Programación en Python aplicada a la Hidráulica (Canales Abiertos de sección variada)

1. PYTHON PARA INGENIERIA - CANALES ABIERTOS
METODO DE NEWTHON RAPHSON:
ECUACION DE MANNING
ECUACION PARA EL METODO DE NEWTON RAPHSON
ECUACIONES PARA CANAL TRAPEZOIDAL
AREA ->
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2. PERIMETRO->
DERIVANDO EL AREA RESPECTO A y:
DERIVANDO EL PERIMETRO RESPECTO A y:
DONDE:
B = ANCHO DE BASE DEL CANAL
Z1 = PENDIENTE IZQUIERDA DE LA SECCION DEL CANAL
Z2 = PENDIENTE DERECHA DE LA SECCION DEL CANAL
yi = TIRANTE NORMAL A CALCULAR
In [3]: import math
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3. In [1]: def trapezoidal(q, b, z1, z2, s, n):
"""
|Retorna una lista con
| - Tirante Normal de la seccion
| - Area hidráulica de la seccion
| - Perímetro mojado
| - Radio Hidráulico
| - Espejo de agua
| - Tirante medio
| - Velocidad media
| - Energía
| - Radio Hidráulico
| - Espejo de agua
| - Tirante medio
| - Velocidad media
| - Energía
| - Numero de Froude
|Entrada de Datos:
| q(m3/s) -> Caudal circulante por el canal
| s -> Pendi☺ente del canal
| n -> Coeficiente de manning
| z1 -> Pendiente izquierda de la seccion del canal
| z2 -> Pendiente derecha de la seccion del canal
| b -> Longitud de base del canal
|Ejemplo:
| >>> seccion_trapezoidal = trapezoidal(0.5, 0.5, 1, 1, 0.01, 0.014)
| >>> print(seccion_trapezoidal)
| >>> (0.28720341374388053, 0.2260875077380789, 1.3123339257528945, 0
.1722789476835105,
| 1.0744068274877612, 0.21043007355670804, 0.5365684383574587, 2
.2115330696610056, 1.5394970620863067)
"""
cte = q*n/s**0.5
yi = 1
yf = 2
error = 0.000000001
cont = 1
while abs(yi-yf) > error:
print("Iteración: ", cont)
yi = yf
area = b*yi + z1*yi**2/2 + z2*yi**2/2
perimeter = b + yi*(z1**2+1)**0.5 + yi*(z2**2+1)**0.5
fy = area**(5/3)/perimeter**(2/3)-cte
d_area = b + z1 * yi + z2 * yi
d_perimeter = (z1**2+1)**0.5 + (z2**2+1)**0.5
dfy = 5*area**(2/3) / perimeter**(2/3) * d_area / 3 - 2 * area**(
5/3) / perimeter**(5/3) * d_perimeter / 3
yf = yi - fy/dfy
cont += 1
if cont > 40:
break
print("yf=", yf)
print("error: ", abs(yf-yi))
a = b*yf + z1*yf**2/2 + z2*yf**2/2
p = b + yf*(z1**2+1)**0.5 + yf*(z2**2+1)**0.5
r = a/p
t = b+z1*yf+z2*yf
ym = a/t
v = q/a
e = yf + v**2/(2*9.806665)
froude = v/(9.806665*a/t)**0.5
return yf, a, p, r, t, ym, e, v, froude
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4. In [3]: seccion_trapezoidal = trapezoidal(0.5, 0.5, 1, 1, 0.01, 0.014)
In [4]: print(seccion_trapezoidal)
ECUACIONES PARA CANAL CIRCULAR
Iteración: 1
yf= 1.1757678260211835
error: 0.8242321739788165
Iteración: 2
yf= 0.6835237825717543
error: 0.4922440434494292
Iteración: 3
yf= 0.41666557810687294
error: 0.26685820446488134
Iteración: 4
yf= 0.30840080977437995
error: 0.10826476833249299
Iteración: 5
yf= 0.2879311733654157
error: 0.020469636408964265
Iteración: 6
yf= 0.2872043201745382
error: 0.0007268531908775078
Iteración: 7
yf= 0.28720341374528957
error: 9.064292486038461e-07
Iteración: 8
yf= 0.28720341374388053
error: 1.4090395517030174e-12
(0.28720341374388053, 0.2260875077380789, 1.3123339257528945, 0.1722789
476835105, 1.0744068274877612, 0.21043007355670804, 0.5365684383574587,
2.2115330696610056, 1.5394970620863067)
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5. AREA ->
PERIMETRO ->
DERIVANDO EL AREA RESPECTO A ->
DERIVANDO EL PERIMETRO RESPECTO A ->
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6. In [1]: def circle(q, d, s, n):
"""
|Retorna una lista con
| - Tirante Normal de la seccion
| - Area hidráulica de la seccion
| - Perímetro mojado
| - Radio Hidráulico
| - Espejo de agua
| - Tirante medio
| - Velocidad media
| - Energía
| - Numero de Froude
| q(m3/s) -> Caudal circulante por el canal
| d -> Longitud del diámetro de la sección
| s -> Pendiente del canal
| n -> Coeficiente de manning
|Ejemplo:
| >>> seccion_circular = circle(0.5, 1.3, 0.001, 0.014)
| >>> print(seccion_circular)
| >>> (0.5332481279157819, 0.5127041358149504, 1.8072573165180572,
| 0.28369183022744604, 1.2788573030089485, 0.4009080095243143,
| 0.5817384458469715, 0.9752213120053009, 0.49183581132307763)
"""
cte = q * n / s ** 0.5
yi = math.pi-0.01
yf = yi+0.01
if q > d/2:
yi = math.pi * 1.95 - 0.01
yf = math.pi * 1.95
error = 0.000000001
cont = 1
while abs(yi - yf) > error:
print("Iteración: ", cont)
yi = yf
area = d**2*(yi-math.sin(yi))/8
perimeter = yi*d/2
fy = area ** (5 / 3) / perimeter ** (2 / 3) - cte
d_area = d**2*(1-math.cos(yi))/8
d_perimeter = d/2
dfy = 5 * area ** (2 / 3) / perimeter ** (2 / 3) * d_area / 3 - 2
* area ** (5 / 3) / perimeter ** (
5 / 3) * d_perimeter / 3
yf = yi - fy / dfy
cont += 1
if cont > 40 or area > (math.pi*d**2/4):
break
print("yf=", yf)
print("error: ", abs(yf - yi))
if area < (math.pi*d**2/4):
h = d*(1-math.cos(yf/2))/2
a = d**2*(yi-math.sin(yi))/8
p = yi*d/2
r = a / p
t = d*math.sin(yf/2)
ym = a / t
v = q / a
e = h + v ** 2 / (2 * 9.806665)
froude = v / (9.806665 * a / t) ** 0.5
return h, a, p, r, t, ym, e, v, froude
else:
return 'Error'
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7. In [4]: seccion_circular = circle(0.5, 1.3, 0.001, 0.014)
In [5]: print(seccion_circular)
ECUACIONES PARA CANAL PARABÓLICO
Iteración: 1
yf= 2.7947684239726702
error: 0.3468242296171229
Iteración: 2
yf= 2.7804381906696047
error: 0.014330233303065487
Iteración: 3
yf= 2.780395871566242
error: 4.23191033629422e-05
Iteración: 4
yf= 2.7803958711917014
error: 3.745403986954443e-10
(0.5332481279157819, 0.5127041358149504, 1.8072573165180572, 0.28369183
022744604, 1.2788573030089485, 0.4009080095243143, 0.5817384458469715,
0.9752213120053009, 0.49183581132307763)
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8. AREA ->
DERIVANDO EL AREA RESPECTO A ->
SI
PERIMETRO ->
DERIVANDO EL PERIMETRO RESPECTO A ->
SI
PERIMETRO ->
DERIVANDO EL PERIMETRO RESPECTO A ->
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9. In [8]: def parabolic(q, t, s, n):
cte = q * n / s ** 0.5
yi = 1
yf = 2
error = 0.000000001
cont = 1
while abs(yi - yf) > error:
print("Iteración: ", cont)
yi = yf
area = 2*t*yi/3
d_area = 2*t/3
perimeter = 0
d_perimeter = 0
if yi/t <= 0.25:
perimeter = t + 8*yi**2/(3*t)
d_perimeter = 16*yi/(3*t)
elif yi/t > 0.25:
x = 4*yi/t
dx = 4/t
perimeter = 0.5*t*((1+x**2)**0.5+math.log(x+(1+x**2)**0.5)/x)
d_perimeter = 0.5*t*(x/(1+x**2)**0.5+1/(x*(1+x**2)**0.5)-math
.log(x+(1+x**2)**0.5)/x**2)*dx
fy = area ** (5 / 3) / perimeter ** (2 / 3) - cte
dfy = 5 * area ** (2 / 3) / perimeter ** (2 / 3) * d_area / 3 - 2
* area ** (5 / 3) / perimeter ** (
5 / 3) * d_perimeter / 3
yf = yi - fy / dfy
cont += 1
if cont > 40:
break
print("yf=", yf)
print("error: ", abs(yf - yi))
a = 2*t*yf/3
p = 0
if yi / t <= 0.25:
p = t + 8 * yf ** 2 / (3 * t)
elif yi / t > 0.25:
x = 4 * yf / t
p = 0.5 * t * ((1 + x ** 2) ** 0.5 + math.log(x + (1 + x ** 2) **
0.5) / x)
r = a / p
ym = a / t
v = q / a
e = yf + v ** 2 / (2 * 9.806665)
froude = v / (9.806665 * a / t) ** 0.5
return yf, a, p, r, ym, e, v, froude
In [9]: para = parabolic(1.5, 1.0, 0.01, 0.014)
print(para)
Iteración: 1
yf= 0.7463737949066422
error: 1.2536262050933578
Iteración: 2
yf= 0.76054974826516
error: 0.014175953358517734
Iteración: 3
yf= 0.7605484112206655
error: 1.3370444944538207e-06
Iteración: 4
yf= 0.7605484112206562
error: 9.325873406851315e-15
(0.7605484112206562, 0.5070322741471042, 1.902217044788166, 0.266548065
86677815, 0.5070322741471042, 1.2067796537800057, 2.9583915590446384, 1
.32671360454801)
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10. ECUACIONES PARA CANAL NATURAL
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12. AREA ->
DERIVANDO EL AREA RESPECTO A ->
PERIMETRO ->
DERIVANDO EL PERIMETRO RESPECTO A ->
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13. In [1]: def coord_yn(c, yn):
if max_matrix(c)[1] > (yn + min_matrix(c)[1]):
c_1 = []
for i in range(len(c)):
if c[i][1] <= (yn + min_matrix(c)[1]):
c_1.append(c[i])
c_1.insert(0, c[c.index(c_1[0]) - 1])
c_1.append(c[c.index(c_1[len(c_1) - 1]) + 1])
return c_1
else:
return c
In [2]: def min_matrix(c):
return min([i[0] for i in c]), min([i[1] for i in c])
def max_matrix(c):
return max([i[0] for i in c]), max([i[1] for i in c])
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14. In [14]: def natural(c, q, s, n):
cte = q * n / s ** 0.5
yi = (max_matrix(c)[1] - min_matrix(c)[1]) / 1.05
yf = yi + 0.01
error = 0.0000000001
cont = 1
y_min = min_matrix(c)[1]
while abs(yi - yf) > error:
yi = yf
area = 0
perimeter = 0
d_area = 0
c_1 = coord_yn(c, yi)
n_c = len(c_1)
for i in range(1, n_c - 2):
area += (yi + y_min) * (c_1[i + 1][0] - c_1[i][0]) - (c_1[i +
1][i] + c_1[i][1]) * (
c_1[i + 1][0] - c_1[i][0]) / 2
perimeter += ((c_1[i + 1][0] - c_1[i][0]) ** 2 + (c_1[i + 1][
1] - c_1[i][1]) ** 2) ** 0.5
d_area += c_1[i + 1][0] - c_1[i][0]
area = area - ((y_min + yi - c_1[1][1]) ** 2 / 2) * (c_1[1][0] -
c_1[0][0]) / (c_1[1][1] - c_1[0][1]) + (
(y_min + yi - c_1[n_c - 2][1]) ** 2 / 2) * (c_1[n_c - 1][
0] - c_1[n_c - 2][0]) / (
c_1[n_c - 1][1] - c_1[n_c - 2][1])
perimeter = perimeter + (yi + y_min - c_1[1][1]) * (
1 + ((c_1[1][0] - c_1[0][0]) / (c_1[1][1] - c_1[0][1])) *
* 2) ** 0.5 + (
yi + y_min - c_1[n_c - 2][1]) * (
1 + ((c_1[n_c - 1][0] - c_1[n_c - 2][0]) / (c
_1[n_c - 1][1] - c_1[n_c - 2][1])) ** 2) ** 0.5
d_area = d_area - (yi+y_min-c_1[1][1])*(c_1[1][0]-c_1[0][0])/(c_1
[1][1]-c_1[0][1]) + (yi+y_min-c_1[n_c-2][1])*(
c_1[n_c-1][0]-c_1[n_c-2][0])/(c_1[n_c-1][1]-c_1[n_c-2][1])
d_perimeter = (1+((c_1[1][0]-c_1[0][0])/(c_1[1][1]-c_1[0][1]))**2
)**0.5 + (1+(
(c_1[n_c-1][0]-c_1[n_c-2][0])/(c_1[n_c-1][1]-c_1[n_c-2][1
])))**0.5
fy = area ** (5 / 3) / perimeter ** (2 / 3) - cte
dfy = 5 * area ** (2 / 3) / perimeter ** (2 / 3) * d_area / 3 - 2
* area ** (5 / 3) / perimeter ** (
5 / 3) * d_perimeter / 3
yf = yi - fy / dfy
if yf>(max_matrix(c)[1]-min_matrix(c)[1]):
yf = 'Error'
break
if cont > 40:
yf = 'Error'
break
cont += 1
if yf != 'Error':
area = 0
perimeter = 0
d_area = 0
c_1 = coord_yn(c, yf)
n_c = len(c_1)
for i in range(1, n_c - 2):
area += (yf + y_min) * (c_1[i + 1][0] - c_1[i][0]) - (c_1[i +
1][i] + c_1[i][1]) * (
c_1[i + 1][0] - c_1[i][0]) / 2
perimeter += ((c_1[i + 1][0] - c_1[i][0]) ** 2 + (c_1[i + 1][
1] - c_1[i][1]) ** 2) ** 0.5
d_area += c_1[i + 1][0] - c_1[i][0]
area = area - ((y_min + yf - c_1[1][1]) ** 2 / 2) * (c_1[1][0] -
c_1[0][0]) / (c_1[1][1] - c_1[0][1]) + (
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15. In [16]: c = [[0,1],[1,0.25],[2,0],[3,1]]
nat = natural(c, 1.5, 0.01, 0.025)
print("yf=", nat[0])
print("area=", nat[1])
print("perimeter=", nat[2])
print("rh=", nat[3])
print("ta=", nat[4])
print("ym o yh=", nat[5])
print("energy=", nat[6])
print("velocity=", nat[7])
print("froude=", nat[8])
yf= 0.634691104739465
area= 0.8097656680009463
perimeter= 2.569517015876971
rh= 0.3151431428542515
ta= 2.1476125777254182
ym o yh= 0.37705388597536815
energy= 0.8096405069611481
velocity= 1.8523877453375155
froude= 0.9633176526809352
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