ME 320 Lab 4
ME 320 Lab 4
ME 320 Lab 4
Ajay Krish
ME 320
April 1, 2016
Table of Contents
1.
INTRODUCTION....................................................................................................................3
1.1 Objective and Utility of Lumped Mass/Convective Heat Transfer........................................3
1.2 Assumptions, Equations and Variables..................................................................................3
2.
EXPERIMENTAL PROCEDURES.........................................................................................6
2.1 Natural Convection................................................................................................................6
2.2 Forced Convection Perpendicular Orientation....................................................................6
2.3 Forced Convection: Effect of Orientation..............................................................................8
3.
4.
CONCLUSION......................................................................................................................17
Main Points................................................................................................................................17
Recommendations......................................................................................................................18
1. INTRODUCTION
1.1 Objective and Utility of Lumped Mass/Convective Heat Transfer
This experiment investigates the difference between natural and forced convection on the cooling
of an aluminum block along with the effects of orientation on heat transfer. Furthermore, analysis
will incorporate the lumped-capacitance method to relate the temperature of an object with time.
This analysis requires utilizing multiple dimensionless constants, such as the average Nusselt
number
Nu
, Reynolds number Re, and the average Rayleigh number Ra.
h=
where
object,
q
A s (T sT )
(1)
As
Ts
heat transfer occurs. In order to determine the heat transfer coefficient, the average Nusselt
number can be used, and is described by the following equation:
3
h L ref
Nu=
kf
where
Lref
(2)
is a reference length and kf is the thermal conductivity of a material. The local heat
transfer coefficient can also be found using a local Nusselt number, described as:
Nu=
where
T
y
h L ref Lref T
=
kf
T sT y
(3)
is evaluated at y=0.
The lumped capacitance method helps determine the reliance of temperature changes on time,
and can be expressed as:
h A
t
T T a
=e mc
T iT a
(4)
coefficient, m is the mass of the object, t is time, T is the temperature of the mass at time t, and
Ta is the surrounding temperature. When dealing with fluids in convection, the fluid properties
are evaluated at a film temperature expressed as
1
T film= (T s +T )
2
(5)
T 0 +2 T a
3
(6)
where T0 = T(0) is the initial temperature of the block. Heat transfer, in this experiment, is dealt
with in two modes, convection and radiation, and are described as:
q conv =hc A (T sT )
(7)
q rad =hr A ( T sT ) =A (T 4s T 4a )
(8)
(9)
hc + hr
h=
(10)
where hc and hr are the convective and radiative heat transfer coefficients respectively. These
equations quantify heat transfer rates in terms of temperature differences and provide a total heat
transfer rate and coefficient based on average heat transfer values.
2. EXPERIMENTAL PROCEDURES
2.1 Natural Convection
1.
2.
3.
4.
5.
exceeded)
5. Take the measurement of air velocity in the middle of the flow
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6. At constant time intervals (20 seconds), record the temperature of the block
7. Keep recording temperatures until steady state is reached or after 10 minutes has passed,
whichever comes first
8. Repeat steps (1)-(7) for different fan speeds.
9. For one of the values of air speed, adjust the anemometer so that it isnt in the middle of
the air flow
10. As the radial distance adjusts, record the air speed
Time (s)
Low Speed Perpendicular
Figure 3. Combined plot of temperature as a function of time for free and forced
convection cases
The forced convection speeds used for tests are 0.76 m/s, 3.21 m/s and 4.9 m/s for low, medium
and high speeds respectively. Furthermore, the perpendicular orientation refers to the setup
shown in Figure 2, where a face of the block is perpendicular to the incoming airflow. Angled
orientation refers to a placement of the block that is at 45 degrees to the incoming airflow. All of
the tests demonstrate an exponential decay for temperature as a function of time. Figure 3 shows
that free convection, even at the lower temperature (313.15 K), takes much longer than forced
convection to reach ambient temperature. However, among forced convection tests, angled and
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perpendicular orientations followed the same cooling profile and reached ambient temperature at
the same time.
Orientation should affect the cooling curve of the cube, although this trend is not reflected much
in Figure 3. Constants for square cylinders in gas crossflow change with the orientation angle, as
do the reference lengths for calculating Reynolds number and Nusselt number. However, since
the cube does not follow the same properties as an infinite cylinder, the effects of orientation
may be diminished.
Question 2
0
-0.2
1000
2000
3000
4000
-0.4
-0.6
-0.8
-1
-1.2
-1.4
-1.6
-1.8
Time (s)
Low Temp Free Convection
the largest slopes in the logarithm plot, meaning temperature of the aluminum block decreases
faster with increased air speed. This phenomenon matches with theoretical understanding of
forced convection.
When air flows over a surface, a boundary layer is created, and over the length of an object, this
boundary layer increases in length. This boundary layer represents the separation that occurs
when fluid flows over a solid object and affects the heat transfer rates between the fluid and the
solid. The larger the boundary layer, the lower the heat transfer rates will be (heat transfer
coefficients will decrease with increasing boundary layers). Higher velocity air flows cause
boundary layer thickness to be smaller relative to those of lower velocity. This can be seen in the
inverse relationship between the boundary layer thickness and the Reynolds number. Smaller
boundary layer thicknesses decrease the room for temperature gradients in the boundary layer
thickness. As a result, the air at the free stream temperature is closer to the objects surface,
leading to higher heat transfer rates.
Question 3
Table 1. Heat transfer coefficients and time constants for free and forced convection tests
h (W/m2K)
6.86
7.01
37.55
30.99
13.26
35.91
30.82
13.90
Time
Constant (s)
2197.80
2222.22
1081.08
1067.62
490.60
481.54
421.05
446.10
The time constants listed in Table 1 are reasonable, as they represent the time required to for the
temperature of the aluminum block to reach about 37% of the difference between the initial
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temperature and ambient temperature. This makes sense in terms of experimental observations,
as our lab group waited for around 43 minutes for the temperature difference between the
aluminum block and ambient temperature to reach 4 C (for 40 C initial temperature), which is
only slightly more than one time constant value to reach a temperature difference of 6 C. Each
condition has its own time constant, however, the free convection time constants are very close,
and the time constants for forced convection are very similar amongst tests with the same air
speed. The heat transfer coefficients in table 1 are very close to the 8.7 W/m2K coefficient
provided in the lab manual. Errors may have arisen from inaccurate temperature recordings that
occur occasionally over long periods of data acquisition. Furthermore, different thermocouples
and different aluminum blocks may have been machined in different ways, causing small
discrepancies in temperature measurements. Another source of error could have come from
having different ambient temperatures. Regardless of these errors, experimental heat transfer
coefficients match very well.
Question 4
74
80
70
63
60
50
54
45
Nu 40
30
28
20
19
10
0
Re
Figure 5. Nusselt number and Reynolds number correlation over different air speeds
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Equation 3 was used to determine the Nusselt number. Figure 5 shows that as air speed increases,
both the Nusselt number and the Reynolds number increase. This makes sense in terms of what
both of these dimensionless parameters represent. The Nusselt number represents the ratio
between the heat transfer coefficient of a free stream fluid to its thermal conductivity. As
mentioned previously, increasing stream velocity increases the heat transfer coefficient,
increasing the Nusselt number. The Reynolds number represents the ratio between inertial forces
to viscous forces. As flow velocity increases, inertial forc, es increase and viscous forces do not
change much, increasing the Reynolds number (this does depend on the type of fluid, however,
in the case of air, there is not much change in viscosity as long as free stream temperature is
maintained). Figure 5 demonstrates that the cube at an angled orientation experiences higher
Reynolds numbers and higher Nusselt numbers. This makes sense when considering the effect of
changing orientation on the distance fluid flows over the cube. When the air travels across the
flat facing cube, it flows over a length L, equal to the side length of the cube. Upon changing the
cubes orientation to be 45 degrees to the incoming flow, air travels a length of
L 2 across
the top length of the cube. This increases both the Reynolds number and Nusselt number as they
are both dependent on the direction of growth of the boundary layer, which is affected by the
orientation length of the cube.
Question 5
The Rayleigh number was calculated using the formula
g ( T 0T a ) L3
R a L=
. The Rayleigh
number is associated with buoyancy-driven flow (free convection). Below a critical Rayleigh
number, heat transfer primarily occurs from conduction, while above the critical value it occurs
from convection. Furthermore, the Rayleigh number can be described as the product of the
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Grashof number and the Prandtl number. The Grashof number describes the relationship between
the buoyance and viscosity within a fluid, while the Prandtl number describes the relationship
between momentum diffusivity and thermal diffusivity.
Table 2. Rayleigh number at different free stream temperatures
Free Convection Initial Temperature
Ra
40C
80984
60C
180211
As seen from Table 1, the Rayleigh number increases with the initial block temperature. This
makes sense with respect to the Grashof number, as the buoyancy forces, which are driven by
density differences in the fluid, increase as a result of higher initial temperature (the higher the
temperature the lower the density of air). As buoyancy forces increase the Grashof number
increases, increasing the Rayleigh number.
Question 6
Table 3. Biot number for each test condition
Flow Condition and Temperature
Free Convection at 60 C
Free Convection at 40 C
Perpendicular Forced Convection (4.90 m/s)
Perpendicular Forced Convection (3.21 m/s)
Perpendicular Forced Convection (0.76 m/s)
Angled Forced Convection (4.90 m/s)
Angled Forced Convection (3.21 m/s)
Angled Forced Convection (0.76 m/s)
Bi
0.000181
0.000185
0.000994
0.00082
0.000351
0.00095
0.000815
0.000368
In order to utilize the lumped capacitance method, the Biot number must be less than 0.1 in all
test cases, which it is. This constraint is important because of what the lumped capacitance
method assumes in its derivation. Conduction through an object of interest is neglected, and only
convection is represented as the mode of heat transfer. In order for this to make sense, the
temperature profile through the object must be essentially non-variant. The Biot number captures
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this assumption as the ratio of the heat transfer coefficient and the thermal conductivity of the
object:
Bi=
h Lc
k . If the thermal conductivity of a material is high, temperature differences
Trial 1 Velocity
(m/s)
5
4.6
4.5
4.9
3.96
Trial 2 Velocity
(m/s)
5
4.38
5.1
4.9
4.3
Velocity non-uniformity would cause boundary layer thicknesses to vary across the cube. This
would lead to a non-uniform heat transfer coefficient distribution across the cube, which could
lead to a temperature gradient across it. However, since the Biot number is so small, the thermal
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conductivity through the aluminum block is very large, so this could diminish the effects of
varying heat transfer coefficients around the cube.
As seen from Table 4, the velocity did not follow a parabolic curve, which is expected from
laminar fluid flow in a tube. However, velocity does vary with radial position, which could cause
a non-uniform heat transfer from forced convection.
Question 9
Table 5. Average Heat transfer coefficients for square cylinder and spheres
Free
Convection
Forced Convection
47.17
Angled
(0.76
m/s)
12.26
Angled
(3.21
m/s)
31.72
36.52
46.07
20.36
44.16
55.84
30.99
37.55
13.90
30.82
35.91
40
C
60 C
Perpendicula
r (0.76 m/s)
Perpendicula
r (3.21 m/s)
Perpendicula
r (4.90 m/s)
hsquare
6.27
13.79
13.79
35.68
hsphere
7.84
17.02
17.02
hcube
7.01
6.86
13.26
Angled
(4.90 m/s)
41.93
The heat transfer coefficients for the infinite circular cylinder approximates the cube in free
convection better than the sphere does (Table 5). Furthermore, the infinite square cylinder
approximates the cube in forced convection better than the sphere (Table 5). This result may be
due to the geometry of the square cylinder being closer to a cube than a sphere. Geometric
similarities may help the square cylinder approximate boundary layer formation in cross flow
better than the sphere would. In order to acquire these values, a series of Nusselt number
correlation had to be used. For a sphere in free convection, equation 9.35 from the book was
used:
Nu D =2+
0.589 R a 1/D 4
[ 1+ ( 0.469/ Pr )9 /16 ]
4/ 9
For an infinite circular cylinder in free convection, equation 9.34 was used:
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Nu D = 0.60+
0.387 R a1D/6
[ 1+ ( 0.559/Pr )9 /16 ]
8 /27
For a sphere in forced convection, equation 7.56 from the book was used:
Nu D =2+ ( 0.4 R e1D/2 +0.06 R e 2/D 3 ) P r 0.4
1 /4
( )
For an infinite cylinder with a square cross section, equation 7.52 and Table 7.3 from the
textbook were used:
1/ 3
Nu D =0.158 R e 0.66
D Pr
However, it is important to note that all of the equations are approximations, and results cannot
be expected to be within 20% of actual heat transfer coefficients. Numerous assumptions, such as
constant fluid properties, are assumed despite a fluid temperature gradient.
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4. CONCLUSION
Main Points
1. Forced convection cools the aluminum block faster than free convection.
2. Orientation and initial temperature did not affect cooling curves or time constants
significantly.
3. Heat transfer coefficients were higher for higher air speeds, however, there was not much
variation for orientation.
4. The Nusselt number and Reynolds number increased for a cube at 45 degrees to incoming
flow, as both numbers relate to a flows boundary layer
5. The Biot number was less than 0.1 for all cases, which affirmed the assumption that
convection is the dominant heat transfer mode over conduction. This allows us to use the
lumped capacitance method for analysis.
6. The square cylinder and circular cylinder are better approximations for cube heat transfer
than the sphere in forced and free flow respectively based on Nusselt number
correlations.
Recommendations
In order to improve this experiment, acquiring a module that records temperature at set time
intervals would be very beneficial. This would remove potential recording error and allow for
multiple test runs with different geometries. Furthermore, using a fluid aside from air would help
visualize boundary layer formation from flow and would help visualize how separation is
affected by geometry.
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