Combine Asynchronous Measurements

A liquid flows from two tanks and is combined into a single flow. The individual flows from each tank are calculated via the measurement of liquid levels in each tank and given as times series. Compute the combined flow from both tanks.

In[1]:=

h1 = TemporalData[TimeSeries, {{ StructuredArray[QuantityArray, {101}, StructuredArray`StructuredData[QuantityArray, CompressedData[" 1:eJwBNQPK/CFib1JlAQAAAGUAAADe5APaz4oAQCSFr48qYgBAQzBMKDxTAEDX ymqz2x8AQNOLnrenpP8/Kmwvk1Ia/z8huN4wvNb+P4rfuUg5z/4/mdIrXD7C /j/Uwimro3P+PxLt/Jvz4/0/qjRGTexP/T+BerQGpvj8P89ndGK45/w/uSuU QGrm/D/3+kpQAbD8P6MLVq3yMPw/4lTjuNuY+z9wTWvQuS77P9Lv9rulD/s/ //pavhEU+z8YLJKVKfT6P+YyAQjbifo/QKWO0djz+T/A5FSVYHn5P9wGcJO5 SPk/s14mlqhM+T8WL8Ql9T/5PzZrds4O7fg/5iLiH0xf+D+CvLZMftj3P+3W qeZulPc/BmCaoeOR9z+FMpKnwpP3P7+KpFA/Wfc/735hFG3Z9j++3QIdd0v2 P87+0M7j8/U/oJ8LlpTl9T+KFYiNavD1P1yaFoaDzfU/ddD7xmhg9T/9saxT N9H0P7+BjKTCZ/Q/jidTHYdJ9D9A+nBKMlf0P9b8SZ1xSfQ/3ZNSCoXy8z/B 8GpwT2jzPxZGDC848PI/HzcSGFq/8j/6I6RCr8nyPx+uXV8fzfI/Mvnt4kWO 8j+lcOh1DA/yP1yDkX7qjPE/7W3IZGBI8T+9GiYLqEnxP152KqElWfE/9Zck XIky8T8wIG3wnMPwP04gUh4EPfA/WedE9gjL7z/8E27037HvPxCrWcsl3e8/ 4hlDFD+97z9nOo3taQjvP+h16WqI/u0/J0/Rkm8u7T+urSnBhPLsP8WXTrap He0/bRLmuLQk7T9tTK7IZp7sP2u9NqEupOs/J3WK3Mq66j8bzo9SOljqP3/m Tig4d+o/OWJ4ihuc6j8Upnb4hkbqP08bxY1wZ+k/QrxSy/Zu6D8Tx6a6S+Xn PwYq0SZz7ec/ndGUjRUl6D+4gEpd7f7nP2Kb8eanROc/FZ293t1I5j+9YnNA IJvlP6EruLryg+U/LLUSnx7C5T+Q8wcHr8blP4TvD3hdOOU/Wl/dx6lF5D+Z /JtJH3rjP4Nyzev7PeM/ZcZaRVl24z93vVHo6Z3jPwwN4rqOP+M/aTuqJv5h 4j9XG7Zkp4HhP0VTVME+HuE/9dGrFA== "], "Meters", {{1}}]]}, { TemporalData`DateSpecification[{2016, 2, 3, 17, 0, 0.}, {2016, 2, 7, 21, 0, 0}, {1, "Hour"}]}, 1, {"Continuous", 1}, {"Discrete", 1}, 1, {ResamplingMethod -> { "Interpolation", InterpolationOrder -> 1}}}, True, 314.1]; h2 = TemporalData[TimeSeries, {{ StructuredArray[QuantityArray, {101}, StructuredArray`StructuredData[QuantityArray, CompressedData[" 1:eJwBNQPK/CFib1JlAQAAAGUAAADe5APaz4oAQCOFr48qYgBAWwPf4N9MAEBS Om4VdZn+PwngPIFrVfw/w0NCHNWZ+z9a09uBZsf8Pxe3P5fCfv4/ss8cEKDm /j9YFSmc6l39P5tC/AfYC/s/AuMHBPbM+T+AxXXML336P+AErK6XMfw/oeO4 kWwU/T+VJkksOBX8P32SqPG33/k/lmsozcI5+D/2xRoL6mD4PyP1dj3D6fk/ Ez6I2lQn+z9rGrkfDLf6P6iE3q7kw/g/DtWOrkXa9j9h6QGSL3n2P1fJJ+hT tPc/mXtDXuwm+T8KWSKioj75P/bpYbSxq/c/KZ3c+IGl9T8gGf0U2sj0PyaB IA5KnfU/Z4uDEF4d9z8a3WXPFav3P6Hb1msJjPY/eG3sElyQ9D9aBVrA4E7z P+VlDz+PrvM/Ot4sO24W9T8yXSYgSf/1P5XHmyBiXPU/5XIOMrWO8z+C/ifc mQbyP4FPhVQb7/E/goxHzU4e8z/yOQXGjkH0P5HPQ+F1F/Q/8hzXvYuU8j+R H7wjQOjwPz7fXelsYvA/ZeCYyYBA8T/t50aD6nryP+dNdqScu/I/ZyQA5B+X 8T/DSwWmlNPvP0LAfkiUEO4/1MyQlYgN7z+Ad8BdLbbwP3USRUfmSvE/SgME ROiN8D+S13+SyP/tP81vGU/Jues/EBlSmGjw6z87i6MVxv3tP3gbv1aYle8/ 9Ca3+aTm7j9ga3NGEj7sP5Wwax17suk/LYTLxm8x6T+9Mlb+gsDqP5sdtOk9 h+w/OLwD2HeK7D9YKMteHXnqP/PR9gS95+c/ogVVcVPR5j94zJwiUcjnP0b7 k3+Mf+k/yzWCkxAK6j9p8Rw1DZ/oP6BooC9NQ+Y/058TsNzJ5D+x4cno0iTl PwgutSnBleY/kbmhed9u5z+9Gd9kjKLmPzSKhEg+reQ/ojbXlLsO4z+zPmb0 IOHiP9R0w+wu4uM/pCGvgmbH5D9reNbsHnrkP/Ko3Wy9DOM/RtvVUBGQ4T+k hbaMlgbhP/MowdSsfuE/XFEVpd8i4j/yDMcEJRjiPxyQlJ3qROE/fqHOrPZG 4D/MgGaimHDfP3Bp5/unEN8/l3mlKQ== "], "Meters", {{1}}]]}, { TemporalData`DateSpecification[{2016, 2, 3, 17, 30, 0.}, {2016, 2, 7, 21, 30, 0}, {1, "Hour"}]}, 1, {"Continuous", 1}, { "Discrete", 1}, 1, { ResamplingMethod -> {"Interpolation", InterpolationOrder -> 1}}}, True, 314.1];

Plot the liquid levels.

In[2]:=

DateListPlot[{h1, h2}, FrameLabel -> Automatic, Joined -> False, Filling -> Axis]

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The measurements are not simultaneous—the time series have different time stamps.

In[3]:=

SameQ[h1["Dates"], h2["Dates"]]

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The net flow from each tank depends on the common discharge coefficient , the area of the cross sections of an outlet, and the liquid level, and is given as the following function.

In[4]:=

flowFun[c_, a_, height_] := Evaluate[c*a* Sqrt[2*UnitConvert[Entity["Planet", "Earth"]["Gravity"]]]* Sqrt[height ]]

Compute the flow from each tank by applying the formula to each time series.

In[5]:=

flow1 = flowFun[.7, Quantity[\[Pi] 0.25^2, "Meters Squared"], h1];

In[6]:=

flow2 = flowFun[.7, Quantity[\[Pi] 0.26^2, "Meters squared"], h2];

Calculate the total flow.

In[7]:=

TotalFlow = flow1 + flow2;

The time stamps of the sum are the union of the time stamps from both time series over a common time range.

In[8]:=

newtimes = Union[Complement[flow1["Dates"], {flow1["FirstDate"]}], Complement[flow2["Dates"], {flow2["LastDate"]}]];

In[9]:=

SameQ[newtimes, TotalFlow["Dates"]]

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Plot the flows.

In[10]:=

DateListPlot[{flow1, flow2, TotalFlow}, FrameLabel -> Automatic, Filling -> Axis, Joined -> False, PlotStyle -> PointSize[.01], PlotLegends -> SwatchLegend[{"flow1", "flow2", "TotalFlow"}]]

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