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Tutorial 1: Introduction and Basic Concepts 1. An aluminum pan whose thermal conductivity is 237 W ·m-1K-1 has a flat bottom with diameter 15 cm and thickness 0.4 cm. Heat is transferred steadily to boiling water in the pan through its bottom at a rate of 1400 W. If the inner surface of the bottom of the pan is at 105°C, determine the temperature of the outer surface of the bottom of the pan. Answer: 106.3°C Solution:

2. The boiling temperature of nitrogen at atmospheric pressure at sea level (1atm) is -196°C. Therefore, nitrogen is commonly used in low temperature scientific studies since the temperature or liquid nitrogen in a tank open to the atmosphere remains constant at - 196°C until the liquid nitrogen in the tank is depleted. Any heat transfer to the tank results in the evaporation of some liquid nitrogen, which has a heat of vaporization of 198 kJ·kg-1 and a density of 810 kg·m3 at 1atm. Consider a 4m-diameter spherical tank initially filled with liquid nitrogen at 1atm and -196°C. The tank is exposed to 20°C ambient air with a heat transfer coefficient of 25 W ·m-2K-1. The temperature of the thin-shelled spherical tank is observed to be almost the same as the temperature of the nitrogen inside. Disregarding

any radiation heat exchange, determine the rate of evaporation of the liquid nitrogen in the tank as a result of the heat transfer from the ambient air. Answer: 1.37 kg·s-1 Solution:

3. The free convection heat transfer coefficient on a thin hot vertical plate suspended in still air can be determined from the change in plate temperature with time as it cools. Assuming the plate is isothermal and radiation exchange with its surroundings is negligible, evaluate the convection coefficient at the instant of time when the plate temperature is 245°C and the change in plate temperature with time (i.e. dT dt ) is -0028 K·s-1. The ambient air temperature is 25°C and the plate measures 0.4×0.4 m with a mass of 4.25 kg and a specific heat of 2770 J·kg-1K-1. Answer: 4.7 W·m-2K-1. Solution:

4. An 800-W iron is left on the iron board with its base exposed to the air at 20°C. The convection heat transfer coefficient between the base surface and the surrounding air is 35 W·m-2K-1. If the base has an emissivity of 0.6 and a surface area of 0.02 m2, determine the temperature of the base of the iron. Answer: 601°C Solution:

Matlab code:

MathCAD code:

Answer:

5. Air at 40°C flows across a long, 25-mm-diameter cylinder with an embedded electrical heater. In a series of tests, measurements were made of the power per unit length, P (W·m-1), required to maintain the cylinder surface temperature at 300°C for different free stream velocities V (m·s-1) of the air. The results are as follows: Air Velocity V, m·s-1

1

2

4

8

12

Power

P , W·m-1

450

658

983

1507

1963

(a) Determine the convection coefficient for each velocity, and display your results graphically. (b) Assuming the dependence of the convection coefficient on the velocity to be of the form h  CV n , determine the parameters C and n from the results of part (a). Solution: Based upon the h definition (see Çengel’s textbook, pages 26, 374 ) we have:   P  L  h( D) L(T  T )  h  Q conv s 

P P P    (D)(Ts  T ) 3.141 0.025  (300  40) 20.42

Air Velocity V, m·s-1

1

2

4

8

12

Power P , W·m h, W·m-2K-1

450

658

983

1507

1963

22.04

32.22

48.14

73.80

96.13

-1

100

h, W/(m2 K)

80

60

40

20

0

h  CV n

0

1

2

3

4

5

6 7 8 9 10 11 12 13 14 15 Air Velo city , m/s

 ln(h)  ln( C)  nln( V) , let’s plot ln(h) vs. ln(V)

The above plot evidences in favor of linear relationship (y = ax+b) between ln(h) and ln(V), therefore we can determine a and b from the plot, i.e. required C and n: n  0.59  h  22.22V 0.59

ln(C)  3.1  C  exp(3.1)  22.22;

150

h , W/(m2 K)

120

h  22.22V 0.59

90

60

30

0

0

1

2

3

4

5

6 7 8 9 10 11 12 13 14 15 Air Velocity, m/s

6. A spherical interplanetary probe of 0.6-m diameter contains electronics that dissipate 170 W. If the probe surface has an emissivity of 0.85 and the probe does not receive radiation from other surfaces, as, for example, from the sun, what is its surface temperature?

Solution:...