packageDarcyFriction

Information

Friction Term

First, the functions for defining the friction force in the waterway are described. The friction force F_f is directed in the opposite direction of the velocity v (the linear velocity average across the cross-section of the pipe) of the fluid. A common expression for friction force in the filled pipes is the following:

$$ F_\mathrm{f} = -\frac{1}{8}\pi\rho LDf_\mathrm{D}v|v| $$

Here, L and D are related to the pipe length and diameter, respectively. f_D is a Darcy friction factor that is a function of Reynolds number N_Re, with the roughness ratio ε/D as a parameter.

Figure: Darcy friction factor as a function of the Reynolds number.

Flow Regimes

The turbulent region (N_Re > 2.3×10³) is a flow regime where the velocity across the pipe has a stochastic nature, and where the velocity v is relatively uniform across the pipe when we average the velocity over some short period of time. The laminar region (N_Re < 2.1×10³) is a flow regime with a regular velocity v which varies as a parabola with the radius of the pipe, with zero velocity at the pipe wall and maximal velocity at the centre of the pipe.

Laminar Flow

Darcy friction factor varies with the roughness of the pipe surface, specified by roughness height ε. For laminar flow in a cylindrical pipe (N_Re < 2.1×10³), the Darcy friction factor f_D can be found using the following expression:

$$ f_\mathrm{D} = \frac{64}{N_\mathrm{Re}} $$

Here, the Reynolds number is found as follows: \(N_\mathrm{Re}=\frac{\rho|v|D}{\mu}\), where μ is the fluid viscosity.

Turbulent Flow

For turbulent flow (N_Re > 2.3×10³), the Darcy friction factor is expressed as:

$$ f_\mathrm{D} = \frac{1}{\left(2\log_{10}\left(\frac{\epsilon}{3.7D} + \frac{5.74}{N_\mathrm{Re}^{0.9}}\right)\right)^2} $$

Transition Zone

In order to define the Darcy friction factor in a region between laminar and turbulent flow regimes, a cubic polynomial interpolation is used between the laminar value at N_Re=2100 and the turbulent value at N_Re=2300, with matching slopes at both endpoints to achieve global differentiability.

Implementation

Based on the presented equations for calculation of the friction force in the waterway, two functions are encoded in class DarcyFriction:

1. fDarcy — calculates the Darcy friction factor. This function has the following inputs: Reynolds number N_Re, pipe diameter D, and pipe roughness height ε. Returns the friction factor f_D.
2. Friction — calculates the actual friction force based on the response from the fDarcy function. This function has the following inputs: linear velocity v, pipe length and diameter L and D, liquid density and viscosity ρ and μ, and pipe roughness height ε. Returns friction force F_f.

Example Code

An example of a Modelica code for defining the Friction function:

function Friction "Friction force with Darcy friction factor"
  import Modelica.Constants.pi;
  input Modelica.SIunits.Velocity v "Flow velocity";
  input Modelica.SIunits.Diameter D "Pipe diameter";
  input Modelica.SIunits.Length L "Pipe length";
  input Modelica.SIunits.Density rho "Density";
  input Modelica.SIunits.DynamicViscosity mu "Dynamic viscosity of water";
  input Modelica.SIunits.Height eps "Pipe roughness height";
  output Modelica.SIunits.Force F_f "Friction force";
protected
  Modelica.SIunits.ReynoldsNumber N_Re "Reynolds number";
  Real f "friction factor";
algorithm
  N_Re:= rho * abs(v) * D / mu;
  f := fDarcy(N_Re, D, eps);
  F_f := 0.5 * pi * f * rho * L * v * abs(v) * D / 4;
end Friction;

Contents

Name	Description
Friction	Friction force with Darcy friction factor
fDarcy	Darcy friction factor