1

Units and Key Constants

2

• Conventional Units

ParameterEnglish Units SI Units

– Distance Feet, Inches Meters, M– Time Seconds Seconds, s– Force Pounds (force), lbf 4.448 Newton, N– Pressure psf, psi Pascal, Pa (1N/1m2)

bar (105Pa)1 ft H2O2.989 kPa

– Mass Pounds (mass), lbm 0.4536 kilogram– Energy Btu Joule, J– Power 1 Hp 0.7457 kWatt

3

Equivalent Systems of Units

4

Important Constants for Air

5

Useful Equivalents

6

• For Liquid Water :

• U.S. Standard Atmosphere - 1976

3/4.62 ftlbm

214.696 101,325lbfpressure Pain

518.67 273otemperature R K

7

Standard Atmosphere

Stratosphere >65,000 ft

59 FTemperature

Altitude

3.202 psia

14.696 psiaPressure

36,089 ft

Altitude

36,089 ft

8

9

10

Thermodynamics Review

11

Thermodynamics Review• Thermodynamic views

– microscopic: collection of particles in random motion. Equilibrium refers to maximum state of disorder

– macroscopic: gas as a continuum. Equilibrium is evidenced by no gradients

• 0th Law of Thermo [thermodynamic definition of temperature]: – When any two bodies are in thermal equilibrium with a third,

they are also in thermal equilibrium with each other. – Correspondingly, when two bodies are in thermal

equilibrium with one another they are said to be at the same temperature.

12

Thermodynamics Review• 1st Law of Thermo [Conservation of energy]: Total work

is same in all adiabatic processes between any two equilibrium states having same kinetic and potential energy.– Introduces idea of stored or internal energy E– dE = dQ - dW

• dW = Work done by system [+]=dWout= - pdV• Some books have dE=dQ+dW [where dW is work done

ON system]• dQ = Heat added to system [+]=dQin

– Heat and work are mutually convertible. Ratio of conversion is called mechanical equivalent of heat J = joule

13

Review of Thermodynamics• Stored energy E components

– Internal energy (U), kinetic energy (mV2/2), potential energy, chemical energy

• Energy definitions– Introduces e = internal energy = e(T, p)– e = e(T) de = Cv(T) dT thermally perfect – e = Cv T calorically perfect

• 2nd law of Thermo – Introduces idea of entropy S– Production of s must be positive– Every natural system, if left undisturbed, will change spontaneously

and approach a state of equilibrium or rest. The property associated with the capability of systems for change is called entropy.

revQdS TdS dE dWT

14

Review of Thermodynamics• Extensive variables – depend on total mass of the system, e.g. M, E, S, V

• Intensive variables – do not depend on total mass of the system, e.g. p, T, s, (1/v)

• Equilibrium (state of maximum disorder) – bodies that are at the same temperature are called in thermal equilibrium.

• Reversible – process from one state to another state during which the whole process is in equilibrium

• Irreversible – all natural or spontaneous processes are irreversible, e.g. effects of viscosity, conduction, etc.

15

Thermodynamic PropertiesPrimitive Derived

2

0 0

0

2k p

T

VE E E E or e e gz

Total or stagnation state

16

1st Law of Thermodynamics• For steady flow, defining:

• We can write:

• and

2

2

0

/ 2 specific kinetic energy specific potential energy

specific internal energy

= + + specific enthalpy

e total spec2

Vgze u

ph e pv e

Ve gz

ific energy

2

0e2Vpv e gz pv

0 0h e pv and h e pv

17

1st Law of Thermodynamics

• Substituting back into 1st law:

– Height term often negligible (not for hydraulic machines)

• Defining total or stagnation enthalpy:

• The first law for open systems is:

2 20 / 2 / 2

out in

E Q W m h V gz m h V gz

20 / 2h h V

0 oout in

Q W m h m h

18

Equation of State• The relation between the thermodynamic properties of a pure substance is

referred to as the equation of state for that substance, i.e. F(p, v, T) = 0

• Ideal (Perfect) Gas– Intermolecular forces are neglected– The ratio pV/T in limit as p 0 is known as the universal gas constant (R).

p /T R = 8.3143e3

– At sufficiently low pressures, for all gases

p/T = R

or

• Real gas: intermolecular forces are important p RT

19

Real Gas

1150 R

20

Real Gas

21

1st & 2nd Law of Thermodynamics

• Gibbs Eqn. relates 2nd law properties to 1st law properties:

Tds pdv deh e pvdh de pdv vdp

dpTds dh

22

Gibbs Equation

• Isentropic form of Gibbs equation:

• and using specific heat at constant pressure:

dpdh

p

p

RTc dT dPP

dT R dPT c P

23

Thermally & Calorically Perfect Gas

• Also, for a thermally perfect gas Cp[T]:

• Calorically perfect gas - Constant Cp

-1 =k= = pTP v

s v p

ck Rc c Rk c c

PdP

TdT

1

2

1

2

1

1PdP

TdT

1.4 3.51 0.4pR Rc R for air

24

Isentropic Flow• For Isentropic Flow [if dQ=0, Adiabatic Gas Law]:

• Precise gas tables available for design work• Thermally Perfect Gas good flows at moderate

temperature.

1 /1 /2 2

1 1

1 /0 0

T P or T CPT P

also

T PT P

25

Common Gases

Gas

Argon 1.67

Helium 1.67

Air 1.40

Hydrogen 1.40

Nitrogen 1.40

Oxygen 1.39

Water vapor 1.33

Carbon dioxide 1.29

Sulfur dioxide 1.29

Butane 1.10

monatomic

diatomic

polyatomic

26

Important Constants for Air

2 2/ 8314.3 / 28.97 287 /

53.35 / 0.24 /1

1716 / 7.73 /1

287 / 1004.5 /1

air

p air

p air

p air

R M m s K

RR ft lb lbm R c Btu lbm R

RR ft lbf slug R c Btu lbf R

RR J kg K c J kg K

27

Gibbs Equation• Rewriting Gibbs Equation:

28

Gibbs Equation• Rewriting Gibbs Equation:

02 022 1

01 01

0

022 1

01

02 2 1

01

1ln ln

,

1 ln

exp 1

p

p

Apply at stagnation state

T Ps sc T P

For adiabatic processes T constant

Ps sc P

P s sP R

29

Mollier Chart for Air

500

1,000

1,500

2,000

2,500

3,000

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

Entropy - BTU/Lbm/deg R

Tem

pera

ture

Deg

R

P=50Atm

20

10

5

2

1

Isobars are not parallel

30

Mollier for Static / Total States

450

650

850

1,050

1,250

1,450

1,650

-0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06

S

T

IdealReal

P in

P out

s

Poin

Poout

V2/2

h02i

h02

h01

2

0 2Vh h

We will soon see