An engineer desires to measure the thermal conductivity of

Assume steady-state, one-dimensional heat conduction through the axisymmetric shape shown below. Assuming constant properties and no internal heat generation, sketch the temperature distribution on T  x coordinates. Briefly explain the shape of your curve.

Assume steady-state, one-dimensional conduction in the axisymmetric object below, which is insulated around its perimeter. If the properties remain constant and no internal heat generation occurs, sketch the heat flux distribution, , and the temperature distribution, T(x). Explain the shapes of your curves. How do your curves depend on the thermal conductivity of the material?

A hot water pipe with outside radius r1 has a temperature T1. A thick insulation, applied to reduce the heat loss, has an outer radius r2 and temperature T2. On T  r coordinates, sketch the temperature distribution in the insulation for one-dimensional, steady-state heat transfer with constant properties. Give a brief explanation, justifying the shape of your curve.

A spherical shell with inner radius r1 and outer radius r2 has surface temperatures T1 and T2, respectively, where T1
T2. Sketch the temperature distribution on T  r coordinates assuming steady-state, one-dimensional conduction with constant properties. Briefly justify the shape of your curve.

Assume steady-state, one-dimensional heat conduction through the symmetric shape shown. Assuming that there is no internal heat generation, derive an expression for the thermal conductivity k(x) for these conditions: A(x)  (1  x), T(x)  300 (1  2x  x3 ), and q  6000 W, where A is in square meters, T in kelvins, and x in meters.

A composite rod consists of two different materials, A and B, each of length 0.5L. The thermal conductivity of Material A is half that of Material B, that is, kA/kB  0.5. Sketch the steady-state temperature and heat flux distributions, T(x) and , respectively. Assume constant properties and no internal heat generation in either material

A solid, truncated cone serves as a support for a system that maintains the top (truncated) face of the cone at a temperature T1, while the base of the cone is at a temperature T2 T1. The thermal conductivity of the solid depends on temperature according to the relation k  k0  aT, where a is a positive constant, and the sides of the cone are well insulated. Do the following quantities increase, decrease, or remain the same with increasing x: the heat transfer rate qx, the heat flux , the thermal conductivity k, and the temperature gradient dT/dx?

To determine the effect of the temperature dependence of the thermal conductivity on the temperature distribution in a solid, consider a material for which this dependence may be represented as where ko is a positive constant and a is a coefficient that may be positive or negative. Sketch the steady-state temperature distribution associated with heat transfer in a plane wall for three cases corresponding to a
0, a  0, and a 0.

A young engineer is asked to design a thermal protection barrier for a sensitive electronic device that might be exposed to irradiation from a high-powered infrared laser. Having learned as a student that a low thermal conductivity material provides good insulating characteristics, the engineer specifies use of a nanostructured aerogel, characterized by a thermal conductivity of ka  0.005 W/m
K, for the protective barrier. The engineers boss questions the wisdom of selecting the aerogel because it has a low thermal conductivity. Consider the sudden laser irradiation of (a) pure aluminum, (b) glass, and (c) aerogel. The laser provides irradiation of G  10 106 W/m2 . The absorptivities of the materials are   0.2, 0.9, and 0.8 for the aluminum, glass, and aerogel, respectively, and the initial temperature of the barrier is Ti  300 K. Explain why the boss is concerned. Hint: All materials experience thermal expansion (or contraction), and local stresses that develop within a material are, to a first approximation, proportional to the local temperature gradient.

A one-dimensional plane wall of thickness 2L  100 mm experiences uniform thermal energy generation of and is convectively cooled at x  50 mm by an ambient fluid characterized by T
 20 C. If the steady-state temperature distribution within the wall is T(x)  a(L2  x2 )  b where a  10 C/m2 and b  30 C, what is the thermal conductivity of the wall? What is the value of the convection heat transfer coefficient, h?

Consider steady-state conditions for one-dimensional conduction in a plane wall having a thermal conductivity k  50 W/m
K and a thickness L  0.25 m, with no internal heat generation. Determine the heat flux and the unknown quantity for each case and sketch the temperature distribution, indicating the direction of the heat flux

Consider a plane wall 100 mm thick and of thermal conductivity 100 W/m
K. Steady-state conditions are known to exist with T1  400 K and T2  600 K. Determine the heat flux and the temperature gradient dT/dx for the coordinate systems shown

A cylinder of radius ro, length L, and thermal conductivity k is immersed in a fluid of convection coefficient h and unknown temperature T
. At a certain instant the temperature distribution in the cylinder is T(r)  a  br2 , where a and b are constants. Obtain expressions for the heat transfer rate at ro and the fluid temperature.

In the two-dimensional body illustrated, the gradient at surface A is found to be
T/
y  30 K/m. What are
T/
y and
T/
x at surface B?

Consider the geometry of Problem 2.14 for the case where the thermal conductivity varies with temperature as k  ko  aT, where ko  10 W/m
K, a  103 W/m
K2 , and T is in kelvins. The gradient at surface B is
T/
x  30 K/m. What is
T/
y at surface A?

Steady-state, one-dimensional conduction occurs in a rod of constant thermal conductivity k and variable crosssectional area Ax(x)  Aoeax, where Ao and a are constants. The lateral surface of the rod is well insulated. (a) Write an expression for the conduction heat rate, qx(x). Use this expression to determine the temperature distribution T(x) and qualitatively sketch the distribution for T(0)
T(L). (b) Now consider conditions for which thermal energy is generated in the rod at a volumetric rate where is a constant. Obtain an expression for qx(x) when the left face (x  0) is well insulated

An apparatus for measuring thermal conductivity employs an electrical heater sandwiched between two identical samples of diameter 30 mm and length 60 mm, which are pressed between plates maintained at a uniform temperature To  77 C by a circulating fluid. A conducting grease is placed between all the surfaces to ensure good thermal contact. Differential thermocouples are imbedded in the samples with a spacing of 15 mm. The lateral sides of the samples are insulated to ensure onedimensional heat transfer through the samples. (a) With two samples of SS316 in the apparatus, the heater draws 0.353 A at 100 V, and the differential thermocouples indicate T1  T2  25.0 C. What is the thermal conductivity of the stainless steel sample material? What is the average temperature of the samples? Compare your result with the thermal conductivity value reported for this material in Table A.1. (b) By mistake, an Armco iron sample is placed in the lower position of the apparatus with one of the SS316 samples from part (a) in the upper portion. For this situation, the heater draws 0.601 A at 100 V, and the differential thermocouples indicate T1  T2  15.0 C. What are the thermal conductivity and average temperature of the Armco iron sample? (c) What is the advantage in constructing the apparatus with two identical samples sandwiching the heater rather than with a single heatersample combination? When would heat leakage out of the lateral surfaces of the samples become significant? Under what conditions would you expect T1  T2 ? 2.18

An engineer desires to measure the thermal conductivity of an aerogel material. It is expected that the aerogel will have an extremely small thermal conductivity (a) Explain why the apparatus of Problem 2.17 cannot be used to obtain an accurate measurement of the aerogels thermal conductivity. (b) The engineer designs a new apparatus for which an electric heater of diameter D  150 mm is sandwiched between two thin plates of aluminum. The steady-state temperatures of the 5-mm-thick aluminum plates, T1 and T2, are measured with thermocouples. Aerogel sheets of thickness t  5 mm are placed outside the aluminum plates, while a coolant with an inlet temperature of Tc,i  25 C maintains the exterior surfaces of the aerogel at a low temperature. The circular aerogel sheets are formed so that they encase the heater and aluminum sheets, providing insulation to minimize radial heat losses. At steady state, T1  T2  55 C, and the heater draws 125 mA at 10 V. Determine the value of the aerogel thermal conductivity ka. (c) Calculate the temperature difference across the thickness of the 5-mm-thick aluminum plates. Comment on whether it is important to know the axial locations at which the temperatures of the aluminum plates are measured. (d) If liquid water is used as the coolant with a total flow rate of (0.5 kg/min for each of the two streams), calculate the outlet temperature of the water, Tc,o

Consider a 300 mm 300 mm window in an aircraft. For a temperature difference of 80 C from the inner to the outer surface of the window, calculate the heat loss through L  10-mm-thick polycarbonate, soda lime glass, and aerogel windows, respectively. The thermal conductivities of the aerogel and polycarbonate are kag  0.014 W/m
K and kpc  0.21 W/m
K, respectively. Evaluate the thermal conductivity of the soda lime glass at 300 K. If the aircraft has 130 windows and the cost to heat the cabin air is $1/kW
h, compare the costs associated with the heat loss through the windows for an 8-hour intercontinental flight

Consider a small but known volume of metal that has a large thermal conductivity. (a) Since the thermal conductivity is large, spatial temperature gradients that develop within the metal in response to mild heating are small. Neglecting spatial temperature gradients, derive a differential equation that could be solved for the temperature of the metal versus time T(t) if the metal is subjected to a fixed surface heat rate q supplied by an electric heater. (b) A student proposes to identify the unknown metal by comparing measured and predicted thermalresponses. Once a match is made, relevant thermophysical properties might be determined, and, in turn, the metal may be identified by comparison to published property data. Will this approach work? Consider aluminum, gold, and silver as the candidate metals.

Use IHT to perform the following tasks. (a) Graph the thermal conductivity of pure copper, 2024 aluminum, and AISI 302 stainless steel over the temperature range 300  T  600 K. Include all data on a single graph, and comment on the trends you observe. (b) Graph the thermal conductivity of helium and air over the temperature range 300  T  800 K. Include the data on a single graph, and comment on the trends you observe. (c) Graph the kinematic viscosity of engine oil, ethylene glycol, and liquid water over the temperature range 300  T  360 K. Include all data on a single graph, and comment on the trends you observe. (d) Graph the thermal conductivity of a water-Al2O3 nanofluid at T  300 K over the volume fraction range 0    0.08. See Example 2.2.

Calculate the thermal conductivity of air, hydrogen, and carbon dioxide at 300 K, assuming ideal gas behavior. Compare your calculated values to values from Table A.4.

A method for determining the thermal conductivity k and the specific heat cp of a material is illustrated in the sketch. Initially the two identical samples of diameter D  60 mm and thickness L  10 mm and the thin heater are at a uniform temperature of Ti  23.00 C, while surrounded by an insulating powder. Suddenly the heater is energized to provide a uniform heat flux on each of the sample interfaces, and the heat flux is maintained constant for a period of time, to. A short time after sudden heating is initiated, the temperature at this interface To is related to the heat flux as To(t)  Ti  2q o
t cpk 1/2 For a particular test run, the electrical heater dissipates 15.0 W for a period of to  120 s, and the temperature at the interface is To(30 s)  24.57 C after 30 s of heating. A long time after the heater is deenergized, t  t0, the samples reach the uniform temperature of To(
)  33.50 C. The density of the sample materials, determined by measurement of volume and mass, is  3965 kg/m3 . D Determine the specific heat and thermal conductivity of the test material. By looking at values of the thermophysical properties in Table A.1 or A.2, identify the test sample material.

Compare and contrast the heat capacity cp of common brick, plain carbon steel, engine oil, water, and soil. Which material provides the greatest amount of thermal energy storage per unit volume? Which material would you expect to have the lowest cost per unit heat capacity? Evaluate properties at 300 K.

A cylindrical rod of stainless steel is insulated on its exterior surface except for the ends. The steady-state temperature distribution is T(x)  a  bx/L, where a  305 K and b  10 K. The diameter and length of the rod are D  20 mm and L  100 mm, respectively. Determine the heat flux along the rod, Hint: The mass of the rod is M  0.248 kg.

At a given instant of time, the temperature distribution within an infinite homogeneous body is given by the function T(x, y, z)  x2  2y2  z 2  xy  2yz Assuming constant properties and no internal heat generation, determine the regions where the temperature changes with time.

A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate qo. There are two stages to the process. In Stage 1, the water is taken from its initial (room) temperature Ti to the boiling point, as heat is transferred from the pan by natural convection. During this stage, a constant value of the convection coef- ficient h may be assumed, while the bulk temperature of the water increases with time, T
 T
(t). In Stage 2, the water has come to a boil, and its temperature remains at a fixed value, T
 Tb, as heating continues. Consider a pan bottom of thickness L and diameter D, with a coordinate system corresponding to x  0 and x  L for the surfaces in contact with the stove and water, respectively. (a) Write the form of the heat equation and the boundary/ initial conditions that determine the variation of temperature with position and time, T(x, t), in the pan bottom during Stage 1. Express your result in terms of the parameters qo, D, L, h, and T
, as well as appropriate properties of the pan material. (b) During Stage 2, the surface of the pan in contact with the water is at a fixed temperature, T(L, t)  TL
Tb. Write the form of the heat equation and boundary conditions that determine the temperature distribution T(x) in the pan bottom. Express your result in terms of the parameters qo, D, L, and TL, as well as appropriate properties of the pan material.

Uniform internal heat generation at is occurring in a cylindrical nuclear reactor fuel rod of 50-mm diameter, and under steady-state conditions the temperature distribution is of the form T(r)  a  br2 , where T is in degrees Celsius and r is in meters, while a  800 C and b  4.167 105 C/m2 . The fuel rod properties are k  30 W/m
K,  1100 kg/m3 , and cp  800 J/kg K. (a) What is the rate of heat transfer per unit length of the rod at r  0 (the centerline) and at r  25 mm (the surface)? (b) If the reactor power level is suddenly increased to q . 2  108 W/m3 , what is the initial time rate of temperature change at r  0 and r  25 mm?

Consider a one-dimensional plane wall with constant properties and uniform internal generation . The left face is insulated, and the right face is held at a uniform temperature. (a) Using the appropriate form of the heat equation, derive an expression for the x-dependence of the steady-state heat flux q(x). (b) Using a finite volume spanning the range 0  x  , derive an expression for q( ) and compare the expression to your result for part (a).

The steady-state temperature distribution in a onedimensional wall of thermal conductivity 50 W/m
K and thickness 50 mm is observed to be T( C)  a  bx2 , where a  200 C, b  2000 C/m2 , and x is in meters. (a) What is the heat generation rate in the wall? (b) Determine the heat fluxes at the two wall faces. In what manner are these heat fluxes related to the heat generation rate?

The temperature distribution across a wall 0.3 m thick at a certain instant of time is T(x)  a  bx  cx2 , where T is in degrees Celsius and x is in meters, a  200 C, b  200 C/m, and c  30 C/m2 . The wall has a thermal conductivity of 1 W/m
K. (a) On a unit surface area basis, determine the rate of heat transfer into and out of the wall and the rate of change of energy stored by the wall. (b) If the cold surface is exposed to a fluid at 100 C, what is the convection coefficient?

A plane wall of thickness 2L  40 mm and thermal conductivity k  5 W/m
K experiences uniform volumetric heat generation at a rate q . , while convection heat transfer occurs at both of its surfaces (x  L,  L), each of which is exposed to a fluid of temperature T
 20 C. Under steady-state conditions, the temperature distribution in the wall is of the form T(x)  a  bx  cx2 where a  82.0 C, b  210 C/m, c  2 104 C/m2 , and x is in meters. The origin of the x-coordinate is at the midplane of the wall. (a) Sketch the temperature distribution and identify significant physical features. (b) What is the volumetric rate of heat generation in the wall? (c) Determine the surface heat fluxes, and How are these fluxes related to the heat generation rate? (d) What are the convection coefficients for the surfaces at x  L and x  L? (e) Obtain an expression for the heat flux distribution Is the heat flux zero at any location? Explain any significant features of the distribution. (f) If the source of the heat generation is suddenly deactivated , what is the rate of change of energy stored in the wall at this instant? (g) What temperature will the wall eventually reach with ? How much energy must be removed by the fluid per unit area of the wall (J/m2 ) to reach this state? The density and specific heat of the wall material are 2600 kg/m3 and 800 J/kg
K, respectively

Temperature distributions within a series of onedimensional plane walls at an initial time, at steady state, and at several intermediate times are as shown. For each case, write the appropriate form of the heat diffusion equation. Also write the equations for the initial condition and the boundary conditions that are applied at x  0 and x  L. If volumetric generation occurs, it is uniform throughout the wall. The properties are constant

One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of 50 mm and a constant thermal conductivity of 5 W/m
K. For these conditions, the temperature distribution has the form T(x)  a  bx  cx2 . The surface at x  0 has a temperature of T(0) To  120 C and experiences convection with a fluid for which T
 20 C and h  500 W/m2
K. The surface at x  L is well insulated. (a) Applying an overall energy balance to the wall, calculate the volumetric energy generation rate . (b) Determine the coefficients a, b, and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution. (c) Consider conditions for which the convection coef- ficient is halved, but the volumetric energy generation rate remains unchanged. Determine the new values of a, b, and c, and use the results to plot the temperature distribution. Hint: recognize that T(0) is no longer 120 C. (d) Under conditions for which the volumetric energy generation rate is doubled, and the convection coef- ficient remains unchanged (h  500 W/m2
K), determine the new values of a, b, and c and plot the corresponding temperature distribution. Referring to the results of parts (b), (c), and (d) as Cases 1, 2, and 3, respectively, compare the temperature distributions for the three cases and discuss the effects of h and on the distributions

Derive the heat diffusion equation, Equation 2.26, for cylindrical coordinates beginning with the differential control volume shown in Figure 2.12.

Derive the heat diffusion equation, Equation 2.29, for spherical coordinates beginning with the differential control volume shown in Figure 2.13

The steady-state temperature distribution in a semitransparent material of thermal conductivity k and thickness L exposed to laser irradiation is of the form where A, a, B, and C are known constants. For this situation, radiation absorption in the material is manifested by a distributed heat generation term, (a) Obtain expressions for the conduction heat fluxes at the front and rear surfaces. (b) Derive an expression for (c) Derive an expression for the rate at which radiation is absorbed in the entire material, per unit surface q(x). x L Laser irradiation Semitransparent medium, T(x) q(x). T(x)   A ka2 eax  Bx  C area. Express your result in terms of the known constants for the temperature distribution, the thermal conductivity of the material, and its thickness.

One-dimensional, steady-state conduction with no energy generation is occurring in a cylindrical shell of inner radius r1 and outer radius r2. Under what condition is the linear temperature distribution shown possible?

One-dimensional, steady-state conduction with no energy generation is occurring in a spherical shell of inner radius r1 and outer radius r2. Under what condition is the linear temperature distribution shown in Problem 2.38 possible?

The steady-state temperature distribution in a onedimensional wall of thermal conductivity k and thickness L is of the form T  ax3  bx2  cx  d. Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces (x  0, L)

One-dimensional, steady-state conduction with no energy generation is occurring in a plane wall of constant thermal conductivity (b) With the temperature at x  0 and the fluid temperature fixed at T(0)  0 C and T
 20 C, respectively, compute and plot the temperature at x  L, T(L), as a function of h for 10  h  100 W/m2
K. Briefly explain your results.

A plane layer of coal of thickness L  1 m experiences uniform volumetric generation at a rate of due to slow oxidation of the coal particles. Averaged over a daily period, the top surface of the layer transfers heat by convection to ambient air for which h  5 W/m2
K and T
 25 C, while receiving solar irradiation in the amount GS  400 W/m2 . Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are each S    0.95. (a) Write the steady-state form of the heat diffusion equation for the layer of coal. Verify that this equation is satisfied by a temperature distribution of the form From this distribution, what can you say about conditions at the bottom surface (x  0)? Sketch the temperature distribution and label key features. (b) Obtain an expression for the rate of heat transfer by conduction per unit area at x  L. Applying an energy balance to a control surface about the top surface of the layer, obtain an expression for Ts. Evaluate Ts and T(0) for the prescribed conditions. (c) Daily average values of GS and h depend on a number of factors, such as time of year, cloud cover, and wind conditions. For h  5 W/m2
K, compute and plot TS and T(0) as a function of GS for 50  GS  500 W/m2 . For GS  400 W/m2 , compute and plot TS and T(0) as a function of h for 5  h  50 W/m2
K

The cylindrical system illustrated has negligible variation of temperature in the r- and z-directions. Assume that r  ro  ri is small compared to ri , and denote the length in the z-direction, normal to the page, as L.(a) Beginning with a properly defined control volume and considering energy generation and storage effects, derive the differential equation that prescribes the variation in temperature with the angular coordinate . Compare your result with Equation 2.26. (b) For steady-state conditions with no internal heat generation and constant properties, determine the temperature distribution T( ) in terms of the constants T1, T2, ri , and ro. Is this distribution linear in ? (c) For the conditions of part (b) write the expression for the heat rate q .

Beginning with a differential control volume in the form of a cylindrical shell, derive the heat diffusion equation for a one-dimensional, cylindrical, radial coordinate system with internal heat generation. Compare your result with Equation 2.26.

Beginning with a differential control volume in the form of a spherical shell, derive the heat diffusion equation for a one-dimensional, spherical, radial coordinate system with internal heat generation. Compare your result with Equation 2.29.

A steam pipe is wrapped with insulation of inner and outer radii ri and ro, respectively. At a particular instant the temperature distribution in the insulation is known to be of the form T(r)  C1 ln
r ro  C2 Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius?

For a long circular tube of inner and outer radii r1 and r2, respectively, uniform temperatures T1 and T2 are maintained at the inner and outer surfaces, while thermal energy generation is occurring within the tube wall (r1 r r2). Consider steady-state conditions for which T1 T2. Is it possible to maintain a linear radial temperature distribution in the wall? If so, what special conditions must exist?

Passage of an electric current through a long conducting rod of radius ri and thermal conductivity kr results in uniform volumetric heating at a rate of . The conducting rod is wrapped in an electrically nonconducting cladding material of outer radius ro and thermal conductivity kc, and convection cooling is provided by an adjoining fluid. For steady-state conditions, write appropriate forms of the heat equations for the rod and cladding. Express appropriate boundary conditions for the solution of these equations.

Two-dimensional, steady-state conduction occurs in a hollow cylindrical solid of thermal conductivity k  16 W/m
K, outer radius ro  1 m and overall length 2zo  5 m, where the origin of the coordinate system is located at the midpoint of the center line. The inner surface of the cylinder is insulated, and the temperature distribution within the cylinder has the form T(r, z)  a  br2  clnr  dz2 , where a  20 C, b  150 C/m2 , c  12 C, d  300 C/m2 and r and z are in meters. (a) Determine the inner radius ri of the cylinder. (b) Obtain an expression for the volumetric rate of heat generation, (c) Determine the axial distribution of the heat flux at the outer surface, What is the heat rate at the outer surface? Is it into or out of the cylinder? (d) Determine the radial distribution of the heat flux at the end faces of the cylinder, and What are the corresponding heat rates? Are they into or out of the cylinder? (e) Verify that your results are consistent with an overall energy balance on the cylinder.

An electric cable of radius r1 and thermal conductivity kc is enclosed by an insulating sleeve whose outer surface is of radius r2 and experiences convection heat transfer and radiation exchange with the adjoining air and large surroundings, respectively. When electric current passes through the cable, thermal energy is generated within the cable at a volumetric rate . (a) Write the steady-state forms of the heat diffusion equation for the insulation and the cable. Verify that these equations are satisfied by the following temperature distributions: Insulation: Cable: Sketch the temperature distribution, T(r), in the cable and the sleeve, labeling key features. (b) Applying Fouriers law, show that the rate of conduction heat transfer per unit length through the sleeve may be expressed as Applying an energy balance to a control surface placed around the cable, obtain an alternative expression for q r, expressing your result in terms of and r1. (c) Applying an energy balance to a control surface placed around the outer surface of the sleeve, obtain an expression from which Ts,2 may be determined as a function of , r1, h, T
, , and Tsur. (d) Consider conditions for which 250 A are passing through a cable having an electric resistance per unit length of R e  0.005 /m, a radius of r1  15 mm, and a thermal conductivity of kc  200 W/m
K. For ks  15 W/m
K, r2  15.5 mm, h  25 W/m2 K,   0.9, T
 25 C, and Tsur  35 C, evaluate the surface temperatures, Ts,1 and Ts,2, as well as the temperature To at the centerline of the cable. (e) With all other conditions remaining the same, compute and plot To, Ts,1, and Ts,2 as a function of r2 for 15.5  r2  20 mm

A spherical shell of inner and outer radii ri and ro, respectively, contains heat-dissipating components, and at a particular instant the temperature distribution in the shell is known to be of the form T(r)  C1 r  C2 Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius?

A chemically reacting mixture is stored in a thin-walled spherical container of radius r1  200 mm, and the exothermic reaction generates heat at a uniform, but temperaturedependent volumetric rate of  o exp(A/To), where o  5000 W/m3 , A  75 K, and To is the mixture temperature in kelvins. The vessel is enclosed by an insulating material of outer radius r2, thermal conductivity k, and emissivity . The outer surface of the insulation experiences convection heat transfer and net radiation exchange with the adjoining air and large surroundings, respectively. (a) Write the steady-state form of the heat diffusion equation for the insulation. Verify that this equation is satisfied by the temperature distribution Sketch the temperature distribution, T(r), labeling key features. (b) Applying Fouriers law, show that the rate of heat transfer by conduction through the insulation may be expressed as Applying an energy balance to a control surface about the container, obtain an alternative expression for qr, expressing your result in terms of and r1 q . (c) Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which Ts,2 may be determined as a function of , r1, h, T
, , and Tsur. (d) The process engineer wishes to maintain a reactor temperature of To  T(r1)  95 C under conditions for which k  0.05 W/m
K, r2  208 mm, h  5 W/m2
K,   0.9, T
 25 C, and Tsur  35 C. What is the actual reactor temperature and the outer surface temperature Ts,2 of the insulation? (e) Compute and plot the variation of Ts,2 with r2 for 201  r2  210 mm. The engineer is concerned about potential burn injuries to personnel who may come into contact with the exposed surface of the insulation. Is increasing the insulation thickness a practical solution to maintaining Ts,2  45 C? What other parameter could be varied to reduce Ts,2?

A thin electrical heater dissipating 4000 W/m2 is sandwiched between two 25-mm-thick plates whose exposed surfaces experience convection with a fluid for which and . The thermophysical properties of the plate material are  2500 kg/m3 , , and . (a) On T  x coordinates, sketch the steady-state temperature distribution for L  x  L. Calculate values of the temperatures at the surfaces, x  L, and the midpoint, x  0. Label this distribution as Case 1, and explain its salient features. (b) Consider conditions for which there is a loss of coolant and existence of a nearly adiabatic condition on the x  L surface. On the T  x coordinates used for part (a), sketch the corresponding steady-state temperature distribution and indicate the temperatures at x  0, L. Label the distribution as Case 2, and explain its key features. (c) With the system operating as described in part (b), the surface x  L also experiences a sudden loss of coolant. This dangerous situation goes undetected for 15 min, at which time the power to the heater is deactivated. Assuming no heat losses from the surfaces of the plates, what is the eventual (t l
), uniform, steady-state temperature distribution in the plates? Show this distribution as Case 3 on your sketch, and explain its key features. Hint: Apply the conservation of energy requirement on a time-interval basis, Eq. 1.12b, for the initial and final conditions corresponding to Case 2 and Case 3, respectively. (d) On T  t coordinates, sketch the temperature history at the plate locations x  0, L during the transient period between the distributions for Cases 2 and 3. Where and when will the temperature in the system achieve a maximum value?

The one-dimensional system of mass M with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature Ti . The electrical heater is suddenly energized, providing a uniform heat flux at the surface x  0. The boundaries at x  L and elsewhere are perfectly insulated. (a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the system. (b) On T  x coordinates, sketch the temperature distributions for the initial condition (t  0) and for several times after the heater is energized. Will a steady-state temperature distribution ever be reached? (c) On q x  t coordinates, sketch the heat flux q x (x, t) at the planes x  0, x  L/2, and x  L as a function of time. (d) After a period of time te has elapsed, the heater power is switched off. Assuming that the insulation is perfect, the system will eventually reach a final uniform temperature Tf . Derive an expression that can be used to determine Tf as a function of the parameters , te, Ti , and the system characteristics M, cp, and As (the heater surface area).

Consider a one-dimensional plane wall of thickness 2L. The surface at x  L is subjected to convective conditions characterized by T
,1, h1, while the surface at x  L is subjected to conditions T
,2, h2. The initial temperature of the wall is To  (T
,1  T
,2)/2 where T
,1
T
,2. (a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature distribution T(x, t) as a function of position and time. (b) On T  x coordinates, sketch the temperature distributions for the initial condition, the steady-state condition, and for two intermediate times for the case h1  h2. (c) On  t coordinates, sketch the heat flux at the planes x  0, L, and L. (d) The value of h1 is now doubled with all other conditions being identical as in parts (a) through (c). On T  x coordinates drawn to the same scale as used in part (b), sketch the temperature distributions for the initial condition, the steady-state condition, and for two intermediate times. Compare the sketch to that of part (b). (e) Using the doubled value of h1, sketch the heat flux at the planes x  0, L, and L on the same plot you prepared for part (c). Compare the two responses.

A large plate of thickness 2L is at a uniform temperature of Ti  200 C, when it is suddenly quenched by dipping it in a liquid bath of temperature T
 20 C. Heat transfer to the liquid is characterized by the convection coefficient h. (a) If x  0 corresponds to the midplane of the wall, on T  x coordinates, sketch the temperature distributions for the following conditions: initial condition (t  0), steady-state condition (t l
), and two intermediate times. (b) On  t coordinates, sketch the variation with time of the heat flux at x  L. (c) If h  100 W/m2
K, what is the heat flux at x  L and t  0? If the wall has a thermal conductivity of k  50 W/m
K what is the corresponding temperature gradient at x  L? (d) Consider a plate of thickness 2L  20 mm with a density of  2770 kg/m3 and a specific heat cp  875 J/kg
K. By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate (J/m2 ) that is transferred to the bath over the time required to reach steady-state conditions. (e) From other considerations, it is known that, during the quenching process, the heat flux at x  L and x  L decays exponentially with time according to the relation,  A exp(Bt), where t is in seconds, A  1.80 104 W/m2 , and B  4.126 103 s 1 . Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process.

The plane wall with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature Ti . Suddenly the surface at x  L is heated by a fluid at T
having a convection heat transfer coefficient h. The boundary at x  0 is perfectly insulated. (a) Write the differential equation, and identify the boundary and initial conditions that could be used to determine the temperature as a function of position and time in the wall. (b) On T  x coordinates, sketch the temperature distributions for the following conditions: initial condition (t  0), steady-state condition (t l
), and two intermediate times. (c) On q x  t coordinates, sketch the heat flux at the locations x  0, x  L. That is, show qualitatively how (0, t) and (L, t) vary with time. (d) Write an expression for the total energy transferred to the wall per unit volume of the wall (J/m3 ).

Consider the steady-state temperature distributions within a composite wall composed of Material A and Material B for the two cases shown. There is no internal generation, and the conduction process is onedimensional. Answer the following questions for each case. Which material has the higher thermal conductivity? Does the thermal conductivity vary significantly with temperature? If so, how? Describe the heat flux distribution through the composite wall. If the thickness and thermal conductivity of each material were both doubled and the boundary temperatures remained the same, what would be the effect on the heat flux distribution? Case 1. Linear temperature distributions exist in both materials, as shown. Case 2. Nonlinear temperature distributions exist in both materials, as shown.

A plane wall has constant properties, no internal heat generation, and is initially at a uniform temperature Ti . Suddenly, the surface at x  L is heated by a fluid at T
having a convection coefficient h. At the same instant, the electrical heater is energized, providing a constant heat flux at x  0. (a) On T  x coordinates, sketch the temperature distributions for the following conditions: initial condition (t  0), steady-state condition (t l
), and for two intermediate times(b) On coordinates, sketch the heat flux corresponding to the four temperature distributions of part (a). (c) On q x  t coordinates, sketch the heat flux at the locations x  0 and x  L. That is, show qualitatively how (0, t) and (L, t) vary with time. (d) Derive an expression for the steady-state temperature at the heater surface, T(0,
), in terms of , T
, k, h, and L.

A plane wall with constant properties is initially at a uniform temperature To. Suddenly, the surface at x  L is exposed to a convection process with a fluid at T
(
To) having a convection coefficient h. Also, suddenly the wall experiences a uniform internal volumetric heating that is sufficiently large to induce a maximum steadystate temperature within the wall, which exceeds that of the fluid. The boundary at x  0 remains at To (a) On T  x coordinates, sketch the temperature distributions for the following conditions: initial condition (t  0), steady-state condition (t l
), and for two intermediate times. Show also the distribution for the special condition when there is no heat flow at the x  L boundary. (b) On q x  t coordinates, sketch the heat flux for the locations x  0 and x  L, that is, and , respectively.

Consider the conditions associated with Problem 2.60, but now with a convection process for which T
To. (a) On T  x coordinates, sketch the temperature distributions for the following conditions: initial condition (t  0), steady-state condition (t l
), and for two intermediate times. Identify key features of the distributions, especially the location of the maximum temperature and the temperature gradient at x  L. (b) On q x  t coordinates, sketch the heat flux for the locations x  0 and x  L, that is, and , respectively. Identify key features of the flux histories.

Consider the steady-state temperature distribution within a composite wall composed of Materials A and B. The conduction process is one-dimensional. Within which material does uniform volumetric generation occur? What is the boundary condition at x  LA? How would the temperature distribution change if the thermal conductivity of Material A were doubled? How would the temperature distribution change if the thermal conductivity of Material B were doubled? Does a contact resistance exist at the interface between the two materials? Sketch the heat flux distribution through the composite wall

A spherical particle of radius r1 experiences uniform thermal generation at a rate of q . . The particle is encapsulated by a spherical shell of outside radius r2 that is cooled by ambient air. The thermal conductivities of the particle and shell are k1 and k2, respectively, where k1  2k2. (a) By applying the conservation of energy principle to spherical control volume A, which is placed at an arbitrary location within the sphere, determine a relationship between the temperature gradient dT/dr and the local radius r, for 0  r  r1. (b) By applying the conservation of energy principle to spherical control volume B, which is placed at an arbitrary location within the spherical shell, determine a relationship between the temperature gradient dT/dr and the local radius r, for r1  r  r2. (c) On T  r coordinates, sketch the temperature distribution over the range 0  r  r2

A long cylindrical rod, initially at a uniform temperature Ti , is suddenly immersed in a large container of liquid at T
Ti . Sketch the temperature distribution within the rod, T(r), at the initial time, at steady state, and at two intermediate times. On the same graph, carefully sketch the temperature distributions that would occur at the same times within a second rod that is the same size as the first rod. The densities and specific heats of the two rods are identical, but the thermal conductivity of the second rod is very large. Which rod will approach steady-state conditions sooner? Write the appropriate boundary conditions that would be applied at r  0 and r  D/2 for either rod

A plane wall of thickness L  0.1 m experiences uniform volumetric heating at a rate q . . One surface of the wall (x  0) is insulated, and the other surface is exposed to a fluid at T
 20 C, with convection heat transfer characterized by h  1000 W/m2
K. Initially, the temperature distribution in the wall is T(x, 0)  a  bx2 , where a  300 C, b  1.0 104 C/m2 , and x is in meters. Suddenly, the volumetric heat generation is deactivated (q .  0 for t 0), while convection heat transfer continues to occur at x  L. The properties of the wall are  7000 kg/m3 , cp  450 J/kg
K, and k  90 W/m
K. (a) Determine the magnitude of the volumetric energy generation rate q . associated with the initial condition (t 0). (b) On T  x coordinates, sketch the temperature distribution for the following conditions: initial condition (t 0), steady-state condition (t l
), and two intermediate conditions. (c) On  t coordinates, sketch the variation with time of the heat flux at the boundary exposed to the convection process, . Calculate the corresponding value of the heat flux at t  0, . (d) Calculate the amount of energy removed from the wall per unit area (J/m2 ) by the fluid stream as the wall cools from its initial to steady-state condition.

A plane wall that is insulated on one side (x  0) is initially at a uniform temperature Ti , when its exposed surface at x  L is suddenly raised to a temperature Ts. (a) Verify that the following equation satisfies the heat equation and boundary conditions: where C1 is a constant and  is the thermal diffusivity. (b) Obtain expressions for the heat flux at x  0 and x  L. (c) Sketch the temperature distribution T(x) at t  0, at t l
, and at an intermediate time. Sketch the variation with time of the heat flux at x  L, . (d) What effect does  have on the thermal response of the material to a change in surface temperature?

A composite one-dimensional plane wall is of overall thickness 2L. Material A spans the domain L  x 0 and experiences an exothermic chemical reaction leading to a uniform volumetric generation rate of q . A. Material B spans the domain 0  x  L and undergoes an endothermic chemical reaction corresponding to a uniform volumetric generation rate of q . B  q . A. The surfaces at x  L are insulated. Sketch the steady-state temperature and heat flux distributions T(x) and q x(x), respectively, over the domain L  x  L for kA  kB, kA  0.5kB, and kA  2kB. Point out the important features of the distributions you have drawn. If , can you sketch the steady-state temperature distribution?

Typically, air is heated in a hair dryer by blowing it across a coiled wire through which an electric current is passed. Thermal energy is generated by electric resistance heating within the wire and is transferred by convection from the surface of the wire to the air. Consider conditions for which the wire is initially at room temperature, Ti , and resistance heating is concurrently initiated with airflow at t  0. (a) For a wire radius ro, an air temperature T
, and a convection coefficient h, write the form of the heat equation and the boundary/initial conditions that govern the transient thermal response, T(r, t), of the wire. (b) If the length and radius of the wire are 500 mm and 1 mm, respectively, what is the volumetric rate of thermal energy generation for a power consumption of Pelec  500 W? What is the convection heat flux under steady-state conditions? (c) On T  r coordinates, sketch the temperature distributions for the following conditions: initial condition (t  0), steady-state condition (t l
), and for two intermediate times. (d) On q r  t coordinates, sketch the variation of the heat flux with time for locations at r  0 and r  ro

The steady-state temperature distribution in a composite plane wall of three different materials, each of constant thermal conductivity, is shown. (a) Comment on the relative magnitudes of and , and of and . (b) Comment on the relative magnitudes of kA and kB, and of kB and kC. (c) Sketch the heat flux as a function of x.