Pipe (MA)

The net flow rate of moist air in the volume of a pipe is equal to:

,

,

,

,

where

The water vapour mass conservation equation relates the mass flow rate of water vapour to the dynamics of the humidity level in the internal volume of humid air:

.

Similarly, the equation of conservation of admixture gas mass relates the mass flow rate of admixture gas to the dynamics of the level of admixture gas in the internal volume of humid air:

.

The equation of conservation of mixture mass relates the mass flow rate of the mixture to the dynamics of pressure, temperature and mass fractions of the internal volume of moist air:

.

Finally, the energy conservation equation relates the energy flow rate to the dynamics of pressure, temperature, and mass fraction of the internal volume of moist air:

.

The equation of state relates the density of the mixture to pressure and temperature:

.

The universal gas constant of the mixture is equal to:

.

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

,

,

where

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

,

,

where

If Reynolds number is less than the value of parameters Laminar flow upper Reynolds number limit, the flow is in laminar regime. If the Reynolds number is greater than the limiting value of the parameter Turbulent flow lower Reynolds number limit, the flow is in the turbulent regime.

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

,

,

where

In the turbulent flow regime, the pressure losses due to viscous friction are:

,

,

where

Darcy coefficients are calculated from the Haaland correlation:

,

,

where is the value of the parameters Internal surface absolute roughness.

When the Reynolds number is between the upper limit of Reynolds number for laminar flow and the parameters values of the lower limit of Reynolds number for turbulent flow, the flow is in a transient state between laminar and turbulent flow regimes. The pressure losses due to viscous friction in the transition state follow a smooth relationship between the losses in the laminar flow regime and the losses in the turbulent flow regime.

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 momentum balance for each half of the pipe between port A and the internal node and between port B and the internal node is considered an adiabatic process. Adiabatic relations:

,

,

where is the specific enthalpy at port A, port B or internal node I, as indicated by the lower index.

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

,

where is the surface area of the pipe, .

If no condensate is formed on the wall surface, assuming exponential temperature distribution along the pipe, convective heat transfer is equal:

,

where

The heat transfer coefficient depends on the Nusselt number:

,

where is the heat transfer coefficient calculated at the average temperature.

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

,

where is the Prandtl number calculated at mean temperature.

The average Reynolds number is equal to:

,

where is the dynamic viscosity estimated at mean temperature.

When the average Reynolds number is between the parameters of the upper limit Reynolds number for laminar flow and the lower limit Reynolds number for turbulent flow, the Nusselt number corresponds to a smooth transition between the Nusselt number values for laminar and turbulent flows.

The equations in this section account for condensation, which occurs when a volume of moist air becomes saturated.

When the volume of moist air reaches saturation, condensation can form. The specific humidity at saturation is equal to:

,

where

The condensation flow rate is equal to:

,

where is the value of the parameters Condensation time constant.

Condensed water is subtracted from the volume of moist air as shown in the mass conservation equations. The energy associated with condensed water is equal to:

,

where is the specific enthalpy of vaporisation, estimated at .

Parameters of change in the amount of moisture and impurity gases are related to each other as follows:

,

,

,

,

.

Humid air units that contain an internal liquid volume (such as chambers, converters and so on) simulate water vapour condensation when that liquid volume becomes fully saturated with water vapour, i.e. at 100% relative humidity. However, water vapour can also condense on a cold surface even if the volume of air as a whole has not yet reached saturation. The ability to model this effect in the Pipe (MA) unit is important because many HVAC systems contain pipes and ducts. If these pipes and ducts are poorly insulated, their surfaces can cool and condensation can form on the wall surface. Note that this effect does not replace condensation that occurs when the volume of moist air reaches 100% relative humidity, both effects can occur simultaneously.

To model the effect of condensation on the cold surface of the pipe in contact with the humid air volume, check the box Condensation on wall surface. In this case, the convective heat transfer equation must account for both visible and latent heat, and the block has an additional equation that calculates the condensation rate of water vapour on the surface.

If the Condensation on wall surface checkbox is checked, the combined convective heat transfer is equal to:

,

where

This equation is similar to the convective heat transfer equation, but the temperature difference has been replaced by the difference in enthalpy of the mixture. Since the enthalpy of the mixture depends on both the temperature and the composition of the moist air, the enthalpy difference of the mixture accounts for both the change in temperature and the change in moisture content. The unit captures both explicit and implicit thermal effects. The exponent and correlation parts of the equation used in calculating the heat transfer coefficient remain the same as before because the model is derived from the analogy between heat and mass exchange.

To simplify the derivation, the enthalpy of the mixture per unit mass of dry air and impurity gas is used in the equation, as opposed to the enthalpy of the mixture per unit mass of the mixture, since the amount of dry air and impurity gas does not change during the condensation of water vapour. To keep the equation consistent, the difference in enthalpy of the mixture is multiplied by the mass flow rate of dry air and impurity gas.

The enthalpy of the mixture per unit mass of dry air and impurity gases at the inlet is equal to:

,

where

The enthalpy of the mixture per unit mass of dry air and impurity gases at the wall is equal to:

,

where

The function in the previous equation provides a switch between dry and wet heat transfer:

The condensation flow rate of water vapour on the wall surface is:

.

This equation is similar to the combined convective heat transfer equation because the amount of water vapour condensing on the wall is the same as the convective mass transfer from the moist air to the pipe wall. The exponential component of the equation is also the same because of the analogy used between heat and mass transfer.

The energy associated with water condensing on the pipe wall is equal to:

,

where is the specific enthalpy of vaporisation at the wall temperature.

The essential part of convective heat transfer between the pipe wall and moist air is:

.

This equation has a plus sign because is negative when the moist air is cooled. Thus, adding , which is a positive value, eliminates the latent part of the heat transfer.

The block then uses this value in the first convective heat transfer equation to calculate the heat transfer at port H.

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

,

.

However, the port pressure variables used in the momentum balance equations, and , are not necessarily the same as the pressures in variables and , because the pipe outlet can reach a sound velocity barrier. The sound barrier occurs when the outlet pressure is low enough. At this point, the flow rate depends only on inlet conditions. Consequently, when the sound barrier is reached, the outlet pressure ( or , whichever is the outlet) cannot decrease further, even if the downstream pressure, represented by or , continues to decrease.

The sound barrier may occur at the outlet of the pipe, but not at the inlet. Hence, if port A is the inlet port, . If port A is an outlet port, then:

Similarly, if port B is an inlet port, then . If port B is an exhaust port, then:

The pressure when the sound barrier is reached at ports A and B is determined from the pulse balance, assuming that the velocity at the outlet is equal to the speed of sound:

,

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