• Home
  • Textbooks
  • Fundamentals of Heat and Mass Transfer
  • Free Convection

Fundamentals of Heat and Mass Transfer

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

Chapter 9

Free Convection - all with Video Answers

Educators

Chapter Questions

05:09

Problem 1

The one-dimensional plane wall of Figure $3.1$ is of thickness $L=75 \mathrm{~mm}$ and thermal conductivity $k=$ $5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The fluid temperatures are $T_{\infty, 1}=200^{\circ} \mathrm{C}$ and $T_{\infty, 2}=100^{\circ} \mathrm{C}$, respectively. Using the minimum and maximum typical values of the convection heat transfer coefficients listed in Table $1.1$, determine the minimum and maximum steady-state heat fluxes through the wall for (i) free convection in gases, (ii) free convection in liquids, (iii) forced convection in gases, (iv) forced convection in liquids, and (v) convection with phase change.

David Morabito
David Morabito
Numerade Educator
03:48

Problem 2

Using the values of density for water in Table A.6, calculate the volumetric thermal expansion coefficient at $300 \mathrm{~K}$ from its definition, Equation $9.4$, and compare your result with the tabulated value.

Keshav Singh
Keshav Singh
Numerade Educator
04:26

Problem 3

Consider an object of characteristic length $0.01 \mathrm{~m}$ and a situation for which the temperature difference is $30^{\circ} \mathrm{C}$. Evaluating thermophysical properties at the prescribed conditions, determine the Rayleigh number for the following fluids: air ( 1 atm, $400 \mathrm{~K})$, helium ( 1 atm, $400 \mathrm{~K})$, glycerin $(285 \mathrm{~K})$, and water $(310 \mathrm{~K})$.

Averell Hause
Averell Hause
Carnegie Mellon University
02:04

Problem 4

To assess the efficacy of different liquids for cooling by natural convection, it is convenient to introduce a fure of merit, $F_{N}$, which combines the influence of all pertinent fluid properties on the convection coefficient. If the Nusselt number is governed by an expression of the form, $N u_{L} \sim R a^{n}$, obtain the corresponding relationship between $F_{N}$ and the fluid properties. For a representative value of $n=0.33$, calculate values of $F_{N}$ for air $(k=0.026$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, \quad \beta=0.0035 \mathrm{~K}^{-1}, \quad \nu=1.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}, \quad P r=$ $0.70)$, water $\left(k=0.600 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=2.7 \times 10^{-4} \mathrm{~K}^{-1}\right.$, $\nu=10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \quad P r=5.0$ ), and a dielectric liquid $\left(k=0.064 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad \beta=0.0014 \mathrm{~K}^{-1}, \quad \nu=10^{-6} \mathrm{~m}^{2} / \mathrm{s}\right.$, $P r=25)$. What fluid is the most effective cooling agent?

Ajay Singhal
Ajay Singhal
Numerade Educator
09:17

Problem 5

In many cases, we are concerned with free convection involving gases that are contained within sealed enclosures. Consider air at $27^{\circ} \mathrm{C}$ and pressures of 1,10 , and 100 bars. Determine the fure of merit described in Problem $9.4$ for each of these three pressures. Which air pressure will provide the most effective cooling? Hint: See Problem 6.22.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:17

Problem 6

The heat transfer rate due to free convection from a vertical surface, $1 \mathrm{~m}$ high and $0.6 \mathrm{~m}$ wide, to quiescent air that is $20 \mathrm{~K}$ colder than the surface is known. What is the ratio of the heat transfer rate for that situation to the rate corresponding to a vertical surface, $0.6 \mathrm{~m}$ high and $1 \mathrm{~m}$ wide, when the quiescent air is $20 \mathrm{~K}$ warmer than the surface? Neglect heat transfer by radiation and any influence of temperature on the relevant thermophysical properties of air.

Mayukh Banik
Mayukh Banik
Numerade Educator
05:13

Problem 7

Consider a large vertical plate with a uniform surface temperature of $130^{\circ} \mathrm{C}$ suspended in quiescent air at $25^{\circ} \mathrm{C}$ and atmospheric pressure.
(a) Estimate the boundary layer thickness at a location $0.25 \mathrm{~m}$ measured from the lower edge.
(b) What is the maximum velocity in the boundary layer at this location and at what position in the boundary layer does the maximum occur?
(c) Using the similarity solution result, Equation $9.19$, determine the heat transfer coefficient $0.25 \mathrm{~m}$ from the lower edge.
(d) At what location on the plate measured from the lower edge will the boundary layer become turbulent?

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:27

Problem 8

For laminar free convection flow on a vertical plate, the recommended values of $C$ and $n$ for use in the correlation of Equation $9.24$ are $0.59$ and 1/4, respectively.
Derive the values of $C$ from the similarity solution, Equation $9.21$, for $\operatorname{Pr}=0.01,1,10$, and 100 .

Chai Santi
Chai Santi
Numerade Educator
01:00

Problem 9

Consider an array of vertical rectangular fins, which is to be used to cool an electronic device mounted in quiescent, atmospheric air at $T_{\infty}=27^{\circ} \mathrm{C}$. Each fin has $L=20 \mathrm{~mm}$ and $H=150 \mathrm{~mm}$ and operates at an approximately uniform temperature of $T_{s}=77^{\circ} \mathrm{C}$.
(a) Viewing each fin surface as a vertical plate in an infinite, quiescent medium, briefly describe why there exists an optimum fin spacing $S$. Using Figure $9.4$, estimate the optimum value of $S$ for the prescribed conditions.
(h) For the optimum value of $S$ and a fin thickness of $t=1.5 \mathrm{~mm}$, estimate the rate of heat transfer from the fins for an array of width $W=355 \mathrm{~mm}$.

Raj Bala
Raj Bala
Numerade Educator
04:36

Problem 10

A number of thin plates are to be cooled by vertically suspending them in a water bath at a temperature of $20^{\circ} \mathrm{C}$. If the plates are initially at $54^{\circ} \mathrm{C}$ and are $0.15 \mathrm{~m}$ long, what minimum spacing would prevent interference between their free convection boundary layers?

Dading Chen
Dading Chen
Numerade Educator
02:17

Problem 11

Beginning with the free convection correlation of the form given by Equation 9.24, show that for air at atmospheric pressure and a film temperature of $400 \mathrm{~K}$, the average heat transfer coefficient for a vertical plate can be expressed as
$$
\begin{array}{ll}
\bar{h}_{L}=1.40\left(\frac{\Delta T}{L}\right)^{1 / 4} & 10^{4}<R a_{L}<10^{9} \\
\bar{h}_{L}=0.98 \Delta T^{1 / 3} & 10^{9}<R a_{L}<10^{13}
\end{array}
$$

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
01:58

Problem 12

A solid object is to be cooled by submerging it in a quiescent fluid, and the associated free convection coefficient is given by $\bar{h}=C \Delta T^{1 / 4}$, where $C$ is a constant and $\Delta T=T-T_{\infty}$.
(a) Using the results of Section $5.3 .3$, obtain an expression for the time required for the object to cool from an initial temperature $T_{i}$ to a final temperature $T_{f}$.
(b) Consider a highly polished, $150-\mathrm{mm}$ square aluminum alloy (2024) plate of $5-\mathrm{mm}$ thickness, initially at $225^{\circ} \mathrm{C}$, and suspended in ambient air at $25^{\circ} \mathrm{C}$. Using the appropriate approximate correlation from Problem 9.11, determine the time required for the plate to reach $80^{\circ} \mathrm{C}$.
(c) Plot the temperature-time history obtained from part (b) and compare with the results from a lumped capacitance analysis using a constant free convection coefficient, $\bar{h}_{o}$. Evaluate $\bar{h}_{o}$ from an appropriate correlation based on an average surface temperature of $\bar{T}=\left(T_{i}+T_{f}\right) / 2$.

Narayan Hari
Narayan Hari
Numerade Educator
02:02

Problem 13

A square aluminum plate $5 \mathrm{~mm}$ thick and $200 \mathrm{~mm}$ on a side is heated while vertically suspended in quiescent air at $40^{\circ} \mathrm{C}$. Determine the average heat transfer coefficient for the plate when its temperature is $15^{\circ} \mathrm{C}$ by two methods: using results from the similarity solution to the boundary layer equations, and using results from an empirical correlation.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:02

Problem 14

An aluminum alloy (2024) plate, heated to a uniform temperature of $227^{\circ} \mathrm{C}$, is allowed to cool while vertically suspended in a room where the ambient air and surroundings are at $27^{\circ} \mathrm{C}$. The plate is $0.3 \mathrm{~m}$ square with a thickness of $15 \mathrm{~mm}$ and an emissivity of $0.25$.
(a) Develop an expression for the time rate of change of the plate temperature, assuming the temperature to be uniform at any time.
(b) Determine the initial rate of cooling (K/s) when the plate temperature is $227^{\circ} \mathrm{C}$.
(c) Justify the uniform plate temperature assumption.
(d) Compute and plot the temperature history of the plate from $t=0$ to the time required to reach a temperature of $30^{\circ} \mathrm{C}$. Compute and plot the corresponding variations in the convection and radiation heat transfer rates.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:17

Problem 15

The plate described in Problem $9.14$ has been used in an experiment to determine the free convection heat transfer coefficient. At an instant of time when the plate temperature was $127^{\circ} \mathrm{C}$, the time rate of change of this temperature was observed to be $-0.0465 \mathrm{~K} / \mathrm{s}$. What is the corresponding free convection heat transfer coefficient? Compare this result with an estimate based on a standard empirical correlation.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
02:50

Problem 16

Determine the average convection heat transfer coefficient for the $2.5-\mathrm{m}$-high vertical walls of a home having respective interior air and wall surface temperatures of (a) 20 and $10^{\circ} \mathrm{C}$ and (b) 27 and $37^{\circ} \mathrm{C}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:01

Problem 17

Consider a vertical plate of dimension $0.25 \mathrm{~m} \times 0.50 \mathrm{~m}$ that is at $T_{s}=100^{\circ} \mathrm{C}$ in a quiescent environment at $T_{\infty}=20^{\circ} \mathrm{C}$. In the interest of minimizing heat transfer from the plate, which orientation, (A) or (B), is preferred? What is the convection heat transfer from the front surface of the plate when it is in the preferred orientation?

Narayan Hari
Narayan Hari
Numerade Educator
04:13

Problem 18

During a winter day, the window of a patio door with a height of $1.8 \mathrm{~m}$ and width of $1.0 \mathrm{~m}$ shows a frost line near its base. The room wall and air temperatures are $15^{\circ} \mathrm{C}$.
(a) Explain why the window would show a frost layer at the base rather than at the top.
(b) Estimate the heat loss through the window due to free convection and radiation. Assume the window has a uniform temperature of $0^{\circ} \mathrm{C}$ and the emissivity of the glass surface is $0.94$. If the room has electric baseboard heating, estimate the corresponding daily cost of the window heat loss for a utility rate of $0.18 \mathrm{\$} / \mathrm{kW} \cdot \mathrm{h}$.

Surendra Kumar
Surendra Kumar
Numerade Educator
04:13

Problem 19

A vertical, thin pane of window glass that is $1 \mathrm{~m}$ on a side separates quiescent room air at $T_{\infty, i}=20^{\circ} \mathrm{C}$ from quiescent ambient air at $T_{\infty, o}=-20^{\circ} \mathrm{C}$. The walls of the room and the external surroundings (landscape, buildings, etc.) are also at $T_{\text {sur }, i}=20^{\circ} \mathrm{C}$ and $T_{\text {sur }, o}=-20^{\circ} \mathrm{C}$, respectively.
If the glass has an emissivity of $\varepsilon=1$, what is its temperature $T$ ? What is the rate of heat loss through the glass?

Surendra Kumar
Surendra Kumar
Numerade Educator
01:12

Problem 20

Consider the conditions of Problem 9.19, but now allow for a difference between the inner and outer surface temperatures, $T_{s, i}$ and $T_{s,,}$, of the window. For a glass thickness and thermal conductivity of $t_{g}=10 \mathrm{~mm}$ and $k_{g}=1.4 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, respectively, evaluate $T_{s, i}$ and $T_{s, o}$. What is the heat loss through the window?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:07

Problem 21

A household oven door of $0.5-\mathrm{m}$ height and $0.7-\mathrm{m}$ width reaches an average surface temperature of $32^{\circ} \mathrm{C}$ during operation. Estimate the heat loss to the room with ambient air at $22^{\circ} \mathrm{C}$. If the door has an emissivity of $1.0$ and the surroundings are also at $22^{\circ} \mathrm{C}$, comment on the heat loss by free convection relative to that by radiation.

Dading Chen
Dading Chen
Numerade Educator
01:12

Problem 22

Consider a vertical, single-pane window of equivalent width and height $(W=L=1 \mathrm{~m})$. The interior surface is exposed to the air and walls of a room, which are each at $18^{\circ} \mathrm{C}$. Under cold ambient conditions for which a thin layer of frost has formed on the inner surface, what is the heat loss through the window? How would your analysis be affected by a frost layer whose thickness is not negligible? During incipience of frost formation, where would you expect the frost to begin to develop on the window? The frost may be assumed to have an emissivity of $\varepsilon=0.90$.

Ajay Singhal
Ajay Singhal
Numerade Educator
02:01

Problem 23

Consider laminar flow about a vertical isothermal plate of length $L$, providing an average heat transfer coefficient of $\bar{h}_{L}$. If the plate is divided into $N$ smaller plates, each of length, $L_{N}=L / N$, determine an expression for the ratio of the heat transfer coefficient averaged over the $N$ plates to the heat transfer coefficient averaged over the single plate, $\bar{h}_{L, N} / \bar{h}_{L, 1}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:17

Problem 24

Consider the conveyor system described in Problem $7.24$, but under conditions for which the conveyor is not moving and the air is quiescent. Radiation effects and interactions between boundary layers on adjoining surfaces may be neglected.
(a) For the prescribed plate dimensions and initial temperature, as well as the prescribed air temperature, what is the initial rate of heat transfer from one of the plates?
(b) How long does it take for a plate to cool from $300^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ ? Comment on the assumption of negligible radiation.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
04:17

Problem 25

A thin-walled container with a hot process fluid at $50^{\circ} \mathrm{C}$ is placed in a quiescent, cold water bath at $10^{\circ} \mathrm{C}$. Heat transfer at the inner and outer surfaces of the container may be approximated by free convection from a vertical plate.
(a) Determine the overall heat transfer coefficient between the hot process fluid and the cold water bath. Assume the properties of the hot process fluid are those of water.
(b) Generate a plot of the overall heat transfer coefficient as a function of the hot process fluid temperature $T_{\infty, h}$ for the range 20 to $60^{\circ} \mathrm{C}$, with all other conditions remaining the same.

Surendra Kumar
Surendra Kumar
Numerade Educator
04:13

Problem 26

Consider an experiment to investigate the transition to turbulent flow in a free convection boundary layer that develops along a vertical plate suspended in a large room. The plate is constructed of a thin heater that is sandwiched between two aluminum plates and may be assumed to be isothermal. The heated plate is $1 \mathrm{~m}$ high and $2 \mathrm{~m}$ wide. The quiescent air and the surroundings are both at $25^{\circ} \mathrm{C}$.
(a) The exposed surfaces of the aluminum plate are painted with a very thin coating of high emissivity $(\varepsilon=0.95)$ paint. Determine the electrical power that must be supplied to the heater to sustain the plate at a temperature of $T_{s}=35^{\circ} \mathrm{C}$. How much of the plate is exposed to turbulent conditions in the free convection boundary layer?
(b) The experimentalist speculates that the roughness of the paint is affecting the transition to turbulence in the boundary layer and decides to remove the paint and polish the aluminum surface ( $\varepsilon=0.05$ ). If the same power is supplied to the plate as in part (a), what is the steady-state plate temperature? How much of the plate is exposed to turbulent conditions in the free convection boundary layer?

Chai Santi
Chai Santi
Numerade Educator
03:20

Problem 27

9.27 The vertical rear window of an automobile is of thickness $L=8 \mathrm{~mm}$ and height $H=0.5 \mathrm{~m}$ and contains fine-meshed heating wires that can induce nearly uniform volumetric heating, $\dot{q}\left(\mathrm{~W} / \mathrm{m}^{3}\right)$.
(a) Consider steady-state conditions for which the interior surface of the window is exposed to quiescent air at $10^{\circ} \mathrm{C}$, while the exterior surface is exposed to air at $-10^{\circ} \mathrm{C}$ moving in parallel flow over the surface with a velocity of $20 \mathrm{~m} / \mathrm{s}$. Determine the volumetric heating rate needed to maintain the interior window surface at $T_{s, i}=15^{\circ} \mathrm{C}$.
(b) The interior and exterior window temperatures, $T_{s, i}$ and $T_{s, \rho}$, depend on the compartment and ambient temperatures, $T_{\infty, i}$ and $T_{\infty, p}$, as well as on the velocity $u_{\infty}$ of air flowing over the exterior surface and the volumetric heating rate $\dot{q}$. Subject to the constraint that $T_{s, i}$ is to be maintained at $15^{\circ} \mathrm{C}$, we wish to develop guidelines for varying the heating rate in response to changes in $T_{\infty,}, T_{\infty \rho,}$, and/or $u_{\infty}$. If $T_{\infty, i}$ is maintained at $10^{\circ} \mathrm{C}$, how will $\dot{q}$ and $T_{s, \rho}$ vary with $T_{\infty, o}$ for $-25 \leq T_{\infty,,} \leq 5^{\circ} \mathrm{C}$ and $u_{\infty}=10,20$, and $30 \mathrm{~m} / \mathrm{s}$ ?
If a constant vehicle speed is maintained, such that $u_{\infty}=30 \mathrm{~m} / \mathrm{s}$, how will $\dot{q}$ and $T_{s, o}$ vary with $T_{\infty, i}$ for $5 \leq T_{\infty, i} \leq 20^{\circ} \mathrm{C}$ and $T_{\infty, o}=-25,-10$, and $5^{\circ} \mathrm{C}$ ?

Nick Auwerda
Nick Auwerda
Numerade Educator
01:40

Problem 28

Determine the maximum allowable uniform heat flux that may be imposed at a wall heating panel $1 \mathrm{~m}$ high if the maximum temperature is not to exceed $37^{\circ} \mathrm{C}$ when the ambient air temperature is $25^{\circ} \mathrm{C}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:36

Problem 29

The components of a vertical circuit board, $150 \mathrm{~mm}$ on a side, dissipate $5 \mathrm{~W}$. The back surface is well insulated and the front surface is exposed to quiescent air at $27^{\circ} \mathrm{C}$.
Assuming a uniform surface heat flux, what is the maximum temperature of the board? What is the temperature of the board for an isothermal surface condition?

Anand Jangid
Anand Jangid
Numerade Educator
03:37

Problem 30

Circuit boards are mounted to interior vertical surfaces of a rectangular duct of height $H=400 \mathrm{~mm}$ and length $L=800 \mathrm{~mm}$. Although the boards are cooled by forced convection heat transfer to air flowing through the duct, not all of the heat dissipated by the electronic components is transferred to the flow. Some of the heat is instead transferred by conduction to the vertical walls of the duct and then by natural convection and radiation to the ambient (atmospheric) air and surroundings, which are at equivalent temperatures of $T_{\infty}=T_{\text {sur }}=20^{\circ} \mathrm{C}$. The walls are metallic and, to a first approximation, may be assumed to be isothermal at a temperature $T_{s}$.
(a) Consider conditions for which the electronic components dissipate $200 \mathrm{~W}$ and air enters the duct at a flow rate of $\dot{m}=0.015 \mathrm{~kg} / \mathrm{s}$ and a temperature of $T_{m, i}=20^{\circ} \mathrm{C}$. If the emissivity of the side walls is $\varepsilon_{s}=0.15$ and the outlet temperature of the air is $T_{m, o}=30^{\circ} \mathrm{C}$, what is the surface temperature $T_{s}$ ?
(b) To reduce the temperature of the electronic components, it is desirable to enhance heat transfer from the side walls. Assuming no change in the airflow conditions, what is the effect on $T_{s}$ of applying a high emissivity coating $\left(\varepsilon_{s}=0.90\right)$ to the side walls?
(c) If there is a loss of airflow while power continues to be dissipated, what are the resulting values of $T_{s}$ for $\varepsilon_{s}=0.15$ and $\varepsilon_{s}=0.90$ ?

Manish Jain
Manish Jain
Numerade Educator
00:59

Problem 31

A refrigerator door has a height and width of $H=1 \mathrm{~m}$ and $W=0.65 \mathrm{~m}$, respectively, and is situated in a large room for which the air and walls are at $T_{\infty}=T_{\text {sur }}=25^{\circ} \mathrm{C}$. The door consists of a layer of polystyrene insulation $(k=0.03 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ sandwiched between thin sheets of steel $(\varepsilon=0.6)$ and polypropylene. Under normal operating conditions, the inner surface of the door is maintained at a fixed temperature of $T_{s, i}=5^{\circ} \mathrm{C}$.
(a) Estimate the heat gain through the door for the worst case condition corresponding to no insulation $(L=0)$.
(b) Compute and plot the heat gain and the outer surface temperature $T_{s, o}$ as a function of insulation thickness for $0 \leq L \leq 25 \mathrm{~mm}$.

Sanjeev Kumar
Sanjeev Kumar
Numerade Educator
08:22

Problem 32

Air at $3 \mathrm{~atm}$ and $100^{\circ} \mathrm{C}$ is discharged from a compressor into a vertical receiver of $2.5-\mathrm{m}$ height and $0.75-\mathrm{m}$ diameter. Assume that the receiver wall has negligible thermal resistance, is at a uniform temperature, and that heat transfer at its inner and outer surfaces is by free convection from a vertical plate. Neglect radiation exchange and any losses from the top.
(a) Estimate the receiver wall temperature and the heat transfer to the ambient air at $25^{\circ} \mathrm{C}$. To facilitate use of the free convection correlations with appropriate film temperatures, assume that the receiver wall temperature is $60^{\circ} \mathrm{C}$.
(b) Were the assumed film temperatures of part (a) reasonable? If not, use an iteration procedure to find consistent values.
(c) Now consider two features of the receiver neglected in the previous analysis: (i) radiation exchange from the exterior surface of emissivity $0.85$ to large surroundings, also at $25^{\circ} \mathrm{C}$; and (ii) the thermal resistance of a 20 -mm-thick wall with a thermal conductivity of $0.25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Represent the system by a thermal circuit and estimate the wall temperatures and the heat transfer rate.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:53

Problem 33

In the central receiver concept of a solar power plant, many heliostats at ground level are used to direct a concentrated solar flux $q_{s}^{\prime \prime}$ to the receiver, which is positioned at the top of a tower. However, even with absorption of all the solar flux by the outer surface of the receiver, losses due to free convection and radiation reduce the collection efficiency below the maximum possible value of $100 \%$. Consider a cylindrical receiver of diameter $D=7 \mathrm{~m}$, length $L=12 \mathrm{~m}$, and emissivity $\varepsilon=0.20$.
(a) If all of the solar flux is absorbed by the receiver and a surface temperature of $T_{s}=800 \mathrm{~K}$ is maintained, what is the rate of heat loss from the receiver? The ambient air is quiescent at a temperature of $T_{\infty}=300 \mathrm{~K}$, and irradiation from the surroundings may be neglected. If the corresponding value of the solar flux is $q_{S}^{\prime \prime}=10^{5} \mathrm{~W} / \mathrm{m}^{2}$, what is the collector efficiency?
(b) The surface temperature of the receiver is affected by design and operating conditions within the power plant. Over the range from 600 to $1000 \mathrm{~K}$, plot the variation of the convection, radiation, and total heat rates as a function of $T_{s}$. For a fixed value of $q_{S}^{\prime \prime}=10^{5} \mathrm{~W} / \mathrm{m}^{2}$, plot the corresponding variation of the receiver efficiency.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
18:58

Problem 34

Consider the transformer of Problem 8.103, whose lateral surface is being maintained at $47^{\circ} \mathrm{C}$ by a forced convection coolant line removing $1000 \mathrm{~W}$. It is desired to explore cooling of the transformer by free convection and radiation, assuming the surface to have an emissivity of $0.80$.
(a) Determine how much power could be removed by free convection and radiation from the lateral and the upper horizontal surfaces when the ambient temperature and the surroundings are at $27^{\circ} \mathrm{C}$.
(b) Vertical fins, $5 \mathrm{~mm}$ thick, $75 \mathrm{~mm}$ wide, and $500 \mathrm{~mm}$ long, can easily be welded to the lateral surface. What is the heat removal rate by free convection if 30 such fins are attached?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:48

Problem 35

Airflow through a long, 0.2-m-square air conditioning duct maintains the outer duct surface temperature at $10^{\circ} \mathrm{C}$. If the horizontal duct is uninsulated and exposed to air at $35^{\circ} \mathrm{C}$ in the crawlspace beneath a home, what is the heat gain per unit length of the duct?

Jincy M Saji
Jincy M Saji
Numerade Educator
03:24

Problem 36

Consider the conditions of Example 9.3, including the effect of adding insulation of thickness $t$ and thermal conductivity $k=0.035 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ to the duct. We wish to now include the effect of radiation on the outer surface temperatures and the total heat loss per unit length of duct.
(a) If $T_{s, 1}=45^{\circ} \mathrm{C}, t=25 \mathrm{~mm}, \varepsilon=1$, and $T_{\text {sur }}=288 \mathrm{~K}$, what are the temperatures of the side, top, and bottom surfaces? What are the corresponding heat losses per unit length of duct?
(b) For the top surface, compute and plot $T_{s, 2}$ and $q^{\prime}$ as a function of insulation thickness for $0 \leq t \leq 50 \mathrm{~mm}$. The exposed duct surface $(t=0)$ may also be assumed to have an emissivity of $\varepsilon=1$.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
02:04

Problem 37

An electrical heater in the form of a horizontal disk of $400-\mathrm{mm}$ diameter is used to heat the bottom of a tank filled with engine oil at a temperature of $5^{\circ} \mathrm{C}$. Calculate the power required to maintain the heater surface temperature at $70^{\circ} \mathrm{C}$.

Dominador Tan
Dominador Tan
Numerade Educator
05:07

Problem 38

Consider a horizontal 6-mm-thick, 100 -mm-long straight fin fabricated from plain carbon steel $(k=57 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\varepsilon=0.5$ ). The base of the fin is maintained at $150^{\circ} \mathrm{C}$, while the quiescent ambient air and the surroundings are at $25^{\circ} \mathrm{C}$. Assume the fin tip is adiabatic.
(a) Estimate the fin heat rate per unit width, $q_{f}^{\prime}$. Use an average fin surface temperature of $125^{\circ} \mathrm{C}$ to estimate the free convection coefficient and the linearized radiation coefficient. How sensitive is your estimate to the choice of the average fin surface temperature?
(b) Generate a plot of $q_{f}^{\prime}$ as a function of the fin emissivity for $0.05 \leq \varepsilon \leq 0.95$. On the same coordinates, show the fraction of the total heat rate due to radiation exchange.

Satpal Satpal
Satpal Satpal
Numerade Educator
View

Problem 39

The thermal conductivity and surface emissivity of a material may be determined by heating its bottom surface and exposing its top surface to quiescent air and large surroundings of equivalent temperatures, $T_{\infty}=T_{\text {sur }}=25^{\circ} \mathrm{C}$. The remaining surfaces of the sample/heater are well insulated.
Consider a sample of thickness $L=25 \mathrm{~mm}$ and a square planform of width $W=250 \mathrm{~mm}$. In an experiment performed under steady-state conditions, temperature measurements made at the lower and upper surface of the sample yield values of $T_{1}=150^{\circ} \mathrm{C}$ and $T_{2}=100^{\circ} \mathrm{C}$, respectively, for a power input of $P_{\text {elec }}=70 \mathrm{~W}$. What are the thermal conductivity and emissivity of the sample?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:30

Problem 40

Convection heat transfer coefficients for a hot horizontal surface facing upward may be determined by a gage whose specific features depend on whether the temperature of the surroundings is known. For configuration A, a copper disk, which is electrically heated from below, is encased in an insulating material such that all of the heat is transferred by convection and radiation from the top surface. If the surface emissivity and the temperatures of the air and surroundings are known, the convection coefficient may be determined from measurement of the electrical power and the surface temperature of the disk. Configuration B is used in situations for which the temperature of the surroundings is not known. A thin, insulating strip separates semicircular disks with independent electrical heaters and different emissivities. If the emissivities and temperature of the air are known, the convection coefficient may be determined from measurement of the electrical power supplied to each of the disks in order to maintain them at a common temperature.
(a) In an application of configuration A to a disk of diameter $D=160 \mathrm{~mm}$ and emissivity $\varepsilon=0.8$, values of $P_{\text {elec }}=10.8 \mathrm{~W}$ and $T=67^{\circ} \mathrm{C}$ are measured for $T_{\infty}=T_{\text {sur }}=27^{\circ} \mathrm{C}$. What is the corresponding value of the average convection coefficient? How does it compare with predictions based on a standard correlation?
(b) Now consider an application of configuration $B$ for which $T_{\infty}=17^{\circ} \mathrm{C}$ and $T_{\text {sur }}$ is unknown. With $D=160 \mathrm{~mm}, \varepsilon_{1}=0.8$, and $\varepsilon_{2}=0.1$, values of $P_{\text {elect, } 1}=9.70 \mathrm{~W}$ and $P_{\text {elec, } 2}=5.67 \mathrm{~W}$ are measured when $T_{1}=T_{2}=77^{\circ} \mathrm{C}$. Determine the corresponding values of the convection coefficient and the temperature of the surroundings. How does the convection coefficient compare with predictions by an appropriate correlation?

Dominique Jan Tan
Dominique Jan Tan
Numerade Educator
03:37

Problem 41

Many laptop computers are equipped with thermal management systems that involve liquid cooling of the central processing unit (CPU), transfer of the heated liquid to the back of the laptop screen assembly, and dissipation of heat from the back of the screen assembly by way of a flat, isothermal heat spreader. The cooled liquid is recirculated to the CPU and the process continues. Consider an aluminum heat spreader that is of width $w=275 \mathrm{~mm}$ and height $L=175 \mathrm{~mm}$. The screen assembly is oriented at an angle $\theta=30^{\circ}$ from the vertical direction, and the heat spreader is attached to the $t=3$-mm-thick plastic housing with a thermally conducting adhesive. The plastic housing has a thermal conductivity of $k=0.21 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and emissivity of $\varepsilon=0.85$. The contact resistance associated with the heat spreaderhousing interface is $R_{t, c}^{\prime \prime}=2.0 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. If the CPU generates, on average, $15 \mathrm{~W}$ of thermal energy,
what is the temperature of the heat spreader when $T_{\infty}=T_{\text {sur }}=23^{\circ} \mathrm{C}$ ? Which thermal resistance (contact, conduction, radiation, or free convection) is the largest?

Manish Jain
Manish Jain
Numerade Educator
05:00

Problem 43

Consider the roof of the refrigerated truck compartment described in Problem 7.20, but under conditions for which the truck is parked $(V=0)$. All other conditions remain unchanged. For $\alpha_{S}=\varepsilon=0.5$, determine the outer surface temperature, $T_{s, o}$, and the heat load imposed on the refrigeration system. Hint: Assume $T_{s, o}>T_{\infty}$ and $R a_{L}>10^{7} .$

Vipender Yadav
Vipender Yadav
Numerade Educator
09:57

Problem 44

The $4 \mathrm{~m} \times 4 \mathrm{~m}$ horizontal roof of an uninsulated aluminum melting furnace is comprised of a $0.08-\mathrm{m}$-thick fireclay brick refractory covered by a 5 -mm-thick steel (AISI 1010) plate. The refractory surface exposed to the furnace gases is maintained at $1700 \mathrm{~K}$ during operation, while the outer surface of the steel is exposed to the air and walls of a large room at $25^{\circ} \mathrm{C}$. The emissivity of the steel is $\varepsilon=0.3$.
(a) What is the rate of heat loss from the roof?
(b) If a $20-\mathrm{mm}$-thick layer of alumina-silica insulation $\left(64 \mathrm{~kg} / \mathrm{m}^{3}\right)$ is placed between the refractory and the steel, what is the new rate of heat loss from the roof? What is the temperature at the inner surface of the insulation?
(c) One of the process engineers claims that the temperature at the inner surface of the insulation found in part (b) is too high for safe, long-term operation. What thickness of fireclay brick would reduce this temperature to $1350 \mathrm{~K}$ ?

Rachel Peterson
Rachel Peterson
Numerade Educator
02:35

Problem 45

At the end of its manufacturing process, a silicon wafer of diameter $D=150 \mathrm{~mm}$, thickness $\delta=1 \mathrm{~mm}$, and emissivity $\varepsilon=0.65$ is at an initial temperature of $T_{i}=325^{\circ} \mathrm{C}$ and is allowed to cool in quiescent, ambient air and large surroundings for which $T_{\infty}=T_{\text {sur }}=25^{\circ} \mathrm{C}$.
(a) What is the initial rate of cooling?
(b) How long does it take for the wafer to reach a temperature of $50^{\circ} \mathrm{C}$ ? Comment on how the relative effects of convection and radiation vary with time during the cooling process.

Manish Jain
Manish Jain
Numerade Educator
03:23

Problem 46

A 200-mm-square, 10-mm-thick tile has the thermophysical properties of Pyrex $(\varepsilon=0.80)$ and emerges
from a curing process at an initial temperature of $T_{i}=140^{\circ} \mathrm{C}$. The backside of the tile is insulated while the upper surface is exposed to ambient air and surroundings at $25^{\circ} \mathrm{C}$.

Anand Jangid
Anand Jangid
Numerade Educator
05:41

Problem 47

Integrated circuit (IC) boards are stacked within a duct and dissipate a total of $500 \mathrm{~W}$. The duct has a square cross section with $w=H=150 \mathrm{~mm}$ and a length of $0.5 \mathrm{~m}$. Air flows into the duct at $25^{\circ} \mathrm{C}$ and $1.2 \mathrm{~m}^{3} / \mathrm{min}$, and the convection coefficient between the air and the inner surfaces of the duct is $\bar{h}_{i}=50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The entire outer surface of the duct, which is anodized with an emissivity of $0.5$, is exposed to ambient air and large surroundings at $25^{\circ} \mathrm{C}$.
Your assignment is to develop a model to estimate the outlet temperature of the air, $T_{m, o}$ and the average surface temperature of the duct, $\bar{T}_{s}$.
(a) Assuming a surface temperature of $37^{\circ} \mathrm{C}$, estimate the average free convection coefficient, $\bar{h}_{o}$, for the outer surface of the duct.
(b) Assuming a surface temperature of $37^{\circ} \mathrm{C}$, estimate the average linearized radiation coefficient, $\bar{h}_{\text {rad }}$, for the outer surface of the duct.
(c) Perform an energy balance on the duct by considering the dissipation of electrical power in the ICs, the rate of change in the energy of air flowing through the duct, and the rate of heat transfer from the air in the duct to the surroundings. Express the last process in terms of thermal resistances between the mean temperature, $\bar{T}_{m}$, of the air in the duct and the temperature of the ambient air and the surroundings.
(d) Substitute numerical values into the expression of part (c) and calculate the air outlet temperature, $T_{m, \sigma^{*}}$. Estimate the corresponding value of $\bar{T}_{s}$. Comment on your results and the assumptions inherent in your model.

Chai Santi
Chai Santi
Numerade Educator
00:56

Problem 48

A highly polished aluminum plate of length $0.5 \mathrm{~m}$ and width $0.2 \mathrm{~m}$ is subjected to an airstream at a temperature of $23^{\circ} \mathrm{C}$ and a velocity of $10 \mathrm{~m} / \mathrm{s}$. Because of upstream conditions, the flow is turbulent over the entire length of the plate. A series of segmented, independently controlled heaters is attached to the lower side of the plate to maintain approximately isothermal conditions over the entire plate. The electrical heater covering the section between the positions $x_{1}=0.2 \mathrm{~m}$ and $x_{2}=0.3 \mathrm{~m}$ is shown in the schematic.
(a) Estimate the electrical power that must be supplied to the designated heater segment to maintain the plate surface temperature at $T_{s}=47^{\circ} \mathrm{C}$.
(b) If the blower that maintains the airstream velocity over the plate malfunctions, but the power to the heaters remains constant, estimate the surface temperature of the designated segment. Assume that the ambient air is extensive, quiescent, and at $23^{\circ} \mathrm{C}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
05:41

Problem 49

The average free convection coefficient for the exterior surfaces of a long, horizontal rectangular duct exposed to a quiescent fluid can be estimated from the Hahn-Didion (H-D) correlation [ASHRAE Proceedings, Part 1, pp. 262-67, 1972]
$$
\overline{N u}_{P}=0.55 R a_{P}^{1 / 4}\left(\frac{H}{P}\right)^{1 / 8} \quad R a_{P} \leq 10^{7}
$$]
where the characteristic length is the half-perimeter, $P=(w+H)$, and $w$ and $H$ are the horizontal width and vertical height, respectively, of the duct. The thermophysical properties are evaluated at the film temperature.
(a) Consider a horizontal $0.15-\mathrm{m}$-square duct with a surface temperature of $35^{\circ} \mathrm{C}$ in ambient air at $15^{\circ} \mathrm{C}$. Calculate the average convection coefficient and the heat rate per unit length using the H-D correlation.
(b) Calculate the average convection coefficient and the heat rate per unit length considering the duct as formed by vertical plates (sides) and horizontal plates (top and bottom). Do you expect this estimate to be lower or higher than that obtained with the H-D correlation? Explain the difference, if any.
(c) Using an appropriate correlation, calculate the average convection coefficient and the heat rate per unit length for a duct of circular cross section having a perimeter equal to the wetted perimeter of the rectangular duct of part (a). Do you expect this estimate to be lower or higher than that obtained with the H-D correlation? Explain the difference, if any.

Chai Santi
Chai Santi
Numerade Educator
01:51

Problem 50

Certain wood stove designs rely exclusively on heat transfer by radiation and natural convection to the surroundings. Consider a stove that forms a cubical enclosure, $L_{s}=1 \mathrm{~m}$ on a side, in a large room. The exterior walls of the stove have an emissivity of $\varepsilon=0.8$ and are at an operating temperature of $T_{s s s}=500 \mathrm{~K}$.
The stove pipe, which may be assumed to be isothermal at an operating temperature of $T_{s, p}=400 \mathrm{~K}$,
has a diameter of $D_{p}=0.25 \mathrm{~m}$ and a height of $L_{p}=2 \mathrm{~m}$, extending from stove to ceiling. The stove is in a large room whose air and walls are at $T_{\infty}=T_{\text {sur }}=300 \mathrm{~K}$. Neglecting heat transfer from the small horizontal section of the pipe and radiation exchange between the pipe and stove, estimate the rate at which heat is transferred from the stove and pipe to the surroundings.

Suzanne W.
Suzanne W.
Numerade Educator
01:31

Problem 51

A plate $1 \mathrm{~m} \times 1 \mathrm{~m}$, inclined at an angle of $45^{\circ}$, is exposed to a net radiation heat flux of $300 \mathrm{~W} / \mathrm{m}^{2}$ at its bottom surface. If the top surface of the plate is well insulated, estimate the temperature the plate reaches when the ambient air is quiescent and at a temperature of $0^{\circ} \mathrm{C}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:12

Problem 52

A horizontal rod $5 \mathrm{~mm}$ in diameter is immersed in water maintained at $18^{\circ} \mathrm{C}$. If the rod surface temperature is $56^{\circ} \mathrm{C}$, estimate the free convection heat transfer rate per unit length of the rod.

Zachary Warner
Zachary Warner
Numerade Educator
02:17

Problem 53

A horizontal uninsulated steam pipe passes through a large room whose walls and ambient air are at $300 \mathrm{~K}$. The pipe of $150-\mathrm{mm}$ diameter has an emissivity of $0.85$ and an outer surface temperature of $400 \mathrm{~K}$. Calculate the heat loss per unit length from the pipe.

Averell Hause
Averell Hause
Carnegie Mellon University
View

Problem 54

As discussed in Section 5.2, the lumped capacitance approximation may be applied if $B_{i}<0.1$, and, when implemented in a conservative fashion for a long cylinder, the characteristic length is the cylinder radius. After its extrusion, a long glass rod of diameter $D=15 \mathrm{~mm}$ is suspended horizontally in a room and cooled from its initial temperature by natural convection and radiation. At what rod temperatures may the lumped capacitance approximation be applied? The temperature of the quiescent air is the same as that of the surroundings, $T_{\infty}=T_{\text {sur }}=27^{\circ} \mathrm{C}$, and the glass emissivity is $\varepsilon=0.94$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:30

Problem 55

Beverage in cans $150 \mathrm{~mm}$ long and $60 \mathrm{~mm}$ in diameter is initially at $27^{\circ} \mathrm{C}$ and is to be cooled by placement in a refrigerator compartment at $4^{\circ} \mathrm{C}$. In the interest of maximizing the cooling rate, should the cans be laid horizontally or vertically in the compartment? As a first approximation, neglect heat transfer from the ends.

Anand Jangid
Anand Jangid
Numerade Educator
02:17

Problem 56

A long, uninsulated steam line with a diameter of $89 \mathrm{~mm}$ and a surface emissivity of $0.8$ transports steam at $200^{\circ} \mathrm{C}$ and is exposed to atmospheric air and large surroundings at an equivalent temperature of $20^{\circ} \mathrm{C}$.
(a) Calculate the heat loss per unit length for a calm day.
(b) Calculate the heat loss on a breezy day when the wind speed is $8 \mathrm{~m} / \mathrm{s}$.
(c) For the conditions of part (a), calculate the heat loss with a 20 -mm-thick layer of insulation ( $k=$ $0.08 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. Would the heat loss change significantly with an appreciable wind speed?

Averell Hause
Averell Hause
Carnegie Mellon University
View

Problem 57

Consider Problem 8.47. A more realistic solution would account for the resistance to heat transfer due to free convection in the paraffin during melting. Assuming the tube surface to have a uniform temperature of $55^{\circ} \mathrm{C}$ and the paraffin to be an infinite, quiescent liquid, determine the convection coefficient associated with the outer surface. Using this result and recognizing that the tube surface temperature is not known, determine the water outlet temperature, the total heat transfer rate, and the time required to completely liquefy the paraffin, for the prescribed conditions. Thermophysical properties associated with the liquid state of the paraffin are $k=0.15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \beta=8 \times 10^{-4} \mathrm{~K}^{-1}, \rho=770 \mathrm{~kg} / \mathrm{m}^{3}$, $\nu=5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$, and $\alpha=8.85 \times 10^{-8} \mathrm{~m}^{2} / \mathrm{s}$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
00:48

Problem 58

A horizontal tube of $12.5-\mathrm{mm}$ diameter with an outer surface temperature of $240^{\circ} \mathrm{C}$ is located in a room with an air temperature of $20^{\circ} \mathrm{C}$. Estimate the heat transfer rate per unit length of the tube due to free convection.

Nikhil Choudhary
Nikhil Choudhary
Numerade Educator
01:37

Problem 59

Saturated steam at 4 bars absolute pressure with a mean velocity of $3 \mathrm{~m} / \mathrm{s}$ flows through a horizontal pipe whose inner and outer diameters are 55 and $65 \mathrm{~mm}$, respectively. The heat transfer coefficient for the steam flow is known to be $11,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) If the pipe is covered with a $25-\mathrm{mm}$-thick layer of $85 \%$ magnesia insulation and is exposed to atmospheric air at $25^{\circ} \mathrm{C}$, determine the rate of heat transfer by free convection to the room per unit length of the pipe. If the steam is saturated at the inlet of the pipe, estimate its quality at the outlet of a pipe $30 \mathrm{~m}$ long.
(b) Net radiation to the surroundings also contributes to heat loss from the pipe. If the insulation has a surface emissivity of $\varepsilon=0.8$ and the surroundings are at $T_{\text {sur }}=T_{\infty}=25^{\circ} \mathrm{C}$, what is the rate of heat transfer to the room per unit length of pipe? What is the quality of the outlet flow?
(c) The heat loss may be reduced by increasing the insulation thickness and/or reducing its emissivity. What is the effect of increasing the insulation thickness to $50 \mathrm{~mm}$ if $\varepsilon=0.8$ ? Of decreasing the emissivity to $0.2$ if the insulation thickness is $25 \mathrm{~mm}$ ? Of reducing the emissivity to $0.2$ and increasing the insulation thickness to $50 \mathrm{~mm}$ ?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:43

Problem 60

A horizontal electrical cable of $25-\mathrm{mm}$ diameter has a heat dissipation rate of $30 \mathrm{~W} / \mathrm{m}$. If the ambient air temperature is $27^{\circ} \mathrm{C}$, estimate the surface temperature of the cable.

Penny Riley
Penny Riley
Numerade Educator
01:33

Problem 61

An electric immersion heater, $10 \mathrm{~mm}$ in diameter and $300 \mathrm{~mm}$ long, is rated at $550 \mathrm{~W}$. If the heater is horizontally positioned in a large tank of water at $20^{\circ} \mathrm{C}$, estimate its surface temperature. Estimate the surface temperature if the heater is accidentally operated in air at $20^{\circ} \mathrm{C}$.

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
00:56

Problem 62

The maximum surface temperature of the $20-\mathrm{mm}-$ diameter shaft of a motor operating in ambient air at $27^{\circ} \mathrm{C}$ should not exceed $87^{\circ} \mathrm{C}$. Because of power dissipation within the motor housing, it is desirable to reject as much heat as possible through the shaft to the ambient air. In this problem, we will investigate several methods for heat removal.
(a) For rotating cylinders, a suitable correlation for estimating the convection coefficient is of the form
$$
\begin{gathered}
\overline{N u}_{D}=0.133 \operatorname{Re}_{D}^{2 / 3} \operatorname{Pr}^{1 / 3} \\
\left(\operatorname{Re}_{D}<4.3 \times 10^{5}, \quad 0.7<\operatorname{Pr}<670\right)
\end{gathered}
$$
where $R e_{D} \equiv \Omega D^{2} / \nu$ and $\Omega$ is the rotational velocity (rad/s). Determine the convection coefficient and the maximum heat rate per unit length as a function of rotational speed in the range from 5000 to $15,000 \mathrm{rpm}$.
(b) Estimate the free convection coefficient and the maximum heat rate per unit length for the stationary shaft. Mixed free and forced convection effects may become significant for $R e_{D}<4.7\left(G r_{D}^{3} / P r\right)^{0.137}$. Are free convection effects important for the range of rotational speeds designated in part (a)?
(c) Assuming the emissivity of the shaft is $0.8$ and the surroundings are at the ambient air temperature, is radiation exchange important?
(d) If ambient air is in cross flow over the shaft, what air velocities are required to remove the heat rates determined in part (a)?

Manik Pulyani
Manik Pulyani
Numerade Educator
05:07

Problem 63

Consider a horizontal pin fin of 6- $\mathrm{mm}$ diameter and $60-\mathrm{mm}$ length fabricated from plain carbon steel $(k=57 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \varepsilon=0.5)$. The base of the fin is maintained at $150^{\circ} \mathrm{C}$, while the quiescent ambient air and the surroundings are at $25^{\circ} \mathrm{C}$. Assume the fin tip is adiabatic.
(a) Estimate the fin heat rate, $q_{f}$. Use an average fin surface temperature of $125^{\circ} \mathrm{C}$ in estimating the free convection coefficient and the linearized radiation coefficient. How sensitive is this estimate to your choice of the average fin surface temperature?
(b) Use the finite-difference method of solution to obtain $q_{f}$ when the convection and radiation coefficients are based on local, rather than average, temperatures for the fin. How does your result compare with the analytical solution of part (a)?

Satpal Satpal
Satpal Satpal
Numerade Educator
03:24

Problem 64

Consider the hot water pipe of Problem 7.56, but under conditions for which the ambient air is not in cross flow over the pipe and is, instead, quiescent. Accounting for the effect of radiation with a pipe emissivity of $\varepsilon_{p}=0.6$, what is the corresponding daily cost of heat loss per unit length of the uninsulated pipe?

Morgan Cheatham
Morgan Cheatham
Numerade Educator
03:24

Problem 65

Common practice in chemical processing plants is to clad pipe insulation with a durable, thick aluminum foil. The functions of the foil are to confine the batt insulation and to reduce heat transfer by radiation to the surroundings. Because of the presence of chlorine (at chlorine or seaside plants), the aluminum foil surface, which is initially bright, becomes etched with in-service time. Typically, the emissivity might change from $0.12$ at installation to $0.36$ with extended service. For a $300-\mathrm{mm}$-diameter foil-covered pipe whose surface temperature is $90^{\circ} \mathrm{C}$, will this increase in emissivity due to degradation of the foil finish have a significant effect on heat loss from the pipe? Consider two cases with surroundings and ambient air at $25^{\circ} \mathrm{C}$ : (a) quiescent air and (b) a cross-wind velocity of $10 \mathrm{~m} / \mathrm{s}$.

Morgan Cheatham
Morgan Cheatham
Numerade Educator
04:56

Problem 66

Consider the electrical heater of Problem 7.49. If the blower were to malfunction, terminating airflow while the heater continued to operate at $1000 \mathrm{~W} / \mathrm{m}$, what temperature would the heater assume? How long would it take to come within $10^{\circ} \mathrm{C}$ of this temperature? Allow for radiation exchange between the heater $(\varepsilon=0.8)$ and the duct walls, which are also at $27^{\circ} \mathrm{C}$.

Yaqub Khan
Yaqub Khan
Numerade Educator
02:34

Problem 67

A computer code is being developed to analyze a 12.5-mm-diameter, cylindrical sensor used to determine ambient air temperature. The sensor experiences free convection while positioned horizontally in quiescent air at $T_{\infty}=27^{\circ} \mathrm{C}$. For the temperature range from 30 to $80^{\circ} \mathrm{C}$, derive an expression for the convection coefficient as a function of only $\Delta T=$ $T_{s}-T_{\infty}$, where $T_{s}$ is the sensor temperature. Evaluate properties at an appropriate film temperature and show what effect this approximation has on the convection coefficient estimate.

Naman Kumar
Naman Kumar
Numerade Educator
09:01

Problem 68

A thin-walled tube of $20-\mathrm{mm}$ diameter passes hot fluid at a mean temperature of $45^{\circ} \mathrm{C}$ in an experimental flow loop. The tube is mounted horizontally in quiescent air at a temperature of $15^{\circ} \mathrm{C}$. To satisfy the stringent temperature control requirements of the experiment, it was decided to wind thin electrical heating tape on the outer surface of the tube to prevent heat loss from the hot fluid to the ambient air.
(a) Neglecting radiation heat loss, calculate the heat flux $q_{e}^{\prime \prime}$ that must be supplied by the electrical tape to ensure a uniform fluid temperature.
(b) Assuming the emissivity of the tape is $0.95$ and the surroundings are also at $15^{\circ} \mathrm{C}$, calculate the required heat flux.
(c) The heat loss may be reduced by wrapping the heating tape in a layer of insulation. For $85 \%$ magnesia insulation $(k=0.050 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ having a surface emissivity of $\varepsilon=0.60$, compute and plot the required heat flux $q_{e}^{\prime \prime}$ as a function of insulation thickness in the range from 0 to $20 \mathrm{~mm}$. For this range, compute and plot the convection and radiation heat rates per unit tube length as a function of insulation thickness.

Averell Hause
Averell Hause
Carnegie Mellon University
05:18

Problem 69

A billet of stainless steel, AISI 316, with a diameter of $150 \mathrm{~mm}$ and a length of $500 \mathrm{~mm}$ emerges from a heat treatment process at $200^{\circ} \mathrm{C}$ and is placed in an unstirred oil bath maintained at $20^{\circ} \mathrm{C}$.
(a) Determine whether it is advisable to position the billet in the bath with its centerline horizontal or vertical in order to decrease the cooling time.
(b) Estimate the time for the billet to cool to $30^{\circ} \mathrm{C}$ for the preferred arrangement.

Dading Chen
Dading Chen
Numerade Educator
02:15

Problem 70

Long stainless steel rods of $50-\mathrm{mm}$ diameter are preheated to a uniform temperature of $1000 \mathrm{~K}$ before being suspended from an overhead conveyor for transport to a hot forming operation. The conveyor is in a large room whose walls and air are at $300 \mathrm{~K}$.
(a) Assuming the linear motion of the rod to have a negligible effect on convection heat transfer from its surface, determine the average convection coefficient at the start of the transport process.
(b) If the surface emissivity of the rod is $\varepsilon=0.40$, what is the effective radiation heat transfer coefficient at the start of the transport process?
(c) Assuming a constant cumulative (radiation plus convection) heat transfer coefficient corresponding to the results of parts (a) and (b), what is the maximum allowable conveyor transit time, if the centerline temperature of the rod must exceed $900 \mathrm{~K}$ for the forming operation? Properties of the steel are $k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $\alpha=5.2 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$.
(d) Heat transfer by convection and radiation are actually decreasing during the transfer operation. Accounting for this reduction, reconsider the conditions of part (c) and obtain a more accurate estimate of the maximum allowable conveyor transit time.

Narayan Hari
Narayan Hari
Numerade Educator
04:40

Problem 71

Hot air flows from a furnace through a $0.15$-m-diameter, thin-walled steel duct with a velocity of $3 \mathrm{~m} / \mathrm{s}$. The duct passes through the crawlspace of a house, and its uninsulated exterior surface is exposed to quiescent air and surroundings at $0^{\circ} \mathrm{C}$.
(a) At a location in the duct for which the mean air temperature is $70^{\circ} \mathrm{C}$, determine the heat loss per unit duct length and the duct wall temperature. The duct outer surface has an emissivity of $0.5$.
(b) If the duct is wrapped with a $25-\mathrm{mm}$-thick layer of $85 \%$ magnesia insulation $(k=0.050 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ having a surface emissivity of $\varepsilon=0.60$, what are the duct wall temperature, the outer surface temperature, and the heat loss per unit length?

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
05:39

Problem 72

A biological fluid moves at a flow rate of $\dot{m}=0.02 \mathrm{~kg} / \mathrm{s}$ through a coiled, thin-walled, 5 -mm-diameter tube submerged in a large water bath maintained at $50^{\circ} \mathrm{C}$. The fluid enters the tube at $25^{\circ} \mathrm{C}$.
(a) Estimate the length of the tube and the number of coil turns required to provide an exit temperature of $T_{m, o}=38^{\circ} \mathrm{C}$ for the biological fluid. Assume that the water bath is an extensive, quiescent medium,
that the coiled tube approximates a horizontal tube, and that the biological fluid has the thermophysical properties of water.
(b) The flow rate through the tube is controlled by a pump that experiences throughput variations of approximately $\pm 10 \%$ at any one setting. This condition is of concern to the project engineer because the corresponding variation of the exit temperature of the biological fluid could influence the downstream process. What variation would you expect in $T_{m, o}$ for a $\pm 10 \%$ change in $\dot{m}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
02:25

Problem 73

Consider a batch process in which $200 \mathrm{~L}$ of a pharmaceutical are heated from $25^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$ by saturated steam condensing at $2.455$ bars as it flows through a coiled tube of $15-\mathrm{mm}$ diameter and $15-\mathrm{m}$ length. At any time during the process, the liquid may be approximated as an infinite, quiescent medium of uniform temperature and may be assumed to have constant properties of $\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=2000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \quad k=0.25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ $\nu=4.0 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \operatorname{Pr}=10$, and $\beta=0.002 \mathrm{~K}^{-1}$. The thermal resistances of the condensing steam and tube wall may be neglected.
(a) What is the initial rate of heat transfer to the pharmaceutical?
(b) Neglecting heat transfer between the tank and its surroundings, how long does it take to heat the pharmaceutical to $70^{\circ} \mathrm{C}$ ? Plot the corresponding variation with time of the fluid temperature and the convection coefficient at the outer surface of the tube. How much steam is condensed during the heating process?

Anand Jangid
Anand Jangid
Numerade Educator
18:21

Problem 74

In the analytical treatment of the fin with uniform cross-sectional area, it was assumed that the convection heat transfer coefficient is constant along the length of the fin. Consider an AISI 316 steel fin of 6-mm diameter and 50-mm length (with insulated tip) operating under conditions for which $T_{b}=125^{\circ} \mathrm{C}, T_{\infty}=27^{\circ} \mathrm{C}$, $T_{\text {sur }}=27^{\circ} \mathrm{C}$, and $\varepsilon=0.6$.
(a) Estimate average values of the fin heat transfer coefficients for free convection $\left(h_{c}\right)$ and radiation exchange $\left(h_{r}\right)$. Use these values to predict the tip temperature and fin effectiveness.
(b) Use a numerical method of solution to estimate the foregoing parameters when the convection and radiation coefficients are based on local, rather than average, values for the fin.

Joseph Lentino
Joseph Lentino
Numerade Educator
06:58

Problem 75

A hot fluid at $35^{\circ} \mathrm{C}$ is to be transported through a tube horizontally positioned in quiescent air at $25^{\circ} \mathrm{C}$. Which of the tube shapes, each of equal cross-sectional area, would you use in order to minimize heat losses to the ambient air by free convection?
Use the following correlation of Lienhard [Int. J. Heat Mass Transfer, 16, 2121, 1973] to approximate the laminar convection coefficient for an immersed body on which the boundary layer does not separate from the surface,
$$
\overline{N u_{l}}=0.52 R a_{l}^{1 / 4}
$$
The characteristic length $l$ is the length of travel of the fluid in the boundary layer across the shape surface. Compare this correlation to that given for a sphere to test its utility.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:53

Problem 76

Consider a 2-mm-diameter sphere immersed in a fluid at $300 \mathrm{~K}$ and $1 \mathrm{~atm}$.
(a) If the fluid around the sphere is quiescent and extensive, show that the conduction limit of heat transfer from the sphere can be expressed as $N u_{D, \text { cond }}=2$. Hint: Begin with the expression for the thermal resistance of a hollow sphere, Equation $3.41$, letting $r_{2} \rightarrow \infty$, and then expressing the result in terms of the Nusselt number.
(b) Considering free convection, at what surface temperature will the Nusselt number be twice that for the conduction limit? Consider air and water as the fluids.
(c) Considering forced convection, at what velocity will the Nusselt number be twice that for the conduction limit? Consider air and water as the fluids.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:02

Problem 77

A sphere of $25-\mathrm{mm}$ diameter contains an embedded electrical heater. Calculate the power required to maintain the surface temperature at $94^{\circ} \mathrm{C}$ when the sphere is exposed to a quiescent medium at $20^{\circ} \mathrm{C}$ for: (a) air at atmospheric pressure, (b) water, and (c) ethylene glycol.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
18:21

Problem 79

Consider the conditions of Problem 9.9, but now view the problem as one involving free convection between vertical, parallel plate channels. What is the optimum fin spacing $S$ ? For this spacing and the prescribed values of $t$ and $W$, what is the rate of heat transfer from the fins?

Joseph Lentino
Joseph Lentino
Numerade Educator
01:27

Problem 80

A vertical array of circuit boards is immersed in quiescent ambient air at $T_{\infty}=17^{\circ} \mathrm{C}$. Although the components protrude from their substrates, it is reasonable, as a first approximation, to assume $A t$ plates with uniform surface heat flux $q_{s}^{\prime \prime}$. Consider boards of length and width $L=W=0.4 \mathrm{~m}$ and spacing $S=25 \mathrm{~mm}$. If the maximum allowable board temperature is $77^{\circ} \mathrm{C}$, what is the maximum allowable power dissipation per board?

Mayukh Banik
Mayukh Banik
Numerade Educator
01:12

Problem 81

Determined to reduce the $\$ 7$ per week cost associated with heat loss through their patio window by convection and radiation, the tenants of Problem $9.18$ cover the inside of the window with a 50 -mm-thick sheet of extruded insulation. Because they are not very handy around the house, the insulation is installed poorly, resulting in an $S=5-\mathrm{mm}$ gap between the extruded insulation and the window pane, allowing the room air to infiltrate into the space between the pane and the insulation.
(a) Determine the window heat loss and associated weekly cost with the ill-fitting insulation in place. The insulation will significantly reduce the radiation losses through the window. Losses will be due almost entirely to convection.
(b) Plot the heat loss through the patio window as a function of the gap spacing for $1 \mathrm{~mm} \leq S \leq 20 \mathrm{~mm}$.

Ajay Singhal
Ajay Singhal
Numerade Educator
03:25

Problem 82

The front door of a dishwasher of width $580 \mathrm{~mm}$ has a vertical air vent that is $500 \mathrm{~mm}$ in height with a $20-\mathrm{mm}$ spacing between the inner tub operating at $52^{\circ} \mathrm{C}$ and an outer plate that is thermally insulated.
(a) Determine the heat loss from the tub surface when the ambient air is $27^{\circ} \mathrm{C}$.
(b) A change in the design of the door provides the opportunity to increase or decrease the $20-\mathrm{mm}$ spacing by $10 \mathrm{~mm}$. What recommendations would you offer with regard to how the change in spacing will alter heat losses?

Supratim Pal
Supratim Pal
Numerade Educator
00:17

Problem 83

A natural convection air heater consists of an array of parallel, equally spaced vertical plates, which may be maintained at a fixed temperature $T_{s}$ by embedded electrical heaters. The plates are of length and width $L=W=300 \mathrm{~mm}$ and are in quiescent, atmospheric air at $T_{\infty}=20^{\circ} \mathrm{C}$. The total width of the array cannot exceed a value of $W_{\text {ar }}=150 \mathrm{~mm}$.
For $T_{s}=75^{\circ} \mathrm{C}$, what is the plate spacing $S$ that maximizes heat transfer from the array? For this spacing, how many plates comprise the array and what is the corresponding rate of heat transfer from the array?

Mayukh Banik
Mayukh Banik
Numerade Educator
06:07

Problem 84

A bank of drying ovens is mounted on a rack in a room with an ambient air temperature of $27^{\circ} \mathrm{C}$. The cubical ovens are $500 \mathrm{~mm}$ to a side, and the spacing between the ovens is $15 \mathrm{~mm}$.
(a) Estimate the heat loss from a facing side of an oven when its surface temperature is $47^{\circ} \mathrm{C}$.
(b) Explore the effect of the spacing on the heat loss. At what spacing is the heat loss a maximum? Describe the boundary layer behavior for this condition. Can this condition be analyzed by treating the side of an oven as an isolated vertical plate?

Dading Chen
Dading Chen
Numerade Educator
04:53

Problem 85

A solar collector consists of a parallel plate channel that is connected to a water storage plenum at the bottom and to a heat sink at the top. The channel is inclined $\theta=30^{\circ}$ from the vertical and has a transparent cover plate. Solar radiation transmitted through the cover plate and the water maintains the isothermal absorber plate at a temperature $T_{s}=67^{\circ} \mathrm{C}$, while water returned to the reservoir from the heat $\operatorname{sink}$ is at $T_{\infty}=27^{\circ} \mathrm{C}$. The system operates as a thermosyphon, for which water flow is driven exclusively by buoyancy forces. The plate spacing and length are $S=15 \mathrm{~mm}$ and $L=1.5 \mathrm{~m}$.
Assuming the cover plate to be adiabatic with respect to convection heat transfer to or from the water, estimate the rate of heat transfer per unit width normal to the flow direction $(\mathrm{W} / \mathrm{m})$ from the absorber plate to the water.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:46

Problem 86

As is evident from the property data of Tables A.3 and A.4, the thermal conductivity of glass at room temperature is more than 50 times larger than that of air. It is therefore desirable to use windows of double-pane construction, for which the two panes of glass enclose an air space. If heat transfer across the air space is by conduction, the corresponding thermal resistance may be increased by increasing the thickness $L$ of the space. However, there are limits to the efficacy of such a measure, since convection currents are induced if $L$ exceeds a critical value, beyond which the thermal resistance decreases.

Consider atmospheric air enclosed by vertical panes at temperatures of $T_{1}=22^{\circ} \mathrm{C}$ and $T_{2}=-20^{\circ} \mathrm{C}$. If the critical Rayleigh number for the onset of convection is $R a_{L} \approx 2000$, what is the maximum allowable spacing for conduction across the air? How is this spacing affected by the temperatures of the panes? How is it affected by the pressure of the air, as, for example, by partial evacuation of the space?

Manish Jain
Manish Jain
Numerade Educator
05:57

Problem 87

A building window pane that is $1.2 \mathrm{~m}$ high and $0.8 \mathrm{~m}$ wide is separated from the ambient air by a storm window of the same height and width. The air space between the two windows is $0.06 \mathrm{~m}$ thick. If the building and storm windows are at 20 and $-10^{\circ} \mathrm{C}$, respectively, what is the rate of heat loss by free convection across the air space?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:20

Problem 88

To reduce heat losses, a horizontal rectangular duct that is $W=0.80 \mathrm{~m}$ wide and $H=0.3 \mathrm{~m}$ high is encased in a metal radiation shield. The duct wall and shield are separated by an air gap of thickness $t=0.06 \mathrm{~m}$. For a duct wall temperature of $T_{d}=40^{\circ} \mathrm{C}$ and a shield temperature of $T_{\mathrm{sh}}=20^{\circ} \mathrm{C}$, determine the convection heat loss per unit length from the duct.

Jincy M Saji
Jincy M Saji
Numerade Educator
01:31

Problem 89

The absorber plate and the adjoining cover plate of a flat-plate solar collector are at 70 and $35^{\circ} \mathrm{C}$, respectively, and are separated by an air space of $0.05 \mathrm{~m}$. What is the rate of free convection heat transfer per unit surface area between the two plates if they are inclined at an angle of $60^{\circ}$ from the horizontal?

Mayukh Banik
Mayukh Banik
Numerade Educator
04:02

Problem 90

Consider a thermal storage system in which the phase change material (paraffin) is housed in a large container whose bottom, horizontal surface is maintained at $T_{s}=50^{\circ} \mathrm{C}$ by warm water delivered from a solar collector.
(a) Neglecting the change in sensible energy of the liquid phase, estimate the amount of paraffin that is melted over a five-hour period beginning with an initial liquid layer at the bottom of the container of thickness $s_{i}=10 \mathrm{~mm}$. The paraffin of Problems $8.47$ and $9.57$ is used as the phase change material and is initially at the phase change temperature, $T_{\text {mup }}=27.4^{\circ} \mathrm{C}$. The bottom surface area of the container is $A=2.5 \mathrm{~m}^{2}$.
(b) Compare the amount of energy needed to melt the paraffin to the amount of energy required to increase the temperature of the same amount of liquid from the phase change temperature to the average liquid temperature, $\left(T_{s}+T_{\text {mp }}\right) / 2$.
(c) Neglecting the change in sensible energy of the liquid phase, estimate the amount of paraffin that would melt over a five-hour time period if the hot plate is placed at the top of the container and $s_{i}=10 \mathrm{~mm}$.

Supratim Pal
Supratim Pal
Numerade Educator
05:13

Problem 91

A rectangular cavity consists of two parallel, $0.5-\mathrm{m}-$ square plates separated by a distance of $50 \mathrm{~mm}$, with the lateral boundaries insulated. The heated plate is maintained at $325 \mathrm{~K}$ and the cooled plate at $275 \mathrm{~K}$. Estimate the heat flux between the surfaces for three orientations of the cavity using the notation of Figure 9.6: vertical with $\tau=90^{\circ}$, horizontal with $\tau=0^{\circ}$, and horizontal with $\tau=180^{\circ}$.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
01:16

Problem 92

Consider a horizontal flat roof section having the same dimensions as a vertical wall section. For both sections, the surfaces exposed to the air gap are at $18^{\circ} \mathrm{C}$ (inside) and $-10^{\circ} \mathrm{C}$ (outside).
(a) Estimate the ratio of the convection heat rate for the horizontal section to that of the vertical section.
(b) What effect will inserting a baffle at the mid-height of the vertical section have on the convection heat rate for that section?

Dominador Tan
Dominador Tan
Numerade Educator
01:26

Problem 93

A 50 -mm-thick air gap separates two horizontal metal plates that form the top surface of an industrial furnace. The bottom plate is at $T_{h}=200^{\circ} \mathrm{C}$ and the top plate is at $T_{c}=50^{\circ} \mathrm{C}$. The plant operator wishes to provide insulation between the plates to minimize heat loss. The relatively hot temperatures preclude use of foamed or felt insulation materials. Evacuated insulation materials
cannot be used due to the harsh industrial environment and their expense. A young engineer suggests that equally spaced, thin horizontal sheets of aluminum foil may be inserted in the gap to eliminate natural convection and minimize heat loss through the air gap.

Narayan Hari
Narayan Hari
Numerade Educator
08:09

Problem 94

The space between the panes of a double-glazed window can be filled with either air or carbon dioxide at atmospheric pressure. The window is $1.5 \mathrm{~m}$ high and the spacing between the panes can be varied. Develop an analysis to predict the convection heat transfer rate across the window as a function of pane spacing and determine, under otherwise identical conditions, whether air or carbon dioxide will yield the smaller rate. Illustrate the results of your analysis for two surface-temperature conditions: winter $\left(-10^{\circ} \mathrm{C}, 20^{\circ} \mathrm{C}\right)$ and summer $\left(35^{\circ} \mathrm{C}, 25^{\circ} \mathrm{C}\right)$.

Lisa Tarman
Lisa Tarman
Numerade Educator
08:09

Problem 95

A vertical, double-pane window, which is $1 \mathrm{~m}$ on a side and has a $25-\mathrm{mm}$ gap filled with atmospheric air, separates quiescent room air at $T_{\infty, i}=20^{\circ} \mathrm{C}$ from quiescent ambient air at $T_{\infty, o}=-20^{\circ} \mathrm{C}$. Radiation exchange between the window panes, as well as between each pane and its surroundings, may be neglected.
(a) Neglecting the thermal resistance associated with conduction heat transfer across each pane, determine the corresponding temperature of each pane and the rate of heat transfer through the window.
(b) Comment on the validity of neglecting the conduction resistance of the panes if each is of thickness $L_{p}=6 \mathrm{~mm}$.

Lisa Tarman
Lisa Tarman
Numerade Educator
01:31

Problem 96

The top surface $(0.5 \mathrm{~m} \times 0.5 \mathrm{~m})$ of an oven is $60^{\circ} \mathrm{C}$ for a particular operating condition when the room air is $23^{\circ} \mathrm{C}$. To reduce heat loss from the oven and to minimize burn hazard, it is proposed to create a $50 \mathrm{~mm}$ air space by adding a cover plate.
(a) Assuming the same oven surface temperature $T_{s}$ for both situations, estimate the reduction in the convection heat loss resulting from installation of the cover plate. What is the temperature of the cover plate?
(b) Explore the effect of the cover plate spacing on the convection heat loss and the cover plate temperature for spacings in the range $5 \leq L \leq 50 \mathrm{~mm}$. Is there an optimum spacing?

Mayukh Banik
Mayukh Banik
Numerade Educator
08:09

Problem 97

Consider window blinds that are installed in the air space between the two panes of a vertical double-pane window. The window is $H=0.5 \mathrm{~m}$ high and $w=0.5 \mathrm{~m}$ wide, and includes $N=19$ individual blinds that are each $L=25 \mathrm{~mm}$ wide. When the blinds are open, 20 smaller, square enclosures are formed along the height of the window. In the closed position, the blinds form a nearly continuous sheet with two $t=12.5 \mathrm{~mm}$ open gaps at the top and bottom of the enclosure. Determine the convection heat transfer rate between the inner pane, which is held at $T_{s, i}=20^{\circ} \mathrm{C}$, and the outer pane, which is at $T_{s, o}=-20^{\circ} \mathrm{C}$, when the blinds are in the open and closed positions, respectively. Explain why the closed blinds have little effect on the convection heat transfer rate across the cavity.

Lisa Tarman
Lisa Tarman
Numerade Educator
02:08

Problem 98

A solar water heater consists of a flat-plate collector that is coupled to a storage tank. The collector consists of a transparent cover plate and an absorber plate that are separated by an air gap.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:24

Problem 99

Consider the cylindrical, 0.12-m-diameter radiation shield of Example $9.5$ that is installed concentric with a $0.10$-m-diameter tube carrying steam. The spacing provides an air gap of $L=10 \mathrm{~mm}$.
(a) Calculate the heat loss per unit length of the tube by convection when a second shield of $0.14-\mathrm{m}$ diameter is installed, with the second shield maintained at $35^{\circ} \mathrm{C}$. Compare the result to that for the single shield of the example.
(b) In the two-shield configuration of part (a), the air gaps formed by the annular concentric tubes are $L=10 \mathrm{~mm}$. Calculate the heat loss per unit length if the gap dimension is $L=15 \mathrm{~mm}$. Do you expect the heat loss to increase or decrease?

Morgan Cheatham
Morgan Cheatham
Numerade Educator
04:45

Problem 100

The effective thermal conductivity $k_{\text {eff }}$ for concentric cylinders and concentric spheres is provided in Equations $9.59$ and $9.62$, respectively. Derive expressions for the critical Rayleigh numbers associated with the cylindrical and spherical geometries, $R a_{c, c r i t}$ and $R a_{s, c r i t}$, respectively, below which $k_{\text {eff }}$ is minimized. Evaluate $R a_{c, c r i t}$ and $R a_{s, c r i t}$ for air, water, and glycerin at a mean temperature of $300 \mathrm{~K}$. For specified inner and outer surface temperatures and inner cylinder or sphere radii, comment on the heat transfer rate for outer cylinder or sphere radii corresponding to $R a_{c, c r i t}$ and $R a_{s, \text {,rit, }}$, respectively.

Sunita Kumari
Sunita Kumari
Numerade Educator
03:36

Problem 101

A solar collector design consists of an inner tube enclosed concentrically in an outer tube that is transparent to solar radiation. The tubes are thin walled with inner and outer diameters of $0.10$ and $0.15 \mathrm{~m}$, respectively. The annular space between the tubes is completely enclosed and filled with air at atmospheric pressure. Under operating conditions for which the inner and outer tube surface temperatures are 70 and $30^{\circ} \mathrm{C}$, respectively, what is the convective heat loss per meter of tube length across the air space?

Neelesh Sharma
Neelesh Sharma
Numerade Educator
04:45

Problem 102

A proposed method to reduce heat losses from a horizontal, isothermal cylinder placed within a large room is to encase it within a larger cylinder, as shown in the schematic, with all surfaces painted with a low emissivity coating.
Air at atmospheric pressure exists within the annular region. For a concentrically located inner cylinder of surface temperature $T_{i}=70^{\circ} \mathrm{C}$ and radius $r_{i}=20 \mathrm{~mm}$, and for an ambient temperature of $T_{\infty}=30^{\circ} \mathrm{C}$, determine the optimal radius of the outer cylinder $r_{o}$ that will minimize the convection heat loss. Compare the convection heat loss from the inner cylinder with the optimally sized outer cylinder in place to the heat loss without the outer cylinder. Is the approach effective?

Sunita Kumari
Sunita Kumari
Numerade Educator
02:15

Problem 103

$\mathbf{9 . 1 0 3}$ It has been proposed to use large banks of rechargeable, lithium ion batteries to power electric vehicles. The cylindrical batteries, each of which is of radius $r_{i}=9 \mathrm{~mm}$ and length $L=65 \mathrm{~mm}$, undergo exothermic electrochemical reactions while being discharged. Since excessively high temperatures damage the batteries, it is proposed to encase them in a phase change material that melts when the batteries discharge (and resolidifies when the batteries are charged; charging is associated with an endothermic electrochemical reaction). Consider the paraffin of Problems $8.47$ and $9.57$.
(a) At an instant in time during the discharge of a battery, liquid paraffin occupies an annular region of outer radius $r_{o}=19 \mathrm{~mm}$ around the battery, which is generating $\dot{E}_{g}=1 \mathrm{~W}$ of thermal energy. Determine the surface temperature of the battery.
(b) At the time of interest in part (a), what is the rate at which the liquid annulus radius is increasing?
(c) Plot the battery surface temperature versus the outer radius of the liquid-filled annulus. Explain the relative insensitivity of the battery surface temperature to the size of the annulus for $15 \mathrm{~mm} \leq$ $r_{o} \leq 30 \mathrm{~mm}$.

Ze-Han Lee
Ze-Han Lee
Numerade Educator
01:52

Problem 104

Free convection occurs between concentric spheres. The inner sphere is of diameter $D_{i}=50 \mathrm{~mm}$ and temperature $T_{i}=50^{\circ} \mathrm{C}$, while the outer sphere is maintained at $T_{o}=20^{\circ} \mathrm{C}$. Air is in the gap between the spheres. What outer sphere diameter is required so that the convection heat transfer from the inner sphere is the same as if it were placed in a large, quiescent environment with air at $T_{\infty}=20^{\circ} \mathrm{C}$ ?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:04

Problem 105

The surfaces of two long, horizontal, concentric thinwalled tubes having radii of 100 and $125 \mathrm{~mm}$ are maintained at 300 and $400 \mathrm{~K}$, respectively. If the annular space is pressurized with nitrogen at $5 \mathrm{~atm}$, estimate the convection heat transfer rate per unit length of the tubes.

Anand Jangid
Anand Jangid
Numerade Educator
03:45

Problem 106

Consider the phase change material (PCM) of Problems $8.47$ and $9.57$. The $\mathrm{PCM}$ is housed in a long, horizontal, and insulated cylindrical enclosure of diameter $D_{c}=200 \mathrm{~mm}$, which in turn includes a concentric, heated inner cylinder of diameter $D_{i}=$ $30 \mathrm{~mm}$. Initially, the PCM is entirely solid and at its phase change temperature. The inner cylinder temperature is suddenly raised to $T_{h}=50^{\circ} \mathrm{C}$. Assuming the PCM melts to form an expanding concentric liquid region about the heated tube such as the one shown in the schematic, determine how long it takes to melt half of the PCM.

Keshav Singh
Keshav Singh
Numerade Educator
View

Problem 107

Liquid nitrogen is stored in a thin-walled spherical vessel of diameter $D_{i}=1 \mathrm{~m}$. The vessel is positioned concentrically within a larger, thin-walled spherical container of diameter $D_{o}=1.10 \mathrm{~m}$, and the intervening cavity is filled with atmospheric helium.
Under normal operating conditions, the inner and outer surface temperatures are $T_{i}=77 \mathrm{~K}$ and $T_{o}=283 \mathrm{~K}$. If the latent heat of vaporization of nitrogen is $2 \times 10^{5} \mathrm{~J} / \mathrm{kg}$, what is the mass rate $m(\mathrm{~kg} / \mathrm{s})$ at which gaseous nitrogen is vented from the system?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
03:45

Problem 108

The human eye contains aqueous humor, which separates the external cornea and the internal iris-lens structure. It is hypothesized that, in some individuals, small flakes of pigment are intermittently liberated from the iris and migrate to, and subsequently damage, the cornea. Approximating the geometry of the enclosure formed by the cornea and iris-lens structure as a pair of concentric hemispheres of outer radius $r_{o}=10 \mathrm{~mm}$ and inner radius $r_{i}=7 \mathrm{~mm}$, respectively, investigate whether free convection can occur in the aqueous humor by evaluating the effective thermal conductivity ratio, $k_{\mathrm{efI}} / k$. If free convection can occur, it is possible that the damaging particles are advected from the iris to the cornea. The iris-lens structure is at the core temperature, $T_{i}=37^{\circ} \mathrm{C}$, while the cornea temperature is measured to be $T_{o}=34^{\circ} \mathrm{C}$. The properties of the aqueous humor are $\rho=990 \mathrm{~kg} / \mathrm{m}^{3}, k=0.58 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $c_{p}=4.2 \times 10^{3} \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=7.1 \times 10^{-4} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, and $\beta=3.2 \times 10^{-4} \mathrm{~K}^{-1}$

Farhanul Hasan
Farhanul Hasan
Numerade Educator
05:10

Problem 109

A horizontal, $25-\mathrm{mm}$ diameter cylinder is maintained at a uniform surface temperature of $35^{\circ} \mathrm{C}$. A fluid with a velocity of $0.05 \mathrm{~m} / \mathrm{s}$ and temperature of $20^{\circ} \mathrm{C}$ is in cross flow over the cylinder. Determine whether heat transfer by free convection will be significant for (i) air, (ii) water, (iii) engine oil, and (iv) mercury.

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
02:17

Problem 110

According to experimental results for parallel airflow over a uniform temperature, heated vertical plate, the effect of free convection on the heat transfer convection coefficient will be $5 \%$ when $G r_{L} / R e_{L}^{2}=0.08$. Consider a heated vertical plate $0.3 \mathrm{~m}$ long, maintained at a surface temperature of $60^{\circ} \mathrm{C}$ in atmospheric air at $25^{\circ} \mathrm{C}$. What is the minimum vertical velocity required of the airflow such that free convection effects will be less than $5 \%$ of the heat transfer rate?

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
01:36

Problem 111

A vertical array of circuit boards of 150 -mm height is to be air cooled such that the board temperature does not exceed $60^{\circ} \mathrm{C}$ when the ambient temperature is $25^{\circ} \mathrm{C}$.
Assuming isothermal surface conditions, determine the allowable electrical power dissipation per board for the cooling arrangements:
(a) Free convection only (no forced airflow).
(b) Airflow with a downward velocity of $0.6 \mathrm{~m} / \mathrm{s}$.
(c) Airflow with an upward velocity of $0.3 \mathrm{~m} / \mathrm{s}$.
(d) Airflow with a velocity (upward or downward) of $5 \mathrm{~m} / \mathrm{s}$.

Anand Jangid
Anand Jangid
Numerade Educator
02:03

Problem 112

A probe, used to measure the velocity of air in a lowspeed wind tunnel, is fabricated of an $L=100 \mathrm{~mm}$ long, $D=8$-mm outside diameter horizontal aluminum tube. Power resistors are inserted into the stationary tube and dissipate $P=1.5 \mathrm{~W}$. The surface temperature of the tube is determined experimentally by measuring the emitted radiation from the exterior of the tube. To maximize surface emission, the exterior of the tube is painted with flat black paint having an emissivity of $\varepsilon=0.95$.
(a) For air at a temperature and cross flow velocity of $T_{\infty}=25^{\circ} \mathrm{C}, V=0.1 \mathrm{~m} / \mathrm{s}$, respectively, determine the surface temperature of the tube. The surroundings temperature is $T_{\text {sur }}=25^{\circ} \mathrm{C}$.
(b) For the conditions of part (a), plot the tube surface temperature versus the cross flow velocity over the range $0.05 \mathrm{~m} / \mathrm{s} \leq V \leq 1 \mathrm{~m} / \mathrm{s}$.

Chai Santi
Chai Santi
Numerade Educator
01:29

Problem 113

A horizontal 100-mm-diameter pipe passing hot oil is to be used in the design of an industrial water heater. Based on a typical water draw rate, the velocity over the pipe is $0.5 \mathrm{~m} / \mathrm{s}$. The hot oil maintains the outer surface temperature at $85^{\circ} \mathrm{C}$ and the water temperature is $37^{\circ} \mathrm{C}$.
Investigate the effect of flow direction on the heat rate (W/m) for (a) horizontal, (b) downward, and (c) upward flow.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:17

Problem 114

Determine the heat transfer rate from the steel plates of Problem $7.24$ accounting for free convection from the plate surfaces. What is the corresponding rate of change of the plate temperature? Plot the heat transfer coefficient associated with free convection, forced convection, and mixed convection for air velocities ranging from $2 \leq u_{\infty} \leq 10 \mathrm{~m} / \mathrm{s}$. The velocity of the plate is small compared to the air velocity.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
01:38

Problem 115

An experiment involves heating a very small sphere that is suspended by a fine string in air with a laser beam in order to induce the highest sphere temperature possible. After inspecting Equation 9.64, a research assistant suggests inducing a uniform downward airflow to exactly offset free convection from the sphere, thereby minimizing heat losses and maximizing the steady-state sphere temperature. In the limiting case of a very small sphere, what is the minimum value of the convection heat transfer coefficient expressed in terms of the sphere diameter and thermal conductivity of the air?

Penny Riley
Penny Riley
Numerade Educator
01:33

Problem 116

Square panels $(250 \mathrm{~mm} \times 250 \mathrm{~mm}$ ) with a decorative, highly reflective plastic finish are cured in an oven at $125^{\circ} \mathrm{C}$ and cooled in quiescent air at $29^{\circ} \mathrm{C}$. Quality considerations dictate that the panels remain horizontal and that the cooling rate be controlled. To increase productivity in the plant, it is proposed to replace the batch cooling method with a conveyor system having a velocity of $0.5 \mathrm{~m} / \mathrm{s}$.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:02

Problem 117

A garment soaked with water is hung up to dry in a warm room at atmospheric pressure. The still air is dry and at a temperature of $40^{\circ} \mathrm{C}$. The garment may be assumed to have a temperature of $25^{\circ} \mathrm{C}$ and a characteristic length of $1 \mathrm{~m}$ in the vertical direction. Estimate the drying rate per unit width of the garment.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:29

Problem 118

A water bath is used to maintain canisters containing experimental biological reactions at a uniform temperature of $37^{\circ} \mathrm{C}$. The top of the bath has a width and length of $0.25 \mathrm{~m}$ and $0.50 \mathrm{~m}$, respectively, and is uncovered to allow easy access for removal or insertion of the canisters. The bath is located in a draft-free laboratory with air at atmospheric pressure, a temperature of $20^{\circ} \mathrm{C}$, and a relative humidity of $60 \%$. The walls of the laboratory are at a uniform temperature of $25^{\circ} \mathrm{C}$.
(a) Estimate the heat loss from the surface of the bath by radiation exchange with the surroundings.
(b) Calculate the Grashof number using Equation $9.65$, which can be applied to natural convection flows driven by temperature and concentration gradients. Use a characteristic length $L$ that is appropriate for the exposed surface of the water bath.
(c) Estimate the free convection heat transfer coefficient using the result for $G r_{L}$ obtained in part (b).
(d) Invoke the heat and mass transfer analogy and use an appropriate correlation to estimate the mass transfer coefficient using $G r_{L}$. Calculate the water evaporation rate on a daily basis and the heat loss by evaporation.
(e) Calculate the total heat loss from the surface, and compare the relative contributions of the sensible, latent, and radiative effects. Review the assumptions made in your analysis, especially those relating to the heat and mass transfer analogy.

Sheryl Ezze
Sheryl Ezze
Numerade Educator
02:36

Problem 119

On a very still morning, the surface temperature of a lake used to cool the condenser of a power plant is $30^{\circ} \mathrm{C}$ while the air temperature is $23^{\circ} \mathrm{C}$ with a relative humidity of $80 \%$. Assume a surroundings temperature of $285 \mathrm{~K}$. The lake is nominally circular in shape with a diameter of approximately $4 \mathrm{~km}$. Determine the heat loss from the surface of the lake by radiation, free convection, and evaporation. This heat loss determines the capacity of the lake to cool the condenser. Justify why the heat transfer correlation you select is useful, even though $R a_{L}$ is outside of its specified range. Hint: See Problem 9.118.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
08:33

Problem 120

Fuel cells similar to the PEM cell of Example $1.5$ operate with a mixture of liquid water and methanol instead of hydrogen; the anode is placed in direct contact with the liquid fuel. Oxygen (species A) is delivered to the exposed cathode by free convection. Hence, no fans or pumps are needed to operate the device. The power output of passive, direct methanol fuel cells (DMFCs) can become mass transfer limited, since the electric current produced by the DMFC is related to the rate at which oxygen is consumed at the cathode by the expression $I=4 n_{\mathrm{A}} \mathcal{F} M_{\mathrm{A}}$, where $\mathcal{F}$ is Faraday's constant, $\mathcal{F}=$ 96,489 coulombs/mol. Consider a passive DMFC with a $120 \mathrm{~mm} \times 120 \mathrm{~mm}$ membrane. Determine the maximum possible electric current produced by the DMFC when the oxygen mass fraction at the cathode is $m_{A, s}=0.10$ for cases where the cathode is facing up or is vertical. As a first approximation and to illustrate the sensitivity of the device to its orientation relative to the vertical direction, assume buoyancy forces are dominated by the difference in density associated with the change in the oxygen mass fraction between the cathode surface and the quiescent environment, which is atmospheric air at $T_{\infty}=25^{\circ} \mathrm{C}$. Assume the quiescent air is composed of nitrogen and oxygen, with an oxygen mass fraction $m_{\Lambda, \infty}=0.233$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator