my thermodynamics cheat sheets

Nasser M. Abbasi

Sumemr 2004 compiled on — Wednesday January 03, 2018 at 09:09 AM

1. all of theormodynamics in one sheet. PDF image

2. polytropic process diagrams PDF image

3. first and second laws diagrams PDF image

4. Gas laws PDF image

all of theormodynamics in one sheet

1

Ideal gas, Any processAny gas, any process

General polytropic process

Ideal Gas Process classification

reversibleirreversible

P1 V1

T1

P2 V2

T2

P Vn

constant

n constant Vn 0 constant P

n 1 constant T

Boundary

Work

Boundary

Work

Boundary

Work

W 0

Boundary Work

Boundary

Work

P2

P1

T2

T1

n

n 1 V1

V2

n

w R T1 lnP1

P2

R T1 ln2

1

P1 1 ln2

1

u2 u1 Cv T2 T1

h2 h1 Cp T2 T1

s2 s11

2 Q

Tsgen

mass constant

s2 s11

2 C0

TdT R ln

2

1

s2 s11

2 Cp0

TdT R ln

P2

P1

Assume

constant

specific

heat

s2 s1 C0ln

T2

T1

R ln2

1

s2 s1 Cp0ln

T2

T1

R lnP2

P1

Table A.6

s2 s1 sT2

0 sT1

0 R lnP2

P1

Using

Table

A.7 or

A.8

WORK

Shaft work (for FLOW

process only)

wn

1 nPe e Pi i

n R

1 nTe Ti

wP2 2 P2 2

1 n

R T2 T1

1 n

wP2 2 P2 2

1 k

R T2 T1

1 k

w P2 2 P2 2

R T2 T1

Shaft work

(for FLOW

process

only)

W 0

Shaft work (for

FLOW process only)

w Pi i lnPe

Pi

R Ti lnPe

Pi

R Ti lne

i

Shaft work (for

FLOW process only)

wk

1 kPe e Pi i

k R

1 kTe Ti

Shaft work (for FLOW

process only)

n 1

n 1

w Pe e Pi i

R Te Ti

Verify this, what volume is this?

dsQ

T

W P dV

1stLaw Q W U

T ds dU P dV

Substitute into

Substitute into

T dS dH V dP

1

2

3

4

5

Gibbs equations

enthlapy law H U P V

so dH dU P dV V dp

Specialized polytropic

processes work formulas

General formulas for reversible compressible processes formulas for

general

polytropic

process

Introduction to Thermodynamics, equations. By Nasser M Abbasiimage2.vsdAugust 2004

Solving

Entropy change determination formulas, for an ideal gas, ANY process type

Entropy change determination formulas,

for an ideal gas, polytropic process type

General

polytropic

relation

s2 s1 Cv0

R

1 nln

T2

T1

s2 s1 Cv0 lnT2

T1

s2 s1 Cp0 lnT2

T1

s2 s1 0

Entropy

changeEntropy

change

Entropy

change

Entropy

change

n=k, constant entropy

wi

e

P

w1

2

P

Shaft Work

Total specific work for steady state flow process where only

shaft work is involved (no boundary work). Valid for ANY

reversible process (do not have to be polytropic)

wtotal i

ev dP

Vi2Ve

2

2 gZi Ze

wi

e

Pwi

e

P wi

e

P

du Cv0 dT

dh Cp0 dT

Cp0 Cv0 R

s2 s1 R lnP2

P1

GAS

ds CpdT

T R dP

P

Solids/Liquids

dP 0 (incompressible), and d 0

dh du

dh C dT

Ideal Gas

h u P

dh du dP

dh du P d v dP

P RT

so

h u RT

dh du R dT

dh CvdT RdT

dh CpdT

Process that causes irreversibility

1. Friction

2. Unrestrained expansion

3. Heat transfer from hot to cold body

4. Mixing of 2 differrent substances

5. i2R loss in electric circuits

6. Hystereris effects

7. Ordinary combustion

h2 h1 C lnT2

T1

h2 h1 Cp lnT2

T1

Entropy change equation

Solids/Liquids Ideal Gas

ds dq

T(by definition, entropy law)

dw duT

dwT

duT

1T

dPv 1T

dCvT

1TP dv v dP Cv

TdT

PT

dv vT

dP CvdTT

but Pv RT, hence

ds R dvv R dP

P Cv

dTT

s2 s1 R lnv2v1

R lnP2

P1 Cv ln

T2

T1 1

How to get this below from the above??

ds dq

T(by definition, entropy law)

dw duT

dwT

duT

1T

dPv 1T

d C T

1T

P dv v dP Cv

TdT

PT

dv vT

dP CvdTT

but dP 0 since incompressible,

and dv is very small, so

ds C dTT

s2 s1 C lnT2

T1

1Q W dE

where E U KE PE

Enthlapy definition

First law

FlowNon-Flow

Q1,2 W1,2 mu2 u1

Or, it can be written as follows (ignoring KE and PE changes to the control mass)

As a Rate equation

Q W dE

dt

Non-steady state (Transient, state change)

Q C.V. m ih KE PEi W C.V.m eh KE PE

e dE

dt

QC.V. mh KE PEi WC.V.mh KE PE

e

QC.V. mhi WC.V.mhe

QC.V. WC.V.mhe hi

Steady state devices: Heat exchanges, Nozzle, Diffuser, Throttle, Turbine, Compressors and Pumps

QC.V. mih KE PEi WC.V.meh KE PE

e m2u2 m1u1

General equation. Valid at any instance of time. Steady or not steady flow.

Usually Simplifies to

QC.V. mihi WC.V.mehe m2u2 m1u1

steady state. mi me m

q w he hi

heh i

h i m1 m2

State 1 State 2

Second law

Non-flow

ms2 s1 Q

T Sgen

s2 s1 q

T sgen

flow

steady transient

0 misi mese Q

T Sgen

0 si se q

T sgen

m2s2 m1s1 misi mese q

T sgen

polytropic process diagrams

2

P1 V1 T1

State 1

P2 V2 T2

State 2

P Vn

constant

Polytropic process

We have a total of 7 unknowns. 3 in state 1, 3 in state 2, and n, the polytrpoic process exponent.If given any 5 out of these 7, then the remaining 2 can be found.For example, if we know T1, P1, V1, n, P2, then we can find V2, and T2

Polytrpoic processby Nasser M Abbasipolytropic.vsdAugust 2004

n=0 const P

n constant VP

v

n=1, const T

n=k, adiapatic, const S

EXPANSION QUADRANT

COMPRESSIONQUADRANT

Ignore this quadrant in real

engineering equipments

Ignore this quadrant in real

engineering equipments

Initial

state

point

n=1 const T

T

s

n=0, const P

EXPANSION QUADRANT

COMPRESSIONQUADRANT

Initial

state

point

n=k const S

Clock wise, n=0,1,k,infinity

Clock wise, n=0,1,k,infinity

n constant V

Ignore this quadrant in real

engineering equipments

Ignore this quadrant in real

engineering equipments

Heat OUT processes

Heat IN processes

Heat IN processes

Heat OUT processes

Process lines to left of adiabatic line means negative Q (i.e. heat OUT), on the right are positive Q (i.e. heat IN) process

first and second laws diagrams

3

Laws of thermodynamics

First law Second law

This is also called the law of conservation of energy

Chapter 5. 1st Law for control mass

Q12 W12 E2 E1

E U PE KE

Chapter 5.5

enthalpy

H U PV

h u Pv

Derived from first Law

by setting P constant

Q W mu2 u1

Q P dV mu2 u1

Q PV2 V1 mu2 u1

q P2 1 u2 u1

q P2 u2 P1 u1

1q2 h2 h1

Note: Even though enthlapy was derived by the

assumption of constant P, it is a property of a

system state regardless of the process that lead

to the state.

Chapter 5.6

Cv Cp

Specific heat is the

amount of heat required

to raise the temp. of a

unit mass by one degree

Cv uT v

Cp hT p

Solids and liquids: Use

average specific heat C.

dh du dP v

dh du v dP P dv

For solids and liquids, dv 0

dh du v dP For Solids and Liquids

Also for solids and liquids, v is very small, hence

dh du For Solids and Liquids

since almost incompressible, hence

Cv Cv For solids and liquids

For solids and liquids

Hence for solids/liquids, dh du C dT

For ideal gas

du Cv0 dT

dh Cp0 dT

Cp0 Cv0 R

Chapter 6

1st Law for control volume

Qin Win mih PE KEi Qout Wout meh PE KEe m2u2 m1u1

Steady

state

Transient

State

Adiabatic process

Wnet he hi

Wnet

heh i

The entropy of an isolated system increases in all real processes and is

conserved in reversible processes.

OR

The AVAILABLE energy of an isolated system decreases in all real

processes (and is conserved in reversible processes)

The 2nd law says that work and heat energy are not the same. It is easier

to convert all of work to heat energy, but not vis versa. i.e. work energy is

more valuable than heat energy. Heat can not be converted completely and

continuously into work.

So, 2nd law dictates the direction at which a system state will change. It

will move to a state of lesser available energy

Entropy is constant in a reversible adiabatic process.

S2 S1 1

2 Q

T Sgen Sgen 0

Sgen 0 reversible process

Q 0 adiabatic process

Hence, S2 S1 for a reversible adiabatic process

Wlost T dSgen

Actual boundary work W12 P dV Wlost

Gibbs relations

T ds du P dv

T ds dh v dp

Solids, Liquids

s2 s1 C lnT2

T1

Ideal Gas

sT0

T0

T Cp0

TdT

s2 s1 sT2

0 sT1

0 R lnP2

P1Using table A.7 or A.8

s2 s1 Cp0 lnT2

T1 R ln

P2

P1Constant Cp,Cv

s2 s1 Cv lnT2

T1 R ln

v 2

v 1Constant Cp,Cv

k Cp0

Cv0

Polytrpic process

PVn constantP2

P1 V1

V2

n

P2

P1 T2

T1

n

n1

specific work (work is moving boundary work, P dv

w12 1

1nP2v2 P1v1 R

1nT2 T1 n 1

w12 P1v1 lnv 2

v 1 RT1 ln

v 2

v 1 RT1 ln

P1

P2n 1

By Nasser M. AbbasiImage1.vsd

Gas laws

4

Solids/liquids

Ideal gas, Any processAny gas, any process

General polytropic process

Ideal Gas Process classification

reversibleirreversible

P1 V1

T1

P2 V2

T2

P Vn

constant

n constant Vn 0 constant P

n 1 constant T

Boundary

Work

Boundary

Work

Boundary

Work

W 0

Boundary Work

Boundary

Work

P2

P1

T2

T1

n

n 1 V1

V2

n

w R T1 lnP1

P2

R T1 ln2

1

P1 1 ln2

1

u2 u1 Cv T2 T1

h2 h1 Cp T2 T1

s2 s11

2 Q

Tsgen

mass constant

s2 s11

2 C0

TdT R ln

2

1

s2 s11

2 Cp0

TdT R ln

P2

P1

Assume

constant

specific

heat

s2 s1 C0ln

T2

T1

R ln2

1

s2 s1 Cp0ln

T2

T1

R lnP2

P1

Table A.6

s2 s1 sT2

0 sT1

0 R lnP2

P1

Using

Table

A.7 or

A.8

WORK

Shaft work (for FLOW

process only)

wn

1 nPe e Pi i

n R

1 nTe Ti

wP2 2 P2 2

1 n

R T2 T1

1 n

wP2 2 P2 2

1 k

R T2 T1

1 k

w P2 2 P2 2

R T2 T1

Shaft work

(for FLOW

process

only)

W 0

Shaft work (for

FLOW process only)

w Pi i lnPe

Pi

R Ti lnPe

Pi

R Ti lne

i

Shaft work (for

FLOW process only)

wk

1 kPe e Pi i

k R

1 kTe Ti

Shaft work (for FLOW

process only)

n 1

n 1

w Pe e Pi i

R Te Ti

Verify this, what volume is this?

dsQ

T

W P dV

1stLaw Q W U

T ds dU P dV

Substitute into

Substitute into

T dS dH V dP

1

2

3

4

5

Gibbs equations

enthlapy law H U P V

so dH dU P dV V dp

Specialized polytropic

processes work formulas

General formulas for reversible compressible processes formulas for

general

polytropic

process

Introduction to Thermodynamics, equations. By Nasser M Abbasiimage2.vsdAugust 2004

Solving

Entropy change determination formulas, for an ideal gas, ANY process type

Entropy change determination formulas,

for an ideal gas, polytropic process type

General

polytropic

relation

s2 s1 Cv0

R

1 nln

T2

T1

s2 s1 Cv0 lnT2

T1

s2 s1 Cp0 lnT2

T1

s2 s1 0

Entropy

changeEntropy

change

Entropy

change

Entropy

change

n=k, constant entropy

wi

e

P

w1

2

P

Shaft Work

Total specific work for steady state flow process where only

shaft work is involved (no boundary work). Valid for ANY

reversible process (do not have to be polytropic)

wtotal i

ev dP

Vi2Ve

2

2 gZi Ze

wi

e

Pwi

e

P wi

e

P

du Cv0 dT

dh Cp0 dT

Cp0 Cv0 R

s2 s1 R lnP2

P1

GASds Cp

dT

T R dP

P

5