Geometric properties
of a parabolic section


Jhonath W. Mejía and Víctor M. Ponce

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ABSTRACT

The purpose of this article is to formulate the equations for the calculation of the geometric properties of a parabolic section, in open-channel hydraulics. These equations are compared with those presented in Chapter 2 of Ven Te Chow's book "Open-channel Nydraulics" (1959).


1.   INTRODUCTION

The calculation of the geometric properties of a parabolic channel section is accomplished by applying normal and line integrals, making use of the parameters that characterize a parabola.


2.  GEOMETRIC PROPERTIES OF A PARABOLIC SECTION

The Manning equation is:

             A
  Q  =  ____  R 2/3  So1/2
             n
(1)

in which Q = flow rate; A = flow area; R = hydraulic radius; So = bottom slope; and n = Manning's roughness. Therefore:

   Q n              A 2/3
  _____  =  A  _____
  So1/2            P 2/3
(2)

in which the section factor Fs is given as follows:

              A 5/3
  Fs  =  ______
             P 2/3
(3)

Therefore:

               A 5
  Fs3  =  _____
               P 2
(4)

The area and perimeter of a parabolic section corresponding to a flow depth yo and top width T are shown in Fig. 1.

Fig. 1  Definition sketch of a parabolic section.

To calculate the area, the following integral is used:

                    T/2
  A  =  2 ∫  (fx2 - fx1dx
                   0
(5)

                    T/2                          T/2
  A  =  2 ∫  (yodx  -  2 ∫  (4Fx 2dx
                   0                              0
(6)

                                          x 3
  A  =  2yo [ x ] 0T/2  -  8F [ ___ ] 0T/2
                                          3
(7)

                      FT 3
  A  =  yoT -  _____
                       3
(8)

in which 4F is a parameter of the parabola defined as follows:

               yo            4yo
  4F  =  ______  =  _____
            (T/2) 2        T 2
(9)

Therefore:

             yo
  F  =  _____
            T 2
(10)

Substituting Eq. 10 in Eq. 8:

                         yo       T 3                    yoT
  A  =  yoT  -   _____  _____  =  yoT  -  _____
                        T 2       3                       3
(11)

Therefore:

            2yoT
  A  =  ______
               3
(12)

For the wetted perimeter, a line integral is used:

                    T/2
  P  =  2 ∫  [1 + (f 'x1) 2] 1/2 dx
                   0
(13)

                    T/2
  P  =  2 ∫  [1 + (8F) 2x 2] 1/2 dx
                   0
(14)

Equation 14 has the form of a known integral:    (a2+x2)1/2dx, and its solution is:

                 x                                  sinh-1 (8Fx)
  P  =  2 [ ___ [1 + (8F) 2x 2] 1/2 + _______________ ] 0T/2
                 2                                        2 (8F)
(15)

                  T/2                                         sinh-1 (8FT/2)
  P  =  2 { ______ [1 + (8F)2(T/2) 2] 1/2 + _________________ }
                    2                                                  2(8F)
(16)

Substituting Eq. 10 in Eq. 16:

                  T/2                                                   sinh-1 (8(yo/T 2)(T/2))
  P  =  2 { ______ [1 + (8(yo/T 2)) 2 (T/2) 2] 1/2 + ________________________ }
                    2                                                            2(8(yo/T 2))
(17)

             T            16yo 2              T 2                   4yo
  P  =  ____ ( 1 + ______ ) 1/2 + ______ sinh-1 ( ______ )
             2              T 2                8yo                    T
(18)

             T                  4yo                     T                     4yo
  P  =  ____ [ ( 1 + ( ______ ) 2 ) 1/2 + ______ sinh-1 ( ______ ) ]
             2                   T                      4yo                    T
(19)

Equation 19 may be expressed in terms of logarithms as follows:

 sinh-1θ = ln [θ+ (1+θ 2) 1/2] (20)

Therefore:

             T                  4yo                   T            4yo                  4yo
  P  =  _____ { [1 + ( _____ ) 2 ] 1/2 + _____ ln [ _____ + ( 1 + ( _____ ) 2 ) 1/2] }
             2                   T                    4yo           T                     T
(21)

Equation 21 is the same formula that appears in Table 2-1 of Ven Te Chow's 1959 book: “Open-channel Hydraulics”.

Figure 2 shows Eq. 19 in graphical form: P = f (T,yo), for 0 < T < 10; 0 < yo < 5:

Fig. 2  Wetted perimeter versus flow depth and top width.

Equation 19 is used to construct the geometric properties of the parabola.

The free surface width T is:

            3A
  T  =  _____
            2yo
(22)

The hydraulic radius R is:

             A
  R  =  _____
             P
(23)

                                                  2yoT
                                                ______
                                                     3
  R  =  ____________________________________________________
             T                  4yo                    T                     4yo
           ____ { [ 1 + ( ______ ) 2 ] 1/2 + ______ sinh-1 ( ______ ) }
             2                   T                      4yo                   T
(24)

            4yo                 4yo                     T                    4yo
  R  =  _____ { [ 1 + ( ______ ) 2 ] 1/2 + ______ sinh-1 ( ______ ) } -1
              3                    T                      4yo                  T
(25)

The hydraulic depth D is:

             A          A
  D  =  ____  =  ____
             T         3A
                       ____
                        2yo
(26)

Therefore:

            2yo
  D  =  _____
              3
(27)

Table 1 summarizes the properties of a parabolic section.

Table 1.   Geometric properties of a parabolic section.
Property Equation Formula
Area Eq. 12           2yoT
  A  =  ______
           3
Wetted perimeter Eq. 19      T                  4yo                     T                     4yo
  P  =  ____ { [ 1 + ( ______ ) 2 ] 1/2 + ______ sinh-1 ( ______ ) }
   2                   T                      4yo                    T
Top
width
Eq. 22            3A
  T  =  ______
            2yo
Hydraulic
radius
Eq. 25    4yo                  4yo                    T                    4yo
  R  =  _____ { [ 1 + ( ______ ) 2 ] 1/2 + ______ sinh-1 ( _____ ) } -1
  3                    T                      4yo                   T
Hydraulic depth Eq. 27            2yo
  D  =  ______
            3


APPENDIX I.   REFERENCES

Chow, V. T. 1959. Open Channel Hydraulics. McGraw-Hill, New York.

Leithold, L. 1998. Calculus, Seventh Edition, Oxford University Press - Harla México, S.A.


APPENDIX II.  NOTATION

The following notation is used in this document:

Q = flow rate;

n = Manning's coefficient;

So = bottom slope;

yo = (center) flow depth;

Fs = section factor;

A = flow area;

P = wetted perimeter;

T = top width;

R = hydraulic radius; and

D = hydraulic depth.


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