1st law control vol

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First Law Analysis for aControl Volume (open system)

Control Volume A control volume is a volume in space in which

we have interest for a particular study oranalysis. The surface of this control volume is

called the control surface. The size and shape of control volume are

arbitrarily chosen to suit our analysis.

The surface may be fixed, or it may be movable(expand or contract).

Mass as well as heat and work can cross thecontrol surface, and the mass in the control

volume (as well as its properties) change withtime.

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Control Volume

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Mass Balance When we have many flows coming in and

going out of the control volume,

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massin massout

controlvolume

ei

C.V. mmdt

dm

So, if the mass inside thecontrol volume changes withtime, it is only because we addsome mass or take out some

mass. There can be NO othermeans by which mass inside a

control volume can change.

The above equation iscalled Conservation of

Mass or Continuity

Equation.

The first law of thermodynamics fora control volume

The first law for a control mass,1Q2 1W2 = E = E2 E1

The rate form of first law for control mass,

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WQdt

dE MC ..

In a similar manner, a control volume willhave rates of heat transfer, rates of work, and(in addition) mass flows.

Mass and Volume Flow Rates

The amount of mass flowing through a cross-section per unit time is called the mass flow

rate. A liquid or gas flows in and out of a control

volume through pipes or ducts. The mass flowrate of a fluid flowing in a pipe is proportionalto the cross-sectional area A of the pipe, thedensity of the fluid, and the velocity of thefluid.

The mass flow rate through a differential areadA is given by,

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dAVmd nWhere nV is the velocity component normal to dA.


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The mass flow rate through the entire cross-

sectional area of the pipe is obtained byintegration:

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dAVmdn

dAVm n

In most practical applications, the flow of fluidthrough a pipe can be approximated to be one-dimensional flow. This means, properties can beassumed to vary in one direction only (the directionof flow). So, the properties are uniform at any cross-section normal to the flow direction.


Here, is density (=1/v), kg/m3 , A is cross-

sectional area normal to flow direction, m2

.The volume of the fluid flowing through a cross-

section per unit time is called the volume flowrate, given by,

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vVAAVdAVm n /

AVdAVVA

n

The mass and volume

flow rates are related

by

)(velAv

VVm

The concept of Flow EnergyFor control volumes, an additional mechanism

can change the energy of a system: mass flowin and out of the control volume.

When mass enters a control volume, the energy ofa control volume increases because the enteringmass carries some energy with it.

Similarly, when some mass leaves the controlvolume, the energy contained within the control

volume decreases because the leaving masstakes out some energy with it.

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Total Energy of a flowing fluid

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gzV

upekeue2

2

The fluidpossesses anadditional formof energy __the flow energy(flow work).

The fluid flowing across the control surfaceenters or leaves with an amount of energy perunit mass as,

whenever a fluid mass enters a control volume atstate i, or exits at state e, there is a boundarymovement work associated with that process.

Flow WorkControl volumes involve flow of mass across the

control surfaces, and some work is required to

push the mass into or out of the control volume.This work is known as the flow work. It is

necessary for maintaining a continuous flowthrough the control volume.

Consider a fluid element of volume V just enteringthe control volume. The fluid immediatelyupstream will force this fluid element to enterthe control volume; thus it can be regarded as an

imaginary piston.11

Flow Work..

If the fluid pressure is P and the cross-sectionalarea of the element is A, the force applied on

the fluid element by the imaginary piston is:F = P A

To push the entire fluid element into the controlvolume, this force acts through a distance X;thus the work done in pushing the fluidelement across the boundary (flow work) is:

WFLOW = P A X = P V

So, the flow work per unit mass is given by,

wFLOW = P v

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Flow work.

The rate of flow work is,

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mPvVPdAVPVFdtdxFW flow /.

For the flow that leaves the control volume,work is being done by the control volume,

eeemvP

and for the mass that enters, the surroundingsdo the rate of work equal to

iii mvP

Flow work

The flow work per unit mass is then Pv, and thetotal energy associated with the flow of mass is,

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The first law equation control volume case,

gz2

2Vhgz

2

2VPvuPve

)eedtdE

e.V .C.V.C.V.

)z2V( hz2

V( hdtdE

e2

ee2ii.V .C.V.C.V.

Flow work

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Here, the work term W accounts for all other typesof work (shaft work, boundary work, etc.) exceptflow work.For a general control volume where we may havemultiple mass streams entering the control

volume, and multiple mass streams leaving the

control volume-

)z2V( hz2

V( hdtdE

e2

ee2ii.V .C.V.C.V.

)z2V( hz2

V( hdtdE

e2

ee2ii.V .C.V.C.V.

)z2V( hz2

V( hdtdE

e2

ee2ii.V .C.V.C.V.

This means that :--------The rate of change of energy inside the control

volume is due to :--A net rate of heat transfer,A net rate of work andThe summation of energy fluxes due to mass

flows into and out of a control volume.

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First Law for a control volume

Note that:

is the net rate of energy changewithin the CV (it is zer o for st eady-state, steady flow systems)

are terms representing fluxes ofenergy into and out of the CV,

which depend on the massfluxes.

are terms representing energytransfer across the CV boundariesnot associated with the massfluxes.

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dtdEcv

2V

m(h gz)2

cv cvQ and W :

The steady stateprocess

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When is a system at equilibrium?

For a system to be at equilibrium there can beno processes occurring within the system.

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i.e. It must be at the same temperature as thatof surroundings.

Steady state is not the same asequilibrium

Equilibrium

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0xT 0

tT

At equilibrium the temperature is thesame throughout the system, and itdoes not change with time.

Steady State

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0x

T0

t

T

At steady state different temperaturescan exist at different points around the

system, but the system does notchange with time.

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First Law Analysisfor a Control

Volume

Contd

Open Systems: Steady State Steady Flow

(SSSF)

Steady-state (SS): The state of a substancewithin Control Volume does not change withrespect to time.

Steady-flow (SF): the total mass within aControl Volume does not change with respectto time.

WE deal with steady-state, steady-flow (SSSF)system now onwards

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Assumptions of the steady statemodel

No change in the mass of fluid in theCV i.e. Mass in = Mass out.

Fluid is uniform in state and incomposition.

State of fluid at any point is same at alltime.

Heat and work interaction betweensystem & surroundings across controlsurface dont change with time.

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Assumption: The state of mass at each point in thecontrolvolumedoes notvarywithtime.

Consequence:

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0..

dt

dmVC 0..

dt

dE VC

So, for the steady state process:Continuity equation:

eimm

..

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.. )2

()2

( VCee

eeii

iiVC WgzV

hmgzV

hmQ

First Law becomes for steady state process,

)2

()2

(

22

....

..

ee

eeii

iiVCVCVC gz

Vhmgz

VhmWQ

dt

dE

When we have only one stream entering thecontrol volume and only one stream leaving,

Continuity equation:

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mmmei

..

22

.. )2

()2

( VCee

eeii

iiVC WgzV

hmgzV

hmQ

First Law

m

Qq VC ..Define and

m

Ww VC ..

wgzV

hgzV

hq ee

eii

i22

22

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STEADY- FLOWENGINEERING

DEVICES

Some common steady flow devices

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Only one input & one output

Single Stream Steady FlowSystem

Nozzles

Diffusers

Turbines

Compressors

Throttling Valve

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ieie

ie zzgVV

hhmWQ2

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Nozzles and Diffusers arecommonly used in jetengines, rockets,spacecrafts and even ingarden hoses.

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Nozzles and Diffusers


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Nozzles

A nozzle is a steady

state device thatincreases thevelocity of a fluidat the exit/ outletpoint at expenseof pressure.

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Nozzles

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There is no means to doany work in a nozzle,

since there are no movingparts.The kinetic energy of thefluid at the nozzle inlet isusually small, and would

be neglected if its value isnot known.

Diffuser A steady state diffuser is a device constructed

to slow down a high-velocity fluid in a manner

that results in an increase in the pressure ofthe fluid at the exit.

In a way, it is an EXACT OPPOSITE of a

nozzle.

There is large kinetic energy at the diffuser

inlet and a small, (but usually not negligible)kinetic energy at the exit.

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NOZZLES & DIFFUSERS. These are assumed to be adiabatic devices

because flow thru these devices are at highvelocities, so there is NO significant change inheat transfer.

Also there is NO work interactions, since theyare used for acceleration & deceleration of flowonly.

Also there is no change in elevation of flowthrough these devices.

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Nozzles and Diffusers

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ieie

ie zzgVV

hhmWQ2

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Is there work interaction in this system? NO

Is there heat transfer?

Does the fluid change elevation? NO

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22

ieie

VVhh

NO

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22

ieie

VVhh

How can you find the mass flow rate

in a nozzle?e

ee

i

ii

v

AV

v

AVm

In a nozzle, enthalpy is converted

into kinetic energy

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)h2(hVe eiIf inlet velocity is neglected,then exit velocity is given by


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Throttling Valves

A throttling process occurswhen a fluid flowing in aline suddenly encounters

a restriction in the flowpassage.

This may be a plate with asmall hole in it, a partiallyclosed valve protrudinginto the flow passage, or itmay be a change to amuch smaller diametertube, such as capillary.

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Throttling ValveThe result of this

restriction is an abruptpressure drop in thefluid, as it is forced to

find its way through asuddenly smallerpassage way.

A steady state throttlingprocess isapproximately apressure drop atconstant enthalpy.

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Throttling Valve

Throttling valves are adiabatic because

there is neither sufficient time & nor largearea for any effective heat transfer to take

place.

Also NO work interaction takes place in

these Valves.

Change in fluid elevation is also ignored.

Usually change in K.E of fluid is alsoneglected.

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ieie

ie zzgVV

hhmWQ2

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Throttling Valve

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Is there work in this system? NO

Is there heat transfer? NO

Does the fluid change elevation? NO

Does the fluid change velocity? Usually it can be ignored

iehh0

Throttling Valves

hin = hout

For ideal gasesh = Cp T

But h = 0

So T = 0

The inlet and outlet temperatures are the same!!!

Throttling Valves are sometimes called asisenthaplic devices.

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Turbine

A turbine is a rotary

steady state machine whose purpose is toproduce shaft work(power) at theexpense of the

pressure of theworking fluid.

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Turbine

Inside the turbine, there are two distinctprocesses:

In the first, the working fluid passes through a

set of nozzles, that expand the fluid to a lowerpressure and higher velocity.

In the second, this high-velocity fluid stream isdirected onto a set of moving (rotating) blades,in which the velocity is reduced before beingdischarged from the outlet. This produces atorque on the rotating shaft, resulting in a shaftwork output.

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The purpose of a steady statecompressor (gas) or pump (liquid) is:

To increase the pressure of a fluid byputting in shaft work (power).

The working fluid enters thecompressor at low pressure, movinginto a set of rotation blades, fromwhich it exits at higher velocity, as aresult of the shaft work on the fluid.The fluid then passes through adiffuser section, in which it is sloweddown in a way that increases itspressure. So, fluid finally exits fromthe compressor at high pressure.

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Compressors

Turbines and Compressors

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A turbine is a device thatproduces work at the expense oftemperature and pressure.

A compressor is a device thatincreases the pressure of a fluid by

adding work to the system.

ieie

ie zzgVV

hhmWQ2

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Turbines and Compressors

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Is there work in this system? Yes!

Is there heat transfer? Usually it can be ignored

Does the fluid change elevation? Usually it can be ignored

Does the kinetic energy change? Usually it can be ignored

iehhmW iehhw

Mixing of Streams

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Mixing Chamber

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Mixing two or more fluids

is a common engineeringprocess.

Mixing

Chamber


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Mixing Chamber

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i

i

iie

e

eegz

Vhmgz

Vhm

netnet WQ 22

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We no longer have only one inlet and one exit stream

Is there any work done? No

Is there any heat transferred? No

Is there a velocity change? No

Is there an elevation change? Usually it can be ignored

iiee hmhm0

Mixing Chamber

Mass Balance

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iemm

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Mixing Chamber

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332211 hmhmhm

321 mmm

Energy

Balance

Mass

Balance

Heat Exchanger

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Heat Exchanger

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A heat exchanger is adevice where twomoving fluids exchangeheat without mixing.


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Heat Exchangers

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Our analysis approach will depend on howwe define your system

1st law control vol

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