## Isothermal and Nonisothermal MEMS Heat Exchanger

*Isothermal & Nonisothermal Heat Exchanger in MEMS*

The example concerns a stainless-steel MEMS heat exchanger, which you can find in lab-on-a-chip devices in biotechnology and in micro-reactors such as for micro fuel cells. This model examines the heat exchanger in 3D, and it involves heat transfer through both convection and conduction. The model solves for the temperature and heat flux in the device and investigate the convective term’s influence on the heat exchange.

### Introduction

The following example builds and solves a conduction and convection heat transfer problem using the Heat Transfer interface. The example concerns a stainless-steel MEMS heat exchanger, which you can find in lab-on-a-chip devices in biotechnology and in micro-reactors such as for micro fuel cells. This application examines the heat exchanger in 3D, and it involves heat transfer through both convection and conduction.

### Model Definition

Figure 1 shows the geometry of the heat exchanger. It is necessary to model only one unit cell because they are all almost identical except for edge effects in the outer cells.

The governing equation for this model is the heat equation for conductive and convective heat transfer

$ \displaystyle \rho {{C}_{p}}\vec{u}.\nabla T+\nabla .\left( {-k\nabla T} \right)=Q$

where C_{p} denotes the specific heat capacity (SI unit: J/(kg·K)), T is the temperature (SI unit: K), k is the thermal conductivity (SI unit: W/(m·K)), ρ is the density (SI unit: kg/ m^{3}), **u** is the velocity vector (SI unit: m/s), and Q is a sink or source term (which you set to zero because there is no production or consumption of heat in the device).

In the solid part of the heat exchanger the velocity vector, **u** = (*u,v,w*) is set to zero in all directions. In the channels the velocity field is defined by an analytical expression that approximates fully-developed laminar flow for a circular cross section. For both the hot and cold streams, you set the velocity components in the x and z directions to zero. For the hot stream, the expression

$ \displaystyle v={{v}_{{\max }}}\left( {1-{{{\left( {\frac{r}{R}} \right)}}^{2}}} \right)$

gives the y-component of the velocity where

- v
_{max}is the maximum velocity (SI unit: m/s), which arises in the middle of the channel - r is the distance from the center of the channel (SI unit: m)
- R is the channel radius (SI unit: m)

You describe velocity in the cold stream in the same manner but in the opposite direction

$ \displaystyle v=-{{v}_{{\max }}}\left( {1-{{{\left( {\frac{r}{R}} \right)}}^{2}}} \right)$

In an extended approach, instead of using the analytical expression for the velocity field, the fluid in the channels can be simulated using the Laminar Flow interface. Here the density is defined as

$ \displaystyle \rho ={{\rho }_{m}}\left( {1-\frac{{T-{{T}_{m}}}}{{{{T}_{m}}}}} \right)$

Where ρ_{m} is the mean density (SI unit: kg/m^{3}), and T_{m} = (T_{cold} + T_{hot})/ 2 is the mean fluid temperature. The boundary conditions are insulating for all outer surfaces except for the inlet and outlet boundaries in the fluid channels. At the inlets, you specify constant temperatures for the cold and hot streams, T_{cold} and T_{hot}, respectively. At the outlets, convection dominates the transport of heat so you apply the convective flux boundary condition:

$ \displaystyle -k\nabla T.\vec{n}=0$

### Results and Discussion

Figure 2 shows the temperature isosurfaces and the heat flux streamlines for the conductive heat flux in the device. The temperature isosurfaces clearly show the convective term’s influence in the channels. Figure 3 displays the corresponding results for the extended application (see Nonisothermal MEMS Heat Exchanger for model description and results). As the plot shows, the temperature distribution is very similar to that in the first study, which can therefore be concluded to be a good approximation of the extended case.

**Source:** COMSOL Blogs, Models, Cyclopedia