Thermodynamics

Thermodynamics Notes

1. Nomenclature
2. Polytropic process
3. Equation of State
4. Relationships between P,V,T
5. Adiabatic process
6. Isothermal Process
7. Internal Energy
8. Constant Volume Process
9. Constant Pressure Process

Introduction

This page provides a limited notes on thermodynamic relationships useful to mechanical engineers.

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Nomenclature

Identifier	Description	Units (typical)
c p	Specific Heat Capacity at Constant pressure	kJ/(kg K)
c v	Specific Heat Capacity at Constant Volume	kJ/(kg K)
P	Absolute Pressure	N / m 2
T	Absolute Temperature	K
V	volume	m 3
m	mass	kg
W	Work Output per unit mass	kJ/kg
M	Molecular Weight	-
R o	Universal Gas Constant = 8,31	kJ /(kg mole.K)
Q	Heat Quantity	kJ
R	Gas Constant = R o / M	kJ /kg.K
U	Internal energy (thermal)	kJ
γ	Ratio cp / cv	-

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Thermodynamic Process Relationships

General Polytropic Process

The majority of frictionless processes for ideal gases are called polytropic processes and are in accordance with the following relationship

PV ⁿ = constant
That is PV ⁿ = c
therefore P = cV ^-n

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Equation of state for and Ideal Gas

PV = mRT

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Thermodynamic Relationships between P,V & T

Consider a piston in a frictionless cylinder

The work done on/by the gas in moving the piston δx = (PA)δx = P δ V = δ W

The gas is assumed to be expanding in balanced resisted reversible process.
The equation of state for an ideal gas is assumed to apply i.e PV = mRT
The total work done in moving the piston from state 1 to state 2 =

For a perfect gas - The relationship between Temperature , Pressure and Volume over a cycle

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Adiabatic Process.

For an adiabatic process with no transfer of heat across the system boundary.(Q = 0 )

Consider a fixed mass of gas in a cylinder which is expanding in a reversible manner...

For an adiabatic process there is no heat transfer.
Therefore applying the first law of thermodynamics ..heat transfer (= 0) = increase in internal energy +external work done by gas...

Therefore the increase in internal energy = - External work done by gas

It is shown below that c_p - c_v = R = c_v ( c_p / c_v -1) and therefore

γ = 1.4 for Air,  H ₂,  O ₂, CO, NO, Hcl

γ = 1.3 for CO ₂, SO ₂,  H ₂O, H ₂S, N ₂O, NH ₃, CL ₂,  CH ₄, C ₂H ₂, C ₂H ₄

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Isothermal Process

In a isothermal process the temperature = constant and therefore

PV = c and P = c / V

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Internal Energy, C_p and C_v

Although it is not possible to determine the absolute value of the internal energy of a substance.  The internal energy change between the initial and final equilibrium states of any process is definite and determinable.

It can be easily proved that the internal energy of a fluid depends on the temperature alone and not upon changes in the pressure or volume.

Heating at constant volume....

If a definite mass of gas (m) at constant volume is a closed system is heated from initial conditions P₁, V, T₁, U₁ to P₂, V , T₂,U₂.   As the volume is fixed then no work has been done.    Then in accordance with the First Law of Thermodynamics (δQ = δU + δW ).

mC_v (T₂ - T₁) = (U₂ - U₁) + 0
or U₂ - U₂ = mC_v (T₂ - T₁)

Heating at constant pressure....

If a definite mass of gas (m) at constant volume is a closed system is heated from initial conditions P, V₁, T₁, U₁ to P, V₁ , T₂,U₂.   As the volume is fixed then no work has been done.    Then in accordance with the First Law of Thermodynamics (δQ = δU + δW ).

mc_p (T₂ - T₁)
= (U₂ - U₁) + P (V₂ - V₁)
= (U₂ - U₁) + mR (T₂ - T₁)

mc_v (T₂ - T₁) = U₂ - U₁ therefore mc_p (T₂ - T₁) = mc_v (T₂ - T₁) + mR (T₂ - T₁) therefore

c_p = c_v + R... and ..
c_p - c_v = R = PV/mT

Thermodynamic Links

1. Thermodynamics..NASA - Glenn Research center at Series of informative notes on Thermodynamics
2. Thermodynaic properties,property relationships and processes..A very detailed clear study of the subject, (3,3 Mbyte download)