# Description

Heat transfer can be defined as energy transfer between systems and occurs only because of temperature differences. Heat flows spontaneously from a hotter body. Electronic devices produce heat as a by-product of normal operation. When electronic current flows through a semiconductor or a passive device, a portion of the power is dissipated as heat energy. In above mentioned example heat was transferred by both conduction and convection. In total, there are three modes of heat transfer, by conduction, by convection and by radiation. Heat transfer in solids occurs by conduction mainly because of temperature gradient in different parts of the material.

Conjugate heat transfer (CHT) refers to the coupling of conduction in solids with the convective and radiative het transfer in the surrounding fluids. The heat transfer between solid and fluid occurs by the process of conduction and then it gets carried away the current of the fluid. When the solid body is sufficiently hot then heat transfer also occurs via radiation, however radiation is not considered in this example. Accurately predicting CHT is an essential part of the design process in many industrial applications including electronics cooling, turbo-machinery and HVAC.

The heat transfer rate in a fluid flow problem depends upon velocity, temperature and Geometry. For better understanding, velocity profile and temperature profile inside the cavity at a steady state are drawn in the next sections and a parametric study at different Fin width is done.

**Problem statement**

Heat transfer between a solid and fluid is investigated in this section. Heat is generated inside the solid at 80W. Flow under investigation is assumed to be incompressible and laminar.

Figure 1.1.: Domain of the case with heat source, inflow on left side and outflow on the right side.

Figure 1.2.: Detail view of the fins geometry(in mm) No. of Fins 26

Schematic diagram of the problem is shown above and the boundary conditions. Walls are isolated and heat flux across them is zero. Heat is generated inside Silicon chip. Zero Dirichlet boundary condition is used for velocity on all walls. The material properties of air at temp 300.0K are used for inlet flow and also for the initial condition.

## Mesh

Case is solved using a increasing grid density, highest at the fins for high resolution which enables in capturing the high variation in temperature gradient.

Figure1.3 Mesh for case-0.50mm Fin size, no. of cells = 51

*35*57.

## Results and discussion

All simulations were run till a stable temperature at the face of a silicon chip is reached signifying that a steady solution is achieved.

Figure 1.4 Temperature at different Fin Size.

As the Fin pitch remains constant at 2.5mm but fin size is increased, it can be seen that heat transfer because of convection is much more in case of a smaller fin size.

Figure 1.5 Pressure at different Fin Size.

It is observed from the pressure and temperature plots that stacking too many fins is not a solution for decreasing the Temp on the heat sink since this prevents the passage of air coming from the inlet to the hottest part of the heat sink.

Figure 1.6 Variation of Temperature with the vertical distance at the centre of geometry.

Above Figure presents the comparison between the Temperature at the centre of geometry for different fin pitch. We can see that there is not much variation in the temperature inside the solids and but there is a sharp decline near the contact between fins and fluid.

Figure 1.7 Average Temperature at the face of the chip. F 0.50 P2.50 refers that the Fin size is 0.5mm and fin pitch is 2.50mm.

Temperature centres around 342K when Fin pitch is increased more than 3mm.

# Comparison with Experimental Data

In the paper ( 1 ) experiment results are compared with the difference in maximum and the minimum temperature obtained in the simulations. Figure 1.8 belows shows the method used for finding the the maximum and the minimum temperature in the Plate Heat Sink.

Final Result obtained after comparison is shown below.

## Comparison with Analytical solution

Re_{d} = \frac{U_{\infty L}}{\nu}
= \frac{2.562 \times 0.04}{1.8 \times 10^{-5}}
= 5693

Nusselt number in case of heat transfer from a rectangular fin can be described from the equation below:

\begin{equation}
Nu_{d} = c Re^{m} Pr^{\frac{1}{3}}
\end{equation}

Here \mathbf{c} is a constant 0.228

\mathbf{Re} is the Reynolds number

\mathbf{Pr} is the Prandtl number (0.71 in case of air

\mathbf{m} depends on the type of fin( 0.731 )

After calculation Nu_{d} is 113.112, with the help of Nu_{d}, heat transfer coefficient can be found easily

By using the formula h = Nu\times \frac{\kappa}{L}, it comes out to be 79.127

\vspace{100pt}

Heat gained by air is equal to the heat released by the Fins.

Heat content of air at inflow =

= \rho A * \textbf{inlet velocity} * \textbf{Specific Heat} *\textbf{inlet temperature(Avg 300K)} \\
= 8753.520182

Heat content of air at outflow =

= \rho* A * \textbf{outlet velocity} * \textbf{Specific Heat} * \textbf{outlet temperature(Avg 309.35) } \\
= 9026.417549

Net energy gained = 272.8973664

Heat transfer equation in solids is as follows:

\frac{Q}{A} = \kappa \frac{T_{0} - T_{1}}{L}

Q is the heat produced in W, A is the area of cross-section of solids. \kappa is the thermal conductivity.

In our case, Q = 80W

As total no. of fins = 22 and 2 sides

A = 0.054 \times 0.064 \times (44-1) = 0.09288m^{2}

L = 0.04m

\kappa = 240 W/m^{2}.K

Temperature difference comes out to be = 37.108K

Average Temperature at CPU face in Simulation = 343.0K

Average Temperature at CPU calculated analytically = 337.108K

Error = 15.94 \%

\textbf{Temperature variation comparison in fin between Simulation and Analytical Solution}

\frac{T-T_{\infty}}{T_{0} - T_{\infty}} = \frac{cosh[(1-\frac{x}{L})\sqrt{\frac{hP}{\kappa A}}L]}{cosh[\sqrt{\frac{hP}{\kappa A}}L]}

**Error** between the above curves comes out to be 7.16/43.0 =

**16.65 %**
## References

[1] R.Mohan and Dr.P.Govindarajan, Thermal Analysis of CPU with variable Heat Sink Base Plate

Thickness using CFD.

[2] Mohan, R. & Govindarajan, P. J Mech Sci Technol (2011) 25: 2003. https://doi.org/10.1007/s12206-011-0531-8.

[3] Ralph Remsburg, Thermal Design of Electronic Equipment.