solution of recession and runoff integrals
TRANSCRIPT
SOLUTION OF RECESSION AND RUNOFF INTEGRALS
By Mohammad Akram Gill,1 M. ASCE
INTRODUCTION
In a publication on border irrigation, Strelkoff (3) presented recession and runoff equations in the integral form. A solution of these integrals was obtained by arithmetic integration and the results were presented graphically. In a similar work, these integrals were solved in algebraic form by the author but these solutions were not published. Later, Reddy (2) computed these very integrals using four points in the Gaussian quadrature technique. Reddy's computed results were similar to those of Strelkoff but required less computing time.
In view of the preceding efforts at solving these integrals, it is considered opportune to present the algebraic solutions which are simpler and more elegant than the numerical results. Algebraic solutions do not require a computer and can be easily handled on any pocket calculator having scientific functions.
RECESSION INTEGRAL AND SOLUTION
Strelkoff deduced the recession integral in the following form:
HfG-4*) » in which tR = recession time at the upstream end of the border; Sy = slope of the water surface; I = average infiltration rate in the border at the beginning of recession; / = length from the downstream end of the border to the point of recession; L = length of the border; and t = recession time. By a change of variables, £ = C3/2l, Eq. 1 is rewritten as
[*•&£{&) <2> in which C = (CuSl/2S5/3)/n; I = a constant; C„ = a coefficient; S0 = bed slope of the border; and n = Manning's coefficient.
The right-hand side of Eq. 2 is similar to the gradually varied flow integral whose solution was considered at length in an earlier publication by the writer (1). Consider the following integral:
. T T F - ( 3 )
'Prof, of Hydr. Engrg., Dept. of Water Resources and Environmental Engrg., Ahmadu Bello Univ., Zaria, Nigeria.
Note.—Discussion open until February 1, 1984. To extend the closing date one month, a written request must be filed with the ASCE Manager of Technical and Professional Publications. The manuscript for this paper was submitted for review and possible publication on October 21, 1982. This paper is part of the Journal of Irrigation and Drainage Engineering, Vol. 109, No. 3, September, 1983. ©ASCE, ISSN 0733-9437/83/0003-0335/$01.00. Paper No. 18210.
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in which £L = C3/2L. Let £ = 03 so that d£, = 362d6 and Eq. 3 reduces to
*
1 362d6 (4)
e3 1 + 62
whose solution is easily obtained as follows:
4> = 3 [(H/3 - e/3) - (tan-1 £ / 3 - tan"1 ^ 3 ) ] (5)
Solution of Eq. 1 is thus given by
t-tR = ^-2lVC (L1'3 - Z1/3) - tan"1 (VCL1/3) + tan"1 (Vc/1/3)] (6)
RUNOFF INTEGRAL AND SOLUTION
The equation of runoff is written as r fa £5/3
Vro(t) = Vro(tR) + • £2/3 dk (7) H 1 + ^
after Strelkoff, in which Vra(tR) = volume of runoff before the beginning of recession; and Vw(t) = volume of runoff at any instant of time after the onset of recession. The runoff integral in Eq. 7 is separated as follows:
r k £5/3
2/3 d%.
Again using £ = 8 simplifies Eq. 8 to
<I>R = e L
+ 0 -0
1 + d%
whose solution is obtained as follows:
<f>R = 3
£4/3 £2/3 •,
— + — - - in (i + e/3) 4 2 2 V S '
£L
which can further be simplified to give
4>R = 3 \ (& -?)-\ &/3 - n + \ <#» - n - \ in Vl + e/3 i + &3'
(8)
(9)
(10)
(11)
ANALYSIS AND CONCLUSION
Some spot checks were made by calculating the values of the recession and runoff integrals using the solutions presented here and comparing them with values in Strelkoff s graphs. The algebraic solutions obtained herein can be used in the border irrigation problems with a clear advantage over the previously proposed numerical solutions.
APPENDIX.—REFERENCES
1. Gill, M. A., "Exact Solution of Gradually Varied Flow," Journal of the Hydraul-
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ics Division, ASCE, Vol. 102, No. HY9, Proc. Paper 12406, Sept., 1976, pp 1353-1364.
2. Reddy, J. M., "Computation of Recession and Runoff Integrals," Journal of the Irrigation and Drainage Division, ASCE, Vol. 106, No. IR4, Proc. Paper 15873, Dec, 1980, pp. 367-370.
3. Strelkoff, T., "Algebraic Computation of Flow in Border Irrigation," Journal of the Irrigation and Drainage Division, ASCE, Vol. 103, No. IR3, Proc. Paper 13236, Sept., 1977, pp. 357-377.
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