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Fundamentals of Heat and Mass Transfer

Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera

Chapter 2

Introduction to Conduction - all with Video Answers

Educators

Chapter Questions

01:29

Problem 1

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.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:29

Problem 2

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, $q_{x}^{\prime \prime}(x)$, and the temperature distribution, $T(x)$. Explain the shapes of your curves. How do your curves depend on the thermal conductivity of the material?

Manik Pulyani
Manik Pulyani
Numerade Educator
03:24

Problem 3

A hot water pipe with outside radius $r_{1}$ has a temperature $T_{1}$. A thick insulation, applied to reduce the heat loss, has an outer radius $r_{2}$ and temperature $T_{2}$. 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.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
00:39

Problem 4

A spherical shell with inner radius $r_{1}$ and outer radius $r_{2}$ has surface temperatures $T_{1}$ and $T_{2}$, respectively, where $T_{1}>T_{2}$. Sketch the temperature distribution on $T-r$ coordinates assuming steady-state, one-dimensional conduction with constant properties. Briefly justify the shape of your curve.

Nikhil Choudhary
Nikhil Choudhary
Numerade Educator
01:47

Problem 5

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: $A(x)=(1-x), \quad T(x)=300$ $\left(1-2 x-x^{3}\right)$, and $q=6000 \mathrm{~W}$, where $A$ is in square meters, $T$ in kelvins, and $x$ in meters.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:19

Problem 6

A composite rod consists of two different materials, $\mathrm{A}$ and $\mathrm{B}$, each of length $0.5 \mathrm{~L}$.
The thermal conductivity of Material $\mathrm{A}$ is half that of Material $\mathrm{B}$, that is, $k_{\mathrm{A}} / k_{\mathrm{B}}=0.5$. Sketch the steady-state temperature and heat flux distributions, $T(x)$ and $q_{x}^{\prime \prime}$, respectively. Assume constant properties and no internal heat generation in either material.

Narayan Hari
Narayan Hari
Numerade Educator
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Problem 7

A solid, truncated cone serves as a support for a system that maintains the top (truncated) face of the cone at a temperature $T_{1}$, while the base of the cone is at a temperature $T_{2}<T_{1}$.
The thermal conductivity of the solid depends on temperature according to the relation $k=k_{0}-a T$, 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 $q_{x}$, the heat flux $q_{x}^{\prime \prime}$, the thermal conductivity $k$, and the temperature gradient $d T / d x$ ?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
02:26

Problem 8

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
$$
k=k_{o}+a T
$$
where $k_{o}$ 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 $a<0$.

Narayan Hari
Narayan Hari
Numerade Educator
04:46

Problem 9

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 $k_{a}=0.005 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, for the protective barrier. The engineer's 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 $G=10 \times 10^{6} \mathrm{~W} / \mathrm{m}^{2}$. The absorptivities of the materials are $\alpha=0.2,0.9$, and $0.8$ for the aluminum, glass, and aerogel, respectively, and the initial temperature of the barrier is $T_{i}=300 \mathrm{~K}$. 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.

Andrew Duncan
Andrew Duncan
Numerade Educator
01:01

Problem 10

A one-dimensional plane wall of thickness $2 L=$ $100 \mathrm{~mm}$ experiences uniform thermal energy generation of $\dot{q}=1000 \mathrm{~W} / \mathrm{m}^{3}$ and is convectively cooled at $x=\pm 50 \mathrm{~mm}$ by an ambient fluid characterized by $T_{\infty}=20^{\circ} \mathrm{C}$. If the steady-state temperature distribution
within the wall is $T(x)=a\left(L^{2}-x^{2}\right)+b$ where $a=10^{\circ} \mathrm{C} / \mathrm{m}^{2}$ and $b=30^{\circ} \mathrm{C}$, what is the thermal conductivity of the wall? What is the value of the convection heat transfer coefficient, $h$ ?

Mayukh Banik
Mayukh Banik
Numerade Educator
01:01

Problem 11

Consider steady-state conditions for one-dimensional conduction in a plane wall having a thermal conductivity $k=50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and a thickness $L=0.25 \mathrm{~m}$, 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.
\begin{tabular}{crcc}
\hline Case & $T_{1}\left({ }^{\circ} \mathrm{C}\right)$ & $T_{2}\left({ }^{\circ} \mathrm{C}\right)$ & $d T / d x(\mathbf{K} / \mathbf{m})$ \\
\hline 1 & 50 & $-20$ & \\
2 & $-30$ & $-10$ & 160 \\
3 & 70 & & $-80$ \\
4 & & 40 & 200 \\
5 & & 30 & \\
\hline
\end{tabular}

Mayukh Banik
Mayukh Banik
Numerade Educator
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Problem 12

Consider a plane wall $100 \mathrm{~mm}$ thick and of thermal conductivity $100 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Steady-state conditions are known to exist with $T_{1}=400 \mathrm{~K}$ and $T_{2}=600 \mathrm{~K}$. Determine the heat flux $q_{x}^{\prime \prime}$ and the temperature gradient $d T / d x$ for the coordinate systems shown.

Ankur S
Ankur S
Numerade Educator
02:34

Problem 13

A cylinder of radius $r_{o}$, length $L$, and thermal conductivity $k$ is immersed in a fluid of convection coefficient $h$ and unknown temperature $T_{\infty}$. At a certain instant the temperature distribution in the cylinder is $T(r)=a+b r^{2}$, where $a$ and $b$ are constants. Obtain expressions for the heat transfer rate at $r_{o}$ and the fluid temperature.

Mahendra K
Mahendra K
Numerade Educator
08:01

Problem 14

In the two-dimensional body illustrated, the gradient at surface $A$ is found to be $\partial T / \partial y=30 \mathrm{~K} / \mathrm{m}$. What are $\partial T / \partial y$ and $\partial T / \partial x$ at surface $B ?$

Mahnoor Amin
Mahnoor Amin
Numerade Educator
01:29

Problem 15

Consider the geometry of Problem $2.14$ for the case where the thermal conductivity varies with temperature as $k=k_{o}+a T$, where $k_{o}=10 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad a=-10^{-3}$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}^{2}$, and $T$ is in kelvins. The gradient at surface $\mathrm{B}$ is $\partial T / \partial x=30 \mathrm{~K} / \mathrm{m}$. What is $\partial T / \partial y$ at surface $\mathrm{A} ?$

James Kiss
James Kiss
Numerade Educator
01:47

Problem 16

Steady-state, one-dimensional conduction occurs in a rod of constant thermal conductivity $k$ and variable crosssectional area $A_{x}(x)=A_{o} e^{a x}$, where $A_{o}$ and $a$ are constants. The lateral surface of the rod is well insulated.
(a) Write an expression for the conduction heat rate, $q_{x}(x)$. 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 $\dot{q}=\dot{q}_{s} \exp (-a x)$, where $\dot{q}_{o}$ is a constant. Obtain an expression for $q_{x}(x)$ when the left face $(x=0)$ is well insulated.

Ajay Singhal
Ajay Singhal
Numerade Educator
05:24

Problem 17

An apparatus for measuring thermal conductivity employs an electrical heater sandwiched between two identical samples of diameter $30 \mathrm{~mm}$ and length $60 \mathrm{~mm}$, which are pressed between plates maintained at a uniform temperature $T_{o}=77^{\circ} \mathrm{C}$ 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 $15 \mathrm{~mm}$. The lateral sides of the samples are insulated to ensure onedimensional heat transfer through the samples.
(a) With two samples of SS 316 in the apparatus, the heater draws $0.353 \mathrm{~A}$ at $100 \mathrm{~V}$, and the differential thermocouples indicate $\Delta T_{1}=\Delta T_{2}=25.0^{\circ} \mathrm{C}$. 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 $0.601 \mathrm{~A}$ at $100 \mathrm{~V}$, and the differential thermocouples indicate $\Delta T_{1}=\Delta T_{2}=$ $15.0^{\circ} \mathrm{C}$. 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 heater-sample combination? When would heat leakage out of the lateral surfaces of the samples become significant? Under what conditions would you expect $\Delta T_{1} \neq \Delta T_{2} ?$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:17

Problem 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 aerogel's thermal conductivity.
(b) The engineer designs a new apparatus for which an electric heater of diameter $D=150 \mathrm{~mm}$ is sandwiched between two thin plates of aluminum. The steady-state temperatures of the 5 -mm-thick aluminum plates, $T_{1}$ and $T_{2}$, are measured with thermocouples. Aerogel sheets of thickness $t=5 \mathrm{~mm}$ are placed outside the aluminum plates, while a coolant with an inlet temperature of $T_{c, i}=25^{\circ} \mathrm{C}$ 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, $T_{1}=T_{2}=55^{\circ} \mathrm{C}$, and the heater draws $125 \mathrm{~mA}$ at $10 \mathrm{~V}$. Determine the value of the aerogel thermal conductivity $k_{a+}$
(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 $\dot{m}=1 \mathrm{~kg} / \mathrm{min}(0.5 \mathrm{~kg} / \mathrm{min}$ for each of the two streams), calculate the outlet temperature of the water, $T_{c, \sigma^{*}}$

Carlos Henrique De Lima
Carlos Henrique De Lima
Numerade Educator
03:46

Problem 19

Consider a $300 \mathrm{~mm} \times 300 \mathrm{~mm}$ window in an aircraft. For a temperature difference of $80^{\circ} \mathrm{C}$ 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 $k_{\mathrm{ag}}=0.014 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $k_{\mathrm{pc}}=0.21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, respectively. Evaluate the thermal conductivity of the soda lime glass at $300 \mathrm{~K}$. If the aircraft has 130 windows and the cost to heat the cabin air is $\$ 1 / \mathrm{kW} \cdot \mathrm{h}$, compare the costs associated with the heat loss through the windows for an 8-hour intercontinental flight.

Manish Jain
Manish Jain
Numerade Educator
03:34

Problem 20

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 thermal responses. 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.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 21

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 $300 \leq T \leq 600 \mathrm{~K}$. 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 $300 \leq T \leq 800 \mathrm{~K}$. 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 $300 \leq T \leq 360 \mathrm{~K}$. Include all data on a single graph, and comment on the trends you observe.
(d) Graph the thermal conductivity of a water- $\mathrm{Al}_{2} \mathrm{O}_{3}$ nanofluid at $T=300 \mathrm{~K}$ over the volume fraction range $0 \leq \varphi \leq 0.08$. See Example 2.2.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:28

Problem 22

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

Penny Riley
Penny Riley
Numerade Educator
09:01

Problem 23

A method for determining the thermal conductivity $k$ and the specific heat $c_{p}$ of a material is illustrated in the sketch. Initially the two identical samples of diameter $D=60 \mathrm{~mm}$ and thickness $L=10 \mathrm{~mm}$ and the thin heater are at a uniform temperature of $T_{i}=23.00^{\circ} \mathrm{C}$, while surrounded by an insulating powder. Suddenly the heater is energized to provide a uniform heat flux $q_{o}^{\prime \prime}$ on each of the sample interfaces, and the heat flux is maintained constant for a period of time, $\Delta t_{o}$. A short time after sudden heating is initiated, the temperature at this interface $T_{o}$ is related to the heat flux as
$$
T_{o}(t)-T_{i}=2 q_{o}^{\prime \prime}\left(\frac{t}{\pi \rho c_{p} k}\right)^{1 / 2}
$$
For a particular test run, the electrical heater dissipates $15.0 \mathrm{~W}$ for a period of $\Delta t_{o}=120 \mathrm{~s}$, and the temperature at the interface is $T_{o}(30 \mathrm{~s})=24.57^{\circ} \mathrm{C}$ after $30 \mathrm{~s}$ of heating. A long time after the heater is deenergized, $t \geqslant \Delta t_{0}$, the samples reach the uniform temperature of $T_{o}(\infty)=33.50^{\circ} \mathrm{C}$. The density of the sample materials, determined by measurement of volume and mass, is $\rho=3965 \mathrm{~kg} / \mathrm{m}^{3}$.
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.

Averell Hause
Averell Hause
Carnegie Mellon University
01:10

Problem 24

Compare and contrast the heat capacity $\rho c_{p}$ 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 $300 \mathrm{~K}$.

Nicole Bylsma
Nicole Bylsma
Numerade Educator
02:15

Problem 25

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

Narayan Hari
Narayan Hari
Numerade Educator
02:10

Problem 26

At a given instant of time, the temperature distribution within an infinite homogeneous body is given by the function
$$
T(x, y, z)=x^{2}-2 y^{2}+z^{2}-x y+2 y z
$$
Assuming constant properties and no internal heat generation, determine the regions where the temperature changes with time.

Lucas Finney
Lucas Finney
Numerade Educator
09:53

Problem 27

A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate $q_{\sigma}$. There are two stages to the process. In Stage 1, the water is taken from its initial (room) temperature $T_{i}$ to the boiling point, as heat is transferred from the pan by natural convection. During this stage, a constant value of the convection coefficient $h$ may be assumed, while the bulk temperature of the water increases with time, $T_{\infty}=T_{\infty}(t)$. In Stage 2, the water has come to a boil, and its temperature remains at a fixed value, $T_{\infty}=T_{b}$, 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 $q_{o}, D, L, h$, and $T_{\infty}$, 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, $T(L, t)=$ $T_{L}>T_{b}$. 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 $q_{o}, D, L$, and $T_{L}$, as well as appropriate properties of the pan material.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:15

Problem 28

Uniform internal heat generation at $\dot{q}=5 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}$ 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 $T(r)=a+b r^{2}$, where $T$ is in degrees Celsius and $r$ is in meters, while $a=800^{\circ} \mathrm{C}$ and $b=-4.167 \times 10^{5}{ }^{\circ} \mathrm{C} / \mathrm{m}^{2}$. The fuel rod properties are $k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=1100 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.
(a) What is the rate of heat transfer per unit length of the rod at $r=0$ (the centerline) and at $r=25 \mathrm{~mm}$ (the surface)?
(b) If the reactor power level is suddenly increased to $\dot{q}_{2}=10^{8} \mathrm{~W} / \mathrm{m}^{3}$, what is the initial time rate of temperature change at $r=0$ and $r=25 \mathrm{~mm}$ ?

Narayan Hari
Narayan Hari
Numerade Educator
02:20

Problem 29

Consider a one-dimensional plane wall with constant properties and uniform internal generation $\dot{q}$. 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 $q^{\prime \prime}(x)$.
(b) Using a finite volume spanning the range $0 \leq$ $x \leq \xi$, derive an expression for $q^{\prime \prime}(\xi)$ and compare the expression to your result for part (a).

Anand Jangid
Anand Jangid
Numerade Educator
02:20

Problem 30

The steady-state temperature distribution in a onedimensional wall of thermal conductivity $50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and thickness $50 \mathrm{~mm}$ is observed to be $T\left({ }^{\circ} \mathrm{C}\right)=a+b x^{2}$, where $a=200^{\circ} \mathrm{C}, b=-2000^{\circ} \mathrm{C} / \mathrm{m}^{2}$, and $x$ is in meters.
(a) What is the heat generation rate $\dot{q}$ 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?

Anand Jangid
Anand Jangid
Numerade Educator
01:01

Problem 31

The temperature distribution across a wall $0.3 \mathrm{~m}$ thick at a certain instant of time is $T(x)=a+b x+c x^{2}$, where $T$ is in degrees Celsius and $x$ is in meters, $a=200^{\circ} \mathrm{C}$, $b=-200^{\circ} \mathrm{C} / \mathrm{m}$, and $c=30^{\circ} \mathrm{C} / \mathrm{m}^{2}$. The wall has a thermal conductivity of $1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.
(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 $100^{\circ} \mathrm{C}$, what is the convection coefficient?

Mayukh Banik
Mayukh Banik
Numerade Educator
03:37

Problem 32

A plane wall of thickness $2 L=40 \mathrm{~mm}$ and thermal conductivity $k=5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ experiences uniform volumetric heat generation at a rate $\dot{q}$, while convection heat transfer occurs at both of its surfaces $(x=-L,+L)$, each of which is exposed to a fluid of temperature $T_{\infty}=20^{\circ} \mathrm{C}$. Under steady-state conditions, the temperature distribution in the wall is of the form $T(x)=a+b x+c x^{2}$ where $a=82.0^{\circ} \mathrm{C}, b=-210^{\circ} \mathrm{C} / \mathrm{m}, c=-2 \times 10^{4 \circ} \mathrm{C} / \mathrm{m}^{2}$, 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 $\dot{q}$ in the wall?
(c) Determine the surface heat fluxes, $q_{x}^{\prime \prime}(-L)$ and $q_{x}^{\prime \prime}(+L)$. 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 $q_{x}^{\prime \prime}(x)$. 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 $(\dot{q}=0)$, what is the rate of change of energy stored in the wall at this instant?
(g) What temperature will the wall eventually reach with $\dot{q}=0$ ? How much energy must be removed by the fluid per unit area of the wall $\left(\mathrm{J} / \mathrm{m}^{2}\right)$ to reach this state? The density and specific heat of the wall material are $2600 \mathrm{~kg} / \mathrm{m}^{3}$ and $800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, respectively.

Manish Jain
Manish Jain
Numerade Educator
02:06

Problem 33

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.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
03:37

Problem 34

One-dimensional, steady-state conduction with uniform internal energy generation occurs in a plane wall with a thickness of $50 \mathrm{~mm}$ and a constant thermal conductivity of $5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. For these conditions, the temperature distribution has the form $T(x)=a+b x+c x^{2}$. The surface at $x=0$ has a temperature of $T(0) \equiv T_{o}=120^{\circ} \mathrm{C}$ and experiences convection with a fluid for which $T_{\infty}=20^{\circ} \mathrm{C}$ and $h=500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The surface at $x=L$ is well insulated.
(a) Applying an overall energy balance to the wall, calculate the volumetric energy generation rate $\dot{q}$.
(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 coefficient 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 $120^{\circ} \mathrm{C}$.
(d) Under conditions for which the volumetric energy generation rate is doubled, and the convection coefficient remains unchanged $\left(h=500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)$, 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 $\dot{q}$ on the distributions.

Manish Jain
Manish Jain
Numerade Educator
05:56

Problem 35

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

Dading Chen
Dading Chen
Numerade Educator
01:38

Problem 36

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

Penny Riley
Penny Riley
Numerade Educator
01:47

Problem 37

The steady-state temperature distribution in a semitransparent material of thermal conductivity $k$ and thickness $L$ exposed to laser irradiation is of the form
$$
T(x)=-\frac{A}{k a^{2}} e^{-a x}+B x+C
$$
(a) Obtain expressions for the conduction heat fluxes at the front and rear surfaces.
(b) Derive an expression for $\dot{q}(x)$.
(c) Derive an expression for the rate at which radiation is absorbed in the entire material, per unit surface
area. Express your result in terms of the known constants for the temperature distribution, the thermal conductivity of the material, and its thickness.
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, $\dot{q}(x)$.

Ajay Singhal
Ajay Singhal
Numerade Educator
01:01

Problem 38

One-dimensional, steady-state conduction with no energy generation is occurring in a cylindrical shell of inner radius $r_{1}$ and outer radius $r_{2}$. Under what condition is the linear temperature distribution shown possible?

Raj Bala
Raj Bala
Numerade Educator
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Problem 39

One-dimensional, steady-state conduction with no energy generation is occurring in a spherical shell of inner radius $r_{1}$ and outer radius $r_{2}$. Under what condition is the linear temperature distribution shown in Problem $2.38$ possible?

Victor Salazar
Victor Salazar
Numerade Educator
01:01

Problem 40

The steady-state temperature distribution in a onedimensional wall of thermal conductivity $k$ and thickness $L$ is of the form $T=a x^{3}+b x^{2}+c x+d .$ Derive expressions for the heat generation rate per unit volume in the wall and the heat fluxes at the two wall faces $(x=0, L)$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:47

Problem 41

One-dimensional, steady-state conduction with no energy generation is occurring in a plane wall of constant thermal conductivity.
(a) Is the prescribed temperature distribution possible? Briefly explain your reasoning.
(b) With the temperature at $x=0$ and the fluid temperature fixed at $T(0)=0^{\circ} \mathrm{C}$ and $T_{\infty}=20^{\circ} \mathrm{C}$, respectively, compute and plot the temperature at $x=L$, $T(L)$, as a function of $h$ for $10 \leq h \leq 100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Briefly explain your results.

Ajay Singhal
Ajay Singhal
Numerade Educator
04:58

Problem 42

A plane layer of coal of thickness $L=1 \mathrm{~m}$ experiences uniform volumetric generation at a rate of $\dot{q}=20 \mathrm{~W} / \mathrm{m}^{3}$ 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 $h=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $T_{\infty}=25^{\circ} \mathrm{C}$, while receiving solar irradiation in the amount $G_{S}=400 \mathrm{~W} / \mathrm{m}^{2}$. Irradiation from the atmosphere may be neglected. The solar absorptivity and emissivity of the surface are each $\alpha_{S}=\varepsilon=0.95 .$
(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
$$
T(x)=T_{s}+\frac{\dot{q} L^{2}}{2 k}\left(1-\frac{x^{2}}{L^{2}}\right)
$$
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 $T_{s}$. Evaluate $T_{s}$ and $T(0)$ for the prescribed conditions.
(c) Daily average values of $G_{S}$ and $h$ depend on a number of factors, such as time of year, cloud cover, and wind conditions. For $h=5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, compute and plot $T_{S}$ and $T(0)$ as a function of $G_{S}$ for $50 \leq$ $G_{S} \leq 500 \mathrm{~W} / \mathrm{m}^{2}$. For $G_{S}=400 \mathrm{~W} / \mathrm{m}^{2}$, compute and plot $T_{S}$ and $T(0)$ as a function of $h$ for $5 \leq h \leq$ $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Jincy M Saji
Jincy M Saji
Numerade Educator
03:24

Problem 43

The cylindrical system illustrated has negligible variation of temperature in the $r$ - and $z$-directions. Assume that $\Delta r=r_{o}-r_{i}$ is small compared to $r_{i}$, 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 $\phi$. Compare your result with Equation 2.26.
(b) For steady-state conditions with no internal heat generation and constant properties, determine the temperature distribution $T(\phi)$ in terms of the constants $T_{1}, T_{2}, r_{i}$, and $r_{\sigma}$. Is this distribution linear in $\phi$ ?
(c) For the conditions of part (b) write the expression for the heat rate $q_{\phi}$.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
05:56

Problem 44

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.

Dading Chen
Dading Chen
Numerade Educator
09:52

Problem 45

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.

Mohammad Mehran
Mohammad Mehran
Numerade Educator
03:24

Problem 46

A steam pipe is wrapped with insulation of inner and outer radii $r_{i}$ and $r_{o}$, respectively. At a particular instant the temperature distribution in the insulation is known to be of the form
$$
T(r)=C_{1} \ln \left(\frac{r}{r_{o}}\right)+C_{2}
$$
Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius?

Morgan Cheatham
Morgan Cheatham
Numerade Educator
06:38

Problem 46

For a long circular tube of inner and outer radii $r_{1}$ and $r_{2}$, respectively, uniform temperatures $T_{1}$ and $T_{2}$ are maintained at the inner and outer surfaces, while thermal energy generation is occurring within the tube wall $\left(r_{1}<r<r_{2}\right)$. Consider steady-state conditions for which $T_{1}<T_{2}$. Is it possible to maintain a linear radial temperature distribution in the wall? If so, what special conditions must exist?

Surendra Kumar
Surendra Kumar
Numerade Educator
01:19

Problem 48

Passage of an electric current through a long conducting rod of radius $r_{i}$ and thermal conductivity $k_{r}$ results in uniform volumetric heating at a rate of $\dot{q}$. The conducting rod is wrapped in an electrically nonconducting cladding material of outer radius $r_{o}$ and thermal conductivity $k_{c}$, 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.

Narayan Hari
Narayan Hari
Numerade Educator
09:01

Problem 49

Two-dimensional, steady-state conduction occurs in a hollow cylindrical solid of thermal conductivity $k=16 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, outer radius $r_{o}=1 \mathrm{~m}$ and overall length $2 z_{o}=5 \mathrm{~m}$, 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 $T(r, z)=a+b r^{2}+c \ln r+d z^{2}$, where $a=$ $-20^{\circ} \mathrm{C}, \quad b=150^{\circ} \mathrm{C} / \mathrm{m}^{2}, c=-12^{\circ} \mathrm{C}, d=-300^{\circ} \mathrm{C} / \mathrm{m}^{2}$ and $r$ and $z$ are in meters.
(a) Determine the inner radius $r_{i}$ of the cylinder.
(b) Obtain an expression for the volumetric rate of heat generation, $\dot{q}\left(\mathrm{~W} / \mathrm{m}^{3}\right)$.
(c) Determine the axial distribution of the heat flux at the outer surface, $q_{r}^{\prime \prime}\left(r_{o}, z\right)$. 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, $q_{r}^{\prime \prime}\left(r,+z_{o}\right)$ and $q_{r}^{\prime \prime}\left(r,-z_{o}\right)$. 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.

Averell Hause
Averell Hause
Carnegie Mellon University
00:56

Problem 50

An electric cable of radius $r_{1}$ and thermal conductivity $k_{c}$ is enclosed by an insulating sleeve whose outer surface is of radius $r_{2}$ 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 $\dot{q}$.
(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: $T(r)=T_{s, 2}+\left(T_{s, 1}-T_{s, 2}\right) \frac{\ln \left(r / r_{2}\right)}{\ln \left(r_{1} / r_{2}\right)}$
Cable: $T(r)=T_{s, 1}+\frac{\dot{q} r_{1}^{2}}{4 k_{c}}\left(1-\frac{r^{2}}{r_{1}^{2}}\right)$
Sketch the temperature distribution, $T(r)$, in the cable and the sleeve, labeling key features.
(b) Applying Fourier's law, show that the rate of conduction heat transfer per unit length through the sleeve may be expressed as
$$
q_{r}^{\prime}=\frac{2 \pi k_{s}\left(T_{s, 1}-T_{s, 2}\right)}{\ln \left(r_{2} / r_{1}\right)}
$$
Applying an energy balance to a control surface placed around the cable, obtain an alternative expression for $q_{r}^{\prime}$, expressing your result in terms of $\dot{q}$ and $r_{1^{*}}$
(c) Applying an energy balance to a control surface placed around the outer surface of the sleeve, obtain an expression from which $T_{s, 2}$ may be determined as a function of $\dot{q}, r_{1}, h, T_{\infty}, \varepsilon$, and $T_{\text {sur- }}$
(d) Consider conditions for which $250 \mathrm{~A}$ are passing through a cable having an electric resistance per unit length of $R_{e}^{\prime}=0.005 \Omega / \mathrm{m}$, a radius of $r_{1}=15 \mathrm{~mm}$, and a thermal conductivity of $k_{c}=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.
For $k_{s}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad r_{2}=15.5 \mathrm{~mm}, \quad h=25$ $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}, \varepsilon=0.9, T_{\text {o }}=25^{\circ} \mathrm{C}$, and $T_{\text {sur }}=35^{\circ} \mathrm{C}$, evaluate the surface temperatures, $T_{s, 1}$ and $T_{s, 2}$, as well as the temperature $T_{o}$ at the centerline of the cable.
(e) With all other conditions remaining the same, compute and plot $T_{o}, T_{s, 1}$, and $T_{s, 2}$ as a function of $r_{2}$ for $15.5 \leq r_{2} \leq 20 \mathrm{~mm}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:12

Problem 51

A spherical shell of inner and outer radii $r_{i}$ and $r_{o}$, respectively, contains heat-dissipating components, and at a particular instant the temperature distribution in the shell is known to be of the form
$$
T(r)=\frac{C_{1}}{r}+C_{2}
$$
Are conditions steady-state or transient? How do the heat flux and heat rate vary with radius?

Dominador Tan
Dominador Tan
Numerade Educator
09:01

Problem 52

A chemically reacting mixture is stored in a thin-walled spherical container of radius $r_{1}=200 \mathrm{~mm}$, and the exothermic reaction generates heat at a uniform, but temperaturedependent volumetric rate of $\dot{q}=\dot{q}_{o} \exp \left(-A / T_{o}\right)$, where $\dot{q}_{o}=5000 \mathrm{~W} / \mathrm{m}^{3}, A=75 \mathrm{~K}$, and $T_{o}$ is the mixture temperature in kelvins. The vessel is enclosed by an insulating material of outer radius $r_{2}$, thermal conductivity $k$, and emissivity $\varepsilon$. 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
$$
T(r)=T_{s, 1}-\left(T_{s, 1}-T_{s, 2}\right)\left[\frac{1-\left(r_{1} / r\right)}{1-\left(r_{1} / r_{2}\right)}\right]
$$
Sketch the temperature distribution, $T(r)$, labeling key features.
(b) Applying Fourier's law, show that the rate of heat transfer by conduction through the insulation may be expressed as
$$
q_{r}=\frac{4 \pi k\left(T_{s, 1}-T_{s, 2}\right)}{\left(1 / r_{1}\right)-\left(1 / r_{2}\right)}
$$
Applying an energy balance to a control surface about the container, obtain an alternative expression for $q_{r}$, expressing your result in terms of $\dot{q}$ and $r_{1}$.
(c) Applying an energy balance to a control surface placed around the outer surface of the insulation, obtain an expression from which $T_{s, 2}$ may be determined as a function of $\dot{q}, r_{1}, h, T_{\infty}, \varepsilon$, and $T_{\text {sur }}$
(d) The process engineer wishes to maintain a reactor temperature of $T_{o}=T\left(r_{1}\right)=95^{\circ} \mathrm{C}$ under conditions for which $k=0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, r_{2}=208 \mathrm{~mm}, h=5$ $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}, \varepsilon=0.9, T_{\infty}=25^{\circ} \mathrm{C}$, and $T_{\text {sur }}=35^{\circ} \mathrm{C}$. What is the actual reactor temperature and the outer surface temperature $T_{s, 2}$ of the insulation?
(e) Compute and plot the variation of $T_{s, 2}$ with $r_{2}$ for $201 \leq r_{2} \leq 210 \mathrm{~mm}$. 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 $T_{s, 2} \leq 45^{\circ} \mathrm{C}$ ? What other parameter could be varied to reduce $T_{s, 2}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
01:00

Problem 53

A thin electrical heater dissipating $4000 \mathrm{~W} / \mathrm{m}^{2}$ is sandwiched between two 25 -mm-thick plates whose exposed surfaces experience convection with a fluid for which $T_{\infty}=20^{\circ} \mathrm{C}$ and $h=400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The thermophysical properties of the plate material are $\rho=2500$ $\mathrm{kg} / \mathrm{m}^{3}, c=700 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.
(a) On $T-x$ coordinates, sketch the steady-state temperature distribution for $-L \leq x \leq+L$. Calculate values of the temperatures at the surfaces, $x=\pm L$, 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 $x=0$, $\pm L$. 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 $15 \mathrm{~min}$, at which time the power to the heater is deactivated. Assuming no heat losses from the surfaces of the plates, what is the eventual $(t \rightarrow \infty)$, 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. $1.12 \mathrm{~b}$, 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 $x=0$, $\pm$ 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?

Raj Bala
Raj Bala
Numerade Educator
01:00

Problem 54

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 $T_{i^{*}}$. The electrical heater is suddenly energized, providing a uniform heat flux $q_{o}^{\prime \prime}$ 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 $(t \leq 0)$ and for several times after the heater is energized. Will a steady-state temperature distribution ever be reached?
(c) On $q_{x}^{\prime \prime}-t$ coordinates, sketch the heat flux $q_{x}^{\prime \prime}(x, t)$ at the planes $x=0, x=L / 2$, and $x=L$ as a function of time.
(d) After a period of time $t_{e}$ has elapsed, the heater power is switched off. Assuming that the insulation is perfect, the system will eventually reach a final uniform temperature $T_{f}$ Derive an expression that can be used to determine $T_{f}$ as a function of the parameters $q_{o}^{\prime \prime}, t_{e}, T_{i}$, and the system characteristics $M, c_{p}$, and $A_{s}$ (the heater surface area).

Raj Bala
Raj Bala
Numerade Educator
01:00

Problem 55

Consider a one-dimensional plane wall of thickness $2 L$. The surface at $x=-L$ is subjected to convective conditions characterized by $T_{\infty, 1}, h_{1}$, while the surface at $x=+L$ is subjected to conditions $T_{\infty, 2}, h_{2}$. The initial temperature of the wall is $T_{0}=\left(T_{\infty, 1}+T_{\infty, 2}\right) / 2$ where $T_{\infty, 1}>T_{\infty, 2}$
(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 $h_{1}=h_{2}$.
(c) On $q_{x}^{\prime \prime}-t$ coordinates, sketch the heat flux $q_{x}^{\prime \prime}(x, t)$ at the planes $x=0,-L$, and $+L$.
(d) The value of $h_{1}$ 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 $h_{1}$, sketch the heat flux $q_{x}^{\prime \prime}(x, t)$ at the planes $x=0,-L$, and $+L$ on the same plot you prepared for part (c). Compare the two responses.

Raj Bala
Raj Bala
Numerade Educator
04:34

Problem 56

A large plate of thickness $2 L$ is at a uniform temperature of $T_{i}=200^{\circ} \mathrm{C}$, when it is suddenly quenched by dipping it in a liquid bath of temperature $T_{\infty}=20^{\circ} \mathrm{C}$. 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 $(t \leq 0)$, steady-state condition $(t \rightarrow \infty)$, and two intermediate times.
(b) On $q_{x}^{\prime \prime}-t$ coordinates, sketch the variation with time of the heat flux at $x=L$.
(c) If $h=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, what is the heat flux at $x=L$ and $t=0$ ? If the wall has a thermal conductivity of $k=50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ what is the corresponding temperature gradient at $x=L$ ?
(d) Consider a plate of thickness $2 L=20 \mathrm{~mm}$ with a density of $\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}$ and a specific heat $c_{p}=875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. By performing an energy balance on the plate, determine the amount of energy per unit surface area of the plate $\left(\mathrm{J} / \mathrm{m}^{2}\right)$ 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, $q_{x}^{\prime \prime}=A \exp (-B t)$, where $t$ is in seconds, $A=1.80 \times 10^{4} \mathrm{~W} / \mathrm{m}^{2}$, and $B=4.126 \times 10^{-3} \mathrm{~s}^{-1}$. Use this information to determine the energy per unit surface area of the plate that is transferred to the fluid during the quenching process.

Manne Andergronde
Manne Andergronde
Numerade Educator
09:42

Problem 57

The plane wall with constant properties and no internal heat generation shown in the figure is initially at a uniform temperature $T_{i}$. Suddenly the surface at $x=L$ is heated by a fluid at $T_{\infty}$ 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 $(t \leq 0)$, steady-state condition $(t \rightarrow \infty)$, and two intermediate times.
(c) On $q_{x}^{\prime \prime}-t$ coordinates, sketch the heat flux at the locations $x=0, x=L$. That is, show qualitatively how $q_{x}^{\prime \prime}(0, t)$ and $q_{x}^{\prime \prime}(L, t)$ vary with time.
(d) Write an expression for the total energy transferred to the wall per unit volume of the wall $\left(\mathrm{J} / \mathrm{m}^{3}\right)$.

Prachita Kush
Prachita Kush
Numerade Educator
08:47

Problem 58

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 $q_{x}^{\prime \prime}(x)$ 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.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
01:00

Problem 59

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

Raj Bala
Raj Bala
Numerade Educator
02:49

Problem 60

A plane wall with constant properties is initially at a uniform temperature $T_{v}$. Suddenly, the surface at $x=L$ is exposed to a convection process with a fluid at $T_{\infty}\left(>T_{a}\right)$ having a convection coefficient $h$. Also, suddenly the wall experiences a uniform internal volumetric heating $\dot{q}$ 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 $T_{a}$.
(a) On $T-x$ coordinates, sketch the temperature distributions for the following conditions: initial condition $(t \leq 0)$, steady-state condition $(t \rightarrow \infty)$, 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 $q_{x}^{\prime \prime}-t$ coordinates, sketch the heat flux for the locations $x=0$ and $x=L$, that is, $q_{x}^{\prime \prime}(0, t)$ and $q_{x}^{\prime \prime}(L, t)$, respectively.

Chai Santi
Chai Santi
Numerade Educator
04:06

Problem 61

Consider the conditions associated with Problem 2.60, but now with a convection process for which $T_{\infty}<T_{o}$.
(a) On $T-x$ coordinates, sketch the temperature distributions for the following conditions: initial condition $(t \leq 0)$, steady-state condition $(t \rightarrow \infty)$, 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 $q_{x}^{\prime \prime}-t$ coordinates, sketch the heat flux for the locations $x=0$ and $x=L$, that is, $q_{x}^{\prime \prime}(0, t)$ and $q_{\mathrm{x}}^{\prime \prime}(L, t)$, respectively. Identify key features of the flux histories.

Joseph Liao
Joseph Liao
Numerade Educator
02:12

Problem 62

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 $x=-L_{\mathrm{A}}$ ? 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 $q_{x}^{\prime \prime}(x)$ through the composite wall.

Mahendra K
Mahendra K
Numerade Educator
03:12

Problem 63

A spherical particle of radius $r_{1}$ experiences uniform thermal generation at a rate of $\dot{q}$. The particle is encapsulated by a spherical shell of outside radius $r_{2}$ that is cooled by ambient air. The thermal conductivities of the particle and shell are $k_{1}$ and $k_{2}$, respectively, where $k_{1}=2 k_{2}$.
(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 $d T / d r$ and the local radius $r$, for $0 \leq r \leq r_{1}$.
(b) By applying the conservation of energy principle to spherical control volume $\mathrm{B}$, which is placed at an arbitrary location within the spherical shell, determine a relationship between the temperature gradient $d T / d r$ and the local radius $r$, for $r_{1} \leq r \leq r_{2}$.
(c) On $T-r$ coordinates, sketch the temperature distribution over the range $0 \leq r \leq r_{2}$.

Dominador Tan
Dominador Tan
Numerade Educator
02:15

Problem 64

A long cylindrical rod, initially at a uniform temperature $T_{i}$, is suddenly immersed in a large container of liquid at $T_{\infty}<T_{i}$. 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.

Narayan Hari
Narayan Hari
Numerade Educator
01:00

Problem 65

A plane wall of thickness $L=0.1 \mathrm{~m}$ experiences uniform volumetric heating at a rate $\dot{q}$. One surface of the wall $(x=0)$ is insulated, and the other surface is exposed to $\mathrm{a}$ fluid at $T_{\infty}=20^{\circ} \mathrm{C}$, with convection heat transfer characterized by $h=1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Initially, the temperature distribution in the wall is $T(x, 0)=a+b x^{2}$, where $a=300^{\circ} \mathrm{C}, b=-1.0 \times 10^{40} \mathrm{C} / \mathrm{m}^{2}$, and $x$ is in meters. Suddenly, the volumetric heat generation is deactivated ( $\dot{q}=0$ for $t \geq 0$ ), while convection heat transfer continues to occur at $x=L$. The properties of the wall are $\rho=7000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=450 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=90 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.
(a) Determine the magnitude of the volumetric energy generation rate $\dot{q}$ 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 $(t \rightarrow \infty)$, and two intermediate conditions.
(c) On $q_{x}^{\prime \prime}-t$ coordinates, sketch the variation with time of the heat flux at the boundary exposed to the convection process, $q_{x}^{\prime \prime}(L, t)$. Calculate the corresponding value of the heat flux at $t=0, q_{x}^{\prime \prime}(L, 0)$.
(d) Calculate the amount of energy removed from the wall per unit area $\left(\mathrm{J} / \mathrm{m}^{2}\right)$ by the fluid stream as the wall cools from its initial to steady-state condition.

Raj Bala
Raj Bala
Numerade Educator
04:06

Problem 66

A plane wall that is insulated on one side $(x=0)$ is initially at a uniform temperature $T_{i}$, when its exposed surface at $x=L$ is suddenly raised to a temperature $T_{s}$.
(a) Verify that the following equation satisfies the heat equation and boundary conditions:
$$
\frac{T(x, t)-T_{s}}{T_{i}-T_{s}}=C_{1} \exp \left(-\frac{\pi^{2}}{4} \frac{\alpha t}{L^{2}}\right) \cos \left(\frac{\pi}{2} \frac{x}{L}\right)
$$
where $C_{1}$ is a constant and $\alpha$ 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 $t \rightarrow \infty$, and at an intermediate time. Sketch the variation with time of the heat flux at $x=L, q_{L}^{\prime \prime}(t)$.
(d) What effect does $\alpha$ have on the thermal response of the material to a change in surface temperature?

Joseph Liao
Joseph Liao
Numerade Educator
05:42

Problem 67

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

Carson Merrill
Carson Merrill
Numerade Educator
06:42

Problem 68

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, $T_{i}$, and resistance heating is concurrently initiated with airflow at $t=0$.
(a) For a wire radius $r_{o}$, an air temperature $T_{\infty}$, 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 $500 \mathrm{~mm}$ and $1 \mathrm{~mm}$, respectively, what is the volumetric rate of thermal energy generation for a power consumption of $P_{\text {elec }}=500 \mathrm{~W}$ ? 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 $(t \leq 0)$, steady-state condition $(t \rightarrow \infty)$, and for two intermediate times.
(d) On $q_{r}^{\prime \prime}-t$ coordinates, sketch the variation of the heat flux with time for locations at $r=0$ and $r=r_{o^{*}}$.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
02:43

Problem 69

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 $q_{2}^{\prime \prime}$ and $q_{3}^{\prime \prime}$, and of $q_{3}^{\prime \prime}$ and $q_{4}^{\prime \prime}$.
(b) Comment on the relative magnitudes of $k_{\mathrm{A}}$ and $k_{\mathrm{B}}$, and of $k_{\mathrm{B}}$ and $k_{\mathrm{C}^{-}}$
(c) Sketch the heat flux as a function of $x$.

Mahendra K
Mahendra K
Numerade Educator