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Fundamentals of Heat and Mass Transfer

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

Chapter 7

External Flow - all with Video Answers

Educators

Chapter Questions

06:06

Problem 1

Consider the following fluids at a film temperature of $300 \mathrm{~K}$ in parallel flow over a flat plate with velocity of $1 \mathrm{~m} / \mathrm{s}$ : atmospheric air, water, engine oil, and mercury.
(a) For each fluid, determine the velocity and thermal boundary layer thicknesses at a distance of $40 \mathrm{~mm}$ from the leading edge.
(b) For each of the prescribed fluids and on the same coordinates, plot the boundary layer thicknesses as a function of distance from the leading edge to a plate length of $40 \mathrm{~mm}$.

Manish Jain
Manish Jain
Numerade Educator
06:06

Problem 2

Engine oil at $100^{\circ} \mathrm{C}$ and a velocity of $0.1 \mathrm{~m} / \mathrm{s}$ flows over both surfaces of a 1 -m-long flat plate maintained at $20^{\circ} \mathrm{C}$. Determine:
(a) The velocity and thermal boundary layer thicknesses at the trailing edge.
(b) The local heat flux and surface shear stress at the trailing edge.
(c) The total drag force and heat transfer per unit width of the plate.
(d) Plot the boundary layer thicknesses and local values of the surface shear stress, convection coefficient, and heat flux as a function of $x$ for $0 \leq x \leq 1 \mathrm{~m}$.

Manish Jain
Manish Jain
Numerade Educator
04:13

Problem 3

Consider steady, parallel flow of atmospheric air over a flat plate. The air has a temperature and free stream velocity of $300 \mathrm{~K}$ and $25 \mathrm{~m} / \mathrm{s}$.
(a) Evaluate the boundary layer thickness at distances of $x=1,10$, and $100 \mathrm{~mm}$ from the leading edge. If a second plate were installed parallel to and at a distance of $3 \mathrm{~mm}$ from the first plate, what is the distance from the leading edge at which boundary layer merger would occur?
(b) Evaluate the surface shear stress and the $y$-velocity component at the outer edge of the boundary layer for the single plate at $x=1,10$, and $100 \mathrm{~mm}$.
(c) Comment on the validity of the boundary layer approximations.

Chai Santi
Chai Santi
Numerade Educator
04:06

Problem 4

Consider a liquid metal $(\operatorname{Pr} \leqslant 1)$, with free stream conditions $u_{\infty}$ and $T_{\infty}$, in parallel flow over an isothermal flat plate at $T_{s}$. Assuming that $u=u_{\infty}$ throughout the thermal boundary layer, write the corresponding form of the boundary layer energy equation. Applying appropriate initial $(x=0)$ and boundary conditions, solve this equation for the boundary layer temperature field, $T(x, y)$. Use the result to obtain an expression for the local Nusselt number $\mathrm{Nu}_{x}$. Hint: This problem is analogous to one-dimensional heat transfer in a semiinfinite medium with a sudden change in surface temperature.

Joseph Liao
Joseph Liao
Numerade Educator
02:43

Problem 5

Consider the velocity boundary layer profile for flow over a flat plate to be of the form $u=C_{1}+C_{2} y$. Applying appropriate boundary conditions, obtain an expression for the velocity profile in terms of the boundary layer thickness $\delta$ and the free stream velocity $u_{\infty}$. Using the integral form of the boundary layer momentum equation (Appendix G), obtain expressions for the boundary layer thickness and the local friction coefficient, expressing your result in terms of the local Reynolds number. Compare your results with those obtained from the exact solution (Section 7.2.1) and the integral solution with a cubic profile (Appendix $G$ ).

James Kiss
James Kiss
Numerade Educator
04:24

Problem 6

Consider a steady, turbulent boundary layer on an isothermal flat plate of temperature $T_{s}$. The boundary layer is "tripped" at the leading edge $x=0$ by a fine wire. Assume constant physical properties and velocity and temperature profiles of the form
$$
\frac{u}{u_{x}}=\left(\frac{y}{\delta}\right)^{1 / 7} \quad \text { and } \quad \frac{T-T_{\infty}}{T_{s}-T_{x}}=1-\left(\frac{y}{\delta_{t}}\right)^{1 / 7}
$$
(a) From experiment it is known that the surface shear stress is related to the boundary layer thickness by an expression of the form
$$
\tau_{s}=0.0228 \rho u_{\infty}^{2}\left(\frac{u_{\infty} \delta}{v}\right)^{-1 / 4}
$$
Beginning with the momentum integral equation (Appendix G), show that
$$
\delta / x=0.376 R e_{x}^{-1 / 5}
$$
Determine the average friction coefficient $\bar{C}_{f x}$
(b) Beginning with the energy integral equation, obtain an expression for the local Nusselt number $N u_{x}$ and use this result to evaluate the average Nusselt number $\overline{N u}$.

Nick Johnson
Nick Johnson
Numerade Educator
02:01

Problem 7

Consider flow over a flat plate for which it is desired to determine the average heat transfer coefficient over the short span $x_{1}$ to $x_{2}, \bar{h}_{1-2}$, where $\left(x_{2}-x_{1}\right) \leqslant L$.
Provide three different expressions that can be used to evaluate $\bar{h}_{1-2}$ in terms of (a) the local coefficient at $x=\left(x_{1}+x_{2}\right) / 2,(b)$ the local coefficients at $x_{1}$ and $x_{2}$, and (c) the average coefficients at $x_{1}$ and $x_{2}$. Indicate which of the expressions is approximate. Considering whether the flow is laminar, turbulent, or mixed, indicate when it is appropriate or inappropriate to use each of the equations.

Narayan Hari
Narayan Hari
Numerade Educator
05:01

Problem 8

A flat plate of width $1 \mathrm{~m}$ is maintained at a uniform surface temperature of $T_{s}=150^{\circ} \mathrm{C}$ by using independently controlled, heat-generating rectangular modules of thickness $a=10 \mathrm{~mm}$ and length $b=50 \mathrm{~mm}$. Each module is insulated from its neighbors, as well as on its back side. Atmospheric air at $25^{\circ} \mathrm{C}$ flows over the plate at a velocity of $30 \mathrm{~m} / \mathrm{s}$. The thermophysical properties of the module are $k=5.2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, c_{p}=320 \mathrm{~J} / \mathrm{kg}+\mathrm{K}$, and $\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}$.
(a) Find the required power generation, $\dot{q}\left(\mathrm{~W} / \mathrm{m}^{3}\right)$, in a module positioned at a distance $700 \mathrm{~mm}$ from the leading edge.
(b) Find the maximum temperature $T_{\max }$ in the heatgenerating module.

Keshav Singh
Keshav Singh
Numerade Educator
04:40

Problem 9

An electric air heater consists of a horizontal array of thin metal strips that are each $10 \mathrm{~mm}$ long in the direction of an airstream that is in parallel flow over the top of the strips. Each strip is $0.2 \mathrm{~m}$ wide, and 25 strips are arranged side by side, forming a continuous and smooth surface over which the air flows at $2 \mathrm{~m} / \mathrm{s}$. During operation, each strip is maintained at $500^{\circ} \mathrm{C}$ and the air is at $25^{\circ} \mathrm{C}$.
(a) What is the rate of convection heat transfer from the first strip? The fifth strip? The tenth strip? All the strips?
(b) For air velocities of 2,5 , and $10 \mathrm{~m} / \mathrm{s}$, determine the convection heat rates for all the locations of part (a). Represent your results in tabular or bar graph form.
(c) Repeat part (b), but under conditions for which the flow is fully turbulent over the entire array of strips.

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
02:01

Problem 10

Consider atmospheric air at $25^{\circ} \mathrm{C}$ and a velocity of $25 \mathrm{~m} / \mathrm{s}$ flowing over both surfaces of a 1 - $\mathrm{m}$-long flat plate that is maintained at $125^{\circ} \mathrm{C}$. Determine the rate of heat transfer per unit width from the plate for values of the critical Reynolds number corresponding to $10^{5}$, $5 \times 10^{5}$, and $10^{6}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:01

Problem 11

Consider laminar, parallel flow past an isothermal flat 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
02:45

Problem 12

Repeat Problem $7.11$ for the case when the boundary layer is tripped to a turbulent condition at its leading edge.

Chai Santi
Chai Santi
Numerade Educator
03:02

Problem 13

Consider a flat plate subject to parallel flow (top and bottom) characterized by $u_{\infty}=5 \mathrm{~m} / \mathrm{s}, T_{\infty}=20^{\circ} \mathrm{C}$.
(a) Determine the average convection heat transfer coefficient, convective heat transfer rate, and drag force associated with an $L=2$-m-long, $w=2-\mathrm{m}$ wide flat plate for airflow and surface temperatures of $T_{s}=50^{\circ} \mathrm{C}$ and $80^{\circ} \mathrm{C}$.
(b) Determine the average convection heat transfer coefficient, convective heat transfer rate, and drag force associated with an $L=0.1$-m-long, $w=0.1$-m-wide flat plate for water flow and surface temperatures of $T_{s}=50^{\circ} \mathrm{C}$ and $80^{\circ} \mathrm{C}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:01

Problem 14

Consider water at $27^{\circ} \mathrm{C}$ in parallel flow over an isother$\mathrm{mal}, 1$-m-long flat plate with a velocity of $2 \mathrm{~m} / \mathrm{s}$.
(a) Plot the variation of the local heat transfer coefficient, $h_{x}(x)$, with distance along the plate for three flow conditions corresponding to transition Reynolds numbers of (i) $5 \times 10^{5}$, (ii) $3 \times 10^{5}$, and (iii) 0 (the flow is fully turbulent).
(b) Plot the variation of the average heat transfer coefficient $\bar{h}_{x}(x)$ with distance for the three flow conditions of part (a).
(c) What are the average heat transfer coefficients for the entire plate $\bar{h}_{L}$ for the three flow conditions of part (a)?

Narayan Hari
Narayan Hari
Numerade Educator
02:01

Problem 15

Explain under what conditions the total rate of heat transfer from an isothermal flat plate of dimensions $L \times 2 L$ would be the same, independent of whether parallel flow over the plate is directed along the side of length $L$ or $2 L$. With a critical Reynolds number of $5 \times 10^{5}$, for what values of $R e_{L}$ would the total heat transfer be independent of orientation?

Narayan Hari
Narayan Hari
Numerade Educator
12:12

Problem 16

In fuel cell stacks, it is desirable to operate under conditions that promote uniform surface temperatures for the electrolytic membranes. This is especially true in hightemperature fuel cells where the membrane is constructed of a brittle ceramic material. Electrochemical reactions in the electrolytic membranes generate thermal energy, while gases flowing above and below the membranes cool it. The stack designer may specify top and bottom flows that are in the same, opposite, or orthogonal directions. A preliminary study of the effect of the relative flow directions is conducted whereby a $150 \mathrm{~mm} \times 150 \mathrm{~mm}$ thin sheet of material, producing a uniform heat flux of $100 \mathrm{~W} / \mathrm{m}^{2}$, is cooled (top and bottom) by air with a free stream temperature and velocity of $25^{\circ} \mathrm{C}$ and $2 \mathrm{~m} / \mathrm{s}$, respectively.
(a) Determine the minimum and maximum local membrane temperatures for top and bottom flows that are in the same, opposite, and orthogonal directions. Which flow configuration minimizes the membrane temperature? Hint: For the opposite and orthogonal flow cases, the boundary layers are subject to boundary conditions that are neither uniform temperature nor uniform heat flux. It is, however, reasonable to expect that the resulting temperatures would be bracketed by your answers based on the constant heat flux and constant temperature boundary conditions.
(b) Plot the surface temperature distribution $T(x)$ for the cases involving flow in the opposite and same directions. Thermal stresses are undesirable and are related to the spatial temperature gradient along the membrane. Which configuration minimizes spatial temperature gradients?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
00:17

Problem 17

Air at a pressure of 1 atm and a temperature of $50^{\circ} \mathrm{C}$ is in parallel flow over the top surface of a flat plate that is heated to a uniform temperature of $100^{\circ} \mathrm{C}$. The plate has a length of $0.20 \mathrm{~m}$ (in the flow direction) and a width of $0.10 \mathrm{~m}$. The Reynolds number based on the plate length is 40,000 . What is the rate of heat transfer from the plate to the air? If the free stream velocity of the air is doubled and the pressure is increased to $10 \mathrm{~atm}$, what is the rate of heat transfer?

Mayukh Banik
Mayukh Banik
Numerade Educator
01:34

Problem 18

Consider the photovoltaic solar panel of Example 3.3. The heat transfer coefficient should no longer be taken to be a specified value.
(a) Determine the silicon temperature and the electric power produced by the solar cell for an air velocity of $4 \mathrm{~m} / \mathrm{s}$ parallel to the long direction, with air and surroundings temperatures of $20^{\circ} \mathrm{C}$. The boundary layer is tripped to a turbulent condition at the leading edge of the panel.
(b) Repeat part (a), except now the panel is oriented with its short side parallel to the airflow, that is, $L=0.1 \mathrm{~m}$ and $w=1 \mathrm{~m}$.
(c) Plot the electric power output and the silicon temperature versus air velocity over the range $0 \leq u_{m}$ $\leq 10 \mathrm{~m} / \mathrm{s}$ for the $L=0.1 \mathrm{~m}$ and $w=1 \mathrm{~m}$ case.

Penny Riley
Penny Riley
Numerade Educator
01:00

Problem 19

Concentration of sunlight onto photovoltaic cells is desired since the concentrating mirrors and lenses are less expensive than the photovoltaic material. Consider the solar photovoltaic cell of Example 3.3. A $100 \mathrm{~mm} \times 100 \mathrm{~mm}$ photovoltaic cell is irradiated with concentrated solar energy. Since the concentrating lens is glass, it absorbs $10 \%$ of the irradiation instead of the top surface of the solar cell, as in Example 3.3. The remaining irradiation is reflected from the system (7\%) or is absorbed in the silicon semiconductor material of the photovoltaic cell $(83 \%)$. The photovoltaic cell is cooled by air directed parallel to its top and bottom surfaces. The air temperature and velocity are $25^{\circ} \mathrm{C}$ and $5 \mathrm{~m} / \mathrm{s}$, respectively, and the bottom surface is coated with a high-emissivity paint, $\varepsilon_{b}=0.95$.
(a) Determine the electric power produced by the photovoltaic cell and the silicon temperature for a square concentrating lens with $L_{\text {lans }}=400 \mathrm{~mm}$, which focuses the irradiation falling on the lens to the smaller area of the photovoltaic cell. Assume the concentrating lens temperature is $25^{\circ} \mathrm{C}$ and does not interfere with boundary layer development over the photovoltaic cell's top surface. The top and bottom boundary layers are both tripped to turbulent conditions at the leading edge of the photovoltaic material.
(b) Determine the electric power output of the photovoltaic cell and the silicon temperature over the range $100 \mathrm{~mm} \leq L_{\text {lens }} \leq 600 \mathrm{~mm}$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
07:12

Problem 20

The roof of a refrigerated truck compartment is of composite construction, consisting of a layer of foamed urethane insulation $\left(t_{2}=50 \mathrm{~mm}, k_{i}=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$ sandwiched between aluminum alloy panels $\left(t_{1}=5 \mathrm{~mm}\right.$, $\left.k_{p}=180 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$. The length and width of the roof are $L=10 \mathrm{~m}$ and W $=3.5 \mathrm{~m}$, respectively, and the temperature of the inner surface is $T_{s, i}=-10^{\circ} \mathrm{C}$. Consider conditions for which the truck is moving at a speed of $V=105 \mathrm{~km} / \mathrm{h}$, the air temperature is $T_{\infty}=32^{\circ} \mathrm{C}$, and the solar irradiation is $G_{S}=750 \mathrm{~W} / \mathrm{m}^{2}$. Turbulent flow may be assumed over the entire length of the roof.
(a) For equivalent values of the solar absorptivity and the emissivity of the outer surface $\left(\alpha_{S}=\varepsilon=0.5\right)$, estimate the average temperature $T_{s, o}$ of the outer surface. What is the corresponding heat load imposed on the refrigeration system?
(b) A special finish $\left(\alpha_{S}=0.15, \varepsilon=0.8\right)$ may be applied to the outer surface. What effect would such an application have on the surface temperature and the heat load?
(c) If, with $\alpha_{S}=\varepsilon=0.5$, the roof is not insulated $\left(t_{2}=0\right)$, what are the corresponding values of the surface temperature and the heat load?

Nick Auwerda
Nick Auwerda
Numerade Educator
03:14

Problem 21

The top surface of a heated compartment consists of very smooth (A) and highly roughened (B) portions, and the surface is placed in an atmospheric airstream.
In the interest of minimizing total convection heat transfer from the surface, which orientation, (1) or (2), is preferred? If $T_{s}=100^{\circ} \mathrm{C}, T_{\infty}=20^{\circ} \mathrm{C}$, and $u_{\infty}=$ $20 \mathrm{~m} / \mathrm{s}$, what is the convection heat transfer from the entire surface for this orientation?

Mayukh Banik
Mayukh Banik
Numerade Educator
02:01

Problem 22

Calculate the value of the average heat transfer coefficient for the plate of Problem $7.21$ when the entire plate is rotated $90^{\circ}$ so that half of the leading edge consists of a very smooth portion (A) and the other half consists of a highly roughened portion (B).

Narayan Hari
Narayan Hari
Numerade Educator
01:47

Problem 23

The proposed design for an anemometer to determine the velocity of an airstream in a wind tunnel is comprised of a thin metallic strip whose ends are supported by stiff rods serving as electrodes for passage of current used to heat the strip. A fine-wire thermocouple is attached to the trailing edge of the strip and serves as the sensor for a system that controls the power to maintain the strip at a constant operating temperature for variable airstream velocities. Design conditions pertain to an airstream at $T_{\infty}=25^{\circ} \mathrm{C}$ and $1 \leq u_{\infty} \leq 50 \mathrm{~m} / \mathrm{s}$, with a strip temperature of $T_{s}=35^{\circ} \mathrm{C}$.
(a) Determine the relationship between the electrical power dissipation per unit width of the strip in the transverse direction, $P^{\prime}(\mathrm{mW} / \mathrm{mm})$, and the airstream velocity. Show this relationship graphically for the specified range of $u_{\infty}$.
(b) If the accuracy with which the temperature of the operating strip can be measured and maintained constant is $\pm 0.2^{\circ} \mathrm{C}$, what is the uncertainty in the airstream velocity?
(c) The proposed design operates in a strip constanttemperature mode for which the airstream velocity is related to the measured power. Consider now an alternative mode wherein the strip is provided with a constant power, say, $30 \mathrm{~mW} / \mathrm{mm}$, and the airstream velocity is related to the measured strip temperature $T_{s^{*}}$. For this mode of operation, show the graphical relationship between the strip temperature and airstream velocity. If the temperature can be measured with an uncertainty of $\pm 0.2^{\circ} \mathrm{C}$, what is the uncertainty in the airstream velocity?
(d) Compare the features associated with each of the anemometer operating modes.

Aadit Sharma
Aadit Sharma
Numerade Educator
05:01

Problem 24

Steel (AISI 1010) plates of thickness $\delta=6 \mathrm{~mm}$ and length $L=1 \mathrm{~m}$ on a side are conveyed from a heat treatment process and are concurrently cooled by atmospheric air of velocity $u_{\infty}=10 \mathrm{~m} / \mathrm{s}$ and $T_{x}=20^{\circ} \mathrm{C}$ in parallel flow over the plates.
For an initial plate temperature of $T_{i}=300^{\circ} \mathrm{C}$, what is the rate of heat transfer from the plate? What is the corresponding rate of change of the plate temperature? The velocity of the air is much larger than that of the plate.

Keshav Singh
Keshav Singh
Numerade Educator
View

Problem 25

Consider weather conditions for which the prevailing wind blows past the penthouse tower on a tall building. The tower length in the wind direction is $10 \mathrm{~m}$ and there are 10 window panels.
(a) Calculate the average convection coefficient for the first, third, and tenth window panels when the wind speed is $5 \mathrm{~m} / \mathrm{s}$. Use a film temperature of $300 \mathrm{~K}$ to evaluate the thermophysical properties required of the correlation. Would this be a suitable value of the film temperature for ambient air temperatures in the range $-15 \leq T_{\infty} \leq 38^{\circ} \mathrm{C}$ ?
(b) For the first, third, and tenth windows, on one graph, plot the variation of the average convection coefficient with wind speed for the range $5 \leq u_{\infty} \leq 100 \mathrm{~km} / \mathrm{h}$. Explain the major features of each curve and their relative magnitudes.

Shu Naito
Shu Naito
Numerade Educator
03:05

Problem 26

The Weather Channel reports that it is a hot, muggy day with an air temperature of $90^{\circ} \mathrm{F}$, a $10 \mathrm{mph}$ breeze out of the southwest, and bright sunshine with a solar insolation of $400 \mathrm{~W} / \mathrm{m}^{2}$. Consider the wall of a metal building over which the prevailing wind blows. The length of the wall in the wind direction is $10 \mathrm{~m}$, and the emissivity is $0.93$. Assume that all the solar irradiation is absorbed, that irradiation from the sky is negligible, and that flow is fully turbulent over the wall. Estimate the average wall temperature.

Lisa Tarman
Lisa Tarman
Numerade Educator
01:00

Problem 28

In the production of sheet metals or plastics, it is customary to cool the material before it leaves the production process for storage or shipment to the customer. Typically, the process is continuous, with a sheet of thickness $\delta$ and width $W$ cooled as it transits the distance $L$ between two rollers at a velocity $V$. In this problem, we consider cooling of an aluminum alloy (2024-T6) by an airstream moving at a velocity $u_{\infty}$ in counter flow over the top surface of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface.
(a) By applying conservation of energy to a differential control surface of length $d x$, which either moves with the sheet or is stationary and through which the sheet passes, derive a differential equation that governs the temperature distribution along the sheet. Because of the low emissivity of the aluminum, radiation effects may be neglected. Express your result in terms of the velocity, thickness, and properties of the sheet $\left(V, \delta, \rho, c_{p}\right)$, the local convection coefficient $h_{x}$ associated with the counter flow, and the air temperature. For a known temperature of the sheet $\left(T_{i}\right)$ at the onset of cooling and a negligible effect of the sheet velocity on boundary layer development, solve the equation to obtain an expression for the outlet temperature $T_{a}$.
(b) For $\delta=2 \mathrm{~mm}, V=0.10 \mathrm{~m} / \mathrm{s}, L=5 \mathrm{~m}, W=1 \mathrm{~m}$, $u_{\infty}=20 \mathrm{~m} / \mathrm{s}, T_{\infty}=20^{\circ} \mathrm{C}$, and $T_{i}=300^{\circ} \mathrm{C}$, what is the outlet temperature $T_{a}$ ?

Raj Bala
Raj Bala
Numerade Educator
01:27

Problem 29

An array of electronic chips is mounted within a sealed rectangular enclosure, and cooling is implemented by attaching an aluminum heat $\operatorname{sink}(k=180 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. The base of the heat sink has dimensions of $w_{1}=w_{2}=$ $100 \mathrm{~mm}$, while the 6 fins are of thickness $t=10 \mathrm{~mm}$ and pitch $S=18 \mathrm{~mm}$. The fin length is $L_{f}=50 \mathrm{~mm}$, and the base of the heat sink has a thickness of $L_{b}=10 \mathrm{~mm}$.
If cooling is implemented by water flow through the heat sink, with $u_{\infty}=3 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=17^{\circ} \mathrm{C}$, what is the base temperature $T_{b}$ of the heat sink when power dissipation by the chips is $P_{\text {elec }}=1800 \mathrm{~W}$ ? The average convection coefficient for surfaces of the fins and the exposed base may be estimated by assuming parallel flow over a flat plate. Properties of the water may be approximated as $k=0.62 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=995 \mathrm{~kg} / \mathrm{m}^{3}$, $c_{p}=4178 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \nu=7.73 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$, and $\operatorname{Pr}=5.2$.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:00

Problem 30

Consider the concentrating photovoltaic apparatus of Problem 7.19. The apparatus is to be installed in a desert environment, so the space between the concentrating lens and top of the photovoltaic cell is enclosed to protect the cell from sand abrasion in windy conditions. Since convection cooling from the top of the cell is reduced by the enclosure, an engineer proposes to cool the photovoltaic cell by attaching an aluminum heat sink to its bottom surface. The heat sink dimensions and material are the same as those of Problem 7.29. A contact resistance of $0.5 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$ exists at the photovoltaic cell/heat sink interface and a dielectric liquid $\left(k=0.064 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=1400 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=1300 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$ $y=10^{-6} \mathrm{~m}^{2} / \mathrm{s}, P r=25$ ) flows between the heat $\operatorname{sink}$ fins at $u_{\infty}=3 \mathrm{~m} / \mathrm{s}, T_{\infty}=25^{\circ} \mathrm{C}$.
(a) Determine the electric power produced by the photovoltaic cell and the silicon temperature for a square concentrating lens with $L_{\text {lens }}=400 \mathrm{~mm}$.
(b) Compare the electric power produced by the photovoltaic cell with the heat sink in place and with the bottom surface cooled directly by the dielectric fluid (i.e., no heat $\operatorname{sink}$ ) for $L_{\text {lens }}=1.5 \mathrm{~m}$.
(c) Determine the electric power output and the silicon temperature over the range $100 \mathrm{~mm} \leq L_{\text {lens }}<$ $3000 \mathrm{~mm}$ with the aluminum heat sink in place.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
11:24

Problem 31

In the production of sheet metals or plastics, it is customary to cool the material before it leaves the production process for storage or shipment to the customer. Typically, the process is continuous, with a sheet of thickness $\delta$ and width $W$ cooled as it transits the distance $L$ between two rollers at a velocity $V$. In this problem, we consider cooling of plain carbon steel by an airstream moving at a velocity $u_{\infty}$ in cross flow over the top and bottom surfaces of the sheet. A turbulence promoter is used to provide turbulent boundary layer development over the entire surface.
(a) By applying conservation of energy to a differential control surface of length $d x$, which either moves with the sheet or is stationary and through which the sheet passes, and assuming a uniform sheet temperature in the direction of airflow, derive a differential equation that governs the temperature distribution, $T(x)$, along the sheet. Consider the effects of radiation, as well as convection, and express your result in terms of the velocity, thickness, and properties of the sheet $\left(V, \delta, \rho, c_{p}, \varepsilon\right)$, the average convection coefficient $\bar{h}_{W}$ associated with the cross flow, and the environmental temperatures $\left(T_{\infty}, T_{\text {sur }}\right)$.
(b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For $\delta=3 \mathrm{~mm}, V=$ $0.10 \mathrm{~m} / \mathrm{s}, L=10 \mathrm{~m}, W=1 \mathrm{~m}, u_{\infty}=20 \mathrm{~m} / \mathrm{s}, T_{\infty}=$ $20^{\circ} \mathrm{C}$, and a sheet temperature of $T_{i}=500^{\circ} \mathrm{C}$ at the onset of cooling, what is the outlet temperature $T_{o}$ ? Assume a negligible effect of the sheet velocity on boundary layer development in the direction of airflow. The density and specific heat of the steel are $\rho=7850 \mathrm{~kg} / \mathrm{m}^{3}$ and $c_{p}=620 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, while properties of the air may be taken to be $k=0.044$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, \nu=4.5 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}, \operatorname{Pr}=0.68$.
(c) Accounting for the effects of radiation, with $\varepsilon=$ $0.70$ and $T_{\text {sur }}=20^{\circ} \mathrm{C}$, numerically integrate the differential equation derived in part (a) to determine the temperature of the sheet at $L=10 \mathrm{~m}$. Explore the effect of $V$ on the temperature distribution along the sheet.

Nathan Prins
Nathan Prins
Numerade Educator
05:18

Problem 32

A steel strip emerges from the hot roll section of a steel mill at a speed of $20 \mathrm{~m} / \mathrm{s}$ and a temperature of $1200 \mathrm{~K}$. Its length and thickness are $L=100 \mathrm{~m}$ and $\delta=$ $0.003 \mathrm{~m}$, respectively, and its density and specific heat are $7900 \mathrm{~kg} / \mathrm{m}^{3}$ and $640 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, respectively.
Accounting for heat transfer from the top and bottom surfaces and neglecting radiation and strip conduction effects, determine the time rate of change of the strip temperature at a distance of $1 \mathrm{~m}$ from the leading edge and at the trailing edge. Determine the distance from the leading edge at which the minimum cooling rate is achieved.

Dading Chen
Dading Chen
Numerade Educator
11:24

Problem 33

In Problem 7.23, an anemometer design was explored, and the assumption was made that the strip temperature was uniform. This is a good assumption when the heat transfer coefficient is low or the strip thermal conductivity high, because then conduction within the strip redistributes the generated heat and makes the strip temperature uniform. However, as the heat transfer coefficient increases or strip thermal conductivity decreases, heat generated at a point in the strip leaves the surface in the vicinity of that point, and the thermal condition is closer to one of uniform surface heat flux.
(a) Develop the calibration equations for both the constant surface temperature and constant heat flux conditions, that is, find the equations that predict the velocity as a function of the power per unit strip width, $P^{\prime}(\mathrm{mW} / \mathrm{mm})$, and the temperature measured at the trailing edge (as in Problem 7.23). Assume laminar flow conditions.
(b) If the true condition is uniform surface heat flux, but the uniform surface temperature calibration is used, what percentage error will be incurred in the velocity determination?
(c) Where could the thermocouple be placed so that the calibration is insensitive to whether the thermal condition is uniform surface temperature or uniform surface heat flux?

Nathan Prins
Nathan Prins
Numerade Educator
02:01

Problem 34

A flat plate of width $1 \mathrm{~m}$ and length $0.2 \mathrm{~m}$ is maintained at a temperature of $32^{\circ} \mathrm{C}$. Ambient fluid at $22^{\circ} \mathrm{C}$ flows across the top of the plate in parallel flow. Determine the average heat transfer coefficient, the convection heat transfer rate from the top of the plate, and the drag force on the plate for the following:
(a) The fluid is water flowing at a velocity of $0.5 \mathrm{~m} / \mathrm{s}$.
(b) The nanofluid of Example $2.2$ is flowing at a velocity of $0.5 \mathrm{~m} / \mathrm{s}$.
(c) Water is flowing at a velocity of $2.5 \mathrm{~m} / \mathrm{s}$.
(d) The nanofluid of Example $2.2$ is flowing at a velocity of $2.5 \mathrm{~m} / \mathrm{s}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:08

Problem 35

One hundred electrical components, each dissipating $25 \mathrm{~W}$, are attached to one surface of a square $(0.2 \mathrm{~m} \times 0.2 \mathrm{~m})$ copper plate, and all the dissipated energy is transferred to water in parallel flow over the opposite surface. A protuberance at the leading edge of the plate acts to trip the boundary layer, and the plate itself may be assumed to be isothermal. The water velocity and temperature are $u_{\infty}=2 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=17^{\circ} \mathrm{C}$, and the water's thermophysical properties may be approximated as $\nu=0.96 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, k=$ $0.620 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\operatorname{Pr}=5.2$.
(a) What is the temperature of the copper plate?
(b) If each component has a plate contact surface area of $1 \mathrm{~cm}^{2}$ and the corresponding contact resistance is $2 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$, what is the component temperature? Neglect the temperature variation across the thickness of the copper plate.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:33

Problem 36

Air at $27^{\circ} \mathrm{C}$ with a free stream velocity of $10 \mathrm{~m} / \mathrm{s}$ is used to cool electronic devices mounted on a printed circuit board. Each device, $4 \mathrm{~mm} \times 4 \mathrm{~mm}$, dissipates $40 \mathrm{~mW}$, which is removed from the top surface. A turbulator is located at the leading edge of the board, causing the boundary layer to be turbulent.
(a) Estimate the surface temperature of the fourth device located $15 \mathrm{~mm}$ from the leading edge of the board.
(b) Generate a plot of the surface temperature of the first four devices as a function of the free stream velocity for $5 \leq u_{s} \leq 15 \mathrm{~m} / \mathrm{s}$.
(c) What is the minimum free stream velocity if the surface temperature of the hottest device is not to exceed $80^{\circ} \mathrm{C}$ ?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
04:06

Problem 37

The boundary layer associated with parallel flow over an isothermal plate may be tripped at any $x$-location by using a fine wire that is stretched across the width of the plate. Determine the value of the critical Reynolds number $R e_{x, c, o p}$ that is associated with the optimal location of the trip wire from the leading edge that will result in maximum heat transfer from the warm plate to the cool fluid.

Joseph Liao
Joseph Liao
Numerade Educator
01:27

Problem 38

Forced air at $25^{\circ} \mathrm{C}$ and $10 \mathrm{~m} / \mathrm{s}$ is used to cool electronic elements mounted on a circuit board. Consider a chip of length $4 \mathrm{~mm}$ and width $4 \mathrm{~mm}$ located $120 \mathrm{~mm}$ from the leading edge. Because the board surface is irregular, the flow is disturbed and the appropriate convection correlation is of the form $N u_{x}=0.04 R e_{x}^{0.85} P r^{0.33}$.
Estimate the surface temperature of the chip, $T_{s}$, if its heat dissipation rate is $30 \mathrm{~mW}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
05:13

Problem 39

Air at atmospheric pressure and a temperature of $25^{\circ} \mathrm{C}$ is in parallel flow at a velocity of $5 \mathrm{~m} / \mathrm{s}$ over a 1 -m-long flat plate that is heated with a uniform heat flux of $1250 \mathrm{~W} / \mathrm{m}^{2}$. Assume the flow is fully turbulent over the length of the plate.
(a) Calculate the plate surface temperature, $T_{s}(L)$, and the local convection coefficient, $h_{x}(L)$, at the trailing edge, $x=L$.
(b) Calculate the average temperature of the plate surface, $\bar{T}_{s}$.
(c) Plot the variation of the surface temperature, $T_{s}(x)$, and the convection coefficient, $h_{x}(x)$, with distance on the same graph. Explain the key features of these distributions.

Vikash Ranjan
Vikash Ranjan
Numerade Educator
09:53

Problem 40

Air at atmospheric pressure and a temperature of $25^{\circ} \mathrm{C}$ is in parallel flow at a velocity of $5 \mathrm{~m} / \mathrm{s}$ over a 1 -m-long flat plate that is heated with a uniform heat flux of $1250 \mathrm{~W} / \mathrm{m}^{2}$. Assume the flow is fully turbulent over the length of the plate.
(a) Calculate the plate surface temperature, $T_{s}(L)$, and the local convection coefficient, $h_{x}(L)$, at the trailing edge, $x=L$.
(b) Calculate the average temperature of the plate surface, $\bar{T}_{s}$.
(c) Plot the variation of the surface temperature, $T_{s}(x)$, and the convection coefficient, $h_{x}(x)$, with distance on the same graph. Explain the key features of these distributions.
Working in groups of two, our students design and perform experiments on forced convection phenomena using the general arrangement shown schematically. The air box consists of two muffin fans, a plenum chamber, and flow straighteners discharging a nearly uniform airstream over the flat test-plate. The objectives of one experiment were to measure the heat transfer coefficient and to compare the results with standard convection correlations. The velocity of the airstream was measured using a thermistorbased anemometer, and thermocouples were used to determine the temperatures of the airstream and the test-plate.

With the airstream from the box fully stabilized at $T_{\infty}=20^{\circ} \mathrm{C}$, an aluminum plate was preheated in a convection oven and quickly mounted in the testplate holder. The subsequent temperature history of the plate was determined from thermocouple measurements, and histories obtained for airstream velocities of 3 and $9 \mathrm{~m} / \mathrm{s}$ were fitted by the following polynomial:
The temperature $T$ and time $t$ have units of ${ }^{\circ} \mathrm{C}$ and $\mathrm{s}$, respectively, and values of the coefficients appropriate for the time interval of the experiments are tabulated as follows:
\begin{tabular}{lcc}
\hline Velocity $(\mathrm{m} / \mathrm{s})$ & 3 & 9 \\
\hline Elapsed Time (s) & 300 & 160 \\
$a\left({ }^{\circ} \mathrm{C}\right)$ & $56.87$ & $57.00$ \\
$b\left({ }^{\circ} \mathrm{C} / \mathrm{s}\right)$ & $-0.1472$ & $-0.2641$ \\
$c\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{2}\right)$ & $3 \times 10^{-4}$ & $9 \times 10^{-4}$ \\
$d\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{3}\right)$ & $-4 \times 10^{-7}$ & $-2 \times 10^{-6}$ \\
$e\left({ }^{\circ} \mathrm{C} / \mathrm{s}^{4}\right)$ & $2 \times 10^{-10}$ & $1 \times 10^{-9}$ \\
\hline
\end{tabular}
The plate is square, $133 \mathrm{~mm}$ to a side, with a thickness of $3.2 \mathrm{~mm}$, and is made from a highly polished aluminum alloy $\left(\rho=2770 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=875 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right.$, $k=177 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$.
(a) Determine the heat transfer coefficients for the two cases, assuming the plate behaves as a spacewise isothermal object.
(b) Evaluate the coefficients $C$ and $m$ for a correlation of the form
$$
\overline{N u_{L}}=C \operatorname{Re}^{m} \operatorname{Pr}^{1 / 3}
$$
Compare this result with a standard flat-plate correlation. Comment on the goodness of the comparison and explain any differences.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:17

Problem 41

Consider atmospheric air at $u_{\infty}=2 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=300 \mathrm{~K}$ in parallel flow over an isothermal flat plate of length $L=1 \mathrm{~m}$ and temperature $T_{s}=350 \mathrm{~K}$.
(a) Compute the local convection coefficient at the leading and trailing edges of the heated plate with and without an unheated starting length of $\xi=1 \mathrm{~m}$.
(b) Compute the average convection coefficient for the plate for the same conditions as part (a).
(c) Plot the variation of the local convection coefficient over the plate with and without an unheated starting length.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
07:03

Problem 42

7.42 Consider a thin, $50 \mathrm{~mm} \times 50 \mathrm{~mm}$ fuel cell similar to that of Example 1.5, with air in parallel flow over its surfaces. Very small-diameter wires are stretched across both sides of the fuel cell at a distance $x=x_{c}$ from the leading edge in order to trip the flow into turbulent conditions. Using an appropriate correlation from Chapter 7 , determine the minimum velocity needed to sustain the fuel cell at $T_{c}=77^{\circ} \mathrm{C}$, and the associated location of the wire. The air and large surroundings are at $T_{s o}=T_{\text {sur }}=27^{\circ} \mathrm{C}$ and the fuel cell dissipates $\dot{E}_{g}=11 \mathrm{~W}$. The fuel cell emissivity is $\varepsilon=0.85$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:31

Problem 43

The cover plate of a flat-plate solar collector is at $15^{\circ} \mathrm{C}$, while ambient air at $10^{\circ} \mathrm{C}$ is in parallel flow over the plate, with $u_{\infty}=2 \mathrm{~m} / \mathrm{s}$.
(a)
(b)
(a) What is the rate of convective heat loss from the plate?
(b) If the plate is installed $2 \mathrm{~m}$ from the leading edge of a roof and flush with the roof surface, what is the rate of convective heat loss?

Mayukh Banik
Mayukh Banik
Numerade Educator
03:11

Problem 44

An array of 10 silicon chips, each of length $L=10 \mathrm{~mm}$ on a side, is insulated on one surface and cooled on the opposite surface by atmospheric air in parallel flow with $T_{\infty}=24^{\circ} \mathrm{C}$ and $u_{\infty}=40 \mathrm{~m} / \mathrm{s}$. When in use, the same electrical power is dissipated in each chip, maintaining a uniform heat flux over the entire cooled surface.
If the temperature of each chip may not exceed $80^{\circ} \mathrm{C}$, what is the maximum allowable power per chip? What is the maximum allowable power if a turbulence
promoter is used to trip the boundary layer at the leading edge? Would it be preferable to orient the array normal, instead of parallel, to the airflow?

Chai Santi
Chai Santi
Numerade Educator
01:27

Problem 45

A square ( $10 \mathrm{~mm} \times 10 \mathrm{~mm}$ ) silicon chip is insulated on one side and cooled on the opposite side by atmospheric air in parallel flow at $u_{\infty}=20 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=$ $24^{\circ} \mathrm{C}$. When in use, electrical power dissipation within the chip maintains a uniform heat flux at the cooled surface. If the chip temperature may not exceed $80^{\circ} \mathrm{C}$ at any point on its surface, what is the maximum allowable power? What is the maximum allowable power if the chip is flush mounted in a substrate that provides for an unheated starting length of $20 \mathrm{~mm}$ ?

Mayukh Banik
Mayukh Banik
Numerade Educator
02:58

Problem 46

Consider the following fluids, each with a velocity of $V=5 \mathrm{~m} / \mathrm{s}$ and a temperature of $T_{\infty}=20^{\circ} \mathrm{C}$, in cross flow over a 10-mm-diameter cylinder maintained at $50^{\circ} \mathrm{C}$ : atmospheric air, saturated water, and engine oil.
(a) Calculate the rate of heat transfer per unit length, $q^{\prime}$, using the Churchill-Bernstein correlation.
(b) Generate a plot of $q^{\prime}$ as a function of fluid velocity for $0.5 \leq V \leq 10 \mathrm{~m} / \mathrm{s}$.

Chai Santi
Chai Santi
Numerade Educator
02:02

Problem 47

A circular pipe of 25 -mm outside diameter is placed in an airstream at $25^{\circ} \mathrm{C}$ and 1 -atm pressure. The air moves in cross flow over the pipe at $15 \mathrm{~m} / \mathrm{s}$, while the outer surface of the pipe is maintained at $100^{\circ} \mathrm{C}$. What is the drag force exerted on the pipe per unit length? What is the rate of heat transfer from the pipe per unit length?

Narayan Hari
Narayan Hari
Numerade Educator
02:50

Problem 48

An $L=1$-m-long vertical copper tube of inner diameter $D_{i}=20 \mathrm{~mm}$ and wall thickness $t=2 \mathrm{~mm}$ contains liquid water at $T_{w}=0^{\circ} \mathrm{C}$. On a winter day, air at $V=3 \mathrm{~m} / \mathrm{s}, T_{\infty}=-20^{\circ} \mathrm{C}$ is in cross flow over the tube.
(a) Determine the heat loss per unit mass from the water (W/kg) when the tube is full of water.
(b) Determine the heat loss from the water (W/kg) when the tube is half full.

Narayan Hari
Narayan Hari
Numerade Educator
02:18

Problem 49

A long, cylindrical, electrical heating element of diameter $D=10 \mathrm{~mm}$, thermal conductivity $k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, density $\rho=2700 \mathrm{~kg} / \mathrm{m}^{3}$, and specific heat $c_{p}=900 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ is installed in a duct for which air moves in cross flow over the heater at a temperature and velocity of $27^{\circ} \mathrm{C}$ and $10 \mathrm{~m} / \mathrm{s}$, respectively.
(a) Neglecting radiation, estimate the steady-state surface temperature when, per unit length of the heater, electrical energy is being dissipated at a rate of $1000 \mathrm{~W} / \mathrm{m}$.
(b) If the heater is activated from an initial temperature of $27^{\circ} \mathrm{C}$, estimate the time required for the surface temperature to come within $10^{\circ} \mathrm{C}$ of its steady-state value.

Keshav Singh
Keshav Singh
Numerade Educator
04:40

Problem 50

Consider the conditions of Problem 7.49, but now allow for radiation exchange between the surface of the heating element $(\varepsilon=0.8)$ and the walls of the duct, which form a large enclosure at $27^{\circ} \mathrm{C}$.
(a) Evaluate the steady-state surface temperature.
(b) If the heater is activated from an initial temperature of $27^{\circ} \mathrm{C}$, estimate the time required for the surface temperature to come within $10^{\circ} \mathrm{C}$ of the steadystate value.
(c) To guard against overheating due to unanticipated excursions in the blower output, the heater controller is designed to maintain a fixed surface temperature of $275^{\circ} \mathrm{C}$. Determine the power dissipation required to maintain this temperature for air velocities in the range $5 \leq V \leq 10 \mathrm{~m} / \mathrm{s}$.

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
01:01

Problem 51

Pin fins are to be specified for use in an industrial cooling application. The fins will be subjected to a gas in cross flow at $V=10 \mathrm{~m} / \mathrm{s}$. The cylindrical fin has a diameter of $D=15 \mathrm{~mm}$, and the cross-sectional area is the same for each configuration shown in the sketch.

Dominador Tan
Dominador Tan
Numerade Educator
01:01

Problem 52

A pin fin of $10-\mathrm{mm}$ diameter dissipates $30 \mathrm{~W}$ by forced convection to air in cross flow with a Reynolds number of 4000 . If the diameter of the fin is doubled and all other conditions remain the same, estimate the fin heat rate. Assume the pin to be infinitely long.

Dominador Tan
Dominador Tan
Numerade Educator
05:07

Problem 53

Air at $27^{\circ} \mathrm{C}$ and a velocity of $5 \mathrm{~m} / \mathrm{s}$ passes over the small region $A_{s}(20 \mathrm{~mm} \times 20 \mathrm{~mm})$ on a large surface, which is maintained at $T_{s}=127^{\circ} \mathrm{C}$. For these conditions, $0.5 \mathrm{~W}$ is removed from the surface $A_{s}$. To increase the heat removal rate, a stainless steel (AISI 304) pin fin of diameter $5 \mathrm{~mm}$ is affixed to $A_{s}$, which is assumed to remain at $T_{s}=127^{\circ} \mathrm{C}$.
(a) Determine the maximum possible heat removal rate through the fin.
(b) What fin length would provide a close approximation to the heat rate found in part (a)? Hint: Refer to Example 3.9.
(c) Determine the fin effectiveness, $\varepsilon_{f}$
(d) What is the percentage increase in the heat rate from $A_{s}$ due to installation of the fin?

Satpal Satpal
Satpal Satpal
Numerade Educator
01:45

Problem 54

To enhance heat transfer from a silicon chip of width $W=4 \mathrm{~mm}$ on a side, a copper pin fin is brazed to the surface of the chip. The pin length and diameter are $L=12 \mathrm{~mm}$ and $D=2 \mathrm{~mm}$, respectively, and atmospheric air at $V=10 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=300 \mathrm{~K}$ is in cross flow over the pin. The surface of the chip, and hence the base of the pin, are maintained at a temperature of $T_{b}=350 \mathrm{~K}$.
(a) Assuming the chip to have a negligible effect on flow over the pin, what is the average convection coefficient for the surface of the pin?
(b) Neglecting radiation and assuming the convection coefficient at the pin tip to equal that calculated in part (a), determine the pin heat transfer rate.
(c) Neglecting radiation and assuming the convection coefficient at the exposed chip surface to equal that calculated in part (a), determine the total rate of heat transfer from the chip.
(d) Independently determine and plot the effect of increasing velocity $(10 \leq V \leq 40 \mathrm{~m} / \mathrm{s})$ and pin diameter $(2 \leq D \leq 4 \mathrm{~mm})$ on the total rate of heat transfer from the chip. What is the heat rate for $V=40 \mathrm{~m} / \mathrm{s}$ and $D=4 \mathrm{~mm} ?$

Anand Jangid
Anand Jangid
Numerade Educator
07:35

Problem 55

Consider the Nichrome wire $\left(D=1 \mathrm{~mm}, \rho_{e}=10^{-6}\right.$ $\Omega \cdot \mathrm{m}, k=25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \varepsilon=0.20$ ) used to fabricate the air heater of Problem $3.86$, but now under conditions for which the convection heat transfer coefficient must be determined.
(a) For atmospheric air at $50^{\circ} \mathrm{C}$ and a cross-flow velocity of $5 \mathrm{~m} / \mathrm{s}$, what are the surface and centerline temperatures of the wire when it carries a current of $25 \mathrm{~A}$ and the housing of the heater is also at $50^{\circ} \mathrm{C}$ ?
(b) Explore the effect of variations in the flow velocity and electrical current on the surface and centerline temperatures of the wire.

Vishal Gupta
Vishal Gupta
Numerade Educator
02:27

Problem 56

Hot water at $50^{\circ} \mathrm{C}$ is routed from one building in which it is generated to an adjoining building in which it is used for space heating. Transfer between the buildings occurs in a steel pipe $(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ of $100-\mathrm{mm}$ outside diameter and 8-mm wall thickness. During the winter, representative environmental conditions involve air at $T_{\infty}=-5^{\circ} \mathrm{C}$ and $V=3 \mathrm{~m} / \mathrm{s}$ in cross flow over the pipe.
(a) If the cost of producing the hot water is $\$ 0.10$ per $\mathrm{kW} \cdot \mathrm{h}$, what is the representative daily cost of heat loss from an uninsulated pipe to the air per meter of pipe length? The convection resistance associated with water flow in the pipe may be neglected.
(b) Determine the savings associated with application of a 10-mm-thick coating of urethane insulation $(k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ to the outer surface of the pipe.

Anand Jangid
Anand Jangid
Numerade Educator
01:20

Problem 57

In a manufacturing process, long aluminum rods of square cross section with $d=25 \mathrm{~mm}$ are cooled from an initial temperature of $T_{i}=400^{\circ} \mathrm{C}$. Which configuration in the sketch should be used to minimize the time needed for the rods to reach a safe-to-handle temperature of $60^{\circ} \mathrm{C}$ when exposed to air in cross flow at $V=8 \mathrm{~m} / \mathrm{s}, T_{\infty}=30^{\circ} \mathrm{C}$ ? What is the required cooling time for the preferred configuration? The emissivity of the rods is $\varepsilon=0.10$ and the surroundings temperature is $T_{\text {sar }}=20^{\circ} \mathrm{C}$.

Dominador Tan
Dominador Tan
Numerade Educator
01:33

Problem 58

A fine wire of diameter $D$ is positioned across a passage to determine flow velocity from heat transfer characteristics. Current is passed through the wire to heat it, and the heat is dissipated to the flowing fluid by convection. The resistance of the wire is determined from electrical measurements, and the temperature is known from the resistance.
(a) For a fluid of arbitrary Prandtl number, develop an expression for its velocity in terms of the difference between the temperature of the wire and the free stream temperature of the fluid.
(b) What is the velocity of an airstream at $1 \mathrm{~atm}$ and $25^{\circ} \mathrm{C}$, if a wire of $0.5-\mathrm{mm}$ diameter achieves a temperature of $40^{\circ} \mathrm{C}$ while dissipating $35 \mathrm{~W} / \mathrm{m}$ ?

Mayukh Banik
Mayukh Banik
Numerade Educator
05:08

Problem 59

To determine air velocity changes, it is proposed to measure the electric current required to maintain a platinum wire of $0.5-\mathrm{mm}$ diameter at a constant temperature of $77^{\circ} \mathrm{C}$ in a stream of air at $27^{\circ} \mathrm{C}$.
(a) Assuming Reynolds numbers in the range $40<R e_{D}<1000$, develop a relationship between the wire current and the velocity of the air that is in cross flow over the wire. Use this result to establish a relation between fractional changes in the current, $\Delta I / I$, and the air velocity, $\Delta V / V$.
(b) Calculate the current required when the air velocity is $10 \mathrm{~m} / \mathrm{s}$ and the electrical resistivity of the platinum wire is $17.1 \times 10^{-5} \Omega \cdot \mathrm{m}$.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
02:55

Problem 60

Fluid velocities can be measured using hot-film sensors, and a common design is one for which the sensing element forms a thin film about the circumference of a quartz rod. The film is typically comprised of a thin $(\sim 100 \mathrm{~nm})$ layer of platinum, whose electrical resistance is proportional to its temperature. Hence, when submerged in a fluid stream, an electric current may be passed through the film to maintain its temperature above that of the fluid. The temperature of the film is controlled by monitoring its electric resistance, and with concurrent measurement of the electric current, the power dissipated in the film may be determined.
Proper operation is assured only if the heat generated in the film is transferred to the fluid, rather than conducted from the film into the quartz rod. Thermally, the film should therefore be strongly coupled to the fluid and weakly coupled to the quartz rod. This condition is satisfied if the Biot number is very large, $B i=\bar{h} D / 2 k \geqslant 1$, where $\bar{h}$ is the convection coefficient between the fluid and the film and $k$ is the thermal conductivity of the rod.
(a) For the following fluids and velocities, calculate and plot the convection coefficient as a function of velocity: (i) water, $0.5 \leq V \leq 5 \mathrm{~m} / \mathrm{s}$; (ii) air, $1 \leq V \leq 20 \mathrm{~m} / \mathrm{s}$.
(b) Comment on the suitability of using this hot-film sensor for the foregoing conditions.

Prabhu Ramji
Prabhu Ramji
Numerade Educator
04:34

Problem 61

Consider use of the hot-film sensor described in Problem $7.60$ to determine the velocity of water entering the cooling system of an electric power plant from an adjoining lake. The sensor is mounted within an intake pipe, and its controls are set to maintain an average hotfilm temperature that is $5^{\circ} \mathrm{C}$ larger than the fluid temperature $\left(T_{s, h f}-T_{\infty}=5^{\circ} \mathrm{C}\right)$.
(a) If an independent measurement of the water temperature yields a value of $T_{\infty}=17^{\circ} \mathrm{C}$, use the Churchill-Bernstein correlation to estimate the velocity of the water under conditions for which the power input to the sensor maintains a heat flux of $q_{\mathrm{hf}}^{\prime \prime}=4 \times 10^{4} \mathrm{~W} / \mathrm{m}^{2}$ from the film to the water.
(b) If the sensor is exposed to the water for an extended period, its surface will be fouled by an accumulation of deposits from the water. Consider conditions for which the deposits form a 0.l-mmthick shell around the sensor and have a thermal conductivity of $k_{d}=2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. For $T_{\infty}=17^{\circ} \mathrm{C}$ and the flow velocity determined in part (a), what heat flux must be supplied to the sensor to maintain its temperature at $T_{s, h f}=22^{\circ} \mathrm{C}$ ? What is the corresponding error in the velocity measurement? Note: Conduction across the deposit may be approximated as that across a plane wall.

Manne Andergronde
Manne Andergronde
Numerade Educator
02:17

Problem 62

Determine the convection heat loss from both the top and the bottom of a flat plate at $T_{s}=80^{\circ} \mathrm{C}$ with air in parallel flow at $T_{\infty}=25^{\circ} \mathrm{C}, u_{\infty}=3 \mathrm{~m} / \mathrm{s}$. The plate is $t=1 \mathrm{~mm}$ thick, $L=25 \mathrm{~mm}$ long, and of depth $w=50 \mathrm{~mm}$. Neglect the heat loss from the edges of the plate. Compare the convection heat loss from the plate to the convection heat loss from an $L_{c}=50$-mm-long cylinder of the same volume as that of the plate. The convective conditions associated with the cylinder are the same as those associated with the plate.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
03:21

Problem 63

Consider two very long, straight fins of uniform cross section, as shown in Figure 3.17. The rectangular fin has dimensions $t=1 \mathrm{~mm}$ and $w=20 \mathrm{~mm}$. The circular pin fin has the same cross-sectional area as the rectangular fin. Both fins are constructed of aluminum with $k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. In both cases, the base temperature is $T_{b}=85^{\circ} \mathrm{C}$. Airflow is directed as shown in the figure, with $T_{\infty}=20^{\circ} \mathrm{C}$ and $u_{\infty}=5 \mathrm{~m} / \mathrm{s}$.
(a) Calculate the heat loss from each fin. Assume that the heat transfer coefficient on the edges of the rectangular fin is equal to the average value on the upper and lower surfaces.
(b) What diameter cylindrical fin would be needed to provide the same fin heat transfer rate as the rectangular cross-section fin?

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
04:36

Problem 64

A computer code is being developed to analyze a temperature sensor of $12.5-\mathrm{mm}$ diameter experiencing cross flow of water with a free stream temperature of $80^{\circ} \mathrm{C}$ and variable velocity. Derive an expression for the convection heat transfer coefficient as a function of the sensor surface temperature $T_{s}$ for the range $20<T_{s}<80^{\circ} \mathrm{C}$ and for velocities $V$ in the range $0.005<V<0.20 \mathrm{~m} / \mathrm{s}$. Use the Zukauskas correlation for the range $40<R e_{D}<1000$ and assume that the Prandtl number of water has a linear temperature dependence.

Dading Chen
Dading Chen
Numerade Educator
03:49

Problem 65

A 25 -mm-diameter, high-tension line has an electrical resistance of $10^{-4} \mathrm{\Omega} / \mathrm{m}$ and is transmitting a current of $1000 \mathrm{~A}$.
(a) If ambient air at $10^{\circ} \mathrm{C}$ and $5 \mathrm{~m} / \mathrm{s}$ is in cross flow over the line, what is its surface temperature?
(b) If the line may be approximated as a solid copper rod, what is its centerline temperature?
(c) Generate a plot that depicts the variation of the surface temperature with air velocity for $1 \leq V \leq$ $10 \mathrm{~m} / \mathrm{s}$.

Shoukat Ali
Shoukat Ali
Other Schools
02:43

Problem 66

An aluminum transmission line with a diameter of $20 \mathrm{~mm}$ has an electrical resistance of $R_{\text {cloc }}^{\prime}=2.636 \times$ $10^{-4} \Omega / \mathrm{m}$ and carries a current of $700 \mathrm{~A}$. The line is subjected to frequent and severe cross winds, increasing the probability of contact between adjacent lines, thereby causing sparks and creating a potential fire hazard for nearby vegetation. The remedy is to insulate the line, but with the adverse effect of increasing the conductor operating temperature.
(a) Calculate the conductor temperature when the air temperature is $20^{\circ} \mathrm{C}$ and the line is subjected to cross flow with a velocity of $10 \mathrm{~m} / \mathrm{s}$.
(b) Calculate the conductor temperature for the same conditions, but with a 2 -mm-thick insulation having a thermal conductivity of $0.15 \mathrm{~W} / \mathrm{m}+\mathrm{K}$.
(c) Calculate and plot the temperatures of the bare and insulated conductors for wind velocities in the range from 2 to $20 \mathrm{~m} / \mathrm{s}$. Comment on features of the curves and the effect of the wind velocity on the conductor temperatures.

Dominador Tan
Dominador Tan
Numerade Educator
06:48

Problem 67

To augment heat transfer between two flowing fluids, it is proposed to insert a 100 -mm-long, 5 -mm-diameter 2024 aluminum pin fin through the wall separating the two fluids. The pin is inserted to a depth of $d$ into fluid 1 . Fluid 1 is air with a mean temperature of $10^{\circ} \mathrm{C}$ and velocity of $10 \mathrm{~m} / \mathrm{s}$. Fluid 2 is air with a mean temperature of $40^{\circ} \mathrm{C}$ and velocity of $3 \mathrm{~m} / \mathrm{s}$.
(a) Determine the rate of heat transfer from the warm air to the cool air through the pin fin for $d=50 \mathrm{~mm}$.
(b) Plot the variation of the heat transfer rate with the insertion distance, $d$. Does an optimal insertion distance exist?

Mohammad Mehran
Mohammad Mehran
Numerade Educator
05:56

Problem 68

An uninsulated steam pipe is used to transport hightemperature steam from one building to another. The pipe is of $0.5-\mathrm{m}$ diameter, has a surface temperature of $150^{\circ} \mathrm{C}$, and is exposed to ambient air at $-10^{\circ} \mathrm{C}$. The air moves in cross flow over the pipe with a velocity of $5 \mathrm{~m} / \mathrm{s}$.
(a) What is the heat loss per unit length of pipe?
(b) Consider the effect of insulating the pipe with a rigid urethane foam $(k=0.026 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$. Evaluate and plot the heat loss as a function of the thickness $\delta$ of the insulation layer for $0 \leq \delta \leq 50 \mathrm{~mm}$.

Dading Chen
Dading Chen
Numerade Educator
01:34

Problem 69

A thermocouple is inserted into a hot air duct to measure the air temperature. The thermocouple $\left(T_{1}\right)$ is soldered to the tip of a steel thermocouple well of length $L=0.15 \mathrm{~m}$ and inner and outer diameters of $D_{i}=5 \mathrm{~mm}$
and $D_{o}=10 \mathrm{~mm}$. A second thermocouple $\left(T_{2}\right)$ is used to measure the duct wall temperature.
Consider conditions for which the air velocity in the duct is $V=3 \mathrm{~m} / \mathrm{s}$ and the two thermocouples register temperatures of $T_{1}=450 \mathrm{~K}$ and $T_{2}=375 \mathrm{~K}$. Neglecting radiation, determine the air temperature $T_{\infty}$. Assume that, for steel, $k=35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and, for air, $\rho=0.774 \mathrm{~kg} / \mathrm{m}^{3}$, $\mu=251 \times 10^{-7} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, \quad k=0.0373 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \quad$ and $P r=0.686 .$

Aadit Sharma
Aadit Sharma
Numerade Educator
03:55

Problem 70

Consider conditions for which a mercury-in-glass thermometer of 4 - $\mathrm{mm}$ diameter is inserted to a length $L$ through the wall of a duct in which air at $77^{\circ} \mathrm{C}$ is flowing. If the stem of the thermometer at the duct wall is at the wall temperature $T_{w}=15^{\circ} \mathrm{C}$, conduction heat transfer through the glass causes the bulb temperature to be lower than that of the airstream.
(a) Develop a relationship for the immersion error, $\Delta T_{i}=T(L)-T_{\infty}$, as a function of air velocity, thermometer diameter, and insertion length $L$.
(b) To what length $L$ must the thermometer be inserted if the immersion error is not to exceed $0.25^{\circ} \mathrm{C}$ when the air velocity is $10 \mathrm{~m} / \mathrm{s}$ ?
(c) Using the insertion length determined in part (b), calculate and plot the immersion error as a function of air velocity for the range 2 to $20 \mathrm{~m} / \mathrm{s}$.
(d) For a given insertion length, will the immersion error increase or decrease if the diameter of the thermometer is increased? Is the immersion error more sensitive to the diameter or air velocity?

Averell Hause
Averell Hause
Carnegie Mellon University
01:20

Problem 71

In a manufacturing process, a long, coated plastic rod $\left(\rho=2200 \mathrm{~kg} / \mathrm{m}^{3}, c=800 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=1 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$ of diameter $D=20 \mathrm{~mm}$ is initially at a uniform temperature of $25^{\circ} \mathrm{C}$ and is suddenly exposed to a cross flow of air at $T_{\infty}=350^{\circ} \mathrm{C}$ and $V=50 \mathrm{~m} / \mathrm{s}$.
(a) How long will it take for the surface of the rod to reach $175^{\circ} \mathrm{C}$, the temperature above which the special coating will cure?
(b) Generate a plot of the time to reach $175^{\circ} \mathrm{C}$ as a function of air velocity for $5 \leq V \leq 50 \mathrm{~m} / \mathrm{s}$.

Dominador Tan
Dominador Tan
Numerade Educator
03:43

Problem 72

In an extrusion process, copper wire emerges from the extruder at a velocity $V_{e}$ and is cooled by convection heat transfer to air in cross flow over the wire, as well as by radiation to the surroundings.
(a) By applying conservation of energy to a differential control surface of length $d x$, which either moves with the wire or is stationary and through which the wire passes, derive a differential equation that governs the temperature distribution, $T(x)$, along the wire. In your derivation, the effect of axial conduction along the wire may be neglected. Express your result in terms of the velocity, diameter, and properties of the wire $\left(V_{e}, D, \rho, c_{p}, \varepsilon\right)$, the convection coefficient associated with the cross flow $(\bar{h})$, and the environmental temperatures $\left(T_{w}, T_{\text {sur }}\right)$.
(b) Neglecting radiation, obtain a closed form solution to the foregoing equation. For $V_{c}=0.2 \mathrm{~m} / \mathrm{s}$, $D=5 \mathrm{~mm}, V=5 \mathrm{~m} / \mathrm{s}, T_{\infty}=25^{\circ} \mathrm{C}$, and an initial wire temperature of $T_{i}=600^{\circ} \mathrm{C}$, compute the temperature $T_{o}$ of the wire at $x=L=5 \mathrm{~m}$. The density and specific heat of the copper are $\rho=8900 \mathrm{~kg} / \mathrm{m}^{3}$ and $c_{p}=400 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, while properties of the air may be taken to be $k=0.037 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \nu=3 \times$ $10^{-5} \mathrm{~m}^{2} / \mathrm{s}$, and $P r=0.69$.
(c) Accounting for the effects of radiation, with $\varepsilon=0.55$ and $T_{\text {sur }}=25^{\circ} \mathrm{C}$, numerically integrate the differential equation derived in part (a) to determine the temperature of the wire at $L=5 \mathrm{~m}$. Explore the effects of $V_{e}$ and $\varepsilon$ on the temperature distribution along the wire.

Kajal Gautam
Kajal Gautam
Numerade Educator
09:45

Problem 73

The objective of an experiment performed by our students is to determine the effect of pin fins on the thermal resistance between a flat plate and an airstream. A $25.9-\mathrm{mm}$-square polished aluminum plate is subjected to an airstream in parallel flow at $T_{\infty}=20^{\circ} \mathrm{C}$ and $u_{\infty}=6 \mathrm{~m} / \mathrm{s}$. An electrical heating patch is attached to the backside of the plate and dissipates $15.5 \mathrm{~W}$ under all conditions. Pin fins of diameter $D=4.8 \mathrm{~mm}$ and length $L=25.4 \mathrm{~mm}$ are fabricated from brass and can be firmly attached to the plate at various locations over its surface. Thermocouples are attached to the plate surface and the tip of one of the fins.
Measured temperatures for five pin-fin configurations are tabulated.
\begin{tabular}{lcc}
\hline \multirow{2}{*}{ Number of Pin Fins } & \multicolumn{2}{c}{ Temperature $\left({ }^{\circ} \mathbf{C}\right)$} \\
\cline { 2 - 3 } & Fin Tip & Plate Base \\
\hline 0 & $-$ & $70.2$ \\
1 & $40.6$ & $67.4$ \\
2 & $39.5$ & $64.7$ \\
5 & $36.4$ & $57.4$ \\
8 & $34.2$ & $52.1$ \\
\hline
\end{tabular}
(a) Using the experimental observations and neglecting the effect of flow interactions between pins, determine the thermal resistance between the plate and the airstream for the five configurations.
(b) Develop a model of the plate-pin fin system and using appropriate convection correlations, predict the thermal resistances for the five configurations. Compare your predictions with the observations and explain any differences.
(c) Use your model to predict the thermal resistances when the airstream velocity is doubled.

Dading Chen
Dading Chen
Numerade Educator
03:11

Problem 74

Air at $25^{\circ} \mathrm{C}$ flows over a $10-\mathrm{mm}$-diameter sphere with a velocity of $25 \mathrm{~m} / \mathrm{s}$, while the surface of the sphere is maintained at $75^{\circ} \mathrm{C}$.
(a) What is the drag force on the sphere?
(b) What is the rate of heat transfer from the sphere?
(c) Generate a plot of the heat transfer from the sphere as a function of the air velocity for the range 1 to $25 \mathrm{~m} / \mathrm{s}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
View

Problem 75

Consider a sphere with a diameter of $20 \mathrm{~mm}$ and a surface temperature of $60^{\circ} \mathrm{C}$ that is immersed in a fluid at a temperature of $30^{\circ} \mathrm{C}$ and a velocity of $2.5 \mathrm{~m} / \mathrm{s}$. Calculate the drag force and the heat rate when the fluid is (a) water and (b) air at atmospheric pressure. Explain why the results for the two fluids are so different.

Victor Salazar
Victor Salazar
Numerade Educator
02:34

Problem 76

Consider the material processing experiment of Problem $5.24$, with atmospheric nitrogen used to implement cooling by convection. However, instead of using a prescribed value of the convection coefficient, compute the coefficient from an appropriate correlation.
(a) Neglecting radiation, determine the time required to cool the sphere from $900^{\circ} \mathrm{C}$ to $300^{\circ} \mathrm{C}$ if the velocity and temperature of the nitrogen are $V=5 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=25^{\circ} \mathrm{C}$.
(b) Accounting for the effects of both convection and radiation, with $\varepsilon=0.6$ and $T_{\text {sur }}=25^{\circ} \mathrm{C}$, determine the time required to cool the sphere. Explore the effects of the flow velocity on your result.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:50

Problem 77

A spherical, underwater instrument pod used to make soundings and to measure conditions in the water has a diameter of $85 \mathrm{~mm}$ and dissipates $300 \mathrm{~W}$.
(a) Estimate the surface temperature of the pod when suspended in a bay where the current is $1 \mathrm{~m} / \mathrm{s}$ and the water temperature is $15^{\circ} \mathrm{C}$.
(b) Inadvertently, the pod is hauled out of the water and suspended in ambient air without deactivating the power. Estimate the surface temperature of the pod if the air temperature is $15^{\circ} \mathrm{C}$ and the wind speed is $3 \mathrm{~m} / \mathrm{s}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:29

Problem 78

Worldwide, over a billion solder balls must be manufactured daily for assembling electronics packages. The uniform droplet spray method uses a piezoelectric device to vibrate a shaft in a pot of molten solder that, in turn, ejects small droplets of solder through a precision-machined nozzle. As they traverse a collection chamber, the droplets cool and solidify. The collection chamber is flooded with an inert gas such as nitrogen to prevent oxidation of the solder ball surfaces.
(a) Molten solder droplets of diameter $130 \mu \mathrm{m}$ are ejected at a velocity of $2 \mathrm{~m} / \mathrm{s}$ at an initial temperature of $225^{\circ} \mathrm{C}$ into gaseous nitrogen that is at $30^{\circ} \mathrm{C}$ and slightly above atmospheric pressure. Determine the terminal velocity of the particles and the distance the particles have traveled when they become completely solidified. The solder properties are $\rho=8230 \mathrm{~kg} / \mathrm{m}^{3}$, $c=240 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=38 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, h_{s f}=42 \mathrm{~kJ} / \mathrm{kg}$. The solder's melting temperature is $183^{\circ} \mathrm{C}$.
(b) The piezoelectric device oscillates at $1.8 \mathrm{kHz}$, producing 1800 particles per second. Determine the separation distance between the particles as they traverse the nitrogen gas and the pot volume needed in order to produce the solder balls continuously for one week.

Manik Pulyani
Manik Pulyani
Numerade Educator
08:02

Problem 79

A spherical workpiece of pure copper with a diameter of $15 \mathrm{~mm}$ and an emissivity of $0.5$ is suspended in a large furnace with walls at a uniform temperature of $600^{\circ} \mathrm{C}$. Air flows over the workpiece at a temperature of $900^{\circ} \mathrm{C}$ and a velocity of $7.5 \mathrm{~m} / \mathrm{s}$.
(a) Determine the steady-state temperature of the workpiece.
(b) Estimate the time required for the workpiece to come within $5^{\circ} \mathrm{C}$ of the steady-state temperature if it is at an initial, uniform temperature of $25^{\circ} \mathrm{C}$.
(c) To decrease the time to heat the workpiece, the air velocity is doubled, with all other conditions remaining the same. Determine the steady-state temperature of the workpiece and the time required for it to come within $5^{\circ} \mathrm{C}$ of this value. Plot on the same graph the workpiece temperature histories for the two velocities.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
06:06

Problem 80

Copper spheres of $20-\mathrm{mm}$ diameter are quenched by being dropped into a tank of water that is maintained at $280 \mathrm{~K}$. The spheres may be assumed to reach the terminal velocity on impact and to drop freely through the water. Estimate the terminal velocity by equating the drag and gravitational forces acting on the sphere. What is the approximate height of the water tank needed to cool the spheres from an initial temperature of $360 \mathrm{~K}$ to a center temperature of $320 \mathrm{~K}$ ?

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
01:43

Problem 81

For the conditions of Problem $7.80$, what are the terminal velocity and the tank height if engine oil at $300 \mathrm{~K}$, rather than water, is used as the coolant?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
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Problem 82

Consider the plasma spray coating process of Problem $5.25$. In addition to the prescribed conditions, the argon plasma jet is known to have a mean velocity of $V=400 \mathrm{~m} / \mathrm{s}$, while the initial velocity of the injected alumina particles may be approximated as zero. The nozzle exit and the substrate are separated by a distance of $L=100 \mathrm{~mm}$, and pertinent properties of the argon plasma may be approximated as $k=0.671 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $c_{p}=1480 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=2.70 \times 10^{-4} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}$, and $\nu=$ $5.6 \times 10^{-3} \mathrm{~m}^{2} / \mathrm{s}$.
(a) Assuming the motion of particles entrained by the plasma jet to be governed by Stokes' law, derive expressions for the particle velocity, $V_{p}(t)$, and its distance of travel from the nozzle exit, $x_{p}(t)$, as a function of time, $t$, where $t=0$ corresponds to particle injection. Evaluate the time-in-flight required for a particle to traverse the separation distance, $x_{p}=L$, and the velocity $V_{p}$ at this time.
(b) Assuming an average relative velocity of $\left(\overline{V-V_{p}}\right)=$ $315 \mathrm{~m} / \mathrm{s}$ during the time-of-flight, estimate the convection coefficient associated with heat transfer from the plasma to the particle. Using this coefficient and assuming an initial particle temperature of $T_{i}=300 \mathrm{~K}$, estimate the time-in-flight required to heat a particle to its melting point, $T_{\text {map }}$, and, once at $T_{\text {mp }}$, for the particle to experience complete melting. Is the prescribed value of $L$ sufficient to ensure complete particle melting before surface impact?

Lainey Roebuck
Lainey Roebuck
Numerade Educator
03:26

Problem 83

Highly reflective aluminum coatings may be formed on the surface of a substrate by impacting the surface with molten drops of aluminum. The droplets are discharged from an injector, proceed through an inert gas (helium), and must still be in a molten state at the time of impact.
$V=3 \mathrm{~m} / \mathrm{s}$, and $T_{i}=1100 \mathrm{~K}$, respectively, traverse a stagnant layer of atmospheric helium that is at a temperature of $T_{\infty}=300 \mathrm{~K}$. What is the maximum allowable thickness of the helium layer needed to ensure that the temperature of droplets impacting the substrate is greater than or equal to the melting point of aluminum $\left(T_{f} \geq T_{\text {mp }}=933 \mathrm{~K}\right)$ ? Properties of the molten aluminum may be approximated as $\rho=2500 \mathrm{~kg} / \mathrm{m}^{3}, c=$ $1200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

Grey Garrett
Grey Garrett
Numerade Educator
19:08

Problem 84

Tissue engineering involves the development of biological substitutes that restore or improve tissue function. Once manufactured, engineered organs can be implanted and grow within the patient, obviating chronic shortages of natural organs that arise when traditional organ transplant procedures are used. Artificial organ manufacture involves two major steps. First, a porous scaffold is fabricated with a specific pore size and pore distribution, as well as overall shape and size. Second, the top surface of the scaffold is seeded with human cells that grow into the pores of the scaffold. The scaffold material is biodegradable and is eventually replaced with healthy tissue. The artificial organ is then ready to be implanted in the patient.

The complex pore shapes, small pore sizes, and unusual organ shapes preclude use of traditional manufacturing methods to fabricate the scaffolds. A method that has been used with success is a solid freeform fabrication technique whereby small spherical drops are directed to a substrate. The drops are initially molten and solidify when they impact the room-temperature substrate. By controlling the location of the droplet deposition, complex scaffolds can be built up, one drop at a time. A device similar to that of Problem $7.78$ is used to generate uniform, 75- $\mu \mathrm{m}$-diameter drops at an initial temperature of $T_{i}=150^{\circ} \mathrm{C}$. The particles are sent through quiescent air at $T_{\infty}=25^{\circ} \mathrm{C}$. The droplet properties are $\rho=2200 \mathrm{~kg} / \mathrm{m}^{3}, c=700 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.
(a) It is desirable for the droplets to exit the nozzle at their terminal velocity. Determine the terminal velocity of the drops.
(b) It is desirable for the droplets to impact the structure at a temperature of $T_{2}=120^{\circ} \mathrm{C}$. What is the required distance between the exit nozzle and the structure, $L$ ?

Eric Goldman
Eric Goldman
Numerade Educator
01:34

Problem 85

A spherical thermocouple junction $1.0 \mathrm{~mm}$ in diameter is inserted in a combustion chamber to measure the temperature $T_{\infty}$ of the products of combustion. The hot gases have a velocity of $V=5 \mathrm{~m} / \mathrm{s}$.
(a) If the thermocouple is at room temperature, $T_{i}$, when it is inserted in the chamber, estimate the time required for the temperature difference, $T_{\infty}-T$, to reach $2 \%$ of the initial temperature difference, $T_{\infty}-T_{i}$. Neglect radiation and conduction through the leads. Properties of the thermocouple junction are approximated as $k=100 \mathrm{~W} / \mathrm{m}+\mathrm{K}, c=385 \mathrm{~J} / \mathrm{kg}+\mathrm{K}$, and $\rho=8920 \mathrm{~kg} / \mathrm{m}^{3}$, while those of the combustion gases may be approximated as $k=0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\nu=50 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$, and $\operatorname{Pr}=0.69$.
(b) If the thermocouple junction has an emissivity of $0.5$ and the cooled walls of the combustor are at $T_{c}=400 \mathrm{~K}$, what is the steady-state temperature of the thermocouple junction if the combustion gases are at $1000 \mathrm{~K}$ ? Conduction through the lead wires may be neglected.
(c) To determine the influence of the gas velocity on the thermocouple measurement error, compute the steady-state temperature of the thermocouple junction for velocities in the range $1 \leq V \leq 25 \mathrm{~m} / \mathrm{s}$. The emissivity of the junction can be controlled through application of a thin coating. To reduce the measurement error, should the emissivity be increased or decreased? For $V=5 \mathrm{~m} / \mathrm{s}$, compute the steadystate junction temperature for emissivities in the range $0.1 \leq \varepsilon \leq 1.0$.

Aadit Sharma
Aadit Sharma
Numerade Educator
01:34

Problem 86

A thermocouple junction is inserted in a large duct to measure the temperature of hot gases flowing through the duct.
(a) If the duct surface temperature $T_{s}$ is less than the gas temperature $T_{g}$, will the thermocouple sense a temperature that is less than, equal to, or greater than $T_{g}$ ? Justify your answer on the basis of a simple analysis.
(b) A thermocouple junction in the shape of a $2-\mathrm{mm}-$ diameter sphere with a surface emissivity of $0.60$ is placed in a gas stream moving at $3 \mathrm{~m} / \mathrm{s}$. If the thermocouple senses a temperature of $320^{\circ} \mathrm{C}$ when the duct surface temperature is $175^{\circ} \mathrm{C}$, what is the actual gas temperature? The gas may be assumed to have the properties of air at atmospheric pressure.
(c) How would changes in velocity and emissivity affect the temperature measurement error? Determine the measurement error for velocities in the range $1 \leq V \leq 25 \mathrm{~m} / \mathrm{s}(\varepsilon=0.6)$ and for emissivities in the range $0.1 \leq \varepsilon \leq 1.0(V=3 \mathrm{~m} / \mathrm{s})$.

Aadit Sharma
Aadit Sharma
Numerade Educator
02:29

Problem 87

Consider temperature measurement in a gas stream using the thermocouple junction described in Problem $7.86(D=2 \mathrm{~mm}, \varepsilon=0.60)$. If the gas velocity and temperature are $3 \mathrm{~m} / \mathrm{s}$ and $500^{\circ} \mathrm{C}$, respectively, what temperature will be indicated by the thermocouple if the duct surface temperature is $200^{\circ} \mathrm{C}$ ? The gas may be assumed to have the properties of atmospheric air. What temperature will be indicated by the thermocouple if the gas pressure is doubled and all other conditions remain the same?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
06:05

Problem 88

A silicon chip $\left(k=150 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=2300 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=\right.$ $700 \mathrm{~J} / \mathrm{kg}+\mathrm{K}), 10 \mathrm{~mm}$ on a side and $1 \mathrm{~mm}$ thick, is connected to a substrate by solder balls $(k=40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\left.\rho=10,000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=150 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ of 1 -mm diameter, and during an accelerated thermal stress test, the system is exposed to the flow of a dielectric liquid $\left(k=0.064 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \nu=10^{-6} \mathrm{~m}^{2} / \mathrm{s}, P r=25\right)$. As first approximations, treat the top and bottom surfaces of the chip as flat plates in turbulent, parallel flow and assume the substrate and lower chip surfaces to have a negligible effect on flow over the solder balls. Also assume point contact between the chip and the solder, thereby neglecting heat transfer by conduction between the components.
(a) The stress test begins with the components at ambient temperature $\left(T_{i}=20^{\circ} \mathrm{C}\right)$ and proceeds with heating by the fluid at $T_{\infty}=80^{\circ} \mathrm{C}$. If the fluid velocity is $V=0.2 \mathrm{~m} / \mathrm{s}$, estimate the ratio of the
time constant of the chip to that of a solder ball. Which component responds more rapidly to the heating process?
(b) The thermal stress acting on the solder joint is proportional to the chip-to-solder temperature difference. What is this temperature difference $0.25 \mathrm{~s}$ after the start of heating?

Paul Gabriel
Paul Gabriel
Numerade Educator
01:54

Problem 89

Repeat Example $7.7$ for a more compact tube bank in which the longitudinal and transverse pitches are $S_{L}=$ $S_{T}=20.5 \mathrm{~mm}$. All other conditions remain the same.

Manish Jain
Manish Jain
Numerade Educator
04:00

Problem 90

A preheater involves the use of condensing steam at $100^{\circ} \mathrm{C}$ on the inside of a bank of tubes to heat air that enters at 1 atm and $25^{\circ} \mathrm{C}$. The air moves at $5 \mathrm{~m} / \mathrm{s}$ in cross flow over the tubes. Each tube is $1 \mathrm{~m}$ long and has an outside diameter of $10 \mathrm{~mm}$. The bank consists of 196 tubes in a square, aligned array for which $S_{T}=S_{L}=15 \mathrm{~mm}$. What is the total rate of heat transfer to the air? What is the pressure drop associated with the airflow?

Mohammad Mehran
Mohammad Mehran
Numerade Educator
03:26

Problem 91

Consider the in-line tube bank of Problem $7.90$ $\left(D=10 \mathrm{~mm}, L=1 \mathrm{~m}\right.$, and $\left.S_{T}=S_{L}=15 \mathrm{~mm}\right)$, with condensing steam used to heat atmospheric air entering the tube bank at $T_{i}=25^{\circ} \mathrm{C}$ and $V=5 \mathrm{~m} / \mathrm{s}$. In this case, however, the desired outlet temperature, not the number of tube rows, is known. What is the minimum value of $N_{L}$ needed to achieve an outlet temperature of $T_{o} \geq$ $75^{\circ} \mathrm{C}$ ? What is the corresponding pressure drop across the tube bank?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:50

Problem 92

A tube bank uses an aligned arrangement of $10-\mathrm{mm}-$ diameter tubes with $S_{T}=S_{L}=20 \mathrm{~mm}$. There are 10 rows of tubes with 50 tubes in each row. Consider an application for which cold water flows through the tubes, maintaining the outer surface temperature at $27^{\circ} \mathrm{C}$, while flue gases at $427^{\circ} \mathrm{C}$ and a velocity of $5 \mathrm{~m} / \mathrm{s}$ are in cross flow over the tubes. The properties of the flue gas may be approximated as those of atmospheric air at $427^{\circ} \mathrm{C}$. What is the total rate of heat transfer per unit length of the tubes in the bank?

Narayan Hari
Narayan Hari
Numerade Educator
04:40

Problem 93

An air duct heater consists of an aligned array of electrical heating elements in which the longitudinal and transverse pitches are $S_{L}=S_{T}=24 \mathrm{~mm}$. There are 3 rows of elements in the flow direction $\left(N_{L}=3\right)$ and 4 elements per row $\left(N_{T}=4\right)$. Atmospheric air with an upstream velocity of $12 \mathrm{~m} / \mathrm{s}$ and a temperature of $25^{\circ} \mathrm{C}$ moves in cross flow over the elements, which have a diameter of $12 \mathrm{~mm}$, a length of $250 \mathrm{~mm}$, and are maintained at a surface temperature of $350^{\circ} \mathrm{C}$.
(a) Determine the total heat transfer to the air and the temperature of the air leaving the duct heater.
(b) Determine the pressure drop across the element bank and the fan power requirement.
(c) Compare the average convection coefficient obtained in your analysis with the value for an isolated (single) element. Explain the difference between the results.
(d) What effect would increasing the longitudinal and transverse pitches to $30 \mathrm{~mm}$ have on the exit temperature of the air, the total heat rate, and the pressure drop?

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
04:00

Problem 94

A tube bank uses an aligned arrangement of $30-\mathrm{mm}$ diameter tubes with $S_{T}=S_{L}=60 \mathrm{~mm}$ and a tube length of $1 \mathrm{~m}$. There are 10 tube rows in the flow direction $\left(N_{L}=10\right)$ and 7 tubes per row $\left(N_{T}=7\right)$. Air with upstream conditions of $T_{s e}=27^{\circ} \mathrm{C}$ and $V=15 \mathrm{~m} / \mathrm{s}$ is in cross flow over the tubes, while a tube wall temperature of $100^{\circ} \mathrm{C}$ is maintained by steam condensation inside the tubes. Determine the temperature of air leaving the tube bank, the pressure drop across the bank, and the fan power requirement.

Mohammad Mehran
Mohammad Mehran
Numerade Educator
00:37

Problem 95

Repeat Problem 7.94, but with $N_{L}=7, N_{T}=10$, and $V=10.5 \mathrm{~m} / \mathrm{s}$.

Sari Ogami
Sari Ogami
Numerade Educator
06:05

Problem 96

Electrical components mounted to each of two isothermal plates are cooled by passing atmospheric air between the plates, and an in-line array of aluminum pin fins is used to enhance heat transfer to the air.
The pins are of diameter $D=2 \mathrm{~mm}$, length $L=$ $100 \mathrm{~mm}$, and thermal conductivity $k=240 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The longitudinal and transverse pitches are $S_{L}=S_{T}=4 \mathrm{~mm}$, with a square array of 625 pins $\left(N_{T}=N_{L}=25\right)$ mounted to square plates that are each of width $W=100 \mathrm{~mm}$ on a side. Air enters the pin array with a velocity of $10 \mathrm{~m} / \mathrm{s}$ and a temperature of $300 \mathrm{~K}$.
(a) Evaluating air properties at $300 \mathrm{~K}$, estimate the average convection coefficient for the array of pin fins.
(b) Assuming a uniform convection coefficient over all heat transfer surfaces (plates and pins), use the result of part (a) to determine the air outlet temperature and total heat rate when the plates are maintained at $350 \mathrm{~K}$. Hint: The air outlet temperature is governed by an exponential relation of the form $\left[\left(T_{s}-T_{o}\right) /\left(T_{s}-T_{i}\right)\right]=\exp \left[-\left(\bar{h} A_{t} \eta_{o}\right) / \dot{m} c_{p}\right]$, where $\dot{m}=\rho V L N_{T} S_{T}$ is the mass flow rate of air passing through the array, $A_{r}$ is the total heat transfer surface area (plates and fins), and $\eta_{o}$ is the overall surface efficiency defined by Equation $3.107 .$

Paul Gabriel
Paul Gabriel
Numerade Educator
01:45

Problem 97

Consider the chip cooling scheme of Problem $3.146$, but with an insulated top wall placed at the pin tips to force airflow across the pin array. Air enters the array at $20^{\circ} \mathrm{C}$ and with a velocity $V$ that may be varied but cannot exceed $10 \mathrm{~m} / \mathrm{s}$ due to pressure drop considerations. The pin fin geometry, which includes the number of pins in the $N \times N$ square array, as well as the pin diameter $D_{p}$ and length $L_{p}$, may also be varied, subject to the constraint that the product $N D_{p}$ not exceed $9 \mathrm{~mm}$. Neglecting heat transfer through the board, assess the effect of changes in air velocity, and hence $h_{o}$, as well as pin fin geometry, on the air outlet temperature and the chip heat rate, if the remaining conditions of Problems $3.146$ and $3.27$, including a maximum allowable chip temperature of $75^{\circ} \mathrm{C}$, remain in effect. Recommend design and operating conditions for which chip cooling is enhanced. Hint: The air outlet temperature is governed by a relation of the form $\left[\left(T_{s}-T_{o}\right) /\right.$ $\left.\left(T_{s}-T_{i}\right)\right]=\exp \left[-\left(\bar{h} A_{t} \eta_{o}\right) / \dot{m} c_{p}\right]$, where $\dot{m}$ is the mass flow rate of air passing through the array, $A_{t}$ is the total heat transfer surface area (chip and pins), and $\eta_{o}$ is the overall surface efficiency defined by Equation 3.107.

Anand Jangid
Anand Jangid
Numerade Educator
02:25

Problem 98

An air-cooled steam condenser is operated with air in cross flow over a square, in-line array of 400 tubes $\left(N_{L}=N_{T}=20\right.$ ), with an outside tube diameter of $20 \mathrm{~mm}$ and longitudinal and transverse pitches of $S_{L}=60 \mathrm{~mm}$ and $S_{T}=30 \mathrm{~mm}$, respectively. Saturated steam at a pressure of $2.455$ bars enters the tubes, and a uniform tube outer surface temperature of $T_{s}=390 \mathrm{~K}$ may be assumed to be maintained as condensation occurs within the tubes.
(a) If the temperature and velocity of the air upstream of the array are $T_{i}=300 \mathrm{~K}$ and $V=4 \mathrm{~m} / \mathrm{s}$, what is the temperature $T_{o}$ of the air that leaves the array? As a first approximation, evaluate the properties of air at $300 \mathrm{~K}$.
(b) If the tubes are $2 \mathrm{~m}$ long, what is the total heat transfer rate for the array? What is the rate at which steam is condensed in $\mathrm{kg} / \mathrm{s}$ ?
(c) Assess the effect of increasing $N_{L}$ by a factor of 2 , while reducing $S_{L}$ to $30 \mathrm{~mm}$. For this configuration, explore the effect of changes in the air velocity.

Anand Jangid
Anand Jangid
Numerade Educator
01:31

Problem 99

Heating and cooling with miniature impinging jets has been proposed for numerous applications. For a single round jet, determine the minimum jet diameter for which Equation $7.71$ may be applied for air at atmospheric pressure (a) at $T_{e}=0^{\circ} \mathrm{C}$ and (b) at $T_{e}=500^{\circ} \mathrm{C}$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:09

Problem 100

A circular transistor of $10-\mathrm{mm}$ diameter is cooled by impingement of an air jet exiting a 2 -mm-diameter round nozzle with a velocity of $20 \mathrm{~m} / \mathrm{s}$ and a temperature of $15^{\circ} \mathrm{C}$. The jet exit and the exposed surface of the transistor are separated by a distance of $10 \mathrm{~mm}$.
If the transistor is well insulated at all but its exposed surface and the surface temperature is not to exceed $85^{\circ} \mathrm{C}$, what is the transistor's maximum allowable operating power?

Anand Jangid
Anand Jangid
Numerade Educator
02:35

Problem 101

A long rectangular plate of AISI 304 stainless steel is initially at $1200 \mathrm{~K}$ and is cooled by an array of slot jets (see Figure 7.17). The nozzle width and pitch are $W=10 \mathrm{~mm}$ and $S=100 \mathrm{~mm}$, respectively, and the nozzle-to-plate separation is $H=200 \mathrm{~mm}$. The plate thickness and width are $t=8 \mathrm{~mm}$ and $L=1 \mathrm{~m}$, respectively. If air exits the nozzles at a temperature of $400 \mathrm{~K}$ and a velocity of $30 \mathrm{~m} / \mathrm{s}$, what is the initial cooling rate of the plate?

Manish Jain
Manish Jain
Numerade Educator
View

Problem 102

A cryogenic probe is used to treat cancerous skin tissue. The probe consists of a single round jet of diameter $D_{e}=2 \mathrm{~mm}$ that issues from a nozzle concentrically situated within a larger, enclosed cylindrical tube of outer diameter $D_{o}=15 \mathrm{~mm}$. The wall thickness of the AISI 302 stainless steel probe is $t=2 \mathrm{~mm}$, and the separation distance between the nozzle and the inner surface of the probe is $H=5 \mathrm{~mm}$.
Assuming the cancerous skin tissue to be a semiinfinite medium with $k_{c}=0.20 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $T_{c}=37^{\circ} \mathrm{C}$ far from the probe location, determine the surface temperature $T_{s}$. Neglect the contact resistance between the probe and the tissue. Cold nitrogen exits the jet at $T_{e}=100 \mathrm{~K}, V_{e}=20 \mathrm{~m} / \mathrm{s}$. Hint: Due to the probe walls, the jet is confined and behaves as if it were one in an array such as in Figure $7.18 c$.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
01:58

Problem 103

Air at $10 \mathrm{~m} / \mathrm{s}$ and $15^{\circ} \mathrm{C}$ is used to cool a square hot molded plastic plate $0.5 \mathrm{~m}$ to a side having a surface temperature of $140^{\circ} \mathrm{C}$. To increase the throughput of the production process, it is proposed to cool the plate using an array of slotted nozzles with width and pitch of $4 \mathrm{~mm}$ and $56 \mathrm{~mm}$, respectively, and a nozzle-toplate separation of $40 \mathrm{~mm}$. The air exits the nozzle at a temperature of $15^{\circ} \mathrm{C}$ and a velocity of $10 \mathrm{~m} / \mathrm{s}$.
(a) Determine the improvement in cooling rate that can be achieved using the slotted nozzle arrangement in lieu of turbulated air at $10 \mathrm{~m} / \mathrm{s}$ and $15^{\circ} \mathrm{C}$ in parallel flow over the plate.
(b) Would the heat rates for both arrangements change significantly if the air velocities were increased by a factor of 2 ?
(c) What is the air mass rate requirement for the slotted nozzle arrangement?

Narayan Hari
Narayan Hari
Numerade Educator
01:31

Problem 104

Consider Problem 7.103, in which the improvement in performance of slot-jet cooling over parallel-flow cooling was demonstrated. Design an optimal round nozzle array, using the same air jet velocity and temperature, $10 \mathrm{~m} / \mathrm{s}$ and $15^{\circ} \mathrm{C}$, respectively, and compare the cooling rates and supply air requirements. Discuss the features associated with each of the three methods relevant to selecting one for this application of cooling the plastic part.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:20

Problem 105

Consider the plasma spraying process of Problems $5.25$ and $7.82$. For a nozzle exit diameter of $D=$ $10 \mathrm{~mm}$ and a substrate radius of $r=25 \mathrm{~mm}$, estimate the rate of heat transfer by convection $q_{\text {cany }}$ from the argon plasma to the substrate, if the substrate temperature is maintained at $300 \mathrm{~K}$. Energy transfer to the substrate is also associated with the release of latent heat $q_{\text {lat }}$, which occurs during solidification of the impacted molten droplets. If the mass rate of droplet impingement is $\dot{m}_{p}=0.02 \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}^{2}$, estimate the rate of latent heat release.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:05

Problem 106

You have been asked to determine the feasibility of using an impinging jet in a soldering operation for electronic assemblies. The schematic illustrates the use of a single, round nozzle to direct high-velocity, hot air to a location where a surface mount joint is to be formed.
For your study, consider a round nozzle with a diameter of $1 \mathrm{~mm}$ located a distance of $2 \mathrm{~mm}$ from the region of the surface mount, which has a diameter of $2.5 \mathrm{~mm}$.
(a) For an air jet velocity of $70 \mathrm{~m} / \mathrm{s}$ and a temperature of $500^{\circ} \mathrm{C}$, estimate the average convection coefficient over the area of the surface mount.
(b) Assume that the surface mount region on the printed circuit board (PCB) can be modeled as a semi-infinite medium, which is initially at a uniform temperature of $25^{\circ} \mathrm{C}$ and suddenly experiences convective heating by the jet. Estimate the time required for the surface to reach $183^{\circ} \mathrm{C}$. The thermophysical properties of a typical solder are $\rho=8333 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=188 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=$ $51 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

Paul Gabriel
Paul Gabriel
Numerade Educator
05:07

Problem 107

Consider the packed bed of aluminum spheres described in Problem $5.12$ under conditions for which the bed is charged by hot air with an inlet velocity of $V=1 \mathrm{~m} / \mathrm{s}$ and temperature of $T_{g, i}=300^{\circ} \mathrm{C}$, but for which the convection coefficient is not prescribed. If the porosity of the bed is $\varepsilon=0.40$ and the initial temperature of the spheres is $T_{i}=25^{\circ} \mathrm{C}$, how long does it take a sphere near the inlet of the bed to accumulate $90 \%$ of its maximum possible energy?

Supratim Pal
Supratim Pal
Numerade Educator
03:16

Problem 108

The use of rock pile thermal energy storage systems has been considered for solar energy and industrial process heat applications. A particular system involves a cylindrical container, $2 \mathrm{~m}$ long by $1 \mathrm{~m}$ in diameter, in which nearly spherical rocks of $0.03-\mathrm{m}$ diameter are packed. The bed has a void space of $0.42$, and the density and specific heat of the rock are $\rho=2300 \mathrm{~kg} / \mathrm{m}^{3}$ and $c_{p}=879 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, respectively. Consider conditions for which atmospheric air is supplied to the rock pile at a steady flow rate of $1 \mathrm{~kg} / \mathrm{s}$ and a temperature of $90^{\circ} \mathrm{C}$. The air flows in the axial direction through the container. If the rock is at a temperature of $25^{\circ} \mathrm{C}$, what is the total rate of heat transfer from the air to the rock pile?

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:32

Problem 109

The cylindrical chamber of a pebble bed nuclear reactor is of length $L=10 \mathrm{~m}$, and diameter $D=3 \mathrm{~m}$. The chamber is filled with spherical uranium oxide pellets of core diameter $D_{p}=50 \mathrm{~mm}$. Each pellet generates thermal energy in its core at a rate of $\dot{E}_{g}$ and is coated with a layer of non-heat-generating graphite, which is of uniform thickness $\delta=5 \mathrm{~mm}$, to form a pebble. The uranium oxide and graphite each have a thermal conductivity of $2 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The packed bed has a porosity of $\varepsilon=0.4$. Pressurized helium at 40 bars is used to absorb the thermal energy from the pebbles. The helium enters the packed bed at $T_{i}=450^{\circ} \mathrm{C}$ with a velocity of $3.2 \mathrm{~m} / \mathrm{s}$. The properties of the helium may be assumed to be $c_{p}=5193 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, $k=0.3355 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, \rho=2.1676 \mathrm{~kg} / \mathrm{m}^{3}, \mu=4.214 \times$ $10^{-5} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, \operatorname{Pr}=0.654$.
(a) For a desired overall thermal energy transfer rate of $q=125 \mathrm{MW}$, determine the mean outlet temperature of the helium leaving the bed, $T_{o}$, and the amount of thermal energy generated by each pellet, $\dot{E}_{g^{*}}$
(b) The amount of energy generated by the fuel decreases if a maximum operating temperature of approximately $2100^{\circ} \mathrm{C}$ is exceeded. Determine the maximum internal temperature of the hottest pellet in the packed bed. For Reynolds numbers in the range $4000 \leq R e_{D} \leq 10,000$, Equation $7.81$ may be replaced by $\varepsilon \bar{j}_{H}=2.876 R e_{D}^{-1}+0.3023 R e_{D}^{-0.35}$.

Rob Ball
Rob Ball
Numerade Educator
16:01

Problem 110

Latent heat capsules consist of a thin-walled spherical shell within which a solid-liquid, phase-change material $(\mathrm{PCM})$ of melting point $T_{\mathrm{mp}}$ and latent heat of fusion $h_{s f}$ is enclosed. As shown schematically, the capsules may be packed in a cylindrical vessel through which there is fluid flow. If the $\mathrm{PCM}$ is in its solid state and $T_{\mathrm{mp}}<T_{i}$, heat is transferred from the fluid to the capsules and latent energy is stored in the PCM as it melts. Conversely, if the PCM is a liquid and $T_{\text {mp }}>T_{i}$, energy is released from the $\mathrm{PCM}$ as it freezes and heat is transferred to the fluid. In either situation, all of the capsules within the packed bed would remain at $T_{\mathrm{mp}}$ through much of the phase change process, in which case the fluid outlet temperature would remain at a fixed value $T_{o^{*}}$.
Consider an application for which air at atmospheric pressure is chilled by passing it through a packed bed $(\varepsilon=0.5)$ of capsules $\left(D_{c}=50 \mathrm{~mm}\right)$ containing an organic compound with a melting point of $T_{\text {mp }}=4^{\circ} \mathrm{C}$. The air enters a cylindrical vessel $\left(L_{v}=D_{v}=0.40 \mathrm{~m}\right)$ at $T_{i}=25^{\circ} \mathrm{C}$ and $V=1.0 \mathrm{~m} / \mathrm{s}$.
(a) If the PCM in each capsule is in the solid state at $T_{\text {map }}$ as melting occurs within the capsule, what is the outlet temperature of the air? If the density and latent heat of fusion of the $\mathrm{PCM}$ are $\rho=$ $1200 \mathrm{~kg} / \mathrm{m}^{3}$ and $h_{s f}=165 \mathrm{~kJ} / \mathrm{kg}$, what is the mass rate $(\mathrm{kg} / \mathrm{s})$ at which the $\mathrm{PCM}$ is converted from solid to liquid in the vessel?
(b) Explore the effect of the inlet air velocity and capsule diameter on the outlet temperature.
(c) At what location in the vessel will complete melting of the PCM in a capsule first occur? Once complete melting begins to occur, how will the outlet temperature vary with time and what is its asymptotic value?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
04:21

Problem 111

The porosity of a packed bed can be decreased by vibrating the containment vessel as the vessel is filled with the particles. The vibration promotes particle settling.
(a) Consider the air chilling process of Problem $7.110 \mathrm{a}$. Determine the outlet air temperature $T_{o}$ and mass rate at which the $\mathrm{PCM}$ is melted for $\varepsilon=0.30$. Assume the total mass of PCM and the mass flow rate of air are unchanged. The length of the containment vessel $L_{v}$ is decreased to compensate for the reduced porosity.
(b) Determine $T_{o}$ and the PCM melting rate for the case where the diameter of the containment vessel $D_{v}$ is decreased to compensate for the reduced porosity. Which containment vessel configuration is preferred?

Niamat Khuda
Niamat Khuda
Numerade Educator
01:52

Problem 112

Consider the packed bed $(\varepsilon=0.5)$ of latent heat capsules $\left(D_{c}=50 \mathrm{~mm}\right)$ described in Problem $7.110$, but now for an application in which ambient air is to be heated by passing it through the bed. In this case the capsules contain an organic compound with a melting point of $T_{\text {mp }}=50^{\circ} \mathrm{C}$, and the air enters the vessel $\left(L_{v}=D_{v}=0.40 \mathrm{~m}\right)$ at $T_{i}=20^{\circ} \mathrm{C}$ and $V=1.0 \mathrm{~m} / \mathrm{s}$.
(a) If the PCM in each capsule is in the liquid state at $T_{\text {mp }}$ as solidification occurs within the capsule, what is the outlet temperature of the air? If the density and latent heat of fusion of the $\mathrm{PCM}$ are $\rho=900 \mathrm{~kg} / \mathrm{m}^{3}$ and $h_{s f}=200 \mathrm{~kJ} / \mathrm{kg}$, what is the mass rate $(\mathrm{kg} / \mathrm{s})$ at which the PCM is converted from liquid to solid in the vessel?
(b) Explore the effect of the inlet air velocity and capsule diameter on the outlet temperature.
(c) At what location in the vessel will complete freezing of the PCM in a capsule first occur? Once complete freezing begins to occur, how will the outlet temperature vary with time and what is its asymptotic value?

Lottie Adams
Lottie Adams
Numerade Educator
09:01

Problem 113

Packed beds of spherical particles can be sintered at high temperature to form permeable, rigid foams. A foam sheet of thickness $t=10 \mathrm{~mm}$ is comprised of sintered bronze spheres, each of diameter $D=0.6 \mathrm{~mm}$. The metal foam has a porosity of $\varepsilon=0.25$, and the foam sheet fills the cross section of an $L=40 \mathrm{~mm} \times W=40 \mathrm{~mm}$ wind tunnel. The upper and lower surfaces of the foam are at temperatures $T_{s}=80^{\circ} \mathrm{C}$, and the two other foam edges (the front edge shown in the schematic and the corresponding back edge) are insulated. Air flows in the wind tunnel at an upstream temperature and velocity of $T_{i}=20^{\circ} \mathrm{C}$ and $V=10 \mathrm{~m} / \mathrm{s}$, respectively.
(a) Assuming the foam is at a uniform temperature $T_{s}$, estimate the convection heat transfer rate to the air. Do you expect the actual heat transfer rate to be equal to, less than, or greater than your estimated value?
(b) Assuming one-dimensional conduction in the $x=$ direction, use an extended surface analysis to estimate the heat transfer rate to the air. To do so, show that the effective perimeter associated with Equation $3.70$ is $P_{\mathrm{eff}}=A_{p, \mathrm{r}} / L$. Determine the effective thermal conductivity of the foam $k_{\text {eff }}$ by using Equation 3.25. Do you expect the actual heat transfer rate to be equal to, less than, or greater than your estimated value?

Averell Hause
Averell Hause
Carnegie Mellon University
02:17

Problem 114

Consider mass loss from a smooth wet flat plate due to forced convection at atmospheric pressure. The plate is $0.5 \mathrm{~m}$ long and $3 \mathrm{~m}$ wide. Dry air at $300 \mathrm{~K}$ and a free stream velocity of $35 \mathrm{~m} / \mathrm{s}$ flows over the surface, which is also at a temperature of $300 \mathrm{~K}$. Estimate the average mass transfer coefficient $\bar{h}_{m}$ and determine the water vapor mass loss rate $(\mathrm{kg} / \mathrm{s})$ from the plate.

Nicholas Mogoi
Nicholas Mogoi
Numerade Educator
00:17

Problem 115

Consider dry, atmospheric air in parallel flow over a $0.5$-m-long plate whose surface is wetted. The air velocity is $35 \mathrm{~m} / \mathrm{s}$, and the air and water are each at a temperature of $300 \mathrm{~K}$.
(a) Estimate the heat loss and evaporation rate per unit width of the plate, $q^{\prime}$ and $n_{A}^{\prime}$, respectively.
(b) Assuming the air temperature remains at $300 \mathrm{~K}$, generate plots of $q^{\prime}$ and $n_{\mathrm{A}}^{\prime}$ for a range of water temperatures from 300 to $350 \mathrm{~K}$, with air velocities of 10,20 , and $35 \mathrm{~m} / \mathrm{s}$.
(c) For the air velocities and air temperature of part (b), determine the water temperatures for which the heat loss will be zero.

Mayukh Banik
Mayukh Banik
Numerade Educator
01:34

Problem 116

A flat plate coated with a volatile substance (species A) is exposed to dry, atmospheric air in parallel flow with $T_{\infty}=20^{\circ} \mathrm{C}$ and $u_{\infty}=8 \mathrm{~m} / \mathrm{s}$. The plate is maintained at a constant temperature of $134^{\circ} \mathrm{C}$ by an electrical heating element, and the substance evaporates from the surface. The plate has a width of $0.25 \mathrm{~m}$ (normal to the plane of the sketch) and is well insulated on the bottom.
The molecular weight and the latent heat of vaporization of species A are $M_{A}=150 \mathrm{~kg} / \mathrm{kmol}$ and $h_{f g}=5.44 \times 10^{6} \mathrm{~J} / \mathrm{kg}$, respectively, and the mass diffusivity is $D_{\mathrm{AB}}=7.75 \times 10^{-7} \mathrm{~m}^{2} / \mathrm{s}$. If the saturated vapor pressure of the substance is $0.12$ atm at $134^{\circ} \mathrm{C}$, what is the electrical power required to maintain steady-state conditions?

Manik Pulyani
Manik Pulyani
Numerade Educator
02:13

Problem 117

Dry air at atmospheric pressure and $350 \mathrm{~K}$, with a free stream velocity of $25 \mathrm{~m} / \mathrm{s}$, flows over a smooth, porous plate $1 \mathrm{~m}$ long.
(a) Assuming the plate to be saturated with liquid water at $350 \mathrm{~K}$, estimate the mass rate of evaporation per unit width of the plate, $n_{\mathrm{A}}^{\prime}(\mathrm{kg} / \mathrm{s} \cdot \mathrm{m})$.
(b) For air and liquid water temperatures of 300,325 , and $350 \mathrm{~K}$, generate plots of $n_{\mathrm{A}}^{\prime}$ as a function of velocity for the range from 1 to $25 \mathrm{~m} / \mathrm{s}$.

Chai Santi
Chai Santi
Numerade Educator
02:35

Problem 118

A scheme for dissipating heat from an array of $N=100$ integrated circuits involves joining the circuits to the bottom of a plate and exposing the top of the plate to a water bath. The water container is of length $L=100 \mathrm{~mm}$ on a side and is exposed to airflow at its top surface. The flow is turbulated by the protruding lip of the side wall.
If the sides and bottom of the container are well insulated from the surroundings and heat is uniformly dissipated in each circuit, at what rate may heat be dissipated from each circuit when the water temperature is maintained at $T_{b}=350 \mathrm{~K}$ ?

Manish Jain
Manish Jain
Numerade Educator
01:05

Problem 119

A series of water-filled trays, each $222 \mathrm{~mm}$ long, experiences an evaporative drying process. Dry air at $T_{\infty}=300 \mathrm{~K}$ flows over the trays with a velocity of $15 \mathrm{~m} / \mathrm{s}$, while radiant heaters maintain the surface temperature at $T_{s}=330 \mathrm{~K}$.
(a) What is the evaporative flux $\left(\mathrm{kg} / \mathrm{s}^{*} \mathrm{~m}^{2}\right)$ at a distance $1 \mathrm{~m}$ from the leading edge?
(b) What is the irradiation (W/m²) that should be supplied to the tray surface at this location to maintain the water temperature at $330 \mathrm{~K}$ ?
(c) Assuming the water temperature is uniform over the tray at this location, what is the evaporation rate $(\mathrm{kg} / \mathrm{s} \cdot \mathrm{m})$ from the tray per unit width of the tray?
(d) What irradiation should be applied to each of the first four trays such that the corresponding evaporation rates are identical to that found in part (c)?

Manik Pulyani
Manik Pulyani
Numerade Educator
06:19

Problem 120

Consider the physical system of Problem $7.119$ (a series of water-filled trays heated radiatively), but under operating conditions for which each tray is $0.25 \mathrm{~m}$ long by $1 \mathrm{~m}$ wide and is uniformly irradiated, with $G=10^{4} \mathrm{~W} / \mathrm{m}^{2}$. Dry air at $T_{\infty}=300 \mathrm{~K}$ continues to flow over the trays at a velocity of $15 \mathrm{~m} / \mathrm{s}$.
(a) What is the rate of water loss $(\mathrm{kg} / \mathrm{s})$ from the first, third, and fourth trays?
(b) Estimate the temperature of the water in each of the designated trays.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
01:47

Problem 121

The apparatus described in Problem $7.40$ is used by our students to experimentally determine convection heat and mass transfer coefficients, to confirm the heat-mass analogy, and to compare measured results with predictions based on standard correlations. The velocity, $V$, of the airstream is measured using a thermistor-based anemometer, and its relative humidity is determined from measurements of the wet and dry bulb temperatures, $T_{\mathrm{wb}}$ and $T_{\mathrm{db}}$, respectively. Thermocouples are attached to the test-plate, which is covered with a sheet of wet paper in the mass transfer experiments.
(a) Convection heat transfer coeffiient. Using the data provided in Problem 7.40, determine the heat transfer coefficients for the two velocities, assuming the plate to behave as a spacewise isothermal object. Evaluate the coefficients $C$ and $m$ for a correlation of the form $\overline{N u}_{L}=C R e^{m} P r^{1 / 3}$. Compare this result with a standard flat-plate correlation. Comment on the goodness of the comparison and provide reasons for any differences.
(b) Convection mass transfer coeffilent. A sheet of water-saturated paper, $133 \mathrm{~mm}$ to a side, was used as the test surface and its mass measured at two different times, $m(t)$ and $m(t+\Delta t)$. Thermocouples were used to monitor the paper temperature as a function of time, from which the average temperature, $\bar{T}_{s}$, was determined. The wet and dry bulb temperatures were $T_{\mathrm{wb}}=13^{\circ} \mathrm{C}$ and $T_{\mathrm{db}}=27^{\circ} \mathrm{C}$, and data recorded for two airstream velocities are as follows:

Aadit Sharma
Aadit Sharma
Numerade Educator
01:10

Problem 122

Dry air at $35^{\circ} \mathrm{C}$ and a velocity of $20 \mathrm{~m} / \mathrm{s}$ flows over a wetted plate of length $500 \mathrm{~mm}$ and width $150 \mathrm{~mm}$. An embedded electrical heater supplies power to maintain the plate surface temperature at $20^{\circ} \mathrm{C}$.
(a) What is the evaporation rate $(\mathrm{kg} / \mathrm{h})$ of water from the plate? What electrical power is required to maintain steady-state conditions?
(b) After a long period of operation, all the water is evaporated from the plate and its surface is dry. For the same free stream conditions and heater power of part (a), estimate the temperature of the plate.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:34

Problem 123

A minivan traveling $90 \mathrm{~km} / \mathrm{h}$ has just passed through a thunderstorm that left a film of water $0.1 \mathrm{~mm}$ thick on the top of the van. The top of the van can be assumed to be a flat plate $6 \mathrm{~m}$ long. Assume isothermal conditions at $27^{\circ} \mathrm{C}$, an ambient air relative humidity of $80 \%$, and turbulent flow over the entire surface. What location on the van top will be the last to dry? What is the water evaporation rate per unit area $\left(\mathrm{kg} / \mathrm{s} \cdot \mathrm{m}^{2}\right)$ at the trailing edge of the van top?
e in the saturated vapor and liquid states are known to be $0.417$ and $900 \mathrm{~kg} / \mathrm{m}^{3}$, respectively.

Keshav Singh
Keshav Singh
Numerade Educator
14:12

Problem 124

Benzene, a known carcinogen, has been spilled on the laboratory floor and has spread to a length of $2 \mathrm{~m}$. If a film $1 \mathrm{~mm}$ deep is formed, how long will it take for the benzene to completely evaporate? Ventilation in the laboratory provides for airflow parallel to the surface at $1 \mathrm{~m} / \mathrm{s}$, and the benzene and air are both at $25^{\circ} \mathrm{C}$. The mass densities of benzen

Chareen Guzman
Chareen Guzman
Numerade Educator
01:10

Problem 125

Atmospheric air of $40 \%$ relative humidity and temperature $T_{\infty}=300 \mathrm{~K}$ is in parallel flow over a series of water-filled trays, with $u_{\infty}=12 \mathrm{~m} / \mathrm{s}$. What is the rate at which energy must be supplied to each of the first three trays to maintain the water at $300 \mathrm{~K}$ ?

Manik Pulyani
Manik Pulyani
Numerade Educator
04:13

Problem 126

A stream of atmospheric air is used to dry a series of biological samples on plates that are each of length $L_{i}=0.25 \mathrm{~m}$ in the direction of the airflow. The air is dry and at a temperature equal to that of the plates $\left(T_{\infty}=T_{s}=50^{\circ} \mathrm{C}\right)$. The air speed is $u_{\infty}=9.1 \mathrm{~m} / \mathrm{s}$.
(a) Sketch the variation of the local convection mass transfer coefficient $h_{m x}$ with distance $x$ from the leading edge. Indicate the specific nature of the $x$ dependence.
(b) Which of the plates will dry the fastest? Calculate the drying rate per meter of width for this plate $(\mathrm{kg} / \mathrm{s}+\mathrm{m})$.
(c) At what rate would heat have to be supplied to the fastest drying plate to maintain it at $T_{s}=50^{\circ} \mathrm{C}$ during the drying process?

Chai Santi
Chai Santi
Numerade Educator
01:15

Problem 127

Condenser cooling water for a power plant is stored in a cooling pond that is $1000 \mathrm{~m}$ long by $500 \mathrm{~m}$ wide. However, because of evaporative losses, it is necessary to periodically add "makeup" water to the pond in order to maintain a suitable water level. Assuming isothermal conditions at $27^{\circ} \mathrm{C}$ for the water and the air, that the free stream air is dry and moving at a velocity of $2 \mathrm{~m} / \mathrm{s}$ in the direction of the $1000-\mathrm{m}$ pond length, and that the boundary layer on the water surface is everywhere turbulent, determine the amount of makeup water that should be added to the pond daily.

Manik Pulyani
Manik Pulyani
Numerade Educator
18:21

Problem 128

Consider the plate conveyor system of Problem $7.24$, but now under conditions for which the plates are being transported from a liquid bath used for surface cleaning. The initial plate temperature is $T_{i}=40^{\circ} \mathrm{C}$, and the surfaces are covered with a thin liquid film. If the air velocity and temperature are $u_{\infty}=1 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=20^{\circ} \mathrm{C}$, respectively, what is the initial rate of heat transfer from the plate? What is the corresponding rate of change of the plate temperature? The latent heat of vaporization of the solvent, the diffusion coefficient associated with transport of its vapor in air, and its saturated vapor density at $40^{\circ} \mathrm{C}$ are $h_{f g}=900 \mathrm{~kJ} / \mathrm{kg}$, $D_{\mathrm{AB}}=10^{-5} \mathrm{~m}^{2} / \mathrm{s}$, and $\rho_{\mathrm{A} \text { sat }}=0.75 \mathrm{~kg} / \mathrm{m}^{3}$, respectively. The velocity of the conveyor can be neglected relative to that of the air.

Joseph Lentino
Joseph Lentino
Numerade Educator
03:05

Problem 129

In a paper-drying process, the paper moves on a conveyor belt at $0.2 \mathrm{~m} / \mathrm{s}$, while dry air from an in-line array of round jets (Figure $7.18 b$ ) impinges normal to its surface. The nozzle diameter and pitch are $D=20 \mathrm{~mm}$ and $S=100 \mathrm{~mm}$, respectively, and the nozzle-to-paper separation is $H=200 \mathrm{~mm}$. Air exits the nozzle at a velocity and temperature of $20 \mathrm{~m} / \mathrm{s}$ and $300 \mathrm{~K}$, while the wet paper is maintained at $300 \mathrm{~K}$. In $\mathrm{kg} / \mathrm{s} \cdot \mathrm{m}^{2}$, what is the average drying rate of the paper?

Amany Waheeb
Amany Waheeb
Numerade Educator
03:55

Problem 130

In a paper mill drying process, a sheet of paper slurry (water-fiber mixture) has a linear velocity of $5 \mathrm{~m} / \mathrm{s}$ as it is rolled. Radiant heaters maintain a sheet temperature of $T_{s}=330 \mathrm{~K}$, as evaporation occurs to dry, ambient air at $300 \mathrm{~K}$ above and below the sheet.
(a) What is the evaporative flux at a distance of $x=1 \mathrm{~m}$ from the leading edge of the roll? What is the corresponding value of the radiant flux (irradiation, $G$ ) that must be supplied to the sheet to maintain its temperature at $330 \mathrm{~K}$ ? The sheet has an absorptivity of $\alpha=1$.
(b) To accelerate the drying and paper production processes, the velocity and temperature of the strip are increased to $10 \mathrm{~m} / \mathrm{s}$ and $340 \mathrm{~K}$, respectively. To maintain a uniform strip temperature, the irradiation $G$ must be varied with $x$ along the strip. For $0 \leq x \leq 1 \mathrm{~m}$, compute and plot the variations $h_{m, N}(x), N_{\Lambda}^{m}(x)$, and $G(x)$.

Ryan Hood
Ryan Hood
Numerade Educator
View

Problem 131

A channel of triangular cross section, which is $25 \mathrm{~m}$ long and $1 \mathrm{~m}$ deep, is used for the storage of water.
The water and the surrounding air are each at a temperature of $25^{\circ} \mathrm{C}$, and the relative humidity of the air is $50 \%$.
(a) If the air moves at a velocity of $5 \mathrm{~m} / \mathrm{s}$ along the length of the channel, what is the rate of water loss due to evaporation from the surface?
(b) Obtain an expression for the rate at which the water depth would decrease with time due to evaporation. For the above conditions, how long would it take for all the water to evaporate?

Victor Salazar
Victor Salazar
Numerade Educator
05:41

Problem 132

Mass transfer experiments have been conducted on a naphthalene cylinder of $18.4-\mathrm{mm}$ diameter and $88.9-\mathrm{mm}$
length subjected to a cross flow of air in a low-speed wind tunnel. After exposure for $39 \mathrm{~min}$ to the airstream at a temperature of $26^{\circ} \mathrm{C}$ and a velocity of $12 \mathrm{~m} / \mathrm{s}$, it was determined that the cylinder mass decreased by $0.35 \mathrm{~g}$. The barometric pressure was recorded at $750.6 \mathrm{~mm} \mathrm{Hg}$. The saturation pressure $p_{\text {sat }}$ of naphthalene vapor in equilibrium with solid naphthalene is given by the relation $p_{\text {sat }}=p \times 10^{E}$, where $E=8.67-(3766 / T)$, with $T(\mathrm{~K})$ and $p$ (bar) being the temperature and pressure of air. Naphthalene has a molecular weight of $128.16 \mathrm{~kg} / \mathrm{kmol}$.
(a) Determine the convection mass transfer coefficient from the experimental observations.
(b) Compare this result with an estimate from an appropriate correlation for the prescribed flow conditions.

Ronald Prasad
Ronald Prasad
Numerade Educator
27:48

Problem 133

Dry air at $1-a t m$ pressure and a velocity of $15 \mathrm{~m} / \mathrm{s}$ is to be humidified by passing it in cross flow over a porous cylinder of diameter $D=40 \mathrm{~mm}$, which is saturated with water.
(a) Assuming the water and air to be at $300 \mathrm{~K}$, calculate the mass rate of water evaporated under steady-state conditions from the cylindrical medium per unit length.
(b) How will the evaporation rate change if the air and water are maintained at a higher temperature? Generate a plot for the temperature range 300 to $350 \mathrm{~K}$ to illustrate the effect of temperature on the evaporation rate.

Chareen Guzman
Chareen Guzman
Numerade Educator
01:10

Problem 134

Dry air at $35^{\circ} \mathrm{C}$ and a velocity of $15 \mathrm{~m} / \mathrm{s}$ flows over a long cylinder of $20-\mathrm{mm}$ diameter. The cylinder is covered with a thin porous coating saturated with water, and an embedded electrical heater supplies power to maintain the coating surface temperature at $20^{\circ} \mathrm{C}$.
(a) What is the evaporation rate of water from the cylinder per unit length $(\mathrm{kg} / \mathrm{h}+\mathrm{m})$ ? What electrical power per unit length of the cylinder $(\mathrm{W} / \mathrm{m})$ is required to maintain steady-state conditions?
(b) After a long period of operation, all the water is evaporated from the coating and its surface is dry. For the same free stream conditions and heater power of part (a), estimate the temperature of the surface.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:15

Problem 135

Dry air at $20^{\circ} \mathrm{C}$ and a velocity of $15 \mathrm{~m} / \mathrm{s}$ flows over a 20 -mm-diameter rod covered with a thin porous coating that is saturated with water. The rod $(k=175 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ is $250 \mathrm{~mm}$ long and its ends are attached to heat sinks maintained at $35^{\circ} \mathrm{C}$.
Perform a steady-state, finite-difference analysis of the rod-porous coating system, considering conduction in the rod as well as energy transfer from the surface by convection heat and mass transfer. Use the analysis to estimate the temperature at the midspan of the rod and the evaporation rate from the surface. (Suggestions: Use 10 nodes to represent the half-length of the system. Estimate the overall average convection heat transfer coefficient based on an average film temperature for the system, and use the heat-mass transfer analogy to determine the average convection mass transfer coefficient. Validate your code by using it to predict a temperature distribution that agrees with the analytical solution for a fin without evaporation.)

Narayan Hari
Narayan Hari
Numerade Educator
07:58

Problem 136

Approximate the human form as an unclothed vertical cylinder of $0.3-\mathrm{m}$ diameter and $1.75-\mathrm{m}$ length with a surface temperature of $30^{\circ} \mathrm{C}$.
(a) Calculate the heat loss in a $10-\mathrm{m} / \mathrm{s}$ wind at $20^{\circ} \mathrm{C}$.
(b) What is the heat loss if the skin is covered with a thin layer of water at $30^{\circ} \mathrm{C}$ and the relative humidity of the air is $60 \%$ ?

Pronoy Sinha
Pronoy Sinha
Numerade Educator
03:00

Problem 137

It has been suggested that heat transfer from a surface can be augmented by wetting it with water. As a specific example, consider a horizontal tube that is exposed to a transverse stream of dry air. You may assume that the tube, which is maintained at a temperature $T_{s}>T_{\infty}$, is completely wetted on the outside with a thin film of water. Derive an equation to determine the extent of heat transfer enhancement due to wetting. Evaluate the enhancement for $V=10 \mathrm{~m} / \mathrm{s}$, $D=10 \mathrm{~mm}, T_{s}=320 \mathrm{~K}$, and $T_{s}=300 \mathrm{~K}$.

Anand Jangid
Anand Jangid
Numerade Educator
04:27

Problem 138

In the first stage of a paper-drying process, a cylinder of diameter $0.15 \mathrm{~m}$ is covered by moisture-soaked paper. The temperature of the paper is maintained at $70^{\circ} \mathrm{C}$ by embedded electrical heaters. Dry air at a velocity of $10 \mathrm{~m} / \mathrm{s}$ and temperature of $20^{\circ} \mathrm{C}$ flows over the cylinder.
(a) Calculate the required electrical power and the evaporation rate per unit length of the cylinder, $q^{\prime}$ and $n_{A}^{\prime}$, respectively.
(b) Generate plots of $q^{\prime}$ and $n_{A}^{\prime}$ as a function of the dry air velocity for $5 \leq V \leq 20 \mathrm{~m} / \mathrm{s}$ and for paper temperatures of 65,70 , and $75^{\circ} \mathrm{C}$.

Chai Santi
Chai Santi
Numerade Educator
01:31

Problem 139

Cylindrical dry-bulb and wet-bulb thermometers are installed in a large-diameter duct to obtain the temperature $T_{\infty}$ and the relative humidity $\phi_{\infty}$ of moist air flowing through the duct at a velocity $V$. The dry-bulb thermometer has a bare glass surface of diameter $D_{\mathrm{db}}$ and emissivity $\varepsilon_{g}$. The wet-bulb thermometer is covered with a thin wick that is saturated with water flowing continuously by capillary action from a bottom reservoir. Its diameter and emissivity are designated as $D_{\text {wb }}$ and $\varepsilon_{w}$. The duct inside surface is at a known temperature $T_{s}$, which is less than $T_{\infty}$. Develop expressions that may be used to obtain $T_{\infty}$ and $\phi_{\infty}$ from knowledge of the dry-bulb and wet-bulb temperatures $T_{\mathrm{db}}$ and $T_{\mathrm{ub}}$ and the foregoing parameters. Determine $T_{\infty}$ and $\phi_{\infty}$ when $T_{\mathrm{db}}=45^{\circ} \mathrm{C}, T_{\mathrm{wb}}=25^{\circ} \mathrm{C}, T_{s}=35^{\circ} \mathrm{C}, p=1 \mathrm{~atm}$, $V=5 \mathrm{~m} / \mathrm{s}, D_{\mathrm{db}}=3 \mathrm{~mm}, D_{\mathrm{wb}}=4 \mathrm{~mm}$, and $\varepsilon_{\mathrm{g}}=\varepsilon_{w}=$ $0.95$. As a first approximation, evaluate the dry- and wet-bulb air properties at 45 and $25^{\circ} \mathrm{C}$, respectively.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:26

Problem 140

The thermal pollution problem is associated with discharging warm water from an electrical power plant or from an industrial source to a natural body of water. Methods for alleviating this problem involve cooling the warm water before allowing the discharge to occur. Two such methods, involving wet cooling towers or spray ponds, rely on heat transfer from the warm water in droplet form to the surrounding atmosphere. To develop an understanding of the mechanisms that contribute to this cooling, consider a spherical droplet of diameter $D$ and temperature $T$, which is moving at a velocity $V$ relative to air at a temperature $T_{\infty}$ and relative humidity $\phi_{\infty}$. The surroundings are characterized by the temperature $T_{\text {sur }}$ Develop expressions for the droplet evaporation and cooling rates. Calculate the evaporation rate $(\mathrm{kg} / \mathrm{s})$ and cooling rate $(\mathrm{K} / \mathrm{s})$ when $D=3 \mathrm{~mm}, V=7 \mathrm{~m} / \mathrm{s}, T=40^{\circ} \mathrm{C}, T_{\infty}=25^{\circ} \mathrm{C}$, $T_{\text {sar }}=15^{\circ} \mathrm{C}$, and $\phi_{\infty}=0.60$. The emissivity of water is $\varepsilon_{w}=0.96$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:15

Problem 141

Cranberries are harvested by flooding the bogs in which they are grown and raking them into troughs for transport. At the processing plant, the surface moisture on the berries is removed as they roll over a fine screen through which warm air is blown. The berries have an average diameter of $15 \mathrm{~mm}$, and the thickness of the water layer is $0.2 \mathrm{~mm}$.
If the velocity and temperature of the heated air are $2 \mathrm{~m} / \mathrm{s}$ and $30^{\circ} \mathrm{C}$, respectively, estimate the time required to dry the berries. Assume that the water film on the berries is also at $30^{\circ} \mathrm{C}$.

Mohammad Mehran
Mohammad Mehran
Numerade Educator
07:50

Problem 142

A spherical drop of water, $0.5 \mathrm{~mm}$ in diameter, is falling at a velocity of $2.15 \mathrm{~m} / \mathrm{s}$ through dry, still air at 1-atm pressure. Estimate the instantaneous rate of evaporation from the drop if the drop surface is at $60^{\circ} \mathrm{C}$ and the air is at $100^{\circ} \mathrm{C}$.

Guilherme Barros
Guilherme Barros
Numerade Educator
01:22

Problem 143

A spherical droplet of alcohol, $0.5 \mathrm{~mm}$ in diameter, is falling freely through quiescent air at a velocity of $1.8 \mathrm{~m} / \mathrm{s}$. The concentration of alcohol vapor at the surface of the droplet is $0.0573 \mathrm{~kg} / \mathrm{m}^{3}$, and the diffusion coefficient for alcohol in air is $10^{-5} \mathrm{~m}^{2} / \mathrm{s}$. Neglecting radiation and assuming steady-state conditions, calculate the surface temperature of the droplet if the ambient air temperature is $300 \mathrm{~K}$. The latent heat of vaporization is $8.42 \times 10^{5} \mathrm{~J} / \mathrm{kg}$.

Shital Rijal
Shital Rijal
Numerade Educator
35:47

Problem 144

As described in Problem 7.84, the second step in tissue engineering is to seed the top surface of the scaffold with human cells that subsequently grow into the pores of the scaffold. A seeding method that has been proposed is to use a droplet generator similar to that of Problem $7.84$ to generate $D_{p}=50 \mu \mathrm{m}$ diameter drops. The material in the droplet generator is a slurry consisting of a mixture of a host liquid and human liver cells. The host liquid has properties similar to water, and the liver cells are spherical with a diameter of $D_{\mathrm{lc}}=20 \mu \mathrm{m}$ and density $\rho_{\mathrm{lc}}=2400 \mathrm{~kg} / \mathrm{m}^{3}$. Droplets are injected into atmospheric air with a relative humidity and temperature of $\phi_{\infty}=0.50$ and $T_{\infty}=25^{\circ} \mathrm{C}$, respectively. The particles are injected with an initial temperature of $T_{i}=25^{\circ} \mathrm{C}$.
(a) It is desirable for each drop to contain one liver cell. Determine the volume fraction, $f$, of liver cells in the slurry and the terminal velocity for a drop containing one liver cell.
(b) The droplet containing one liver cell is injected at its terminal velocity. Determine the time of flight for a distance between the ejector nozzle and the scaffold of $L=4 \mathrm{~mm}$.
(c) Determine the initial evaporation rate from the droplet.
(d) The tissue engineer is concerned that evaporation will change the mass of the droplet and, in turn, will affect its time of flight and the precision with which the seeds can be placed on the scaffold. Estimate the maximum change in mass due to evaporation during the time of flight. Compare the variation of mass due to evaporation to the variation associated with there being one to three liver cells per droplet. Does evaporation or the liver cell population per droplet influence the variability of the droplet mass most significantly?

Donald Albin
Donald Albin
Numerade Educator
09:28

Problem 145

Motile bacteria are equipped with flagella that are rotated by tiny, biological electrochemical engines which, in turn, propel the bacteria through a host liquid. Consider a nominally spherical Escherichia coli bacterium that is of diameter $D=2 \mu \mathrm{m}$. The bacterium is in a water-based solution at $37^{\circ} \mathrm{C}$ containing a nutrient which is characterized by a binary diffusion coefficient of $D_{\mathrm{AB}}=0.7 \times 10^{-9} \mathrm{~m}^{2} / \mathrm{s}$ and a food energy value of $\mathcal{N}=16,000 \mathrm{~kJ} / \mathrm{kg}$. There is a nutrient density difference between the fluid and the shell of the bacterium of $\Delta \rho_{\mathrm{A}}=860 \times 10^{-12} \mathrm{~kg} / \mathrm{m}^{3}$. Assuming a propulsion efficiency of $\eta=0.5$, determine the maximum speed of the E. coli. Report your answer in body diameters per second.

Supratim Pal
Supratim Pal
Numerade Educator
01:22

Problem 146

In a home furnace humidification system, water droplets of diameter $D$ are discharged in a direction opposing the motion of warm air emerging from the heater. The air is humidified by evaporation from the droplets, and the excess water is collected on a splash plate, from which it is routed to a drain.
Consider conditions for which air enters the heater at a temperature and relative humidity of $17^{\circ} \mathrm{C}$ and $70 \%$, respectively, and leaves the heater at a temperature of $47^{\circ} \mathrm{C}$. The droplet diameter is $1 \mathrm{~mm}$, and the relative velocity between the droplets and the heated air is $15 \mathrm{~m} / \mathrm{s}$. During the time-of-flight, the change in droplet diameter may be neglected and the droplet temperature may be assumed to remain at $47^{\circ} \mathrm{C}$. What is the rate of evaporation from a single droplet?

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:43

Problem 147

Evaporation of liquid fuel droplets is often studied in the laboratory by using a porous sphere technique in which the fuel is supplied at a rate just sufficient to maintain a completely wetted surface on the sphere.
Consider the use of kerosene at $300 \mathrm{~K}$ with a porous sphere of 1 -mm diameter. At this temperature the kerosene has a saturated vapor density of $0.015 \mathrm{~kg} / \mathrm{m}^{3}$ and a latent heat of vaporization of $300 \mathrm{~kJ} / \mathrm{kg}$. The mass diffusivity for the vapor-air mixture is $10^{-5} \mathrm{~m}^{2} / \mathrm{s}$. If dry, atmospheric air at $V=15 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=300 \mathrm{~K}$ flows over the sphere, what is the minimum mass rate at which kerosene must be supplied to maintain a wetted surface? For this condition, by how much must $T_{\infty}$ actually exceed $T_{s}$ to maintain the wetted surface at $300 \mathrm{~K}$ ?

Manik Pulyani
Manik Pulyani
Numerade Educator
04:00

Problem 148

Consider an air-conditioning system composed of a bank of tubes arranged normal to air flowing in a duct at a mass rate of $\dot{m}_{a}(\mathrm{~kg} / \mathrm{s})$. A coolant flowing through the tubes is able to maintain the surface temperature of the tubes at a constant value of $T_{s}<T_{a, i}$, where $T_{a, i}$ is the inlet air temperature (upstream of the tube bank). It has been suggested that air cooling may be enhanced if a thin, uniform film of water is maintained on the outer surface of each of the tubes.
(a) Assuming the water film to be at the temperature $T_{s}$, develop an expression for the ratio of the amount of cooling that occurs with the water film to the amount of cooling that occurs without the film. The amount of cooling may be defined as $T_{a, i}-T_{a, \rho}$, where $T_{a, o}$ is the outlet air temperature (downstream of the tube bank). The upstream air may be assumed to be dry, and the driving potentials for convection heat and mass transfer may be approximated as $\left(T_{a, i}-T_{s}\right)$ and $\rho_{\Lambda, \text { sat }}\left(T_{s}\right)$, respectively. Note: The total rate of heat loss from the air may be expressed as $q=\dot{m}_{a} c_{p, a}\left(T_{a, i}-T_{a, a}\right)$. Estimate the value of this ratio under conditions for which $T_{a, i}=35^{\circ} \mathrm{C}$ and $T_{s}=10^{\circ} \mathrm{C}$.
(b) Consider a tube bank that is 5 rows deep, with 12 tubes in a row. Each tube is $0.5 \mathrm{~m}$ long, with an outside diameter of $8 \mathrm{~mm}$, and a staggered arrangement is used for which $S_{T}=S_{L}=24 \mathrm{~mm}$. Under conditions for which $\dot{m}_{a}=0.5 \mathrm{~kg} / \mathrm{s}, V=3 \mathrm{~m} / \mathrm{s}$, $T_{a, i}=35^{\circ} \mathrm{C}$, and $T_{s}=10^{\circ} \mathrm{C}$, what is the value of
$T_{a, \theta}$ if the tubes are wetted? What is the specific humidity of the air leaving the tube bank?

Paul Gabriel
Paul Gabriel
Numerade Educator
02:22

Problem 149

In a paper-drying process, the paper moves on a conveyor belt at $0.2 \mathrm{~m} / \mathrm{s}$, while dry air from an array of slot jets (Figure 7.17) impinges normal to its surface. The nozzle width and pitch are $W=10 \mathrm{~mm}$ and $S=100 \mathrm{~mm}$, respectively, and the nozzle-to-plate separation is $H=200 \mathrm{~mm}$. The wet paper is of width $L=1 \mathrm{~m}$ and is maintained at $300 \mathrm{~K}$, while the air exits the nozzles at a temperature of $300 \mathrm{~K}$ and a velocity of $20 \mathrm{~m} / \mathrm{s}$. In $\mathrm{kg} / \mathrm{s}^{*} \mathrm{~m}^{2}$, what is the average drying rate per unit surface area of the paper?

Amany Waheeb
Amany Waheeb
Numerade Educator