Files-5-Exams Quizzes Examples Problems Me315

5/27/2018 Files-5-Exams Quizzes Examples Problems Me315

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xamples andProblems


2/44

Problem 1.31

Chips of width L= 15 mm on a side are mountedto a substrate that is installed in an enclosurewhose walls and air are maintained at atemperature of Tsur = T = 25

oC. The chips havean emissivity of = 0.60 and a maximum allowable

temperature of Ts= 85oC.a) If heat is rejected from the chips by radiation and

natural convection, what is the maximum operatingpower of each chip? The convection coefficient

depends on the chip-to-air temperature differenceand may be approximated as h = C(Ts T)

1/4,where C= 4.2 W/m2.K5/4.


3/44

Problem 1.31 (Contd.)

b) If a fan is used to maintain air flowthrough the enclosure and heat transfer

is by forced convection, with h = 250

W/m2

.K, what is the maximumoperating power?


4/44

Problem 1.44

Radioactive wastes are packed in a long, thin-

walled cylindrical container. The wastesgenerate thermal energy non-uniformlyaccording to the relation , where isthe local rate of energy generation per unit

volume, is a constant, and ro is the radius ofthe container. Steady-state conditions aremaintained by submerging the container in aliquid that is at T and provides a uniform

convection coefficient h.Obtain an expression for the total rate at whichenergy is generated in a unit length of thecontainer. Use this result to obtain an expression

for the temperature Tsof the container wall.

21 orroqq q

oq


5/44

Problem 1.63A rectangular forced air heating duct is suspended

from the ceiling of a basement whose air and wallsare at a temperature of T= Tsur= 5oC. The duct is

15 m long, and its cross-section is 350 mm 200mm.

a) For an uninsulated duct whose average surface

temperature is 50oC, estimate the rate of heat lossfrom the duct. The surface emissivity andconvection coefficient are approximately 0.5 and 4W/m2.K, respectively.

b) If heated air enters the duct at 58oC and a velocityof 4 m/s and the heat loss corresponds to the resultof part (a), what is the outlet temperature? Thedensity and specific heat of the air may beassumed to be r = 1.10 kg/m3 and cp = 1008J/kg.K, respectively.


6/44

Problem 2.22

Uniform internal heat generation at W/m3

is occurring in a cylindrical nuclear reactor fuel

rod of 50-mm diameter, and under steady-state

conditions the temperature distribution is of theform T(r) = a + br2, where T is in degrees

Celsius and ris in meters, while a= 800oC and b

= -4.167 105oC/m2. The fuel rod properties are

k = 30 W/m.k, r = 1100 kg/m3, and cp = 800J/kg.K.

7105q


7/44

Problem 2.22 (contd.)

a) What is the rate of heat transfer per unit length

of the rod at r= 0 (the centerline) and r= 25

mm (the surface)?

b) If the reactor power level is suddenly increasedto W/m3, what is the initial time rate of

temperature change at r= 0 and r= 25 mm?

8102 q


8/44

Problem 2.26 (a) & (b)

One-dimensional, steady-state conduction with

uniform internal energy generation occurs in a

plane wall with a thickness of 50 mm and a

constant thermal conductivity of 5 W/m.K. Forthese conditions, the temperature distribution

has the form, T(x) = a+ bx+ cx2. The surface at

x = 0 has a temperature of T(0) = To = 120oC

and experiences convection with a fluid forwhich T = 20

oC and h = 500 W/m2.K. The

surface atx= Lis well insulated.


9/44


a) Applying an overall energy balance to

the wall, calculate the internal energy

generation rate

b) Determine the coefficients a, b, and cby

applying the boundary conditions to the

prescribed temperature distribution. Use

the results to calculate and plot thetemperature distribution.

.q


10/44

Problem 3.2 (a)

The rear window of an automobile isdefogged by passing warm air over its

inner surface.

a) If the warm air is at T,i = 40o

C and thecorresponding convection coefficient is hi =

30 W/m2.K, what are the inner and outer

surface temperatures of 4-mm-thick window

glass, if the outside ambient air temperatureis T,o = -10

oC and the associated

convection coefficient is ho= 65 W/m2.K?


11/44

Problem 3.29The diagram shows a conical section fabricated

from pure aluminum. It is of circular crosssection having diameter D = ax1/2, where a =0.5 m1/2. The small end is located at x1 = 25mm and the large end atx2= 125 mm. The end

temperatures are T1= 600 K and T2= 400 K,while the lateral surface is well insulated.

a) Derive an expression for the temperaturedistribution T(x) in symbolic form, assuming 1-D

conditions. Sketch the temperature distribution.

b) Calculate the heat rate qx.


12/44

Example 3.5

The possible existence of an optimum insulation

thickness for radial systems is suggested by the

presence of competing effects associated with an

increase in this thickness. In particular, although the

conduction resistance increases with the addition ofinsulation, the convection resistance decreases due

to increasing outer surface area. Hence there may

exist an insulation thickness that minimizes heat

loss by maximizing the total resistance to heattransfer. Resolve this issue by considering the

following system.


13/44

Example 3.5 (Contd.)

1. A thin-walled copper tube of radius riis used to

transport a low-temperature refrigerant and is

at a temperature Tithat is less than that of the

ambient air at Taround the tube. Is there an

optimum thickness associated with applicationof insulation to the tube?


14/44

Problem 3.73

Consider 1-D conduction in a plane composite

wall. The outer surfaces are exposed to a fluid at

25oC and a convection heat transfer coefficient of

1000 W/m2.K. The middle wall B experiences

uniform heat generation , while there is no

generation in walls A and C. The temperatures at

the interfaces are T1= 261oC and T2= 211

oC.

a) Assuming a negligible contact resistance at the

interfaces, determine the volumetric heat

generation and the thermal conductivity kB.

B

q

Bq


15/44


b) Plot the temperature distribution, showing its

important features.

c) Consider conditions corresponding to a loss of

coolant at the exposed surface of material A ( h=

0). Determine T1and T2and plot the temperaturedistribution throughout the system.


16/44

Problem 3.101

A thin flat plate of length L, thickness t, and width

W L is thermally joined to two large heat sinksthat are maintained at a temperature To. The

bottom of the plate is well insulated, while the net

heat flux to the top surface of the plate is known to

have a uniform value of .

a) Derive the differential equation that determines the

steady-state temperature distribution T(x) in the

plate.

b) Solve the foregoing equation for the temperature

distribution, and obtain an expression for the rate

of heat transfer from the plate to the heat sinks.

''

oq


17/44

Problem 3.102

Consider the flat plate of problem 3.101, but

with the heat sinks at different temperatures,

T(0) = To and T(L) = TL, and with the bottom

surface no longer insulated. Convection heat

transfer is now allowed to occur between thissurface and a fluid at T, with a convection

coefficient h.

a) Derive the differential equation that determines

the steady-state temperature distribution T(x) in

the plate.


18/44

Problem 3.134

As more and more components are placed

on a single integrated circuit (chip), the

amount of heat that is dissipated continues to

increase. However, this increase is limited by

the maximum allowable chip operating

temperature, which is approximately 75oC. To

maximize heat dissipation, it is proposed that

a 4 4 array of copper pin fins bemetallurgically joined to the outer surface of a

square chip that is 12.7 mm on a side.


19/44



20/44


a) Sketch the equivalent thermal circuit for the

pin-chip-board assembly, assuming one-dimensional, steady-state conditions andnegligible contact resistance between the pinsand the chip. In variable form, label

appropriate resistances, temperatures, andheat rates.

b) For the conditions prescribed in Problem 3.27,what is the maximum rate at which heat can bedissipated in the chip when the pins are inplace? That is, what is the value of qcfor Tc=75oC? The pin diameter and length are Dp =1.5 mm and L

p= 15 mm.


21/44

Problem 4.10

A pipeline, used for transport of crude oil, isburied in the earth such that its centerline is a

distance of 1.5 m below the surface. The pipe

has an outer diameter of 0.5 m and is insulated

with a layer of cellular glass 100 mm thick. What

is the heat loss per unit length of pipe under

conditions for which heated oil at 120oC flows

through the pipe and the surface of the earth isat temperature of 0oC?


22/44

Problem 4.23

A hole of diameter D= 0.25 m is drilled through

the center of a solid block of square crosssection with w = 1 m on a side. The hole is

drilled along the length, l = 2 m, of the block,

which has a thermal conductivity of k = 150

W/m.K. The outer surfaces are exposed to

ambient air, with T,2= 25oC and h2= 4 W/m

2.K,

while hot oil flowing through the hole is

characterized by T,1 = 300o

C and h1 = 50W/m2.K. Determine the corresponding heat rate

and surface temperatures.


23/44

2ndMajor Exam (062)

An aluminum heated plate is being cooled by

air flowing over both sides and parallel to theplate as shown in the figure below with T=25oCand h=30 W/m2K. At time t = 0, the plate is200C.

i. Find the Biot number and check the validity oflumped analysis

ii. Find the plate temperature at t=10sec.

iii. Find the time rate of change of the platetemperature at t=0.

(Hint: Neglect radiation)

Properties of Aluminum: k=200W/m.K, c=900 J/kg.K,r=2700kg/m3.


24/44

2ndMajor Exam (062)

U=12m/s

T=25C

L=1m

L=1m

=5mm

Air

Aluminum Plate


25/44

Problem 5.111 (modified)

A plane wall of thickness 20 mm is insulated onthe left face and subjected to convection

condition on the right face, as shown below.


26/44

Problem 5.111 (modified, contd.)

a) Consider the 5-node network shownschematically. Write the implicit form of the

finite-difference equations for the network and

determine temperature distributions for t = 50,

100, and 500 s using a time increment oft= 1s.

b) Use the one-term approximation given in

section 5.5 to obtain the temperature at thesame location and times as in (a). Compare the

two results.


27/44

Solution of Problem 5.111

mLx 005.04/020.04/

6115

005.0500

k

xhBi


28/44

Explicit method

Node 1:

Node 2:

Node 3:

Node 4:

Node 5:

pTFopTFopT1

21

2

21

1

pTFopTpTFopT2

2131

12

pTFopTpTFopT3

2142

13

pTFopTpTFopT4

21

53

1

4

TFoBip

TFoBiFop

TFop

T 25

2214

21

5


29/44

Stability condition:

&

021 Fo

0221 FoBiFo

45.2

12

1

tBi

Fo


30/44

Implicit method

Node 1:

Node 2:

Node 3:

Node 4:

Node 5:

pTpTFopTFo1

1

2

21

1

21

pTpTpTFopTFo2

13

11

12

21

pTpTpTFopTFo3

14

12

13

21

pTpTpTFopTFo4

15

1

3

1

421

TFoBip

Tp

TFop

TFoBiFo 25

1

42

15

221


31/44

Problem 6.26

Forced air at T= 25oC and V = 10 m/s is used to

cool electronic elements on a circuit board. Onesuch element is a chip, 4 mm by 4 mm, located 120mm from the leading edge of the board.Experiments have revealed that flow over the boardis disturbed by the elements and that convection

heat transfer is correlated by an expression of theform

Estimate the surface temperature of the chip if it is

dissipating 30 mW.

31Pr

85.0Re04.0 xxNu


32/44

Problem 8.13

Consider a cylindrical nuclear fuel rod of length L

and diameter Dthat is encased in a concentric tube.

Pressurized water flows through the annular region

between the rod and the tube at a rate , and the

outer surface of the tube is well insulated. Heatgeneration occurs within the fuel rod, and the

volumetric generation rate is known to vary

sinusoidally with distance along the rod. That is

, where (W/m3) is a constant. Auniform convection coefficient h may be assumed to

exist between the surface of the rod and the water.

m

,sin Lxqxq o oq


33/44


a) Obtain expressions for the local heat flux q(x)

and the total heat transfer q from the fuel rod to

the water.

b) Obtain an expression for the variation of the

mean temperature Tm(x) of the water with

distancexalong the tube.

c) Obtain an expression for the variation of the

rod surface temperature Ts(x) with distance xalong the tube. Develop an expression for thex

location at which this temperature is

maximized.


34/44

Problem 8.31

To cool a summer home without using a vapor

compression refrigeration cycle, air is routed

through a plastic pipe (k= 0.15 W/m.K, Di= 0.15

m, Do = 0.17 m) that is submerged in anadjoining body of water. The water temperature

is nominally at T = 17oC, and a convection

coefficient of ho= 1500 W/m2.K is maintained at

the outer surface of the pipe.


35/44


If air from the home enters the pipe at a

temperature of Tm,i = 29oC and volumetric flow

rate of Vi = 0.025 m3/s, what pipe length L is

needed to provide a discharge temperature ofTm,o= 21

oC? What is the fan power required to

move the air through this length of pipe if its

inner surface is smooth?


36/44

Problem 7.88

A tube bank uses an aligned arrangement of 30-

mm-diameter tubes with ST= SL= 60 mm and a

tube length of 1 m. There are 10 tube rows in the

flow direction (NL= 10) and 7 tubes per row (NT

= 7). Air with upstream conditions of T= 27oCand V= 15 m/s is in cross flow over the tubes,

while a tube wall temperature of 100oC is

maintained by steam condensation inside thetubes. Determine the temperature of air leaving

the tube bank, the pressure drop across the

bank, and the fan power requirement.


37/44

Problem 11.7The condenser of a steam power plant contains N=

1000 brass tubes (kt= 110 W/m.K), each of inner

and outer diameters, Di= 25 mm and Do= 28 mm,

respectively. Steam condensation on the outer

surfaces of the tubes is characterized by aconvection coefficient of ho= 10,000 W/m

2.K.

a) If cooling water from a large lake is pumped

through the condenser tubes at , what

is the overall heat transfer coefficient Uobased on

the outer surface area of a tube? Properties of the

water may be approximated as = 9.60 10-4

N.s/m2, k= 0.60 W/m.K, and Pr = 6.6.

skgmc /400


38/44

Problem 11.7(contd.)

b) If, after extended operation, fouling provides a

resistance of R''f,i = 10-4 m2.K/W, at the inner

surface, what is the value of Uo?

c) If water is extracted from the lake at 15oC and

10 kg/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 is4180 J/kg.K.


39/44

Problem 11.44

A shell-and-tube heat exchanger is to heat

10,000 kg/h of water from 16 to 84oC by hotengine oil flowing through the shell. The oil

makes a single shell pass, entering at 160oC

and leaving at 94o

C, with an average heattransfer coefficient of 400 W/m2.K. The water

flows through 11 brass tubes of 22.9-mm inside

diameter and 25.5-mm outside diameter, with

each tube making four passesthrough the shell.

(a) Assuming fully developed flow for the water,

determine the required tube length per pass.


40/44

Problem 11.25In a diary operation, milk at a flow rate of 250

liter/hour and a cow-body temperature of 38.6oCmust be chilled to a safe-to-store temperature of

13oC or less. Ground water at 10oC is available at a

flow rate of 0.72 m3/h. The density and specific heat

of milk are 10300 kg/m3

and 3860 J/kg.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 W/m2.K.


41/44


b) Determine the outlet temperature of the water.

c) Using the value of UA found in part (a),

determine the milk outlet temperature if the

water flow rate is doubled. What is the outlettemperature if the flow rate is halved?


42/44

Problem 12.29

The spectral, hemispherical emissivity of

tungsten may be approximated by the

distribution depicted below. Consider a

cylindrical tungsten filament that is of diameter

D = 0.8 mm and length L = 20 mm. Thefilament is enclosed in an evacuated bulb and

is heated by an electrical current to a steady-

state temperature of 2900 K.

a) What is the total hemispherical emissivity

when the filament temperature is 2900 K?


43/44


b) Assuming the surroundings are at 300 K,what is the initial rate of cooling of the

filament when the current is switched off?


44/44

Problem 13.50

A very long electrical conductor 10 mm in

diameter is concentric with a cooled cylindrical

tube 50 mm in diameter whose surface is diffuse

and gray with an emissivity of 0.9 andtemperature of 27oC. The electrical conductor

has a diffuse, gray surface with an emissivity of

0.6 and is dissipating 6.0 W per meter of length.

Assuming that the space between the twosurfaces is evacuated, calculate the surface

temperature of the conductor.

Files-5-Exams Quizzes Examples Problems Me315

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