heat exchanger ni · cells. This model examines the heat exchanger in 3D, and it involves heat transfer through both convection and conduction. Model Definition—Heat Exchanger.

Solved with COMSOL Multiphysics 4.2

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Non - I s o t h e rma l Hea t E x c h ang e r

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 microreactors such as for micro fuel cells. This model examines the heat exchanger in 3D, and it involves heat transfer through both convection and conduction.

Model Definition—Heat Exchanger

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.

cold stream

hot stream

Figure 1: Depiction of the modeled part of the heat exchanger (left).

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

(1)

where Cp denotes the specific heat capacity (J/(kg·K)), T is the temperature (K), k is the thermal conductivity (W/(m·K)), is the density (kg/m3), u is the velocity vector (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).

k T– Q Cpu T–=

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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

(2)

gives the y-component of the velocity where

• vmax is the maximum velocity (m/s), which arises in the middle of the channel

• r is the distance from the center of the channel (m)

• R is the channel radius (m)

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

(3)

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

(4)

where m the mean density (kg/m3), T is the temperature (K), and Tm(TcoldThot)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, Tcold and Thot, respectively. At the outlets, convection dominates the transport of heat so you apply the convective flux boundary condition:

v vmax 1 rR---- 2

– =

v vmax 1 rR---- 2

– –=

m 1T Tm–

Tm------------------–

=

k T n– 0=

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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 model. 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.

Figure 2: Isotherms and conductive heat flux streamlines in the cell unit’s geometry.



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Figure 3: Extended model results; isotherms and conductive heat flux streamlines in the cell unit’s geometry.

Model Library path: Heat_Transfer_Module/Tutorial_Models/heat_exchanger_ni

Modeling Instruction

M O D E L W I Z A R D

1 Go to the Model Wizard window.

2 Click Next.

3 In the Add physics tree, select Fluid Flow>Non-Isothermal Flow>Laminar Flow (nitf).

4 Click Add Selected.

5 Click Next.

6 In the Studies tree, select Preset Studies>Stationary.



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7 Click Finish.

G L O B A L D E F I N I T I O N S

Parameters1 In the Model Builder window, right-click Global Definitions and choose Parameters.

2 Go to the Settings window for Parameters.

3 Locate the Parameters section. In the Parameters table, enter the following settings:

G E O M E T R Y 1

1 In the Model Builder window, click Model 1 (mod1)>Geometry 1.

2 Go to the Settings window for Geometry.

3 Locate the Units section. From the Length unit list, select µm.

Work Plane 1 1 Right-click Model 1 (mod1)>Geometry 1 and choose Work Plane.

2 Go to the Settings window for Work Plane.

3 Locate the Work Plane section. From the Plane list, select zx-plane.

Rectangle 1 1 In the Model Builder window, right-click Geometry and choose Rectangle.

2 Go to the Settings window for Rectangle.

3 Locate the Size section. In the Width edit field, type 300.

4 In the Height edit field, type 100.

Circle 1 1 In the Model Builder window, right-click Work Plane 1 (wp1)>Geometry and choose

Circle.

2 Go to the Settings window for Circle.

3 Locate the Size and Shape section. In the Radius edit field, type R.

NAME EXPRESSION DESCRIPTION

R 50[um] Channel radius

v_mean 2.5[mm/s] Mean velocity

T_hot 330[K] Temperature, hot channel

T_cold 300[K] Temperature, cold channel

T_mean (T_hot+T_cold)/2 Mean temperature

rho_mean_w 1000[kg/m^3] Fluid mean density



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4 Locate the Position section. In the x edit field, type 100.

Circle 2 1 In the Model Builder window, right-click Work Plane 1 (wp1)>Geometry and choose

Circle.

2 Go to the Settings window for Circle.

3 Locate the Size and Shape section. In the Radius edit field, type R.

4 Locate the Position section. In the x edit field, type 200.

5 In the y edit field, type 100.

Union 1 1 In the Model Builder window, right-click Work Plane 1 (wp1)>Geometry and choose

Boolean Operations>Union.

2 Select the objects r1, c1, and c2 only.

Rectangle 21 In the Model Builder window, right-click Work Plane 1 (wp1)>Geometry and choose

Rectangle.


3 Locate the Size section. In the Width edit field, type 2*R.

4 In the Height edit field, type R.

5 Locate the Position section. In the x edit field, type 100-R.

6 In the y edit field, type -R.

Rectangle 3 1 In the Model Builder window, right-click Work Plane 1 (wp1)>Geometry and choose

Rectangle.


3 Locate the Size section. In the Width edit field, type 2*R.

4 In the Height edit field, type R.

5 Locate the Position section. In the x edit field, type 200-R.

6 In the y edit field, type 100.

Compose 11 In the Model Builder window, right-click Work Plane 1 (wp1)>Geometry and choose

Boolean Operations>Compose.

2 Select the objects uni1, r2, and r3 only.



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3 Go to the Settings window for Compose.

4 Locate the Compose section. In the Set formula edit field, type uni1-r2-r3.

Extrude 1 1 In the Model Builder window, right-click Work Plane 1 (wp1) and choose Extrude.

2 Go to the Settings window for Extrude.

3 Locate the Distances from Work Plane section. In the associated table, enter the following settings:

Form Union In the Model Builder window, right-click Form Union (fin) and choose Build Selected.

D E F I N I T I O N S

Solid1 In the Model Builder window, right-click Model 1 (mod1)>Definitions and choose

Selections>Explicit.

2 Right-click Explicit 1 and choose Rename.

3 Go to the Rename Explicit dialog box and type solid in the New name edit field.

4 Click OK.

5 Select Domain 1 only.

Channels1 In the Model Builder window, right-click Definitions and choose Explicit.

2 Select Domains 2 and 3 only.


4 Go to the Rename Explicit dialog box and type channels in the New name edit field.

5 Click OK.

Hot channel1 Right-click Definitions and choose Explicit.



4 Go to the Rename Explicit dialog box and type hot channel in the New name edit field.

DISTANCES (µm)

400



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5 Click OK.

Cold channel1 Right-click Definitions and choose Explicit.



4 Go to the Rename Explicit dialog box and type cold channel in the New name edit field.

5 Click OK.

Channel walls1 Right-click Definitions and choose Explicit.

2 Go to the Settings window for Explicit.

3 Locate the Input Entities section. From the Geometric entity level list, select Boundary.

4 Select Boundaries 6, 8, 12, and 13 only.


6 Go to the Rename Explicit dialog box and type channel walls in the New name edit field.

7 Click OK.

Variables 11 Right-click Definitions and choose Variables.

2 Go to the Settings window for Variables.

3 Locate the Geometric Entity Selection section. From the Geometric entity level list, select Domain.

4 From the Selection list, select channels.

5 Locate the Variables section. In the Variables table, enter the following settings:

M A T E R I A L S

1 In the Model Builder window, right-click Model 1 (mod1)>Materials and choose Open

Material Browser.

2 Go to the Material Browser window.

NAME EXPRESSION DESCRIPTION

rho_w rho_mean_w*(1-(T-T_mean)/T_mean)

Fluid density



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3 Locate the Materials section. In the Materials tree, select Built-In>Steel AISI 4340.

4 Right-click and choose Add Material to Model from the menu.

Steel AISI 43401 In the Model Builder window, click Steel AISI 4340.

2 Go to the Settings window for Material.

3 Locate the Geometric Entity Selection section. From the Selection list, select solid.

4 In the Model Builder window, right-click Materials and choose Open Material Browser.

5 Go to the Material Browser window.

6 Locate the Materials section. In the Materials tree, select Built-In>Water, liquid.

7 Right-click and choose Add Material to Model from the menu.

Water, liquid1 In the Model Builder window, click Water, liquid.

2 Go to the Settings window for Material.

3 Locate the Geometric Entity Selection section. From the Selection list, select channels.

N O N - I S O T H E R M A L F L O W ( N I T F )

Fluid 11 In the Model Builder window, expand the Model 1 (mod1)>Non-Isothermal Flow (nitf)

node, then click Fluid 1.

2 Go to the Settings window for Fluid.

3 Locate the Thermodynamics section. From the list, select User defined. In the associated edit field, type rho_w.

4 From the list, select User defined. In the associated edit field, type 1.

Heat Transfer in Solids 11 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose Heat

Transfer in Solids.

2 Go to the Settings window for Heat Transfer in Solids.

3 Locate the Domain Selection section. From the Selection list, select solid.

Inlet 11 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the

boundary condition Laminar Flow>Inlet.

2 Select Boundary 5 only.



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3 Go to the Settings window for Inlet.

4 Locate the Boundary Condition section. From the Boundary condition list, select Laminar inflow.

5 Locate the Laminar Inflow section. In the Uav edit field, type v_mean.

Inlet 21 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the

boundary condition Laminar Flow>Inlet.


3 Go to the Settings window for Inlet.

4 Locate the Boundary Condition section. From the Boundary condition list, select Laminar inflow.

5 Locate the Laminar Inflow section. In the Uav edit field, type v_mean.

Outlet 11 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the

boundary condition Laminar Flow>Outlet.

2 Select Boundaries 11 and 14 only.

Symmetry 11 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the

boundary condition Laminar Flow>Symmetry.


Temperature 11 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the

boundary condition Heat Transfer>Temperature.


3 Go to the Settings window for Temperature.

4 Locate the Temperature section. In the T0 edit field, type T_hot.

Temperature 21 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the

boundary condition Heat Transfer>Temperature.


3 Go to the Settings window for Temperature.

4 Locate the Temperature section. In the T0 edit field, type T_cold.

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Outflow 11 In the Model Builder window, right-click Non-Isothermal Flow (nitf) and choose the

boundary condition Heat Transfer>Outflow.


M E S H 1

Free Triangular 11 In the Model Builder window, right-click Model 1 (mod1)>Mesh 1 and choose More

Operations>Free Triangular.

2 Go to the Settings window for Free Triangular.

3 Locate the Boundary Selection section. From the Geometric entity level list, select Boundary.

4 From the Selection list, select channel walls.

Size 11 Right-click Free Triangular 1 and choose Size.

2 Go to the Settings window for Size.

3 Locate the Element Size section. Click the Custom button.

4 Locate the Element Size Parameters section. Select the Maximum element size check box.

5 In the associated edit field, type 10[um].

Free Tetrahedral 11 In the Model Builder window, right-click Mesh 1 and choose Free Tetrahedral.

2 Right-click Mesh 1 and choose Build All.

S T U D Y 1

1 In the Model Builder window, right-click Study 1 and choose Show Default Solver.

2 Expand the Study 1>Solver Configurations node.

Solver 11 In the Model Builder window, expand the Study 1>Solver Configurations>Solver 1

node, then click Dependent Variables 1.

2 Go to the Settings window for Dependent Variables.

3 Locate the Scaling section. From the Method list, select None.

4 In the Model Builder window, right-click Study 1 and choose Compute.



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R E S U L T S

Velocity (nitf)The first default plot shows the velocity magnitude on slices.

Temperature (nitf)To reproduce Figure 3, proceed as follows:

3D Plot Group 31 In the Model Builder window, right-click Results and choose 3D Plot Group.

2 Right-click Results>3D Plot Group 3 and choose Isosurface.

3 Go to the Settings window for Isosurface.

4 In the upper-right corner of the Expression section, click Replace Expression.

5 From the menu, choose Non-Isothermal Flow (Heat Transfer)>Temperature (T).

6 Locate the Levels section. From the Entry method list, select Levels.

7 In the Levels edit field, type range(301,2,329).

8 Locate the Coloring and Style section. From the Color table list, select ThermalLight.

9 In the Model Builder window, right-click 3D Plot Group 3 and choose Streamline.

10 Go to the Settings window for Streamline.

11 In the upper-right corner of the Expression section, click Replace Expression.

12 From the menu, choose Non-Isothermal Flow (Heat Transfer)>Conductive heat flux

(nitf.dfluxx,...,nitf.dfluxz).

13 Locate the Streamline Positioning section. From the Positioning list, select Start point

controlled.

14 Locate the Coloring and Style section. From the Line type list, select Tube.

15 Select the Radius scale factor check box.

16 Click the Plot button.

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heat exchanger ni · cells. This model examines the heat exchanger in 3D, and it involves heat transfer through both convection and conduction. Model Definition—Heat Exchanger.

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