Thermodynamics - the formulae

I don't know how useful this may be but I have summarised the formula I found in my son's lecture notes on thermodynamics:

This is a very brief summary of the definitions and formulae for Thermodynamics, Emphasis has been laid on two things, consistent use of symbols for every variable and repeated definition of the SI units of measurement (Pa, kJ, kg, sec, m, mol)

1 The definition of systems

A closed system = flexible + constant mass
An open system = fixed boundary + variable mass (or moving mass)

2 Main new parameters

u = Internal Energy = temperature energy + latent energy + chemical energy + nuclear energy (kJ/kg)
h = Enthalpy = u + pressure energy P*V (kJ/kg)
s = Entropy = thermal disorder (kJ/kg)

3 The dryness factor for mixtures of liquid and vapour phases

x (0-1 or %) = mv/mt = Mass Vapour / Mass total

4 The gas law

A very fundamental relating Pressure and Volume of a gas to constants and temperature

PV = nRoT

P = Pressure (Pa)
V = Volume (m3)
n = Amount of substance (moles)
Ro = Universal gas constant 8.3144472 (J/kmol)
T = Temperature (K)

The Universal gas constant is the same for all gasses, it is related to the Specific gas constant like this

Ro = RM

M = Molar mass, or mass of one mole of the substance
R = Sp gas constant (kJ/kgK)

So, substituting

PV = nRMT

and this related to the mass by

nM = m

m = mass (kg)

So, in terms of mass

PV = mRT

It is useful also to relate the gas equation to the density

d = m/V

d = Density (kg/m3)

So

d = P/RT

5 Processes can be

Isochromic P/T = constant, P1/T1 = P2/T2
Isobaric V/T = constant, V1/T1 = V2/T2
Isothermal PV = constant P1V1 = P2V2
Polytropic P1V1n = P2V2n...

6 1st Law of Thermodynamics, two cases

Steady Flow for constant volume and constant pressure systems,

Q - W = m[(u2 - u1) + (P2V2 - P1V1) + (v2^2 - v1^2)/2000) + (g(z2 - z1)/1000)]

Q - W = m[(h2 - h1) + (P2V2 - P1V1) + (v2^2 - v1^2)/2000) + (g(z2 - z1)/1000)]

Energy - Work Done = Internal energy + pressure energy + kinetic energy + potential energy

Q = Heat (kJ)
W = Work (kJ)
v = Velocity (m/s)
g = Acceleration of gravity = 9.81 m/s2
z = height (m)

The third and forth terms are divided by 2000 & 1000 to get the units to kJ/kg

Non-Flow

Q - W = m(u2 - u1)

7 Specific Heat C

Constant Volume u2 - u1 = Cv(T2 - T1)
Constant Pressure h2 - h1 = Cp(T2 - T1)

Cv = Specific heat constant volume
Cp = Specific heat constant pressure

Where

Cp - Cv = R

Cv and Cp are used like this

u = Cv * T (kJ/kg)
h = Cp*T + PV (kJ/kg)

8 Conduction

In a solid

Q = kA(T1 - T2)/y,

k = conductivity (W/m2K)
y = thickness
A = area

For a Solid-Fluid interface

Q = hA(T1 - T2)

h = heat transfer coefficient (W/m2K)

9 Radiation of a body (power output)

E = εσA (T1 - T2)^4

E = Energy flux (J/s/m2 or W/m2)
ε = emissivity (black body = 1), but emissivity depends on wavelength
σ = Stefan-Boltzman constant 5.76 x 10^8 (W/m2K)
A = Area (m2)

Take care that the result J is in Joules, not the SI unit of kJ

10 2nd Law of Thermodynamics

Entropy increase dS = ΔQ/T = heat supplied / temperature

For gasses at

Constant volume

S2 - S1 = Cv ln(T2/T1) + R ln (V2/V1)

Constant pressure

S2 - S1 = Cp ln(T2/T1) + R ln(P2/P1)

and

S2 - S1 = Cp ln(V2/V2) + R ln(P2/P1)