Chapter 20, Radiative heat transfer Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

World energy consumption, averaged over a year, is currently around 18 'TW $\left(18 \times 10^{12} \mathrm{~W}\right)$. The average solar irradiance at Earth's surface is around $20 \mathrm{MJ} / \mathrm{m}^2$ per day. What fraction of the surface area of the Farth would be required to provide all energy needs from solar power, if the conversion efficiency is $10 \%$ ?

Matthew Muscat

Numerade Educator

01:13

Problem 2

In global economic calculations, the effects of policy decisions have to be estimated for timescales of many decades. Typically the amount of any given benefit or cost is multiplied by a factor $\alpha(t)$ which gets smaller with time $t$. What value or functional form would you consider appropriate for such a factor? Try to discover what factors are being used to frame public policy in your country, and in world economic meetings.

Carson Merrill

Numerade Educator

02:29

Problem 3

A thermocouple is used to measure the temperature of air inside a long pipe with black walls. The pipe walls are at $100^{\circ} \mathrm{C}$, and the air is at $20^{\circ} \mathrm{C}$. Treating the thermocouple as black, what temperature will it indicate if it exchanges heat primarily by radiation and by convection in the air? (The convection may be modelled by Newton's law of cooling, with a coefficient $h=14 \mathrm{~W} / \mathrm{m}^2 \mathrm{~K}$.) What will be the temperature reading if the thermocouple is surrounded by a tightly fitting shiny envelope of emissivity 0.05 ?

Sri Datta Vikas Buchemmavari

Numerade Educator

Problem 4

Show that the solid angle subtended by a sphere of radius $r$ at a distance $R$ from its centre is $4 \pi r^2 / R^2$. Hence, by reasoning directly from the geometry, show that for a spherical body at the centre of a spherical cavity, the proportion of the radiation emitted from the cavity wall that hits the sphere is $r^2 / R^2$, if the emission is isotropic. Derive the same result using the shape factor reciprocity relation.

Check back soon!

02:16

Problem 5

A hot sphere of radius $R_0$ is mounted in the centre of an cvacuated spherical cavity of radius $R_1$, which is maintained at temperature $T_1$. Show that, if a thin spherical shield hanging freely is interposed half way between the sphere and the wall of the cavity, the rate of loss of heat by radiation from the sphere is reduced by a factor $R_2^2 /\left(R_5^2+R_5^2\right)$, where $R_r$ is the radius of the shield. All the surfaces can be assumed to be black.

Mahendra Rathore

Numerade Educator

02:12

Problem 6

A small spherical meteor is at a distance from the Sun of 50 Sun radï. Estimate its temperature, given that the surface temperature of the Sun is $6000 \mathrm{~K}$.

Ajay Singhal

Numerade Educator

01:38

Problem 7

A large evacuated chamber is maintained at a constant temperature $T_0$. Two black bodies $A$ and $B$ are situated in the enclosure, not touching, with body $A$ maintained at temperature $T_1\left(T_1>T_0\right)$. When $B$ is at $T_0$ the rate of loss of heat from $A$ is $W_1$ and when $B$ is at $T_1$ the rate of heat loss from $A$ is $W_2$. Show that $A$ experiences no net loss of heat by radiation when the temperature of $B$ is $T_2$, given by
$$
T_2^4=\frac{W_1 T_1^4-W_2 T_0^4}{W_1-W_2}
$$

Ajay Singhal

Numerade Educator

03:46

Problem 8

A long cylinder of radius $R_1$ and emissivity $\epsilon$ is positioned inside an evacuated enclosure whose walls are held at temperature $T_0$ and are black. The cylinder attains a steady-state temperature $T_1$ when power $W_1$ per unit length is dissipated in it. Derive an expression for $T_1^*$ in terms of Stefan's constant and the parameters specificd. An intermediate cylinder of external radius $R_2=1.2 R_1$, whose surfaces are black, is next suspended in the enclosure, surrounding the inner cylinder and centred on the same axis. It is now found that the power dissipation in the inner cylinder needed to maintain it at the same steadystate temperature $T_1$ is reduced to $75 \%$ of its former value. Deduce the value of $\epsilon$.

Khoobchandra Agrawal

Numerade Educator