Reference solution ===================== Calculation method ------------------ Calculating vapour pressure from temperature ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ We assume the linear saturation curve. So it is written: .. image:: images/Object_5.svg :width: 96 :height: 25 .. _RefImage_Object_5.svg: .. _RefEquation 2.1.1-1: eq 2.1.1-1 [:ref:`R7.01.11 `] equation [:ref:`éq 3.2.1-2 <éq 3.2.1-2>`] then gives: .. image:: images/Object_6.svg :width: 96 :height: 25 .. _RefImage_Object_6.svg: .. _RefEquation 2.1.1-2: Eq 2.1.1-2 We write that the total mass of water is preserved (because there is no water flow at the edge) and we get: .. image:: images/Object_7.svg :width: 96 :height: 25 .. _RefImage_Object_7.svg: .. _RefEquation 2.1.1-3: Eq 2.1.1-3 [:ref:`R7.01.11 `] equation [:ref:`éq 4.4-1 <éq 4.4-1>`] also gives .. image:: images/Object_8.svg :width: 96 :height: 25 .. _RefImage_Object_8.svg: .. _RefEquation 2.1.1-4: Eq 2.1.1-4 The coupling of equations [:ref:`éq 2.1.1-3 <éq 2.1.1-3>`] and [:ref:`éq 2.1.1-4 <éq 2.1.1-4>`], to which must be added the ideal gas equation for steam, is a highly non-linear system that we will solve in a small disturbance, which allows it to be linearized. All calculations done, we get: .. image:: images/Object_9.svg :width: 96 :height: 25 .. _RefImage_Object_9.svg: .. _RefEquation 2.1.1-5: Eq 2.1.1-5 Temperature calculation ~~~~~~~~~~~~~~~~~~~~~~~~~~ [:ref:`R7.01.11 `] equation [:ref:`éq 3.2.4.3-1 <éq 3.2.4.3-1>`] gives: .. image:: images/Object_10.svg :width: 96 :height: 25 .. _RefImage_Object_10.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_11.svg :width: 96 :height: 25 .. _RefImage_Object_11.svg: .. _RefEquation 2.1.2-2: eq 2.1.2-2 So we get: .. image:: images/Object_12.svg :width: 96 :height: 25 .. _RefImage_Object_12.svg: .. _RefEquation 2.1.2-3: Eq 2.1.2-3 In this problem, .. image:: images/Object_13.svg :width: 96 :height: 25 .. _RefImage_Object_13.svg: is nothing but the heat provided per unit volume. By calling .. image:: images/Object_14.svg :width: 96 :height: 25 .. _RefImage_Object_14.svg: the total volume of the room and .. image:: images/Object_15.svg :width: 96 :height: 25 .. _RefImage_Object_15.svg: its lateral surface and .. image:: images/Object_16.svg :width: 96 :height: 25 .. _RefImage_Object_16.svg: the application time of the flows: .. image:: images/Object_17.svg :width: 96 :height: 25 .. _RefImage_Object_17.svg: .. _RefEquation 2.1.2-4: Eq 2.1.2-4 System to be solved ~~~~~~~~~~~~~~~~~~~~ .. image:: images/Object_18.svg :width: 96 :height: 25 .. _RefImage_Object_18.svg: eq 2.1.2-5 Benchmark results ---------------------- The value of the temperature, liquid pressure and vapor pressure, solution of the system (10) is given with the data summarized in paragraph [:ref:`§1.2 <§1.2>`] and recalled below. To calculate the calorific capacities, the following relationships are used: .. image:: images/Object_19.svg :width: 96 :height: 25 .. _RefImage_Object_19.svg: .. image:: images/Object_20.svg :width: 96 :height: 25 .. _RefImage_Object_20.svg: .. image:: images/Object_21.svg :width: 96 :height: 25 .. _RefImage_Object_21.svg: , the latter relationship being true because the grain expansion coefficient is zero. +----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+ | | | | | | | | + .. image:: images/Object_22.svg + .. image:: images/Object_23.svg + .. image:: images/Object_24.svg + .. image:: images/Object_25.svg + .. image:: images/Object_26.svg + .. image:: images/Object_27.svg + .. image:: images/Object_28.svg + | :width: 96 | :width: 276 | :width: 272 | :width: 481 | :width: 501 | :width: 176 | :width: 102 | + :height: 25 + :height: 58 + :height: 56 + :height: 111 + :height: 117 + :height: 28 + :height: 43 + | | | | | | | | + + + + + + (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_29.svg + .. image:: images/Object_30.svg + .. image:: images/Object_31.svg + .. image:: images/Object_32.svg + .. image:: images/Object_33.svg + .. image:: images/Object_34.svg + .. image:: images/Object_35.svg + | :width: 96 | :width: 276 | :width: 272 | :width: 481 | :width: 501 | :width: 176 | :width: 102 | + :height: 25 + :height: 58 + :height: 56 + :height: 111 + :height: 117 + :height: 28 + :height: 43 + | | | | | | | | + + + (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_36.svg + .. image:: images/Object_37.svg + .. image:: images/Object_38.svg + .. image:: images/Object_39.svg + + + + | :width: 96 | :width: 276 | :width: 272 | :width: 481 | | | | + :height: 25 + :height: 58 + :height: 56 + :height: 111 + + + + | | | | | | | | + + + + + + + + | | | | | | | | +----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+ |1.00E+06 |100 |6,00E+04 |1.00E+06 | | | | +----------------------------------------------------------------------------------+-----------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------+--------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+------------------------------------------------------------------------------------------------+ After resolution, the following results are obtained: +------------------------------------------------------------------------------------+--------+ | |425 | + .. image:: images/Object_40.svg + + | :width: 96 | | + :height: 25 + + | | | + + + | | | +------------------------------------------------------------------------------------+--------+ | |-1,4E+06| + .. image:: images/Object_41.svg + + | :width: 96 | | + :height: 25 + + | | | + l + + | | | +------------------------------------------------------------------------------------+--------+ | |2E+00 | + .. image:: images/Object_42.svg + + | :width: 96 | | + :height: 25 + + | | | + + + | | | +------------------------------------------------------------------------------------+--------+ Uncertainties ------------ The uncertainties are quite large because the analytical solution is an approximate solution due to the linearization of the equations.