One-Dimensional, Steady-State Conduction

For one-dimensional, steady-state conduction in a plane wall with no heat generation and constant thermal conductivity, how does the temperature vary with position?

What is the correct formula for the thermal resistance for conduction, Rt,cond, in a plane wall of thickness L, area A, and thermal conductivity k?

A plane wall is 0.2 meters thick and has a thermal conductivity of 50 W/m K. If the wall area is 5 square meters, what is its thermal resistance for conduction?

What is the principal cause of thermal contact resistance when two solid surfaces are pressed together?

For one-dimensional, steady-state radial conduction through a hollow cylinder with no heat generation, the heat transfer rate (qr) is:

What is the critical radius of insulation (rcr) for a cylindrical pipe with insulation thermal conductivity 'k' and outer surface convection coefficient 'h'?

A long pipe is insulated with a material having a thermal conductivity of 0.055 W/m K. The outer surface convection coefficient is 5 W/m^2 K. What is the critical radius of insulation for this pipe?

How does the temperature profile change in a plane wall with constant thermal conductivity when uniform heat generation is introduced, assuming the surface temperatures are held constant?

In a solid cylinder with uniform heat generation q_dot and constant surface temperature Ts, where does the maximum temperature occur?

What is the primary purpose of adding fins to a surface?

What does the term 'fin effectiveness' represent?

Under what condition is the use of fins generally considered justified from a performance standpoint?

What is the correct expression for the thermal resistance of a hollow sphere for one-dimensional, steady-state radial conduction, with inner radius r1, outer radius r2, and thermal conductivity k?

A hollow spherical container has an inner radius of 0.25 m and an outer radius of 0.275 m. The container is made of a material with a thermal conductivity of 0.0017 W/m K. What is the conduction resistance of the container wall?

For a very long fin, where it can be assumed the tip temperature approaches the ambient fluid temperature, what is the temperature distribution T(x)?

In the 'alternative conduction analysis' for steady-state, one-dimensional heat transfer, what key assumption allows for the direct integration of Fourier's law without first solving the heat equation?

In a symmetrically cooled plane wall of thickness 2L with uniform heat generation q_dot, the heat flux at the midplane (x=0) is:

A long copper rod (k=398 W/m K) of diameter 5 mm is assumed to be an infinitely long fin. It is exposed to air with a convection coefficient of 100 W/m^2 K. What is the value of the fin parameter 'm'?

To what does the Pennes bioheat equation presented in the chapter reduce if the metabolic heat generation and blood perfusion terms are both zero?

When analyzing a composite wall with materials A, B, and C in series, with heat flowing from fluid 1 to fluid 4, the heat transfer rate (qx) can be related to the temperature drop across material B (T2 - T3) by:

What is the tip condition for a fin that is described as 'adiabatic'?

Which of the following would enhance fin effectiveness for an infinitely long fin, according to the expression epsilon_f = (kP / (h*Ac))^0.5?

A composite wall has a layer of material A (k_A = 20 W/m K, L_A = 0.30 m) and a layer of material C (k_C = 50 W/m K, L_C = 0.15 m). A 0.15 m thick layer of material B is between them. The inner surface temperature is 600 C, the outer surface is 20 C, and the oven air is 800 C with h=25 W/m^2 K. What is the heat flux at the inner surface (x=0)?

The overall surface efficiency, eta_o, for a fin array is defined as:

What is the perfusion term in the Pennes bioheat equation intended to account for?

For a fin with an adiabatic tip, the heat transfer rate q_f is given by M*tanh(mL). What does the term M represent?

A composite cylindrical wall consists of an inner layer A and an outer layer B. In the equivalent thermal circuit, the resistance for conduction through layer A is given by:

The heat rate (q) from a fin is related to the fin thermal resistance (Rt,f) and the temperature difference between the base (Tb) and the fluid (T_inf) by which expression?

If adding a layer of insulation to a small-diameter pipe increases the heat loss, it means that the initial radius of the pipe was:

A plane wall of thickness 2L = 40 mm and thermal conductivity k = 5 W/m K has a steady-state temperature distribution of T(x) = 82.0 - 210x - 20000x^2, where x is in meters from the midplane. What is the volumetric heat generation rate q_dot?

The fin efficiency, eta_f, is defined as the ratio q_f / q_max. What is q_max?

A conical section is fabricated from pyroceram (k = 3.46 W/m K). The diameter is D = 0.25x. The small end is at x1 = 50 mm and the large end is at x2 = 250 mm. The end temperatures are T1 = 400 K and T2 = 600 K. What is the heat rate qx through the cone?

In a composite system, how are series and parallel thermal resistances typically represented in an analysis?

The addition of insulation is always beneficial for reducing heat transfer from a plane wall because:

What is the heat transfer rate per unit length for a very long fin with parameter M = 8.3 W?

If a material experiences uniform volumetric heat generation q_dot, can it be represented by a single thermal circuit resistance?

What is the overall heat transfer coefficient, U, for a composite plane wall with a total thermal resistance R_tot and area A?

For the bioheat equation, the perfusion term q_dot_p = omega*rho_b*c_b*(T_a - T) acts as a heat sink when:

A rectangular fin's efficiency can be approximated using a corrected length, Lc, to account for heat loss from the tip. For a fin of length L and thickness t, what is the corrected length?

A long, thin-walled copper tube has a radius ri and is at a temperature Ti. To minimize heat loss to ambient air at T_inf, insulation is added. If the initial radius ri is greater than the critical radius rcr, what is the effect of adding insulation?

A silicon chip and an 8-mm-thick aluminum substrate are separated by a 0.02-mm-thick epoxy joint. The thermal conductivity of the aluminum is 239 W/m K. The contact resistance of the epoxy is 0.9 x 10^-4 m^2 K/W. What is the total thermal resistance of the substrate and joint per unit area?

In a series composite wall, if one assumes that surfaces normal to the x-direction of heat flow are isothermal, how are the resistances of parallel layers F and G combined?

A long conducting rod has a temperature-dependent thermal conductivity given by k(T) = k_o + aT. For one-dimensional, steady-state heat transfer with no generation, what is the integrated form of Fourier's law between x0 (at T0) and x1 (at T1)?

The critical insulation radius for a sphere is given as 2k/h. If insulation with k = 0.04 W/m K is applied to a sphere in an environment with h = 10 W/m^2 K, what is the critical radius?

What is the primary assumption of the 'lumped capacitance method' for transient conduction analysis, as introduced in Chapter 5 but relevant to the limitations of steady-state analysis?

For a plane wall of thickness L, the condition for which the internal conduction resistance is equal to the surface convection resistance is:

A human body is in an environment where the insulation of a snow suit is the dominant thermal resistance. The suit has a thickness of 4.4 mm and thermal conductivity of 0.014 W/m K, covering an area of 1.8 m^2. What is the approximate thermal resistance of the suit?

In a series-parallel composite wall, as shown in Figure 3.3, what assumption is made if the thermal circuit is drawn with surfaces parallel to the x-direction treated as adiabatic?

A solid, truncated cone has its sides well-insulated. If heat flows under steady-state from the large end (x2) to the small end (x1), how does the magnitude of the temperature gradient |dT/dx| change along the direction of heat flow (from x2 to x1)?

Which condition is NOT a requirement for the 'alternative conduction analysis' (direct integration of Fourier's law) to be applicable?