Modeling of a heated pipe in Modelica/Dymola

Question

I'm currently studying chemical engineering and for my Bachelor thesis, I'm supposed to model a heated pipe that can be used in a superheater by connecting two pipes via a heatport together. Even though I made a big effort on understanding how I code correctly in Modelica, my code is still not working and I'm getting pretty desperate.

So the model basically has to be applicable for both fluid water and overheated steam, so just one-phase flow in instationary conditions. Heat transfer is supposed to happen convectively. Also, I neglect pressure losses due to friction in this model.

Here´s my idea of how the model is supposed to work: I'm pretty much trying to build a model like the one in the MSL, "Dynamic Pipe", just way more easier so that students who work on the same topic are able to understand my code quickly. So I splitted the pipe into a number of nodes n, the first volume being a inlet state, so basically that state does not really belong to the pipe. After that, the balance equations apply. I´m not quite sure about the momentum equations, so any help on them is highly appreciated. Convective heat transfer is defined by the Model "Convection" from the MSL, Thermal.HeatTransfer.Components. When testing the model with a flow source, a boundary with fixed pressure and a fixed temperature at the wall, I also get the error "Failed to reduce the DAE index" and I have absolutely no idea what that means.

Also, here is my code:

        model Pipe_base3
      //Import

      import Modelica.SIunits.*;
      import Modelica.Constants.pi;
      replaceable package Medium =
          Modelica.Media.Interfaces.PartialTwoPhaseMedium                          annotation (choicesAllMatching = true);

  parameter Integer n=2;
  parameter Integer np=1;

  // Geometry==================================================================//

  parameter Diameter d_pipe = 0.05 "Inner diameter of pipe"
                      annotation (Dialog(tab="Geometry"));
  parameter Length L = 1 "Length of unit"
                   annotation (Dialog(tab="Geometry"));
  parameter Area A_hex = pi * d_pipe * L
    "Shell surface of pipe for heat exchange"                                                 annotation (Dialog(tab="Geometry"));
  parameter Area A_q = (pi/4)*d_pipe^2
                       annotation (Dialog(tab="Geometry"));

  //Initialisation=============================================================//

  parameter Medium.Temperature T_start = 403.15 annotation (Dialog(tab="Initialization"));
  parameter Medium.SpecificEnthalpy h_start = Medium.specificEnthalpy_pT(p_start, T_start) annotation (Dialog(tab="Initialization"));
  parameter AbsolutePressure p_start = Medium.saturationPressure(T_start) annotation (Dialog(tab="Initialization"));
  parameter Medium.MassFlowRate m_flow_start = 0.5 annotation (Dialog(tab="Initialization"));

  //Temperature, pressure, energy==============================================//

  Medium.Temperature T[n+1]( each start=T_start, fixed=false);
  Medium.SpecificEnthalpy h[n+1]( each start=h_start, fixed=false);
  Medium.AbsolutePressure p[n+1](each start=p_start, fixed=false);

  HeatFlowRate Q_flow[n](fixed = false);
  Energy U[n](min=0);
  Energy KE[n]; //Kinetic Energy
  Medium.ThermodynamicState state[n+1];

  // Nondimensional Variables + HeatTransfer===================================//

  Medium.PrandtlNumber Pr[n](fixed=false);
  ReynoldsNumber Re[n](fixed=false);
  Real Xi[n];
  NusseltNumber Nu[n];
  CoefficientOfHeatTransfer alpha[n];

  // Thermodynamic properties==================================================//

  Medium.SpecificInternalEnergy u[n](fixed=false);
  Medium.DynamicViscosity eta[n];
  Density rho[n+1];
  Medium.SpecificHeatCapacity cp[n];
  Medium.ThermalConductivity lambda_fluid[n];

  //Segmental properties

  Mass ms[n]; //Mass per Segment
  MassFlowRate m_flow[n+1]( each start=m_flow_start/np, fixed=false);
  Velocity w[n+1](fixed=false);

  // Momentum

  Force F_p[n];
  Momentum I[n];
  Force Ib_flow[n];

  parameter Boolean init = false;

  Modelica.Fluid.Interfaces.FluidPort_a fluidin( redeclare package Medium = Medium, m_flow(start = m_flow_start, min = 0), p(start = p_start))
    annotation (Placement(transformation(extent={{-90,-100},{-70,-80}}),
        iconTransformation(extent={{-90,-100},{-70,-80}})));
  Modelica.Fluid.Interfaces.FluidPort_b fluidout( redeclare package Medium = Medium, m_flow(start = -m_flow_start, max = 0), p(start = p_start), h_outflow(start=h_start))
    annotation (Placement(transformation(extent={{70,-100},{90,-80}}),
        iconTransformation(extent={{70,-100},{90,-80}})));
  Modelica.Thermal.HeatTransfer.Interfaces.HeatPort_a[n] heatport
    annotation (Placement(transformation(extent={{-10,60},{10,80}}),
        iconTransformation(extent={{-10,60},{10,80}})));
  Modelica.Blocks.Interfaces.RealOutput[n] alpha_output annotation (Placement(
        transformation(extent={{-100,38},{-140,78}}), iconTransformation(extent={{-100,
            38},{-140,78}})));

protected 
  parameter Volume vn = (A_q * L) / n; //Volume per segment
  parameter Real x[n] = linspace((L/n), L, n);
  parameter Length length = L/n;

initial equation 

    for i in 1:(n+1) loop
      //h[i] = Medium.specificEnthalpy_pTX(p_start, T_start, {1});
      p[i] = p_start;
    end for;

equation 

  //Port equations=============================================================//

  fluidout.p = p[n];
  //fluidin.p-fluidout.p=p[1]-p[n+1];
  fluidout.h_outflow = h[n];
  fluidout.m_flow = -m_flow[n+1];

  //===========================================================================//

  h[1]=inStream(fluidin.h_outflow);
  p[1]=fluidin.p;
  state[1]=Medium.setState_ph(p[1],h[1]);
  T[1]=Medium.temperature(state[1]);
  rho[1]=Medium.density(state[1]);
  m_flow[1]=fluidin.m_flow/np;
  m_flow[1]=A_q*rho[1]*w[1];

  for i in 1:(n) loop

    // Heatport equations======================================================//

    T[i] = heatport[i].T;
    Q_flow[i] = heatport[i].Q_flow;

    // Momentum Balance =======================================================//

    der(I[i]) = Ib_flow[i] - F_p[i];
    I[i]=m_flow[i]*length;
    Ib_flow[i] = (p[i+1]*w[i+1]*w[i+1] - p[i]*w[i]*w[i])*A_q*np;
    F_p[i] = (A_q*p[i+1]-A_q*p[i]);

    // Energy Balance=========================================================//

    U[i] = ms[i] * u[i];
    KE[i] = 0.5*ms[i]*w[i+1]*w[i+1];

    der(U[i]+KE[i])=m_flow[i]*(h[i]+0.5*w[i]) - m_flow[i+1]*(h[i+1]+0.5*w[i+1]) + Q_flow[i];

    der(rho[i+1])= -((rho[i+1]-rho[i])*w[i+1] + (w[i+1]-w[i])*rho[i+1]); //Konti


     ms[i]=vn*rho[i+1];

     T[i+1]=Medium.temperature(state[i+1]);

     state[i+1] = Medium.setState_ph(p[i+1], h[i+1], 1); //Sets thermodynamic state from which other properties can be determined
     u[i] = Medium.specificInternalEnergy(state[i+1]);
     cp[i] = Medium.specificHeatCapacityCp(state[i+1]);
     rho[i+1] = Medium.density(state[i+1]);
     eta[i] = Medium.dynamicViscosity(state[i+1]);
     lambda_fluid[i] = Medium.thermalConductivity(state[i+1]);


     Re[i] * eta[i] = (rho[i+1] * abs(w[i+1]) * d_pipe);
     Pr[i] *lambda_fluid[i] = (eta[i] * cp[i]);
     Xi[i] = (1.8 * log10(abs(Re[i])+1) - 1.5)^(-2);
     Nu[i] = ((Xi[i]/8)*Re[i]*Pr[i])/(1+12.7*sqrt(Xi[i]/8)*((Pr[i])^(2/3)-1))*(1+(1/3)*(d_pipe/x[i])^(2/3));

     Nu[i] = Modelica.Fluid.Pipes.BaseClasses.CharacteristicNumbers.NusseltNumber(alpha[i], d_pipe, lambda_fluid[i]);

     alpha_output[i] = alpha[i] * (A_hex/n);

     m_flow[i+1] = A_q * w[i+1] * rho[i+1];

    // der(p[i]) = - w[i]*der(w[i]) * rho[i];

    // 0 = m_flow[i-1] - m_flow[i];

    // der(rho[i]) = -((rho[i]-rho[i-1])*w[i] + (v[i]-v[i-1])*rho[i]);

    //m_flow[i] = A_q * w[i] * rho[i]; //Calculation of flow velocity

    //ms[i] = vn * rho[i]; //Mass per segment

    //Calculation of thermodynamic properties for each segment=================//



    //Heat Transfer============================================================//



  end for;

  fluidin.h_outflow = h[1]; //

  annotation (Icon(coordinateSystem(preserveAspectRatio=false, extent={{-100,-100},
            {100,100}}), graphics={Line(
          points={{-80,-80},{-80,94},{-80,100},{0,20},{80,100},{80,-80}},
          color={0,0,255},
          smooth=Smooth.None), Line(
          points={{-60,-60},{-60,-48},{-60,0},{60,0},{60,-60},{48,-40},{72,-40},
              {60,-60}},
          color={0,0,255},
          smooth=Smooth.None)}), __Dymola_selections);
end Pipe_base3;

Thank you so much in advance!

Answer 1

I was in the same situation when I started using Modelica: I wanted the features of Modelica.Fluid.Pipes.DynamicPipe but with less complexity (I wanted the code to be more readable and less hierarchical). So, like you, I started building my own pipe model from scratch. However, because I wanted to be able to replace the pressure drop and heat transfer correlations and have great flexibility I ended up with a model of nearly the same complexity as Modelica.Fluid.Pipes.DynamicPipe .

My recommendation to you is to

1. build your own simple dynamic pipe model without any complex features. This will only be usable for educational purposes (eg letting other students understand your coding principles)
2. learn how to use Modelica.Fluid.Pipes.DynamicPipe for problems where you need vary model complexity (number of segment, replaceable pressure drop and heat transfer methods etc.). Modelica.Fluid.Examples.HeatExchanger is an example of how you can use Modelica.Fluid.Pipes.DynamicPipe to model a heat exchanger like the one you request.

Here I've shared an example of a very simple dynamic pipe that can be used as a heat exchanger. The pipe is made from n pipe segments and takes advantage of the fact that you can instantiate an array of components and connect the elements in a for loop.

As for the momentum balance, the correct/complete way is to account for the change in momentum by summing all the forces acting on each control volume (Newton's Second law). However, in most lumped models a steady-state momentum balance is adequate which reduces the equation to a linear or quadratic relation between mass flow rate and pressure drop. Modelica.Fluid.Pipes.DynamicPipe has a number of different presssure/flow correlations to choose from.

Best regards,

Rene Just Nielsen

Answer 2

I have built a small example/test that uses your model. It should be a very simple application of your model. Unfortunately I get the same error message:

Cannot find differentiation function: Modelica.Media.Water.IF97_Utilities.waterBaseProp_ph(boundary1.p, pipe_base3_1.h[2], 0, 1) with respect to time

Index reduction basically means that the model contains equations that have no unknown. This is solved by differentiation of these equations with respect to time (which can happen multiple times). For more information you can check https://www.inf.ethz.ch/personal/cellier/Lect/NSDS/Ppt/nsds_ppt_engl.html
especially lecture 16 and probably the ones before it :)

Therefore the Modelica tool will have to know how to do this differentiation. For equations this is usually done automatically, but for functions it has to be specified by the developer. It seems this is not done for

Modelica.Media.Water.IF97_Utilities.waterBaseProp_ph()

which is why you get the error message.

There are basically two possibilities to solve this problem:

1. You change your model to get rid of or revise the constraint equation (the one which has no unknown). It should be the one shown in the error message:

    der(pipe_base3_1.rho[2]) = ...  

2. You add the function for differentiation to the medium (I'm not much into the Fluid/Media so I have no idea how complicated that is, so I would try to go with 1. first). How this can be done is shown in https://modelica.org/documents/ModelicaSpec33Revision1.pdf section 12.7

Here is the code of the example:

 model PipeTest Pipe_base3 pipe_base3_1(redeclare package Medium = Modelica.Media.Water.WaterIF97_R1pT) annotation (Placement(transformation(extent={{-10,-10},{10,10}}))); Modelica.Fluid.Sources.FixedBoundary boundary( nPorts=1, p=100000, redeclare package Medium = Modelica.Media.Water.WaterIF97_R1pT) annotation (Placement(transformation(extent={{-60,-40},{-40,-20}}))); Modelica.Fluid.Sources.FixedBoundary boundary1( nPorts=1, p=100000, redeclare package Medium = Modelica.Media.Water.WaterIF97_R1pT) annotation (Placement(transformation(extent={{60,-40},{40,-20}}))); Modelica.Thermal.HeatTransfer.Sources.FixedHeatFlow fixedHeatFlow[2](Q_flow={0,0}) annotation (Placement(transformation(extent={{-40,20},{-20,40}}))); equation connect(boundary.ports[1], pipe_base3_1.fluidin) annotation (Line(points={{-40,-30},{-8,-30},{-8,-9}}, color={0,127,255})); connect(boundary1.ports[1], pipe_base3_1.fluidout) annotation (Line(points={{40,-30},{8,-30},{8,-9}}, color={0,127,255})); connect(fixedHeatFlow.port, pipe_base3_1.heatport) annotation (Line(points={{-20,30},{0,30},{0,7}}, color={191,0,0})); annotation ( Icon(coordinateSystem(preserveAspectRatio=false)), Diagram(coordinateSystem(preserveAspectRatio=false)), uses(Modelica(version="3.2.2"))); end PipeTest; 

Hope this helps...