Derive an expression for the temperature distribution in a sphere of radius $r$ with uniform heat generation $\dot{q}$ and constant surface temperature $T_w$.

Consider steady one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection to the ambient air at $T_{\infty};$ with a convection coefficient of h, and by radiation to the surrounding surfaces at an average temperature of $T_{surr}$ The nodal network of the fin consists of nodes 0 (at the base) , 1 (in the middle) , and 2 (at the fin tip) with a uniform nodal spacing of $\Delta x.$ Using the energy balance approach, obtain the finite difference formulation of this problem to determine $T_1$ and $T_2$ for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in $^\circ C.$

Consider steady one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection with the ambient air at $T_{\infty}$ $(in ^\circ C)$ with a convection coefficient of h, and by radiation to the surrounding surfaces at an average temperature of $T_{surr} (in K).$ The nodal network of the fin consists of nodes 0 (at the base) , 1 (in the middle) and 2(at the fin tip) with a uniform nodal spacing of $\Delta x.$ Using the energy balance approach, obtain the finite difference formulation of this problem for the case of a specified temperature at the fin base and convection and radiation heat transfer at the fin tip.

Consider a composite wall that includes an 8-mm-thick hardwood siding, 40-mm by 130-mm hardwood studs on 0.65-m centers with glass fiber insulation (paper faced, $28 \mathrm{kg} / \mathrm{m}^{3}$), and a 12-mm layer of gypsum (vermiculite) wall board. What is the thermal resistance associated with a wall that is 2.5 m high by 6.5 m wide (having 10 studs, each 2.5 m high)? Assume surfaces normal to the x-direction are isothermal.

A solid steel sphere (AISI 1010), 300 mm in diameter, is coated with a dielectric material layer of thickness 2 mm and thermal conductivity $0.04 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$. The coated sphere is initially at a uniform temperature of $500^{\circ} \mathrm{C}$ and is suddenly quenched in a large oil bath for which $T_{\infty}=100^{\circ} \mathrm{C}$ and $h=3300 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Estimate the time required for the coated sphere temperature to reach $140^{\circ} \mathrm{C}$. Hint: Neglect the effect of energy storage in the dielectric material, since its thermal capacitance $(\rho c V)$ is small compared to that of the steel sphere.

Consider a thin electrical heater attached to a plate and backed by insulation. Initially, the heater and plate are at the temperature of the ambient air, $T_{\infty}$. Suddenly, the power to the heater is activated, yielding a constant heat flux $q_{o}^{\prime \prime}\left(\mathrm{W} / \mathrm{m}^{2}\right)$ at the inner surface of the plate. (a) Sketch and label, on T x coordinates, the temperature distributions: initial, steady-state, and at two initial, steady-state, and at two intermediate times. (b) Sketch the heat flux at the outer surface $q_{x}^{\prime \prime}(L, t)$ as a function of time.

A technique for measuring convection heat transfer coefficients involves bonding one surface of a thin metallic foil to an insulating material and exposing the other surface to the fluid flow conditions of interest. By passing an electric current through the foil, heat is dissipated uniformly within the foil and the corresponding flux, $P_{\text {elec }}^{n}$, may be inferred from related voltage and current measurements. If the insulation thickness L and thermal conductivity k are known and the fluid, foil, and insulation temperatures $\left(T_{\infty}, T_{s}, T_{b}\right)$ are measured, the convection coefficient may be determined. Consider conditions for which $T_{\infty}=T_{b}=25^{\circ} \mathrm{C}, \quad P_{\mathrm{elec}}^{\prime \prime}=2000 \mathrm{W} / \mathrm{m}^{2}$ L=10 mm, and $k=0.040 \mathrm{W} / \mathrm{m} \cdot \mathrm{K}$.

Which equation below is used to determine the heat flux for conduction?

(a) $-k \mathrm{~A} \frac{d T}{d x}$

(b) $-\mathrm{kgrad} T$

(c) $h\left(T_2-T_1\right)$

(d) $\varepsilon \sigma T^4$

(e) None of them

The steady-state temperature distribution in a one-dimensional wall of thermal conductivity 50 W/m K and thickness 50 mm is observed to be T(°C) = a + bx², where a = 200°C, b = -2000°C/m², and x is in meters.

a. What is the heat generation rate q in the wall?

b. Determine the heat fluxes at the two wall faces. In what manner are these heat fluxes related to the heat generation rate?

The temperature controller for a clothes dryer consists of a bimetallic switch mounted on an electrical heater attached to a wall-mounted insulation pad. The switch is set to open at $70^{\circ} \mathrm{C}$, the maximum dryer air temperature. To operate the dryer at a lower air temperature, sufficient power is supplied to the heater such that the switch reaches $70^{\circ} \mathrm{C}\left(T_{\mathrm{set}}\right)$ when the air temperature T is less than $T_{\mathrm{set}}$. If the convection heat transfer coefficient between the air and the exposed switch surface of $30 \mathrm{mm}^{2}$ is $25 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, how much heater power $P_{e}$ is required when the desired dryer air temperature is $T_{\infty}=50^{\circ} \mathrm{C}$?

A plane layer of coal of thickness L=1 m experiences uniform volumetric generation at a rate of $q=20 \mathrm{W} / \mathrm{m}^{3}$ due to slow oxidation of the coal particles. Averaged over a daily period, the top surface of the layer transfers heat by convection to ambient air for which $h=5 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $T_{\infty}=25^{\circ} \mathrm{C}$, while receiving solar irradiation in the amount $G_{S}=400 \mathrm{W} / \mathrm{m}^{2}$. Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are each $\alpha_{S}=\varepsilon=0.95$. (a) Write the steady-state form of the heat diffusion equation for the layer of coal. Verify that this equation is satisfied by a temperature distribution of the form $T(x)=T_{s}+\frac{q L^{2}}{2 k}\left(1-\frac{x^{2}}{L^{2}}\right)$. From this distribution, what can you say about conditions at the bottom surface (x=0)? Sketch the temperature distribution and label key features. (b) Obtain an expression for the rate of heat transfer by conduction per unit area at x=L. Applying an energy balance to a control surface about the top surface of the layer, obtain an expression for $T_{s}$. Evaluate $T_{s}$ and T(0) for the prescribed conditions. (c) Daily average values of $G_{S}$ and h depend on a number of factors, such as time of year, cloud cover, and wind conditions. For $h=5 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, compute and plot $T_{S}$ and T(0) as a function of $G_{S}$ for $50 \leq G_{S} \leq 500 \mathrm{W} / \mathrm{m}^{2}$. For $G_{S}=400 \mathrm{W} / \mathrm{m}^{2}$, compute and plot $T_{S}$ and T(0) as a function of h for $5 \leq h \leq 50 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Why is the insulated-tip solution important for the fin difficulties?

One side of a copper block $5 \mathrm{~cm}$ thick is maintained at $250^{\circ} \mathrm{C}$. The other side is covered with a layer of fiberglass $2.5 \mathrm{~cm}$ thick. The outside of the fiberglass is maintained at $35^{\circ} \mathrm{C}$, and the total heat flow through the copper-fiberglass combination is $52 \mathrm{~kW}$. What is the area of the slab?

At steady state, air at 200 kPa, 325 K, and mass flow rate of 0.5 kg/s enters an insulated duct having differing inlet and exit cross-sectional areas. The inlet cross-sectional area is $6\: cm^2$. At the duct exit, the pressure of the air is 100 kPa and the velocity is 250 m/s. Neglecting potential energy effects and modeling air as an ideal gas with constant $c_{p}= 1.008 \: kJ/kg \cdot K$, determine

$$\begin{array}{l l } \text{(a) the velocity of the air at the inlet, in m/s.}\\ \text{(b) the temperature of the air at the exit, in K.}\\ \text{(c) the exit cross-sectional area, in $cm^2$}\\ \end{array}$$