Heat Transfer Ch 2

True or false: In rectangular coordinates, the magnitude of heat transfer rate in the y-direction is given by Q˙y = −κAy∂T/∂y. Here, Ay is the heat conduction area perpendicular to the y-direction.

True

Heat Flux is a .....

vector quantity

If T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat, ρ is the density, and x is the linear distance, then the equation ∂2T∂x2 + e˙genk = ρck ∂T∂t reduces to ______ under the steady-state condition.

∂2T/∂x2 + e˙gen/k = 0

Identify the Poisson equation for heat transfer from the following, where T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, and x, y, and z are the distances.

∂2T/∂x2 + ∂2T/∂y2 + ∂2T/∂z2 + e˙gen/k = 0

In the case of constant thermal conductivity, the general heat conduction equation in rectangular coordinates is given by the equation ______.

∂2T/∂x2 + ∂2T/∂y2 + ∂2T/∂z2 + e˙gen/k = 1/α ∂T/∂t

If T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat, ρ is the density, and x is the linear distance, then the equation ∂2T/∂x2 + e˙gen/k = ρc/k ∂T∂t reduces to ______ under the transient, no heat generation condition.

∂2T/∂x2 = ρc/k ∂T/∂t

The one-dimensional transient heat conduction equation for a sphere with variable conductivity is ______, where r is the radius, T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat of the medium, and ρρ is the density of the medium.

1/r2 ∂/∂r(r2k∂/T∂r) + e˙gen= ρc∂T/∂t

The one-dimensional transient heat conduction equation for a sphere with variable conductivity is ______, where r is the radius, T is the temperature, e˙gene˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat of the medium, and ρρ is the density of the medium.

1/r2 ∂/∂r(r2k∂T/∂r) + e˙gen= ρc∂T/∂t

The one-dimensional transient heat conduction equation for a sphere with constant conductivity is ______, where r is the radius, T is the temperature, e˙gene˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat of the medium, and ρρ is the density of the medium.

1/r2 ∂/∂r(r2∂T/∂r) + e˙gen/k = 1/α ∂T/∂t

The thermal conductivity (k) of a material is 50 W/m⋅K, the density is 2700 kg/m3, and the specific heat is 904 J/kg⋅K. The thermal diffusivity (α) of the material is _____.

2.049 × 10-5 m2/s

A 1800-W hair dryer contains a resistance wire of 80 cm2 surface area. The heat flux on the outer surface of the wire as a result of heat generation is ____ W/cm2.

22.5

The resistance wire of a 1400-W hair dryer is 80 cm long and has a diameter of 0.3 cm. The rate of heat generation in the wire per unit volume is _____ W/cm3.

248

For a surface that involves convection, radiation, and specified heat flux simultaneously. The boundary condition is obtained using a surface energy balance, expressed as _____.

(Heat transfer to the surface in all modes) = (Heat transfer from the surface in all modes)

If the thermal conductivity (k) of a brick is 5 W/m··K, the cross sectional area (A) of the brick is 10 m2, and the temperature gradient (dT/dx) is 2 K/m, calculate the rate of heat conduction (Q˙cond) using Fourier's law of heat conduction in one-dimension.

-100 W

Identify the examples where the heat conduction can be approximated to be one-dimensional.

-Heat conduction through an electrical resistance wire -Heat conduction through a cylindrical nuclear fuel element -Heat conduction through a metal plate at the bottom of a pressing iron -Heat conduction through a cast-iron steam pipe

From Fourier's law of heat conduction, the boundary condition at a boundary is obtained by setting the specified heat flux equal to -k∂T/∂x at that boundary. Identify the correct statements about the specified heat flux. (Select all that apply.)

-It is negative if the heat flux is in the negative direction of the coordinate axis. -It is positive if the heat flux is in the positive direction of the coordinate axis.

Which of the following statements are true of a heat transfer vector at a point P on an isothermal surface?

-The heat transfer vector at a point on the surface is perpendicular to the surface. -The heat transfer vector points in the direction of decreasing temperature.

Identify the assumptions that are made in the case of heat conduction in a solar heated wall.

-Thermal conductivity is constant. -Heat transfer is steady. -Heat transfer is one-dimensional.

Identify the assumptions that are made in case of heat conduction in a spherical shell.

-Thermal conductivity is constant. -There is thermal symmetry in the medium -Heat transfer is steady.

Identify the necessary requirements at an interface of the boundary conditions. Neglect thermal contact resistance.

-Two bodies in contact must have the same temperature at the area of contact -An interface cannot store any energy

If r is the radius, k is the thermal conductivity, ρ is the density, c is the specific heat, T is the temperature, t is the time, z is the distance, φ is the angle, e˙gen is the total rate of heat generation, then the general heat conduction equation in cylindrical coordinate is given by _______.

1/r ∂/∂r(kr∂T/∂r) + 1/r2 ∂/∂φ (k∂T/∂φ) + ∂/∂z(k∂T/∂z) + e˙gen = ρc∂T/∂t

The constant conductivity one-dimensional heat conduction equation for a cylinder is 1/r ∂/∂r(r∂T/∂r) + e˙gen/k = 1/α ∂T/∂t, where r is the radius, T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, and α is the thermal diffusivity. For transient, no heat generation, the equation reduces to _______.

1/r ∂/∂r(r∂T/∂r) = 1/α ∂T/∂t

A metal sphere initially at a uniform temperature is cooled uniformly from the entire outer surface in ambient air. If there is no heat generated, which of the following differential equations describes the variation of the temperature in the sphere during cooling?

1/r2 ∂/∂r (r2k∂T/∂r) = ρc ∂T/∂t

The one-dimensional transient heat conduction equation for a sphere with constant conductivity is ______, where r is the radius, T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat of the medium, and ρ is the density of the medium.

1/r2 ∂/∂r(r2 ∂T/∂r) + e˙gen/k = 1/α ∂T/∂t

The constant conductivity one-dimensional heat conduction equation for a sphere is 1/r2 ∂/∂r(r2∂T/∂r) + e˙gen/k= 1/α ∂T/∂t, where r is the radius, T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, and α is the thermal diffusivity. For transient, no heat generation, the equation reduces to _______.

1/r2 ∂/∂r(r2 ∂T/∂r) = 1/α ∂T/∂t

The resistance wire of a 2000-W hair dryer is 80 cm long and has a diameter of 0.4 cm. The rate of heat generation in the wire per unit volume is _____ W/cm3

199 W/cm^3 e˙gen = E˙gen/Vwire

Consider a steam pipe of length L = 20 m, inner radius r1 = 4 cm, outer radius r2 = 6 cm, and thermal conductivity k = 20 W/m·K. The rate of heat loss (Q˙) from the steam pipe is ______ kW if the temperature difference between the surfaces of the pipe is ∆T = 90°C.

558

Consider that a large plane wall of thickness L = 0.2 m and the two sides of the wall are maintained at constant temperatures of T1 = 120°C and T2 = 50°C, respectively. Determine the value of temperature at x = 0.1 m.

85°C

An engineer, while solving a heat conduction problem, obtains the general solution of the applicable differential equation. Identify the next step that he has to perform for solving the problem.

Apply the boundary conditions.

True or false: The constant conductivity one-dimensional heat conduction equation for a cylinder is 1/r ∂/∂r(r∂T/∂r) + e˙gen/k = 1/α ∂T/∂t, where r is the radius, T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, and α is the thermal diffusivity. For transient, no heat generation, the equation reduces to 1/α ∂T/∂t = 0.

False

True or false: The temperature distribution in a medium depends on the conditions at the boundaries of the medium but not on the heat transfer mechanism inside the medium.

False

Which of the following is true of heat transfer?

Heat transfer rate has direction as well as magnitude.

The temperature at a point in a medium is to be specified. This requires that the location of the point be specified. Identify the true statement about specifying the location of the point at which the temperature should be specified.

It can be done in either the rectangular, cylindrical, or spherical coordinate systems.

Calculate the rate of heat conduction (Q˙) through a spherical shell if thermal conductivity (k) is 45 W/m·K, inner radius (r1) is 0.08 m, outer radius (r2) is 0.10 m, and difference in temperature (∆T) is 100°C.

Q˙ = 4πkr1r2∆T/(r2 − r1) Q˙ = 4 × 3.14 × 45 W/m·Km·K × 0.08 m × 0.10 m 100°C/(0.10 m − 0.08 m) Q˙ = 22.6 kW

Under steady conditions, the energy balance for a solid can be expressed as _____, where Q˙Q˙ is the rate of heat transfer, e˙gene˙gen is the heat generated at a constant rate per unit volume, and V is the volume of the solid.

Q˙ = e˙genV

Under steady conditions, the heat transfer rate for a solid, from newton's law of cooling, can be expressed as ____, where Q˙ is the rate of heat transfer, h is the heat transfer coefficient, As is the cross-sectional area of the solid, and Ts is the surface temperature of the solid, and T∞ is the temperatures of the surrounding medium. Assume Ts>T∞.

Q˙ = hAs(Ts - T∞)

If k is the thermal conductivity of the material, A is the area of cross section, n is the normal of the isothermal surface at a point, and T is the temperature, then, Fourier's law of heat conduction in one-dimension is given by the equation _____.

Q˙cond = −kA ∂T/∂n

If k is the thermal conductivity of the material, A is the area of cross section, and dT/dx is the temperature gradient, then Fourier's law of heat conduction in one-dimension is given by the equation ______.

Q˙cond = −kAdT/dx

Temperature .....

Scalar quantity

For one-dimensional heat transfer through a plane wall of thickness L, the specified temperature boundary conditions can be expressed as _____, where T1 is the specified temperatures at surfaces at x = 0 at time t.

T(0, t) = T1

Identify the general relation for the temperature distribution inside the pipe under steady conditions if r is the radial distance, r1 and r2 are the inner and outer radii, respectively, and T1 and T2 are the inner and outer temperatures, respectively.

T(r) = ln(r/r1)/ln(r2/r1) (T2 - T1) + T1

Calculate the magnitude of rate of heat conduction (Q˙) in a wall if the area of cross section (A) is 15 m2, thickness (L) is 0.2 m, difference is temperature (∆T) is 300 K, and thermal conductivity (k) is 1.2 W/m·K.

The rate of heat conduction through the wall is Q˙ = kA∆T/L Q˙ = 1.2 W/m·K × 15 m2 (300°C/0.2m) Q˙ = 27000 W

Identify the correct statement about the specified temperature boundary condition.

The specified temperature boundary condition is a constant for steady heat conduction.

A system at standard conditions is placed in a room. If the temperature difference in the system increases, what happens to the rate of heat transfer?

The system's rate of heat transfer increases.

In the constant conductivity equation ∂2T/∂x2 + e˙gen/k = 1/α ∂T/∂x, what is α?

Thermal diffusivity

True or False: The boundary condition on a perfectly insulated surface is expressed as k∂T(0, t)/∂x=0, when the first derivative of temperature with respect to the space variable is in the direction normal to the insulated surface.

True

True or false: In two-dimensional heat transfer, the temperature of a medium varies mainly in two primary directions and the heat transfer in the third direction is negligible.

True

True or false: The Laplace equation for heat transfer is ∂2T/∂x2 + ∂2T/∂y2 + ∂2T/∂z2 = 0.

True

The unit of heat flux (q˙) defined from Fourier's law of heat conduction is _______.

Watt/meter2

The unit of heat generation in an electrical wire is _____.

Watts/meter^3

On an insulated surface, the first derivative of temperature with respect to the space variable (the temperature gradient) in the direction normal to the insulated surface is

Zero

The constant conductivity one-dimensional heat conduction equation for a sphere is 1/r2 ∂/∂r(r2∂T/∂r) + e˙gen/k= 1/α ∂T/∂t, where r is the radius, T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, and α is the thermal diffusivity. For steady-state, no heat generation, the equation reduces to _______.

d/dr(r2dT/dr)= 0

If T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat, ρ is the density, and x is the linear distance, then the equation ∂2T/∂x2 + e˙gen/k = ρc/k ∂T∂t reduces to ______ under the steady-state, no heat generation condition.

d2T/dx2 = 0

The temperature of a resistance wire rises rapidly when electric current passes through it as a result of ______.

electrical energy being converted to heat

The heat generation (e˙gen) in an electrical wire of radius roro and length L can be expressed as _____, where l is the electric current and Re is the electrical resistance of the wire.

e˙gen = l^2Re/(πro^2L)

The radiation boundary condition involves the _____ power of temperature, and thus it is a nonlinear condition.

fourth

The surface energy balance equation states that the heat conduction at the surface in a selected direction equals the _____.

heat convection at the surface in the same direction

In the constant conductivity equation ∂2T/∂x2 + e˙gen/κ = ρc/κ ∂T/∂x, ρc/κ is equivalent 1/α, T is the temperature, e˙gen is the rate of heat generation per unit volume, k is the thermal conductivity, c is the specific heat, ρ is the density, and x is the linear distance. Here, α is the thermal diffusivity which indicates _______.

how fast heat propagates through a material

In the constant conductivity equation ∂2T/∂x2 + e˙gen/κ = 1/α ∂T/∂x, the thermal diffusivity (α) can be calculated using the formula _________, where k is the thermal conductivity, ρ is the density of the medium, and c is the specific heat capacity of the medium.

k/ρc

The variation in the thermal conductivity of a material with a temperature in the temperature range of interest can often be approximated as a(n) _____ function.

linear

A(n) _____ process in a system is one in which the temperature or heat flux varies uniformly with time but not with position.

lumped

Heat conduction through a large plane wall such as the wall of a house can be approximated as being _____.

one-dimensional

Consider a large wall of thickness L = 0.6 m and thermal conductivity k = 1.2 W/m·K in space. A temperature difference of ∆T = 9.5 K is maintained between the surfaces of the wall. The magnitude of rate of heat conduction through the wall per unit area (q˙) is ________ when steady operating conditions are reached.

q˙ = k∆T/L q˙ = 1.2 W/m·K (9.5 K/0.6 m) q˙ = 19 W/m2

A(n) _____ process in a system is one in which the temperature or heat flux does not vary with time at any point within the medium.

steady

In the case of cylindrical bodies having thermal symmetry about the center line, the thermal symmetry boundary condition requires that the first derivative of _____ with respect to r (the radial variable) be zero at the centerline.

temperature

The driving force for any form of heat transfer is the _____.

temperature difference