INTERCOMPANY MEMORANDUM
CAL CHEM CORPORATION
To:
CHE Juniors
From:
CHE faculty
Laboratory Managers
Subject:
Date: Winter Quarter
File: CHE 333
Extended Surface Heat Transfer
Heat Transfer Along A Cylindrical Fin
In this heat transport lab you will study and perform calculations for extended surface heat
transfer. As part of the experiment you will be using automated data collection instruments and
thermocouples.
We will be interested in the performance of an aluminum pin fin available in our laboratory. You
should determine the temperature distribution for both free and forced convection flows and
compare the experimental measurements with the predicted values.
Introduction
Consider the area A on the surface shown in Figure 1 where heat is being transfer from the
surface at a fixed temperature Ts to the surrounding fluid at a temperature T with a heat transfer
coefficient h. The heat transfer rate may be increased by increasing the convection coefficient h,
reducing the fluid temperature T, or adding materials to the area A.
Plate
As
As
A
A
L
Surface
Figure 1. Use of extended surface or fin to enhance heat transfer.
Look on the plane side-view of the surface and the surface with fin. The heat transfer rate
without the fin from area A to the surrounding fluid is
qc = hA(Ts  T)
With the fin attached to the area A, the heat transfer to the surrounding fluid must first be
transferred by conduction from area A to the fin
1
qf =  kA
T
x
=
x 0

As
0
h(T ( x)  T )dAs
where dAs = Pdx and P = perimeter of the fin.
For the extended surface to enhance the heat transfer rate, the ratio of heat transfer with and
without the fin must be greater than one
T
q
x x 0
f  f =
=
qc
hA(Ts  T )
 kA

As
0
h(T ( x)  T )dAs
hA(Ts  T )
(1)
f is called the fin effectiveness. For the fin to be cost effective, the fin effectiveness should be
greater than 2. The temperature profile along the fin must be determined before the fin
effectiveness can be calculated. Consider the cylindrical extended surface with diameter D
shown in Figure 2. To simplify the analysis, we will assume one-dimensional heat transfer in the
x direction, steady state, no heat generation, no radiation, constant heat transfer coefficient, and
constant physical properties.
T
T(x)
Tb
A
L
x
dx
Figure 2. A cylindrical fin with convective end.
An energy balance will be applied to a differential control volume, xA, shown in Figure 2.
Since temperature is dependent on x, a differential distance along x must be chosen. The surface
area of the control volume is
As = xP = xD
From the energy balance applied to the control volume xA
qx – (qx+x + qc) = 0
Divide the equation by x and take the limit as x 0
limit  q x  q x  x 
limit  qc 
–
=0


Δx  0 
Δx  0  x 
x

2
–
dq x
dq c
–
=0
dx
dx
dqc = hdAs(T(x)  T)
The energy equation becomes
dq x
dA
– h s (T(x)  T) = 0
dx
dx
dT
Substituting Fourier's law qx = – kA
where A is the cross-sectional area normal to the xdx
direction, the energy equation becomes
–
d
dx
since As = Px,
dAs
 dT 
kA dx  – h dx (T(x)  T) = 0
dAs
=P
dx
For constant k and A, the energy equation becomes a second order ordinary differential equation
(ODE) with constant coefficients.
hP
d 2T
–
(T(x)  T) = 0
2
kA
dx
(2)
The above equation is a non-homogeneous ODE which can be made homogeneous by
introducing a new variable  = T(x)  T
hP
d 2
–
 =0
2
kA
dx
Let m2 =
(3)
hP
, the solution to the homogenous ODE can be written as
kA
 = B1sinh(mx) + B2cosh(mx)
(4)
The constants B1 and B2 can be evaluated using the following boundary conditions
at x = 0, T = Tb   = b
d h
dT
at x = L,  k
= h(T  T)  
= 
dx
dx k
to obtain the temperature distribution along the pin fin
3
(5a)
(5b)

=
b
h
sinh m( L  x )
mk
h
cosh mL 
sinh mL
mk
cosh m( L  x ) 
(6)
and the fin heat transfer rate
h
cosh mL
mk
qf = M
h
cosh mL 
sinh mL
mk
sinh mL 
(7)
where
0.5
 = T  T ,
b = Tb  T,
M = b(hPkA)0.5,
P = perimeter = D,
 4h 
m= 
 ,
 kD 
D 2
A=
4
The heat transfer coefficient for a long, horizontal cylinder can be estimated from appropriate
empirical correlations for free and forced convection flow1. For forced convection, the heat
transfer coefficient may be estimated from
NuD = 0.3 + [0.62 ReD1/2Pr1/3[1 + (0.4/Pr)2/3]-1/4][1 + (ReD/282,000)5/8]4/5 (8)
This equation is valid for cross flow and ReDPr > 0.2. The physical properties should be
evaluated at the film temperature Tf = 0.5(Ts + T). For free convection,
1/ 6


0.387 Ra D
NuD = 0.60 
9 / 16 8 / 27 
[1  (0.559 / Pr) ] 

2
This equation is valid for Rayleigh number RaD < 10
(9)
12
where RaD = GrLPr =
g  (Ts  T ) D 3
.

The physical properties should be evaluated at the film temperature Tf = 0.5(Ts + T).  is the
expansion coefficient that depends on the fluid. For an ideal gas,  = p/RT, the expansion
coefficient can be determined
=
1
1   
1 p
=

 =
2
T
  T  p  RT
(10)
4
Fin performance is assessed by two factors: Fin Effectiveness, f, and the Fin Efficiency, f. Fin
effectiveness is defined as the ratio of the fin heat transfer rate to the heat transfer rate that would
exist without the fin as given by equation (1) earlier.
f =
qf
hAc b
(11)
Fin efficiency is defined as the ratio of the actual amount of heat transferred to the amount of
heat that would be transferred if the entire fin was at the base temperature.
f =
qf
(12)
hAf  b
For this experiment you will determine the temperature distribution, the amount of energy
transferred to the air, the fin effectiveness, and the fin efficiency for both forced and free
convection.
Procedure:
For Free Convection :
1. Turn on the Varriac. Check to make sure that the heater connected to the fin is connected
with the varriac.
2. Record the fin temperatures using the DAC express software. Instructions for using the
software is given in appendix C. Stop recording the temperatures when the system reaches
steady state.
3. Read and record the ambient air temperature and pressure. Record the humidity using the
wet bulb thermometer.
For Forced Convection :
4. Turn the air blower on. Adjust the varriac so that the based temperature of the fin has
approximately the same value as in free convection.
5. Measure the air velocity by placing the wind velocity meter near the fin. It is suggested that
at least 5 readings over different x position along the fin be taken to obtain an average
value.
6. Wait until the system reach steady state and record the temperatures along the fin.
Repeat steps 1-6 for at least two more settings. Turn everything off and clean up.
Note: What is the criterion for a steady state temperature?
Report :
Your report should:
5
1] Derive equations 6 & 7.
2] Determine the temperature distribution for both free and forced convection.
3] Compare the predicted values with the experimental values. Note: the predicted and
experimental values have the same base temperature.
4] Determine the fin effectiveness and fin efficiency for both free and forced convection.
References:
1) Incropera and De Witt, Fundamentals of Heat and Mass Transfer, Wiley 2002.
2) Chapman, A. J., Heat Transfer, McMillan Publishing Co., 1985.
3) Any CRC Handbook of Physics and Chemistry.
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