Fundamental of Heat Transfer Chapter 2 Problem 5 Answer Key

PROBLEM 2.5 KNOWN: End-face temperatures and temperature dependence of k for a truncated cone. FIND: Variation with axial distance along the cone of q q k, and dT / dx. x x , , ′ ′ SCHEMATIC: ASSUMPTIONS: (1) One-dimensional conduction in x (negligible temperature gradients in the r direction), (2) Steady-state conditions, (3) Adiabatic sides, (4) No internal heat generation. ANALYSIS: For the prescribed conditions, it follows from conservation of energy, Eq. 1.11c, that for a differential control volume, . E E or q q in out x x+dx = = Hence q x is independent of x. Since A(x) increases with increasing x, it follows that ( ) x x q q /A x ′′ = decreases with increasing x. Since T decreases with increasing x, k increases with increasing x. Hence, from Fourier’s law, Eq. 2.2, ′′ = − q k dT dx x , it follows that | dT/dx | decreases with increasing x. COMMENT: How is the analysis changed if a has a negative value? r

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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 5 Answer Key

PROBLEM 2.5 KNOWN: End-face temperatures and temperature dependence of k for a truncated cone. FIND: Variation with axial distance along the cone of q q k, and dT / dx.x x, , SCHEMATIC:

ASSUMPTIONS: (1) One-dimensional conduction in x (negligible temperature gradients in the r direction), (2) Steady-state conditions, (3) Adiabatic sides, (4) No internal heat generation. ANALYSIS: For the prescribed conditions, it follows from conservation of energy, Eq. 1.11c, that for a differential control volume, .E E or q qin out x x+dx= = Hence qx is independent of x. Since A(x) increases with increasing x, it follows that ( )x xq q / A x = decreases with increasing x. Since T decreases with increasing x, k increases with increasing x. Hence, from Fouriers law, Eq. 2.2,

= q k dTdxx

,

it follows that | dT/dx | decreases with increasing x. COMMENT: How is the analysis changed if a has a negative value?

rr

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