Topic 2

Thermodynamics

1 Why Thermodynamics?

• To tell us the relationship between the p,pressure, temperature and density of a gas as it moves. • Altogether there are 6 variables we need to describe the state of a gas… • p pressure ()(Pa) •  density (kg/m3), or specific volume v=1/ • T temperature (K) • e internal energy per unit mass ( ) • h enthalpy ( ), he+p/ • s entropy ( ) • To relate these variables we use 1. The Kinetic Theory of Gases, 2. The 1st Law of Thermodynamics, 3. Specific Heats 4. The 2nd Law of Thermodynamics 2 1. Kinetic Theory of Gases

• Assumes • Gas is a collection of molecules in random motion • Molecules bounce of each other like hard spheres • Gives two sets of relations… Neglects forces between molecules

Equation of State

Gas constant R = 287 J/kg/K for air

• Gases that obey these relations are called “Thermally Perfect” 3 Thermally Perfect

• We will always assume gases are thermally perfect in this class, but this isn’t always true…

Ceases to be valid at high densities. Why?

4 2. 1st Law of Thermodynamics “Energy is Conserved” W E  Q  W E Q Change in Amount of work Amount of energy done by pressure heat added and viscous forces Collection of identified Work = gas molecules (System) p V (<0)

Surface area S

5 In terms of enthalpy… dq  de  pdv

6 3. Specific Heats • Specific heat – amount of heat needed to raise the q q temperature of 1kg by 1 Kelvin C   • Two kinds: T T

• Specific heat at constant volume Cv  (q / T )v

• Specific heat at constant pressure C p  (q / T ) p

Pratt and Whitney JT58 Test (Powered SR71)7 dq  de  pdv dq  dh  vdp With the energy equation…

8 de dh Cv  Cp  dT Calorically Perfect Gas dT A gas that is thermally perfect and for which the… • …specific heats are constant with temperature, so…

• We will assume calorically perfect gases in this class, but…

9 4. 2nd Law of Thermodynamics Shock in a Converging Diverging Nozzle

Bourgoing & Benay (2005), ONERA, France Schlieren visualization 10 Sensitive to in‐plane index of ref. gradient Processes and Entropy

• Adiabatic Process ‐ no heat added or removed. • Reversible Process –no dissipative phenomena occur. • Isentropic Process – adiabatic and reversible Constant entropy. Entropy –measure of disorder in the gas molecules. • 2nd Law of Thermo – dissipative processes may only increase entropy, but adding or removing heat may increase or decrease entropy by an amount Q/T • So…

11 The Entropy Equation…

For a reversible process… Tds  dq

1st Law dq  de  pdv

Therma lly de  CvdT perfect gas p  RT / v

12   C / C R  C p  Cv p v …The Entropy Equation

13 Isentropic Relations

 1 s  s T     2 1  log  2  1   C e T    v  1  2  

• If s2 – s1 = 0 then…

14 Summary Calorically All gases Thermally perfect perfect e  e(T ) • p  RT KTG h  h(T ) dq  de  pdv st • 1 Law dq  dh  vdp C  de / dT e  C T • Specific Cv  (e / T )v v v

heats Cp  (h / T ) p Cp  dh / dT h  CpT

  1 nd s2  s1 T2  1  • 2 Law  loge     C T    v  1  2  

nd 1  • 2 Law  1  1 2  T2  p2  T2  Isentropic       1  T1  p1  T1  15