilibrium. For such cases, nonequilibrium thermal model is more realistic. In the case of the radiator, this issue is important since the temperature difference between the solid (fins) and

the fluid (air flowing through fins) is the driving mechanism for the heat transfer [4]. Therefore, the nonequilibrium model includes two energy equations (known as also two-equation energy model): one is for the fluid domain and the other is for the solid domain. The coupling of these two models is via the term which represents the heat transfer between the fluid and the solid domains. The conservation equations for the two energy model can be written as [2]:

$$
\frac{\partial}{\partial t} \left( \gamma \rho\_f E\_f \right) + \nabla \cdot \left( \vec{V} \left( \rho\_f E\_f + p \right) \right) = \nabla \cdot \left[ \gamma k\_f \nabla T\_f - \left( \sum\_i h\_i I\_i \right) + \left( \stackrel{\leftarrow}{\tau} \vec{V} \right) \right] + S\_f^k + h\_{jk} A\_{jk} \left( T\_s - T\_f \right) \tag{6}
$$

$$\frac{\partial}{\partial t}\left(\left(1-\gamma\right)\rho\_s E\_s\right) = \nabla \cdot \left(\left(1-\gamma\right)k\_s \nabla T\_s\right) + S\_s^\natural + h\_{jk}A\_{jk}\left(T\_f - T\_s\right) \tag{7}$$

where subscripts *`s'* and *`f'* stand for solid and fluid, respectively. *E* is the total energy, *T* is the temperature, *k* is the thermal conductivity, *S* is the energy source term and ( ) stands for the effect of enthalpy transport due to the diffusion of species. The last term in both of the equations is the coupling term which models heat transfer between the fluid and solid domains. In this coupling term,ℎ denotes heat transfer coefficient for the fluid/solid interface and denotes the interfacial area density that is the ratio of the area of the fluid/solid interface and the volume of the porous zone.

Through Eqs. (3)–(7), there are many parameters which are material's property and fixed once the materials for the fluid and the solid are selected. On the other hand, there are some parameters (i.e. porous parameters) which are functions of material, geometry and the flow condition. These parameters are *γ*, *α*, *C*2, ℎ and . Among these, *γ* and are purely geometric parameters and can be determined once the geometry of the porous structure is known. In the case of a radiator modeling, once the geometry of the fins is set, these two parameters can be determined beforehand. The other parameters are flow-dependent, meaning that they need to be determined for a specific flow condition. At this point, these parameters can be determined through some analytical expressions [9–11], experimental results (e.g. wind tunnel testing) [4, 12], empirical correlations [13] and/or computational models [14–21] typically valid for a representative unit cell. All these approaches were implemented in the literature for different studies for the analysis of micro/macro heat sinks and HXs.

#### **1.2. Computational modeling of heat exchangers**

where *CF* is the dimensionless form-drag constant and *ρ* is the density of the fluid. The first term denotes the viscous characteristics of porous flow and the second term (also called Forchheimer term) denotes the inertial characteristics. Lastly, Forchheimer-Brinkman model includes additional Laplacian term in addition to Forchheimer's equation. Forchheimer-

where is the effective viscosity. In general, added Laplacian term (also known as Brinkman term) resolves effects of the flow characteristics in a thin boundary layer at the near wall regions. Strictly speaking, the last term becomes important for large porosity (ratio of the fluid volume to the solid volume in a porous medium) values which means the effect is negligible for many practical applications where typically porosity value is relatively small. Eq. (3) without the quadratic term is known as extended Darcy (or Brinkman) model. Therefore, Forchheimer-Brinkman model is the most general model, but the inclusion of the Brinkman and Forchheimer term on the left-hand side can be questionable since the Brinkman term is appropriate for large porosity values, yet there exists uncertainty about the validity of the

Velocity definition in porous modeling is specified by using two different descriptions: superficial formulation and physical velocity formulation. Superficial velocity formulation does not take the porosity into account during the evaluation of the continuity, momentum and energy equations. On the other hand, physical velocity formulation includes porosity during the calculation of transport equations [8]. The continuity and momentum transport

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where *γ* is the porosity, *C*2 is the inertial coefficient for porous domain and is the body force

Besides flow modeling, heat transfer modeling for porous flow is described by using two models which are (i) equilibrium model and (ii) nonequilibrium model. Equilibrium model (one-equation energy model) is used when the porous medium and fluid phase are in thermal equilibrium. However, in most cases, fluid phase and porous medium are not in thermal equilibrium. For such cases, nonequilibrium thermal model is more realistic. In the case of the radiator, this issue is important since the temperature difference between the solid (fins) and

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Brinkman model is expressed as [7]:

246 Heat Exchangers– Design, Experiment and Simulation

Forchheimer term at larger porosity values [7].

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term.

Porous modeling can be implemented to any geometry which would resemble a porous structure. Moreover, if the porous structure has repetitive nature, the porous coefficients can be obtained through a detailed modeling of representative unit cell through analytical, experimental or computational means. Heat sinks are very good examples of this case and porous modeling approach has been implemented for the analysis of micro/macro heat sinks [9–12]. A two-equation energy model has been implemented to analyze a straight-finned heat sink [9] together with Darcy's model and implemented to perform an optimization for an internally finned tube [10] and to discuss the effect of aspect ratio and effective thermal conductivity on the thermal performance of a micro-heat sink [11] together with extended Darcy's model. The heat sinks contain a regular structure; therefore, there is a chance to derive analytical expressions to estimate the porous parameters [9–11].

Considering the HXs with complex fin structures, computational modeling is even more challenging; therefore, the computational models typically focus on specific subcomponents of HXs such as a representative unit cell for the fin structure [14–21], radiator fan [22] and inlet manifold [23, 24]. The thermal performance of a HX can be achieved by simply increasing the performance of the fin structure alone. A fin structure with higher heat transfer together with less pressure drop can significantly enhance the performance of the entire system. To investigate the thermal performance of a fin structure, experimental [14–18] and/or computational models [14, 17, 19–21] can be realized for different fin geometries. Moreover, improving the flow maldistribution at the inlet manifold may also increase the thermal performance. Computational modeling of the flow maldistribution may lead to performance enhancement for HXs [23, 24].

Analyzing subcomponents may lead to qualitative conclusion for the thermal performance of an HX, however to estimate the thermal performance quantitatively, a rigorous 3-D modeling of the entire HX is required. Since a rigorous modeling is not computationally feasible, a 2D model [4], hydraulic and thermal resistances-based models [12, 25] and 3D mesoscale models (considering macro control volumes) have been introduced in the literature to predict the thermal performance quantitatively [26–29]. A 2D model was developed to compare the equilibrium model (one-equation thermal model) and nonequilibrium thermal model (twoequation energy model) for a relatively small size matrix type HX [4]. A resistance-based model was implemented to predict the hydrodynamic and thermal performance of a carbon-foam– finned HX which combined many different correlations from the literature to predict the hydrodynamic and thermal resistances [25]. The success of the model strongly depends on the accuracy of the porous parameters. For this particular example, the model was proven to predict the hydrodynamic and the thermal performance within ±15% of the experimental data. A Compact Heat Exchanger Simulation Software (CHESS) has been developed [26–28] as a rating and design tool for industrial use based on the empirical correlation of the porous parameters to analyze the fin-and-tube part of a vehicle radiators (excluding inlet and outlet manifolds). It was demonstrated that by using CHESS, the thermal performance of different vehicle radiators was predicted within ±15% of the experimental values. Alternatively, a porous modeling-based CFD model for fluid flow and meso-scale ε-NTU-based modeling for thermal characteristic was utilized for an air-to-air cross-flow HX [29] to investigate the effect of the maldistribution on the thermal performance. A 3D CFD model coupled with porous medium approach has been developed to investigate the hydrodynamic performance of a plate-fin HX in which the porous parameters were also determined using a detailed CFD model on the unit cell [30].

A full-size 3D thermal modeling of a relatively small compact HX was conducted with different fin configurations and the heat transfer and friction factor parameters which can be used in conjunction with the LMTD or ε-NTU method [5]. Since the size of the HX was small, the meshing was not a problem and the computational model was utilized for the design of an inlet manifold for a better performance. Considering the size of the vehicle radiator, this approach is not an option. Thermal and structural analysis of a heavy-duty truck radiator which had finned structure both on the liquid and air side has been performed using Commercial CFD software, FLUENT® [31]. Forchheimer's relation was used for the porous modeling together with the experimental data. One-equation energy model was used together with the averaged equivalent thermal conductivity. The local heat transfer coefficients and pressure distribution gathered from the thermal analysis were used as a boundary condition for finite element structural analysis through which the thermal stresses and strains were obtained.

One alternative to all these approaches can be the modeling of the vehicle radiator with porous medium approach where the porous parameters are also deduced from a rigorous CFD modeling on a unit cell. Moreover, this procedure may be performed with a commercial CFD software which would have very strong meshing, solving and post-processing capabilities. However, implementation of the two-temperature energy equation is crucial for an accurate prediction especially for the vehicle radiators. This may not be straightforward with a commercial software. At this point, FLUENT® may be a viable solution since the two-temperature model capability has been included in version 14.5. More recently, a computational modeling of a fin-and-tube-type vehicle radiator has been conducted based on two-temperature model and the cooling capacity of a heavy-duty vehicle radiator has been estimated without any need for empirical and/or experimental data [32]. In the upcoming section, the computational methodology of such a computational model is outlined. This approach may allow CFD modeling to be an efficient rating and design tool for vehicle radiators. Although the proposed computational methodology is discussed for a vehicle radiator, it may also be implemented to any compact HX with repetitive fin structures which is an important problem for many industrial applications.
