Heat Transfer Practice Problems

A food container has a total area of 0.242 m2. A warm liquid at 42°C is placed in the container. The temperature of the outer wall after thermal equilibrium has been established is found to be 35°C. The container has a thickness of 2.4 mm, and it loses heat to the surroundings at a rate of 191 W. Using this data, determine the thermal conductivity of the material making the container.

An oven has its surface fitted with insulating 0.12 m² rock wool that is 2.0 cm thick. At one instance of the oven's operation, the inner surface of the rock wool is at a temperature of 150°C and the outer surface has a temperature equal to room temperature (25°C). The thermal conductivity of rock wool is 0.040 W/m•K. Determine the heat current flowing through the rock wool insulation. 

A 1.20 cm chromium cube (emissivity = 0.28) is used for heating. It is enclosed in a container that is nearly a vacuum, whose walls have a temperature of 283.0 K. Determine the rate of power supply to the cube that will keep it at a temperature of 811 K. Assume conduction by supports is negligible.

During a room heating exercise, a spherical 2.0 L pot is filled with boiling water at 100°C. The emissivity of the pot is 0.55 and the room temperature is 14.0°C. Determine the rate at which the water will lose heat through radiation.

A craftsman fabricates a 50.0 cm long aluminum bar that has a cross-sectional area of 5.07cm². If the bar is used to transfer heat between reservoirs at T_H = 100.0°C and T_C = 25.0°C. Determine the steady-state temperature gradient for the bar. 

A 0.400m long aluminum rod is joined end to end with a 0.750m copper rod for a heat transfer experiment. All sections on the compound rod have a cross-sectional area of 0.00860 m². The copper end is inserted in boiling water while the aluminum end is inserted in an ice-water mixture. The experiment is conducted at atmospheric pressure. The rod is wrapped to prevent heat loss to the surroundings. Determine the mass of ice that will melt in 10 min using the heat supplied by the rod. 

A shipping container is to be thermally insulated. The architect uses 2.5 cm granulated cork to insulate the container from outside. On the inside, the container is coated with a 2.0 cm thick layer of felt. The values of k for granulated cork and felt are 0.048 W/m•K and 0.040 W/m•K, respectively. The interior temperature is 18.0°C while the exterior temperature is -8.0°C. Determine the temperature at the interface of the materials. Assume the materials are in contact.

Imagine that a spherical iron ball which has a diameter of 5.0 cm is heated and begins to radiate energy at a rate of 850 W. Assuming that the emissivity (e) of iron is 0.7, at what wavelength, in units of μm, does its emission spectrum peak?

Imagine that a student is investigating a spherical blackbody object that has a radius of 1.5 cm and an emission peak wavelength of 4.0 μm. Assuming that the emissivity is 1, calculate what the power radiated will be.

A metal plate 25 cm long having a rectangular cross-section of 3.0 cm x 2.0 cm has its ends maintained at 10°C and 150°C. The rate of heat conduction through the rod is 48.4 × 10⁴ J/h. Determine the material of the metal plate.

You planned to roast meat on fire. The iron rod that will be used to hold the meat is to be placed horizontally above the fire but unknowingly one end of the rod slipped off the ring that is used to keep the rod horizontal and touched the fire, with the other end still touching the ring. The rod is now in a slanting position. The radius of the rod is 40 mm and the length is 55 cm. The temperature of the fire is 100°C. Determine the time required to transfer 160 J of energy from the fire to the ring. Assume the surrounding temperature to be 30°C.

Liquid helium is used in superconductivity research to cool superconducting materials below their critical temperature, allowing them to conduct electricity with zero resistance. The liquid helium is stored in a vacuum-insulated container designed to store liquid helium at ultra-low temperatures, typically below -269°C (4 K). It is made of a double-walled stainless steel vessel with an evacuated space in between the walls to minimize heat transfer. The helium container is equipped with a pressure relief valve and a liquid helium withdrawal system. Consider a cylindrical container with a cross-sectional area of 0.015 m² and a length of 0.25 m. The effective emissivity between the inner vessel containing liquid helium and the radiation shield is 0.2. Liquid helium has a density of 125 kg/m³ and a heat of vaporization of 2.1 × 10⁴ J/kg. Determine the mass of liquid helium inside the completely filled cylinder.

A main-sequence star is a star that fuses hydrogen into helium in its core. The energy produced radiates outward and sustains the star's luminosity. The power radiated by a main sequence star is determined mainly by its mass and radius, with more massive and larger stars being more luminous. The power radiated by a main-sequence star of mass M is given by P = {1.5(M/M_s)^3.5}P_s, where M_s is the mass of our sun, and P_s is the power radiated by our sun. Consider a hypothetical main sequence star "V" with a mass of 2.1M_s and a radius of 2.4R_s. Determine the surface temperature of "V". The sun's surface temperature is approximately 5780 K. Assume that both V and our sun have the same emissivity.

A thermally insulated storage container has an outer layer made of 4.0 cm corkboard and an inside layer made of 2.5 cm thick fiberglass. Thermal conductivity, k for corkboard, and fiberglass are 0.043 W/m•K and 0.036 W/m•K, respectively. The inside temperature is 15.0°C while the external temperature is -8.0°C. Determine the rate at which heat flows through the container's wall per square meter.

If an office building has a volume of 1300 m³, how much air is inside at 1 atm atmospheric pressure and 17°C temperature? [Hint: Universal gas constant (R)=8.315 J/mol⋅K and

Molar mass of air (M)=29 g/mol(29×10^-3 kg/mol).

Considering that 4.0 kg of ethanol is initially at 15.0°C, how much will its temperature increase with 8300 J of heat? [Hint: Specific heat capacity of ethanol is 2440 J/kg·°C].

How long will it take to raise the temperature of 300 mL of liquid from 20°C to 70°C using a portable coffee warmer rated at 400 W? [Hint: Specific heat capacity of water is 4182 J/kg·°C].

Calculate the rate of heat conduction along a 65-cm-long iron rod with a diameter of 3.0 cm, where one end is kept at 470°C, and the other end is embedded in water at 23°C. [Hint: k=80 j/m.s.°C].

If the temperature within a planet's crust rises by 1.0°C for every 32 m, calculate the heat transferred from the interior to the surface of the planet in 1 hour. Assume the radius of the planet is 6.7 × 10⁶ m and the thermal conductivity of the planet's crust is 0.82 J/s.m·°C.

Suppose the sun shines directly on a flower petal that has an area of 45 cm² and a mass of 3.0×10^-4 kg. Calculate what the rate of energy absorption by the petal from the sun will be. Assume that the petal has an emissivity of 0.75 and a specific heat capacity of 0.86 kcal/kg·K. [Hint: The sun's energy reaching the Earth's surface per unit time per unit area = 1000 W/m²]

Suppose 152 J of heat is conducted through a brick wall with a surface area of 1.6 m² every second. Given a temperature difference of 21.0 ^∘C across the wall, estimate the wall's average thickness. The thermal conductivity of the brick is 1.31 J/s^.m^.∘C.

Imagine you are a scientist studying the ability of the body of a chimpanzee to regulate its temperature. Your task is to calculate the rate of conduction of heat from the inside of the body of chimpanzees to their skin. The biologists in your team have provided you with some basic values like the thickness of the tissues of chimpanzees is 3.0 cm, the temperature of their skin is at 33 degrees, the temperature of their inside is at 36 degrees, and the base surface area is $1.6〖 m〗^2$. You have asked to compare the estimated heat conductance with a measured value of about 200 W. This measured value represents the amount of heat that needs to be dissipated by a chimpanzee performing tricks.

Consider the value for thermal conductivity of chimpanzee tissue usually denoted by $k=(0.5 W)/mK.$

The surface temperature of an LED lamp rated at 120W reaches 90°C when the ambient room temperature is 22°C. Considering only radiation and assuming that 70% of the total electrical input gets converted into thermal output, estimate the expected surface temperature of a 240W LED lamp.

A deep-sea diver jumps in the sea wearing a standard diving wet suit that has a thickness of 0.7 cm and a total surface area of 1.8 m². The surface temperature of the standard diving suit is -2°C, and when it is on the skin, its temperature is 37°C. Determine the heat rate flow by conduction through the wet suit., assuming (a) it is dry and the thermal conductivity K is that of a standard diving suit. Also, if the heat rate flow is calculated by conduction through the wet suit. When (b) the suit is moist, and the value of k is equal to that of water. Also, consider if the suit has been compressed to a thickness of 0.35 cm.

A solar panel of area 60cm² and mass 5.0 kg directly faces the Sun on a clear day. The solar panel has an emissivity of 0.90 and a specific heat of 1.00 kcal/kg × K. Will the temperature rise continue for hours? Why or why not?"

An astronaut is wearing a special space suit that has a water layer between the gap of his skin and the suit. The thickness of the water layer is 0.8 mm. The initial temperature of the water is 17°C and the surface area of the suit that covers the astronaut's skin with a water layer is 1.2 m². Gradually the water layer is warmed to the body temperature of 37°C of the astronaut. Determine how much body heat the water layer absorbs for this temperature rise (in the units of Cakes = 350 kcal).

(i) Calculate the power radiated by an aluminum ball with an emissivity of 0.1 and a radius of 15 cm at a temperature of 50°C. (ii) Determine the net rate of energy flow out of the ball, if it is placed in a box where the sides of the box are kept at 10°C.

Consider two solid cylinders, one made of silver and the other of nickel, each having identical lengths and cross-sectional areas, which are joined end to end as illustrated. The silver end is kept at a constant temperature of 251°C, while the nickel end is maintained at 25.0°C. Determine the temperature at the interface where the two cylinders meet. The thermal conductivities of silver and nickel are k_silver = 429 W/mK and k_nickel = 90.0 W/mK respectively.

As you go toward the center, the Martian crust warms gradually, gaining about 0.50 °C every 25 meters. It has a thermal conductivity of 0.039 J/s·m·°C. The Sun, at the heart of our solar system, provides Mars with 590 watts per square meter (perpendicular to the solar rays). Compare the heat conducted from the core of Mars to its surface over 2.0 hours with the heat received from the Sun by the Martian surface in the same amount of time. [Hint: The radius of Mars = 3389.5 km]

How many days will it take for 2.5-W bulb to use up 2$×$10⁶ Joules of energy, assuming continuous operation?

(i) Consider a star with a radius of 5.0 × 10⁸ m. Calculate the total power radiated by the star, assuming it acts as a perfect blackbody at a temperature of T= 6000 K. (ii) Determine the power per unit area received by a planet orbiting this star at a distance of 2.0 × 10¹¹ m as shown in the figure.

A copper pellet weighing 15 g traveling at 250 m/s penetrates a thin metal barrier and exits at 180 m/s. Given that 60% of the generated heat is absorbed by the pellet and its initial temperature is 25°C, determine whether any part of the pellet will melt, and if so, calculate what amount of the pellet will melt. Not that the specific heat capacity of copper is c = 387 J/kg·K.

In an operation theatre, vets are treating a 35 kg monkey whose body metabolism has shut down. The monkey's temperature has dropped to 32.1°C from 37.1°C in some time. Given that, the rate of heat loss for a monkey through conduction and radiation is about 190 W, determine what time it took for the body temperature of the monkey to drop. Assume that the specific heat capacity for the monkey's body is 4000 J/kg⋅°C.

A physicist is doing some measurements on a sunflower that is directly facing the Sun on a bright sunny day. He finds that a petal of the sunflower weighs about 1.0 × 10^-4 kg, and its specific heat capacity is 0.92 kcal/kg·K. Afterward, he calculates that the petal absorbs heat at a rate of 1.8 W. If his calculations and measurements are correct, what should be the rate of temperature rise of the sunflower petal? [Hint: Assume that 1 kcal = 4200 J.]

Astronomers are observing a planet of radius r = 3.6 × 10⁶ m that is orbiting a star. They have determined that the power of the star reaching a unit area (perpendicular to the light rays) at the planet is 1500 W. (i) Calculate the rate at which the star's energy is delivered to the entire planet. (ii) If the planet had an emissivity of 1.0, and it radiated the same amount of energy in space it received from the star at the same rate, what would be its surface temperature? [Hint: In part (i) take the area A = πr² and in part (ii) take the area A = 4πr²]

A 15 kg piece of solid gallium is kept at its melting point of 30°C inside a sealed box made of a special type of foam. The dimensions of the box are 32 cm × 42 cm × 52 cm. The thickness of the box walls is 1.0 mm. Given that the conductivity of the foam is three times that of air and the temperature of the surroundings of the box outside is 42°C, approximate the time required by the gallium inside the box to melt completely. [Hint: The thermal conductivity of air is 0.025 W/m°C, and the latent heat of fusion of gallium is 8.0 × 10⁴ J/kg.]