Benchmark solution ===================== Calculation method ------------------ Calculating vapour pressure from temperature ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We assume the linear saturation curve. So it is written: .. image:: images/Object_8.svg :width: 96 :height: 25 .. _RefImage_Object_8.svg: .. _RefEquation 2.1.1-1: eq 2.1.1-1 The equation [:ref:`éq 2.2.3.3-2 <éq 2.2.3.3-2>`] from the reference document [:ref:`R7.01.11 `] then gives: .. image:: images/Object_9.svg :width: 96 :height: 25 .. _RefImage_Object_9.svg: .. _RefEquation 2.1.1-2: Eq 2.1.1-2 We write that the total mass of water and the total mass of air are conserved (because there is no water or gas flow at the edge) and we get: .. image:: images/Object_10.svg :width: 96 :height: 25 .. _RefImage_Object_10.svg: .. image:: images/Object_11.svg :width: 96 :height: 25 .. _RefImage_Object_11.svg: .. _RefEquation 2.1.1-3: Eq 2.1.1-3 .. image:: images/Object_12.svg :width: 96 :height: 25 .. _RefImage_Object_12.svg: .. image:: images/Object_13.svg :width: 96 :height: 25 .. _RefImage_Object_13.svg: .. _RefEquation 2.1.1-4: Eq 2.1.1-4 [:ref:`R7.01.11 `] [:ref:`éq 4.1.4-1 <éq 4.1.4-1>`] also gives: .. image:: images/Object_14.svg :width: 96 :height: 25 .. _RefImage_Object_14.svg: .. _RefEquation 2.1.1-5: Eq 2.1.1-5 The coupling of equations [:ref:`éq 2.1.1-3 <éq 2.1.1-3>`], [:ref:`éq 2.1.1-4 <éq 2.1.1-4>`] and [:ref:`éq 2.1.1-5 <éq 2.1.1-5>`], to which must be added the ideal gas equation for steam, dry air and dissolved air as well as Henry's law is a highly non-linear system that we will solve in small disturbances, which allows it to be linearized. All calculations done, we get: .. image:: images/Object_15.svg :width: 96 :height: 25 .. _RefImage_Object_15.svg: Eq 2.1.1-6 Temperature calculation ~~~~~~~~~~~~~~~~~~~~~~~~~~~ Equation [:ref:`éq 3.2.4.3-1 <éq 3.2.4.3-1>`] from the reference document [:ref:`R7.01.11 `] gives: .. image:: images/Object_16.svg :width: 96 :height: 25 .. _RefImage_Object_16.svg: .. _RefEquation 2.1.2-1: Eq 2.1.2-1 (since the other expansion coefficients are zero). Equation [:ref:`éq 3.2.4.3-2 <éq 3.2.4.3-2>`] gives: .. image:: images/Object_17.svg :width: 96 :height: 25 .. _RefImage_Object_17.svg: .. _RefEquation 2.1.2-2: eq 2.1.2-2 So we get: .. image:: images/Object_18.svg :width: 96 :height: 25 .. _RefImage_Object_18.svg: .. _RefEquation 2.1.2-3: Eq 2.1.2-3 In this problem, .. image:: images/Object_19.svg :width: 96 :height: 25 .. _RefImage_Object_19.svg: is nothing but the heat provided per unit volume. By calling .. image:: images/Object_20.svg :width: 96 :height: 25 .. _RefImage_Object_20.svg: the total volume of the room and .. image:: images/Object_21.svg :width: 96 :height: 25 .. _RefImage_Object_21.svg: its lateral surface and .. image:: images/Object_22.svg :width: 96 :height: 25 .. _RefImage_Object_22.svg: the application time of the flows: .. image:: images/Object_23.svg :width: 96 :height: 25 .. _RefImage_Object_23.svg: .. _RefEquation 2.1.2-4: Eq 2.1.2-4 System to be solved ~~~~~~~~~~~~~~~~~~~~ .. image:: images/Object_24.svg :width: 96 :height: 25 .. _RefImage_Object_24.svg: .. _RefEquation 2.1.2-5: eq 2.1.2-5 +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | | | | | | | | + .. image:: images/Object_25.svg + .. image:: images/Object_26.svg + .. image:: images/Object_27.svg + .. image:: images/Object_28.svg + .. image:: images/Object_29.svg + .. image:: images/Object_30.svg + .. image:: images/Object_31.svg + | :width: 96 | :width: 276 | :width: 165 | :width: 259 | :width: 165 | :width: 366 | :width: 540 | + :height: 25 + :height: 106 + :height: 25 + :height: 26 + :height: 24 + :height: 25 + :height: 111 + | | | | | | | | + + + + + + (calculated) + + | | | | | | | | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |5,00E-01 |-1,00E-12 |3,00E+02 |3,70E+03 |2.50E+06 |2,67E-02 |1.00E+03 | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | | | | | | | | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | | | | | | | | + .. image:: images/Object_32.svg + .. image:: images/Object_33.svg + .. image:: images/Object_34.svg + .. image:: images/Object_35.svg + .. image:: images/Object_36.svg + .. image:: images/Object_37.svg + .. image:: images/Object_38.svg + | :width: 96 | :width: 276 | :width: 165 | :width: 259 | :width: 165 | :width: 366 | :width: 540 | + :height: 25 + :height: 106 + :height: 25 + :height: 26 + :height: 24 + :height: 25 + :height: 111 + | | | | | | | | + + + (calculated) + + l + + (calculated) + | | | | | | | | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |2.20E+03 |3,00E-01 |2,93E+03 |1.05E+03 |4,18E+03 |1.90E+03 |2.78E+06 | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | | | | | | | | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ | | | | | | | | + .. image:: images/Object_39.svg + .. image:: images/Object_40.svg + .. image:: images/Object_41.svg + .. image:: images/Object_42.svg + + + + | :width: 96 | :width: 276 | :width: 165 | :width: 259 | | | | + :height: 25 + :height: 106 + :height: 25 + :height: 26 + + + + | | | | | | | | + + + + + + + + | | | | | | | | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ |1.00E+06 |10 |400 |1.00E+04 | | | | +----------------------------------------------------------------------------------+------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+-------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+-------------------------------------------------------------------------------------------------+ The following results are obtained: After solving this system, we obtain: .. image:: images/Object_43.svg :width: 96 :height: 25 .. _RefImage_Object_43.svg: This gives in terms of Aster result (increment): .. csv-table:: ":math:`\mathit{PRE1}` "," :math:`\mathit{PRE2}` "," :math:`\mathit{DT}` "," :math:`\mathit{PVP}(\mathit{V3})`" "9.95E4", "7.5E1", "1.44E-1", "2.94E1", "2.94E1" Uncertainties ------------ The uncertainties are quite large because the analytical solution is an approximate solution due to the linearization of the equations.