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…

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

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter2: Steady Heat Conduction

Section: Chapter Questions

Problem 2.29P: 2.29 In a cylindrical fuel rod of a nuclear reactor, heat is generated internally according to the...

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Concept explainers

Conduction and Convection

When energy travels from one object to another it is said to have undergone heat transfer. Heat transfer is nothing but the transfer of thermal energy or internal energy from one thing to another, like e.g., when a vessel with water, kept on top of a burner, heats up because of the flame underneath it.

Question

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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