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R.M.K COLLEGE OF ENGINEERING
AND TECHNOLOGY
RSM NAGAR, PUDUVOYAL-601206
DEPARTMENT OF MECHANICAL ENGINEERING
CE6451 – FLUID MECHANICS & MACHINERY
III SEM MECHANICAL ENGINEERING
Regulation 2013
FORMULA BOOK
PREPARED BY
C.BIBIN / R.ASHOK KUMAR / N.SADASIVAN
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 2
PROPERTIES OF FLUID:
MASS DESNITY (ρ):
𝜌 =
𝑚
𝑉
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
m Mass Kg
V Volume m3
SPECIFIC VOLUME (v):
𝑣 =
𝑉
𝑚
=
1
𝜌
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
m Mass Kg
V Volume m3
𝑣 Specific Volume 𝑚3
𝑘𝑔⁄
SPECIFIC WEIGTH or WEIGTH DENSITY (w):
𝑤 =
𝑊
𝑉
=
𝑚𝑔
𝑉
= 𝜌𝑔
𝑆𝑖𝑛𝑐𝑒 𝑊 = 𝑚𝑔 𝑎𝑛𝑑 𝜌 = 𝑚
𝑉⁄
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
m Mass Kg
V Volume m3
UNIT – I – FLUID PROPERTIES AND FLOW
CHARACTERISTICS
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 3
𝑤 Specific Weight 𝑁
𝑚3⁄
g Acceleration due to gravity 𝑚
𝑠2⁄
SPECIFIC GRAVITY (S):
𝑆 =
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑢𝑖𝑑
𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑙𝑢𝑖𝑑
𝑆 =
𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑢𝑖𝑑
𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑙𝑢𝑖𝑑
Symbol Description Unit
𝑆 Specific Gravity No unit
𝜌 Density or Mass Density 𝑘𝑔
𝑚3⁄
𝑤 Specific Weight 𝑁
𝑚3⁄
𝑤 𝑤𝑎𝑡𝑒𝑟
Specific Weight of
Standard Fluid (Water) =
9.81
𝑁
𝑚3⁄
𝜌 𝑤𝑎𝑡𝑒𝑟
Mass Density of Standard
Fluid (Water) = 1000
𝑘𝑔
𝑚3⁄
VISCOSITY (μ):
𝜏 𝛼
𝑑𝑢
𝑑𝑦
𝜏 = 𝜇
𝑑𝑢
𝑑𝑦
Symbol Description Unit
𝜏 Shear Stress 𝑁
𝑚2⁄
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
𝑑𝑢 Change in Velocity 𝑚
𝑠⁄
𝑑𝑦 Change in Distance 𝑚
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 4
DYNAMIC VISCOSITY (μ):
𝜇 =
𝜏
𝑑𝑢
𝑑𝑦⁄
Symbol Description Unit
𝜏 Shear Stress 𝑁
𝑚2⁄
𝜇 Dynamic Viscosity 𝑁 − 𝑠
𝑚2⁄
𝑑𝑢 Change in Velocity 𝑚
𝑠⁄
𝑑𝑦 Change in Distance 𝑚
𝑑𝑢
𝑑𝑦⁄ Rate of Shear Strain 1
𝑠⁄
Unit Conversion:
1
𝑁𝑠
𝑚2
= 10 𝑝𝑜𝑖𝑠𝑒
1 𝐶𝑒𝑛𝑡𝑖𝑝𝑜𝑖𝑠𝑒 =
1
100
𝑝𝑜𝑖𝑠𝑒
1 𝑝𝑜𝑖𝑠𝑒 = 0.1
𝑁𝑠
𝑚2
KINEMATIC VISCOSITY (γ):
𝛾 =
𝜇
𝜌
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝜇 Dynamic Viscosity 𝑁 − 𝑠
𝑚2⁄
γ Kinematic Viscosity 𝑚2
𝑠⁄
Unit Conversion:
1 𝑠𝑡𝑜𝑘𝑒 = 10−4 𝑚2
𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 5
1 𝐶𝑒𝑛𝑡𝑖𝑠𝑡𝑜𝑘𝑒 =
1
100
𝑠𝑡𝑜𝑘𝑒
VISCOSITY PROBLEMS FOR PLATE TYPE:
FORCE (F):
𝜏 =
𝐹
𝐴
Symbol Description Unit
𝜏 Shear Stress 𝑁
𝑚2⁄
F Force N
A Area of the plate 𝑚2
POWER (P):
𝑃 = 𝐹 ∗ 𝑑𝑢
Symbol Description Unit
𝑃 Power 𝑊
F Force N
𝑑𝑢 Change in Velocity 𝑚
𝑠⁄
VISCOSITY PROBLEMS FOR SHAFT TYPE:
VELOCITY OF SHAFT (u):
𝑢 =
𝜋𝐷𝑁
60
Symbol Description Unit
𝐷 Diameter of Shaft 𝑚
N Speed of Shaft Rpm
𝑢 Velocity 𝑚
𝑠⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 6
FORCE (F):
𝜏 =
𝐹
𝐴
𝜏 = 𝜋𝐷𝐿
Symbol Description Unit
𝜏 Shear Stress 𝑁
𝑚2⁄
F Force N
A Circumference of Shaft 𝑚2
𝐷 Diameter of Shaft 𝑚
𝐿 Length of Shaft 𝑚
TORQUE ON SHAFT (T):
𝑇 = 𝐹 ∗
𝐷
2
Symbol Description Unit
𝑇 Torque 𝑁 − 𝑚
F Force N
𝐷 Diameter of Shaft 𝑚
POWER ON SHAFT (P):
𝑃 =
2𝜋𝑁𝑇
60
Symbol Description Unit
𝑃 Power 𝑊
𝑇 Torque 𝑁 − 𝑚
N Speed of Shaft Rpm
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 7
VISCOSITY PROBLEMS FOR CONICAL BEARING:
ANGULAR VELOCITY (ω):
𝜔 =
2𝜋𝑁
60
Symbol Description Unit
𝜔 Angular Velocity 𝑟𝑎𝑑
𝑠𝑒𝑐⁄
N Speed of Shaft Rpm
ANGLE (θ):
𝑡𝑎𝑛𝜃 =
𝑟1 − 𝑟2
𝐻
Symbol Description Unit
𝑟1 Outer Radius 𝑚
𝑟2 Inner Radius 𝑚
𝐻 Height 𝑚
POWER (P):
𝑃 =
2𝜋𝑁𝑇
60
Symbol Description Unit
𝑃 Power 𝑊
𝑇 Torque 𝑁 − 𝑚
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 8
N Speed of Shaft Rpm
THICKNESS OF OIL (h):
𝑇 =
𝜋𝜇𝜔
2ℎ𝑠𝑖𝑛𝜃
( 𝑟1
4
− 𝑟2
4)
Symbol Description Unit
𝜇 Dynamic Viscosity 𝑁 − 𝑠
𝑚2⁄
𝑇 Torque 𝑁 − 𝑚
𝜔 Angular Velocity 𝑟𝑎𝑑
𝑠𝑒𝑐⁄
ℎ Thickness of Oil 𝑚
𝑟1 Outer Radius 𝑚
𝑟2 Inner Radius 𝑚
CAPILLARITY:
HEIGHT OF LIQUID IN TUBE (h):
ℎ =
4𝜎𝑐𝑜𝑠𝜃
𝜌𝑔𝑑
Symbol Description Unit
ℎ Height of Liquid in Tube 𝑚
𝜎 Surface Tension 𝑁
𝑚⁄
𝜃
Angle of Contact between
Liquid and Tube
𝑟𝑎𝑑
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
𝑑 Diameter of Tube 𝑚
SURFACE TENSION:
PRESSURE IN LIQUID DROPLET (P):
𝑃 =
4𝜎
𝑑
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 9
Symbol Description Unit
𝑃 Pressure 𝑁
𝑚2⁄
𝜎 Surface Tension 𝑁
𝑚⁄
𝑑 Diameter of Droplet 𝑚
PRESSURE IN BUBBLE (P):
𝑃 =
8𝜎
𝑑
Symbol Description Unit
𝑃 Pressure 𝑁
𝑚2⁄
𝜎 Surface Tension 𝑁
𝑚⁄
𝑑 Diameter of Bubble 𝑚
PRESSURE IN LIQUID JET (P):
𝑃 =
2𝜎
𝑑
Symbol Description Unit
𝑃 Pressure 𝑁
𝑚2⁄
𝜎 Surface Tension 𝑁
𝑚⁄
𝑑 Diameter of Jet 𝑚
CONTINUITY EQUATION:
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
+
𝜕𝑤
𝜕𝑧
= 0 [ 𝐹𝑜𝑟 3 − 𝐷 𝑓𝑙𝑜𝑤]
𝜕𝑢
𝜕𝑥
+
𝜕𝑣
𝜕𝑦
+ = 0 [ 𝐹𝑜𝑟 2 − 𝐷 𝑓𝑙𝑜𝑤]
𝜕
𝜕𝑟
( 𝑟𝑢 𝑟) +
𝜕
𝜕𝜃
( 𝑢 𝜃) = 0[ 𝐹𝑜𝑟 𝑝𝑜𝑙𝑎𝑟 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠]
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 10
BERNOULLI’S EQUATION:
𝜕𝑃
𝜌
+ 𝑣. 𝑑𝑣 + 𝑔. 𝑑𝑧 = 0
𝑃1
𝜌𝑔
+
𝑣1
2
2𝑔
+ 𝑧1 =
𝑃2
𝜌𝑔
+
𝑣2
2
2𝑔
+ 𝑧2 + ℎ 𝑓
Symbol Description Unit
𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁
𝑚2⁄
𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚
𝑠⁄
𝑧1 & 𝑧2
Datum Head at Section 1 &
2
𝑚
ℎ 𝑓 Head Loss 𝑚
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
COEFFICIENT OF DISCHARGE:
𝐶 𝑑 =
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
COEFFICIENT OF VELOCITY:
𝐶𝑣 =
𝑣 𝐴𝑐𝑡𝑢𝑎𝑙
𝑣 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
DISCHARGE OF VENTURIMETER AND ORIFICEMETER:
𝑄 = 𝐶 𝑑
𝑎1 𝑎2
√( 𝑎1
2 − 𝑎1
2)
√2𝑔ℎ
Symbol Description Unit
𝑎1 & 𝑎2 Area at Section 1 & 2 𝑚2
ℎ
Pressure Difference
between Section 1 & 2
(
𝑃1− 𝑃2
𝜌𝑔
)
𝑚
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 11
𝐶 𝑑 Coefficient of Discharge
𝑥
Difference in Mercury
Level
𝑚
ℎ = 𝑥 (1 −
𝑆 𝑚
𝑆
) [ 𝑤ℎ𝑒𝑛 𝑆 > 𝑆 𝑚]
ℎ = 𝑥 (
𝑆 𝑚
𝑆
− 1) [ 𝑤ℎ𝑒𝑛 𝑆 𝑚 > 𝑆]
ℎ = (
𝑃1
𝜌𝑔
+ 𝑍1) − (
𝑃2
𝜌𝑔
+ 𝑍2) [ 𝐼𝑛𝑐𝑙𝑖𝑛𝑒𝑑 𝑉𝑒𝑛𝑡𝑢𝑟𝑖𝑚𝑒𝑡𝑒𝑟]
MOMENTUM EQUATION:
𝐹 =
𝑑 (𝑚𝑣)
𝑑𝑡
FORCE ACTING IN X – DIRECTION:
𝐹𝑥 = 𝜌𝑄 ( 𝑣1 − 𝑣2 𝑐𝑜𝑠𝜃) + 𝑃1 𝐴1 − 𝑃2 𝐴2 𝑐𝑜𝑠𝜃
FORCE ACTING IN Y – DIRECTION:
𝐹𝑦 = 𝜌𝑄 (− 𝑣2 𝑠𝑖𝑛𝜃) − 𝑃2 𝐴2 𝑠𝑖𝑛𝜃
Symbol Description Unit
𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁
𝑚2⁄
𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚
𝑠⁄
𝐴1 & 𝐴2 Area at Section 1 & 2 𝑚
𝜃 Angle of the Bend 𝐷𝑒𝑔𝑟𝑒𝑒
𝑄 Discharge 𝑚3
𝑠⁄
RESULTANT FORCE:
𝐹𝑅 = √𝐹𝑥
2
+ 𝐹𝑦
2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 12
ANGLE MADE BY RESULTANT FORCE:
𝑡𝑎𝑛𝜃 =
𝐹𝑦
𝐹𝑥
MOMENT OF MOMENTUM EQUATION:
𝑇 = 𝜌𝑄 ( 𝑣2 𝑟2 − 𝑣1 𝑟1)
Symbol Description Unit
𝑇 Torque 𝑁 − 𝑚
𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚
𝑠⁄
𝑟1 & 𝑟2
Radius of Curvature at
Section 1 & 2
𝑚
𝑄 Discharge 𝑚3
𝑠⁄
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 13
TOTAL ENERGY LINE (TEL):
𝑇𝐸𝐿 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 + 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐻𝑒𝑎𝑑 + 𝐷𝑎𝑡𝑢𝑚 𝐻𝑒𝑎𝑑
𝑇𝐸𝐿 =
𝑃
𝜌𝑔
+
𝑣2
2𝑔
+ 𝑍
Symbol Description Unit
𝑃 Pressure 𝑁
𝑚2⁄
𝑣 Velocity 𝑚
𝑠⁄
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝑍 Datum Head 𝑚
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
HYDRAULIC ENERGY LINE (HEL):
𝐻𝐸𝐿 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 + 𝐷𝑎𝑡𝑢𝑚 𝐻𝑒𝑎𝑑
𝑇𝐸𝐿 =
𝑃
𝜌𝑔
+ 𝑍
HAGEN POISEUILLE’S EQUATION:
SHEAR STRESS:
𝜏 = −
𝜕𝑝
𝜕𝑥
∗
𝑟
2
Symbol Description Unit
𝜏 Shear Stress 𝑁
𝑚2⁄
𝜕𝑝
𝜕𝑥
Pressure Gradient 𝑁
𝑚3⁄
𝑟 Radius of pipe 𝑚
UNIT – II – FLOW THROUGH CIRCULAR CONDUITS
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 14
VELOCITY:
𝑢 = −
1
4𝜇
∗
𝜕𝑝
𝜕𝑥
∗ (𝑅2
− 𝑟2
)
Symbol Description Unit
𝑢 Velocity of Fluid in Pipe 𝑚
𝑠⁄
𝜕𝑝
𝜕𝑥
Pressure Gradient 𝑁
𝑚3⁄
𝑟 Radius of pipe 𝑚
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
MAXIMUM VELOCITY:
𝑢 = −
1
4𝜇
∗
𝜕𝑝
𝜕𝑥
∗ (𝑅2
)
Symbol Description Unit
𝑢 Velocity of Fluid in Pipe 𝑚
𝑠⁄
𝜕𝑝
𝜕𝑥
Pressure Gradient 𝑁
𝑚3⁄
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
AVERAGE VELOCITY:
𝑢̅ = −
1
4𝜇
∗
𝜕𝑝
𝜕𝑥
∗ (𝑅2
)
Symbol Description Unit
𝑢̅
Average Velocity of Fluid
in Pipe
𝑚
𝑠⁄
𝜕𝑝
𝜕𝑥
Pressure Gradient 𝑁
𝑚3⁄
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
RATIO BETWEEN MAXIMUM VELOCITY AND AVERAGE VELOCITY:
𝑢 𝑚𝑎𝑥
𝑢̅
= 2
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 15
DISCHARGE:
𝑢 = −
1
8𝜇
∗
𝜕𝑝
𝜕𝑥
∗ 𝜋 ∗ 𝑅4
Symbol Description Unit
𝑢 Velocity of Fluid in Pipe 𝑚
𝑠⁄
𝜕𝑝
𝜕𝑥
Pressure Gradient 𝑁
𝑚3⁄
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
PRESSURE DIFFERENCE:
𝑃1 − 𝑃2 =
32𝜇𝑢̅𝐿
𝐷2
Symbol Description Unit
𝑢̅
Average Velocity of Fluid
in Pipe
𝑚
𝑠⁄
𝐿 Length of Pipe 𝑚
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
𝐷 Diameter of Pipe 𝑚
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
LOSS OF HEAD:
ℎ 𝑓 =
𝑃1 − 𝑃2
𝜌𝑔
=
32𝜇𝑢̅𝐿
𝜌𝑔𝐷2
[ 𝑓𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤]
DARCY WEISBACH EQUATION:
ℎ 𝑓 =
𝑃1 − 𝑃2
𝜌𝑔
=
4𝑓𝐿𝑣2
2𝑔𝑑
[ 𝑓𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤]
Symbol Description Unit
𝑣 Velocity of Fluid in Pipe 𝑚
𝑠⁄
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𝐿 Length of Pipe 𝑚
𝑓 Friction Factor
𝑑 Diameter of Pipe 𝑚
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
REYNOLD’S NUMBER:
𝑅 𝑒 =
𝜌𝑣𝑑
𝜇
𝑓 =
0.079
𝑅 𝑒
0.25
[ 𝐹𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝐹𝑙𝑜𝑤]
𝑓 =
16
𝑅 𝑒
[ 𝐹𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝐹𝑙𝑜𝑤]
𝑅 𝑒 < 2000 𝑇ℎ𝑒𝑛 𝑡ℎ𝑒 𝐹𝑙𝑜𝑤 𝑖𝑠 𝐿𝑎𝑚𝑖𝑛𝑎𝑟
𝑅 𝑒 > 2000 𝑇ℎ𝑒𝑛 𝑡ℎ𝑒 𝐹𝑙𝑜𝑤 𝑖𝑠 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡
Symbol Description Unit
𝑣 Velocity of Fluid in Pipe 𝑚
𝑠⁄
𝑑 Diameter of Pipe 𝑚
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
MAJOR LOSS IN PIPES:
ℎ 𝑓 =
32𝜇𝑢̅𝐿
𝜌𝑔𝑑2
[ 𝑓𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤]
ℎ 𝑓 =
4𝑓𝐿𝑣2
2𝑔𝑑
[ 𝑓𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤]
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Symbol Description Unit
𝑢̅ & 𝑣 Velocity of Fluid in Pipe 𝑚
𝑠⁄
𝑑 Diameter of Pipe 𝑚
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
𝑙 Length of Pipe 𝑚
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
𝑓 Friction Factor
MINOR LOSS IN PIPES:
LOSS DUE TO SUDDEN ENLARGEMENT:
ℎ 𝑒 =
( 𝑣1 − 𝑣2)2
2𝑔
Symbol Description Unit
𝑣1 & 𝑣2
Velocity of Fluid in Pipe at
Inlet and Outlet
𝑚
𝑠⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
LOSS DUE TO SUDDEN CONTRACTION:
ℎ 𝑐 =
𝐾𝑣2
2𝑔
𝐾 = (
1
𝐶𝑐
− 1)
2
ℎ 𝑐 =
0.5𝑣2
2𝑔
[ 𝐼𝑓 𝐶𝑐 𝑛𝑜𝑡 𝑔𝑖𝑣𝑒𝑛]
Symbol Description Unit
𝑣 Velocity of Fluid at Outlet 𝑚
𝑠⁄
𝐶𝑐
Coefficient of Contraction
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𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
LOSS AT ENTRANCE OF PIPE:
ℎ𝑖 =
0.5𝑣2
2𝑔
Symbol Description Unit
𝑣 Velocity of Fluid at Inlet 𝑚
𝑠⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
LOSS AT EXIT OF PIPE:
ℎ 𝑜 =
𝑣2
2𝑔
Symbol Description Unit
𝑣 Velocity of Fluid at Outlet 𝑚
𝑠⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
LOSS DUE TO GRADUAL CONTRACTION:
ℎ 𝑒 =
𝐾( 𝑣1 − 𝑣2)2
2𝑔
Symbol Description Unit
𝑣1 & 𝑣2
Velocity of Fluid in Pipe at
Inlet and Outlet
𝑚
𝑠⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
𝐾 Coefficient of Contraction
LOSS AT BEND OF PIPE:
ℎ 𝑏 =
𝐾𝑣2
2𝑔
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Symbol Description Unit
𝑣 Velocity of Flow 𝑚
𝑠⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
𝐾 Coefficient of Bend
LOSS AT DUE TO VARIOUS FITTINGS:
ℎ 𝑣 =
𝐾𝑣2
2𝑔
Symbol Description Unit
𝑣 Velocity of Flow 𝑚
𝑠⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
𝐾 Coefficient of Fittings
LOSS AT DUE TO OBSTRUCTION:
ℎ 𝑣 =
𝑣2
2𝑔
(
𝐴
𝐶𝑐 ( 𝐴 − 𝑎)
− 1)
𝐶𝑐 =
𝐴 𝑐
( 𝐴 − 𝑎)
Symbol Description Unit
𝑣 Velocity of Flow 𝑚
𝑠⁄
𝐴 Area of Pipe 𝑚2
𝑎 Area of Obstruction 𝑚2
𝐴 𝑐
Area of Vena Contraction 𝑚2
WHEN PIPES ARE CONNECTED IN SERIES:
DISCHARGE:
𝑄 = 𝑄1 = 𝑄2
𝑄 = 𝐴1 𝑣1 = 𝐴2 𝑣2
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HEAD LOSS:
ℎ 𝑓 = ℎ 𝑓1 + ℎ 𝑓2
ℎ 𝑓 =
4𝑓𝑙1 𝑣1
2
2𝑔𝑑1
+
4𝑓𝑙2 𝑣2
2
2𝑔𝑑2
Symbol Description Unit
𝑣1 & 𝑣2
Velocity of Flow at Pipe 1
& 2
𝑚
𝑠⁄
𝐴1& 𝐴2 Area of Pipe 1 & 2 𝑚2
𝑑1& 𝑑2 Diameter of Pipe 1 & 2 𝑚
𝑙1& 𝑙2 Length of Pipe 1 & 2 𝑚
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
𝑓 Friction Factor
WHEN PIPES ARE CONNECTED IN PARALLEL:
DISCHARGE:
𝑄 = 𝑄1 + 𝑄2
𝑄 = 𝐴1 𝑣1 + 𝐴2 𝑣2
HEAD LOSS:
ℎ 𝑓 = ℎ 𝑓1 = ℎ 𝑓2
ℎ 𝑓 =
4𝑓𝑙1 𝑣1
2
2𝑔𝑑1
=
4𝑓𝑙2 𝑣2
2
2𝑔𝑑2
Symbol Description Unit
𝑣1 & 𝑣2
Velocity of Flow at Pipe 1
& 2
𝑚
𝑠⁄
𝐴1& 𝐴2 Area of Pipe 1 & 2 𝑚2
𝑑1& 𝑑2 Diameter of Pipe 1 & 2 𝑚
𝑙1& 𝑙2 Length of Pipe 1 & 2 𝑚
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𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
𝑓 Friction Factor
EQUIVALENT PIPE:
𝐿
𝐷5
=
𝐿1
𝐷1
5 +
𝐿2
𝐷2
5 +
𝐿3
𝐷3
5 + ⋯ +
𝐿 𝑛
𝐷 𝑛
5
Symbol Description Unit
𝐷 Diameter of Pipe 𝑚
𝐿 Length of Pipe 𝑚
BOUNDARY LAYER:
DISPLACEMENT THICKNESS:
𝛿∗
= ∫ (1 −
𝑢
𝑈
)
𝛿
0
𝑑𝑦
MOMENTUM THICKNESS:
𝜃 = ∫
𝑢
𝑈
(1 −
𝑢
𝑈
)
𝛿
0
𝑑𝑦
MOMENTUM THICKNESS:
𝛿∗∗
= ∫
𝑢
𝑈
(1 −
𝑢2
𝑈2
)
𝛿
0
𝑑𝑦
Symbol Description Unit
𝑢
𝑈
Velocity Distribution
𝛿 Boundary layer thickness
SHEAR STRESS:
𝜏0
𝜌𝑈2
=
𝜕𝜃
𝜕𝑥
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𝜃 = ∫
𝑢
𝑈
(1 −
𝑢
𝑈
)
𝛿
0
𝑑𝑦
DRAG FORCE:
𝐹 𝐷 = ∫ 𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 ∗ 𝐴𝑟𝑒𝑎
𝐿
0
𝐹 𝐷 = ∫ 𝜏0 ∗ 𝑏 ∗ 𝑑𝑥
𝐿
0
LOCAL COEFFICIENT OF DRAG:
𝐶 𝐷
∗
=
𝜏0
1
2
𝜌𝑈2
AVERAGE COEFFICIENT OF DRAG:
𝐶 𝐷 =
𝐹 𝐷
1
2
𝜌𝐴𝑈2
Symbol Description Unit
𝜏0 Shear Stress 𝑁
𝑚2⁄
𝑏 Width of Plate 𝑚
𝑈 Free Stream Velocity 𝑚
𝑠⁄
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝐴 Area 𝑚2
𝐹 𝐷 Drag Force 𝑁
BLASIUS’S SOLUTION:
BOUNDARY LAYER THICKNESS:
𝛿 =
4.91𝑥
√ 𝑅 𝑒𝑥
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LOCAL COEFFICIENT OF DRAG:
𝐶 𝐷
∗
=
0.664
√ 𝑅 𝑒𝑥
AVERAGE COEFFICIENT OF DRAG:
𝐶 𝐷 =
1.328
√ 𝑅 𝑒𝐿
Symbol Description Unit
𝑅 𝑒𝑥
Reynold’s Number at
distance x
𝑅 𝑒𝐿
Reynold’s Number at
distance L
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UNITS:
Physical Quantity Symbol Unit Dimensions
Length L m L
Mass M Kg M
Time T Sec T
Area A m2
L2
Volume V m3
L3
Diameter D m L
Head H m L
Roughness k M L
Velocity v m/s LT-1
Angular Velocity ω rad/sec T-1
Acceleration a m/s2
LT-2
Angular Acceleration α rad/sec2
T-2
Speed N Rpm T-1
Discharge Q m3
/s L3
T-1
Kinematic Viscosity γ cm2
/s L2
T-1
Dynamic Viscosity μ N-s/m2
ML-1
T-1
Force F N MLT-2
Weight W N MLT-2
Thrust T N MLT-2
Density ρ Kg/ m3
ML-3
Pressure P N/m2
ML-1
T-2
UNIT – III – DIMENSIONAL ANALYSIS
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Physical Quantity Symbol Unit Dimensions
Specific Weight w N/m3
ML-2
T-2
Young’s Modulus E N/m2
ML-1
T-2
Bulk Modulus K N/m2
ML-1
T-2
Shear Stress τ N/m2
ML-1
T-2
Surface Tension σ N/m MT-2
Energy / Work W/E J = N-m ML2
T-2
Torque T N-m ML-2
T-2
Power P W=J/s ML-2
T-3
Momentum M Kg m/s MLT-1
Efficiency η No Unit Dimensionless
SIMILARITY:
GEOMETRIC SIMILARITY:
𝐿 𝑝
𝐿 𝑚
=
𝑏 𝑝
𝑏 𝑚
=
𝐷 𝑝
𝐷 𝑚
= 𝐿 𝑟
𝐴 𝑝
𝐴 𝑚
=
𝐿 𝑝
𝐿 𝑚
∗
𝑏 𝑝
𝑏 𝑚
= 𝐿 𝑟 ∗ 𝐿 𝑟 = 𝐿 𝑟
2
𝑉𝑝
𝑉𝑚
=
𝐿 𝑝
𝐿 𝑚
∗
𝑏 𝑝
𝑏 𝑚
∗
𝑡 𝑝
𝑡 𝑚
= 𝐿 𝑟 ∗ 𝐿 𝑟 ∗ 𝐿 𝑟 = 𝐿 𝑟
3
Symbol Description Unit
𝐿 𝑝&𝐿 𝑚
Length of Prototype &
Model
𝑚
𝑏 𝑝&𝑏 𝑚
Breadth of Prototype &
Model
𝑚
𝐷 𝑝&𝐷 𝑚
Diameter of Prototype &
Model
𝑚
𝑡 𝑝&𝑡 𝑚
Thickness of Prototype &
Model
𝑚
𝐴 𝑝&𝐴 𝑚 Area of Prototype & Model 𝑚2
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𝑉𝑝&𝑉𝑚
Volume of Prototype &
Model
𝑚3
𝐿 𝑟 Length Ratio
KINEMATIC SIMILARITY:
𝑣 𝑝
𝑣 𝑚
= 𝑣𝑟
𝑎 𝑝
𝑎 𝑚
= 𝑎 𝑟
Symbol Description Unit
𝑣 𝑝&𝑣 𝑚
Velocity of Prototype &
Model
𝑚
𝑠⁄
𝑎 𝑝&𝑎 𝑚
Acceleration of Prototype
& Model
𝑚
𝑠2⁄
𝑣𝑟 Velocity Ratio
𝑎 𝑟 Acceleration Ratio
DYNAMIC SIMILARITY:
( 𝐹𝑖) 𝑝
( 𝐹𝑖) 𝑚
=
( 𝐹𝑣) 𝑝
( 𝐹𝑣) 𝑚
=
(𝐹𝑔)
𝑝
(𝐹𝑔)
𝑚
= 𝐹𝑟
Symbol Description Unit
( 𝐹𝑖) 𝑝& ( 𝐹𝑖) 𝑚
Inertia Force of Prototype
& Model
𝑁
( 𝐹𝑣) 𝑝& ( 𝐹𝑣) 𝑚
Viscous Force of Prototype
& Model
𝑁
(𝐹𝑔)
𝑝
& (𝐹𝑔)
𝑚
Gravity Force of Prototype
& Model
𝑁
𝐹𝑟 Force Ratio
DIMENSIONLESS NUMBER:
REYNOLD’S NUMBER:
𝑅 𝑒 =
𝜌𝑣𝐷
𝜇
(𝑜𝑟)
𝜌𝑣𝐿
𝜇
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Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑣 Velocity 𝑚
𝑠⁄
𝜇 Viscosity 𝑁 − 𝑠
𝑚2⁄
𝐷 Diameter 𝑚
𝐿 Length 𝑚
FROUDE’S NUMBER:
𝐹𝑒 =
𝑣
√ 𝐿𝑔
Symbol Description Unit
𝑣 Velocity 𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐿 Length 𝑚
FROUDE’S NUMBER:
𝐹𝑒 =
𝑣
√ 𝐿𝑔
Symbol Description Unit
𝑣 Velocity 𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐿 Length 𝑚
EULER’S NUMBER:
𝐸 𝑢 =
𝑣
√ 𝑝
𝜌⁄
Symbol Description Unit
𝑣 Velocity 𝑚
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
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𝑝 Pressure 𝑁
𝑚2⁄
WEBER’S NUMBER:
𝑊𝑒 =
𝑣
√ 𝜎
𝜌𝐿⁄
Symbol Description Unit
𝑣 Velocity 𝑚
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
𝐿 Length 𝑚
𝜎 Surface Tension 𝑁
𝑚⁄
MACH’S NUMBER:
𝑊𝑒 =
𝑣
√ 𝐾
𝜌⁄
Symbol Description Unit
𝑣 Velocity 𝑚
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
𝐾 Elastic Stress 𝑁
𝑚2⁄
REYNOLD’S MODEL LAW:
TIME RATIO:
𝐹𝑟 = 𝑚 𝑟 𝑎 𝑟
𝐹𝑟 = 𝑚 𝑟
𝑣𝑟
𝑇𝑟
DISCHARGE RATIO:
𝑄 𝑟 = 𝜌𝑟 𝐿 𝑟
2
𝑣𝑟
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Symbol Description Unit
𝐹𝑟 Force Ratio
𝑚 𝑟 Mass Ratio
𝑣𝑟 Velocity Ratio
𝑇𝑟 Time Ratio
𝐿 𝑟 Length Ratio
𝜌𝑟 Density Ratio
FROUDE’S MODEL LAW:
TIME RATIO:
𝑇𝑟 = √ 𝐿 𝑟
ACCELERATION RATIO:
𝑎 𝑟 = 1
DISCHARGE RATIO:
𝑄 𝑟 = ( 𝐿 𝑟)
5
2⁄
FORCE RATIO:
𝐹𝑟 = ( 𝐿 𝑟)3
PRESSURE RATIO:
𝐹𝑟 = 𝐿 𝑟
ENERGY RATIO:
𝐸𝑟 = ( 𝐿 𝑟)4
MOMENTUM RATIO:
𝑀𝑟 = ( 𝐿 𝑟)3
∗ √ 𝐿 𝑟
TORQUE RATIO:
𝑇𝑟 = ( 𝐿 𝑟)4
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POWER RATIO:
𝑃𝑟 = ( 𝐿 𝑟)
7
2⁄
Symbol Description Unit
𝐿 𝑟 Length Ratio
DISTORTED MODELS:
( 𝐿 𝑟) 𝐻 =
𝐿𝑖𝑛𝑒𝑎𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑃𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒
𝐿𝑖𝑛𝑒𝑎𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑀𝑜𝑑𝑒𝑙
( 𝐿 𝑟) 𝐻 =
𝐿 𝑝
𝐿 𝑚
=
𝐵𝑝
𝐵 𝑚
( 𝐿 𝑟) 𝑉 =
𝐿𝑖𝑛𝑒𝑎𝑟 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑃𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒
𝐿𝑖𝑛𝑒𝑎𝑟 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑀𝑜𝑑𝑒𝑙
( 𝐿 𝑟) 𝑉 =
ℎ 𝑝
ℎ 𝑚
VELOCITY RATIO:
𝑣𝑟 = √(𝐿 𝑟) 𝑉
AREA RATIO:
𝐴 𝑟 = ( 𝐿 𝑟) 𝐻 ∗ ( 𝐿 𝑟) 𝑉
DISCAHRGE RATIO:
𝑄 𝑟 = ( 𝐿 𝑟) 𝐻 ∗ [( 𝐿 𝑟) 𝑉]
3
2⁄
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CENTRIFUGAL PUMP:
VELOCITY TRIANGLE DIAGRAM:
Symbol Description Unit
𝑢1&𝑢2
Tangential Velocity of
Impeller at Inlet & Outlet
𝑚
𝑠⁄
𝑣 𝑟1&𝑣 𝑟2
Relative Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑣1&𝑣2
Absolute Velocity at Inlet
& Outlet
𝑚
𝑠⁄
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
UNIT – IV – PUMPS
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𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝛽
Angle made by Absolute
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
𝜙
Angle made by Relative
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
TANGENTIAL VELOCITY AT INLET:
𝑢1 =
𝜋𝑑1 𝑁
60
Symbol Description Unit
𝑑1
Inlet (or) Internal Diameter
of Impeller
𝑚
𝑁 Speed of Impeller 𝑟𝑝𝑚
TANGENTIAL VELOCITY AT OUTLET:
𝑢2 =
𝜋𝑑2 𝑁
60
Symbol Description Unit
𝑑2
Oulet (or) External
Diameter of Impeller
𝑚
𝑁 Speed of Impeller 𝑟𝑝𝑚
FROM INLET VELOCITY TRIANGLE DIAGRAM:
𝑡𝑎𝑛𝜃 =
𝑣𝑓1
𝑢1
Symbol Description Unit
𝑢1
Tangential Velocity of
Impeller at Inlet
𝑚
𝑠⁄
𝑣1 Absolute Velocity at Inlet 𝑚
𝑠⁄
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𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
∵ 𝛼 = 90°
𝑣1 = 𝑣𝑓1
FROM OUTLET VELOCITY TRIANGLE DIAGRAM:
𝑡𝑎𝑛𝜙 =
𝑣𝑓2
𝑢2 − 𝑣 𝑤2
𝑣2 = √𝑣𝑓2
2 + 𝑣 𝑤2
2
𝑡𝑎𝑛𝛽 =
𝑣𝑓2
𝑣 𝑤2
Symbol Description Unit
𝑢2
Tangential Velocity of
Impeller at Outlet
𝑚
𝑠⁄
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑣2 Absolute Velocity at Outlet 𝑚
𝑠⁄
𝑣𝑓2 Flow Velocity at Outlet 𝑚
𝑠⁄
𝜙
Angle made by Relative
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
𝛽
Angle made by Absolute
Velocity at Outlet with the
Degree
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Direction of Motion of
Vane
DISCHARGE:
𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2
Symbol Description Unit
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑑1&𝑑2
Diameter of Impeller at
Inlet & Outlet
𝑚
𝑏1&𝑏2
Width of Impeller at Inlet
& Outlet
𝑚
𝑄 Discharge 𝑚3
𝑠⁄
WORK DONE BY AN IMPELLER PER SECOND:
𝑊 =
𝜌𝑔𝑄
𝑔
𝑣 𝑤2 𝑢2
Symbol Description Unit
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑢2
Tangential Velocity at
Outlet
𝑚
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
WORK DONE BY AN IMPELLER PER UNIT WEIGHT OF WATER:
𝑊 =
𝑣 𝑤2 𝑢2
𝑔
Symbol Description Unit
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑢2
Tangential Velocity at
Outlet
𝑚
𝑠⁄
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𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
MANOMETRIC EFFICIENCY:
𝜂 𝑚 =
𝑔𝐻
𝑣 𝑤2 𝑢2
Symbol Description Unit
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑢2
Tangential Velocity at
Outlet
𝑚
𝑠⁄
𝐻 Manometric Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
POWER REQUIRED BY THE PUMP:
𝑃 = 𝜌𝑄𝑣 𝑤2 𝑢2
Symbol Description Unit
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑢2
Tangential Velocity at
Outlet
𝑚
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑃 Power 𝑘𝑊
MINIMUM SPEED TO START THE PUMP:
𝑁 𝑚𝑖𝑛 =
120 ∗ 𝜂 𝑚 ∗ 𝑣 𝑤2 ∗ 𝑑2
𝜋 (𝑑2
2
− 𝑑1
2
)
Symbol Description Unit
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑑1&𝑑2
Diameter of Impeller at
Inlet & Outlet
𝑚
𝜂 𝑚 Manometric Efficiency
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OVERALL EFFICIENCY:
𝜂 𝑜 =
𝐼𝑚𝑝𝑒𝑙𝑙𝑒𝑟 𝑃𝑜𝑤𝑒𝑟
𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟
=
𝜌𝑔𝑄𝐻
𝑆. 𝑃
𝜂 𝑜 = 𝜂 𝑚𝑎𝑛𝑜 ∗ 𝜂 𝑚𝑒𝑐ℎ ∗ 𝜂 𝑣𝑜𝑙
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Manometric Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
MECHANICAL EFFICIENCY:
𝜂 𝑚𝑒𝑐ℎ =
𝜌𝑔𝑄𝐻
𝑆. 𝑃
∗
𝑣 𝑤2 𝑢2
𝑔𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Manometric Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝑆. 𝑃 Shaft Power 𝑊
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑢2
Tangential Velocity at
Outlet
𝑚
𝑠⁄
POWER OF PUMP:
𝑃 = 𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
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𝐻 Manometric Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
HYDRAULIC EFFICIENCY:
𝜂ℎ𝑦𝑑 =
𝐴𝑐𝑡𝑢𝑎𝑙 𝐿𝑖𝑓𝑡
𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐿𝑖𝑓𝑡
=
𝐴𝑐𝑡𝑢𝑎𝑙 𝐻𝑒𝑎𝑑
𝐼𝑑𝑒𝑎𝑙 𝐻𝑒𝑎𝑑
IDEAL HEAD:
𝑃𝐼 = 𝜌𝑔(𝑄 + 𝑞)𝐻𝑖
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝑞 Leakage of Water 𝑚3
𝑠⁄
𝐻𝑖 Ideal Head 𝑚
𝑃𝐼 Power at Input 𝑊
TORQUE EXERTED BY IMPELLER:
𝑇 =
𝜌𝑔𝑄
𝑔
∗ 𝑣 𝑤2 ∗ 𝑅2
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑅2
Radius of Impeller at
Outlet
𝑚
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SPECIFIC SPEED:
𝑁𝑠 =
𝑁√ 𝑄
𝐻
3
4⁄
𝑁𝑠 =
𝑁√ 𝑃
𝐻
5
4⁄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Head 𝑚
𝑃 Power 𝑘𝑊
𝑁 Speed 𝑟𝑝𝑚
𝑁𝑠 Specific Speed
SPEED RATIO:
𝐾 𝑢 =
𝑢2
√2𝑔𝐻
𝐾 𝑢 = 0.95 − 1.25
Symbol Description Unit
𝑢2
Tangential Velocity at
Outlet
𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾 𝑢 Speed Ratio
FLOW RATIO:
𝐾𝑓 =
𝑣𝑓2
√2𝑔𝐻
𝐾𝑓 = 0.1 − 0.25
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Symbol Description Unit
𝑣𝑓2 Flow Velocity at Outlet 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾𝑓 Flow Ratio
RECIPROCATING PUMP:
DISCHARGE:
𝑄 =
𝐴𝐿𝑁
60
𝐴 =
𝜋
4
𝐷2 [ 𝐹𝑜𝑟 𝑆𝑖𝑛𝑔𝑙𝑒 𝐴𝑐𝑡𝑖𝑛𝑔 𝑃𝑢𝑚𝑝]
𝐴 = [
𝜋
4
𝐷2
+
𝜋
4
( 𝐷2
− 𝑑2)] [ 𝐹𝑜𝑟 𝐷𝑜𝑢𝑏𝑙𝑒 𝐴𝑐𝑡𝑖𝑛𝑔 𝑃𝑢𝑚𝑝]
Symbol Description Unit
𝐴 Area of Cylinder 𝑚2
𝐿 Stroke Length 𝑚
𝑁 Speed 𝑟𝑝𝑚
𝐷
Diameter of Cylinder or
Bore
𝑚
𝑑 Diameter of Piston Rod 𝑚
WEIGHT OF THE WATER DELIVERED PER SECOND:
𝑊 = 𝜌𝑔𝑄
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝑊 Weight of Water 𝑁
𝑠⁄
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WORK DONE BY RECIPROCATING PUMP:
𝑊 = 𝜌𝑔𝑄𝐻
𝐻 = ℎ 𝑠 + ℎ 𝑑
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
𝑊 Work Done 𝑊
ℎ 𝑠 Suction Head 𝑚
ℎ 𝑑 Delivery Head 𝑚
POWER DEVELOPED BY RECIPROCATING PUMP:
𝑃 = 𝜌𝑔𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
POWER REQUIRED TO DRIVE THE PUMP:
𝑃 = 𝜌𝑔𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
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SLIP OF RECIPROCATING PUMP:
𝑆 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 − 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
Symbol Description Unit
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3
𝑠⁄
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3
𝑠⁄
COEFFICENT OF DISCHARGE:
𝐶 𝑑 =
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
Symbol Description Unit
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3
𝑠⁄
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3
𝑠⁄
PERCENTAGE OF SLIP IN RECIPROCATING PUMP:
% 𝑜𝑓 𝑆𝑙𝑖𝑝 =
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 − 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
% 𝑜𝑓 𝑆𝑙𝑖𝑝 = 1 − 𝐶 𝑑
Symbol Description Unit
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3
𝑠⁄
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3
𝑠⁄
𝐶 𝑑 Coefficient of Discharge
VOLUMETRIC EFFICIENCY:
𝜂 𝑉𝑜𝑙 =
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
= 𝐶 𝑑
Symbol Description Unit
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3
𝑠⁄
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙
Theoretical Discharge 𝑚3
𝑠⁄
𝐶 𝑑
Coefficient of Discharge
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MECHANICAL EFFICIENCY:
𝜂 𝑚𝑒𝑐ℎ =
𝑃𝑜𝑤𝑒𝑟 𝐷𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑏𝑦 𝑃𝑢𝑚𝑝
𝑃𝑜𝑤𝑒𝑟 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑡𝑜 𝐷𝑟𝑖𝑣𝑒 𝑡ℎ𝑒 𝑃𝑢𝑚𝑝
𝜂 𝑚𝑒𝑐ℎ =
𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑃𝑢𝑚𝑝
𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑀𝑜𝑡𝑜𝑟
𝜂 𝑚𝑒𝑐ℎ =
𝜌𝑔𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻
𝜌𝑔𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3
𝑠⁄
𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
ACCELERATION HEAD:
ℎ 𝑎𝑠 =
𝑙 𝑠
𝑔
∗
𝐴
𝑎 𝑠
∗ 𝜔2
∗ 𝑟 ∗ 𝑐𝑜𝑠𝜃 [ 𝐴𝑡 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒]
ℎ 𝑑𝑠 =
𝑙 𝑑
𝑔
∗
𝐴
𝑎 𝑑
∗ 𝜔2
∗ 𝑟 ∗ 𝑐𝑜𝑠𝜃 [ 𝐴𝑡 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒]
𝐴 =
𝜋
4
𝐷2
𝑎 𝑠 =
𝜋
4
𝑑 𝑠
2
𝑎 𝑑 =
𝜋
4
𝑑 𝑑
2
𝜔 =
2𝜋𝑁
60
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𝑟 =
𝐿
2
Symbol Description Unit
𝑙 𝑠 Length of Suction Pipe 𝑚
𝑙 𝑑 Length of Delivery Pipe 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐴 Area of Cylinder 𝑚2
𝑎 𝑠 Area of Suction Pipe 𝑚2
𝑎 𝑑 Area of Delivery Pipe 𝑚2
𝜔 Angular Speed 𝑟𝑎𝑑
𝑠⁄
𝑟 Radius of Crank 𝑚
𝜃 Angle of Crank 𝑑𝑒𝑔𝑟𝑒𝑒
𝐷
Diameter of Cylinder or
Bore
𝑚
𝑑 𝑠 Diameter of Suction Pipe 𝑚
𝑑 𝑑 Diameter of Delivery Pipe 𝑚
𝑁 Speed of Crank 𝑟𝑝𝑚
𝐿 Stroke Length 𝑚
PRESSURE HEAD:
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = ℎ 𝑠 + ℎ 𝑎𝑠 [ 𝐹𝑜𝑟 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒]
𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = ℎ 𝑑 + ℎ 𝑎𝑑 [ 𝐹𝑜𝑟 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒]
Symbol Description Unit
ℎ 𝑠 Suction Head 𝑚
ℎ 𝑑 Delivery Head 𝑚
ℎ 𝑎𝑠
Acceleration Head at
Suction
𝑚
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ℎ 𝑎𝑑
Acceleration Head at
Delivery
𝑚
ABSOLUTE PRESSURE HEAD:
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑
= 𝐻 𝑎𝑡𝑚 − (ℎ 𝑠 + ℎ 𝑎𝑠) [ 𝐹𝑜𝑟 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒]
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑
= 𝐻 𝑎𝑡𝑚 + (ℎ 𝑑 + ℎ 𝑎𝑑 ) [ 𝐹𝑜𝑟 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒]
Symbol Description Unit
ℎ 𝑠 Suction Head 𝑚
ℎ 𝑑 Delivery Head 𝑚
ℎ 𝑎𝑠
Acceleration Head at
Suction
𝑚
ℎ 𝑎𝑑
Acceleration Head at
Delivery
𝑚
𝐻 𝑎𝑡𝑚
Atmospheric Pressure
Head
𝑚
SEPARATION HEAD:
𝑃𝑠𝑒𝑝 = 𝜌𝑔ℎ 𝑆𝑒𝑝
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
ℎ 𝑠𝑒𝑝 Separation Head 𝑚
𝑃𝑠𝑒𝑝 Separation Pressure 𝑁
𝑚2⁄
HEAD LOSS WITHOUT AIR VESSEL:
ℎ 𝑓𝑊𝑂𝐴 =
4𝑓𝑙 𝑑 𝑣2
2𝑔𝑑 𝑑
Symbol Description Unit
𝑓 Friction Factor
𝑙 𝑑 Length of Delivery Pipe 𝑚
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𝑣
Velocity without Air
Vessel
𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝑑 𝑑 Diameter of Delivery Pipe 𝑚
VELOCITY WITHOUT AIR VESSEL:
𝑣 =
𝐴
𝑎 𝑑
∗ 𝜔 ∗ 𝑟
𝐴 =
𝜋
4
𝐷2
𝑎 𝑑 =
𝜋
4
𝑑 𝑑
2
𝜔 =
2𝜋𝑁
60
𝑟 =
𝐿
2
Symbol Description Unit
𝐴 Area of Cylinder 𝑚2
𝑎 𝑑 Area of Delivery Pipe 𝑚2
𝜔 Angular Speed 𝑟𝑎𝑑
𝑠⁄
𝑟 Radius of Crank 𝑚
𝐷
Diameter of Cylinder or
Bore
𝑚
𝑑 𝑑 Diameter of Delivery Pipe 𝑚
𝑁 Speed of Crank 𝑟𝑝𝑚
𝐿 Stroke Length 𝑚
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HEAD LOSS WITH AIR VESSEL:
ℎ 𝑓𝑊𝐴 =
4𝑓𝑙 𝑑 𝑣2
2𝑔𝑑 𝑑
Symbol Description Unit
𝑓 Friction Factor
𝑙 𝑑 Length of Delivery Pipe 𝑚
𝑣 Velocity with Air Vessel 𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝑑 𝑑 Diameter of Delivery Pipe 𝑚
VELOCITY WITH AIR VESSEL:
𝑣 =
𝐴
𝑎 𝑑
∗ 𝜔 ∗
𝑟
𝜋
𝐴 =
𝜋
4
𝐷2
𝑎 𝑑 =
𝜋
4
𝑑 𝑑
2
𝜔 =
2𝜋𝑁
60
𝑟 =
𝐿
2
Symbol Description Unit
𝐴 Area of Cylinder 𝑚2
𝑎 𝑑 Area of Delivery Pipe 𝑚2
𝜔 Angular Speed 𝑟𝑎𝑑
𝑠⁄
𝑟 Radius of Crank 𝑚
𝐷
Diameter of Cylinder or
Bore
𝑚
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𝑑 𝑑 Diameter of Delivery Pipe 𝑚
𝑁 Speed of Crank 𝑟𝑝𝑚
𝐿 Stroke Length 𝑚
POWER SAVED BY AIR VESSEL:
𝑃 = 𝜌𝑔𝑄 (
2
3
ℎ 𝑓𝑊𝑂𝐴 − ℎ 𝑓𝑊𝐴)
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
ℎ 𝑓𝑊𝑂𝐴
Head Loss Without Air
Vessel
𝑚
ℎ 𝑓𝑊𝐴 Head Loss With Air Vessel 𝑚
POWER REQUIRED TO DRIVE THE PUMP:
𝑃 = 𝜌𝑔𝑄 (ℎ 𝑠 + ℎ 𝑑 +
2
3
ℎ 𝑓𝑠𝑊𝑂𝐴 + ℎ 𝑓𝑑𝑊𝐴)
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
ℎ 𝑠 Suction Head 𝑚
ℎ 𝑑 Delivery Head 𝑚
ℎ 𝑓𝑠𝑊𝑂𝐴
Head Loss Without Air
Vessel at Suction
𝑚
ℎ 𝑓𝑑𝑊𝐴
Head Loss With Air Vessel
at Delivery
𝑚
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PELTON WHEEL:
Symbol Description Unit
𝑢1&𝑢2
Tangential Velocity of
Runner at Inlet & Outlet
𝑚
𝑠⁄
𝑣 𝑟1&𝑣 𝑟2
Relative Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑉1&𝑉2
Absolute Velocity at Inlet
& Outlet
𝑚
𝑠⁄
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝛽
Angle made by Absolute
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
𝜙
Angle made by Relative
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
UNIT – V – TURBINES
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TANGENTIAL VELOCITY AT INLET AND OUTLET (OR) VELOCITY OF
WHEEL:
𝑢 =
𝜋𝐷𝑁
60
Symbol Description Unit
𝐷 Diameter of Runner 𝑚
𝑁 Speed of Impeller 𝑟𝑝𝑚
VELOCITY OF JET:
𝑉1 = 𝐶𝑣√2𝑔𝐻
𝐶𝑣 = 0.97 − 0.99
Symbol Description Unit
𝐶𝑣 Coefficient of Velocity
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
VELOCITY OF WHEEL:
𝑢 = 𝑘 𝑢√2𝑔𝐻
𝑘 𝑢 = 0.43 − 0.45
Symbol Description Unit
𝑘 𝑢 Speed Ratio
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
FROM INLET VELOCITY TRIANGLE DIAGRAM:
𝑉 𝑤1 = 𝑉1
𝑉 𝑤1 = 𝑢1 + 𝑉𝑟1
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Symbol Description Unit
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑉1 Absolute Velocity at Inlet 𝑚
𝑠⁄
FROM OUTLET VELOCITY TRIANGLE DIAGRAM:
cos 𝜙 =
𝑢2 + 𝑣 𝑤2
𝑣 𝑟2
tan 𝜙 =
𝑣𝑓2
𝑢2 + 𝑣 𝑤2
sin 𝜙 =
𝑣𝑓2
𝑣 𝑟2
tan 𝛽 =
𝑣𝑓2
𝑣 𝑤2
Symbol Description Unit
𝑢2
Tangential Velocity of
Runner at Outlet
𝑚
𝑠⁄
𝑣 𝑟2 Relative Velocity at Outlet 𝑚
𝑠⁄
𝑣 𝑤2 Whirl Velocity at Outlet 𝑚
𝑠⁄
𝑣𝑓2 Flow Velocity at Outlet 𝑚
𝑠⁄
WORK DONE BY JET PER SECOND:
𝑊 = 𝜌𝑄 [ 𝑣 𝑤1 + 𝑣 𝑤2] 𝑢
Symbol Description Unit
𝑢
Tangential Velocity of
Runner
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
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HYDRAULIC EFFICIENCY:
𝜂ℎ𝑦𝑑 =
2[ 𝑣 𝑤1 + 𝑣 𝑤2] 𝑢
𝑉1
2
Symbol Description Unit
𝑢
Tangential Velocity of
Runner
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑉1 Absolute Velocity at Inlet 𝑚
𝑠⁄
OVERALL EFFICIENCY:
𝜂 𝑜 =
𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟
𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟
𝜂 𝑜 =
𝑆. 𝑃
𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
𝑆. 𝑃 Shaft Power 𝑊
DISCHARGE OF SINGLE JET:
𝑞 =
𝜋
4
∗ 𝑑2
∗ 𝑉1
Symbol Description Unit
𝑑 Diameter of Jet 𝑚
𝑉1 Absolute Velocity at Inlet 𝑚
𝑠⁄
𝑞 Discharge of Single Jet 𝑚3
𝑠⁄
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NUMBER OF JET:
𝑛 =
𝑄
𝑞
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝑞 Discharge of Single Jet 𝑚3
𝑠⁄
NUMBER OF BUCKET:
𝑍 = 15 +
𝐷
2𝑑
Symbol Description Unit
𝑑 Diameter of Jet 𝑚
𝐷 Diameter of Runner 𝑚
DIMENSIONS OF BUCKET:
𝐴𝑥𝑖𝑎𝑙 𝑊𝑖𝑑𝑡ℎ 𝐵 = 4.5𝑑
𝑅𝑎𝑑𝑖𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ 𝐿 = 2.5𝑑
𝐷𝑒𝑝𝑡ℎ 𝑜𝑓 𝐵𝑢𝑐𝑘𝑒𝑡 𝑇 = 𝑑
Symbol Description Unit
𝑑 Diameter of Jet 𝑚
KINETIC ENERGY OF JET:
𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 =
1
2
𝑚 𝑉1
2
𝑆𝑖𝑛𝑐𝑒 𝑚 = 𝜌𝐴𝑉
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 =
1
2
𝜌 ∗ 𝐴 ∗ 𝑉1 ∗ 𝑉1
2
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𝑆𝑖𝑛𝑐𝑒 𝑄 = 𝐴𝑉
𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 =
1
2
𝜌 ∗ 𝑄 ∗ 𝑉1
2
POWER LOST IN NOZZLE:
𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 = 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑁𝑜𝑧𝑧𝑙𝑒
POWER LOST IN RUNNER:
𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟
= 𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑓𝑡 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑁𝑜𝑧𝑧𝑙𝑒
+ 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑅𝑢𝑛𝑛𝑒𝑟
+ 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝐷𝑢𝑒 𝑡𝑜 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒
RESULTANT FORCE ON BUCKET:
𝐹 = 𝜌𝑄 [ 𝑣 𝑤1 + 𝑣 𝑤2]
Symbol Description Unit
𝐹 Resultant Force on Bucket 𝑁
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
TORQUE:
𝑇 = 𝐹 ∗
𝐷
2
Symbol Description Unit
𝐹 Resultant Force on Bucket 𝑁
𝐷 Diameter of Runner 𝑚
𝑇 Torque 𝑁 − 𝑚
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POWER:
𝑃 =
2𝜋𝑁𝑇
60
Symbol Description Unit
𝑃 Power 𝑊
𝑇 Torque 𝑁 − 𝑚
N Speed of Shaft Rpm
SPECIFIC SPEED:
𝑁𝑠 =
𝑁√ 𝑄
𝐻
3
4⁄
𝑁𝑠 =
𝑁√ 𝑃
𝐻
5
4⁄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Head 𝑚
𝑃 Power 𝑘𝑊
𝑁 Speed 𝑟𝑝𝑚
𝑁𝑠 Specific Speed
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REACTION TURBINE:
INWARD FLOW REACTION TURBINE:
Symbol Description Unit
𝑢1&𝑢2
Tangential Velocity of
Runner at Inlet & Outlet
𝑚
𝑠⁄
𝑣 𝑟1&𝑣 𝑟2
Relative Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑉1&𝑉2
Absolute Velocity at Inlet
& Outlet
𝑚
𝑠⁄
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜙
Angle made by Relative
Velocity at Outlet with the
Degree
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Direction of Motion of
Vane
TANGENTIAL VELOCITY AT INLET:
𝑢1 =
𝜋𝑑1 𝑁
60
Symbol Description Unit
𝑑1
Inlet (or) External
Diameter
𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
TANGENTIAL VELOCITY AT OUTLET:
𝑢2 =
𝜋𝑑2 𝑁
60
Symbol Description Unit
𝑑2
Outlet (or) Internal
Diameter
𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
FROM INLET VELOCITY TRIANGLE DIAGRAM:
sin 𝛼 =
𝑣𝑓1
𝑉1
cos 𝛼 =
𝑣 𝑤1
𝑉1
tan 𝛼 =
𝑣𝑓1
𝑣 𝑤1
sin 𝜃 =
𝑣𝑓1
𝑣 𝑟1
cos 𝜃 =
𝑣 𝑤1 − 𝑢1
𝑣 𝑟1
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tan 𝜃 =
𝑣𝑓1
𝑣 𝑤1 − 𝑢1
Symbol Description Unit
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑉1 Absolute Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
RELATIVE VELOCITY AT INLET:
𝑣 𝑟1 = √ 𝑣𝑓1
2 + ( 𝑣 𝑤1 − 𝑢1)2
Symbol Description Unit
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
DISCHARGE:
𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2
𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2
Symbol Description Unit
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𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑑1&𝑑2
Diameter of Impeller at
Inlet & Outlet
𝑚
𝑏1&𝑏2
Width of Impeller at Inlet
& Outlet
𝑚
𝑄 Discharge 𝑚3
𝑠⁄
𝐴 Area of Runner 𝑚2
𝐴 𝑓1&𝐴 𝑓2
Area of Flow at Inlet &
Outlet
𝑚
𝑠⁄
MASS OF WATER FLOWING THROUGH THE RUNNER:
𝑚 = 𝜌 𝑄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
HEAD AT INLET OF TURBINE:
𝐻 =
1
𝑔
∗ 𝑣 𝑤1 ∗ 𝑢1 +
𝑣𝑓1
2
2𝑔
Symbol Description Unit
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE:
𝑃 = 𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
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𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
POWER DEVELOPED BY TURBINE:
𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
HYDRAULIC EFFICIENCY:
𝜂ℎ𝑦𝑑 =
𝑣 𝑤1 𝑢1
𝑔𝐻
𝜂ℎ𝑦𝑑 =
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡
Symbol Description Unit
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
OVERALL EFFICIENCY:
𝜂 𝑜 =
𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟
𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟
𝜂 𝑜 =
𝑆. 𝑃
𝜌𝑔𝑄𝐻
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Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
𝑆. 𝑃 Shaft Power 𝑊
SPEED RATIO:
𝐾 𝑢 =
𝑢
√2𝑔𝐻
𝐾 𝑢 = 0.6 − 0.9
Symbol Description Unit
𝑢 Tangential Velocity 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾 𝑢 Speed Ratio
FLOW RATIO:
𝐾𝑓 =
𝑣𝑓1
√2𝑔𝐻
𝐾𝑓 = 0.15 − 0.3
Symbol Description Unit
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾𝑓 Flow Ratio
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SPECIFIC SPEED:
𝑁𝑠 =
𝑁√ 𝑄
𝐻
3
4⁄
𝑁𝑠 =
𝑁√ 𝑃
𝐻
5
4⁄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Head 𝑚
𝑃 Power 𝑘𝑊
𝑁 Speed 𝑟𝑝𝑚
𝑁𝑠 Specific Speed
OUTWARD FLOW REACTION TURBINE:
Symbol Description Unit
𝑢1&𝑢2
Tangential Velocity of
Runner at Inlet & Outlet
𝑚
𝑠⁄
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𝑣 𝑟1&𝑣 𝑟2
Relative Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑉1&𝑉2
Absolute Velocity at Inlet
& Outlet
𝑚
𝑠⁄
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜙
Angle made by Relative
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
TANGENTIAL VELOCITY AT INLET:
𝑢1 =
𝜋𝑑1 𝑁
60
Symbol Description Unit
𝑑1 Inlet (or) Internal Diameter 𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
TANGENTIAL VELOCITY AT OUTLET:
𝑢2 =
𝜋𝑑2 𝑁
60
Symbol Description Unit
𝑑2
Outlet (or) External
Diameter
𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
FROM INLET VELOCITY TRIANGLE DIAGRAM:
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sin 𝛼 =
𝑣𝑓1
𝑉1
cos 𝛼 =
𝑣 𝑤1
𝑉1
tan 𝛼 =
𝑣𝑓1
𝑣 𝑤1
sin 𝜃 =
𝑣𝑓1
𝑣 𝑟1
cos 𝜃 =
𝑣 𝑤1 − 𝑢1
𝑣 𝑟1
tan 𝜃 =
𝑣𝑓1
𝑣 𝑤1 − 𝑢1
Symbol
Description Unit
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑉1 Absolute Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
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RELATIVE VELOCITY AT INLET:
𝑣 𝑟1 = √ 𝑣𝑓1
2 + ( 𝑣 𝑤1 − 𝑢1)2
Symbol Description Unit
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
DISCHARGE:
𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2
𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2
Symbol Description Unit
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑑1&𝑑2
Diameter of Impeller at
Inlet & Outlet
𝑚
𝑏1&𝑏2
Width of Impeller at Inlet
& Outlet
𝑚
𝑄 Discharge 𝑚3
𝑠⁄
𝐴 Area of Runner 𝑚2
𝐴 𝑓1&𝐴 𝑓2
Area of Flow at Inlet &
Outlet
𝑚
𝑠⁄
MASS OF WATER FLOWING THROUGH THE RUNNER:
𝑚 = 𝜌 𝑄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
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INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE:
𝑃 = 𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
POWER DEVELOPED BY TURBINE:
𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
HYDRAULIC EFFICIENCY:
𝜂ℎ𝑦𝑑 =
𝑣 𝑤1 𝑢1
𝑔𝐻
𝜂ℎ𝑦𝑑 =
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡
Symbol Description Unit
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
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OVERALL EFFICIENCY:
𝜂 𝑜 =
𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟
𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟
𝜂 𝑜 =
𝑆. 𝑃
𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
𝑆. 𝑃 Shaft Power 𝑊
SPEED RATIO:
𝐾 𝑢 =
𝑢
√2𝑔𝐻
𝐾 𝑢 = 0.6 − 0.9
Symbol Description Unit
𝑢 Tangential Velocity 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾 𝑢 Speed Ratio
FLOW RATIO:
𝐾𝑓 =
𝑣𝑓1
√2𝑔𝐻
𝐾𝑓 = 0.15 − 0.3
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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Symbol Description Unit
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾𝑓 Flow Ratio
SPECIFIC SPEED:
𝑁𝑠 =
𝑁√ 𝑄
𝐻
3
4⁄
𝑁𝑠 =
𝑁√ 𝑃
𝐻
5
4⁄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Head 𝑚
𝑃 Power 𝑘𝑊
𝑁 Speed 𝑟𝑝𝑚
𝑁𝑠 Specific Speed
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FRANCIS TURBINE:
Symbol Description Unit
𝑢1&𝑢2
Tangential Velocity of
Runner at Inlet & Outlet
𝑚
𝑠⁄
𝑣 𝑟1&𝑣 𝑟2
Relative Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑉1&𝑉2
Absolute Velocity at Inlet
& Outlet
𝑚
𝑠⁄
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜙
Angle made by Relative
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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TANGENTIAL VELOCITY AT INLET:
𝑢1 =
𝜋𝑑1 𝑁
60
Symbol Description Unit
𝑑1
Inlet (or) External
Diameter
𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
TANGENTIAL VELOCITY AT OUTLET:
𝑢2 =
𝜋𝑑2 𝑁
60
Symbol Description Unit
𝑑2
Outlet (or) Internal
Diameter
𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
FROM INLET VELOCITY TRIANGLE DIAGRAM:
sin 𝛼 =
𝑣𝑓1
𝑉1
cos 𝛼 =
𝑣 𝑤1
𝑉1
tan 𝛼 =
𝑣𝑓1
𝑣 𝑤1
sin 𝜃 =
𝑣𝑓1
𝑣 𝑟1
cos 𝜃 =
𝑣 𝑤1 − 𝑢1
𝑣 𝑟1
tan 𝜃 =
𝑣𝑓1
𝑣 𝑤1 − 𝑢1
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Symbol Description Unit
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑉1 Absolute Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
RELATIVE VELOCITY AT INLET:
𝑣 𝑟1 = √ 𝑣𝑓1
2 + ( 𝑣 𝑤1 − 𝑢1)2
Symbol Description Unit
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
DISCHARGE:
𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2
𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2
Symbol Description Unit
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑑1&𝑑2
Diameter of Impeller at
Inlet & Outlet
𝑚
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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𝑏1&𝑏2
Width of Impeller at Inlet
& Outlet
𝑚
𝑄 Discharge 𝑚3
𝑠⁄
𝐴 Area of Runner 𝑚2
𝐴 𝑓1&𝐴 𝑓2
Area of Flow at Inlet &
Outlet
𝑚
𝑠⁄
CIRCUMFERENTIAL AREA OF RUNNER:
𝐴 = 𝜋𝑑1 𝑏1 = 𝜋𝑑2 𝑏2
Symbol Description Unit
𝑑1&𝑑2
Diameter of Impeller at
Inlet & Outlet
𝑚
𝑏1&𝑏2
Width of Impeller at Inlet
& Outlet
𝑚
𝐴
Circumferential Area of
Runner
𝑚2
MASS OF WATER FLOWING THROUGH THE RUNNER:
𝑚 = 𝜌 𝑄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE:
𝑃 = 𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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POWER DEVELOPED BY TURBINE:
𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
HYDRAULIC EFFICIENCY:
𝜂ℎ𝑦𝑑 =
𝑣 𝑤1 𝑢1
𝑔𝐻
𝜂ℎ𝑦𝑑 =
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡
Symbol Description Unit
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
OVERALL EFFICIENCY:
𝜂 𝑜 =
𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟
𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟
𝜂 𝑜 =
𝑆. 𝑃
𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
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𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
𝑆. 𝑃 Shaft Power 𝑊
SPEED RATIO:
𝐾 𝑢 =
𝑢
√2𝑔𝐻
𝐾 𝑢 = 0.6 − 0.9
Symbol Description Unit
𝑢 Tangential Velocity 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾 𝑢 Speed Ratio
FLOW RATIO:
𝐾𝑓 =
𝑣𝑓1
√2𝑔𝐻
𝐾𝑓 = 0.15 − 0.3
Symbol Description Unit
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾𝑓 Flow Ratio
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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BREADTH RATIO:
𝑛 =
𝑏1
𝑑1
𝑛 = 0.1 − 0.4
Symbol Description Unit
𝑏1 Width of Runner at Inlet 𝑚
𝑑1 Diameter of Runner at Inlet 𝑚
𝑛 Breadth Ratio
SPECIFIC SPEED:
𝑁𝑠 =
𝑁√ 𝑄
𝐻
3
4⁄
𝑁𝑠 =
𝑁√ 𝑃
𝐻
5
4⁄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Head 𝑚
𝑃 Power 𝑘𝑊
𝑁 Speed 𝑟𝑝𝑚
𝑁𝑠 Specific Speed
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KAPLAN TURBINE:
Symbol Description Unit
𝑢1&𝑢2
Tangential Velocity of
Runner at Inlet & Outlet
𝑚
𝑠⁄
𝑣 𝑟1&𝑣 𝑟2
Relative Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑣 𝑤1&𝑣 𝑤2
Whirl Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝑉1&𝑉2
Absolute Velocity at Inlet
& Outlet
𝑚
𝑠⁄
𝑣𝑓1&𝑣𝑓2
Flow Velocity at Inlet &
Outlet
𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜙
Angle made by Relative
Velocity at Outlet with the
Direction of Motion of
Vane
Degree
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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TANGENTIAL VELOCITY AT INLET:
𝑢1 =
𝜋𝐷 𝑜 𝑁
60
Symbol Description Unit
𝐷 𝑜
Inlet (or) External
Diameter
𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
TANGENTIAL VELOCITY AT OUTLET:
𝑢2 =
𝜋𝐷 𝑏 𝑁
60
=
𝜋𝐷ℎ 𝑁
60
Symbol Description Unit
𝐷 𝑏 𝑜𝑟 𝐷ℎ
Outlet (or) Boss (or) Hub
Diameter
𝑚
𝑁 Speed of Turbine 𝑟𝑝𝑚
FROM INLET VELOCITY TRIANGLE DIAGRAM:
sin 𝛼 =
𝑣𝑓1
𝑉1
cos 𝛼 =
𝑣 𝑤1
𝑉1
tan 𝛼 =
𝑣𝑓1
𝑣 𝑤1
sin 𝜃 =
𝑣𝑓1
𝑣 𝑟1
cos 𝜃 =
𝑣 𝑤1 − 𝑢1
𝑣 𝑟1
tan 𝜃 =
𝑣𝑓1
𝑣 𝑤1 − 𝑢1
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Symbol Description Unit
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑉1 Absolute Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝛼
Angle made by Absolute
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
𝜃
Angle made by Relative
Velocity at Inlet with the
Direction of Motion of
Vane
Degree
RELATIVE VELOCITY AT INLET:
𝑣 𝑟1 = √ 𝑣𝑓1
2 + ( 𝑣 𝑤1 − 𝑢1)2
Symbol Description Unit
𝑣 𝑟1 Relative Velocity at Inlet 𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
DISCHARGE:
𝑄 =
𝜋
4
[𝐷0
2
− 𝐷 𝑏
2
]𝑣𝑓1
Symbol Description Unit
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝐷0
Inlet (or) External
Diameter
𝑚
𝐷 𝑏 𝑜𝑟 𝐷ℎ
Outlet (or) Boss (or) Hub
Diameter
𝑚
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𝑄 Discharge 𝑚3
𝑠⁄
CIRCUMFERENTIAL AREA OF RUNNER:
𝐴 =
𝜋
4
[𝐷0
2
− 𝐷 𝑏
2
]
Symbol Description Unit
𝐷0
Inlet (or) External
Diameter
𝑚
𝐷 𝑏 𝑜𝑟 𝐷ℎ
Outlet (or) Boss (or) Hub
Diameter
𝑚
𝐴
Circumferential Area of
Runner
𝑚2
MASS OF WATER FLOWING THROUGH THE RUNNER:
𝑚 = 𝜌 𝑄
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝜌 Density 𝑘𝑔
𝑚3⁄
INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE:
𝑃 = 𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
POWER DEVELOPED BY TURBINE:
𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
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𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
HYDRAULIC EFFICIENCY:
𝜂ℎ𝑦𝑑 =
𝑣 𝑤1 𝑢1
𝑔𝐻
𝜂ℎ𝑦𝑑 =
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡
Symbol Description Unit
𝑢1
Tangential Velocity of
Runner at Inlet
𝑚
𝑠⁄
𝑣 𝑤1 Whirl Velocity at Inlet 𝑚
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
OVERALL EFFICIENCY:
𝜂 𝑜 =
𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟
𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟
𝜂 𝑜 =
𝑆. 𝑃
𝜌𝑔𝑄𝐻
Symbol Description Unit
𝜌 Density 𝑘𝑔
𝑚3⁄
𝑄 Discharge 𝑚3
𝑠⁄
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐻 Head 𝑚
𝑆. 𝑃 Shaft Power 𝑊
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SPEED RATIO:
𝐾 𝑢 =
𝑢
√2𝑔𝐻
𝐾 𝑢 = 0.6 − 0.9
Symbol Description Unit
𝑢 Tangential Velocity 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾 𝑢 Speed Ratio
FLOW RATIO:
𝐾𝑓 =
𝑣𝑓1
√2𝑔𝐻
𝐾𝑓 = 0.15 − 0.3
Symbol Description Unit
𝑣𝑓1 Flow Velocity at Inlet 𝑚
𝑠⁄
𝐻 Head 𝑚
𝑔
Acceleration due to
Gravity
𝑚
𝑠2⁄
𝐾𝑓 Flow Ratio
SPECIFIC SPEED:
𝑁𝑠 =
𝑁√ 𝑄
𝐻
3
4⁄
𝑁𝑠 =
𝑁√ 𝑃
𝐻
5
4⁄
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 81
Symbol Description Unit
𝑄 Discharge 𝑚3
𝑠⁄
𝐻 Head 𝑚
𝑃 Power 𝑘𝑊
𝑁 Speed 𝑟𝑝𝑚
𝑁𝑠 Specific Speed
DRAFT TUBE:
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 82
Symbol Description Unit
𝑉1&𝑉2 Velocity at Inlet & Outlet 𝑚
𝑠⁄
𝐻𝑠
Vertical Height of Draft
Tube Above Tail Race
𝑚
𝑦
Distance of Bottom of
Draft Tube from Tail Race
𝑚
FROM BERNOULLI’S EQUATION:
𝑃1
𝜌𝑔
+
𝑉1
2
2𝑔
+ 𝑧1 =
𝑃2
𝜌𝑔
+
𝑉2
2
2𝑔
+ 𝑧2 + ℎ 𝑓
Symbol Description Unit
𝑃1 & 𝑃2
Pressure at Inlet & Outlet
of Draft Tube
𝑁
𝑚2⁄
𝑉1 & 𝑉2
Velocity at Inlet & Outlet
of Draft Tube
𝑚
𝑠⁄
𝑧1 & 𝑧2
Datum Head Inlet & Outlet
of Draft Tube
𝑚
ℎ 𝑓 Head Loss 𝑚
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
LENGTH OF DRAFT TUBE:
𝐿 = 𝐻𝑠 + 𝑦
Symbol Description Unit
𝐿 Length of Draft Tube 𝑚
𝐻𝑠
Vertical Height of Draft
Tube Above Tail Race
𝑚
𝑦
Distance of Bottom of
Draft Tube from Tail Race
𝑚
EFFICIENCY OF DRAFT TUBE:
𝜂 𝑑 =
(
𝑉1
2
2𝑔
−
𝑉2
2
2𝑔
) − ℎ 𝑓
𝑉1
2
2𝑔
R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK
Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 83
Symbol Description Unit
𝑉1 & 𝑉2
Velocity at Inlet & Outlet
of Draft Tube
𝑚
𝑠⁄
ℎ 𝑓 Head Loss 𝑚
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄
HYDRAULIC EFFICIENCY OF DRAFT TUBE:
𝜂ℎ𝑦𝑑 =
𝐻𝑒𝑎𝑑 𝑈𝑡𝑖𝑙𝑖𝑧𝑒𝑑 𝑏𝑦 𝑇𝑢𝑟𝑏𝑖𝑛𝑒
𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 𝑜𝑓 𝑇𝑢𝑟𝑏𝑖𝑛𝑒
𝜂ℎ𝑦𝑑 =
𝐻 − ℎ 𝑓𝑡 − ℎ 𝑓𝑑 −
𝑉2
2
2𝑔
𝑃1
𝜌𝑔
+
𝑉1
2
2𝑔
+ 𝑧1
Symbol Description Unit
𝑃1
Pressure at Inlet of Draft
Tube
𝑁
𝑚2⁄
𝑉1 & 𝑉2
Velocity at Inlet & Outlet
of Draft Tube
𝑚
𝑠⁄
𝑧1
Datum Head Inlet of Draft
Tube
𝑚
ℎ 𝑓 Head Loss 𝑚
𝜌 Density of Liquid 𝑘𝑔
𝑚3⁄
𝑔 Acceleration due to gravity 𝑚
𝑠2⁄

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FLUID MECHANICS AND MACHINERY FORMULA BOOK

  • 1. R.M.K COLLEGE OF ENGINEERING AND TECHNOLOGY RSM NAGAR, PUDUVOYAL-601206 DEPARTMENT OF MECHANICAL ENGINEERING CE6451 – FLUID MECHANICS & MACHINERY III SEM MECHANICAL ENGINEERING Regulation 2013 FORMULA BOOK PREPARED BY C.BIBIN / R.ASHOK KUMAR / N.SADASIVAN
  • 2. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 2 PROPERTIES OF FLUID: MASS DESNITY (ρ): 𝜌 = 𝑚 𝑉 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ m Mass Kg V Volume m3 SPECIFIC VOLUME (v): 𝑣 = 𝑉 𝑚 = 1 𝜌 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ m Mass Kg V Volume m3 𝑣 Specific Volume 𝑚3 𝑘𝑔⁄ SPECIFIC WEIGTH or WEIGTH DENSITY (w): 𝑤 = 𝑊 𝑉 = 𝑚𝑔 𝑉 = 𝜌𝑔 𝑆𝑖𝑛𝑐𝑒 𝑊 = 𝑚𝑔 𝑎𝑛𝑑 𝜌 = 𝑚 𝑉⁄ Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ m Mass Kg V Volume m3 UNIT – I – FLUID PROPERTIES AND FLOW CHARACTERISTICS
  • 3. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 3 𝑤 Specific Weight 𝑁 𝑚3⁄ g Acceleration due to gravity 𝑚 𝑠2⁄ SPECIFIC GRAVITY (S): 𝑆 = 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑢𝑖𝑑 𝑆𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑊𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑙𝑢𝑖𝑑 𝑆 = 𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑔𝑖𝑣𝑒𝑛 𝑓𝑙𝑢𝑖𝑑 𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑙𝑢𝑖𝑑 Symbol Description Unit 𝑆 Specific Gravity No unit 𝜌 Density or Mass Density 𝑘𝑔 𝑚3⁄ 𝑤 Specific Weight 𝑁 𝑚3⁄ 𝑤 𝑤𝑎𝑡𝑒𝑟 Specific Weight of Standard Fluid (Water) = 9.81 𝑁 𝑚3⁄ 𝜌 𝑤𝑎𝑡𝑒𝑟 Mass Density of Standard Fluid (Water) = 1000 𝑘𝑔 𝑚3⁄ VISCOSITY (μ): 𝜏 𝛼 𝑑𝑢 𝑑𝑦 𝜏 = 𝜇 𝑑𝑢 𝑑𝑦 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑑𝑢 Change in Velocity 𝑚 𝑠⁄ 𝑑𝑦 Change in Distance 𝑚
  • 4. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 4 DYNAMIC VISCOSITY (μ): 𝜇 = 𝜏 𝑑𝑢 𝑑𝑦⁄ Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ 𝜇 Dynamic Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑑𝑢 Change in Velocity 𝑚 𝑠⁄ 𝑑𝑦 Change in Distance 𝑚 𝑑𝑢 𝑑𝑦⁄ Rate of Shear Strain 1 𝑠⁄ Unit Conversion: 1 𝑁𝑠 𝑚2 = 10 𝑝𝑜𝑖𝑠𝑒 1 𝐶𝑒𝑛𝑡𝑖𝑝𝑜𝑖𝑠𝑒 = 1 100 𝑝𝑜𝑖𝑠𝑒 1 𝑝𝑜𝑖𝑠𝑒 = 0.1 𝑁𝑠 𝑚2 KINEMATIC VISCOSITY (γ): 𝛾 = 𝜇 𝜌 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝜇 Dynamic Viscosity 𝑁 − 𝑠 𝑚2⁄ γ Kinematic Viscosity 𝑚2 𝑠⁄ Unit Conversion: 1 𝑠𝑡𝑜𝑘𝑒 = 10−4 𝑚2 𝑠⁄
  • 5. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 5 1 𝐶𝑒𝑛𝑡𝑖𝑠𝑡𝑜𝑘𝑒 = 1 100 𝑠𝑡𝑜𝑘𝑒 VISCOSITY PROBLEMS FOR PLATE TYPE: FORCE (F): 𝜏 = 𝐹 𝐴 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ F Force N A Area of the plate 𝑚2 POWER (P): 𝑃 = 𝐹 ∗ 𝑑𝑢 Symbol Description Unit 𝑃 Power 𝑊 F Force N 𝑑𝑢 Change in Velocity 𝑚 𝑠⁄ VISCOSITY PROBLEMS FOR SHAFT TYPE: VELOCITY OF SHAFT (u): 𝑢 = 𝜋𝐷𝑁 60 Symbol Description Unit 𝐷 Diameter of Shaft 𝑚 N Speed of Shaft Rpm 𝑢 Velocity 𝑚 𝑠⁄
  • 6. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 6 FORCE (F): 𝜏 = 𝐹 𝐴 𝜏 = 𝜋𝐷𝐿 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ F Force N A Circumference of Shaft 𝑚2 𝐷 Diameter of Shaft 𝑚 𝐿 Length of Shaft 𝑚 TORQUE ON SHAFT (T): 𝑇 = 𝐹 ∗ 𝐷 2 Symbol Description Unit 𝑇 Torque 𝑁 − 𝑚 F Force N 𝐷 Diameter of Shaft 𝑚 POWER ON SHAFT (P): 𝑃 = 2𝜋𝑁𝑇 60 Symbol Description Unit 𝑃 Power 𝑊 𝑇 Torque 𝑁 − 𝑚 N Speed of Shaft Rpm
  • 7. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 7 VISCOSITY PROBLEMS FOR CONICAL BEARING: ANGULAR VELOCITY (ω): 𝜔 = 2𝜋𝑁 60 Symbol Description Unit 𝜔 Angular Velocity 𝑟𝑎𝑑 𝑠𝑒𝑐⁄ N Speed of Shaft Rpm ANGLE (θ): 𝑡𝑎𝑛𝜃 = 𝑟1 − 𝑟2 𝐻 Symbol Description Unit 𝑟1 Outer Radius 𝑚 𝑟2 Inner Radius 𝑚 𝐻 Height 𝑚 POWER (P): 𝑃 = 2𝜋𝑁𝑇 60 Symbol Description Unit 𝑃 Power 𝑊 𝑇 Torque 𝑁 − 𝑚
  • 8. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 8 N Speed of Shaft Rpm THICKNESS OF OIL (h): 𝑇 = 𝜋𝜇𝜔 2ℎ𝑠𝑖𝑛𝜃 ( 𝑟1 4 − 𝑟2 4) Symbol Description Unit 𝜇 Dynamic Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑇 Torque 𝑁 − 𝑚 𝜔 Angular Velocity 𝑟𝑎𝑑 𝑠𝑒𝑐⁄ ℎ Thickness of Oil 𝑚 𝑟1 Outer Radius 𝑚 𝑟2 Inner Radius 𝑚 CAPILLARITY: HEIGHT OF LIQUID IN TUBE (h): ℎ = 4𝜎𝑐𝑜𝑠𝜃 𝜌𝑔𝑑 Symbol Description Unit ℎ Height of Liquid in Tube 𝑚 𝜎 Surface Tension 𝑁 𝑚⁄ 𝜃 Angle of Contact between Liquid and Tube 𝑟𝑎𝑑 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝑑 Diameter of Tube 𝑚 SURFACE TENSION: PRESSURE IN LIQUID DROPLET (P): 𝑃 = 4𝜎 𝑑
  • 9. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 9 Symbol Description Unit 𝑃 Pressure 𝑁 𝑚2⁄ 𝜎 Surface Tension 𝑁 𝑚⁄ 𝑑 Diameter of Droplet 𝑚 PRESSURE IN BUBBLE (P): 𝑃 = 8𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 𝑚2⁄ 𝜎 Surface Tension 𝑁 𝑚⁄ 𝑑 Diameter of Bubble 𝑚 PRESSURE IN LIQUID JET (P): 𝑃 = 2𝜎 𝑑 Symbol Description Unit 𝑃 Pressure 𝑁 𝑚2⁄ 𝜎 Surface Tension 𝑁 𝑚⁄ 𝑑 Diameter of Jet 𝑚 CONTINUITY EQUATION: 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + 𝜕𝑤 𝜕𝑧 = 0 [ 𝐹𝑜𝑟 3 − 𝐷 𝑓𝑙𝑜𝑤] 𝜕𝑢 𝜕𝑥 + 𝜕𝑣 𝜕𝑦 + = 0 [ 𝐹𝑜𝑟 2 − 𝐷 𝑓𝑙𝑜𝑤] 𝜕 𝜕𝑟 ( 𝑟𝑢 𝑟) + 𝜕 𝜕𝜃 ( 𝑢 𝜃) = 0[ 𝐹𝑜𝑟 𝑝𝑜𝑙𝑎𝑟 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠]
  • 10. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 10 BERNOULLI’S EQUATION: 𝜕𝑃 𝜌 + 𝑣. 𝑑𝑣 + 𝑔. 𝑑𝑧 = 0 𝑃1 𝜌𝑔 + 𝑣1 2 2𝑔 + 𝑧1 = 𝑃2 𝜌𝑔 + 𝑣2 2 2𝑔 + 𝑧2 + ℎ 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 𝑚2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚 𝑠⁄ 𝑧1 & 𝑧2 Datum Head at Section 1 & 2 𝑚 ℎ 𝑓 Head Loss 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ COEFFICIENT OF DISCHARGE: 𝐶 𝑑 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 COEFFICIENT OF VELOCITY: 𝐶𝑣 = 𝑣 𝐴𝑐𝑡𝑢𝑎𝑙 𝑣 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 DISCHARGE OF VENTURIMETER AND ORIFICEMETER: 𝑄 = 𝐶 𝑑 𝑎1 𝑎2 √( 𝑎1 2 − 𝑎1 2) √2𝑔ℎ Symbol Description Unit 𝑎1 & 𝑎2 Area at Section 1 & 2 𝑚2 ℎ Pressure Difference between Section 1 & 2 ( 𝑃1− 𝑃2 𝜌𝑔 ) 𝑚
  • 11. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 11 𝐶 𝑑 Coefficient of Discharge 𝑥 Difference in Mercury Level 𝑚 ℎ = 𝑥 (1 − 𝑆 𝑚 𝑆 ) [ 𝑤ℎ𝑒𝑛 𝑆 > 𝑆 𝑚] ℎ = 𝑥 ( 𝑆 𝑚 𝑆 − 1) [ 𝑤ℎ𝑒𝑛 𝑆 𝑚 > 𝑆] ℎ = ( 𝑃1 𝜌𝑔 + 𝑍1) − ( 𝑃2 𝜌𝑔 + 𝑍2) [ 𝐼𝑛𝑐𝑙𝑖𝑛𝑒𝑑 𝑉𝑒𝑛𝑡𝑢𝑟𝑖𝑚𝑒𝑡𝑒𝑟] MOMENTUM EQUATION: 𝐹 = 𝑑 (𝑚𝑣) 𝑑𝑡 FORCE ACTING IN X – DIRECTION: 𝐹𝑥 = 𝜌𝑄 ( 𝑣1 − 𝑣2 𝑐𝑜𝑠𝜃) + 𝑃1 𝐴1 − 𝑃2 𝐴2 𝑐𝑜𝑠𝜃 FORCE ACTING IN Y – DIRECTION: 𝐹𝑦 = 𝜌𝑄 (− 𝑣2 𝑠𝑖𝑛𝜃) − 𝑃2 𝐴2 𝑠𝑖𝑛𝜃 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Section 1 & 2 𝑁 𝑚2⁄ 𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚 𝑠⁄ 𝐴1 & 𝐴2 Area at Section 1 & 2 𝑚 𝜃 Angle of the Bend 𝐷𝑒𝑔𝑟𝑒𝑒 𝑄 Discharge 𝑚3 𝑠⁄ RESULTANT FORCE: 𝐹𝑅 = √𝐹𝑥 2 + 𝐹𝑦 2
  • 12. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 12 ANGLE MADE BY RESULTANT FORCE: 𝑡𝑎𝑛𝜃 = 𝐹𝑦 𝐹𝑥 MOMENT OF MOMENTUM EQUATION: 𝑇 = 𝜌𝑄 ( 𝑣2 𝑟2 − 𝑣1 𝑟1) Symbol Description Unit 𝑇 Torque 𝑁 − 𝑚 𝑣1 & 𝑣2 Velocity at Section 1 & 2 𝑚 𝑠⁄ 𝑟1 & 𝑟2 Radius of Curvature at Section 1 & 2 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄
  • 13. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 13 TOTAL ENERGY LINE (TEL): 𝑇𝐸𝐿 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 + 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐻𝑒𝑎𝑑 + 𝐷𝑎𝑡𝑢𝑚 𝐻𝑒𝑎𝑑 𝑇𝐸𝐿 = 𝑃 𝜌𝑔 + 𝑣2 2𝑔 + 𝑍 Symbol Description Unit 𝑃 Pressure 𝑁 𝑚2⁄ 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑍 Datum Head 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ HYDRAULIC ENERGY LINE (HEL): 𝐻𝐸𝐿 = 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 + 𝐷𝑎𝑡𝑢𝑚 𝐻𝑒𝑎𝑑 𝑇𝐸𝐿 = 𝑃 𝜌𝑔 + 𝑍 HAGEN POISEUILLE’S EQUATION: SHEAR STRESS: 𝜏 = − 𝜕𝑝 𝜕𝑥 ∗ 𝑟 2 Symbol Description Unit 𝜏 Shear Stress 𝑁 𝑚2⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝑟 Radius of pipe 𝑚 UNIT – II – FLOW THROUGH CIRCULAR CONDUITS
  • 14. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 14 VELOCITY: 𝑢 = − 1 4𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ (𝑅2 − 𝑟2 ) Symbol Description Unit 𝑢 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝑟 Radius of pipe 𝑚 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ MAXIMUM VELOCITY: 𝑢 = − 1 4𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ (𝑅2 ) Symbol Description Unit 𝑢 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ AVERAGE VELOCITY: 𝑢̅ = − 1 4𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ (𝑅2 ) Symbol Description Unit 𝑢̅ Average Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ RATIO BETWEEN MAXIMUM VELOCITY AND AVERAGE VELOCITY: 𝑢 𝑚𝑎𝑥 𝑢̅ = 2
  • 15. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 15 DISCHARGE: 𝑢 = − 1 8𝜇 ∗ 𝜕𝑝 𝜕𝑥 ∗ 𝜋 ∗ 𝑅4 Symbol Description Unit 𝑢 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝜕𝑝 𝜕𝑥 Pressure Gradient 𝑁 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ PRESSURE DIFFERENCE: 𝑃1 − 𝑃2 = 32𝜇𝑢̅𝐿 𝐷2 Symbol Description Unit 𝑢̅ Average Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝐿 Length of Pipe 𝑚 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝐷 Diameter of Pipe 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS OF HEAD: ℎ 𝑓 = 𝑃1 − 𝑃2 𝜌𝑔 = 32𝜇𝑢̅𝐿 𝜌𝑔𝐷2 [ 𝑓𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤] DARCY WEISBACH EQUATION: ℎ 𝑓 = 𝑃1 − 𝑃2 𝜌𝑔 = 4𝑓𝐿𝑣2 2𝑔𝑑 [ 𝑓𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤] Symbol Description Unit 𝑣 Velocity of Fluid in Pipe 𝑚 𝑠⁄
  • 16. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 16 𝐿 Length of Pipe 𝑚 𝑓 Friction Factor 𝑑 Diameter of Pipe 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ REYNOLD’S NUMBER: 𝑅 𝑒 = 𝜌𝑣𝑑 𝜇 𝑓 = 0.079 𝑅 𝑒 0.25 [ 𝐹𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝐹𝑙𝑜𝑤] 𝑓 = 16 𝑅 𝑒 [ 𝐹𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝐹𝑙𝑜𝑤] 𝑅 𝑒 < 2000 𝑇ℎ𝑒𝑛 𝑡ℎ𝑒 𝐹𝑙𝑜𝑤 𝑖𝑠 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑅 𝑒 > 2000 𝑇ℎ𝑒𝑛 𝑡ℎ𝑒 𝐹𝑙𝑜𝑤 𝑖𝑠 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 Symbol Description Unit 𝑣 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝑑 Diameter of Pipe 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ MAJOR LOSS IN PIPES: ℎ 𝑓 = 32𝜇𝑢̅𝐿 𝜌𝑔𝑑2 [ 𝑓𝑜𝑟 𝐿𝑎𝑚𝑖𝑛𝑎𝑟 𝑓𝑙𝑜𝑤] ℎ 𝑓 = 4𝑓𝐿𝑣2 2𝑔𝑑 [ 𝑓𝑜𝑟 𝑇𝑢𝑟𝑏𝑢𝑙𝑒𝑛𝑡 𝑓𝑙𝑜𝑤]
  • 17. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 17 Symbol Description Unit 𝑢̅ & 𝑣 Velocity of Fluid in Pipe 𝑚 𝑠⁄ 𝑑 Diameter of Pipe 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝑙 Length of Pipe 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝑓 Friction Factor MINOR LOSS IN PIPES: LOSS DUE TO SUDDEN ENLARGEMENT: ℎ 𝑒 = ( 𝑣1 − 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS DUE TO SUDDEN CONTRACTION: ℎ 𝑐 = 𝐾𝑣2 2𝑔 𝐾 = ( 1 𝐶𝑐 − 1) 2 ℎ 𝑐 = 0.5𝑣2 2𝑔 [ 𝐼𝑓 𝐶𝑐 𝑛𝑜𝑡 𝑔𝑖𝑣𝑒𝑛] Symbol Description Unit 𝑣 Velocity of Fluid at Outlet 𝑚 𝑠⁄ 𝐶𝑐 Coefficient of Contraction
  • 18. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 18 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS AT ENTRANCE OF PIPE: ℎ𝑖 = 0.5𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS AT EXIT OF PIPE: ℎ 𝑜 = 𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Fluid at Outlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LOSS DUE TO GRADUAL CONTRACTION: ℎ 𝑒 = 𝐾( 𝑣1 − 𝑣2)2 2𝑔 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Fluid in Pipe at Inlet and Outlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝐾 Coefficient of Contraction LOSS AT BEND OF PIPE: ℎ 𝑏 = 𝐾𝑣2 2𝑔
  • 19. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 19 Symbol Description Unit 𝑣 Velocity of Flow 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝐾 Coefficient of Bend LOSS AT DUE TO VARIOUS FITTINGS: ℎ 𝑣 = 𝐾𝑣2 2𝑔 Symbol Description Unit 𝑣 Velocity of Flow 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝐾 Coefficient of Fittings LOSS AT DUE TO OBSTRUCTION: ℎ 𝑣 = 𝑣2 2𝑔 ( 𝐴 𝐶𝑐 ( 𝐴 − 𝑎) − 1) 𝐶𝑐 = 𝐴 𝑐 ( 𝐴 − 𝑎) Symbol Description Unit 𝑣 Velocity of Flow 𝑚 𝑠⁄ 𝐴 Area of Pipe 𝑚2 𝑎 Area of Obstruction 𝑚2 𝐴 𝑐 Area of Vena Contraction 𝑚2 WHEN PIPES ARE CONNECTED IN SERIES: DISCHARGE: 𝑄 = 𝑄1 = 𝑄2 𝑄 = 𝐴1 𝑣1 = 𝐴2 𝑣2
  • 20. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 20 HEAD LOSS: ℎ 𝑓 = ℎ 𝑓1 + ℎ 𝑓2 ℎ 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 + 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 𝑚 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 𝑚2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 𝑚 𝑙1& 𝑙2 Length of Pipe 1 & 2 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝑓 Friction Factor WHEN PIPES ARE CONNECTED IN PARALLEL: DISCHARGE: 𝑄 = 𝑄1 + 𝑄2 𝑄 = 𝐴1 𝑣1 + 𝐴2 𝑣2 HEAD LOSS: ℎ 𝑓 = ℎ 𝑓1 = ℎ 𝑓2 ℎ 𝑓 = 4𝑓𝑙1 𝑣1 2 2𝑔𝑑1 = 4𝑓𝑙2 𝑣2 2 2𝑔𝑑2 Symbol Description Unit 𝑣1 & 𝑣2 Velocity of Flow at Pipe 1 & 2 𝑚 𝑠⁄ 𝐴1& 𝐴2 Area of Pipe 1 & 2 𝑚2 𝑑1& 𝑑2 Diameter of Pipe 1 & 2 𝑚 𝑙1& 𝑙2 Length of Pipe 1 & 2 𝑚
  • 21. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 21 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ 𝑓 Friction Factor EQUIVALENT PIPE: 𝐿 𝐷5 = 𝐿1 𝐷1 5 + 𝐿2 𝐷2 5 + 𝐿3 𝐷3 5 + ⋯ + 𝐿 𝑛 𝐷 𝑛 5 Symbol Description Unit 𝐷 Diameter of Pipe 𝑚 𝐿 Length of Pipe 𝑚 BOUNDARY LAYER: DISPLACEMENT THICKNESS: 𝛿∗ = ∫ (1 − 𝑢 𝑈 ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: 𝜃 = ∫ 𝑢 𝑈 (1 − 𝑢 𝑈 ) 𝛿 0 𝑑𝑦 MOMENTUM THICKNESS: 𝛿∗∗ = ∫ 𝑢 𝑈 (1 − 𝑢2 𝑈2 ) 𝛿 0 𝑑𝑦 Symbol Description Unit 𝑢 𝑈 Velocity Distribution 𝛿 Boundary layer thickness SHEAR STRESS: 𝜏0 𝜌𝑈2 = 𝜕𝜃 𝜕𝑥
  • 22. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 22 𝜃 = ∫ 𝑢 𝑈 (1 − 𝑢 𝑈 ) 𝛿 0 𝑑𝑦 DRAG FORCE: 𝐹 𝐷 = ∫ 𝑆ℎ𝑒𝑎𝑟 𝑆𝑡𝑟𝑒𝑠𝑠 ∗ 𝐴𝑟𝑒𝑎 𝐿 0 𝐹 𝐷 = ∫ 𝜏0 ∗ 𝑏 ∗ 𝑑𝑥 𝐿 0 LOCAL COEFFICIENT OF DRAG: 𝐶 𝐷 ∗ = 𝜏0 1 2 𝜌𝑈2 AVERAGE COEFFICIENT OF DRAG: 𝐶 𝐷 = 𝐹 𝐷 1 2 𝜌𝐴𝑈2 Symbol Description Unit 𝜏0 Shear Stress 𝑁 𝑚2⁄ 𝑏 Width of Plate 𝑚 𝑈 Free Stream Velocity 𝑚 𝑠⁄ 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝐴 Area 𝑚2 𝐹 𝐷 Drag Force 𝑁 BLASIUS’S SOLUTION: BOUNDARY LAYER THICKNESS: 𝛿 = 4.91𝑥 √ 𝑅 𝑒𝑥
  • 23. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 23 LOCAL COEFFICIENT OF DRAG: 𝐶 𝐷 ∗ = 0.664 √ 𝑅 𝑒𝑥 AVERAGE COEFFICIENT OF DRAG: 𝐶 𝐷 = 1.328 √ 𝑅 𝑒𝐿 Symbol Description Unit 𝑅 𝑒𝑥 Reynold’s Number at distance x 𝑅 𝑒𝐿 Reynold’s Number at distance L
  • 24. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 24 UNITS: Physical Quantity Symbol Unit Dimensions Length L m L Mass M Kg M Time T Sec T Area A m2 L2 Volume V m3 L3 Diameter D m L Head H m L Roughness k M L Velocity v m/s LT-1 Angular Velocity ω rad/sec T-1 Acceleration a m/s2 LT-2 Angular Acceleration α rad/sec2 T-2 Speed N Rpm T-1 Discharge Q m3 /s L3 T-1 Kinematic Viscosity γ cm2 /s L2 T-1 Dynamic Viscosity μ N-s/m2 ML-1 T-1 Force F N MLT-2 Weight W N MLT-2 Thrust T N MLT-2 Density ρ Kg/ m3 ML-3 Pressure P N/m2 ML-1 T-2 UNIT – III – DIMENSIONAL ANALYSIS
  • 25. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 25 Physical Quantity Symbol Unit Dimensions Specific Weight w N/m3 ML-2 T-2 Young’s Modulus E N/m2 ML-1 T-2 Bulk Modulus K N/m2 ML-1 T-2 Shear Stress τ N/m2 ML-1 T-2 Surface Tension σ N/m MT-2 Energy / Work W/E J = N-m ML2 T-2 Torque T N-m ML-2 T-2 Power P W=J/s ML-2 T-3 Momentum M Kg m/s MLT-1 Efficiency η No Unit Dimensionless SIMILARITY: GEOMETRIC SIMILARITY: 𝐿 𝑝 𝐿 𝑚 = 𝑏 𝑝 𝑏 𝑚 = 𝐷 𝑝 𝐷 𝑚 = 𝐿 𝑟 𝐴 𝑝 𝐴 𝑚 = 𝐿 𝑝 𝐿 𝑚 ∗ 𝑏 𝑝 𝑏 𝑚 = 𝐿 𝑟 ∗ 𝐿 𝑟 = 𝐿 𝑟 2 𝑉𝑝 𝑉𝑚 = 𝐿 𝑝 𝐿 𝑚 ∗ 𝑏 𝑝 𝑏 𝑚 ∗ 𝑡 𝑝 𝑡 𝑚 = 𝐿 𝑟 ∗ 𝐿 𝑟 ∗ 𝐿 𝑟 = 𝐿 𝑟 3 Symbol Description Unit 𝐿 𝑝&𝐿 𝑚 Length of Prototype & Model 𝑚 𝑏 𝑝&𝑏 𝑚 Breadth of Prototype & Model 𝑚 𝐷 𝑝&𝐷 𝑚 Diameter of Prototype & Model 𝑚 𝑡 𝑝&𝑡 𝑚 Thickness of Prototype & Model 𝑚 𝐴 𝑝&𝐴 𝑚 Area of Prototype & Model 𝑚2
  • 26. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 26 𝑉𝑝&𝑉𝑚 Volume of Prototype & Model 𝑚3 𝐿 𝑟 Length Ratio KINEMATIC SIMILARITY: 𝑣 𝑝 𝑣 𝑚 = 𝑣𝑟 𝑎 𝑝 𝑎 𝑚 = 𝑎 𝑟 Symbol Description Unit 𝑣 𝑝&𝑣 𝑚 Velocity of Prototype & Model 𝑚 𝑠⁄ 𝑎 𝑝&𝑎 𝑚 Acceleration of Prototype & Model 𝑚 𝑠2⁄ 𝑣𝑟 Velocity Ratio 𝑎 𝑟 Acceleration Ratio DYNAMIC SIMILARITY: ( 𝐹𝑖) 𝑝 ( 𝐹𝑖) 𝑚 = ( 𝐹𝑣) 𝑝 ( 𝐹𝑣) 𝑚 = (𝐹𝑔) 𝑝 (𝐹𝑔) 𝑚 = 𝐹𝑟 Symbol Description Unit ( 𝐹𝑖) 𝑝& ( 𝐹𝑖) 𝑚 Inertia Force of Prototype & Model 𝑁 ( 𝐹𝑣) 𝑝& ( 𝐹𝑣) 𝑚 Viscous Force of Prototype & Model 𝑁 (𝐹𝑔) 𝑝 & (𝐹𝑔) 𝑚 Gravity Force of Prototype & Model 𝑁 𝐹𝑟 Force Ratio DIMENSIONLESS NUMBER: REYNOLD’S NUMBER: 𝑅 𝑒 = 𝜌𝑣𝐷 𝜇 (𝑜𝑟) 𝜌𝑣𝐿 𝜇
  • 27. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 27 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑣 Velocity 𝑚 𝑠⁄ 𝜇 Viscosity 𝑁 − 𝑠 𝑚2⁄ 𝐷 Diameter 𝑚 𝐿 Length 𝑚 FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐿 Length 𝑚 FROUDE’S NUMBER: 𝐹𝑒 = 𝑣 √ 𝐿𝑔 Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐿 Length 𝑚 EULER’S NUMBER: 𝐸 𝑢 = 𝑣 √ 𝑝 𝜌⁄ Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄
  • 28. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 28 𝑝 Pressure 𝑁 𝑚2⁄ WEBER’S NUMBER: 𝑊𝑒 = 𝑣 √ 𝜎 𝜌𝐿⁄ Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝐿 Length 𝑚 𝜎 Surface Tension 𝑁 𝑚⁄ MACH’S NUMBER: 𝑊𝑒 = 𝑣 √ 𝐾 𝜌⁄ Symbol Description Unit 𝑣 Velocity 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝐾 Elastic Stress 𝑁 𝑚2⁄ REYNOLD’S MODEL LAW: TIME RATIO: 𝐹𝑟 = 𝑚 𝑟 𝑎 𝑟 𝐹𝑟 = 𝑚 𝑟 𝑣𝑟 𝑇𝑟 DISCHARGE RATIO: 𝑄 𝑟 = 𝜌𝑟 𝐿 𝑟 2 𝑣𝑟
  • 29. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 29 Symbol Description Unit 𝐹𝑟 Force Ratio 𝑚 𝑟 Mass Ratio 𝑣𝑟 Velocity Ratio 𝑇𝑟 Time Ratio 𝐿 𝑟 Length Ratio 𝜌𝑟 Density Ratio FROUDE’S MODEL LAW: TIME RATIO: 𝑇𝑟 = √ 𝐿 𝑟 ACCELERATION RATIO: 𝑎 𝑟 = 1 DISCHARGE RATIO: 𝑄 𝑟 = ( 𝐿 𝑟) 5 2⁄ FORCE RATIO: 𝐹𝑟 = ( 𝐿 𝑟)3 PRESSURE RATIO: 𝐹𝑟 = 𝐿 𝑟 ENERGY RATIO: 𝐸𝑟 = ( 𝐿 𝑟)4 MOMENTUM RATIO: 𝑀𝑟 = ( 𝐿 𝑟)3 ∗ √ 𝐿 𝑟 TORQUE RATIO: 𝑇𝑟 = ( 𝐿 𝑟)4
  • 30. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 30 POWER RATIO: 𝑃𝑟 = ( 𝐿 𝑟) 7 2⁄ Symbol Description Unit 𝐿 𝑟 Length Ratio DISTORTED MODELS: ( 𝐿 𝑟) 𝐻 = 𝐿𝑖𝑛𝑒𝑎𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑃𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒 𝐿𝑖𝑛𝑒𝑎𝑟 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑀𝑜𝑑𝑒𝑙 ( 𝐿 𝑟) 𝐻 = 𝐿 𝑝 𝐿 𝑚 = 𝐵𝑝 𝐵 𝑚 ( 𝐿 𝑟) 𝑉 = 𝐿𝑖𝑛𝑒𝑎𝑟 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑃𝑟𝑜𝑡𝑜𝑡𝑦𝑝𝑒 𝐿𝑖𝑛𝑒𝑎𝑟 𝑉𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝐷𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑠 𝑜𝑓 𝑀𝑜𝑑𝑒𝑙 ( 𝐿 𝑟) 𝑉 = ℎ 𝑝 ℎ 𝑚 VELOCITY RATIO: 𝑣𝑟 = √(𝐿 𝑟) 𝑉 AREA RATIO: 𝐴 𝑟 = ( 𝐿 𝑟) 𝐻 ∗ ( 𝐿 𝑟) 𝑉 DISCAHRGE RATIO: 𝑄 𝑟 = ( 𝐿 𝑟) 𝐻 ∗ [( 𝐿 𝑟) 𝑉] 3 2⁄
  • 31. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 31 CENTRIFUGAL PUMP: VELOCITY TRIANGLE DIAGRAM: Symbol Description Unit 𝑢1&𝑢2 Tangential Velocity of Impeller at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣1&𝑣2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree UNIT – IV – PUMPS
  • 32. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 32 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝛽 Angle made by Absolute Velocity at Outlet with the Direction of Motion of Vane Degree 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter of Impeller 𝑚 𝑁 Speed of Impeller 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Oulet (or) External Diameter of Impeller 𝑚 𝑁 Speed of Impeller 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: 𝑡𝑎𝑛𝜃 = 𝑣𝑓1 𝑢1 Symbol Description Unit 𝑢1 Tangential Velocity of Impeller at Inlet 𝑚 𝑠⁄ 𝑣1 Absolute Velocity at Inlet 𝑚 𝑠⁄
  • 33. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 33 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree ∵ 𝛼 = 90° 𝑣1 = 𝑣𝑓1 FROM OUTLET VELOCITY TRIANGLE DIAGRAM: 𝑡𝑎𝑛𝜙 = 𝑣𝑓2 𝑢2 − 𝑣 𝑤2 𝑣2 = √𝑣𝑓2 2 + 𝑣 𝑤2 2 𝑡𝑎𝑛𝛽 = 𝑣𝑓2 𝑣 𝑤2 Symbol Description Unit 𝑢2 Tangential Velocity of Impeller at Outlet 𝑚 𝑠⁄ 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑣2 Absolute Velocity at Outlet 𝑚 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet 𝑚 𝑠⁄ 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree 𝛽 Angle made by Absolute Velocity at Outlet with the Degree
  • 34. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 34 Direction of Motion of Vane DISCHARGE: 𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ WORK DONE BY AN IMPELLER PER SECOND: 𝑊 = 𝜌𝑔𝑄 𝑔 𝑣 𝑤2 𝑢2 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ WORK DONE BY AN IMPELLER PER UNIT WEIGHT OF WATER: 𝑊 = 𝑣 𝑤2 𝑢2 𝑔 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄
  • 35. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 35 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ MANOMETRIC EFFICIENCY: 𝜂 𝑚 = 𝑔𝐻 𝑣 𝑤2 𝑢2 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝐻 Manometric Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ POWER REQUIRED BY THE PUMP: 𝑃 = 𝜌𝑄𝑣 𝑤2 𝑢2 Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑃 Power 𝑘𝑊 MINIMUM SPEED TO START THE PUMP: 𝑁 𝑚𝑖𝑛 = 120 ∗ 𝜂 𝑚 ∗ 𝑣 𝑤2 ∗ 𝑑2 𝜋 (𝑑2 2 − 𝑑1 2 ) Symbol Description Unit 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝜂 𝑚 Manometric Efficiency
  • 36. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 36 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝐼𝑚𝑝𝑒𝑙𝑙𝑒𝑟 𝑃𝑜𝑤𝑒𝑟 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 = 𝜌𝑔𝑄𝐻 𝑆. 𝑃 𝜂 𝑜 = 𝜂 𝑚𝑎𝑛𝑜 ∗ 𝜂 𝑚𝑒𝑐ℎ ∗ 𝜂 𝑣𝑜𝑙 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Manometric Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ MECHANICAL EFFICIENCY: 𝜂 𝑚𝑒𝑐ℎ = 𝜌𝑔𝑄𝐻 𝑆. 𝑃 ∗ 𝑣 𝑤2 𝑢2 𝑔𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Manometric Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑆. 𝑃 Shaft Power 𝑊 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ POWER OF PUMP: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄
  • 37. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 37 𝐻 Manometric Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝐴𝑐𝑡𝑢𝑎𝑙 𝐿𝑖𝑓𝑡 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐿𝑖𝑓𝑡 = 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻𝑒𝑎𝑑 𝐼𝑑𝑒𝑎𝑙 𝐻𝑒𝑎𝑑 IDEAL HEAD: 𝑃𝐼 = 𝜌𝑔(𝑄 + 𝑞)𝐻𝑖 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑞 Leakage of Water 𝑚3 𝑠⁄ 𝐻𝑖 Ideal Head 𝑚 𝑃𝐼 Power at Input 𝑊 TORQUE EXERTED BY IMPELLER: 𝑇 = 𝜌𝑔𝑄 𝑔 ∗ 𝑣 𝑤2 ∗ 𝑅2 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑅2 Radius of Impeller at Outlet 𝑚
  • 38. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 38 SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed SPEED RATIO: 𝐾 𝑢 = 𝑢2 √2𝑔𝐻 𝐾 𝑢 = 0.95 − 1.25 Symbol Description Unit 𝑢2 Tangential Velocity at Outlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓2 √2𝑔𝐻 𝐾𝑓 = 0.1 − 0.25
  • 39. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 39 Symbol Description Unit 𝑣𝑓2 Flow Velocity at Outlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio RECIPROCATING PUMP: DISCHARGE: 𝑄 = 𝐴𝐿𝑁 60 𝐴 = 𝜋 4 𝐷2 [ 𝐹𝑜𝑟 𝑆𝑖𝑛𝑔𝑙𝑒 𝐴𝑐𝑡𝑖𝑛𝑔 𝑃𝑢𝑚𝑝] 𝐴 = [ 𝜋 4 𝐷2 + 𝜋 4 ( 𝐷2 − 𝑑2)] [ 𝐹𝑜𝑟 𝐷𝑜𝑢𝑏𝑙𝑒 𝐴𝑐𝑡𝑖𝑛𝑔 𝑃𝑢𝑚𝑝] Symbol Description Unit 𝐴 Area of Cylinder 𝑚2 𝐿 Stroke Length 𝑚 𝑁 Speed 𝑟𝑝𝑚 𝐷 Diameter of Cylinder or Bore 𝑚 𝑑 Diameter of Piston Rod 𝑚 WEIGHT OF THE WATER DELIVERED PER SECOND: 𝑊 = 𝜌𝑔𝑄 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑊 Weight of Water 𝑁 𝑠⁄
  • 40. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 40 WORK DONE BY RECIPROCATING PUMP: 𝑊 = 𝜌𝑔𝑄𝐻 𝐻 = ℎ 𝑠 + ℎ 𝑑 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑊 Work Done 𝑊 ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 POWER DEVELOPED BY RECIPROCATING PUMP: 𝑃 = 𝜌𝑔𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = 𝜌𝑔𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚
  • 41. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 41 SLIP OF RECIPROCATING PUMP: 𝑆 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 − 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ COEFFICENT OF DISCHARGE: 𝐶 𝑑 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ PERCENTAGE OF SLIP IN RECIPROCATING PUMP: % 𝑜𝑓 𝑆𝑙𝑖𝑝 = 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 − 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 % 𝑜𝑓 𝑆𝑙𝑖𝑝 = 1 − 𝐶 𝑑 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝐶 𝑑 Coefficient of Discharge VOLUMETRIC EFFICIENCY: 𝜂 𝑉𝑜𝑙 = 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 = 𝐶 𝑑 Symbol Description Unit 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝐶 𝑑 Coefficient of Discharge
  • 42. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 42 MECHANICAL EFFICIENCY: 𝜂 𝑚𝑒𝑐ℎ = 𝑃𝑜𝑤𝑒𝑟 𝐷𝑒𝑣𝑒𝑙𝑜𝑝𝑒𝑑 𝑏𝑦 𝑃𝑢𝑚𝑝 𝑃𝑜𝑤𝑒𝑟 𝑅𝑒𝑞𝑢𝑖𝑟𝑒𝑑 𝑡𝑜 𝐷𝑟𝑖𝑣𝑒 𝑡ℎ𝑒 𝑃𝑢𝑚𝑝 𝜂 𝑚𝑒𝑐ℎ = 𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑃𝑢𝑚𝑝 𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑀𝑜𝑡𝑜𝑟 𝜂 𝑚𝑒𝑐ℎ = 𝜌𝑔𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 𝐻 𝜌𝑔𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 𝐴𝑐𝑡𝑢𝑎𝑙 Actual Discharge 𝑚3 𝑠⁄ 𝑄 𝑇ℎ𝑒𝑜𝑟𝑒𝑡𝑖𝑐𝑎𝑙 Theoretical Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 ACCELERATION HEAD: ℎ 𝑎𝑠 = 𝑙 𝑠 𝑔 ∗ 𝐴 𝑎 𝑠 ∗ 𝜔2 ∗ 𝑟 ∗ 𝑐𝑜𝑠𝜃 [ 𝐴𝑡 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒] ℎ 𝑑𝑠 = 𝑙 𝑑 𝑔 ∗ 𝐴 𝑎 𝑑 ∗ 𝜔2 ∗ 𝑟 ∗ 𝑐𝑜𝑠𝜃 [ 𝐴𝑡 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒] 𝐴 = 𝜋 4 𝐷2 𝑎 𝑠 = 𝜋 4 𝑑 𝑠 2 𝑎 𝑑 = 𝜋 4 𝑑 𝑑 2 𝜔 = 2𝜋𝑁 60
  • 43. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 43 𝑟 = 𝐿 2 Symbol Description Unit 𝑙 𝑠 Length of Suction Pipe 𝑚 𝑙 𝑑 Length of Delivery Pipe 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐴 Area of Cylinder 𝑚2 𝑎 𝑠 Area of Suction Pipe 𝑚2 𝑎 𝑑 Area of Delivery Pipe 𝑚2 𝜔 Angular Speed 𝑟𝑎𝑑 𝑠⁄ 𝑟 Radius of Crank 𝑚 𝜃 Angle of Crank 𝑑𝑒𝑔𝑟𝑒𝑒 𝐷 Diameter of Cylinder or Bore 𝑚 𝑑 𝑠 Diameter of Suction Pipe 𝑚 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 𝑁 Speed of Crank 𝑟𝑝𝑚 𝐿 Stroke Length 𝑚 PRESSURE HEAD: 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = ℎ 𝑠 + ℎ 𝑎𝑠 [ 𝐹𝑜𝑟 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒] 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = ℎ 𝑑 + ℎ 𝑎𝑑 [ 𝐹𝑜𝑟 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒] Symbol Description Unit ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 ℎ 𝑎𝑠 Acceleration Head at Suction 𝑚
  • 44. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 44 ℎ 𝑎𝑑 Acceleration Head at Delivery 𝑚 ABSOLUTE PRESSURE HEAD: 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = 𝐻 𝑎𝑡𝑚 − (ℎ 𝑠 + ℎ 𝑎𝑠) [ 𝐹𝑜𝑟 𝑆𝑢𝑐𝑡𝑖𝑜𝑛 𝑆𝑡𝑟𝑜𝑘𝑒] 𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐻𝑒𝑎𝑑 = 𝐻 𝑎𝑡𝑚 + (ℎ 𝑑 + ℎ 𝑎𝑑 ) [ 𝐹𝑜𝑟 𝐷𝑒𝑙𝑖𝑣𝑒𝑟𝑦 𝑆𝑡𝑟𝑜𝑘𝑒] Symbol Description Unit ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 ℎ 𝑎𝑠 Acceleration Head at Suction 𝑚 ℎ 𝑎𝑑 Acceleration Head at Delivery 𝑚 𝐻 𝑎𝑡𝑚 Atmospheric Pressure Head 𝑚 SEPARATION HEAD: 𝑃𝑠𝑒𝑝 = 𝜌𝑔ℎ 𝑆𝑒𝑝 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ ℎ 𝑠𝑒𝑝 Separation Head 𝑚 𝑃𝑠𝑒𝑝 Separation Pressure 𝑁 𝑚2⁄ HEAD LOSS WITHOUT AIR VESSEL: ℎ 𝑓𝑊𝑂𝐴 = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe 𝑚
  • 45. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 45 𝑣 Velocity without Air Vessel 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 VELOCITY WITHOUT AIR VESSEL: 𝑣 = 𝐴 𝑎 𝑑 ∗ 𝜔 ∗ 𝑟 𝐴 = 𝜋 4 𝐷2 𝑎 𝑑 = 𝜋 4 𝑑 𝑑 2 𝜔 = 2𝜋𝑁 60 𝑟 = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder 𝑚2 𝑎 𝑑 Area of Delivery Pipe 𝑚2 𝜔 Angular Speed 𝑟𝑎𝑑 𝑠⁄ 𝑟 Radius of Crank 𝑚 𝐷 Diameter of Cylinder or Bore 𝑚 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 𝑁 Speed of Crank 𝑟𝑝𝑚 𝐿 Stroke Length 𝑚
  • 46. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 46 HEAD LOSS WITH AIR VESSEL: ℎ 𝑓𝑊𝐴 = 4𝑓𝑙 𝑑 𝑣2 2𝑔𝑑 𝑑 Symbol Description Unit 𝑓 Friction Factor 𝑙 𝑑 Length of Delivery Pipe 𝑚 𝑣 Velocity with Air Vessel 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 VELOCITY WITH AIR VESSEL: 𝑣 = 𝐴 𝑎 𝑑 ∗ 𝜔 ∗ 𝑟 𝜋 𝐴 = 𝜋 4 𝐷2 𝑎 𝑑 = 𝜋 4 𝑑 𝑑 2 𝜔 = 2𝜋𝑁 60 𝑟 = 𝐿 2 Symbol Description Unit 𝐴 Area of Cylinder 𝑚2 𝑎 𝑑 Area of Delivery Pipe 𝑚2 𝜔 Angular Speed 𝑟𝑎𝑑 𝑠⁄ 𝑟 Radius of Crank 𝑚 𝐷 Diameter of Cylinder or Bore 𝑚
  • 47. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 47 𝑑 𝑑 Diameter of Delivery Pipe 𝑚 𝑁 Speed of Crank 𝑟𝑝𝑚 𝐿 Stroke Length 𝑚 POWER SAVED BY AIR VESSEL: 𝑃 = 𝜌𝑔𝑄 ( 2 3 ℎ 𝑓𝑊𝑂𝐴 − ℎ 𝑓𝑊𝐴) Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ ℎ 𝑓𝑊𝑂𝐴 Head Loss Without Air Vessel 𝑚 ℎ 𝑓𝑊𝐴 Head Loss With Air Vessel 𝑚 POWER REQUIRED TO DRIVE THE PUMP: 𝑃 = 𝜌𝑔𝑄 (ℎ 𝑠 + ℎ 𝑑 + 2 3 ℎ 𝑓𝑠𝑊𝑂𝐴 + ℎ 𝑓𝑑𝑊𝐴) Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ ℎ 𝑠 Suction Head 𝑚 ℎ 𝑑 Delivery Head 𝑚 ℎ 𝑓𝑠𝑊𝑂𝐴 Head Loss Without Air Vessel at Suction 𝑚 ℎ 𝑓𝑑𝑊𝐴 Head Loss With Air Vessel at Delivery 𝑚
  • 48. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 48 PELTON WHEEL: Symbol Description Unit 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛽 Angle made by Absolute Velocity at Outlet with the Direction of Motion of Vane Degree 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree UNIT – V – TURBINES
  • 49. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 49 TANGENTIAL VELOCITY AT INLET AND OUTLET (OR) VELOCITY OF WHEEL: 𝑢 = 𝜋𝐷𝑁 60 Symbol Description Unit 𝐷 Diameter of Runner 𝑚 𝑁 Speed of Impeller 𝑟𝑝𝑚 VELOCITY OF JET: 𝑉1 = 𝐶𝑣√2𝑔𝐻 𝐶𝑣 = 0.97 − 0.99 Symbol Description Unit 𝐶𝑣 Coefficient of Velocity 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 VELOCITY OF WHEEL: 𝑢 = 𝑘 𝑢√2𝑔𝐻 𝑘 𝑢 = 0.43 − 0.45 Symbol Description Unit 𝑘 𝑢 Speed Ratio 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: 𝑉 𝑤1 = 𝑉1 𝑉 𝑤1 = 𝑢1 + 𝑉𝑟1
  • 50. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 50 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ FROM OUTLET VELOCITY TRIANGLE DIAGRAM: cos 𝜙 = 𝑢2 + 𝑣 𝑤2 𝑣 𝑟2 tan 𝜙 = 𝑣𝑓2 𝑢2 + 𝑣 𝑤2 sin 𝜙 = 𝑣𝑓2 𝑣 𝑟2 tan 𝛽 = 𝑣𝑓2 𝑣 𝑤2 Symbol Description Unit 𝑢2 Tangential Velocity of Runner at Outlet 𝑚 𝑠⁄ 𝑣 𝑟2 Relative Velocity at Outlet 𝑚 𝑠⁄ 𝑣 𝑤2 Whirl Velocity at Outlet 𝑚 𝑠⁄ 𝑣𝑓2 Flow Velocity at Outlet 𝑚 𝑠⁄ WORK DONE BY JET PER SECOND: 𝑊 = 𝜌𝑄 [ 𝑣 𝑤1 + 𝑣 𝑤2] 𝑢 Symbol Description Unit 𝑢 Tangential Velocity of Runner 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄
  • 51. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 51 HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 2[ 𝑣 𝑤1 + 𝑣 𝑤2] 𝑢 𝑉1 2 Symbol Description Unit 𝑢 Tangential Velocity of Runner 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 DISCHARGE OF SINGLE JET: 𝑞 = 𝜋 4 ∗ 𝑑2 ∗ 𝑉1 Symbol Description Unit 𝑑 Diameter of Jet 𝑚 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑞 Discharge of Single Jet 𝑚3 𝑠⁄
  • 52. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 52 NUMBER OF JET: 𝑛 = 𝑄 𝑞 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝑞 Discharge of Single Jet 𝑚3 𝑠⁄ NUMBER OF BUCKET: 𝑍 = 15 + 𝐷 2𝑑 Symbol Description Unit 𝑑 Diameter of Jet 𝑚 𝐷 Diameter of Runner 𝑚 DIMENSIONS OF BUCKET: 𝐴𝑥𝑖𝑎𝑙 𝑊𝑖𝑑𝑡ℎ 𝐵 = 4.5𝑑 𝑅𝑎𝑑𝑖𝑎𝑙 𝐿𝑒𝑛𝑔𝑡ℎ 𝐿 = 2.5𝑑 𝐷𝑒𝑝𝑡ℎ 𝑜𝑓 𝐵𝑢𝑐𝑘𝑒𝑡 𝑇 = 𝑑 Symbol Description Unit 𝑑 Diameter of Jet 𝑚 KINETIC ENERGY OF JET: 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 = 1 2 𝑚 𝑉1 2 𝑆𝑖𝑛𝑐𝑒 𝑚 = 𝜌𝐴𝑉 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 = 1 2 𝜌 ∗ 𝐴 ∗ 𝑉1 ∗ 𝑉1 2
  • 53. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 53 𝑆𝑖𝑛𝑐𝑒 𝑄 = 𝐴𝑉 𝑇ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝐾. 𝐸 𝑜𝑓 𝐽𝑒𝑡 = 1 2 𝜌 ∗ 𝑄 ∗ 𝑉1 2 POWER LOST IN NOZZLE: 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 = 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑁𝑜𝑧𝑧𝑙𝑒 POWER LOST IN RUNNER: 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 = 𝑃𝑜𝑤𝑒𝑟 𝑜𝑓 𝑆ℎ𝑎𝑓𝑡 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑁𝑜𝑧𝑧𝑙𝑒 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝑖𝑛 𝑅𝑢𝑛𝑛𝑒𝑟 + 𝑃𝑜𝑤𝑒𝑟 𝐿𝑜𝑠𝑡 𝐷𝑢𝑒 𝑡𝑜 𝑀𝑒𝑐ℎ𝑎𝑛𝑖𝑐𝑎𝑙 𝑅𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 RESULTANT FORCE ON BUCKET: 𝐹 = 𝜌𝑄 [ 𝑣 𝑤1 + 𝑣 𝑤2] Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ TORQUE: 𝑇 = 𝐹 ∗ 𝐷 2 Symbol Description Unit 𝐹 Resultant Force on Bucket 𝑁 𝐷 Diameter of Runner 𝑚 𝑇 Torque 𝑁 − 𝑚
  • 54. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 54 POWER: 𝑃 = 2𝜋𝑁𝑇 60 Symbol Description Unit 𝑃 Power 𝑊 𝑇 Torque 𝑁 − 𝑚 N Speed of Shaft Rpm SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed
  • 55. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 55 REACTION TURBINE: INWARD FLOW REACTION TURBINE: Symbol Description Unit 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 Angle made by Relative Velocity at Outlet with the Degree
  • 56. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 56 Direction of Motion of Vane TANGENTIAL VELOCITY AT INLET: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1
  • 57. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 57 tan 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit
  • 58. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 58 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝐴 Area of Runner 𝑚2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet 𝑚 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ HEAD AT INLET OF TURBINE: 𝐻 = 1 𝑔 ∗ 𝑣 𝑤1 ∗ 𝑢1 + 𝑣𝑓1 2 2𝑔 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄
  • 59. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 59 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻
  • 60. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 60 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 SPEED RATIO: 𝐾 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio
  • 61. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 61 SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed OUTWARD FLOW REACTION TURBINE: Symbol Description Unit 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄
  • 62. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 62 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree TANGENTIAL VELOCITY AT INLET: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) Internal Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM:
  • 63. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 63 sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1 tan 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree
  • 64. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 64 RELATIVE VELOCITY AT INLET: 𝑣 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝐴 Area of Runner 𝑚2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet 𝑚 𝑠⁄ MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄
  • 65. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 65 INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚
  • 66. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 66 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 SPEED RATIO: 𝐾 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 0.3
  • 67. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 67 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed
  • 68. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 68 FRANCIS TURBINE: Symbol Description Unit 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree
  • 69. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 69 TANGENTIAL VELOCITY AT INLET: 𝑢1 = 𝜋𝑑1 𝑁 60 Symbol Description Unit 𝑑1 Inlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝑑2 𝑁 60 Symbol Description Unit 𝑑2 Outlet (or) Internal Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1 tan 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1
  • 70. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 70 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋𝑑1 𝑏1 𝑣𝑓1 = 𝜋𝑑2 𝑏2 𝑣𝑓2 𝑄 = 𝐴𝑣𝑓1 = 𝐴𝑣𝑓2 = 𝐴 𝑓1 𝑣𝑓1 = 𝐴 𝑓2 𝑣𝑓2 Symbol Description Unit 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚
  • 71. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 71 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝑄 Discharge 𝑚3 𝑠⁄ 𝐴 Area of Runner 𝑚2 𝐴 𝑓1&𝐴 𝑓2 Area of Flow at Inlet & Outlet 𝑚 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = 𝜋𝑑1 𝑏1 = 𝜋𝑑2 𝑏2 Symbol Description Unit 𝑑1&𝑑2 Diameter of Impeller at Inlet & Outlet 𝑚 𝑏1&𝑏2 Width of Impeller at Inlet & Outlet 𝑚 𝐴 Circumferential Area of Runner 𝑚2 MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚
  • 72. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 72 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄
  • 73. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 73 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊 SPEED RATIO: 𝐾 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio
  • 74. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 74 BREADTH RATIO: 𝑛 = 𝑏1 𝑑1 𝑛 = 0.1 − 0.4 Symbol Description Unit 𝑏1 Width of Runner at Inlet 𝑚 𝑑1 Diameter of Runner at Inlet 𝑚 𝑛 Breadth Ratio SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄ Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed
  • 75. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 75 KAPLAN TURBINE: Symbol Description Unit 𝑢1&𝑢2 Tangential Velocity of Runner at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑟1&𝑣 𝑟2 Relative Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣 𝑤1&𝑣 𝑤2 Whirl Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑉1&𝑉2 Absolute Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝑣𝑓1&𝑣𝑓2 Flow Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree 𝜙 Angle made by Relative Velocity at Outlet with the Direction of Motion of Vane Degree
  • 76. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 76 TANGENTIAL VELOCITY AT INLET: 𝑢1 = 𝜋𝐷 𝑜 𝑁 60 Symbol Description Unit 𝐷 𝑜 Inlet (or) External Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 TANGENTIAL VELOCITY AT OUTLET: 𝑢2 = 𝜋𝐷 𝑏 𝑁 60 = 𝜋𝐷ℎ 𝑁 60 Symbol Description Unit 𝐷 𝑏 𝑜𝑟 𝐷ℎ Outlet (or) Boss (or) Hub Diameter 𝑚 𝑁 Speed of Turbine 𝑟𝑝𝑚 FROM INLET VELOCITY TRIANGLE DIAGRAM: sin 𝛼 = 𝑣𝑓1 𝑉1 cos 𝛼 = 𝑣 𝑤1 𝑉1 tan 𝛼 = 𝑣𝑓1 𝑣 𝑤1 sin 𝜃 = 𝑣𝑓1 𝑣 𝑟1 cos 𝜃 = 𝑣 𝑤1 − 𝑢1 𝑣 𝑟1 tan 𝜃 = 𝑣𝑓1 𝑣 𝑤1 − 𝑢1
  • 77. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 77 Symbol Description Unit 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑉1 Absolute Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝛼 Angle made by Absolute Velocity at Inlet with the Direction of Motion of Vane Degree 𝜃 Angle made by Relative Velocity at Inlet with the Direction of Motion of Vane Degree RELATIVE VELOCITY AT INLET: 𝑣 𝑟1 = √ 𝑣𝑓1 2 + ( 𝑣 𝑤1 − 𝑢1)2 Symbol Description Unit 𝑣 𝑟1 Relative Velocity at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ DISCHARGE: 𝑄 = 𝜋 4 [𝐷0 2 − 𝐷 𝑏 2 ]𝑣𝑓1 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐷0 Inlet (or) External Diameter 𝑚 𝐷 𝑏 𝑜𝑟 𝐷ℎ Outlet (or) Boss (or) Hub Diameter 𝑚
  • 78. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 78 𝑄 Discharge 𝑚3 𝑠⁄ CIRCUMFERENTIAL AREA OF RUNNER: 𝐴 = 𝜋 4 [𝐷0 2 − 𝐷 𝑏 2 ] Symbol Description Unit 𝐷0 Inlet (or) External Diameter 𝑚 𝐷 𝑏 𝑜𝑟 𝐷ℎ Outlet (or) Boss (or) Hub Diameter 𝑚 𝐴 Circumferential Area of Runner 𝑚2 MASS OF WATER FLOWING THROUGH THE RUNNER: 𝑚 = 𝜌 𝑄 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝜌 Density 𝑘𝑔 𝑚3⁄ INPUT POWER TO TURBINE (OR) POWER GIVEN TO TURBINE: 𝑃 = 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 POWER DEVELOPED BY TURBINE: 𝑃 = 𝜌 ∗ 𝑄 ∗ 𝑣 𝑤1 ∗ 𝑢1 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄
  • 79. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 79 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ HYDRAULIC EFFICIENCY: 𝜂ℎ𝑦𝑑 = 𝑣 𝑤1 𝑢1 𝑔𝐻 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 − 𝐻𝑒𝑎𝑑 𝐿𝑜𝑠𝑠 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 Symbol Description Unit 𝑢1 Tangential Velocity of Runner at Inlet 𝑚 𝑠⁄ 𝑣 𝑤1 Whirl Velocity at Inlet 𝑚 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 OVERALL EFFICIENCY: 𝜂 𝑜 = 𝑆ℎ𝑎𝑓𝑡 𝑃𝑜𝑤𝑒𝑟 𝐼𝑛𝑝𝑢𝑡 𝑃𝑜𝑤𝑒𝑟 𝜂 𝑜 = 𝑆. 𝑃 𝜌𝑔𝑄𝐻 Symbol Description Unit 𝜌 Density 𝑘𝑔 𝑚3⁄ 𝑄 Discharge 𝑚3 𝑠⁄ 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐻 Head 𝑚 𝑆. 𝑃 Shaft Power 𝑊
  • 80. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 80 SPEED RATIO: 𝐾 𝑢 = 𝑢 √2𝑔𝐻 𝐾 𝑢 = 0.6 − 0.9 Symbol Description Unit 𝑢 Tangential Velocity 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾 𝑢 Speed Ratio FLOW RATIO: 𝐾𝑓 = 𝑣𝑓1 √2𝑔𝐻 𝐾𝑓 = 0.15 − 0.3 Symbol Description Unit 𝑣𝑓1 Flow Velocity at Inlet 𝑚 𝑠⁄ 𝐻 Head 𝑚 𝑔 Acceleration due to Gravity 𝑚 𝑠2⁄ 𝐾𝑓 Flow Ratio SPECIFIC SPEED: 𝑁𝑠 = 𝑁√ 𝑄 𝐻 3 4⁄ 𝑁𝑠 = 𝑁√ 𝑃 𝐻 5 4⁄
  • 81. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 81 Symbol Description Unit 𝑄 Discharge 𝑚3 𝑠⁄ 𝐻 Head 𝑚 𝑃 Power 𝑘𝑊 𝑁 Speed 𝑟𝑝𝑚 𝑁𝑠 Specific Speed DRAFT TUBE:
  • 82. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 82 Symbol Description Unit 𝑉1&𝑉2 Velocity at Inlet & Outlet 𝑚 𝑠⁄ 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race 𝑚 𝑦 Distance of Bottom of Draft Tube from Tail Race 𝑚 FROM BERNOULLI’S EQUATION: 𝑃1 𝜌𝑔 + 𝑉1 2 2𝑔 + 𝑧1 = 𝑃2 𝜌𝑔 + 𝑉2 2 2𝑔 + 𝑧2 + ℎ 𝑓 Symbol Description Unit 𝑃1 & 𝑃2 Pressure at Inlet & Outlet of Draft Tube 𝑁 𝑚2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube 𝑚 𝑠⁄ 𝑧1 & 𝑧2 Datum Head Inlet & Outlet of Draft Tube 𝑚 ℎ 𝑓 Head Loss 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ LENGTH OF DRAFT TUBE: 𝐿 = 𝐻𝑠 + 𝑦 Symbol Description Unit 𝐿 Length of Draft Tube 𝑚 𝐻𝑠 Vertical Height of Draft Tube Above Tail Race 𝑚 𝑦 Distance of Bottom of Draft Tube from Tail Race 𝑚 EFFICIENCY OF DRAFT TUBE: 𝜂 𝑑 = ( 𝑉1 2 2𝑔 − 𝑉2 2 2𝑔 ) − ℎ 𝑓 𝑉1 2 2𝑔
  • 83. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017 CE6451 – FLUID MECHANICS AND MACHINERY FORMULA BOOK Prepared By BIBIN.C / ASHOK KUMAR.R / SADASIVAN . N (AP / Mech) 83 Symbol Description Unit 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube 𝑚 𝑠⁄ ℎ 𝑓 Head Loss 𝑚 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄ HYDRAULIC EFFICIENCY OF DRAFT TUBE: 𝜂ℎ𝑦𝑑 = 𝐻𝑒𝑎𝑑 𝑈𝑡𝑖𝑙𝑖𝑧𝑒𝑑 𝑏𝑦 𝑇𝑢𝑟𝑏𝑖𝑛𝑒 𝐻𝑒𝑎𝑑 𝐼𝑛𝑙𝑒𝑡 𝑜𝑓 𝑇𝑢𝑟𝑏𝑖𝑛𝑒 𝜂ℎ𝑦𝑑 = 𝐻 − ℎ 𝑓𝑡 − ℎ 𝑓𝑑 − 𝑉2 2 2𝑔 𝑃1 𝜌𝑔 + 𝑉1 2 2𝑔 + 𝑧1 Symbol Description Unit 𝑃1 Pressure at Inlet of Draft Tube 𝑁 𝑚2⁄ 𝑉1 & 𝑉2 Velocity at Inlet & Outlet of Draft Tube 𝑚 𝑠⁄ 𝑧1 Datum Head Inlet of Draft Tube 𝑚 ℎ 𝑓 Head Loss 𝑚 𝜌 Density of Liquid 𝑘𝑔 𝑚3⁄ 𝑔 Acceleration due to gravity 𝑚 𝑠2⁄