3-Zone Pipe (2P)
Conduit for the transport of a phase-changing fluid with heat transfer
Since R2018b
Libraries:
Simscape /
Fluids /
Two-Phase Fluid /
Pipes & Fittings
Description
The 3-Zone Pipe (2P) block models a pipe with a phase-changing fluid. Each fluid phase is called a zone, which is a fractional value between 0 and 1. Zones do not mix. The block uses a boundary-following model to track the sub-cooled liquid (L), vapor-liquid mixture (M), and super-heated vapor (V) in three zones. The relative amount of space a zone occupies in the system is called a zone length fraction within the system.
Port H is a thermal port that represents the environmental temperature. The rate of heat transfer between the fluid and the environment depends on the fluid phase of each zone. The pipe wall is modeled within the block and the pipe wall temperature in each zone may be different. The pressure and temperature are influenced by fluid dynamic compressibility and the fluid zone thermal capacity.
Heat Transfer Between the Fluid and the Wall
The convective heat transfer coefficient between the fluid and the wall, αF, varies per zone according to the Nusselt number:
where:
Nu
is the zone Nusselt number.k the average fluid thermal conductivity.
DH is the pipe Hydraulic diameter, the equivalent diameter of a non-circular pipe.
The Nusselt number used in the heat transfer coefficient is the greater of the turbulent- and laminar-flow Nusselt numbers.
For turbulent flows in the subcooled liquid or superheated vapor zones, the Nusselt number is calculated with the Gnielinski correlation:
where:
Re
is the zone average Reynolds number.Pr
is the zone average Prandtl number.f is the Darcy friction factor, calculated from the Haaland correlation:
where ε is the wall Internal surface absolute roughness.
For turbulent flows in the liquid-vapor mixture zone, the Nusselt number is calculated with the Cavallini-Zecchin correlation:
Where:
ReSL is the Reynolds number of the saturated liquid.
PrSL is the Prandtl number of the saturated liquid.
ρSL is the density of the saturated liquid.
ρSV is the density of the saturated vapor.
a= 0.05, b = 0.8, and c= 0.33.
When fins are modeled on the pipe internal surface, the heat transfer coefficient is:
where:
ηInt is the Internal fin efficiency.
sInt is the Ratio of internal fins surface area to no-fin surface area.
For laminar flows, the Nusselt number is set by the Laminar flow Nusselt number parameter.
When the Heat transfer coefficient model parameter is Colburn equation
, the block calculates the Nusselt number for the
subcooled liquid and superheated vapor zones by using the empirical
Colburn equation
where a, b, and c are values in the Coefficients [a, b, c] for a*Re^b*Pr^c in liquid zone and Coefficients [a, b, c] for a*Re^b*Pr^c in vapor zone parameters.
The block calculates the Nusselt number for liquid-vapor mixture zones by using the Cavallini-Zecchin equation with the variables in the Coefficients [a, b, c] for a*Re^b*Pr^c in mixture zone parameter.
The heat transfer rate from the fluid is based on the change in specific enthalpy in each zone:
where is the mass flow rate for heat transfer. It is the pipe inlet mass flow rate, either A or B, depending on the direction of fluid flow.
In the liquid and vapor zones, the change in specific enthalpy is defined as:
where:
cp is the specific heat of the liquid or vapor.
TH is the environmental temperature.
TI is the liquid inlet temperature.
z is the fluid zone length fraction.
αE is the heat transfer coefficient between the wall and the environment.
SW is wall surface area:
where:
A is the pipe Cross-sectional area.
L is the Pipe length.
Note that this wall surface area does not include fin area, which is defined in the Ratio of external fins surface area to no-fin surface area and Ratio of internal fins surface area to no-fin surface area parameters. Fins are set in proportion to the wall surface area. A value of
0
means there are no fins on the pipe wall.
In the liquid-vapor mixture zone, the change in specific enthalpy is calculated as:
where TS is the fluid saturation temperature. It is assumed that the liquid-vapor mixture is always at this temperature.
The total heat transfer between the fluid and the pipe wall is the sum of the heat transfer in each fluid phase:
The heat transfer rate between the fluid and the pipe in the liquid zone is:
where TW,L is the temperature of the wall surrounding the liquid zone.
The heat transfer rate between the fluid and the pipe in the mixture zone is:
The heat transfer rate between the fluid and the pipe in the vapor zone is:
where TW,V is the temperature of the wall surrounding the vapor zone.
Heat Transfer Between the Wall and the Environment
If the pipe wall has a finite thickness, the heat transfer coefficient between the wall and the environment, αE, is defined by:
where αW is the heat transfer coefficient due to conduction through the wall:
and where:
kW is the Wall thermal conductivity.
tW is the Wall thickness .
αExt is the External environment heat transfer coefficient.
ηExt is the External fin efficiency.
sExt is the Ratio of external fins surface area to no-fin surface area.
If the wall does not have thermal mass, the heat transfer coefficient between the wall and environment equals the heat transfer coefficient of the environment, αExt.
The heat transfer rate between each wall zone and the environment is:
The total heat transfer between the wall and the environment is:
The heat transfer rate depends on the thermal mass of the wall, CW:
where:
cp,W is the Wall specific heat.
ρW is the Wall density.
The governing equations for heat transfer between the fluid and the external environment are, for the liquid zone:
for the mixture zone:
and for the vapor zone:
Momentum Balance
The pressure differential over the pipe consists of two factors: changes in pressure due to changes in density, and changes in pressure due to friction at the pipe walls.
For turbulent flows, when the Reynolds number is above the Turbulent flow lower Reynolds number limit, pressure loss is calculated in terms of the Darcy friction factor. The pressure differential between port A and the internal node I is:
where:
ρI is the fluid density at internal node I.
ρA* is the fluid density at port A. This is the same as ρA when the flow is steady-state; when the flow is transient, it is calculated from the fluid internal state with the adiabatic expression:
where:
h is the average specific enthalpy,
ρ is the average density,
This is due to the fact that heat transfer calculation takes place at the internal node I.
A is the mass flow rate through port A.
L is the Pipe length.
LAdd is the Aggregate equivalent length of local resistances, which is the equivalent length of a tube that introduces the same amount of loss as the sum of the losses due to other local resistances.
Note that the Darcy friction factor is dependent on the Reynolds number, and is calculated at both ports.
The pressure differential between port B and internal node I is:
where:
ρB* is the fluid density at port B. This is the same as ρB when the flow is steady-state; when the flow is transient, it is calculated from the fluid internal state with the adiabatic expression:
B is the mass flow rate through port B.
For laminar flows, when the Reynolds number is below the Laminar flow upper Reynolds number limit, the pressure loss due to friction is calculated in terms of the Laminar friction constant for Darcy friction factor, λ. The pressure differential between port A and internal node I is:
where μ is the average fluid dynamic viscosity:
The pressure differential between port B and internal node I is:
For transitional flows, the pressure differential due to viscous friction is a smoothed blend between the values for laminar and turbulent pressure losses.
Mass Balance
The total mass accumulation rate is defined as:
where M is the total fluid mass in the pipe. In terms of fluid zones, the mass accumulation rate is a function of the change in density, ρ, with respect to pressure, p, and the fluid specific internal energy, u:
where:
uout is the specific internal energy after all heat transfer has occurred.
V is the total fluid volume, or the volume of the pipe.
Energy Balance
The energy conservation equation is:
where:
ϕA is the energy flow rate at port A.
ϕB is the energy flow rate at port B.
QF is the heat transfer rate between the fluid and the wall.
Assumptions and Limitations
The pipe wall is perfectly rigid.
The flow is fully developed. Friction losses and heat transfer do not include entrance effects.
Fluid inertia is negligible.
The effect of gravity is negligible.
When the pressure is above the fluid critical pressure, large values of thermal fluid properties (such as Prandtl number, thermal conductivity, and specific heat) may not accurately reflect the heat exchange in the pipe.
Ports
Output
Conserving
Parameters
References
[1] White, F.M., Fluid Mechanics, 7th Ed, Section 6.8. McGraw-Hill, 2011.
[2] Çengel, Y.A., Heat and Mass Transfer—A Practical Approach, 3rd Ed, Section 8.5. McGraw-Hill, 2007.