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Thermo final cheat sheet, Cheat Sheet of Thermodynamics

Cheat Sheet

Typology: Cheat Sheet

2014/2015

Uploaded on 09/22/2015

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Adiabatic: ฮ”๐‘„=ฮ”๐‘ =0 Isentropic: ฮ”๐‘ =0
Unit Conversions
- Temperature
๐‘‡(โ„ƒ)=๐‘‡(๐พ)โˆ’273.15
๐‘‡(โ„‰)=1.8โˆ—๐‘‡(โ„ƒ)+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
๐œŒ๐ป2๐‘‚ = 1000 kg/m^3 =
1.94 slug/ft^3
๐›พ๐ป2๐‘‚ = 9800 N/m^3 = 62.4
lbf/ft^3
๐›พ๐ป๐‘” = 133,200 N/m^3
- Mass
Slug = 32.174 lbmass =
lbforce*s^2/ft
1st Law
โˆ†๐ธ=โˆ†๐พ๐ธ+โˆ†๐‘ƒ๐ธ+โˆ†๐‘ˆ= ๐‘„2
1โˆ’ ๐‘Š2
1
Rate Form: ๐ธ๓ฐ‡—=๐‘„๓ฐ‡—โˆ’๐‘Š๓ฐ‡—
Enthalpy
๐ป=๐‘ˆ+๐‘ƒโˆ—๐‘‰
โ„Ž=๐‘ข+๐‘ƒโˆ—๐‘ฃ
Mixtures
๐‘‹=๐‘€๐‘ฃ๐‘Ž๐‘
๐‘€=๐‘ฅโˆ’๐‘ฅ๐‘“
๐‘ฅ๐‘”โˆ’๐‘ฅ๐‘“ (๐‘ฅ=๐‘ฃ,๐‘ข,โ„Ž)
๐›ฝ=(1โˆ’๐‘‹)๐›ฝ๐‘“+๐‘‹โˆ—๐›ฝ๐‘”
Ideal Gas
๐‘ƒ๐‘ฃ๎ชง=๐‘…๏Œค๐‘‡ , ๐‘…๏Œค is universal
๐‘ƒ๐‘ฃ=๐‘…๐‘‡ , where ๐‘…= ๐‘…๏Œค
๐‘€๐‘ค
๐‘ƒ=๐œŒ๐‘…๐‘‡
๐‘ƒ๐‘‰=๐‘š๐‘…๐‘‡
๐‘…๏Œค=
{
8.314 ๐‘˜๐ฝ
๐พ๐‘š๐‘œ๐‘™โˆ—๐พ
1.986 ๐ต๐‘ก๐‘ข
๐‘™๐‘๐‘š๐‘œ๐‘™โˆ— ยฐ๐‘…
1545 ๐‘“๐‘กโˆ—๐‘™๐‘๐‘“
๐‘™๐‘๐‘š๐‘œ๐‘™โˆ—ยฐ๐‘…
Polytropic Ideal Gas
๐‘‡2
๐‘‡1=(๐‘ƒ2
๐‘ƒ1)๐‘›โˆ’1
๐‘›=(๐‘‰1
๐‘‰2)๐‘›โˆ’1
๐‘Š2={๐‘š๐‘…(๐‘‡2โˆ’๐‘‡1)
(1โˆ’๐‘›) , ๐‘›โ‰ 1
๐‘š๐‘…๐‘‡ln(๐‘‰2
๐‘‰1), ๐‘›=1
1
Polytropic Process
๐‘ƒโˆ—๐‘‰๐‘›=๐‘ ๐‘œ๐‘Ÿ ๐‘ƒโˆ—๐‘ฃ๐‘›=๐‘
- Volume expansion work
๐‘Š2={(๐‘ƒ2๐‘‰2โˆ’๐‘ƒ1๐‘‰1)
(1โˆ’๐‘›) , ๐‘›โ‰ 1
๐‘ƒ1๐‘‰1ln(๐‘‰2
๐‘‰1), ๐‘›=1
1
Mass Conservation
๐‘‘๐‘€๐‘Ž๐‘ ๐‘ ๐‘๐‘ฃ
๐‘‘๐‘ก =โˆ‘๐‘š๐‘–
๓ฐ‡— โˆ’โˆ‘๐‘š๐‘’
๓ฐ‡—
๐‘š๓ฐ‡— =๐œŒ๐ด|๐‘‰|=๐ด|๐‘‰|
๐‘ฃ
Volumetric flow rate = ๐ด|๐‘‰|
Flow work = (๐‘ƒโˆ—๐ด)โˆ—|๐‘‰|
Compressed Liquid Approx.
๐‘ฅ(๐‘‡,๐‘ƒ)โ‰ˆ๐‘ฅ๐‘“(๐‘‡) , x = (v,u,h,s)
Velocity^2 to kJ/kg
๐‘š2
๐‘ 2โˆ—๐‘โˆ—๐‘ 2
๐‘˜๐‘”โˆ—๐‘šโˆ—๐ฝ
๐‘โˆ—๐‘šโˆ—๐‘˜๐ฝ
103 ๐ฝ
Energy Conservation
๐‘‘๐ธ๐‘๐‘ฃ
๐‘‘๐‘ก =๐‘„๐‘๐‘ฃ
๓ฐ‡—โˆ’๐‘Š๐‘๐‘ฃ
๓ฐ‡—+โˆ‘๐‘š๐‘–
๓ฐ‡— (โ„Ž๐‘–+1
2|๐‘‰๐‘–|2+๐‘”โˆ—๐‘ง๐‘–)
๐‘–โˆ’โˆ‘๐‘š๐‘’
๓ฐ‡— (โ„Ž๐‘’+1
2|๐‘‰๐‘’|2+๐‘”โˆ—๐‘ง๐‘’)
๐‘’
1 Inlet, 1 Exit: 0=๐‘„๐‘๐‘ฃ
๓ฐ‡—โˆ’๐‘Š๐‘๐‘ฃ
๓ฐ‡—+๐‘š๓ฐ‡— [(โ„Ž๐‘–โˆ’โ„Ž๐‘’)+1
2(|๐‘‰๐‘–|2โˆ’|๐‘‰๐‘’|2)+๐‘”(๐‘ง๐‘–โˆ’๐‘ง๐‘’)]
Entropy
๐‘‘๐‘†=(๐›ฟ๐‘„
๐‘‡)๐‘Ÿ๐‘’๐‘ฃ , ๐‘†=๐‘ โˆ—๐‘š
๐‘†2โˆ’๐‘†1=โˆซ(๐›ฟ๐‘„
๐‘‡)๐‘
2
1
๐‘‡โˆ—๐‘‘๐‘ =๐‘‘๐‘ข+๐‘ƒโˆ—๐‘‘๐‘ฃ
๐‘‡โˆ—๐‘‘๐‘ =๐‘‘โ„Žโˆ’๐‘ฃโˆ—๐‘‘๐‘ƒ
Rev: ๐‘„2=โˆซ๐‘‡๐‘‘๐‘ 
2
1=๐‘š๐‘‡(๐‘ 2โˆ’๐‘ 1)
1
Entropy and Ideal Gas
ฮ”๐‘ =๐‘ (๐‘‡2,๐‘ƒ2)โˆ’๐‘ (๐‘‡1,๐‘ƒ1)=โˆซ๐ถ๐‘ฃ(๐‘‡)
๐‘‡๐‘‘๐‘ก+๐‘…โˆ—ln
๐‘‡2
๐‘‡1(๐‘ฃ2
๐‘ฃ1 )
ฮ”๐‘ =๐‘ (๐‘‡2,๐‘ƒ2)โˆ’๐‘ (๐‘‡1,๐‘ƒ1)=โˆซ๐ถ๐‘(๐‘‡)
๐‘‡๐‘‘๐‘กโˆ’๐‘…โˆ—ln
๐‘‡2
๐‘‡1(๐‘ƒ2
๐‘ƒ1 )
ฮ”๐‘ =๐‘ ยฐ(๐‘‡2)โˆ’๐‘ ยฐ(๐‘‡1)โˆ’๐‘…โˆ—ln(๐‘ƒ2
๐‘ƒ1)
Entropy Balance (Cycle)
๐‘‘๐‘†๐‘๐‘ฃ
๐‘‘๐‘ก =โˆ‘๐‘„๓ฐ‡—
๐‘‡+โˆ‘๐‘š๓ฐ‡—๐‘–โˆ—๐‘ ๐‘–โˆ’โˆ‘๐‘š๐‘’
๓ฐ‡— โˆ—๐‘ ๐‘’+๐œŽ๓ฐ‡—๐‘๐‘ฃ
1 in, 1 out: 0=โˆ‘๐‘„๓ฐ‡—
๐‘‡+๐‘š๓ฐ‡—(๐‘ 1โˆ’๐‘ 2)+๐œŽ๓ฐ‡—๐‘๐‘ฃ
ฮ”๐‘ =๐‘ 2โˆ’๐‘ 1=1
๐‘š๓ฐ‡—(โˆ‘๐‘„๓ฐ‡—
๐‘‡+๐œŽ๓ฐ‡—๐‘๐‘ฃ)
Entropy Balance (Process)
๐‘†2โˆ’๐‘†1=โˆซ(๐›ฟ๐‘„
๐‘‡)๐‘
2
1+๐œŽ
๐‘†2โˆ’๐‘†1=โˆ‘๐‘„
๐‘‡+๐œŽ
๐‘‘๐‘†
๐‘‘๐‘ก=โˆ‘๐‘„๓ฐ‡—
๐‘‡+๐œŽ๓ฐ‡—
2nd Law and Entropy
โˆฎ(๐›ฟ๐‘„
๐‘‡)๐‘โ‰ค0
๐‘๐‘ฆ๐‘๐‘™๐‘’
S production: ๐œŽ=โˆ’โˆฎ(๐›ฟ๐‘„
๐‘‡)๐‘โ‰ฅ0
๐œŽ>0 : irreversible cycle
๐œŽ=0 : reversible cycle
๐œŽ<0 : impossible
Isentropic Efficiency
Turbine: ๐œ‚๐‘ก=๐‘Š๓ฐ‡—๐‘ก๐‘š๓ฐ‡—
โ„
(๐‘Š๓ฐ‡—๐‘ก๐‘š๓ฐ‡—
โ„)๐‘ =โ„Ž1โˆ’โ„Ž2
โ„Ž1โˆ’โ„Ž2๐‘  โ‰ค1
Nozzle: ๐œ‚๐‘›=1
2|๐‘‰2|2
(1
2|๐‘‰2|2)๐‘ โ‰ค1
Compressor/Pump: ๐œ‚๐‘/๐‘=(๐‘Š
๓ฐ‡—๐‘ก๐‘š๓ฐ‡—)
โ„๐‘ 
(๐‘Š๓ฐ‡—๐‘ก๐‘š๓ฐ‡—
โ„)๐‘Žโ‰ค1
2nd Law and Cycle COPโ€™s
๐ถ๐‘‚๐‘ƒ๐‘–๐‘Ÿ๐‘Ÿ๐‘’๐‘ฃ<๐ถ๐‘‚๐‘ƒ๐‘Ÿ๐‘’๐‘ฃ=๐ถ๐‘‚๐‘ƒ๐‘š๐‘Ž๐‘ฅ
๐ถ๐‘‚๐‘ƒ๐‘Ÿ๐‘’๐‘ฃ,1=๐ถ๐‘‚๐‘ƒ๐‘Ÿ๐‘’๐‘ฃ,2
Power: ๐œ‚๐‘Ÿ๐‘’๐‘ฃ=๐‘Š๓ฐ‡—
๐‘„๓ฐ‡—๐ป=๐‘„๐ปโˆ’๐‘„๐ถ
๐‘„๐ป=1โˆ’๐‘‡๐‘
๐‘‡โ„Ž
Refrig: ๐›ฝ๐‘Ÿ๐‘’๐‘ฃ=๐‘„๓ฐ‡—๐‘–๐‘›
๐‘Š๓ฐ‡—=๐‘„๓ฐ‡—๐‘–๐‘›
๐‘„๓ฐ‡—๐‘œ๐‘ข๐‘กโˆ’๐‘„๓ฐ‡—๐‘–๐‘› =๐‘‡๐ถ
๐‘‡๐ปโˆ’๐‘‡๐ถ
Heat Pump: ๐›พ๐‘Ÿ๐‘’๐‘ฃ=๐‘„๓ฐ‡—๐‘œ๐‘ข๐‘ก
๐‘Š๓ฐ‡—=๐‘„๓ฐ‡—๐‘œ๐‘ข๐‘ก
๐‘„๓ฐ‡—๐‘œ๐‘ข๐‘กโˆ’๐‘„๓ฐ‡—๐‘–๐‘› =๐‘‡๐ป
๐‘‡๐ปโˆ’๐‘‡๐ถ
Real gas
๐‘ง=๐‘ƒ๐‘‰
๐‘…๐‘‡ , where z =
compressibility factor
Work and Power
๐‘Š=โˆซ๐นโˆ™๐‘‘๐‘Ÿ
- Volume expansion work:
๐‘Š=โˆซ๐‘ƒ ๐‘‘๐‘‰
๐‘Š=๐‘šโˆ—โˆซ๐‘ƒ ๐‘‘๐‘ฃ
๐‘ƒ๐‘œ๐‘ค๐‘’๐‘Ÿ=๐‘‘๐‘Š
๐‘‘๐‘ก
Energy
๐พ๐ธ=1
2๐‘š๐‘ฃ2 ๐‘ƒ๐ธ=๐‘š๐‘”๐‘ง
1st Law and Cycles
Power: ๐‘„๐ปโˆ’๐‘„๐ถ=๐‘Š๐‘๐‘ฆ๐‘๐‘™๐‘’
R+HP: ๐‘„๐‘œ๐‘ข๐‘กโˆ’๐‘„๐‘–๐‘› =๐‘Š๐‘๐‘ฆ๐‘๐‘™๐‘’
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Unit Conversions Adiabatic : ฮ”๐‘„ = ฮ”๐‘  = 0 Isentropic : ฮ”๐‘  = 0

  • Temperature

T(ยบR) = T(ยบF) + 459.

  • Pressure

1 bar = 10

5

Pa = 100 kPa

P

atm

= 101 kPa = 14.

lbf/in^

6895 Pa = 1 psi

  • Energy + Power

1 Btu = 778 ft*lbf

1 hp = 550 ft*lbf/s = 2545

Btu/hr

N = kg*m/s^

  • Volume

1 m = 3.28084 ft

  • Fluids

๐ป 2

๐‘‚

= 1000 kg/m^3 =

1.94 slug/ft^

๐ป 2

๐‘‚

= 9800 N/m^3 = 62.

lbf/ft^

๐ป๐‘”

= 133,200 N/m^ 3

  • Mass

Slug = 32.174 lb mass

lb force

*s^2/ft

st

Law

1 2

1 2

Rate Form: ๐ธ

Enthalpy

Mixtures

๐‘ฃ๐‘Ž๐‘

๐‘“

๐‘”

๐‘“

๐‘“

๐‘”

Ideal Gas

is universal

๐‘ƒ๐‘ฃ = ๐‘…๐‘‡ , where ๐‘… =

๐‘…

ฬ…

๐‘€ ๐‘ค

๐‘˜๐ฝ

๐พ๐‘š๐‘œ๐‘™โˆ—๐พ

๐ต๐‘ก๐‘ข

๐‘™๐‘๐‘š๐‘œ๐‘™โˆ— ยฐ๐‘…

๐‘“๐‘กโˆ—๐‘™๐‘๐‘“

๐‘™๐‘๐‘š๐‘œ๐‘™โˆ—ยฐ๐‘…

Polytropic Ideal Gas

2

1

2

1

๐‘›โˆ’ 1

๐‘›

1

2

๐‘›โˆ’ 1

2

๐‘š๐‘…

( ๐‘‡ 2

โˆ’๐‘‡ 1

)

( 1 โˆ’๐‘›)

๐‘š๐‘…๐‘‡ ln (

๐‘‰ 2

๐‘‰ 1

1

Polytropic Process

๐‘›

๐‘›

  • Volume expansion work

2

(๐‘ƒ

2

๐‘‰

2

โˆ’๐‘ƒ

1

๐‘‰

1

)

( 1 โˆ’๐‘›)

1

1

ln (

๐‘‰

2

๐‘‰

1

1

Mass Conservation

๐‘‘๐‘€๐‘Ž๐‘ ๐‘ 

๐‘๐‘ฃ

๐‘‘๐‘ก

๐‘–

๐‘’

๐ด|๐‘‰|

๐‘ฃ

Volumetric flow rate = ๐ด|๐‘‰|

Flow work = (๐‘ƒ โˆ— ๐ด) โˆ— |๐‘‰|

Compressed Liquid Approx.

๐‘“

, x = (v,u,h,s)

Velocity^2 to kJ/kg

๐‘š

2

๐‘ 

2

๐‘โˆ—๐‘ 

2

๐‘˜๐‘”โˆ—๐‘š

๐ฝ

๐‘โˆ—๐‘š

๐‘˜๐ฝ

10

3

๐ฝ

Specific Heat (heat capacity)

C used when incompressible,

otherwise C v

and C p

๐‘ฃ

๐‘‡ 2

๐‘‡ 1

๐‘

๐‘‡

2

๐‘‡

1

2

1

Constant C v

approximation

๐‘ฃ

๐‘ฃ

1

๐‘ฃ

2

๐‘

๐‘ฃ

Energy Conservation

๐‘‘๐ธ

๐‘๐‘ฃ

๐‘‘๐‘ก

๐‘๐‘ฃ

๐‘๐‘ฃ

๐‘–

๐‘–

1

2

๐‘–

2

๐‘–

๐‘–

๐‘’

๐‘’

1

2

๐‘’

2

๐‘’

๐‘’

1 Inlet, 1 Exit: 0 = ๐‘„

๐‘๐‘ฃ

๐‘๐‘ฃ

+ ๐‘šฬ‡ [(โ„Ž

๐‘–

๐‘’

1

2

๐‘–

2

๐‘’

2

๐‘–

๐‘’

)]

Entropy

๐›ฟ๐‘„

๐‘‡

๐‘Ÿ๐‘’๐‘ฃ

2

1

๐›ฟ๐‘„

๐‘‡

๐‘

2

1

Rev: ๐‘„

2

2

1

2

1

1

Entropy and Ideal Gas

2

2

1

1

๐ถ

๐‘ฃ

(๐‘‡)

๐‘‡

๐‘‘๐‘ก + ๐‘… โˆ— ln

๐‘‡

2

๐‘‡ 1

๐‘ฃ

2

๐‘ฃ

1

2

2

1

1

๐ถ

๐‘

(๐‘‡)

๐‘‡

๐‘‘๐‘ก โˆ’ ๐‘… โˆ— ln

๐‘‡

2

๐‘‡ 1

๐‘ƒ 2

๐‘ƒ 1

2

1

) โˆ’ ๐‘… โˆ— ln (

๐‘ƒ 2

๐‘ƒ 1

Entropy Balance (Cycle)

๐‘‘๐‘†

๐‘๐‘ฃ

๐‘‘๐‘ก

๐‘„

ฬ‡

๐‘‡

๐‘–

๐‘–

๐‘’

๐‘’

๐‘๐‘ฃ

1 in, 1 out: 0 = โˆ‘

๐‘„

ฬ‡

๐‘‡

1

2

๐‘๐‘ฃ

2

1

1

๐‘šฬ‡

๐‘„

ฬ‡

๐‘‡

๐‘๐‘ฃ

Entropy Balance (Process)

2

1

๐›ฟ๐‘„

๐‘‡

๐‘

2

1

2

1

๐‘„

๐‘‡

๐‘‘๐‘†

๐‘‘๐‘ก

๐‘„

ฬ‡

๐‘‡

nd

Law and Entropy

๐›ฟ๐‘„

๐‘‡

๐‘

๐‘๐‘ฆ๐‘๐‘™๐‘’

S production: ๐œŽ = โˆ’

๐›ฟ๐‘„

๐‘‡

๐‘

๐œŽ > 0 : irreversible cycle

๐œŽ = 0 : reversible cycle

๐œŽ < 0 : impossible

Isentropic Efficiency

Turbine: ๐œ‚

๐‘ก

๐‘Š

ฬ‡

๐‘ก

โ„๐‘šฬ‡

(๐‘Š

ฬ‡

๐‘ก

โ„๐‘š ฬ‡)

๐‘ 

โ„Ž

1

โˆ’โ„Ž

2

โ„Ž

1

โˆ’โ„Ž

2 ๐‘ 

Nozzle: ๐œ‚

๐‘›

1

2

|๐‘‰

2

|

2

(

1

2

|๐‘‰

2

|

2

)

๐‘ 

Compressor/Pump: ๐œ‚

๐‘/๐‘

(๐‘Š

ฬ‡

๐‘ก

โ„๐‘š ฬ‡ )

๐‘ 

(๐‘Š

ฬ‡

๐‘ก

โ„๐‘š ฬ‡)

๐‘Ž

nd

Law and Cycle COPโ€™s

๐‘–๐‘Ÿ๐‘Ÿ๐‘’๐‘ฃ

๐‘Ÿ๐‘’๐‘ฃ

๐‘š๐‘Ž๐‘ฅ

๐‘Ÿ๐‘’๐‘ฃ, 1

๐‘Ÿ๐‘’๐‘ฃ, 2

Power: ๐œ‚

๐‘Ÿ๐‘’๐‘ฃ

๐‘Š

ฬ‡

๐‘„

ฬ‡

๐ป

๐‘„ ๐ป

โˆ’๐‘„ ๐ถ

๐‘„ ๐ป

๐‘‡ ๐‘

๐‘‡ โ„Ž

Refrig: ๐›ฝ

๐‘Ÿ๐‘’๐‘ฃ

๐‘„

ฬ‡

๐‘–๐‘›

๐‘Š

ฬ‡

๐‘„

ฬ‡

๐‘–๐‘›

๐‘„

ฬ‡

๐‘œ๐‘ข๐‘ก

โˆ’๐‘„

ฬ‡

๐‘–๐‘›

๐‘‡ ๐ถ

๐‘‡ ๐ป

โˆ’๐‘‡ ๐ถ

Heat Pump: ๐›พ

๐‘Ÿ๐‘’๐‘ฃ

๐‘„

ฬ‡

๐‘œ๐‘ข๐‘ก

๐‘Š

ฬ‡

๐‘„

ฬ‡

๐‘œ๐‘ข๐‘ก

๐‘„

ฬ‡

๐‘œ๐‘ข๐‘ก

โˆ’๐‘„

ฬ‡

๐‘–๐‘›

๐‘‡

๐ป

๐‘‡

๐ป

โˆ’๐‘‡

๐ถ

Real gas

๐‘ƒ๐‘‰

๐‘…๐‘‡

, where z =

compressibility factor

Work and Power

  • Volume expansion work:

๐‘‘๐‘Š

๐‘‘๐‘ก

Energy

1

2

2

st

Law and Cycles

Power: ๐‘„

๐ป

๐ถ

๐‘๐‘ฆ๐‘๐‘™๐‘’

R+HP: ๐‘„

๐‘œ๐‘ข๐‘ก

๐‘–๐‘›

๐‘๐‘ฆ๐‘๐‘™๐‘’

Ideal Rankine Cycle

T:

๐‘Š

ฬ‡

๐‘ก

๐‘š

ฬ‡

1

2

C:

๐‘„

ฬ‡

๐‘œ๐‘ข๐‘ก

๐‘š

ฬ‡

2

3

P:

๐‘Š

ฬ‡

๐‘

๐‘š

ฬ‡

4

3

4

3

3

4

3

B:

๐‘„

ฬ‡

๐‘–๐‘›

๐‘šฬ‡

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‘Š

ฬ‡

๐‘ก

โˆ’๐‘Š

ฬ‡

๐‘

๐‘„

ฬ‡

๐‘–๐‘›

โ„Ž 2

โˆ’โ„Ž 3

โ„Ž 1

โˆ’โ„Ž 4

๐‘Š

ฬ‡

๐‘

๐‘Š

ฬ‡

๐‘ก

โ„Ž 4

โˆ’โ„Ž 3

โ„Ž 1

โˆ’โ„Ž 2

Carnot Refrigeration Cycle

Com:

๐‘Š

ฬ‡

๐‘๐‘œ๐‘š

๐‘šฬ‡

2

1

Con: ๐‘„

๐‘œ๐‘ข๐‘ก

๐ป

3

2

T:

๐‘Š

ฬ‡

๐‘ก

๐‘šฬ‡

3

4

E: ๐‘„

๐‘–๐‘›

๐‘

1

4

๐‘„

ฬ‡

๐‘–๐‘›

๐‘Š

ฬ‡

๐‘๐‘œ๐‘š

โˆ’๐‘Š

ฬ‡

๐‘ก

๐‘„

ฬ‡

๐‘–๐‘›

๐‘„

ฬ‡

๐‘œ๐‘ข๐‘ก

โˆ’ ๐‘„

ฬ‡

๐‘–๐‘›

๐‘‡

๐ถ

๐‘‡

๐ป

โˆ’๐‘‡

๐ถ

Ideal Vapor-Comp Refrigeration

V: โ„Ž

4

3

others same as Carnot

Fluid Statics

h is depth, z is height

0

0

๐‘”๐‘Ž๐‘ข๐‘”๐‘’

๐‘Ž

Pascalโ€™s Principle: same depth in same

fluid = same pressure

Specific Gravity: ๐‘†๐บ =

๐œŒ

๐œŒ ๐ป

2

๐‘‚

|

4โ„ƒ

๐›พ

๐›พ ๐ป

2

๐‘‚

Archimedes (buoyancy): ๐น

๐ต

weight of fluid displaced

Submerged Gate

๐‘…

๐‘

โˆ— ๐ด, h c

= depth of center of gate

๐‘…

๐ผ

๐‘ฅ๐‘

๐‘ฆ

๐‘

โˆ—๐ด

๐‘

y R

= distance from free surface to resultant force

I

xc

= moment of inertia of gate

y c

= 1/2 the length of the gate.

Rectangle: ๐ผ

๐‘ฅ๐‘

1

12

3

Circle: ๐ผ

๐‘ฅ๐‘

๐œ‹

4

4

Newtonโ€™s Law of Viscosity

No slip condition

๐œ = shear stress , ๐œ‡ = viscosity (Pa*s) ,

๐›พฬ‡ = rate of shear strain = speed = du/dy

Non-uniform flow

Solve for u (speed) equation

Volumetric flow rate = |๐‘‰|๐ด =

๐ด

Bernoulli Equation

Along streamlines:

|๐‘‰|

2

2

๐‘ƒ

๐œŒ

Momentum Conservation

B

๐‘ 

๐œ•

๐œ•๐‘ก

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.

๐‘ฃ โˆ™ ๐‘› = |๐‘ฃ||๐‘›| cos(๐œƒ) = |๐‘ฃ|cos(๐œƒ)