Conduction with Internal Heat Generation: Interactive Learning Quizzes

Question 1

A long cylindrical fuel rod of radius R generates heat uniformly within its volume at a rate of q (W/m³). Assuming steady-state conditions and negligible heat loss from the ends of the rod, what is the temperature distribution T(r) within the rod, where r is the radial distance from the center?

Accepted Answer

T(r) = T_s + (q/(4k))(R² - r²)

Answer

T(r) = T_s - (q/k)(R - r)

Answer

T(r) = T_s + (q/(2πk))(ln(R/r))

Question 2

Which of the following dimensionless numbers is particularly important in characterizing heat transfer problems with internal heat generation?

Accepted Answer

Biot Number

Answer

Prandtl Number

Answer

Reynolds Number

Question 3

In the context of heat conduction with internal heat generation, what is the physical significance of the term q/k, where q is the heat generation rate per unit volume and k is the thermal conductivity?

Accepted Answer

Represents the rate of temperature change due to internal heat generation.

Answer

Indicates the total heat generated within the material.

Answer

Represents the heat flux at the surface of the material.

Question 4

Consider a long, cylindrical wire carrying an electric current.  The wire generates heat internally due to electrical resistance. Which of the following boundary conditions is most appropriate for analyzing the temperature distribution within the wire?

Accepted Answer

Convective heat transfer from the wire surface to the surrounding air.

Answer

Constant temperature at the center of the wire.

Answer

Adiabatic condition at the wire surface.

Question 5

A slab of material with internal heat generation is perfectly insulated on one side.  The temperature of the insulated side will be:

Accepted Answer

Higher than the other side of the slab

Answer

The same as the other side of the slab

Answer

Lower than the other side of the slab

Question 6

How does increasing the thermal conductivity of a material affect its maximum temperature when subjected to internal heat generation?

Accepted Answer

Decreases the maximum temperature

Answer

Increases the maximum temperature

Answer

Has no effect on the maximum temperature

Question 7

Which of the following statements is NOT true about heat conduction with internal heat generation?

Accepted Answer

The temperature distribution is always linear within the material.

Answer

The analysis often involves solving the heat diffusion equation with a source term.

Answer

The rate of heat generation can be spatially dependent.

Question 8

A cylindrical nuclear fuel rod generates heat uniformly at a rate of q̇ (W/m³). If the rod has a radius of r and thermal conductivity k, what is the temperature difference between the center and the surface of the rod?

Accepted Answer

q̇ * r² / (4k)

Answer

q̇ * r / (4k)

Answer

q̇ * r² / (2k)

Answer

q̇ * r / (2k)

Question 9

A long, thin wire with uniform internal heat generation is exposed to a surrounding fluid at temperature T∞. The heat transfer coefficient between the wire and the fluid is h. What is the governing equation for the temperature distribution in the wire?

Accepted Answer

k(d²T/dx²) + q̇ = h(T - T∞)

Answer

k(dT/dx) + q̇ = h(T - T∞)

Answer

k(d²T/dx²) = q̇ - h(T - T∞)

Answer

k(d²T/dx²) = q̇ + h(T - T∞)

Question 10

In a system with internal heat generation, what is the effect of increasing the thermal conductivity of the material on the temperature difference between the center and the surface?

Accepted Answer

The temperature difference decreases.

Answer

The effect is unpredictable.

Answer

The temperature difference remains unchanged.

Answer

The temperature difference increases.

Question 11

A thick-walled spherical shell with uniform internal heat generation is subjected to a constant surface temperature. Which of the following methods is most suitable for determining the temperature distribution in the shell?

Accepted Answer

Solving the steady-state heat conduction equation with appropriate boundary conditions.

Answer

Applying the Biot number criteria.

Answer

Using the lumped capacitance method.

Question 12

A long, cylindrical wire carrying an electric current experiences Joule heating, which is an example of:

Accepted Answer

Internal heat generation.

Answer

Radiative heat transfer.

Answer

Convective heat transfer.

Answer

Thermal conductivity.

Question 13

A thick metal plate with internal heat generation is exposed to a cool air stream. What is the effect of increasing the air velocity on the temperature distribution in the plate?

Accepted Answer

The temperature difference between the center and the surface decreases.

Answer

The temperature difference remains unchanged.

Answer

The temperature difference increases.

Question 14

A long, thin rod with uniform internal heat generation is clamped at both ends. What is the maximum temperature in the rod?

Accepted Answer

It occurs at the midpoint of the rod.

Answer

It depends on the specific heat generation rate.

Answer

It occurs at the clamped ends.

Answer

It is uniform throughout the rod.

Question 15

A composite wall consists of two layers with different thermal conductivities and internal heat generation rates. How does the temperature profile vary across the composite wall?

Accepted Answer

The temperature profile is non-linear and depends on the thermal conductivity and heat generation rate of each layer.

Answer

The temperature profile is linear and uniform across the composite wall.

Question 16

A nuclear reactor generates heat at a constant rate of 100 W/m³. The reactor has a cylindrical shape with a diameter of 1 m and a height of 2 m. What is the temperature at the center of the reactor if the surface temperature is maintained at 100 °C?

Accepted Answer

200 °C

Answer

250 °C

Answer

150 °C

Question 17

An electrical heating element is embedded in a solid material. The element generates heat at a rate of 100 W/m². The material has a thermal conductivity of 20 W/(m·K) and a thickness of 1 cm. Determine the temperature difference between the two faces of the material if the heat flux is perpendicular to the faces.

Accepted Answer

5 °C

Answer

15 °C

Answer

20 °C

Answer

10 °C

Question 18

A spherical reactor vessel has an internal heat generation rate of 100 W/m³. The vessel has a diameter of 2 m and a surface temperature of 120 °C. Determine the temperature at the center of the vessel.

Accepted Answer

240 °C

Answer

360 °C

Answer

300 °C

Answer

180 °C

Question 19

A hollow cylindrical rod has an inner radius of 1 cm and an outer radius of 2 cm. The rod generates heat at a rate of 100 W/m³ throughout its volume. If the inner surface of the rod is maintained at 100 °C and the outer surface is insulated, determine the temperature at the outer surface.

Accepted Answer

150 °C

Answer

210 °C

Answer

180 °C

Answer

120 °C

Question 20

Consider a rectangular slab with dimensions of 10 cm x 10 cm x 2 cm. Heat is generated within the slab at a rate of 100 W/m³. If the top and bottom surfaces are perfectly insulated and the left and right surfaces are maintained at 100 °C, what is the temperature at the center of the slab?

Accepted Answer

150 °C

Answer

120 °C

Answer

210 °C

Answer

180 °C

Question 21

A fuel rod in a nuclear reactor has a diameter of 1 cm and a length of 1 m. The heat generation rate within the rod is 100 W/m³. If the surface temperature of the rod is maintained at 300 °C, determine the temperature at the center of the rod.

Accepted Answer

350 °C

Answer

380 °C

Answer

320 °C

Question 22

A composite wall consists of two layers, each with a thickness of 10 cm. The thermal conductivity of the first layer is 10 W/(m·K) and the thermal conductivity of the second layer is 20 W/(m·K). Heat is generated within the first layer at a rate of 100 W/m³. If the temperature at the outer surface of the second layer is maintained at 100 °C, what is the temperature at the interface between the two layers?

Accepted Answer

150 °C

Answer

210 °C

Answer

180 °C

Answer

120 °C

Question 23

A spherical shell with an inner radius of 10 cm and an outer radius of 20 cm generates heat at a rate of 100 W/m³. The inner surface of the shell is maintained at 100 °C and the outer surface is insulated. Determine the temperature at the outer surface.

Accepted Answer

150 °C

Answer

180 °C

Answer

120 °C

Answer

210 °C

Question 24

A rectangular prism with dimensions of 10 cm x 20 cm x 30 cm generates heat at a rate of 100 W/m³ throughout its volume. If all surfaces of the prism are perfectly insulated, determine the maximum temperature within the prism.

Accepted Answer

150 °C

Answer

210 °C

Answer

180 °C

Answer

120 °C

Question 25

A cylindrical rod with a diameter of 1 cm and a length of 1 m has a heat generation rate of 100 W/m³. The rod is placed in a fluid with a convective heat transfer coefficient of 10 W/(m²·K) and a fluid temperature of 20 °C. Determine the surface temperature of the rod.

Accepted Answer

30 °C

Answer

40 °C

Answer

25 °C

Answer

35 °C

Question 26

Which of the following is the governing equation for heat conduction with internal heat generation?

Accepted Answer

∇²T + q/k = 0

Answer

∇T² + q/k = 0

Answer

∇²T - q/k = 0

Answer

∇T² - q/k = 0

Question 27

In a rod with internal heat generation, the temperature distribution is:

Accepted Answer

Parabolic

Answer

Linear

Answer

Exponential

Answer

Hyperbolic

Question 28

Which of the following boundary conditions is appropriate for a rod with internal heat generation and insulated ends?

Accepted Answer

dT/dx = 0 at x = 0, L

Answer

dT/dx + h(T - Tamb) = 0 at x = 0, L

Answer

T = Tamb at x = 0, L

Answer

T = 0 at x = 0, L

Question 29

The total heat transfer rate through a rod with internal heat generation is given by:

Accepted Answer

-kA(dT/dx) |_0^L + q''V

Answer

kA(dT/dx) |_0^L - q''V

Answer

kA(dT/dx) |_0^L + q''V

Answer

-kA(dT/dx) |_0^L - q''V

Question 30

Which of the following is an application of heat conduction with internal heat generation?

Accepted Answer

Nuclear reactor

Answer

Heat exchanger

Answer

Solar panel

Answer

Convection heater

Question 31

In a composite rod with internal heat generation, the temperature discontinuity at the interface is caused by:

Accepted Answer

Difference in thermal conductivities

Answer

Difference in cross-sectional areas

Answer

Difference in heat generation rates

Answer

Difference in boundary conditions

Question 32

The Biot number is a dimensionless parameter that represents the ratio of:

Accepted Answer

Conduction resistance to convection resistance

Answer

Surface area to volume ratio

Answer

Heat generation rate to conduction rate

Answer

Convection resistance to conduction resistance

Question 33

Which of the following assumptions is made in the analysis of conduction with internal heat generation?

Accepted Answer

Steady-state conditions

Answer

One-dimensional heat flow

Answer

Transient conditions

Answer

Isotropic material properties

Question 34

The critical radius of insulation for a rod with internal heat generation is the radius at which:

Accepted Answer

The surface temperature reaches the melting point

Answer

The temperature gradient is zero

Answer

The internal heat generation rate is balanced by the heat loss

Answer

The heat transfer rate is maximum

Question 35

In a cylindrical rod with internal heat generation, the radial temperature distribution is:

Accepted Answer

Bessel function of the first kind

Answer

Power function

Answer

Exponential function