In the context of micro- and nanoscale heat transfer in a thin film, what happens to the thermal conductivity as the film thickness (L) becomes very small compared to the mean free path (mfp) of the energy carriers?

Correct answer: The thermal conductivity becomes anisotropic (kx is not equal to ky) and is reduced from the bulk value.

Explanation

This question addresses the modern topic of nanoscale effects on thermal conductivity, a key concept for microelectronics and advanced materials.

Other questions

Question 1

How is Fourier's law of conduction characterized in its origin?

Question 2

In the general vector statement of Fourier's law, q'' = -k∇T, what is the relationship between the heat flux vector (q'') and an isothermal surface?

Question 3

For a material to be considered isotropic in the context of heat conduction, what must be true about its thermal conductivity (k)?

Question 4

What are the two primary mechanisms for the transport of thermal energy in a solid?

Question 5

In pure metals, which contribution to conduction heat transfer is typically dominant?

Question 6

What does thermal diffusivity (alpha) represent?

Question 7

Calculate the thermal diffusivity (alpha) for paraffin at 300 K, given k = 0.24 W/m·K, rho = 900 kg/m^3, and cp = 2890 J/kg·K.

Question 8

What is the general form of the heat diffusion equation derived from?

Question 9

In the one-dimensional, steady-state heat equation with no energy generation, d/dx(k dT/dx) = 0, what is the important implication for the heat flux?

Question 10

What physical situation is represented by a Dirichlet boundary condition?

Question 11

A wall has a temperature distribution T(x) = 900 - 300x - 50x^2. The wall has a uniform heat generation (q_dot) of 1000 W/m^3, thermal conductivity (k) of 40 W/m·K, and volumetric heat capacity (rho*cp) of 6.4 x 10^6 J/m^3·K. What is the time rate of temperature change (dT/dt) in the wall?

Question 12

What is a Neumann boundary condition?

Question 13

An adiabatic surface is a special case of which type of boundary condition?

Question 14

What is the general trend for the thermal conductivity of a nonmetallic liquid as temperature increases?

Question 15

Why is the thermal conductivity of a gas generally much smaller than that of a solid?

Question 16

What is the physical significance of the energy storage term (E_dot_st) in a control volume energy balance?

Question 18

For which of the following materials from Table 2.1 is the critical film thickness (L_crit,x), below which microscale effects on conduction become significant, the smallest?

Question 19

The heat equation for cylindrical coordinates contains the term (1/r) * d/dr(r * dT/dr). Why is the additional 'r' term present inside the derivative compared to the plane wall equation?

Question 20

What is the primary purpose of solving the heat diffusion equation?

Question 21

For steady-state conditions in a medium with no heat generation, what must be true about the temperature at a particular location?

Question 22

What does a material with a large thermal diffusivity (alpha) indicate about its thermal response?

Question 23

In the derivation of the heat diffusion equation, the conduction heat rate at a surface x+dx is expressed using a Taylor series expansion. What is the assumption made in this step?

Question 24

How does the thermal conductivity of a gas typically change with increasing temperature and increasing molecular weight?

Question 25

For a long copper bar described in Example 2.3, initially at a uniform temperature To, what is the initial condition for solving the temperature distribution T(x, t)?

Question 26

What is the volumetric heat capacity (rho * cp) of a material, and what does it measure?

Question 27

Why is the minus sign necessary in Fourier's law, q'' = -k(dT/dx)?

Question 28

In the steady-state heat conduction experiment shown in Figure 2.1, if the cross-sectional area A and temperature difference ΔT are held constant, how does the heat transfer rate qx relate to the rod length Δx?

Question 29

For a plane wall under steady-state, one-dimensional conduction with constant thermal conductivity and no heat generation, what is the shape of the temperature distribution?

Question 30

Why might a crystalline, nonmetallic solid like diamond have a higher thermal conductivity than a good metallic conductor like aluminum?

Question 31

What type of boundary condition is represented by the equation -k(dT/dx)|x=L = h[T(L,t) - T_infinity]?

Question 32

What is the key feature of heat transfer in an 'effective thermal conductivity' for an insulation system?

Question 33

Calculate the time rate of change of energy storage, E_dot_st, for a wall with an area of 10 m^2 and length of 1 m, given q_in = 120 kW, q_out = 160 kW, and a uniform heat generation q_dot = 1000 W/m^3.

Question 34

What is the physical difference between the energy generation term (E_dot_g) and the energy storage term (E_dot_st) in the heat equation?

Question 35

For a transient conduction problem, how many initial conditions are required to solve the heat equation?

Question 36

For a three-dimensional conduction problem in Cartesian coordinates, how many boundary conditions must be specified?

Question 37

What conditions are required for the heat diffusion equation to be valid according to the discussion on microscale effects?

Question 38

Calculate the thermal diffusivity (alpha) for silicon carbide at 1000 K, given k = 87 W/m·K, rho = 3160 kg/m^3, and cp = 1195 J/kg·K.

Question 39

The heat flux vector can be resolved into Cartesian components q''_x, q''_y, and q''_z. How is the x-component, q''_x, related to the temperature field T(x,y,z)?

Question 40

How does the pressure generally affect the thermal conductivity of a gas, according to the text?

Question 41

What is the general simplified form of the heat equation if the thermal conductivity 'k' is constant?

Question 42

If heat transfer by conduction through a medium occurs under steady-state conditions, will the temperature at a particular location vary with time?

Question 43

A plane wall experiences one-dimensional conduction. Under what conditions will the temperature distribution be linear?

Question 44

What is the primary heat conduction mechanism in a nonconducting solid?

Question 45

In the general heat diffusion equation, what does the term 'q_dot' represent?

Question 46

If a material has a very small thermal diffusivity (alpha), how will it behave when subjected to a change in thermal conditions?

Question 47

What kind of temperature distribution is implied in a solid cylinder under steady-state, one-dimensional radial conduction with no heat generation?

Question 48

A boundary condition of the 'third kind' involves which physical process at the surface?

Question 49

A plane wall of thickness 0.25 m and thermal conductivity 50 W/m·K has surface temperatures of T1 = 50 C and T2 = -20 C. What is the heat flux?

Question 50

In what physical scenario is the alternative approach of directly integrating Fourier's law (qx * integral(dx/A(x)) = - integral(k(T)dT)) a valid method for analysis?