Гидравлические системы 

Моделирование изменения уровня воды в трёх ёмкостях в случае, когда в третьей ёмкости имеется течь.

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Давление на дне ёмкостей 1, 2 и 3 обозначены символами , и соответственно. Труба, выходящая из ёмкости 1, разветвляется, ведя к ёмкостям 2 и 3. Давление в месте раздвоения трубы обозначим . Скорость потока жидкости зависит от разности давлений и от геометрии соединительных труб.

In[1]:=

X flowRate1B = (Subscript[p, 1][t] - Subscript[p, B][t]) (\[Pi] Subscript[d, p]^4)/( 128 \[Mu] Subscript[L, B]); flowRateB2 = (Subscript[p, B][t] - Subscript[p, 2][t]) (\[Pi] Subscript[d, p]^4)/( 128 \[Mu] Subscript[L, 1]); flowRateB3 = (Subscript[p, B][t] - Subscript[p, 3][t]) (\[Pi] Subscript[d, p]^4)/( 128 \[Mu] Subscript[L, 2]);

Вся жидкость, вытекающая с ёмкости 1, должна оказаться в ёмкости 2 или ёмкости 3. Это условие выражается в виде алгебраической связи. Высота жидкости в ёмкостях связана с давлением на дне ёмкости законом Торичелли.

In[2]:=

X flowRateConstraint = (flowRate1B == flowRateB2 + flowRateB3); pressureEqns = {Subscript[p, 1][t] == \[Rho] g Subscript[h, 1][t], Subscript[p, 2][t] == \[Rho] g Subscript[h, 2][t], Subscript[p, 3][t] == \[Rho] g Subscript[h, 3][t]};

Скорость, с которой жидкость вытекает из или втекает в ёмкость (т.е. скорость течения), является прямо пропорциональной скорости изменения уровня воды в ёмкости. Это соотношение следует из закона сохранения массы в несжимаемой жидкости.

In[3]:=

X massConservation = {Subscript[a, 1] Subscript[h, 1]'[t] == -flowRate1B, Subscript[a, 2] Subscript[h, 2]'[t] == flowRateB2, Subscript[a, 3] Subscript[h, 3]'[t] == flowRateB3 - Subscript[a, 3] Subscript[h, 3][t]/10};

Зададим размеры и параметры модели. Площади поперечного сечения ёмкостей 1, 2 и 3 обозначены , и соответственно. Плотность и вязкость жидкости обозначены через (kg/m³) и (kg/m-s) соответственно.

In[4]:=

X params = {Subscript[a, 1] -> 1, Subscript[a, 2] -> 1, Subscript[a, 3] -> 1, Subscript[L, B] -> 0.1, Subscript[L, 1] -> 0.1, Subscript[L, 2] -> 0.2, Subscript[d, p] -> 0.2, \[Rho] -> .2, \[Mu] -> 2 10^-3, g -> 9.81};

Зададим начальные уровни жидкости в каждой ёмкости.

In[5]:=

X ics = {Subscript[h, 1][0] == 1, Subscript[h, 2][0] == 0, Subscript[h, 3][0] == 0};

Решим уравнения и визуализируем решение.

In[6]:=

X tankSol = NDSolve[{pressureEqns, massConservation, flowRateConstraint, ics} /. params, {Subscript[h, 1], Subscript[h, 2], Subscript[h, 3]}, {t, 0, 90}];

In[7]:=

X Plot[Evaluate[{Subscript[h, 1][t], Subscript[h, 2][t], Subscript[h, 3][t]} /. tankSol], {t, 0, 90}, PlotRange -> All, ImageSize -> 300, PlotStyle -> {Red, Blue, Magenta}, PlotLegends -> LineLegend[Automatic, {"Tank 1", "Tank 2", "Tank 3"}, LegendFunction -> "Panel", LegendMargins -> 10]]

Out[7]=