# Conjugate Heat Transfer: Rectangular Fins

## Overview

The goal of this case study is to validate the Conjugate Heat Transfer analysis type on SimScale platform, particularly focusing on applications with volumetric heat generation, i.e. assignment of heat sources to solid regions. Validation is performed against experiments data and analytical calculations of Ventola et al. [1] who analyzed the thermo-fluid dynamic behavior of air-cooled, unshrouded plate-fin heat sinks (PFHS) under forced convection. The source of heat in that study is a transistor enclosed in a plastic casing.

Heat generated by the transistor spreads through the aluminum body of the heat sink, which raises its temperature. Air with lower temperature, on the other side, is forced over the heat sink and between the fins in order to cool the solid. Sufficient airflow is usually ensured by a cooling fan. In the present case, both the heat conduction through solid material and convective heat transfer to the air are simulated, hence the word Conjugate in the name. Present case can be a reference for any cooling/heating of material studies which deal with similar physics.

In the SimScale workbench, heat sources can be added to the simulation in the Advanced Concepts sub-menu as showed in Fig.1.

Heat sources can be assigned in two ways:

• Heat source [W]: This option allows assignment of total power input to the selected entity.
• Volumetric heat source [W/m3]: This option assigns power per unit volume to the selected entity.

Both workflows are validated in the present study. Below you can find links to SimScale projects with corresponding simulations.

Rectangular fins – Coarse

Rectangular fins – Intermediate

Recntangular fins – Fine

## Geometry

Fig.2. Geometric model of the heat sink and its dimensions [right figure is from Ventola et al. [1]]

The geometry of a heat sink analyzed in this work is showed in Fig.2. It is characterized by its length (L), height (H), thickness(t), number (N), spacing between neighboring fins (p), baseplate width (W) and thickness (tb). All relevant parameters are showed in Table 1.

Table 1: Heat sink dimensions (see Fig.2.)
T H t N p W tb
57.2 mm 21.8 mm 1 mm 14 2.1 mm 41.4 mm 8.4 mm

Except for contact surface area between the heat sink and the heat source, which is  As=1.555 cm2, other spatial dimensions of the transistor heat source are not mentioned in Ventola et al. [1]. For that reason, an assumption is made that the heat source is cubic in shape, with a side length of a=12.45 mm. This ensures that the area requirement of the contact face is satisfied.

## Analysis Type and Domain

Tool Type : OPENFOAM®

Analysis Type : Conjugate heat transfer (Laminar)

The computational domain’s cross-section dimensions correspond to the experimental setup, where channel width and height are 7.47 cm and 13.07 cm, respectively. The distance between the heat sink and inlet and outlet surfaces is decided based on the standard CFD guidelines, as showed on Fig.3.

Laminar analysis type is chosen, since in the most relevant part of the domain for this study, which is between heat sink’s fins, a flow with low Reynolds numbers develops. In this case Re<1000 is expected.

Mesh and Element types :

A hex-dominant unstructured grid was generated with appropriate refinements in the areas near the heat sink surface and regions where high gradients in the air flow variables are expected. Figure 4. shows the mesh over the whole computational domain, while Fig.5. shows refinements around the heat sink.

## Simulation Setup

The heat sink is made out of extruded aluminum alloy with conductivity κs=209 W/m/K. The cooling fluid is air, while the transistor heat source specifications given by the manufacturer are:

• Contact area between the element and the heat sink As=1.555 cm2 [1]
• Thermal resistance between them Rjc=0.5 K/W [1] .

No material properties of the heat source are provided, so making an assumption of cubic heat source with side length of a=12.45mm, together with the given thermal resistance, the conductivity of the heat source is defined as:

(1)$\kappa_{hs} = \frac{\frac{1}{2} a}{R_{jc} A_s},$

with its value being κhs=80 K/W. Additionally, the following boundary conditions are prescribed in the simulation:

• No slip boundary conditions at all solid walls
• Constant inlet temperature and velocity
• Outlet pressure of 101325 Pa
• Zero temperature gradient at all walls.
• Thermal coupling at all solid-solid and solid-fluid interfaces.

Automatic relaxation was used in all cases.

## Results

In the reference study [1], experimental and analytical results are expressed as overall heat resistance between the heat source and the ambient air Rja. For illustration purposes, the complete array of thermal resistances between junction and air is shown in Fig.6.

Therefore, Rja is composed of four components (Fig.6.):

Rja=Rjc+Rcs+Rsa+Rspr

where where Rjc, Rcs, Rsa, and Rspr are the junction-to-case, case-to-sink, sink-to-ambient, and spreading resistances, respectively.

A comparison of the simulation results for intermediate mesh fineness (see project link: Rectangular fins – Intermediate at the bottom of the page) with experimental and analytical values for different inlet velocities can be seen in Table 2., as well as on Fig.7.

Quantities in the table and result plots are:

• va : inlet air velocity
• Ta : ambient air temperature
• P : thermal power (output of the heat source)
• Tj,e: junction temperature – experiment
• Rja,e: junction-to-air thermal resistance – experiment
• Rja,t: junction-to-air thermal resistance – analytical calculation
• Tj,s : junction temperature – simulation
• Rja,s: junction-to-air thermal resistance – simulation

The value of Tj,s  is obtained in post-processing using the following relation:
$T_{j,s} = T_{interface} + P \cdot R_{jc}$

where Tinterface is the area average of temperature on the surface connecting heat sink and heat source, P is thermal power output of the heat source and Rjc is thermal resistance between them, a provided by the manufacturer (see Simulation Setup). Tinterface is obtained using a Result Control Item in SimScale Workbench.

Overall thermal resistance between cooling air and the junction, is obtained by:

$R_{ja,s} = \frac{T_{j,s} – T_a}{P}$

Table 2: Simulation results compared to experimental and analytical values [1]
va [m/s] Ta [K] P [W] Tj,e [K] Rja,e [K/W] Rja,t [K/W] Tj,s [K] Rja,s [K/W]
5.6 296.95 56.64 371.25 1.312 1.203 367 1.237
7.2 297.35 71.4 384.15 1.216 1.142 381.67 1.181
8.8 297.95 82.36 391.85 1.140 1.1 392.73 1.151
10.2 298.25 87.32 395.05 1.109 1.071 396.24 1.122
11.5 298.95 85.07 391.15 1.084 1.05 392.85 1.104
12.8 299.25 76.3 377.45 1.025 1.031 382.17 1.087
13.9 299.65 60.24 357.55 0.961 1.017 364.16 1.071

A trend of thermal resistance decrease can be observed. This can be attributed to increased convective heat transfer from heat sink surface to the stream of air, which occurs with higher fluid velocities. Overall thermal resistance depends on both convective heat transfer and conduction through the solid material. If only one of the two heat transfer modes is intensified, the overall thermal resistance drops. Once the material is chosen, and an object is designed and manufactured, conduction through solid can’t be influenced any more. Only way to enhance cooling at that point is through enhancing the convective heat transfer.

A mesh convergence study was performed to assess grid independence of simulation results. Mesh specifications and cell counts can be found in Table 3. A case with largest difference between experimental and simulation results is chosen for the mesh convergence study (va=13.9 [m/s]). Simulations were done in projects with corresponding names: Rectangular fins – CoarseRectangular fins – Intermediate and Rectangular fins – Fine, links to which you can find at the bottom of this page.

Table 3: Mesh Metrics
Fineness Mesh Operation Number of cells Mesh Type
Coarse Hex-dominant parametric 2 904 901 3D hexahedral
Medium Hex-dominant parametric 5 128 288 3D hexahedral
Fine Hex-dominant parametric 11 615 447 3D hexahedral

For Coarse and Fine meshes, simulations were completed for all experimental conditions (see Table 2.), results of which are in Fig.9.

Fig.9. Comparison of junction-to-air thermal resistance for Coarse and Fine meshes

In sections Geometry and Simulation Setup assumptions that had to be made about dimensions and conductivity of the heat source are explained. With only thermal resistance between heat source and heat sink Rjc given by the authors [1] , conductivity and thickness of transistor heat source were calculated using equation (1). However, one can argue there is still a degree of freedom in doing that. Do we assume the thickness and calculate the conductivity, or vice-versa?

A study was conducted in order to make sure that the result is invariant to thickness and conductivity changes by varying both parameters in such way that thermal resistance Rjc remains 0.5 K/W, as it was in the experiment. It was found that as long as this condition is satisfied there is no significant difference in the result (up to ± 0.15%).

Similarly, different numerical settings were compared, with both first order and second order schemes and with or without non-orthogonal correctors. Results for temperature and overall thermal resistance Rja,s varied up to ± 0.25%.

All simulation setups and results are available at project links. Sensitivity studies were performed on the coarsest mesh (Project: CHT: Rectangular fins – Coarse) having in mind the mesh convergence study, which demonstrated its validity (Fig.8. and 9.).

## Disclaimer

This offering is not approved or endorsed by OpenCFD Limited, producer and distributor of the OpenFOAM software and owner of the OPENFOAM® and OpenCFD® trade marks. OPENFOAM® is a registered trade mark of OpenCFD Limited, producer and distributor of the OpenFOAM software.