The problem of solving Manning's formula is that it is an implicit formula, for the water depth variable, that defines the R (Hydraulic Radius).
A common approach for solving this equation is using numerical methods, as the Newton-Raphson method. An implementation for the resolution of the Manning Formula for pipes, by the Newton-Raphson approximation, is shown below.
def manningTub(depth, args):
Q = args[0]
diamMM = args[1]
nMann = args[2]
slope = args[3]
diamM = diamMM / 1000.0
angle = 2*acos(1-2*depth/diamM)
area = diamM*diamM/8.0*(angle-sin(angle))
radius = diamM/4*(1-sin(angle)/angle)
mannR = (Q*nMann/slope**0.5)-(area*radius**(2.0/3.0))
return mannR
def solve(f, x0, h, args):
lastX = x0
nextX = lastX + 10* h # "different than lastX so loop starts OK
while (abs(lastX - nextX) > h): # this is how you terminate the loop - note use of abs()
newY = f(nextX,args) # just for debug... see what happens
print "f(", nextX, ") = ", newY # print out progress... again just debug
lastX = nextX
nextX = lastX - newY / derivative(f, lastX, h, args) # update estimate using N-R
return nextX
def derivative(f, x, h, args):
return (f(x+h,args) - f(x-h,args)) / (2.0*h) # might want to return a small non-zero if ==0
#### EXAMPLE CALL
xFound = solve(manningTub, initDepth, 0.001, [flowRate,diam,nMann,enSlope]) # call the solver
