f5a43e4b-c7a4-4e95-a8e8-2977729f53ec 241 5 7 decimal

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) ％ 1100 / :

cumec m 3^s / :

MCM 106^m 3^* :

GWHr GW hr * :

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

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data FLOW DATA Year 1 Year 2 Year 3 January 110180193 February 102118109 March 978899 April 847991 May 705682 June 625274 July 454768 August 673543 September 826030 October 1347548 November 2059849 December 14212763134 mat :

extract numeric data columns from  above table & insert units inflow data 2 data rows 2 data cols submatrix cumec * :

Stack into one colums;this is 
required for probability plotting I inflow mat2sys sys2mat :

I Mean cumec 87.89

Arrange data in ascending order II I sort reverse :

II Mean cumec 87.89

II Min cumec 30

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

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 66.66667 69.44444 72.22222 75 77.77778 80.55556 83.33333 86.11111 88.88889 91.66667 94.44444 97.22222 100361 mat

3. Fitting function and Flow Duration Curve (FDC)

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Fitting function for data x Fit Pr II x ainterp cumec * :

Example 25 Fit cumec 109

Mean Flow (at 30% probability) Q.s P.s Fit : cumec 99.31

Q.s cumec 99.312

Q.min II Min : cumec 30

Function to plot PP30 P.s 0 P.s I max 22 mat :

Function to plot QQ30 0 Q.s Pr max Q.s 22 mat :

QQ30 0 99.312 m 3^s / * 100 99.312 m 3^s / * 22 mat

PP30 30030205 m 3^s / * 22 mat

QQ.min 0 Q.min Pr max Q.min 22 mat : 030 m 3^s / * 10030 m 3^s / * 22 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 
DEVELOPED BY "MRASKA" FOR SMATH FORUM 1 f 1 g x.1 x.2 n hatch dx x.2 x.1 - n / (: h x.1 x.1 f x.1 x.1 g 22 mat : x x.1 x.2 dx - x.1 dx + range h h x dx 2 / + x dx 2 / + g x dx 2 / + x dx 2 / + f x dx + x dx + f x dx + x dx + g 42 mat stack : for h 41 line :

Define x Fit2 0 :

Define x Fit4 Q.s :

Define x Fit6 0 :

MyHatch1 x Fit4 x Fit6 0 P.s 15 hatch eval :

MyHatch2 x Fit2 x Fit P.s 10030 hatch eval :

x Fit QQ30 MyHatch1 MyHatch2 PP30 QQ.min 6 1 sys

P.s 30

Q.s cumec 99.31

Q.min cumec 30

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

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Q.s cumec 99.31

Plant Capacity PW1 Q.s h * g.e * σ.w * η * : MW 10.23

Area (A1) of Rectangle 
from 0 to Ps (30%) A1 P.s Q.s * : cumec 2979.36

Area (A2) from 
30% to the end A2 x Fit x P.s Pr max int : cumec 4720.93

Total shaded area At A1 A2 + : cumec 7700.29

Annual Energy is given by total shaded area E1 At 100 / 365 * day * h * g.e * σ.w * η * : GWHr 69.46

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: Q_t 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 11 t / - ln - ln - :

t K 0.78 t y * 0.45 - :

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

T1 y 4.78

T1 K 3.28

Mean monthly flow once in 10 years T1 Q1 cumec 227.11

Flow from Flow Duration Curve P1 Fit cumec 212.8

5. Alternatively, assume Turbine operates from full rated flow (Q_s) to 1/3rd of Q_s

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1/3rd Flow Q2 Q.s 3 / : cumec 33.104

Minimum flow Q.low II min : cumec 30

Fitting function for Q2 x Fit3 Q2 :

Example 20 Fit3 cumec 33.104

To calculate probability of Q2 flow P2 x Fit (x Fit3 - x 0100 solve : 98.1

Example P2 Fit cumec 33.104

Function for plotting Z4 P.s P2 n2 myHatch2 :

QQ.s2 0 Q2 P2 Q2 22 mat :

PP2 P2 0 P2 II max 22 mat :

Area (A22) from 
30% to the 'tt' A22 x Fit x P.s P2 int : cumec 4661.07

Annual Energy is given by shaded area E2 A1 A22 + (100 / 365 * day * h * g.e * σ.w * η * : GWHr 68.92

Compare with E1 GWHr 69.46

MyHatch3 x Fit2 x Fit P.s P2 30 hatch eval :

MyHatch4 x Fit4 x Fit6 0 P.s 15 hatch eval :

x Fit QQ30 MyHatch4 MyHatch3 PP30 PP2 QQ.s2 7 1 sys

Assumed 
Probability P.s 30

Prob at 
1/3 rd flow P2 98.1

Mean flow at 
prob = Ps Q.s cumec 99.31

1/3 rd Flow Q2 cumec 33.104

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