Chapter 4, Heat Transfer and Phase Transitions Video Solutions, Competitive Physics: Thermodynamics, Electromagnetism and Relativity

Problem 1

Two concentric, hollow spheres have radii $r_1$ and $r_2$ respectively with $r_1<r_2$. Denote their instantaneous temperatures as $T_1$ and $T_2$. If the space between them is filled with a material with thermal conductivity $k$ and negligible heat capacity, determine the instantaneous heat flux between the two spheres. Using the previous result, find $T_1(t)$ and $T_2(t)$ if the heat capacities of the spheres are $C_1$ and $C_2$ and if their initial temperatures are $T_{10}$ and $T_{20}$.

Surendra Kumar

Numerade Educator

04:45

Problem 2

Suppose that the cylindrical shell in Section 4.2 is now covered with felt that has a uniform thermal conductivity $k_2$ and an outer radius $r_2$. Let the thermal conductivity of the cylindrical shell, with inner and outer radii $r_0$ and $r_1$, be $k_1$. The inner surface of the cylindrical shell is maintained at $T_0$ while the outer surface of the felt is maintained at $T_1$. Determine the heat flux across cylindrical layers.

Sunita Kumari

Numerade Educator

Problem 3

Consider a long cylindrical wire with a radius $R$ and thermal conductivity $k$. A current runs through it such that each unit volume of the wire produces $p$ amount of heat per unit time. If the temperature of the cylindrical surface of the wire is maintained at $T_0$, determine the temperature distribution in the wire $T(r)$ as a function of its radial coordinate $r$.

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01:48

Problem 4

$n_0$ moles of an ideal gas fill a container of constant cross sectional area $A$ and length $l$. It is known that the thermal conductivity of a section of ideal gas is proportional to the square root of its temperature $k=c \sqrt{T}$. If the ends of the container are maintained at temperatures $T_1$ and $T_2$ respectively, determine the pressure of the gas at steady state. Assume one-dimensional heat flow in the direction perpendicular to the cross section of the container.

Surendra Kumar

Numerade Educator

05:47

Problem 5

A truncated cone has two circular surfaces of radii $r_0$ and $r_1, r_0<r_1$, which are maintained at temperatures $T_0$ and $T_1$ respectively. The perpendicular distance between these two surfaces is $h$. Find the heat flux in the direction of the axis. Assume that $r_1-r_0 \ll h$ such that the half-angle of the cone is small. Where is this assumption necessary in your working?

Dading Chen

Numerade Educator

05:31

Problem 6

The $N$ vertices of a homogeneous regular $N$-gon are maintained at temperatures $T_1, T_2, \ldots, T_N$ respectively by an external agency. Determine the steady state temperature of the centroid.

Lucía Guerrero

Numerade Educator

02:43

Problem 7

Three slabs (filled with black) have thermal conductivities $k_1, k_2$ and $k_3$, cross sectional areas $A_1=A_2=A$ and $A_3$ and lengths $l_1, l_2$ and $l_3$. They are connected by tubes filled with gases (shaded gray) of heat transfer coefficient $h$ as shown in the figure below. The cross sectional areas of the gas tubes are not given and are irrelevant. If the left end of the left slab and the right end of the right slab are maintained at temperatures $T_{1 l}$ and $T_{2 r}$, determine the condition for the middle slab to have a uniform temperature at steady state.
(GRAPH CAN'T COPY)

Mahendra K

Numerade Educator

04:34

Problem 8

A small, black plate of area $A$ is stationary at a large distance away from the Sun which is a spherical black body with radius $r_s$, mass $M$ and constant temperature $T_s$. Determine the mass of the plate, $m$. Neglect all other gravitational effects and assume that the surface of the plate is perpendicular to the line joining the center of the Sun to it.

Keshav Singh

Numerade Educator

12:15

Problem 9

A space station takes the form of a black sphere in outer space with surroundings at zero absolute temperature. Due to the operation of the space station, its internal appliances produce a certain amount of power that is conducted isotropically within the sphere. If the equilibrium temperature of the space station under such circumstances is $T$, determine the new equilibrium temperature $T^{\prime}$ of the space station after a black spherical shell, of a slightly larger radius than the space station, is used to envelope the space station. What if $N$ thin black shields are used? What if a single thin shield, made of an opaque gray material of emissivity $\varepsilon$, is used?

Ren Jie Tuieng

Numerade Educator

Problem 10

Two plates of emissivities $\varepsilon_1$ and $\varepsilon_2$ are oriented parallel to each other. The first plate is opaque while the second plate has a reflectivity $r$. If the two plates are maintained at temperatures $T_1$ and $T_2$ respectively, determine the heat flux transmitted across the second plate.

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Problem 11

Two large, gray and opaque plates with emissivities $\varepsilon_1$ and $\varepsilon_3$ are oriented parallel to each other and are maintained at temperatures $T_1$ and $T_3$ respectively. Now, another plate of equal emissivity, absorptivity and transmittivity is placed between the two plates. Determine the equilibrium temperature of this plate, $T_2$.

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02:16

Problem 12

In this problem, we will model the effect of an atmosphere on Earth. Suppose that the Sun is a black body with temperature $T_1$ and radius $r_1$. The Earth is a sphere that is located at a distance $R$ from the Sun and has a radius $r_3$. The emissivity of the Earth is $\varepsilon_3$.
(1) If there is no atmosphere on Earth, determine the temperature of the Earth at equilibrium, $T_3$.
(2) Now, we consider the effects of an atmosphere. Model the atmosphere as a spherical shell of gas, with an emissivity $\varepsilon_2$ and outer radius $r_2>r_3$, surrounding the Earth. At thermal equilibrium, its absorptivity for both ultraviolet and infrared light is $\varepsilon_2$. The atmosphere transmits a fraction $t$ of ultraviolet light but is completely opaque to infrared. Assuming that the Sun emits ultraviolet light while the Earth emits and re-emits infrared, determine the temperature of the atmosphere $T_2$ and the Earth, $T_3$, at thermodynamic equilibrium. Assume that the atmosphere is a perfect thermal conductor such that all incident radiation is instantaneously evenly distributed across it.

Keshav Singh

Numerade Educator

01:44

Problem 13

A flat, circular ring has an inner radius and outer radius. If the ring is now heated such that it undergoes isotropic expansion, does the area of the hole in the middle increase or decrease?

Dading Chen

Numerade Educator

01:27

Problem 14

Spherical ball A is hung down from a massless, inextensible string that is connected to a wall. Spherical ball B lies motionless on a horizontal floor. The same quantity of heat $Q$ is supplied to both balls. Assuming no heat losses, are the final temperatures of the balls the same? If not, estimate the difference in the final temperatures in terms of parameters of your choice. (International Physics Olympiad)

Ajay Singhal

Numerade Educator

06:50

Problem 15

Consider a container with a piston that contains a certain amount of gaseous and liquid states of the same substance. The piston is first fixed and the system is at equilibrium at temperature $T$. The latent heat of vaporization per mole of gas of this configuration is determined to be $L$. Now, consider the case where the massive piston is not fixed and is instead, balanced by the difference between the interior pressure and atmospheric pressure. The system is initially at equilibrium at temperature $T$. Determine the latent heat of vaporization per mole of gas of this new configuration in terms of $L$ and $T$. Assume that the gaseous form of the substance is ideal and attains thermodynamic equilibrium at every instance.

Keshav Singh

Numerade Educator

02:20

Problem 16

A motionless cylindrical vessel of cross sectional area $A$ in outer space initially contains an ideal gas of total mass $M$ and initial pressure $p \ll p_s$ where $p_s$ is the saturation pressure at its current temperature (which is above the triple point temperature but below the critical temperature). The vessel is then given a constant acceleration $a$ along its cylindrical axis while its temperature is maintained. Determine the mass of liquid $m$ formed by condensation due to this motion after the system has equilibrated.

Mahendra K

Numerade Educator

01:21

Problem 17

$1 \mathrm{~kg}$ of ice at $0^{\circ} \mathrm{C}$ floats in $5 \mathrm{~kg}$ of water at $50^{\circ} \mathrm{C}$. The whole system is thermally isolated. Determine the change in entropy of the whole system when thermal equilibrium has been reached. The specific heat capacity of water is $4.2 \mathrm{kJkg}^{-1} \mathrm{~K}^{-1}$ and the latent heat of fusion of ice is $333 \mathrm{kJkg}^{-1}$.

Manne Andergronde

Numerade Educator

15:05

Problem 18

Model the atmosphere as a spherical shell of uniform gas with molar mass $\mu$ and uniform temperature $T_a$ that envelopes the spherical, uniform Earth. If the atmospheric pressure and boiling point of water at the surface of the Earth is $p_0$ and $T_0$ respectively, determine the boiling point of water at a small height $z$ from the surface of the Earth. Assume that the latent heat of vaporization is a constant $L$ in this regime and that the specific volume of water vapor is much larger than that of water.

Mark Scythian

Numerade Educator

06:49

Problem 19

A closed container of constant volume currently contains certain amount of gaseous and liquid states of the same substance at equilibrium. If the current temperature of the system is $T$ and the specific latent heat of vaporization in the current state is $L$, determine the fractional change in the moles of gaseous molecules due to evaporation if the equilibrium temperature of the system is slightly increased by $\Delta T \ll T$. The specific volume of the gas can be assumed to be much greater than that of the liquid. The gas molecules have molar mass $\mu$.

Oswaldo Jiménez

Numerade Educator