Two Dimensional Steady Conduc4on

Size: px

Start display at page:

Download "Two Dimensional Steady Conduc4on"

Susan Burns
5 years ago
Views:

1 1 Two Dimensional Steady Conduc4on

2 Shape Factors 2 Shape factors are a simple means of accoun4ng for the geometric factors affec4ng conduc4on in 1D, 2D and 3D systems. The shape factor S is defined such that: q = Sk(T 1 T 2 ) The shape factor is related to the thermal resistance through: R = 1 S k Shape factors are determined from analy4c solu4ons to Laplace s equa4on and using the temperature field to calculate S or R. 2 T x T y T z 2 = 0 Many tabulated results exist for S.

3 Shape Factors 3 Shape factors are convenient for determining thermal resistance in simple systems such as:

4 Shape Factors 4 Shape factors are convenient for determining thermal resistance in complex systems such as:

5 Shape Factors 5 Shape factors can easily be used in thermal circuits: where R Total = 1 h i A i + 1 Sk + 1 h o A o q = T i T o R Total Here is a sample of shape factors from your text:

6 6

7 7

8 Shape Factors Many shape factors are tabulated in a number of Heat Transfer Handbooks. Here is a sample of some useful shape factors taken from the Handbook of Heat Transfer, McGraw- Hill, 1985: 8

9 Shape Factors 9

10 Shape Factors 10

11 Shape Factors 11 Shape factors can also be approximated from many one dimensional solu4ons. Other solu4ons can be used to approximate situa4ons where no shape factor is available. Two rules for shape factor equivalence : Rule #1 Preserve the smaller surface area (inside or outside). Rule #2 Preserve the volume of the conduc4ve material in the system.

12 Example Calculate the heat transfer rate from the cubical oven enclosure having an outside side length of W=5 m, and a wall thickness of t=0.35 m. The thermal conduc4vity of the wall is 1.4 W/mK, and there is an outside heat transfer coefficient of h o =5 W/m 2 k and an ambient temperature of 25 C. The inside surface is constant at 1100 C. Use two approaches: 1) Find appropriate shape factors from the table in your text or from the tables provided. 2) Using the two modelling rules approximate the cubical enclosure as a spherical enclosure.

Example - 2 13 Compare the shape factor for the system composed of a circular

13 Example Compare the shape factor for the system composed of a circular hole in a square bar, with that approximated using a circular hole in a circular bar.

14 Example Compare the shape factor for a spherical enclosure in an infinite medium to that of a cubical enclosure in an infinite medium. Using the solu4on for the spherical enclosure approximate the shape factor for the cubical enclosure and compare the error. Why? So that we can test the two rule method!

15 Example You are asked to analyze the liquid cooled heat sink, which is produced by machining triangular grooves in two matching plates which eventually form a diamond shaped holes when the plates are assembled as shown in the sketch. Calculate the overall heat transfer coefficient for the system shown below assuming that the internal convec4ve heat transfer coefficient is 1000 W/m 2 K. You will need to approximate the shape factor using sound physical principles as no exact solu4on is available to this configura4on. Assume that the plate is aluminum with a thermal conduc4vity of k = 180 W/mK. If the surface temperature is limited to T s = 50 C and coolant has a mean bulk temperature of 30 C, what heat transfer rate is permissible, when the sink contains four channels of length 15 cm?

AURANGABADKAR. Table 1. Table showing HTC for different notches at base temp of 60 0 C. Table 2: Comparison of HTC by CFD and HTC by experiment

AURANGABADKAR. Table 1. Table showing HTC for different notches at base temp of 60 0 C. Table 2: Comparison of HTC by CFD and HTC by experiment HEAT TRANSFER ANALYSIS AND OPTIMIZATION OF FINS BY VARIATION IN GEOMETRY 1 MAYANK JAIN, 2 MAHENDRA SANKHALA, 3 KANHAIYA PATIDAR, 4 LOKESH AURANGABADKAR 1,2,3 Student, Medicaps University, Indore (M.P.)