# Porous Modelling Approach for complex Flow and Heat Transfer in CFD

**Abstract **

In this article, we have discussed about modelling the flow and heat transfer through complex geometries with porous media approach, to reduce the simulation complexities and time taken without compromising on the accuracy of the simulation. The methodology includes studying of complex geometry, ascertaining porous media inputs such as inertial resistance, porosity, viscous resistance, interfacial area density based on the geometry and modelling the complex regions as porous media with calculated inputs. A case study on the micro channel cooler (MCC) demonstrated with the porous modelling approach.

**Introduction**

Porous medium is a material containing pores or voids which may act as passage for fluid flow. The fluid flow through the porous media will undergo pressure drop and may enhance heat transfer. The most common industrial applications of porous medium are Microchannel Coolers, Fuel cells, Catalytic converters, Filters, tube banks, food products, etc. The fluid flow and heat transfer study through such industrial products are very important for efficient design. However modelling and meshing of porous media is very complex and time consuming. Hence In this article, we have detailed our Porous modelling approach for CFD Simulation of Porous media in ANSYS fluent.

Fig 1 Industrial Applications of porous media (a) Fuel cell stack (b) Catalytic converter (c) Micro Channel Cooler |

**Methodology:**

**Concept of Porous Media approach:**

** **The complexity of modelling porous media as such in any application is very high, as modeling and meshing of thousands of micro holes are cumbersome and time consuming. Therefore, solving the flow field at the pore scale is impractical. Hence by means of volumetric averaging the micro-scale properties (pore size and pore space geometry) of the porous medium by an equivalent continuum on a larger scale described by new properties (porosity, permeability, relative permeability). This concept of representative elementary volume (REV) is fundamental to the mathematical description of fluid flow and transport in porous media. On the one hand, REV must be large enough to avoid undesirable fluctuations of averaged properties and on the other hand it must be small enough to render the spatial dependency of these properties. Thus the flow field is computed for the porous domain in REV scale with new averaged properties.

Fig 2 Concept of porous media approach |

Porous modelling approach requires properties such as porosity, inertial and viscous resistance coefficients, interfacial area density, etc. Which are calculated as follows based on the properties of micro voids.

a.Porosity:

The porosity is the ratio of volume of voids over total volume.

b.Interfacial Area Density:

Interfacial area density calculated based on the ratio of wetted surface area over the sample volume of porous media.

c.Inertial and Viscous resistance:

The characteristic curve of pressure drop vs flow rate is essential for calculating the inertial and viscous resistance. Typical pressure drop vs flow rate characteristic curve shown in fig 3.

The pressure drop across porous media is defined by following equation,

ΔP = Si * L, where ΔP is pressure drop across porous media and L is its thickness.

, Viscous loss term + Inertial loss term

The above equation can be generalized as ΔP = an u2 + b u, where constants a and b are inertial & viscous coefficients respectively.

The trend line equation “y = 0.2719 x^{2 }+ 4.8521 x” is derived for the pressure drop curve as shown in Fig 4, where values of a & b are 0.2719 and 4.8521 respectively. Hence, the inertial resistance coefficient (C_{ij}) and viscous resistance coefficient (D_{ij}) calculated by equating a & b values.

**Porous Modelling in Fluent:**

**(a) Flow characteristics of Porous Media:**

In Fluent, porous media model by default uses the **Superficial Velocity Formulation** for fluid flows. However, superficial velocity values within a porous region remain the same as those outside the porous region. It cannot predict the velocity increase in porous zones, which limits the accuracy of the Simulation. Therefore, it becomes necessary to solve for the true or **physical velocity** throughout the flow field rather than the superficial velocity.

**(b)Heat transfer modelling in porous media:**

Ansys Fluent has two different approaches to model energy transport in porous media, Local Thermal Equilibrium (**LTE**) and Local Thermal Non-Equilibrium (**LTNE**)

In LTE model, porous media and fluid flow are assumed to be in thermal equilibrium, which causes the solid phase temperature equal to that of the fluid phase. This means that the local fluid temperature is approximately equal to the porous matrix temperature. The assumption of Local thermal equilibrium is not valid when there is a substantial difference between the thermal conductivities of the two phases.

In LTNE model, porous media and fluid flow are assumed to be not in thermal equilibrium. This requires additional inputs to account for the convective mode of energy transfer between the two considered phases. When LTNE model is enabled in Fluent, porous solid zone will be created and it requires the thermal boundary conditions such as heat transfer coefficient of the wetted area of Porous zone (h_{fs}) & interfacial area density (A_{fs }– the ratio of the wetted area and the volume of the porous zone).

**Porous modelling of microchannel cooler:**

In this article, we have demonstrated the porous modelling approach with Micro channel coolers (MCC). MCC is predominantly used in electronic cooling applications, to cool the heat generated by electronic components. MCC is assembled with electronic components, which has thousands of tiny holes though which coolant is passed through to cool the electronic components.

Fig. 4 shows sample CAD model of MCC with micro channels. To illustrate the porous modelling approach, single MCC passage (Solid and Fluid Zones) is considered for CFD Simulation along with micro channels modelled as porous zones. The electronic components not modelled in the Simulation, only electronic heat source applied on the bottom surface of MCC.

Fig 4 CAD & CFD model of Microchannel cooler |

The flow though porous media is modelled with superficial velocity formulation and heat transfer is modelled with Local Thermal Non Equilibrium (LTNE) Model. The inputs for porous modelling like Porosity, interfacial area density, Heat transfer Coefficient, are calculated based on the geometry of micro channels. Other inputs like inertial & Viscous coefficients are calculated based on pressure drop curve across micro channels.

In Fluent there are two approaches under LTNE, Approach 1, only interaction between porous solid and porous fluid zones and Approach 2, interaction between porous solid and adjacent solid. As we enable LTNE model under porous zone condition, fluent creates an overlapping solid porous zone in addition to the porous fluid zone. By default, the porous solid zone has heat interaction with porous fluid zone alone and it doesn’t participate in the heat transfer with adjacent solid zones (Approach 1). But to predict the heat transfer through MCC appropriately, we need to account for the heat transfer between porous solid and adjacent solid zones. In Approach 2, interface is created between porous solid zone and adjacent solid zones, to enable the heat transfer between porous solid and adjacent solid zones.

**Results and Discussion:**

The porous modelling simulation results are compared between Approach 1 & 2 as shown below. Fig 5 shows the Planes (Plane 1, 2 & 3) used for post processing.

Fig 6 shows Pressure distribution in Three Planes

Pressure Distribution is same for Approach 1 & 2. pressure is decreasing from Inlet to Outlet as flow passes through micro channels.

Fig 7 & Fig 8, shows Velocity Contour and Vectors in Three Planes

Velocity Distribution is same for Approach 1 & 2. Flow streamline indicates the flow pattern, which highlightes the recirculation zones in the outlet

Fig 9 shows the Temperature distribution across solid regions for Approach 1. From the contour plot it is clearly evident that there is no temperature rise in porous solid zone, due to heat transfer from adjacent solid zone to porous solid. The peak temperature on Solid zone is ~288 ^{o}C, which is unrealistic.

Fig 10 shows the Temperature distribution across solid regions for Approach 2, the temperature rise in porous solid and hence to MCC solid zone above is clearly due to heat transfer from adjacent solid zone to porous solid zone. Peak temperature of Solid zone is ~96 ^{o}C, which is realistic. Fig 9 and Fig 10 clearly shows the advantage of Approach 2 over Approach 1.

Fig 11 & 12 shows the flow recirculation in the outlet area, which enhances the heat transfer in the outlet area. This causes the solid temperature on outlet area is less than inlet area as shown in Fig 9 & 10.

**Conclusion:**

Overall, we conclude that CFD Simulation of flow and Heat transfer through complex geometries can be simplified with the Porous modelling Approach without compromising on the accuracy of the simulation. Simulation Cycle time with porous modelling approach is very less and hence improves the design time and time to market. Ansys Fluent Porous modelling approach with LTNE model and Approach 2 holds good for modelling the flow and heat transfer through porous media.

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