# 3-Zone Pipe (2P)

Pipe with phase-changing fluid in a two-phase fluid network

**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 block models the pipe wall and the pipe wall
temperature in each zone may be different. The fluid dynamic compressibility and the
fluid zone thermal capacity impact the fluid pressure and temperature.

### Heat Transfer Between the Fluid and the Wall

The convective heat transfer coefficient between the fluid and the wall,
*α _{F}*, varies per zone according to
the Nusselt number

$${\alpha}_{F}=\frac{\text{Nu}k}{{D}_{\text{H}}},$$

where:

`Nu`

is the zone Nusselt number.*k*the average fluid thermal conductivity.*D*is the value of the_{H}**Hydraulic diameter**parameter, 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

$$\text{Nu}=\frac{\frac{f}{8}\left(\text{Re}-1000\right)\text{Pr}}{1+12.7{\left(\text{}\frac{f}{8}\right)}^{1/2}\left({\text{Pr}}_{}^{2/3}-1\right)},$$

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$$\frac{1}{\sqrt{{f}_{}}}=-1.8\text{log}\left[{\left(\frac{\frac{\epsilon}{{D}_{\text{H}}}}{3.7}\right)}^{1.11}+\frac{6.9}{{\text{Re}}_{}}\right],$$

where

*ε*is the value of the**Internal surface absolute roughness**parameter.

For turbulent flows in the liquid-vapor mixture zone, the Nusselt number is calculated with the Cavallini-Zecchin correlation

$$\text{Nu}=\frac{{\text{aRe}}_{\text{SL}}^{b}{\text{Pr}}_{\text{SL}}^{c}\left\{{\left[\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right){x}_{\text{Out}}+1\right]}^{1+b}-{\left[\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right){x}_{\text{In}}+1\right]}^{1+b}\right\}}{\left(1+b\right)\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right)({x}_{\text{Out}}-{x}_{\text{In}})},$$

where:

*Re*is the Reynolds number of the saturated liquid._{SL}*Pr*is the Prandtl number of the saturated liquid._{SL}*ρ*is the density of the saturated liquid._{SL}*ρ*is the density of the saturated vapor._{SV}*a*= 0.05,*b*= 0.8, and*c*= 0.33.

When fins are modeled on the pipe internal surface, the heat transfer coefficient is

$${\alpha}_{F}=\frac{\text{Nu}k}{{D}_{\text{H}}}\left(1+{\eta}_{\text{Int}}{s}_{\text{Int}}\right),$$

where:

*η*_{Int}is the value of the**Internal fin efficiency**parameter.*s*_{Int}is the value of the**Ratio of internal fins surface area to no-fin surface area**parameter.

For laminar flows, the Nusselt number is the value of the **Laminar flow
Nusselt number** parameter.

**Empirical Nusselt Number Formulation**

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

$$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c},$$

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.

**Specific Enthalpy**

The heat transfer rate from the fluid is based on the change in specific enthalpy in each zone

$$Q={\dot{m}}_{Q}\left(\Delta {h}_{\text{L}}+\Delta {h}_{\text{M}}+\Delta {h}_{\text{V}}\right),$$

where $${\dot{m}}_{Q}$$ is the mass flow rate for heat transfer. It is the pipe inlet
mass flow rate, either $$\dot{m}$$_{A} or $$\dot{m}$$_{B}, depending on the direction of fluid
flow.

In the liquid and vapor zones, the change in specific enthalpy is defined as

$$\Delta h={c}_{\text{p}}\left({T}_{H}-{T}_{\text{I}}\right)\left[1-\text{exp}\left(-\frac{z{S}_{\text{W}}}{{\dot{m}}_{Q}{c}_{\text{p}}\left({\alpha}_{F}^{-1}+{\alpha}_{\text{E}}^{-1}\right)}\right)\right],$$

where:

*c*is the specific heat of the liquid or vapor._{p}*T*is the environmental temperature._{H}*T*is the liquid inlet temperature._{I}*z*is the fluid zone length fraction.*α*is the heat transfer coefficient between the wall and the environment._{E}*S*is wall surface area_{W}$${S}_{\text{W}}=\frac{4A}{{D}_{\text{H}}}L,$$

where:

*A*is the value of the**Cross-sectional area**parameter.*L*is the value of the**Pipe length**parameter.

The wall surface area does not include fin area, which is set by 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

$$\Delta h=\left({T}_{\text{H}}-{T}_{\text{S}}\right)\frac{z{S}_{\text{W}}}{{\dot{m}}_{Q}\left({\alpha}_{F}^{-1}+{\alpha}_{\text{E}}^{-1}\right)},$$

where *T _{S}* is the
fluid saturation temperature. It is assumed that the liquid-vapor mixture is
always at this temperature.

**Heat Transfer Rate**

The total heat transfer between the fluid and the pipe wall is the sum of the heat transfer in each fluid phase

$${Q}_{\text{F}}={Q}_{\text{F,L}}+{Q}_{\text{F,V}}+{Q}_{\text{F,M}}.$$

The heat transfer rate between the fluid and the pipe in the liquid zone is

$${Q}_{\text{F,L}}={\dot{m}}_{\text{Q}}{c}_{\text{p,L}}\left[{T}_{\text{W,L}}-\text{min}\left({T}_{\text{I}},{T}_{\text{S}}\right)\right]\left[1-\text{exp}\left(-\frac{{z}_{\text{L}}{S}_{\text{W}}{\alpha}_{\text{L}}}{{\dot{m}}_{\text{Q}}{c}_{\text{p,L}}}\right)\right].$$

where *T _{W,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

$${Q}_{\text{F,M}}=\left({T}_{\text{H}}-{T}_{\text{Sat}}\right){z}_{\text{M}}{S}_{\text{W}}{\alpha}_{\text{M}}.$$

The heat transfer rate between the fluid and the pipe in the vapor zone is

$${Q}_{\text{F,V}}={\dot{m}}_{\text{Q}}{c}_{\text{p,V}}\left[{T}_{\text{W,V}}-\text{min}\left({T}_{\text{I}},{T}_{\text{Sat}}\right)\right]\left[1-\text{exp}\left(-\frac{{z}_{\text{V}}{S}_{\text{W}}{\alpha}_{\text{V}}}{{\dot{m}}_{\text{Q}}{c}_{\text{p,V}}}\right)\right],$$

where *T _{W,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, *α _{E}*, is defined by

$$\frac{1}{{\alpha}_{\text{E}}}=\frac{1}{{\alpha}_{\text{W}}}+\frac{1}{{\alpha}_{\text{Ext}}\left(1+{\eta}_{\text{Ext}}{s}_{\text{Ext}}\right)},$$

where *α*_{W} is the heat
transfer coefficient due to conduction through the wall

$${\alpha}_{\text{W}}=\frac{{k}_{\text{W}}}{{D}_{\text{H}}\text{ln}\left(1+\frac{{t}_{\text{W}}}{{D}_{\text{H}}}\right)},$$

and where:

*k*is the value of the_{W}**Wall thermal conductivity**parameter.*t*is the value of the_{W}**Wall thickness**parameter.*α*is the value of the_{Ext}**External environment heat transfer coefficient**parameter.*η*is the value of the_{Ext}**External fin efficiency**parameter.*s*is the value of the_{Ext}**Ratio of external fins surface area to no-fin surface area**parameter.

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,
*α*_{Ext}.

**Heat Transfer Rate**

The heat transfer rate between each wall zone and the environment is

$${Q}_{\text{H,zone}}=\left({T}_{\text{H}}-{T}_{\text{W}}\right)z{S}_{\text{W}}{\alpha}_{\text{E}}.$$

The total heat transfer between the wall and the environment is

$${Q}_{\text{H}}={Q}_{\text{H,L}}+{Q}_{\text{H,V}}+{Q}_{\text{H,M}}.$$

**Governing Differential Equations**

The heat transfer rate depends on the thermal mass of the wall,
*C _{W}*

$${C}_{\text{W}}={c}_{\text{p,W}}{\rho}_{\text{W}}{S}_{\text{W}}\left({t}_{\text{W}}+\frac{{t}_{\text{W}}^{2}}{{D}_{\text{H}}}\right),$$

where:

*c*_{p,W}is the value of the**Wall specific heat**parameter.*ρ*is the value of the_{W}**Wall density**parameter.

The governing equations for heat transfer between the fluid and the external environment for the liquid zone are

$${Q}_{\text{H,L}}-{Q}_{\text{F,L}}={C}_{\text{W}}\left[{z}_{\text{L}}\frac{d{T}_{\text{W,L}}}{dt}+\text{max}\left(\frac{d{z}_{\text{L}}}{dt},0\right)\left({T}_{\text{W,L}}-{T}_{\text{W,M}}\right)\right],$$

for the mixture zone are

$${Q}_{\text{H,M}}-{Q}_{\text{F,M}}={C}_{\text{W}}\left[{z}_{\text{M}}\frac{d{T}_{\text{W,M}}}{dt}+\text{min}\left(\frac{d{z}_{\text{L}}}{dt},0\right)\left({T}_{\text{W,L}}-{T}_{\text{W,M}}\right)+\text{min}\left(\frac{d{z}_{\text{V}}}{dt},0\right)\left({T}_{\text{W,V}}-{T}_{\text{W,M}}\right)\right],$$

and for the vapor zone are

$${Q}_{\text{H,V}}-{Q}_{\text{F,V}}={C}_{\text{W}}\left[{z}_{\text{V}}\frac{d{T}_{\text{W,V}}}{dt}+\text{max}\left(\frac{d{z}_{\text{V}}}{dt},0\right)\left({T}_{\text{W,V}}-{T}_{\text{W,M}}\right)\right].$$

### Momentum Balance

Two factors determine the pressure differential over the pipe: the 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 value of the
**Turbulent flow lower Reynolds number limit** parameter, the
block calculates the pressure loss in terms of the Darcy friction factor. The
pressure differential between port **A** and the internal node I is

$${p}_{\text{A}}-{p}_{\text{I}}=\left(\frac{1}{{\rho}_{\text{I}}}-\frac{1}{{\rho}_{\text{A}}^{*}}\right){\left(\frac{{\dot{m}}_{\text{A}}}{S}\right)}^{2}+\frac{{f}_{\text{A}}{\dot{m}}_{\text{A}}\left|{\dot{m}}_{\text{A}}\right|}{2{\rho}_{I}{D}_{\text{H}}{S}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right)+\frac{\Delta {p}_{hydrostatic}}{2},$$

where:

*ρ*is the fluid density at internal node I._{I}*ρ*is the fluid density at port_{A}***A**, which is the same as*ρ*when the flow is steady-state. When the flow is transient, the block calculates_{A}*ρ*from the fluid internal state at the internal node I with the adiabatic expression_{A}*$${u}_{\text{A}}^{*}+\frac{{p}_{\text{A}}}{{\rho}_{\text{A}}^{*}}+\frac{1}{2}{\left(\frac{{\dot{m}}_{\text{A}}}{{\rho}_{\text{A}}^{*}S}\right)}^{2}=h+\frac{1}{2}{\left(\frac{{\dot{m}}_{\text{A}}}{\rho S}\right)}^{2},$$

where:

*h*is the average specific enthalpy, $$h={h}_{\text{L}}{z}_{\text{L}}+{h}_{\text{V}}{z}_{\text{V}}+{h}_{\text{M}}{z}_{\text{M}}.$$*ρ*is the average density, $$\rho ={\rho}_{\text{L}}{z}_{\text{L}}+{\rho}_{\text{M}}{z}_{\text{M}}+{\rho}_{\text{V}}{z}_{\text{V}}.$$

$$\dot{m}$$

_{A}is the mass flow rate through port**A**.*L*is the value of the**Pipe length**parameter.*L*is the value of the_{Add}**Aggregate equivalent length of local resistances**parameter, 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.$$\Delta {p}_{hydrostatic}=\frac{M}{V}g\Delta z$$ is the hydrostatic pressure, where:

*M*is the total fluid mass in the pipe.*V*is the total fluid volume which is the volume of the pipe.*g*is the value of the**Gravitational acceleration**parameter.*Δz*is the value of the**Elevation gain from port A to port B**parameter.

The Darcy friction factor depends on the Reynolds number, which the block calculates at both ports.

The pressure differential between port **B** and internal node I is

$${p}_{\text{B}}-{p}_{\text{I}}=\left(\frac{1}{{\rho}_{\text{I}}}-\frac{1}{{\rho}_{\text{B}}^{*}}\right){\left(\frac{{\dot{m}}_{\text{B}}}{S}\right)}^{2}+\frac{{f}_{\text{B}}{\dot{m}}_{\text{B}}\left|{\dot{m}}_{\text{B}}\right|}{2{\rho}_{I}{D}_{\text{H}}{S}_{}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right)-\frac{\Delta {p}_{hydrostatic}}{2},$$

where:

*ρ*is the fluid density at port_{B}***B**, which is the same as*ρ*when the flow is steady-state. When the flow is transient, the block calculates_{B}*ρ*from the fluid internal state with the adiabatic expression_{B}*$${u}_{\text{B}}^{*}+\frac{{p}_{\text{B}}}{{\rho}_{\text{B}}^{*}}+\frac{1}{2}{\left(\frac{{\dot{m}}_{\text{B}}}{{\rho}_{\text{B}}^{*}S}\right)}^{2}=h+\frac{1}{2}{\left(\frac{{\dot{m}}_{\text{B}}}{\rho S}\right)}^{2}.$$

$$\dot{m}$$

_{B}is the mass flow rate through port**B**.

For laminar flows, when the Reynolds number is below the value of the
**Laminar flow upper Reynolds number limit** parameter, the
block calculates the pressure loss due to friction in terms of the **Laminar
friction constant for Darcy friction factor** parameter,
*λ*. The pressure differential between port
**A** and internal node I is

$${p}_{\text{A}}-{p}_{\text{I}}=\left(\frac{1}{{\rho}_{\text{I}}}-\frac{1}{{\rho}_{\text{A}}^{*}}\right){\left(\frac{{\dot{m}}_{\text{A}}}{S}\right)}^{2}+\frac{\lambda \mu {\dot{m}}_{\text{A}}}{2{\rho}_{I}{D}_{\text{H}}^{2}S}\left(\frac{L+{L}_{\text{Add}}}{2}\right)+\frac{\Delta {p}_{hydrostatic}}{2},$$

where *μ* is the average fluid dynamic viscosity

$$\mu ={\mu}_{\text{L}}{z}_{\text{L}}+{\mu}_{M}{z}_{\text{M}}+{\mu}_{\text{V}}{z}_{\text{V}}.$$

The pressure differential between port **B** and internal node I is

$${p}_{\text{B}}-{p}_{\text{I}}=\left(\frac{1}{{\rho}_{\text{I}}}-\frac{1}{{\rho}_{\text{B}}^{*}}\right){\left(\frac{{\dot{m}}_{\text{B}}}{S}\right)}^{2}+\frac{\lambda \mu {\dot{m}}_{\text{B}}}{2{\rho}_{I}{D}_{\text{H}}^{2}S}\left(\frac{L+{L}_{\text{Add}}}{2}\right)-\frac{\Delta {p}_{hydrostatic}}{2}.$$

For transitional flows, the block smooths the pressure differential due to viscous friction between the values for laminar and turbulent pressure losses.

### Mass Balance

The total mass accumulation rate is

$$\frac{dM}{dt}={\dot{m}}_{\text{A}}+{\dot{m}}_{\text{B}}.$$

In terms of the 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*

$$\frac{dM}{dt}=\left[{\left(\frac{\partial \rho}{\partial p}\right)}_{u}\frac{dp}{dt}+{\left(\frac{\partial \rho}{\partial u}\right)}_{p}\frac{d{u}_{out}}{dt}+{\rho}_{\text{L}}\frac{d{z}_{\text{L}}}{dt}+{\rho}_{\text{M}}\frac{d{z}_{\text{M}}}{dt}+{\rho}_{\text{V}}\frac{d{z}_{\text{V}}}{dt}\right]V,$$

where *u _{out}* is the
specific internal energy after all heat transfer has occurred.

### Energy Balance

The energy conservation equation is

$$M\frac{d{u}_{out}}{dt}+\left({\dot{m}}_{A}+{\dot{m}}_{B}\right){u}_{out}={\varphi}_{\text{A}}+{\varphi}_{\text{B}}+{Q}_{\text{F}}-{\dot{m}}_{avg}g\Delta z,$$

where:

*ϕ*is the energy flow rate at port_{A}**A**.*ϕ*is the energy flow rate at port_{B}**B**.*Q*is the heat transfer rate between the fluid and the wall._{F}$${\dot{m}}_{avg}=\frac{{\dot{m}}_{A}-{\dot{m}}_{B}}{2}.$$

### 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 block models gravitational effects at a bulk-system level and does not model the separation of liquid and vapor within the pipe due to gravity.

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*, 7^{th} Ed, Section 6.8. McGraw-Hill,
2011.

[2] Çengel, Y.A., *Heat
and Mass Transfer—A Practical Approach*, 3^{rd}
Ed, Section 8.5. McGraw-Hill, 2007.

## Extended Capabilities

## Version History

**Introduced in R2018b**