1.

Dane:

B=450m K=0,8m/d l=250m x₁=100m

x₂=150m ,x₃=200m

1. Obliczamy q.

q = K * I * f

$q = K*\frac{{h_{1}}^{2} - {h_{2}}^{2}}{2l}$

h₁ = 150 − 125, 5 =  24, 5m 

h₂ = 135 − 125, 5 = 9, 5m

$q = K*\frac{{h_{1}}^{2} - {h_{2}}^{2}}{2l}$

2. Obliczamy Q

$q = 0,8*\frac{{24,5}^{2} - {9,5}^{2}}{2*250} = 0,82\ \frac{m^{3}}{d}$

Q = B * q 

$Q = 450*0,82 = 369\frac{m^{3}}{d}\ $

3. Obliczamy położenie zwierciadła wody w odległości x

q = K * I * f

$q = K*\frac{{h_{1}}^{2} - {h_{2}}^{2}}{2l}$

$K*\frac{{h_{1}}^{2} - {h_{2}}^{2}}{2l} = K*\frac{{h_{x}}^{2} - {h_{2}}^{2}}{2x}$

$\mathbf{h}_{\mathbf{x}}\mathbf{=}\sqrt{{\mathbf{h}_{\mathbf{2}}}^{\mathbf{2}}\mathbf{+}\frac{\mathbf{x}}{\mathbf{l}}\mathbf{*}\left( {\mathbf{h}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{-}{\mathbf{h}_{\mathbf{2}}}^{\mathbf{2}} \right)}$

a) Dla x₁=100m

${h_{x}}_{1} = \sqrt{{9,5}^{2} + \frac{100}{250}*\left( {24,5}^{2} - {9,5}^{2} \right)}$

h_x1 = 17, 2 m

b) Dla x₂=150m

${h_{x}}_{2} = \sqrt{{9,5}^{2} + \frac{150}{250}*\left( {24,5}^{2} - {9,5}^{2} \right)}$

h_x2 = 19, 9 m

c) Dla x₃=200m

${h_{x}}_{3} = \sqrt{{9,5}^{2} + \frac{200}{250}*\left( {24,5}^{2} - {9,5}^{2} \right)}$

h_x3 = 22, 32 m

H_x = H_p + h_x

H_p = 125, 5m

H_x1 = 125, 5 + 17, 2 = 142, 7 m, n, p, m

H_x2 = 125, 5 + 19, 9 = 145, 4 m, n, p, m

H_x3 = 125, 5 + 22, 32 = 147, 82 m, n, p, m

2.

Dane:

B=250m l=80m K₁=8m/d

K₂=4m/d H₁=133m H₂=130m h₁=9m h₂=10m

q = K_sr * I * f

K₁ > K₂

$K_{sr} = \frac{K_{1} - K_{2}}{\ln\frac{K_{1}}{K_{2}}}$

$I = \frac{H_{1} - H_{2}}{l}$

$f = \frac{h_{1} - h_{2}}{2}$

$q = K_{sr}*\frac{H_{1} - H_{2}}{l}*\frac{h_{1} - h_{2}}{2}$

$K_{sr} = \frac{8 - 4}{\ln\frac{8}{4}} = \frac{84}{\ln 2} = 5,77\ \frac{m}{d}$

$q = 5,77*\frac{133 - 130}{80}*\frac{9 + 10}{2} = 2,05\frac{m^{2}}{d}\ $

$Q = B*q = 250*2,05 = 5,15\frac{m^{3}}{d}\ $

3.Okreslić przepływ strumienia beznaporowego o szerokości B=200m

Dane:

K₁ = 10 m/d, K₂ = 4 m/d, h₁ = , h₂ = , l₁ = 30 m, l₂ = 70 m

Dane:

K₁ = 10 m/d, K₂ = 4 m/d, h₁ = , h₂ = , l₁ =

l₂ = 70 m

q = K_sr * I * f

$K_{sr} = \frac{l_{1} + l_{2}}{\frac{l_{1}}{K_{1}} + \frac{l_{2}}{K_{2}}} = 4,88\frac{m}{d}$

$I = \frac{h_{1} - h_{2}}{{l_{1} + l}_{2}} = 0,03$

$f = \frac{h_{1} - h_{2}}{2} = 13,5m$

$q = \frac{{h_{1}}^{2} - {h_{2}}^{2}}{2*\left( \frac{l_{1}}{K_{1}} + \frac{l_{2}}{K_{2}} \right)} = 1,98\ \frac{m^{2}}{d}$

$Q = B*q = \ 396\frac{m^{3}}{d}$

4.

DANE

K₁=7,3 m/d

K₂=5,4m/d, l=380m, m=2m, H₁=154,1 m.n.p.m, H₂=150,5 m.n.p.m

h₁^′ = h₁ − m = 2, 9m

h₂^′ = h₂ − m = 4, 8m

q_sr = q₁ + q₂

$q_{sr} = K_{1}*\frac{H_{1} - H_{2}}{l}*m + K_{2}*\frac{H_{1} - H_{2}}{l}*\frac{{h_{1}}^{'} + {h_{2}}^{'}}{2}$

$q_{sr} = 7,3\frac{154,1 - 150,5}{380}*2 + 5,4*\frac{154,1 - 150,5}{380}*\frac{2,90 + 4,20}{2} = 0,32m^{2}/d$

5.Obliczyć dopływ strumienia beznaporowego do rzeki

K₁ = 7,3m/d, K₂ = 5,4m/d, l=500m, B=150m

Szukane:q,Q

H₁=149m H₂=152m h₁=15m h₂=8m

h₁^'=h₁-m₁^'=10m

h₂^'= h₂-m₂^'=5m

q = K_sr * I * f

$m_{1} = \frac{{m_{1}}^{'} + {m_{2}}^{"}}{2} = 4m$

$m_{2} = \frac{m_{1} + m_{2}}{2} = 4m$

$I = \frac{H_{1} - H_{2}}{l}$

$f = \frac{h_{1} - h_{2}}{2}$

$K_{sr} = \frac{K_{1}m_{1} - K_{2}m_{2}}{m_{1} + m_{2}} = 6,06m/d$

$q = K_{sr}*\frac{H_{1} - H_{2}}{l}*\frac{h_{1} - h_{2}}{2}$

$q = - 0,42\ \frac{m^{2}}{d}$

$Q = B*q = 150*0,42 = 63\frac{m^{3}}{d}\ $

6.

q = K * m * I

$q = K*m*\frac{H_{1} - H_{2}}{l}$

H₁ = 316, 8 − 20, 7 = 296, 1

296, 1 − 255, 5 = 40,6m

H₂ = 315 − 7, 3 = 307, 7

307, 7 − 255, 5 = 52,2m

$q = 10,2*27,5*\frac{52,2 - 40,6}{920} = 3,54\ \frac{m^{2}}{d}$

T = K * m

$T = 10,2*27,5 = 280,5\frac{m^{2}}{d}$

$q = K*m*\frac{H_{1} - h_{x}}{x} = K*m*\frac{H_{1} - H_{2}}{l}$

$T*\frac{H_{1} - h_{x}}{x} = T*\frac{H_{1} - H_{2}}{l}$

$\mathbf{h}_{\mathbf{x}}\mathbf{=}\mathbf{H}_{\mathbf{1}}\mathbf{-}\frac{\mathbf{x}}{\mathbf{l}}\mathbf{*}\left( {\mathbf{H}_{\mathbf{1}}}^{\mathbf{2}}\mathbf{-}{\mathbf{H}_{\mathbf{2}}}^{\mathbf{2}} \right)$

a)x₁=200m

${h_{x}}_{1} = 52,2 - \frac{200}{920}*\left( {52,2}^{2} - {40,6}^{2} \right) = 49,69m$

H₁ = 255, 5 + 49, 69 = 305, 19 m.n.p.m

b)x₂=400m

${h_{x}}_{2} = 52,2 - \frac{400}{920}*\left( {52,2}^{2} - {40,6}^{2} \right) = 47,15m$

H₂ = 255, 5 + 47, 15 = 302, 65 m.n.p.m

7

q₁ + q₂ = q₃

q_I = q_II

$q_{1} = \ K_{1}*\frac{h_{1} - m}{x}*\frac{h_{1} - m}{2} = K_{1}*\frac{{{(h}_{1} - m)}^{2}}{2x}$

$q_{2} = \ K_{2}*\frac{h_{1} - m}{x}*m = K_{2m}*\frac{h_{1} - m}{x}$

${q_{3} = K}_{2}*\frac{{m - h}_{2}}{l - x}*\frac{{m + h}_{2}}{2} = K_{2}*\frac{{{m^{2} - h}_{2}}^{2}}{2(l - x)}$

h₁ = 19m

h₂ = 4m

$K_{1}*\frac{{{(h}_{1} - m)}^{2}}{2x} + K_{2m}*\frac{h_{1} - m}{x} = K_{2}*\frac{{{m^{2} - h}_{2}}^{2}}{2\left( l - x \right)}\ \ \ /*2x\left( l - x \right)$

$x = \frac{l\left( h_{1} - m \right)\lbrack K_{1}*\left( h_{1} - m \right) + 2K_{2}m\rbrack}{\left( h_{1} - m \right)\left\lbrack K_{1}*\left( h_{1} - m \right) + 2K_{2}m \right\rbrack + K_{2}({{m^{2} - h}_{2}}^{2})}$

x = 516, 13m

q = q₁ = q₂

$q_{1} = T_{1}*I_{1} = T_{1}*\frac{H_{1} - H_{x}}{\frac{l}{2}}$

$q_{2} = T_{2}*I_{2} = T_{2}*\frac{H_{1x} - H_{2}}{\frac{l}{2}}$

$I_{1} = \frac{H_{1} - H_{x}}{l_{1}}$

$I_{2} = \frac{H_{x} - H_{2}}{l_{2}}$

$T_{1}*\frac{H_{1} - H_{x}}{\frac{l}{2}} = T_{2}*\frac{H_{1} - H_{2}}{\frac{l}{2}}\ \ \ /\ *\frac{l}{2}$

T₁ * (H₁ − H_x)= T₂ * (H_x − H₂)

T₁H₁ − T₁H_x = T₂H_x − T₂H₂

T₁H_x + T₂H_x = T₁H₁ + T₂H₂ /  * T₁T₂

$H_{x} = \frac{T_{1}H_{1} + T_{2}H_{2\ }}{T_{1} + T_{2}}$

$H_{x} = \frac{35*93,7 + 110*88,9}{35 + 110} = 90,06\ m.n.p.m$

$I_{1} = \frac{H_{1} - H_{x}}{l_{1}} = \frac{93,7 - 90,06}{250} = 1,46*10^{- 2}$

$I_{2} = \frac{H_{x} - H_{2}}{l_{2}} = \frac{90,06 - 88,9}{250} = 4,64*10^{- 3}$

q = q₁ = q₂

$q_{1} = T_{1}I_{1} = 35*1,46*10^{- 2} = 0,51\ \frac{m^{2}}{d}$

$q_{2} = T_{2}I_{2} = 110*4,64*10^{- 3} = 0,51\ \frac{m^{2}}{d}$

a)Dla x₁=100m

$T_{1}*\frac{H_{1} - H_{x_{1}}}{x_{1}} = T_{1}*\frac{H_{1} - H_{x}}{l_{1}}\ \ \ /*x_{1}$

$H_{1} - H_{x_{1}} = \frac{x_{1}}{l_{1}}*\left( H_{1} - H_{x} \right)$

$H_{x_{1}} = H_{1} + \frac{x_{1}}{l_{1}}*\left( H_{1} - H_{x} \right)$

$H_{x_{1}} = 93,7 + \frac{100}{250}*\left( 93,7 - 90,06 \right) = 92,24\ m.n.p.m$