This document describes an experiment to study the effects of buoyancy and momentum on thermal plumes. Warm dyed water was pumped into a tank of cooler water at two different flow rates, creating thermal plumes. A thermocouple array measured temperature profiles in the plumes, which were used to calculate reduced gravities, buoyancy fluxes, and momentum fluxes. The Morton length scale was determined for each flow rate, showing one plume was momentum-dominated and one was buoyancy-dominated. Temperature profiles, normalized reduced buoyancy plots, and centerline reduced buoyancy plots were analyzed to characterize the thermal structure and test scaling laws of the plumes.
4.
List of Tables
Table Description Page
A Calibration of Flow Rate Data 28
B Temperature of Thermocouples vs. Distance Away from Jet 28
C
Week 2 Raw Data of Thermocouples at Various Positions at 20 mL/min
29
D
Week 2 Raw Data of Densities and Reduced Gravities
30
E
Week 2 Raw Data of Buoyancy Flux, Momentum Flux, Normalized
Plume Radius
30
F
Week 2 Raw Data of Normalized Plume Buoyancy, Morton Length Scale
30
G
Week 3 Raw Data of Thermocouples at Various Positions at 10 mL/min
31
H
Week 3 Raw Data of Densities and Reduced Gravities
32
I
Week 3 Raw Data of Buoyancy Flux, Momentum Flux, Normalized
Plume Radius
32
J
Week 3 Raw Data of Normalized Plume Buoyancy, Morton Length Scale
32
3
5.
List of Figures
Figure Description Page
1 Experimental Setup of Lab 8
2a Calibration of Flow Rate from Peristaltic Pump 11
2b Calibration of Flow Rate from Peristaltic Pump Incorrect 11
3 Thermocouple Temperatures vs. Height Above Nozzle Exit 12
4 Bulk Mean Temperature Week 2 13
5 Normalized Reduced Buoyancy vs. Normalized Plume 14
6 Normalized Reduced Buoyancy vs. Nor. Plume Rad. Sq. 14
7 Normalized Plume Centerline Reduced vs. Nor. Height 14
8 Bulk Mean Temperature Week 3 15
9 Normalized Reduced Buoyancy vs. Nor. Plume Radius 16
10 Normalized Reduced Buoyancy vs. Nor. Plume Rad. Sq. 16
11 Normalized Plume Centerline Reduced Buoy. vs. Nor. Hght 16
4
7.
Theory
When a lighter fluid is submerged in a surrounding denser fluid, that lighter fluid exerts
an upward buoyancy force continually as it rises. This buoyancy force is equivalent to the
reduced gravity, which is found through the difference in density of the bulk, homogenous water
temperature ( ) and the heated water temperature ). It is convenient to define localρw ρ( h
buoyancy, or reduced gravity as:
gg′ = * ρw
ρ − ρh w
(1)
The hot water discharging from the plume is a continuous momentum source part of
which is coming from the buoyancy flux. When the volume flow rate Q is constant and known,
the plume buoyancy flux, B, can be determined by the product of the volume flow rate, Q, and
the reduced gravity g’ of the plume:
B = g’ * Q (2)
The buoyancy flux, B, can also be expressed as the integral across the section of the
plume of the product of the vertical velocity, w, with the buoyancy, g’. Since what drives a
plume is its heat flux, defined as the amount of heat being discharged through the exit hole per
unit time, as the plume rises, it entrains ambient fluid. But this effect does not change the heat
flux carried by the plume, thus by virtue of conservation of heat, the buoyancy flux remains
unchanged with height and is the same at level z as it was at the start of the plume.
Then the momentum flux, M, depends on the volume flow rate, Q, of the source and the
cross sectional area, A, of the nozzle:
M = A
Q2
(3)
The temperature acquired by the thermocouples can then be used to find the density of
the surrounding water using an equation that relates temperature to density:
6
8.
1000 [1 T )]ρT = * − (T+288.9414)
508929.2 (T+68.12963)* * ( − 3.98632
(4)
where the density of the water is (kg/m3
) and temperature is T (°C).ρT
Through this experiment, the concept of thermal plumes is better understood by seeing
that a buoyant fluid rising up creates a buoyant flux and a momentum flux, which contributes to
the momentum force. Plumes that exhibit these characteristics are called forced plumes and
buoyant jets. Near the inlet, the plume behaves more like a jet where the flow is dominated by
momentum due to the jet’s flow rate, Q. As the it rises, however, the plume transitions to be
having like a buoyant plume, where the flow is dominated mainly by the buoyancy of the
warmer fluid. This jet to plume transition can be express as a length scale, called the Morton
Length scale:
LM = B(1/2)
M(3/4)
(5)
where M refers to the momentum flux and B refers to the buoyancy flux. For smaller , whenLM
z < , the plume can be said to be dominated by momentum, also known as a jet. For higherLM
, when z > , the plume is dominated by buoyancy, also known as a buoyant plume.LM LM
In order to assess the thermal structure of the plume, a plot is made using a normalized
reduced buoyancy (6) across the normalized plume radius (7) over different heights:
B zg′
−(2/3) (5/3) (6)
/zr (7)
Another plot to see the structure is by plotting the normalized plume centerline reduced
buoyancy (8) against the normalized height (9), where M is the momentum flux at nozzle exit:
zM Bg′
(1/2) (−1)
(8)
/Lz M (9)
7
9.
Then a power law curve is used to test the selfsimilarity of the thermal plume, which
indicates that the structure of the flow is geometrically the same with height, as mentioned
earlier:
zM B C(z/L )g′
(1/2) (−1)
= M
D
(10)
where through the decay of the normalized centerline reduced buoyancy, the coefficient C and
the power law exponent D can be determined.
A Gaussian curve is used to show the radial profiles of the normalized reduced buoyancy
for below the critical region and also above:/Lz M
= B zg′
−(2/3) (5/3) exp(− (r/z)A0 B0
2
(11)
The theory is that through a controlled experiment such as this, a real life plume featured
in environmental fluids, which occur whenever a persistent source of buoyancy creates a rising
motion of the buoyant fluid upward, can be better understood.
Experimental Procedure
Figure 1. Experimental Setup of Lab
Week 1 Procedures
The first week of the experiment was to calibrate the flow rate from the peristaltic pump,
calibrate temperature signals from the thermocouples, determine buoyancy and momentum
8
14.
Week 2 Data
Due to the incorrect data analysis in week 1, 18 RPM is being used in this week’s
experiment. Converting 18 RPM into SI unit, Q = . A new updated reduced.667 0 m /s2 × 1 −7 3
gravity was found using the adjusted temperature readings of the jet and bulk and was found to
be = 6.64 . Given the jet has a diameter of 1.32mm, M and B are calculated usingg′o 0* 1 −2
s2
m
equation (2) and (3), to be B = 1.77 and M = 5.21 respectively. With these 0 0* 1 −8
s3
m4
0 * 1 −8
s2
m4
new values of buoyancy and momentum flux, the Morton length scale can be calculated to be
around 0.0259 m using Equation (5).
Figure 4. Bulk Mean Temperature
The main task of week 2 was to measure the temperature of water around the plume at a
relatively high RPM. Using the raw data provided in Appendix Table (C), Figure 4 shows the
change of the bulk temperature of the water.
13
16.
Week 3 Data
Week 3 follows similar procedures as week 2, but with a lower flow rate of 10 mL/min,
(also incorrect due to the mistake in week 1, which equals to 1.504 ). In this part of0 m /s× 1 −7 3
the lab a new updated reduced gravity was calculated again at = 3.31 . A new set ofg′o 0* 1 −2
s2
m
buoyancy and momentum flux were also calculated with the new flow rate value. Buoyancy flux
was found to be B = 4.970 while the Momentum Flux was M = 1.56 . Using 0* 1 −9
s3
m4
0 * 1 −8
s2
m4
a relatively lower flow rate, resulted in a more buoyancy dominating effect for the plot. The
Morton length was then calculated to be , a slightly lower value than from week 2. Figure (8)
below shows the bulk water temperature change of week 3.
Figure 8. Bulk Mean Temperature Week 3
15
19.
equation, an RPM of 18 was used to produce a plume, which had a volumetric flow rate, Q, of
16.012 mL/min. In calculating the time it took to pump 30 mL, a handheld timer was utilized as
well as a graduated cylinder, both which have an error value due to the fact that a human reaction
can only react so quickly as well as using visual judgement.
Error in time: +/ 0.5 sec to account for human reaction time
Q verage δQ of all 4 points 0.9 mL/min δ = Q × √(t
δt
)
2
+ ( V
δV
)
2
⇒ A =
Flow Rate: 16.0 +/ 0.9 mL/min
With an error of 5.6%, the measurements are slightly out of acceptable range but will still be
taken as accurate for the sake of the experiment. If we correctly achieved a flow rate of
approximately 20 mL/min using the correct calibration, the error would only be about 4.5%
which is acceptable.
The next part looked to calculate the values of the buoyancy and momentum flux using
Equations (1), (2) and (3). Before using starting the plume and reading the temperatures, the
plume base needed to be readjusted and moved to make sure that the jet was right above the
centerline of the 6 thermocouples. Since the base of the jet was not dense or heavy enough to
stay still once the tank began filling with water, a team member had to keep his hands on the jet
base as the bulk water filled up the tank to prevent the base from moving. The team member
continued to keep his arms in the tank as the tank water became fully quiescent, which slightly
raised the temperature of the bulk water due to the team member’s warm arm temperature.
Although it would have been ideal for the bulk water to retain its original temperature, since his
arm was not in the tank for a significant amount of time, the warming of the bulk water was
insignificant.
After setting the pump at the respective 18 RPM, it was given a few minutes for the
18
21.
Using this reduced gravity, the buoyancy flux, B, was then calculated using Equation (2) to be,
. .78 .13 0 B = 1 × 10−8
± 0 × 1 −8
s3
m4
B .78 0 .3 0 .3% δ = B
√(Q
δQ
)
2
+ (g′
δg′
)
2
= 1 × 1 −8
√(16
0.9
)
2
+ (0.067
0.003
)
2
= 1 × 1 −9
⇒ 7
Then lastly, the momentum flux, M was calculated using Equation (3), found to be
..2 .4 0M = 5 × 10−8
± 0 × 1 −8
s2
m4
M .20 .4 0 .7% δ = M
√2(Q
δQ
)
2
= 5 × 10−8
√2(16
0.9
)
2
= 0 × 1 −8
⇒ 7
With the three calculated values having a low error percentage, the calculations appeared to be
accurate and reasonable.
Another source of error could have come from the physical material of the the
thermocouples. Each of the six thermocouples had a brass slug soldered to its tip. Since the jet
water could have essentially touched the first thermocouple closest to the jet, this could have
resulted in the shape of the plume to change. Adding mass on the thermocouple junction is not
desirable since the joints have a larger volume, which would make it inaccurate in measuring the
temperature of just the plume. It is also less desirable since contact with the larger bottom
thermocouple will negatively affect the momentum of the plume. Ideally, decreasing the size of
the brass slug would have increased the accuracy of the temperature data, since it would not have
affected the shape of the plume as much.
Another observation was the drastic temperature decrease of the thermocouples as it was
exposed to air when emptying out the tank. As the thermocouples emerge from the water and are
exposed to the air, the temperature gradually drops from room temperature to about 16° C.This
phenomenon can be explained with two different reasonings. One being that once the
thermocouples were exposed to air, some of the water molecules left on the tips began
20
25.
.031 .002 .45% errorg′ = 0 ± 0 s2
m
⇒ 6
.7 0 .4 0 .51% errorB = 4 × 1 −9
± 0 × 1 −9
s3
m4
⇒ 8
M .65 .2 0 δ = M
√2(Q
δQ
)
2
= 1 × 10−8
√2( 9
0.9
)
2
= 0 × 1 −8
.6 0 .2 0 2.5% errorM = 1 × 1 −8
± 0 × 1 −8
s2
m4
⇒ 1
Using these new values of buoyancy and momentum flux, a new Morton Length was calculated
to be 0.0213 +/ 0.0009 m.
The Morton Length scale in week 3 of the experiment shows a shorter length than that of
week 2, which is expected since the flow rate of week 3 was smaller than that of week 2. This
lower value of the Morton length scale indicates that the flow becomes more buoyancy driven at
a shorter distance and that momentum runs out closer to the nozzle than in week 2.
Then just like in week 2 of the experiment, plots and fits were made with a power law
curve and Gaussian curve, to calculate for the unknown A0, B0, C, and D.
A0 = 9.9659 … B0 = 30.35 … C = 0.0772 … D = 1.4694
Through this experiment a closer analysis of the physical structure of a plume was done.
Although different errors were made throughout the experiment, the experiment accomplished its
objectives in showing when and how a plume is more buoyancy or momentum dominated as
well as how temperature distribution occurs within a plume.
Conclusion
This experiment showed how the characteristics and structures of turbulent plumes are
dictated when dominated by either buoyancy or momentum. After calibrating the peristaltic
pump and thermocouples, a flow rate of 16.0 ± 0.9 mL/min was used to measure the vertical
temperature profile through the plume. The temperature profile and flow rate was then used to
calculate a buoyancy flux of B = 1.78 ± 0.13 and momentum flux of M = 5.2 ± 0.40* 1 −8
s3
m4
24
26.
. Plotting the temperature profile showed that the temperature increased at higher0 * 1 −8
s2
m4
thermocouple positions, which was the opposite of what was expected and may indicate
inaccurate thermocouple positioning.
Another turbulent plume with a high momentum flux of 5.2 ± 0.4 compared to0 * 1 −8
s2
m4
the smaller buoyancy flux of 1.78 ± 0.11 was generated in the next part of the 0* 1 −8
s3
m4
experiment. These fluxes were used to calculate a Morton length scale of 0.0259 ± 0.0009LM =
m, showing that this plume is momentumdominated. The normalized plume centerline reduced
buoyancy was plotted against normalized height and fitted with a power curve, which has a
power coefficient of C = 0.4845 and a power exponent of D = 0.3073. The normalized reduced
buoyancy was also plotted against the normalized plume radius and fitted with a Gaussian curve
that had constants of = 3.903 and = 22.41.A0 B0
The last part of the experiment was done on another plume with a lower flow rate of 9.0
± 0.9 mL/min. This plume differed from the last plume because it had a lower momentum flux of
1.6 ± 0.2 and higher buoyancy flux of 4.7 ± 0.4 , ultimately resulting in a0 * 1 −8
s2
m4
0* 1 −9
s3
m4
smaller Morton length scale of m. The smaller Morton length scale proved.0213 ± 0.0009LM = 0
that this plume was more buoyancydominated. Like the calculations done on the previous
plume, a power curve fitted to the normalized plume centerline reduced buoyancy resulted in a
power coefficient of C = 0.0772 and power exponent of D = 1.4694. The Gaussian curve fitted to
the normalized reduced buoyancy gave constant values of = 9.9659 and = 30.35.A0 B0
25
