Pipe (G)

The law of conservation of mass relates mass flow rates to the dynamics of pressure and temperature of the internal unit representing the volume of gas:

where

The law of conservation of energy relates the expenditure of energy and heat to the dynamics of pressure and temperature of the internal unit representing the volume of gas:

where

Partial derivatives of mass M and internal energy U The gas pressure and temperature at constant volume depend on the model of the gas properties. For models of an ideal and semi-ideal gas, the equations are as follows:

,

,

,

,

where

For a real gas model, partial derivatives of mass M and internal energy U gases with respect to pressure and temperature at constant volume are equal to:

,

,

,

,

where

The momentum balance for each half of the pipe simulates a pressure drop due to the momentum of the gas flow and viscous friction:

,

,

where

The heat exchange with the pipe wall through the H port is added to the energy of the gas represented by the internal node through the energy conservation equation. Therefore, the pulse balance for each half of the pipe between port A and the inner node and between port B and the inner node is considered an adiabatic process. Adiabatic relations:

,

,

where — specific enthalpy at port A, port B, or internal node I, as indicated by the subscript.

Pressure loss due to viscous friction and they depend on the flow regime. The Reynolds numbers for each half of the pipe are defined as:

,

,

where

If the Reynolds number is less than the parameter value Laminar flow upper Reynolds number limit, then the flow is in laminar mode. If the Reynolds number is greater than the limit value of the parameter Turbulent flow lower Reynolds number limit, then the current is in a turbulent mode.

In the laminar flow regime, the pressure loss due to viscous friction is:

,

,

where

In the turbulent flow regime, the pressure loss due to viscous friction is:

,

,

where — the Darcy coefficient on port A or B, as indicated by the subscript.

The Darcy coefficients are calculated from the Haaland correlation:

,

,

where — parameter value Internal surface absolute roughness.

When the Reynolds number is between the upper limit of the Reynolds number for laminar flow (parameter value Laminar flow upper Reynolds number limit) and the value of the lower limit of the Reynolds number for a turbulent flow (the value of the parameter Turbulent flow lower Reynolds number limit), the flow is in a transition state between laminar and turbulent flow. Pressure losses due to viscous friction in the transition region smoothly change between losses in the laminar flow regime and losses in the turbulent flow regime.

The equation of convective heat transfer between the pipe wall and the internal gas volume:

where — pipe surface area, .

Assuming an exponential temperature distribution along the pipe, the convective heat transfer is

where

Heat transfer coefficient depends on the Nusselt number:

where — the coefficient of thermal conductivity calculated at an average temperature.

The Nusselt number depends on the flow regime. The Nusselt number in the laminar flow mode is constant and is equal to the value of the parameter Nusselt number for laminar flow heat transfer. The Nusselt number in the turbulent flow regime is calculated from the Gnelinsky equation:

where tem:[Pr_("avg")] is the Prandtl number calculated at an average temperature.

The average Reynolds number is

where — dynamic viscosity, estimated at an average temperature.

When the mean is between the upper limit of the Reynolds number for laminar flow (parameter value Laminar flow upper Reynolds number limit) and the value of the lower limit of the Reynolds number for the turbulent flow (the value of the parameter Turbulent flow lower Reynolds number limit), the Nusselt number smoothly varies between the values of the Nusselt number for laminar and turbulent flows.

The mass flow rate when the gas pressure from the pipe decreases at ports A and B is determined by the following expressions:

,

,

where and — the speed of sound on ports A and B respectively.

The pressure at subcritical flow at port A or B is equal to the value of the corresponding variable:

,

.

At critical flow, the pressure at ports A and B is obtained by substituting mass flow rates at critical flow into the momentum balance equations for the pipe:

,

,

where and They represent pressure losses due to viscous friction under the assumption that the flow goes into critical mode. They are calculated similarly and with the replacement of the mass flow values at ports A and B with the mass flow values of the critical flow.

Depending on whether the flow has become critical, the unit assigns the pressure value in the critical or subcritical flow as the actual pressure at the port. The flow may become critical at the outlet of the pipe, but not at the inlet. Therefore, if , then port A is the input and . If , then port A is the output.

Similarly, if , then port B is the input and . If , then port B is the output.

Use the parameter group Initial Targets to set the priority and initial target values for the block parameter variables before modeling. For more information, see Configuring physical blocks using target values.