Chapter 2, Problem 2.34P

One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of $5\text{W/m}\cdot \text{K}.$ For these conditions, the temperature distribution has the form $T\left(x\right)=a+bx+c{x}^2.$ The surface at $x=0$ has a temperature of $T\left(0\right)\equiv T_o=120°\text{C}$ and experiences convection with a fluid for which $T_{\infty }=20°\text{C}$ and $h=500{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ The surface at $x=L$ is well insulated.

1. Applying an overall energy balance to the wall, calculate the volumetric energy generation rate $\stackrel{.}{q}.$
2. Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution.
3. Consider conditions for which the convection coefficient is halved, but the volumetric energy generation rate remains unchanged. Determine the new values of a. b. and c. and use the results to plot the temperature distribution. Hint: recognize that $T\left(0\right)$ is no longer $120°\text{C}\text{.}$
4. Under conditions for which the volumetric energy generation rate is doubled, and the convection coefficient remains unchanged $\left(h=500{\text{W/m}}^{\text{2}}\cdot \text{K}\right),$ determine the new values of a. b, and c and plot the corresponding temperature distribution. Referring to the results of parts (b), (c), and (d) as Cases 1, 2, and 3. respectively. compare the temperature distributions for the three cases and discuss the effects of h and on the distributions.