Adiabatic: ฮ๐=ฮ๐ =0 Isentropic: ฮ๐ =0
Unit Conversions
- Temperature
๐(โ)=๐(๐พ)โ273.15
๐(โ)=1.8โ๐(โ)+32
T(ยบR) = T(ยบF) + 459.67
- Pressure
1 bar = 105 Pa = 100 kPa
Patm = 101 kPa = 14.7
lbf/in^2
6895 Pa = 1 psi
- Energy + Power
1 Btu = 778 ft*lbf
1 hp = 550 ft*lbf/s = 2545
Btu/hr
N = kg*m/s^2
- Volume
1 m = 3.28084 ft
- Fluids
๐๐ป2๐ = 1000 kg/m^3 =
1.94 slug/ft^3
๐พ๐ป2๐ = 9800 N/m^3 = 62.4
lbf/ft^3
๐พ๐ป๐ = 133,200 N/m^3
- Mass
Slug = 32.174 lbmass =
lbforce*s^2/ft
1st Law
โ๐ธ=โ๐พ๐ธ+โ๐๐ธ+โ๐= ๐2
1โ ๐2
1
Rate Form: ๐ธ๓ฐ=๐๓ฐโ๐๓ฐ
Enthalpy
๐ป=๐+๐โ๐
โ=๐ข+๐โ๐ฃ
Mixtures
๐=๐๐ฃ๐๐
๐=๐ฅโ๐ฅ๐
๐ฅ๐โ๐ฅ๐ (๐ฅ=๐ฃ,๐ข,โ)
๐ฝ=(1โ๐)๐ฝ๐+๐โ๐ฝ๐
Ideal Gas
๐๐ฃ๎ชง=๐
๏ค๐ , ๐
๏ค is universal
๐๐ฃ=๐
๐ , where ๐
= ๐
๏ค
๐๐ค
๐=๐๐
๐
๐๐=๐๐
๐
๐
๏ค=
{
8.314 ๐๐ฝ
๐พ๐๐๐โ๐พ
1.986 ๐ต๐ก๐ข
๐๐๐๐๐โ ยฐ๐
1545 ๐๐กโ๐๐๐
๐๐๐๐๐โยฐ๐
Polytropic Ideal Gas
๐2
๐1=(๐2
๐1)๐โ1
๐=(๐1
๐2)๐โ1
๐2={๐๐
(๐2โ๐1)
(1โ๐) , ๐โ 1
๐๐
๐ln(๐2
๐1), ๐=1
1
Polytropic Process
๐โ๐๐=๐ ๐๐ ๐โ๐ฃ๐=๐
- Volume expansion work
๐2={(๐2๐2โ๐1๐1)
(1โ๐) , ๐โ 1
๐1๐1ln(๐2
๐1), ๐=1
1
Mass Conservation
๐๐๐๐ ๐ ๐๐ฃ
๐๐ก =โ๐๐
๓ฐ โโ๐๐
๓ฐ
๐๓ฐ =๐๐ด|๐|=๐ด|๐|
๐ฃ
Volumetric flow rate = ๐ด|๐|
Flow work = (๐โ๐ด)โ|๐|
Compressed Liquid Approx.
๐ฅ(๐,๐)โ๐ฅ๐(๐) , x = (v,u,h,s)
Velocity^2 to kJ/kg
๐2
๐ 2โ๐โ๐ 2
๐๐โ๐โ๐ฝ
๐โ๐โ๐๐ฝ
103 ๐ฝ
Specific Heat (heat capacity)
C used when incompressible,
otherwise Cv and Cp
โ๐ข=โซ ๐ถ๐ฃ(๐) ๐๐
๐2
๐1
โโ=โซ ๐ถ๐(๐) ๐๐
๐2
๐1+๐ฃ(๐2โ๐1)
Constant Cv approximation
๐ถ๐ฃโ๐ถ๐ฃ(๐1)+๐ถ๐ฃ(๐2)
2
๐ถ๐=๐ถ๐ฃ+๐
Energy Conservation
๐๐ธ๐๐ฃ
๐๐ก =๐๐๐ฃ
๓ฐโ๐๐๐ฃ
๓ฐ+โ๐๐
๓ฐ (โ๐+1
2|๐๐|2+๐โ๐ง๐)
๐โโ๐๐
๓ฐ (โ๐+1
2|๐๐|2+๐โ๐ง๐)
๐
1 Inlet, 1 Exit: 0=๐๐๐ฃ
๓ฐโ๐๐๐ฃ
๓ฐ+๐๓ฐ [(โ๐โโ๐)+1
2(|๐๐|2โ|๐๐|2)+๐(๐ง๐โ๐ง๐)]
Entropy
๐๐=(๐ฟ๐
๐)๐๐๐ฃ , ๐=๐ โ๐
๐2โ๐1=โซ(๐ฟ๐
๐)๐
2
1
๐โ๐๐ =๐๐ข+๐โ๐๐ฃ
๐โ๐๐ =๐โโ๐ฃโ๐๐
Rev: ๐2=โซ๐๐๐
2
1=๐๐(๐ 2โ๐ 1)
1
Entropy and Ideal Gas
ฮ๐ =๐ (๐2,๐2)โ๐ (๐1,๐1)=โซ๐ถ๐ฃ(๐)
๐๐๐ก+๐
โln
๐2
๐1(๐ฃ2
๐ฃ1 )
ฮ๐ =๐ (๐2,๐2)โ๐ (๐1,๐1)=โซ๐ถ๐(๐)
๐๐๐กโ๐
โln
๐2
๐1(๐2
๐1 )
ฮ๐ =๐ ยฐ(๐2)โ๐ ยฐ(๐1)โ๐
โln(๐2
๐1)
Entropy Balance (Cycle)
๐๐๐๐ฃ
๐๐ก =โ๐๓ฐ
๐+โ๐๓ฐ๐โ๐ ๐โโ๐๐
๓ฐ โ๐ ๐+๐๓ฐ๐๐ฃ
1 in, 1 out: 0=โ๐๓ฐ
๐+๐๓ฐ(๐ 1โ๐ 2)+๐๓ฐ๐๐ฃ
ฮ๐ =๐ 2โ๐ 1=1
๐๓ฐ(โ๐๓ฐ
๐+๐๓ฐ๐๐ฃ)
Entropy Balance (Process)
๐2โ๐1=โซ(๐ฟ๐
๐)๐
2
1+๐
๐2โ๐1=โ๐
๐+๐
๐๐
๐๐ก=โ๐๓ฐ
๐+๐๓ฐ
2nd Law and Entropy
โฎ(๐ฟ๐
๐)๐โค0
๐๐ฆ๐๐๐
S production: ๐=โโฎ(๐ฟ๐
๐)๐โฅ0
๐>0 : irreversible cycle
๐=0 : reversible cycle
๐<0 : impossible
Isentropic Efficiency
Turbine: ๐๐ก=๐๓ฐ๐ก๐๓ฐ
โ
(๐๓ฐ๐ก๐๓ฐ
โ)๐ =โ1โโ2
โ1โโ2๐ โค1
Nozzle: ๐๐=1
2|๐2|2
(1
2|๐2|2)๐ โค1
Compressor/Pump: ๐๐/๐=(๐
๓ฐ๐ก๐๓ฐ)
โ๐
(๐๓ฐ๐ก๐๓ฐ
โ)๐โค1
2nd Law and Cycle COPโs
๐ถ๐๐๐๐๐๐๐ฃ<๐ถ๐๐๐๐๐ฃ=๐ถ๐๐๐๐๐ฅ
๐ถ๐๐๐๐๐ฃ,1=๐ถ๐๐๐๐๐ฃ,2
Power: ๐๐๐๐ฃ=๐๓ฐ
๐๓ฐ๐ป=๐๐ปโ๐๐ถ
๐๐ป=1โ๐๐
๐โ
Refrig: ๐ฝ๐๐๐ฃ=๐๓ฐ๐๐
๐๓ฐ=๐๓ฐ๐๐
๐๓ฐ๐๐ข๐กโ๐๓ฐ๐๐ =๐๐ถ
๐๐ปโ๐๐ถ
Heat Pump: ๐พ๐๐๐ฃ=๐๓ฐ๐๐ข๐ก
๐๓ฐ=๐๓ฐ๐๐ข๐ก
๐๓ฐ๐๐ข๐กโ๐๓ฐ๐๐ =๐๐ป
๐๐ปโ๐๐ถ
Real gas
๐ง=๐๐
๐
๐ , where z =
compressibility factor
Work and Power
๐=โซ๐นโ๐๐
- Volume expansion work:
๐=โซ๐ ๐๐
๐=๐โโซ๐ ๐๐ฃ
๐๐๐ค๐๐=๐๐
๐๐ก
Energy
๐พ๐ธ=1
2๐๐ฃ2 ๐๐ธ=๐๐๐ง
1st Law and Cycles
Power: ๐๐ปโ๐๐ถ=๐๐๐ฆ๐๐๐
R+HP: ๐๐๐ข๐กโ๐๐๐ =๐๐๐ฆ๐๐๐
