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Sat512: Advanced Heat Transfer Question Paper
Sat512: Advanced Heat Transfer
Course:Master Of Science
Institution: Kenyatta University question papers
Exam Year:
KENYATTA UNIVERSITY
UNIVERSITY EXAMINATIONS 2007/2008
SECOND SEMESTER EXAMINATION FOR THE DEGREE OF MASTER OF
SCIENCE
SAT 512:
ADVANCED HEAT TRANSFER
DATE: Monday, 9th June, 2008
TIME: 9.00 a.m. – 12.00 noon
------------------------------------------------------------------------------------------------------------
INSTRUCTIONS:
Do any THREE (3) questions.
All questions carry equal marks.
Stefan-Boltzmann constant
8
?
2
?
4
? ? 67
.
5
10
?
x
Wm K
Q.1
A composite solid wall is arranged as shown below Figure Q.1:-
The upper and lower surfaces as well as the two surfaces parallel to the plane of
the page are insulated so that heat transfer within the composite is from right to
left or left to right only. The composite has unit thickness in the direction normal
to the page. Outer surfaces of layers A and B are exposed to a fluid at
temperature T ? 1 and a convective heat coefficient h1. For steady – state
conditions use the data:
H1 = H2 = 1.25m; L1 = L2 = 10 cm; L3 = L4 = 15 cm
KA = 20Wm-1K-1; KB = 15Wm-1K-1; KG = 5Wm-1K-1
KD = 2Wm-1K-1; KE = 0.5Wm-1K-1
T ,
? = 10oC
T ?, = 160o
1
2
H1 = 60Wm-2K-1
h2 = 40Wm-2K-1
a)
Sketch the thermal circuit of the wall and show how the resistances can be
determined.
b)
Determine:-
i)
The rate of heat transfer across the wall.
ii)
The temperatures T1 and T4
Q.2
a)
Show that for purely radial conduction of heat in a right circular cylinder
of thermal conductivity K that
Q
T
? r2 ?
1 – T2 =
log
e ?
?
2 K
?
? r1 ?
Where T is the temperature at a corresponding radius r, and Q is the rate of
heat transmission.
(6 marks)
b)
A cylindrical conductor of radius r, which is at a uniform temperature, is
covered with electrical insulation of thermal conductivity K. The heat
transfer coefficient between the surface of the insulation and the
surrounding air is h:-
i)
Determine the outside radius of the insulation for the heat transfer
2
from the conductor to be a maximum for the given conductor and
the air temperature.
(6 marks)
ii)
Show that the insulation will act as thermal insulation only if its
outside radius is greater than the value of r2 given by
r
?hr ? ?r ?
1
1
??
?ln 2
?
? 1
?
r
K
r
(5 marks)
2
?
? ? 1 ?
iii)
If r1 = 3mm, h = 10W/ (m2K) and
K = 0.1 W/(mk);
Find the value of r2.
(3 marks)
Q.3
a)
It is desired to cut down the radiation loss between two parallel surfaces
by inserting a sheet of aluminium foil midway between them. The
temperatures of the two surfaces are maintained at 40oC and 5oC, and the
emissivities of both surfaces is 0.85. The emissivity of aluminium foil is
0.05. Calculate the percentage reduction in heat loss by radiation using
the aluminium foil, assuming that the surface temperatures are the same in
both cases and that all surfaces are grey. Neglect end effects.
(10 marks)
b)
Two parallel plates 0.5m by 1m are spaced 0.5m apart. One plate is
maintained 1000oC and the other at 500oC. The emissivities of the plates
are 0.2 and 0. 5 respectively. The plates are located in a very large
room, the walls of which are maintained at 27oC. The plates exchange
heat with each other and with the room, but only the plate surfaces facing
each other are to be considered in the analysis.
Find the net transfer of heat to each plate and to the room. (10 marks)
Q.4
a)
Show that in Natural Convection heat transfer that NU = ? ? .
Gr Pr? using
variables ?, ?, K,Cp,?, g
? ,L where ? is the coefficient of cubical
expansion,
3
1
?
?
d
?
and g is the gravitational acceleration.
(6
dT ?
marks)
b)
For forced-convection heat transfer problems the heat transfer coefficient
h
is found to depend only on the fluid viscosity ?,density? , the thermal
conductivity of the fluid K, the specific heat capacity of the fluid C, a
characteristic dimension of the solid surface L, the temperature difference
between the surface and the fluid ? , and athe fluid velocity U.
i)
Show, by method of dimensions, that one form of relationship may
K
??c ?ul u2 ?
be expressed as h?
? ? ,
,
?
l
? K ?
?
c ?
or Nu = ??Pr,Re, Ec?
hl
Nu = Nusselt Number =
K
Pr = Prandtl Number = c
? K
?ul
Re = Reynolds Number =
?
Ec = Eckert Number = u2 c
(7 marks)
ii)
The heat transfer coefficient for a forced-convection air cooler
system is to be estimated from a test on a one third scale model
using air. If the velocity of the air in the cooler is to be 12 m/s,
calculate the corresponding air velocity in the model test.
If, in the model test, the heat transfer rate is measured as 33.33 KW
when the cooling surface area is 4m2 and the temperature
difference between the air and the surface is 25K, calculate the
heat transfer coefficient for the prototype cooler.
Assume that compressibility effects are negligible and that fluid
properties are the same for both model and prototype.
4
(7 marks)
Q.5
a)
Heat flows through a bar whose surface is insulated. In this boundary –
value problem the bar has a length of 3 units and a diffusivity of 2 units.
Its ends are kept at temperature zero units and its initial temperature
U (x, o) is given. By method of separation of variables or otherwise,
using the equation.
2
?u
?
?
u
2
,
0 ? x? ,
3 t ?0
2
?t
?x
Find the temperature at position x at time t i.e. find U ?x,t?
Given that U ? ,
O t??U ? ,
3 t??0
U ?x,0?5Sin 4 x
? ?3Sin8 x
? ? 2Sin10 x
? for
u ?x,t?? M .
(10 marks)
b)
Derive the finite difference scheme for the heat flow in a body of
homogeneous material which is governed by the equation
2
2
? u
?
?
u
4
, 0? x? 1
2
2
?y
?x
U ? ,
0 y? 0 ?
with boundary conditions
U ?
? y ?
,
1 y?
0
2y?
and initial conditions U ?x,0??Sin? x
? u
?
?
?
?
x,o? x
?
y
? ?
Take the x – step h = 0.2, the y – step
K = 0.01 and find the solution along y = 0.03.
(10 marks)
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5
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