# System-Level Heat Exchanger (TL-G)

Heat exchanger based on performance data between thermal liquid and gas networks

*Since R2023b*

**Libraries:**

Simscape /
Fluids /
Heat Exchangers /
Thermal Liquid - Gas

## Description

The System-Level Heat Exchanger (TL-G) block models a heat exchanger based on performance data between a thermal liquid and a gas 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 block sizes the heat exchanger to match the specified performance at the nominal operating condition at steady state.

This block is similar to the Heat Exchanger (G-TL) block, but uses a different parameterization model. The table compares the two blocks:

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

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, shell and tube, 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 |

### Heat Transfer

The block divides the gas flow and the thermal liquid flow each into three segments of equal size. The block calculates heat transfer between the fluids 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

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

where:

*Q*is the heat flow rate from the wall that is the heat transfer surface to the gas in the segment._{seg,G}*Q*is the heat flow rate from the wall to the thermal liquid in the segment._{seg,TL}

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

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

where:

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

The heat flow rate from the wall to the gas in the segment is

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

where:

*UA*is the heat transfer conductance for the gas in the segment._{seg,G}*T*is the average temperature for the gas in the segment._{seg,G}

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

### Thermal Liquid Heat Transfer Correlation

The heat transfer conductance on the thermal liquid side of the heat exchanger 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.

### Gas Heat Transfer Correlation

The heat transfer conductance on the gas side of the heat exchanger is

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

where:

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

The average Reynolds number is

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

where:

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

**Note**

The *D _{ref,G}* and

*S*terms are included in this equation for unit calculation purposes only, to make

_{ref,G}*Re*nondimensional. The values of

_{seg,G}*D*and

_{ref,G}*S*are arbitrary because the

_{ref,G}*G*calculation overrides these values.

_{G}### 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*and_{A,TL}*p*are the pressures at ports_{B,TL}**A1**and**B1**, respectively.*p*is internal thermal liquid pressure at which the block calculates heat transfer._{TL}$${\dot{m}}_{A,TL}$$ and $${\dot{m}}_{B,TL}$$ are the mass flow rates into ports

**A1**and**B1**, respectively.*ρ*is the average thermal liquid density over all segments._{avg,TL}$${\dot{m}}_{thres,TL}$$ 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*, so that_{TL}*p*–_{A,TL}*p*matches the nominal pressure loss at the nominal mass flow rate._{B,TL}

The pressure losses on the gas side are

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

where:

*p*and_{A,G}*p*_{B,G}are the pressures at ports**A2**and**B2**, respectively.*p*is the internal gas pressure at which the block calculates heat transfer._{G}*ṁ*and_{A,G}*ṁ*are the mass flow rates into ports_{B,G}**A2**and**B2**, respectively.*ρ*is the average gas density over all segments._{avg,G}*ṁ*is the laminar threshold for pressure loss, approximated as 1e-4 of the nominal mass flow rate. The block calculates the pressure loss coefficient,_{thres,G}*K*, so that_{G}*p*–_{A,G}*p*matches the nominal pressure loss at the nominal mass flow rate._{B,G}

### 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}. Consequently,
*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 the specific internal energy with respect to pressure for the segment.

$$\frac{\partial {u}_{seg,TL}}{\partial T}$$ is the partial derivative of the 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 $${\dot{m}}_{B,TL}$$.

### Gas Mass and Energy Conservation

The mass conservation for the overall gas flow is

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

where:

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

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

*T*is the temperature for the segment._{seg,G}*V*is the total gas volume._{G}

The summation is over all segments.

**Note**

Although the block divides the gas flow into *N*=3 segments
for heat transfer calculations, it assumes all segments are at the same internal
pressure, *p _{G}*. Consequently,

*p*is outside of the summation.

_{G}The energy conservation equation for each segment is

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

where:

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

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

*M*is the total gas mass._{G}*ṁ*and_{seg,in,G}*ṁ*are the mass flow rates into and out of the segment._{seg,out,G}*Φ*and_{seg,in,G}*Φ*are the energy flow rates into and out of the segment._{seg,out,G}

The block assumes the mass flow rates between segments are linearly distributed between the
values of *ṁ _{A,G}* and

*ṁ*.

_{B,G}## 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 R2023b**