Heat and Mass Transfer: Fundamentals and Applications

Yunus A. Cengel, Afshin Jahanshahi Ghajar

Chapter 4

Transient Heat Conduction - all with Video Answers

Educators

Chapter Questions

Problem 1

What is the physical significance of the Biot number? Is the Biot number more likely to be larger for highly conducting solids or poorly conducting ones?

Nathan Silvano

Nathan Silvano

Numerade Educator

Problem 2

What is lumped system analysis? When is it applicable?

Averell Hause

Carnegie Mellon University

Problem 3

In what medium is the lumped system analysis more likely to be applicable: in water or in air? Why?

Anand Jangid

Numerade Educator

Problem 4

For which solid is the lumped system analysis more likely to be applicable: an actual apple or a golden apple of the same size? Why?

Anand Jangid

Numerade Educator

Problem 5

For which kinds of bodies made of the same material is the lumped system analysis more likely to be applicable: slender ones or well-rounded ones of the same volume? Why?

Anand Jangid

Numerade Educator

Problem 6

Consider heat transfer between two identical hot solid bodies and the air surrounding them. The first solid is being cooled by a fan while the second one is allowed to cool naturally. For which solid is the lumped system analysis more likely to be applicable? Why?

David Collins

Numerade Educator

Problem 7

Consider heat transfer between two identical hot solid bodies and their environments. The first solid is dropped in a large container filled with water, while the second one is allowed to cool naturally in the air. For which solid is the lumped system analysis more likely to be applicable? Why?

David Collins

Numerade Educator

Problem 8

Consider a hot baked potato on a plate. The temperature of the potato is observed to drop by $4^{\circ} \mathrm{C}$ during the first minute. Will the temperature drop during the second minute be less than, equal to, or more than $4^{\circ} \mathrm{C}$ ? Why?

Narayan Hari

Numerade Educator

Problem 9

Consider a potato being baked in an oven that is maintained at a constant temperature. The temperature of the potato is observed to rise by $5^{\circ} \mathrm{C}$ during the first minute. Will the temperature rise during the second minute be less than, equal to, or more than $5^{\circ} \mathrm{C}$ ? Why?

Narayan Hari

Numerade Educator

Problem 10

Consider two identical 4-kg pieces of roast beef. The first piece is baked as a whole, while the second is baked after being cut into two equal pieces in the same oven. Will there be any difference between the cooking times of the whole and cut roasts? Why?

WM

William Mead

Numerade Educator

Problem 11

Consider a sphere and a cylinder of equal volume made of copper. Both the sphere and the cylinder are initially at the same temperature and are exposed to convection in the same environment. Which do you think will cool faster, the cylinder or the sphere? Why?

Mahendra Rathore

Mahendra Rathore

Numerade Educator

Problem 12

Obtain relations for the characteristic lengths of a large plane wall of thickness $2 L$, a very long cylinder of radius $r_{o}$ and a sphere of radius $r_{o}$.

Chris Trentman

Numerade Educator

Problem 13

Obtain a relation for the time required for a lumped system to reach the average temperature $\frac{1}{2}\left(T_{i}+T_{\infty}\right)$, where $T_{i}$ is the initial temperature and $T_{\infty}$ is the temperature of the environment.

Bahar Tehranipoor

Bahar Tehranipoor

Numerade Educator

Problem 14

A brick of $203 \times 102 \times 57 \mathrm{~mm}$ in dimension is being burned in a kiln to $1100^{\circ} \mathrm{C}$ and then allowed to cool in a room with ambient air temperature of $30^{\circ} \mathrm{C}$ and convection heat transfer coefficient of $5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the brick has properties of $\rho=1920 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=790 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=0.90 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, determine the time required to cool the brick to a temperature difference of $5^{\circ} \mathrm{C}$ from the ambient air temperature.

Nicholas Mogoi

Numerade Educator

Problem 15

In a device for cryogenic process, a section of an ASTM A203 B steel plate is occasionally exposed to very cold fluid. The plate has a thickness of $1 \mathrm{~cm}$ with a thermal conductivity of $52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, a specific heat of $470 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a density of $7.9 \mathrm{~g} / \mathrm{cm}^{3}$. The plate is situated such that both surfaces are exposed to cryogenic fluid at $-50^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $50 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$. The minimum temperature suitable for the ASTM A203 B steel plate is $-30^{\circ} \mathrm{C}$ (ASME Code for Process Piping, B31.3-2014, Table A-1M). If the plate has an initial temperature of $20^{\circ} \mathrm{C}$, determine how long it will take to reach the minimum suitable temperature stipulated by the ASME Code for Process Piping.

Keshav Singh

Numerade Educator

Problem 16

Consider an 800-W iron whose base plate is made of $0.5-\mathrm{cm}$-thick aluminum alloy $2024-\mathrm{T} 6\left(\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}\right.$, $c_{p}=875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \alpha=7.3 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}$ ). The base plate has a surface area of $0.03 \mathrm{~m}^{2}$. Initially, the iron is in thermal equilibrium with the ambient air at $22^{\circ} \mathrm{C}$. Taking the heat transfer coefficient at the surface of the base plate to be $12 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$ and assuming 85 percent of the heat generated in the resistance wires is transferred to the plate, determine how long it will take for the plate temperature to reach $140^{\circ} \mathrm{C}$. Is it realistic to assume the plate temperature to be uniform at all times?

Keshav Singh

Numerade Educator

Problem 17

Reconsider Prob. 4-16. Using appropriate software, investigate the effects of the heat transfer coefficient and the final plate temperature on the time it will take for the plate to reach this temperature. Let the heat transfer coefficient vary from $5 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ to $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and the temperature from $30^{\circ} \mathrm{C}$ to $200^{\circ} \mathrm{C}$. Plot the time as functions of the heat transfer coefficient and the temperature, and discuss the results.

Bret Rosen

Numerade Educator

Problem 18

Metal plates $\left(k=180 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=2800 \mathrm{~kg} / \mathrm{m}^{3}\right.$, and $\left.c_{p}=880 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ with a thickness of $1 \mathrm{~cm}$ are being heated in an oven for $2 \mathrm{~min}$. Air in the oven is maintained at $800^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $200 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$. If the initial temperature of the plates is $20^{\circ} \mathrm{C}$, determine the temperature of the plates when they are removed from the oven.

Keshav Singh

Numerade Educator

Problem 19

A 5 -mm-thick stainless steel strip $(k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\rho=8000 \mathrm{~kg} / \mathrm{m}^{3}$, and $\left.c_{p}=570 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ is being heat treated as it moves through a furnace at a speed of $1 \mathrm{~cm} / \mathrm{s}$. The air temperature in the furnace is maintained at $900^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the furnace length is $3 \mathrm{~m}$ and the stainless steel strip enters it at $20^{\circ} \mathrm{C}$, determine the temperature of the strip as it exits the furnace.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 20

Two plates are held together by square stainless steel (ASTM A479 904L) bars with thickness of
$12 \mathrm{~mm}$. Each bar has a thermal conductivity of $12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, a specific heat of $500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a density of $7.9 \mathrm{~g} / \mathrm{cm}^{3}$.
The bars are situated in between the two plates, and they are occasionally submerged in hot liquid at $300^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $96 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The length of each bar that is exposed to the hot liquid is $25 \mathrm{~mm}$ with an initial temperature of $20^{\circ} \mathrm{C}$. The maximum use temperature allowed for ASTM A479 904L bars is $260^{\circ} \mathrm{C}$ (ASME Code for Process Piping, ASME B31.3-2014, Table A-1M). If the bars are submerged in the hot liquid for $5 \mathrm{~min}$, will they be in compliance with the ASME code? How long will it take for the bars to reach the maximum use temperature?

Arun Bana

Numerade Educator

Problem 21

A batch of 2 -cm-thick stainless steel plates $\left(k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8000 \mathrm{~kg} / \mathrm{m}^{3}\right.$, and $\left.c_{p}=570 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ are conveyed through a furnace to be heat treated. The plates enter the furnace at $18^{\circ} \mathrm{C}$, and they travel a distance of $3 \mathrm{~m}$ inside the furnace. The air temperature in the furnace is maintained at $950^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Using appropriate software, determine how the velocity of the plates affects the temperature of the plates at the end of the heat treatment. Let the velocity of the plates vary from 5 to $60 \mathrm{~mm} / \mathrm{s}$, and plot the temperature of the plates at the furnace exit as a function of the velocity.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 22

A 6-mm-thick stainless steel strip $(k=21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, oven at a temperature of $500^{\circ} \mathrm{C}$ is allowed to cool within a buffer zone distance of $5 \mathrm{~m}$. To prevent thermal burns to workers who are handling the strip at the end of the buffer zone, the surface temperature of the strip should be cooled to $45^{\circ} \mathrm{C}$. If the air temperature in the buffer zone is $15^{\circ} \mathrm{C}$ and the convection heat transfer coefficient is $120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the maximum speed of the stainless steel strip.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

After heat treatment, the $2-\mathrm{cm}$-thick metal ${ }_{\mathrm{a}}$ length of $10 \mathrm{~m}$. The plates enter the cooling chamber at an initial temperature of $500^{\circ} \mathrm{C}$. The cooling chamber maintains temperature of $10^{\circ} \mathrm{C}$, and the convection heat transfer coefficient is given as a function of the air velocity blowing over the plates $h=33 V^{0.8}$, where $h$ is in $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $V$ is in $\mathrm{m} / \mathrm{s}$. To prevent any incident of thermal burn, it is necessary for the plates to exit the cooling chamber at a temperature below $50^{\circ} \mathrm{C}$. In designing the cooling process to meet this safety criterion, use appropriate software to investigate the effect of the air velocity on the temperature of the plates at the exit of the cooling chamber. Let the air velocity vary from 0 to $40 \mathrm{~m} / \mathrm{s}$, and plot the temperatures of the plates exiting the cooling chamber as a function of air velocity at the moving plate speed of 2,5 , and $8 \mathrm{~cm} / \mathrm{s}$.

Dominique Jan Tan

Dominique Jan Tan

Numerade Educator

Problem 24

A long copper rod of diameter $2.0 \mathrm{~cm}$ is initially at a uniform temperature of $100^{\circ} \mathrm{C}$. It is now exposed to an airstream at $20^{\circ} \mathrm{C}$ with a heat transfer coefficient of $200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. How long would it take for the copper rod to cool to an average temperature of $25^{\circ} \mathrm{C}$ ?

Anand Jangid

Numerade Educator

Problem 25

Springs in automobile suspension systems are made of steel rods heated and wound into coils while ductile. Consider steel rods $\left(\rho=7832 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=434 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $k=63.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ ) with diameter of $2.5 \mathrm{~cm}$ and length of $1.27$ $\mathrm{m}$. The steel rods are heated in an oven with a uniform convection heat transfer coefficient of $20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The steel rods were heated from an initial temperature of $20^{\circ} \mathrm{C}$ to the desired temperature of $450^{\circ} \mathrm{C}$ before being wound into coils. Determine the ambient temperature in the oven if the steel rods were to be heated to the desired temperature within $10 \mathrm{~min}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 26

Steel rods $\left(\rho=7832 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=434 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $k=63.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ ) are heated in a furnace to $850^{\circ} \mathrm{C}$ and then quenched in a water bath at $50^{\circ} \mathrm{C}$ for a period of $40 \mathrm{~s}$ as part of a hardening process. The convection heat transfer coefficient is $650 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$. If the steel rods have diameter of $40 \mathrm{~mm}$ and length of $2 \mathrm{~m}$, determine their average temperature when they are taken out of the water bath.

Keshav Singh

Numerade Educator

Problem 27

Two plates are held together by bolts made of ASTM B98 (UNS C65500) copper-silicon. The bolts are $3 \mathrm{~mm}$ in diameter; they have a thermal conductivity of $36 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, a specific heat of $377 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a density of $8.55 \mathrm{~g} / \mathrm{cm}^{3}$. The bolts are situated in between the two plates, and they are occasionally exposed to hot steam at $200^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $50 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$. The length of each bolt exposed to the steam is $25 \mathrm{~mm}$. According to the ASME Code for Process Piping (ASME B31.3-2014, Table A-2M), the maximum temperature allowed for ASTM $\mathrm{B} 98$ bolts is $149^{\circ} \mathrm{C}$. If the initial temperature of the bolts is $20^{\circ} \mathrm{C}$, how long can the bolts be in the hot steam before the maximum use temperature is reached?

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 28

Two components in cryogenic equipment are held together by stainless steel (ASTM A437 B4B) bolts with diameter of $25 \mathrm{~mm}$. The bolts have a thermal conductivity of $23.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, a specific heat of $460 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a density of $7.8 \mathrm{~g} / \mathrm{cm}^{3}$. When the cryogenic fluid flows between the two components, the bolts are submerged in the cold fluid. The length of each bolt submerged in the cryogenic fluid is $5 \mathrm{~cm}$, and the initial temperature of the bolts is $10^{\circ} \mathrm{C}$. The cryogenic fluid has a temperature of $-40^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The ASME Code for Process Piping limits the minimum suitable temperature for ASTM A437 B4B bolts to $-30^{\circ} \mathrm{C}$ (ASME B31.3-2014, Table A-2M). If the bolts are exposed to the cold fluid for $12 \mathrm{~min}$, will they still comply with the ASME code? How long will it take for the bolts to reach the minimum suitable temperature?

Lainey Roebuck

Numerade Educator

Problem 29

To warm up some milk for a baby, a mother pours milk into a thin-walled cylindrical container whose diameter is $6 \mathrm{~cm}$. The height of the milk in the container is $7 \mathrm{~cm}$. She then places the container into a large pan filled with hot water at $70^{\circ} \mathrm{C}$. The milk is stirred constantly so that its temperature is uniform at all times. If the heat transfer coefficient between the water and the container is $120 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine how long it will take for the milk to warm up from $3^{\circ} \mathrm{C}$ to $38^{\circ} \mathrm{C}$. Assume the entire surface area of the cylindrical container (including the top and bottom) is in thermal contact with the hot water. Take the properties of the milk to be the same as those of water. Can the milk in this case be treated as a lumped system?
Why? Answer: $4.50$ min

Kurt Kleinberg

Numerade Educator

Problem 30

A person is found dead at 5 p.m. in a room whose temperature is $20^{\circ} \mathrm{C}$. The temperature of the body is measured to be $25^{\circ} \mathrm{C}$ when found, and the heat transfer coefficient is estimated to be $8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Modeling the body as a $30-\mathrm{cm}$ diameter, $1.70$-m-long cylinder and using the lumped system analysis as a rough approximation, estimate the time of death of that person.

Carson Merrill

Numerade Educator

Problem 31

The temperature of a gas stream is to be measured by a thermocouple whose junction can be approximated as a $1.2$-mm-diameter sphere. The properties of the junction are $k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8500 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=320 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and the heat transfer coefficient between the junction and the gas is $h=110 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine how long it will take for the thermocouple to read 99 percent of the initial temperature difference.

Prabhu Ramji

Numerade Educator

Problem 32

In an experiment, the temperature of a hot gas stream is to be measured by a thermocouple with a spherical junction. Due to the nature of this experiment, the response time of the thermocouple to register 99 percent of the initial temperature difference must be within $5 \mathrm{~s}$. The properties of the thermocouple junction are $k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8500 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=320 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. If the heat transfer coefficient between the thermocouple junction and the gas is $250 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the diameter of the junction.

Prabhu Ramji

Numerade Educator

Problem 33

A thermocouple with a spherical junction diameter of $0.5 \mathrm{~mm}$ is used for measuring the temperature of hot airflow in a circular duct. The convection heat transfer coefficient of the airflow can be related with the diameter $(D)$ of the spherical junction and the average airflow velocity $(V)$ as $h=2.2(V / D)^{0.5}$, where $D, h$, and $V$ are in $\mathrm{m}, \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$, and $\mathrm{m} / \mathrm{s}$, respectively. The properties of the thermocouple junction are $k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\rho=8500 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=320 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Determine the minimum airflow velocity that the thermocouple can use, if the maximum response time of the thermocouple to register 99 percent of the initial temperature difference is $5 \mathrm{~s}$.

Aadit Sharma

Numerade Educator

Problem 34

Pulverized coal particles are used in oxy-fuel combustion power plants for electricity generation. Consider a situation where coal particles are suspended in hot air flowing through a heated tube, where the convection heat transfer coefficient is $100 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$. If the average surface area and volume of the coal particles are $3.1 \mathrm{~mm}^{2}$ and $0.5 \mathrm{~mm}^{3}$, respectively, determine how much time it would take to heat the coal particles to two-thirds of the initial temperature difference.

Prabhu Ramji

Numerade Educator

Problem 35

Oxy-fuel combustion power plants use pulverized coal particles as fuel to burn in a pure oxygen environment to generate electricity. Before entering the furnace, pulverized spherical coal particles with an average diameter of $300 \mu \mathrm{m}$ are transported at $2 \mathrm{~m} / \mathrm{s}$ through a $3-\mathrm{m}$-long heated tube while suspended in hot air. The air temperature in the tube is $900^{\circ} \mathrm{C}$, and the average convection heat transfer coefficient is 250 $\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the temperature of the coal particles at the exit of the heated tube if the initial temperature of the particles is $20^{\circ} \mathrm{C}$.

Manik Pulyani

Numerade Educator

Problem 36

Plasma spraying is a process used for coating a material surface with a protective layer to prevent the material from degradation. In a plasma spraying process, the protective layer in powder form is injected into a plasma jet. The powder is then heated to molten droplets and propelled onto the material surface. Once deposited on the material surface, the molten droplets solidify and form a layer of protective coating. Consider a plasma spraying process using alumina $(k=30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\rho=3970 \mathrm{~kg} / \mathrm{m}^{3}$, and $\left.c_{p}=800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ powder that is injected into a plasma jet at $T_{\infty}=15,000^{\circ} \mathrm{C}$ and $h=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The alumina powder is made of spherical particles with an average diameter of $60 \mu \mathrm{m}$ and a melting point at $2300^{\circ} \mathrm{C}$. Determine the amount of time it would take for the particles, with an initial temperature of $20^{\circ} \mathrm{C}$, to reach their melting point from the moment they are injected into the plasma jet.

Lainey Roebuck

Numerade Educator

Problem 37

Consider a spherical shell satellite with outer diameter of $4 \mathrm{~m}$ and shell thickness of $10 \mathrm{~mm}$ that is reentering the atmosphere. The shell satellite is made of stainless steel with properties of $\rho=8238 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=468 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=13.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. During the reentry, the effective atmosphere temperature surrounding the satellite is $1250^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $130 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$. If the initial temperature of the shell is $10^{\circ} \mathrm{C}$, determine the shell temperature after $5 \mathrm{~min}$ of reentry. Assume heat transfer occurs only on the satellite shell.

Dominador Tan

Numerade Educator

Problem 38

Carbon steel balls $\left(\rho=7833 \mathrm{~kg} / \mathrm{m}^{3}, k=54 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, $c_{p}=0.465 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, and $\left.\alpha=1.474 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right) 8 \mathrm{~mm}$ in diameter are annealed by heating them first to $900^{\circ} \mathrm{C}$ in a furnace and then allowing them to cool slowly to $100^{\circ} \mathrm{C}$ in ambient air at $35^{\circ} \mathrm{C}$. If the average heat transfer coefficient is $75 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine how long the annealing process will take. If 2500 balls are to be annealed per hour, determine the total rate of heat transfer from the balls to the ambient air.

Vipender Yadav

Numerade Educator

Problem 39

Reconsider Prob. 4-38. Using appropriate software, investigate the effect of the initial temperature of the balls on the annealing time and the total rate of heat transfer. Let the temperature vary from $500^{\circ} \mathrm{C}$ to $1000^{\circ} \mathrm{C}$. Plot the time and the total rate of heat transfer as a function of the initial temperature, and discuss the results.

Vipender Yadav

Numerade Educator

Problem 40

In a manufacturing facility, 2-in-diameter brass balls $\left(k=64.1 \mathrm{Btw} / \mathrm{h} \cdot \mathrm{ft}{ }^{\circ} \mathrm{F}, \rho=532 \mathrm{lbm} / \mathrm{ft}^{3}\right.$, and $\left.c_{p}=0.092 \mathrm{Btu} / \mathrm{lbm} \cdot{ }^{\circ} \mathrm{F}\right)$ initially at $250^{\circ} \mathrm{F}$ are quenched in a water bath at $120^{\circ} \mathrm{F}$ for a period of $2 \mathrm{~min}$ at a rate of $120 \mathrm{balls}$ per minute. If the convection heat transfer coefficient is $42 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2},{ }^{\circ} \mathrm{F}$, determine $(a)$ the temperature of the balls after quenching and $(b)$ the rate at which heat needs to be removed from the water in order to keep its temperature constant at $120^{\circ} \mathrm{F}

Anand Jangid

Numerade Educator

Problem 41

Consider a sphere of diameter $5 \mathrm{~cm}$, a cube of side length $5 \mathrm{~cm}$, and a rectangular prism of dimension $4 \mathrm{~cm} \times 5 \mathrm{~cm} \times 6 \mathrm{~cm}$, all initially at $0^{\circ} \mathrm{C}$ and all made of silver $\left(k=429 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=10,500 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=0.235 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)$. Now all three of these geometries are exposed to ambient air at $33^{\circ} \mathrm{C}$ on all of their surfaces with a heat transfer coefficient of $12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine how long it will take for the temperature of each geometry to rise to $25^{\circ} \mathrm{C}$.

Yaqub Khan

Numerade Educator

Problem 42

An electronic device dissipating $18 \mathrm{~W}$ has a mass of $20 \mathrm{~g}$, a specific heat of $850 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a surface area of $4 \mathrm{~cm}^{2}$. The device is lightly used, and it is on for $5 \mathrm{~min}$ and then off for several hours, during which it cools to the ambient temperature of $25^{\circ} \mathrm{C}$. Taking the heat transfer coefficient to be $12 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the temperature of the device at the end of the 5-min operating period. What would your answer be if the device were attached to an aluminum heat sink having a mass of $200 \mathrm{~g}$ and a surface area of $80 \mathrm{~cm}^{2}$ ? Assume the device and the heat sink to be nearly isothermal.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 43

An egg is to be cooked to a certain level of doneness by being dropped into boiling water. Can the cooking time be shortened by turning up the heat and bringing water to a more rapid boiling?

Eduard Sanchez

Numerade Educator

Problem 44

What is an infinitely long cylinder? When is it proper to treat an actual cylinder as being infinitely long, and when is it not? For example, is it proper to use this model when finding the temperatures near the bottom or top surfaces of a cylinder?
Explain.

Lucas Finney

Numerade Educator

Problem 45

What is the physical significance of the Fourier number? Will the Fourier number for a specified heat transfer problem double when the time is doubled?

Anand Jangid

Numerade Educator

Problem 46

Can the one-term approximate solutions for a plane wall exposed to convection on both sides be used for a plane wall with one side exposed to convection while the other side is insulated? Explain.

David Zhang

Numerade Educator

Problem 47

How can we use the one-term approximate solutions when the surface temperature of the geometry is specified instead of the temperature of the surrounding medium and the convection heat transfer coefficient?

Jincy M Saji

Numerade Educator

Problem 48

The Biot number during a heat transfer process between a sphere and its surroundings is determined to be $0.02$. Would you use lumped system analysis or the one-term approximate solutions when determining the midpoint temperature of the sphere? Why?

Anand Jangid

Numerade Educator

Problem 49

A body at an initial temperature of $T_{i}$ is brought into a medium at a constant temperature of $T_{\infty}$. How can you determine the maximum possible amount of heat transfer between the body and the surrounding medium?

Gio Maya

Numerade Educator

Problem 50

A hot brass plate is having its upper surface cooled by an impinging jet of air at temperature of $15^{\circ} \mathrm{C}$ and convection heat transfer coefficient of $220 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The 10 -cm-thick brass plate $\left(\rho=8530 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=380 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, and $\alpha=33.9 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) has a uniform initial temperature of $650^{\circ} \mathrm{C}$, and the bottom surface of the plate is insulated. Determine the temperature at the center plane of the brass plate after 3 min of cooling. Solve this problem using the analytical one-term approximation method.

Keshav Singh

Numerade Educator

Problem 51

In a meat processing plant, 2-cm-thick steaks $\left(k=0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$ and $\left.\alpha=0.91 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ that are initially at $25^{\circ} \mathrm{C}$ are to be cooled by passing them through a refrigeration room at $-11^{\circ} \mathrm{C}$. The heat transfer coefficient on both sides of the steaks is $9 \mathrm{~W} / \mathrm{m}^{2}$. K. If both surfaces of the steaks are to be cooled to $2^{\circ} \mathrm{C}$, determine how long the steaks should be kept in the refrigeration room. Solve this problem using the analytical one-term approximation method.

AG

Ankit Gupta

Numerade Educator

Problem 52

A large plate made of stainless steel (ASTM A240 $410 S$ ) that has a thickness of $5 \mathrm{~cm}$ is part of the equipment for a fluid flow process at high temperature. The plate has a thermal conductivity of $26.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, a specific heat of $460 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a density of $7.73 \mathrm{~g} / \mathrm{cm}^{3}$. Both surfaces of the plate are occasionally exposed to hot fluid at $700^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $323 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The maximum use temperature for the ASTM A240 410S stainless steel plate is $649^{\circ} \mathrm{C}$ (ASME Code for Process Piping, initial temperature of $20^{\circ} \mathrm{C}$, how long can the plate be exposed to the hot fluid before reaching its maximum use temperature?

Manish Jain

Numerade Educator

Problem 53

A $10-\mathrm{cm}$-thick aluminum plate $\left(\alpha=97.1 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ is being heated in liquid with temperature of $500^{\circ} \mathrm{C}$. The aluminum plate has a uniform initial temperature of $25^{\circ} \mathrm{C}$. If the surface temperature of the aluminum plate is approximately the liquid temperature, determine the temperature at the center plane of the aluminum plate after 15 s of heating. Solve this problem using the analytical one-term approximation method.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 54

In a production facility, 3-cm-thick large brass plates $\left(k=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=8530 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=380 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $\alpha=33.9 \times$ $10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) that are initially at a uniform temperature of $25^{\circ} \mathrm{C}$ are heated by passing them through an oven maintained at $700^{\circ} \mathrm{C}$. The plates remain in the oven for a period of $10 \mathrm{~min}$. Taking the convection heat transfer coefficient to be $h=80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the surface temperature of the plates when they come out of the oven. Solve this problem using the analytical one-term approximation method. Can this problem be solved using lumped system analysis? Justify your answer.

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 55

Reconsider Prob. 4-54. Using appropriate software, investigate the effects of the temperature of the oven and the heating time on the final surface temperature of the plates. Let the oven temperature vary from $500^{\circ} \mathrm{C}$ to $900^{\circ} \mathrm{C}$ and the time from $2 \mathrm{~min}$ to $30 \mathrm{~min}$. Plot the surface temperature as the functions of the oven temperature and the time, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 56

Layers of $23-\mathrm{cm}$-thick meat slabs $(k=0.47 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=0.13 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) initially at a uniform temperature of $7^{\circ} \mathrm{C}$ are to be frozen by refrigerated air at $-30^{\circ} \mathrm{C}$ flowing at a velocity of $1.4 \mathrm{~m} / \mathrm{s}$. The average heat transfer coefficient between the meat and the air is $20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming the size of the meat slabs to be large relative to their thickness, determine how long it will take for the center temperature of the slabs to drop to $-18^{\circ} \mathrm{C}$. Also, determine the surface temperature of the meat slab at that time.

Hast Aggarwal

Numerade Educator

Problem 57

A large ASTM A203 B steel plate, with a thickness of $7 \mathrm{~cm}$, in a cryogenic process is suddenly exposed to very cold fluid at $-50^{\circ} \mathrm{C}$ with $h=594 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The plate has a thermal conductivity of $52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, a specific heat of $470 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a density of $7.9 \mathrm{~g} / \mathrm{cm}^{3}$. The ASME Code for Process Piping limits the minimum suitable temperature for ASTM A203 B steel plate to $-30^{\circ} \mathrm{C}$ (ASME B31.32014 , Table A-1M). If the initial temperature of the plate is $20^{\circ} \mathrm{C}$ and the plate is exposed to the cryogenic fluid for $6 \mathrm{~min}$, would it still comply with the ASME code?

Keshav Singh

Numerade Educator

Problem 58

A heated 6-mm-thick Pyroceram plate $\left(\rho=2600 \mathrm{~kg} / \mathrm{m}^{3}\right.$, $c_{p}=808 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=3.98 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\left.\alpha=1.89 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ is being cooled in a room with air temperature of $25^{\circ} \mathrm{C}$ and convection heat transfer coefficient of $13.3 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The heated Pyroceram plate had an initial temperature of $500^{\circ} \mathrm{C}$, and it is allowed to cool for $286 \mathrm{~s}$. If the mass of the Pyroceram plate is $10 \mathrm{~kg}$, determine the heat transfer from the Pyroceram plate during the cooling process using the analytical one-term approximation method.

Vipender Yadav

Numerade Educator

Problem 59

After a long, hard week on the books, you and your friend are ready to relax and enjoy the weekend. You take a steak $50 \mathrm{~mm}$ thick from the freezer. (a) How long (in hours) do you have to let the good times roll before the steak has thawed? Assume that the steak is initially at $-8^{\circ} \mathrm{C}$, that it thaws when the temperature at the center of the steak reaches $4^{\circ} \mathrm{C}$, and that the room temperature is $22^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $10 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Neglect the heat of fusion associated with the melting phase change. Treat the steak as a one-dimensional plane wall having the following properties: $\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}$, $c_{p}=4472 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=0.625 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. (b) How much energy per unit area (in J/m²) has been removed from the steak during this period of thawing? (c) Show whether or not the thawing of this steak can be analyzed by neglecting the internal thermal resistance of the steak.

AG

Ankit Gupta

Numerade Educator

Problem 60

A long Pyroceram rod $\left(\rho=2600 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=808 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, $k=3.98 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=1.89 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) with diameter of $10 \mathrm{~mm}$ has an initial uniform temperature of $1000^{\circ} \mathrm{C}$. The Pyroceram rod is allowed to cool in an ambient temperature of $25^{\circ} \mathrm{C}$ and convection heat transfer coefficient of $80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the Pyroceram rod is allowed to cool for $3 \mathrm{~min}$, determine the temperature at the center of the rod using the analytical one-term approximation method.

Narayan Hari

Numerade Educator

Problem 61

A long cylindrical wood log $(k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=1.28 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ) is $10 \mathrm{~cm}$ in diameter and is initially at a uniform temperature of $25^{\circ} \mathrm{C}$. It is exposed to hot gases at $525^{\circ} \mathrm{C}$ in a fireplace with a heat transfer coefficient of $13.6 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$ on the surface. If the ignition temperature of the wood is $375^{\circ} \mathrm{C}$, determine how long it will be before the log ignites. Solve this problem using the analytical one-term approximation method.

Averell Hause

Carnegie Mellon University

Problem 62

Long cylindrical AISI stainless steel rods $\left(k=7.74 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{\circ} \mathrm{F}\right.$ and $\left.\alpha=0.135 \mathrm{ft}^{2} / \mathrm{h}\right)$ of 4 -in diameter are heat treated by drawing them at a velocity of $7 \mathrm{ft} / \mathrm{min}$ through a 21 -ft-long oven maintained at $1700^{\circ} \mathrm{F}$. The heat transfer coefficient in the oven is $20 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}$. If the rods enter the oven at $70^{\circ} \mathrm{F}$, determine their centerline temperature when they leave. Solve this problem using the analytical one-term approximation method.

Keshav Singh

Numerade Educator

Problem 63

A long iron rod $\left(\rho=7870 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=447 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, $k=80.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=23.1 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) with diameter of $25 \mathrm{~mm}$ is initially heated to a uniform temperature of $700^{\circ} \mathrm{C}$. The iron rod is then quenched in a large water bath that is maintained at constant temperature of $50^{\circ} \mathrm{C}$ and with a convection heat transfer coefficient of $128 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the time required for the iron rod surface temperature to $c 00 \mathrm{l}^{\circ}$ to $200^{\circ} \mathrm{C}$. Solve this problem using the analytical one-term approximation method.

Narayan Hari

Numerade Educator

Problem 64

A 2-cm-diameter plastic rod has a thermocouple inserted to measure temperature at the center of the rod. The plastic rod $\left(\rho=1190 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=1465 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $k=0.19$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ ) was initially heated to a uniform temperature of $70^{\circ} \mathrm{C}$ and allowed to be cooled in ambient air at $25^{\circ} \mathrm{C}$. After $1388 \mathrm{~s}$ of cooling, the thermocouple measured the temperature at the center of the rod to be $30^{\circ} \mathrm{C}$. Determine the convection heat transfer coefficient for this process. Solve this problem using the analytical one-term approximation method.

Narayan Hari

Numerade Educator

Problem 65

A $65-\mathrm{kg}$ beef carcass $(k=0.47 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=0.13 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) initially at a uniform temperature of $37^{\circ} \mathrm{C}$ is to be cooled by refrigerated air at $-10^{\circ} \mathrm{C}$ flowing at a velocity of $1.2 \mathrm{~m} / \mathrm{s}$. The average heat transfer coefficient between the carcass and the air is $22 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Treating the carcass as a cylinder of diameter $24 \mathrm{~cm}$ and height $1.4 \mathrm{~m}$ and disregarding heat transfer from the base and top surfaces, determine how long it will take for the center temperature of the carcass to drop to $4^{\circ} \mathrm{C}$. Also, determine if any part of the carcass will freeze during this process. Solve this problem using the analytical one-term approximation method. Answer: $12.2 \mathrm{~h}$

Chinmai Managoli

Chinmai Managoli

Numerade Educator

Problem 66

A long nickel alloy (ASTM B335) cylindrical rod is used as a component in high-temperature process equipment. The rod has a diameter of $5 \mathrm{~cm}$; its thermal conductivity, specific heat, and density are $11 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, 380 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $9.3 \mathrm{~g} / \mathrm{cm}^{3}$, respectively. Occasionally, the rod is submerged in hot fluid for several minutes, where the fluid temperature is $500^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $440 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The ASME Code for Process Piping limits the maximum use temperature for ASTM B335 rod to $427^{\circ} \mathrm{C}$ (ASME B31.32014 , Table A-1M). If the initial temperature of the rod is $20^{\circ} \mathrm{C}$, how long can the rod be submerged in the hot fluid before reaching its maximum use temperature?

Narayan Hari

Numerade Educator

Problem 67

A 30-cm-diameter, 4-m-high cylindrical column of a house made of concrete $\left(k=0.79 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=5.94 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right.$, $\rho=1600 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=0.84 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$ ) cooled to $14^{\circ} \mathrm{C}$ during a cold night is heated again during the day by being exposed to ambient air at an average temperature of $28^{\circ} \mathrm{C}$ with an average heat transfer coefficient of $14 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$. Using the analytical one-term approximation method, determine $(a)$ how long it will take for the column surface temperature to rise to $27^{\circ} \mathrm{C}$, (b) the amount of heat transfer until the center temperature reaches to $28^{\circ} \mathrm{C}$, and $(c)$ the amount of heat transfer until the surface temperature reaches $27^{\circ} \mathrm{C}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 68

A long 35-cm-diameter cylindrical shaft made of stainless steel $304\left(k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=7900 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=477 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) comes out of an oven at a uniform temperature of $500^{\circ} \mathrm{C}$. The shaft is then allowed to cool slowly in a chamber at $150^{\circ} \mathrm{C}$ with an average convection heat transfer coefficient of $h=60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the temperature at the center of the shaft 20 min after the start of the cooling process. Also, determine the heat transfer per unit length of the shaft during this time period. Solve this problem using the analytical one-term approximation method. Answers: $486^{\circ} \mathrm{C}, 22,270 \mathrm{~kJ}$

Vipender Yadav

Numerade Educator

Problem 69

Reconsider Prob. 4-68. Using appropriate software, investigate the effect of the cooling time on the final center temperature of the shaft and the amount of heat transfer. Let the time vary from $5 \mathrm{~min}$ to $60 \mathrm{~min}$. Plot the center temperature and the heat transfer as a function of the time, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 70

Steel rods, $2 \mathrm{~m}$ in length and $60 \mathrm{~mm}$ in diameter, are being drawn through an oven that maintains a temperature of $800^{\circ} \mathrm{C}$ and convection heat transfer coefficient of $128 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The steel rods $\left(\rho=7832 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=434 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=63.9\right.$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=18.8 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) were initially in uniform temperature of $30^{\circ} \mathrm{C}$. Using the analytical one-term approximation method, determine the amount of heat transferred to the steel rod after $133 \mathrm{~s}$ of heating.

Keshav Singh

Numerade Educator

Problem 71

For heat transfer purposes, an egg can be considered to be a $5.5-\mathrm{cm}$-diameter sphere having the properties of water. An egg that is initially at $4.3^{\circ} \mathrm{C}$ is dropped into boiling water at $100^{\circ} \mathrm{C}$. The heat transfer coefficient at the surface of the egg is estimated to be $800 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$. If the egg is considered cooked when its center temperature reaches $71^{\circ} \mathrm{C}$, determine how long the egg should be kept in the boiling water. Solve this problem using the analytical one-term approximation method.

Brandy Heflin

Numerade Educator

Problem 72

Citrus fruits are very susceptible to cold weather, and extended exposure to subfreezing temperatures can destroy them. Consider an 8 -cm-diameter orange that is initially at $15^{\circ} \mathrm{C}$. A cold front moves in one night, and the ambient temperature suddenly drops to $-6^{\circ} \mathrm{C}$, with a heat transfer coefficient of $15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Using the properties of water for the orange and assuming the ambient conditions remain constant for $4 \mathrm{~h}$ before the cold front moves out, determine if any part of the orange will freeze that night. Solve this problem using the analytical one-term approximation method.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

Chickens with an average mass of $1.7 \mathrm{~kg}(k=0.45$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=0.13 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) initially at a uniform temperature of $15^{\circ} \mathrm{C}$ are to be chilled in agitated brine at $-7^{\circ} \mathrm{C}$. The average heat transfer coefficient between the chicken and the brine is determined experimentally to be $440 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Taking the average density of the chicken to be $0.95 \mathrm{~g} / \mathrm{cm}^{3}$ and treating the chicken as a spherical lump, determine the center and the surface temperatures of the chicken in $2 \mathrm{~h}$ and $45 \mathrm{~min}$. Also, determine if any part of the chicken will freeze during this process. Solve this problem using the analytical one-term approximation method.

Nathan Nowack

Numerade Educator

Problem 74

Hailstones are formed in high-altitude clouds at $253 \mathrm{~K}$. Consider a hailstone with diameter of $20 \mathrm{~mm}$ that is falling through air at $15^{\circ} \mathrm{C}$ with convection heat transfer coefficient of $163 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming the hailstone can be modeled as a sphere and has properties of ice at $253 \mathrm{~K}$, determine how long it takes to reach melting point at the surface of the falling hailstone. Solve this problem using the analytical one-term approximation method.

Salamat Ali

Numerade Educator

Problem 75

An ordinary egg can be approximated as a $5.5$-cm-diameter sphere whose properties are roughly $k=0.6$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=0.14 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$. The egg is initially at a uniform temperature of $4^{\circ} \mathrm{C}$ and is dropped into boiling water at $97^{\circ} \mathrm{C}$. Taking the convection heat transfer coefficient to be $h=1400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine how long it will take for the center of the egg to reach $70^{\circ} \mathrm{C}$. Solve this problem using the analytical one-term approximation method.

Saman Zulfiqar

Numerade Educator

Problem 76

Reconsider Prob. 4-75. Using appropriate software, investigate the effect of the final center temperature of the egg on the time it will take for the center to reach this temperature. Let the temperature vary from $50^{\circ} \mathrm{C}$ to $95^{\circ} \mathrm{C}$. Plot the time versus the temperature, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 77

Oranges of $2.5$-in-diameter $\left(k=0.26 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.$ and $\alpha=1.4 \times 10^{-6} \mathrm{ft}^{2} / \mathrm{s}$ ) initially at a uniform temperature of $78^{\circ} \mathrm{F}$ are to be cooled by refrigerated air at $25^{\circ} \mathrm{F}$ flowing at a velocity of $1 \mathrm{ft} / \mathrm{s}$. The average heat transfer coefficient between the oranges and the air is experimentally determined to be $4.6 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}$. Determine how long it will take for the center temperature of the oranges to drop to $40^{\circ} \mathrm{F}$. Also, determine if any part of the oranges will freeze during this process. Solve this problem using the analytical one-term approximation method.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 78

White potatoes $\left(k=0.50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$ and $\left.\alpha=0.13 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ that are initially at a uniform temperature of $20^{\circ} \mathrm{C}$ and have an average diameter of $6 \mathrm{~cm}$ are to be cooled by refrigerated air at $2^{\circ} \mathrm{C}$ flowing at a velocity of $4 \mathrm{~m} / \mathrm{s}$. The average heat transfer coefficient between the potatoes and the air is experimentally determined to be $19 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine how long it will take for the center temperature of the potatoes to drop to $6^{\circ} \mathrm{C}$. Also, determine if any part of the potatoes will experience chilling injury during this process. Solve this problem using the analytical one-term approximation method.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 79

An experiment is to be conducted to determine the heat transfer coefficient on the surfaces of tomatoes that are placed in cold water at $7^{\circ} \mathrm{C}$. The tomatoes $(k=0.59 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\left.\alpha=0.141 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \rho=999 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.99 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)$ with an initial uniform temperature of $30^{\circ} \mathrm{C}$ are spherical with a diameter of $8 \mathrm{~cm}$. After $2 \mathrm{~h}$, the temperatures at the center and the surface of the tomatoes are measured to be $10.0^{\circ} \mathrm{C}$ and $7.1^{\circ} \mathrm{C}$, respectively. Using the analytical one-term approximation method, determine the heat transfer coefficient and the amount of heat transfer during this period if there are eight such tomatoes in water.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 80

A person puts a few apples into the freezer at $-15^{\circ} \mathrm{C}$ to cool them quickly for guests who are about to arrive. Initially, the apples are at a uniform temperature of $25^{\circ} \mathrm{C}$, and the heat transfer coefficient on the surfaces is $8 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Treating the apples as 9 -cm-diameter spheres and taking their properties to be $\rho=840 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.81 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.418 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=1.3 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$, determine the center and surface temperatures of the apples in $1 \mathrm{~h}$. Also, determine the amount of heat transfer from each apple. Solve this problem using the analytical one-term approximation method.

Vipender Yadav

Numerade Educator

Problem 81

Reconsider Prob. 4-80. Using appropriate software, investigate the effect of the initial temperature of the apples on the final center and surface temperatures and the amount of heat transfer. Let the initial temperature vary from $2^{\circ} \mathrm{C}$ to $30^{\circ} \mathrm{C}$. Plot the center temperature, the surface temperature, and the amount of heat transfer as a function of the initial temperature, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 82

A $9-\mathrm{cm}$-diameter potato $\left(\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3900 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, $k=0.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=1.4 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ) that is initially at a uniform temperature of $25^{\circ} \mathrm{C}$ is baked in an oven at $170^{\circ} \mathrm{C}$ until a temperature sensor inserted into the center of the potato indicates a reading of $70^{\circ} \mathrm{C}$. The potato is then taken out of the oven and wrapped in thick towels so that almost no heat is lost from the baked potato. Assuming the heat transfer coefficient in the oven to be $40 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$, determine $(a)$ how long the potato is baked in the oven and $(b)$ the final equilibrium temperature of the potato after it is wrapped. Solve this problem using the analytical one-term approximation method.

Rashmi Sinha

Numerade Educator

Problem 83

In Betty Crocker's Cookbook, it is stated that it takes $2 \mathrm{~h} \mathrm{} 45 \mathrm{~min}$ to roast a $3.2-\mathrm{kg}$ rib initially at $4.5^{\circ} \mathrm{C}$ to "rare" in an oven maintained at $163^{\circ} \mathrm{C}$. It is recommended that a meat thermometer be used to monitor the cooking, and the rib is considered rare when the thermometer inserted into the center of the thickest part of the meat registers $60^{\circ} \mathrm{C}$. The rib can be treated as a homogeneous spherical object with the properties $\rho=1200 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.1 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=0.91 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$. Determine $(a)$ the heat transfer coefficient at the surface of the rib; $(b)$ the temperature of the outer surface of the rib when it is done; and $(c)$ the amount of heat transferred to the rib. $(d)$ Using the values obtained, predict how long it will take to roast this rib to "medium" level, which occurs when the innermost temperature of the rib reaches $71^{\circ} \mathrm{C}$. Compare your result to the listed value of $3 \mathrm{~h} \mathrm{} 20 \mathrm{~min} .$
If the roast rib is to be set on the counter for about $15 \mathrm{~min}$ before it is sliced, it is recommended that the rib be taken out of the oven when the thermometer registers about $4^{\circ} \mathrm{C}$ below the indicated value because the rib will continue cooking even after it is taken out of the oven. Do you agree with this recommendation? Solve this problem using the analytical one-term approximation method. Answers: (a) $156.9 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}$, (b) $159.5^{\circ} \mathrm{C}$, (c) $1629 \mathrm{~kJ}$, (d) $3.0 \mathrm{~h}$

Erica Bischoff

Numerade Educator

Problem 84

Repeat Prob. 4 83 for a roast rib that is to be "welldone" instead of "rare." A rib is considered to be well-done when its center temperature reaches $77^{\circ} \mathrm{C}$, and the roasting in this case takes about $4 \mathrm{~h} \mathrm{} 15 \mathrm{~min}$.

Nick Johnson

Numerade Educator

Problem 85

Under what conditions can a plane wall be treated as a semi-infinite mediaum?

Katie Mcalpine

Numerade Educator

Problem 86

What is a semi-infinite medium? Give examples of solid bodies that can be treated as semi-infinite media for heat transfer purposes.

Catherine Lemar

Catherine Lemar

Numerade Educator

Problem 87

Consider a hot semi-infinite solid at an initial temperature of $T_{i}$ that is exposed to convection to a cooler medium at a constant temperature of $T_{\infty}$, with a heat transfer coefficient of $h$. Explain how you can determine the total amount of heat transfer from the solid up to a specified time $t_{v}$.

Vipender Yadav

Numerade Educator

Problem 88

The walls of a furnace are made of $1.5$-ft-thick concrete $\left(k=0.64 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft} \cdot{ }^{\circ} \mathrm{F}\right.$ and $\left.\alpha=0.023 \mathrm{ft}^{2} / \mathrm{h}\right)$. Initially, the furnace and the surrounding air are in thermal equilibrium at $70^{\circ} \mathrm{F}$. The furnace is then fired, and the inner surfaces of the furnace are subjected to hot gases at $1800^{\circ} \mathrm{F}$ with a very large heat transfer coefficient. Determine how long it will take for the temperature of the outer surface of the furnace walls to rise to $70.1^{\circ} \mathrm{F}$. Answer: $3.0 \mathrm{~h}$

Anand Jangid

Numerade Educator

Problem 89

Consider a curing kiln whose walls are made of $30-\mathrm{cm}-$ thick concrete with a thermal diffusivity of $\alpha=0.23 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}$. Initially, the kiln and its walls are in equilibrium with the surroundings at $6^{\circ} \mathrm{C}$. Then all the doors are closed and the kiln is heated by steam so that the temperature of the inner surface of the walls is raised to $42^{\circ} \mathrm{C}$ and the temperature is maintained at that level for $2.5 \mathrm{~h}$. The curing kiln is then opened and exposed to the atmospheric air after the steam flow is turned off. If the outer surfaces of the walls of the kiln were insulated, would it save any energy that day during the period the kiln was used for curing for $2.5 \mathrm{~h}$ only, or would it make no difference? Base your answer on calculations.

Mayukh Banik

Numerade Educator

Problem 90

In areas where the air temperature remains below $0^{\circ} \mathrm{C}$ for prolonged periods of time, the freezing of water in underground pipes is a major concern. Fortunately, the soil remains relatively warm during those periods, and it takes weeks for the subfreezing temperatures to reach the water mains in the ground. Thus, the soil effectively serves as an insulation to protect the water from the freezing atmospheric temperatures in winter.
The ground at a particular location is covered with snowpack at $-8^{\circ} \mathrm{C}$ for a continuous period of 60 days, and the average soil properties at that location are $k=0.35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$. Assuming an initial uniform temperature of $8^{\circ} \mathrm{C}$ for the ground, determine the minimum burial depth to prevent the water pipes from freezing.

Anand Jangid

Numerade Educator

Problem 91

A large cast iron container $(k=52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=1.70 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}$ ) with $4-\mathrm{cm}$-thick walls is initially at a uniform temperature of $0^{\circ} \mathrm{C}$ and is filled with ice at $0^{\circ} \mathrm{C}$. Now the outer surfaces of the container are exposed to hot water at $55^{\circ} \mathrm{C}$ with a very large heat transfer coefficient. Determine how long it will be before the ice inside the container starts melting. Also, taking the heat transfer coefficient on the inner surface of the container to be $250 \mathrm{~W} / \mathrm{m}^{2}$, $\mathrm{K}$, determine the rate of heat transfer to the ice through a $1.2-\mathrm{m}$-wide and $2-\mathrm{m}$-high section of the wall when steady operating conditions are reached. Assume the ice starts melting when its inner surface temperature rises to $0.1^{\circ} \mathrm{C}$.

Manish Jain

Numerade Educator

Problem 92

A highway made of asphalt is initially at a uniform temperature of $55^{\circ} \mathrm{C}$. Suddenly the highway surface temperature is reduced to $25^{\circ} \mathrm{C}$ by rain. Determine the temperature at the depth of $3 \mathrm{~cm}$ from the highway surface and the heat flux transferred from the highway after $60 \mathrm{~min}$. Assume the highway surface temperature is maintained at $25^{\circ} \mathrm{C}$. Answers: $53.6^{\circ} \mathrm{C}, 98 \mathrm{~W} / \mathrm{m}^{2}$

Mohammad Mehran

Mohammad Mehran

Numerade Educator

Problem 93

A thick aluminum block initially at $20^{\circ} \mathrm{C}$ is subjected to constant heat flux of $4000 \mathrm{~W} / \mathrm{m}^{2}$ by an electric resistance heater whose top surface is insulated. Determine how much the surface temperature of the block will rise after $30 \mathrm{~min}$.

Anand Jangid

Numerade Educator

Problem 94

Refractory bricks are used as linings for furnaces, and they generally have low thermal conductivity to minimize heat loss through the furnace walls. Consider a thick furnace wall lined with refractory bricks $(k=1.0 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=5.08 \times$ $10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ), where initially the wall has a uniform temperature of $15^{\circ} \mathrm{C}$. If the wall surface is subjected to uniform heat flux of $20 \mathrm{~kW} / \mathrm{m}^{2}$, determine the temperature at the depth of $10 \mathrm{~cm}$ from the surface after an hour of heating time.

Mayank Tripathi

Mayank Tripathi

Numerade Educator

Problem 95

A thick wall made of refractory bricks $\left(k=1.0 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$ and $\left.\alpha=5.08 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ has a uniform initial temperature of $15^{\circ} \mathrm{C}$. The wall surface is subjected to uniform heat flux of $20 \mathrm{~kW} / \mathrm{m}^{2}$. Using appropriate software, investigate the effect of heating time on the temperature at the wall surface and at $x=1 \mathrm{~cm}$ and $x=5 \mathrm{~cm}$ from the surface. Let the heating time vary from 10 to $3600 \mathrm{~s}$, and plot the temperatures at $x=0,1$, and $5 \mathrm{~cm}$ from the wall surface as a function of heating time.

Mayukh Banik

Numerade Educator

Problem 96

A stainless steel slab $(k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\left.\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ and a copper slab $(k=401$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ and $\left.\alpha=117 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ are subjected to uniform heat flux of $10 \mathrm{~kW} / \mathrm{m}^{2}$ at the surface. Both slabs have a uniform initial temperature of $20^{\circ} \mathrm{C}$. Using appropriate software, investigate the effect of time on the temperatures of both materials at the depth of $8 \mathrm{~cm}$ from the surface. By varying the time of exposure to the heat flux from 5 to $300 \mathrm{~s}$, plot the temperatures at a depth of $x=8 \mathrm{~cm}$ from the surface as a function of time.

Surjit Tewari

Numerade Educator

Problem 97

Thick slabs of stainless steel $(k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\left.\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ and copper $(k=401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=117 \times$ $10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) are subjected to uniform heat flux of $8 \mathrm{~kW} / \mathrm{m}^{2}$ at the surface. The two slabs have a uniform initial temperature of $20^{\circ} \mathrm{C}$. Determine the temperatures of both slabs, at $1 \mathrm{~cm}$ from the surface, after $60 \mathrm{~s}$ of exposure to the heat flux.

Narayan Hari

Numerade Educator

Problem 98

A thick wood slab $(k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=1.28 \times$ $10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ) that is initially at a uniform temperature of $25^{\circ} \mathrm{C}$ is exposed to hot gases at $550^{\circ} \mathrm{C}$ for a period of $5 \mathrm{~min}$. The heat transfer coefficient between the gases and the wood slab is $35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the ignition temperature of the wood is $450^{\circ} \mathrm{C}$, determine if the wood will ignite.

Yaqub Khan

Numerade Educator

Problem 99

An ASTM F441 chlorinated polyvinyl chloride (CPVC) tube is embedded in a thick metal plate.
The metal plate has a thermal conductivity of $26.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, a specific heat of $460 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and a density of $7.73 \mathrm{~g} / \mathrm{cm}^{3}$. The upper surface of the plate is occasionally exposed to hot fluid at $300^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $200 \mathrm{~W} /$ $\mathrm{m}^{2} \cdot \mathrm{K}$. The distance measured from the plate's upper surface to the tube surface is $L=2 \mathrm{~cm}$. The ASME Code for Process Piping limits the maximum use temperature for ASTM F441 CPVC tube to $93.3^{\circ} \mathrm{C}$ (ASME B31.3-2014, Table B-1). If the initial temperature of the plate is $20^{\circ} \mathrm{C}$ and the plate's upper surface is exposed to the hot fluid for $10 \mathrm{~min}$, would the CPVC tube embedded in the plate still comply with the ASME code?

Eileen Sullivan

Eileen Sullivan

Numerade Educator

Problem 100

The soil temperature in the upper layers of the earth varies with the variations in the atmospheric conditions. Before a cold front moves in, the earth at a location is initially at a uniform temperature of $15^{\circ} \mathrm{C}$. Then the area is subjected to a temperature of $-8^{\circ} \mathrm{C}$ and high winds that result in a convection heat transfer coefficient of $40 \mathrm{~W} / \mathrm{m}^{2}$ - $\mathrm{K}$ on the earth's surface for a period of $10 \mathrm{~h}$. Taking the properties of the soil at that location to be $k=0.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=1.6 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}$, determine the soil temperature at distances 0,10 , 20, and $50 \mathrm{~cm}$ from the earth's surface at the end of this 10-h period.

Averell Hause

Carnegie Mellon University

Problem 101

Reconsider Prob. 4-100. Using appropriate software, plot the soil temperature as a function of the distance from the earth's surface as the distance varies from $0 \mathrm{~m}$ to $1 \mathrm{~m}$, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 102

We often cut a watermelon in half and put it into the freezer to cool it quickly. But usually we forget to check on it and end up having a watermelon with a frozen layer on the top. To avoid this potential problem, a person wants to set a timer so that it will go off when the temperature of the exposed surface of the watermelon drops to $3^{\circ} \mathrm{C}$. Consider a $25-\mathrm{cm}$-diameter spherical watermelon that is cut into two equal parts and put into a freezer at $-12^{\circ} \mathrm{C}$. Initially, the entire watermelon is at a uniform temperature of $25^{\circ} \mathrm{C}$, and the heat transfer coefficient on the surfaces is $22 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming the watermelon to have the properties of water, determine how long it will take for the center of the exposed cut surfaces of the watermelon to drop to $3^{\circ} \mathrm{C}$.

Penny Riley

Numerade Educator

Problem 103

In a vacuum chamber, a thick slab is placed under an array of laser diodes with an output constant pulse. A thermocouple is inserted inside the slab at $25 \mathrm{~mm}$ from the surface, and the slab has an initial uniform temperature of $20^{\circ} \mathrm{C}$. The known properties of the slab are $k=63.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $a=18.8 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$. If the thermocouple measured a temperature of $130^{\circ} \mathrm{C}$ after the slab surface has been exposed to the laser pulse, for $30 \mathrm{~s}$, determine $(a)$ the amount of energy per unit surface area directed on the slab surface and (b) the thermocouple reading after $60 \mathrm{~s}$ has elapsed.

Ren Jie Tuieng

Numerade Educator

Problem 104

Thick slabs of stainless steel $(k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\left.\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ and copper $(k=401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\left.\alpha=117 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ are placed under an array of laser diodes, which supply an energy pulse of $5 \times 10^{7} \mathrm{~J} / \mathrm{m}^{2}$ instantaneously at $t=0$ to both materials. The two slabs have a uniform initial temperature of $20^{\circ} \mathrm{C}$. Determine the temperatures of both slabs at $5 \mathrm{~cm}$ from the surface and $60 \mathrm{~s}$ after receiving an energy pulse from the laser diodes.

Surjit Tewari

Numerade Educator

Problem 105

A stainless steel slab $(k=14.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\left.\alpha=3.95 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ and a copper slab $\left(k=401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$ and $\left.\alpha=117 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ are placed under an array of laser diodes, which supply an energy pulse of $5 \times 10^{7} \mathrm{~J} / \mathrm{m}^{2}$ instantaneously at $t=0$ to both materials. The two slabs have a uniform initial temperature of $20^{\circ} \mathrm{C}$. Using appropriate software, investigate the effect of time on the temperatures of both materials at the depth of $5 \mathrm{~cm}$ from the surface. By varying the time from 1 to $80 \mathrm{~s}$ after the slabs have received the energy pulse, plot the temperatures at $5 \mathrm{~cm}$ from the surface as a function of time.

Surendra Kumar

Numerade Educator

Problem 106

A barefooted person whose feet are at $32^{\circ} \mathrm{C}$ steps on a large aluminum block at $20^{\circ} \mathrm{C}$. Treating both the feet and the aluminum block as semi-infinite solids, determine the contact surface temperature. What would your answer be if the person stepped on a wood block instead? At room temperature, the $\sqrt{k \rho c_{p}}$ value is $24 \mathrm{~kJ} / \mathrm{m}^{2},{ }^{\circ} \mathrm{C}$ for aluminum, $0.38 \mathrm{~kJ} / \mathrm{m}^{2},{ }^{\circ} \mathrm{C}$ for wood, and $1.1 \mathrm{~kJ} / \mathrm{m}^{2},{ }^{\circ} \mathrm{C}$ for human flesh.

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 107

What is the product solution method? How is it used to determine the transient temperature distribution in a twodimensional system?

Shahab Ullah

Numerade Educator

Problem 108

How is the product solution used to determine the variation of temperature with time and position in threedimensional systems?

Adriano Chikande

Adriano Chikande

Numerade Educator

Problem 109

A short cylinder initially at a uniform temperature $T_{i}$ is subjected to convection from all of its surfaces to a medium at temperature $T_{\infty}$. Explain how you can determine the temperature of the midpoint of the cylinder at a specified time $t$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 110

Consider a short cylinder whose top and bottom surfaces are insulated. The cylinder is initially at a uniform temperature $T_{i}$ and is subjected to convection from its side surface to a medium at temperature $T_{\infty}$ with a heat transfer coefficient of $h$. Is the heat transfer in this short cylinder oneor two-dimensional? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 111

Consider a cubic block whose sides are $5 \mathrm{~cm}$ long and a cylindrical block whose height and diameter are also $5 \mathrm{~cm}$. Both blocks are initially at $20^{\circ} \mathrm{C}$ and are made of granite $\left(k=2.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$ and $\left.\alpha=1.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$. Now both blocks are exposed to hot gases at $500^{\circ} \mathrm{C}$ in a furnace on all of their surfaces with a heat transfer coefficient of $40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the center temperature of each geometry after 10 , 20 , and $60 \mathrm{~min}$. Solve this problem using the analytical oneterm approximation method.

Satpal Satpal

Numerade Educator

Problem 112

Repeat Prob. 4-111 with the heat transfer coefficient at the top and the bottom surfaces of each block being doubled to $80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Farnaz Mohseni

Numerade Educator

Problem 113

A long square stainless steel (ASTM A479 904L) bar is part of a component in high-temperature process equipment. The bar has a thickness of $5 \mathrm{~cm}$, and its specific heat, thermal conductivity, and density are $500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, $12 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $7.9 \mathrm{~g} / \mathrm{cm}^{3}$, respectively. Occasionally, the bar is submerged in hot fluid at $300^{\circ} \mathrm{C}$ with a convection heat transfer coefficient of $288 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. According to the ASME Code for Process Piping, the maximum use temperature for ASTM A479 904L bar is $260^{\circ} \mathrm{C}$ (ASME B31.3-2014, Table A-1M). If the initial temperature of the bar is $20^{\circ} \mathrm{C}$ and the bar is submerged in the hot fluid for $7 \mathrm{~min}$, would it be in compliance with the ASME code? How long will it take for the bar to reach the maximum use temperature? Solve this problem using the analytical one-term approximation method.

Arun Bana

Numerade Educator

Problem 114

A hot dog can be considered to be a cylinder 5 in long and $0.8$ in in diameter whose properties are $\rho=61.2 \mathrm{lbm} / \mathrm{ft}^{3}$, $c_{p}=0.93 \mathrm{Btu} / \mathrm{bm} \cdot{ }^{\circ} \mathrm{F}, k=0.44 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}{ }^{\circ} \mathrm{F}$, and $\alpha=0.0077 \mathrm{ft}^{2} / \mathrm{h}$. A hot dog initially at $40^{\circ} \mathrm{F}$ is dropped into boiling water at $212^{\circ} \mathrm{F}$. If the heat transfer coefficient at the surface of the hot dog is estimated to be $120 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}+{ }^{\circ} \mathrm{F}$, determine the center temperature of the hot dog after 5,10 , and 15 min by treating the hot dog as $(a)$ a finite cylinder and (b) an infinitely long cylinder. Solve this problem using the analytical one-term approximation method.

Keshav Singh

Numerade Educator

Problem 115

A 6-cm-high rectangular ice block $(k=2.22 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=0.124 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ) initially at $-18^{\circ} \mathrm{C}$ is placed on a table on its square base $4 \mathrm{~cm} \times 4 \mathrm{~cm}$ in size in a room at $18^{\circ} \mathrm{C}$. The heat transfer coefficient on the exposed surfaces of the ice block is $12 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$. Disregarding any heat transfer from the base to the table, determine how long it will be before the ice block starts melting. Where on the ice block will the first liquid droplets appear? Solve this problem using the analytical one-term approximation method.

Paul Gabriel

Numerade Educator

Problem 116

Reconsider Prob. 4-115. Using appropriate software, investigate the effect of the initial temperature of the ice block on the time period before the ice block starts melting. Let the initial temperature vary from $-26^{\circ} \mathrm{C}$ to $-4^{\circ} \mathrm{C}$. Plot the time versus the initial temperature, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 117

A 2-cm-high cylindrical ice block $(k=2.22 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\left.\alpha=0.124 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ is placed on a table on its base of diameter $2 \mathrm{~cm}$ in a room at $24^{\circ} \mathrm{C}$. The heat transfer coefficient on the exposed surfaces of the ice block is $13 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, and heat transfer from the base of the ice block to the table is negligible. If the ice block is not to start melting at any point for at least $3 \mathrm{~h}$, determine what the initial temperature of the ice block should be. Solve this problem using the analytical oneterm approximation method.

Manish Jain

Numerade Educator

Problem 118

A short brass cylinder $\left(\rho=8530 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=0.389 \mathrm{~kJ} /\right.$ $\mathrm{kg} \cdot \mathrm{K}, k=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\left.\alpha=3.39 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right)$ of diameter $4 \mathrm{~cm}$ and height $20 \mathrm{~cm}$ is initially at a uniform temperature of $150^{\circ} \mathrm{C}$. The cylinder is now placed in atmospheric air at $20^{\circ} \mathrm{C}$, where heat transfer takes place by convection with a heat transfer coefficient of $40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Calculate $(a)$ the center temperature of the cylinder; $(b)$ the center temperature of the top surface of the cylinder; and (c) the total heat transfer from the cylinder $15 \mathrm{~min}$ after the start of the cooling. Solve this problem using the analytical one-term approximation method.

Keshav Singh

Numerade Educator

Problem 119

Reconsider Prob. 4-118. Using appropriate software, investigate the effect of the cooling time on the center temperature of the cylinder, the center temperature of the top surface of the cylinder, and the total heat transfer. Let the time vary from $5 \mathrm{~min}$ to $60 \mathrm{~min}$. Plot the center temperature of the cylinder, the center temperature of the top surface, and the total heat transfer as a function of the time, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 120

A semi-infinite aluminum cylinder $(k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\alpha=9.71 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}$ ) of diameter $D=15 \mathrm{~cm}$ is initially at a uniform temperature of $T_{i}=115^{\circ} \mathrm{C}$. The cylinder is now placed in water at $10^{\circ} \mathrm{C}$, where heat transfer takes place by convection with a heat transfer coefficient of $h=140 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the temperature at the center of the cylinder $5 \mathrm{~cm}$ from the end surface 8 min after the start of cooling. Solve this problem using the analytical one-term approximation method.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 121

A 30 -cm-long cylindrical aluminum block $\left(\rho=2702 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=0.896 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=236 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, and $\left.\alpha=9.75 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right), 15 \mathrm{~cm}$ in diameter, is initially at a uniform temperature of $20^{\circ} \mathrm{C}$. The block is to be heated in a furnace at $1200^{\circ} \mathrm{C}$ until its center temperature rises to $300^{\circ} \mathrm{C}$. If the heat transfer coefficient on all surfaces of the block is $80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine how long the block should be kept in the furnace. Also, determine the amount of heat transfer from the aluminum block if it is allowed to cool in the room until its temperature drops to $20^{\circ} \mathrm{C}$ throughout. Solve this problem using the analytical one-term approximation method.

Manish Jain

Numerade Educator

Problem 122

Repeat Prob. 4-121 for the case where the aluminum block is inserted into the furnace on a low-conductivity material so that the heat transfer to or from the bottom surface of the block is negligible.

Narayan Hari

Numerade Educator

Problem 123

Reconsider Prob. 4-121. Using appropriate software, investigate the effect of the final center temperature of the block on the heating time and the amount of heat transfer. Let the final center temperature vary from $50^{\circ} \mathrm{C}$ to $1000^{\circ} \mathrm{C}$. Plot the time and the heat transfer as a function of the final center temperature, and discuss the results.

Jincy M Saji

Numerade Educator

Problem 124

What are the common kinds of microorganisms? What undesirable changes do microorganisms cause in foods?

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Rabia Bibi

Numerade Educator

Problem 125

How does refrigeration prevent or delay the spoilage of foods? Why does freezing extend the storage life of foods for months?

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Rabia Bibi

Numerade Educator

Problem 126

What are the environmental factors that affect the growth rate of microorganisms in foods?

Narayan Hari

Numerade Educator

Problem 127

What is the effect of cooking on the microorganisms in foods? Why is it important that the internal temperature of a roast in an oven be raised above $70^{\circ} \mathrm{C}$ ?

Matthew Mcvay

Numerade Educator

Problem 128

How can the contamination of foods with microorganisms be prevented or minimized? How can the growth of microorganisms in foods be retarded? How can the microorganisms in foods be destroyed?

Qudsiya Anis

Numerade Educator

Problem 129

How does $(a)$ the air motion and $(b)$ the relative humidity of the environment affect the growth of microorganisms in foods?

Keshav Singh

Numerade Educator

Problem 130

The cooling of a beef carcass from $37^{\circ} \mathrm{C}$ to $5^{\circ} \mathrm{C}$ with refrigerated air at $0^{\circ} \mathrm{C}$ in a chilling room takes about $48 \mathrm{~h}$. To reduce the cooling time, it is proposed to cool the carcass with refrigerated air at $-10^{\circ} \mathrm{C}$. How would you evaluate this proposal?

Vipender Yadav

Numerade Educator

Problem 131

Consider the freezing of packaged meat in boxes with refrigerated air. How do $(a)$ the temperature of air, $(b)$ the velocity of air, $(c)$ the capacity of the refrigeration system, and $(d)$ the size of the meat boxes affect the freezing time?

Vipender Yadav

Numerade Educator

Problem 132

How does the rate of freezing affect the tenderness, color, and the drip of meat during thawing?

David Collins

Numerade Educator

Problem 133

It is claimed that beef can be stored for up to two years at $-23^{\circ} \mathrm{C}$ but no more than one year at $-12^{\circ} \mathrm{C}$. Is this claim reasonable? Explain.

Jessica Bisagno

Jessica Bisagno

Numerade Educator

Problem 134

What is a refrigerated shipping dock? How does it reduce the refrigeration load of the cold storage rooms?

Keshav Singh

Numerade Educator

Problem 135

How does immersion chilling of poultry compare to forced-air chilling with respect to $(a)$ cooling time, $(b)$ moisture loss of poultry, and (c) microbial growth?

Lottie Adams

Numerade Educator

Problem 136

What is the proper storage temperature of frozen poultry? What are the primary methods of freezing for poultry?

Vipender Yadav

Numerade Educator

Problem 137

What are the factors that affect the quality of frozen fish?

Alexander Cheng

Alexander Cheng

Numerade Educator

Problem 138

The chilling room of a meat plant is $15 \mathrm{~m} \times$ $18 \mathrm{~m} \times 5.5 \mathrm{~m}$ in size and has a capacity of 350 beef carcasses. The power consumed by the fans and the lights in the chilling room are 22 and $2 \mathrm{~kW}$, respectively, and the room gains heat through its envelope at a rate of $14 \mathrm{~kW}$. The average mass of beef carcasses is $220 \mathrm{~kg}$. The carcasses enter the chilling room at $35^{\circ} \mathrm{C}$, after they are washed to facilitate evaporative cooling, and are cooled to $16^{\circ} \mathrm{C}$ in $12 \mathrm{~h}$. The air enters the chilling room at $-2.2^{\circ} \mathrm{C}$ and leaves at $0.5^{\circ} \mathrm{C}$. Determine $(a)$ the refrigeration load of the chilling room and $(b)$ the volume flow rate of air. The average specific heats of beef carcasses and air are $3.14$ and $1.0 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$, respectively, and the density of air can be taken to be $1.28 \mathrm{~kg} / \mathrm{m}^{3}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 139

Chickens with an average mass of $2.2 \mathrm{~kg}$ and average specific heat of $3.54 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$ are to be cooled by chilled water that enters a continuous-flow-type immersion chiller at $0.5^{\circ} \mathrm{C}$. Chickens are dropped into the chiller at a uniform temperature of $15^{\circ} \mathrm{C}$ at a rate of 500 chickens per hour and are cooled to an average temperature of $3^{\circ} \mathrm{C}$ before they are taken out. The chiller gains heat from the surroundings at a rate of $210 \mathrm{~kJ} / \mathrm{min}$. Determine $(a)$ the rate of heat removal from the chicken, in $\mathrm{kW}$, and ( $b$ ) the mass flow rate of water, in $\mathrm{kg} / \mathrm{s}$, if the temperature rise of water is not to exceed $2^{\circ} \mathrm{C}$.

Naman Kumar

Numerade Educator

Problem 140

Chickens with a water content of 74 percent, an initial temperature of $32^{\circ} \mathrm{F}$, and a mass of about $7.5 \mathrm{lbm}$ are to be frozen by refrigerated air at $-40^{\circ} \mathrm{F}$. Using Fig. $4-50$, determine how long it will take to reduce the inner surface temperature of chickens to $25^{\circ} \mathrm{F}$. What would your answer be if the air temperature were $-80^{\circ} \mathrm{F}$ ?

Jincy M Saji

Numerade Educator

Problem 141

Turkeys with a water content of 64 percent that are initially at $1^{\circ} \mathrm{C}$ and have a mass of about $7 \mathrm{~kg}$ are to be frozen by submerging them into brine at $-29^{\circ} \mathrm{C}$. Using Fig. $4-51$, determine how long it will take to reduce the temperature of the turkey breast at a depth of $3.8 \mathrm{~cm}$ to $-188^{\circ} \mathrm{C}$. If the temperature at a depth of $3.8 \mathrm{~cm}$ in the breast represents the average temperature of the turkey, determine the amount of heat transfer per turkey assuming $(a)$ the entire water content of the turkey is frozen and (b) only 90 percent of the water content of the turkey is frozen at $-18^{\circ} \mathrm{C}$. Take the specific heats of turkey to be $2.98$ and $1.65 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$ above and below the freezing point of $-2.8^{\circ} \mathrm{C}$, respectively, and the latent heat of fusion of turkey to be $214 \mathrm{~kJ} / \mathrm{kg}$. Answers: (a) $1753 \mathrm{~kJ}$, (b) $1617 \mathrm{~kJ}$

Naman Kumar

Numerade Educator

Problem 142

Large steel plates $1.0 \mathrm{~cm}$ in thickness are quenched from $600^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ by submerging them in an oil reservoir held at $30^{\circ} \mathrm{C}$. The average heat transfer coefficient for both faces of steel plates is $400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Average steel properties are $k=45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=7800 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=470 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Calculate the quench time for steel plates.

Naman Kumar

Numerade Educator

Problem 143

A long roll of 2 -m-wide and $0.5$-cm-thick 1-Mn manganese steel plate coming off a furnace at $820^{\circ} \mathrm{C}$ is to be quenched in an oil bath $\left(c_{p}=2.0 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)$ at $45^{\circ} \mathrm{C}$. The metal sheet is moving at a steady velocity of $20 \mathrm{~m} / \mathrm{min}$, and the oil bath is $9 \mathrm{~m}$ long. Taking the convection heat transfer coefficient on both sides of the plate to be $860 \mathrm{~W} / \mathrm{m}^{2}$. $\mathrm{K}$, determine the temperature of the sheet metal when it leaves the oil bath. Also, determine the required rate of heat removal from the oil to keep its temperature constant at $45^{\circ} \mathrm{C}$.

Dading Chen

Numerade Educator

Problem 144

Long aluminum wires of diameter $3 \mathrm{~mm}\left(\rho=2702 \mathrm{~kg} / \mathrm{m}^{3}\right.$, $c_{p}=0.896 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=236 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\left.\alpha=9.75 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right)$ are extruded at a temperature of $350^{\circ} \mathrm{C}$ and exposed to atmospheric air at $30^{\circ} \mathrm{C}$ with a heat transfer coefficient of $35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. (a) Determine how long it will take for the wire temperature to drop to $50^{\circ} \mathrm{C}$. (b) If the wire is extruded at a velocity of 10 $\mathrm{m} / \mathrm{min}$, determine how far the wire travels after extrusion by the time its temperature drops to $50^{\circ} \mathrm{C}$. What change in the cooling process would you propose to shorten this distance?
(c) Assuming the aluminum wire leaves the extrusion room at $50^{\circ} \mathrm{C}$, determine the rate of heat transfer from the wire to the extrusion room. Answers: (a) $144 \mathrm{~s}$, (b) $24 \mathrm{~m}$, (c) $856 \mathrm{~W}$

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 145

Repeat Prob. 4-144 for a copper wire $\left(\rho=8950 \mathrm{~kg} / \mathrm{m}^{3}\right.$, $c_{p}=0.383 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=386 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=1.13 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}$ ).

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 146

Aluminum wires $4 \mathrm{~mm}$ in diameter are produced by extrusion. The wires leave the extruder at an average temperature of $350^{\circ} \mathrm{C}$ and at a linear rate of $10 \mathrm{~m} / \mathrm{min}$. Before leaving the extrusion room, the wires are cooled to an average temperature of $50^{\circ} \mathrm{C}$ by transferring heat to the surrounding air at $25^{\circ} \mathrm{C}$ with a heat transfer coefficient of $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Calculate the necessary length of the wire cooling section in the extrusion room.

Hariprasad Annamalai

Hariprasad Annamalai

Numerade Educator

Problem 147

During a picnic on a hot summer day, the only available drinks were those at the ambient temperature of $90^{\circ} \mathrm{F}$. In an effort to cool a 12 -fluid-oz drink in a can, which is 5 in high and has a diameter of $2.5 \mathrm{in}$, a person grabs the can and starts shaking it in the iced water of the chest at $32^{\circ} \mathrm{F}$. The temperature of the drink can be assumed to be uniform at all times, and the heat transfer coefficient between the iced water and the aluminum can is $30 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}{ }^{\circ} \mathrm{F}$. Using the properties of water for the drink, estimate how long it will take for the canned drink to cool to $40^{\circ} \mathrm{F}$. Solve this problem using lumped system analysis. Is the lumped system analysis applicable to this problem? Why?

Matthew Biollo

Numerade Educator

Problem 148

Two metal rods are being heated in an oven with uniform ambient temperature of $1000^{\circ} \mathrm{C}$ and convection heat transfer coefficient of $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Rod $\mathrm{A}$ is made of aluminum $\left(\rho=2702 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=903 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $\left.k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$ and $\mathrm{rod}$ $\mathrm{B}$ is made of stainless steel $\left(\rho=8238 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=468 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $k=13.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. Both rods have a diameter of $25 \mathrm{~mm}$ and a length of $1 \mathrm{~m}$. If the initial temperature of both rods is $15^{\circ} \mathrm{C}$, determine the average temperatures of both rods after $5 \mathrm{~min}$.

Rashmi Sinha

Numerade Educator

Problem 149

Stainless steel ball bearings $\left(\rho=8085 \mathrm{~kg} / \mathrm{m}^{3}\right.$, $k=15.1 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}, \quad c_{p}=0.480 \mathrm{KJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, \quad$ and $\quad \alpha=3.91 \times$ $10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) having a diameter of $1.2 \mathrm{~cm}$ are to be quenched in water. The balls leave the oven at a uniform temperature of $900^{\circ} \mathrm{C}$ and are exposed to air at $30^{\circ} \mathrm{C}$ for a while before they are dropped into the water. If the temperature of the balls is not to fall below $850^{\circ} \mathrm{C}$ prior to quenching and the heat transfer coefficient in the air is $125 \mathrm{~W} / \mathrm{m}^{2},{ }^{\circ} \mathrm{C}$, determine how long they can stand in the air before being dropped into the water.

Keshav Singh

Numerade Educator

Problem 150

In an annealing process, a 50-mm-thick stainless steel plate $\left(\rho=8238 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=468 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=13.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, and $\alpha=3.48 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) was reheated in a furnace from an initial uniform temperature of $230^{\circ} \mathrm{C}$. The ambient temperature inside the furnace is uniform at $1000^{\circ} \mathrm{C}$, and the convection heat transfer coefficient is $215 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the entire stainless steel plate is to be heated to at least $600^{\circ} \mathrm{C}$, determine the time that the plate should be heated in the furnace using the analytical one-term approximation method.

Vipender Yadav

Numerade Educator

Problem 151

During a fire, the trunks of some dry oak trees $\left(k=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$ and $\left.\alpha=1.28 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ that are initially at a uniform temperature of $30^{\circ} \mathrm{C}$ are exposed to hot gases at $600^{\circ} \mathrm{C}$ for a period of $4 \mathrm{~h}$, with a heat transfer coefficient of $65 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ on the surface. The ignition temperature of the trees is $410^{\circ} \mathrm{C}$. Treating the trunks of the trees as long cylindrical rods of diameter $20 \mathrm{~cm}$, determine if these dry trees will ignite as the fire sweeps through them. Solve this problem using the analytical one-term approximation method.

Lainey Roebuck

Numerade Educator

Problem 152

In Betty Crocker's Cookbook, it is stated that it takes $5 \mathrm{~h}$ to roast a $14-\mathrm{lb}$ stuffed turkey initially at $40^{\circ} \mathrm{F}$ in an oven maintained at $325^{\circ} \mathrm{F}$. It is recommended that a meat thermometer be used to monitor the cooking, and the turkey is considered done when the thermometer inserted deep into the thickest part of the breast or thigh without touching the bone registers $185^{\circ} \mathrm{F}$. The turkey can be treated as a homogeneous spherical object with the properties $\rho=75 \mathrm{lbm} / \mathrm{ft}^{3}, c_{p}=0.98 \mathrm{Btu} / \mathrm{lbm}-{ }^{\circ} \mathrm{F}$, $k=0.26 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{\circ} \mathrm{F}$, and $\alpha=0.0035 \mathrm{ft}^{2} / \mathrm{h}$. Assuming the tip of the thermometer is at one-third radial distance from the center of the turkey, determine $(a)$ the average heat transfer coefficient at the surface of the turkey, $(b)$ the temperature of the skin of the turkey when it is done, and (c) the total amount of heat transferred to the turkey in the oven. Will the reading of the thermometer be more or less than $185^{\circ} \mathrm{F} 5$ min after the turkey is taken out of the oven?

Nick Johnson

Numerade Educator

Problem 153

A watermelon initially at $35^{\circ} \mathrm{C}$ is to be cooled by dropping it into a lake at $15^{\circ} \mathrm{C}$. After $4 \mathrm{~h}$ and $40 \mathrm{~min}$ of cooling, the center temperature of the watermelon is measured to be $20^{\circ} \mathrm{C}$. Treating the watermelon as a $20-\mathrm{cm}$-diameter sphere and using the properties $k=0.618 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$, $\rho=995 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=4.18 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$, determine the average heat transfer coefficient and the surface temperature of the watermelon at the end of the cooling period. Solve this problem using the analytical one-term approximation method.

Yaqub Khan

Numerade Educator

Problem 154

Spherical glass beads coming out of a kiln are allowed to cool in a room temperature of $30^{\circ} \mathrm{C}$. A glass bead with a diameter of $10 \mathrm{~mm}$ and an initial temperature of $400^{\circ} \mathrm{C}$ is allowed to cool for $3 \mathrm{~min}$. If the convection heat transfer coefficient is $28 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the temperature at the center of the glass bead using the analytical one-term approximation method. The glass bead has properties of $\rho=2800 \mathrm{~kg} / \mathrm{m}^{3}$, $c_{p}=750 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=0.7 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

Anand Jangid

Numerade Educator

Problem 155

Water mains must be placed at sufficient depth below the earth's surface to avoid freezing during extended periods of subfreezing temperatures. Determine the minimum depth at which the water main must be placed at a location where the soil is initially at $15^{\circ} \mathrm{C}$ and the earth's surface temperature under the worst conditions is expected to remain at $-10^{\circ} \mathrm{C}$ for 75 days. Take the properties of soil at that location to be $k=0.7 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=1.4 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}$. Answer: $7.05 \mathrm{~m}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 156

A 40 -cm-thick brick wall $(k=0.72 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=1.6 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) is heated to an average temperature of $18^{\circ} \mathrm{C}$ by the heating system and the solar radiation incident on it during the day. During the night, the outer surface of the wall is exposed to cold air at $-3^{\circ} \mathrm{C}$ with an average heat transfer coefficient of $20 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the wall temperatures at distances 15,30 , and $40 \mathrm{~cm}$ from the outer surface for a period of $2 \mathrm{~h}$.

Anand Jangid

Numerade Educator

Problem 157

In a volcano eruption, lava at $1200^{\circ} \mathrm{C}$ is found flowing on the ground. The ground was initially at $15^{\circ} \mathrm{C}$, and the lava flow has a convection heat transfer coefficient of $3500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the ground surface $(a)$ temperature and $(b)$ heat flux after 2 s of lava flow. Assume the ground has properties of dry soil.

cm

Charles Magnusen

Numerade Educator

Problem 158

A large heated steel block $\left(\rho=7832 \mathrm{~kg} / \mathrm{m}^{3}\right.$, $c_{p}=434 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=63.9 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\left.\alpha=18.8 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ is allowed to cool in a room at $25^{\circ} \mathrm{C}$. The steel block has an initial temperature of $450^{\circ} \mathrm{C}$, and the convection heat transfer coefficient is $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming that the steel block can be treated as a quarter-infinite medium, determine the temperature at the edge of the steel block after $10 \mathrm{~min}$ of cooling.

Varsha Aggarwal

Varsha Aggarwal

Numerade Educator

Problem 159

A large iron slab $\left(\rho=7870 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=447 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $k=80.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ was initially heated to a uniform temperature of $150^{\circ} \mathrm{C}$ and then placed on a concrete floor $\left(\rho=1600 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=840 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, and $\left.k=0.79 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$. The concrete floor was initially at a uniform temperature of $30^{\circ} \mathrm{C}$. Determine $(a)$ the surface temperature between the iron slab and concrete floor and $(b)$ the temperature of the concrete floor at the depth of $25 \mathrm{~mm}$, if the surface temperature remains constant after $15 \mathrm{~min}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 160

A hot dog can be considered to be a $12-\mathrm{cm}$-long cylinder whose diameter is $2 \mathrm{~cm}$ and whose properties are $\rho=980 \mathrm{~kg} / \mathrm{m}^{3}$, $c_{p}=3.9 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.76 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=2 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$. A hot dog initially at $5^{\circ} \mathrm{C}$ is dropped into boiling water at $100^{\circ} \mathrm{C}$. The heat transfer coefficient at the surface of the hot dog is estimated to be $600 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the hot $\operatorname{dog}$ is considered cooked when its center temperature reaches $80^{\circ} \mathrm{C}$, determine how long it will take to cook it in the boiling water. Solve this problem using the analytical one-term approximation method.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 161

Consider the engine block of a car made of cast iron $\left(k=52 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$ and $\left.\alpha=1.7 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\right)$. The engine can be considered to be a rectangular block whose sides are $80 \mathrm{~cm}$, $40 \mathrm{~cm}$, and $40 \mathrm{~cm}$. The engine is at a temperature of $150^{\circ} \mathrm{C}$ when it is turned off. The engine is then exposed to atmospheric air at $17^{\circ} \mathrm{C}$ with a heat transfer coefficient of $6 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$. Determine $(a)$ the center temperature of the top surface whose sides are $80 \mathrm{~cm}$ and $40 \mathrm{~cm}$ and $(b)$ the corner temperature after 45 min of cooling. Solve this problem using the analytical one-term approximation method.

Satpal Satpal

Numerade Educator

Problem 162

A man is found dead in a room at $12^{\circ} \mathrm{C}$. The surface temperature on his waist is measured to be $23^{\circ} \mathrm{C}$, and the heat transfer coefficient is estimated to be $9 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Modeling the body as a $28-\mathrm{cm}$ diameter, $1.80$-m-long cylinder, estimate how long it has been since he died. Take the properties of the body to be $k=0.62 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=0.15 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$, and assume the initial temperature of the body to be $36^{\circ} \mathrm{C}$. Solve this problem using the analytical one-term approximation method.

Stanley Enemuo

Numerade Educator

Problem 163

The Biot number can be thought of as the ratio of
(a) the conduction thermal resistance to the convective thermal resistance
(b) the convective thermal resistance to the conduction thermal resistance
(c) the thermal energy storage capacity to the conduction thermal resistance
(d) the thermal energy storage capacity to the convection thermal resistance
(e) none of the above

Anand Jangid

Numerade Educator

Problem 164

Lumped system analysis of transient heat conduction situations is valid when the Biot number is
(a) very small
(b) approximately one
(c) very large
(d) any real number
(e) cannot say unless the Fourier number is also known

Anand Jangid

Numerade Educator

Problem 165

Polyvinylchloride automotive body panels $(k=0.092$ $\left.\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=1.05 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, \rho=1714 \mathrm{~kg} / \mathrm{m}^{3}\right), 1 \mathrm{~mm}$ thick, emerge from an injection molder at $120^{\circ} \mathrm{C}$. They need to be cooled to $40^{\circ} \mathrm{C}$ by exposing both sides of the panels to $20^{\circ} \mathrm{C}$ air before they can be handled. If the convective heat transfer coefficient is $15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and radiation is not considered, the time that the panels must be exposed to air before they can be handled is
(a) $0.8 \mathrm{~min}$
(b) $1.6 \mathrm{~min}$
(c) $2.4 \mathrm{~min}$
(d) $3.1 \mathrm{~min}$
(e) $5.6 \mathrm{~min}$

Eric Mockensturm

Eric Mockensturm

Numerade Educator

Problem 166

A steel casting cools to 90 percent of the original temperature difference in $30 \mathrm{~min}$ in still air. The time it takes to cool this same casting to 90 percent of the original temperature difference in a moving air stream whose convective heat transfer coefficient is 5 times that of still air is
(a) $3 \mathrm{~min}$
(b) $6 \mathrm{~min}$
(c) $9 \mathrm{~min}$
(d) $12 \mathrm{~min}$
(e) $15 \mathrm{~min}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 167

An 18-cm-long, 16-cm-wide, and 12-cm-high hot iron block $\left(\rho=7870 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=447 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ initially at $20^{\circ} \mathrm{C}$ is placed in an oven for heat treatment. The heat transfer coefficient on the surface of the block is $100 \mathrm{~W} / \mathrm{m}^{2} . \mathrm{K}$. If the temperature of the block must rise to $750^{\circ} \mathrm{C}$ in a 25 -min period, the oven must be maintained at
(a) $750^{\circ} \mathrm{C}$
(b) $830^{\circ} \mathrm{C}$
(c) $875^{\circ} \mathrm{C}$
(d) $910^{\circ} \mathrm{C}$
(e) $1000^{\circ} \mathrm{C}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 168

A $10-\mathrm{cm}$-inner-diameter, 30-cm-long can filled with water initially at $25^{\circ} \mathrm{C}$ is put into a household refrigerator at $3^{\circ} \mathrm{C}$. The heat transfer coefficient on the surface of the can is $14 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Assuming that the temperature of the water remains uniform during the cooling process, the time it takes for the water temperature to drop to $5^{\circ} \mathrm{C}$ is
(a) $0.55 \mathrm{~h}$
(b) $1.17 \mathrm{~h}$
(c) $2.09 \mathrm{~h}$
(d) $3.60 \mathrm{~h}$
(e) $4.97 \mathrm{~h}$

Surendra Kumar

Numerade Educator

Problem 169

A 6-cm-diameter, 13-cm-high canned drink $(\rho=977$ $\left.\mathrm{kg} / \mathrm{m}^{3}, k=0.607 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ initially at $25^{\circ} \mathrm{C}$ is to be cooled to $5^{\circ} \mathrm{C}$ by dropping it into iced water at $0^{\circ} \mathrm{C}$. Total surface area and volume of the drink are $A_{s}=301.6 \mathrm{~cm}^{2}$ and $V=367.6 \mathrm{~cm}^{3}$. If the heat transfer coefficient is $120 \mathrm{~W} / \mathrm{m}^{2}, \mathrm{~K}$, determine how long it will take for the drink to cool to $5^{\circ} \mathrm{C}$. Assume the can is agitated in water, and thus the temperature of the drink changes uniformly with time.
(a) $1.5 \mathrm{~min}$
(b) $8.7 \mathrm{~min}$
(c) $11.1 \mathrm{~min}$
(d) $26.6 \mathrm{~min}$
(e) $6.7 \mathrm{~min}$

Paul Gabriel

Numerade Educator

Problem 170

Copper balls $\left(\rho=8933 \mathrm{~kg} / \mathrm{m}^{3}, \quad k=401 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, $c_{p}=385 \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}, \alpha=1.166 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}$ ) initially at $180^{\circ} \mathrm{C}$ are allowed to cool in air at $30^{\circ} \mathrm{C}$ for a period of $2 \mathrm{~min}$. If the balls have a diameter of $2 \mathrm{~cm}$ and the heat transfer coefficient is $80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, the center temperature of the balls at the end of cooling is
(a) $78^{\circ} \mathrm{C}$
(b) $95^{\circ} \mathrm{C}$
(c) $118^{\circ} \mathrm{C}$
(d) $134^{\circ} \mathrm{C}$
(e) $151^{\circ} \mathrm{C}$

Anand Jangid

Numerade Educator

Problem 171

Carbon steel balls $\left(\rho=7830 \mathrm{~kg} / \mathrm{m}^{3}, k=64 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, $\left.c_{p}=434 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ initially at $200^{\circ} \mathrm{C}$ are quenched in an oil bath at $20^{\circ} \mathrm{C}$ for a period of $3 \mathrm{~min}$. If the balls have a diameter of $5 \mathrm{~cm}$ and the convection heat transfer coefficient is $450 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, the center temperature of the balls after quenching will be (Hint: Check the Biot number.)
(a) $30.3^{\circ} \mathrm{C}$
(b) $46.1^{\circ} \mathrm{C}$
(c) $55.4^{\circ} \mathrm{C}$
(d) $68.9^{\circ} \mathrm{C}$
(e) $79.4^{\circ} \mathrm{C}$

Vipender Yadav

Numerade Educator

Problem 172

In a production facility, large plates made of stainless steel $\left(k=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=3.91 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right)$ of $40 \mathrm{~cm}$ thickness are taken out of an oven at a uniform temperature of $750^{\circ} \mathrm{C}$. The plates are placed in a water bath that is kept at a constant temperature of $20^{\circ} \mathrm{C}$ with a heat transfer coefficient of $600 \mathrm{~W} /$ $\mathrm{m}^{2} \cdot \mathrm{K}$. The time it takes for the surface temperature of the plates to drop to $120^{\circ} \mathrm{C}$ is
(a) $0.6 \mathrm{~h}$
(b) $0.8 \mathrm{~h}$
(c) $1.4 \mathrm{~h}$
(d) $2.6 \mathrm{~h}$
(e) $3.2 \mathrm{~h}$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 173

A long 18-cm-diameter bar made of hardwood $(k=$ $\left.0.159 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=1.75 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ is exposed to air at $30^{\circ} \mathrm{C}$ with a heat transfer coefficient of $8.83 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the center temperature of the bar is measured to be $15^{\circ} \mathrm{C}$ after $3 \mathrm{~h}$, the initial temperature of the bar is
(a) $11.9^{\circ} \mathrm{C}$
(b) $4.9^{\circ} \mathrm{C}$
(c) $1.7^{\circ} \mathrm{C}$
(d) $0^{\circ} \mathrm{C}$
(e) $-9.2^{\circ} \mathrm{C}$

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 174

Consider a 7.6-cm-long and 3-cm-diameter cylindrical lamb meat chunk $\left(\rho=1030 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.49 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right.$, $k=0.456 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=1.3 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ). Fifteen such meat chunks initially at $2^{\circ} \mathrm{C}$ are dropped into boiling water at $95^{\circ} \mathrm{C}$ with a heat transfer coefficient of $1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The amount of heat transfer during the first $8 \mathrm{~min}$ of cooking is
(a) $71 \mathrm{~kJ}$
(b) $227 \mathrm{~kJ}$
(c) $238 \mathrm{~kJ}$
(d) $269 \mathrm{~kJ}$
(e) $307 \mathrm{~kJ}$

Vipender Yadav

Numerade Educator

Problem 175

Consider a 7.6-cm-diameter cylindrical lamb meat chunk $\left(\rho=1030 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=3.49 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.456 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right.$, $\alpha=1.3 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ ). Such a meat chunk intially at $2^{\circ} \mathrm{C}$ is dropped into boiling water at $95^{\circ} \mathrm{C}$ with a heat transfer coefficient of $1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The time it takes for the center temperature of the meat chunk to rise to $75^{\circ} \mathrm{C}$ is
(a) $136 \mathrm{~min}$
(b) $21.2 \mathrm{~min}$
(c) $13.6 \mathrm{~min}$
(d) $11.0 \mathrm{~min}$
(e) $8.5 \mathrm{~min}$

Yaqub Khan

Numerade Educator

Problem 176

A potato may be approximated as a $5.7-\mathrm{cm}$ solid sphere with the properties $\rho=910 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.25 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$, $k=0.68 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=1.76 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$. Twelve such potatoes initially at $25^{\circ} \mathrm{C}$ are to be cooked by placing them in an oven maintained at $250^{\circ} \mathrm{C}$ with a heat transfer coefficient of $95 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The amount of heat transfer to the potatoes by the time the center temperature reaches $90^{\circ} \mathrm{C}$ is
(a) $1012 \mathrm{~kJ}$
(b) $1366 \mathrm{~kJ}$
(c) $1788 \mathrm{~kJ}$
(d) $2046 \mathrm{~kJ}$
(e) $3270 \mathrm{~kJ}$

Vipender Yadav

Numerade Educator

Problem 177

A small chicken $(k=0.45 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad \alpha=0.15 \times$ $10^{-6} \mathrm{~m}^{2} / \mathrm{s}$ ) can be approximated as an $11.25-\mathrm{cm}$-diameter solid sphere. The chicken is initially at a uniform temperature of $8^{\circ} \mathrm{C}$ and is to be cooked in an oven maintained at $220^{\circ} \mathrm{C}$ with a heat transfer coefficient of $80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. With this idealization, the temperature at the center of the chicken after a 90 -min period is
(a) $25^{\circ} \mathrm{C}$
(b) $61^{\circ} \mathrm{C}$
(c) $89^{\circ} \mathrm{C}$
(d) $122^{\circ} \mathrm{C}$
(e) $168^{\circ} \mathrm{C}$

Saman Zulfiqar

Numerade Educator

Problem 178

A potato may be approximated as a $5.7-\mathrm{cm}$-diameter solid sphere with the properties $\rho=910 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4.25 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}$, $k=0.68 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\alpha=1.76 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$. Twelve such potatoes initially at $25^{\circ} \mathrm{C}$ are to be cooked by placing them in an oven maintained at $250^{\circ} \mathrm{C}$ with a heat transfer coefficient of $95 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The amount of heat transfer to the potatoes during a 30 -min period is
(a) $77 \mathrm{~kJ}$
(b) $483 \mathrm{~kJ}$
(c) $927 \mathrm{~kJ}$
(d) $970 \mathrm{~kJ}$
(e) $1012 \mathrm{~kJ}$

Vipender Yadav

Numerade Educator

Problem 179

When water, as in a pond or lake, is heated by warm air above it, it remains stable, does not move, and forms a warm layer of water on top of a cold layer. Consider a deep lake $\left(k=0.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=4.179 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}\right)$ that is initially at a uniform temperature of $2^{\circ} \mathrm{C}$ and has its surface temperature suddenly increased to $20^{\circ} \mathrm{C}$ by a spring weather front. The temperature of the water $1 \mathrm{~m}$ below the surface $400 \mathrm{~h}$ after this change is
(a) $2.1^{\circ} \mathrm{C}$
(b) $4.2^{\circ} \mathrm{C}$
(c) $6.3^{\circ} \mathrm{C}$
(d) $8.4^{\circ} \mathrm{C}$
(e) $10.2^{\circ} \mathrm{C}$

Ashok Prajapati

Ashok Prajapati

Numerade Educator

Problem 180

A large chunk of tissue at $35^{\circ} \mathrm{C}$ with a thermal diffusivity of $1 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$ is dropped into iced water. The water is well-stirred so that the surface temperature of the tissue drops to $0^{\circ} \mathrm{C}$ at time zero and remains at $0^{\circ} \mathrm{C}$ at all times. The temperature of the tissue after 4 min at a depth of $1 \mathrm{~cm}$ is
(a) $5^{\circ} \mathrm{C}$
(b) $30^{\circ} \mathrm{C}$
(c) $25^{\circ} \mathrm{C}$
(d) $20^{\circ} \mathrm{C}$
(e) $10^{\circ} \mathrm{C}$

Ren Jie Tuieng

Numerade Educator

Problem 181

The 35-cm-thick roof of a large room made of concrete $\left(k=0.79 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \alpha=5.88 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}\right)$ is initially at a uniform temperature of $15^{\circ} \mathrm{C}$. After a heavy snowstorm, the outer surface of the roof remains covered with snow at $-5^{\circ} \mathrm{C}$. The roof temperature at $12 \mathrm{~cm}$ distance from the outer surface after $2 \mathrm{~h}$ is
(a) $13^{\circ} \mathrm{C}$
(b) $11^{\circ} \mathrm{C}$
(c) $7^{\circ} \mathrm{C}$
(d) $3^{\circ} \mathrm{C}$
(e) $-5^{\circ} \mathrm{C}$

Saman Zulfiqar

Numerade Educator

Problem 182

Conduct the following experiment at home to determine the combined convection and radiation heat transfer coefficient at the surface of an apple exposed to the room air. You will need two thermometers and a clock.
First, weigh the apple and measure its diameter. You can measure its volume by placing it in a large measuring cup halfway filled with water, and measuring the change in volume when it is completely immersed in the water. Refrigerate the apple overnight so that it is at a uniform temperature in the morning, and measure the air temperature in the kitchen. Then take the apple out and stick one of the thermometers to its middle and the other just under the skin. Record both temperatures every $5 \mathrm{~min}$ for an hour. Using these two temperatures, calculate the heat transfer coefficient for each interval and take their average. The result is the combined convection and radiation heat transfer coefficient for this heat transfer process. Using your experimental data, also calculate the thermal conductivity and thermal diffusivity of the apple and compare them to the values given above.

Nick Auwerda

Numerade Educator

Problem 183

Repeat Prob. 4-182 using a banana instead of an apple. The thermal properties of bananas are practically the same as those of apples.

Kratika Bhadauria

Kratika Bhadauria

Numerade Educator

Problem 184

Conduct the following experiment to determine the time constant for a can of soda and then predict the temperature of the soda at different times. Leave the soda in the refrigerator overnight. Measure the air temperature in the kitchen and the temperature of the soda while it is still in the refrigerator by taping the sensor of the thermometer to the outer surface of the can. Then take the soda out and measure its temperature again in $5 \mathrm{~min}$. Using these values, calculate the exponent $b$. Using this $b$-value, predict the temperatures of the soda in 10 , $15,20,30$, and $60 \mathrm{~min}$ and compare the results with the actual temperature measurements. Do you think the lumped system analysis is valid in this case?

Darshan Maheshwari

Darshan Maheshwari

Numerade Educator

Problem 185

Citrus trees are very susceptible to cold weather, and extended exposure to subfreezing temperatures can destroy the crop. In order to protect the trees from occasional cold fronts with subfreezing temperatures, tree growers in Florida usually install water sprinklers on the trees. When the temperature drops below a certain level, the sprinklers spray water on the trees and their fruits to protect them against the damage the subfreezing temperatures can cause. Explain the basic mechanism behind this protection measure, and write an essay on how the system works in practice.

Jennifer Stoner

Jennifer Stoner

Numerade Educator