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Example 3: Advanced 2-D Conduction with Free Convection

Consider the square shown in Fig. 1.

The left, top and right face are

exposed to air at 22.22°C. You

would like to look at the temperature

at nodes 1-12.

The bottom face (nodes 13-16) will

 be maintained at a fixed temperature

(cases 1-3). Use the finite difference

method to calculate the temperatures

at nodes 1-12.

The thermal conductivity of thematerial is assumed to be constant at

10 W/m 

· °C. and the spacing

 between nodes is 0.1 m.

The free convection coefficients are

as follows:

hright face = 5 W/m2

· °C

htop face = 10 W/m2· °C

hleft face = 7 W/m2

· °C

Figure 1: Example 3

Fixed Temperature Cases 1-3

T bottom,Case 1 = 50 °C

T bottom,Case 2 = 75 °C

T bottom,Case3 = 100 °C

Simulation techniques can be used to help with solving the temperature at nodes 1-12 for cases

1-3. A simulation tool has been provided to illustrate how quickly each case can be solved using

simulation. Please read the information page about the simple2D tool  before continuing on with

Example 3.

Example 3 Solution:

 Now that you are familiar with simple2D enter the conditions outlined in Example 3 and

compare your answers for cases 1-3 with the answers provided below.

http://www.users.muohio.edu/sommerad/NSF%20Files/Part%204%20Using%20simple2D.pdfhttp://www.users.muohio.edu/sommerad/NSF%20Files/Part%204%20Using%20simple2D.pdfhttp://www.users.muohio.edu/sommerad/NSF%20Files/Part%204%20Using%20simple2D.pdfhttp://www.users.muohio.edu/sommerad/NSF%20Files/Part%204%20Using%20simple2D.pdf

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Case 1: Set the bottom face of the square to 50°C.  

40.7962 41.4066 41.4066 40.7962

42.9723 43.6305 43.6305 42.9723

45.9072 46.5126 46.5126 45.9072

50 50 50 50

Temperatures (°C) at nodes 1-16 (case 1)

Case 2: Set the bottom face of the square to 75°C.

57.5135 58.6732 58.6732 57.5135

61.6479 62.8984 62.8984 61.6479

67.2239 68.3741 68.3741 67.2239

75 75 75 75

Temperatures (°C) at nodes 1-16 (case 2)

Case 3: Set the bottom face of the square to 100°C.

74.2308 75.9398 75.9398 74.2308

80.3234 82.1663 82.1663 80.3234

88.5407 90.2357 90.2357 88.5407

100 100 100 100

Temperatures (°C) at nodes 1-16 (case 3)