Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent

Information

This library provides pressure loss factors of a pipe segment (orifice, bending etc.) with a minimum amount of data. If available, data can be provided for both flow directions, i.e., flow from port_a to port_b and from port_b to port_a, as well as for the laminar and the turbulent region. It is also an option to provide the loss factor only for the turbulent region for a flow from port_a to port_b. Basically, the pressure drop is defined by the following equation:

   Δp = 0.5*ζ*ρ*v*|v|
      = 0.5*ζ/A^2 * (1/ρ) * m_flow*|m_flow|
      = 8*ζ/(π^2*D^4*ρ) * m_flow*|m_flow|

where

• Δp is the pressure drop: Δp = port_a.p - port_b.p
• v is the mean velocity.
• ρ is the density.
• ζ is the loss factor that depends on the geometry of the pipe. In the turbulent flow regime, it is assumed that ζ is constant and is given by "zeta1" and "zeta2" depending on the flow direction.
• D is the diameter of the pipe segment. If this is not a circular cross section, D = 4*A/P, where A is the cross section area and P is the wetted perimeter.

Package Content

Name	Description
LossFactorData	Data structure defining constant loss factor data for dp = zeta*rho*v*|v|/2 and functions providing the data for some loss types
massFlowRate_dp	Return mass flow rate from constant loss factor data and pressure drop (m_flow = f(dp))
massFlowRate_dp_and_Re	Return mass flow rate from constant loss factor data, pressure drop and Re (m_flow = f(dp))
pressureLoss_m_flow	Return pressure drop from constant loss factor and mass flow rate (dp = f(m_flow))
pressureLoss_m_flow_and_Re	Return pressure drop from constant loss factor, mass flow rate and Re (dp = f(m_flow))
BaseModel	Generic pressure drop component with constant turbulent loss factor data and without an icon
TestWallFriction	Pressure drop in pipe due to wall friction (only for test purposes; if needed use Pipes.StaticPipe instead)
BaseModelNonconstantCrossSectionArea	Generic pressure drop component with constant turbulent loss factor data and without an icon, for non-constant cross section area
pressureLoss_m_flow_totalPressure	Return pressure drop from constant loss factor and mass flow rate (dp = f(m_flow))

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.LossFactorData

Information

This record defines the pressure loss factors of a pipe segment (orifice, bending etc.) with a minimum amount of data. If available, data should be provided for both flow directions, i.e., flow from port_a to port_b and from port_b to port_a, as well as for the laminar and the turbulent region. It is also an option to provide the loss factor only for the turbulent region for a flow from port_a to port_b.

The following equations are used:

   Δp = 0.5*ζ*ρ*v*|v|
      = 0.5*ζ/A^2 * (1/ρ) * m_flow*|m_flow|
      = 8*ζ/(π^2*D^4*ρ) * m_flow*|m_flow|
        Re = |v|*D*ρ/μ

where

• Δp is the pressure drop: Δp = port_a.p - port_b.p
• v is the mean velocity.
• ρ is the density.
• ζ is the loss factor that depends on the geometry of the pipe. In the turbulent flow regime, it is assumed that ζ is constant and is given by "zeta1" and "zeta2" depending on the flow direction.
  When the Reynolds number Re is below "Re_turbulent", the flow is laminar for small flow velocities. For higher velocities there is a transition region from laminar to turbulent flow. The loss factor for laminar flow at small velocities is defined by the often occurring approximation c0/Re. If c0 is different for the two flow directions, the mean value has to be used (c0 = (c0_ab + c0_ba)/2).
• The equation "Δp = 0.5*ζ*ρ*v*|v|" is either with respect to port_a or to port_b, depending on the definition of the particular loss factor ζ (in some references loss factors are defined with respect to port_a, in other references with respect to port_b).
• Re = |v|*D_Re*ρ/μ = |m_flow|*D_Re/(A_Re*μ) is the Reynolds number at the smallest cross section area. This is often at port_a or at port_b, but can also be between the two ports. In the record, the diameter D_Re of this smallest cross section area has to be provided, as well, as Re_turbulent, the absolute value of the Reynolds number at which the turbulent flow starts. If Re_turbulent is different for the two flow directions, use the smaller value as Re_turbulent.
• D is the diameter of the pipe. If the pipe has not a circular cross section, D = 4*A/P, where A is the cross section area and P is the wetted perimeter.
• A is the cross section area with A = π(D/2)^2.
• μ is the dynamic viscosity.

The laminar and the transition region is usually of not much technical interest because the operating point is mostly in the turbulent regime. For simplification and for numerical reasons, this whole region is described by two polynomials of third order, one polynomial for m_flow ≥ 0 and one for m_flow < 0. The polynomials start at Re = |m_flow|*4/(π*D_Re*μ), where D_Re is the smallest diameter between port_a and port_b. The common derivative of the two polynomials at Re = 0 is computed from the equation "c0/Re". Note, the pressure drop equation above in the laminar region is always defined with respect to the smallest diameter D_Re.

If no data for c0 is available, the derivative at Re = 0 is computed in such a way, that the second derivatives of the two polynomials are identical at Re = 0. The polynomials are constructed, such that they smoothly touch the characteristic curves in the turbulent regions. The whole characteristic is therefore continuous and has a finite, continuous first derivative everywhere. In some cases, the constructed polynomials would "vibrate". This is avoided by reducing the derivative at Re=0 in such a way that the polynomials are guaranteed to be monotonically increasing. The used sufficient criteria for monotonicity follows from:

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

Parameters

Name	Description
diameter_a	Diameter at port_a [m]
diameter_b	Diameter at port_b [m]
zeta1	Loss factor for flow port_a -> port_b
zeta2	Loss factor for flow port_b -> port_a
Re_turbulent	Loss factors suited for Re >= Re_turbulent [1]
D_Re	Diameter used to compute Re [m]
zeta1_at_a	dp = zeta1*(if zeta1_at_a then rho_a*v_a^2/2 else rho_b*v_b^2/2)
zeta2_at_a	dp = -zeta2*(if zeta2_at_a then rho_a*v_a^2/2 else rho_b*v_b^2/2)
zetaLaminarKnown	= true, if zeta = c0/Re in laminar region
c0	zeta = c0/Re; dp = zeta*rho_Re*v_Re^2/2, Re=v_Re*D_Re*rho_Re/mu_Re)

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.massFlowRate_dp

Information

Compute mass flow rate from constant loss factor and pressure drop (m_flow = f(dp)). For small pressure drops (dp < dp_small), the characteristic is approximated by a polynomial in order to have a finite derivative at zero mass flow rate.

Inputs

Name	Description
dp	Pressure drop (dp = port_a.p - port_b.p) [Pa]
rho_a	Density at port_a [kg/m3]
rho_b	Density at port_b [kg/m3]
data	Constant loss factors for both flow directions
dp_small	Turbulent flow if |dp| >= dp_small [Pa]

Outputs

Name	Description
m_flow	Mass flow rate from port_a to port_b [kg/s]

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.massFlowRate_dp_and_Re

Information

Compute mass flow rate from constant loss factor and pressure drop (m_flow = f(dp)). If the Reynolds-number Re ≥ data.Re_turbulent, the flow is treated as a turbulent flow with constant loss factor zeta. If the Reynolds-number Re < data.Re_turbulent, the flow is laminar and/or in a transition region between laminar and turbulent. This region is approximated by two polynomials of third order, one polynomial for m_flow ≥ 0 and one for m_flow < 0. The common derivative of the two polynomials at Re = 0 is computed from the equation "data.c0/Re".

If no data for c0 is available, the derivative at Re = 0 is computed in such a way, that the second derivatives of the two polynomials are identical at Re = 0. The polynomials are constructed, such that they smoothly touch the characteristic curves in the turbulent regions. The whole characteristic is therefore continuous and has a finite, continuous first derivative everywhere. In some cases, the constructed polynomials would "vibrate". This is avoided by reducing the derivative at Re=0 in such a way that the polynomials are guaranteed to be monotonically increasing. The used sufficient criteria for monotonicity follows from:

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

Inputs

Name	Description
dp	Pressure drop (dp = port_a.p - port_b.p) [Pa]
rho_a	Density at port_a [kg/m3]
rho_b	Density at port_b [kg/m3]
mu_a	Dynamic viscosity at port_a [Pa.s]
mu_b	Dynamic viscosity at port_b [Pa.s]
data	Constant loss factors for both flow directions

Outputs

Name	Description
m_flow	Mass flow rate from port_a to port_b [kg/s]

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.pressureLoss_m_flow

Information

Compute pressure drop from constant loss factor and mass flow rate (dp = f(m_flow)). For small mass flow rates(|m_flow| < m_flow_small), the characteristic is approximated by a polynomial in order to have a finite derivative at zero mass flow rate.

Inputs

Name	Description
m_flow	Mass flow rate from port_a to port_b [kg/s]
rho_a	Density at port_a [kg/m3]
rho_b	Density at port_b [kg/m3]
data	Constant loss factors for both flow directions
m_flow_small	Turbulent flow if |m_flow| >= m_flow_small [kg/s]

Outputs

Name	Description
dp	Pressure drop (dp = port_a.p - port_b.p) [Pa]

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.pressureLoss_m_flow_and_Re

Information

Compute pressure drop from constant loss factor and mass flow rate (dp = f(m_flow)). If the Reynolds-number Re ≥ data.Re_turbulent, the flow is treated as a turbulent flow with constant loss factor zeta. If the Reynolds-number Re < data.Re_turbulent, the flow is laminar and/or in a transition region between laminar and turbulent. This region is approximated by two polynomials of third order, one polynomial for m_flow ≥ 0 and one for m_flow < 0. The common derivative of the two polynomials at Re = 0 is computed from the equation "data.c0/Re".

If no data for c0 is available, the derivative at Re = 0 is computed in such a way, that the second derivatives of the two polynomials are identical at Re = 0. The polynomials are constructed, such that they smoothly touch the characteristic curves in the turbulent regions. The whole characteristic is therefore continuous and has a finite, continuous first derivative everywhere. In some cases, the constructed polynomials would "vibrate". This is avoided by reducing the derivative at Re=0 in such a way that the polynomials are guaranteed to be monotonically increasing. The used sufficient criteria for monotonicity follows from:

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

Inputs

Name	Description
m_flow	Mass flow rate from port_a to port_b [kg/s]
rho_a	Density at port_a [kg/m3]
rho_b	Density at port_b [kg/m3]
mu_a	Dynamic viscosity at port_a [Pa.s]
mu_b	Dynamic viscosity at port_b [Pa.s]
data	Constant loss factors for both flow directions

Outputs

Name	Description
dp	Pressure drop (dp = port_a.p - port_b.p) [Pa]

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.BaseModel

Information

This model computes the pressure loss of a pipe segment (orifice, bending etc.) with a minimum amount of data provided via parameter data. If available, data should be provided for both flow directions, i.e., flow from port_a to port_b and from port_b to port_a, as well as for the laminar and the turbulent region. It is also an option to provide the loss factor only for the turbulent region for a flow from port_a to port_b.

The following equations are used:

   Δp = 0.5*ζ*ρ*v*|v|
      = 0.5*ζ/A^2 * (1/ρ) * m_flow*|m_flow|
        Re = |v|*D*ρ/μ

where

• Δp is the pressure drop: Δp = port_a.p - port_b.p
• v is the mean velocity.
• ρ is the density.
• ζ is the loss factor that depends on the geometry of the pipe. In the turbulent flow regime, it is assumed that ζ is constant and is given by "zeta1" and "zeta2" depending on the flow direction.
  When the Reynolds number Re is below "Re_turbulent", the flow is laminar for small flow velocities. For higher velocities there is a transition region from laminar to turbulent flow. The loss factor for laminar flow at small velocities is defined by the often occurring approximation c0/Re. If c0 is different for the two flow directions, the mean value has to be used (c0 = (c0_ab + c0_ba)/2).
• The equation "Δp = 0.5*ζ*ρ*v*|v|" is either with respect to port_a or to port_b, depending on the definition of the particular loss factor ζ (in some references loss factors are defined with respect to port_a, in other references with respect to port_b).
• Re = |v|*D_Re*ρ/μ = |m_flow|*D_Re/(A_Re*μ) is the Reynolds number at the smallest cross section area. This is often at port_a or at port_b, but can also be between the two ports. In the record, the diameter D_Re of this smallest cross section area has to be provided, as well, as Re_turbulent, the absolute value of the Reynolds number at which the turbulent flow starts. If Re_turbulent is different for the two flow directions, use the smaller value as Re_turbulent.
• D is the diameter of the pipe. If the pipe has not a circular cross section, D = 4*A/P, where A is the cross section area and P is the wetted perimeter.
• A is the cross section area with A = π(D/2)^2.
• μ is the dynamic viscosity.

The laminar and the transition region is usually of not much technical interest because the operating point is mostly in the turbulent regime. For simplification and for numerical reasons, this whole region is described by two polynomials of third order, one polynomial for m_flow ≥ 0 and one for m_flow < 0. The polynomials start at Re = |m_flow|*4/(π*D_Re*μ), where D_Re is the smallest diameter between port_a and port_b. The common derivative of the two polynomials at Re = 0 is computed from the equation "c0/Re". Note, the pressure drop equation above in the laminar region is always defined with respect to the smallest diameter D_Re.

If no data for c0 is available, the derivative at Re = 0 is computed in such a way, that the second derivatives of the two polynomials are identical at Re = 0. The polynomials are constructed, such that they smoothly touch the characteristic curves in the turbulent regions. The whole characteristic is therefore continuous and has a finite, continuous first derivative everywhere. In some cases, the constructed polynomials would "vibrate". This is avoided by reducing the derivative at Re=0 in such a way that the polynomials are guaranteed to be monotonically increasing. The used sufficient criteria for monotonicity follows from:

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

Parameters

Name	Description
replaceable package Medium	Medium in the component
pathLength	Length flow path [m]
data	Loss factor data
Nominal operating point
m_flow_nominal	Nominal mass flow rate [kg/s]
Assumptions
allowFlowReversal	= true to allow flow reversal, false restricts to design direction (port_a -> port_b)
Dynamics
momentumDynamics	Formulation of momentum balance
Advanced
dp_start	Guess value of dp = port_a.p - port_b.p [Pa]
m_flow_start	Guess value of m_flow = port_a.m_flow [kg/s]
m_flow_small	Small mass flow rate for regularization of zero flow [kg/s]
use_Re	= true, if turbulent region is defined by Re, otherwise by m_flow_small
from_dp	= true, use m_flow = f(dp) else dp = f(m_flow)
Diagnostics
show_T	= true, if temperatures at port_a and port_b are computed
show_V_flow	= true, if volume flow rate at inflowing port is computed
show_Re	= true, if Reynolds number is included for plotting

Connectors

Name	Description
port_a	Fluid connector a (positive design flow direction is from port_a to port_b)
port_b	Fluid connector b (positive design flow direction is from port_a to port_b)

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.TestWallFriction

Information

Parameters

Name	Description
replaceable package Medium	Medium in the component
data	Loss factor data
length	Length of pipe [m]
diameter	Inner diameter of pipe [m]
roughness	Absolute roughness of pipe (> 0 required, details see info layer) [m]
Nominal operating point
m_flow_nominal	Nominal mass flow rate [kg/s]
Assumptions
allowFlowReversal	= true to allow flow reversal, false restricts to design direction (port_a -> port_b)
Advanced
dp_start	Guess value of dp = port_a.p - port_b.p [Pa]
m_flow_start	Guess value of m_flow = port_a.m_flow [kg/s]
m_flow_small	Small mass flow rate for regularization of zero flow [kg/s]
use_Re	= true, if turbulent region is defined by Re, otherwise by m_flow_small
from_dp	= true, use m_flow = f(dp) else dp = f(m_flow)
Diagnostics
show_T	= true, if temperatures at port_a and port_b are computed
show_V_flow	= true, if volume flow rate at inflowing port is computed
show_Re	= true, if Reynolds number is included for plotting

Connectors

Name	Description
port_a	Fluid connector a (positive design flow direction is from port_a to port_b)
port_b	Fluid connector b (positive design flow direction is from port_a to port_b)

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.BaseModelNonconstantCrossSectionArea

Information

This model computes the pressure loss of a pipe segment (orifice, bending etc.) with a minimum amount of data provided via parameter data. If available, data should be provided for both flow directions, i.e., flow from port_a to port_b and from port_b to port_a, as well as for the laminar and the turbulent region. It is also an option to provide the loss factor only for the turbulent region for a flow from port_a to port_b.

The following equations are used:

   Δp = 0.5*ζ*ρ*v*|v|
      = 0.5*ζ/A^2 * (1/ρ) * m_flow*|m_flow|
        Re = |v|*D*ρ/μ

where

• Δp is the pressure drop: Δp = port_a.p - port_b.p
• v is the mean velocity.
• ρ is the density.
• ζ is the loss factor that depends on the geometry of the pipe. In the turbulent flow regime, it is assumed that ζ is constant and is given by "zeta1" and "zeta2" depending on the flow direction.
  When the Reynolds number Re is below "Re_turbulent", the flow is laminar for small flow velocities. For higher velocities there is a transition region from laminar to turbulent flow. The loss factor for laminar flow at small velocities is defined by the often occurring approximation c0/Re. If c0 is different for the two flow directions, the mean value has to be used (c0 = (c0_ab + c0_ba)/2).
• The equation "Δp = 0.5*ζ*ρ*v*|v|" is either with respect to port_a or to port_b, depending on the definition of the particular loss factor ζ (in some references loss factors are defined with respect to port_a, in other references with respect to port_b).
• Re = |v|*D_Re*ρ/μ = |m_flow|*D_Re/(A_Re*μ) is the Reynolds number at the smallest cross section area. This is often at port_a or at port_b, but can also be between the two ports. In the record, the diameter D_Re of this smallest cross section area has to be provided, as well, as Re_turbulent, the absolute value of the Reynolds number at which the turbulent flow starts. If Re_turbulent is different for the two flow directions, use the smaller value as Re_turbulent.
• D is the diameter of the pipe. If the pipe has not a circular cross section, D = 4*A/P, where A is the cross section area and P is the wetted perimeter.
• A is the cross section area with A = π(D/2)^2.
• μ is the dynamic viscosity.

The laminar and the transition region is usually of not much technical interest because the operating point is mostly in the turbulent regime. For simplification and for numerical reasons, this whole region is described by two polynomials of third order, one polynomial for m_flow ≥ 0 and one for m_flow < 0. The polynomials start at Re = |m_flow|*4/(π*D_Re*μ), where D_Re is the smallest diameter between port_a and port_b. The common derivative of the two polynomials at Re = 0 is computed from the equation "c0/Re". Note, the pressure drop equation above in the laminar region is always defined with respect to the smallest diameter D_Re.

If no data for c0 is available, the derivative at Re = 0 is computed in such a way, that the second derivatives of the two polynomials are identical at Re = 0. The polynomials are constructed, such that they smoothly touch the characteristic curves in the turbulent regions. The whole characteristic is therefore continuous and has a finite, continuous first derivative everywhere. In some cases, the constructed polynomials would "vibrate". This is avoided by reducing the derivative at Re=0 in such a way that the polynomials are guaranteed to be monotonically increasing. The used sufficient criteria for monotonicity follows from:

Fritsch F.N. and Carlson R.E. (1980):

Monotone piecewise cubic interpolation. SIAM J. Numerc. Anal., Vol. 17, No. 2, April 1980, pp. 238-246

Parameters

Name	Description
replaceable package Medium	Medium in the component
pathLength	Length flow path [m]
data	Loss factor data
Nominal operating point
m_flow_nominal	Nominal mass flow rate [kg/s]
Assumptions
allowFlowReversal	= true to allow flow reversal, false restricts to design direction (port_a -> port_b)
Dynamics
momentumDynamics	Formulation of momentum balance
Advanced
dp_start	Guess value of dp = port_a.p - port_b.p [Pa]
m_flow_start	Guess value of m_flow = port_a.m_flow [kg/s]
m_flow_small	Small mass flow rate for regularization of zero flow [kg/s]
Diagnostics
show_T	= true, if temperatures at port_a and port_b are computed
show_V_flow	= true, if volume flow rate at inflowing port is computed
show_Re	= true, if Reynolds number is included for plotting
show_totalPressures	= true, if total pressures are included for plotting
show_portVelocities	= true, if port velocities are included for plotting

Connectors

Name	Description
port_a	Fluid connector a (positive design flow direction is from port_a to port_b)
port_b	Fluid connector b (positive design flow direction is from port_a to port_b)

Modelica.Fluid.Fittings.BaseClasses.QuadraticTurbulent.pressureLoss_m_flow_totalPressure

Information

Compute pressure drop from constant loss factor and mass flow rate (dp = f(m_flow)). For small mass flow rates(|m_flow| < m_flow_small), the characteristic is approximated by a polynomial in order to have a finite derivative at zero mass flow rate.

Inputs

Name	Description
m_flow	Mass flow rate from port_a to port_b [kg/s]
rho_a_des	Density at port_a, mass flow in design direction a -> b [kg/m3]
rho_b_des	Density at port_b, mass flow in design direction a -> b [kg/m3]
rho_b_nondes	Density at port_b, mass flow against design direction a <- b [kg/m3]
rho_a_nondes	Density at port_a, mass flow against design direction a <- b [kg/m3]
data	Constant loss factors for both flow directions
m_flow_small	Turbulent flow if |m_flow| >= m_flow_small [kg/s]

Outputs

Name	Description
dp	Pressure drop (dp = port_a.p - port_b.p) [Pa]