• Home
  • Textbooks
  • Fundamentals of Heat and Mass Transfer
  • Heat Exchangers

Fundamentals of Heat and Mass Transfer

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

Chapter 11

Heat Exchangers - all with Video Answers

Educators

Chapter Questions

02:50

Problem 1

In a fire-tube boiler, hot products of combustion flowing through an array of thin-walled tubes are used to boil water flowing over the tubes. At the time of installation, the overall heat transfer coefficient was $400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. After 1 year of use, the inner and outer tube surfaces are fouled, with corresponding fouling factors of $R_{f, i}^{N}=0.0015$ and $R_{f, w}^{*}=0.0005 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$, respectively. Should the boiler be scheduled for cleaning of the tube surfaces?

Narayan Hari
Narayan Hari
Numerade Educator
04:00

Problem 2

A type-302 stainless steel tube of inner and outer diameters $D_{i}=22 \mathrm{~mm}$ and $D_{v}=27 \mathrm{~mm}$, respectively, is used in a cross-flow heat exchanger. The fouling factors, $R_{f}^{\prime \prime}$, for the inner and outer surfaces are estimated to be $0.0004$ and $0.0002 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$, respectively.
(a) Determine the overall heat transfer coefficient based on the outside area of the tube, $U_{a}$. Compare the
thermal resistances due to convection, tube wall conduction, and fouling.
(b) Instead of air flowing over the tube, consider a situation for which the cross-flow fluid is water at $15^{\circ} \mathrm{C}$ with a velocity of $V_{a}=1 \mathrm{~m} / \mathrm{s}$. Determine the overall heat transfer coefficient based on the outside area of the tube, $U_{o}$ Compare the thermal resistances due to convection, tube wall conduction, and fouling.
(c) For the water-air conditions of part (a) and mean velocities, $u_{m, i}$, of $0.2,0.5$, and $1.0 \mathrm{~m} / \mathrm{s}$, plot the overall heat transfer coefficient as a function of the cross-flow velocity for $5 \leq V_{a} \leq 30 \mathrm{~m} / \mathrm{s}$.
(d) For the water-water conditions of part (b) and cross-flow velocities, $V_{a}$ of 1,3 , and $8 \mathrm{~m} / \mathrm{s}$, plot the overall heat transfer coefficient as a function of the mean velocity for $0.5 \leq u_{m, i} \leq 2.5 \mathrm{~m} / \mathrm{s}$.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:00

Problem 3

11.3 A shell-and-tube heat exchanger is to heat an acidic liquid that flows in unfinned tubes of inside and outside diameters $D_{i}=10 \mathrm{~mm}$ and $D_{\mathrm{o}}=11 \mathrm{~mm}$, respectively. A hot gas flows on the shell side. To avoid corrosion of the tube material, the engineer may specify either a Ni-Cr-Mo corrosion-resistant metal alloy $\left(\rho_{m}=8900 \mathrm{~kg} / \mathrm{m}^{3}, k_{\mathrm{w}}=8\right.$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K})$ or a polyvinylidene fluoride (PVDF) plastic $\left(\rho_{p}=1780 \mathrm{~kg} / \mathrm{m}^{3}, k_{p}=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$. The inner and outer heat transfer coefficients are $h_{j}=1500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and $h_{v}=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively.
(a) Determine the ratio of plastic to metal tube surface areas needed to transfer the same amount of heat.
(b) Determine the ratio of plastic to metal mass associated with the two heat exchanger designs.
(c) The cost of the metal alloy per unit mass is three times that of the plastic. Determine which tube material should be specified on the basis of cost.
11.4 A steel tube $(k=50 \mathrm{~W} / \mathrm{m}-\mathrm{K})$ of inner and outer diameters $D_{i}=20 \mathrm{~mm}$ and $D_{o}=26 \mathrm{~mm}$, respectively, is used to transfer heat from hot gases flowing over the tube $\left(h_{\mathrm{h}}=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)$ to cold water flowing through the tube $\left(h_{c}=8000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)$. What is the cold-side overall heat transfer coefficient $U_{c}$ ? To enhance heat transfer, 16 straight fins of rectangular profile are installed longitudinally along the outer surface of the tube. The fins are equally spaced around the circumference of the tube, each having a thickness of $2 \mathrm{~mm}$ and a length of $15 \mathrm{~mm}$. What is the corresponding overall heat transfer coefficient $U_{c}$ ?

Paul Gabriel
Paul Gabriel
Numerade Educator
03:00

Problem 4

A steel tube $(k=50 \mathrm{~W} / \mathrm{m}$ - $\mathrm{K})$ of inner and outer diameters $D_{i}=20 \mathrm{~mm}$ and $D_{a}=26 \mathrm{~mm}$, respectively, is used to transfer heat from hot gases flowing over the tube $\left(h_{\mathrm{h}}=200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\right)$ to cold water flowing through the tube $\left(h_{c}=8000 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}\right)$. What is the cold-side overall heat transfer coefficient $U_{c}$ ? To enhance heat transfer, 16 straight fins of rectangular profile are installed longitudinally along the outer surface of the tube. The fins are equally spaced around the circumference of the tube, each having a thickness of $2 \mathrm{~mm}$ and a length of $15 \mathrm{~mm}$. What is the corresponding overall heat transfer coefficient $U_{c}$ ?

Anand Jangid
Anand Jangid
Numerade Educator
02:50

Problem 5

A heat recovery device involves transferring energy from the hot flue gases passing through an annular region to pressurized water flowing through the inner tube of the annulus. The inner tube has inner and outer diameters of 24 and $30 \mathrm{~mm}$ and is connected by eight struts to an insulated outer tube of $60-\mathrm{mm}$ diameter. Each strut is $3 \mathrm{~mm}$ thick and is integrally fabricated with the inner tube from carbon steel $(k=50 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$.
Consider conditions for which water at $300 \mathrm{~K}$ flows through the inner tube at $0.161 \mathrm{~kg} / \mathrm{s}$ while flue gases at $800 \mathrm{~K}$ flow through the annulus, maintaining a convection coefficient of $100 \mathrm{~W} / \mathrm{m}^{2}-\mathrm{K}$ on both the struts and the outer surface of the inner tube. What is the rate of heat transfer per unit length of tube from gas to the water?

Narayan Hari
Narayan Hari
Numerade Educator
01:30

Problem 6

A novel design for a condenser consists of a tube of thermal conductivity $200 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ with longitudinal fins snugly fitted into a larger tube. Condensing refrigerant at $45^{\circ} \mathrm{C}$ flows axially through the inner tube, while water at a flow rate of $0.012 \mathrm{~kg} / \mathrm{s}$ passes through the six channels around the inner tube. The pertinent diameters are $D_{1}=10 \mathrm{~mm}, D_{2}=14 \mathrm{~mm}$, and $D_{3}=50 \mathrm{~mm}$, while the fin thickness is $t=2 \mathrm{~mm}$. Assume that the convection coefficient associated with the condensing refrigerant is extremely large.
Determine the heat removal rate per unit tube length in a section of the tube for which the water is at $15^{\circ} \mathrm{C}$.

Jincy M Saji
Jincy M Saji
Numerade Educator
02:25

Problem 7

The condenser of a steam power plant contains $N=1000$ brass tubes $\left(k_{\mathrm{t}}=110 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$, each of inner and outer diameters, $D_{i}=25 \mathrm{~mm}$ and $D_{o}=$ $28 \mathrm{~mm}$, respectively. Steam condensation on the outer surfaces of the tubes is characterized by a convection coefficient of $h_{o}=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) If cooling water from a large lake is pumped through the condenser tubes at $m_{c}=400 \mathrm{~kg} / \mathrm{s}$, what is the overall heat transfer coefficient $U_{o}$ based on the outer surface area of a tube? Properties of the water may be approximated as $\mu=9.60 \times$ $10^{-4} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}, k=0.60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\mathrm{Pr}=6.6 .$
(b) If, after extended operation, fouling provides a resistance of $R_{f, i}^{\prime}=10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$, at the inner surface, what is the value of $U_{o}$ ?
(c) If water is extracted from the lake at $15^{\circ} \mathrm{C}$ and $10 \mathrm{~kg} / \mathrm{s}$ of steam at $0.0622$ bars are to be condensed, what is the corresponding temperature of the water leaving the condenser? The specific heat of the water is $4180 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.

Anand Jangid
Anand Jangid
Numerade Educator
03:55

Problem 8

Thin-walled aluminum tubes of diameter $D=10 \mathrm{~mm}$ are used in the condenser of an air conditioner. Under normal operating conditions, a convection coefficient of $h_{i}=5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ is associated with condensation on the inner surface of the tubes, while a coefficient of $h_{o}=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ is maintained by airflow over the tubes.
(a) What is the overall heat transfer coefficient if the tubes are unfinned?
(b) What is the overall heat transfer coefficient based on the inner surface, $U_{i}$, if aluminum annular fins of thickness $t=1.5 \mathrm{~mm}$, outer diameter $D_{o}=20 \mathrm{~mm}$, and pitch $S=3.5 \mathrm{~mm}$ are added to the outer surface? Base your calculations on a 1-m-long section of tube. Subject to the requirements that $t \geq 1 \mathrm{~mm}$ and $(S-t) \geq 1.5 \mathrm{~mm}$, explore the effect of variations in $t$ and $S$ on $U_{i}$. What combination of $t$ and $S$ would yield the best heat transfer performance?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
02:50

Problem 9

A finned-tube, cross-flow heat exchanger is to use the exhaust of a gas turbine to heat pressurized water. Laboratory measurements are performed on a prototype version of the exchanger, which has a surface area of $10 \mathrm{~m}^{2}$, to determine the overall heat transfer coefficient as a function of operating conditions. Measurements made under particular conditions, for which $\dot{h}_{k}=2 \mathrm{~kg} / \mathrm{s}$, $T_{h u}=325^{\circ} \mathrm{C}, \dot{m}_{c}=0.5 \mathrm{~kg} / \mathrm{s}$, and $T_{c i}=25^{\circ} \mathrm{C}$, reveal a water outlet temperature of $T_{c \omega}=150^{\circ} \mathrm{C}$. What is the overall heat transfer coefficient of the exchanger?

Narayan Hari
Narayan Hari
Numerade Educator
01:07

Problem 10

Water at a rate of $45,500 \mathrm{~kg} / \mathrm{h}$ is heated from 80 to $150^{\circ} \mathrm{C}$ in a heat exchanger having two shell passes and eight tube passes with a total surface area of $925 \mathrm{~m}^{2}$. Hot exhaust gases having approximately the same thermophysical properties as air enter at $350^{\circ} \mathrm{C}$ and exit at $175^{\circ} \mathrm{C}$. Determine the overall heat transfer coefficient.

Dominador Tan
Dominador Tan
Numerade Educator
04:14

Problem 11

A novel heat exchanger concept consists of a large number of extruded polypropylene sheets $(k=0.17$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K}$ ), each having a fin-like geometry, that are subsequently stacked and melted together to form the heat exchanger core. Besides being inexpensive, the heat exchanger can be easily recycled at the end of its life. Carbon dioxide at a mean temperature of $10^{\circ} \mathrm{C}$ and pressure of $2 \mathrm{~atm}$ flows in the cool channels at a mean velocity of $u_{m}=0.1 \mathrm{~m} / \mathrm{s}$. Air at $30^{\circ} \mathrm{C}$ and $2 \mathrm{~atm}$ flows at $0.2 \mathrm{~m} / \mathrm{s}$ in the warm channels. Neglecting the thermal contact resistance at the welded interface, determine the product of the overall heat transfer coefficient and heat transfer area, UA, for a heat exchanger core consisting of 200 cool channels and 200 warm channels.

Dading Chen
Dading Chen
Numerade Educator
00:54

Problem 12

The properties and flow rates for the hot and cold fluids of a heat exchanger are shown in the following table. Which fluid limits the heat transfer rate of the exchanger? Explain your choice.

Dading Chen
Dading Chen
Numerade Educator
01:05

Problem 13

A process fluid having a specific heat of $3500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ and flowing at $2 \mathrm{~kg} / \mathrm{s}$ is to be cooled from $80^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$ with chilled water, which is supplied at a temperature of $15^{\circ} \mathrm{C}$ and a flow rate of $2.5 \mathrm{~kg} / \mathrm{s}$. Assuming an overall heat transfer coefficient of $2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, calculate the required heat transfer areas for the following exchanger configurations: (a) parallel flow, (b) counterflow, (c) shell-and-tube, one shell pass and two tube passes, and (d) cross-flow, single pass, both fluids unmixed. Compare the results of your analysis. Your work can be reduced by using IHT.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:07

Problem 14

A shell-and-tube exchanger (two shells, four tube passes) is used to heat $10,000 \mathrm{~kg} / \mathrm{h}$ of pressurized water from 35 to $120^{\circ} \mathrm{C}$ with $5000 \mathrm{~kg} / \mathrm{h}$ pressurized water entering the exchanger at $300^{\circ} \mathrm{C}$. If the overall heat transfer coefficient is $1500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the required heat exchanger area.

Dominador Tan
Dominador Tan
Numerade Educator
03:53

Problem 15

Consider the heat exchanger of Problem 11.14. After several years of operation, it is observed that the outlet temperature of the cold water reaches only $95^{\circ} \mathrm{C}$ rather than the desired $120^{\circ} \mathrm{C}$ for the same flow rates and inlet temperatures of the fluids. Determine the cumulative (inner and outer surface) fouling factor that is the cause of the poorer performance.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:03

Problem 16

The hot and cold inlet temperatures to a concentric tube heat exchanger are $T_{h i}=200^{\circ} \mathrm{C}, T_{c, i}=100^{\circ} \mathrm{C}$, respectively. The outlet temperatures are $T_{k, o}=110^{\circ} \mathrm{C}$ and $T_{\omega_{0}}=125^{\circ} \mathrm{C}$. Is the heat exchanger operating in a parallel flow or in a counterflow configuration? What is the heat exchanger effectiveness? What is the NTU? Phase change does not occur in either fluid.

Keshav Singh
Keshav Singh
Numerade Educator
04:00

Problem 17

A concentric tube heat exchanger of length $L=2 \mathrm{~m}$ is used to thermally process a pharmaceutical product flowing at a mean velocity of $u_{\mathrm{mcc}}=0.1 \mathrm{~m} / \mathrm{s}$ with an inlet temperature of $T_{c, i}=20^{\circ} \mathrm{C}$. The inner tube of diameter $D_{i}=10 \mathrm{~mm}$ is thin walled, and the exterior of the outer tube $\left(D_{o}=20 \mathrm{~mm}\right)$ is well insulated. Water flows in the annular region between the tubes at a mean velocity of $u_{\mathrm{mhh}}=0.2 \mathrm{~m} / \mathrm{s}$ with an inlet temperature of $T_{h, i}=60^{\circ} \mathrm{C}$. Properties of the pharmaceutical product are $\nu=10 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, \quad k=0.25 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\rho=1100 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=2460 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Evaluate water properties at $\bar{T}_{\mathrm{h}}=50^{\circ} \mathrm{C}$.
(a) Determine the value of the overall heat transfer coefficient $U$.
(b) Determine the mean outlet temperature of the pharmaceutical product when the exchanger operates in the counterflow mode.
(c) Determine the mean outlet temperature of the pharmaceutical product when the exchanger operates in the parallel-flow mode.

Paul Gabriel
Paul Gabriel
Numerade Educator
04:03

Problem 18

A counterflow, concentric tube heat exchanger is designed to heat water from 20 to $80^{\circ} \mathrm{C}$ using hot oil, which is supplied to the annulus at $160^{\circ} \mathrm{C}$ and discharged at $140^{\circ} \mathrm{C}$. The thin-walled inner tube has a diameter of $D_{i}=20 \mathrm{~mm}$, and the overall heat transfer coefficient is $500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The design condition calls for a total heat transfer rate of $3000 \mathrm{~W}$.
(a) What is the length of the heat exchanger?
(b) After 3 years of operation, performance is degraded by fouling on the water side of the exchanger, and the water outlet temperature is only $65^{\circ} \mathrm{C}$ for the same fluid flow rates and inlet temperatures. What are the corresponding values of the heat transfer rate, the outlet temperature of the oil, the overall heat transfer coefficient, and the water-side fouling factor, $R_{f,}^{n}$ ?

Penny Riley
Penny Riley
Numerade Educator
01:30

Problem 19

Consider the counterflow, concentric tube heat exchanger of Example 11.1. The designer wishes to consider the effect of the cooling water flow rate on the tube length. All other conditions, including the outlet oil temperature of $60^{\circ} \mathrm{C}$, remain the same.
(a) From the analysis of Example 11.1, we saw that the overall coefficient $U$ is dominated by the hotside convection coefficient. Assuming the water properties are independent of temperature, calculate $U$ as a function of the water flow rate. Justify a constant value of $U$ in the calculations of part (b).
(b) Calculate and plot the required exchanger tube length $L$ and the water outlet temperature $T_{\omega o}$ as a function of the cooling water flow rate for $0.15 \leq \dot{m}_{c} \leq 0.30 \mathrm{~kg} / \mathrm{s}$.

Jincy M Saji
Jincy M Saji
Numerade Educator
03:55

Problem 20

Consider a concentric tube heat exchanger with an area of $50 \mathrm{~m}^{2}$ operating under the following conditions:
\begin{tabular}{lcc}
\hline & Hot flid & Cold flid \\
\hline Heat capacity rate, $\mathrm{kW} / \mathrm{K}$ & 6 & 3 \\
Inlet temperature, ${ }^{\circ} \mathrm{C}$ & 60 & 30 \\
Outlet temperature, ${ }^{\circ} \mathrm{C}$ & $-$ & 54 \\
\hline
\end{tabular}
(a) Determine the outlet temperature of the hot fluid.
(b) Is the heat exchanger operating in counterflow or parallel flow, or can't you tell from the available information?
(c) Calculate the overall heat transfer coefficient.
(d) Calculate the effectiveness of this exchanger.
(e) What would be the effectiveness of this exchanger if its length were made very large?

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:39

Problem 21

As part of a senior project, a student was given the assignment to design a heat exchanger that meets the following specifications:
\begin{tabular}{lccc}
\hline & $\dot{m}(\mathrm{~kg} / \mathrm{s})$ & $T_{m, i}\left({ }^{\circ} \mathrm{C}\right)$ & $T_{m, \theta}\left({ }^{\circ} \mathrm{C}\right)$ \\
\hline Hot water & 28 & 90 & $-$ \\
Cold water & 27 & 34 & 60 \\
\hline
\end{tabular}
Like many real-world situations, the customer hasn't revealed, or doesn't know, additional requirements that would allow you to proceed directly to a final configuration. At the outset, it is helpful to make a first-cut design based upon simplifying assumptions, which can be evaluated to determine what additional requirements and trade-offs should be considered by the customer.
(a) Design a heat exchanger to meet the foregoing specifications. List and explain your assumptions. Hint: Begin by finding the required value for $U A$ and using representative values of $U$ to determine $A$.
(b) Evaluate your design by identifying what features and configurations could be explored with your customer in order to develop more complete specifications.

Penny Riley
Penny Riley
Numerade Educator
02:16

Problem 22

A shell-and-tube heat exchanger must be designed to heat $2.5 \mathrm{~kg} / \mathrm{s}$ of water from 15 to $85^{\circ} \mathrm{C}$. The heating is to be accomplished by passing hot engine oil, which is available at $160^{\circ} \mathrm{C}$, through the shell side of the exchanger. The oil is known to provide an average convection coefficient of $h_{o}=400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ on the outside of the tubes. Ten tubes pass the water through the shell. Each tube is thin walled, of diameter $D=25 \mathrm{~mm}$, and makes eight passes through the shell. If the oil leaves the exchanger at $100^{\circ} \mathrm{C}$, what is its flow rate? How long must the tubes be to accomplish the desired heating?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:03

Problem 23

A concentric tube heat exchanger for cooling lubricating oil is comprised of a thin-walled inner tube of 25 -mm diameter carrying water and an outer tube of $45-\mathrm{mm}$ diameter carrying the oil. The exchanger operates in counterflow with an overall heat transfer coefficient of $60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and the tabulated average properties.
\begin{array}{lcc}
\hline \text { Properties } & \text { Water } & \text { Oil } \\
\hline \rho\left(\mathrm{kg} / \mathrm{m}^{3}\right) & 1000 & 800 \\
c_{p}(\mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}) & 4200 & 1900 \\
\nu\left(\mathrm{m}^{2} / \mathrm{s}\right) & 7 \times 10^{-7} & 1 \times 10^{-5} \\
k(\mathrm{~W} / \mathrm{m} \cdot \mathrm{K}) & 0.64 & 0.134 \\
\operatorname{Pr} & 4.7 & 140 \\
\hline
\end{array}
(a) If the outlet temperature of the oil is $60^{\circ} \mathrm{C}$, determine the total heat transfer and the outlet temperature of the water.
(b) Determine the length required for the heat exchanger.

Penny Riley
Penny Riley
Numerade Educator
03:55

Problem 24

A counterflow, concentric tube heat exchanger used for engine cooling has been in service for an extended period of time. The heat transfer surface area of the exchanger is $5 \mathrm{~m}^{2}$, and the design value of the overall convection coefficient is $38 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. During a test run, engine oil flowing at $0.1 \mathrm{~kg} / \mathrm{s}$ is cooled from $110^{\circ} \mathrm{C}$ to $66^{\circ} \mathrm{C}$ by water supplied at a temperature of $25^{\circ} \mathrm{C}$ and a flow rate of $0.2 \mathrm{~kg} / \mathrm{s}$. Determine whether fouling has occurred during the service period. If so, calculate the fouling factor, $R_{f}^{\prime \prime}\left(\mathrm{m}^{2} \cdot \mathrm{K} / \mathrm{W}\right)$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
09:50

Problem 25

An automobile radiator may be viewed as a cross-flow heat exchanger with both fluids unmixed. Water, which has a flow rate of $0.05 \mathrm{~kg} / \mathrm{s}$, enters the radiator at $400 \mathrm{~K}$ and is to leave at $330 \mathrm{~K}$. The water is cooled by air that enters at $0.75 \mathrm{~kg} / \mathrm{s}$ and $300 \mathrm{~K}$.
(a) If the overall heat transfer coefficient is $200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, what is the required heat transfer surface area?
(b) A manufacturing engineer claims ridges can be stamped on the finned surface of the exchanger, which could greatly increase the overall heat transfer coefficient. With all other conditions remaining the same and the heat transfer surface area determined from part (a), generate a plot of the air and water outlet temperatures as a function of $U$ for $200 \leq U \leq 400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. What benefits result from increasing the overall convection coefficient for this application?

Jincy M Saji
Jincy M Saji
Numerade Educator
08:22

Problem 26

Hot air for a large-scale drying operation is to be produced by routing the air over a tube bank (unmixed), while products of combustion are routed through the tubes. The surface area of the cross-flow heat exchanger is $A=25 \mathrm{~m}^{2}$, and for the proposed operating conditions, the manufacturer specifies an overall heat transfer coefficient of $U=35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The air and the combustion gases may each be assumed to have a specific heat of $c_{p}=1040 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Consider conditions for which combustion gases flowing at $1 \mathrm{~kg} / \mathrm{s}$ enter the heat exchanger at $800 \mathrm{~K}$, while air at $5 \mathrm{~kg} / \mathrm{s}$ has an inlet temperature of $300 \mathrm{~K}$.
(a) What are the air and gas outlet temperatures?
(b) After extended operation, deposits on the inner tube surfaces are expected to provide a fouling resistance of $R_{f}^{N}=0.004 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. Should operation be suspended in order to clean the tubes?
(c) The heat exchanger performance may be improved by increasing the surface area and/or the
overall heat transfer coefficient. Explore the effect of such changes on the air outlet temperature for $500 \leq U A \leq 2500 \mathrm{~W} / \mathrm{K}$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:13

Problem 27

In a dairy operation, milk at a flow rate of $250 \mathrm{~L} / \mathrm{h}$ and a cow-body temperature of $38.6^{\circ} \mathrm{C}$ must be chilled to a safe-to-store temperature of $13^{\circ} \mathrm{C}$ or less. Ground water at $10^{\circ} \mathrm{C}$ is available at a flow rate of $0.72 \mathrm{~m}^{3} / \mathrm{h}$. The density and specific heat of milk are $1030 \mathrm{~kg} / \mathrm{m}^{3}$ and $3860 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, respectively.
(a) Determine the UA product of a counterflow heat exchanger required for the chilling process. Determine the length of the exchanger if the inner pipe has a 50 -mm diameter and the overall heat transfer coefficient is $U=1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(b) Determine the outlet temperature of the water.
(c) Using the value of $U A$ found in part (a), determine the milk outlet temperature if the water flow rate is doubled. What is the outlet temperature if the flow rate is halved?

Anand Jangid
Anand Jangid
Numerade Educator
15:49

Problem 28

A shell-and-tube heat exchanger with one shell pass and two tube passes is used as a regenerator, to preheat milk before it is pasteurized. Cold milk enters the regenerator at $T_{c i}=5^{\circ} \mathrm{C}$, while hot milk, which has completed the pasteurization process, enters at $T_{h, i}=$ $70^{\circ} \mathrm{C}$. After leaving the regenerator, the heated milk enters a second heat exchanger, which raises its temperature from $T_{c, 0}$ to $70^{\circ} \mathrm{C}$.
(a) A regenerator is to be used in a pasteurization process for which the flow rate of the milk is $\dot{m}_{c}=\dot{m}_{h}=5 \mathrm{~kg} / \mathrm{s}$. For this flow rate, the manufacturer of the regenerator specifies an overall heat transfer coefficient of $2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. If the desired effectiveness of the regenerator is $0.5$, what is the requisite heat transfer area? What are the corresponding rate of heat recovery and the fluid outlet temperatures? Refer to Problem $11.27$ for the properties of milk.
(b) If the hot fluid in the secondary heat exchanger derives its energy from the combustion of natural
gas and the burner has an efficiency of $90 \%$, what would be the annual savings in energy and fuel costs associated with installation of the regenerator? The facility operates continuously throughout the year, and the cost of natural gas is $C_{m g}=\$ 0.02 / \mathrm{MJ}$.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
03:24

Problem 29

A twin-tube, counterflow heat exchanger operates with balanced flow rates of $0.003 \mathrm{~kg} / \mathrm{s}$ for the hot and cold airstreams. The cold stream enters at $280 \mathrm{~K}$ and must be heated to $340 \mathrm{~K}$ using hot air at $360 \mathrm{~K}$. The average pressure of the airstreams is $1 \mathrm{~atm}$ and the maximum allowable pressure drop for the cold air is $10 \mathrm{kPa}$. The tube walls may be assumed to act as fins, each with an efficiency of $100 \%$.
(a) Determine the tube diameter $D$ and length $L$ that satisfy the prescribed heat transfer and pressure drop requirements.
(b) For the diameter $D$ and length $L$ found in part (a), generate plots of the cold stream outlet temperature, the heat transfer rate, and pressure drop as a function of balanced flow rates in the range from $0.002$ to $0.004 \mathrm{~kg} / \mathrm{s}$. Comment on your results.

Narayan Hari
Narayan Hari
Numerade Educator
01:37

Problem 30

A 5 -m-long, twin-tube, counterflow heat exchanger, such as that illustrated in Problem 11.29, is used to heat air for a drying operation. Each tube is made from plain carbon steel $(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ and has an inner diameter and wall thickness of $50 \mathrm{~mm}$ and $4 \mathrm{~mm}$, respectively. The thermal resistance per unit length of the brazed joint connecting the tubes is $R_{t}^{\prime}=0.01 \mathrm{~m} \cdot \mathrm{K} / \mathrm{W}$. Consider conditions for which air enters one tube at a pressure of $5 \mathrm{~atm}$, a temperature of $17^{\circ} \mathrm{C}$, and flow rate of $0.030 \mathrm{~kg} / \mathrm{s}$, while saturated steam at $2.455$ bar condenses in the other tube. The convection coefficient for condensation may be approximated as $5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. What is the air outlet temperature? What is the mass rate at which condensate leaves the system? Hint: Account for the effects of circumferential conduction in the tubes by treating them as extended surfaces.

Manik Pulyani
Manik Pulyani
Numerade Educator
04:00

Problem 31

Hot water for an industrial washing operation is produced by recovering heat from the flue gases of a furnace. A cross-flow heat exchanger is used, with the gases passing over the tubes and the water making a single pass through the tubes. The steel tubes $(k=60 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})$ have inner and outer diameters of $D_{i}=15 \mathrm{~mm}$ and $D_{o}=20 \mathrm{~mm}$, while the staggered tube array has longitudinal and transverse pitches of $S_{T}=S_{L}=40 \mathrm{~mm}$. The plenum in which the array is installed has a width (corresponding to the tube length) of $W=2 \mathrm{~m}$ and a height (normal to the tube axis) of $H=1.2 \mathrm{~m}$. The number of tubes in the transverse plane is therefore $N_{T} \approx H / S_{T}=30$. The gas properties may be approximated as those of atmospheric air, and the convection coefficient associated with water flow in the tubes may be approximated as $3000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) If $50 \mathrm{~kg} / \mathrm{s}$ of water are to be heated from 290 to $350 \mathrm{~K}$ by $40 \mathrm{~kg} / \mathrm{s}$ of flue gases entering the exchanger at $700 \mathrm{~K}$, what is the gas outlet temperature and how many tube rows $N_{L}$ are required?
(b) The water outlet temperature may be controlled by varying the gas flow rate and/or inlet temperature. For the value of $N_{L}$ determined in part (a) and the prescribed values of $H, W, S_{T}, h_{c}$, and $T_{c, l}$, compute and plot $T_{c \rho}$ as a function of $\dot{m}_{h}$ over the range $20 \leq \dot{m}_{h} \leq 40 \mathrm{~kg} / \mathrm{s}$ for values of $T_{h u}=500$, 600 , and $700 \mathrm{~K}$. Also plot the corresponding variations of $T_{h \rho}$. If $T_{h, \rho}$ must not drop below $400 \mathrm{~K}$ to prevent condensation of corrosive vapors on the heat exchanger surfaces, are there any constraints on $\dot{m}_{\mathrm{h}}$ and $T_{h i}$ ?

Paul Gabriel
Paul Gabriel
Numerade Educator
03:58

Problem 32

A single-pass, cross-flow heat exchanger uses hot exhaust gases (mixed) to heat water (unmixed) from 30 to $80^{\circ} \mathrm{C}$ at a rate of $3 \mathrm{~kg} / \mathrm{s}$. The exhaust gases, having thermophysical properties similar to air, enter and exit the exchanger at 225 and $100^{\circ} \mathrm{C}$, respectively. If the overall heat transfer coefficient is $200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, estimate the required surface area.

Dading Chen
Dading Chen
Numerade Educator
06:58

Problem 33

Consider the fluid conditions and overall heat transfer coefficient of Problem $11.32$ for a concentric tube heat exchanger operating in parallel flow. The thin-walled separator tube has a diameter of $100 \mathrm{~mm}$.
(a) Determine the required length for the exchanger.
(b) Assuming water flow inside the separator tube to be fully developed, estimate the convection heat transfer coefficient.
(c) Using the overall coefficient and the inlet temperatures from Problem 11.32, plot the heat transfer rate and fluid outlet temperatures as a function of the tube length for $60 \leq L \leq 400 \mathrm{~m}$ and the parallelflow configuration.
(d) If the exchanger were operated in counterflow with the same overall coefficient and inlet temperatures, what would be the reduction in the required length relative to the value found in part (a)?
(e) For the counterflow configuration, plot the effectiveness and fluid outlet temperatures as a function of the tube length for $60 \leq L \leq 400 \mathrm{~m}$.

Amany Waheeb
Amany Waheeb
Numerade Educator
09:50

Problem 34

The compartment heater of an automobile exchanges heat between warm radiator fluid and cooler outside air. The flow rate of water is large compared to the air, and the effectiveness, $\varepsilon$, of the heater is known to depend on the flow rate of air according to the relation, $\varepsilon \sim \dot{m}_{\text {air }}^{-0.2}$.
(a) If the fan is switched to high and $\dot{m}_{\text {air }}$ is doubled, determine the percentage increase in the heat added to the car, if fluid inlet temperatures remain the same.
(b) For the low-speed fan condition, the heater warms outdoor air from 0 to $30^{\circ} \mathrm{C}$. When the fan is turned to medium, the airflow rate increases $50 \%$ and the heat transfer increases $20 \%$. Find the new outlet temperature.

Jincy M Saji
Jincy M Saji
Numerade Educator
04:41

Problem 35

A counterflow, twin-tube heat exchanger is made by brazing two circular nickel tubes, each $40 \mathrm{~m}$ long, together as shown below. Hot water flows through the smaller tube of $10-\mathrm{mm}$ diameter and air at atmospheric pressure flows through the larger tube of 30 mm diameter. Both tubes have a wall thickness of $2 \mathrm{~mm}$. The thermal contact conductance per unit length of the brazed joint is $100 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. The mass flow rates of the water and air are $0.04$ and $0.12 \mathrm{~kg} / \mathrm{s}$, respectively. The inlet temperatures of the water and air are 85 and $23^{\circ} \mathrm{C}$, respectively.
Employ the $s-\mathrm{NTU}$ method to determine the outlet temperature of the air. Hint: Account for the effects of circumferential conduction in the walls of the tubes by treating them as extended surfaces.

Salamat Ali
Salamat Ali
Numerade Educator
03:41

Problem 36

Consider a coupled shell-in-tube heat exchange device consisting of two identical heat exchangers $A$ and $B$.
Air flows on the shell side of heat exchanger A, entering at $T_{h, i, \mathrm{~A}}=520 \mathrm{~K}$ and $\dot{m}_{h, \mathrm{~A}}=10 \mathrm{~kg} / \mathrm{s}$. Ammonia flows in the shell of heat exchanger $B$, entering at $T_{c, i \mathrm{~B}}=280 \mathrm{~K}, m_{c, \mathrm{~B}}=5 \mathrm{~kg} / \mathrm{s}$. The tube-side flow is common to both heat exchangers and consists of water at a flow rate $\dot{m}_{c, \mathrm{~A}}=\dot{m}_{\hat{h, B}}$ with two tube passes. The UA product increases with water flow rate for heat exchanger A as expressed by the relation $U A_{\mathrm{A}}=a+$ $b \dot{m}_{c, \mathrm{~A}}$ where $a=6000 \mathrm{~W} / \mathrm{K}$ and $b=100 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. For heat exchanger $\mathrm{B}, U A_{\mathrm{B}}=1.2 U A_{\mathrm{A}}$.
(a) For $\dot{m}_{c, \mathrm{~A}}=\dot{m}_{k, \mathrm{~B}}=1 \mathrm{~kg} / \mathrm{s}$, determine the outlet air and ammonia temperatures, as well as the heat transfer rate.
(b) The plant engineer wishes to fine-tune the heat exchanger performance by installing a variablespeed pump to allow adjustment of the water flow rate. Plot the outlet air and outlet ammonia temperatures versus the water flow rate over the range $0 \mathrm{~kg} / \mathrm{s} \leq \dot{m}_{c \mathrm{~A}}=m_{\mathrm{h}, \mathrm{B}} \leq 2 \mathrm{~kg} / \mathrm{s}$.

Lottie Adams
Lottie Adams
Numerade Educator
08:25

Problem 37

Consider Problem 11.36.
(a) For $\dot{m}_{c \mathrm{CA}}=\dot{m}_{\mathrm{h}, \mathrm{B}}=10 \mathrm{~kg} / \mathrm{s}$, determine the outlet air and ammonia temperatures, as well as the heat transfer rate.
(b) Plot the outlet air and outlet ammonia temperatures versus the water flow rate over the range $5 \mathrm{~kg} / \mathrm{s} \leq \dot{m}_{c, \mathrm{~A}}=m_{h, \mathrm{~B}} \leq 50 \mathrm{~kg} / \mathrm{s}$.

Keshav Singh
Keshav Singh
Numerade Educator
08:45

Problem 38

For health reasons, public spaces require the continuous exchange of a specified mass of stale indoor air with fresh outdoor air. To conserve energy during the heating season, it is expedient to recover the thermal energy in the exhausted, warm indoor air and transfer it to the incoming, cold fresh air. A coupled singlepass, cross-flow heat exchanger with both fluids unmixed is installed in the intake and return ducts of a heating system as shown in the schematic. Water containing an anti-freeze agent is used as the working fluid in the coupled heat exchange device, which is composed of individual heat exchangers $A$ and B. Hence, heat is transferred from the warm stale air to the cold fresh air by way of the pumped water.
Consider a specified air mass flow rate (in each duct) of $m=1.50 \mathrm{~kg} / \mathrm{s}$, an overall heat transfer coefficient-area product of $U A=2500 \mathrm{~W} / \mathrm{K}$ (for each heat exchanger), an outdoor temperature of $T_{c, i, A}=-4^{\circ} \mathrm{C}$ and an indoor temperature of $T_{h, i, B}=$ $23^{\circ} \mathrm{C}$. Since the warm air has been humidified, excessive heat transfer can result in unwanted condensation in the ductwork. What water flow rate is necessary to maximize heat transfer while ensuring the outlet temperature associated with heat exchanger $\mathrm{B}$ does not fall below the dew point temperature, $T_{h, 0, B}=T_{\mathrm{dp}}=$ $13^{\circ} \mathrm{C}$ ? Hint: Assume the maximum heat capacity rate is associated with the air.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:05

Problem 39

A cross-flow heat exchanger used in a cardiopulmonary bypass procedure cools blood flowing at $5 \mathrm{~L} / \mathrm{min}$ from a body temperature of $37^{\circ} \mathrm{C}$ to $25^{\circ} \mathrm{C}$ in order to induce body hypothermia, which reduces metabolic and oxygen requirements. The coolant is ice water at $0^{\circ} \mathrm{C}$, and its flow rate is adjusted to provide an outlet temperature of $15^{\circ} \mathrm{C}$. The heat exchanger operates with both fluids unmixed, and the overall heat transfer coefficient is $750 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The density and specific heat of the blood are $1050 \mathrm{~kg} / \mathrm{m}^{3}$ and $3740 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, respectively.
a) Determine the heat transfer rate for the exchanger.
b) Calculate the water flow rate.
c) What is the surface area of the heat exchanger?
d) Calculate and plot the blood and water outlet temperatures as a function of the water flow rate for the range 2 to $4 \mathrm{~L} / \mathrm{min}$, assuming all other parameters remain unchanged. Comment on how the changes in the outlet temperatures are affected by changes in the water flow rate. Explain this behavior and why it is an advantage for this application.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:37

Problem 40

Saturated steam at $0.14$ bar is condensed in a shell-andtube heat exchanger with one shell pass and two tube passes consisting of 130 brass tubes, each with a length per pass of $2 \mathrm{~m}$. The tubes have inner and outer diameters of $13.4$ and $15.9 \mathrm{~mm}$, respectively. Cooling water enters the tubes at $20^{\circ} \mathrm{C}$ with a mean velocity of $1.25 \mathrm{~m} / \mathrm{s}$. The heat transfer coefficient for condensation on the outer surfaces of the tubes is $13,500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) Determine the overall heat transfer coefficient, the cooling water outlet temperature, and the steam condensation rate.
(b) With all other conditions remaining the same, but accounting for changes in the overall coefficient, plot the cooling water outlet temperature and the steam condensation rate as a function of the water flow rate for $10 \leq m_{c} \leq 30 \mathrm{~kg} / \mathrm{s}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
22:13

Problem 44

Saturated water vapor leaves a steam turbine at a flow rate of $1.5 \mathrm{~kg} / \mathrm{s}$ and a pressure of $0.51$ bar. The vapor is to be completely condensed to saturated liquid in a shell-and-tube heat exchanger that uses city water as the cold fluid. The water enters the thin-walled tubes at $17^{\circ} \mathrm{C}$ and is to leave at $57^{\circ} \mathrm{C}$. Assuming an overall heat transfer coefficient of $2000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the required heat exchanger surface area and the water flow rate. After extended operation, fouling causes the overall heat transfer coefficient to decrease to $1000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, and to completely condense the vapor, there must be an attendant reduction in the vapor flow rate. For the same water inlet temperature and flow rate, what is the new vapor flow rate required for complete condensation?

Gordon Ayadju
Gordon Ayadju
Numerade Educator
02:18

Problem 45

A two-fluid heat exchanger has inlet and outlet temperatures of 65 and $40^{\circ} \mathrm{C}$ for the hot fluid and 15 and $30^{\circ} \mathrm{C}$ for the cold fluid. Can you tell whether this exchanger is operating under counterflow or parallelflow conditions? Determine the effectiveness of the heat exchanger.

Manish Jain
Manish Jain
Numerade Educator
01:16

Problem 46

The human brain is especially sensitive to elevated temperatures. The cool blood in the veins leaving the face and neck and returning to the heart may contribute to thermal regulation of the brain by cooling the arterial blood flowing to the brain. Consider a vein and artery running between the chest and the base of the skull for a distance $L=250 \mathrm{~mm}$, with mass flow rates of $3 \times 10^{-3} \mathrm{~kg} / \mathrm{s}$ in opposite directions in the two vessels. The vessels are of diameter $D=5 \mathrm{~mm}$ and are separated by a distance $w=7 \mathrm{~mm}$. The thermal conductivity of the surrounding tissue is $k_{\mathrm{r}}=0.5 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. If the arterial blood enters at $37^{\circ} \mathrm{C}$ and the venous blood enters at $27^{\circ} \mathrm{C}$, at what temperature will the arterial blood exit? If the arterial blood becomes overheated, and the body responds by halving the blood flow rate, how much hotter can the entering arterial blood be and still maintain its exit temperature below $37^{\circ} \mathrm{C}$ ? Hint: If we assume that all the heat leaving the artery enters the vein, then heat transfer between the two vessels can be modeled using a relationship found in Table 4.1. Approximate the blood properties as those of water.

Nick Auwerda
Nick Auwerda
Numerade Educator
03:14

Problem 47

Consider a very long, concentric tube heat exchanger having hot and cold water inlet temperatures of 85 and $15^{\circ} \mathrm{C}$. The flow rate of the hot water is twice that of the cold water. Assuming equivalent hot and cold water specific heats, determine the hot water outlet temperature for the following modes of operation: (a) counterflow and (b) parallel flow.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
08:45

Problem 48

A plate-fin heat exchanger is used to condense a saturated refrigerant vapor in an air-conditioning system. The vapor has a saturation temperature of $45^{\circ} \mathrm{C}$, and a condensation rate of $0.015 \mathrm{~kg} / \mathrm{s}$ is dictated by system performance requirements. The frontal area of the condenser is fixed at $A_{\mathrm{fr}}=0.25 \mathrm{~m}^{2}$ by installation requirements, and a value of $h_{f g}=135 \mathrm{~kJ} / \mathrm{kg}$ may be assumed for the refrigerant.
(a) The condenser design is to be based on a nominal air inlet temperature of $T_{c, i}=30^{\circ} \mathrm{C}$ and nominal air inlet velocity of $V=2 \mathrm{~m} / \mathrm{s}$ for which the manufacturer of the heat exchanger core indicates an overall coefficient of $U=50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. What is the corresponding value of the heat transfer surface area required to achieve the prescribed condensation rate? What is the air outlet temperature?
(b) From the manufacturer of the heat exchanger core, it is also known that $U \propto V^{0 . t}$. During daily operation the air inlet temperature is not controllable and may vary from 27 to $38^{\circ} \mathrm{C}$. If the heat exchanger area is fixed by the result of part (a), what is the range of air velocities needed to maintain the prescribed condensation rate? Plot the velocity as a function of the air inlet temperature.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
02:16

Problem 49

A shell-and-tube heat exchanger is to heat $10,000 \mathrm{~kg} / \mathrm{h}$ of water from 16 to $84^{\circ} \mathrm{C}$ by hot engine oil flowing through the shell. The oil makes a single shell pass, entering at $160^{\circ} \mathrm{C}$ and leaving at $94^{\circ} \mathrm{C}$, with an average heat transfer coefficient of $400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. The water flows through 11 brass tubes of $22.9-\mathrm{mm}$ inside diameter and 25.4-mm outside diameter, with each tube making four passes through the shell.
(a) Assuming fully developed flow for the water, determine the required tube length per pass.
(b) For the tube length found in part (a), plot the effectiveness, fluid outlet temperatures, and water-side convection coefficient as a function of the water flow rate for $5000 \leq m_{c} \leq 15,000 \mathrm{~kg} / \mathrm{h}$, with all other conditions remaining the same.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
04:00

Problem 50

In a supercomputer, signal propagation delays are reduced by resorting to high-density circuit arrangements which are cooled by immersing them in a special dielectric liquid. The fluid is pumped in a closed loop through the computer and an adjoining shell-and-tube heat exchanger having one shell and two tube passes.
During normal operation, heat generated within the computer is transferred to the dielectric fluid passing through the computer at a flow rate of $m_{f}=4.81 \mathrm{~kg} / \mathrm{s}$. In turn, the fluid passes through the tubes of the heat exchanger and the heat is transferred to water passing over the tubes. The dielectric fluid may be assumed to have constant properties of $c_{p}=1040 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=7.65 \times 10^{-4} \mathrm{~kg} / \mathrm{s} \cdot \mathrm{m}$, $k=0.058 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\operatorname{Pr}=14$. During normal operation, chilled water at a flow rate of $\dot{m}_{w}=2.5 \mathrm{~kg} / \mathrm{s}$ and an inlet temperature of $T_{w, i}=5^{\circ} \mathrm{C}$ passes over the tubes. The water has a specific heat of $4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ and provides an average convection coefficient of $10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ over the outer surface of the tubes.
(a) If the heat exchanger consists of 72 thin-walled tubes, each of 10-mm diameter, and fully developed flow is assumed to exist within the tubes, what is the convection coefficient associated with flow through the tubes?
(b) If the dielectric fluid enters the heat exchanger at $T_{f,}=25^{\circ} \mathrm{C}$ and is to leave at $T_{f e}=15^{\circ} \mathrm{C}$, what is the required tube length per pass?
(c) For the exchanger with the tube length per pass determined in part (b), plot the outlet temperature of the dielectric fluid as a function of its flow rate for $4 \leq m_{f} \leq 6 \mathrm{~kg} / \mathrm{s}$. Account for corresponding changes in the overall heat transfer coefficient, but assume all other conditions to remain the same.
(d) The site specialist for the computer facilities is concerned about changes in the performance of the water chiller supplying the cold water $\left(\dot{m}_{w}\right.$, $\left.T_{w j}\right)$ and their effect on the outlet temperature $T_{f o}$ of the dielectric fluid. With all other conditions remaining the same, determine the effect of a $\pm 10 \%$ change in the cold water flow rate on $T_{f, a^{\circ}}$
(e) Repeat the performance analysis of part (d) to determine the effect of $a \pm 3^{\circ} \mathrm{C}$ change in the water inlet temperature on $T_{f, 0}$, with all other conditions remaining the same.

Paul Gabriel
Paul Gabriel
Numerade Educator
14:10

Problem 51

Untapped geothermal sites in the United States have the estimated potential to deliver $100,000 \mathrm{MW}$ (electric) of new, clean energy. The key component in a geothermal power plant is a heat exchanger that transfers thermal energy from hot, geothermal brine to a second fluid that is evaporated in the heat exchanger. The cooled brinc is reinjected into the gcothermal well after it exits the heat exchange, while the vapor exiting the heat exchanger serves as the working fluid of a Rankine cycle. Consider a geothermal power plant designed to deliver $P=25 \mathrm{MW}$ (electric) operating at a thermal efficiency of $\eta=0.20$. Pressurized hot brine at $T_{h i}=200^{\circ} \mathrm{C}$ is sent to the tube side of a shell-andtube heat exchanger, while the Rankine cycle's working fluid enters the shell side at $T_{c, i}=45^{\circ} \mathrm{C}$. The brine is reinjected into the well at $T_{h_{i o}}=80^{\circ} \mathrm{C}$.
(a) Assuming the brine has the properties of water, determine the required brine flow rate, the required effectiveness of the heat exchanger, and the required heat transfer surface area. The overall heat transfer coefficient is $U=4000 \mathrm{~W} / \mathrm{m}^{2}$.
(b) Over time, the brine fouls the heat transfer surfaces, resulting in $U=2000 \mathrm{~W} / \mathrm{m}^{2}$. For the operating conditions of part (a), determine the electric power generated by the geothermal plant under fouled heat exchanger conditions.

Mahnoor Amin
Mahnoor Amin
Numerade Educator
07:12

Problem 52

An energy storage system is proposed to absorb thermal energy collected during the day with a solar collector and release thermal energy at night to heat a building. The key component of the system is a shelland-tube heat exchanger with the shell side filled with $n$-octadecane (see Problem 8.47).
(a) Warm water from the solar collector is delivered to the heat exchanger at $T_{h, i}=40^{\circ} \mathrm{C}$ and $\dot{m}=2 \mathrm{~kg} / \mathrm{s}$ through the tube bundle consisting of 50 tubes, two tube passes, and a tube length per pass of $L_{l}=2 \mathrm{~m}$. The thin-walled, metal tubes are of diameter $D=25 \mathrm{~mm}$. Free convection exists within the molten $n$-octadecane, providing an average heat transfer coefficient of $h_{o}=25 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ on the outside of each tube. Determine the volume of $n$ octadecane that is melted over a 12 -h period. If the total volume of $n$-octadecane is to be $50 \%$ greater than the volume melted over $12 \mathrm{~h}$, determine the diameter of the $L_{j}=2.2$-m-long shell.
(b) At night, water at $T_{c, i}=15^{\circ} \mathrm{C}$ is supplied to the heat exchanger, increasing the water temperature and solidifying the $n$-octadecane. Do you expect the heat transfer rate to be the same, greater than, or less than the heat transfer rate in part (a)? Explain your reasoning.

Nick Auwerda
Nick Auwerda
Numerade Educator
02:16

Problem 53

A shell-and-tube heat exchanger consists of 135 thinwalled tubes in a double-pass arrangement, each of $12.5$ - $\mathrm{mm}$ diameter with a total surface area of $47.5 \mathrm{~m}^{2}$. Water (the tube-side fluid) enters the heat exchanger at $15^{\circ} \mathrm{C}$ and $6.5 \mathrm{~kg} / \mathrm{s}$ and is heated by exhaust gas entering at $200^{\circ} \mathrm{C}$ and $5 \mathrm{~kg} / \mathrm{s}$. The gas may be assumed to have the properties of atmospheric air, and the overall heat transfer coefficient is approximately $200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) What are the gas and water outlet temperatures?
(b) Assuming fully developed flow, what is the tubeside convection coefficient?
(c) With all other conditions remaining the same, plot the effectiveness and fluid outlet temperatures as a function of the water flow rate over the range from 6 to $12 \mathrm{~kg} / \mathrm{s}$.
(d) What gas inlet temperature is required for the exchanger to supply $10 \mathrm{~kg} / \mathrm{s}$ of hot water at an outlet temperature of $42^{\circ} \mathrm{C}$, all other conditions remaining the same? What is the effectiveness for this operating condition?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:59

Problem 54

An ocean thermal energy conversion system is being proposed for electric power generation. Such a system is based on the standard power cycle for which the working fluid is evaporated, passed through a turbine, and subsequently condensed. The system is to be used in very special locations for which the oceanic water temperature near the surface is approximately $300 \mathrm{~K}$, while the temperature at reasonable depths is approximately $280 \mathrm{~K}$. The warmer water is used as a heat source to evaporate the working fluid, while the colder water is used as a heat sink for condensation of the fluid. Consider a power plant that is to generate $2 \mathrm{MW}$ of electricity at an efficiency (electric power output per heat input) of $3 \%$. The evaporator is a heat exchanger consisting of a single shell with many tubes executing two passes. If the working fluid is evaporated at its phase change temperature of $290 \mathrm{~K}$, with ocean water entering at $300 \mathrm{~K}$ and leaving at $292 \mathrm{~K}$, what is the heat exchanger area required for the evaporator? What flow rate must be maintained for the water passing through the evaporator? The overall heat transfer coefficient may be approximated as $1200 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Manik Pulyani
Manik Pulyani
Numerade Educator
03:58

Problem 55

A single-pass, cross-flow heat exchanger with both fluids unmixed is being used to heat water $\left(m_{c}=2 \mathrm{~kg} / \mathrm{s}\right.$, $c_{p}=4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ ) from $20^{\circ} \mathrm{C}$ to $100^{\circ} \mathrm{C}$ with hot exhaust gases $\left(c_{p}=1200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ entering at $320^{\circ} \mathrm{C}$. What mass flow rate of exhaust gases is required? Assume that UA is equal to its design value of $4700 \mathrm{~W} / \mathrm{K}$, independent of the gas mass flow rate.

Dading Chen
Dading Chen
Numerade Educator
01:37

Problem 56

Saturated process steam at 1 atm is condensed in a shell-and-tube heat exchanger (one shell, two tube passes). Cooling water enters the tubes at $15^{\circ} \mathrm{C}$ with an average velocity of $3.5 \mathrm{~m} / \mathrm{s}$. The tubes are thin walled and made of copper with a diameter of $14 \mathrm{~mm}$ and length of $0.5 \mathrm{~m}$. The convective heat transfer coefficient for condensation on the outer surface of the tubes is $21,800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) Find the number of tubes/pass required to condense $2.3 \mathrm{~kg} / \mathrm{s}$ of steam.
(b) Find the outlet water temperature.
(c) Find the maximum possible condensation rate that could be achieved with this heat exchanger using the same water flow rate and inlet temperature.
(d) Using the heat transfer surface area found in part (a), plot the water outlet temperature and steam condensation rate for water mean velocities in the range from 1 to $5 \mathrm{~m} / \mathrm{s}$. Assume that the shell-side convection coefficient remains unchanged.

Manik Pulyani
Manik Pulyani
Numerade Educator
View

Problem 57

The chief engineer at a university that is constructing a large number of new student dormitories decides to install a counterflow concentric tube heat exchanger on each of the dormitory shower drains. The thinwalled copper drains are of diameter $D_{i}=50 \mathrm{~mm}$. Wastewater from the shower enters the heat exchanger at $T_{h, i}=38^{\circ} \mathrm{C}$ while fresh water enters the dormitory at $T_{c, l}=10^{\circ} \mathrm{C}$. The wastewater flows down the vertical wall of the drain in a thin, falling $f$ m , providing $h_{h}=10,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) If the annular gap is $d=10 \mathrm{~mm}$, the heat exchanger length is $L=1 \mathrm{~m}$, and the water flow rate is $\dot{m}=10 \mathrm{~kg} / \mathrm{min}$, determine the heat transfer rate and the outlet temperature of the warmed fresh water.
(b) If a helical spring is installed in the annular gap so the fresh water is forced to follow a spiral path from the inlet to the fresh water outlet, resulting in $h_{c}=9050 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, determine the heat transfer rate and the outlet temperature of the fresh water.
(c) Based on the result for part (b), calculate the daily savings if 15,000 students each take a 10 -minute shower per day and the cost of water heating is $\$ 0.07 / \mathrm{kW} \cdot \mathrm{h}$.

Victor Salazar
Victor Salazar
Numerade Educator
01:07

Problem 58

A shell-and-tube heat exchanger with one shell pass and 20 tube passes uses hot water on the tube side to heat oil on the shell side. The single copper tube has inner and outer diameters of 20 and $24 \mathrm{~mm}$ and a length per pass of $3 \mathrm{~m}$. The water enters at $87^{\circ} \mathrm{C}$ and $0.2 \mathrm{~kg} / \mathrm{s}$ and leaves at $27^{\circ} \mathrm{C}$. Inlet and outlet temperatures of the oil are 7 and $37^{\circ} \mathrm{C}$. What is the average convection coefficient for the tube outer surface?

Dominador Tan
Dominador Tan
Numerade Educator
04:03

Problem 59

The oil in an engine is cooled by air in a cross-flow heat exchanger where both fluids are unmixed. Atmospheric air enters at $30^{\circ} \mathrm{C}$ and $0.53 \mathrm{~kg} / \mathrm{s}$. Oil at $0.026 \mathrm{~kg} / \mathrm{s}$ enters at $75^{\circ} \mathrm{C}$ and flows through a tube of 10-mm diameter. Assuming fully developed flow and constant wall heat flux, estimate the oil-side heat transfer coefficient. If the overall convection coefficient is $53 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and the total heat transfer area is $1 \mathrm{~m}^{2}$, determine the effectiveness. What is the exit temperature of the oil?

Penny Riley
Penny Riley
Numerade Educator
09:01

Problem 60

A recuperator is a heat exchanger that heats the air used in a combustion process by extracting energy from the products of combustion (the flue gas). Consider using a single-pass, cross-flow heat exchanger as a recuperator.
Eighty $(80)$ silicon carbide ceramic tubes $(k=20$ $\mathrm{W} / \mathrm{m} \cdot \mathrm{K})$ of inner and outer diameters equal to 55 and $80 \mathrm{~mm}$, respectively, and of length $L=1.4 \mathrm{~m}$ are arranged as an aligned tube bank of longitudinal and transverse pitches $S_{L}=100 \mathrm{~mm}$ and $S_{T}=120 \mathrm{~mm}$, respectively. Cold air is in cross flow over the tube bank with upstream conditions of $V=1 \mathrm{~m} / \mathrm{s}$ and $T_{c i}=300 \mathrm{~K}$, while hot flue gases of inlet temperature $T_{\mathrm{h}, \mathrm{I}}=1400 \mathrm{~K}$ pass through the tubes. The tube outer surface is clean, while the inner surface is characterized by a fouling factor of $R_{f}^{N \prime}=2 \times 10^{-4} \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. The air and flue gas flow rates are $\dot{m}_{c}=1.0 \mathrm{~kg} / \mathrm{s}$ and $m_{\mathrm{h}}=1.05 \mathrm{~kg} / \mathrm{s}$, respectively. As first approximations, (1) evaluate all required air properties at $1 \mathrm{~atm}$ and $300 \mathrm{~K},(2)$ assume the flue gas to have the properties of air at $1 \mathrm{~atm}$ and $1400 \mathrm{~K}$, and (3) assume the tube wall temperature to be at $800 \mathrm{~K}$ for the purpose of treating the effect of variable properties on convection heat transfer.
(a) If there is a $1 \%$ fuel savings associated with each $10^{\circ} \mathrm{C}$ increase in the temperature of the combustion air $\left(T_{c o}\right)$ above $300 \mathrm{~K}$, what is the percentage fuel savings for the prescribed conditions?
(b) The performance of the recuperator is strongly influenced by the product of the overall heat transfer coefficient and the total surface area, UA. Compute and plot $T_{c, \infty}$ and the percentage fuel savings as a function of UA for $300 \leq U A \leq 600 \mathrm{~W} / \mathrm{K}$. Without changing the flow rates, what measures may be taken to increase $U A^{*}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
09:36

Problem 61

Consider operation of the furnace-recuperator combination of Problem $11.60$ under conditions for which chemical energy is converted to thermal energy in the combustor at a rate of $q_{c o m b}=2.0 \times 10^{h} \mathrm{~W}$ and energy is transferred from the combustion gases to the load in the furnace at a rate of $q_{\text {boad }}=1.4 \times 10^{6} \mathrm{~W}$. Assuming equivalent flow rates $\left(\dot{m}_{c}=\dot{m}_{\mathrm{h}}=1.0 \mathrm{~kg} / \mathrm{s}\right)$ and specific heats $\left(c_{p, c}=c_{p, \mathrm{~h}}=1200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\right)$ for the cold air and flue gases in the recuperator, determine $T_{h, l}, T_{h, p}$ and $T_{c \rho}$ when $T_{c, i}=300 \mathrm{~K}$ and the recuperator has an effectiveness of $\varepsilon=0.30$. What value of the effectiveness would be needed to achieve a combustor air inlet temperature of $800 \mathrm{~K}$ ?

Ahmed Kamel
Ahmed Kamel
Numerade Educator
01:56

Problem 62

It is proposed that the exhaust gas from a natural gas-powered electric generation plant be used to generate steam in a shell-and-tube heat exchanger with one shell and one tube pass. The steel tubes have a thermal conductivity of $40 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, an inner diameter of $50 \mathrm{~mm}$, and a wall thickness of $4 \mathrm{~mm}$. The exhaust gas, whose flow rate is $2 \mathrm{~kg} / \mathrm{s}$, enters the heat exchanger at $400^{\circ} \mathrm{C}$ and must leave at $215^{\circ} \mathrm{C}$. To limit the pressure drop within the tubes, the tube gas velocity should not exceed $25 \mathrm{~m} / \mathrm{s}$. If saturated water at $11.7$ bar is supplied to the shell side of the exchanger, determine the required number of tubes

Manik Pulyani
Manik Pulyani
Numerade Educator
18:47

Problem 63

A recuperator is a heat exchanger that heats air used in a combustion process by extracting energy from the products of combustion. It can be used to increase the efficiency of a gas turbine by increasing the temperature of air entering the combustor.
Consider a system for which the recuperator is a crossflow heat exchanger with both fluids unmixed and the flow rates associated with the turbine exhaust and the air are $\dot{m}_{h}=6.5 \mathrm{~kg} / \mathrm{s}$ and $\dot{m}_{c}=6.2 \mathrm{~kg} / \mathrm{s}$, respectively. The corresponding value of the overall heat transfer coefficient is $U=100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) If the gas and air inlet temperatures are $T_{\mathrm{hi}, i}=700 \mathrm{~K}$ and $T_{c, i}=300 \mathrm{~K}$, respectively, what heat transfer surface area is needed to provide an air outlet temperature of $T_{c, o}=500 \mathrm{~K}$ ? Both the air and the products of combustion may be assumed to have a specific heat of $1040 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.
(b) For the prescribed conditions, compute and plot the air outlet temperature as a function of the heat transfer surface area.

Saman Zulfiqar
Saman Zulfiqar
Numerade Educator
01:05

Problem 64

A concentric tube heat exchanger uses water, which is available at $15^{\circ} \mathrm{C}$, to cool ethylene glycol from 100 to $60^{\circ} \mathrm{C}$. The water and glycol flow rates are each $0.5 \mathrm{~kg} / \mathrm{s}$. What are the maximum possible heat transfer rate and effectiveness of the exchanger? Which is preferred, a parallel-flow or counterflow mode of operation?

Manik Pulyani
Manik Pulyani
Numerade Educator
06:28

Problem 65

Water is used for both fluids (unmixed) flowing through a single-pass, cross-flow heat exchanger. The hot water enters at $90^{\circ} \mathrm{C}$ and $10,000 \mathrm{~kg} / \mathrm{h}$, while the cold water enters at $10^{\circ} \mathrm{C}$ and $20,000 \mathrm{~kg} / \mathrm{h}$. If the effectiveness of the exchanger is $60 \%$, determine the cold water exit temperature.

Uma Kumari
Uma Kumari
Numerade Educator
01:56

Problem 66

A cross-flow heat exchanger consists of a bundle of 32 tubes in a $0.6-\mathrm{m}^{2}$ duct. Hot water at $150^{\circ} \mathrm{C}$ and a mean velocity of $0.5 \mathrm{~m} / \mathrm{s}$ enters the tubes having inner and outer diameters of $10.2$ and $12.5 \mathrm{~mm}$. Atmospheric air at $10^{\circ} \mathrm{C}$ enters the exchanger with a volumetric flow rate of $1.0 \mathrm{~m}^{3} / \mathrm{s}$. The convection heat transfer coefficient on the tube outer surfaces is $400 \mathrm{~W} / \mathrm{m}^{2}$ - K. Estimate the fluid outlet temperatures.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
07:14

Problem 67

Exhaust gas from a furnace is used to preheat the combustion air supplied to the furnace burners. The gas, which has a flow rate of $15 \mathrm{~kg} / \mathrm{s}$ and an inlet temperature of $1100 \mathrm{~K}$, passes through a bundle of tubes, while the air, which has a flow rate of $10 \mathrm{~kg} / \mathrm{s}$ and an inlet temperature of $300 \mathrm{~K}$, is in cross flow over the tubes. The tubes are unfinned, and the overall heat transfer coefficient is $100 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Determine the total tube surface area required to achieve an air outlet temperature of $850 \mathrm{~K}$. The exhaust gas and the air may each be assumed to have a specific heat of $1075 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.

Dading Chen
Dading Chen
Numerade Educator
01:27

Problem 68

Derive Equation 11.35a. Hint: See Section 8.3.3.

Ajay Singhal
Ajay Singhal
Numerade Educator
16:16

Problem 69

A liquefied natural gas (LNG) regasification facility utilizes a vertical heat exchanger or vaporizer that consists of a shell with a single-pass tube bundle used to convert the fuel to its vapor form for subsequent delivery through a land-based pipeline. Pressurized LNG is off-loaded from an oceangoing tanker to the bottom of the vaporizer at $T_{c, i}=-155^{\circ} \mathrm{C}$ and $\dot{m}_{\mathrm{LNG}}=150 \mathrm{~kg} / \mathrm{s}$ and flows through the shell. The pressurized LNG has a vaporization temperature of $T_{f}=-75^{\circ} \mathrm{C}$ and specific heat $c_{p l}=4200 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. The specific heat of the vaporized natural gas is $c_{p, v}=2210 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ while the gas has a latent heat of vaporization of $h_{f g}=575 \mathrm{~kJ} / \mathrm{kg}$. The LNG is heated with seawater flowing through the tubes, also introduced at the bottom of the vaporizer, that is available at $T_{h, i}=20^{\circ} \mathrm{C}$ with a specific heat of $c_{\mu \mathrm{Sw}}=3985 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. If the gas is to leave the vaporizer at $T_{c o}=8^{\circ} \mathrm{C}$ and the seawater is to exit the device at $T_{\text {hot }}=10^{\circ} \mathrm{C}$, determine the required vaporizer heat transfer area. Hint: Divide the vaporizer into three sections, as shown in the schematic, with $U_{\mathrm{A}}=150 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, $U_{\mathrm{B}}=260 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, and $U_{\mathrm{C}}=40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.

Andrew C
Andrew C
Numerade Educator
01:22

Problem 70

Work Problem $11.69$ for the situation where the seawater is introduced to the top of the vaporizer, resulting in counterflowing natural gas and seawater.

Hossam Mohamed
Hossam Mohamed
Numerade Educator
09:01

Problem 71

Cooling of outdoor electronic equipment such as in telecommunications towers is difficult due to seasonal and diurnal variations of the air temperature, and potential fouling of heat exchange surfaces due to dust accumulation or insect nesting. A concept to provide a nearly constant sink temperature in a hermetically sealed environment is shown below. The cool surface is maintained at nearly constant groundwater temperature $\left(T_{1}=5^{\circ} \mathrm{C}\right)$ while the hot surface is subjected to a constant heat load from the electronic equipment $\left(q_{2}=50 \mathrm{~W}, T_{2}\right)$. Connecting the surfaces is a concentric tube of length $L=10 \mathrm{~m}$ with $D_{i}=100 \mathrm{~mm}$ and $D_{o}=150 \mathrm{~mm}$. A fan moves air at a mass flow rate of $m=0.0325 \mathrm{~kg} / \mathrm{s}$ and dissipates $P=10 \mathrm{~W}$ of thermal energy. Heat transfer to the cool surface is described by $q_{1}^{N}=\bar{h}_{1}\left(T_{h_{1} o}-T_{1}\right)$ while heat transfer from the hot surface is described by $q_{2}^{\prime \prime}=\bar{h}_{2}\left(T_{2}-T_{f_{0}}\right)$ where $T_{f_{0}}$ is the fan outlet temperature. The values of $\bar{h}_{1}$ and $h_{2}$ are 40 and $60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$, respectively. To isolate the electronics from ambient temperature variations, the entire device is insulated at its outer surfaces. The design engineer is concerned that conduction through the wall of the inner tube may adversely affect the device performance. Determine the value of $T_{2}$ for the limiting cases of (i) no conduction resistance in the inner tube wall and (ii) infinite conduction resistance in the inner tube wall. Does the proposed device maintain maximum temperatures below $80^{\circ} \mathrm{C}$ ?

Averell Hause
Averell Hause
Carnegie Mellon University
01:07

Problem 72

A shell-and-tube heat exchanger consisting of one shell pass and two tube passes is used to transfer heat from an ethylene glycol-water solution (shell side)
supplied from a rooftop solar collector to pure water (tube side) used for household purposes. The tubes are of inner and outer diameters $D_{i}=3.6 \mathrm{~mm}$ and $D_{o}=3.8 \mathrm{~mm}$, respectively. Each of the 100 tubes is $0.8 \mathrm{~m}$ long ( $0.4 \mathrm{~m}$ per pass), and the heat transfer coefficient associated with the ethylene glycol-water mixture is $h_{o}=11,000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) For pure copper tubes, calculate the heat transfer rate from the ethylene glycol-water solution $\left(\dot{m}=2.5 \mathrm{~kg} / \mathrm{s}, T_{h, i}=80^{\circ} \mathrm{C}\right)$ to the pure water $(\dot{m}=$ $2.5 \mathrm{~kg} / \mathrm{s}, T_{c, i}=20^{\circ} \mathrm{C}$ ). Determine the outlet temperatures of both streams of fluid. The density and specific heat of the ethylene glycol-water mixture are $1040 \mathrm{~kg} / \mathrm{m}^{3}$ and $3660 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$, respectively.
(b) It is proposed to replace the copper tube bundle with a bundle composed of high-temperature nylon tubes of the same diameter and tube wall thickness. The nylon is characterized by a thermal conductivity of $k_{n}=0.31 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$. Determine the tube length required to transfer the same amount of energy as in part (a).

Dominador Tan
Dominador Tan
Numerade Educator
01:37

Problem 73

In analyzing thermodynamic cycles involving heat exchangers, it is useful to express the heat rate in terms of an overall thermal resistance $R_{r}$ and the inlet temperatures of the hot and cold fluids,
$$
q=\frac{\left(T_{\mathrm{h}, i}-T_{c, j}\right)}{R_{t}}
$$
The heat transfer rate can also be expressed in terms of the rate equations,
$$
q=U A \Delta T_{\mathrm{lm}}=\frac{1}{R_{\mathrm{lm}}} \Delta T_{\mathrm{lm}}
$$
(a) Derive a relation for $R_{\mathrm{lm}} / R_{\mathrm{t}}$ for a parallel- $d w$ heat exchanger in terms of a single dimensionless parameter $B$, which does not involve any fluid temperatures but only $U, A, C_{\mathrm{h}}, C_{c}$ (or $C_{\min }, C_{\max }$ ).
(b) Calculate and plot $R_{\operatorname{lm}} / R_{T}$ for values of $B=0.1$, $1.0$, and 5.0. What conclusions can be drawn from the plot?

Manik Pulyani
Manik Pulyani
Numerade Educator
12:12

Problem 74

The power needed to overcome wind and friction drag associated with an automobile traveling at a constant velocity of $25 \mathrm{~m} / \mathrm{s}$ is $9 \mathrm{~kW}$.
(a) Determine the required heat transfer area of the radiator if the vehicle is equipped with an internal combustion engine operating at an efficiency of $21 \%$. (Assume $79 \%$ of the energy generated by the engine is in the form of waste heat removed by the radiator.) The inlet and outlet mean temperatures of the water with respect to the radiator are $T_{m, i}=400 \mathrm{~K}$ and $T_{\text {em }, ~}=330 \mathrm{~K}$, respectively. Cooling air is available at $3 \mathrm{~kg} / \mathrm{s}$ and $300 \mathrm{~K}$. The radiator may be analyzed as a cross-flow heat exchanger with both fluids unmixed with an overall heat transfer coefficient of $400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(b) Determine the required water mass flow rate and heat transfer area of the radiator if the vehicle is equipped with a fuel cell operating at $50 \%$ efficiency. The fuel cell operating temperature is limited to approximately $85^{\circ} \mathrm{C}$, so the inlet and outlet mean temperatures of the water with respect to the radiator are $T_{m, i}=355 \mathrm{~K}$ and $T_{m e}=330 \mathrm{~K}$, respectively. The air inlet temperature is as in part (a). Assume the flow rate of air is proportional to the surface area of the radiator. Hint: Iteration is required.
(c) Determine the required heat transfer area of the radiator and the outlet mean temperature of the water for the fuel cell-equipped vehicle if the mass flow rate of the water is the same as in part (a).

Eric Mockensturm
Eric Mockensturm
Numerade Educator
15:00

Problem 75

An air conditioner operating between indoor and outdoor temperatures of 23 and $43^{\circ} \mathrm{C}$, respectively, removes $5 \mathrm{~kW}$ from a building. The air conditioner can be modeled as a reversed Carnot heat engine with refrigerant as the working fluid. The efficiency of the motor for the compressor and fan is $80 \%$, and $0.2 \mathrm{~kW}$ is required to operate the fan.
(a) Assuming negligible thermal resistances (Problem 11.73) between the refrigerant in the condenser and the outside air and between the refrigerant in the evaporator and the inside air, calculate the power required by the motor.
(b) If the thermal resistances between the refrigerant and the air in the evaporator and condenser sections are the same, $3 \times 10^{-3} \mathrm{~K} / \mathrm{W}$, determine the temperature required by the refrigerant in each section. Calculate the power required by the motor.

Gordon Ayadju
Gordon Ayadju
Numerade Educator
19:12

Problem 76

In a Rankine power system, $1.5 \mathrm{~kg} / \mathrm{s}$ of steam leaves the turbine as saturated vapor at $0.51$ bar. The steam is condensed to saturated liquid by passing it over the tubes of a shell-and-tube heat exchanger, while liquid water, having an inlet temperature of $T_{c,}=280 \mathrm{~K}$, is passed through the tubes. The condenser contains 100 thinwalled tubes, each of 10-mm diameter, and the total water flow rate through the tubes is $15 \mathrm{~kg} / \mathrm{s}$. The average convection coefficient associated with condensation on the outer surface of the tubes may be approximated as $\bar{h}_{0}=5000 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Appropriate property values for the liquid water are $c_{p}=4178 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}, \mu=700 \times 10^{-6}$ $\mathrm{kg} / \mathrm{s} \cdot \mathrm{m}, k=0.628 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, and $\operatorname{Pr}=4.6$.
(a) What is the water outlet temperature?
(b) What is the required tube length (per tube)?
(c) After extended use, deposits accumulating on the inner and outer tube surfaces provide a cumulative fouling factor of $0.0003 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}$. For the prescribed inlet conditions and the computed tube length, what mass fraction of the vapor is condensed?
(d) For the tube length computed in part (b) and the fouling factor prescribed in part (c), explore the extent to which the water flow rate and inlet temperature may be varied (within physically plausible ranges) to improve the condenser performance. Represent your results graphically, and draw appropriate conclusions.

Eric Mockensturm
Eric Mockensturm
Numerade Educator
01:30

Problem 77

Consider a Rankine cycle with saturated steam leaving the boiler at a pressure of $2 \mathrm{MPa}$ and a condenser pressure of $10 \mathrm{kPa}$.
(a) Calculate the thermal efficiency of the ideal Rankine cycle for these operating conditions.
(b) If the net reversible work for the cycle is $0.5 \mathrm{MW}$, calculate the required flow rate of cooling water supplied to the condenser at $15^{\circ} \mathrm{C}$ with an allowable temperature rise of $10^{\circ} \mathrm{C}$.
(c) Design a shell-and-tube heat exchanger (one-shell, multiple-tube passes) that will meet the heat rate and temperature conditions required of the condenser. Your design should specify the number of tubes and their diameter and length.

Jincy M Saji
Jincy M Saji
Numerade Educator
04:12

Problem 78

Consider the Rankine cycle of Problem 11.77, which rejects $2.3 \mathrm{MW}$ to the condenser, which is supplied with a cooling water flow rate of $70 \mathrm{~kg} / \mathrm{s}$ at $15^{\circ} \mathrm{C}$.
(a) Calculate UA, a parameter that is indicative of the size of the condenser required for this operating condition.
(b) Consider now the situation where the overall heat transfer coefficient for the condenser, $U$, is reduced by $10 \%$ because of fouling. Determine the reduction in the thermal efficiency of the cycle caused by fouling, assuming that the cooling water flow rate and water temperature remain the same and that the condenser is operated at the same steam pressure.

Shahab Ullah
Shahab Ullah
Numerade Educator
00:54

Problem 79

Consider a concentric tube heat exchanger characterized by a uniform overall heat transfer coefficient and operating under the following conditions:
\begin{tabular}{lccrc}
\hline & $\dot{m}$ $(\mathbf{k g} / \mathbf{s})$ & $c_{p}$ $(\mathbf{J} / \mathbf{k g} \cdot \mathbf{K})$ & $T_{i}$ $(\boldsymbol{C})$ & $T_{o}$ $(\mathbf{C})$ \\
\hline Cold fluid & $0.125$ & 4200 & 40 & 95 \\
Hot fluid & $0.125$ & 2100 & 210 & $-$ \\
\hline
\end{tabular}
What is the maximum possible heat transfer rate? What is the heat exchanger effectiveness? Should the heat exchanger be operated in parallel flow or in counterflow? What is the ratio of the required areas for these two flow conditions?

Dading Chen
Dading Chen
Numerade Educator
18:47

Problem 80

The floor space of any facility that houses shell-andtube heat exchangers must be sufficiently large so the tube bundle can be serviced easily. A rule of thumb is that the floor space must be at least $2.5$ times the length of the tube bundle so that the bundle can be completely removed from the shell (hence the absolute minimum floor space is twice the tube bundle length) and subsequently cleaned, repaired, or replaced easily (associated with the extra half bundle length floor space). The room in which the heat exchanger of Problem $11.22$ is to be installed is $8 \mathrm{~m}$ long and, therefore, the 4.7-m-long heat exchanger is too large for the facility. Will a shell-and-tube heat exchanger with two shells, one above the other, be sufficiently small to fit into the facility? Each shell has 10 tubes and 8 tube passes.

Consider the influence of a finite sheet thickness in Example 11.2, when there are 40 gaps.
(a) Determine the exterior dimension, $L$, of the heat exchanger core for a sheet thickness of $t=0.8 \mathrm{~mm}$ for pure aluminum $\left(k_{\mathrm{al}}=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$ and polyvinylidene fluoride $\left(\mathrm{PVDF}, \quad k_{\mathrm{pv}}=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$ sheets. Neglect the thickness of the top and bottom exterior plates.
(b) Plot the heat exchanger core dimension as a function of the sheet thickness for aluminum and PVDF over the range $0 \leq t \leq 1 \mathrm{~mm}$.

Niamat Khuda
Niamat Khuda
Numerade Educator
01:14

Problem 81

Consider the influence of a finite sheet thickness in Example 11.2, when there are 40 gaps.
(a) Determine the exterior dimension, $L$, of the heat exchanger core for a sheet thickness of $t=0.8 \mathrm{~mm}$ for pure aluminum $\left(k_{a l}=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\right)$ and polyvinylidene fluoride (PVDF, $k_{\mathrm{pv}}=0.17 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$ ) sheets. Neglect the thickness of the top and bottom exterior plates.
(b) Plot the heat exchanger core dimension as a function of the sheet thickness for aluminum and PVDF over the range $0 \leq t \leq 1 \mathrm{~mm}$.

Mayukh Banik
Mayukh Banik
Numerade Educator
02:16

Problem 82

Hot exhaust gases are used in a shell-and-tube exchanger to heat $2.5 \mathrm{~kg} / \mathrm{s}$ of water from 35 to $85^{\circ} \mathrm{C}$. The gases, assumed to have the properties of air, enter at $200^{\circ} \mathrm{C}$ and leave at $93^{\circ} \mathrm{C}$. The overall heat transfer coefficient is $180 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Using the effectiveness-NTU method, calculate the area of the heat exchanger.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:06

Problem 83

In open heart surgery under hypothermic conditions, the patient's blood is cooled before the surgery and rewarmed afterward. It is proposed that a concentric tube, counterflow heat exchanger of length $0.5 \mathrm{~m}$ be used for this purpose, with the thin-walled inner tube having a diameter of $55 \mathrm{~mm}$. The specific heat of the blood is $3500 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$.
(a) If water at $T_{h j}=60^{\circ} \mathrm{C}$ and $\dot{m}_{h}=0.10 \mathrm{~kg} / \mathrm{s}$ is used to heat blood entering the exchanger at $T_{c A}=18^{\circ} \mathrm{C}$ and $\dot{m}_{c}=0.05 \mathrm{~kg} / \mathrm{s}$, what is the temperature of the blood leaving the exchanger? The overall heat transfer coefficient is $500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(b) The surgeon may wish to control the heat rate $q$ and the outlet temperature $T_{c, 0}$ of the blood by altering the flow rate and/or inlet temperature of the water during the rewarming process. To assist in the development of an appropriate controller for the prescribed values of $\hat{m}_{c}$ and $T_{c \jmath}$, compute and plot $q$ and $T_{c, \rho}$ as a function of $\dot{m}_{h}$ for $0.05 \leq \dot{m}_{\mathrm{h}} \leq 0.20 \mathrm{~kg} / \mathrm{s}$ and values of $T_{h, l}=50,60$, and $70^{\circ} \mathrm{C}$. Since the dominant influence on the overall heat transfer coefficient is associated with the blood flow conditions, the value of $U$ may be assumed to remain at $500 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$. Should certain operating conditions be excluded?

Jacquelyn Trost
Jacquelyn Trost
Numerade Educator
01:07

Problem 84

Ethylene glycol and water, at 60 and $10^{\circ} \mathrm{C}$, respectively, enter a shell-and-tube heat exchanger for which the total heat transfer area is $15 \mathrm{~m}^{2}$. With ethylene glycol and water flow rates of 2 and $5 \mathrm{~kg} / \mathrm{s}$, respectively, the overall heat transfer coefficient is $800 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$.
(a) Determine the rate of heat transfer and the fluid outlet temperatures.
(b) Assuming all other conditions to remain the same, plot the effectiveness and fluid outlet temperatures as a function of the flow rate of ethylene glycol for $0.5 \leq \dot{m}_{h} \leq 5 \mathrm{~kg} / \mathrm{s}$.

Dominador Tan
Dominador Tan
Numerade Educator
02:25

Problem 85

A boiler used to generate saturated steam is in the form of an unfinned, cross-flow heat exchanger, with water flowing through the tubes and a high-temperature gas in cross flow over the tubes. The gas, which has a specific heat of $1120 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ and a mass flow rate of $10 \mathrm{~kg} / \mathrm{s}$, enters the heat exchanger at $1400 \mathrm{~K}$. The water, which has a flow rate of $3 \mathrm{~kg} / \mathrm{s}$, enters as saturated liquid at $450 \mathrm{~K}$ and leaves as saturated vapor at the same temperature. If the overall heat transfer coefficient is $50 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}$ and there are 500 tubes, each of $0.025-\mathrm{m}$ diameter, what is the required tube length?

Anand Jangid
Anand Jangid
Numerade Educator
02:35

Problem 86

Waste heat from the exhaust gas of an industrial furnace is recovered by mounting a bank of unfinned tubes in the furnace stack. Pressurized water at a flow rate of $0.025 \mathrm{~kg} / \mathrm{s}$ makes a single pass through each of the tubes, while the exhaust gas, which has an upstream velocity of $5.0 \mathrm{~m} / \mathrm{s}$, moves in cross flow over the tubes at $2.25 \mathrm{~kg} / \mathrm{s}$. The tube bank consists of a square array of 100 thin-walled tubes $(10 \times 10)$, each $25 \mathrm{~mm}$ in diameter and $4 \mathrm{~m}$ long. The tubes are aligned with a transverse pitch of $50 \mathrm{~mm}$. The inlet temperatures of the water and the exhaust gas are 300 and $800 \mathrm{~K}$, respectively. The water flow is fully developed, and the gas properties may be assumed to be those of atmospheric air.
(a) What is the overall heat transfer coefficient?
(b) What are the fluid outlet temperatures?
(c) Operation of the heat exchanger may vary according to the demand for hot water. For the prescribed heat exchanger design and inlet conditions, compute and plot the rate of heat recovery and the fluid outlet temperatures as a function of water flow rate per tube for $0.02 \leq \dot{m}_{c, 1} \leq 0.20 \mathrm{~kg} / \mathrm{s}$.

Lottie Adams
Lottie Adams
Numerade Educator
02:25

Problem 87

A heat exchanger consists of a bank of 1200 thinwalled tubes with air in cross flow over the tubes. The tubes are arranged in-line, with 40 longitudinal rows (along the direction of airflow) and 30 transverse rows. The tubes are $0.07 \mathrm{~m}$ in diameter and $2 \mathrm{~m}$ long, with transverse and longitudinal pitches of $0.14 \mathrm{~m}$. The hot fluid flowing through the tubes consists of saturated steam condensing at $400 \mathrm{~K}$. The convection coefficient of the condensing steam is much larger than that of the air.
(a) If air enters the heat exchanger at $\dot{m}_{c}=12 \mathrm{~kg} / \mathrm{s}$, $300 \mathrm{~K}$, and $1 \mathrm{~atm}$, what is its outlet temperature?
(b) The condensation rate may be controlled by varying the airflow rate. Compute and plot the air outlet temperature, the heat rate, and the condensation rate as a function of flow rate for $10 \leq \dot{m}_{c} \leq 50 \mathrm{~kg} / \mathrm{s}$.

Anand Jangid
Anand Jangid
Numerade Educator
02:46

Problem 88

Derive the expression for the modified effectiveness $\varepsilon^{*}$, given in Comment 4 of Example 11.8.

Adriano Chikande
Adriano Chikande
Numerade Educator
04:22

Problem 89

Consider Problem 3.144a.
(a) Using an appropriate correlation from Chapter 8, determine the air inlet velocity for each channel in the heat sink. Assume laminar flow and evaluate air properties at $T=300 \mathrm{~K}$.
(b) Accounting for the increase in air temperature as it flows through the heat sink, determine the chip power $q_{c}$ and the outlet temperature of the air exiting each channel. Assume the airflow along the outer surfaces provides a similar cooling effect as airflow in the channels.
(c) If the air velocity is reduced by half, determine the chip power and the air outlet temperature.

Dading Chen
Dading Chen
Numerade Educator
03:53

Problem 90

Work Problem 7.29, taking into account the increase in temperature of the water as it flows through the heat sink. Properties of water are listed in Problem 7.29, along with $\rho=995 \mathrm{~kg} / \mathrm{m}^{3}$ and $c_{p}=4178 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Hint: Assume the water does not escape through the upper surface of the heat sink and that the boundary layers on each fin surface do not merge, allowing evaluation of the heat transfer coefficient using a correlation from Chapter 7. Also, see Problem 11.68.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
13:04

Problem 91

The heat sink of Problem $7.29$ is considered for an application in which the power dissipation is only $70 \mathrm{~W}$, and the engineer proposes to use air at $T_{m}=20^{\circ} \mathrm{C}$ for cooling. Taking into account the increase in temperature of the air as it flows through the heat sink, plot the allowable power dissipation and the air exit temperature as a function of the air velocity over the range $1 \mathrm{~m} / \mathrm{s} \leq u_{\infty} \leq 5 \mathrm{~m} / \mathrm{s}$, with the constraint that the base temperature not exceed $T_{b}=70^{\circ} \mathrm{C}$. Properties of the air may be approximated as $k=0.027 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}$, $\nu=16.4 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}, P r=0.706, \rho=1.145 \mathrm{~kg} / \mathrm{m}^{3}$, and $c_{p}=1007 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$. Hint: Assume the air does not escape through the upper surface of the heat sink, use a correlation for internal flow, and see Problem 11.68.

Brianna Orr
Brianna Orr
Numerade Educator
01:13

Problem 92

Solve Problem $8.109$ a using the effectiveness-NTU method.

James Kiss
James Kiss
Numerade Educator
02:22

Problem 93

Consider Problem 7.113. Estimate the heat transfer rate to the air, accounting for both the increase in the air temperature as it flows through the foam and the thermal resistance associated with conduction in the foam in the $x$-direction. Do you expect the actual heat transfer rate to the air to be equal to, less than, or greater than the value you have calculated?

Manish Jain
Manish Jain
Numerade Educator
01:27

Problem 94

The metallic foam of Problem $7.113$ is brazed to the surface of a silicon chip of width $W=25 \mathrm{~mm}$ on a side. The foam heat sink is $L=10 \mathrm{~mm}$ tall. Air at $T_{i}=27^{\circ} \mathrm{C}, V=5 \mathrm{~m} / \mathrm{s}$ impinges on the foam heat sink while the chip surface is maintained at $70^{\circ} \mathrm{C}$. Determine the heat transfer rate from the chip. To calculate a conservative estimate of the heat transfer rate, neglect convection and radiation from the top and sides of the heat sink.

Mayukh Banik
Mayukh Banik
Numerade Educator