f5a43e4b-c7a4-4e95-a8e8-2977729f53ec 297 3 5 decimal
Wilson Ex 6.11-R1 - &[PAGENUM] / &[COUNT]
Example 6.11 - E.M. Wilson - Engineering Hydrology - Fourth Edition - Solutions to Problems
 
(a). Draw a flow duration curve from the tabulated data below - the mean monthly discharge (cumec) of a source of water. If a hydro-power plant is to be developed at the site to which the data refers and where the head available is 15 m, what would be a reasonable first estimate of the plant capacity and annual energy production on the basis of the data? Justify your choice.
 
(b). What is the mean monthly flow with a return period of once in 10 years?
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1. Basic Data
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percent (#1)

％ 1 100 / :

Units definition

Assume 30% probability for average flow

P.s 30 :

Head available

h 15 m * :

Assumed Plant efficiency

η 70 ％ * :

Density of water

σ.w 1000 kg m 3 ^ / * :

Return period =10 years (120 months) for part (b)

T1 120 :
2. Flow Data (mean monthly flow)
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extract numeric data columns from above table & insert units

inflow data 2 data rows 2 data cols submatrix cumec * : cumec 110 180 193 102 118 109 97 88 99 84 79 91 70 56 82 62 52 74 45 47 68 67 35 43 82 60 30 134 75 48 205 98 49 142 127 63 12 3 mat

Stack into one colums;this is required for probability plotting

I inflow mat2sys sys2mat : cumec 87.89

Arrange data in ascending order

II I sort reverse : cumec 87.89 cumec 30

Create probability vector

j 1 II length range Pr j el 100 j * II rows / eval : for

XYPlot.XLimMin 1 el 0 : 100 1 1 mat 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

Settings for X-Y Plot
]]> 220 1 1 mat
3. Fitting function and Flow Duration Curve (FDC)
]]> m 3 ^ s /

Stream flow in decending order

II cumec 205 193 180 142 134 127 118 110 109 102 99 98 97 91 88 84 82 82 79 75 74 70 68 23 1 mat

Corresponding Probability

Pr 2.77778 5.55556 8.33333 11.11111 13.88889 16.66667 19.44444 22.22222 25 27.77778 30.55556 33.33333 36.11111 38.88889 41.66667 44.44444 47.22222 50 52.77778 55.55556 58.33333 61.11111 63.88889 23 1 mat

Fitting function for data

x Fit data 1 col data 2 col x ainterp :

Example

25 Fit cumec 109 s m 3 ^ / * x fit x c *

Mean Flow (at 30% probability)

Q.s P.s Fit : cumec 99.31 s m 3 ^ / *

Minimum flow

Q.min II min : cumec 30

Function to plot

PP30 P.s 0 P.s I max 2 2 mat : 30 0 30 205 m 3 ^ s / * 2 2 mat

Function to plot

QQ30 0 Q.s Pr max Q.s 2 2 mat : 0 99.312373 100 99.312373 2 2 mat 2.778 205 5.556 193 8.333 180 11.111 142 13.889 134 16.667 127 19.444 118 22.222 110 25 109 27.778 102 30.556 99 33.333 98 36.111 97 38.889 91 41.667 88 44.444 84 47.222 82 50 82 52.778 79 55.556 75 58.333 74 61.111 70 63.889 68 66.667 67 69.444 63 72.222 62 75 60 77.778 56 80.556 52 83.333 49 86.111 48 88.889 47 91.667 45 94.444 43 97.222 35 100 30 36 2 mat 1 1 mat x Fit x c * x fit 2 1 sys

Function to plot

QQ.min 0 Q.min Pr max Q.min 2 2 mat : 0 30 m 3 ^ s / * 100 30 m 3 ^ s / * 2 2 mat

Erzeugt eine Matrix für die Darstellung vertikaler Linien zwischen den Funktionen f(x) und g(x) im Bereich von x1 bis x2 mit 2*n Linien.

Generates a matrix for 2*n vertical lines (hatching) between functions f(x) and g(x) in x-range from x1 to x2. (from SNIPPET Manager)

 f g x.1 x.2 n hatch dx x.2 x.1 - n / : hh x.1 x.1 f x.1 x.1 g 2 2 mat : x x.1 x.2 dx - ( x.1 dx + range hh hh x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 4 2 mat stack : for hh 4 1 line eval :

Generates a matrix for 2*n vertical lines (hatching) between functions f(x) and g(x) in x-range from x1 to x2. (from SNIPPET Manager)

1 f 1 g x.1 x.2 n hatch dx x.2 x.1 - n / eval : hh x.1 x.1 f x.1 x.1 g 2 2 mat eval : x x.1 x.2 dx - x.1 dx + range hh hh x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 4 2 mat stack eval : for hh 4 1 line :

Define

x Fit2 0 :

Define

x Fit4 Q.s :

Define

x Fit6 0 : 99.312 x Fit2 30 x fit QQ30 MyHatch1 MyHatch2 PP30 QQ.min 6 1 sys 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x Fit QQ30 MyHatch1 MyHatch2 PP30 QQ.min 6 1 sys 30 30 cumec 87.9 cumec 99.31 s m 3 ^ / * cumec 99.31 s m 3 ^ / * iVBORw0KGgoAAAANSUhEUgAAACkAAAAiCAYAAADCp/A1AAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAADwSURBVFhH7ZXRCcQgDEAdwVH6nYGcx2lcxmFyVmPxirEcDbnekQcBW7V5GLVu2zZ0zj02yM/hkyFZk7yNSUphklKYpBT/L5lCnYwuJHrDkTFCG3s5dMINyYShzAOA8oFQnhhS6Elq6ErW5EUuRwQuOfVBzIesoiSVr2Yc2wvUJU+rl+NecsB9wVi0JZvUsA/HsnLoSlJ5IZZWpx2i93cnVCWZVWvX0aLkmpKt1HXSNNiS60nOSt1Z9RXUJC8S9VWe9mtJtn23+LvQfj3uzP48i9UhO0FzPjg4X8AkpTBJKUxSCpOU4nckvffd9pHhvccX1CUx0frVVi4AAAAASUVORK5CYII=

Q.min cumec 30 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 cumec 30

4(a). Plant Capacity and Annual Energy (assuming Turbine operates at rated flow Qs )
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where

Q.s cumec 99.31 s m 3 ^ / *

Plant Capacity

PW1 Q.s h * g.e * σ.w * η * : MW 0.01 s m 4 ^ / *

Area (A1) of Rectangle from 0 to Ps (30%)

A1 P.s Q.s * : cumec 2979.37 s m 3 ^ / *

Area (A2) from 30% to the end

A2 x Fit x P.s Pr max int : cumec 4720.93 s m 3 ^ / *

Total shaded area

At A1 A2 + : cumec 7700.3 s m 3 ^ / *

Annual Energy is given by total shaded area

E1 At 100 / 365 * day * h * g.e * σ.w * η * : GWHr 0.05 s m 4 ^ / *
4(b). Mean monthly flow once in 10 years (120 months)
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Mean value of series

μ I Mean : cumec 87.89

Std deviation of series

σ I StdDev : cumec 42.43
Ref: Wilson - page-237, Gumbel equation: Qt for an infinite sample is defined as follows:
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Flow at 't' return period

t Q1 μ t K σ * + :

y for an infinite sample

t y 1 1 t / - ln - ln - :
In this case, the sample size=36 and assume it as infinite
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given: 10 yrs=120 months

T1 120

Hence, probability

P1 100 T1 / : 0.83 4.78 3.28

Mean monthly flow once in 10 years

T1 Q1 cumec 227.11

Flow from Flow Duration Curve

P1 Fit cumec 212.8 s m 3 ^ / *
5. Alternatively, assume Turbine operates from full rated flow (Qs) to 1/3rd of Qs
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1/3rd Flow

Q2 Q.s 3 / : cumec 33.1041 s m 3 ^ / *

Minimum flow

Q.low II min : cumec 30

Fitting function for Q2

x Fit3 Q2 :

Example

20 Fit3 cumec 33.1041 s m 3 ^ / *

To calculate probability of Q2 flow

P2 x Fit ( x Fit3 - x 0 100 solve : 98.1

Example

P2 Fit cumec 33.1041 s m 3 ^ / *

Function for plotting

Z4 P.s P2 n2 myHatch2 :

Area (A22) from 30% to the 'tt'

A22 x Fit x P.s P2 int : cumec 4661.06 s m 3 ^ / *

Annual Energy is given by shaded area

E2 A1 A22 + ( 100 / 365 * day * h * g.e * σ.w * η * : GWHr 0.05 s m 4 ^ / *

Compare with

E1 GWHr 0.05 s m 4 ^ / * x Fit QQ30 MyHatch4 MyHatch3 PP30 PP2 QQ.s2 7 1 sys 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x Fit QQ30 MyHatch4 MyHatch3 PP30 PP2 QQ.s2 7 1 sys

Prob at 1/3 rd flow

P2 98.1

Assumed Probability

P.s 30

Prob at 1/3 rd flow

P2 98.1

Assumed Probability

P.s 30 cumec 87.9

Mean flow at prob = Ps

Q.s cumec 99.31 s m 3 ^ / *

Mean flow at prob = Ps

Q.s cumec 99.31 s m 3 ^ / *
A1
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A3
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1/3 rd Flow

Q2 cumec 33.1 s m 3 ^ / *
A1
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A3
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