First Law for A Control Volume

P M V SubbaraoProfessor

Mechanical Engineering Department

Modeling of True Engineering Systems…..

Laws of Nature for A Control Mass

Conservation of Mass : constantCMm

Conservation of Momentum : tFVmVm CMCMCMCM

12

First law of thermodynamics :212112 WQEE CMCM

Rate Equations for Laws of Nature for A Control Mass

Conservation of Mass : 0dt

dmCM

Conservation of Momentum :

Fdt

Vmd CMCM

First law of thermodynamics : WQdt

dECM

The Family of Thermodynamic Systems

Engineering Symptoms of Civilization

The Onset of Civilization

The Pinnacle of Civilization

An Important Innovation

Development of Reactors

CVs for Day to Day Use

Supply & Use of LPG through Cylinders

Domestic Using of LPG

Control Mass or Control Volume

A Representation for Engineering Convenience

CM & CV Representation of a Device

• Control Mass representation of Can:

• The total Deodorant

• Control Volume representation of Can:

• Deodorant in side the can

• At time t = 0, (before spray).

• The total mass of Deodorant =Mass of Deodorant in side the can

• Control mass is same as control volume

CM & CV Representation of a Device

• At t = t (after spraying)

• Control mass = control volume + spray

• It is possible to relate CM and CV of a device at any instant!

• Principle of Conservation mass says that the rate of change of mass for a control mass is always zero.

• What about control volume?

The Thermodynamic Control Volume

• In real engineering devices, we are usually interested in a region of space, i.e, control volume and not particular control mass.

• The laws of nature are connected to Control Mass.

• Therefore, we need to transform Laws of Conservation for a control mass to a control volume.

• This is accomplished through the use of Reynolds Transport Theorem.

• Specially derived in thermodynamics for CV .

Flowing Fluid Through A CV

• A typical control volume for flow in an funnel-shaped pipe is bounded by the pipe wall and the broken lines.

• At time t0, all the fluid (control mass) is inside the control volume.

The fluid that was in the control volume at time t0 will be seen at

time t0 +t as:           .

The control volume at time t0+t           .

There will be differences between the fluid (control mass) and the control volume at time t0 +t        .

The control mass at time t0 +t        .

I

II

• Consider a control mass and a control volume (C.V.) as follows:

• the control mass occupies region I and C.V. (region II) at time t0.

• Fluid particles of region – I are trying to enter C.V. (II) at time t0.

II

III

• the same control mass occupies regions (II+III) at t0 + t

• Fluid particles of I will enter CV-II in a time t.

•Few more fluid particles which belong to CV – II at t0 will occupy III at time t0 + t.

A Generalized Functional Model for CV

The control volume may move as time passes.I

II

At time t0

II

III At time t0+t

CVV

I is trying to enter CV at time t0

III has left CV at time t0+t

Reynolds' Transport Theorem

For and infinitesimal time duration

• The rate of change of property B of the system.

t

B

t

B

t

BB

t

B tI

ot

ttIII

ot

tCVttCV

ot

CM

ot

0000 limlimlimlim

IIIICVCM BB

dt

dB

dt

dB

The above mentioned change has occurred over a time t, thereforeTime averaged change in any general property of a control mass, BCM is

t

B

t

B

t

BB

t

B tIttIIItCVttCVCM

0000

A Simple Accounting !!!

Conservation of Mass

• Let B = mass of the system, m.

inoutCVCM mm

dt

dm

dt

dm

The rate of change of mass in a control mass should be zero.

0 inoutCV mm

dt

dm

Conservation of Momentum

• Let B = momentum of the system, mV.

inoutCVCM VmVm

dt

Vmd

dt

Vmd

The rate of change of momentum for a control mass should be equal to resultant external force.

FVmVmdt

Vmdinout

CV

First Law of Thermodynamics

• Let B = E, Energy of the system, me.

inoutCVCM mm

dt

med

dt

med

The rate of change of energy of a control mass should be equal to difference of work and heat transfer rates.

WQmmdt

medinout

CV

Rate Equations for Laws of Nature : Control Mass

Conservation of Mass : 0dt

dmCM

Conservation of Momentum :

Fdt

Vmd CMCM

First law of thermodynamics : WQdt

dECM

First Law for A Control Volume

• Conservation of mass:

0 inoutCV mm

dt

dm

• Conservation of energy:

WQdt

dEinout

CV

Wdt

dEQ out

CVin

FVmVmdt

Vmdinout

CV

• Conservation of momentum:

More Mathematical Definitions for A CV

V

CV dVm V

CV dVeE

inletinlet A

in

A

in AdVAdVm

..

outletoutlet A

out

A

out AdVAdVm

..

inletinlet A

inin

A

ininin AdVAdVm

..

inletoutlet A

outout

A

outoutout AdVAdVm

..

Thermodynamic Nature of Variables of CV

• Incoming and outgoing mass flow rates are steady.• Properties of incoming and outgoing flows are

homogeneous and invariant.• Properties of CV can be inhomogeneous and variant.• Following features for CV are possible.• Inhomogeneous and variant : Difficult to solve using

thermodynamics.• Homogeneous and invariant : A trivial situation for a CV.

No heat or work interactions required.• Inhomogeneous and invariant: Steady State System. • Rate of work and heat interactions must be invariant too.• Homogeneous and variant: Transient System. • Rate of work and heat interactions are variant.

Applications of CV Analysis

A means to estimate the size of engineering devices.

First Law for CV:Steady State Steady Flow

• Conservation of mass:

00 inoutCV mm

dt

dm

• Conservation of energy:

Wdt

dEQ out

CVin

Properties of CV are Invariant:

WQdt

dEoutin

CV 0

NO accumulation or depletion of mass of a CV.

NO addition or removal of energy for a CV.

Rate of Work and Heat Transfers : SSSF

Both rate of heat transfer and rate of work transfer are invariant.

The work done per unit mass and heat transfer per unit mass are invariant.

The specific work transfer at various parts of a CV can be different.

The specific heat transfer at various parts of a CV can be different.

CVCV wmW

Let : mmm outin

CVCV qmQ

The Steam Power PlantExecutes a Thermodynamic Cycle using an assembly of CVs

Complex Engineering Control Volume : SSSF

CV

out

out

in

inCV WgzV

hmgzV

hmQ

22

22

SSSF: Conservation of mass

outin mm

First Law :

CVoutoutininCV WmmQ

Comparison of A control mass and SSSF CV during a change of state

Consider compression processes using CM and CV devices.

Reciprocating compressor : A Control Mass : mCM

Initial State : p1,v1 and T1. Final State : p2,v2 and T2.

constant21

12

CMm

vv

pp

Centrifugal compressor : A Control Volume @SSSF

mmm outin

m mInlet State : pin,vin and Tin.

Outlet State : pout,vout and Tout.

constant

CV

outin

inout

m

vv

pp

Various parts of A CV are at different states during SSSF process !!

Salient Features of CV @SSSF Process

• Rate of mass inflow = Rate mass outflow.• Work done per unit mass is invariant.• Heat transfer per unit mass is invariant.• Change of state or process is not for the CV!• The incoming fluid changes its state from inlet to exit

conditions.• A CM possesses different states at different time intervals.• A CV possesses All states at any time but at different

spatial locations.• A CV with SSSF process is an inhomogeneous device.• A CV can work continuously without changing its state.• A CV lowers the importance of time !