Buildings.Fluid.BaseClasses.FlowModels

Flow models for pressure drop calculations

Information

This package contains basic flow models that are used by the various models that compute pressure drop.

Assumption and limitations

Because the density does not change signficantly in heating, ventilation and air conditioning systems for buildings, the flow models compute the pressure drop based on the mass flow rate and not the volume flow rate. This typically leads to simpler equations because it does not require the mass density, which changes when the flow is reversed. Although, for conceptual design of building energy system, there is in general not enough information available that would warrant a more detailed pressure drop calculation. If a more detailed computation of the flow resistance is needed, then a user can use models from the Modelica.Fluid library.

All functions have an argument m_flow_turbulent that determines where the flow transitions to fully turbulent flow. For smaller mass flow rates, the power-law relation is replaced by a function that has a finite slope near zero pressure drop. This is done for numerical reasons, and to approximate laminar flow, although the implementation does not use a linear function.

Implementation

The four main functions are

• Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp,
• Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow,
• Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp, and
• Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow.

These functions compute the mass flow rate or the pressure drop, respectively. The first two functions assume that the flow resistance is quadratic in the mass flow rate, and the other two functions allow for a flow exponent between 1 and 2. All these functions are two times continuously differentiable. First and second order derivatives are provided in the function that have the suffix _der and _der2.

For the powerLaw functions, the coefficients that are an argument to the functions can be computed using the function Buildings.Fluid.BaseClasses.FlowModels.powerLawData. This computation is done outside the above functions because the arguments generally are all parameters, and hence precomputing them avoid repetitive evaluation.

Extends from Modelica.Icons.BasesPackage (Icon for packages containing base classes).

Package Content

Name	Description
basicFlowFunction_dp	Function that computes mass flow rate for given pressure drop
basicFlowFunction_dp_der	1st derivative of function that computes mass flow rate for given pressure drop
basicFlowFunction_dp_der2	2nd derivative of function that computes mass flow rate for given pressure drop
basicFlowFunction_m_flow	Function that computes pressure drop for given mass flow rate
basicFlowFunction_m_flow_der	1st derivative of function that computes pressure drop for given mass flow rate
basicFlowFunction_m_flow_der2	2nd derivative of function that computes pressure drop for given mass flow rate
powerLawData	Function that computes the coefficients used by powerLaw_dp and powerLaw_m_flow
powerLaw_dp	Power law used in pressure drop equations when the flow exponent is constant and may be different from 2
powerLaw_dp_der	1st derivative of function that computes mass flow rate for given pressure drop
powerLaw_dp_der2	2nd derivative of function that computes mass flow rate for given pressure drop
powerLaw_m_flow	Inverse of power law used in pressure drop equations when the flow exponent is constant and may be different from 2
powerLaw_m_flow_der	1st derivative of function that computes pressure drop for given mass flow rate
powerLaw_m_flow_der2	2nd derivative of function that computes pressure drop for given mass flow rate
Validation	Collection of validation models

Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp

Function that computes mass flow rate for given pressure drop

Information

Function that computes the pressure drop of flow elements as

ṁ = sign(Δp) k √ Δp  

with regularization near the origin. Therefore, the flow coefficient is

k = ṁ ⁄ √ Δp  

The input m_flow_turbulent determines the location of the regularization.

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp_der

1st derivative of function that computes mass flow rate for given pressure drop

Information

Function that implements the first order derivative of Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp, assuming a constant flow coefficient.

When called with dp_der=der(dp), this function returns the time derivative of m_flow. When called with dp_der=1, this function returns the derivative of m_flow with respect to dp.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp_der2

2nd derivative of function that computes mass flow rate for given pressure drop

Information

Function that implements the second order derivative of Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_dp, assuming a constant flow coefficient.

When called with dp_der=der(dp) and dp_der2=der(dp_der), this function returns the second order derivative of m_flow with respect to time. When called with dp_der=1 and dp_der2=0, this function returns the second order derivative of m_flow with respect to dp.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow

Function that computes pressure drop for given mass flow rate

Information

Function that computes the pressure drop of flow elements as

Δp = sign(ṁ) (ṁ ⁄ k)²

with regularization near the origin. Therefore, the flow coefficient is

k = ṁ ⁄ √ Δp  

The input m_flow_turbulent determines the location of the regularization.

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow_der

1st derivative of function that computes pressure drop for given mass flow rate

Information

Function that implements the first order derivative of Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow, assuming a constant flow coefficient.

When called with m_flow_der=der(m_flow), this function returns the time derivative of dp. When called with m_flow_der=1, this function returns the derivative of dp with respect to m_flow.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow_der2

2nd derivative of function that computes pressure drop for given mass flow rate

Information

Function that implements the second order derivative of Buildings.Fluid.BaseClasses.FlowModels.basicFlowFunction_m_flow, assuming a constant flow coefficient.

When called with m_flow_der=der(m_flow) and m_flow_der2=der(m_flow_der), this function returns the second order derivative of dp with respect to time. When called with m_flow_der=1 and m_flow_der2=0, this function returns the second order derivative of dp with respect to m_flow.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.powerLawData

Function that computes the coefficients used by powerLaw_dp and powerLaw_m_flow

Information

This function computes the coefficients that are used by the functions Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp and Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow.

The coefficients dp_turbulent, m, a1, a3 and a5 are used by Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp, while the coefficients C, b1, b3 and b5 are used by Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow.

The coefficients only depend on the k, n and m_flow_turbulent which often are parameters in a model. In this case, this function allows to compute the coefficients only once as parameters and then pass them to the above functions, which avoids recomputing them during the time step simulation.

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp

Power law used in pressure drop equations when the flow exponent is constant and may be different from 2

Information

This model describes the mass flow rate and pressure difference relation of a flow resistance in the form

ṁ = k sign(Δp) |Δp|^1/n

where ṁ is the mass flow rate, k > 0 is a flow coefficient Δ p is the pressure drop and n ∈ [1, 2] is a flow exponent. The equation is regularized for |Δp| < Δp_t, where Δp_t is a parameter that is computed from the input m_flow_turbulent. For laminar flow, set n=1 and for turbulent flow, set n=2.

The polynomial coefficients a1, a3 and a5, the flow exponent m and the pressure difference dp_turbulent are computed by the function Buildings.Fluid.BaseClasses.FlowModels.powerLawData and passed as inputs. As these quantities only depend on the parameters k, n and m_flow_turbulent, they can be computed once as parameters rather than at each function evaluation.

The model is used for the fluid flow models that are neither fully laminar nor fully turbulent. It is identical to Buildings.Airflow.Multizone.BaseClasses.powerLawFixedM except that it is formulated for mass flow rate rather than volume flow rate.

Note regarding arguments

This function takes as inputs not only the coefficients dp_turbulent, m, a1, a3 and a5 that are used in its own implementation, but also the coefficients C, b1, b3 and b5 that are used by its inverse function Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow. These additional arguments are needed so that the inverse annotation can pass the input arguments of this function directly to its inverse function. Therefore, this function and its inverse function Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow have the same input arguments, except that this function takes the pressure difference dp as the first argument while its inverse takes the mass flow rate m_flow as the first argument. The coefficients C, b1, b3 and b5 are computed by the function Buildings.Fluid.BaseClasses.FlowModels.powerLawData.

Implementation

For |Δp| < Δp_t, the equation is regularized so that it is twice continuously differentiable in Δp, and that it has an infinite number of continuous derivatives in n and in k.

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp_der

1st derivative of function that computes mass flow rate for given pressure drop

Information

Function that implements the first order derivative of Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp with respect to the pressure difference dp, assuming constant flow coefficients.

When called with dp_der=der(dp), this function returns the time derivative of m_flow. When called with dp_der=1, this function returns the derivative of m_flow with respect to dp.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp_der2

2nd derivative of function that computes mass flow rate for given pressure drop

Information

Function that implements the second order derivative of Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp with respect to the pressure difference dp, assuming constant flow coefficients.

When called with dp_der=der(dp) and dp_der2=der(dp_der), this function returns the second order derivative of m_flow with respect to time. When called with dp_der=1 and dp_der2=0, this function returns the second order derivative of m_flow with respect to dp.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow

Inverse of power law used in pressure drop equations when the flow exponent is constant and may be different from 2

Information

This model describes the pressure difference and mass flow rate relation of a flow resistance in the form

Δp = sign(ṁ) (ṁ ⁄ k)ⁿ

where Δp is the pressure drop, k > 0 is a flow coefficient, ṁ is the mass flow rate, and n ∈ [1, 2] is a flow exponent. The equation is regularized for |ṁ| < ṁ_t, where ṁ_t is a parameter. For laminar flow, set n=1 and for turbulent flow, set n=2.

The coefficient C and the polynomial coefficients b1, b3 and b5 are computed by the function Buildings.Fluid.BaseClasses.FlowModels.powerLawData and passed as inputs. As these quantities only depend on the parameters k, n and m_flow_turbulent, they can be computed once as parameters rather than at each function evaluation.

The model is used for the fluid flow models that are neither fully laminar nor fully turbulent.

Note regarding arguments

This function takes as inputs not only the coefficients C, b1, b3 and b5 that are used in its own implementation, but also the coefficients dp_turbulent, m, a1, a3 and a5 that are used by its inverse function Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp. These additional arguments are needed so that the inverse annotation can pass the input arguments of this function directly to its inverse function. Therefore, this function and its inverse function Buildings.Fluid.BaseClasses.FlowModels.powerLaw_dp have the same input arguments, except that this function takes the mass flow rate m_flow as the first argument while its inverse takes the pressure difference dp as the first argument. The coefficients dp_turbulent, m, a1, a3 and a5 are computed by the function Buildings.Fluid.BaseClasses.FlowModels.powerLawData.

Implementation

For |ṁ| < ṁ_t, the equation is regularized so that it is twice continuously differentiable in ṁ, and that it has an infinite number of continuous derivatives in n and in k.

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow_der

1st derivative of function that computes pressure drop for given mass flow rate

Information

Function that implements the first order derivative of Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow with respect to the mass flow rate m_flow, assuming constant flow coefficients.

When called with m_flow_der=der(m_flow), this function returns the time derivative of dp. When called with m_flow_der=1, this function returns the derivative of dp with respect to m_flow.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition

Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow_der2

2nd derivative of function that computes pressure drop for given mass flow rate

Information

Function that implements the second order derivative of Buildings.Fluid.BaseClasses.FlowModels.powerLaw_m_flow with respect to the mass flow rate m_flow, assuming constant flow coefficients.

When called with m_flow_der=der(m_flow) and m_flow_der2=der(m_flow_der), this function returns the second order derivative of dp with respect to time. When called with m_flow_der=1 and m_flow_der2=0, this function returns the second order derivative of dp with respect to m_flow.

Extends from Modelica.Icons.Function (Icon for functions).

Inputs

Outputs

Modelica definition