Fundamental of Heat Transfer Chapter 2 Problem 4 Answer Key

PROBLEM 2.4 KNOWN: Symmetric shape with prescribed variation in cross-sectional area, temperature distribution and heat rate. FIND: Expression for the thermal conductivity, k. SCHEMATIC: ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in x-direction, (3) No internal heat generation. ANALYSIS: Applying the energy balance, Eq. 1.11c, to the system, it follows that, since , E E in out = ( ) x q Constant f x. = ≠ Using Fourier’s law, Eq. 2.1, with appropriate expressions for A x and T, yields ( ) ( ) x x 2 3 dT q k A dx d K 6000W=-k 1-x m 300 1 2x-x . dx m =− ⎡ ⎤ ⋅ ⋅ − ⎢ ⎥ ⎣ ⎦ Solving for k and recognizing its units are W/m⋅K, ( ) ( ) ( ) ( ) 2 2 -6000 20 k= . 1 x 2 3x 1-x 300 2 3x = ⎡ ⎤ − + − − ⎢ ⎥ ⎣ ⎦ < COMMENTS: (1) At x = 0, k = 10W/m⋅K and k → ∞ as x → 1. (2) Recognize that the 1-D assumption is an approximation which becomes more inappropriate as the area change with x, and hence two-dimensional effects, become more pronounced.

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7th Edition, Heat TransferWritten by: Incropera, DeWitt, Bergman, LavineConduction and basic concepts such as Fourier's Law, Heat Diffusion Equation, Boundary and Initial Conditions.

Transcript of Fundamental of Heat Transfer Chapter 2 Problem 4 Answer Key

PROBLEM 2.4 KNOWN: Symmetric shape with prescribed variation in cross-sectional area, temperature distribution and heat rate. FIND: Expression for the thermal conductivity, k. SCHEMATIC:

ASSUMPTIONS: (1) Steady-state conditions, (2) One-dimensional conduction in x-direction, (3) No internal heat generation. ANALYSIS: Applying the energy balance, Eq. 1.11c, to the system, it follows that, since ,E Ein out=

( )xq Constant f x .= Using Fouriers law, Eq. 2.1, with appropriate expressions for Ax and T, yields

( ) ( )x x

2 3

dTq k Adx

d K6000W=-k 1-x m 300 1 2x-x .dx m

=

Solving for k and recognizing its units are W/mK,

( ) ( ) ( )( )22-6000 20k= .1 x 2 3x1-x 300 2 3x = + < COMMENTS: (1) At x = 0, k = 10W/mK and k as x 1. (2) Recognize that the 1-D assumption is an approximation which becomes more inappropriate as the area change with x, and hence two-dimensional effects, become more pronounced.

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