 Open Access
 Total Downloads : 2700
 Authors : Harinarayan Tiwari, Pritikana Das, Anish Kumar Bharti
 Paper ID : IJERTV1IS8545
 Volume & Issue : Volume 01, Issue 08 (October 2012)
 Published (First Online): 29102012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
MATLAB Programming Solution For Critical And Normal Depth In Trapezoidal Channels
MATLAB Programming Solution For Critical And Normal Depth In Trapezoidal Channels
Harinarayan Tiwari1,Pritikana Das2, Anish Kumar Bharti3

Research Scholar, Department of Water Resources Development and Management, IIT Roorkee

Research Scholar, Department of Centre of Transportation, IIT Roorkee

Research Scholar, Department of Civil Engineering, IIT Roorkee
Critical depth and Normal depth are an essential parameter in the analysis of varied flow in open channels. The governing equations for critical depth are implicit and no analytical solutions exist. In trapezoidal channel, the governing equations are highly nonlinear in the normal and critical flow depths and thus solution of the implicit equations involves numerical methods. So many solutions already exist in the form of empirical relations and tabulated form. Equations are lengthy and chances of getting error are more. As in advance computing age MATLAB programming got importance for these types of computational problems. In this study result obtained by programming is compared with different explicit solutions and method of genetic algorithm. It is found that result obtained from recent study is up to millimeter of accuracy from results obtained from genetic algorithm and explicit solution suggested by Ali and Essa.
Critical depth and Normal depth calculation is an important task for hydraulic engineers in the design of conveyance open channels for irrigation, drainage, and water supply projects. The flow depth corresponding to the minimum specific energy for a given discharge in an open channel is known as the critical depth. The critical depth concept is used as the basis for flow characterization, i.e., sub or supercritical. The flow at critical condition of an open channel is unstable. At critical condition, a small change in specific energy will cause abrupt fluctuation in water depth of the channel. This is because the specific energy curve is almost vertical at critical state. Therefore, if the design depth of the channel is near or equal to critical depth of the channel, the shape of the channel must be altered to avoid a large fluctuation in water depth. Open channel design is often based on the assumption of uniform flow and normal depth [5]. Because of the implicit nature of governing equations, direct solution
of normal depth is also not possible as critical depth and one has to resort to tedious iterative techniques [6]. But any iterative technique is easy to use with the help of programming interface. In this current study the calculation of critical and normal depth has been done with the help of MATLAB programming. Solution established by MATLAB programming is compared with some explicit solution for critical and normal depth [13] and the critical depth of trapezoidal section is compared with genetic algorithm method [4].
Newtons method use derivative calculus to find the roots of a function or relation by first taking an approximation and then improving the accuracy of that approximation until the root is found. Newton's Method is used to find the root of an equation provided that the function f[x] is equal to zero. f: [a, b] R is a differentiable function defined on the interval [a, b] with values in the real numbers R. Better approximation, xn+1 can be calculated from current approximation xn and the definition of the derivative at a given point that it is the slope of a tangent at that point. Here, f ' denotes the derivative of the function f. Then by simple algebra we can derive
n+1 n
x =x – f(xn) f'(xn)
(1)
Flow Chart for solving the problem
The process was started with some arbitrary initial value variable.
Basic Equations and Functions
Specific energy (E) in an open channel is calculated from Eq. (2)
2
E=ycos2+ Q2
2gA
(2)
Where, = Bed slope; =kinetic energy correction factor; y=flow depth; A=flow area; Q=discharge; and g=gravitational acceleration. By differentiating we get the Eq. (3).
3
Q2dA dE =cos2 dy dy gA
(3)
For the minimum specific energy first
derivative with respect to variable y should be zero. If the slope of the channel is small (less than 10%) than cos = 1 and the governing equation for the critical depth as in Eq. (4).
3
Q2T =1 (4)
gA
Where T = Top Width of the channel (dA=Tdy). For trapezoidal section both top width and area is the function of y and it cannot be separated for the solution of y. Manual calculation is time taking and cumbersome to calculate the critical depth for the trapezoidal section [1]. Especially where the practical design is concern MATLAB coding will serve the purpose by using Newton method of derivative for finding the root of function. For calculating critical depth trapezoidal section, function and its derivative is defined as Eq. (5) & Eq. (6)
A3 2 Q
F y = – (5)
Where; A= F1(y) & T = F2(y)
MATLAB Programming Code:

For Critical Depth
Input Parameters for Trapezoidal cross section Q= input('discharge in cumec=');
b= input('bottom width of channel in m='); z= input('side slope H to V=');
a= input('kinetic energy correction factor='); yinitial=0.01; %Initial Guess to start the iterations
% Critical Depths Calculation yc(1)=yinitial;
ic=1; dyc(1)=1e2;
while (abs(dyc(ic))>1e4) Ac(ic)=b*yc(ic)+z*(yc(ic))^2; Tc(ic)=b+2*z*yc(ic); Pc(ic)=b+2*(z^2+1)^(0.50)*yc(ic);
Rc(ic)=Ac(ic)/(Pc(ic));
Dc(ic)=Ac(ic)/(b+2*z*yc(ic));
fc(ic)=Ac(ic)^(3/2)*Tc(ic)^(1/2)
3
Q/sqrt(9.81/a); %
F' y =
(6)
ffc(ic)=Ac(ic)^(3/2)*(1/2)*Tc(ic)^(
For calculating normal depth trapezoidal section, function and its derivative is defined as Eq. (7) & Eq. (8)
3/2)*2*z+Tc(ic)^(1/2)*(3/2)*Ac(ic)^(1/2)*Tc(ic); % f' Derivative
yc(ic+1)=yc(ic)fc(ic)/ffc(ic);
=
(7)
dyc(ic+1)=fc(ic)/ffc(ic);
Where; P= wetted perimeter; R= Hydraulic Radius (A/P); = Mannings Coefficient; s= Bed Slope; z= Side Slope (H: V) and A= F1(y); P = F2(y); T= F3(y) & R = F4(y).
F' y = + (8)
ic=ic+1;
end criticaldepth=yc(ic);

For Normal depth

% Input Parameters for Trapezoidal cross section Q= input('discharge in cumec=');
b= input('bottom width of channel in m='); z= input('side slope H to V=');
n= input('manning coefficient='); S= input('Longitudinal slope=');
yinitial=0.01; %Initial Guess to start the iterations
% Normal Depth Calculation yn(1)=yinitial;
in=1; dyn(1)=1e2;
while (abs(dyn(in))>1e4) An(in)=b*yn(in)+z*(yn(in))^2; Tn(in)=b+2*z*yn(in); Pn(in)=b+2*(z^2+1)^(0.50)*yn(in);
Rn(in)=An(in)/(Pn(in));
Dn(in)=An(in)/(b+2*z*yn(in)); fn(in)=sqrt(S)*An(in)*Rn(in)^(2/3)*n^(1)Q; ffn(in)=(sqrt(S)*n^(
1))*((Rn(in)^(2/3)*Tn(in))+(Tn(in)/Pn(in)) ((2*yn(in)*Rn(in))/Pn(in)));
yn(in+1)=yn(in)fn(in)/ffn(in);
dyn(in+1)=fn(in)/ffn(in); in=in+1;
end Normaldepth=yn(in);
Results
An open channel having a flow of 17 m3/s and bottom width of the channel is 6 m. Mannings coefficient is taken as 0.0145 with side slope 2:1 (Horizontal: Vertical). Longitudinal slope of the channel is taken as 0.002 ad kinetic energy correction factor is taken as 1. Critical depth and Normal depth for above mentioned problem obtained from some researchers by explicit solution and genetic algorithm is given in Table below.
Table 1 Critical Depth and Normal Depth
Suggested Solutions 
Critical Depth (m) 
Ali R. Vatankhaha [3] 
0.8487 
Ali R. Vatankhaha,Said M. Easa [2] 
0.8465 
Wang et al. [1] 
0.8557 
A Kanani et al.(GA Method) [4] 
0.8462 
Proposed Methodology (MATLAB Programming) 
0.8468 
Normal Depth (m) 

Ali R. Vatankhaha [3] 
0.665 
Proposed Methodology (MATLAB Programming) 
0.6647 
Order of accuracy found from Ali & Easa [2], Genetic Algorithm (GA) from Kanani et al. [4] and MATLAB programming is up to millimeter. Critical depth obtained from Ali & Easa
[2] explicit solution is 0.8465 m and Kanani et al. [4] found the critical depth from the GA method is 0.8462 m. From the proposed MATLABprogramming it is seen that critical depth is 0.8468

Normal depth from Ali [3] 0.665 m for the same above mentioned problem is same as 0.6647 m.
References

Wang ZZ, Yuan S, Wu CL. An inquiry on a formula for calculating critical depth of open channel with trapezoidal cross section. J Hydraul Eng 1999;43(4):147

Vatankhah Ali R, Easa Said M. Explicit solutions for critical and normal depths in channels with different shapes. Flow Meas Instrum 2011;22(1):439.

Ali R. Vatankhah, Explicit solutions for critical and normal depths in trapezoidal and parabolic open channels. (Article in press), http://dx.doi.org/10.1016/j.asej.2012.05.002

A. Kanani; M. Bakhtiari; S. M. Borghei; and D.S. Jeng, Evolutionary Algorithms for the Determination of Critical Depths in Conduits, Journal of Irrigation and Drainage Engineering, Vol. 134, No. 6, December 1, 2008. Â©ASCE, ISSN 0733 9437/2008/6847852

David C. Froehlich, Width and depth constrained best Trapezoidal section, Journal of Irrigation and Drainage Engineering, Vol. 120, No. 4, July/August, 1994. @ASCE, ISSN 07339437/94/0004 0828

K. BabaeyanKoopaei ,Dimensionless curves for normaldepth calculations in Canal sections Journal of Irrigation and Drainage Engineering, Vol. 127, No. 6, November/December, 2001. @ASCE, ISSN 07339437/01/000603860389
