System-Level Heat Exchanger (2P-2P)
Heat exchanger between two two-phase fluid networks, with model based on performance data
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Simscape / Fluids / Fluid Network Interfaces / Heat Exchangers
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 model is based on performance data from the heat exchanger datasheet, rather than on the detailed geometry of the exchanger, and therefore you can use this block when geometry data is unavailable. 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 two-phase fluid 1 flow and the two-phase fluid 2 flow are each divided into three segments of equal size. Heat transfer between the fluids is calculated in each segment. For simplicity, the equation for one segment is shown here.
If the wall thermal mass is off, then the heat balance in the heat exchanger is
where:
Qseg,2P1 is the heat flow rate from the wall (that is, the heat transfer surface) to the two-phase fluid 1 in the segment.
Qseg,2P2 is the heat flow rate from the wall to the two-phase fluid 2 in the segment.
If the wall thermal mass is on, then the heat balance in the heat exchanger is
where:
Mwall is the mass of the wall.
cpwall is the specific heat of the wall.
N = 3 is the number of segments.
Tseg,wall is the average wall temperature in the segment.
t is time.
The heat flow rate from the wall to the two-phase fluid 1 in the segment is
where:
UAseg,2P1 is the weighted-average heat transfer conductance for the two-phase fluid 1 in the segment.
Tseg,2P1 is the weighted-average fluid temperature for the two-phase fluid 1 in the segment.
The heat flow rate from the wall to the two-phase fluid 2 in the segment is
where:
UAseg,2P2 is the weighted-average heat transfer conductance for the two-phase fluid 2 in the segment.
Tseg,2P2 is the weighted-average fluid temperature for the two-phase fluid 2 in the segment.
Two-Phase Fluid 1 Heat Transfer Correlation
If the segment is subcooled liquid, then the heat transfer conductance is
where:
aL,2P1, b2P1, and c2P1 are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the Correlation Coefficients section.
Reseg,L,2P1 is the average liquid Reynolds number for the segment.
Prseg,L,2P1 is the average liquid Prandtl number for the segment.
kseg,L,2P1 is the average liquid thermal conductivity for the segment.
G2P1 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
where:
is the mass flow rate through the segment.
μseg,L,2P1 is the average liquid dynamic viscosity for the segment.
Dref,2P1 is an arbitrary reference diameter.
Sref,2P1 is an arbitrary reference flow area.
Note
The Dref,2P1 and Sref,2P1 terms are included in this equation for unit calculation purposes only, to make Reseg,L,2P1 nondimensional. The values of Dref,2P1 and Sref,2P1 are arbitrary because the G2P1 calculation overrides these values.
Similarly, if the segment is superheated vapor, then the heat transfer conductance is
where:
aV,2P1, b2P1, and c2P1 are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the Correlation Coefficients section.
Reseg,V,2P1 is the average vapor Reynolds number for the segment.
Prseg,V,2P1 is the average vapor Prandtl number for the segment.
kseg,V,2P1 is the average vapor thermal conductivity for the segment.
The average vapor Reynolds number is
where μseg,V,2P1 is the average vapor dynamic viscosity for the segment.
If the segment is liquid-vapor mixture, then the heat transfer conductance is
where:
aM,2P1, b2P1, and c2P1 are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the Correlation Coefficients section.
Reseg,SL,2P1 is the saturated liquid Reynolds number for the segment.
Prseg,SL,2P1 is the saturated liquid Prandtl number for the segment.
kseg,SL,2P1 is the saturated liquid thermal conductivity for the segment.
CZ is the Cavallini and Zecchin term.
The saturated liquid Reynolds number is
where μseg,SL,2P1 is the saturated liquid dynamic viscosity for the segment.
The Cavallini and Zecchin term is
where:
νseg,SL,2P1 is the saturated liquid specific volume for the segment.
νseg,SV,2P1 is the saturated vapor specific volume for the segment.
xseg,in,2P1 is the vapor quality at the segment inlet.
xseg,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 xseg,in,2P1 to xseg,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 (wL, wV, and wM) 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.
where:
hseg,in,2P1 is the specific enthalpy at the segment inlet.
hseg,out,2P1 is the specific enthalpy at the segment outlet.
hseg,SL,2P1 is the saturated liquid specific enthalpy for the segment.
hseg,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
The weighted-average fluid 1 temperature for the segment is
where:
Tseg,L,2P1 is the average liquid temperature for the segment.
Tseg,V,2P1 is the average vapor temperature for the segment.
Tseg,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
where:
aL,2P2, bL,2P2, and cL,2P2 are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the Correlation Coefficients section.
Reseg,L,2P2 is the average liquid Reynolds number for the segment.
Prseg,L,2P2 is the average liquid Prandtl number for the segment.
kseg,L,2P2 is the average liquid thermal conductivity for the segment.
G2P2 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
where:
is the mass flow rate through the segment.
μseg,L,2P2 is the average liquid dynamic viscosity for the segment.
Dref,2P2 is an arbitrary reference diameter.
Sref,2P2 is an arbitrary reference flow area.
Note
The Dref,2P2 and Sref,2P2 terms are included in this equation for unit calculation purposes only, to make Reseg,L,2P2 nondimensional. The values of Dref,2P and Sref,2P2 are arbitrary because the G2P2 calculation overrides these values.
Similarly, if the segment is superheated vapor, then the heat transfer conductance is
where:
aV,2P2, bV,2P2, and cV,2P2 are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the Correlation Coefficients section.
Reseg,V,2P2 is the average vapor Reynolds number for the segment.
Prseg,V,2P2 is the average vapor Prandtl number for the segment.
kseg,V,2P2 is the average vapor thermal conductivity for the segment.
The average vapor Reynolds number is
where μseg,V,2P2 is the average vapor dynamic viscosity for the segment.
If the segment is liquid-vapor mixture, then the heat transfer conductance is
where:
aM,2P2, bL,2P2, and cL,2P2 are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the Correlation Coefficients section.
Reseg,SL,2P2 is the saturated liquid Reynolds number for the segment.
Prseg,SL,2P2 is the saturated liquid Prandtl number for the segment.
kseg,SL,2P2 is the saturated liquid thermal conductivity for the segment.
CZ is the Cavallini and Zecchin term.
The saturated liquid Reynolds number is
where μseg,SL,2P2 is the saturated liquid dynamic viscosity for the segment.
The Cavallini and Zecchin term is
where:
νseg,SL,2P2 is the saturated liquid specific volume for the segment.
νseg,SV,2P2 is the saturated vapor specific volume for the segment.
xseg,in,2P2 is the vapor quality at the segment inlet.
xseg,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 xseg,in,2P2 to xseg,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 (wL, wV, and wM) 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.
where:
hseg,in,2P2 is the specific enthalpy at the segment inlet.
hseg,out,2P2 is the specific enthalpy at the segment outlet.
hseg,SL,2P2 is the saturated liquid specific enthalpy for the segment.
hseg,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
The weighted-average fluid 2 temperature for the segment is
where:
Tseg,L,2P2 is the average liquid temperature for the segment.
Tseg,V,2P2 is the average vapor temperature for the segment.
Tseg,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
where:
pA,2P1 and pB,2P1 are the pressures at ports A1 and B1, respectively.
p2P1 is internal two-phase fluid 1 pressure at which the heat transfer is calculated.
and are the mass flow rates into ports A1 and B1, respectively.
ρavg,2P1 is the average two-phase fluid 1 density over all segments.
is the laminar threshold for pressure loss, approximated as 1e-4 of the nominal mass flow rate. The block calculates the pressure loss coefficient, K2P1, so that pA,2P1 – pB,2P1 matches the nominal pressure loss at the nominal mass flow rate.
The pressure losses on the two-phase fluid 2 side are
where:
pA,2P2 and pB,2P2 are the pressures at ports A2 and B2, respectively.
p2P2 is internal two-phase fluid 2 pressure at which the heat transfer is calculated.
and are the mass flow rates into ports A2 and B2, respectively.
ρavg,2P2 is the average two-phase fluid 2 density over all segments.
is the laminar threshold for pressure loss, approximated as 1e-4 of the nominal mass flow rate. The block calculates the pressure loss coefficient, K2P2, so that pA,2P2 – pB,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
where:
is the partial derivative of density with respect to pressure for the segment.
is the partial derivative of density with respect to specific internal energy for the segment.
useg,2P1 is the specific internal energy for the segment.
V2P1 is the total two-phase fluid 1 volume.
The summation is over all segments.
Note
Although the two-phase fluid 1 flow is divided into N=3 segments for heat transfer calculations, all segments are assumed to be at the same internal pressure, p2P1. That is why p2P1 is outside of the summation.
The energy conservation equation for each segment is
where:
M2P1 is the total two-phase fluid 1 mass.
and are the mass flow rates into and out of the segment.
Φseg,in,2p1 and Φseg,out,2p1 are the energy flow rates into and out of the segment.
The mass flow rates between segments are assumed to be linearly distributed between the values of and .
Two-Phase Fluid 2 Mass and Energy Conservation
The mass conservation equation for the overall two-phase fluid 2 flow is
where:
is the partial derivative of density with respect to pressure for the segment.
is the partial derivative of density with respect to specific internal energy for the segment.
useg,2P2 is the specific internal energy for the segment.
V2P2 is the total two-phase fluid 2 volume.
The summation is over all segments.
Note
Although the two-phase fluid 2 flow is divided into N=3 segments for heat transfer calculations, all segments are assumed to be at the same internal pressure, p2P2. That is why p2P2 is outside of the summation.
The energy conservation equation for each segment is
where:
M2P2 is the total two-phase fluid 2 mass.
and are the mass flow rates into and out of the segment.
Φseg,in,2p2 and Φseg,out,2p2 are the energy flow rates into and out of the segment.
The mass flow rates between segments are assumed to be linearly distributed between the values of and .
Ports
Output
Conserving
Parameters
Extended Capabilities
Version History
Introduced in R2021b