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Tutorial 1: Heat conduction in a slab 2007 Cornell University BEE453, Professor Ashim Datta Authored by Vineet Rakesh and Frank Kung Software: COMSOL 3.3

Tutorial 1: Heat conduction in a slab ............................................................................................... 1 Problem Specification .................................................................................................................. 1 Step 1: Specifying the Problem Type .......................................................................................... 2 Step 2: Creating the Geometry .................................................................................................... 4 Step 3: Meshing ........................................................................................................................... 5 Step 4: Defining Material Properties and Initial Condition ........................................................... 7 Step 5: Defining Boundary Conditions......................................................................................... 8 Step 6: Specifying Solver Parameters ......................................................................................... 9 Step 7: Postprocessing .............................................................................................................. 11 Step 8: Save and Exit ................................................................................................................ 15

Problem Specification This is the first of a series of examples intended for a gentle introduction to COMSOL. Our

goal here is to compute transient temperatures profiles for 1-D heat conduction in the slab

below during the heating process. The temperature in both bottom and top surfaces of the

slab are kept constant and equal to 90 0C and 20 0C respectively. The sides of the slab are

assumed to be insulated. The thickness of the slab is 4 cm. The initial temperature is 20 C.

The density is 900 kg/m3. The thermal conductivity is 0.55 W/mK. The specific heat is 3800

J/kg K.

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Step 1: Specifying the Problem Type The problem in this case is transient heat conduction in a 1D setting. We will first set

COMSOL up for this type of problem

The model you specify determine the Governing Equations that will be used. Starting from the

energy transport equation:

source

p

diffusion

pconvectiontransient

cQ

xT

ck

xTu

tT

ρρ+

∂∂

=∂∂

+∂∂

2

2

(1)

For our problem, the temperatures are dependant on time; there is no fluid flow and no source

terms. The only mode of heat transfer is by diffusion. So the problem is a transient diffusion

problem with no convection and heat source. So the governing equation changes to:

2

2

xT

ck

tT

p ∂∂

=∂∂

ρ (2)

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1. Start COMSOL by double

clicking on the COMSOL

Multiphysics icon on the

Desktop

2. Select 2D next to Space

Dimension

(Note: COMSOL can do 1D

problems, however to give

you a better understanding of

COMSOL we’ll model the

problem as 2D)

3. Single Click on COMSOL

Multiphysics >> Heat Transfer

>> Conduction >> Transient

Analysis. Transient Analysis

under conduction is selected

as we intend to solve a time

dependent conduction

problem (Equation 2).

4. Click on the Settings Tab

5. Set the Unit system to SI

6. Click OK. COMSOL Window

opens up.

7. Under File, click on Save

as…

8. Create your own folder using

your NetID in the My

Documents folder and save

your work there. Specify the

file name (e.g. cond.mph) and

save it as .mph file.

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Step 2: Creating the Geometry The geometry in this case is a 1D slab that is 4 cm high (along y axis). We are modeling the

problem as 2D and so we assume the dimension of 2 cm in the other direction (i.e. along x

axis). In this example (in contrast with the drug delivery example) we will draw out the slab

directly without specifying the grid.

1. Click on Draw >> Specify Objects >>

Rectangle. Rectangle window opens

up.

2. Specify width as 0.02 and height as

0.04. These are the dimensions of

the slab in m.

3. Click on OK.

4. Click on Zoom Extents to fit the

geometry in the window.

The geometry that is created is shown in

the figure.

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Step 3: Meshing Meshing is dividing the geometry into small elements. We can mesh the face directly, but in

this case, we will mesh the edges first and then the face. This method is used to control the

number of elements in certain parts of the geometry like the boundaries and interfaces. In

many cases we need a finer mesh near the boundary and so we mesh the edge accordingly

and what we get is a non-uniform mesh. However in this case we mesh the geometry with a

uniform mesh with a spacing of 0.002 between the nodes.

1. Under Mesh, click on Mapped

Mesh Parameters…

2. Click on the Boundary Tab

3. In Boundary Selection, select 1

and 4 by left clicking and holding

the Ctrl key. We will specify 20

elements each on the left and

right edges.

4. Check the box for Constrained

edge element distribution

5. Click on Number of edge

elements, and type in 20 in the

box below.

6. For boundary 2 and 3, use

number of edge elements as 10.

We specify 10 elements each on

the top and bottom edges.

7. Press the ‘Remesh’ Button on the

bottom. The mesh that is

obtained is shown in the figure.

8. Click ‘Ok’

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9. Your screen should now look like

this.

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Step 4: Defining Material Properties and Initial Condition We are solving the energy equation and so we need to provide the solver with the appropriate

material property values required for the analysis. These properties are:

(i) Density (ρ): The density of the material of the slab is 900 kgm-3

(ii) Thermal Conductivity (k): The thermal conductivity is 0.55 W(mK)-1

(iii) Specific Heat (cp): The specific heat is 3800 J (kgK)-1

The Slab is initially at a temperature of 200C initially. We will also specify this in the software in

this step.

1. Under Physics, click on

Subdomain Settings…

2. Click on 1 to select the slab.

3. Left click on the text field next

to Thermal Conductivity and

type 0.55 4. Left click on the text field next

to Density and type 900

5. Left click on the text field next

to Heat Capacity and type 3800

6. Click on the Init Tab

7. In the box under Initial Value, fill

in 293. The slab is initially at a

temperature of 20 0C (=293 K).

8. Click Ok

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Step 5: Defining Boundary Conditions The boundary conditions for the problem are:

On the bottom boundary: Temperature = 900C

On the top boundary: Temperature = 200C

On the left boundary: The left boundary is insulated. Therefore, Heat Flux =0.

On the right boundary: The right boundary is insulated. Therefore, Heat Flux =0.

We now specify these boundary conditions to the solver.

The default boundary condition for this solver is thermal insulation so we simply need to change

the boundary conditions for the top and bottom

1. Under Physics, click on

Boundary Settings…

2. Click on 2 in the Boundary

Selection box

3. Change the Boundary

Condition to Temperature

4. In T0, change the

temperature to 90. The

bottom of the slab is at

900C (=363 K).

5. Repeat for side 3 using

293. The top of the slab is

at 200C (=293 K).

6. Click on OK

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Step 6: Specifying Solver Parameters We now specify the time integration method for the time dependant problem. We use the

default values for solver settings for this problem as well. To select fixed time steps, we need

to go the “Time Stepping” Tab in the Solver Parameters window. However, we do not do this

for the problem. Variable time stepping is used as the default for this problem.

1. Under Solve, click on

Solver Parameters.

2. Under the General Tab,

select Transient under

Analysis if it is not

already selected

3. Select Time dependent

under Solver.

4. In the Times: box, type

in 0:10:5400. This tells

the solver to start at 0

seconds, then save the

solution every 10

seconds until it reaches

5400 seconds.

5. Click Ok

6. Click on Get Initial Value

under Solve. This step

initializes the solver with

the value provided when

the initial conditions

were specified (Step 4).

7. Under Solve, click on

Solver Manager.

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8. Click on the Solve For

tab.

9. Select T for temperature

if it is not already

selected. By selecting T,

we are directing the

solver to solve for the

temperature.

10. Click on the Output tab.

11. Select T for temperature

if it is not already

selected. By selecting T,

we are directing the

program to save the

temperature values.

12. Press Solve. Once

Solve is pressed, the

solver solves the

transient heat transfer

equation (Equation 2 on

page 2). It will take

approximately 1-2 min

to solve. We are now

ready to do post

processing (looking at

the results).

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Step 7: Postprocessing Post-processing is viewing the results obtained on running the simulations of the problem. We

will generate graphs and charts based upon our simulation.

Displaying the Mesh

1. To display the mesh, simply click on

the Mesh Mode button. Mesh mode

can also be selected by clicking on

Mesh >> Mesh Mode.

The mesh is shown in the figure below.

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Plot Temperature vs. Time at a particular coordinate We will now plot the temperature history at point (0.01, 0.01) in the interior of the slab to see

how the temperature varies with time at that location.

1. Under Postprocessing, click on

Cross-Section Plot Parameters… (It

may take some time to open up).

2. Click on the Point tab,

3. Make sure Temperature is selected in

the Predefined quantities section.

4. In Coordinates, type in 0.01 for x and

0.01 for y.

5. Press OK

The temperature history plot obtained for

the point (0.01, 0.01) is shown in the

figure below.

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Obtain the surface plot at the last time step We now plot the temperature contour in the slab at the end time (i.e. after 1.5 hrs or 5400 s).

1. Under Postprocessing

click on Plot Parameters.

2. Click on the General Tab

3. Check the box for Surface

under Plot Type.

4. Next to Solution at time:

select 5400.

5. Click on the Surface Tab.

6. Select Temperature next

to Predefined quantities, if

it is not already selected.

7. Click on OK.

The contour plot that is

obtained is shown on the next

page.

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Step 8: Save and Exit Now, before we end the session we need to save the files for future use.

1. Go to File

2. Click on Save

3. Go to File >> Exit