# System-Level Heat Exchanger (2P-2P)

Heat exchanger based on performance data between two two-phase fluid networks

*Since R2021b*

**Libraries:**

Simscape /
Fluids /
Heat Exchangers /
Two-Phase Fluid

## Description

The System-Level Heat Exchanger (2P-2P) block models a heat exchanger between two distinct two-phase fluid networks. Each network has its own set of fluid properties.

The block uses performance data from the heat exchanger datasheet, rather than the detailed geometry of the exchanger. Either or both sides of the heat exchanger can condense or vaporize fluid as a result of the heat exchange. You can also use this block as an internal heat exchanger in a refrigeration system. An internal heat exchanger improves refrigeration system efficiency by providing additional heat exchange between the outlet of the condenser and the outlet of the evaporator.

You parameterize the block by the nominal operating condition. The heat exchanger is sized to match the specified performance at the nominal operating condition at steady state.

Each side of the heat exchanger approximates the liquid zone, mixture zone, and vapor zone based on the change in enthalpy along the flow path.

### Heat Transfer

The block divides the two-phase fluid 1 flow and the two-phase fluid 2 flow each into three segments of equal size and calculates heat transfer between the fluids is in each segment. For simplicity, the equation in this section are for one segment.

If you clear the **Wall thermal mass** check box, then the heat balance
in the heat exchanger is

$${Q}_{seg,2P1}+{Q}_{seg,2P2}=0,$$

where:

*Q*_{seg,2P1}is the heat flow rate from the wall that is the heat transfer surface to two-phase fluid 1 in the segment.*Q*_{seg,2P2}is the heat flow rate from the wall to two-phase fluid 2 in the segment.

If you select **Wall thermal mass**, then the heat balance in the heat
exchanger is

$${Q}_{seg,2P1}+{Q}_{seg,2P2}=-\frac{{M}_{wall}{c}_{{p}_{wall}}}{N}\frac{d{T}_{seg,wall}}{dt},$$

where:

*M*_{wall}is the mass of the wall.*c*_{pwall}is the specific heat of the wall.*N*= 3 is the number of segments.*T*_{seg,wall}is the average wall temperature in the segment.*t*is time.

The heat flow rate from the wall to two-phase fluid 1 in the segment is

$${Q}_{seg,2P1}=U{A}_{seg,2P1}\left({T}_{seg,wall}-{T}_{seg,2P1}\right),$$

where:

*UA*is the weighted-average heat transfer conductance for two-phase fluid 1 in the segment._{seg,2P1}*T*is the weighted-average fluid temperature for two-phase fluid 1 in the segment._{seg,2P1}

The heat flow rate from the wall to the two-phase fluid 2 in the segment is

$${Q}_{seg,2P2}=U{A}_{seg,2P2}\left({T}_{seg,wall}-{T}_{seg,2P2}\right),$$

where:

*UA*is the weighted-average heat transfer conductance for two-phase fluid 2 in the segment._{seg,2P2}*T*is the weighted-average fluid temperature for two-phase fluid 2 in the segment._{seg,2P2}

### Two-Phase Fluid 1 Heat Transfer Correlation

The block calculates the heat transfer conductance in both two-phase fluids by using the same expressions.

If the segment is subcooled liquid, then the heat transfer conductance is

$$U{A}_{seg,L,2P1}={a}_{L,2P1}{\left({\mathrm{Re}}_{seg,L,2P1}\right)}^{{b}_{2P1}}{\left({\mathrm{Pr}}_{seg,L,2P1}\right)}^{{c}_{2P1}}{k}_{seg,L,2P1}\frac{{G}_{2P1}}{N},$$

where:

*a*_{L,2P1},*b*_{2P1}, and*c*_{2P1}are the coefficients of the Nusselt number correlation. These coefficients are block parameters in the**Correlation Coefficients**section.*Re*_{seg,L,2P1}is the average liquid Reynolds number for the segment.*Pr*_{seg,L,2P1}is the average liquid Prandtl number for the segment.*k*_{seg,L,2P1}is the average liquid thermal conductivity for the segment.*G*_{2P1}is the geometry scale factor for the two-phase fluid 1 side of the heat exchanger. The block calculates the geometry scale factor so that the total heat transfer over all segments matches the specified performance at the nominal operating conditions.

The average liquid Reynolds number is

$${\mathrm{Re}}_{seg,L,2P1}=\frac{{\dot{m}}_{seg,2P1}{D}_{ref,2P1}}{{\mu}_{seg,L,2P1}{S}_{ref,2P1}},$$

where:

$${\dot{m}}_{seg,2P1}$$ is the mass flow rate through the segment.

*μ*_{seg,L,2P1}is the average liquid dynamic viscosity for the segment.*D*_{ref,2P1}is an arbitrary reference diameter.*S*_{ref,2P1}is an arbitrary reference flow area.

**Note**

The *D*_{ref,2P1} and
*S*_{ref,2P1} terms are included in this equation
for unit calculation purposes only, to make
*Re*_{seg,L,2P1} nondimensional. The values of
*D*_{ref,2P1} and
*S*_{ref,2P1} are arbitrary because the
*G*_{2P1} calculation overrides these values.

Similarly, if the segment is superheated vapor, then the heat transfer conductance is

$$U{A}_{seg,V,2P1}={a}_{V,2P1}{\left({\mathrm{Re}}_{seg,V,2P1}\right)}^{{b}_{2P1}}{\left({\mathrm{Pr}}_{seg,V,2P1}\right)}^{{c}_{2P1}}{k}_{seg,V,2P1}\frac{{G}_{2P1}}{N},$$

where:

*a*_{V,2P1},*b*_{2P1}, and*c*_{2P1}are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the**Correlation Coefficients**section.*Re*_{seg,V,2P1}is the average vapor Reynolds number for the segment.*Pr*_{seg,V,2P1}is the average vapor Prandtl number for the segment.*k*_{seg,V,2P1}is the average vapor thermal conductivity for the segment.

The average vapor Reynolds number is

$${\mathrm{Re}}_{seg,V,2P1}=\frac{{\dot{m}}_{seg,2P1}{D}_{ref,2P1}}{{\mu}_{seg,V,2P1}{S}_{ref,2P1}},$$

where *μ*_{seg,V,2P1} is the average vapor dynamic
viscosity for the segment.

If the segment is liquid-vapor mixture, then the heat transfer conductance is

$$U{A}_{seg,M,2P1}={a}_{M,2P1}{\left({\mathrm{Re}}_{seg,SL,2P1}\right)}^{{b}_{2P1}}CZ{\left({\mathrm{Pr}}_{seg,SL,2P1}\right)}^{{c}_{2P1}}{k}_{seg,SL,2P1}\frac{{G}_{2P1}}{N},$$

where:

*a*_{M,2P1},*b*_{2P1}, and*c*_{2P1}are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the**Correlation Coefficients**section.*Re*_{seg,SL,2P1}is the saturated liquid Reynolds number for the segment.*Pr*_{seg,SL,2P1}is the saturated liquid Prandtl number for the segment.*k*_{seg,SL,2P1}is the saturated liquid thermal conductivity for the segment.*CZ*is the Cavallini and Zecchin term.

The saturated liquid Reynolds number is

$${\mathrm{Re}}_{seg,SL,2P1}=\frac{{\dot{m}}_{seg,2P1}{D}_{ref,2P1}}{{\mu}_{seg,SL,2P1}{S}_{ref,2P1}},$$

where *μ*_{seg,SL,2P1} is the saturated liquid
dynamic viscosity for the segment.

The Cavallini and Zecchin term is

$$CZ=\frac{{\left(\left(\sqrt{\frac{{\nu}_{seg,SV,2P1}}{{\nu}_{seg,SL,2P1}}}-1\right)\left({x}_{seg,out,2P1}+1\right)\right)}^{1+{b}_{2P1}}-{\left(\left(\sqrt{\frac{{\nu}_{seg,SV,2P1}}{{\nu}_{seg,SL,2P1}}}-1\right)\left({x}_{seg,in,2P1}+1\right)\right)}^{1+{b}_{2P1}}}{\left(1+{b}_{2P1}\right)\left(\sqrt{\frac{{\nu}_{seg,SV,2P1}}{{\nu}_{seg,SL,2P1}}}-1\right)\left({x}_{seg,out,2P1}-{x}_{seg,in,2P1}\right)},$$

where:

*ν*_{seg,SL,2P1}is the saturated liquid specific volume for the segment.*ν*_{seg,SV,2P1}is the saturated vapor specific volume for the segment.*x*_{seg,in,2P1}is the vapor quality at the segment inlet.*x*_{seg,out,2P1}is the vapor quality at the segment outlet.

The expression is based on the work of Cavallini and Zecchin [5], which
derives a heat transfer coefficient correlation at a local vapor quality
*x*. Equations for the liquid-vapor mixture are obtained by averaging
Cavallini and Zecchin’s correlation over the segment from
*x*_{seg,in,2P1} to
*x*_{seg,out,2P1}.

### Two-Phase Fluid 1 Weighted Average

The two-phase fluid flow through a segment may not be entirely represented as either
subcooled liquid, superheated vapor, or liquid-vapor mixture. Instead, each segment may
consist of a combination of these. The block approximates this condition by computing
weighting factors (*w*_{L},
*w*_{V}, and
*w*_{M}) based on the change in specific enthalpy
across the segment and the saturated liquid and vapor specific enthalpies. The block assumes
that the specific enthalpy across the segment varies piecewise linearly from inlet to
outlet, with the breakpoints corresponding to the saturation boundaries for liquid and
vapor. The zone with a larger heat transfer coefficient has a steeper slope than the zone
with a lower heat transfer coefficient.

$$\begin{array}{l}{w}_{L}=\frac{{\Delta}_{L}}{{\Delta}_{L}+{\Delta}_{M}+{\Delta}_{V}}\\ {w}_{V}=\frac{{\Delta}_{V}}{{\Delta}_{L}+{\Delta}_{M}+{\Delta}_{V}}\\ {w}_{M}=1-{w}_{L}-{w}_{V}\end{array}$$

$$\begin{array}{l}{\Delta}_{L}=\left|\mathrm{min}\left({h}_{seg,out,2P1},{h}_{seg,SL,2P1}\right)-\mathrm{min}\left({h}_{seg,in,2P1},{h}_{seg,SL,2P1}\right)\right|\cdot U{A}_{seg,M,2P1}\cdot U{A}_{seg,V,2P1}\\ {\Delta}_{M}=\left|\mathrm{min}\left(\mathrm{max}\left({h}_{seg,out,2P1},{h}_{seg,SL,2P1}\right),{h}_{seg,SV,2P1}\right)-\mathrm{min}\left(\mathrm{max}\left({h}_{seg,in,2P1},{h}_{seg,SL,2P1}\right),{h}_{seg,SV,2P1}\right)\right|\cdot U{A}_{seg,L,2P1}\cdot U{A}_{seg,V,2P1}\\ {\Delta}_{V}=\left|\mathrm{max}\left({h}_{seg,out,2P1},{h}_{seg,SV,2P1}\right)-\mathrm{max}\left({h}_{seg,in,2P1},{h}_{seg,SV,2P1}\right)\right|\cdot U{A}_{seg,L,2P1}\cdot U{A}_{seg,M,2P1}\end{array}$$

where:

*h*_{seg,in,2P1}is the specific enthalpy at the segment inlet.*h*_{seg,out,2P1}is the specific enthalpy at the segment outlet.*h*_{seg,SL,2P1}is the saturated liquid specific enthalpy for the segment.*h*_{seg,SV,2P1}is the saturated vapor specific enthalpy for the segment.

The weighted-average two-phase fluid 1 heat transfer conductance for the segment is therefore

$$U{A}_{seg,2P1}={w}_{L}\left(U{A}_{seg,L,2P1}\right)+{w}_{V}\left(U{A}_{seg,V,2P1}\right)+{w}_{M}\left(U{A}_{seg,M,2P1}\right).$$

The weighted-average fluid 1 temperature for the segment is

$${T}_{seg,2P1}=\frac{{w}_{L}\left(U{A}_{seg,L,2P1}\right){T}_{seg,L,2P1}+{w}_{V}\left(U{A}_{seg,V,2P1}\right){T}_{seg,V,2P1}+{w}_{M}\left(U{A}_{seg,M,2P1}\right){T}_{seg,M,2P1}}{U{A}_{seg,2P1}},$$

where:

*T*_{seg,L,2P1}is the average liquid temperature for the segment.*T*_{seg,V,2P1}is the average vapor temperature for the segment.*T*_{seg,M,2P1}is the average mixture temperature for the segment, which is the saturated liquid temperature.

### Two-Phase Fluid 2 Heat Transfer Correlation

If the segment is subcooled liquid, then the heat transfer conductance is

$$U{A}_{seg,L,2P2}={a}_{L,2P2}{\left({\mathrm{Re}}_{seg,L,2P2}\right)}^{{b}_{2P2}}{\left({\mathrm{Pr}}_{seg,L,2P2}\right)}^{{c}_{2P2}}{k}_{seg,L,2P2}\frac{{G}_{2P2}}{N},$$

where:

*a*_{L,2P2},*b*_{L,2P2}, and*c*_{L,2P2}are the coefficients of the Nusselt number correlation. These coefficients are block parameters in the**Correlation Coefficients**section.*Re*_{seg,L,2P2}is the average liquid Reynolds number for the segment.*Pr*_{seg,L,2P2}is the average liquid Prandtl number for the segment.*k*_{seg,L,2P2}is the average liquid thermal conductivity for the segment.*G*_{2P2}is the geometry scale factor for the two-phase fluid 2 side of the heat exchanger. The block calculates the geometry scale factor so that the total heat transfer over all segments matches the specified performance at the nominal operating conditions.

The average liquid Reynolds number is

$${\mathrm{Re}}_{seg,L,2P2}=\frac{{\dot{m}}_{seg,2P2}{D}_{ref,2P2}}{{\mu}_{seg,L,2P2}{S}_{ref,2P2}},$$

where:

$${\dot{m}}_{seg,2P2}$$ is the mass flow rate through the segment.

*μ*_{seg,L,2P2}is the average liquid dynamic viscosity for the segment.*D*_{ref,2P2}is an arbitrary reference diameter.*S*_{ref,2P2}is an arbitrary reference flow area.

**Note**

The *D*_{ref,2P2} and
*S*_{ref,2P2} terms are included in this equation
for unit calculation purposes only, to make
*Re*_{seg,L,2P2} nondimensional. The values of
*D*_{ref,2P} and
*S*_{ref,2P2} are arbitrary because the
*G*_{2P2} calculation overrides these values.

Similarly, if the segment is superheated vapor, then the heat transfer conductance is

$$U{A}_{seg,V,2P2}={a}_{V,2P2}{\left({\mathrm{Re}}_{seg,V,2P2}\right)}^{{b}_{2P2}}{\left({\mathrm{Pr}}_{seg,V,2P2}\right)}^{{c}_{2P2}}{k}_{seg,V,2P2}\frac{{G}_{2P2}}{N},$$

where:

*a*_{V,2P2},*b*_{V,2P2}, and*c*_{V,2P2}are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the**Correlation Coefficients**section.*Re*_{seg,V,2P2}is the average vapor Reynolds number for the segment.*Pr*_{seg,V,2P2}is the average vapor Prandtl number for the segment.*k*_{seg,V,2P2}is the average vapor thermal conductivity for the segment.

The average vapor Reynolds number is

$${\mathrm{Re}}_{seg,V,2P2}=\frac{{\dot{m}}_{seg,2P2}{D}_{ref,2P2}}{{\mu}_{seg,V,2P2}{S}_{ref,2P2}},$$

where *μ*_{seg,V,2P2} is the average vapor dynamic
viscosity for the segment.

If the segment is liquid-vapor mixture, then the heat transfer conductance is

$$U{A}_{seg,M,2P2}={a}_{M,2P2}{\left({\mathrm{Re}}_{seg,SL,2P2}\right)}^{{b}_{2P2}}CZ{\left({\mathrm{Pr}}_{seg,SL,2P2}\right)}^{{c}_{2P2}}{k}_{seg,SL,2P2}\frac{{G}_{2P2}}{N},$$

where:

*a*_{M,2P2},*b*_{L,2P2}, and*c*_{L,2P2}are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the**Correlation Coefficients**section.*Re*_{seg,SL,2P2}is the saturated liquid Reynolds number for the segment.*Pr*_{seg,SL,2P2}is the saturated liquid Prandtl number for the segment.*k*_{seg,SL,2P2}is the saturated liquid thermal conductivity for the segment.*CZ*is the Cavallini and Zecchin term.

The saturated liquid Reynolds number is

$${\mathrm{Re}}_{seg,SL,2P2}=\frac{{\dot{m}}_{seg,2P2}{D}_{ref,2P2}}{{\mu}_{seg,SL,2P2}{S}_{ref,2P2}},$$

where *μ*_{seg,SL,2P2} is the saturated liquid
dynamic viscosity for the segment.

The Cavallini and Zecchin term is

$$CZ=\frac{{\left(\left(\sqrt{\frac{{\nu}_{seg,SV,2P2}}{{\nu}_{seg,SL,2P2}}}-1\right)\left({x}_{seg,out,2P2}+1\right)\right)}^{1+{b}_{2P2}}-{\left(\left(\sqrt{\frac{{\nu}_{seg,SV,2P2}}{{\nu}_{seg,SL,2P2}}}-1\right)\left({x}_{seg,in,2P2}+1\right)\right)}^{1+{b}_{2P2}}}{\left(1+{b}_{2P2}\right)\left(\sqrt{\frac{{\nu}_{seg,SV,2P2}}{{\nu}_{seg,SL,2P2}}}-1\right)\left({x}_{seg,out,2P2}-{x}_{seg,in,2P2}\right)},$$

where:

*ν*_{seg,SL,2P2}is the saturated liquid specific volume for the segment.*ν*_{seg,SV,2P2}is the saturated vapor specific volume for the segment.*x*_{seg,in,2P2}is the vapor quality at the segment inlet.*x*_{seg,out,2P2}is the vapor quality at the segment outlet.

The expression is based on the work of Cavallini and Zecchin [5], which
derives a heat transfer coefficient correlation at a local vapor quality
*x*. Equations for the liquid-vapor mixture are obtained by averaging
Cavallini and Zecchin’s correlation over the segment from
*x*_{seg,in,2P2} to
*x*_{seg,out,2P2}.

### Two-Phase Fluid 2 Weighted Average

The two-phase fluid flow through a segment may not be entirely represented as either
subcooled liquid, superheated vapor, or liquid-vapor mixture. Instead, each segment may
consist of a combination of these. The block approximates this condition by computing
weighting factors (*w*_{L},
*w*_{V}, and
*w*_{M}) based on the change in specific enthalpy
across the segment and the saturated liquid and vapor specific enthalpies. The block assumes
that the specific enthalpy across the segment varies piecewise linearly from inlet to
outlet, with the breakpoints corresponding to the saturation boundaries for liquid and
vapor. The zone with a larger heat transfer coefficient has a steeper slope than the zone
with a lower heat transfer coefficient.

$$\begin{array}{l}{w}_{L}=\frac{{\Delta}_{L}}{{\Delta}_{L}+{\Delta}_{M}+{\Delta}_{V}}\\ {w}_{V}=\frac{{\Delta}_{V}}{{\Delta}_{L}+{\Delta}_{M}+{\Delta}_{V}}\\ {w}_{M}=1-{w}_{L}-{w}_{V}\end{array}$$

$$\begin{array}{l}{\Delta}_{L}=\left|\mathrm{min}\left({h}_{seg,out,2P2},{h}_{seg,SL,2P2}\right)-\mathrm{min}\left({h}_{seg,in,2P2},{h}_{seg,SL,2P2}\right)\right|\cdot U{A}_{seg,M,2P2}\cdot U{A}_{seg,V,2P2}\\ {\Delta}_{M}=\left|\mathrm{min}\left(\mathrm{max}\left({h}_{seg,out,2P2},{h}_{seg,SL,2P2}\right),{h}_{seg,SV,2P2}\right)-\mathrm{min}\left(\mathrm{max}\left({h}_{seg,in,2P2},{h}_{seg,SL,2P2}\right),{h}_{seg,SV,2P2}\right)\right|\cdot U{A}_{seg,L,2P2}\cdot U{A}_{seg,V,2P2}\\ {\Delta}_{V}=\left|\mathrm{max}\left({h}_{seg,out,2P2},{h}_{seg,SV,2P2}\right)-\mathrm{max}\left({h}_{seg,in,2P2},{h}_{seg,SV,2P2}\right)\right|\cdot U{A}_{seg,L,2P2}\cdot U{A}_{seg,M,2P2}\end{array}$$

where:

*h*_{seg,in,2P2}is the specific enthalpy at the segment inlet.*h*_{seg,out,2P2}is the specific enthalpy at the segment outlet.*h*_{seg,SL,2P2}is the saturated liquid specific enthalpy for the segment.*h*_{seg,SV,2P2}is the saturated vapor specific enthalpy for the segment.

The weighted-average two-phase fluid 2 heat transfer conductance for the segment is therefore

$$U{A}_{seg,2P2}={w}_{L}\left(U{A}_{seg,L,2P2}\right)+{w}_{V}\left(U{A}_{seg,V,2P2}\right)+{w}_{M}\left(U{A}_{seg,M,2P2}\right).$$

The weighted-average fluid 2 temperature for the segment is

$${T}_{seg,2P2}=\frac{{w}_{L}\left(U{A}_{seg,L,2P2}\right){T}_{seg,L,2P2}+{w}_{V}\left(U{A}_{seg,V,2P2}\right){T}_{seg,V,2P2}+{w}_{M}\left(U{A}_{seg,M,2P2}\right){T}_{seg,M,2P2}}{U{A}_{seg,2P2}},$$

where:

*T*_{seg,L,2P2}is the average liquid temperature for the segment.*T*_{seg,V,2P2}is the average vapor temperature for the segment.*T*_{seg,M,2P2}is the average mixture temperature for the segment, which is the saturated liquid temperature.

### Pressure Loss

The pressure losses on the two-phase fluid 1 side are

$$\begin{array}{l}{p}_{A,2P1}-{p}_{2P1}=\frac{{K}_{2P1}}{2}\frac{{\dot{m}}_{A,2P1}\sqrt{{\dot{m}}^{2}{}_{A,2P1}+{\dot{m}}^{2}{}_{thres,2P1}}}{2{\rho}_{avg,2P1}}\\ {p}_{B,2P1}-{p}_{2P1}=\frac{{K}_{2P1}}{2}\frac{{\dot{m}}_{B,2P1}\sqrt{{\dot{m}}^{2}{}_{B,2P1}+{\dot{m}}^{2}{}_{thres,2P1}}}{2{\rho}_{avg,2P1}}\end{array}$$

where:

*p*_{A,2P1}and*p*_{B,2P1}are the pressures at ports**A1**and**B1**, respectively.*p*_{2P1}is internal two-phase fluid 1 pressure at which the heat transfer is calculated.$${\dot{m}}_{A,2P1}$$ and $${\dot{m}}_{B,2P1}$$ are the mass flow rates into ports

**A1**and**B1**, respectively.*ρ*_{avg,2P1}is the average two-phase fluid 1 density over all segments.$${\dot{m}}_{thres,2P1}$$ is the laminar threshold for pressure loss, approximated as 1e-4 of the nominal mass flow rate. The block calculates the pressure loss coefficient,

*K*_{2P1}, so that*p*_{A,2P1}–*p*_{B,2P1}matches the nominal pressure loss at the nominal mass flow rate.

The pressure losses on the two-phase fluid 2 side are

$$\begin{array}{l}{p}_{A,2P2}-{p}_{2P2}=\frac{{K}_{2P2}}{2}\frac{{\dot{m}}_{A,2P2}\sqrt{{\dot{m}}^{2}{}_{A,2P2}+{\dot{m}}^{2}{}_{thres,2P2}}}{2{\rho}_{avg,2P2}}\\ {p}_{B,2P2}-{p}_{2P2}=\frac{{K}_{2P2}}{2}\frac{{\dot{m}}_{B,2P2}\sqrt{{\dot{m}}^{2}{}_{B,2P2}+{\dot{m}}^{2}{}_{thres,2P2}}}{2{\rho}_{avg,2P2}}\end{array}$$

where:

*p*_{A,2P2}and*p*_{B,2P2}are the pressures at ports**A2**and**B2**, respectively.*p*_{2P2}is internal two-phase fluid 2 pressure at which the heat transfer is calculated.$${\dot{m}}_{A,2P2}$$ and $${\dot{m}}_{B,2P2}$$ are the mass flow rates into ports

**A2**and**B2**, respectively.*ρ*_{avg,2P2}is the average two-phase fluid 2 density over all segments.$${\dot{m}}_{thres,2P2}$$ is the laminar threshold for pressure loss, approximated as 1e-4 of the nominal mass flow rate. The block calculates the pressure loss coefficient,

*K*_{2P2}, so that*p*_{A,2P2}–*p*_{B,2P2}matches the nominal pressure loss at the nominal mass flow rate.

### Two-Phase Fluid 1 Mass and Energy Conservation

The mass conservation equation for the overall two-phase fluid 1 flow is

$$\left(\frac{d{p}_{2P1}}{dt}{\displaystyle \sum _{segments}\left(\frac{\partial {\rho}_{seg,2P1}}{\partial p}\right)}+{\displaystyle \sum _{segments}\left(\frac{d{u}_{seg,2P1}}{dt}\frac{\partial {\rho}_{seg,2P1}}{\partial u}\right)}\right)\frac{{V}_{2P1}}{N}={\dot{m}}_{A,2P1}+{\dot{m}}_{B,2P1},$$

where:

$$\frac{\partial {\rho}_{seg,2P1}}{\partial p}$$ is the partial derivative of density with respect to pressure for the segment.

$$\frac{\partial {\rho}_{seg,2P1}}{\partial u}$$ is the partial derivative of density with respect to specific internal energy for the segment.

*u*_{seg,2P1}is the specific internal energy for the segment.*V*_{2P1}is the total two-phase fluid 1 volume.

The summation is over all segments.

**Note**

Although the block divides the two-phase fluid 1 flow into *N*=3
segments for heat transfer calculations, it assumes all segments are at the same internal
pressure, *p*_{2P1}. Consequentially,
*p*_{2P1} is outside of the summation.

The energy conservation equation for each segment is

$$\frac{d{u}_{seg,2P1}}{dt}\frac{{M}_{2P1}}{N}+{u}_{seg,2P1}\left({\dot{m}}_{seg,in,2P1}-{\dot{m}}_{seg,out,2P1}\right)={\Phi}_{seg,in,2P1}-{\Phi}_{seg,out,2P1}+{Q}_{seg,2P1},$$

where:

*M*_{2P1}is the total two-phase fluid 1 mass.$${\dot{m}}_{seg,in,2P1}$$ and $${\dot{m}}_{seg,out,2P1}$$ are the mass flow rates into and out of the segment.

*Φ*_{seg,in,2p1}and*Φ*_{seg,out,2p1}are the energy flow rates into and out of the segment.

The block assumes the mass flow rates between segments are linearly distributed between the values of $${\dot{m}}_{A,2P1}$$ and $${\dot{m}}_{B,2P1}$$.

### Two-Phase Fluid 2 Mass and Energy Conservation

The mass conservation equation for the overall two-phase fluid 2 flow is

$$\left(\frac{d{p}_{2P2}}{dt}{\displaystyle \sum _{segments}\left(\frac{\partial {\rho}_{seg,2P2}}{\partial p}\right)}+{\displaystyle \sum _{segments}\left(\frac{d{u}_{seg,2P2}}{dt}\frac{\partial {\rho}_{seg,2P2}}{\partial u}\right)}\right)\frac{{V}_{2P2}}{N}={\dot{m}}_{A,2P2}+{\dot{m}}_{B,2P2},$$

where:

$$\frac{\partial {\rho}_{seg,2P2}}{\partial p}$$ is the partial derivative of density with respect to pressure for the segment.

$$\frac{\partial {\rho}_{seg,2P2}}{\partial u}$$ is the partial derivative of density with respect to specific internal energy for the segment.

*u*_{seg,2P2}is the specific internal energy for the segment.*V*_{2P2}is the total two-phase fluid 2 volume.

The summation is over all segments.

**Note**

Although the block divides the two-phase fluid 2 flow into *N*=3
segments for heat transfer calculations, it assumes all segments are at the same internal
pressure, *p*_{2P2}. Consequentially,
*p*_{2P2} is outside of the summation.

The energy conservation equation for each segment is

$$\frac{d{u}_{seg,2P2}}{dt}\frac{{M}_{2P2}}{N}+{u}_{seg,2P2}\left({\dot{m}}_{seg,in,2P2}-{\dot{m}}_{seg,out,2P2}\right)={\Phi}_{seg,in,2P2}-{\Phi}_{seg,out,2P2}+{Q}_{seg,2P2},$$

where:

*M*_{2P2}is the total two-phase fluid 2 mass.$${\dot{m}}_{seg,in,2P2}$$ and $${\dot{m}}_{seg,out,2P2}$$ are the mass flow rates into and out of the segment.

*Φ*_{seg,in,2p2}and*Φ*_{seg,out,2p2}are the energy flow rates into and out of the segment.

The block assumes the mass flow rates between segments are linearly distributed between the values of $${\dot{m}}_{A,2P2}$$ and $${\dot{m}}_{B,2P2}$$.

## Examples

## Ports

### Output

### Conserving

## Parameters

## References

[1]
*Ashrae Handbook: Fundamentals.* Atlanta: Ashrae,
2013.

[2] Çengel, Yunus A. *Heat and Mass Transfer: A Practical Approach*. 3rd ed. McGraw-Hill
Series in Mechanical Engineering. Boston: McGraw-Hill, 2007.

[3] Mitchell, John W., and James E.
Braun. *Principles of Heating, Ventilation, and Air Conditioning in
Buildings*. Hoboken, NJ: Wiley, 2013.

[4] Shah, R. K., and Dušan P.
Sekulić. *Fundamentals of Heat Exchanger Design*. Hoboken,
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[5] Cavallini, Alberto, and Roberto
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*Proceeding of International Heat Transfer Conference 5*,
309–13. Tokyo, Japan: Begellhouse, 1974. https://doi.org/10.1615/IHTC5.1220.

## Extended Capabilities

## Version History

**Introduced in R2021b**