A one-dimensional plane wall of thickness 2l = 100 mm experiences uniform thermal energy generation of q˙ = 800 w/m3 and is convectively cooled at x = ±50 mm by an ambient fluid characterized by [infinity] t[infinity] = 26.0°c. if the steady-state temperature distribution within the wall is t (x) = a (l2-x2) + b where a = 10°c/m2 and b = 30°c, what is the thermal conductivity of the wall? what is the value of the convection heat transfer coefficient, h?

Question

Physics

Graves

12 November, 00:56

A one-dimensional plane wall of thickness 2l = 100 mm experiences uniform thermal energy generation of q˙ = 800 w/m3 and is convectively cooled at x = ±50 mm by an ambient fluid characterized by [infinity] t[infinity] = 26.0°c. if the steady-state temperature distribution within the wall is t (x) = a (l2-x2) + b where a = 10°c/m2 and b = 30°c, what is the thermal conductivity of the wall? what is the value of the convection heat transfer coefficient, h?

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Boris · Answer 1

The thermal conductivity of the wall = 40W/m. C

h = 10 W/m^2. C

Explanation:

The heat conduction equation is given by:

d^2T / dx^2 + egen / K = 0

The thermal conductivity of the wall can be calculated using:

K = egen / 2a = 800/2*10

K = 800/20 = 40W/m. C

Applying energy balance at the wall surface

"qL = "qconv

-K = (dT/dx) L = h (TL - Tinfinity)

The convention heat transfer coefficient will be:

h = - k * (-2aL) / (TL - Tinfinty)

h = (2 * 40 * 10 * 0.05) / (30-26)

h = 40/4 = 10W/m^2. C

From the given temperature distribution

t (x) = 10 (L^2-X^2) + 30 = 30°

T (L) = (L^2 - L^2) + 30 = 30°

dT / dx = - 2aL

d^2T / dx^2 = - 2a