# System-Level Heat Exchanger (TL-MA)

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

*Since R2022a*

**Libraries:**

Simscape /
Fluids /
Heat Exchangers /
Thermal Liquid - Moist Air

## Description

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

The block uses performance data from the heat exchanger datasheet, rather than the detailed geometry of the exchanger. You can 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 compares the two blocks:

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 |

You can select between parallel, counter, or cross flow configurations | You can 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 block divides the moist air flow and the thermal liquid 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

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

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 you select **Wall thermal mass**, 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 humidity 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 are 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 are 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 block assumes that the condensed water is 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 block divides the thermal liquid flow into *N*=3
segments for heat transfer calculations, it assumes all segments are at the same internal
pressure, *p _{TL}*. Consequentially,

*p*is outside of the summation.

_{TL}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 block assumes the mass flow rates between segments are 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

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

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*, also referred to as 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.$$\dot{m}$$

_{w,seg,conv}is the rate of condensation on the wall surface for the segment.$$\dot{m}$$

_{d,seg,evap}is the rate of water droplet evaporation for the segment.

The summation is over all segments.

**Note**

Although the block divides the moist air flow into *N*=3 segments for
heat transfer calculations, it assumes all segments are at the same internal pressure,
*p*_{MA}. Consequentially,
*p*_{MA} is outside of the summation.

The energy conservation equation for each segment is

$$\begin{array}{l}\frac{d{T}_{seg,MA}}{dt}\left({\overline{c}}_{seg,MA}-{R}_{seg}\right)\frac{{M}_{MA}}{N}+{u}_{a,seg,MA}\left({\dot{m}}_{seg,in,MA}-{\dot{m}}_{seg,out,MA}\right)+\left({u}_{w,seg,MA}-{u}_{a,seg,MA}\right)\left({\dot{m}}_{w,seg,in,MA}-{\dot{m}}_{w,seg,out,MA}\right)+\\ \left({u}_{g,seg,MA}-{u}_{a,seg,MA}\right)\left({\dot{m}}_{g,seg,in,MA}-{\dot{m}}_{g,seg,out,MA}\right)+{h}_{d}\left({\dot{m}}_{d,seg,in,MA}-{\dot{m}}_{d,seg,out,MA}\right)\\ ={\Phi}_{seg,in,MA}-{\Phi}_{seg,out,MA}+{Q}_{seg,MA}-\left(1-{\lambda}_{d}\right)\left({\dot{m}}_{w,seg,cond}{h}_{d}+{\dot{m}}_{w,seg,conv}{h}_{l,wall}\right),\end{array}$$

where:

*u*is the dry air specific internal energy for the segment._{a,seg,MA}*u*is the trace gas specific internal energy for the segment._{g,seg,MA}*u*is the water vapor specific internal energy for the segment._{w,seg,MA}*M*is the total moist air mass._{MA}$${\dot{m}}_{seg,in,MA}$$ and $${\dot{m}}_{seg,out,MA}$$ are the moist air mass flow rates into and out of the segment.

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

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

$${\dot{m}}_{d,seg,in,MA}$$ and $${\dot{m}}_{d,seg,out,MA}$$ are the water droplets mass flow rate into and out of the segment.

*R*is the segment volume specific gas constant._{seg}*Φ*_{seg,in,MA}and*Φ*_{seg,out,MA}are the energy flow rates into and out of the segment.*h*is the water droplet specific enthalpy in the segment._{d}*λ*is the value of the_{d}**Fraction of condensate entrained as water droplets**parameter.

The block assumes the mass flow rates between segments are 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

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

The trace gas mass conservation equation for each segment is

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

The water droplet mass balance conservation equation for each segment is

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

where *r _{d,seg}* is the mass ratio
of the water droplets to the moist air in the fluid volume in the segment.

**Note**

If the **Trace gas model** parameter is
`None`

in the Moist Air Properties
(MA) block, the moist air network does not model trace gas. In this
case, in the System-Level Heat Exchanger (TL-MA) block, the conservation equation for trace gas is set to
0.

If you clear the **Enable entrained water droplets** in the
Moist Air Properties (MA) block, the moist air network
does not model entrained water droplets. In this case, in the System-Level Heat Exchanger (TL-MA) block, the
conservation equation for water droplets is set to 0.

### Assumptions and Limitations

When determining the heat exchanger size, the block ignores the value of the

**Fraction of condensate entrained as water droplets**parameter and assumes that the condensate is not entrained as droplets.

## 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,
NJ: John Wiley & Sons, 2003.

[5] Cavallini, Alberto, and Roberto
Zecchin. “A DIMENSIONLESS CORRELATION FOR HEAT TRANSFER IN FORCED CONVECTION CONDENSATION.” In
*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 R2022a**