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ME3122

NATIONAL UNIVERSITY OF SINGAPORE

ME3122 HEAT TRANSFER

(Semester I : AY2012/2013)

Time Allowed : 2 Hours

INSTRUCTIONS TO CANDIDATES:

1. This examination paper contains FOUR (4) questions and comprises FOURTEEN(14)printed pages.

2. Answer ALL FOUR (4)questions.

3. All questions carry equal marks.

4. This is a CLOSED-BOOK EXAMINATION.

5. Handbook of heat transfer equations, tables and charts are provided.

6. Programmable calculators are NOT allowed for this examination.

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PAGE 2 ME3122

QUESTION 1

The figure below shows the cross-section of a low temperature stage comprising 3

components Component A made of Mat A, one thin heater layer in the middle of

Component A, and Component B made of Mat B. Dimensions and properties of the low

temperature stage are summarized in the table below. (The whole structure has a constantdepth (D) of 5 cm.) The whole setup is connected to a tank filled with liquid nitrogen, which

can be considered as a heat reservoir with a constant temperature of 77 K. Normally, during

the operation, the low temperature stage is sealed in vacuum, and the surroundings are in

room temperature (25C).

Dimension k(W/m-K)

Heater layer 5 cm (W) 0.05 cm (T) 5 cm (D) 5 0.3Component A 5 cm (W) 2 cm (T) 5 cm (D) 400 0.02Component B 2 cm (W) 15 cm (T) 5 cm (D) 2 0.9

(a) When the heater is not switched on, the temperature of the whole low temperaturestage can be assumed to be 77 K. Using this temperature, calculate the total rate of

radiative heat transfer between the low temperature stage (Components A and B, you

can ignore the heater layer in this subquestion) and the surrounding. What does the

negative sign of your result mean?

(4 marks)

(b) Assume that Component A (with the heater layer) is a 1D composite plane wall, inwhich radiative heat transfer through the side walls of Component A can be

neglected. For simplicity, you can assume constant heat fluxesT

q andB

q at the top

and bottom surfaces of Component A (ignore nonuniformity at the bottom surface).The heater is switched on to heat Components A and B to roughly 100 K.

(14 marks)

(i) Assuming an abrupt temperature drop from 100 K to 77 K at the top surface of

Component A, estimate the heat flux Tq if thermal conductance is 1 kW m-2

K-1

.

(2 marks)

Mat A

Mat B

Heater layer

Reservoir 77K

vacuumW

T

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PAGE 3 ME3122

(ii) Using an appropriate control volume, estimate the total rate of heat transfer

through the bottom surface ( AqB , where A is total area of the bottomsurface). Estimate Bq . (In this subquestion, you can assume that the whole ofComponent B is at 100 K.)

(5 marks)

(iii) Under this steady-state condition, what is the rate of heat generation by theheater layer?

(2 marks)

(iv) Neglecting heat transfer through the side walls, sketch the temperature profilein Component A (including the heater and the top surface).

(5 marks)

(c) Assume that the bottom surface of Component A is at 100 K. Estimate the total rate ofheat transfer through Component B. In this subquestion, you CANNOT ignore heat

transfer through the side walls of Component B and CANNOT assume that thetemperature of Component B is uniform. Justify the assumptions made throughout the

estimation. (Tip: you could use the fin equation.)

(7 marks)

QUESTION 2

(a) Air at 1 atmospheric pressure and 58oC enters a thin-walled circular copper tube at anaverage velocity of 3.5 m/s. The copper tube has an inner diameter of 20 mm and a

length of 2.5 m, and it is wrapped by electrical heating elements that provide a

uniform heat flux over the entire length. An air bulk temperature of 146oC is required

at the exit. Neglecting entrance effect, determine:

(i) the convective heat transfer coefficient between the tube and air;(6 marks)

(ii) the uniform heat flux required; and

(6 marks)

(iii) the exit surface temperature of the tube.

(5 marks)

State any assumptions made in your calculations.

(b) Instead of applying a uniform heat flux, if the surface temperature of the tube is nowmaintained at a uniform temperature of 160

oC over its entire length, determine the

length of tube that is required to achieve the same air flow parameters as in (a).

(8 marks)

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QUESTION 3

In a gas-to-gas heat recovery unit, air is preheated from 25C to 255C at the rate of 20 kg/sby waste gas available at the rate of 20 kg/s at 400C. The air preheater is essentially a shell-and-tube heat exchanger with one shell and two tube passes where the gas moves with a

mean velocity of 15 m/s through copper tubes (kwall= 400 W/mK) having outer and innerdiameters of 55 mm and 53 mm respectively, and air flows across the bank of tubes with a

mean velocity of 10 m/s. For cross-flow, the following equation may be used

Nu = 0.27 Re0.63

Pr0.36

(a) Determine the overall heat transfer coefficient. (Hint: You can assume the wall to bethin)

(8 marks)

(b) Determine the required heating surface.

(7 marks)

(c) Determine the number of tubes required.(5 marks)

(d) Determine the length of tubes per pass.(5 marks)

Given:

Properties of air at 140C: a= 0.844 kg/m3, cpa= 1.01 kJ/kgK,

ka= 0.0352 W/mK, a= 28.3 10-6

m2/s and Pra= 0.684

Properties of gas at 295C: g= 0.622 kg/m3, cpg= 1.11 kJ/kgK,kg= 0.0454 W/mK, g= 41.2 10

-6m2/s and Prg= 0.660

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PAGE 5 ME3122

QUESTION 4

A hemispherical cavity of radius 1.0 m is covered with a plate having an opening of 0.2 m

diameter drilled at its centre. The inner surface of the plate is maintained at 700 K by a heater

embedded in the surface. Let the inner surface of the plate be 1, the surface of the hemisphere

be 2, and the virtual surface of the opening be 3. The surface can be assumed to be black andthe hemisphere taken to be well insulated.

(a) Determine F22.(12 marks)

(b) Determine the temperature of the surface of the hemisphere.(8 marks)

(c) Determine the power input to the heater.(5 marks)

1.0 m

3

21

700 K

0.2 m

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PAGE 6 ME3122

INFORMATION SHEETS

1stlaw of thermodynamics:

Conduction:

Convection:

Radiation:

Control Volume:

Surface volume:

Solids:

Free electrons:

Gases:

Joule heating:

Interfaces:

Cartesian:

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Cylindrical:

Spherical:

Heat wave speed:

Error function:

Erf(0)=0; Erf()=1; Erf(-)=-1; Erf(0.48)=0.5

Two semi-infinite solids touch:

Semi-infinite solids, surface temp at Ts

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Fin Efficiency:

Fin Effectiveness:

Lumped Capacitance Method:

,

, , ,

SUMMARY ON FORCED CONVECTION

External flow over Isothermal flat plate with uniform temperature Tw= constant

laminar flow:

local

3/1xx PrRe332.0

k

xhNu 21x Rex5x105

0.6 Pr 60

average3/121

LLL PrRe664.0Nu2kLhNu

turbulent flow for x > xcr:

local

3/1xx PrRe0296.0

k

xhNu 54x 5x105Rex1x107

0.6 Pr 60

average

3/154

LLL PrRe037.0Nu4

5

k

LhNu

mixed laminar-turbulent flow over length L:

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average

53/154L

3/121

cr

3/154

cr

54

LL

105Pr)871Re037.0(

PrRe664.0Pr)Re(Re037.0k

LhNu

crRefor

valid for 5x105ReL1x10

70.6 Pr 60

Total heat transfer rate:

)TT(AhQ ww

External flow over flat plate with uniform heat flux qw=constant

laminar flow:

local

3/121 PrRe453.0 xx

xk

xhNu

Rex5x105

0.6Pr 60

average3/121

LLL PrRe906.0Nu2

k

LhNu

turbulent flow for x > xcr:

local

3/154

xx

x PrRe0308.0k

xhNu

5x105Rex1x10

7

0.6 Pr 60

average3/154

LLL PrRe0385.0Nu

4

5

k

LhNu

mixed laminar-turbulent flow over length L:

average

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53/154L

3/121

cr

3/154

cr

54

LL

105Pr755Re0385.0

PrRe906.0Pr)Re(Re0385.0k

LhNu

crRefor

valid for 5x105ReL1x107 0.6 Pr 60

Total heat transfer rate:

wwAqQ

External flow over flat plate with uniform heat flux qw=constant

Wall temperature distribution:

Local )()(

xh

qTxT

x

ww

average

L

xturx

x

lamx

wL

x

ww

c

c

dxh

dxhL

qdx

hL

qTT

,0

,0

11

1

laminar flow: Rex5x105

0.6Pr 60

average)PrRe68.0()

2

3( 3/121

L

w

L

ww

L

k

q

h

qTT

turbulent flow from x> xcr:

5x105Rex1x10

7

average

)PrRe037.0()5

6

(3/154

L

w

L

w

w

L

k

q

h

qTT

mixed laminar-turbulent flow over length L:

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PAGE 12 ME3122

5cr2

8

5431

5421

2

5431

105ReforRe

10335.3

Re037.0

1

)Pr(

Re037.0

1

Re68.0

1

Re

Re

Re037.0

1

)Pr(

LL

w

crcrL

cr

L

w

w

k

Lq

k

LqTT

valid for 5x105ReL1x10

70.6Pr60

Internal Flow in Smooth Circular Tube/Pipe

Laminar flowin isothermal tube with constant temp Tw :

average 2000Refor66.3 Dh

k

DhNuD

Laminar flowin tube with constant wall heat flux qw:

average 2000Refor36.4 Dh

k

DhNuD

where

hch

Du

P

AD DRe

4

Turbulent flowin smooth circular tube/pipe (for both isothermal wall or constant heat-flux

wall):

Dittus Boelter equation:

average

n

DDk

hNu PrRe023.0

D 8.0h

for ReD 2000 0.6Pr 100

n = 0.4 for heating of fluid (Tw > Tb)

n = 0.3 for cooling of fluid (Tw < Tb)

Photons:

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Solid angle:

Convection:

Spectral Intensity:

Diffuse emitter:

Blackbody:

Weins displacement law:

Real surfaces:

Semitransparent medium:

View factor:

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Radiation exchange:

Radiation network approach:

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