Thermal Energy Transfers: notes and practice questions

An electric ski lift is powered by a motor at the base station. The motor is connected to a 750 V DC power supply by a cable with a total resistance of 0.15 Ω. When operating at full capacity, the motor draws a constant current of 400 A.

(a) Determine the potential difference across the terminals of the motor.

(b) The motor has an efficiency of 92%. Calculate the useful mechanical power output of the motor.

The ski lift carries skiers up a slope of length 1800 m that rises by a vertical height of 500 m. There are 50 chairs on the ascending side. Each empty chair has a mass of 25 kg and carries, on average, 1.5 skiers of average mass 75 kg. A constant resistive force of 12 kN opposes the motion.

(c) Determine the total upward force the motor must provide via the cable to maintain a constant speed.

(d) Estimate the maximum speed, $v$, of the ski lift.

(e) The lift operates continuously. Estimate the maximum number of skiers that can be transported to the top station in one hour.

(f) In an emergency stop, a brake is applied to a large solid steel disc, bringing the lift to a halt from its maximum speed. The total mass of the moving system (chairs, skiers, and cable) is 18 000 kg. Assume all the kinetic energy of the system is converted into thermal energy in the brake disc.

Calculate the temperature rise of the disc.

Data for this question:
Brake disc radius = 0.75 m
Brake disc thickness = 0.10 m
Density of steel = 7850 kg m⁻³
Specific heat capacity of steel = 450 J kg⁻¹ K⁻¹

(g) The speed of a chair is monitored using a radar device at the base station that emits microwaves of frequency 30 GHz. It measures the waves reflected from a chair as it moves away. The frequency shift, $Δf$, for a source moving directly away from a stationary observer can be approximated by the formula $Δf/f ≈ v/c$. In this radar measurement, this formula gives a good approximation for the shift detected.

Calculate the expected frequency shift.

An exoplanet named Xylos orbits a star named Aethel. Scientists are studying its atmospheric conditions and the properties of its host star.

(a) State what is meant by the stellar constant for Xylos.

(b) The following data are given for exoplanet Xylos:

Average albedo of Xylos $= 0.40$

Average orbital distance from Aethel $= 2.0 \times 10^{11} \text{ m}$

Average global surface temperature of Xylos $= 250 \text{ K}$

Luminosity of star Aethel $= 2.0 \times 10^{26} \text{ W}$

(i) Outline the physical mechanism by which some of the infrared radiation emitted by the surface of Xylos is absorbed by its atmospheric gases and re-radiated back towards the surface.

(ii) Show that the average global intensity of radiation absorbed by the surface of Xylos is about $60 \text{ W m}^{-2}$.

(iii) Determine the average intensity re-radiated by Xylos's atmosphere towards its surface. Assume that the emissivity of the surface is $0.95$.

(c) Calculate the total power (luminosity) radiated by the star Aethel, based on the stellar constant at Xylos's orbit.

(d) A possible fusion reaction occurring in stars like Aethel is the deuterium-tritium (D-T) fusion reaction:

$^2_1\text{H} + ^3_1\text{H} \rightarrow ^4_2\text{He} + ^1_0\text{n}$

Relevant atomic masses are:

Mass of deuterium ($^2_1\text{H}$) $= 2.013553 \text{ u}$

Mass of tritium ($^3_1\text{H}$) $= 3.015501 \text{ u}$

Mass of helium-4 ($^4_2\text{He}$) $= 4.001506 \text{ u}$

Mass of neutron ($^1_0\text{n}$) $= 1.008665 \text{ u}$

(i) Calculate, in $\text{MeV}$, the energy released in the reaction.

(ii) Outline the role of fusion reactions in maintaining a stable radius for a star like Aethel.

(iii) Outline how the presence of hydrogen in Aethel can be confirmed empirically.

(e) Aethel has a surface temperature of $5200 \text{ K}$ and a luminosity $0.5$ times that of the Sun.

(i) State the star type of Aethel.

(ii) Discuss how nuclear fusion processes in a red dwarf star differ from those in the Sun.

In a blacksmith's workshop, a 2.00 kg iron horseshoe is heated in a furnace. It is then plunged into a large, insulated barrel containing 5.00 kg of water at 15.0 °C. The system reaches a final equilibrium temperature of 100 °C, by which time 0.120 kg of the water has turned to steam.

(a) Calculate the initial temperature of the iron horseshoe.

Data:

Specific heat capacity of water = 4200 J kg⁻¹ °C⁻¹

Specific latent heat of vaporization of water = 2.26 × 10⁶ J kg⁻¹

Specific heat capacity of iron = 450 J kg⁻¹ °C⁻¹

(b) The barrel is made of wood, but in reality, it is not a perfect thermal insulator and will absorb some energy. Suggest and explain how this would affect the calculated value for the initial temperature of the horseshoe, assuming the calculation does not account for the barrel.

A rocky exoplanet, Planet X, orbits a star. The solar constant for Planet X is $1360 \text{ W m}^{-2}$. The average albedo of Planet X is $0.15$.

State what is meant by an albedo of $0.15$.

Show that the average power absorbed per square metre of Planet X's surface is about $290 \text{ W m}^{-2}$.

Assuming Planet X has no atmosphere and acts as a perfect black body, calculate its equilibrium surface temperature.

Later observations reveal that Planet X has an atmosphere containing carbon dioxide.

Outline how the presence of carbon dioxide in the atmosphere leads to an increase in the surface temperature of the planet.

The surface of Planet X is covered by a large proportion of dark oceans. As the planet warms, some of the oceans evaporate, increasing the cloud cover in the atmosphere.

Discuss how this increase in cloud cover could affect the surface temperature of Planet X.

(a) A weather balloon is filled with hydrogen gas at a pressure of 1.0 x 10$^5$ Pa and a temperature of 290 K. The volume of the balloon is 5.0 m$^3$. Assume that this sample of hydrogen gas behaves as an ideal gas.

The molar mass of hydrogen (H$_2$) is 2.016 g mol$^{-1}$. Show that the mass of a hydrogen molecule is approximately 3.35 x 10$^{-27}$ kg.

(b) Estimate the average speed of the hydrogen molecules in the balloon.

(c) Calculate the number of hydrogen molecules in the balloon.

(d.i) A hydrogen molecule has an approximate volume of 2.0 x 10$^{-30}$ m$^3$. Calculate the ratio $\frac{\text{total volume of hydrogen molecules}}{\text{volume of hydrogen gas}}$.

(d.ii) Explain, using your answer to (d)(i) and with reference to the kinetic model, why this sample of hydrogen can be assumed to be an ideal gas.

An exoplanet orbits a star that has a surface temperature of $5800\text{ K}$. The exoplanet has an average surface temperature of $290\text{ K}$ and an atmosphere containing a high concentration of methane ($\text{CH}_4$) gas.

Outline the differences between the radiation incident on the exoplanet's atmosphere and the radiation emitted by its surface.

Discuss the mechanism of the greenhouse effect on this exoplanet. In your answer, you must refer to:

• the interaction between methane molecules and radiation

• the energy balance of the exoplanet's surface

A closed, thermally insulated laboratory contains an electric oven and a refrigerator. Both appliances are connected to the laboratory's electrical supply.

The oven is switched on with its door open. Outline the energy transfers that lead to the laboratory warming up.

The oven is switched off. The refrigerator is switched on with its door open.

Discuss, with reference to the first law of thermodynamics, the effect of this on the temperature of the laboratory.

Venus orbits the Sun at a distance where the intensity of solar radiation (the solar constant for Venus) is $S = 2600 \text{ W m}^{-2}$. The average albedo of Venus is $a = 0.75$ due to its thick cloud cover.

(a) Define the solar constant.

(b) Venus has a radius $R$. Show that for a planet to be in thermal equilibrium, its surface temperature $T$ is related to the solar constant $S$ and albedo $a$ by the expression $4\sigma T^4 = S(1-a)$, assuming the planet behaves as a black body.

(c) Using the expression from (b), calculate the surface temperature Venus would have if it were a black body.

(d) The actual mean surface temperature of Venus is approximately $735 \text{ K}$. Outline the primary mechanism responsible for this much higher temperature.

(e) A simple model treats Venus as a grey body with an effective emissivity $\epsilon$. Calculate the emissivity of Venus.

The Arctic region is experiencing significant changes due to global warming, including the melting of large areas of sea ice. This has a considerable impact on the Earth's energy balance.

(a) Explain what is meant by albedo.

(b) The table shows typical albedo values for two surfaces found in the Arctic.

Explain, by reference to the properties of the surfaces, the difference in these albedo values.

(c) Climate models predict a continued decrease in the extent of Arctic sea ice. Outline how this decrease is expected to affect the net energy absorbed by the Arctic region.

A satellite measures the solar radiation incident on a $1.0 \text{ km}^2$ area of the Arctic. The average intensity of the incident radiation is $340 \text{ W m}^{-2}$.

(d) (i) Calculate the power reflected from this area if it is completely covered by sea ice with an albedo of $0.60$.

(ii) The sea ice in this area melts completely, exposing open ocean water with an albedo of $0.08$. Determine the change in the power absorbed by this area.