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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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 constant depth (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 (25 C). Reservoir 77K Heater layer
Mat A Mat B
vacuum
W T
Heater layer Component A Component B
Dimension 5 cm (W) 0.05 cm (T) 5 cm (D) 5 cm (W) 2 cm (T) 5 cm (D) 2 cm (W) 15 cm (T) 5 cm (D)
k (W/m-K) 5 400 2
0.3 0.02 0.9
(a)
When the heater is not switched on, the temperature of the whole low temperature stage 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, in which radiative heat transfer through the side walls of Component A can be neglected. For simplicity, you can assume constant heat fluxes qT and q B 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 -2 Component A, estimate the heat flux qT if thermal conductance is 1 kW m -1 K . (2 marks)
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(ii)
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Using an appropriate control volume, estimate the total rate of heat transfer through the bottom surface ( q B A , where A is total area of the bottom surface). Estimate q B . (In this subquestion, you can assume that the whole of Component B is at 100 K.) (5 marks)
(c)
(iii)
Under this steady-state condition, what is the rate of heat generation by the heater layer? (2 marks)
(iv)
Neglecting heat transfer through the side walls, sketch the temperature profile in Component A (including the heater and the top surface). (5 marks)
Assume that the bottom surface of Component A is at 100 K. Estimate the total rate of heat transfer through Component B. In this subquestion, you CANNOT ignore heat transfer through the side walls of Component B and CANNOT assume that the temperature of Component B is uniform. Justify the assumptions made throughout the estimation. (Tip: you could use the fin equation.) (7 marks)
QUESTION 2
(a)
o
Air at 1 atmospheric pressure and 58 C enters a thin-walled circular copper tube at an average 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 146 oC 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 now o maintained at a uniform temperature of 160 C 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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ME3122
QUESTION 3
In a gas-to-gas heat recovery unit, air is preheated from 25 C to 255 C at the rate of 20 kg/s by waste gas available at the rate of 20 kg/s at 400 C. The air preheater is essentially a shelland-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 ( k wall = 400 W/mK) having outer and inner diameters 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 0.63
Nu = 0.27 Re
0.36
Pr
(a)
Determine the overall heat transfer coefficient. (Hint: You can assume the wall to be thin) (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 140 C: a = 0.844 kg/m , c pa = 1.01 kJ/kg K, -6 2 k a = 0.0352 W/m K, a = 28.3 10 m /s and Pr a = 0.684 3
Properties of gas at 295 C: g = 0.622 kg/m , c pg = 1.11 kJ/kg K, k g = 0.0454 W/m K, g = 41.2 10-6 m2/s and Pr g = 0.660 3
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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 and the hemisphere taken to be well insulated. (a)
Determine F 22. (12 marks)
(b)
Determine the temperature of the surface of the hemisphere. (8 marks)
(c)
Determine the power input to the heater. (5 marks)
0.2 m 3
700 K
1
1.0 m
2
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INFORMATION SHEETS st
1 law 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 T s
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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
Nux
hx x
0.332 Re x1 2Pr 1/ 3
k
Rex ≤ 5x10
5
0.6 ≤ Pr ≤ 60
average
NuL
hL k
2 NuL 0.664 Re1L2 Pr 1 / 3
turbulent flow for x > x cr:
local
Nux
hxx k
0.0296 Re x4 5Pr 1/3
5
5x10 ≤ Rex≤ 1x10 0.6 ≤ Pr ≤60
average
NuL
hL 5 NuL 0.037 Re4L5 Pr 1/ 3 k 4
mixed laminar-turbulent flow over length L:
7
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average
NuL
hL k
45 ) Pr 1/ 3 0.664 Re1cr 2 Pr 1/ 3 0.037(Re4L 5 Recr
(0.037 Re4L 5 871) Pr 1/3 valid for
5
for Recr 5105
7
5x10 ≤ReL≤1x10 0.6 ≤ Pr ≤60
Total heat transfer rate:
Q hAw (Tw T) External flow over flat plate with uniform heat flux qw=constant laminar flow:
local
Nu x
h x x k
0.453 Re x1 2 Pr 1 / 3
Rex ≤ 5x10
5
0.6≤ Pr ≤60
average
NuL
hL k
2 NuL 0.906 Re1L2 Pr 1/ 3
turbulent flow for x > x cr:
local
Nux
hx x k
0.0308Re4x 5 Pr 1/ 3
5x10
5
≤ Rex≤ 1x10
0.6 ≤ Pr ≤60
average
NuL
hL k
5
NuL 0.0385 Re4L 5 Pr 1/ 3 4
mixed laminar-turbulent flow over length L:
average
7
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NuL
hL k
45 0.0385 (Re4L5 Recr ) Pr 1/ 3 0.906 Re1cr 2 Pr 1/ 3
0.0385 ReL4 5755 Pr 1/ 3 valid for
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5
5x10 ≤ ReL≤ 1x10
7
for Recr 510
5
0.6 ≤ Pr ≤60
Total heat transfer rate:
Q qw Aw External flow over flat plate with uniform heat flux qw=constant Wall temperature distribution:
Local
average
T w ( x) T
T w T
laminar flow: Rex≤ 5x10
average
T w
h x ( x)
qw
L
1
dx L h 0
5
T
qw
x
qw xc 1
dx dx 0 x L h x,lam h x,tur c
0.6≤Pr ≤60
qw
3 ( h L ) 2
qw k (0.68 Re L1 2 Pr 1 / 3 ) L
turbulent flow from x> x cr: 5
5x10 ≤ Rex≤ 1x10
average
T w
T
7
qw
6 ( h L ) 5
qw k (0.037 Re L4 5 Pr 1/ 3 ) L
mixed laminar-turbulent flow over length L:
L
1
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qw L 1 T w T (k Pr 1 3 ) 0.037Re L4 5 qw L 1 (k Pr 1 3 ) 0.037Re L4 5 valid for
5
5x10 ≤ReL≤1x10
7
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Re cr Re L
2
1 1 1 2 4 5 0.68Recr 0.037Recr
3.335108
Re L2
for Recr 5 105
0.6≤Pr ≤60
Internal Flow in Smooth Circular Tube/Pipe Laminar flow in isothermal tube with constant temp Tw :
average
Nu D
h Dh k
3.66
for Re D
2000
Laminar flow in tube with constant wall heat flux q w :
average
h Dh
Nu D
where
Dh
k
4.36 4 Ac
for Re D Re D
P
2000
u Dh
Turbulent flow in smooth circular tube/pipe (for both isothermal wall or constant heat-flux wall):
Dittus Boelter equation:
average
Nu D
h Dh
for ReD
Photons:
k ≥
0.023 Re D0.8 Pr n
2000
0.6≤ Pr ≤100
n = 0.4 for heating of fluid
(T w > T b)
n = 0.3 for cooling of fluid
(T w < T b)
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Solid angle:
Convection:
Spectral Intensity:
Diffuse emitter:
Blackbody:
Wein’s displacement law:
Real surfaces:
Semitransparent medium:
View factor:
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Radiation exchange:
Radiation network approach:
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ME3122
