Plume Lecture 2010

8/8/2019 Plume Lecture 2010

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MECHANICAL AND ENVIRONMENTAL ENGINEERING

LABORATORY

PLUME DYNAMICS

MAE126A/171A

Professor Paul Linden


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Figure 1: Plumes rising in the atmosphere.


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DEFINITION OF A TURBULENT PLUME

A turbulent plume is the flow above a sustained source of

buoyancyFor example the flow above a chimney from which hot gases are

being discharged

Gas is released at a volume flow rate Qs at a temperature Ts abovethe temperature Ta of the surrounding air. (The subscript s refers

to the stack or source.)

These gases will rise, partly as a result of the upward momentum

of the flow from the stack, but mainly because the gas is buoyant

compared to the surrounding air.


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Relation between density and temperature of a gas

As the temperature increases gas molecules become more energetic

and their mean-free-paths increase. Consequently, the volume

containing a given number of molecules (equivalently, a given

mass of gas) increases and the density decreases. Hence the density

mass/volume,

decreases.

Perfect gas equation

Many gases, including air are well described by the perfect gas

equation = p/RT,

where p is the pressure, T Kelvin is the temperature and R is the

universal gas constant.


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Entrainment

Entrainment is a process whereby fluid is engulfed into a flow. In aturbulent flow, entrainment occurs by engulfing the surrounding

fluid by the turbulent eddies. This is a random are inherently

unpredictable process.

Entrainment mixes the plume fluid with the surrounding fluid

reducing the concentration of pollutants in the plume.

e.g. flow from a car exhaust otherwise you would breathe CO at

a concentration that would kill you as you walk along a street.


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In a uniform environment,

as a plume rises it entrains cooler air, so that the gas within the

plume gets cooler

the buoyancy force decreases as the plume rises, and the

plume slows down

additional fluid is carried upwards by the plume so the

volume flux Q(z) increases with height


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REDUCED GRAVITY

The buoyancy of the emitted gases is defined in terms of the

reduced gravity g

of the gas which is defined by

g ga s

s,

wheres is the density of the fluid leaving the stack,

a is thedensity of the surrounding air, and g = 9.81 ms2 is the

acceleration of gravity.

The dimensions of reduced gravity

gdim = LT2.


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Using the perfect gas equation

= p/RT,

the reduced gravity g can be determined in terms of temperature.

g = p/RTa p/RTsp/RTs

,

which after rearranging gives

g gTs

TaTa

,

where the subscripts have the same meaning as above.

All temperatures are in Kelvin, so, e.g. the reduced gravity of gases

released at 177C into air at 27C is given by

g = 9.81150

300= 4.9ms2.


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BUOYANCY FLUX

The buoyancy flux B of the plume is defined by

B gQ,

where Q is the volume flow rate (m3

s1

).

Hence, in the above example, if the flow rate Q = 10 m3s1, then

B = 49m4s3.


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In almost all practical cases, B is large enough that the flow is

turbulent. (This is even the case for the plume above a cigarette.)The flow inside the plume is unsteady and has randomness.

Dynamically it contains vorticity, while the surrounding stationary

air is irrotational (and has no vorticity).

As a plume rises it entrains ambient air through a process called

turbulent entrainment. It is not possible to describe or calculate

this entrainment process exactly, and so we model it as discussed

below.


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Figure 3: The buoyancy flux into across the two horizontal planes is thesame otherwise the mass within the volume will increase or decrease.


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THE STRUCTURE OF THE PLUME

For a steady plume one with properties that do not change with

time the buoyancy flux B is constant.

Self similarity

A flow is said to be self similar if the structure of the flow is

geometrically the same at all locations.

e.g. similar triangles are essentially the same shape but at different

scales.

Self similarity is an important concept in physics, as it arises from

situations where systems evolve without remembering their

initial states. For example, if you look at a snowflake and thensmaller and smaller portions of it, the structure is reproduced at

each scale. Often such shapes are fractal, in that their dimension is

non-integer.


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A plume is an example of a self similar flow, since the structure ofthe flow is the same at different distances above the source. The

width b(z) increases with distance z from the source, but when the

properties are rescaled using b as the scale they are the same.

The properties of the plume depend on

the (constant) buoyancy flux B

the distance z above the source

The dimensions of the parameters are

Bdim = L4T3,

zdim = L.


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Self similarity means that the vertical velocity can be written in the

following form

w(r, z) = w0(z)f1(r/b),

where w0 is the vertical velocity on the centerline r = 0 of the

plume, and the function f1(r/b) describes the dependence of the

velocity on the distance r from the center.

To determine the behavior of the centerline velocity, we seek acombination ofB and z that has dimensions of velocity (from the

above decomposition the function f is dimensionless). Obviously,

this leads to

w0 = C1B1/3z1/3,

where C1 is a (dimensionless) constant.


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Similarly the reduced gravity

g(r, z) = g0(z)f2(r/b),

and dimensional analysis gives

g0

= C2B2/3z5/3.

Finally, the width b(z) of the plume is

b = C3z.

Thus the mean shape of the plume is a cone, and the radiusincreases linearly with height.


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From these results we see that the volume flux in the plume

Q(z) = 2

0

w(r, z)rdr,

= C4B1/3z5/3,

and is an increasing function of height.


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Details of the buoyancy and velocity fields

The profiles of the buoyancy and velocity across the plume are

only known from experimental measurements. Measurements of

the time-average of the density and velocity at different distances rfrom the plume centerline at a given vertical distance from the

source are found to be well approximated by the Gaussian curves

g(r, z) = g0

(z)er2/b2 ,

w(r, z) = w0(z)er2/b2 ,

where the subscript zero refers to values measured along the

centerline and b is the plume width.


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THE FILLING BOX

In an enclosed space, such as a room, the plume adds heat

continuously and the air within the space increases continually

provided there are no heat losses. This section investigates how

this increase in temperature occurs.

Figure 4: Schematic of the early stage of a filling box.

temperature


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the initial flow of the air in the plume rises and spreads out as

a hot layer across the ceiling.

with time this warm layer descends as more air is carriedupwards in the plume

at this later stage the plume is entraining air in this layer that is

warmer than that originally in the room.

Consequently, the air arrives at the ceiling hotter than it did at the

start. This process continues so that the plume arriving at the

ceiling is always warmer than at a previous time. Hence, the air

spreading across the ceiling gets hotter with time, and so thetemperature of the warm layer is greatest at the top.


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Figure 5: Later stage of a filling box showing the establishment of a

stable density stratification.


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A fluid which has the warmest, and less dense fluid, at the top is

called stably stratified.

Eventually the initial warm layer descends to the floor, since the

ambient air is continually entrained into the plume by entrainment

and carried to the ceiling. This is called the filling box mechanism.


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In practice it is not necessary to have a completely closed space for

this process to occur. Provided the air is stably stratified, the plume

rise will stop at some height. This is because, although the plumeis initially warmer than the atmosphere, and therefore rises, as a

result of entrainment, the air in the plume gets cooler as it rises.

If the atmosphere is stably stratified, the surrounding air increases

in temperature with height this is called an inversion. So with

height the plume is getting cooler and the air around it is getting

warmer. In most cases, the plume reaches a height where these two

temperatures are equal, and the plume has zero buoyancy. Theplume stops rising and spreads out horizontally.


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If there are many plumes each competing for surrounding air to

entrain the horizontal flow is constrained in the same way as it is

by vertical walls in an enclosed space. Thus in a situation where

there are multiple plumes in an inversion such as occurs, e.g. inLos Angeles the filling box mechanism will recirculate the

pollutants in the plumes and bring them down to ground level.


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On a clear night, the ground cools by long wave radiation which

carries the heat to space. Thus the air near the ground becomescooler than that above, so that it is stably stratified. Since this is

opposite to the usual case found during the day, this situation is

known as an inversion.

This intense radiative cooling is why deserts are cold at night, even

when it is hot during the day, and why in moist atmospheres (such

as coastal regions) there is condensation on cars etc. at night.


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The experiments

The objectives of the experiments are

1. To measure the properties of a turbulent plume

2. To verify self-similarity and determine the profiles of buoyancy

3. To observe the filling box mechanism


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The plume will be set up by a source of salt water in a tank of fresh

water. Since salt water is denser than fresh water the plume isnegatively buoyant, and so the source is placed at the top of the

tank. In this case the flow is simply reversed in direction compared

to a plume rising from a chimney stack.

The fluid equations do not distinguish between gases and liquids

provided they are Newtonian. The only difference is in the values

of the molecular properties the viscosity and the diffusion

coefficient .


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Figure 6: The tank and the peristaltic pump for the salt supply.


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Figure 7: The traverse holding the conductivity probe.


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Figure 8: A close-up of the conductivity probe.


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Figure 9: The probe electronics and the probe-pump.


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1. Week 1 Calibration

Objective

To calibrate the conductivity probe and determine the plumebuoyancy flux

Method

The plume buoyancy flux is given by B = gQ at the source.Calibrate the peristaltic pump to determine the flowrate Q, and

determine g of an initial salt solution.

The conductivity probe is calibrated against g

by placing it in saltsolutions of different strengths.


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Week 2 Plume measurements

Objective

To measure the radial profile of density at 3 different verticallocations beneath the plume source.

To test for self-similarity and determine the unknown constant C2.

MethodUse the traverse to place the probe tip at various locations on a

horizontal cut through the plume. Measure the conductivity at

each location and determine the mean and rms of the signal. Use

the results to fit the theoretical curves to the data, and determine

C2.


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REFERENCES

LINDEN, P.F. 2000 Convection in the environment. Perspectives in

Fluid Mechanics, Eds. Batchelor, G.K, Moffatt, H.K. & Worster, M.G,

Cambridge University Press, pp. 287343.

TURNER, J.S. 1973 Buoyancy effects in fluids. Cambridge University

Press.

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Plume Lecture 2010