# System-Level Heat Exchanger (TL-MA)

Heat exchanger between thermal liquid and moist air networks, with model based on performance data

**Library:**Simscape / Fluids / Fluid Network Interfaces / Heat Exchangers

## Description

The System-Level Heat Exchanger (TL-MA) block models a heat exchanger between a thermal liquid network and a moist air network.

The block model is based on performance data from the heat exchanger datasheet, rather than on the detailed geometry of the exchanger, and therefore lets you easily adjust the size and performance of the heat exchanger during design iterations, or model heat exchangers with uncommon geometries. You can also use this block to model heat exchangers with a certain level of performance at an early design stage, when detailed geometry data is not yet available.

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.

The Moist Air 2 side models water vapor condensation based on convective water vapor mass transfer with the heat transfer surface. Condensed water is removed from the moist air flow.

This block is similar to the Heat Exchanger (TL-MA) block but uses a different parameterization model. The table provides a comparison of the two blocks, to help you choose the right block for your application.

Heat Exchanger (TL-MA) | System-Level Heat Exchanger (TL-MA) |
---|---|

Block parameters are based on the heat exchanger geometry | Block parameters are based on performance and operating conditions |

Heat exchanger geometry may be limited by the available geometry parameter options | Model is independent of the specific heat exchanger geometry |

You can adjust the block for different performance requirements by tuning geometry parameters, such as fin sizes and tube lengths | You can adjust the block for different performance requirements by directly specifying the desired heat and mass flow rates |

Lets you select between parallel, counter, or cross flow configurations | Lets you select between parallel, counter, or cross flow arrangement at nominal operating conditions, to help with sizing |

Predictively accurate results over a wide range of operating conditions, subject to the applicability of the E-NTU equations and the heat transfer coefficient correlations | Very accurate results around the specified operating condition; accuracy may decrease far away from the specified operating conditions |

Heat transfer calculations account for the variation of temperature along the flow path by using the E-NTU model | Heat transfer calculations approximate the variation of temperature along the flow path by dividing it into three segments |

Accounts for water vapor condensation and the latent heat on the moist air flow | Accounts for water vapor condensation and the latent heat on the moist air flow |

Does not model the wall thermal mass; you can approximate the effect by connecting a pipe block with a thermal mass downstream | Includes an option to model the wall thermal mass |

### Heat Transfer

The thermal liquid flow and the moist air 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

$${Q}_{seg,TL}+{Q}_{seg,MA}=0,$$

where:

*Q*_{seg,TL}is the heat flow rate from the wall (that is, the heat transfer surface) to the thermal liquid in the segment.*Q*_{seg,MA}is the heat flow rate from the wall to the moist air in the segment.

If the wall thermal mass is on, then the heat balance in the heat exchanger is

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

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 the thermal liquid in the segment is

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

where:

*UA*_{seg,TL}is the heat transfer conductance for the thermal liquid in the segment.*T*_{seg,TL}is the average liquid temperature in the segment.

The heat flow rate from the wall to the moist air in the segment is

$${Q}_{seg,MA}=\frac{U{A}_{seg,MA}}{{\overline{c}}_{{p}_{seg,MA}}}\left({\overline{h}}_{seg,wall}-{\overline{h}}_{seg,MA}\right)+{\dot{m}}_{w,seg,cond}{h}_{l,wall},$$

where:

*UA*_{seg,MA}is the heat transfer conductance for the moist air in the segment.$${\overline{c}}_{{p}_{seg,MA}}$$ is the moist air mixture specific heat per unit mass of dry air and trace gas in the segment.

$${\overline{h}}_{seg,wall}$$ is the moist air mixture enthalpy per unit mass of dry air and trace gas at the average wall segment temperature.

$${\overline{h}}_{seg,MA}$$ is the moist air mixture enthalpy per unit mass of dry air and trace gas in the segment.

$${\dot{m}}_{w,seg,cond}$$ is the rate of water vapor condensation on the wall surface.

*h*_{l,wall}is the specific enthalpy of liquid water at the average wall segment temperature.

Using mixture enthalpy in this equation accounts for both differences in temperature and differences in moisture due to condensation [3].

**Note**

For the moist air quantities, the bar above the symbols indicates that they are quantities for mixture divided by the mass of dry air and trace gas only, as opposed to dividing by the mass of the whole mixture. The whole mixture includes dry air, water vapor, and trace gas.

### Thermal Liquid Heat Transfer Correlation

The heat transfer conductance is

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

where:

*a*_{TL},*b*_{TL}, and*c*_{TL}are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the**Correlation Coefficients**section.*Re*_{seg,TL}is the average Reynolds number for the segment.*Pr*_{seg,TL}is the average Prandtl number for the segment.*k*_{seg,TL}is the average thermal conductivity for the segment.*G*_{TL}is the geometry scale factor for the thermal liquid 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 Reynolds number is

$${\mathrm{Re}}_{seg,\text{TL}}=\frac{{\dot{m}}_{seg,\text{TL}}{D}_{ref,\text{TL}}}{{\mu}_{seg,\text{TL}}{S}_{ref,\text{TL}}},$$

where:

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

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

**Note**

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

### Moist Air Heat Transfer Correlation

The heat transfer conductance is

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

where:

*a*_{MA},*b*_{MA}, and*c*_{MA}are the coefficients of the Nusselt number correlation. These coefficients appear as block parameters in the**Correlation Coefficients**section.*Re*_{seg,MA}is the average Reynolds number for the segment.*Pr*_{seg,MA}is the average Prandtl number for the segment.*k*_{seg,MA}is the average thermal conductivity for the segment.*G*_{MA}is the geometry scale factor for the moist air 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 Reynolds number is

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

where:

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

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

**Note**

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

### Moist Air Condensation

The equation describing the heat flow rate from the wall to the moist air in the segment (the last equation in the Heat Transfer section) uses the average moist air mixture enthalpy, $${\overline{h}}_{seg,MA}$$, and the wall segment moist air mixture enthalpy, $${\overline{h}}_{seg,wall}$$.

The average moist air mixture enthalpy is based on the temperature and humidity of the moist air flow through the segment:

$${\overline{h}}_{seg,MA}={h}_{seg,ag,MA}+{W}_{seg,MA}{h}_{seg,w,MA},$$

where:

*h*_{seg,ag,MA}is the average specific enthalpy of dry air and trace gas for the segment.*h*_{seg,w,MA}is the average specific enthalpy of water vapor for the segment.*W*_{seg,MA}is the humidity ratio of the segment.

The wall segment moist air mixture enthalpy is based on the temperature and humidity at the wall segment:

$${\overline{h}}_{seg,wall}={h}_{seg,ag,wall}+{W}_{seg,wall}{h}_{seg,w,wall},$$

where:

*h*_{seg,ag,wall}is the specific enthalpy of dry air and trace gas at the wall segment temperature.*h*_{seg,w,wall}is the specific enthalpy of water vapor at the wall segment temperature.*W*_{seg,wall}is the humidity ratio at the wall segment:$${W}_{seg,wall}=\mathrm{min}\left({W}_{seg,MA},{W}_{seg,s,wall}\right),$$

where

*W*_{seg,s,wall}is the saturated humidity ratio at the wall segment temperature. In other words, the humidity ratio at the wall is the same as the humidity ratio of the moist air flow but not more than the maximum that can be supported at the wall segment temperature.

When *W*_{seg,s,wall} <
*W*_{seg,MA}, water vapor condensation occurs on the
wall surface. The rate of water vapor condensation is

$${\dot{m}}_{w,seg,cond}=\frac{U{A}_{seg,MA}}{{\overline{c}}_{{p}_{seg,MA}}}\left({W}_{seg,MA}-{W}_{seg,wall}\right).$$

The condensed water is assumed to be drained from the wall surface and is thus removed from the moist air flow downstream.

### Pressure Loss

The pressure losses on the thermal liquid side are

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

where:

*p*_{A,TL}and*p*_{B,TL}are the pressures at ports**A2**and**B2**, respectively.*p*_{TL}is internal thermal liquid pressure at which the heat transfer is calculated.$${\dot{m}}_{A,TL}$$ and $${\dot{m}}_{B,TL}$$ are the mass flow rates into ports

**A2**and**B2**, respectively.*ρ*_{avg,TL}is the average thermal liquid density over all segments.$${\dot{m}}_{thres,TL}$$ 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*_{TL}, so that*p*_{A,TL}–*p*_{B,TL}matches the nominal pressure loss at the nominal mass flow rate.

The pressure losses on the moist air side are

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

where:

*p*_{A,MA}and*p*_{B,MA}are the pressures at ports**A2**and**B2**, respectively.*p*_{MA}is internal moist air pressure at which the heat transfer is calculated.$${\dot{m}}_{A,MA}$$ and $${\dot{m}}_{B,MA}$$ are the mass flow rates into ports

**A2**and**B2**, respectively.*ρ*_{avg,MA}is the average moist air density over all segments.$${\dot{m}}_{thres,MA}$$ 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*_{MA}, so that*p*_{A,MA}–*p*_{B,MA}matches the nominal pressure loss at the nominal mass flow rate.

### Thermal Liquid Mass and Energy Conservation

The mass conservation for the overall thermal liquid flow is

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

where:

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

$$\frac{\partial {\rho}_{seg,TL}}{\partial T}$$ is the partial derivative of density with respect to temperature for the segment.

*T*_{seg,TL}is the temperature for the segment.*V*_{TL}is the total thermal liquid volume.

The summation is over all segments.

**Note**

Although the thermal liquid flow is divided into *N*=3 segments for
heat transfer calculations, all segments are assumed to be at the same internal pressure,
*p*_{TL}. That is why
*p*_{TL} is outside of the summation.

The energy conservation equation for each segment is

$$\begin{array}{l}\left(\frac{d{p}_{TL}}{dt}\frac{\partial {u}_{seg,TL}}{\partial p}+\frac{d{T}_{seg,TL}}{dt}\frac{\partial {u}_{seg,TL}}{\partial T}\right)\frac{{M}_{TL}}{N}+{u}_{seg,TL}\left({\dot{m}}_{seg,in,TL}-{\dot{m}}_{seg,out,TL}\right)=\\ {\Phi}_{seg,in,TL}-{\Phi}_{seg,out,TL}+{Q}_{seg,TL},\end{array}$$

where:

$$\frac{\partial {u}_{seg,TL}}{\partial p}$$ is the partial derivative of specific internal energy with respect to pressure for the segment.

$$\frac{\partial {u}_{seg,TL}}{\partial T}$$ is the partial derivative of specific internal energy with respect to temperature for the segment.

*M*_{TL}is the total thermal liquid mass.$${\dot{m}}_{seg,in,TL}$$ and $${\dot{m}}_{seg,out,TL}$$ are the mass flow rates into and out of the segment.

*Φ*_{seg,in,TL}and*Φ*_{seg,out,TL}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$${\dot{m}}_{A,TL}$$ and $${\dot{m}}_{B,TL}$$.

### Moist Air Mass and Energy Conservation

The mass conservation for the overall moist air mixture flow is

$$\begin{array}{l}\left(\frac{d{p}_{MA}}{dt}{\displaystyle \sum _{segments}\left(\frac{\partial {\rho}_{seg,MA}}{\partial p}\right)}+{\displaystyle \sum _{segments}\left(\frac{d{T}_{seg,MA}}{dt}\frac{\partial {\rho}_{seg,MA}}{\partial T}+\frac{d{x}_{w,seg,MA}}{dt}\frac{\partial {\rho}_{seg,MA}}{\partial {x}_{w}}+\frac{d{x}_{g,seg,MA}}{dt}\frac{\partial {\rho}_{seg,MA}}{\partial {x}_{g}}\right)}\right)\frac{{V}_{MA}}{N}=\\ {\dot{m}}_{A,MA}+{\dot{m}}_{B,MA}-{\displaystyle \sum _{segments}\left({\dot{m}}_{w,seg,cond}\right)},\end{array}$$

where:

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

$$\frac{\partial {\rho}_{seg,MA}}{\partial T}$$ is the partial derivative of density with respect to temperature for the segment.

$$\frac{\partial {\rho}_{seg,MA}}{\partial {x}_{w}}$$ is the partial derivative of density with respect to specific humidity for the segment.

$$\frac{\partial {\rho}_{seg,MA}}{\partial {x}_{g}}$$ is the partial derivative of density with respect to trace gas mass fraction for the segment.

*x*_{w,seg,MA}is the specific humidity, that is, the water vapor mass fraction, for the segment.*x*_{g,seg,MA}is the trace gas mass fraction for the segment.*V*_{MA}is the total moist air volume.

The summation is over all segments.

**Note**

Although the moist air flow is divided into *N*=3 segments for heat
transfer calculations, all segments are assumed to be at the same internal pressure,
*p*_{MA}. That is why
*p*_{MA} is outside of the summation.

The energy conservation equation for each segment is

$$\begin{array}{l}\left(\frac{d{T}_{seg,MA}}{dt}\frac{\partial {u}_{seg,MA}}{\partial T}+\frac{d{x}_{w,seg,MA}}{dt}\frac{\partial {u}_{seg,MA}}{\partial {x}_{w}}+\frac{d{x}_{g,seg,MA}}{dt}\frac{\partial {u}_{seg,MA}}{\partial {x}_{g}}\right)\frac{{M}_{MA}}{N}+{u}_{seg,MA}\left({\dot{m}}_{seg,in,MA}-{\dot{m}}_{seg,out,MA}\right)=\\ {\Phi}_{seg,in,MA}-{\Phi}_{seg,out,MA}+{Q}_{seg,MA}-{\dot{m}}_{w,seg,cond}{h}_{l,wall},\end{array}$$

where:

$$\frac{\partial {u}_{seg,MA}}{\partial T}$$ is the partial derivative of specific internal energy with respect to temperature for the segment.

$$\frac{\partial {u}_{seg,MA}}{\partial {x}_{w}}$$ is the partial derivative of specific internal energy with respect to specific humidity for the segment.

$$\frac{\partial {u}_{seg,MA}}{\partial {x}_{g}}$$ is the partial derivative of specific internal energy with respect to trace gas mass fraction for the segment.

*u*_{seg,2P}is the specific internal energy for the segment.*M*_{MA}is the total moist air mass.$${\dot{m}}_{seg,in,MA}$$ and $${\dot{m}}_{seg,out,MA}$$ are the mass flow rates into and out of the segment.

*Φ*_{seg,in,MA}and*Φ*_{seg,out,MA}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$${\dot{m}}_{A,MA}$$ and $${\dot{m}}_{B,MA}$$.

The water vapor mass conservation equation for each segment is

$$\begin{array}{l}\frac{d{x}_{w,seg,MA}}{dt}\frac{{M}_{MA}}{N}+{x}_{w,seg,MA}\left({\dot{m}}_{seg,in,MA}-{\dot{m}}_{seg,out,MA}\right)=\\ {\dot{m}}_{w,seg,in,MA}-{\dot{m}}_{w,seg,out,MA}-{\dot{m}}_{w,seg,cond},\end{array}$$

where $${\dot{m}}_{w,seg,in,MA}$$ and $${\dot{m}}_{w,seg,out,MA}$$ are the water vapor mass flow rates into and out of the segment.

The trace gas mass conservation equation for each segment is

$$\begin{array}{l}\frac{d{x}_{g,seg,MA}}{dt}\frac{{M}_{MA}}{N}+{x}_{g,seg,MA}\left({\dot{m}}_{seg,in,MA}-{\dot{m}}_{seg,out,MA}\right)=\\ {\dot{m}}_{g,seg,in,MA}-{\dot{m}}_{g,seg,out,MA},\end{array}$$

where $${\dot{m}}_{g,seg,in,MA}$$ and $${\dot{m}}_{g,seg,out,MA}$$ are the trace gas mass flow rates into and out of the segment.

## Ports

### Output

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2022a**