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Determine F12 and F21 for the following configurations using the reciprocity theorem and other basic shape factor relations. Do not use tables or charts.
- Long duct
(a)
The shape factors.
Answer to Problem 13.1P
The shape factors are 1 and 0.212.
Explanation of Solution
Formula used:
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Here,
Calculation:
The geometrical shape is given as,
Figure (1)
The above geometry shows that inner surface completely intercepted by the outer surface. Therefore, the value of shape factors can be calculated as,
Conclusion:
Therefore, the shape factors are 1 and 0.212.
(b)
The shape factors.
Answer to Problem 13.1P
The shape factors are 0.5 and 0.25.
Explanation of Solution
Formula used:
The summation rule is given as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Calculation:
The geometrical shape is given as,
Figure (2)
The above geometry is a convex geometry, therefore the values of shape factor are obtained as,
Further obtaining the values of shape factor by summation rule as,
Further obtaining the values as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Conclusion:
Therefore, the shape factors are 0.5 and 0.25.
(c)
The shape factors.
Answer to Problem 13.1P
The shape factors are 1, 0.637 and 0.363.
Explanation of Solution
Formula used:
The summation rule is given as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
The summation rule is given as,
Calculation:
The geometrical shape is given as,
Figure (3)
The above geometry have a flat geometry at bottom, therefore the values of shape factor are obtained as,
Further obtaining the values of shape factor by summation rule as,
The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,
Further obtain the values of shape factor as,
Conclusion:
Therefore, the shape factors are 1, 0.637 and 0.363
(d)
The shape factors.
Answer to Problem 13.1P
The shape factors are 0.5 and 0.707.
Explanation of Solution
Formula used:
The summation rule is given as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Calculation:
The geometrical shape is given as,
Figure (4)
The above geometry have a flat geometry at bottom, therefore the values of shape factor are obtained as,
Further obtaining the values of shape factor by summation rule as,
The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,
Conclusion:
Therefore, the shape factors are 0.5 and 0.707.
(e)
The shape factors.
Answer to Problem 13.1P
The shape factors are 0.5 and 0.
Explanation of Solution
Formula used:
The summation rule is given as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Calculation:
The geometrical shape is given as,
Figure (5)
The above geometry have a flat geometry at bottom, therefore the values of shape factor are obtained as,
Further obtaining the values of shape factor by summation rule as,
The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,
Conclusion:
Therefore, the shape factors are 0.5 and 0.
(f)
The shape factors.
Answer to Problem 13.1P
The shape factors are 1 and 0.125.
Explanation of Solution
Formula used:
The summation rule is given as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Calculation:
The geometrical shape is given as,
Figure (6)
The above geometry have flat disk surface, therefore the values of shape factor are obtained as,
Further obtaining the values of shape factor by summation rule as,
The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,
Conclusion:
Therefore, the shape factors are 1 and 0.25.
(g)
The shape factors.
Answer to Problem 13.1P
The shape factors are 0.5 and 0.637.
Explanation of Solution
Formula used:
The summation rule is given as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Calculation:
The geometrical shape is given as,
Figure (7)
The above geometry have flat disk surface, therefore the values of shape factor are obtained as,
Further obtaining the values of shape factor by summation rule as,
The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,
Conclusion:
Therefore, the shape factors are 0.5 and 0.637.
(h)
The shape factors.
Answer to Problem 13.1P
The shape factors are 1 and
Explanation of Solution
Formula used:
The summation rule is given as,
The expression for the shape factor for surface 2 with respect to surface 1 is given as,
Calculation:
The geometrical shape is given as,
Figure (8)
The above geometry have spherical surface, therefore the values of shape factor are obtained as,
Further obtaining the values of shape factor by summation rule as,
The expression for the shape factor for surface 2 with respect to surface 1 can be calculated as,
Conclusion:
Therefore, the shape factors are 1 and
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