A plane wall of thickness 2L = 30 mm and thermal conductivity k = 7 W/m-K experiences uniform volumetric heat generation at arate q, while convection heat transfer occurs at both of its surfaces (x = − L, + L), each of which is exposed to a fluid oftemperature T = 20°C. Under steady-state conditions, the temperature distribution in the wall is of the formT(x) = a + bx + cx² where a = 82.0°C, b = -210°C/m, c = -2x 10°C/m², and x is in meters. The origin of the x-coordinate is atthe midplane of the wall.(a) What is the volumetric rate à of heat generation in the wall?(b) Determine the surface heat fluxes, q" (L)and q ( + L).(c) What are the convection coefficients for the surfaces at x = - Land x = + L?The volumetric rate of heat generation in the wall, in W/m³:q = iW/m³The surface heat flux, in W/m²:qx ( - L) = iThe surface heat flux, in W/m²:q (+ L) = iW/m²W/m²The convection coefficients for the surface at x = - L, in W/m²-K:h(- L) = iW/m².KThe convection coefficients for the surface at x = + L, in W/m².K:h(+ L) = iW/m².K

Answered: (a) What is the volumetric rate à of…

Elements Of Electromagnetics

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A plane wall of thickness 2L = 30 mm and thermal conductivity k = 7 W/m-K experiences uniform volumetric heat generation at a
rate q, while convection heat transfer occurs at both of its surfaces (x = − L, + L), each of which is exposed to a fluid of
temperature T = 20°C. Under steady-state conditions, the temperature distribution in the wall is of the form
T(x) = a + bx + cx² where a = 82.0°C, b = -210°C/m, c = -2x 10°C/m², and x is in meters. The origin of the x-coordinate is at
the midplane of the wall.
(a) What is the volumetric rate à of heat generation in the wall?
(b) Determine the surface heat fluxes, q" (L)and q ( + L).
(c) What are the convection coefficients for the surfaces at x = - Land x = + L?
The volumetric rate of heat generation in the wall, in W/m³:
q = i
W/m³
The surface heat flux, in W/m²:
qx ( - L) = i
The surface heat flux, in W/m²:
q (+ L) = i
W/m²
W/m²
The convection coefficients for the surface at x = - L, in W/m²-K:
h(- L) = i
W/m².K
The convection coefficients for the surface at x = + L, in W/m².K:
h(+ L) = i
W/m².K

Expert Solution

Step 1

Given:

The thickness of the wall is, $2 L = 30 m m, L = 15 m m = 0.015 m$

The thermal conductivity of wall is, $k = 7 W / m - K$

Part (a)

The 1-0 heat conduction with heat generation can be represented as,

$\frac{d^{2} T}{d x^{2}} + \frac{q_{g}}{k} = 0$ .....................(1)

We have steady state condition

$T (x) = a + b x + c x^{2}$

Differentiate the above equation we have

$\frac{d T}{d x} = b + 2 c x$

Again differentiate above equation,

$\frac{d^{2} T}{d x} = 2 c$

Substitute above value in equation (1)

$2 c + \frac{q_{g}}{k} = 0 q_{g} = - 2 c \times k$

Substitute $- 2 \times 10^{4} {}^{°}C / m^{2}$ for c and $7 W / m - k$ for k in the above equation,

$q_{g} = - 2 (- 2 \times 10^{4}) \times 7 q_{g} = 2.8 \times 10^{5} \frac{W}{m^{3}}$

Thus the heat generation is, $2.8 \times 10^{5} W / m^{3}$ .

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(a) What is the volumetric rate à of heat generation in the wall? (b) Determine the surface heat fluxes, q (L)and q (+ L).