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Thermodynamics Formula, Lecture notes of Thermodynamics

the branch of physical science that deals with the relations between heat and other forms of energy (such as mechanical, electrical, or chemical energy), and, by extension, of the relationships between all forms of energy.

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Thermodynamics
MTX 220 Formules
Chapter 2 – Concepts & Definitions
Formule Units
Pressure
F
PA
=
Pa
Units
2
1 1 /Pa N m
=
5
1 10 0.1bar Pa Mpa
= =
1 101325atm Pa
=
Specific Volume
V
vm
=
3
/m kg
Density
m
V
ρ
=
1
v
ρ
=
Static Pressure Variation
P gh
ρ
=
,
↑= = +
Pa
Absolute Temperature
( ) ( ) 273.15T K T C
= ° +
Chapter 3 – Properties of a Pure Substance
Formule Units
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
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Thermodynamics

MTX 220 Formules

Chapter 2 – Concepts & Definitions

Formule Units

Pressure

F

P

A

Pa

  • Units 2

1 Pa = 1 N /m

5

1 bar = 10 Pa =0.1Mpa

1 atm = 101325 Pa

Specific Volume

V

v

m

3

m /kg

Density

m

V

ρ =

v

ρ =

3

kg /m

Static Pressure Variation

∆ P = ρgh ↑= − ,↓ = +

Pa

Absolute Temperature

T K( ) = T ( °C ) +273.

Chapter 3 – Properties of a Pure Substance

Formule Units

Quality

vapor

tot

m

x

m

(vapour mass fraction)

liquid

tot

m

x

m

(Liquid mass fraction)

Specific Volume

f fg

v = v +xv

3

m /kg

Average Specific Volume

f g

v = − x v +xv

(only two phase

mixture)

3

m /kg

Ideal –gas law

c

P << P

c

T << T Z^ =^1

  • Equations

Pv = RT PV = mRT =nRT

  • Universal Gas Constant

R = 8.

kJ /kmol K

  • Gas Constant

R

R

M

M

= molekulêre mass

kJ /kg K

Compressibility Factor

Z

Pv =ZRT

Reduced Properties

r

c

P

P

P

,

r

c

T

T

T

Chapter 4 – Work & Heat

Formule Units

Conduction Heat Transfer

dT

Q kA

dx

,

k

=conductivity

W

Convection Heat Transfer

Q = hA T∆

,

h

=convection coefficient

W

Radiation Heat Transfer 4 4

( ) s amb

Q = εσA T −T

W

Terminology

Q

= heat

1 2

Q

= heat transferred during the process between state 1 and state 2

Q

= rate of heat transfer

W

= work

1 2

W

= work done during the change from state 1 to state 2

W

= rate of work = Power. 1 W=1 J/s

Chapter 5 – The First Law of Thermodynamics

Formule Units

Total Energy

E = U + KE + PE → dE = dU+ d KE( +) d PE( ) J

Energy

2 1 1 2 1 2

dE = δ Q − δW → E − E = Q − W J

Kinetic Energy 2

KE = 0.5mV J

Potential Energy

2 1 2 1

PE = mgZ → PE − PE = mg Z( −Z ) J

Internal Energy

liq vap liq f vap g

U = U + U → mu = m u +m u

Specific Internal Energy

of

Saturated Steam

(two-phase mass

average)

f g

f fg

u x u xu

u u xu

kJ /kg

Total Energy 2 2

2 1

2 1 2 1 1 2 1 2

m V V

U U mg Z Z Q W

J

Specific Energy 2

e = u + 0.5V +gZ

Enthalpy

H = U +PV

Specific Enthalpy

h = u + Pv kJ^ /kg

For Ideal Gasses

Pv = RT and u =f T( )

  • Enthalpy

h = u + Pv = u +RT

  • R Constant

u = f t( ) → h =f T( )

Specific Enthalpy for

Saturation State

(two-phase mass

average)

f g

f fg

h x h xh

h h xh

kJ /kg

Chapter 6 – First-Law Analysis for A Control Volume

Formule Units

Volume Flow Rate

V = V dA =AV

(using average velocity)

Mass Flow Rate

V

m VdA AV A

v

= ρ = ρ = ∫

(using average values)

kg /s

Power

p

W =mC T

& V

v

W =mC T

& V V

m

v

W

Flow Work Rate

flow

W = PV =mPv

Flow Direction From higher P to lower P unless significant KE or PE

  • Total

Enthalpy

2 1

tot

h = h + V +gZ

Instantaneous

Process

  • Continuity

Equation

C V.. i e

m = m − m ∑ ∑

  • Energy

Equation

C V.. C V.. C V.. i tot i e tot e

E = Q − W + m h − m h ∑ ∑

 First Law

( )

2 2 1 1 ( )

i i i e e e

dE

Q m h V gZ m h V gZ W

dt

∑ ∑

Steady State

Process

A steady-state has no storage effects, with all properties

constant with time

  • No Storage ....

C V C V

m = E =

  • Continuity

Equation

i e

m = m ∑ ∑

(in = out)

  • Energy

Equation

C V.. i tot i C V.. e tot e

Q + m h = W + m h ∑ ∑

(in = out)  First Law

( )

2 2 1 1 ( )

i i i e e e

→ Q + m h + V + gZ = W + m h + V +gZ ∑ ∑

  • Specific

Heat

Transfer

C V..

Q

q

m

kJ /kg

  • Specific

Work

C V..

W

w

m

kJ /kg

  • SS Single

Flow Eq.

tot i tot e

q + h = w +h

(in = out)

Transient Process Change in mass (storage) such as filling or emptying of a

container.

  • Continuity

Equation

2 1 i e

m − m = m − m ∑ ∑

  • Energy

Equation

2 1 C V. C V.. i tot i e tot e

E − E = Q −W + m h − m h ∑ ∑

( ) ( )

2 2

2 1 2 2 2 2 1 1 1 1

E − E = m u + V + gZ − m u + V +gZ

( ) ( )

2 2

. 2 2 2 2 1 1 2 1.. ..

C V i tot i e tot e C V

C V

Q m h m h m u V gZ m u V gZ W

∑ ∑

Chapter 7 – The Second Law of Thermodynamics

Formule Units

All

W Q ,

can also be rates

W ,Q

& &

  • Real

Refrigerator

L L

REF Carnot REF

REF H L

Q T

W T T

β = ≤ β =

Absolute Temp.

L L

H H

T Q

T Q

Chapter 8 – Entropy

Formule Units

Inequality of Clausis

Q

T

Ñ∫

Entropy

rev

Q

dS

T

kJ /kgK

Change of Entropy 2

2 1

1 rev

Q

S S

T

 δ 

kJ /kgK

Specific Entropy

f g

s = − x s +xs

f fg

s = s +xs

kJ /kgK

Entropy Change

  • Carnot Cycle Isothermal Heat Transfer:

2

1 2

2 1

1

H H

Q

S S Q

T T

− = δ = ∫

Reversible Adiabatic (Isentropic Process):

rev

Q

dS

T

Reversible Isothermal Process:

4

3 4

4 3

3 rev^ L

Q Q

S S

T T

 δ 

− = =  

Reversible Adiabatic (Isentropic Process): Entropy

decrease in

process 3-4 = the entropy increase

in process 1-2.

  • Reversible

Heat-Transfer

Process

2 2

1 2

2 1

1 1

(^1 1) fg

fg

rev

h Q q

s s s Q

m T mT T T

δ

δ

∫ ∫

Gibbs Equations

Tds = du +Pdv

Tds = dh −vdP

Entropy Generation

gen

Q

dS S

T

irr gen

δW = PdV −T δS

2 2

2 1 1 2

1 1

gen

Q

S S dS S

T

δ

∫ ∫

Entropy Balance Eq.

VEntropy = + in − out +gen

Ideal Gas Undergoing

an Isentropic Process

2 2

2 1 0

1 1

0 ln ln

T P

s s Cp R

T P

0

2 2

1 1

R

Cp

T P

T P

but

0 0

0 0

p v^1

p p

C C

R k

C C k

,

0

0

p

v

C

k

C

= ratio of

specific heats

1

2 1 2 1

1 2 1 2

k k

T v P v

T v P v

Special case of polytropic process where k = n:

k

Pv =const

Reversible Polytropic

Process for Ideal Gas

1 1 2 2

n n n

PV = const = PV =PV

1 1

2 1 2 2 1

1 2 1 1 2

n n n n

P V T P V

P V T P V

− −

  • Work 2 2

2 2 1 1 2 1

1 2

1 1

n

dV PV PV mR T T

W PdV const

V n n

∫ ∫

  • Values for n Isobaric process:

Isothermal Process:

Isentropic Process:

Isochronic Process:

n = ∞ , v =const

Chapter 9 – Second-Law Analysis for a Control Volume

Formule Unit

s

2

nd Law Expressed as a

Change of Entropy

c m..

gen

dS Q

S

dt T

Entropy Balance Eq.

rate of change = + in − out +generation

C V.. C V..

i i e e gen

dS Q

m s m s S

dt T

∑ ∑ ∑

where

....

C V c v A A B B

S = ρsdV = m s = m s + m s +

and

..

gen gen gen A gen B

S = ρs dV = S + S + ∫

Steady State Process

.. 0

C V

dS

dt

.. ..

C V

e e i i gen

C V

Q

m s m s S

T

∑ ∑ ∑

  • Continuity eq.

i e

m& = m& =m&

.. ..

C V

e i gen

C V

Q

m s s S

T

  • Reversible

Polytropic

Process

for Ideal Gas

e

n n

i

w = − vdP and Pv = const =C

( ) ( )

1

e e

n i i

e e i i e i

dP

w vdP C

P

n nR

P v Pv T T

n n

∫ ∫

  • Isothermal

Process (n=1) ln

e e

e

i i

i i^ i

dP P

w vdP C Pv

P P

∫ ∫

Principle of the

Increase of Entropy

.. 0

net C V surr

gen

dS dS dS

S

dt dt dt

Efficiency

  • Turbine

a i e

s i es

w h h

w h h

η

Turbine work is out

  • Compressor

(Pump)

s i es

a i e

w h h

w h h

η

Compressor work is in

  • Cooled

Compressor

T

w

w

η =

  • Nozzle

2

2

e

es

V

V

η =

Kinetic energy is out