MACE30431
TWO HOURS
THE UNIVERSITY OF MANCHESTER
HEAT TRANSFER
15th January 2020
09:45 – 11:45
Answer ALL FOUR questions.
Note that the questions do NOT carry equal marks.
The data sheets at the back of this exam paper may be used as required.
Electronic calculators may be used, provided that they cannot store text.
© The University of Manchester, 2020
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MACE30431
Q1.
(a)
[Total Marks 37]
A fin attached to a plane plate is in the form of a uniform cross-sectioned cylinder. The
fin and plate are composed of aluminium with thermal conductivity k = 245W/mK. The
fin is of length L = 30cm and diameter D = 3cm. The plate has a uniform temperature Tb
= 800C. The plate and fin are surrounded by a fluid with bulk temperature T∞ = 200C. The
convective heat transfer coefficient h = 100W/m2K.
(i)
By means of a steady-state heat balance analysis on a small disc-shaped section of
the fin, obtain the equation for the temperature variation in the x direction along
the fin d 2Θ  m2Θ  0 , where Θ( x)  T ( x) T and m2 = 4h/kD.
[10 marks]
dx2
(ii)
Assume that the solution to the differential equation in (i) is


Θ  cosh  m ( L  x )  . Demonstrate that this solution obeys the two boundary
Θb
cosh  mL 
conditions of a continuous temperature variation at the base of the fin and a zero
heat transfer rate from the tip of the fin.
[4 marks]
(iii)
Show that the total heat transfer rate from the fin is Q f 
hpkA c Θb tanh  mL  ,
where perimeter p = πD and cross-sectional area Ac = πD2/4.
(iv)
(b)
[3 marks]
Evaluate the number of fins that would need to be attached to a square metre of
the plate in order for the total heat transfer rate from the fins and the remaining
plate surface to be approximately eight times that of the plate without fins.
[10 marks]
The fin in part (a) emerges from the manufacturing process at a temperature T0 = 1000C.
It is placed in an environment at TF = 200C to cool. In this environment h = 40W/m2K.
(i)
Demonstrate that the cooling process can be modelled by the lumped-heatcapacity method. In this analysis assume that the characteristic length scale is
equal to the cylinder radius.
[3 marks]
(ii)
For aluminium, specific heat capacity c = 921J/kgK and density ρ = 2700kg/m3.
Evaluate the time taken for the fin to cool to 400C. To evaluate the surface area of
the fin, the end surfaces may be neglected.
[7 marks]
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MACE30431
Q2. [Total Marks 23]
Two rectangular plates 1m by 50cm in size are placed perpendicularly along the longer edge.
Surface 1 is heated on the outside by a heating coil and is composed of a material with emissivity
1 = 0.4, while surface 2 has a temperature T2 = 700K and is insulated on the outside. The two
surfaces are in radiant balance with a large room at T3 = 300K. Consider radiation only from the
inside surfaces of the plates.
(a) Evaluate all the view factors for this system.
[4 marks]
(b) Calculate all the resistances to radiation heat transfer.
[5 marks]
(c) Draw the equivalent electrical network, showing resistances R, and the positions of the
radiosities J and black body emissive powers Eb.
[4 marks]
(d) Determine the temperature T1 of the uninsulated surface and the rate of heat transfer to
the room surfaces.
[10 marks]
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Q3.
[Total Marks 20]
(a) Write down Newton’s Law of Cooling and identify the variables involved.
[4 marks]
(b) As shown in Figure Q3.1, a fine wire which is 150mm long and 10mm in diameter, D,
is exposed to an air stream which flows across it at a free stream velocity, U∞, of 10m/s.
The free stream temperature is 15oC. The wire is maintained at a uniform temperature
of 150oC, by passing electrical current through it. For a cylinder in cross-flow, the
average value of the Nusselt number around its circumference, based on the cylinder
diameter, NuD, is given by:
NuD = C (ReD Pr)n
With C=0.68 and n=0.466
Where
ReD [≡U∞D/ν] is the flow Reynolds number, based on the cylinder
diameter
Pr is the fluid Prandtl number
The relevant air properties have the following values:
Thermal conductivity k = 0.02984 W/m.K
Kinematic viscosity ν = 2.073x10-5 m2/s
Prandtl number
Pr = 0.707
From the above information calculate the electrical power needed to maintain this
cylinder temperature.
[8 marks]
(c) The free stream velocity subsequently drops to zero, which makes natural convection the
mode of heat transfer between cylinder and the surroundings. Assuming that the thermal
properties and cylinder and ambient temperature remain the same as in (b) above,
calculate the electrical power needed.
[8 marks]
For natural convection over a horizontal cylinder, the average value of the Nusselt
number around its circumference, based on the cylinder diameter, NuD, is given by:
with C=0.53 and m=1/4
Where
GrD = gβ(TW-T∞)D3/ν2
β the coefficient of thermal expansion, which for ideal gases is the inverse
of the absolute film temperature.
Figure Q3.1 Heated cylinder in cross-flow
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Q4.
[Total Marks 20]
A tubular cross flow heat exchanger, which consists of an array of 400 tubes, of internal
diameter of 10mm and external diameter of 12mm is to be used as a condenser. Saturated
water vapour at 100oC enters the tubes and air flows across the tubes with an inlet
temperature of 20oC and mass flow rate of 10kg/s. The value of the overall heat transfer
coefficient U, based on the tubes’ outer diameter is 50W/m2K. For the air stream, the specific
heat capacity cP, is 1.009kJ/kgK and the water latent heat of vapourisation at 100 oC, hfg, has
the value of 2257kJ/kg.
(a) What is the maximum rate of condensation possible for this combination of fluid
streams?
[4 marks]
(b) Calculate the length of the tubes required, if the required rate of condensation is 0.1kg/s.
[10 marks]
(c) If the heat exchanger available, with the same number and type of tubes as described
above has 10 m-long tubes, what will be the resulting condensation rate?
[6 marks]
END OF EXAMINATION PAPER
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MACE30431
Heat Transfer Data Sheet
MACE30431
1
Heat Conduction
1.1
Fourier’s Law – plane geometry
1.2
1-D unsteady heat conduction with internal heat generation – plane geometry
(
1.3
Fourier’s Law – cylindrical geometry
1.4
1-D unsteady heat conduction with internal heat generation – cylindrical geometry
(
1.5
Thermal resistance R of a plane wall of thickness L, area A and thermal conductivity k
L
R
kA
1.6
Thermal resistance R of a cylindrical wall of internal and external radii ri and ro
respectively, length L and thermal conductivity k
ln( rro )
i
R
2 Lk
1.7
Thermal resistance R between a solid surface of area A and a fluid when the convection
heat transfer coefficient is h
R 1
hA
1.8
Pipe critical radius of insulation ro,critt for an insulator of thermal conductivity k in a
medium with convection heat transfer coefficient h
ro ,crit  k
h
1.9
1-D heat conduction equation for a fin of varying cross-sectional are Ac, with surface area
As and thermal conductivity k in a medium at bulk temperature T with convection heat
transfer coefficient h
d 2T
1 dA c dT
h dA s


(T  T  )  0
2
dx
A c dx
kA c dx
dx
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1.10
Fin effectiveness
with convection heat transfer coefficient h, fin base area
Ac ,b , temperature difference Θb  Tb  T and total heat loss
1.11
Fin efficiency
.
with convection heat transfer coefficient h, total fin surface
area A f , temperature difference Θb  Tb  T and total heat loss
1.12
.
Unsteady heat conduction from/to a body of characteristic length L or ro, with thermal
conductivity k in a medium with convection heat transfer coefficient h at bulk
temperature T∞.
Biot number Bi  hs , where in general s is defined as V/A, where V is the volume and A
k
the surface area of the body, but may be L or ro.
Lumped-heat-capacity method, appropriate if Bi < 0.1. The temperature of the body T is
given by the equation T  T 
T0  T

e
hA
 Vc
t
, where T0 is the temperature of the body at t
= 0, ρ is the density of the material and c is the thermal capacity.
If Bi > 0.1, see attached Heisler and Grober charts, where   k and the Fourier
c
number   t *  Fo  kt .
 cs 2
2.
Heat Convection
2.1.1 Nusselt number NuL for a surface of characteristic length L, in a medium with thermal
conductivity k and convection heat transfer coefficient h
hL
Nu L 
k
2.1.2 Reynolds number ReL for a body of characteristic length L in a fluid with bulk velocity U
and kinematic viscosity ν
ReL 
2.1.3
UL

Grashof number Gr – uniform wall temperature
g TL3
GrL 
2
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Grashof number Grx – uniform wall heat flux
Gr x 
g q w x 4
k 2
where g is the acceleration due to gravity,  is the coefficient of thermal expansion, k is
the fluid thermal conductivity, T is the difference between the wall and fluid bulk
temperatures, q w is the wall heat transfer rate per unit area, ν is the fluid kinematic
viscosity, L is the overall length and x is the local distance.
2.2
Local Nusselt number for laminar flow over a flat plate at a distance x from the leading
edge with Prandtl number Pr and Rex < 5x105
Nu x  0.332Re1x/ 2 Pr1/ 3 (uniform wall temperature)
Nu x  0.453 Re1x/ 2 Pr1 / 3 (uniform wall heat flux)
2.3
Local Nusselt number for turbulent flow over a flat plate at a distance x from the leading
edge with Prandtl number Pr
Nux  0.029Re0x.8 Pr1/ 3
2.4
Average Nusselt number for fully-developed laminar flow in a straight pipe
Nu D  3.66 (uniform wall temperature)
Nu D  4.36 (uniform wall heat flux)
2.5
Average Nusselt number for fully-developed turbulent flow in a straight pipe
0 .8 1 / 3
Nu D  0.023 Re D
Pr
2.6
Average Nusselt number for a tube in cross flow
n
Nu D  CRe D
Pr 1 / 3
For air (Pr = 1)
ReD 0.4 - 4 4 - 40 40 - 4000 4x103 – 4x104 4x104 – 4x105
C
0.990 0.911 0.683
0.193
0.0266
n
0.330 0.385 0.466
0.618
0.805
For liquids (Pr > 5) and 10-1 < ReD < 105
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0.52
Nu D  (0.35  0.56 Re D
) Pr 0.3
MACE30431
2.7
Average Nusselt number for free convection from a vertical surface of length L at
uniform temperature
Nu L  C GrL Pr m
GrL Pr
C
m
2.8
104 – 109 109 – 1013
0.59
0.021
0.25
0.4
Local Nusselt number for free convection from a vertical surface with uniform wall heat
flux, where x is the vertical height
Nu x  C Grx Pr m
Grx
105 – 1011 2x1013 – 1016
C
m
2.9
0.60
0.20
0.17
0.25
Average Nusselt number for free convection from a horizontal cylinder at uniform
temperature
Nu D  C GrD Pr m
GrD Pr
C
m
104 – 109 109 – 1012
0.53
0.13
0.25
0.333
3.
Heat Exchanger Analysis
3.1
Number of transfer units NTU, with overall heat flux coefficient U, heat exchanger area
A and Cmin the minimum capacity rate (mcp)min
N  NTU 
3.2
UA
Cmin
Capacity rate ratio C
C
C min
C max
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3.3
Heat exchanger effectiveness ε is defined as the ratio of the actual heat transfer to the
maximum possible heat transfer. The latter quantity is given by
Flow geometry
Double pipe:
Parallel flow
Relation
1  exp N (1  C )
1 C
1  exp N (1  C )

1  C exp N (1  C ) 

Counter flow
Counter flow, C = 1
Cross flow:
Both fluids unmixed

N
N 1
 exp( NCn)  1 

Cn


  1  exp 
n  N 0.22
Both fluids mixed
Cmax mixed, Cmin unmixed
Cmax unmixed, Cmin mixed
Shell and tube:
One shell pass, 2, 4, 6 tube
passes

1
C
1
 

 
 1  exp( N ) 1  exp( NC ) N 

1
1
(1  exp C (1  exp( N ))
C
 1
1  exp( NC )
 C

  1  exp 





1  exp  N (1  C 2 ) 0.5 

  21  C  (1  C 2 ) 0.5 

1  exp  N (1  C 2 ) 0.5 
1  exp(  N )
All exchangers with C = 0
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3.4
Expressions for heat exchanger NTU values
Flow geometry
Double pipe:
Parallel flow
Relation
Counter flow
Counter flow, C = 1
N
 ln1  (1  C ) 
1 C
N
1
  1 
ln
C  1  C  1 
N
Cross flow:
Cmax mixed, Cmin unmixed

1 
 1

N   ln 1  ln(1  C )
 C

Cmax unmixed, Cmin mixed
N
Shell and tube:
One shell pass, 2, 4, 6 tube passes
All exchangers with C = 0
1
ln1  C ln(1   )
C
2

 1  C  (1  C 2 ) 0.5 

N  (1 C 2 ) 0.5  ln  

 2  1  C  (1  C 2 ) 0.5 
 

N   ln(1   )
4
Radiation
4.1
Thermal power Eb emitted per unit surface area of a black surface at temperature T
Eb  T 4
4.2
  5 . 67  10  8 W / m 2 K 4
Surface thermal resistance R of a grey surface of area A and emissivity ε
R
4.3
1
A
Inter-surface thermal resistance to radiation between two surfaces of areas A1 and A2 and
view factors F12 and F21 (see following pages)
R
1
1

A1 F12 A2 F21
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View factors for
aligned parallel
rectangles
View factors for
perpendicular
rectangles with a
common edge
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View factors
between two
coaxial parallel
discs
(a)
(b)
View factors for two concentric cylinders of finite length:
(a) outer cylinder to inner cylinder,
(b) outer cylinder to itself
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MACE30431
Heisler Chart 1: Centre plane temperature as a function of time for a plane wall of length L
Heisler Chart 2: Temperature
distribution in a plane wall of length
L.
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Heisler Chart 1: Centre line temperature as a function of time for a cylinder of radius r0.
Heisler Chart 2: Temperature
distribution in a cylinder of radius r0.
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Heisler Chart 1: Centre point temperature as a function of time for a sphere of radius r0.
Heisler Chart 2: Temperature distribution in a sphere of radius r0.
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(i) Grober chart for plane wall of length L
(ii)
(iii)
Grober chart for cylindrical wall of radius r0
Grober chart for a sphere of radius r0.
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