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Fundamentals of Heat and Mass Transfer

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

Chapter 8

Internal Flow - all with Video Answers

Educators

Chapter Questions

06:59

Problem 1

Fully developed conditions are known to exist for water flowing through a $25-\mathrm{mm}$-diameter tube at $0.01 \mathrm{~kg} / \mathrm{s}$ and $27^{\circ} \mathrm{C}$. What is the maximum velocity of the water in the tube? What is the pressure gradient associated with the flow?

Satpal Satpal
Satpal Satpal
Numerade Educator
01:21

Problem 2

What is the pressure drop associated with water at $27^{\circ} \mathrm{C}$ flowing with a mean velocity of $0.2 \mathrm{~m} / \mathrm{s}$ through a 600 - $\mathrm{m}$-long cast iron pipe of $0.15-\mathrm{m}$ inside diameter?

James Kiss
James Kiss
Numerade Educator
01:45

Problem 3

Water at $27^{\circ} \mathrm{C}$ flows with a mean velocity of $1 \mathrm{~m} / \mathrm{s}$ through a $1-\mathrm{km}$-long pipe of $0.25-\mathrm{m}$ inside diameter.
(a) Determine the pressure drop over the pipe length and the corresponding pump power requirement, if the pipe surface is smooth.
(b) If the pipe is made of cast iron and its surface is clean, determine the pressure drop and pump power requirement.
(c) For the smooth pipe condition, generate a plot of pressure drop and pump power requirement for mean velocities in the range from $0.05$ to $1.5 \mathrm{~m} / \mathrm{s}$.

James Kiss
James Kiss
Numerade Educator
04:27

Problem 4

An engine oil cooler consists of a bundle of 25 smooth tubes, each of length $L=2.5 \mathrm{~m}$ and diameter $D=10 \mathrm{~mm}$.
(a) If oil at $300 \mathrm{~K}$ and a total flow rate of $24 \mathrm{~kg} / \mathrm{s}$ is in fully developed flow through the tubes, what is the pressure drop and the pump power requirement?
(b) Compute and plot the pressure drop and pump power requirement as a function of flow rate for $10 \leq \dot{m} \leq 30 \mathrm{~kg} / \mathrm{s}$.

Alexander Allen
Alexander Allen
Numerade Educator
01:18

Problem 5

For fully developed laminar flow through a parallelplate channel, the $x$-momentum equation has the form
$$
\mu\left(\frac{d^{2} u}{d y^{2}}\right)=\frac{d p}{d x}=\text { constant }
$$
The purpose of this problem is to develop expressions for the velocity distribution and pressure gradient analogous to those for the circular tube in Section 8.1.
(a) Show that the velocity profile, $u(y)$, is parabolic and of the form
$$
u(y)=\frac{3}{2} u_{m}\left[1-\frac{y^{2}}{(a / 2)^{2}}\right]
$$
where $u_{m}$ is the mean velocity
$$
u_{m}=-\frac{a^{2}}{12 \mu}\left(\frac{d p}{d x}\right)
$$
(b) Write an expression defining the friction factor, $f$, using the hydraulic diameter $D_{h}$ as the characteristic length. What is the hydraulic diameter for the parallel-plate channel?
(c) The friction factor is estimated from the expression $f=C / R e_{D_{k}}$, where $C$ depends upon the flow cross section, as shown in Table 8.1. What is the coefficient $C$ for the parallel-plate channel?
(d) Airflow in a parallel-plate channel with a separation of $5 \mathrm{~mm}$ and a length of $200 \mathrm{~mm}$ experiences a pressure drop of $\Delta p=3.75 \mathrm{~N} / \mathrm{m}^{2}$. Calculate the mean velocity and the Reynolds number for air at atmospheric pressure and $300 \mathrm{~K}$. Is the assumption of fully developed flow reasonable for this application? If not, what is the effect on the estimate for $u_{m}$ ?

Chai Santi
Chai Santi
Numerade Educator
03:45

Problem 6

Consider pressurized water, engine oil (unused), and NaK $(22 \% / 78 \%)$ flowing in a 20 -mm-diameter tube.
(a) Determine the mean velocity, the hydrodynamic entry length, and the thermal entry length for each of the fluids when the fluid temperature is $366 \mathrm{~K}$ and the flow rate is $0.01 \mathrm{~kg} / \mathrm{s}$.
(b) Determine the mass flow rate, the hydrodynamic entry length, and the thermal entry length for water and engine oil at 300 and $400 \mathrm{~K}$ and a mean velocity of $0.02 \mathrm{~m} / \mathrm{s}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:56

Problem 7

Velocity and temperature profiles for laminar flow in a tube of radius $r_{o}=10 \mathrm{~mm}$ have the form
$$
\begin{aligned}
&u(r)=0.1\left[1-\left(r / r_{o}\right)^{2}\right] \\
&T(r)=344.8+75.0\left(r / r_{o}\right)^{2}-18.8\left(r / r_{o}\right)^{4}
\end{aligned}
$$
with units of $\mathrm{m} / \mathrm{s}$ and $\mathrm{K}$, respectively. Determine the corresponding value of the mean (or bulk) temperature, $T_{\text {m }}$, at this axial position.

Hast Aggarwal
Hast Aggarwal
Numerade Educator
01:23

Problem 8

At a particular axial station, velocity and temperature profiles for laminar flow in a parallel plate channel have the form
$$
\begin{aligned}
&u(y)=0.75\left[1-\left(y / y_{o}\right)^{2}\right] \\
&T(y)=5.0+95.66\left(y / y_{o}\right)^{2}-47.83\left(y / y_{o}\right)^{4}
\end{aligned}
$$
with units of $\mathrm{m} / \mathrm{s}$ and ${ }^{\circ} \mathrm{C}$, respectively.
Determine corresponding values of the mean velocity, $u_{m}$, and mean (or bulk) temperature, $T_{m}$. Plot the velocity and temperature distributions. Do your values of $u_{m}$ and $T_{m}$ appear reasonable?

Chai Santi
Chai Santi
Numerade Educator
08:14

Problem 9

In Chapter 1, it was stated that for incompressible liquids, flow work could usually be neglected in the steady-flow energy equation (Equation 1.12d). In the trans-Alaska pipeline, the high viscosity of the oil and long distances cause significant pressure drops, and it is reasonable to question whether flow work would be significant. Consider an $L=100 \mathrm{~km}$ length of pipe of diameter $D=1.2 \mathrm{~m}$, with oil flow rate $\dot{m}=500 \mathrm{~kg} / \mathrm{s}$. The oil properties are $\rho=900 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=2000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=0.765$ $\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}$. Calculate the pressure drop, the flow work, and the temperature rise caused by the flow work.

Sheh Lit Chang
Sheh Lit Chang
University of Washington
09:03

Problem 10

When viscous dissipation is included, Equation $8.48$ (multiplied by $\rho c_{p}$ ) becomes
$$
\rho c_{p} u \frac{\partial T}{\partial x}=\frac{k}{r} \frac{\partial}{\partial r}\left(r \frac{\partial T}{\partial r}\right)+\mu\left(\frac{d u}{d r}\right)^{2}
$$
This problem explores the importance of viscous dissipation. The conditions under consideration are laminar, fully developed flow in a circular pipe, with $u$ given by Equation 8.15.
(a) By integrating the left-hand side over a section of a pipe of length $L$ and radius $r_{o}$, show that this term yields the right-hand side of Equation 8.34.
(b) Integrate the viscous dissipation term over the same volume.
(c) Find the temperature rise caused by viscous dissipation by equating the two terms calculated above. Use the same conditions as in Problem 8.9.

Ajay Singhal
Ajay Singhal
Numerade Educator
06:58

Problem 11

Consider a circular tube of diameter $D$ and length $L$, with a mass flow rate of $\dot{m}$.
(a) For constant heat flux conditions, derive an expression for the ratio of the temperature difference between the tube wall at the tube exit and the inlet temperature, $T_{s}(x=L)-T_{m, i}$, to the total heat transfer rate to the fluid $q$. Express your result in terms of $\dot{m}, L$, the local Nusselt number at the tube exit $N u_{D}(x=L)$, and relevant fluid properties.
(b) Repeat part (a) for constant surface temperature conditions. Express your result in terms of $\dot{m}, L$, the average Nusselt number from the tube inlet to the tube exit $\overline{N u}_{D}$, and relevant fluid properties.

Amany Waheeb
Amany Waheeb
Numerade Educator
04:19

Problem 12

Water enters a tube at $27^{\circ} \mathrm{C}$ with a flow rate of $450 \mathrm{~kg} / \mathrm{h}$. The heat transfer from the tube wall to the fluid is given as $q_{s}^{\prime}(\mathrm{W} / \mathrm{m})=a x$, where the coefficient $a$ is $20 \mathrm{~W} / \mathrm{m}^{2}$ and $x(\mathrm{~m})$ is the axial distance from the tube entrance.
(a) Beginning with a properly defined differential control volume in the tube, derive an expression for the temperature distribution $T_{m}(x)$ of the water.
(b) What is the outlet temperature of the water for a heated section $30 \mathrm{~m}$ long?
(c) Sketch the mean fluid temperature, $T_{m}(x)$, and the tube wall temperature, $T_{s}(x)$, as a function of distance along the tube for fully developed and developing flow conditions.
(d) What value of a uniform wall heat flux, $q_{s}^{\prime \prime}$ (instead of $q_{s}^{\prime}=a x$ ), would provide the same fluid outlet temperature as that determined in part (b)? For this type of heating, sketch the temperature distributions requested in part (c).

Kevin Luu
Kevin Luu
Numerade Educator
06:58

Problem 13

Consider flow in a circular tube. Within the test section length (between 1 and 2 ) a constant heat flux $q_{s}^{\prime \prime}$ is maintained.
(a) For the following two cases, sketch the surface temperature $T_{s}(x)$ and the fluid mean temperature $T_{m}(x)$ as a function of distance along the test section $x$. In case A, flow is hydrodynamically and thermally fully developed. In case B, flow is not developed.
(b) Assuming that the surface flux $q_{s}^{\prime \prime}$ and the inlet mean temperature $T_{m, 1}$ are identical for both cases, will
the exit mean temperature $T_{m, 2}$ for case A be greater than, equal to, or less than $T_{m, 2}$ for case B? Briefly explain why.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:15

Problem 14

Consider a cylindrical nuclear fuel rod of length $L$ and diameter $D$ that is encased in a concentric tube. Pressurized water flows through the annular region between the rod and the tube at a rate $\dot{m}$, and the outer surface of the tube is well insulated. Heat generation occurs within the fuel rod, and the volumetric generation rate is known to vary sinusoidally with distance along the rod. That is, $\dot{q}(x)=\dot{q}_{o} \sin (\pi x / L)$, where $\dot{q}_{o}\left(\mathrm{~W} / \mathrm{m}^{3}\right)$ is a constant. A uniform convection coefficient $h$ may be assumed to exist between the surface of the rod and the water.
(a) Obtain expressions for the local heat flux $q^{\prime \prime}(x)$ and the total heat transfer $q$ from the fuel rod to the water.
(b) Obtain an expression for the variation of the mean temperature $T_{m}(x)$ of the water with distance $x$ along the tube.
(c) Obtain an expression for the variation of the rod surface temperature $T_{s}(x)$ with distance $x$ along the tube. Develop an expression for the $x$-location at which this temperature is maximized.

Narayan Hari
Narayan Hari
Numerade Educator
View

Problem 15

Consider the laminar thermal boundary layer development near the entrance of the tube shown in Figure 8.4. When the hydrodynamic boundary layer is thin relative to the tube diameter, the inviscid flow region has a uniform velocity that is approximately equal to the mean velocity $u_{m}$. Hence the boundary layer development is similar to what would occur for a flat plate.
(a) Beginning with Equation 7.23, derive an expression for the local Nusselt number $N u_{D}$, as a function of the Prandtl number $P r$ and the inverse Graetz number $G z_{D}^{-1}$. Plot the expression using the coordinates shown in Figure 8.10 $a$ for $P r=0.7$.
(b) Beginning with Equation 7.30, derive an expression for the average Nusselt number $\overline{N u}_{D}$, as a function of the Prandtl number $P r$ and the inverse Graetz number $G z_{D}^{-1}$. Compare your results with the Nusselt number for the combined entrance length in the limit of small $x$.

Victor Salazar
Victor Salazar
Numerade Educator
06:58

Problem 16

In a particular application involving fluid flow at a rate $\dot{m}$ through a circular tube of length $L$ and diameter $D$,
the surface heat flux is known to have a sinusoidal variation with $x$, which is of the form $q_{s}^{\prime \prime}(x)=q_{s, m}^{\prime \prime} \sin (\pi x / L)$. The maximum flux, $q_{s, m}^{n}$, is a known constant, and the fluid enters the tube at a known temperature, $T_{m, i}$ Assuming the convection coefficient to be constant, how do the mean temperature of the fluid and the surface temperature vary with $x$ ?

Amany Waheeb
Amany Waheeb
Numerade Educator
01:31

Problem 17

A flat-plate solar collector is used to heat atmospheric air flowing through a rectangular channel. The bottom surface of the channel is well insulated, while the top surface is subjected to a uniform heat flux $q_{o}^{\prime \prime}$, which is due to the net effect of solar radiation absorption and heat exchange between the absorber and cover plates.
(a) Beginning with an appropriate differential control volume, obtain an equation that could be used to determine the mean air temperature $T_{m}(x)$ as a function of distance along the channel. Solve this equation to obtain an expression for the mean temperature of the air leaving the collector.
(b) With air inlet conditions of $\dot{m}=0.1 \mathrm{~kg} / \mathrm{s}$ and $T_{m, i}=40^{\circ} \mathrm{C}$, what is the air outlet temperature if $L=3 \mathrm{~m}, w=1 \mathrm{~m}$, and $q_{o}^{\prime \prime}=700 \mathrm{~W} / \mathrm{m}^{2}$ ? The specific heat of air is $c_{p}=1008 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
04:00

Problem 18

Atmospheric air enters the heated section of a circular tube at a flow rate of $0.005 \mathrm{~kg} / \mathrm{s}$ and a temperature of $20^{\circ} \mathrm{C}$. The tube is of diameter $D=50 \mathrm{~mm}$, and fully developed conditions with $h=25 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$ exist over the entire length of $L=3 \mathrm{~m}$.
(a) For the case of uniform surface heat flux at $q_{s}^{\prime \prime}=1000 \mathrm{~W} / \mathrm{m}^{2}$, determine the total heat transfer rate $q$ and the mean temperature of the air leaving the tube $T_{m \rho^{-}}$What is the value of the surface temperature at the tube inlet $T_{s, i}$ and outlet $T_{s, \rho}$ ? Sketch the axial variation of $T_{s}$ and $T_{m}$. On the same figure, also sketch (qualitatively) the axial variation of $T_{s}$ and $T_{m}$ for the more realistic case in which the local convection coefficient varies with $x$.
(b) If the surface heat flux varies linearly with $x$, such that $q_{s}^{\prime \prime}\left(\mathrm{W} / \mathrm{m}^{2}\right)=500 x(\mathrm{~m})$, what are the values of $q, T_{m, o}, T_{s, j}$, and $T_{s, o}$ ? Sketch the axial variation of $T_{s}$ and $T_{m-}$ On the same figure, also sketch (qualitatively) the axial variation of $T_{s}$ and $T_{m}$ for the more realistic case in which the local convection coefficient varies with $x$.
(c) For the two heating conditions of parts (a) and (b), plot the mean fluid and surface temperatures, $T_{m}(x)$ and $T_{s}(x)$, respectively, as functions of distance along the tube. What effect will a fourfold increase in the convection coefficient have on the temperature distributions?
(d) For each type of heating process, what heat fluxes are required to achieve an air outlet temperature of $125^{\circ} \mathrm{C}$ ? Plot the temperature distributions.

Paul Gabriel
Paul Gabriel
Numerade Educator
03:55

Problem 19

Fluid enters a tube with a flow rate of $0.015 \mathrm{~kg} / \mathrm{s}$ and an inlet temperature of $20^{\circ} \mathrm{C}$. The tube, which has a length of $6 \mathrm{~m}$ and diameter of $15 \mathrm{~mm}$, has a surface temperature of $30^{\circ} \mathrm{C}$.
(a) Determine the heat transfer rate to the fluid if it is water.
(b) Determine the heat transfer rate for the nanofluid of Example 2.2.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:27

Problem 20

Water at $300 \mathrm{~K}$ and a flow rate of $5 \mathrm{~kg} / \mathrm{s}$ enters a black, thin-walled tube, which passes through a large furnace whose walls and air are at a temperature of $700 \mathrm{~K}$. The diameter and length of the tube are $0.25 \mathrm{~m}$ and $8 \mathrm{~m}$, respectively. Convection coefficients associated with water flow through the tube and airflow over the tube are $300 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively.
(a) Write an expression for the linearized radiation coefficient corresponding to radiation exchange between the outer surface of the pipe and the furnace walls. Explain how to calculate this coefficient if the surface temperature of the tube is represented by the arithmetic mean of its inlet and outlet values.
(b) Determine the outlet temperature of the water, $T_{m, o^{\circ}}$

Anand Jangid
Anand Jangid
Numerade Educator
03:08

Problem 21

Slug flow is an idealized tube flow condition for which the velocity is assumed to be uniform over the entire tube cross section. For the case of laminar slug flow with a uniform surface heat flux, determine the form of the fully developed temperature distribution $T(r)$ and the Nusselt number $N u_{D}$.

Chai Santi
Chai Santi
Numerade Educator
01:59

Problem 22

Superimposing a control volume that is differential in $x$ on the tube flow conditions of Figure $8.8$, derive Equation $8.45 a$.

Chai Santi
Chai Santi
Numerade Educator
04:00

Problem 23

An experimental nuclear core simulation apparatus consists of a long thin-walled metallic tube of diameter $D$ and length $L$, which is electrically heated to produce the sinusoidal heat flux distribution
$$
q_{s}^{\prime \prime}(x)=q_{o}^{\prime \prime} \sin \left(\frac{\pi x}{L}\right)
$$
where $x$ is the distance measured from the tube inlet. Fluid at an inlet temperature $T_{m, i}$ flows through the tube at a rate of $\dot{m}$. Assuming the flow is turbulent and fully developed over the entire length of the tube, develop expressions for:
(a) the total rate of heat transfer, $q$, from the tube to the fluid;
(b) the fluid outlet temperature, $T_{m, o} ;$
(c) the axial distribution of the wall temperature, $T_{s}(x)$; and
(d) the magnitude and position of the highest wall temperature.
(e) Consider a $40-\mathrm{mm}$-diameter tube of $4-\mathrm{m}$ length with a sinusoidal heat flux distribution for which $q_{o}^{\prime \prime}=10,000 \mathrm{~W} / \mathrm{m}^{2}$. Fluid passing through the tube has a flow rate of $0.025 \mathrm{~kg} / \mathrm{s}$, a specific heat of $4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, an entrance temperature of $25^{\circ} \mathrm{C}$, and a convection coefficient of $1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Plot the mean fluid and surface temperatures as a function of distance along the tube. Identify important features of the distributions. Explore the effect of $\pm 25 \%$ changes in the convection coefficient and the heat flux on the distributions.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:00

Problem 24

Water at $20^{\circ} \mathrm{C}$ and a flow rate of $0.1 \mathrm{~kg} / \mathrm{s}$ enters a heated, thin-walled tube with a diameter of $15 \mathrm{~mm}$ and length of $2 \mathrm{~m}$. The wall heat flux provided by the heating elements depends on the wall temperature according to the relation
$$
q_{s}^{\prime \prime}(x)=q_{s, o}^{\prime \prime}\left[1+\alpha\left(T_{s}-T_{\mathrm{ref}}\right)\right]
$$
where $q_{s, \rho}^{\prime \prime}=10^{4} \mathrm{~W} / \mathrm{m}^{2}, \alpha=0.2 \mathrm{~K}^{-1}, T_{\text {ref }}=20^{\circ} \mathrm{C}$, and $T_{s}$ is the wall temperature in ${ }^{\circ} \mathrm{C}$. Assume fully developed flow and thermal conditions with a convection coefficient of $3000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) Beginning with a properly defined differential control volume in the tube, derive expressions for the variation of the water, $T_{m}(x)$, and the wall, $T_{s}(x)$,
temperatures as a function of distance from the tube inlet.
(b) Using a numerical integration scheme, calculate and plot the temperature distributions, $T_{m}(x)$ and $T_{s}(x)$, on the same graph. Identify and comment on the main features of the distributions. Hint: The $I H T$ integral function $D E R\left(T_{m}, x\right)$ can be used to perform the integration along the length of the tube.
(c) Calculate the total rate of heat transfer to the water.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:03

Problem 25

Engine oil is heated by flowing through a circular tube of diameter $D=50 \mathrm{~mm}$ and length $L=25 \mathrm{~m}$ and whose surface is maintained at $150^{\circ} \mathrm{C}$.
(a) If the flow rate and inlet temperature of the oil are $0.5 \mathrm{~kg} / \mathrm{s}$ and $20^{\circ} \mathrm{C}$, what is the outlet temperature $T_{m, o}$ ? What is the total heat transfer rate $q$ for the tube?
(b) For flow rates in the range $0.5 \leq \dot{m} \leq 2.0 \mathrm{~kg} / \mathrm{s}$, compute and plot the variations of $T_{m, o}$ and $q$ with $\dot{m}$. For what flow rate(s) are $q$ and $T_{m, \rho}$ maximized? Explain your results.

Penny Riley
Penny Riley
Numerade Educator
04:27

Problem 26

Engine oil flows through a $25-\mathrm{mm}$-diameter tube at a rate of $0.5 \mathrm{~kg} / \mathrm{s}$. The oil enters the tube at a temperature of $25^{\circ} \mathrm{C}$, while the tube surface temperature is maintained at $100^{\circ} \mathrm{C}$.
(a) Determine the oil outlet temperature for a $5-\mathrm{m}$ and for a 100 -m long tube. For each case, compare the log mean temperature difference to the arithmetic mean temperature difference.
(b) For $5 \leq L \leq 100 \mathrm{~m}$, compute and plot the average Nusselt number $\overline{N u}_{D}$ and the oil outlet temperature as a function of $L$.

Alexander Allen
Alexander Allen
Numerade Educator
09:01

Problem 27

In the final stages of production, a pharmaceutical is sterilized by heating it from 25 to $75^{\circ} \mathrm{C}$ as it moves at $0.2 \mathrm{~m} / \mathrm{s}$ through a straight thin-walled stainless steel tube of $12.7=\mathrm{mm}$ diameter. A uniform heat flux is maintained by an electric resistance heater wrapped around the outer surface of the tube. If the tube is $10 \mathrm{~m}$ long, what is the required heat flux? If fluid enters the tube with a fully developed velocity profile and a uniform temperature profile, what is the surface temperature at the tube exit and at a distance of $0.5 \mathrm{~m}$ from the entrance? Fluid properties may be approximated as $\rho=$ $1000 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, m=2 \times 10^{-3} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}$, $k=0.8 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $P r=10$.

Averell Hause
Averell Hause
Carnegie Mellon University
01:31

Problem 28

An oil preheater consists of a single tube of $10-\mathrm{mm}$ diameter and $5-\mathrm{m}$ length, with its surface maintained at $175^{\circ} \mathrm{C}$ by swirling combustion gases. The engine oil (new) enters at $75^{\circ} \mathrm{C}$. What flow rate must be supplied to maintain an oil outlet temperature of $100^{\circ} \mathrm{C}$ ? What is the corresponding heat transfer rate?

Naman Kumar
Naman Kumar
Numerade Educator
03:24

Problem 29

Engine oil flows at a rate of $1 \mathrm{~kg} / \mathrm{s}$ through a $5-\mathrm{mm}-$ diameter straight tube. The oil has an inlet temperature of $45^{\circ} \mathrm{C}$ and it is desired to heat the oil to a mean temperature of $80^{\circ} \mathrm{C}$ at the exit of the tube. The surface of the tube is maintained at $150^{\circ} \mathrm{C}$. Determine the required length of the tube. Hint: Calculate the Reynolds numbers at the entrance and exit of the tube before proceeding with your analysis.

Narayan Hari
Narayan Hari
Numerade Educator
03:04

Problem 30

Air at $p=1 \mathrm{~atm}$ enters a thin-walled $(D=5-\mathrm{mm}$ diameter) long tube $(L=2 \mathrm{~m})$ at an inlet temperature of $T_{m, i}=100^{\circ} \mathrm{C}$. A constant heat flux is applied to the air from the tube surface. The air mass flow rate is $\dot{m}=135 \times 10^{-6} \mathrm{~kg} / \mathrm{s}$.
(a) If the tube surface temperature at the exit is $T_{s, o}=160^{\circ} \mathrm{C}$, determine the heat rate entering the tube. Evaluate properties at $T=400 \mathrm{~K}$.
(b) If the tube length of part (a) were reduced to $L=0.2 \mathrm{~m}$, how would flow conditions at the tube exit be affected? Would the value of the heat transfer coefficient at the tube exit be greater than, equal to, or smaller than the heat transfer coefficient for part (a)?
(c) If the flow rate of part (a) were increased by a factor of 10 , would there be a difference in flow conditions at the tube exit? Would the value of the heat transfer coefficient at the tube exit be greater than, equal to, or smaller than the heat transfer coefficient for part (a)?

Anand Jangid
Anand Jangid
Numerade Educator
02:27

Problem 31

To cool a summer home without using a vaporcompression refrigeration cycle, air is routed through a plastic pipe $\left(k=0.15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, D_{i}=0.15 \mathrm{~m}, D_{o}=\right.$ $0.17 \mathrm{~m}$ ) that is submerged in an adjoining body of water. The water temperature is nominally at $T_{\infty}=17^{\circ} \mathrm{C}$, and a convection coefficient of $h_{o}=1500 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$ is maintained at the outer surface of the pipe.
If air from the home enters the pipe at a temperature of $T_{m, i}=29^{\circ} \mathrm{C}$ and a volumetric flow rate of $\dot{\forall}_{i}=0.025 \mathrm{~m}^{3} / \mathrm{s}$,
what pipe length $L$ is needed to provide a discharge temperature of $T_{\text {m?o }}=21^{\circ} \mathrm{C}$ ? What is the fan power required to move the air through this length of pipe if its inner surface is smooth?

Anand Jangid
Anand Jangid
Numerade Educator
09:53

Problem 32

Batch processes are often used in chemical and pharmaceutical operations to achieve a desired chemical composition for the final product. Related heat transfer processes are typically transient, involving a liquid of fixed volume that may be heated from room temperature to a desired process temperature, or cooled from the process temperature to room temperature. Consider a batch process for which a pharmaceutical (the cold fluid, $c$ ) is poured into an insulated, highly agitated vessel (a stirred reactor) and heated by passing a hot fluid ( $h$ ) through a submerged heat exchanger coil of thinwalled tubing and surface area $A_{s}$. The flow rate, $\dot{m}_{h}$, mean inlet temperature, $T_{h i}$, and specific heat, $c_{p, h}$, of the hot fluid are known, as are the initial temperature, $T_{c, i}<T_{h, i}$, the volume, $V_{c}$, mass density, $\rho_{c}$, and specific heat, $c_{v, c}$, of the pharmaceutical. Heat transfer from the hot fluid to the pharmaceutical is governed by an overall heat transfer coefficient $U$.
(a) Starting from basic principles, derive expressions that can be used to determine the variation of $T_{c}$ and $T_{h, o}$ with time during the heating process. Hint: Two equations may be written for the rate of heat transfer, $q(t)$, to the pharmaceutical, one based on the logmean temperature difference and the other on an energy balance for flow of the hot fluid through the tube. Equate these expressions to determine $T_{\text {ho }}(t)$ as a function of $T_{c}(t)$ and prescribed parameters. Use the expression for $T_{h, o}(t)$ and the energy balance for flow through the tube with an energy balance for a control volume containing the pharmaceutical to obtain an expression for $T_{c}(t)$.
(b) Consider a pharmaceutical of volume $V_{c}=1 \mathrm{~m}^{3}$, density $\rho_{c}=1100 \mathrm{~kg} / \mathrm{m}^{3}$, specific heat $c_{v, c}=2000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and an initial temperature of $T_{c i}=25^{\circ} \mathrm{C}$. A coiled tube of length $L=40 \mathrm{~m}$, diameter $D=50 \mathrm{~mm}$, and coil diameter $C=500 \mathrm{~mm}$ is submerged in the vessel, and hot fluid enters the tubing at $T_{h, i}=200^{\circ} \mathrm{C}$ and $\dot{m}_{h}=2.4 \mathrm{~kg} / \mathrm{s}$. The convection coefficient at the outer surface of the tubing may be approximated as $h_{o}=1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, and the fluid properties are $c_{p, h}=2500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu_{\mathrm{h}}=0.002 \mathrm{~N}+\mathrm{s} / \mathrm{m}^{2}, k_{h}=$ $0.260 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $P r_{\mathrm{h}}=20$. For the foregoing conditions, compute and plot the pharmaceutical temperature $T_{c}$ and the outlet temperature $T_{h o o}$ as a function of time over the range $0 \leq t \leq 3600 \mathrm{~s}$. How long does it take to reach a batch temperature of $T_{c}=160^{\circ} \mathrm{C}$ ? The process operator may control the heating time by varying $\dot{m}_{h b}$. For $1 \leq \dot{m}_{h} \leq 5$ $\mathrm{kg} / \mathrm{s}$, explore the effect of the flow rate on the time $t_{c}$ required to reach a value of $T_{c}=160^{\circ} \mathrm{C}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:44

Problem 33

The evaporator section of a heat pump is installed in a large tank of water, which is used as a heat source during the winter. As energy is extracted from the water, it begins to freeze, creating an ice/water bath at $0^{\circ} \mathrm{C}$, which may be used for air conditioning during the summer. Consider summer cooling conditions for which air is passed through an array of copper tubes, each of inside diameter $D=50 \mathrm{~mm}$, submerged in the bath.
(a) If air enters each tube at a mean temperature of $T_{m, i}=24^{\circ} \mathrm{C}$ and a flow rate of $\dot{m}=0.01 \mathrm{~kg} / \mathrm{s}$, what tube length $L$ is needed to provide an exit temperature of $T_{m \rho}=14^{\circ} \mathrm{C}$ ? With 10 tubes passing through a tank of total volume $V=10 \mathrm{~m}^{3}$, which initially contains $80 \%$ ice by volume, how long would it take to completely melt the ice? The density and latent heat of fusion of ice are $920 \mathrm{~kg} / \mathrm{m}^{3}$ and $3.34 \times 10^{5} \mathrm{~J} / \mathrm{kg}$, respectively.
(b) The air outlet temperature may be regulated by adjusting the tube mass flow rate. For the tube length determined in part (a), compute and plot $T_{m \rho}$ as a function of $\dot{m}$ for $0.005 \leq \dot{m} \leq 0.05 \mathrm{~kg} / \mathrm{s}$. If the dwelling cooled by this system requires approximately $0.05 \mathrm{~kg} / \mathrm{s}$ of air at $16^{\circ} \mathrm{C}$, what design and operating conditions should be prescribed for the system?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
View

Problem 34

A liquid food product is processed in a continuousflow sterilizer. The liquid enters the sterilizer at a temperature and flow rate of $T_{m, i, h}=20^{\circ} \mathrm{C}, \dot{m}=1 \mathrm{~kg} / \mathrm{s}$, respectively. A time-at-temperature constraint requires that the product be held at a mean temperature of $T_{m}=90^{\circ} \mathrm{C}$ for $10 \mathrm{~s}$ to kill bacteria, while a second constraint is that the local product temperature cannot exceed $T_{\max }=230^{\circ} \mathrm{C}$ in order to preserve a pleasing taste. The sterilizer consists of an upstream, $L_{k}=5 \mathrm{~m}$ heating section characterized by a uniform heat flux,
an intermediate insulated sterilizing section, and a downstream cooling section of length $L_{c}=10 \mathrm{~m}$. The cooling section is composed of an uninsulated tube exposed to a quiescent environment at $T_{\infty}=20^{\circ} \mathrm{C}$. The thin-walled tubing is of diameter $D=40 \mathrm{~mm}$. Food properties are similar to those of liquid water at $T=330 \mathrm{~K}$.
(a) What heat flux is required in the heating section to ensure a maximum mean product temperature of $T_{m}=90^{\circ} \mathrm{C}$ ?
(b) Determine the location and value of the maximum local product temperature. Is the second constraint satisfied?
(c) Determine the minimum length of the sterilizing section needed to satisfy the time-at-temperature constraint.
(d) Sketch the axial distribution of the mean, surface, and centerline temperatures from the inlet of the heating section to the outlet of the cooling section.

Lainey Roebuck
Lainey Roebuck
Numerade Educator
08:04

Problem 35

Water flowing at $2 \mathrm{~kg} / \mathrm{s}$ through a $40-\mathrm{mm}$-diameter tube is to be heated from 25 to $75^{\circ} \mathrm{C}$ by maintaining the tube surface temperature at $100^{\circ} \mathrm{C}$.
(a) What is the required tube length for these conditions?
(b) To design a water heating system, we wish to consider using tube diameters in the range from 30 to $50 \mathrm{~mm}$. What are the required tube lengths for water flow rates of 1,2 , and $3 \mathrm{~kg} / \mathrm{s}$ ? Represent this design information graphically.
(c) Plot the pressure gradient as a function of tube diameter for the three flow rates. Assume the tube wall is smooth.

Ajay Singhal
Ajay Singhal
Numerade Educator
00:53

Problem 36

Consider the conditions associated with the hot water pipe of Problem $7.56$, but now account for the convection resistance associated with water flow at a mean velocity of $u_{m}=0.5 \mathrm{~m} / \mathrm{s}$ in the pipe. What is the corresponding daily cost of heat loss per meter of the uninsulated pipe?

Mayukh Banik
Mayukh Banik
Numerade Educator
01:36

Problem 37

A thick-walled, stainless steel (AISI 316) pipe of inside and outside diameters $D_{i}=20 \mathrm{~mm}$ and $D_{o}=40 \mathrm{~mm}$ is heated electrically to provide a uniform heat generation rate of $\dot{q}=10^{6} \mathrm{~W} / \mathrm{m}^{3}$. The outer surface of the pipe is insulated, while water flows through the pipe at a rate of $\dot{m}=0.1 \mathrm{~kg} / \mathrm{s}$.

Nick Johnson
Nick Johnson
Numerade Educator
04:00

Problem 38

An air heater for an industrial application consists of an insulated, concentric tube annulus, for which air flows through a thin-walled inner tube. Saturated steam flows through the outer annulus, and condensation of the steam maintains a uniform temperature $T_{s}$ on the tube surface.
Consider conditions for which air enters a 50 -mmdiameter tube at a pressure of $5 \mathrm{~atm}$, a temperature of $T_{m, i}=17^{\circ} \mathrm{C}$, and a flow rate of $\dot{m}=0.03 \mathrm{~kg} / \mathrm{s}$, while saturated steam at $2.455$ bars condenses on the outer surface of the tube. If the length of the annulus is $L=5 \mathrm{~m}$, what are the outlet temperature $T_{m, o}$ and pressure $p_{o}$ of the air? What is the mass rate at which condensate leaves the annulus?

Mohammad Mehran
Mohammad Mehran
Numerade Educator
03:29

Problem 39

Consider fully developed conditions in a circular tube with constant surface temperature $T_{s}<T_{m^{*}}$ Determine whether a small- or large-diameter tube is more effective in minimizing heat loss from the flowing fluid characterized by a mass flow rate of $\dot{m}$. Consider both laminar and turbulent conditions.

Ma Ednelyn Lim
Ma Ednelyn Lim
Numerade Educator
02:51

Problem 40

Consider the encased pipe of Problem $4.29$, but now allow for the difference between the mean temperature of the fluid, which changes along the pipe length, and that of the pipe.
(a) For the prescribed values of $k, D, w, h$, and $T_{\infty}$ and a pipe of length $L=100 \mathrm{~m}$, what is the outlet temperature $T_{m, o}$ of water that enters the pipe at a temperature of $T_{m, i}=90^{\circ} \mathrm{C}$ and a flow rate of $\dot{m}=2 \mathrm{~kg} / \mathrm{s}$ ?
(b) What is the pressure drop of the water and the corresponding pump power requirement?
(c) Subject to the constraint that the width of the duct is fixed at $w=0.30 \mathrm{~m}$, explore the effects of the flow rate and the pipe diameter on the outlet temperature.

Nick Johnson
Nick Johnson
Numerade Educator
06:58

Problem 41

Water flows through a thick-walled tube with an inner diameter of $12 \mathrm{~mm}$ and a length of $8 \mathrm{~m}$. The tube is immersed in a well-stirred, hot reaction tank maintained at $85^{\circ} \mathrm{C}$, and the conduction resistance of the tube wall (based on the inner surface area) is $R_{\text {cd }}^{\prime \prime}=0.002 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. The inlet temperature of the process fluid is $T_{m, i}=20^{\circ} \mathrm{C}$, and the flow rate is $33 \mathrm{~kg} / \mathrm{h}$.
(a) Estimate the outlet temperature of the process fluid, $T_{\text {mo. }}$ Assume, and then justify, fully developed flow and thermal conditions within the tube.
(b) Do you expect $T_{m, o}$ to increase or decrease if combined thermal and hydrodynamic entry conditions exist within the tube? Estimate the outlet temperature of the water for this condition.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:08

Problem 42

Atmospheric air enters a 10-m-long, 150 -mm-diameter uninsulated heating duct at $60^{\circ} \mathrm{C}$ and $0.04 \mathrm{~kg} / \mathrm{s}$. The duct surface temperature is approximately constant at $T_{s}=15^{\circ} \mathrm{C}$.
(a) What are the outlet air temperature, the heat rate $q$, and pressure drop $\Delta p$ for these conditions?
(b) To illustrate the tradeoff between heat transfer rate and pressure drop considerations, calculate $q$ and $\Delta p$ for diameters in the range from $0.1$ to $0.2 \mathrm{~m}$. In your analysis, maintain the total surface area, $A_{s}=\pi D L$, at the value computed for part (a). Plot $q, \Delta p$, and $L$ as a function of the duct diameter.

Chai Santi
Chai Santi
Numerade Educator
02:25

Problem 43

\mathrm{NaK}(45 \% / 55 \%)$, which is an alloy of sodium and potassium, is used to cool fast neutron nuclear reactors. The NaK flows at a rate of $m=1 \mathrm{~kg} / \mathrm{s}$ through a $D=50-\mathrm{mm}$ diameter tube that has a surface temperature of $T_{s}=450 \mathrm{~K}$. The NaK enters the tube at $T_{m, i}=332 \mathrm{~K}$ and exits at an outlet temperature of $T_{m, o}=400 \mathrm{~K}$. Determine the tube length $L$ and the local convective heat flux at the tube exit.

Anand Jangid
Anand Jangid
Numerade Educator
07:41

Problem 45

Liquid mercury at $0.5 \mathrm{~kg} / \mathrm{s}$ is to be heated from 300 to $400 \mathrm{~K}$ by passing it through a $50-\mathrm{mm}$-diameter tube whose surface is maintained at $450 \mathrm{~K}$. Calculate the required tube length by using an appropriate liquid metal convection heat transfer correlation. Compare your result with that which would have been obtained by using a correlation appropriate for $P r \geq 0.7$.

Adriano Chikande
Adriano Chikande
Numerade Educator
03:04

Problem 46

The surface of a 50 -mm-diameter, thin-walled tube is maintained at $100^{\circ} \mathrm{C}$. In one case air is in cross flow over the tube with a temperature of $25^{\circ} \mathrm{C}$ and a velocity of $30 \mathrm{~m} / \mathrm{s}$. In another case air is in fully developed flow through the tube with a temperature of $25^{\circ} \mathrm{C}$ and a mean velocity of $30 \mathrm{~m} / \mathrm{s}$. Compare the heat flux from the tube to the air for the two cases.

Anand Jangid
Anand Jangid
Numerade Educator
09:01

Problem 47

Consider a horizontal, thin-walled circular tube of diameter $D=0.025 \mathrm{~m}$ submerged in a container of $n$ octadecane (paraffin), which is used to store thermal energy. As hot water flows through the tube, heat is transferred to the paraffin, converting it from the solid to liquid state at the phase change temperature of $T_{z}=27.4^{\circ} \mathrm{C}$. The latent heat of fusion and density of paraffin are $h_{\text {ff }}=244 \mathrm{~kJ} / \mathrm{kg}$ and $\rho=770 \mathrm{~kg} / \mathrm{m}^{3}$, respectively, and thermophysical properties of the water may be taken as $c_{p}=4.185 \mathrm{~kJ} / \mathrm{kg} \cdot \mathrm{K}, k=0.653 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\mu=467 \times 10^{-6} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}$, and $\operatorname{Pr}=2.99$
(a) Assuming the tube surface to have a uniform temperature corresponding to that of the phase change, determine the water outlet temperature and total heat transfer rate for a water flow rate of $0.1 \mathrm{~kg} / \mathrm{s}$ and an inlet temperature of $60^{\circ} \mathrm{C}$. If $H=W=0.25 \mathrm{~m}$, how long would it take to completely liquefy the paraffin, from an initial state for which all the paraffin is solid and at $27.4^{\circ} \mathrm{C}$ ?
(b) The liquefaction process can be accelerated by increasing the flow rate of the water. Compute and plot the heat rate and outlet temperature as a function of flow rate for $0.1 \leq \dot{m} \leq 0.5 \mathrm{~kg} / \mathrm{s}$. How long would it take to melt the paraffin for $\dot{m}=0.5 \mathrm{~kg} / \mathrm{s}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
03:41

Problem 48

Consider pressurized liquid water flowing at $\dot{m}=0.1 \mathrm{~kg} / \mathrm{s}$ in a circular tube of diameter $D=0.1 \mathrm{~m}$ and length $L=6 \mathrm{~m}$.
(a) If the water enters at $T_{m, i}=500 \mathrm{~K}$ and the surface temperature of the tube is $T_{s}=510 \mathrm{~K}$, determine the water outlet temperature $T_{\text {m,o. }}$.
(b) If the water enters at $T_{m, i}=300 \mathrm{~K}$ and the surface temperature of the tube is $T_{s}=310 \mathrm{~K}$, determine the water outlet temperature $T_{\text {m, } \sigma}$.
(c) If the water enters at $T_{m, i}=300 \mathrm{~K}$ and the surface temperature of the tube is $T_{s}=647 \mathrm{~K}$, discuss whether the flow is laminar or turbulent.

Kajal Gautam
Kajal Gautam
Numerade Educator
01:30

Problem 49

Cooling water flows through the $25.4-\mathrm{mm}$-diameter thin-walled tubes of a steam condenser at $1 \mathrm{~m} / \mathrm{s}$, and a surface temperature of $350 \mathrm{~K}$ is maintained by the condensing steam. The water inlet temperature is $290 \mathrm{~K}$, and the tubes are $5 \mathrm{~m}$ long.
(a) What is the water outlet temperature? Evaluate water properties at an assumed average mean temperature, $\bar{T}_{\mathrm{m}}=300 \mathrm{~K}$.
(b) Was the assumed value for $\bar{T}_{m}$ reasonable? If not, repeat the calculation using properties evaluated at a more appropriate temperature.
(c) A range of tube lengths from 4 to $7 \mathrm{~m}$ is available to the engineer designing this condenser. Generate a plot to show what coolant mean velocities are possible if the water outlet temperature is to remain at the value found for part (b). All other conditions remain the same.

Jincy M Saji
Jincy M Saji
Numerade Educator
16:41

Problem 50

The air passage for cooling a gas turbine vane can be approximated as a tube of $3-\mathrm{mm}$ diameter and $75-\mathrm{mm}$ length. The operating temperature of the vane is $650^{\circ} \mathrm{C}$, and air enters the tube at $427^{\circ} \mathrm{C}$.
(a) For an airflow rate of $0.18 \mathrm{~kg} / \mathrm{h}$, calculate the air outlet temperature and the heat removed from the vane.
(b) Generate a plot of the air outlet temperature as a function of flow rate for $0.1 \leq \dot{m} \leq 0.6 \mathrm{~kg} / \mathrm{h}$. Compare this result with those for vanes having 2 - and 4-mm-diameter tubes, with all other conditions remaining the same.

Gordon Ayadju
Gordon Ayadju
Numerade Educator
01:31

Problem 51

The core of a high-temperature, gas-cooled nuclear reactor has coolant tubes of $20-\mathrm{mm}$ diameter and $780-\mathrm{mm}$ length. Helium enters at $600 \mathrm{~K}$ and exits at $1000 \mathrm{~K}$ when the flow rate is $8 \times 10^{-3} \mathrm{~kg} / \mathrm{s}$ per tube.
(a) Determine the uniform tube wall surface temperature for these conditions.
(b) If the coolant gas is air, determine the required flow rate if the heat removal rate and tube wall surface temperature remain the same. What is the outlet temperature of the air?

Simran Hiranandani
Simran Hiranandani
Numerade Educator
03:31

Problem 52

Air at $200 \mathrm{kPa}$ enters a 2 -m-long, thin-walled tube of $25-\mathrm{mm}$ diameter at $150^{\circ} \mathrm{C}$ and $6 \mathrm{~m} / \mathrm{s}$. Steam at 20 bars condenses on the outer surface.
(a) Determine the outlet temperature and pressure drop of the air, as well as the rate of heat transfer to the air.
(b) Calculate the parameters of part (a) if the pressure of the air is doubled.

James Kiss
James Kiss
Numerade Educator
01:30

Problem 53

Heated air required for a food-drying process is generated by passing ambient air at $20^{\circ} \mathrm{C}$ through long, circular tubes $(D=50 \mathrm{~mm}, L=5 \mathrm{~m})$ housed in a steam condenser. Saturated steam at atmospheric pressure condenses on the outer surface of the tubes, maintaining a uniform surface temperature of $100^{\circ} \mathrm{C}$.
(a) If an airflow rate of $0.01 \mathrm{~kg} / \mathrm{s}$ is maintained in each tube, determine the air outlet temperature $T_{m, o}$ and the total heat rate $q$ for the tube.
(b) The air outlet temperature may be controlled by adjusting the tube mass flow rate. Compute and plot $T_{m \rho}$ as a function of $\dot{m}$ for $0.005 \leq \dot{m} \leq$ $0.050 \mathrm{~kg} / \mathrm{s}$. If a particular drying process requires approximately $1 \mathrm{~kg} / \mathrm{s}$ of air at $75^{\circ} \mathrm{C}$, what design and operating conditions should be prescribed for the air heater, subject to the constraint that the tube diameter and length be fixed at $50 \mathrm{~mm}$ and $5 \mathrm{~m}$, respectively?

Jincy M Saji
Jincy M Saji
Numerade Educator
04:00

Problem 54

Consider laminar flow of a fluid with $P r=4$ that undergoes a combined entrance process within a constant surface temperature tube of length $L<x_{\mathrm{fd}, t}$ with a flow rate of $\dot{m}$. An engineer suggests that the total heat transfer rate can be improved if the tube is divided into $N$ shorter tubes, each of length $L_{N}=L N$ with a flow rate of $\dot{m} / N$. Determine an expression for the ratio of the heat transfer coefficient averaged over the $N$ tubes, each experiencing a combined entrance process, to the heat transfer coefficient averaged over the single tube, $\bar{h}_{D, N} / \bar{h}_{D, 1}$
supplied at an inlet temperature $T_{m, i}$ and a total mass flow rate $\dot{m}$ (for the entire heat $\operatorname{sink}$ ).
(a) Assuming that $q_{c}^{\prime \prime}$ is dispersed in the heat sink such that a uniform heat flux $q_{s}^{\prime \prime}$ is maintained at the surface of each channel, obtain expressions for the longitudinal distributions of the mean fluid, $T_{m}(x)$, and surface, $T_{s}(x)$, temperatures in each channel. Assume laminar, fully developed flow throughout each channel, and express your results in terms of $\dot{m}, q_{c}^{\prime \prime}, C_{1}, D$, and/or $L$, as well as appropriate thermophysical properties.
(b) For $L=12 \mathrm{~mm}, D=1 \mathrm{~mm}, C_{1}=2, q_{k}^{\prime \prime}=20 \mathrm{~W} / \mathrm{cm}^{2}$, $\dot{m}=0.010 \mathrm{~kg} / \mathrm{s}$, and $T_{m, i}=290 \mathrm{~K}$, compute and plot the temperature distributions $T_{m}(x)$ and $T_{s}(x)$.
(c) A common objective in designing such heat sinks is to maximize $q_{c}^{\prime \prime}$ while maintaining the heat sink at an acceptable temperature. Subject to prescribed values of $L=12 \mathrm{~mm}$ and $T_{m, i}=290 \mathrm{~K}$ and the constraint that $T_{s, \max } \leq 50^{\circ} \mathrm{C}$, explore the effect on $q_{c}^{\prime \prime}$ of variations in heat sink design and operating conditions.

Paul Gabriel
Paul Gabriel
Numerade Educator
02:08

Problem 56

One way to cool chips mounted on the circuit boards of a computer is to encapsulate the boards in metal frames that provide efficient pathways for conduction to supporting cold plates. Heat generated by the chips is then dissipated by transfer to water flowing through passages drilled in the plates. Because the plates are made from a metal of large thermal conductivity (typically aluminium or copper), they may be assumed to be at a temperature, $T_{s, c p^{-}}$
(a) Consider circuit boards attached to cold plates of height $H=750 \mathrm{~mm}$ and width $L=600 \mathrm{~mm}$, each with $N=10$ holes of diameter $D=10 \mathrm{~mm}$. If operating conditions maintain plate temperatures of $T_{\text {s.tp }}=32^{\circ} \mathrm{C}$ with water flow at $\dot{m}_{1}=0.2 \mathrm{~kg} / \mathrm{s}$ per passage and $T_{m, i}=7^{\circ} \mathrm{C}$, how much heat may be dissipated by the circuit boards?
(b) To enhance cooling, thereby allowing increased power generation without an attendant increase in system temperatures, a hybrid cooling scheme may be used. The scheme involves forced airflow over the encapsulated circuit boards, as well as water flow through the cold plates. Consider conditions for which $N_{\mathrm{cb}}=10$ circuit boards of width $W=350 \mathrm{~mm}$ are attached to the cold plates and their average surface temperature is $T_{s, \text { do }}=47^{\circ} \mathrm{C}$ when $T_{s, \text { ep }}=32^{\circ} \mathrm{C}$. If air is in parallel flow over the plates with $u_{\infty}=10 \mathrm{~m} / \mathrm{s}$ and $T_{\infty}=7^{\circ} \mathrm{C}$, how much of the heat generated by the circuit boards is transferred to the air?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:49

Problem 57

Refrigerant- $134 \mathrm{a}$ is being transported at $0.1 \mathrm{~kg} / \mathrm{s}$ through a Teflon tube of inside diameter $D_{i}=25 \mathrm{~mm}$ and outside diameter $D_{o}=28 \mathrm{~mm}$, while atmospheric air at $V=25 \mathrm{~m} / \mathrm{s}$ and $300 \mathrm{~K}$ is in cross flow over the tube. What is the heat transfer per unit length of tube to Refrigerant-134a at $240 \mathrm{~K}$ ?

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
03:55

Problem 58

Oil at $150^{\circ} \mathrm{C}$ flows slowly through a long, thin-walled pipe of $30-\mathrm{mm}$ inner diameter. The pipe is suspended in a room for which the air temperature is $20^{\circ} \mathrm{C}$ and the convection coefficient at the outer tube surface is $11 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Estimate the heat loss per unit length of tube.
8.59 Exhaust gases from a wire processing oven are discharged into a tall stack, and the gas and stack surface temperatures at the outlet of the stack must be estimated. Knowledge of the outlet gas temperature $T_{m, o}$ is useful

Eric Mockensturm
Eric Mockensturm
Numerade Educator
05:00

Problem 59

Exhaust gases from a wire processing oven are discharged into a tall stack, and the gas and stack surface temperatures at the outlet of the stack must be estimated. Knowledge of the outlet gas temperature $T_{m, o}$ is useful
for predicting the dispersion of effluents in the thermal plume, while knowledge of the outlet stack surface temperature $T_{s, a}$ indicates whether condensation of the gas products will occur. The thin-walled, cylindrical stack is $0.5 \mathrm{~m}$ in diameter and $6.0 \mathrm{~m}$ high. The exhaust gas flow rate is $0.5 \mathrm{~kg} / \mathrm{s}$, and the inlet temperature is $600^{\circ} \mathrm{C}$.
(a) Consider conditions for which the ambient air temperature and wind velocity are $4^{\circ} \mathrm{C}$ and $5 \mathrm{~m} / \mathrm{s}$, respectively. Approximating the thermophysical properties of the gas as those of atmospheric air, estimate the outlet gas and stack surface temperatures for the given conditions.
(b) The gas outlet temperature is sensitive to variations in the ambient air temperature and wind velocity. For $T_{\infty}=-25^{\circ} \mathrm{C}, 5^{\circ} \mathrm{C}$, and $35^{\circ} \mathrm{C}$, compute and plot the gas outlet temperature as a function of wind velocity for $2 \leq V \leq 10 \mathrm{~m} / \mathrm{s}$.

Niamat Khuda
Niamat Khuda
Numerade Educator
03:41

Problem 60

A hot fluid passes through a thin-walled tube of $10-\mathrm{mm}$ diameter and 1-m length, and a coolant at $T_{\infty}=25^{\circ} \mathrm{C}$ is in cross flow over the tube. When the flow rate is $\dot{m}=18 \mathrm{~kg} / \mathrm{h}$ and the inlet temperature is $T_{m, i}=85^{\circ} \mathrm{C}$, the outlet temperature is $T_{m \rho}=78^{\circ} \mathrm{C}$.
Assuming fully developed flow and thermal conditions in the tube, determine the outlet temperature, $T_{m, a}$ if the flow rate is increased by a factor of 2 . That is, $\dot{m}=36 \mathrm{~kg} / \mathrm{h}$, with all other conditions the same. The thermophysical properties of the hot fluid are $\rho=$ $1079 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=2637 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=0.0034 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, and $k=0.261 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

Kajal Gautam
Kajal Gautam
Numerade Educator
04:00

Problem 61

Consider a thin-walled tube of $10-\mathrm{mm}$ diameter and 2 -m length. Water enters the tube from a large reservoir at $\dot{m}=0.2 \mathrm{~kg} / \mathrm{s}$ and $T_{m, i}=47^{\circ} \mathrm{C}$.
(a) If the tube surface is maintained at a uniform temperature of $27^{\circ} \mathrm{C}$, what is the outlet temperature of the water, $T_{m, o}$ ? To obtain the properties of water, assume an average mean temperature of $\bar{T}_{m}=300 \mathrm{~K}$.
(b) What is the exit temperature of the water if it is heated by passing air at $T_{\infty}=100^{\circ} \mathrm{C}$ and $V=10 \mathrm{~m} / \mathrm{s}$ in cross flow over the tube? The properties of air may be evaluated at an assumed film temperature of $T_{f}=350 \mathrm{~K}$.
(c) In the foregoing calculations, were the assumed values of $\bar{T}_{m}$ and $T_{f}$ appropriate? If not, use properly evaluated properties and recompute $T_{m, o}$ for the conditions of part (b).

Paul Gabriel
Paul Gabriel
Numerade Educator
02:31

Problem 62

8.62 through a thin-walled tube of diameter $D=50 \mathrm{~mm}$ and maintaining a coolant at $T_{\infty}=15^{\circ} \mathrm{C}$ in cross flow over the tube.
(a) What is the required tube length if the coolant is air and its velocity is $V=20 \mathrm{~m} / \mathrm{s}$ ?
(b) What is the tube length if the coolant is water and $V=2 \mathrm{~m} / \mathrm{s}$ ?

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
04:00

Problem 63

The problem of heat losses from a fluid moving through a buried pipeline has received considerable attention. Practical applications include the trans-Alaska pipeline, as well as power plant steam and water distribution lines. Consider a steel pipe of diameter $D$ that is used to transport oil flowing at a rate $\dot{m}_{o}$ through a cold region. The pipe is covered with a layer of insulation of thickness $t$ and thermal conductivity $k_{i}$ and is buried in soil to a depth $z$ (distance from the soil surface to the pipe centerline). Each section of pipe is of length $L$ and extends between pumping stations in which the oil is heated to ensure low viscosity and hence low pump power requirements. The temperature of the oil entering the pipe from a pumping station and the temperature of the ground above the pipe are designated as $T_{m, i}$ and $T_{s}$, respectively, and are known.
Consider conditions for which the oil (o) properties may be approximated as $\rho_{o}=900 \mathrm{~kg} / \mathrm{m}^{3}, c_{p, o}=2000$ $\mathrm{J} / \mathrm{kg} \cdot \mathrm{K}, \quad \nu_{o}=8.5 \times 10^{-4} \mathrm{~m}^{2} / \mathrm{s}, \quad k_{o}=0.140 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$,
$P r_{o}=10^{4}$; the oil flow rate is $\dot{m}_{o}=500 \mathrm{~kg} / \mathrm{s}$; and the pipe diameter is $1.2 \mathrm{~m}$.
(a) Expressing your results in terms of $D, L, z, t, \dot{m}_{o}$, $T_{m, i}$ and $T_{s}$, as well as the appropriate oil $(o)$, insulation ( $i$ ), and soil $(s)$ properties, obtain all the expressions needed to estimate the temperature $T_{m \rho o}$ of the oil leaving the pipe.
(b) If $T_{s}=-40^{\circ} \mathrm{C}, T_{m, i}=120^{\circ} \mathrm{C}, t=0.15 \mathrm{~m}, k_{i}=0.05$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, k_{s}=0.5 \mathrm{~W} / \mathrm{m}+\mathrm{K}, z=3 \mathrm{~m}$, and $L=100 \mathrm{~km}$, what is the value of $T_{m \rho}$ ? What is the total rate of heat transfer $q$ from a section of the pipeline?
(c) The operations manager wants to know the tradeoff between the burial depth of the pipe and insulation thickness on the heat loss from the pipe. Develop a graphical representation of this design information.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:05

Problem 64

To maintain pump power requirements per unit flow rate below an acceptable level, operation of the oil pipeline of Problem $8.63$ is subject to the constraint that the oil exit temperature $T_{m, \rho}$ exceed $110^{\circ} \mathrm{C}$. For the values of $T_{m, i}, T_{s}$, $D, t_{i}, z, L$, and $k_{i}$ prescribed in Problem $8.63$, operating parameters that are variable and affect $T_{m o}$ are the thermal conductivity of the soil and the flow rate of the oil. Depending on soil composition and moisture and the demand for oil, representative variations are $0.25 \leq k_{s} \leq$ $1.0 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and $250 \leq \dot{m}_{o} \leq 500 \mathrm{~kg} / \mathrm{s}$. Using the properties prescribed in Problem 8.63, determine the effect of the foregoing variations on $T_{m \rho}$ and the total heat rate $q$. What is the worst case operating condition? If necessary, what adjustments could be made to ensure that $T_{m a} \geq$ $110^{\circ} \mathrm{C}$ for the worst case conditions?

Chai Santi
Chai Santi
Numerade Educator
04:41

Problem 65

Consider a thin-walled, metallic tube of length $L=1 \mathrm{~m}$ and inside diameter $D_{i}=3 \mathrm{~mm}$. Water enters the tube at $\dot{m}=0.015 \mathrm{~kg} / \mathrm{s}$ and $T_{m, i}=97^{\circ} \mathrm{C}$.
(a) What is the outlet temperature of the water if the tube surface temperature is maintained at $27^{\circ} \mathrm{C}$ ?
(b) If a $0.5-\mathrm{mm}$-thick layer of insulation of $k=0.05$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ is applied to the tube and its outer surface is maintained at $27^{\circ} \mathrm{C}$, what is the outlet temperature of the water?
(c) If the outer surface of the insulation is no longer maintained at $27^{\circ} \mathrm{C}$ but is allowed to exchange heat by free convection with ambient air at $27^{\circ} \mathrm{C}$, what is the outlet temperature of the water? The free convection heat transfer coefficient is $5 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$.

Salamat Ali
Salamat Ali
Numerade Educator
06:58

Problem 66

A circular tube of diameter $D=0.2 \mathrm{~mm}$ and length $L=$ $100 \mathrm{~mm}$ imposes a constant heat flux of $q^{\prime \prime}=20 \times 10^{3}$ $\mathrm{W} / \mathrm{m}^{2}$ on a fluid with a mass flow rate of $\dot{m}=0.1 \mathrm{~g} / \mathrm{s}$. For an inlet temperature of $T_{m, i}=29^{\circ} \mathrm{C}$, determine the tube wall temperature at $x=L$ for pure water. Evaluate fluid properties at $\bar{T}=300 \mathrm{~K}$. For the same conditions, determine the tube wall temperature at $x=L$ for the nanofluid of Example $2.2$.

Amany Waheeb
Amany Waheeb
Numerade Educator
01:09

Problem 67

Repeat Problem $8.66$ for a circular tube of diameter $D=2 \mathrm{~mm}$, an applied heat flux of $q^{\prime \prime}=200,000 \mathrm{~W} / \mathrm{m}^{2}$, and a mass flow rate of $\dot{m}=10 \mathrm{~g} / \mathrm{s}$.

Narayan Hari
Narayan Hari
Numerade Educator
02:50

Problem 68

Heat is to be removed from a reaction vessel operating at $75^{\circ} \mathrm{C}$ by supplying water at $27^{\circ} \mathrm{C}$ and $0.12 \mathrm{~kg} / \mathrm{s}$ through a thin-walled tube of $15-\mathrm{mm}$ diameter. The convection coefficient between the tube outer surface and the fluid in the vessel is $3000 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$.
(a) If the outlet water temperature cannot exceed $47^{\circ} \mathrm{C}$, what is the maximum rate of heat transfer from the vessel?
(b) What tube length is required to accomplish the heat transfer rate of part (a)?

Narayan Hari
Narayan Hari
Numerade Educator
02:50

Problem 69

A heating contractor must heat $0.2 \mathrm{~kg} / \mathrm{s}$ of water from $15^{\circ} \mathrm{C}$ to $35^{\circ} \mathrm{C}$ using hot gases in cross flow over a thinwalled tube.
Your assignment is to develop a series of design graphs that can be used to demonstrate acceptable combinations of tube dimensions ( $D$ and $L$ ) and of hot gas conditions ( $T_{\infty}$ and $V$ ) that satisfy this requirement. In your analysis, consider the following parameter ranges: $D=20,30$, or $40 \mathrm{~mm} ; L=3,4$, or $6 \mathrm{~m} ; T_{\infty}=250,375$, or $500^{\circ} \mathrm{C}$; and $20 \leq V \leq 40 \mathrm{~m} / \mathrm{s}$.

Narayan Hari
Narayan Hari
Numerade Educator
07:54

Problem 69

A thin-walled tube with a diameter of $6 \mathrm{~mm}$ and length of $20 \mathrm{~m}$ is used to carry exhaust gas from a smoke stack to the laboratory in a nearby building for analysis. The gas enters the tube at $200^{\circ} \mathrm{C}$ and with a mass flow rate of $0.003 \mathrm{~kg} / \mathrm{s}$. Autumn winds at a temperature of $15^{\circ} \mathrm{C}$ blow directly across the tube at a velocity of $5 \mathrm{~m} / \mathrm{s}$. Assume the thermophysical properties of the exhaust gas are those of air.
(a) Estimate the average heat transfer coefficient for the exhaust gas flowing inside the tube.
(b) Estimate the heat transfer coefficient for the air flowing across the outside of the tube.
(c) Estimate the overall heat transfer coefficient $U$ and the temperature of the exhaust gas when it reaches the laboratory.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:24

Problem 71

A 50 -mm-diameter, thin-walled metal pipe covered with a 25 -mm-thick layer of insulation $(0.085 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ and carrying superheated steam at atmospheric pressure is suspended from the ceiling of a large room. The steam temperature entering the pipe is $120^{\circ} \mathrm{C}$, and the air temperature is $20^{\circ} \mathrm{C}$. The convection heat transfer coefficient on the outer surface of the covered pipe is $10 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$. If the velocity of the steam is $10 \mathrm{~m} / \mathrm{s}$, at what point along the pipe will the steam begin condensing?

Sarah Mccrumb
Sarah Mccrumb
Numerade Educator
04:40

Problem 72

A thin-walled, uninsulated $0.3$-m-diameter duct is used to route chilled air at $0.05 \mathrm{~kg} / \mathrm{s}$ through the attic of a large commercial building. The attic air is at $37^{\circ} \mathrm{C}$, and natural circulation provides a convection coefficient of $2 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$ at the outer surface of the duct. If chilled air enters a $15-\mathrm{m}$-long duct at $7^{\circ} \mathrm{C}$, what is its exit temperature and the rate of heat gain? Properties of the chilled air may be evaluated at an assumed average temperature of $300 \mathrm{~K}$.

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
07:58

Problem 73

Pressurized water at $T_{m, i}=200^{\circ} \mathrm{C}$ is pumped at $\dot{m}=2 \mathrm{~kg} / \mathrm{s}$ from a power plant to a nearby industrial user through a thin-walled, round pipe of inside diameter $D=1 \mathrm{~m}$. The pipe is covered with a layer of insulation of thickness $t=0.15 \mathrm{~m}$ and thermal conductivity $k=0.05 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The pipe, which is of length $L=500 \mathrm{~m}$, is exposed to a cross flow of air at $T_{\infty}=-10^{\circ} \mathrm{C}$ and $V=4 \mathrm{~m} / \mathrm{s}$. Obtain a differential equation that could be used to solve for the variation of the mixed mean temperature of the water $T_{m}(x)$ with the axial coordinate. As a first approximation, the internal flow may be assumed to be fully developed throughout the pipe. Express your results in terms of $\dot{m}, V, T_{\infty}, D, t, k$, and appropriate water $(w)$ and air $(a)$ properties. Evaluate the heat loss per unit length of the pipe at the inlet. What is the mean temperature of the water at the outlet?

Ivan Kochetkov
Ivan Kochetkov
Numerade Educator
02:50

Problem 74

Water at $290 \mathrm{~K}$ and $0.2 \mathrm{~kg} / \mathrm{s}$ flows through a Teflon tube $(k=0.35 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ of inner and outer radii equal to 10 and $13 \mathrm{~mm}$, respectively. A thin electrical heating tape wrapped around the outer surface of the tube delivers a uniform surface heat flux of $2000 \mathrm{~W} / \mathrm{m}^{2}$, while a convection coefficient of $25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ is maintained on the outer surface of the tape by ambient air at $300 \mathrm{~K}$. What is the fraction of the power dissipated by the tape,
which is transferred to the water? What is the outer surface temperature of the Teflon tube?

Narayan Hari
Narayan Hari
Numerade Educator
01:47

Problem 75

The temperature of flue gases flowing through the large stack of a boiler is measured by means of a thermocouple enclosed within a cylindrical tube as shown. The tube axis is oriented normal to the gas flow, and the thermocouple senses a temperature $T_{t}$ corresponding to that of the tube surface. The gas flow rate and temperature are designated as $\dot{m}_{g}$ and $T_{g}$, respectively, and the gas flow may be assumed to be fully developed. The stack is fabricated from sheet metal that is at a uniform temperature $T_{s}$ and is exposed to ambient air at $T_{\infty}$ and large surroundings at $T_{\text {gur }}$. The convection coefficient associated with the outer surface of the duct is designated as $h_{o}$, while those associated with the inner surface of the duct and the tube surface are designated as $h_{i}$ and $h_{t}$, respectively. The tube and duct surface emissivities are designated as $\varepsilon_{t}$ and $\varepsilon_{s}$, respectively.
(a) Neglecting conduction losses along the thermocouple tube, develop an analysis that could be used to predict the error $\left(T_{g}-T_{t}\right)$ in the temperature measurement.
(b) Assuming the flue gas to have the properties of atmospheric air, evaluate the error for $T_{\mathrm{s}}=300^{\circ} \mathrm{C}$, $D_{s}=0.6 \mathrm{~m}, D_{t}=10 \mathrm{~mm}, \dot{m}_{g}=1 \mathrm{~kg} / \mathrm{s}, T_{\mathrm{s}}=T_{\text {sur }}=$ $27^{\circ} \mathrm{C}, \varepsilon_{t}=\varepsilon_{s}=0.8$, and $h_{o}=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Aadit Sharma
Aadit Sharma
Numerade Educator
09:13

Problem 77

Consider the ground source heat pump of Problem $5.100$ under winter conditions for which the liquid is discharged from the heat pump into high-density polyethylene tubing of thickness $t=8 \mathrm{~mm}$ and thermal conductivity $k=0.47 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The tubing is routed through soil that maintains a uniform temperature of approximately $10^{\circ} \mathrm{C}$ at the tube outer surface. The properties of the fluid may be approximated as those of water.
(a) For a tube inner diameter and flow rate of $D_{i}=$ $25 \mathrm{~mm}$ and $\dot{m}=0.03 \mathrm{~kg} / \mathrm{s}$ and a fluid inlet temperature of $T_{m, i}=0^{\circ} \mathrm{C}$, determine the tube outlet temperature (heat pump inlet temperature), $T_{m, o}$ as a function of the tube length $L$ for $10 \leq L \leq 50 \mathrm{~m}$.
(b) Recommend an appropriate length for the system. How would your recommendation be affected by variations in the liquid flow rate?

Keshav Singh
Keshav Singh
Numerade Educator
01:33

Problem 78

For a sharp-edged inlet and a combined entry region, the average Nusselt number may be computed from Equation 8.63, with $C=24 R e_{D}^{-0.23}$ and $m=0.815-$ $2.08 \times 10^{-6} R_{D}[23]$. Determine $\overline{N u}_{D} / N u_{D, \text { fd }}$ at $x / D=10$ and 60 for $R e_{D}=10^{4}$ and $10^{5}$.

Chai Santi
Chai Santi
Numerade Educator
06:58

Problem 79

Fluid enters a thin-walled tube of $5-\mathrm{mm}$ diameter and $2-\mathrm{m}$ length with a flow rate of $0.04 \mathrm{~kg} / \mathrm{s}$ and a temperature of $T_{m, i}=85^{\circ} \mathrm{C}$. The tube surface is maintained at a temperature of $T_{s}=25^{\circ} \mathrm{C}$, and for this operating condition, the outlet temperature is $T_{m, o}=31.1^{\circ} \mathrm{C}$. What is the outlet temperature if the flow rate is doubled? Fully developed, turbulent flow may be assumed to exist in both cases, and the fluid properties may be assumed to be independent of temperature.

Amany Waheeb
Amany Waheeb
Numerade Educator
02:15

Problem 80

Air at $3 \times 10^{-4} \mathrm{~kg} / \mathrm{s}$ and $27^{\circ} \mathrm{C}$ enters a rectangular duct that is $1 \mathrm{~m}$ long and $4 \mathrm{~mm} \times 16 \mathrm{~mm}$ on a side. A uniform heat flux of $600 \mathrm{~W} / \mathrm{m}^{2}$ is imposed on the duct surface. What is the temperature of the air and of the duct surface at the outlet?

Anand Jangid
Anand Jangid
Numerade Educator
01:13

Problem 81

Air at $25^{\circ} \mathrm{C}$ flows at $30 \times 10^{-6} \mathrm{~kg} / \mathrm{s}$ within $100-\mathrm{mm}$ long channels used to cool a high thermal conductivity metal mold. Assume the flow is hydrodynamically and thermally fully developed.
(a) Determine the heat transferred to the air for a circular channel $(D=10 \mathrm{~mm})$ when the mold temperature is $50^{\circ} \mathrm{C}$ (case A).
(b) Using new manufacturing methods (see Problem 8.105), channels of complex cross section can be readily fabricated within metal objects, such as molds. Consider air flowing under the same conditions as in case A, except now the channel is segmented into six smaller triangular sections. The flow area of case A is equal to the total flow area of case B. Determine the heat transferred to the air for the segmented channel.
(c) Compare the pressure drops for cases A and B.

Aadit Sharma
Aadit Sharma
Numerade Educator
03:30

Problem 82

A cold plate is an active cooling device that is attached to a heat-generating system in order to dissipate the heat while maintaining the system at an acceptable temperature. It is typically fabricated from a material of high thermal conductivity, $k_{\text {cp, }}$, within which channels are machined and a coolant is passed. Consider a copper cold plate of height $H$ and width $W$ on a side, within which water passes through square channels of width $w=h$. The transverse spacing between channels $\delta$ is twice the spacing between the sidewall of an outer channel and the sidewall of the cold plate.
Consider conditions for which equivalent heat-generating systems are attached to the top and bottom of the cold plate, maintaining the corresponding surfaces at the same temperature $T_{s}$. The mean velocity and inlet temperature of the coolant are $u_{m}$ and $T_{m i}$, respectively.
(a) Assuming fully developed turbulent flow throughout each channel, obtain a system of equations that may be used to evaluate the total rate of heat transfer to the cold plate, $q$, and the outlet temperature of the water, $T_{m, o}$, in terms of the specified parameters.
(b) Consider a cold plate of width $W=100 \mathrm{~mm}$ and height $H=10 \mathrm{~mm}$, with 10 square channels of width $w=6 \mathrm{~mm}$ and a spacing of $\delta=4 \mathrm{~mm}$ between channels. Water enters the channels at a temperature of $T_{m, i}=300 \mathrm{~K}$ and a velocity of $u_{m}=2 \mathrm{~m} / \mathrm{s}$. If the top and bottom cold plate surfaces are at $T_{s}=360 \mathrm{~K}$, what is the outlet water temperature and the total rate of heat transfer to the cold plate? The thermal conductivity of the copper is $400 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, while average properties of the water may be taken to be $\rho=984 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4184 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=489 \times$ $10^{-6} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.65 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $P r=3.15$. Is this a good cold plate design? How could its performance be improved?

Dominique Jan Tan
Dominique Jan Tan
Numerade Educator
01:05

Problem 83

The cold plate design of Problem $8.82$ has not been optimized with respect to selection of the channel width, and we wish to explore conditions for which the rate of heat transfer may be enhanced. Assume that the width and height of the copper cold plate are fixed at $W=100 \mathrm{~mm}$
and $H=10 \mathrm{~mm}$, while the channel height and spacing between channels are fixed at $h=6 \mathrm{~mm}$ and $\delta=4 \mathrm{~mm}$. The mean velocity and inlet temperature of the water are maintained at $u_{m}=2 \mathrm{~m} / \mathrm{s}$ and $T_{m, i}=300 \mathrm{~K}$, while equivalent heat-generating systems attached to the top and bottom of the cold plate maintain the corresponding surfaces at $360 \mathrm{~K}$. Evaluate the effect of changing the channel width, and hence the number of channels, on the rate of heat transfer to the cold plate. Include consideration of the limiting case for which $w=96 \mathrm{~mm}$ (one channel).

Samantha Lincroft
Samantha Lincroft
Numerade Educator
08:42

Problem 84

A device that recovers heat from high-temperature combustion products involves passing the combustion gas between parallel plates, each of which is maintained at $350 \mathrm{~K}$ by water flow on the opposite surface. The plate separation is $40 \mathrm{~mm}$, and the gas flow is fully developed. The gas may be assumed to have the properties of atmospheric air, and its mean temperature and velocity are $1000 \mathrm{~K}$ and $60 \mathrm{~m} / \mathrm{s}$, respectively.
(a) What is the heat flux at the plate surface?
(b) If a third plate, $20 \mathrm{~mm}$ thick, is suspended midway between the original plates, what is the surface heat flux for the original plates? Assume the temperature and fiw rate of the gas to be unchanged and radiation effects to be negligible.

Vipender Yadav
Vipender Yadav
Numerade Educator
06:54

Problem 85

Air at $1 \mathrm{~atm}$ and $285 \mathrm{~K}$ enters a 2 -m-long rectangular duct with cross section $75 \mathrm{~mm} \times 150 \mathrm{~mm}$. The duct is maintained at a constant surface temperature of $400 \mathrm{~K}$, and the air mass flow rate is $0.10 \mathrm{~kg} / \mathrm{s}$. Determine the heat transfer rate from the duct to the air and the air outlet temperature.

Vipender Yadav
Vipender Yadav
Numerade Educator
09:01

Problem 86

A double-wall heat exchanger is used to transfer heat between liquids flowing through semicircular copper tubes. Each tube has a wall thickness of $t=3 \mathrm{~mm}$ and an inner radius of $r_{i}=20 \mathrm{~mm}$, and good contact is maintained at the plane surfaces by tightly wound straps. The tube outer surfaces are well insulated.
(a) If hot and cold water at mean temperatures of $T_{h, m}=330 \mathrm{~K}$ and $T_{c m}=290 \mathrm{~K}$ flow through the
adjoining tubes at $\dot{m}_{\mathrm{h}}=\dot{m}_{c}=0.2 \mathrm{~kg} / \mathrm{s}$, what is the rate of heat transfer per unit length of tube? The wall contact resistance is $10^{-5} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. Approximate the properties of both the hot and cold water as $\mu=800 \times 10^{-6} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, \quad k=0.625 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\operatorname{Pr}=5.35$. Hint: Heat transfer is enhanced by conduction through the semicircular portions of the tube walls, and each portion may be subdivided into two straight fins with adiabatic tips.
(b) Using the thermal model developed for part (a), determine the heat transfer rate per unit length when the fluids are ethylene glycol. Also, what effect will fabricating the exchanger from an aluminum alloy have on the heat rate? Will increasing the thickness of the tube walls have a beneficial effect?

Averell Hause
Averell Hause
Carnegie Mellon University
06:58

Problem 87

Consider laminar, fully developed flow in a channel of constant surface temperature $T_{s}$. For a given mass flow rate and channel length, determine which rectangular channel, $b / a=1.0,1.43$, or $2.0$, will provide the highest heat transfer rate. Is this heat transfer rate greater than, equal to, or less than the heat transfer rate associated with a circular tube?

Amany Waheeb
Amany Waheeb
Numerade Educator
03:27

Problem 88

You have been asked to perform a feasibility study on the design of a blood warmer to be used during the transfusion of blood to a patient. This exchanger is to heat blood taken from the bank at $10^{\circ} \mathrm{C}$ to $37^{\circ} \mathrm{C}$ at a flow rate of $200 \mathrm{ml} / \mathrm{min}$. The blood passes through a rectangular cross-section tube, $6.4 \mathrm{~mm} \times 1.6 \mathrm{~mm}$, which is sandwiched between two plates held at a constant temperature of $40^{\circ} \mathrm{C}$.
(a) Compute the length of the tubing required to achieve the desired outlet conditions at the specified flow rate. Assume the flow is fully developed and the blood has the same properties as water.
(b) Assess your assumptions and indicate whether your analysis over- or underestimates the necessary length.

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Numerade Educator
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Problem 89

A coolant flows through a rectangular channel (gallery) within the body of a mold used to form metal injection parts. The gallery dimensions are $a=90 \mathrm{~mm}$ and $b=9.5 \mathrm{~mm}$, and the fluid flow rate is $1.3 \times 10^{-3} \mathrm{~m}^{3} / \mathrm{s}$. The coolant temperature is $15^{\circ} \mathrm{C}$, and the mold wall is at an approximately uniform temperature of $140^{\circ} \mathrm{C}$.
To minimize corrosion damage to the expensive mold, it is customary to use a heat transfer fluid such as ethylene glycol, rather than process water. Compare the convection coefficients of water and ethylene glycol for this application. What is the tradeoff between thermal performance and minimizing corrosion?

Susan Hallstrom
Susan Hallstrom
Numerade Educator
08:34

Problem 90

An electronic circuit board dissipating $50 \mathrm{~W}$ is sandwiched between two ducted, forced-air-cooled heat sinks. The sinks are $150 \mathrm{~mm}$ in length and have 20 rectangular passages $6 \mathrm{~mm} \times 25 \mathrm{~mm}$. Atmospheric air at a volumetric flow rate of $0.060 \mathrm{~m}^{3} / \mathrm{s}$ and $27^{\circ} \mathrm{C}$ is drawn through the sinks by a blower. Estimate the operating temperature of the board and the pressure drop across the sinks.

Dading Chen
Dading Chen
Numerade Educator
03:33

Problem 91

To slow down large prime movers like locomotives, a process termed dynamic electric braking is used to switch the traction motor to a generator mode in which mechanical power from the drive wheels is absorbed and used to generate electrical current. As shown in the schematic, the electric power is passed through a resistor grid $(a)$, which consists of an array of metallic blades electrically connected in series (b). The blade material is a high-temperature, high electrical resistivity alloy, and the electrical power is dissipated as heat by internal volumetric generation. To cool the blades, a motor-fan moves high-velocity air through the grid.
(a) Treating the space between the blades as a rectangular channel of $220-\mathrm{mm} \times 4-\mathrm{mm}$ cross section and $70-\mathrm{mm}$ length, estimate the heat removal rate per blade if the airstream has an inlet temperature and velocity of $25^{\circ} \mathrm{C}$ and $50 \mathrm{~m} / \mathrm{s}$, respectively, while the blade has an operating temperature of $600^{\circ} \mathrm{C}$.
(b) On a locomotive pulling a 10 -car train, there may be 2000 of these blades. Based on your result from part (a), how long will it take to slow a train whose total mass is $10^{6} \mathrm{~kg}$ from a speed of $120 \mathrm{~km} / \mathrm{h}$ to $50 \mathrm{~km} / \mathrm{h}$ using dynamic electric braking?

Prabhu Ramji
Prabhu Ramji
Numerade Educator
04:22

Problem 92

A printed circuit board (PCB) is cooled by laminar, fully developed airflow in adjoining, parallel-plate channels of length $L$ and separation distance $a$. The channels may be assumed to be of infinite extent in the transverse direction, and the upper and lower surfaces are insulated. The temperature $T_{s}$ of the $\mathrm{PCB}$ board is uniform, and airflow with an inlet temperature of $T_{m, i}$ is driven by a pressure difference $\Delta p$.
Calculate the average heat removal rate per unit area $\left(\mathrm{W} / \mathrm{m}^{2}\right)$ from the $\mathrm{PCB}$.

Dading Chen
Dading Chen
Numerade Educator
01:56

Problem 93

Water at $\dot{m}=0.02 \mathrm{~kg} / \mathrm{s}$ and $T_{m, i}=20^{\circ} \mathrm{C}$ enters an annular region formed by an inner tube of diameter $D_{i}=25 \mathrm{~mm}$ and an outer tube of diameter $D_{o}=100 \mathrm{~mm}$. Saturated steam flows through the inner tube, maintaining its surface at a uniform temperature of $T_{s, i}=100^{\circ} \mathrm{C}$, while the outer surface of the outer tube is well insulated. If fully developed conditions may be assumed throughout the annulus, how long must the system be to provide an outlet water temperature of $75^{\circ} \mathrm{C}$ ? What is the heat flux from the inner tube at the outlet?

Manik Pulyani
Manik Pulyani
Numerade Educator
01:06

Problem 94

For the conditions of Problem $8.93$, how long must the annulus be if the water flow rate is $0.30 \mathrm{~kg} / \mathrm{s}$ instead of $0.02 \mathrm{~kg} / \mathrm{s}$ ?

James Kiss
James Kiss
Numerade Educator
05:47

Problem 95

Referring to Figure 8.11, consider conditions in an annulus having an outer surface that is insulated $\left(q_{o}^{\prime \prime}=0\right)$ and a uniform heat flux $q_{i}^{\prime \prime}$ at the inner surface. Fully developed, laminar flow may be assumed to exist.
(a) Determine the velocity profile $u(r)$ in the annular region.
(b) Determine the temperature profile $T(r)$ and obtain an expression for the Nusselt number $N u_{i}$ associated with the inner surface.

Dading Chen
Dading Chen
Numerade Educator
04:00

Problem 96

Consider the air heater of Problem $8.38$, but now with airflow through the annulus and steam flow through the inner tube. For the prescribed conditions and an outer tube diameter of $D_{o}=65 \mathrm{~mm}$, determine the outlet temperature and pressure of the air, as well as the mass rate of steam condensation.

Mohammad Mehran
Mohammad Mehran
Numerade Educator
03:37

Problem 97

Consider a concentric tube annulus for which the inner and outer diameters are 25 and $50 \mathrm{~mm}$. Water enters the annular region at $0.04 \mathrm{~kg} / \mathrm{s}$ and $25^{\circ} \mathrm{C}$. If the inner tube wall is heated electrically at a rate (per unit length) of $q^{\prime}=4000 \mathrm{~W} / \mathrm{m}$, while the outer tube wall is insulated, how long must the tubes be for the water to achieve an outlet temperature of $85^{\circ} \mathrm{C}$ ? What is the inner tube surface temperature at the outlet, where fully developed conditions may be assumed?

Surendra Kumar
Surendra Kumar
Numerade Educator
04:58

Problem 98

It is common practice to recover waste heat from an oilor gas-fired furnace by using the exhaust gases to preheat the combustion air. A device commonly used for this purpose consists of a concentric pipe arrangement for which the exhaust gases are passed through the inner pipe, while the cooler combustion air flows through an annular passage around the pipe.

Consider conditions for which there is a uniform heat transfer rate per unit length, $q_{i}^{\prime}=1.25 \times 10^{5} \mathrm{~W} / \mathrm{m}$, from the exhaust gases to the pipe inner surface, while air flows through the annular passage at a rate of $\dot{m}_{a}=2.1 \mathrm{~kg} / \mathrm{s}$. The thin-walled inner pipe is of diameter $D_{i}=2 \mathrm{~m}$, while the outer pipe, which is well insulated from the surroundings, is of diameter $D_{o}=2.05 \mathrm{~m}$. The air properties may be taken to be $c_{p}=1030 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=270 \times 10^{-7} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.041$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$, and $P r=0.68$.
(a) If air enters at $T_{a, 1}=300 \mathrm{~K}$ and $L=7 \mathrm{~m}$, what is the air outlet temperature $T_{a, 2}$ ?
(b) If the airflow is fully developed throughout the annular region, what is the temperature of the inner pipe at the inlet $\left(T_{s, i, 1}\right)$ and outlet $\left(T_{s, i, 2}\right)$ sections of the device? What is the outer surface temperature $T_{s, \theta, 1}$ at the inlet?

Jincy M Saji
Jincy M Saji
Numerade Educator
10:53

Problem 99

A concentric tube arrangement, for which the inner and outer diameters are $80 \mathrm{~mm}$ and $100 \mathrm{~mm}$, respectively, is used to remove heat from a biochemical reaction occurring in a 1-m-long settling tank. Heat is generated uniformly within the tank at a rate of $10^{5} \mathrm{~W} / \mathrm{m}^{3}$, and water is supplied to the annular region at a rate of $0.2 \mathrm{~kg} / \mathrm{s}$.
(a) Determine the inlet temperature of the supply water that will maintain an average tank surface temperature of $37^{\circ} \mathrm{C}$. Assume fully developed flow and thermal conditions. Is this assumption reasonable?
(b) It is desired to have a slight, axial temperature gradient on the tank surface, since the rate of the biochemical reaction is highly temperature dependent. Sketch the axial variation of the water and surface temperatures along the flow direction for the following two cases: (i) the fully developed conditions of part (a), and (ii) conditions for which entrance effects are important. Comment on features of the temperature distributions. What change to the system or operating conditions would you make to reduce the surface temperature gradient?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
13:04

Problem 100

Consider the air cooling system and conditions of Problem 8.31, but with a prescribed pipe length of $L=15 \mathrm{~m}$.
(a) What is the air outlet temperature, $T_{\text {m?o }}$ ? What is the fan power requirement?
(b) The convection coefficient associated with airflow in the pipe may be increased twofold by inserting a coiled spring along the length of the pipe to disrupt flow conditions near the inner surface. If such a heat transfer enhancement scheme is adopted, what is the attendant value of $T_{m, o}$ ? Use of the insert would not come without a corresponding increase in the fan power requirement. What is the power requirement if the friction factor is increased by $50 \%$ ?
(c) After extended exposure to the water, a thin coating of organic matter forms on the outer surface of the pipe, and its thermal resistance (for a unit area of the outer surface) is $R_{r, o}^{p}=0.050 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. What is the corresponding value of $T_{m, o}$ without the insert of part (b)?

Brianna Orr
Brianna Orr
Numerade Educator
04:00

Problem 101

Consider sterilization of the pharmaceutical product of Problem 8.27. To avoid any possibility of heating the product to an unacceptably high temperature, atmospheric steam is condensed on the exterior of the tube instead of using the resistance heater, providing a uniform surface temperature, $T_{s}=100^{\circ} \mathrm{C}$.
(a) For the conditions of Problem 8.27, determine the required length of straight tube, $L_{s}$, that would be needed to increase the mean temperature of the pharmaceutical product from $25^{\circ} \mathrm{C}$ to $75^{\circ} \mathrm{C}$.
(b) Consider replacing the straight tube with a coiled tube characterized by a coil diameter $C=100 \mathrm{~mm}$ and a coil pitch $S=25 \mathrm{~mm}$. Determine the overall
length of the coiled tube, $L_{\mathrm{c}}$ (i.e., the product of the tube pitch and the number of coils), necessary to increase the mean temperature of the pharmaceutical to the desired value.
(c) Calculate the pressure drop through the straight tube and through the coiled tube.
(d) Calculate the steam condensation rate.

Paul Gabriel
Paul Gabriel
Numerade Educator
02:52

Problem 102

An engineer proposes to insert a solid rod of diameter $D_{i}$ into a circular tube of diameter $D_{o}$ to enhance heat transfer from the flowing fluid of temperature $T_{m}$ to the outer tube wall of temperature $T_{s, \sigma^{\circ}}$ Assuming laminar flow, calculate the ratio of the heat flux from the fluid to the outer tube wall with the rod to the heat flux without the rod, $q_{o}^{\prime \prime} / q_{o, w o}^{\prime \prime}$, for $D_{i} / D_{o}=0,0.10,0.25$ and $0.50$. The rod is placed concentrically within the tube.

Dading Chen
Dading Chen
Numerade Educator
01:30

Problem 103

An electrical power transformer of diameter $230 \mathrm{~mm}$ and height $500 \mathrm{~mm}$ dissipates $1000 \mathrm{~W}$. It is desired to maintain its surface temperature at $47^{\circ} \mathrm{C}$ by supplying ethylene glycol at $24^{\circ} \mathrm{C}$ through thin-walled tubing of $20-\mathrm{mm}$ diameter welded to the lateral surface of the transformer. All the heat dissipated by the transformer is assumed to be transferred to the ethylene glycol.
Assuming the maximum allowable temperature rise of the coolant to be $6^{\circ} \mathrm{C}$, determine the required coolant flow rate, the total length of tubing, and the coil pitch $S$ between turns of the tubing.

Jincy M Saji
Jincy M Saji
Numerade Educator
01:05

Problem 104

A bayonet cooler is used to reduce the temperature of a pharmaceutical fluid. The pharmaceutical fluid flows through the cooler, which is fabricated of $10-\mathrm{mm}-$ diameter, thin-walled tubing with two 250 -mm-long straight sections and a coil with six and a half turns and a coil diameter of $75 \mathrm{~mm}$. A coolant flows outside the cooler, with a convection coefficient at the outside surface of $h_{o}=500 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$ and a coolant temperature of $20^{\circ} \mathrm{C}$. Consider the situation where the pharmaceutical fluid enters at $90^{\circ} \mathrm{C}$ with a mass flow rate of $0.005 \mathrm{~kg} / \mathrm{s}$. The pharmaceutical has the following properties: $\rho=1200 \mathrm{~kg} / \mathrm{m}^{3}, \quad \mu=4 \times 10^{-3} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, $c_{p}=2000 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $k=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
18:47

Problem 105

8.105 The mold used in an injection molding process consists of a top half and a bottom half. Each half is $60 \mathrm{~mm} \times 60 \mathrm{~mm} \times 20 \mathrm{~mm}$ and is constructed of metal $\left(\rho=7800 \mathrm{~kg} / \mathrm{m}^{3}, \quad c=450 \mathrm{~J} / \mathrm{kg}+\mathrm{K}\right)$. The cold mold $\left(100^{\circ} \mathrm{C}\right)$ is to be heated to $200^{\circ} \mathrm{C}$ with pressurized water (available at $275^{\circ} \mathrm{C}$ and a total flow rate of $0.02 \mathrm{~kg} / \mathrm{s}$ ) prior to injecting the thermoplastic material. The injection takes only a fraction of a second, and the hot mold $\left(200^{\circ} \mathrm{C}\right)$ is subsequently cooled with cold water (available at $25^{\circ} \mathrm{C}$ and a total flow rate of $0.02 \mathrm{~kg} / \mathrm{s})$ prior to ejecting the molded part. After part ejection, which also takes a fraction of a second, the process is repeated.
(a) In conventional mold design, straight cooling (heating) passages are bored through the mold in a location where the passages will not interfere with the molded part. Determine the initial heating rate and the initial cooling rate of the mold when five 5 -mm-diameter, 60-mm-long passages are bored in each half of the mold (10 passages total). The velocity distribution of the water is fully developed at the entrance of each passage in the hot (or cold) mold.
(b) New additive manufacturing processes, known as selective freeform fabrication, or $S F F$, are used to construct molds that are configured with conformal cooling passages. Consider the same mold as before, but now a 5 -mm-diameter, coiled, conformal cooling passage is designed within each half of the SFF-manufactured mold. Each of the two coiled passages has $N=2$ turns. The coiled passage does not interfere with the molded part. The conformal channels have a coil diameter $C=50 \mathrm{~mm}$. The total water flow remains the same as in part (a) $(0.01 \mathrm{~kg} / \mathrm{s}$ per coil). Determine the initial heating rate and the initial cooling rate of the mold.
(c) Compare the surface areas of the conventional and conformal cooling passages. Compare the rate at which the mold temperature changes for molds configured with the conventional and conformal heating and cooling passages. Which cooling passage, conventional or conformal, will enable production of more parts per day? Neglect the presence of the thermoplastic material.

Niamat Khuda
Niamat Khuda
Numerade Educator
03:47

Problem 106

8.106 Consider the pharmaceutical product of Problem 8.27. Prior to finalizing the manufacturing process, test trials are performed to experimentally determine the dependence of the shelf life of the drug as a function of the sterilization temperature. Hence, the sterilization temperature must be carefully controlled in the trials. To promote good mixing of the pharmaceutical and, in turn, relatively uniform outlet temperatures across the exit tube area, experiments are performed using a device that is constructed of two interwoven coiled tubes, each of 10 -mm diameter. The thin-walled tubing is welded to a solid high thermal conductivity rod of diameter $D_{r}=40 \mathrm{~mm}$. One tube carries the pharmaceutical product at a mean velocity of $u_{p}=0.1 \mathrm{~m} / \mathrm{s}$ and inlet temperature of $25^{\circ} \mathrm{C}$, while the second tube carries pressurized liquid water at $u_{w}=0.12 \mathrm{~m} / \mathrm{s}$ with an inlet temperature of $127^{\circ} \mathrm{C}$. The tubes do not contact each other but are each welded to the solid metal rod, with each tube making 20 turns around the rod. The exterior of the apparatus is well insulated.
(a) Determine the outlet temperature of the pharmaceutical product. Evaluate the liquid water properties at $380 \mathrm{~K}$.
(b) Investigate the sensitivity of the pharmaceutical's outlet temperature to the velocity of the pressurized water over the range $0.10<u_{w}<0.25 \mathrm{~m} / \mathrm{s}$.

Rashmi Sinha
Rashmi Sinha
Numerade Educator
01:30

Problem 107

An extremely effective method of cooling high-powerdensity silicon chips involves etching microchannels in the back (noncircuit) surface of the chip. The channels are covered with a silicon cap, and cooling is maintained by passing water through the channels.
Consider a chip that is $10 \mathrm{~mm} \times 10 \mathrm{~mm}$ on a side and in which fifty 10 -mm-long rectangular microchannels, each of width $W=50 \mu \mathrm{m}$ and height $H=200 \mu \mathrm{m}$, have been etched. Consider operating conditions for which water enters each microchannel at a temperature of $290 \mathrm{~K}$ and a flow rate of $10^{-4} \mathrm{~kg} / \mathrm{s}$, while the chip and cap are at a uniform temperature of $350 \mathrm{~K}$. Assuming fully developed flow in the channel and that all the heat dissipated by the circuits is transferred to the water, determine the water outlet temperature and the chip power dissipation. Water properties may be evaluated at $300 \mathrm{~K}$.

Jincy M Saji
Jincy M Saji
Numerade Educator
03:32

Problem 108

An ideal gas flows within a small diameter tube. Derive an expression for the transition density of the gas $\rho_{c}$ below which microscale effects must be accounted for. Express your result in terms of the gas molecule diameter, universal gas constant, Boltzmann's constant, and the tube diameter. Evaluate the transition density for a $D=10-\mu \mathrm{m}$-diameter tube for hydrogen, air, and carbon dioxide. Compare the calculated transition densities with the gas density at atmospheric pressure and $T=23^{\circ} \mathrm{C}$.

Gaurav Gupta
Gaurav Gupta
Numerade Educator
01:00

Problem 109

Consider the microchannel cooling arrangement of Problem 8.107. However, instead of assuming the entire chip and cap to be at a uniform temperature, adopt a more conservative (and realistic) approach that prescribes a temperature of $T_{s}=350 \mathrm{~K}$ at the base of the channels $(x=0$ ) and allows for a decrease in temperature with increasing $x$ along the side walls of each channel.
(a) For the operating conditions prescribed in Problem $8.107$ and a chip thermal conductivity of $k_{\mathrm{ch}}=$ $140 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, determine the water outlet temperature and the chip power dissipation. Heat transfer from the sides of the chip to the surroundings and from the side walls of a channel to the cap may be neglected. Note that the spacing between channels, $\delta=S-W$, is twice the spacing between the side wall of an outer channel and the outer surface of the chip. The channel pitch is $S=L / N$, where $L=10 \mathrm{~mm}$ is the chip width and $N=50$ is the number of channels.
(b) The channel geometry prescribed in Problem $8.107$ and considered in part (a) is not optimized, and larger heat rates may be dissipated by adjusting related dimensions. Consider the effect of reducing the pitch to a value of $S=100 \mu \mathrm{m}$, while retaining a width of $W=50 \mu \mathrm{m}$ and a flow rate per channel of $\dot{m}_{1}=10^{-4} \mathrm{~kg} / \mathrm{s}$.

Raj Bala
Raj Bala
Numerade Educator
03:35

Problem 110

The onset of turbulence in a gas flowing within a circular tube occurs at $R e_{D, c}=2300$, while a transition from incompressible to compressible flow occurs at a critical Mach number of $M a_{c}=0.3$. Determine the critical tube diameter $D_{c}$, below which incompressible turbulent flow and heat transfer cannot exist for (i) air, (ii) $\mathrm{CO}_{2}$, (iii) He. Evaluate properties at atmospheric pressure and a temperature of $T=300 \mathrm{~K}$.

Chai Santi
Chai Santi
Numerade Educator
01:34

Problem 111

Due to its comparatively large thermal conductivity, water is a preferred fluid for convection cooling. However, in applications involving electronic devices, water must not come into contact with the devices, which would therefore have to be hermetically sealed. To circumvent related design and operational complexities and to ensure that the devices are not rendered inoperable by contact with the coolant, a dielectric fluid is commonly used in lieu of water. Many gases have excellent dielectric characteristics, and despite its poor heat transfer properties, air is the common choice for electronic cooling. However, there is an alternative, which involves a class of perflorinated liquids that are excellent dielectrics and have heat transfer properties superior to those of gases.

Consider the microchannel chip cooling application of Problem $8.109$ but now for a perfluorinated liquid with properties of $c_{p}=1050 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=0.065$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, \mu=0.0012 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, and $P r=15$.
(a) For channel dimensions of $H=200 \mu \mathrm{m}, W=50$ $\mu \mathrm{m}$, and $S=20 \mu \mathrm{m}$, a chip thermal conductivity of $k_{\mathrm{ch}}=140 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ and width $L=10 \mathrm{~mm}$, a channel base temperature $(x=0)$ of $T_{s}=350 \mathrm{~K}$, a channel inlet temperature of $T_{m, i}=290 \mathrm{~K}$, and a flow rate of $\dot{m}_{1}=10^{-4} \mathrm{~kg} / \mathrm{s}$ per channel, determine the outlet temperature and the chip power dissipation for the dielectric liquid.
(b) Consider the foregoing conditions, but with air at a flow rate of $\dot{m}_{1}=10^{-6} \mathrm{~kg} / \mathrm{s}$ used as the coolant.
Using properties of $c_{p}=1007 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=0.0263$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$, and $\mu=185 \times 10^{-7} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, determine the air outlet temperature and the chip power dissipation.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:58

Problem 112

Many of the solid surfaces for which values of the thermal and momentum accommodation coefficients have been measured are quite different from those used in micro- and nanodevices. Plot the Nusselt number $N u_{D}$ associated with fully developed laminar flow with constant surface heat flux versus tube diameter for $1 \mu \mathrm{m} \leq D \leq 1 \mathrm{~mm}$ and (i) $\alpha_{s}=1, \alpha_{p}=1$, (ii) $\alpha_{t}=0.1, \alpha_{p}=0.1$, (iii) $\alpha_{t}=1, \alpha_{p}=0.1$, and (iv) $\alpha_{t}=0.1, \alpha_{p}=1$. For tubes of what diameter do the accommodation coefficients begin to influence convection heat transfer? For which combination of $\alpha_{t}$ and $\alpha_{p}$ does the Nusselt number exhibit the least sensitivity to changes in the diameter of the tube? Which combination results in Nusselt numbers greater than the conventional fully developed laminar value for constant heat flux conditions, $N u_{D}=4.36$ ? Which combination is associated with the smallest Nusselt numbers? What can you say about the ability to predict convection heat transfer coefficients in a small-scale device if the accommodation coefficients are not known for material from which the device is fabricated? Use properties of air at atmospheric pressure and $T=300 \mathrm{~K}$.

Narayan Hari
Narayan Hari
Numerade Educator
06:05

Problem 113

A novel scheme for dissipating heat from the chips of a multichip array involves machining coolant channels in the ceramic substrate to which the chips are attached. The square chips $\left(L_{c}=5 \mathrm{~mm}\right)$ are aligned above each of the channels, with longitudinal and transverse pitches of $S_{L}=S_{T}=20 \mathrm{~mm}$. Water flows through the square cross section $(W=5 \mathrm{~mm}$ ) of each channel with a mean velocity of $u_{m}=1 \mathrm{~m} / \mathrm{s}$, and its properties may be approximated as $\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}$, $c_{p}=4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=855 \times 10^{-6} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, k=0.610$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$, and $\operatorname{Pr}=5.8$. Symmetry in the transverse direction dictates the existence of equivalent conditions for each substrate section of length $L_{s}$ and width $S_{T}$.
(a) Consider a substrate whose length in the flow direction is $L_{s}=200 \mathrm{~mm}$, thereby providing a total of $N_{L}=10$ chips attached in-line above each flow channel. To a good approximation, all the heat dissipated by the chips above a channel may be assumed to be transferred to the water flowing through the channel. If each chip dissipates $5 \mathrm{~W}$, what is the temperature rise of the water passing through the channel?
(b) The chip-substrate contact resistance is $R_{\mathrm{t}, c}^{\mathrm{r}}=$ $0.5 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$, and the three-dimensional conduction resistance for the $L_{s} \times S_{T}$ substrate section is $R_{\text {cond }}=0.120 \mathrm{~K} / \mathrm{W}$. If water enters the substrate at $25^{\circ} \mathrm{C}$ and is in fully developed flow, estimate the temperature $T_{c}$ of the chips and the temperature $T_{s}$ of the substrate channel surface.

Paul Gabriel
Paul Gabriel
Numerade Educator
01:45

Problem 114

Consider air flowing in a small-diameter steel tube. Graph the Nusselt number associated with fully developed laminar flow with constant surface heat flux for tube diameters ranging from $1 \mu \mathrm{m} \leq D \leq 1 \mathrm{~mm}$. Evaluate air properties at $T=350 \mathrm{~K}$ and atmospheric pressure. The thermal and momentum accommodation coefficients are $\alpha_{t}=0.92$ and $\alpha_{p}=0.87$, respectively. Compare the Nusselt number you calculate to the value provided in Equation $8.53, N u_{D}=4.36$.

Narayan Hari
Narayan Hari
Numerade Educator
09:01

Problem 115

An experiment is designed to study microscale forced convection. Water at $T_{\text {mi }}=300 \mathrm{~K}$ is to be heated in a straight, circular glass tube with a $50-\mu \mathrm{m}$ inner diameter and a wall thickness of $1 \mathrm{~mm}$. Warm water at $T_{\infty}=350 \mathrm{~K}, V=2 \mathrm{~m} / \mathrm{s}$ is in cross flow over the exterior tube surface. The experiment is to be designed to cover the operating range $1 \leq R e_{D} \leq 2000$, where $R e_{D}$ is the Reynolds number associated with the internal flow.
(a) Determine the tube length $L$ that meets a design requirement that the tube be twice as long as the thermal entrance length associated with the highest Reynolds number of interest. Evaluate water properties at $305 \mathrm{~K}$.
(b) Determine the water outlet temperature, $T_{\text {mo }}$ that is expected to be associated with $R e_{D}=2000$. Evaluate the heating water (water in cross flow over the tube) properties at $330 \mathrm{~K}$.
(c) Calculate the pressure drop from the entrance to the exit of the tube for $R e_{D}=2000$.
(d) Based on the calculated flow rate and pressure drop in the tube, estimate the height of a column of water (at $300 \mathrm{~K}$ ) needed to supply the necessary pressure at the tube entrance and the time needed to collect $0.1$ liter of water. Discuss how the outlet temperature of the water flowing from the tube, $T_{m, o}$, might be measured.

Averell Hause
Averell Hause
Carnegie Mellon University
03:00

Problem 116

Determine the tube diameter that corresponds to a $10 \%$ reduction in the convection heat transfer coefficient for thermal and momentum accommodation coefficients of $\alpha_{t}=0.92$ and $\alpha_{p}=0.89$, respectively. Determine the channel spacing, $a$, that is associated with a $10 \%$ reduction in $h$ using the same accommodation coefficients. The gas is air at $T=350 \mathrm{~K}$ and atmospheric pressure for both the tube and the parallel plate configurations. The flow is laminar and fully developed with constant surface heat flux.

Anand Jangid
Anand Jangid
Numerade Educator
06:24

Problem 117

An experiment is devised to measure liquid flow and convective heat transfer rates in microscale channels. The mass flow rate through a channel is determined by measuring the amount of liquid that has flowed through the channel and dividing by the duration of the experiment. The mean temperature of the outlet fluid is also measured. To minimize the time needed to perform the experiment (that is, to collect a significant amount of liquid so that its mass and temperature can be accurately measured), arrays of microchannels are typically used. Consider an array of microchannels of
circular cross section, each with a nominal diameter of $50 \mu \mathrm{m}$, fabricated into a copper block. The channels are $20 \mathrm{~mm}$ long, and the block is held at $310 \mathrm{~K}$. Water at an inlet temperature of $300 \mathrm{~K}$ is forced into the channels from a pressurized plenum, so that a pressure difference of $2.5 \times 10^{6} \mathrm{~Pa}$ exists from the entrance to the exit of each channel.

In many microscale systems, the characteristic dimensions are similar to the tolerances that can be controlled during the manufacture of the experimental apparatus. Hence, careful consideration of the effect of machining tolerances must be made when interpreting the experimental results.
(a) Consider the case in which three microchannels are machined in the copper block. The channel diameters exhibit some deviation due to manufacturing constraints and are of actual diameter $45 \mu \mathrm{m}, 50 \mu \mathrm{m}$, and $55 \mu \mathrm{m}$, respectively. Calculate the mass flow rate through each of the three channels, along with the mean outlet temperature of each channel.
(b) If the water exiting each of the three channels is collected and mixed in a single container, calculate the average flow rate through each of the three channels and the average mixed temperature of the water that is collected from all three channels.
(c) The enthusiastic experimentalist uses the average flow rate and the average mixed outlet temperature to analyze the performance of the average $(50 \mu \mathrm{m})$ diameter channel and concludes that flow rates and heat transfer coefficients are increased and decreased, respectively, by about $5 \%$ when forced convection occurs in microchannels. Comment on the validity of the experimentalist's conclusion.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
06:32

Problem 118

In the processing of very long plastic tubes of $2-\mathrm{mm}$ inside diameter, air flows inside the tubing with a Reynolds number of 1000 . The interior layer of the plastic material evaporates into the air under fully developed conditions. Both plastic and air are at $400 \mathrm{~K}$, and the Schmidt number for the mixture of the plastic vapor and air is $2.0$. Determine the convection mass transfer coefficient.

Satpal Satpal
Satpal Satpal
Numerade Educator
04:21

Problem 119

Air at $300 \mathrm{~K}$ and a flow rate of $3 \mathrm{~kg} / \mathrm{h}$ passes upward through a 30 -mm tube, as shown in the sketch. A thin film of water, also at $300 \mathrm{~K}$, slowly falls downward on the inner surface of the tube. Determine the convection mass transfer coefficient for this situation.

Uma Kumari
Uma Kumari
Numerade Educator
01:13

Problem 120

What is the convection mass transfer coefficient associated with fully developed atmospheric airflow at $27^{\circ} \mathrm{C}$ and $0.04 \mathrm{~kg} / \mathrm{s}$ through a $50-\mathrm{mm}$-diameter tube whose surface has been coated with a thin layer of naphthalene? Determine the velocity and concentration entry lengths.

Mayukh Banik
Mayukh Banik
Numerade Educator
03:35

Problem 121

Air flowing through a tube of $75-\mathrm{mm}$ diameter passes over a 150 -mm-long roughened section that is constructed from naphthalene having the properties $M=128.16 \mathrm{~kg} / \mathrm{kmol}$ and $p_{\mathrm{sat}}(300 \mathrm{~K})=1.31 \times 10^{-4} \mathrm{bar}$. The air is at $1 \mathrm{~atm}$ and $300 \mathrm{~K}$, and the Reynolds number is $R e_{D}=35,000$. In an experiment for which flow was maintained for $3 \mathrm{~h}$, mass loss due to sublimation from the roughened surface was determined to be $0.01 \mathrm{~kg}$. What is the associated convection mass transfer coefficient? What would be the corresponding convection heat transfer coefficient? Contrast these results with those predicted by conventional smooth tube correlations.

Chai Santi
Chai Santi
Numerade Educator
03:14

Problem 122

Dry air at $35^{\circ} \mathrm{C}$ and a velocity of $10 \mathrm{~m} / \mathrm{s}$ flows over a thin-walled tube of $20-\mathrm{mm}$ diameter and $200-\mathrm{mm}$ length, having a fibrous coating that is water-saturated.
To maintain an approximately uniform surface temperature of $27^{\circ} \mathrm{C}$, water at a prescribed flow rate and temperature passes through the tube.
(a) Considering the heat and mass transfer processes on the external surface of the tube, determine the heat rate from the tube.
(b) For a flow rate of $0.025 \mathrm{~kg} / \mathrm{s}$, determine the inlet temperature, $T_{\mathrm{mm},}$, at which water must be supplied to the tube.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:02

Problem 123

Consider gas flow of mass density $\rho$ and rate $\dot{m}$ through a tube whose inner surface is coated with a liquid or a sublimable solid of uniform vapor density $\rho_{\Lambda, S}$ Derive Equation $8.86$ for variation of the mean vapor density, $\rho_{\Lambda, m}$, with distance $x$ from the tube entrance and Equation $8.83$ for the total rate of vapor transfer from a tube of length $L$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:12

Problem 124

Atmospheric air at $25^{\circ} \mathrm{C}$ and $3 \times 10^{-4} \mathrm{~kg} / \mathrm{s}$ flows through a 10-mm-diameter, 1-m-long circular tube whose inner surface is wetted with a water film. Determine the water vapor density at the tube outlet, assuming the inlet air to be dry. What is the rate at which vapor is added to the air?

Penny Riley
Penny Riley
Numerade Educator
04:08

Problem 125

Air at $25^{\circ} \mathrm{C}$ and $1 \mathrm{~atm}$ is in fully developed flow at $\dot{m}=10^{-3} \mathrm{~kg} / \mathrm{s}$ through a $10-\mathrm{mm}$-diameter circular tube whose inner surface is wetted with water. Determine the tube length required for the water vapor in the air to reach $99 \%$ of saturation. The inlet air is dry.

Chai Santi
Chai Santi
Numerade Educator
01:56

Problem 126

A humidifier consists of a bundle of vertical tubes, each of $20-\mathrm{mm}$ diameter, through which dry atmospheric air is in fully developed flow at $10^{-3} \mathrm{~kg} / \mathrm{s}$ and $298 \mathrm{~K}$. The inner tube surface is wetted with a water film. Determine the tube length required for the water vapor to reach $99 \%$ of saturation. What is the rate at which energy must be supplied to each tube to maintain its temperature at $298 \mathrm{~K}$ ?

Manik Pulyani
Manik Pulyani
Numerade Educator
05:27

Problem 127

The final step of a manufacturing process in which a protective coating is applied to the inner surface of a circular tube involves passage of dry, atmosphere air through the tube to remove a residual liquid associated with the process. Consider a coated 5-m-long tube with an inner diameter of $50 \mathrm{~mm}$. The tube is maintained at a temperature of $300 \mathrm{~K}$, and the residual liquid exists as a thin film whose corresponding vapor pressure is $15 \mathrm{~mm}$ Hg. The molecular weight and diffusion coefficient of the vapor are $\Lambda_{\alpha}=70 \mathrm{~kg} / \mathrm{kmol}$ and $D_{A B}=10^{-5} \mathrm{~m}^{2} / \mathrm{s}$, respectively. Air enters the tube at a mean velocity of $0.5 \mathrm{~m} / \mathrm{s}$ and a temperature of $300 \mathrm{~K}$.
(a) Estimate the partial pressure and mass density of vapor in the air exiting the tube.
(b) What is the rate of liquid removal from the tube in $\mathrm{kg} / \mathrm{s}$ ?

Kudakwashe Mapiki
Kudakwashe Mapiki
Numerade Educator
04:49

Problem 128

Dry air is inhaled at a rate of $10 \mathrm{liter} / \mathrm{min}$ through a trachea with a diameter of $20 \mathrm{~mm}$ and a length of $125 \mathrm{~mm}$. The inner surface of the trachea is at a normal body temperature of $37^{\circ} \mathrm{C}$ and may be assumed to be saturated with water.
(a) Assuming steady, fully developed flow in the trachea, estimate the mass transfer convection coefficient.
(b) Estimate the daily water loss (liter/day) associated with evaporation in the trachea.

James Kiss
James Kiss
Numerade Educator
08:36

Problem 129

A mass transfer operation is preceded by laminar flow of a gaseous species B through a circular tube that is sufficiently long to achieve a fully developed velocity profile. Once the fully developed condition is reached, the gas enters a section of the tube that is wetted with a liquid film (A). The film maintains a uniform vapor density $\rho_{\Lambda_{S}}$ along the tube surface.
(a) Write the differential equation and boundary conditions that govern the species A mass density distribution, $\rho_{A}(x, r)$, for $x>0$.
(b) What is the heat transfer analog to this problem? From this analog, write an expression for the average Sherwood number associated with mass exchange over the region $0 \leq x \leq L$.
(c) Beginning with application of conservation of species to a differential control volume of extent $\pi r_{o}^{2} d x$, derive an expression (Equation 8.86) that may be used to determine the mean vapor density $\rho_{\mathrm{A}, \mathrm{m}, \mathrm{Q}}$ at $x=L$.
(d) Consider conditions for which species $\mathrm{B}$ is air at $25^{\circ} \mathrm{C}$ and $1 \mathrm{~atm}$ and the liquid film consists of water, also at $25^{\circ} \mathrm{C}$. The flow rate is $\dot{m}=2.5 \times 10^{-4} \mathrm{~kg} / \mathrm{s}$, and the tube diameter is $D=10 \mathrm{~mm}$. What is the mean vapor density at the tube outlet if $L=1 \mathrm{~m}$ ?

Sheh Lit Chang
Sheh Lit Chang
University of Washington