A plane wall of thickness 2L = 60 mm and thermal conductivity k= 5W/m.K experiences uniform volumetric heat generation at a rate qdot , while convection heat transfer occurs at both of its surfaces (x=-L, +L), each of which is exposed to a fluid of temperature T[infinity] = 30 oC . Under steady-state conditions, the temperature distribution in the wall is of the form T(x) = a + bx+ cx2 where a = 860C, b=-2000C /m, c=-2× 104 0C /m2, and x is in meters. The origin of the x-coordinate is at the midplane of the wall. a) Sketch the temperature distribution and identify significant physical features.

b) What is the volumetric rate of heat generation in the wall?

c) Determine the surface heat fluxes at x=-L, +L, and How are these fluxes related to the heat generation rate?

d) What are the convection coefficients for the surfaces at x=-L and x=+L?

Explanation:

A plane wall of thickness 2L=40 mm and thermal conductivity k=5W/m⋅Kk=5W/m⋅K experiences uniform volumetric heat generation at a rateq  

˙

q

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∘CT  

∞

=20  

∘

C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x)=a+bx+cx2T(x)=a+bx+cx  

2

 where a=82.0∘C,b=−210∘C/m,c=−2×104C/m2a=82.0  

∘

C,b=−210  

∘

C/m,c=−2×10  

4

C/m  

2

, and x is in meters. The origin of the x-coordinate is at the midplane of the wall. (a) Sketch the temperature distribution and identify significant physical features. (b) What is the volumetric rate of heat generation q in the wall? (c) Determine the surface heat fluxes, q

′′

x

(−L)q  

x

′′

(−L) and q

′′

x

(+L)q  

x

′′

(+L). How are these fluxes related to the heat generation rate? (d) What are the convection coefficients for the surfaces at x=-L and x=+L? (e) Obtain an expression for the heat flux distribution q

′′

x

(x)q  

x

′′

(x). Is the heat flux zero at any location? Explain any significant features of the distribution. (f) If the source of the heat generation is suddenly deactivated (q=0), what is the rate of change of energy stored in the wall at this instant? (g) What temperature will the wall eventually reach with q=0? How much energy must be removed by the fluid per unit area of the wall (J/m2)(J/m  

2

) to reach this state? The density and specific heat of the wall material are 2600kg/m32600kg/m  

3

 and 800J/kg⋅K800J/kg⋅K, respectively.