Consider a pin fin with variable conductivity k(T), constant cross sectional area Ac and constant perimeter, P. Develop the difference equations for steady one-dimensional conduction in the fin and suggest a method for solving the equations. The fin is exposed to ambient temperature Ta through a heat transfer coefficient h. The fin tip is insulated and the fin root is at temperature To.
GIVEN
• Fin with variable thermal conductivity, k(T)
FIND
(a) Difference equation (b) Solution method
SKETCH
For the control volume centered over the interior node i, an energy balance gives
The thermal conductivities are given in
For the node at the root T1 = To.
At the tip, an energy balance gives
These equations can be written in tridiagonal form,
Note that kright, kleft, and kN depend on the nodal temperatures. To solve the system of equations, it will be necessary to (1) Guess at the nodal temperatures (2) Calculate the values for kright, kleft, and kN (3) Calculate the matrix coefficients ai, bi, ci and di, 1 ? i ? N (4) Solve for the nodal temperatures by inverting the matrix as in (5) Repeat steps 2 through 4 until the nodal temperatures cease to change
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a. the same as f. b. higher than f. c. lower than f. d. unrelated to f.
An area of 1.00 × 102 cm2 is how many square meters?
A) 1.00 m2 B) 1.00 × 102 m2 C) 1.00 × 10-2 m2 D) 1.00 × 104 m2 E) 1.00 × 10-3 m2
How fast do plates move on Earth?
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Show that for fully developed laminar flow between two flat plates spaced 2a apart, the Nusselt number based on the ‘bulk mean’ temperature and the passage spacing is 4.12 if the temperature of both walls varies linearly with the distance x, i.e., ?T/?x = C. The ‘bulk mean’ temperature is defined as
GIVEN
• Fully developed laminar flow between two flat plates
• Spacing = 2a & ?T/?x = C
• Bulk mean temperature as defined above
FIND
• Show that the Nusselt number based on the bulk mean temperature = 4.12
ASSUMPTION
• Steady state
• Constant and uniform property values
• Fluid temperature varies linearly with x (This corresponds to a constant heat flux boundary)
SKETCH
