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Fundamentals of Heat and Mass Transfer

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

Chapter 10

Boiling and Condensation - all with Video Answers

Educators

Chapter Questions

00:35

Problem 1

Show that, for water at 1-atm pressure with $T_{s}-T_{\text {st }}=$ $10^{\circ} \mathrm{C}$, the Jakob number is much less than unity. What is the physical significance of this result? Verify that this conclusion applies to other fluids.

David Collins
David Collins
Numerade Educator
03:32

Problem 2

The surface of a horizontal, 7 -mm-diameter cylinder is maintained at an excess temperature of $5^{\circ} \mathrm{C}$ in saturated water at $1 \mathrm{~atm}$. Estimate the heat flux using an appropriate free convection correlation and compare your result to the boiling curve of Figure 10.4. Repeat the calculation for a horizontal, $7-\mu \mathrm{m}$-diameter wire at the same excess temperature. What can you say about the general applicability of Figure $10.4$ to all situations involving boiling of water at $1 \mathrm{~atm}$ ?

Narayan Hari
Narayan Hari
Numerade Educator
02:45

Problem 3

The role of surface tension in bubble formation can be demonstrated by considering a spherical bubble of pure saturated vapor in mechanical and thermal equilibrium with its superheated liquid.
(a) Beginning with an appropriate free-body diagram of the bubble, perform a force balance to obtain an expression of the bubble radius,
$$
r_{b}=\frac{2 \sigma}{p_{\mathrm{sta}}-p_{l}}
$$
where $p_{\text {sat }}$ is the pressure of the saturated vapor and $p_{l}$ is the pressure of the superheated liquid outside the bubble.
(b) On a $p-v$ diagram, represent the bubble and liquid states. Discuss what changes in these conditions will cause the bubble to grow or collapse.
(c) Calculate the bubble size under equilibrium conditions for which the vapor is saturated at $101^{\circ} \mathrm{C}$ and the liquid pressure corresponds to a saturation temperature of $100^{\circ} \mathrm{C}$.

VS
Vivek Singh
Numerade Educator
18:21

Problem 4

Estimate the heat transfer coefficient, $h$, associated with Points $A, B, C, D$, and $E$ in Figure 10.4. Which point is associated with the largest value of $h$ ? Which point corresponds to the smallest value of $h$ ? Determine the thickness of the vapor blanket at the Leidenfrost point, neglecting radiation heat transfer through the blanket. Assume the solid is a flat surface.

Joseph Lentino
Joseph Lentino
Numerade Educator
01:33

Problem 5

A long, 1-mm-diameter wire passes an electrical current dissipating $3150 \mathrm{~W} / \mathrm{m}$ and reaches a surface temperature of $126^{\circ} \mathrm{C}$ when submerged in water at $1 \mathrm{~atm}$. What is the boiling heat transfer coefficient? Estimate the value of the correlation coefficient $C_{s, f}$

Mayukh Banik
Mayukh Banik
Numerade Educator
03:04

Problem 6

Estimate the nucleate pool boiling heat transfer coefficient for water boiling at atmospheric pressure on the outer surface of a platinum-plated $10-\mathrm{mm}-$ diameter tube maintained $10^{\circ} \mathrm{C}$ above the saturation temperature.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:51

Problem 11

Water at atmospheric pressure boils on the surface of a large horizontal copper tube. The heat flux is $90 \%$ of the critical value. The tube surface is initially scored; however, over time the effects of scoring diminish and the boiling eventually exhibits behavior similar to that associated with a polished surface. Determine the tube surface temperature immediately after installation and after prolonged service.

Surendra Kumar
Surendra Kumar
Numerade Educator
04:04

Problem 12

The bottom of a copper pan, $150 \mathrm{~mm}$ in diameter, is maintained at $115^{\circ} \mathrm{C}$ by the heating element of an electric range. Estimate the power required to boil the water in this pan. Determine the evaporation rate. What is the ratio of the surface heat flux to the critical heat flux? What pan temperature is required to achieve the critical heat flux?

Banhishikha Sinha
Banhishikha Sinha
Numerade Educator
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Problem 13

A nickel-coated heater element with a thickness of $15 \mathrm{~mm}$ and a thermal conductivity of $50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ is exposed to saturated water at atmospheric pressure. A thermocouple is attached to the back surface, which is well insulated. Measurements at a particular operating condition yield an electrical power dissipation in the heater element of $6.950 \times 10^{7} \mathrm{~W} / \mathrm{m}^{3}$ and a temperature of $T_{o}=266.4^{\circ} \mathrm{C}$.
(a) From the foregoing data, calculate the surface temperature, $T_{s}$, and the heat flux at the exposed surface.
(b) Using the surface heat flux determined in part (a), estimate the surface temperature by applying an appropriate boiling correlation.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
05:34

Problem 14

Advances in very large scale integration (VLSI) of electronic devices on a chip are often restricted by the ability to cool the chip. For mainframe computers, an array of several hundred chips, each of area $25 \mathrm{~mm}^{2}$, may be mounted on a ceramic substrate. A method of cooling the array is by immersion in a low boiling point fluid such as refrigerant R-134a. At 1 atm and $247 \mathrm{~K}$, properties of the saturated liquid are $\mu=1.46 \times 10^{-4} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, $c_{p}=1551 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $\operatorname{Pr}=3.2$. Assume values of $C_{s, f}=0.004$ and $n=1.7$.
(a) Estimate the power dissipated by a single chip if it is operating at $50 \%$ of the critical heat flux. What is the corresponding value of the chip temperature?
(b) Compute and plot the chip temperature as a function of surface heat flux for $0.25 \leq q_{s}^{\prime \prime} / q_{\max }^{\prime \prime} \leq 0.90$.

Stanley Enemuo
Stanley Enemuo
Numerade Educator
01:32

Problem 15

Saturated ethylene glycol at $1 \mathrm{~atm}$ is heated by a horizontal chromium-plated surface which has a diameter of $200 \mathrm{~mm}$ and is maintained at $480 \mathrm{~K}$. Estimate the heating power requirement and the rate of evaporation. What fraction is the power requirement of the maximum power associated with the critical heat flux? At $470 \mathrm{~K}$, properties of the saturated liquid are $\mu=0.38 \times 10^{-3}$ $\mathrm{N} \cdot \mathrm{s} / \mathrm{m}^{2}, c_{p}=3280 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, and $P r=8$.7. The saturated vapor density is $\rho=1.66 \mathrm{~kg} / \mathrm{m}^{3}$. Assume nucleate boiling constants of $C_{s, f}=0.01$ and $n=1.0$.

Manik Pulyani
Manik Pulyani
Numerade Educator
02:38

Problem 16

Copper tubes $25 \mathrm{~mm}$ in diameter and $0.75 \mathrm{~m}$ long are used to boil saturated water at 1 atm.
(a) If the tubes are operated at $75 \%$ of the critical heat flux, how many tubes are needed to provide a vapor production rate of $750 \mathrm{~kg} / \mathrm{h}$ ? What is the corresponding tube surface temperature?
(b) Compute and plot the tube surface temperature as a function of heat flux for $0.25 \leq q_{s}^{\prime \prime} / q_{\max }^{\prime \prime}<0.90$. On the same graph, plot the corresponding number of tubes needed to provide the prescribed vapor production rate.

Anand Jangid
Anand Jangid
Numerade Educator
04:12

Problem 17

Consider a gas-fired boiler in which five coiled, thinwalled, copper tubes of $25-\mathrm{mm}$ diameter and $8-\mathrm{m}$ length are submerged in pressurized water at $4.37$ bars. The walls of the tubes are scored and may be assumed to be isothermal. Combustion gases enter each of the tubes at a temperature of $T_{m, i}=700^{\circ} \mathrm{C}$ and a flow rate of $\dot{m}=0.08 \mathrm{~kg} / \mathrm{s}$, respectively.
(a) Determine the tube wall temperature $T_{s}$ and the gas outlet temperature $T_{m, o}$ for the prescribed conditions. As a first approximation, the properties of the combustion gases may be taken as those of air at $700 \mathrm{~K}$.
(b) Over time the effects of scoring diminish, leading to behavior similar to that of a polished copper surface. Determine the wall temperature and gas outlet temperature for the aged condition.

Anatole Borisov
Anatole Borisov
Numerade Educator
02:22

Problem 18

Estimate the current at which a 1 -mm-diameter nickel wire will burn out when submerged in water at atmospheric pressure. The electrical resistance of the wire is $0.129 \Omega / \mathrm{m}$.

Hubert Agamasu
Hubert Agamasu
Numerade Educator
01:33

Problem 19

Estimate the power $\left(\mathrm{W} / \mathrm{m}^{2}\right)$ required to maintain a brass plate at $\Delta T_{e}=15^{\circ} \mathrm{C}$ while boiling saturated water at $1 \mathrm{~atm}$. What is the power requirement if the water is pressurized to $10 \mathrm{~atm}$ ? At what fraction of the critical heat flux is the plate operating?

Ajay Singhal
Ajay Singhal
Numerade Educator
13:40

Problem 20

A dielectric fluid at atmospheric pressure is heated with a $0.5$-mm-diameter, horizontal platinum wire. Determine the temperature of the wire when the wire is heated at $50 \%$ of the critical heat flux. The properties of the fluid are $c_{p l}=1300 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, h_{f,}=142 \mathrm{~kJ} / \mathrm{kg}, k_{l}=0.075 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $v_{l}=0.32 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \rho_{l}=1400 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{v}=7.2 \mathrm{~kg} / \mathrm{m}^{3}$, $\sigma=12.4 \times 10^{-3} \mathrm{~N} / \mathrm{m}, T_{\text {sot }}=34^{\circ} \mathrm{C}$. Assume the nucleate boiling constants are $C_{s f}=0.005$ and $n=1.7$. For small horizontal cylinders, the critical heat flux is found by multiplying the value associated with large horizontal cylinders by a correction factor $F$, where $F=0.89+2.27$ $\exp \left(-3.44 \mathrm{Co}^{-1 / 2}\right)$. The Confinement number is based on the radius of the cylinder, and the range of applicability for the correction factor is $1.3 \leq \mathrm{Co} \leq 6.7[11]$.

Bruce White
Bruce White
Numerade Educator
03:53

Problem 21

It has been demonstrated experimentally that the critical heat flux is highly dependent on pressure, primarily through the pressure dependence of the fluid surface tension and latent heat of vaporization. Using Equation $10.6$, calculate values of $q_{\max }^{\prime \prime}$ for water on a large horizontal surface as a function of pressure. Demonstrate that the peak critical heat flux occurs at approximately one-third the critical pressure ( $p_{c}=221$ bars). Since all common fluids have this characteristic, suggest what coordinates should be used to plot critical heat flux-pressure values to obtain a universal curve.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:42

Problem 22

In applying dimensional analysis, Kutateladze [9] postulated that the critical heat flux varies with the heat of vaporization, vapor density, surface tension, and the bubble diameter parameter given in Equation 10.4a. Verify that dimensional analysis would yield the following expression for the critical heat flux:
$$
q_{\max }^{\prime \prime}=C h_{f_{g}} \rho_{v}^{1 / 2} D_{b}^{-1 / 2} \sigma^{1 / 2}
$$

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:29

Problem 23

A silicon chip of thickness $L=2.5 \mathrm{~mm}$ and thermal conductivity $k_{s}=135 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ is cooled by boiling a saturated fluorocarbon liquid $\left(T_{\text {sat }}=57^{\circ} \mathrm{C}\right)$ on its surface. The electronic circuits on the bottom of the chip produce a uniform heat flux of $q_{o}^{\prime \prime}=5 \times 10^{4} \mathrm{~W} / \mathrm{m}^{2}$, while the sides of the chip are perfectly insulated.

Properties of the saturated fluorocarbon are $c_{p, l}=$ $1100 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, h_{f g}=84,400 \mathrm{~J} / \mathrm{kg}, \rho_{l}=1619.2 \mathrm{~kg} / \mathrm{m}^{3}$, $\rho_{v}=13.4 \mathrm{~kg} / \mathrm{m}^{3}, \sigma=8.1 \times 10^{-3} \mathrm{~N} / \mathrm{m}, \mu_{1}=440 \times 10^{-6}$ $\mathrm{kg} / \mathrm{m} \cdot \mathrm{s}$, and $P r_{I}=9.01$. In addition, the nucleate boiling constants are $C_{s, f}=0.005$ and $n=1.7$.
(a) What is the steady-state temperature $T_{o}$ at the bottom of the chip? If, during testing of the chip, $q_{o}^{\prime \prime}$ is increased to $90 \%$ of the critical heat flux, what is the new steady-state value of $T_{o}$ ?
(b) Compute and plot the chip surface temperatures (top and bottom) as a function of heat flux for $0.20 \leq q_{o}^{\prime \prime} / q_{\max }^{\prime \prime} \leq 0.90$. If the maximum allowable chip temperature is $80^{\circ} \mathrm{C}$, what is the maximum allowable value of $q_{e}^{\text {"? }}$ ?

Manik Pulyani
Manik Pulyani
Numerade Educator
05:27

Problem 24

What is the critical heat flux for boiling water at 1 atm on a large horizontal surface on the surface of the moon, where the gravitational acceleration is one-sixth that of the earth?

Sri Datta Vikas Buchemmavari
Sri Datta Vikas Buchemmavari
Numerade Educator
02:06

Problem 25

A heater for boiling a saturated liquid consists of two concentric stainless steel tubes packed with dense boron
nitride powder. Electrical current is passed through the inner tube, creating uniform volumetric heating $\dot{q}$ $\left(\mathrm{W} / \mathrm{m}^{3}\right)$. The exposed surface of the outer tube is in contact with the liquid and the boiling heat flux is given as
$$
q_{s}^{\prime \prime}=C\left(T_{s}-T_{\text {sat }}\right)^{3}
$$
It is feared that under high-power operation the stainless steel tubes would severely oxidize if temperatures exceed $T_{s s, x}$ or that the boron nitride would deteriorate if its temperature exceeds $T_{\mathrm{bn}, x^{*}}$ Presuming that the saturation temperature of the liquid $\left(T_{\text {sat }}\right)$ and the boiling surface temperature $\left(T_{x}\right)$ are prescribed, derive expressions for the maximum temperatures in the stainless steel (ss) tubes and in the boron nitride (bn). Express your results in terms of geometric parameters $\left(r_{1}, r_{2}\right.$, $\left.r_{3}, r_{4}\right)$, thermal conductivities $\left(k_{\mathrm{ss}}, k_{\mathrm{ba}}\right)$, and the boiling parameters $\left(C, T_{\text {sial }}, T_{s}\right)$.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
03:09

Problem 26

A device for performing boiling experiments consists of a copper bar $(k=400 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$, which is exposed to a boiling liquid at one end, encapsulates an electrical heater at the other end, and is well insulated from its surroundings at all but the exposed surface. Thermocouples inserted in the bar are used to measure temperatures at distances of $x_{1}=10 \mathrm{~mm}$ and $x_{2}=25 \mathrm{~mm}$ from the surface.
(a) An experiment is performed to determine the boiling characteristics of a special coating applied to the exposed surface. Under steady-state conditions, nucleate boiling is maintained in saturated water at atmospheric pressure, and values of $T_{1}=133.7^{\circ} \mathrm{C}$ and $T_{2}=158.6^{\circ} \mathrm{C}$ are recorded. If $n=1$, what value of the coefficient $C_{s, f}$ is associated with the Rohsenow correlation?
(b) Assuming applicability of the Rohsenow correlation with the value of $C_{s, f}$ determined from part (a), compute and plot the excess temperature $\Delta T_{e}$ as a function of the boiling heat flux for $10^{5} \leq q_{s}^{\prime \prime} \leq 10^{6} \mathrm{~W} / \mathrm{m}^{2}$. What are the corresponding values of $T_{1}$ and $T_{2}$ for $q_{s}^{\prime \prime}=10^{6} \mathrm{~W} / \mathrm{m}^{2}$ ? If $q_{s}^{\prime \prime}$ were increased to $1.5 \times 10^{6} \mathrm{~W} / \mathrm{m}^{2}$, could the foregoing results be extrapolated to infer the corresponding values of $\Delta T_{e}, T_{1}$, and $T_{2}$ ?

James L
James L
Numerade Educator
02:34

Problem 27

A small copper sphere, initially at a uniform, elevated temperature $T(0)=T_{i}$, is suddenly immersed in a large fluid bath maintained at $T_{\text {sat- }}$. The initial temperature of the sphere exceeds the Leidenfrost point corresponding to the temperature $T_{D}$ of Figure $10.4$.
(a) Sketch the variation of the average sphere temperature, $\bar{T}(t)$, with time during the quenching process. Indicate on this sketch the temperatures $T_{i}, T_{D}$, and $T_{\text {sat }}$, as well as the regimes of film, transition, and nucleate boiling and the regime of single-phase convection. Identify key features of the temperature history.
(b) At what time(s) in this cooling process do you expect the surface temperature of the sphere to deviate most from its center temperature? Explain your answer.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:04

Problem 28

A sphere made of aluminum alloy 2024 with a diameter of $20 \mathrm{~mm}$ and a uniform temperature of $500^{\circ} \mathrm{C}$ is suddenly immersed in a saturated water bath maintained at atmospheric pressure. The surface of the sphere has an emissivity of $0.25$.
(a) Calculate the total heat transfer coefficient for the initial condition. What fraction of the total coefficient is contributed by radiation?
(b) Estimate the temperature of the sphere $30 \mathrm{~s}$ after it is immersed in the bath.

Surendra Kumar
Surendra Kumar
Numerade Educator
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Problem 29

A disk-shaped turbine rotor is heat-treated by quenching in water at $p=1 \mathrm{~atm}$. Initially, the rotor is at a uniform temperature of $T_{i}=1100^{\circ} \mathrm{C}$ and the water is at its boiling point as the rotor is lowered into the quenching bath by a harness.
(a) Assuming lumped-capacitance behavior and constant properties for the rotor, carefully plot the rotor temperature versus time, pointing out important features of your $T(t)$ curve. The rotor is in Orientation A.
(b) If the rotor is reoriented so that its large surfaces are horizontal (Orientation B), would the rotor temperature decrease more rapidly or less rapidly relative to Orientation A?

Victor Salazar
Victor Salazar
Numerade Educator
01:38

Problem 30

A steel bar, $20 \mathrm{~mm}$ in diameter and $200 \mathrm{~mm}$ long, with an emissivity of $0.9$, is removed from a furnace at $455^{\circ} \mathrm{C}$ and suddenly submerged horizontally in a water bath under atmospheric pressure. Estimate the initial heat transfer rate from the bar.

Narayan Hari
Narayan Hari
Numerade Educator
01:43

Problem 31

Electrical current passes through a horizontal, 2-mmdiameter conductor of emissivity $0.5$ when immersed in water under atmospheric pressure.
(a) Estimate the power dissipation per unit length of the conductor required to maintain the surface temperature at $555^{\circ} \mathrm{C}$.
(b) For conductor diameters of $1.5,2.0$, and $2.5 \mathrm{~mm}$, compute and plot the power dissipation per unit length as a function of surface temperature for $250 \leq T_{s} \leq 650^{\circ} \mathrm{C}$. On a separate figure, plot the percentage contribution of radiation as a function of $T_{s}$.

Penny Riley
Penny Riley
Numerade Educator
01:33

Problem 32

Consider a horizontal, $D=1$-mm-diameter platinum wire suspended in saturated water at atmospheric pressure. The wire is heated by an electrical current. Determine the heat flux from the wire at the instant when the surface of the wire reaches its melting point. Determine the corresponding centerline temperature of the wire. Due to oxidation at very high temperature, the wire emissivity is $\varepsilon=0.80$ when it burns out. The water vapor properties at the film temperature of $1209 \mathrm{~K}$ are $\rho_{v}=0.189 \mathrm{~kg} / \mathrm{m}^{3}, c_{p, v}=2404 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, $v_{v}=231 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, k_{v}=0.113 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
02:18

Problem 33

A heater element of 5 -mm diameter is maintained at a surface temperature of $350^{\circ} \mathrm{C}$ when immersed horizontally in water under atmospheric pressure. The element sheath is stainless steel with a mechanically polished finish having an emissivity of $0.25$.
(a) Calculate the electrical power dissipation and the rate of vapor production per unit heater length.
(b) If the heater were operated at the same power dissipation rate in the nucleate boiling regime, what temperature would the surface achieve? Calculate the rate of vapor production per unit length for this operating condition.
(c) Sketch the boiling curve and represent the two operating conditions of parts (a) and (b). Compare the results of your analysis. If the heater element is operated in the power-controlled mode, explain how you would achieve these two operating conditions beginning with a cold element.

Keshav Singh
Keshav Singh
Numerade Educator
01:27

Problem 34

The thermal energy generated by a silicon chip increases in proportion to its clock speed. The silicon chip of Problem $10.23$ is designed to operate in the nucleate boiling regime at approximately $30 \%$ of the critical heat flux. A sudden surge in the chip's clock speed triggers film boiling, after which the clock speed and power dissipation return to their design values.
(a) In which boiling regime does the chip operate after the power dissipation returns to its design value?
(b) To return to the nucleate boiling regime, how much must the clock speed be reduced relative to the design value?

Mayukh Banik
Mayukh Banik
Numerade Educator
03:32

Problem 35

A cylinder of $120-\mathrm{mm}$ diameter at $1000 \mathrm{~K}$ is quenched in saturated water at $1 \mathrm{~atm}$. Describe the quenching process and estimate the maximum heat removal rate per unit length during the process.

Narayan Hari
Narayan Hari
Numerade Educator
02:35

Problem 36

A l-mm-diameter horizontal platinum wire of emissivity $\varepsilon=0.25$ is operated in saturated water at 1 -atm pressure.
(a) What is the surface heat flux if the surface temperature is $T_{s}=800 \mathrm{~K}$ ?
(b) For emissivities of $0.1,0.25$, and $0.95$, generate a $\log -\log$ plot of the heat flux as a function of surface excess temperature, $\Delta T_{e} \equiv T_{s}-T_{\text {stit }}$, for $150 \leq \Delta T_{e} \leq 550 \mathrm{~K}$. Show the critical heat flux and the Leidenfrost point on your plot. Separately, plot the percentage contribution of radiation to the total heat flux for $150 \leq \Delta T_{e} \leq 550 \mathrm{~K}$.

Manish Jain
Manish Jain
Numerade Educator
02:23

Problem 37

As strip steel leaves the last set of rollers in a hot rolling mill, it is quenched by planar water jets before being coiled. Due to the large plate temperatures, film boiling is achieved shortly downstream of the jet impingement region.
Consider conditions for which the strip steel beneath the vapor blanket is at a temperature of $907 \mathrm{~K}$ and has an emissivity of $0.35$. Neglecting the effects of the strip and jet motions and assuming boiling within the film to be approximated by that associated with a large horizontal cylinder of 1 - $m$ diameter, estimate the rate of heat transfer per unit surface area from the strip to the wall jet.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:34

Problem 38

A polished copper sphere of $10-\mathrm{mm}$ diameter, initially at a prescribed elevated temperature $T_{i}$, is quenched in a saturated ( $1 \mathrm{~atm}$ ) water bath. Using the lumped capacitance method of Section 5.3.3, estimate the time for the sphere to cool (a) from $T_{i}=130^{\circ} \mathrm{C}$ to $110^{\circ} \mathrm{C}$ and (b) from $T_{i}=550^{\circ} \mathrm{C}$ to $220^{\circ} \mathrm{C}$. Make use of the average sphere temperatures in evaluating properties. Plot the temperature history for each quenching process.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:56

Problem 39

A tube of $2-\mathrm{mm}$ diameter is used to heat saturated water at $1 \mathrm{~atm}$, which is in cross flow over the tube. Calculate and plot the critical heat flux as a function of water velocity over the range 0 to $2 \mathrm{~m} / \mathrm{s}$. On your plot, identify the pool boiling region and the transition region between the low- and high-velocity ranges. Hint: Problem $10.20$ contains relevant information for pool boiling on small-diameter cylinders.

Chai Santi
Chai Santi
Numerade Educator
03:04

Problem 40

Saturated water at $1 \mathrm{~atm}$ and velocity $2 \mathrm{~m} / \mathrm{s}$ flows over a cylindrical heating element of diameter $5 \mathrm{~mm}$. What is the maximum heating rate $(\mathrm{W} / \mathrm{m})$ for nucleate boiling?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
01:37

Problem 41

A vertical steel tube carries water at a pressure of 10 bars. Saturated liquid water is pumped into the $D=0.1$-m-diameter tube at its bottom end $(x=0)$ with a mean velocity of $u_{m}=0.05 \mathrm{~m} / \mathrm{s}$. The tube is exposed to combusting pulverized coal, providing a uniform heat flux of $q^{\prime \prime}=100,000 \mathrm{~W} / \mathrm{m}^{2}$.
(a) Determine the tube wall temperature and the quality of the flowing water at $x=15 \mathrm{~m}$. Assume $G_{s, f}=1$.
(b) Determine the tube wall temperature at a location beyond $x=15 \mathrm{~m}$ where single-phase flow of the
vapor exists at a mean temperature of $T_{\text {sait }}$. Assume the vapor at this location is also at a pressure of 10 bars.
(c) Plot the tube wall temperature in the range $-5 \mathrm{~m}$ $\leq x \leq 30 \mathrm{~m}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
04:38

Problem 42

Consider refrigerant R-134a flowing in a smooth, horizontal, 10-mm-inner-diameter tube of wall thickness $2 \mathrm{~mm}$. The refrigerant is at a saturation temperature of $15^{\circ} \mathrm{C}$ (for which $\rho_{v \text { sat }}=23.75 \mathrm{~kg} / \mathrm{m}^{3}$ ) and flows at a rate of $0.01 \mathrm{~kg} / \mathrm{s}$. Determine the maximum wall temperature associated with a heat flux of $10^{5} \mathrm{~W} / \mathrm{m}^{2}$ at the inner wall at a location $0.4 \mathrm{~m}$ downstream from the onset of boiling for tubes fabricated of (a) pure copper and (b) AISI 316 stainless steel.

Dading Chen
Dading Chen
Numerade Educator
03:42

Problem 43

Determine the tube diameter associated with $p=1 \mathrm{~atm}$ and a critical Confinement number of $0.5$ for ethanol. mercury, water, R-134a, and the dielectric fluid of Problem 10.23.

Prashant Bana
Prashant Bana
Numerade Educator
03:34

Problem 44

Saturated steam at $0.1$ bar condenses with a convection coefficient of $6800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ on the outside of a brass tube having inner and outer diameters of $16.5$ and $19 \mathrm{~mm}$, respectively. The convection coefficient for water flowing inside the tube is $5200 \mathrm{~W} / \mathrm{m}^{2}+\mathrm{K}$. Estimate the steam condensation rate per unit length of the tube when the mean water temperature is $30^{\circ} \mathrm{C}$.

Keshav Singh
Keshav Singh
Numerade Educator
01:11

Problem 45

Consider a container exposed to a saturated vapor, $T_{\text {sat }}$, having a cold bottom surface, $T_{s}<T_{\text {sat }}$, and with insulated sidewalls.
Assuming a linear temperature distribution for the liquid, perform a surface energy balance on the liquid-vapor interface to obtain the following expression for the growth rate of the liquid layer:
$$
\delta(t)=\left[\frac{2 k_{l}\left(T_{\text {sat }}-T_{s}\right)}{\rho_{l} h_{f g}} t\right]^{1 / 2}
$$
Calculate the thickness of the liquid layer formed in $1 \mathrm{~h}$ for a $200-\mathrm{mm}^{2}$ bottom surface maintained at $80^{\circ} \mathrm{C}$ and exposed to saturated steam at $1 \mathrm{~atm}$. Compare

Manik Pulyani
Manik Pulyani
Numerade Educator
01:21

Problem 46

Saturated steam at 1 atm condenses on the outer surface of a vertical, 100 -mm-diameter pipe $1 \mathrm{~m}$ long, having a uniform surface temperature of $94^{\circ} \mathrm{C}$. Estimate the total condensation rate and the heat transfer rate to the pipe.

Dading Chen
Dading Chen
Numerade Educator
03:41

Problem 47

Determine the total condensation rate and the heat transfer rate for Problem $10.46$ when the steam is saturated at $1.5$ bars.

Dading Chen
Dading Chen
Numerade Educator
02:01

Problem 48

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

Narayan Hari
Narayan Hari
Numerade Educator
02:38

Problem 49

A vertical plate $500 \mathrm{~mm}$ high and $200 \mathrm{~mm}$ wide is to be used to condense saturated steam at 1 atm.
(a) At what surface temperature must the plate be maintained to achieve a condensation rate of $\dot{m}=25 \mathrm{~kg} / \mathrm{h}$ ?
(b) Compute and plot the surface temperature as a function of condensation rate for $15 \leq \dot{m} \leq 50 \mathrm{~kg} / \mathrm{h}$.
(c) On the same graph and for the same range of $\dot{m}$, plot the surface temperature as a function of condensation rate if the plate is $200 \mathrm{~mm}$ high and $500 \mathrm{~mm}$ wide.

Anand Jangid
Anand Jangid
Numerade Educator
01:21

Problem 51

A $2 \mathrm{~m} \times 2 \mathrm{~m}$ vertical plate is exposed on one side to saturated steam at atmospheric pressure and on the other side to cooling water that maintains a plate temperature of $50^{\circ} \mathrm{C}$.
(a) What is the rate of heat transfer to the coolant? What is the rate at which steam condenses on the plate?
(b) For plates inclined at an angle $\theta$ from the vertical, the average convection coefficient for condensation on the upper surface, $\bar{h}_{L \text { (incl), }}$, may be approximated by an expression of the form, $\bar{h}_{L \text { (incl) }}=$ $(\cos \theta)^{1 / 4} \cdot \bar{h}_{L \text { vert) }}$, where $\bar{h}_{L \text { (vert) }}$ is the average coefficient for the vertical orientation. If the $2 \mathrm{~m} \times 2 \mathrm{~m}$ plate is inclined $45^{\circ}$ from the normal, what are the rates of heat transfer and condensation?

Dading Chen
Dading Chen
Numerade Educator
01:21

Problem 52

A vertical plate $2.5 \mathrm{~m}$ high, maintained at a uniform temperature of $54^{\circ} \mathrm{C}$, is exposed to saturated steam at atmospheric pressure.
(a) Estimate the condensation and heat transfer rates per unit width of the plate.
(b) If the plate height were halved, would the flow regime stay the same or change?
(c) For $54 \leq T_{s} \leq 90^{\circ} \mathrm{C}$, plot the condensation rate as a function of plate temperature for the two plate heights of parts (a) and (b).

Dading Chen
Dading Chen
Numerade Educator
01:42

Problem 53

Two configurations are being considered in the design of a condensing system for steam at 1 atm employing a vertical plate maintained at $90^{\circ} \mathrm{C}$. The first configuration is a single vertical plate $L \times w$ and the second consists of two vertical plates $(L / 2) \times w$, where $L$ and $w$ are the vertical and horizontal dimensions, respectively. Which configuration would you choose?

Manik Pulyani
Manik Pulyani
Numerade Educator
04:41

Problem 54

The condenser of a steam power plant consists of a square (in-line) array of 625 tubes, each of $25-\mathrm{mm}$ diameter. Consider conditions for which saturated steam at $0.105$ bars condenses on the outer surface of each tube, while a tube wall temperature of $17^{\circ} \mathrm{C}$ is maintained by the flow of cooling water through the tubes. What is the rate of heat transfer to the water per unit length of the tube array? What is the corresponding condensation rate?

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
04:41

Problem 55

The condenser of a steam power plant consists of AISI 302 stainless steel tubes $\left(k_{s}=15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$, each of outer and inner diameters $D_{o}=30 \mathrm{~mm}$ and $D_{i}=$ $26 \mathrm{~mm}$, respectively. Saturated steam at $0.135$ bar condenses on the outer surface of a tube, while water at a mean temperature of $T_{m}=290 \mathrm{~K}$ is in fully developed flow through the tube.
(a) For a water flow rate of $\dot{m}=0.25 \mathrm{~kg} / \mathrm{s}$, what is the outer surface temperature $T_{s, o}$ of the tube and the rates of heat transfer and steam condensation per unit tube length? As a first estimate, you may evaluate the properties of the liquid film at the saturation temperature. If one wishes to increase the transfer rates, what is the limiting factor that should be addressed?
(b) Explore the effect of the water flow rate on $T_{s, o}$ and the rate of heat transfer per unit length.

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
10:04

Problem 56

Saturated vapor from a chemical process condenses at a slow rate on the inner surface of a vertical, thinwalled cylindrical container of length $L$ and diameter D. The container wall is maintained at a uniform temperature $T_{s}$ by flowing cold water across its outer surface.
Derive an expression for the time, $t_{f}$, required to fill the container with condensate, assuming that the condensate film is laminar. Express your result in terms of $D, L,\left(T_{\text {sat }}-T_{s}\right), g$, and appropriate fluid properties.
Derive an expression for the time, $t_{f}$, required to fill the container with condensate, assuming that the condensate film is laminar. Express your result in terms of $D, L,\left(T_{\text {sat }}-T_{s}\right), g$, and appropriate fluid properties.

Eduard Sanchez
Eduard Sanchez
Numerade Educator
03:41

Problem 57

Determine the total condensation rate and heat transfer rate for the process of Problem 10.46 when the pipe is oriented at angles of $\theta=0,30,45$, and $60^{\circ}$ from the horizontal.

Dading Chen
Dading Chen
Numerade Educator
01:14

Problem 58

A horizontal tube of $50-\mathrm{mm}$ outer diameter, with a surface temperature of $34^{\circ} \mathrm{C}$, is exposed to steam at $0.2$ bar. Estimate the condensation rate and heat transfer rate per unit length of the tube.

Naman Kumar
Naman Kumar
Numerade Educator
01:39

Problem 59

The tube of Problem $10.58$ is modified by milling sharp-cornered grooves around its periphery, as in Figure $10.15$. The 2 -mm-deep grooves are each $2 \mathrm{~mm}$ wide with a pitch of $S=4 \mathrm{~mm}$. Estimate the minimum condensation and heat transfer rates per unit length that would be expected for the modified tube. How much is the performance enhanced relative to the original tube of Problem 10.58?

Penny Riley
Penny Riley
Numerade Educator
02:38

Problem 60

A horizontal tube $1 \mathrm{~m}$ long with a surface temperature of $70^{\circ} \mathrm{C}$ is used to condense saturated steam at $1 \mathrm{~atm}$.
(a) What diameter is required to achieve a condensation rate of $125 \mathrm{~kg} / \mathrm{h}$ ?
(b) Plot the condensation rate as a function of surface temperature for $70 \leq T_{s} \leq 90^{\circ} \mathrm{C}$ and tube diameters of $125,150,175 \mathrm{~mm}$.

Anand Jangid
Anand Jangid
Numerade Educator
02:38

Problem 61

Saturated steam at a pressure of $0.1$ bar is condensed over a square array of 100 tubes each of diameter $8 \mathrm{~mm}$.
(a) If the tube surfaces are maintained at $27^{\circ} \mathrm{C}$, estimate the condensation rate per unit tube length.
(b) Subject to the requirement that the total number of tubes and the tube diameter are fixed at 100 and $8 \mathrm{~mm}$, respectively, what options are available for increasing the condensation rate? Assess these options quantitatively.

Anand Jangid
Anand Jangid
Numerade Educator
04:41

Problem 62

A thin-walled concentric tube heat exchanger of $0.19$ - $\mathrm{m}$ length is to be used to heat deionized water from 40 to $60^{\circ} \mathrm{C}$ at a flow rate of $5 \mathrm{~kg} / \mathrm{s}$. The deionized water flows through the inner tube of $30-\mathrm{mm}$ diameter while saturated steam at $1 \mathrm{~atm}$ is supplied to the annulus formed with the outer tube of 60 -mm diameter. The thermophysical properties of the deionized water are $\rho=982.3 \mathrm{~kg} / \mathrm{m}^{3}, c_{p}=4181 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, k=0.643 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\mu=548 \times 10^{-6} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, and $P r=3.56$. Estimate the convection coefficients for both sides of the tube and determine the inner tube wall outlet temperature. Does condensation provide a fairly uniform inner tube wall temperature equal approximately to the saturation temperature of the steam?

Hariprasad Annamalai
Hariprasad Annamalai
Numerade Educator
04:00

Problem 63

A technique for cooling a multichip module involves submerging the module in a saturated fluorocarbon liquid. Vapor generated due to boiling at the module surface is condensed on the outer surface of copper tubing suspended in the vapor space above the liquid. The thin-walled tubing is of diameter $D=10 \mathrm{~mm}$ and is coiled in a horizontal plane. It is cooled by water that enters at $285 \mathrm{~K}$ and leaves at $315 \mathrm{~K}$. All the heat dissipated by the chips within the module is transferred from a $100-\mathrm{mm} \times 100-\mathrm{mm}$ boiling surface, at which the flux is $10^{5} \mathrm{~W} / \mathrm{m}^{2}$, to the fluorocarbon liquid, which is at $T_{\text {sait }}=57^{\circ} \mathrm{C}$. Liquid properties are $k_{l}=0.0537$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}, c_{p, l}=1100 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, h_{f g}^{\prime} \approx h_{f g}=84,400 \mathrm{~J} / \mathrm{kg}$, $\rho_{l}=1619.2 \mathrm{~kg} / \mathrm{m}^{3}, \rho_{v}=13.4 \mathrm{~kg} / \mathrm{m}^{3}, \sigma=8.1 \times 10^{-3}$ $\mathrm{N} / \mathrm{m}, \mu_{l}=440 \times 10^{-6} \mathrm{~kg} / \mathrm{m} \cdot \mathrm{s}$, and $P r_{l}=9$.
(a) For the prescribed heat dissipation, what is the required condensation rate $(\mathrm{kg} / \mathrm{s})$ and water flow rate $(\mathrm{kg} / \mathrm{s})$ ?
(b) Assuming fully developed flow throughout the tube, determine the tube surface temperature at the coil inlet and outlet.
(c) Assuming a uniform tube surface temperature of $T_{s}=53.0^{\circ} \mathrm{C}$, determine the required length of the coil.

Paul Gabriel
Paul Gabriel
Numerade Educator
03:43

Problem 64

Determine the rate of condensation on a $100-\mathrm{mm}-$ diameter sphere with a surface temperature of $150^{\circ} \mathrm{C}$ in saturated ethylene glycol vapor at 1 atm. Approximate the liquid properties as those corresponding to saturated conditions at $373 \mathrm{~K}$ (Table A.5).

Charles Thomas
Charles Thomas
Numerade Educator
03:07

Problem 65

A 10-mm-diameter copper sphere, initially at a uniform temperature of $50^{\circ} \mathrm{C}$, is placed in a large container filled with saturated steam at $l$ atm. Using the lumped capacitance method, estimate the time required for the sphere to reach an equilibrium condition. How much condensate $(\mathrm{kg})$ was formed during this period?

Narayan Hari
Narayan Hari
Numerade Educator
02:46

Problem 66

The Clean Air Act prohibited the production of chlorofluorocarbons (CFCs) in the United States as of 1996. One widely used $\mathrm{CFC}$, refrigerant $\mathrm{R}-12$, has been replaced by $\mathrm{R}-134 \mathrm{a}$ in many applications because of their similar properties, including a low boiling point at atmospheric pressure, $T_{\text {sat }}=243 \mathrm{~K}$ and $246.9 \mathrm{~K}$ for $\mathrm{R}-12$ and $\mathrm{R}-134 \mathrm{a}$, respectively. Compare the performance of these two refrigerants under the following conditions. The saturated refrigerant vapor at $310 \mathrm{~K}$ is condensed as it flows through a 30 -mmdiameter, 0.8-m-long tube whose wall temperature is maintained at $290 \mathrm{~K}$. If vapor enters the tube at a flow rate of $0.010 \mathrm{~kg} / \mathrm{s}$, what is the rate of condensation and the flow rate of vapor leaving the tube? The relevant properties of $\mathrm{R}-12$ at $T_{\text {et }}=310 \mathrm{~K}$ are $\rho_{e}=50.1 \mathrm{~kg} / \mathrm{m}^{3}$, $h_{f s}=160 \mathrm{~kJ} / \mathrm{kg}$, and $\mu_{v}=150 \times 10^{-7} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$ and those of liquid $\mathrm{R}-12$ at $T_{f}=300 \mathrm{~K}$ are $\rho_{l}=1306$ $\mathrm{kg} / \mathrm{m}^{3}, c_{p,}=978 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu_{I}=2.54 \times 10^{-4} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$, $k_{l}=0.072 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The properties of the saturated $\mathrm{R}-134 \mathrm{a}$ vapor are $\rho_{v}=46.1 \mathrm{~kg} / \mathrm{m}^{3}, h_{f 8}=166 \mathrm{~kJ} / \mathrm{kg}$, and $\mu_{v}=136 \times 10^{-7} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$.

Vipender Yadav
Vipender Yadav
Numerade Educator
03:41

Problem 67

Saturated steam at $1.5$ bars condenses inside a horizontal, 75-mm-diameter pipe whose surface is maintained at $100^{\circ} \mathrm{C}$. Assuming low vapor velocities and film condensation, estimate the heat transfer coefficient and the condensation rate per unit length of the pipe.

Dading Chen
Dading Chen
Numerade Educator
02:59

Problem 68

Consider the situation of Problem $10.67$ at relatively high vapor velocities, with a fluid mass flow rate of $\dot{m}=2.5 \mathrm{~kg} / \mathrm{s}$.
(a) Determine the heat transfer coefficient and condensation rate per unit length of tube for a mass fraction of vapor of $X=0.2$.
(b) Plot the heat transfer coefficient and the condensation rate for $0.1 \leq X \leq 0.3$.

Averell Hause
Averell Hause
Carnegie Mellon University
04:48

Problem 69

Refrigerant R-22 with a mass flow rate of $m=8.75 \times$ $10^{-3} \mathrm{~kg} / \mathrm{s}$ is condensed inside a 7-mm-diameter tube. Annular flow is observed. The saturation temperature of the pressurized refrigerant is $T_{\text {sur }}=45^{\circ} \mathrm{C}$, and the wall temperature is $T_{n}=40^{\circ} \mathrm{C}$. Vapor properties are $\rho_{v}=77 \mathrm{~kg} / \mathrm{m}^{3}$ and $\mu_{e}=15 \times 10^{-6} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}$.
(a) Determine the heat transfer coefficient and the heat transfer and condensation rates per unit length at a quality of $X=0.5$.
(b) Plot the condensation rate per unit length over the range $0.2<X<0.8$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:41

Problem 70

Consider Problem 10.44. In an effort to increase the condensation rate, an engineer proposes to apply an $L=100-\mu \mathrm{m}$-thick Teflon coating to the exterior surface of the brass tube to promote dropwise condensation. Estimate the new condensation convection coefficient and the steam condensation rate per unit length of the tube after application of the coating. Comment on the proposed scheme's effect on the condensation rate (the condensation rate per unit length in Problem $10.44$ is approximately $1 \times 10^{-3} \mathrm{~kg} / \mathrm{s}$ ).

Dading Chen
Dading Chen
Numerade Educator
01:57

Problem 71

10.71 Wetting of some metallic surfaces can be inhibited by means of ion implantation of the surface prior to its use, thereby promoting dropwise condensation. The degree of wetting inhibition and, in turn, the efficacy of the implantation process vary from metal to metal. Consider a vertical metal plate that is exposed to saturated steam at atmospheric pressure. The plate is $t=1 \mathrm{~mm}$ thick, and its vertical and horizontal dimensions are $L=250 \mathrm{~mm}$ and $b=100 \mathrm{~mm}$, respectively. The temperature of the plate surface that is exposed to the steam is found to be $T_{s}=90^{\circ} \mathrm{C}$ when the opposite surface of the metal plate is held at a cold temperature, $T_{\alpha^{*}}$
(a) Determine $T_{c}$ for 2024-T6 aluminum. Assume the ion-implantation process does not promote dropwise condensation for this metal.
(b) Determine $T_{c}$ for AISI 302 stainless steel, assuming the ion-implantation process is effective in promoting dropwise condensation.

Nick Johnson
Nick Johnson
Numerade Educator
06:19

Problem 72

A passive technique for cooling heat-dissipating integrated circuits involves submerging the ICs in a low boiling point dielectric fluid. Vapor generated in cooling the circuits is condensed on vertical plates suspended in the vapor cavity above the liquid. The temperature of the plates is maintained below the saturation temperature, and during steady-state operation a balance is established between the rate of heat transfer to the condenser plates and the rate of heat dissipation by the ICs.
Consider conditions for which the $25-\mathrm{mm}^{2}$ surface area of each IC is submerged in a fluorocarbon liquid for which $T_{\mathrm{ser}}=50^{\circ} \mathrm{C}, \rho_{l}=1700 \mathrm{~kg} / \mathrm{m}^{3}, c_{p l}=1005 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, $\mu_{l}=6.80 \times 10^{-4} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}, k_{l}=0.062 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}, P r_{l}=$ $11.0, \quad \sigma=0.013 \mathrm{~N} / \mathrm{m}, h_{\text {f }}=1.05 \times 10^{5} \mathrm{~J} / \mathrm{kg}, C_{x, f}=$ $0.004$, and $n=1.7$. If the integrated circuits are operated at a surface temperature of $T_{x}=75^{\circ} \mathrm{C}$, what is the rate at which heat is dissipated by each circuit? If the condenser plates are of height $H=50 \mathrm{~mm}$ and are maintained at a temperature of $T_{c}=15^{\circ} \mathrm{C}$ by an internal coolant, how much condenser surface area must be provided to balance the heat generated by 500 integrated circuits?

Km Neeraj
Km Neeraj
Numerade Educator
01:05

Problem 73

A thermosyphon consists of a closed container that absorbs heat along its boiling section and rejects heat along its condensation section. Consider a thermosyphon made from a thin-walled mechanically polished stainless steel cylinder of diameter D. Heat supplied to the thermosyphon boils saturated water at atmospheric pressure on the surfaces of the lower boiling section of length $L_{b}$ and is then rejected by condensing vapor into a thin film, which falls by gavily alung the wall of the sualeusation sectiva of length $L_{c}$ back into the boiling section. The two sections are separated by an insulated section of length $L_{\mathrm{i}}$. The top surface of the condensation section may be treated as being insulated. The thermosyphon dimensions are $D=20 \mathrm{~mm}, L_{b}=20 \mathrm{~mm}, L_{c}=40 \mathrm{~mm}$, and $L_{i}=40 \mathrm{~mm}$.
(a) Find the mean surface temperature, $T_{x, b}$, of the boiling surface if the nucleate boiling heat flux is to be maintained at $30 \%$ of the critical heat flux.
(b) Find the total condensation flow rate, $\dot{m}$, and the mean surface temperature of the condensation section, $T_{x, c}$

Aadit Sharma
Aadit Sharma
Numerade Educator
01:30

Problem 74

A novel scheme for cooling computer chips uses a thermosyphon containing a saturated fluorocarbon. The chip is brazed to the bottom of a cuplike container, within which heat is dissipated by boiling and subsequently transferred to an external coolant (water) via condensation on the inner surface of a thin-walled tube.
The nucleate boiling constants and the properties of the fluorocarbon are provided in Problem 10.23. In addition, $k_{l}=0.054 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$.
(a) If the chip operates under steady-state conditions and its surface heat flux is maintained at $90 \%$ of the critical heat flux, what is its temperature T? What is the total power dissipation if the chip width is $L_{c}=20 \mathrm{~mm}$ on a side?
(b) If the tube diameter is $D=30 \mathrm{~mm}$ and its surface is maintained at $T_{s}=25^{\circ} \mathrm{C}$ by the water, what tube length $L$ is required to maintain the designated conditions?

Jincy M Saji
Jincy M Saji
Numerade Educator
09:16

Problem 75

A condenser-boiler section contains a $2-\mathrm{m} \times 2-\mathrm{m}$ copper plate operating at a uniform temperature of $T_{x}=100^{\circ} \mathrm{C}$ and separating saturated steam, which is condensing, from a saturated liquid-X, which experiences nucleate pool boiling. A portion of the boiling curve for liquid- $X$ is shown as follows. Both saturated steam and saturated liquid- $X$ are supplied to the system, while water condensate and vapor-X are removed by means not shown in the sketch. At a pressure of 1 bar, fluid- $X$ has a saturation temperature and a latent heat of vaporization of $T_{\text {sat }}=80^{\circ} \mathrm{C}$ and $h_{f g}=700,000 \mathrm{~J} / \mathrm{kg}$, respectively.
(a) Estimate the rates of evaporation and condensation $(\mathrm{kg} / \mathrm{s})$ for the two fluids.
(b) Determine the saturation temperature $T_{\text {sat }}$ and pressure $p$ for the steam, assuming that film condensation occurs.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
04:47

Problem 76

A thin-walled cylindrical container of diameter $D$ and height $L$ is filled to a height $y$ with a low boiling point liquid (A) at $T_{\text {e.a. }}$. The container is located in a large chamber filled with the vapor of a high boiling point fluid (B). Vapor-B condenses into a laminar film on the outer surface of the cylindrical container, extending from the location of the liquid-A free surface. The condensation process sustains nucleate boiling in liquid-A along the container wall according to the relation $q^{n}=$ $C\left(T_{x}-T_{\text {sat }}\right)^{3}$, where $C$ is a known empirical constant.
(a) For the portion of the wall covered with the condensate film, derive an equation for the average temperature of the container wall, $T_{x}$. Assume that the properties of fluids $A$ and $B$ are known.
(b) At what rate is heat supplied to liquid-A?
(c) Assuming the container is initially filled completely with liquid, that is, $y=L$, derive an expression for the time required to evaporate all the liquid in the container.

Prachi Joshi
Prachi Joshi
Numerade Educator
09:57

Problem 77

It has been proposed that the very hot air trapped inside the attic of a house in the summer may be used as the energy source for a passive water heater installed in the attic. Energy costs associated with heating the cool water and air conditioning the house are both reduced. Ten thermosyphons, similar to that of Problem $10.73$, are inserted in the bottom of a well-insulated water heater. Each thermosyphon has a condensing section that is $L_{c}=50 \mathrm{~mm}$ long, an insulated section that is of length $L_{i}=40 \mathrm{~mm}$, and a boiling section that is $L_{w}=30 \mathrm{~mm}$ long. The diameter of each thermosyphon is $D=20 \mathrm{~mm}$. The working fluid within the thermosyphons is water at a pressure of $p=0.047$ bars.
(a) Determine the heating rate delivered by the 10 thermosyphons when boiling occurs at $25 \%$ of the CHF. What are the mean temperatures of the boiling and condensing sections?
(b) At night the attic air temperature drops below the temperature of the water. Estimate the heat loss from the hot water tank to the cool attic, assuming losses through the tank insulation are negligible and the stainless steel tube wall thickness of each thermosyphon is very small.

Rachel Peterson
Rachel Peterson
Numerade Educator
03:04

Problem 107

Plot the nucleate boiling heat flux for saturated water at atmospheric pressure on a large, horizontal polished copper plate, over the excess temperature range $5^{\circ} \mathrm{C} \leq$ $\Delta T_{e} \leq 30^{\circ} \mathrm{C}$. Compare your results with Figure 10.4. Also find the excess temperature corresponding to the critical heat flux.

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
02:38

Problem 108

10.8 A simple expression to account for the effect of pressure on the nucleate boiling convection coefficient in water $\left(\mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)$ is
$$
h=C\left(\Delta T_{c}\right)^{n}\left(\frac{p}{p_{a}}\right)^{0.4}
$$
where $p$ and $p_{a}$ are the system pressure and standard atmospheric pressure, respectively. For a horizontal plate and the range $15<q_{s}^{\prime \prime}<235 \mathrm{~kW} / \mathrm{m}^{2}, C=5.56$ and $n=3$. Units of $\Delta T_{e}$ are kelvins. Compare predictions from this expression with the Rohsenow correlation $\left(C_{x, f}=0.013, n=1\right)$ for pressures of 2 and 5 bars with $\Delta T_{e}=10^{\circ} \mathrm{C}$.

James Kiss
James Kiss
Numerade Educator
06:23

Problem 109

In Example 10.1 we considered conditions for which vigorous boiling occurs in a pan of water, and we determined the electric power (heat rate) required to maintain a prescribed temperature for the bottom of the pan. However, the electric power is, in fact, the control (independent) variable, from which the temperature of the pan follows.
(a) For nucleate boiling in the copper pan of Example $10.1$, compute and plot the temperature of the pan as a function of the heat rate for $1 \leq q \leq$ $100 \mathrm{~kW}$.
(b) If the water is initially at room temperature, it must, of course, be heated for a period of time before it will boil. Consider conditions shortly after heating is initiated and the water is at $20^{\circ} \mathrm{C}$. Estimate the temperature of the pan bottom for a heat rate of $8 \mathrm{~kW}$.

Susan Hallstrom
Susan Hallstrom
Numerade Educator
05:57

Problem 110

Calculate the critical heat flux on a large horizontal surface for the following fluids at 1 atm: mercury, ethanol, and refrigerant R-134a. Compare these results to the critical heat flux for water at $1 \mathrm{~atm}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator