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Unit Conversions Adiabatic : ฮ๐ = ฮ๐ = 0 Isentropic : ฮ๐ = 0
1 bar = 10
5
Pa = 100 kPa
atm
= 101 kPa = 14.
lbf/in^
6895 Pa = 1 psi
1 Btu = 778 ft*lbf
1 hp = 550 ft*lbf/s = 2545
Btu/hr
N = kg*m/s^
1 m = 3.28084 ft
๐ป 2
๐
= 1000 kg/m^3 =
1.94 slug/ft^
๐ป 2
๐
= 9800 N/m^3 = 62.
lbf/ft^
๐ป๐
= 133,200 N/m^ 3
Slug = 32.174 lb mass
lb force
*s^2/ft
st
Law
1 2
1 2
Rate Form: ๐ธ
Enthalpy
Mixtures
๐ฃ๐๐
๐
๐
๐
๐
๐
Ideal Gas
is universal
๐๐ฃ = ๐ ๐ , where ๐ =
๐
ฬ
๐ ๐ค
๐๐ฝ
๐พ๐๐๐โ๐พ
๐ต๐ก๐ข
๐๐๐๐๐โ ยฐ๐
๐๐กโ๐๐๐
๐๐๐๐๐โยฐ๐
Polytropic Ideal Gas
2
1
2
1
๐โ 1
๐
1
2
๐โ 1
2
๐๐
( ๐ 2
โ๐ 1
)
( 1 โ๐)
๐๐ ๐ ln (
๐ 2
๐ 1
1
Polytropic Process
๐
๐
2
(๐
2
๐
2
โ๐
1
๐
1
)
( 1 โ๐)
1
1
ln (
๐
2
๐
1
1
Mass Conservation
๐๐๐๐ ๐
๐๐ฃ
๐๐ก
๐
๐
๐ด|๐|
๐ฃ
Volumetric flow rate = ๐ด|๐|
Flow work = (๐ โ ๐ด) โ |๐|
Compressed Liquid Approx.
๐
, x = (v,u,h,s)
Velocity^2 to kJ/kg
๐
2
๐
2
๐โ๐
2
๐๐โ๐
๐ฝ
๐โ๐
๐๐ฝ
10
3
๐ฝ
Specific Heat (heat capacity)
C used when incompressible,
otherwise C v
and C p
๐ฃ
๐ 2
๐ 1
๐
๐
2
๐
1
2
1
Constant C v
approximation
๐ฃ
๐ฃ
1
๐ฃ
2
๐
๐ฃ
Energy Conservation
๐๐ธ
๐๐ฃ
๐๐ก
๐๐ฃ
๐๐ฃ
๐
๐
1
2
๐
2
๐
๐
๐
๐
1
2
๐
2
๐
๐
1 Inlet, 1 Exit: 0 = ๐
๐๐ฃ
๐๐ฃ
๐
๐
1
2
๐
2
๐
2
๐
๐
Entropy
๐ฟ๐
๐
๐๐๐ฃ
2
1
๐ฟ๐
๐
๐
2
1
Rev: ๐
2
2
1
2
1
1
Entropy and Ideal Gas
2
2
1
1
๐ถ
๐ฃ
(๐)
๐
๐๐ก + ๐ โ ln
๐
2
๐ 1
๐ฃ
2
๐ฃ
1
2
2
1
1
๐ถ
๐
(๐)
๐
๐๐ก โ ๐ โ ln
๐
2
๐ 1
๐ 2
๐ 1
2
1
) โ ๐ โ ln (
๐ 2
๐ 1
Entropy Balance (Cycle)
๐๐
๐๐ฃ
๐๐ก
๐
ฬ
๐
๐
๐
๐
๐
๐๐ฃ
1 in, 1 out: 0 = โ
๐
ฬ
๐
1
2
๐๐ฃ
2
1
1
๐ฬ
๐
ฬ
๐
๐๐ฃ
Entropy Balance (Process)
2
1
๐ฟ๐
๐
๐
2
1
2
1
๐
๐
๐๐
๐๐ก
๐
ฬ
๐
nd
๐ฟ๐
๐
๐
๐๐ฆ๐๐๐
๐ฟ๐
๐
๐
Isentropic Efficiency
๐ก
๐
ฬ
๐ก
โ๐ฬ
(๐
ฬ
๐ก
โ๐ ฬ)
๐
โ
1
โโ
2
โ
1
โโ
2 ๐
๐
1
2
|๐
2
|
2
(
1
2
|๐
2
|
2
)
๐
๐/๐
(๐
ฬ
๐ก
โ๐ ฬ )
๐
(๐
ฬ
๐ก
โ๐ ฬ)
๐
nd
Law and Cycle COPโs
๐๐๐๐๐ฃ
๐๐๐ฃ
๐๐๐ฅ
๐๐๐ฃ, 1
๐๐๐ฃ, 2
Power: ๐
๐๐๐ฃ
๐
ฬ
๐
ฬ
๐ป
๐ ๐ป
โ๐ ๐ถ
๐ ๐ป
๐ ๐
๐ โ
Refrig: ๐ฝ
๐๐๐ฃ
๐
ฬ
๐๐
๐
ฬ
๐
ฬ
๐๐
๐
ฬ
๐๐ข๐ก
โ๐
ฬ
๐๐
๐ ๐ถ
๐ ๐ป
โ๐ ๐ถ
Heat Pump: ๐พ
๐๐๐ฃ
๐
ฬ
๐๐ข๐ก
๐
ฬ
๐
ฬ
๐๐ข๐ก
๐
ฬ
๐๐ข๐ก
โ๐
ฬ
๐๐
๐
๐ป
๐
๐ป
โ๐
๐ถ
Real gas
๐๐
๐ ๐
, where z =
compressibility factor
Work and Power
๐๐
๐๐ก
Energy
1
2
2
st
Law and Cycles
Power: ๐
๐ป
๐ถ
๐๐ฆ๐๐๐
๐๐ข๐ก
๐๐
๐๐ฆ๐๐๐
Ideal Rankine Cycle
๐
ฬ
๐ก
๐
ฬ
1
2
๐
ฬ
๐๐ข๐ก
๐
ฬ
2
3
๐
ฬ
๐
๐
ฬ
4
3
4
3
3
4
3
๐
ฬ
๐๐
๐ฬ
1
4
In second P equation above, use v in m
3
/kg and P in
kPa to get h 4
directly in kJ/kg
๐
ฬ
๐ก
โ๐
ฬ
๐
๐
ฬ
๐๐
โ 2
โโ 3
โ 1
โโ 4
๐
ฬ
๐
๐
ฬ
๐ก
โ 4
โโ 3
โ 1
โโ 2
Carnot Refrigeration Cycle
Com:
๐
ฬ
๐๐๐
๐ฬ
2
1
Con: ๐
๐๐ข๐ก
๐ป
3
2
๐
ฬ
๐ก
๐ฬ
3
4
๐๐
๐
1
4
๐
ฬ
๐๐
๐
ฬ
๐๐๐
โ๐
ฬ
๐ก
๐
ฬ
๐๐
๐
ฬ
๐๐ข๐ก
โ ๐
ฬ
๐๐
๐
๐ถ
๐
๐ป
โ๐
๐ถ
4
3
Fluid Statics
h is depth, z is height
0
0
๐๐๐ข๐๐
๐
Pascalโs Principle: same depth in same
fluid = same pressure
Specific Gravity: ๐๐บ =
๐
๐ ๐ป
2
๐
|
4โ
๐พ
๐พ ๐ป
2
๐
Archimedes (buoyancy): ๐น
๐ต
weight of fluid displaced
Submerged Gate
๐
๐
โ ๐ด, h c
= depth of center of gate
๐
๐ผ
๐ฅ๐
๐ฆ
๐
โ๐ด
๐
y R
= distance from free surface to resultant force
xc
= moment of inertia of gate
y c
= 1/2 the length of the gate.
Rectangle: ๐ผ
๐ฅ๐
1
12
3
Circle: ๐ผ
๐ฅ๐
๐
4
4
Newtonโs Law of Viscosity
No slip condition
๐ = shear stress , ๐ = viscosity (Pa*s) ,
๐พฬ = rate of shear strain = speed = du/dy
Non-uniform flow
Solve for u (speed) equation
Volumetric flow rate = |๐|๐ด =
๐ด
Bernoulli Equation
Along streamlines:
|๐|
2
2
๐
๐
Momentum Conservation
B
๐
๐
๐๐ก
๐.๐ .
๐
is external force or F = P*A (P always acts inward)
๐ is unit outward normal with respect to the c.v.
๐ฃ โ ๐ = |๐ฃ||๐| cos(๐) = |๐ฃ|cos(๐)