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INME 4032
Table of Contents
Principle
3
Objective
3
Background
3

Newton’s law of cooling
3
Experimental Setup

5
Description of the Combined Convection and Radiation Heat
Transfer Equipment:
5
Useful Data
6
Procedure
7
1. Free convection experiments

Observations
8

Analysis of results
8

Comparison to theoretical correlations
9
2. Forced convection experiments
10

Observations
10

Analysis of results
10

Comparison to theoretical correlations
11
Discussion
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INME 4032
University of Puerto Rico
Mayagüez Campus
Department of Mechanical Engineering
INME 4032 - LABORATORY II
Spring 2004
Instructor: Guillermo Araya
Experiment 2: Natural And Forced Convection Experiment
Principle
This experiment is designed to illustrate the Newton’s law of cooling by convection and
to understand how the heat transfer coefficient is obtained experimentally. Natural and
forced convection over a heated cylinder is analyzed and experimental results are
compared with standard correlations.
Objective
Determine the heat transfer coefficient for a flow around a cylinder under free and
forced convection. Understand the correlation between Nu, Reynolds and Rayleigh
numbers. Compare with standard correlation from textbooks on heat transfer. The effect
of thermal radiation is also included.
Background
Newton’s law of cooling
For convective heat transfer, the rate equation is known as Newton’s law of cooling and
is expressed as:
q  h (Ts  T )
Where Ts is the surface temperature, T the fluid temperature, h the convection heat
transfer coefficient and q  the convective heat flux. The heat transfer coefficient h is a
function of the fluid flow, so, it is influenced by the surface geometry, the fluid motion in
the boundary layer and the fluid properties as well.
From the normalized momentum and energy equation in the boundary layer:
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INME 4032
*
U *
P *
1  2U *
* U
U
V
 * 
Re L y * 2
x *
y *
x
*
*
T *
1  2T *
* T
U
V

x *
y * Re L Pr y * 2
*
Momentum equation
Energy equation
Independently of the solution of these equations for a particular case, the functional
form for U* and T* can be written as:
U* = f(x*,y*,ReL, dp*/dx*)
and
T* = f(x*,y*,ReL, Pr, dp*/dx*)
Heat transfer, due to the no-slip condition at the wall surface of the boundary layer,
occurs by conduction;
q s  k f
"
T
y
y 0
By combining with the Newton’s law of cooling, we obtain:
kf
h
Since T* was defined as T * 
T
y
y 0
Ts  T
T  Ts
T  Ts
h can be written in terms of the dimensionless temperature profile T*
h
k f (T  Ts ) T *
L(Ts  T ) y *
y * 0
k f T *

L y *
y* 0
This expression suggests defining a dimensionless parameter;
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INME 4032
hL T *
Nu 
 *
kf
y
y*  0
From the dimensionless temperature profiles, we can imply a functional form for the
Nusselt number,
Nu = f(x*,ReL*,Pr,dp*/dx*)
To calculate an average heat transfer coefficient, we have to integrate over x*, so the
average Nusselt number becomes independent of x*. For a prescribed geometry,
dp *
is
dx *
specified and
Nu L  f (Re L , Pr)
This means that the Nusselt number, for a prescribed geometry is a universal function of
the Reynolds and Prandtl numbers.
Doing a similar analysis for free convection, it can be shown that, Nu  f (Gr , Pr)
or
Nu  f (Ra , Pr)
Where Gr is the Grashof number and Ra is the Rayleigh number. The Rayleigh number
is simply the product of Grashof and Prandtl numbers (Ra = Gr Pr)
Then, for free convection the Nusselt number is a universal function of the Grashof and
Prandtl numbers or Rayleigh and Prandtl numbers.
Experimental setup
Description of the Combined Convection and Radiation Heat Transfer Equipment:
The combined convection and radiation heat transfer equipment allows investigate the
heat transfer of a radiant cylinder located in flow of air (cross flow) and the effect of
increasing the surface temperature. The unit allows investigation of both natural
convection with radiation and forced convection. The mounting arrangement is designed
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INME 4032
such that heat loss by conduction through the wall of the duct is minimized. A
thermocouple (T10) is attached to the surface of the cylinder. The surface of the cylinder
is coated with a matt black finished, which gives an emissivity close to 1.0. The cylinder
mounting allows the cylinder and thermocouple (T10) position to be turned 360° and
locked in any position using a screw. An index mark on the end of the mounting allows
the actual position of the surface to be determined. The cylinder can reach in excess
600°C when operated at maximum voltage and in still air. However the recommended
maximum for the normal operation is 500°C.
Useful Data:
Cylinder diameter D = 0.01 m
Cylinder heated length L = 0.07 m
Effective air velocity local to cylinder due to blockage effect Ue = (1.22) (Ua )
Physical Properties of Air at Atmospheric Pressure
T
K
300
V
1.568E-5
k
W/mK
0.02624
Pr
0.708
350
2.076E-5
0.03003
0.697
400
2.590E-5
0.03365
0.689
450
2.886E-5
0.03707
0.683
500
3.790E-5
0.04038
0 .6 8
550
4.434E-5
0.04360
0 .6 8
600
5.134E-5
0.04659
0 .6 8
m2/s
Where:
T is the absolute temperature, V is the Dynamic viscosity of air, k is the thermal
conductivity and Pr is the Prandtl number.
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INME 4032
Combined Convection and Radiation Heat
Schematic Diagram showing the Combined
Transfer Equipment
Convection and Radiation Heat Transfer
Equipment
Procedure
a) Connect instruments to the heat transfer unit
b) Measure the reading for the surface temperature of the cylinder, the temperature
and velocity of the air flow and the power supplied by the heater.
c) Repeat steps 1 and 2 for different velocities the air flow and power input.
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1. Free convection experiments
Observations
Se t
1
2
3
4
V
V o lts
4
8
12
16
I
Amp
T9
C
T10
C
hr
W/m2K
hC1th
W/m2K
hC2th
W/m2K
Analysis of results
Se t
1
2
3
4
Qinput
W
4
8
12
16
The total heat input is:
Qinput = VI
The heat transfer rate by radiation is:
Qrad =   A (Ts4 – Ta4) = hr A (Ts – Ta)
So,
hr 
 (Ts4  Ta4 )
Ts  Ta
The heat transfer rate by convection is:
Qconv = Qinput - Qrad
From Newton’s law of cooling
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INME 4032
Qconv  hc A(Ts  Ta )
And
hc 
Qconv
A(Ts  Ta )
Comparison to theoretical correlations
For an isothermal long horizontal cylinder, Morgan suggests a correlation of the form,
Nu D 
hD
 cRa nD
k
(1)
c and n are coefficients that depend on the Rayleigh number
Rayleigh number
10-10 – 10-2
10-2 – 102
102 – 104
104 – 107
107 – 1012
c
0.675
1 .0 2
0.850
0.480
0.125
n
0.058
0.148
0.188
0.250
0.333
The Rayleigh number is calculated from,
Ra 
g(Ts  Ta )D 3
Pr
2
where

1
Tfilm
and
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INME 4032
Tfilm 
Ts  Ta
2
Churchill and Chu recommend a single correlation for a wide range of Rayleigh number,

0.387Ra 1 / 6

Nu D  0.60 

1  (0.559 / Pr) 9 / 16




8 / 27 



2
Ra  1012
(2)
From correlation (1) and (2) we can determine hC1th and hC2th and compare with hc
obtained from the experiment.
Forced convection
Observations
Se t
1
2
3
4
5
6
7
V
V o lts
20
20
20
20
20
20
20
I
Amp
Va
m/s
0 .5
1
2
3
4
5
6
T9
C
T10
C
Analysis of results
Se t
Qinput
W
hr
W/m2K
hC
W/m2K
Re
-
Nu1
-
Nu2
-
hC1th
-
hC2th
-
1
2
3
4
5
6
7
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INME 4032
The total heat input is:
Qinput = VI
The heat transfer rate by radiation is:
Qrad =   A (Ts4 – Ta4) = hr A (Ts – Ta)
So, hr 
 (Ts4  Ta4 )
Ts  Ta
The heat transfer rate by convection is:
Qconv = Qinput - Qrad
From Newton’s law of cooling
Qconv  hc A(Ts  Ta )
and
hc 
Q conv
A(Ts  Ta )
Comparison with theoretical correlations
For an isothermal long horizontal cylinder, Hilper suggests,
Nu D 
hD
 C Re mD Pr 1 / 3
k
(3)
where C and m are coefficient that depend on the Reynolds number:
ReD
0.4-4
4-40
40-4000
4000-400000
40000-400000
C
0.989
0.911
0.683
0.193
0.027
m
0.330
0.385
0.466
0.618
0.805
All properties are evaluated at the film temperature
Tfilm 
Ts  Ta
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Churchill and Bernstein proposed the following correlation for Re Pr>0.2
5/8
0.62 Re 1 / 2 Pr 1 / 3   Re D  
Nu D  0.3 
1 
 
1/ 4 
  0.4  2 / 3    282000  
 
1  
  Pr  
4/5
(4)
where all properties are evaluated at the film temperature.
From correlation (3) and (4) we can determine hC1th and hC2th and compare with hc
obtained from the experiment.
Discussion
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INME 4032