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ENVIRONMENTAL FLUID MECHANICS

–

Mass, Momentum & Energy Balanceswith some examples

Benoit Cushman-RoisinThayer School of Engineering

Dartmouth College

To extract useful quantitative information from a system, it is necessary to apply the laws of physics. The most important physical laws are:

Conservation of mass → mass of carrying fluid (air or water)→ mass of carried contaminant (ex. Mercury)→ mass of a relevant quantity (ex. dissolved oxygen)

Conservation of momentum (Newton’s Second Law)→ mass x acceleration = Σ forces

(Forces act as momentum sources.)

Conservation of energy → total energy (mechanical + thermal)→ usually thermal energy dominates,

and the energy budget devolves into a heat budget.

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And, there are two broad approaches:

1. The usual method is to establish the budgets for a small volume

and then shrink that volume to zero.

The result is a set of partial differential equations.

2. An alternative, which is often practical but overlooked,

is to perform the budgets on a finite volume of the fluid.

The result gives a coarse-grain picture of reality, but it can be extremely useful if done judiciously.

Inside:Chemical reactionsBiological processes

Physical settling

Inflow Outflow

In the environment, we are not dealing only with local processes, such as chemical reactions; we are mostly dealing with interconnected systems, implying that any sub-system is connected to other sub-systems through fluxes in and out.

We must therefore reckon with material balances in the presence of open systems.

Exchangewith theatmosphere

Waterexample

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Finite-volume budget methodology

Step 1: Choose a control volumeThis can be the entire system or only a well-chosen piece of it.

Step 2: Select the quantity for which the budget is to be madeThis can be mass of fluid, mass of a contaminant, or the fluid’s momentum or energy.

Step 3: Consider all contributions, positive and negativeAdd imports and subtract exports through boundaries.Add sources and subtract sinks within the control volume.

Step 4: Extract useful information from the resulting equationSolve the equation for the salient unknown in the budget equation.

Examples of practical control volumes

urban airshed

entire lake

section of a jet

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Sometimes it is difficult to know where to place the boundaries of the control volume…

(Phoenix, Arizona)

The choice of a control volume is more an art than a science, for there are countless possibilities. A good level of intuition is required, which can only be developed with practice.

Notwithstanding this, there are a couple of basic rules that apply:

1. A control volume ought to be practical, in the sense that it should yield budget equations with the minimum number of unknowns of the problem.

2. Its boundaries need to be clearly defined so that there is no ambiguity about what is inside and what is outside and what are the fluxes in and out of the domain.

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Let’s do a budget for a generic quantity, c.

Think of it as the concentration of a contaminant:

cVmV

mc

fluid carrying of volume

substance of mass

In words, the budget is:

Accumulation = imports – exports

+ sources – sinks

Accumulation = imports – exports

+ sources – sinks

Now, we can fill in the details of the budget.

dt

dcV

dt

tcVdttcVdt

tmdttmdt

tdtt

)()(

)()(

duration

)(at amount )(at amount onAccumulati

and construct each term on a “per-time” basis.

The term on the left is then expressed as:

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Notion of flux

The control volume is almost never bounded on all sides by impermeable boundaries. Thus, fluid enters the volume across some of its boundaries and leaves through some others, carrying with it mass, momentum, energy and any dissolved or suspended substance.

The quantity that describes such exchange between the control volume and its surroundings is the flux.

The flux of any quantity (mass, momentum, energy, dissolved or suspended substance) is defined as the amount of that quantity that crosses a boundary per unit area and per unit time:

uc

q

velocityionconcentrat

duration timeareaboundary

volume

volume

boundary thecrossingquantity

duration timeareaboundary

boundary thecrossingquantity flux

nucqnuu ˆˆ

For oblique flow:where is the perpendicular velocity component

is the 3D velocity vector

and is the outward normal unit vector.

u

n̂

u

Because the exports through the boundaries can be counted as negative imports, a single summation suffices for all imports and exports.

Per unit time, this sum is:

where the summation over the index i covers all the inlets and outlets.

iiii

iii

ii

Anuc

Auc

Aq

)ˆ(

boundaries through exports Imports

The determination of the sources and sinks inside the control volume requires the specification of the quantity for which the budget is performed (mass, momentum, energy, etc.) and a knowledge of the mechanisms by which this quantity may be generated or depleted. For the moment, let us subsume all sources and sinks in a single term

S volumecontrol e within thsinks Sources

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This image cannot currently be displayed.

Now with all pieces together, the budget becomes:

SAucdt

dcV iii

In cases when system properties vary continuously inside the control volume and along its boundaries, the total amount of the quantity is to be expressed as a volume integral of its concentration, and the summation over all inlets and outlets must be replaced by an integration over all open portions of the surface enclosing the control volume:

dVsdAnucdVcdt

d)ˆ(

where dV is an elementary piece of volume, dA an element of boundary surface with unit outward normal vector through which passes vector velocity , and s the source/sink per volume.

n̂ u

Mass conservation

To state mass conservation, we use

And we state that there are no sources or sinks.

The budget becomes:

densityvolume

massc

dAnudVdt

d

Audt

dV iii

)ˆ(

or

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Air and water in the environmental systems behave as incompressible fluids. Thus,

constant i

And the mass budget reduces to a volume budget:

outlets

outinlets

inoutlets

outoutinlets

inin QQAuAu

Naturally !

0)ˆ(

0

dAnu

Au ii

More practically, when inlets and outlets can be counted and characterized each by a single volumetric flux Q:

Example of mass conservation

m/yr 7.8km1500

/yrkm )7487(2

3

evap

u

Consider Lake Nasser behind the Aswan High Dam in Egypt, which is fed by the Nile River.

Annual averages of the upstream and downstream river flow rates are 87 km3/yr and 74 km3/yr, respectively, indicating a loss of water from the lake. Assuming that there is no ground seepage, the loss can only be caused by evaporation at the surface. Knowing that the lake surface area is 1500 km2, we can determine the rate of evaporation by performing a water budget for the lake:

River inflow = River outflow + Evaporation

87 km3/yr = 74 km3/yr + uevap A

It follows:

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Momentum budget

Now, for the quantity c , we use the momentum per volume:

And forces act as sources (if accelerating) or sinks (if decelerating).

Forces are:

uV

um

pressure

gravity

friction

Momentum budget is then:

f

g

p

F

gVF

AnpF

ˆ (acting on external surfaces only)

(acting within the volume)

(left unspecified at this stage)

fFgVAnp

AuuAuuudt

dV

surfaces

outoutoutlets

outoutinininlets

inin

ˆ

Again, we invoke the assumption of incompressibility (= o = constant):

o

f

oo

FgVAn

pAuuAuu

dt

udV

surfacesoutoutout

outletsininin

inlets

ˆ

Except in the gravitational term !

Example: Turbulent discharge jet

Statistically steady state:

Uniform outside pressure → pressure term is self-cancelling

No friction on outside surfaces and Internal friction self-cancelling:

Horizontal jet → No gravitational term

0dt

ud

0fF

The momentum budget reduces to:

outoutoutlets

ininininlets

AuuAuu out

entering momentum flux = exiting momentum flux

One-dimensional flow further simplifies the equation

12

12

22

22

21

212

221

21 u

R

RuRuRuAuAu

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So, velocity down a turbulent jet decreases inversely to distance.

There seems to be a problem, however: How can this be compatible with mass conservation?

Indeed, conservation of mass implies conservation of volume, and velocity should be inversely proportional to the cross-sectional area and thus to the square of radius and the square of distance, shouldn’t it?

oo uxR

Rxuu

R

Ru

)()(1

2

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Laboratory experiments show that the radius of a turbulent jet increases proportionally to distance:

005

)(

5)(

ux

Rxu

xxR

x

R

factor determinedexperimentally

The answer lies in the fact that the jet entrains ambient fluid.

We can calculate this entrainment by doing a mass (volume) budget:

A consequence of this is dilution. Should the discharge contain some concentration co of a contaminant, this concentration falls off downstream.

Performing a material balance for the contaminant yields:

)0()()( 222 xEuRxuxR oo E rate of entrainment ≠

0

20

2

ambient0022

5

)(5

5

)0()()()(

cx

Rc

cuRxcx

uRx

cxEcuRxcxuxR

o

oooo

o

=0

We find that the concentration decreases inversely with distance and, surprisingly, is independent of the exit velocity.

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Second Example: Uniform river flow down a sloping channel

Assumptions:

• Statistically steady state

• Uniform flow→ momentum in = momentum out

• Atmospheric pressure all along the way

This leaves only two forces that need to balance each other:

Downslope gravity and friction along the bottom.

hf

f

RSgP

ASg

S

dxPgdxA

)()sin)((0

withh

P

AR

S

h

radius hydraulic

slopesin

Take quadratic form for bottom stress: SRC

SRgUUC h

D

hDf .const2

(Chézy formula)

Bernoulli Principle

An interesting and very important corollary of the momentum budget is the so-called Bernoulli principle, which is obtained when the momentum budget is applied to an infinitesimal cylinder aligned with a streamline in a steady-state flow.

Mass conservation:

dsUdUR

UdsRURdUUR

n

n

2

)2()())(( 22

ds

dURUn 2

(1)

Momentum budget:

sin)(

))(()(

)2()())((

2

22

2222

gdsR

dppRpR

UUdsRURdUUR nooo

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constant2

1

0

sin2

2

zgpU

ds

dzg

ds

dp

ds

dUU

dsgdpdUUdUU

o

o

oo

After mathematical simplifications and elimination of Un with Equation (1), we get

ds

dz slopesindivide by ds and

gather terms on left side

integrate with

respect to s

Kinetic energy “Pressure” energy Potential energy

(all per unit volume)

The expression is called the Bernoulli functionzgpUo 2

2

1

Assuming = constantalong the streamline!

Application of mass, momentum and energy budgets to windmill efficiency: The Betz limit

Is there a limit to the power that we can extract from the wind?

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Consider that the wind cannot be brought to a complete stop because air needs to be evacuated to make room for new air to flow. The best we can expect is to slow down the wind some (0 < U2 < U1).

Obviously, the windmill exerts a resistance to the air flow. Therefore, the pressure on its front

surface (pfront) is higher than atmospheric (patm). Upon approaching, the wind of strength U1

faces an adverse pressure gradient and slows down to an intermediate value U.

Having extracted energy from the wind, the air pressure drops in the rear (down to pback),

which is lower than atmospheric. The flow decelerates further, to a speed U2, as it continues

to face an adverse pressure gradient before recovering atmospheric pressure (patm).

back2

atm22

front2

atm2

1

2211

2

1

2

12

1

2

1

pUpU

pUpU

UAUAUA

pUUUU

ppUU

))((2

1

)(2

1

2121

backfront22

21

pAUUAUUA 111222 )()(

UA UA

pAUUUA )( 21

2))((

2

1)( 21

212121

UUUUUUUUUU

Mass conservation:

Bernoulli in frontup to rotor:

Bernoulli in rearpast rotor:

Across the rotor itself: Downstream momentum = Upstream momentum – braking pressure force

(1)

(2)

Eliminate p between Equations (1) and (2)

Mass conservation

subtract

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Having established this relation between the velocities, let us now get to the power extracted from the wind.

1 with )1)(1(2

1

2

))(()()(2

2availablePower

extractedPower

Efficiency ePerformanc oft Coefficien

2

1

2

1

rate massmass

rateenergy kinetic availablePower

time

ntdisplaceme force

time

workextractedPower

1

2231

22

2121

31

212

122

212

1

31

312

1

311

21

U

U

U

UUUU

U

UUUU

U

Up

UA

UAp

C

UAUAU

KE

UAp

p

pressure difference x area

velocity

Note: U1 in this expression, because it is the strength of the local wind

Vary to optimize this expression and find:3

1for %3.59

27

16max

pC

)(2

1

)(2

1

21

22

21

UU

UU

The optimal windmill, giving 59% efficiency:

How actual windmill designs perform

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Hydrostatic balance

If the fluid is at rest, there is no momentum and no friction, leaving an equation with only the pressure and gravitational forces:

gVAnp

surfaces

ˆ

This simply says that the various pressures on the sides need to support the weight of the fluid within the control volume.

Removal of the hydrostatic balance

The part of the pressure caused by the main o contribution of the density

can be eliminated leaving a smaller (dynamic) pressure in the pressure force and a density anomaly in the gravitational term

gVAnp o

surfaces

chydrostati ˆ

o

f

oo

FgVAn

pAuuAuu

dt

udV

0

surfaces

dynamicoutoutout

outletsininin

inlets

ˆ

where pdynamic = actual p – phydrostatic

Energy conservation

There are various forms of energy: kinetic, potential, “pressure”, and thermal.

In environmental fluid mechanics, the thermal energy dwarfs the other three.

For example: Heating 1 kg of water by 1oC takes 4184 J;Heating 1 kg of air by 1oC takes 1005 J (at constant pressure);1 kg of air or water moving at 10 m/s has 50 J of kinetic energy;1 kg of air or water falling down 50 m releases 490 J of potential energy.

The result is that the energy budget reduces in good first approximation to a heat budget:

Accumulation of heat = Heat entering – Heat exiting + Heat sources – Heat sinks

Furthermore, sources and sinks are rarely internal but occur most often at boundaries. Thus,

Accumulation of heat = Heat entering – Heat exiting

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Heat content = Thermal energy = Internal energy

TCm v

Per unit volume: TCvo

Budget:

)()()()()( outoutoutlets

outinininlets

in AuTCAuTCTCdt

dV vovovo

This means that temperature is conserved along the flow (no source and sinks).

Finally since temperature and density anomaly are related by the equation of state:

,

this means that density anomaly is conserved along the flow.

(This justifies bringing the density anomaly inside the streamwise derivative in establishing the Bernoulli principle, several slides ago.)

)(anomaly oo TT

Conclusions and word of caution

Finite-volume budget equations do not provide a detailed picture of a system, only some basic relations and properties of it.

These relations and properties, however, can be extremely useful.

The usefulness of the information obtained from budget analysis is only as good as the choice of the control volume…

… and only as good as the attending dynamical simplifications that can be made (such as considering only averaged velocities through cross-sections).

In other words, the tool is only as useful as you are smart.