FUNDAMENTALS OF THERMODYNAMICS AND HEAT TRANSFER … · Fundamentals of Thermodynamics and Heat...

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FUNDAMENTALS OF THERMODYNAMICS AND HEAT

TRANSFER

Lecture 2: Evaluating properties

Fundamentals of Thermodynamicsand Heat Transfer

Solid Liquid

Gas Plasma

STATES OF THE MATTER

Phase is a quantity is of matter that is homogeneous throughout inboth chemical composition and physical structure. Homogeneity inphysical structure means that the matter is all solid, all liquid or allvapour. System can contain one or more phases.



Substance thatcontracts on freezing

Substance thatexpands on freezing









Equations of State -- Perfect Gas Laws

Characteristics of perfect (ideal) gasobeys gas laws (Clapeyron, Avogadro, Dalton)constant specific heat dependent only on kind of process, independent of T,pReal gas regarded ideal if under low pressure and high temperatureOnly elastic collisions, particles as material points

From this can be derivedBoyle – Mariotte law (1676), pv=const and constant TCharles’s law v/T=const at constant pAmagat’s lawAvogadro’s principle (1811)the combined gas law (p1V1/T1 = p2V2/T2)




Compressibility factor Z

For a perfect gas, pV/nRT = 1= pV / RTThe ratio pV/ RT is called the compressibility factor, Z.

For real gases, Z can be either greater or less than 1.

gasperfectvolumemolarvolumemolarZ =

Low pressures, Z=1, all gases ideal

At high pressures, Vreal >Videal ; Z>1 repulsive forces dominate, less compressibility

Low pressure, Vreal <Videal, Z< 1 attractive forces dominate

Low temperatures -> molecules moving less rapidly more influenced by attractive forces


Phase transitionsIn most our considerations we assumed ideal gas, where particlesdo not interact. Recalling the gas equation: pv = nRT. In case of constant temperature T = const the distribution of a product pV with p should be following;

p

pV0o

-103o

-130o

In real that is rather different.


Dla rzeczywistych gazów wykresy te wyglądają następująco;

p

pV

10

16o

-35o C

-75o

-103o

-130 o

air

p

pV

107Pa

1 5

200o C

100o

0oCO2

For two gases different results are obtained. In case of highertemperatures the discrepancies from ideal gas are smaller.


Rariefied real gas is similar to ideal gas. Molecules of differentgases have different dimensions and different types of forcesacting on them. Gases where inter-particle forces are present are named non-idealgases.

There are many approaches to define a non-ideal gas. Van-der-Waals (1879) postulated the following definition. Van der WaalsModel reads;

TRbVVap mmm

=−⎟⎟⎠

⎞⎜⎜⎝

⎛+ )(2

(7.27)

The constant b is related to volume occupied by particles. Theterm 1/V2 results from existence of inter-molecular forces, whereas a is a simple gas proportionality constant. The index m denotes the molar quantities.


Van der Waals isotherms for CO2 :

For temperatures above criticalisotherm the curves follow idealgas isotherms.

For a decreasing volume thepressure varies not along thecurve EDCA, but a straight lineECA.

That is due to the fact that incase of real gases in E condensation commences, whichterminates at point A.

Pressure at E is a vapourpressure. It reaches maximumvalue at a critical point K.

molar volume

pres

sure

p

Ideal gas

coexistenceregion


Ideal vs. Real Gases


Other Equations of State

Redlich-Kwong Peng-Robinson

Both are quantitative in region where gas liquefies

Berthelot, Dieterici and others with more than ten parameters can give good fits !!!with seven free parameters, you can describe an elephant …

)(2/1 BVVTA

BVRTp

mmm +−

−=

)()( βββα

β −++−

−=

mmmm VVVVRTp


Principle of Corresponding States

All gases have the same properties if they are compared at corresponding conditionsDefine reduced variables

For homework you will write the vdw eqn in terms of the reduced variables

cRcRcR VVVTTTPPP /;/,/ ===

Compression factor plotted using reduced variables. Different curves are different TR


Virial Equation of State

...)()(1 232 +++==V

TBV

TBRTVPZ VV

...)()(1 232 +++== PPBPPB

RTVPZ PP

)(2 TBV = 0 at Boyle temperature

Most fundamental and theoretically sound

Polynomial expansion – Viral Expansion

Used to summarize P, V, T data

Also allow derivation of exact correspondence between virial coefficients and intermolecular interactions


1.2 Relations between classical mechanics andthermodynamics

Problems of classical mechanics encompass such problems as: force, mass, distance and others. Force can be regarded as something whichis pulling or pushing, but mathematically is represented as a vector. Mechanics is based on a Newton’s II law:

)( vmdtdF rr

∑ =

In description of mechanics the free body is used, which isinfluenced by forces in accordance to 2nd Law of Dynamics.

Mechanical system is definced by spatial coordinated and velocity.


Work

Work is defined as amount of energy supplied by theacting force on a specified distance and is defined as:

∫ ⋅=c

sdFW rr(1.1)

Fθ

ds

cA linear integral definesthe work in the directionof action of force

∫ ∫ ⋅⋅=⋅=c c

dsFsdFW θcosrr

That expression containsthe determination of thesign of work.Work is positive whenthe force anddisplacement have thesame direction.Work is negative whenthese have the oppositedirections.


Work in the gravitational field

m

PF

h W=P·h

Force F which has to be applied to lift the mass m is equal to the weight P of a body.

∫∫ ==⋅=hh

omghdsPsdFW

0

rr

mgPF ==

Work

(2.1)


Work of electrical current

I(A)

The cell V

R

Completion of workmeans that there hasto exist the forcedisplacing the charge in adequate direction.

Work of electrical current: tIVW Δ⋅⋅= (2.2)

How the sign of work can be determined? If the cell will be ourthermodynamical system then the work done on the coil will be negative. If the coil will be our thermodynamical system then itwill be positive.


B

Work of the magnetic field

X

z

y

I

v

Its known that:

BvqFrrr

×= (2.3)

ds

For the element ofconductor length ds we have:

∫∫ =⋅= sdIdtsddtIvq rr

r

(2.4)∫ −=×=

ljlBIBsdIF

0

rrrrhence

kr

jr

ir


Work performed per unit of time (power) is equal to:

vBsdIvFWl

trrrrr

⋅⎟⎠⎞⎜

⎝⎛ ×=⋅= ∫0

hence

( ) ( ) vBlIjvjlBIWt −=⋅⋅−=rr

(2.5)

A negative sign of work is obtained in the case when the conductoris regarded as a thermodynamical system. We calculate workperformed on the conductor by the magnetic field. In order to move to conductor there ought to be applied external force equal inthe extent and directed in the other direction. The workperformed by that force would be equal to;

vBlIWt = (2.5a)


pA

Work of gas compression or expansion in the piston

F = pA

Displacement of the piston by Δx causes execution ofwork

xFdW Δ⋅=Knowing that the piston has a surface A, after rearrangement

dVpA

dVApdW ⋅=⋅= or

∫ ⋅=2

1

V

VdVpW (2.6)

Executed work is positive when dV is positive. That is a workexecuted on the piston by the gas pressure.Gas in the piston is a closed system.


Work is positive when the piston is our thermodynamical system.If our system is gas, then work supplied to that system will be negative. Then;

∫ ⋅−=2

1

V

VdVpW (2.6a)

The sign of work depends on selection of thermodynamical system!!!

Graphical presentation of workperformed by expanding gas ispresented aside. The paths frompoint 1 to point 2 can be different (different processes). Work is denoted as area belowthe relevant path.

1

2

V1 V2

p

V

Friction and other non quasistatic processes (where thesystem does not pass throughsubsequent equilibrium states) have not been considered.


Internal energy

Recapitulating the work is a form of energy, which can pass theboundaries of the system. In order for the work to be acomplishedthere must exist the interaction between the system and itssurroundings. The work depends on the kind of process and can be traced when the system passes from one state to another.

We ought to recall that in mechanics, electromagnetism, etc. we came across with the term potential energy.

Potential energy of gravitation is a work required to lift theweight above the reference level.

Kinetic energy can be calculated by determination of workrequired to provide it with some speed.


Work carried out by the force imposing some velocity on somedistance can be written in the form:

EKvmdvvm

dsdtdvmsdFW

==⋅⋅=

=⋅⋅=⋅=

∫

∫ ∫2

21

rr

(2.7)

Acceleration from one velocity to another renders cumulation ofexecuted work in the form of kinetic energy.

)(21 2

12212

2

1

vvmdvvmEKEKv

v−==− ∫

Identical situation is with potential energy.

WEPEPEP ==− Δ21 (2.8)


In equation (2.8) the change of potential energy is possible onlyby the preserving forces.

First law of thermodynamics enables to generalise the meaning ofpotential energy.

I II

EIEII

Q=0

Energy is a function of state!!

Let’s consider the system under transition from state I to state II. We assume that the system is absolutely isolated so there is no heat transfer. The only interaction with surroundings is through thework which does not change the potential energy.

Wad


The change of energy of the system is through the supply ofadiabatic work by all forces acting on the system.

adIII WEEE ==− Δ (2.9)

It stems from experiments that adiabatic work between twostates is always the same. If the system performs work than isexhausts the internal resources.

We hence assume that energy E is a property of the system anddepends only on the state of the system.

If we allow same transition from state I to state II by removinginsulation then the change of internal energy is the same as thesestates are the same. However, in that case the system canexchange heat with surroundings.


adWQW =+Inaczej

EQW Δ=+

(2.10)

(2.11)

Heat is defined as positive when it is supplied to the system andnegative when rejected from the system.

Our considerations related the closed systems.


The Laws of Thermodynamics

1. Law of Thermodynamics (energy conservation)

change in heat added to/subtracted work doneinternal energy = from the system - by/on the system of the system

Δ U = Q - W

Q > 0 : heat is added to systemQ < 0 : heat is subtracted from systemW > 0 : work done by system on surroundingsW < 0 : work done on system by surroundings

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