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

Hydraulic losses in a penstock

Hydraulic losses in a penstock reduce the effective head in proportion to the length and approximately as the square of the water velocity.

Here you can download the way of calculation in xls

Saddles, supporting blocks and expansion joints

By Ahmad Suhendra

The saddles are designed to support the weight of the penstock full of water, but not to resist significant longitudinal forces. The vertical component of the weight to be supported, in kN, has a value of :

F1 = (Wp + Ww) * L Cos θ
where :
Wp = weight of pipe per meter (kN/m)
Ww = weight of water per meter of pipe (kN/m)
L = length of pipe between mid points of each span (m)
θ = Angle of pipe with horizontal

The design of support rings is based on the elastic theory of thin cylindrical shells. The pipe shell is subject to beam and hoop stresses, and the loads are transmitted to the support ring by shear. If penstocks are continuously supported at a number of points, the bending moment at any point of penstock may be calculated assuming that it is a continuous beam, and using the corresponding equation. The rings are welded to the pipe shell with two full length fillet welds and are tied together with diaphragm plates.

The span between supports L is determined by the value of the maximum permissible deflection L/65000. Therefore the maximum length between supports is given by the equation:

L= 182.61 * [(Dp + 0.0147)^4 - Dp^4)]^(1/3) / (Wp + Ww)

Example :
What is The vertical component of the weight to be supported if such data is given below:

1.Diameter of pipe (Dp) = 0.636 m
2.Pipe thickness (e) = 0.005 m (5 mm)
3.Density of pipe (ρ steel) =7.9 ton/m^3
4.Density of water (ρ water) = 1 ton/m^3
5.Angle of pipe with horizontal (θ )= 5 deg


I. Wp (weight of pipe per meter)
= phi * ( Dp + e) * e * ρ steel
= 3.14 (0.636 + 0.005)* 0.005 * 7.9
= 0.079 ton/m

II. Ww (weight of water per meter of pipe)
= [(phi x ( Dp^2) / 4] * ρ water
= [(3.14 x 0.636^2)/4] * 1
= 0.32 ton /m

III. Total weight (Wp + Ww)
= 0.399 ton/m (Wtotal)
= 0.399 * 9.81
= 3.914 kN/m

IV. The maximum length between supports.
L= 182.61 * [(Dp + 0.0147)^4 - Dp^4)]^(1/3) / (Wp + Ww)
= 182.61 * [(0.636 + 0.0147)^4 - 0.636^4]^(1/3) / (3.914)
= 182.61 * [0. 01566 ]^(1/3) / (3.914)
= 182.61 *[0.2502]/3.914
= 11.67 m

V. The vertical component of the weight to be supported.
F1 = (Wp + Ww) * L Cos θ
=3.914 kN/m * 11.67 m* Cos 5
= 45.503 kN

Ref :
ESHA (European Small Hydropower Association),”Layman’s Handbook on How To Develop a Small Hydro Site,”2nd ed, 1998 http://www.scribd.com/doc/8885765/Layman-Handbook-for-hydro-electric-power-plants

Photo
http://www.fr.aps-sales.com/documentos/downloads/HydroPower%20and%20penstock%20applications.pdf

Resistance Coefficient ( Ke ) For Entrance

Inward projecting----Sharp edged---Slightly Rounded---Well Rounded

he = Ke * V₂ ² / (2*g)

he = Entrance losses (m)
Ke = Depends upon the shape of the intake opening
V2 = The average velocity (m/s) of water in penstock
g = Gravitational constant (9.8 m/s^2)

Ref :
ESHA (European Small Hydropower Association),”Layman’s Handbook on How To Develop a Small Hydro Site,”2nd ed, 1998 http://www.scribd.com/doc/8885765/Layman-Handbook-for-hydro-electric-power-plants

Resistance Coefficient K for Sudden Expansion-Contraction

The losses through these fitting are generally evaluated by first obtaining

ß = d2 / d1

Important Note:

the resulting K values as tabled below are based on the flow velocity in the larger pipe,

if the flow velocity in the small pipe is used to evaluate the head loss then the K values tabled below should be multiplied by 

( ß)^4 = (d2 / d1) ^4

Head loss (h):

h_expansion(he) = Ke*(v2)^2 / (2*g)

h_contraction(hc) = Kc*(v2)^2 / (2*g)

v2= Average velocity (m/s) of water in small pipe

g =  Constant of gravity 9.8 m/s^2

Table of Ke & Kc against β = d2 / d1

β	Ke	Kc
0.15	1887.42	965.43
0.2	576	300
0.25	225	120
0.3	102.23	56.17
0.35	51.31	29.24
0.4	27.56	16.41
0.45	15.51	9.72
0.5	9	6
0.55	5.32	3.81
0.6	3.16	2.47
0.65	1.87	1.62
0.7	1.08	1.06
0.75	0.6	0.69
0.8	0.32	0.44
0.85	0.15	0.27
0.9	0.06	0.14
0.95	0.01	0.06
1	0	0

Ref : http://www.roymech.co.uk/Related/Fluids/Fluids_Pipe.html

The head losses in the penstock

By Ahmad Suhendra

The various head losses which occur between reservoir and turbine are as follows:

1. Trashrack (or screen) losses

2. Entrance losses

3. Losses due to pipe friction

4. Bend losses

5. Losses in valve and fittings.

6. Losses in sudden contraction and expansion

Head loss due to installation of a trashrack
ht = [Kt* (t / b) ^ (4 / 3)* (Vo ^ 2 * Sin α)] / (2 * g)
Kt= depends upon the shape of the screen
Vo=Approach velocity (m/s)

Head loss due to entrance
he = Ke * V ^2 / (2 * g)
Ke = depends upon the shape of the intake opening
V = The average velocity of water in penstock
[Permissible velocity in Penstocks,V(m/s) =0.125 (2 g H)^0.5,
Ref : USBR (1961) (P J Bier)]

Head loss due to friction in the penstock
hf = [(10.29* n ^ 2 *Qp ^ 2) / Dp^5.333] *Lp
n = manning's roughness coefficient depends upon the type of pipe

Head loss due to the installation of the bend
hb= Kb * V^2 / (2 * g)
Kb= depends upon the shape of the bend and the condition
of the inside surface
V= The average velocity of water in penstock

Head loss due to Fitting and Valve
hv = Kv *V ^ 2 / (2 * g)
Kv= depends upon the type of fitting and valve
V= The average velocity of water in penstock

Head loss due to sudden contraction and expansion
hc = Kc *V ^ 2 / (2 * g)
Kc= depends upon the type of sudden contraction and expansion
V= The average velocity of water in small pipe(1.273 Q /Dp^2 m/s)

Total Head Loss

h_total = ht + he + hf + hb + hv + hc

Where:

• g = constant of gravity 9.8 m/s^2
• Qp = flow in penstock (m^3/s)
• Dp = inside diameter of penstock (m)
• Lp= lenght of penstock (m)
  
• k = resistance coefficient
  
• t = screen thickness (mm)
• b = width between bars (mm)
• α = angle of inclination from horizontal (deg)

Ref:

1. http://www.iaa.ncku.edu.tw/~aeromems/Mott/ch10.pdf
2. ESHA (European Small Hydropower Association),”Layman’s Handbook on How To Develop a Small Hydro Site,”2nd ed, 1998

Loss coefficients for pipe bends are commonly used in MHP

The losses in figure as shown left , vary according to the R/D ratio and the deflection angle of the bend. An R/D ratio of six results in the lowest head loss, although only a slight decrease is indicated for R/D ratios greater than four. As the fabrication cost of a bend increases with increasing radius and length, there appears to be no economic advantage in using R/D ratios greater than five.

Ref:
BURIED STEEL PENSTOCKS SECOND EDITION 1998
Published by Construction Marketing Committee, AMERICAN IRON AND STEEL INSTITUTE
In cooperation with and editorial collaboration by STEEL PLATE FABRICATORS ASSOCIATION, INC

Manning coefficient (n)for several commercial pipes

By European Small Hydropower Association (ESHA)
Over the years many empirical formulae, based on accumulated experience, have been developed. They are, in general, not based on sound physical principles and even, occasionally, lack dimensional coherence, but are intuitively based on the belief that the friction on a closed full pipe is:

1. Independent of the water pressure
2. Linearly proportional to its length
3. Inversely proportional to a certain power of its diameter
4. Proportional to a certain exponent of the water velocity
5. In turbulent flows it is influenced by the wall roughness

One of these formulae, widely used to estimate the flow in open channels, but also applicable to closed pipes, is that developed by Manning

Q= (1/n)* A^(5/3)* S^(1/2)* P^(-2/3)

Where n is the Manning roughness coefficient, P is the wetted perimeter (m), A is cross-sectional area of the pipe (m^2) and S is the hydraulic gradient or head loss by linear meter.

Applying the above formulae to a full closed circular cross section pipe:
S= 10.29 n^2 * Q^2 *D^(-5.333)
S= hf / L (head loss)
hf = Head loss (m)
Q = flow in penstock (m^3/s)
D = inside diameter of penstock (m)
L= lenght of pipe
Manning coefficient n for several commercial pipes

Types of Pipe	n
Welded steel Polyethylene(PE) PVC Asbestos cement Ductile iron Cast iron Wood-stave(new) Concrete (steel forms smooth finish)	0.012 0.009 0.009 0.011 0.015 0.014 0.012 0.014

Ref : ESHA (European Small Hydropower Association),”Layman’s Handbook on How To Develop a Small Hydro Site,”2nd ed, 1998. http://www.scribd.com/doc/8885765/Layman-Handbook-for-hydro-electric-power-plants

Calculation of penstock diameter

By Ahmad Suhendra

Penstock serves to drain the water into the turbine, because water power is a combination of head (H) and flow (Q). Water will flow down and create pressure on the end of the pipe that provides power to rotate the turbine,

Example

• Penstock length (Lp) = 15 m
• Flow (Qp) = 2.0 m³/sec
• H gross = 9 m
• Manning Coef (n) = 0.012 (value of the Manning roughness coefficient for Mild steel )

What is the diameter and head loss of the penstock ?
1. Penstock diameter (Dp)

• Dp = [C Qp / V ]^0.5 
• C (constants) = 1.273 = 4 / (phi)
• V (water velocity ) = 1 - 2.8 m / sec.
• Taken, V = 1.6 m / sec.)*
• Dp = [1.273 (2.0 / 1.6)] ^0.5 = 1.261 m ( penstock diameter)

2. Head loss

Assumed there are only head loss due to friction in penstock (applying the manning formulae), then :

• Head loss = [(10.29 n ² Q_p² )/ D_p^5.333] L_p 
• = 0.114 m
• Percent of Head loss = (0.114 m / 9 m ) * 100 % = 1.27 %)*

Note )* :

• Generally, for economic reasons the percent of head loss between 5 % to 10 %
• [Permissible velocity in Penstocks,V(m/s) = 0.125 (2 g H)^0.5, Ref : USBR (1961) (P J Bier)]
• If using Sarkaria's eq ==>; Dp = 3.55 ((Qp^2/(2 g H ))^0.25 = 1.377 m

Ref : ESHA (European Small Hydropower Association),”Layman’s Handbook on How To Develop a Small Hydro Site,”2nd ed, 1998

Welded Stell Penstocks (Engineering Monograph)

United States Department Of The Interior Bureau Of Reclamation

THIS MONOGRAPH will assist designers in the solution of problems in design and construction of safe penstocks which may be fabricated in accordance with modern manufacturing procedures. Certain rules relative to materials, stresses, and tests might be considered unnecessarily conservative. Safety is of paramount importance, however, and penstocks designed and constructed according to these rules have given satisfactory service through years of operation.

Welded Steel Penstocks presents information concerning modern design and construction methods for pressure vessels applied to penstocks for hydroelectric power plants. The data are based on some 40 years experience in penstock construction by the Bureau of Reclamation. During this period many of the largest penstocks in service today were designed and constructed.

Click here