Chapter 10, Heat flow and thermal relaxation Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

A church has limestone walls of thickness $30 \mathrm{~cm}$. The outside air temperature varies daily between $10^{\circ} \mathrm{C}$ and $25^{\circ} \mathrm{C}$, in a sinusoidal fashion with a maximum value at noon. Using a single flat wall model, estimate the temperature fluctuation inside the church, and the time of day when the interior is at its warmest. [Limestone has thermal conductivity $1.5 \mathrm{Wm}^{-1} \mathrm{~K}^{-1}$, specific heat capacity $910 \mathrm{JK}^{-1} \mathrm{~kg}^{-1}$, and density $\left.2180 \mathrm{kgm}^{-3}.\right]$

Salamat Ali

Numerade Educator

02:03

Problem 2

A cylindrical wire of radius $a$, resistivity $\rho$, and thermal conductivity $\kappa$ carries a current $I$. In steady state, the temperature at the surface of the wire is $T_0$. Show that if the current is carried uniformly, then the temperature inside the wire at radius $r$ is
$$T(r)=T_0+\frac{\rho I^2}{4 \pi^2 a^4 \kappa}\left(a^2-r^2\right).$$

Ajay Singhal

Numerade Educator

07:17

Problem 3

Newton's law of cooling states that when a gas transports heat away from a solid or liquid surface by convection and conduction together, the net rate of transport is proportional to the area of contact multiplied by the temperature difference $\Delta T$ between the surface and the bulk of the gas. This amounts to the following statement about the heat flux:
$$\mathbf{J}=\mathbf{k} \Delta T$$
where $\mathbf{k}$ is a coefficient whose magnitude depends on the conditions and whose direction is normal to the surface. This is an empirical rule, and a good approximation when $\Delta T$ is not too large. If the wire considered in question (10.2) is air-cooled, find an expression for $T_0$ in terms of $k$ and properties of the wire.

Eric Goldman

Numerade Educator

04:13

Problem 4

A cup of black coffee is initially at temperature $80^{\circ} \mathrm{C}$ in a room of ambient temperature $25^{\circ} \mathrm{C}$. The temperature difference between the cup and the room decays exponentially with time constant 10 minutes. A small jug of milk in the same room has initial temperature $5^{\circ} \mathrm{C}$ and its temperature difference with the room changes exponentially with a time constant of 10 seconds. The heat capacities of milk and coffee are $100 \mathrm{~J} / \mathrm{K}$ and $1000 \mathrm{~J} / \mathrm{K}$ respectively. At what moment should the milk be added to the coffee in order to obtain milky coffee at the highest temperature?

Linda Hand

Numerade Educator