TRANSIENT ANALYSIS OF PENSTOCK
The structural design of penstock mainly refers to the thickness calculation for the desired hoop stress in case of exposed pipe and hoop and external stresses for buried pipe.
The thickness of pipe shall be checked by either Indian design code of IS 11639 (Part 2): 1995 or other similar codes.
Allowable stress
a. According to ASME code, a penstock may be designed under the following conditions
(i) Normal condition- Maximum static head plus pressure rise due to normal operation
Allowable stress = ultimate tensile strength/3 or 2/3 (0.5) * minimum yield stress, whichever is lesser.
(ii) Intermittent Condition-Condition during filling & draining the penstock
Allowable stress = Ultimate tensile strength/2.25 (2.5) or 0.8 (2/3) * yield strength, whichever is smaller
(iii) Emergency condition- Gate closure
Allowable stress = Ultimate tensile strength /1.5 (2) or 0.8* yield strength, whichever is smaller
(iv) Exceptional condition
Allowable stress < Minimum yield stress.
b. According to Indian Standard (IS 11639: 1995)
(i) Normal condition- Maximum static head plus pressure rise due to normal operation
Allowable stress = ultimate tensile strength/3 or 0.6* minimum yield stress, whichever is lesser.
(ii) Intermittent Condition-Condition during filling & draining the penstock
Allowable stress = 0.4* Ultimate tensile strength or 2/3 of yield strength, whichever is smaller
(iii) Emergency condition- Gate closure
Allowable stress = 2/3* Ultimate tensile strength or 0.9* yield strength, whichever is smaller
(iv) Exceptional condition
Allowable stress < Minimum yield stress
c. According to American Water Works Association (AWWA)
(i) Allowable working pressure, allowable stress = 50% of Yield strength
(ii) Allowable transient, allowable stress = 75% of Yield strength
Loading condition
As per IS 11639 (Part 2): 1995
a) Normal condition: Maximum static head plus pressure rise due to normal operation or head at transient maximum surge whichever is higher.
b) Intermittent Condition: Condition during filling & draining the penstocks and maximum surge in combination with pressure rise during normal operation.
c) Emergency condition- Condition includes partial gate closure in critical time of penstock (2L/a seconds) at maximum rate, and the cushioning stroke being inoperative in one unit
d) Exceptional condition-Condition includes slam shut, malfunctioning of control equipment in the most adverse manner resulting in odd situation of extreme loading. This should not be taken as a design criterion.
WATER HAMMER
Propagation of wave is given by transient formula
Where,
\(γ\) = Specific weight of water = 9.8 kN/m3
\(k\) = Bulk modulus of water = 2.04 x 106 kN/m2
\(E\) = Elastic material of pipe material
\(D\) = Diameter of pipe
\(g\) = Acceleration due to gravity = 9.8 m2/s
As per IS 11639 (Part 2): 1995, the pressure wave velocity in a steel penstock carrying water may be computed as
Where,
\(D\) = Diameter of pipe in m
\(t\) = Thickness of pipe in mm
Thickness of pipe to calculate rise in level.
Allievi Method
Design process
1) Calculate the individual wave velocities for different length and pipe diameter from Equations above
2) Calculate average wave velocity from following formula
3) Calculate average flow velocity
4) Calculate travelling time of wave
5) Calculate increase of height of water due to wave
Where,
Where,
\(\rho\) = Pipe coefficient of Allievi
\(\theta\) = Closing time of closure
\(h\) = Rise of water level due to hammer
\(H\) = Head at that point (Closure) = Net head = Total Head-Head loss
\(V_m\) = Average velocity of pipe
\(T\) = Closing time of closure
\(\alpha_m\) = Average wave velocity in pipes
6) Calculation of surge at each point (Bends, reducer, valve etc.)
Surge level at a point = Normal water level + hi
Surge height at a point = Surge level – Pipe center line level
7) Calculation of thickness
i) Minimum thickness of pipe for handling
Where,
\(t\) = Thickness in mm
\(D\) = Diameter of pipe in mm
As per Indian standard (Clause 7.2 of IS 11639: 1995)
Where,
\(R\) = Radius of pie in m
ii) Assume pipe thickness
iii) Calculation of circumferential stress
Where,
\(P\) = Water pressure =\( γ H\)
\(D\) = Diameter of pipe
\(\epsilon_1\) = Internal corrosion thickness
\(\epsilon_2\) = External corrosion thickness
\(t\) = Thickness of pipe
\(η\) = Welding efficiency = 0.9 to 1.0
iv) Calculation of axial stress
Where,
\(ν\) = Poisson's ratio
v) Calculation of combined circumferential stress
If \(\sigma_a>sigma_y\) and \(\sigma_a>\sigma_g\), The assumed thickness of pipe is safe.
Design for external pressure (As per IS 11639 Part 2: 1995)
A) Vaughan’s formula
Where,
\(K\) = ratio of pipe diameter to plate thickness
\(E_s\) = modulus of elasticity of steel on N/m2
\(mu_s\) = Poisson’s ratio of steel
\(P_{cr}\) = critical external pressure at buckling in N/m2
\(Y_o\) = initial gap between steel lining and concrete in m
\(R\) = radius of steel liner in m
\(F_y\) = yield point stress in steel in N/m2
B) Amstutz’s formula
And
Where,
\(E'=\frac{E_s}{1-mu_s^2}\)
\(f_y'=\frac{f_y}{\sqrt(1-μ_s+μ_s^2) }\)
\(f_y\) = yield stress in steel in N/m2
\(E_s\) = modulus of elasticity of steel in N/m2
\(mu_s\) = Poisson’s ratio of steel
\(R\) = Radius of liner in m
\(Y_o\) = initial gap between steel lining and concrete in m
\(f_n\) = allowable stress in steel in N/m2
\(P_{cr}\) = critical external pressure at buckling in N/m2
\(t\) = thickness of steel liner in m.
The maximum external pressure on the penstock should not exceed two-thirds times the critical external pressure calculated according to Vaughan’s equation or Amstutz’s equation. If the maximum pressure exceeds the value equal to two-thirds times the critical external pressure for unstiffened shell, stiffener should be provided to prevent buckling.