Modelica.Fluid.Pipes.BaseClasses.WallFriction.Detailed.Internal

Package Content

Name	Description
m_flow_of_dp_fric	Calculate mass flow rate as function of pressure drop due to friction
dp_fric_of_m_flow	Calculate pressure drop due to friction as function of mass flow rate

Modelica.Fluid.Pipes.BaseClasses.WallFriction.Detailed.Internal.m_flow_of_dp_fric

Inputs

Type	Name	Default	Description
Pressure	dp_fric		Pressure loss due to friction (dp = port_a.p - port_b.p) [Pa]
Density	rho_a		Density at port_a [kg/m3]
Density	rho_b		Density at port_b [kg/m3]
DynamicViscosity	mu_a		Dynamic viscosity at port_a (dummy if use_mu = false) [Pa.s]
DynamicViscosity	mu_b		Dynamic viscosity at port_b (dummy if use_mu = false) [Pa.s]
Length	length		Length of pipe [m]
Diameter	diameter		Inner (hydraulic) diameter of pipe [m]
ReynoldsNumber	Re1		Boundary between laminar regime and transition [1]
ReynoldsNumber	Re2		Boundary between transition and turbulent regime [1]
Real	Delta		Relative roughness

Outputs

Type	Name	Description
MassFlowRate	m_flow	Mass flow rate from port_a to port_b [kg/s]
Real	dm_flow_ddp_fric	Derivative of mass flow rate with dp_fric

Modelica definition

function m_flow_of_dp_fric 
  "Calculate mass flow rate as function of pressure drop due to friction"

  input SI.Pressure dp_fric 
    "Pressure loss due to friction (dp = port_a.p - port_b.p)";
  input SI.Density rho_a "Density at port_a";
  input SI.Density rho_b "Density at port_b";
  input SI.DynamicViscosity mu_a 
    "Dynamic viscosity at port_a (dummy if use_mu = false)";
  input SI.DynamicViscosity mu_b 
    "Dynamic viscosity at port_b (dummy if use_mu = false)";
  input SI.Length length "Length of pipe";
  input SI.Diameter diameter "Inner (hydraulic) diameter of pipe";
  input SI.ReynoldsNumber Re1 "Boundary between laminar regime and transition";
  input SI.ReynoldsNumber Re2 
    "Boundary between transition and turbulent regime";
  input Real Delta "Relative roughness";
  output SI.MassFlowRate m_flow "Mass flow rate from port_a to port_b";
  output Real dm_flow_ddp_fric "Derivative of mass flow rate with dp_fric";

protected 
  function interpolateInRegion2_withDerivative 
    "Interpolation in log-log space using a cubic Hermite polynomial, where x=log10(lambda2), y=log10(Re)"

    input Real lambda2 "Known independent variable";
    input SI.ReynoldsNumber Re1 
      "Boundary between laminar regime and transition";
    input SI.ReynoldsNumber Re2 
      "Boundary between transition and turbulent regime";
    input Real Delta "Relative roughness";
    input SI.Pressure dp_fric 
      "Pressure loss due to friction (dp = port_a.p - port_b.p)";
    output SI.ReynoldsNumber Re "Unknown return variable";
    output Real dRe_ddp "Derivative of return value";
    // point lg(lambda2(Re1)) with derivative at lg(Re1)
  protected 
    Real x1=log10(64*Re1);
    Real y1=log10(Re1);
    Real y1d=1;

    // Point lg(lambda2(Re2)) with derivative at lg(Re2)
    Real aux2=Delta/3.7 + 5.74/Re2^0.9;
    Real aux3=log10(aux2);
    Real L2=0.25*(Re2/aux3)^2;
    Real aux4=2.51/sqrt(L2) + 0.27*Delta;
    Real aux5=-2*sqrt(L2)*log10(aux4);
    Real x2=log10(L2);
    Real y2=log10(aux5);
    Real y2d=0.5 + (2.51/ln(10))/(aux5*aux4);

    // Point of interest in transformed space
    Real x=log10(lambda2);
    Real y;
    Real dy_dx "Derivative in transformed space";
  algorithm 
    // Interpolation
    (y, dy_dx) := Utilities.cubicHermite_withDerivative(x, x1, x2, y1, y2, y1d, y2d);

    // Return value
    Re := 10^y;

    // Derivative of return value
    dRe_ddp := Re/abs(dp_fric)*dy_dx;
  end interpolateInRegion2_withDerivative;

  SI.DynamicViscosity mu "Upstream viscosity";
  SI.Density rho "Upstream density";
  Real lambda2 "Modified friction coefficient (= lambda*Re^2)";
  SI.ReynoldsNumber Re "Reynolds number";
  Real dRe_ddp "dRe/ddp";
  Real aux1;
  Real aux2;

algorithm 
  // Determine upstream density and upstream viscosity
  if dp_fric >= 0 then
    rho := rho_a;
    mu  := mu_a;
  else
    rho := rho_b;
    mu  := mu_b;
  end if;

  // Positive mass flow rate
  lambda2 := abs(dp_fric)*2*diameter^3*rho/(length*mu*mu) 
    "Known as lambda2=f(dp)";

  aux1:=(2*diameter^3*rho)/(length*mu^2);

  // Determine Re and dRe/ddp under the assumption of laminar flow
  Re := lambda2/64 "Hagen-Poiseuille";
  dRe_ddp := aux1/64 "Hagen-Poiseuille";

  // Modify Re, if turbulent flow
  if Re > Re1 then
    Re :=-2*sqrt(lambda2)*log10(2.51/sqrt(lambda2) + 0.27*Delta) 
      "Colebrook-White";
    aux2 := sqrt(aux1*abs(dp_fric));
    dRe_ddp := 1/ln(10)*(-2*ln(2.51/aux2+0.27*Delta)*aux1/(2*aux2)+2*2.51/(2*abs(dp_fric)*(2.51/aux2+0.27*Delta)));
    if Re < Re2 then
      (Re, dRe_ddp) := interpolateInRegion2_withDerivative(lambda2, Re1, Re2, Delta, dp_fric);
    end if;
  end if;

  // Determine mass flow rate
  m_flow := (pi*diameter/4)*mu*(if dp_fric >= 0 then Re else -Re);
  // Determine derivative of mass flow rate with dp_fric
  dm_flow_ddp_fric := (pi*diameter*mu)/4*dRe_ddp;
end m_flow_of_dp_fric;

Modelica.Fluid.Pipes.BaseClasses.WallFriction.Detailed.Internal.dp_fric_of_m_flow

Inputs

Type	Name	Default	Description
MassFlowRate	m_flow		Mass flow rate from port_a to port_b [kg/s]
Density	rho_a		Density at port_a [kg/m3]
Density	rho_b		Density at port_b [kg/m3]
DynamicViscosity	mu_a		Dynamic viscosity at port_a (dummy if use_mu = false) [Pa.s]
DynamicViscosity	mu_b		Dynamic viscosity at port_b (dummy if use_mu = false) [Pa.s]
Length	length		Length of pipe [m]
Diameter	diameter		Inner (hydraulic) diameter of pipe [m]
ReynoldsNumber	Re1		Boundary between laminar regime and transition [1]
ReynoldsNumber	Re2		Boundary between transition and turbulent regime [1]
Real	Delta		Relative roughness

Outputs

Type	Name	Description
Pressure	dp_fric	Pressure loss due to friction (dp_fric = port_a.p - port_b.p - dp_grav) [Pa]
Real	ddp_fric_dm_flow	Derivative of pressure drop with mass flow rate

Modelica definition

function dp_fric_of_m_flow 
  "Calculate pressure drop due to friction as function of mass flow rate"

  input SI.MassFlowRate m_flow "Mass flow rate from port_a to port_b";
  input SI.Density rho_a "Density at port_a";
  input SI.Density rho_b "Density at port_b";
  input SI.DynamicViscosity mu_a 
    "Dynamic viscosity at port_a (dummy if use_mu = false)";
  input SI.DynamicViscosity mu_b 
    "Dynamic viscosity at port_b (dummy if use_mu = false)";
  input SI.Length length "Length of pipe";
  input SI.Diameter diameter "Inner (hydraulic) diameter of pipe";
  input SI.ReynoldsNumber Re1 "Boundary between laminar regime and transition";
  input SI.ReynoldsNumber Re2 
    "Boundary between transition and turbulent regime";
  input Real Delta "Relative roughness";
  output SI.Pressure dp_fric 
    "Pressure loss due to friction (dp_fric = port_a.p - port_b.p - dp_grav)";
  output Real ddp_fric_dm_flow 
    "Derivative of pressure drop with mass flow rate";

protected 
  function interpolateInRegion2 
    "Interpolation in log-log space using a cubic Hermite polynomial, where x=log10(Re), y=log10(lambda2)"

    input SI.ReynoldsNumber Re "Known independent variable";
    input SI.ReynoldsNumber Re1 
      "Boundary between laminar regime and transition";
    input SI.ReynoldsNumber Re2 
      "Boundary between transition and turbulent regime";
    input Real Delta "Relative roughness";
    input SI.MassFlowRate m_flow "Mass flow rate from port_a to port_b";
    output Real lambda2 "Unknown return value";
    output Real dlambda2_dm_flow "Derivative of return value";
    // point lg(lambda2(Re1)) with derivative at lg(Re1)
  protected 
    Real x1 = log10(Re1);
    Real y1 = log10(64*Re1);
    Real y1d = 1;

    // Point lg(lambda2(Re2)) with derivative at lg(Re2)
    Real aux2 = Delta/3.7 + 5.74/Re2^0.9;
    Real aux3 = log10(aux2);
    Real L2 = 0.25*(Re2/aux3)^2;
    Real x2 = log10(Re2);
    Real y2 = log10(L2);
    Real y2d = 2+(2*5.74*0.9)/(ln(aux2)*Re2^0.9*aux2);

    // Point of interest in transformed space
    Real x=log10(Re);
    Real y;
    Real dy_dx "Derivative in transformed space";
  algorithm 
    // Interpolation
    (y, dy_dx) := Utilities.cubicHermite_withDerivative(x, x1, x2, y1, y2, y1d, y2d);

    // Return value
    lambda2 := 10^y;

    // Derivative of return value
    dlambda2_dm_flow := lambda2/abs(m_flow)*dy_dx;
  end interpolateInRegion2;

  SI.DynamicViscosity mu "Upstream viscosity";
  SI.Density rho "Upstream density";
  SI.ReynoldsNumber Re "Reynolds number";
  Real lambda2 "Modified friction coefficient (= lambda*Re^2)";
  Real dlambda2_dm_flow "dlambda2/dm_flow";
  Real aux1;
  Real aux2;

algorithm 
  // Determine upstream density and upstream viscosity
  if m_flow >= 0 then
    rho := rho_a;
    mu  := mu_a;
  else
    rho := rho_b;
    mu  := mu_b;
  end if;

  // Determine Reynolds number
  Re :=(4/pi)*abs(m_flow)/(diameter*mu);

  aux1 := 4/(pi*diameter*mu);

  // Use correlation for lambda2 depending on actual conditions
  if Re <= Re1 then
    lambda2 := 64*Re "Hagen-Poiseuille";
    dlambda2_dm_flow := 64*aux1 "Hagen-Poiseuille";
  elseif Re >= Re2 then
    lambda2 := 0.25*(Re/log10(Delta/3.7 + 5.74/Re^0.9))^2 "Swamee-Jain";
    aux2 := Delta/3.7+5.74/((aux1*abs(m_flow))^0.9);
    dlambda2_dm_flow := 0.5*aux1*Re*ln(10)^2*(1/(ln(aux2)^2)+(5.74*0.9)/(ln(aux2)^3*Re^0.9*aux2)) 
      "Swamee-Jain";
  else
    (lambda2, dlambda2_dm_flow) := interpolateInRegion2(Re, Re1, Re2, Delta, m_flow);
  end if;

  // Compute pressure drop from lambda2
  dp_fric :=length*mu*mu/(2*rho*diameter*diameter*diameter)*
       (if m_flow >= 0 then lambda2 else -lambda2);

  // Compute derivative from dlambda2/dm_flow
  ddp_fric_dm_flow := (length*mu^2)/(2*diameter^3*rho)*dlambda2_dm_flow;
end dp_fric_of_m_flow;