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⁄
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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
66. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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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
67. 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
68. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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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
69. 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
70. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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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
71. 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
72. 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⁄
73. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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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
74. 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
75. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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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
76. 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
78. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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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⁄
79. R.M.K COLLEGE OF ENGG AND TECH / AQ / R2013/ CE6451 / III / MECH / JUNE 2017 – NOV 2017
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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⁄
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⁄