ME 320: Lab 2
Summary & Report
Name: Vismay Vyas
Section: AB8
TA: Radhe Jha
Date: 10/15/2023
1
Introduction
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Objective and Motivations
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•
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Utilization of “lumped” mass assumption assumes
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Uniform spatial distribution of temperature due to large thermal conductivities and small cross-sectional areas
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Simple geometries and constant material properties such as thermal diffusivity (α)
Observing cooling behavior in brass, aluminum, and ceramic blocks
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Investigation of free/natural and forced convection on cooling of varied material blocks
2 convection types, 3 materials, 3 wind velocities (0, 1, 3, 5 m/s), and 2 block orientations (0°, 45°) tested
Motivation to understand effects of variable quantities on heat transfer
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Estimating time constant (τ), Nusselt (Nu), heat transfer coefficient (h), Reynolds (Re), and Rayleigh numbers (Ra)
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Testing lumped mass assumption validity
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Develop correlations between above calculated quantities and comparisons to known correlations
2
Experimental Equipment
Figure 1. Experimental apparatus consisting of fan, flow straightener, and empirical
tools to determine voltage supplied, air velocity, temperature readings, etc.
3
Experimental Methods
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Specimen heating via regulated oven
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•
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Convective fluid/air velocity set to 0 m/s, 1 m/s, 3 m/s, and 5 m/s
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Using the fan power supply, the voltage to the fan is varied linearly changing the output velocity.
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Anemometer probe held in steady hover at fixed distance through accessible hole in the flow straightener
Heated samples then transferred to end of fan apparatus
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Heated specimens are secured in the flow path in both orientations (0°, 45°) behind flow straightener
Temperature data collection
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Metal samples are heated to at least 70°C while ceramic samples are heated to at least 50°C
Thermocouple readings collected every 30 seconds for 24 samples or until thermal equilibrium is reached
Procedure is then repeated until for all desired materials, speeds, and orientations
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21 total test sets collected between both free (0 m/s) and forced convection (1 m/s, 3 m/s, 5 m/s)
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Free convection tests do not require the additional test per material for orientation effects
4
Experimental Results
• Plotting temperature points over 720 s
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Exponential/logarithmic decay
• Agrees with expected trends
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Newtons law of cooling
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Given large time t, asymptotic approach to
ambient t
Figure 2. All collected data for Temperature against Time
5
Experimental Results
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Biot Number:
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Reynolds Number:
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Ratio of inertial forces to viscous forces within a moving fluid
Nusselt Number:
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Ratio of internal conduction to external convection
Ratio of convective heat transfer to fluid conduction heat transfer
Rayleigh Number:
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Ratio of buoyancy to thermal diffusivity
The Biot number determines the validity of the lumped mass assumption by looking
at the relative internal conduction of the material in questions and comparing it to
the external convection/heat transfer at the surface. When this number depresses
underneath the accepted standard of 0.1, then the conduction within the material is
said to be incredibly quick to the point where you can assume a uniform spatial
temperature distribution. The Reynolds number characterizes the flow of a fluid. By
using the fluid viscosity, fluid velocity, density, and a characteristic length, one can
determine whether flow be laminar (< 2000 Re) or turbulent (4000 Re < ) or even a
combination of both when in between the accepted bounds. The point of this is due
to the nature of laminar and turbulent flow and the assumptions that can or can't be
made depending on the flow of interest. The Nusselt number represents a ratio of
convective heat transfer to conductive heat transfer. This is done by using the
convective heat transfer coefficient, characteristic length, and thermal conductivity.
The Rayleigh number is a dimensionless number associated with buoyancy driven
flow also known as free/natural convection. For example, in our test for natural
convection, the hot block heats up air around it while being cooled by the air around
it through convection which creates a difference in density between the heated air
and the ambient air creating a buoyant-like interaction where the hot air rises. It
essentially dictates whether natural convection will occur by using gravity, the
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coefficient of thermal expansion, kinematic viscosity, thermal diffusivity, a
temperature difference, and a characteristic length like always.
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Experimental Results
Free/Natural
1 m/s
Br.
Al.
Cer.
Br.
90°
Br.
45°
3 m/s
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
Br.
90°
5 m/s
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
Br.
90°
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
0.002
0.001
0.214
0.003
0.004
0.002
0.003
0.302
0.409
0.007
0.007
0.005
0.004
0.557
0.619
0.011
0.002
0.006
0.001
0.793
0.204
Reynolds
--
--
--
1344.1
1344.
1
1344.1
1344.
1
1365.
6
1365.6
4032.
3
4032.3
4032.
3
4032.
3
4096.
8
4096.
8
6720.
4
6720.
4
6720.
4
6720.
4
6828.
0
6828.
0
Nusselt
--
--
--
12.66
20.26
14.85
21.52
16.85
22.82
32.00
33.03
38.99
35.66
31.08
34.52
49.80
9.20
53.14
10.09
44.25
11.37
Rayleigh
33609
136858
1016853
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
--
Biot
Table 1: Biot, Reynolds, Nusselt, and Rayleigh numbers for each material, wind speed, and orientation tested.
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Experimental Results
Figure 3. All collected data for Logarithmic dimensionless Temperature vs Time
The curves for every one of the 21 plots graphed above show linear tendencies at
some point. An inspection of the calculated Biot numbers for each case points to the
same conclusion that most of these appear to be linear and the ones that best
represent the linear nature with an R^2 value of 1 have low enough Biot numbers
that the lumped mass assumption may be used. Some specific cases that don’t meet
this criteria include Ceramic at every speed in every orientation. Though close,
ceramic is not a great material to use for this assumption to be considered valid and
this is also proven by the fact that it is not a metal/conductor meaning it has a lower
thermal conductivity than any of its counter examples while still having the same
characteristic length used (block size). This results in significantly higher biot numbers
and less uniform temperature distributions as the internal conduction cannot be
considered instantaneously.
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Experimental Results
Free/Natural
h
τ
1 m/s
3 m/s
5 m/s
Br.
Al.
Cer.
Br.
90°
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
Br.
90°
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
Br.
90°
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
10.86
10.56
12.30
13.25
21.20
15.55
22.52
17.36
23.51
33.49
34.57
40.82
37.33
32.03
35.56
52.13
09.63
55.62
10.56
45.59
11.71
259.1
1402.5
190.1
1001.0
184.8
719.4
1243.8
1001.0
684.9
1019.4
636.9
680.3
469.5
485.4
358.4
403.2
390.6
259.1
283.3
263.2
237.0
Table 2: Convective heat transfer coefficients and time constants for all collected data sets
Its evident that each case has a unique time constant but the time constant
represents an amount of time before a certain amount of heat is lost/temperature is
changed so similar time constants between different cases (same material with other
different variables or different material also) means that those cases both reject heat
at a similar rate.
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Experimental Results
Cer 5, 90
Al 5, 90
Br 5, 90
Al 3, 90
Al 3, 45
Cer 3, 45
Br 3, 45
Br 3, 90
Cer 3, 90
Cer 1, 45
Al 1, 45
Br 1, 45
Cer 1, 90
Al 1, 90
Br 1, 90
Cer 5, 45
Al 5, 45
Br 5, 45
Figure 4. Nusselt vs Reynolds for all forced convection data sets
The cube edge length was used as the characteristic length for both numbers keeping
the relationship between the important parts of the numbers the same while simply
scaling its position on both axes in the coordinate plane. The Nusselt number
generally increases with the Reynolds number except for a specific orientation in
turbulent flow. As Figure 4 shows, the Nusselt numbers for a specific air velocity is
generally concentrated within a small range of values for both laminar and turbulent
flows. For some reason, the 45 degree orientation breaks this consistency at high
speed flow regimes which leads me to believe that other factors such as part
geometry can play a role in the correlation between these numbers.
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Experimental Results
Free/Natural
Biot
1 m/s
3 m/s
5 m/s
Br.
Al.
Cer.
Br.
90°
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
Br.
90°
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
Br.
90°
Br.
45°
Al.
90°
Al.
45°
Cer.
90°
Cer.
45°
0.002
0.001
0.214
0.003
0.004
0.002
0.003
0.302
0.409
0.007
0.007
0.005
0.004
0.557
0.619
0.011
0.002
0.006
0.001
0.793
0.204
Table 3: Biot numbers from Table 1 with a red cell showing invalidity of lumped-mass assumption (Bi > 0.1)
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Experimental Results
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Forced convection fluid velocity assumed to be uniform
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Applied voltage and fan rotational speed likely have minimal error that can be overlooked
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Mixing of stationary ambient air with forced flow can lead to impurity and inconsistency
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Tape and miniscule pores in the flow straightener
Laminar vs Turbulent flow and the effect on heat transfer
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Laminar Regime
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More uniform velocity profile at lower velocities as seen by calculated Reynolds numbers
Turbulent Regime
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Less uniform velocity profile at higher velocities as seen by calculated Reynolds numbers
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Leads to more heat transfer but harder to predict
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Experimental Results
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Comparison of convective heat transfer coefficients for spheres and infinite cylinders
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Calculated using Nu which can be determined through other numbers such as Pr and Re
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Re found from part geometry depending on characteristic
length in Figure 5
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Nu found from any relationship seen in Figure 6
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Infinite cylinder more closely represents the square block in
both free and forced convection
Figure 6: Nusselt Number Relationships
Figure 5: Geometries for Reynolds Number
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Experimental Results
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Fluid flow is primarily characterized by the Reynolds number
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Laminar flow (Re < 2000)
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Primarily dominated by viscous forces
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Clean, smooth flow
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No vortex shedding
Figure 9. Laminar Flow (Re < 2000)
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Figure 10. Transitional Flow (2000 < Re < 4000)
Turbulent flow (4000 < Re)
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Primarily dominated by inertial forces
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Dirty, random flow
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Von Karmon Vortices and shedding
Figure 11. Turbulent Flow (4000 < Re)
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Conclusions and Recommendations
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Lumped Mass Assumption
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Biot number represents a ratio of internal conduction to external convection
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Primary indicator for determining if there is fast enough conduction to assume uniform temperature within object
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Br and Al tested valid for the assumption with Bi < 0.1 (metals = conductors = high conductivity)
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Ceramic tested invalid for the assumption with Bi > 0.1 (nonmetal = poor conductor = low conductivity)
Nusselt/Reynolds correlation
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Further study into specific correlation will reveal the reason for inconsistency at 5 m/s, 45° orientation
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thins
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Further Study
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Time was a massively limiting factor in this experiment along with semi-operational equipment
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Some data gathering was rushed (initial temperatures taken to be 60°C instead of 50°C or 70°C due to time limitations
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Repeatability in more controlled environments
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Less residual/ambient impacts, more trials, data collection at more frequent intervals than 30 seconds
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References
The Board of Trustees of the University of Illinois, (2023). Lumped-Mass/Convection Heat Transfer Investigation.
Bergman, T. L., & Lavine, A. S. (2019). Fundamentals of heat and mass transfer. Wiley.
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