DASSL solver in OpenModelica for an exchange between exterior and phase change material - differential-equations

Question

I want to model heat transfer between the exterior and a phase change material on OpenModelica. The system of equations obtained is a system of algebraic differential equations. I have already discretized my equations in space (according to x and y) and I use the DASSL integrator to determine the time derivatives. After verification, I get a consistent temperature (T_xy) whatever the temperatures at the following grid cells (x+1, x+2, x+3, y+1, y+2, y+3). However, the liquid fraction obtained is not the one expected. When T_xy is lower than the melting temperature of the MCP (T_fusion in Octadecanol), it should be zero. When T_xy > T_fusion, it must be equal to 1. At T_fusion, it must be between 0 and 1.
These are the models used to simulate PCM_ex_test (with the inputs T_xy = 60°C and the inputs fl_xy = 1):
model PCM_ex
// Modèles intégrant les propriétés thermophysiques des fluides caloporteurs, du MCP et des ailettes
Octadecanol PCM;
Ailettes Fin;
// Déclaration des paramètres
parameter Modelica.SIunits.Temperature T_init = 15+273.15;
//parameter Real fl_init = 0;
parameter Modelica.SIunits.Length H_MCP = 0.4;
parameter Modelica.SIunits.Length P_MCP = 0.4;
parameter Modelica.SIunits.Length L_MCP = 0.4;
parameter Modelica.SIunits.Mass M_MCP = 50;
parameter Modelica.SIunits.Volume V_MCP = M_MCP / PCM.rho_MCP;
parameter Modelica.SIunits.DimensionlessRatio ratio_a = V_MCP / (V_MCP + Fin.V_ailettes);
parameter Modelica.SIunits.CoefficientOfHeatTransfer h_ext = 10;
// Déclaration des paramètres de discrétisation
parameter Real Delta_x = 0.01;
parameter Real Delta_y = 0.01;
// Variables d'entrée
Modelica.Blocks.Interfaces.RealInput T_ext annotation(
Placement(visible = true, transformation(origin = {-52, 26}, extent = {{-12, -12}, {12, 12}}, rotation = 0), iconTransformation(origin = {-47, 33}, extent = {{-7, -7}, {7, 7}}, rotation = 0)));
Modelica.Blocks.Interfaces.RealInput T_xy_plus_3x "Maille x+3 du n,m" annotation(
Placement(visible = true, transformation(origin = {49, -35}, extent = {{-9, -9}, {9, 9}}, rotation = 180), iconTransformation(origin = {45, -35}, extent = {{5, -5}, {-5, 5}}, rotation = 0)));
Modelica.Blocks.Interfaces.RealInput T_xy_plus_2x "Maille x+2 du n,m" annotation(
Placement(visible = true, transformation(origin = {49, -25}, extent = {{9, -9}, {-9, 9}}, rotation = 0), iconTransformation(origin = {45, -25}, extent = {{5, -5}, {-5, 5}}, rotation = 0)));
Modelica.Blocks.Interfaces.RealInput T_xy_plus_1x "Maille x+1 du n,m" annotation(
Placement(visible = true, transformation(origin = {49, -15}, extent = {{9, -9}, {-9, 9}}, rotation = 0), iconTransformation(origin = {45, -15}, extent = {{5, -5}, {-5, 5}}, rotation = 0)));
Modelica.Blocks.Interfaces.RealInput fl_xy_plus_3x "Maille x+3 du n,m" annotation(
Placement(visible = true, transformation(origin = {70, -36}, extent = {{-10, -10}, {10, 10}}, rotation = 180), iconTransformation(origin = {63, -35}, extent = {{-5, -5}, {5, 5}}, rotation = 180)));
Modelica.Blocks.Interfaces.RealInput fl_xy_plus_2x "Maille x+2 du n,m" annotation(
Placement(visible = true, transformation(origin = {70, -26}, extent = {{10, -10}, {-10, 10}}, rotation = 0), iconTransformation(origin = {63, -25}, extent = {{-5, -5}, {5, 5}}, rotation = 180)));
Modelica.Blocks.Interfaces.RealInput fl_xy_plus_1x "Maille x+1 du n,m" annotation(
Placement(visible = true, transformation(origin = {70, -16}, extent = {{10, -10}, {-10, 10}}, rotation = 0), iconTransformation(origin = {63, -15}, extent = {{-5, -5}, {5, 5}}, rotation = 180)));
Modelica.Blocks.Interfaces.RealInput T_xy_plus_3y "Maille y+3 du n,m" annotation(
Placement(visible = true, transformation(origin = {31, 49}, extent = {{-9, -9}, {9, 9}}, rotation = -90), iconTransformation(origin = {33, 45}, extent = {{-5, -5}, {5, 5}}, rotation = -90)));
Modelica.Blocks.Interfaces.RealInput T_xy_plus_2y "Maille y+2 du n,m" annotation(
Placement(visible = true, transformation(origin = {21, 49}, extent = {{-9, -9}, {9, 9}}, rotation = -90), iconTransformation(origin = {23, 45}, extent = {{-5, -5}, {5, 5}}, rotation = -90)));
Modelica.Blocks.Interfaces.RealInput T_xy_plus_1y "Maille y+1 du n,m" annotation(
Placement(visible = true, transformation(origin = {11, 49}, extent = {{-9, -9}, {9, 9}}, rotation = 270), iconTransformation(origin = {13, 45}, extent = {{5, -5}, {-5, 5}}, rotation = 90)));
Modelica.Blocks.Interfaces.RealInput fl_xy_plus_3y "Maille y+3 du n,m" annotation(
Placement(visible = true, transformation(origin = {31, 67}, extent = {{-9, -9}, {9, 9}}, rotation = -90), iconTransformation(origin = {33, 63}, extent = {{-5, -5}, {5, 5}}, rotation = -90)));
Modelica.Blocks.Interfaces.RealInput fl_xy_plus_2y "Maille y+2 du n,m" annotation(
Placement(visible = true, transformation(origin = {21, 67}, extent = {{-9, -9}, {9, 9}}, rotation = -90), iconTransformation(origin = {23, 63}, extent = {{-5, -5}, {5, 5}}, rotation = -90)));
Modelica.Blocks.Interfaces.RealInput fl_xy_plus_1y "Maille y+1 du n,m" annotation(
Placement(visible = true, transformation(origin = {11, 67}, extent = {{-9, -9}, {9, 9}}, rotation = -90), iconTransformation(origin = {13, 63}, extent = {{5, 5}, {-5, -5}}, rotation = 90)));
// Constantes du modèle
Real C1, C2, C3, C4, C5;
// Déclaration des matrices (variables)
Modelica.Blocks.Interfaces.RealOutput T_xy(start = T_init) annotation(
Placement(visible = true, transformation(origin = {-12, -52}, extent = {{-12, -12}, {12, 12}}, rotation = -90), iconTransformation(origin = {-13, -47}, extent = {{-7, -7}, {7, 7}}, rotation = -90)));
Modelica.Blocks.Interfaces.RealOutput fl_xy(start = 0) annotation(
Placement(visible = true, transformation(origin = {-26, -52}, extent = {{-12, -12}, {12, 12}}, rotation = -90), iconTransformation(origin = {-29, -47}, extent = {{-7, -7}, {7, 7}}, rotation = -90)));
equation
C1 = (ratio_a * PCM.ks_MCP + (1 - ratio_a) * Fin.k_ailettes) / (2 * Delta_x);
C2 = ratio_a * (PCM.kl_MCP - PCM.ks_MCP) / (2 * Delta_x);
C3 = 4 * T_xy_plus_1x - T_xy_plus_2x;
C4 = 3 * ratio_a * (PCM.ks_MCP - PCM.kl_MCP) / 4 * ((- T_xy_plus_2x + 4 * T_xy_plus_1x - 3 * T_xy) / Delta_x^2 + (- T_xy_plus_2y + 4* T_xy_plus_1y - 3 * T_xy) / Delta_y^2) + ratio_a * (PCM.kl_MCP - PCM.ks_MCP) * ((2 * T_xy - 5 * T_xy_plus_1x + 4 * T_xy_plus_2x - T_xy_plus_3x) / Delta_x^2 + (2 * T_xy - 5 * T_xy_plus_1y + 4 * T_xy_plus_2y - T_xy_plus_3y) / Delta_y^2);
C5 = ratio_a * (PCM.ks_MCP - PCM.kl_MCP) * ((-fl_xy_plus_2x + 4 * fl_xy_plus_1x) / (2 * Delta_x) * (-T_xy_plus_2x + 4 * T_xy_plus_1x - 3 * T_xy) / (2 * Delta_x) + (-fl_xy_plus_2y + 4 * fl_xy_plus_1y) / (2 * Delta_y) * (-T_xy_plus_2y + 4 * T_xy_plus_1y - 3 * T_xy) / (2 * Delta_y)) - (ratio_a * PCM.ks_MCP + (1 - ratio_a) * Fin.k_ailettes) * ((2 * T_xy - 5 * T_xy_plus_1x + 4 * T_xy_plus_2x - T_xy_plus_3x) / Delta_x^2 + (2 * T_xy - 5 * T_xy_plus_1y + 4 * T_xy_plus_2y - T_xy_plus_3y) / Delta_y^2);
// Calcul de la fraction liquide
der(fl_xy) = (fl_xy * (C4 + PCM.rho_MCP * (PCM.cp_MCP_s - PCM.cp_MCP_l) * der(T_xy)) - C5 - PCM.rho_MCP * PCM.cp_MCP_s * der(T_xy)) / (PCM.rho_MCP * PCM.h_latent + PCM.rho_MCP * (PCM.cp_MCP_s * (PCM.T_fusion - T_xy) + PCM.cp_MCP_l * (T_xy - PCM.T_fusion)));
// Calcul de la température
T_xy = (h_ext * T_ext + C1 * C3 + C2 * C3 * fl_xy) / (h_ext + 3 * C1 + 3 * C2 * fl_xy);
annotation(
Diagram(graphics = {Rectangle(fillColor = {0, 170, 0}, fillPattern = FillPattern.Solid, extent = {{-40, 40}, {40, -40}})}),
Icon(graphics = {Rectangle(fillColor = {0, 170, 0}, fillPattern = FillPattern.Solid, extent = {{-40, 40}, {40, -40}}), Text(origin = {-1, 1}, extent = {{-13, 7}, {13, -7}}, textString = "%name")}));
end PCM_ex;
record Octadecanol
parameter Modelica.SIunits.Temperature T_fusion = 25+273.15;
//Melting temperature
parameter Modelica.SIunits.ThermalConductivity ks_MCP = 0.301;
parameter Modelica.SIunits.ThermalConductivity kl_MCP = 0.205;
parameter Modelica.SIunits.Density rho_MCP = 850;
parameter Modelica.SIunits.SpecificHeatCapacity cp_MCP_s = 1750;
parameter Modelica.SIunits.SpecificHeatCapacity cp_MCP_l = 2150;
type Heat = Real(unit = "J.kg-1");
parameter Heat h_latent = 226000;
end Octadecanol;
record Ailettes
parameter Modelica.SIunits.ThermalConductivity k_ailettes = 20;
parameter Modelica.SIunits.Volume V_ailettes = 0.2;
parameter Modelica.SIunits.Density rho_ailettes = 1200;
end Ailettes;
model PCM_ex_test
MCPBat_V8.PCM.PCM_ex pCM_ex annotation(
Placement(visible = true, transformation(origin = {0, 0}, extent = {{-50, -50}, {50, 50}}, rotation = 0)));
Modelica.Blocks.Sources.RealExpression T_ext(y = 10+273.15) annotation(
Placement(visible = true, transformation(origin = {-56, 16}, extent = {{-10, -10}, {10, 10}}, rotation = 0)));
Modelica.Blocks.Sources.RealExpression T_xy_plus1y(y = 60+273.15) annotation(
Placement(visible = true, transformation(origin = {-14, 56}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
Modelica.Blocks.Sources.RealExpression T_xy_plus2y(y = 60+273.15) annotation(
Placement(visible = true, transformation(origin = {-4, 56}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
Modelica.Blocks.Sources.RealExpression T_xy_plus3y(y = 60+273.15) annotation(
Placement(visible = true, transformation(origin = {6, 56}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
Modelica.Blocks.Sources.RealExpression T_xy_plus3x(y = 60+273.15) annotation(
Placement(visible = true, transformation(origin = {76, -6}, extent = {{-10, -10}, {10, 10}}, rotation = 180)));
Modelica.Blocks.Sources.RealExpression T_xy_plus2x(y = 60+273.15) annotation(
Placement(visible = true, transformation(origin = {76, 4}, extent = {{-10, -10}, {10, 10}}, rotation = 180)));
Modelica.Blocks.Sources.RealExpression T_xy_plus1x(y = 60+273.15) annotation(
Placement(visible = true, transformation(origin = {76, 14}, extent = {{-10, -10}, {10, 10}}, rotation = 180)));
Modelica.Blocks.Sources.RealExpression fl_xy_plus1y(y = 1) annotation(
Placement(visible = true, transformation(origin = {18, 56}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
Modelica.Blocks.Sources.RealExpression fl_xy_plus2y(y = 1) annotation(
Placement(visible = true, transformation(origin = {28, 56}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
Modelica.Blocks.Sources.RealExpression fl_xy_plus3y(y = 1) annotation(
Placement(visible = true, transformation(origin = {38, 56}, extent = {{-10, -10}, {10, 10}}, rotation = -90)));
Modelica.Blocks.Sources.RealExpression fl_xy_plus1x(y = 1) annotation(
Placement(visible = true, transformation(origin = {76, -18}, extent = {{-10, -10}, {10, 10}}, rotation = 180)));
Modelica.Blocks.Sources.RealExpression fl_xy_plus2x(y = 1) annotation(
Placement(visible = true, transformation(origin = {76, -28}, extent = {{-10, -10}, {10, 10}}, rotation = 180)));
Modelica.Blocks.Sources.RealExpression fl_xy_plus3x(y = 1) annotation(
Placement(visible = true, transformation(origin = {76, -38}, extent = {{-10, -10}, {10, 10}}, rotation = 180)));
equation
connect(T_xy_plus1x.y, pCM_ex.T_xy_plus_1x) annotation(
Line(points = {{66, 14}, {22, 14}, {22, -8}}, color = {0, 0, 127}));
connect(T_xy_plus2x.y, pCM_ex.T_xy_plus_2x) annotation(
Line(points = {{66, 4}, {22, 4}, {22, -12}}, color = {0, 0, 127}));
connect(T_xy_plus3x.y, pCM_ex.T_xy_plus_3x) annotation(
Line(points = {{66, -6}, {22, -6}, {22, -18}}, color = {0, 0, 127}));
connect(fl_xy_plus1x.y, pCM_ex.fl_xy_plus_1x) annotation(
Line(points = {{66, -18}, {32, -18}, {32, -8}}, color = {0, 0, 127}));
connect(fl_xy_plus2x.y, pCM_ex.fl_xy_plus_2x) annotation(
Line(points = {{66, -28}, {32, -28}, {32, -12}}, color = {0, 0, 127}));
connect(fl_xy_plus3x.y, pCM_ex.fl_xy_plus_3x) annotation(
Line(points = {{66, -38}, {32, -38}, {32, -18}}, color = {0, 0, 127}));
connect(T_ext.y, pCM_ex.T_ext) annotation(
Line(points = {{-44, 16}, {-24, 16}}, color = {0, 0, 127}));
connect(T_xy_plus1y.y, pCM_ex.T_xy_plus_1y) annotation(
Line(points = {{-14, 46}, {6, 46}, {6, 22}}, color = {0, 0, 127}));
connect(T_xy_plus2y.y, pCM_ex.T_xy_plus_2y) annotation(
Line(points = {{-4, 46}, {12, 46}, {12, 22}}, color = {0, 0, 127}));
connect(T_xy_plus3y.y, pCM_ex.T_xy_plus_3y) annotation(
Line(points = {{6, 46}, {16, 46}, {16, 22}}, color = {0, 0, 127}));
connect(fl_xy_plus1y.y, pCM_ex.fl_xy_plus_1y) annotation(
Line(points = {{18, 46}, {6, 46}, {6, 32}}, color = {0, 0, 127}));
connect(fl_xy_plus2y.y, pCM_ex.fl_xy_plus_2y) annotation(
Line(points = {{28, 46}, {12, 46}, {12, 32}}, color = {0, 0, 127}));
connect(fl_xy_plus3y.y, pCM_ex.fl_xy_plus_3y) annotation(
Line(points = {{38, 45}, {16, 45}, {16, 32}}, color = {0, 0, 127}));
end PCM_ex_test;
So, I wonder if I am using the DASSL solver in OpenModelica, and the time derivatives.
If not, the other option would be to add tests to match the three cases presented above.
Can someone better explain to me how the time derivatives and the DASSL solver work in OpenModelica and give me a solution if possible.
Thanks in advance,
Have a nice day