PENSTOCK THICKNESS CALCULATION (CASE STUDY )

A hydroelectric plant is planned to be built by utilizing the waste water from the reservoir of 3.5 m3 / s, which flowed into a penstock with a diameter of 1600 mm and the thickness (e) 9 mm. As you know, the more thick penstock then the price will also be more expensive.

Try to analyze whether the specifications of the pipe can still be revised, especially for the thickness of pipe used, whether the thickness can be reduced to (e) 6 mm, thus decreasing the cost of piping.

Existing data:

• The material used is Mild steel ( rolled welded steel pipe)
• The diameter of the pipe /penstock (Dp) = 1.6 m (1600 mm)
• Pipe thickness (e) = 9 mm(which will be in the analysis)
• Penstock length (Lp) = 66 m
• Flow of water (Qp) = 3.5m3/det
• High gross ( H gross) = 7.3 m
• Head Loss (H loss)= 0.3 m
• Net Head = 7 m
• Water velocity in thepenstock (V) = 1.74 m / s
• Efficiency of penstock (pipe eff) = H net /  H gross * 100% = 96%
• visc µ = 0.00114 kg / (m.det)
• Density of  water = 1000 kg/m3
• K (bulk modulus of water) = 2.1*10^9 N/m2
• ts (Tensile strength pipe) = 400 * 10^6 N/m2
• E (modulus of elasticity) = 206 * 10^9 (N/m2)
• In this design, penstock is used having a thickness (e) = 9 mm

Working pressure (P work) of the penstock (Mild steel )
 The desired pressure of 1.5 * Gross Head (Hg)

Note: ...[3]

The addition of pressure on the penstock :

Head up to 50 m is not more 50%

Head of 50 to150 is no more than 25%

Head up to 250 m is not more 15%

Then :
Working pressure (P)   = 1.5 *  7.3  = 10.95 m
                                    = 1.5 * (7.3 / 10) = 1.095 kgf/cm2 (Bars)
                                    = 1.095 kg/cm2 *10000 cm2/m2 * 9.81 m/s2
                                    = 107,419 N/m2 (pascal)
                                    = 107,419 N/m2 * (kN/1000N) * (m2/1000000 mm2)
                                    = 0.000107419 kN/mm2

The formula for a thin tube ( if  Dp / e > 20)...[3]
Where:
e     = thickness of penstock in mm
es    = extra thickness for corrosion (1-3 mm )...[3]
Working pressure (P) = 0.000107419 kN/mm2
Dp = diameter of 1600 mm penstock
ts (Tensile strength)  = 400* 10^6 N/m2
                               = (400 * 10^6 N/m2) * (kN/1000N) * (m2/1000.000 mm2)
                               = 0.400 kN/mm2
Minimum penstock thickness (e)
e = (P * Dp) / (2 ts) + es...[2]
        Taken extra thick for corrosion es) = 3 mm
e = (0.000107419 kN/mm2x 1600 mm) / (2 x 0.400 kN/mm2) + 3 mm
e = 3.21 mm

Check the use of penstock thickness formula
Category of penstock used , Dp / e = 1600 / 3.21
                                                      = 498 > 20 (a thin tube, the formulas is ok)

The impact of pipe handling in transportation, laying, deformation, etc., it is necessary to add more rapidly the penstock thickness (in the wills of 3 mm). So thick of penstock (e) is = 3+3.21= 6.21 mm, the thickness of the penstock taken at least 6 mm (see the availability of the thickness of the penstock in the market)

Effect of Water Hammer
In the design of penstock also must take into account the effects of water and control the speed lacing.

If the H / L> 5, the surge tank is required ...[1]
In this design:
H / L   = H gross / length of pipe (L)
           = 7.3 / 66
           = 0.11 < 5
 (Not required surge tank but the effects of  water hammer still be calculated)
The thickness of the penstock (e) = 6 mm is to be used

At wills:
% Closure of the valve flow (z) = 50%
With the closing time (T close)  = 4 seconds (fast enough)
Corrosion allowed (es)             = 3 mm
Overall safety factor (SF)         = 4

Calculation:
The speed of water waves:
C wave = [(10^ (-3)* K) / (1 + (K* Dp / E* e)]^ (0.5) ...[2]
where:

• K = bulk modulus of water 2.1x10^ 9 N/m2
• E = modulus of elasticity of pipe material 206 * 10^9 (N/m2)
• D = pipe diameter 1600 mm
• e = wall thickness 6 mm
• L = length of pipe, 66 m

By entering values:
C.wave         = [(10^(-3)* 2.1*10^9) / (1 + (2.1*10^9 x 1600 / (206 * 10^9 * 6)] ^0.5
                    = 751.5 m / s
Critical closing time of the penstock (Tc) 

The time it takes the pressure wave (pressure wave) to return again to the valve after the sudden closure, known as the critical time.

Tc = 2 L / C...[2]
     = 2 x 66 m / 751.5 m / s.
     = 0.716 seconds
T.Close  (4 sec)> Tc  (0.716 sec)... [4]

Kc   = L* z * V / ( g * Water density * H.gross *  T Close)
        = 66 m * 50 * 1.74 m / s / (9.8 * 1000 kg/m3 m/s2 * 7.3 mx 4 s)
        = 0.2
Surge pressure (H.surge )...[4]

H.surge       = H gross * [(Kc / 2) + ((Kc + (Kc^2 / 4)) ^0.5]
                  = 7.3 * [0.2 / 2 + ((0.2 + (0.22/ 4))^ 0.5
                  = 4.07 m
H.total        = H.surge + H.gross
                  = 4.07 m + 7.3 m
                  = 11.37 m (exceeds the pressure of work, a total of 11.37 m >P work 10.95 m)

For a Total Head ( H.total ) of 11.37 m , the required minimum thicknessof the penstock (e)
e     = (H.total * Dp * SF / 83700) + es
       = (11.37 m * 1600 mm * 4 / 83700) + 3 mm
       = 3, 87 mm ( penstock with a thickness of 6 mm is adequate)

Ref :

1. AHEC/MNRE/SHP Standards/ Civil Works –  Guidelines For Layout Of Small Hydro Plants /Feb 2008. (p-77)
2. ESHA (European Small Hydropower Association),”Layman’s Handbook on How To Develop a
   Small Hydro Site,”2nd ed, 1998 (p-144/145)
3. Patty O.F., Tenaga Air, Erlangga, Jakarta1995 (p-62/64)
4. STEEL PENSTOCK LOSSES & THICKNESS CALCULATION (p-1)
   http://www.energyservices.lk/pdf/techspecs/vh_w_b/pensteel.pdf