M E 320
Professor John M. Cimbala
Lecture 26
Today, we will:
•
•
•
Discuss dimensional analysis of turbines
Do an example problem – dimensional analysis with turbines
Discuss piping networks – how to deal with pipes in series or in parallel
b. Dimensionless parameters in turbine performance
We perform exactly the same dimensional analysis for turbines as we did for pumps. Result:
V
gH
bhp
Dimensionless Parameters: CQ =
CH = 2 2
CP =
3
ωD
ω D
ρω 3 D 5
Capacity coefficient
Head coefficient
Power coefficient
Example: Scaling up a hydroturbine
Given:
An existing hydroturbine (A): Fluid is water at 20oC, DA = 1.95 m,
n A = 120 rpm, bhpA = 220 MW, and VA = 335 m 3 /s at H A = 72.4 m . We are designing a new
turbine (B) that is geometrically similar, still uses water at 20oC, and nB = 120 rpm, but
H B = 97.4 m . [Dam B has a higher gross head available than Dam A.]
To do:
(a) Calculate DB and VB for operation of turbine B at a homologous point.
(b) Calculate bhpB and estimate the turbine efficiency of both turbines.
Solution:
(a) At homologous points, the two turbines are dynamically similar. Apply the affinity laws:
⎛ ω ⎞ HB
⎛ n ⎞ H B
gH
gH
= DA ⎜ A ⎟
CH , A = 2 A 2 = CH , B = 2 B 2 → solve for DB = DA ⎜ A ⎟
ω A DA
ωB DB
⎝ ωB ⎠ H A
⎝ n B ⎠ H A
Plug in numbers:
3
3
⎛
⎞⎛
⎞
⎛
⎞⎛
⎞
ω
D
n
D
VA
VB
B
B
B
B
Similarly, CQ , A = ω D 3 = CQ , B = ω D 3 → VB = VA ⎜ ω ⎟⎜ D ⎟ = VA ⎜ n ⎟⎜ D ⎟
A A
B B
⎝ A ⎠⎝ A ⎠
⎝ A ⎠⎝ A ⎠
Plug in numbers:
(b) Similarly, CP , A
bhp A
bhpB
C
=
=
=
P,B
ρ Aω A3 DA5
ρ Bω B 3 DB 5
3
⎛ ρ ⎞⎛ n ⎞ ⎛ D ⎞
→ bhpB = bhp A ⎜ B ⎟⎜ B ⎟ ⎜ B ⎟
⎝ ρ A ⎠⎝ n A ⎠ ⎝ DA ⎠
Plug in numbers:
Finally, the efficiency is calculated for each turbine:
bhp A
220,000,000 W
⎛ N ⋅ m ⎞⎛ kg ⋅ m ⎞
η turbine, A =
=
⎜
⎟⎜ 2
⎟=
3
⎛
ρ A gH AVA ⎛
kg ⎞⎛
m⎞
m ⎞ ⎝ W ⋅ s ⎠⎝ s ⋅ N ⎠
⎟
⎜1000 3 ⎟⎜ 9.81 2 ⎟ 72.4 m ⎜ 335
m ⎠⎝
s ⎠
s ⎠
⎝
⎝
η turbine, B =
bhpB
461,820,979 W
⎛ N ⋅ m ⎞⎛ kg ⋅ m ⎞
=
⎜
⎟⎜ 2
⎟=
3
⎛
ρ B gH BVB ⎛
kg ⎞⎛
m⎞
m ⎞ ⎝ W ⋅ s ⎠⎝ s ⋅ N ⎠
⎟
⎜1000 3 ⎟⎜ 9.81 2 ⎟ 97.4 m ⎜ 522.728
m ⎠⎝
s ⎠
s ⎠
⎝
⎝
5