Problem 157
A long electrical resistance wire of radius \(r_{1}=0.25 \mathrm{~cm}\) has a thermal conductivity $k_{\text {wirc }}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Heat is generated uniformly in the wire as a result of resistance heating at a constant rate of \(0.5 \mathrm{~W} / \mathrm{cm}^{3}\). The wire is covered with polyethylene insulation with a thickness of \(0.25 \mathrm{~cm}\) and thermal conductivity of $k_{\text {ins }}=0.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The outer surface of the insulation is subjected to free convection in air at \(20^{\circ} \mathrm{C}\) and a convection heat transfer coefficient of \(2 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Formulate the temperature profiles for the wire and the polyethylene insulation. Use the temperature profiles to determine the temperature at the interface of the wire and the insulation and the temperature at the center of the wire. The ASTM D1351 standard specifies that thermoplastic polyethylene insulation is suitable for use on electrical wire that operates at temperatures up to \(75^{\circ} \mathrm{C}\). Under these conditions, does the polyethylene insulation for the wire meet the ASTM D1351 standard?
Problem 158
In a manufacturing plant, a quench hardening process is used to treat steel ball bearings ( $c=500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\(, \)\rho=7900 \mathrm{~kg} / \mathrm{m}^{3}\( ) of \)25 \mathrm{~mm}$ in diameter. After being heated to a prescribed temperature, the steel ball bearings are quenched. Determine the rate of heat loss if the rate of temperature decrease in the ball bearings at a given instant during the quenching process is \(50 \mathrm{~K} / \mathrm{s}\).
Problem 159
Consider a spherical reactor of \(5-\mathrm{cm}\) diameter operating at steady conditions with a temperature variation that can be expressed in the form of \(T(r)=a-b r^{2}\), where \(a=850^{\circ} \mathrm{C}\) and $b=5 \times 10^{5} \mathrm{~K} / \mathrm{m}^{2}\(. The reactor is made of material with \)c=200 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, k=40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=9000 \mathrm{~kg} / \mathrm{m}^{3}$. If the heat generation of the reactor is suddenly set to $9 \mathrm{MW} / \mathrm{m}^{3}$, determine the time rate of temperature change in the reactor. Is the heat generation of the reactor suddenly increased or decreased to $9 \mathrm{MW} / \mathrm{m}^{3}$ from its steady operating condition?
Problem 160
Consider a cylindrical shell of length \(L\), inner radius \(r_{1}\), and outer radius \(r_{2}\) whose thermal conductivity varies in a specified temperature range as \(k(T)=k_{0}\left(1+\beta T^{2}\right)\) where \(k_{0}\) and \(\beta\) are two specified constants. The inner surface of the shell is maintained at a constant temperature of \(T_{1}\) while the outer surface is maintained at \(T_{2}\). Assuming steady onedimensional heat transfer, obtain a relation for the heat transfer rate through the shell.
Problem 161
A pipe is used for transporting boiling water in which the inner surface is at \(100^{\circ} \mathrm{C}\). The pipe is situated where the ambient temperature is \(20^{\circ} \mathrm{C}\) and the convection heat transfer coefficient is $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The pipe has a wall thickness of \(3 \mathrm{~mm}\) and an inner diameter of \(25 \mathrm{~mm}\), and it has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where $k_{0}=1.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.003 \mathrm{~K}^{-1}\(, and \)T\( is in \)\mathrm{K}$. Determine the outer surface temperature of the pipe.
Problem 162
A metal spherical tank is filled with chemicals undergoing an exothermic reaction. The reaction provides a uniform heat flux on the inner surface of the tank. The tank has an inner diameter of \(5 \mathrm{~m}\), and its wall thickness is \(10 \mathrm{~mm}\). The tank wall has a variable thermal conductivity given as \(k(T)=k_{0}(1+\beta T)\), where $k_{0}=9.1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=0.0018 \mathrm{~K}^{-1}\(, and \)T$ is in \(\mathrm{K}\). The area surrounding the tank has an ambient temperature of \(15^{\circ} \mathrm{C}\), the tank's outer surface experiences convection heat transfer with a coefficient of \(80 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}\). Determine the heat flux on the tank's inner surface if the inner surface temperature is \(120^{\circ} \mathrm{C}\).
Problem 163
The heat conduction equation in a medium is given in its simplest form as $$ \frac{1}{r} \frac{d}{d r}\left(r k \frac{d T}{d r}\right)+\hat{e}_{\mathrm{gen}}=0 $$ Select the wrong statement below. (a) The medium is of cylindrical shape. (b) The thermal conductivity of the medium is constant. (c) Heat transfer through the medium is steady. (d) There is heat generation within the medium. (e) Heat conduction through the medium is one-dimensional.
Problem 164
Consider a medium in which the heat conduction equation is given in its simplest forms as $$ \frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial T}{\partial r}\right)=\frac{1}{\alpha} \frac{\partial T}{\partial t} $$ (a) Is heat transfer steady or transient? (b) Is heat transfer one-, two-, or three-dimensional? (c) Is there heat generation in the medium? (d) Is the thermal conductivity of the medium constant or variable? (e) Is the medium a plane wall, a cylinder, or a sphere? (f) Is this differential equation for heat conduction linear or nonlinear?
Problem 166
A solar heat flux \(\dot{q}_{s}\) is incident on a sidewalk whose thermal conductivity is \(k\), solar absorptivity is \(\alpha_{s^{+}}\)and convective heat transfer coefficient is \(h\). Taking the positive \(x\)-direction to be toward the sky and disregarding radiation exchange with the surrounding surfaces, the correct boundary condition for this sidewalk surface is (a) \(-k \frac{d T}{d x}=\alpha_{s} \dot{q}_{s}\) (b) \(-k \frac{d T}{d x}=h\left(T-T_{\infty}\right)\) (c) \(-k \frac{d T}{d x}=h\left(T-T_{o c}\right)-\alpha_{s} \dot{q}_{s}\) (d) \(h\left(T-T_{\infty}\right)=\alpha_{s} \dot{q}_{s}\) (e) None of them
Problem 167
A plane wall of thickness \(L\) is subjected to convection at both surfaces with ambient temperature \(T_{\infty \text { el }}\) and heat transfer coefficient \(h_{1}\) at the inner surface and corresponding \(T_{\infty 2}\) and \(h_{2}\) values at the outer surface. Taking the positive direction of \(x\) to be from the inner surface to the outer surface, the correct expression for the convection boundary condition is (a) \(k \frac{d T(0)}{d x}=h_{1}\left[T(0)-T_{\infty 1}\right]\) (b) $k \frac{d T(L)}{d x}=h_{2}\left[T(L)-T_{\infty 22}\right]$ (c) $-k \frac{d T(0)}{d x}=h_{1}\left(T_{\infty 1}-T_{\infty 22}\right)(d)-k \frac{d T(L)}{d x}=h_{2}\left(T_{\infty \infty 1}-T_{\infty 22}\right)$ (e) None of them
