# Heat Exchanger Interface (TL)

Thermal interface between a thermal liquid and its surroundings

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

## Description

The Heat Exchanger Interface (TL) block models the pressure drop and temperature change between the thermal liquid inlet and outlet of a thermal interface. Combine with the E-NTU Heat Transfer block to model the heat transfer rate across the interface between two fluids.

### Mass Balance

The form of the mass balance equation depends on the dynamic compressibility setting. If
the **Fluid dynamic compressibility** parameter is set to
`Off`

, the mass balance equation is

$${\dot{m}}_{A}+{\dot{m}}_{B}=0,$$

where:

*$${\dot{m}}_{A}$$*and*$${\dot{m}}_{B}$$*are the mass flow rates into the interface through ports A and B.

If the **Fluid dynamic compressibility** parameter is set to
`On`

, the mass balance equation is

$${\dot{m}}_{A}+{\dot{m}}_{B}=\left(\frac{dp}{dt}\frac{1}{\beta}-\frac{dT}{dt}\alpha \right)\rho V,$$

where:

*p*is the pressure of the thermal liquid volume.*T*is the temperature of the thermal liquid volume.*α*is the isobaric thermal expansion coefficient of the thermal liquid volume.*β*is the isothermal bulk modulus of the thermal liquid volume.*ρ*is the mass density of the thermal liquid volume.*V*is the volume of thermal liquid in the heat exchanger interface.

### Momentum Balance

The momentum balance in the heat exchanger interface depends on the fluid dynamic
compressibility setting. If the **Fluid dynamic
compressibility** parameter is set to `On`

, the momentum
balance factors in the internal pressure of the heat exchanger interface explicitly. The
momentum balance in the half volume between port A and the internal interface node is
computed as

$${p}_{A}-p=\Delta {p}_{\text{Loss,A}},$$

while in the half volume between port B and the internal interface node it is computed as

$${p}_{B}-p=\Delta {p}_{\text{Loss,B}},$$

where:

*p*_{A}and*p*_{B}are the pressures at ports A and B.*p*is the pressure in the internal node of the interface volume.*Δp*_{Loss,A}and*Δp*_{Loss,B}are the pressure losses between port A and the internal interface node and between port B and the internal interface node.

If the **Fluid dynamic compressibility** parameter is set
to `Off`

, the momentum balance in the interface volume is computed
directly between ports A and B as

$${p}_{A}-{p}_{B}=\Delta {p}_{Loss,A}-\Delta {p}_{Loss,B}.$$

### Pressure Loss Calculations

The exact form of the pressure loss terms depends on the **Pressure loss
parameterization** setting in the block dialog box. If the pressure loss
parameterization is set to `Constant loss coefficient`

, the
pressure loss in the half volume adjacent to port A is

$$\Delta {p}_{Loss,A}=\{\begin{array}{ll}{\dot{m}}_{A}{\mu}_{A}{\left(CP\right)}_{Loss}{\mathrm{Re}}_{L}\frac{1}{4{D}_{h,p}{\rho}_{A}{S}_{Min}},\hfill & {\mathrm{Re}}_{A}\le {\mathrm{Re}}_{L}\hfill \\ {\left(CP\right)}_{Loss}\frac{{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|}{4{\rho}_{A}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{A}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

while in the half volume adjacent to port B it is

$$\Delta {p}_{Loss,B}=\{\begin{array}{ll}{\dot{m}}_{B}{\mu}_{B}{\left(CP\right)}_{Loss}{\mathrm{Re}}_{L}\frac{1}{4{D}_{h,p}{\rho}_{B}{S}_{Min}},\hfill & {\mathrm{Re}}_{B}\le {\mathrm{Re}}_{L}\hfill \\ {\left(CP\right)}_{Loss}\frac{{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|}{4{\rho}_{B}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{B}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

where:

*μ*_{A}and*μ*_{B}are the fluid dynamic viscosities at ports A and B.*CP*_{Loss}is the**Pressure loss coefficient**parameter specified in the block dialog box.*Re*_{L}is the Reynolds number upper bound for the laminar flow regime.*Re*_{T}is the Reynolds number lower bound for the turbulent flow regime.*D*_{h,p}is the hydraulic diameter for pressure loss calculations.*ρ*_{A}and*ρ*_{B}are the fluid mass densities at ports A and B.*S*_{Min}is the total minimum free-flow area.

If the pressure loss parameterization is set to ```
Correlations for
tubes
```

, the pressure loss in the half volume adjacent to port A is

$$\Delta {p}_{Loss,A}=\{\begin{array}{ll}{\dot{m}}_{A}{\mu}_{A}\lambda \frac{\left({L}_{press}+{L}_{add}\right)}{4{D}_{h,p}{\rho}_{A}{S}_{Min}},\hfill & {\mathrm{Re}}_{A}\le {\mathrm{Re}}_{L}\hfill \\ {f}_{T,A}\frac{\left({L}_{press}+{L}_{add}\right)}{4{D}_{h,p}}\frac{{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|}{{\rho}_{A}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{A}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

while in the half volume adjacent to port B it is

$$\Delta {p}_{Loss,B}=\{\begin{array}{ll}{\dot{m}}_{B}{\mu}_{B}\lambda \frac{\left({L}_{press}+{L}_{add}\right)}{4{D}_{h,p}{\rho}_{B}{S}_{Min}},\hfill & {\mathrm{Re}}_{B}\le {\mathrm{Re}}_{L}\hfill \\ {f}_{T,B}\frac{\left({L}_{press}+{L}_{add}\right)}{4{D}_{h,p}}\frac{{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|}{{\rho}_{B}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{B}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

where:

*L*_{press}is the flow path length from inlet to outlet.*L*_{add}is the aggregate equivalent length of local resistances.*f*_{T,A}and*f*_{T,B}are the turbulent-regime Darcy friction factors at ports A and B.

The Darcy friction factor in the half volume adjacent to port A is

$${f}_{T,A}=\frac{1}{{\left[-1.8{\mathrm{log}}_{10}{\left(\frac{6.9}{{\mathrm{Re}}_{A}}+\frac{r}{3.7{D}_{h,p}}\right)}^{1.11}\right]}^{2}},$$

while in the half volume adjacent to port B it is

$${f}_{T,B}=\frac{1}{{\left[-1.8{\mathrm{log}}_{10}{\left(\frac{6.9}{{\mathrm{Re}}_{B}}+\frac{r}{3.7{D}_{h,p}}\right)}^{1.11}\right]}^{2}},$$

where:

*r*is the internal surface absolute roughness.

If the pressure loss parameterization is set to ```
Tabulated data —
Darcy friction factor vs. Reynolds number
```

, the pressure loss in the half
volume adjacent to port A is

$$\Delta {p}_{Loss,A}=\{\begin{array}{ll}{\dot{m}}_{A}{\mu}_{A}\lambda \frac{{L}_{press}}{4{D}_{h,p}^{2}{\rho}_{A}{S}_{Min}},\hfill & {\mathrm{Re}}_{A}\le {\mathrm{Re}}_{L}\hfill \\ f\left({\mathrm{Re}}_{A}\right)\frac{{L}_{press}}{4{D}_{h,p}}\frac{{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|}{{\rho}_{A}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{A}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

while in the half volume adjacent to port B it is

$$\Delta {p}_{Loss,B}=\{\begin{array}{ll}{\dot{m}}_{B}{\mu}_{B}\lambda \frac{{L}_{press}}{4{D}_{h,p}^{2}{\rho}_{B}{S}_{Min}},\hfill & {\mathrm{Re}}_{B}\le {\mathrm{Re}}_{L}\hfill \\ f\left({\mathrm{Re}}_{B}\right)\frac{{L}_{press}}{4{D}_{h,p}}\frac{{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|}{{\rho}_{B}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{B}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

where:

*λ*is the shape factor for laminar flow viscous friction.*f*(*Re*_{A}) and*f*(*Re*_{B}) are the Darcy friction factors at ports A and B. The block obtains the friction factors from tabulated data specified relative to the Reynolds number.

If the pressure loss parameterization is set to ```
Tabulated data —
Euler number vs. Reynolds number
```

, the pressure loss in the half volume
adjacent to port A is

$$\Delta {p}_{Loss,A}=\{\begin{array}{ll}{\dot{m}}_{A}{\mu}_{A}\text{Eu}\left({\mathrm{Re}}_{L}\right){\mathrm{Re}}_{L}\frac{1}{4{D}_{h,p}{\rho}_{A}{S}_{Min}},\hfill & {\mathrm{Re}}_{A}\le {\mathrm{Re}}_{L}\hfill \\ Eu\left({\mathrm{Re}}_{A}\right)\frac{{\dot{m}}_{A}\left|{\dot{m}}_{A}\right|}{4{\rho}_{A}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{A}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

while in the half volume adjacent to port B it is

$$\Delta {p}_{Loss,B}=\{\begin{array}{ll}{\dot{m}}_{B}{\mu}_{B}\text{Eu}\left({\mathrm{Re}}_{L}\right){\mathrm{Re}}_{L}\frac{1}{4{D}_{h,p}{\rho}_{B}{S}_{Min}},\hfill & {\mathrm{Re}}_{B}\le {\mathrm{Re}}_{L}\hfill \\ Eu\left({\mathrm{Re}}_{B}\right)\frac{{\dot{m}}_{B}\left|{\dot{m}}_{B}\right|}{4{\rho}_{B}{S}_{Min}^{2}},\hfill & {\mathrm{Re}}_{B}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

where:

*Eu*(*Re*_{L}) is the Euler number at the Reynolds number upper bound for laminar flows.*Eu*(*Re*_{A}) and*Eu*(*Re*_{B}) are the Euler numbers at ports A and B. The block obtains the Euler numbers from tabulated data specified relative to the Reynolds number.

### Energy Balance

The energy balance in the heat exchanger interface depends on the fluid dynamic
compressibility setting. If the **Fluid dynamic
compressibility** parameter is set to `On`

, the energy
balance is

$$\frac{dp}{dt}\frac{dU}{dp}+\frac{dT}{dt}\frac{dU}{dT}={\varphi}_{A}+{\varphi}_{B}+{Q}_{H},$$

where:

*U*is the internal energy contained in the volume of the heat exchanger interface.*Φ*_{A}and*Φ*_{B}are the energy flow rates through ports**A**and**B**into the volume of the heat exchanger interface.*Q*_{H}is the heat flow rate through port**H**, representing the interface wall, into the volume of the heat exchange interface.

The internal energy derivatives are defined as

$$\frac{dU}{dp}=\left[\frac{1}{\beta}\left(\rho u+p\right)-T\alpha \right]V$$

and

$$\frac{dU}{dT}=\left[{c}_{p}-\alpha \left(u+\frac{p}{\rho}\right)\right]\rho V,$$

where *u* is the specific internal energy of the thermal
liquid, or the internal energy contained in a unit mass of the same.

If the **Fluid dynamic compressibility** parameter is set
to `Off`

, the thermal liquid density is treated as a constant. The
bulk modulus is then effectively infinite and the thermal expansion coefficient zero. The
pressure and temperature derivatives of the compressible case vanish and the energy balance
is restated as

$$\frac{dE}{dt}={\varphi}_{A}+{\varphi}_{B}+{Q}_{H},$$

where *E* is the total internal energy of the
incompressible thermal liquid, or

$$E=\rho uV.$$

### Heat Transfer Correlations

The block calculates and outputs the liquid-wall heat transfer coefficient value. The
calculation depends on the **Heat transfer coefficient specification**
setting in the block dialog box. If the heat transfer coefficient specification is
`Constant heat transfer coefficient`

, the heat transfer
coefficient is simply the constant value specified in the block dialog box,

$${h}_{L-W}={h}_{Const},$$

where:

*h*_{L-W}is the liquid-wall heat transfer coefficient.*h*_{Const}is the**Liquid-wall heat transfer coefficient value**specified in the block dialog box.

For all other heat transfer coefficient parameterizations, the heat transfer coefficient is defined as the arithmetic average of the port heat transfer coefficients:

$${h}_{L-W}=\frac{{h}_{A}+{h}_{B}}{2},$$

where:

*h*_{A}and*h*_{B}are the liquid-wall heat transfer coefficients at ports A and B.

The heat transfer coefficient at port A is

$${h}_{A}=\frac{N{u}_{A}{k}_{A}}{{D}_{h,heat}},$$

while at port B it is

$${h}_{B}=\frac{N{u}_{B}{k}_{B}}{{D}_{h,heat}},$$

where:

*Nu*_{A}and*Nu*_{B}are the Nusselt numbers at ports A and B.*k*_{A}and*k*_{B}are the thermal conductivities at ports A and B.*D*_{h,heat}is the hydraulic diameter for heat transfer calculations.

The hydraulic diameter used in heat transfer calculations is defined as

$${D}_{h,heat}=\frac{4{S}_{Min}{L}_{heat}}{{S}_{heat}},$$

where:

*L*_{heat}is the flow path length used in heat transfer calculations.*S*_{heat}is the total heat transfer surface area.

### Nusselt Number Calculations

The Nusselt number calculation depends on the **Heat transfer coefficient
specification** setting in the block dialog box. If the heat transfer
specification is set to `Correlations for tubes`

, the Nusselt
number at port A is

$$N{u}_{A}=\{\begin{array}{ll}{\text{Nu}}_{L},\hfill & {\mathrm{Re}}_{A}\le {\mathrm{Re}}_{L}\hfill \\ \frac{\left(\raisebox{1ex}{${f}_{T,A}$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\right)\left({\mathrm{Re}}_{A}-1000\right){\mathrm{Pr}}_{A}}{1+12.7{\left(\raisebox{1ex}{${f}_{T,A}$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\right)}^{1/2}\left({\mathrm{Pr}}_{B}^{2/3}-1\right)},\hfill & {\mathrm{Re}}_{A}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

while at port B it is

$$N{u}_{B}=\{\begin{array}{ll}{\text{Nu}}_{L},\hfill & {\mathrm{Re}}_{B}\le {\mathrm{Re}}_{L}\hfill \\ \frac{\left(\raisebox{1ex}{${f}_{T,B}$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\right)\left({\mathrm{Re}}_{B}-1000\right){\mathrm{Pr}}_{B}}{1+12.7{\left(\raisebox{1ex}{${f}_{T,B}$}\!\left/ \!\raisebox{-1ex}{$8$}\right.\right)}^{1/2}\left({\mathrm{Pr}}_{B}^{2/3}-1\right)},\hfill & {\mathrm{Re}}_{B}\ge {\mathrm{Re}}_{T}\hfill \end{array},$$

where:

*Nu*_{L}is the**Nusselt number for laminar flow heat transfer**value specified in the block dialog box.*Pr*_{A}and*Pr*_{B}are the Prandtl numbers at ports A and B.

If the heat transfer specification is set to ```
Tabulated data —
Colburn data vs. Reynolds number
```

, the Nusselt number at port A is

$$N{u}_{A}=j\left({\mathrm{Re}}_{A,heat}\right){\mathrm{Re}}_{A,heat}{\mathrm{Pr}}_{A}^{1/3},$$

while at port B it is

$$N{u}_{B}=j\left({\mathrm{Re}}_{B,heat}\right){\mathrm{Re}}_{B,heat}{\mathrm{Pr}}_{B}^{1/3},$$

where:

*j*(*Re*_{A,heat}) and*j*(*Re*_{B,heat}) are the Colburn numbers at ports A and B. The block obtains the Colburn numbers from tabulated data provided as a function of the Reynolds number.*Re*_{A,heat}and*Re*_{B,heat}are the Reynolds numbers based on the hydraulic diameters for heat transfer calculations at ports A and B. This parameter is defined at port A as$${\mathrm{Re}}_{A,heat}=\frac{{\dot{m}}_{A}{D}_{h,heat}}{{S}_{Min}{\mu}_{A}},$$

and at port B as

$${\mathrm{Re}}_{B}=\frac{{\dot{m}}_{B}{D}_{h,heat}}{{S}_{Min}{\mu}_{B}}.$$

If the heat transfer specification is set to ```
Tabulated data —
Nusselt number vs. Reynolds number & Prandtl number
```

, the Nusselt number
at port A is

$$N{u}_{A}=Nu\left({\mathrm{Re}}_{A,heat},{\mathrm{Pr}}_{A}\right),$$

while at port B it is

$$N{u}_{B}=Nu\left({\mathrm{Re}}_{B,heat},{\mathrm{Pr}}_{B}\right).$$

### Hydraulic Diameter Calculations

The hydraulic diameter used in heat transfer calculations can differ from the hydraulic diameter employed in pressure loss calculations, and are different if the heated and friction perimeters are not the same. For a concentric pipe heat exchanger with an annular cross-section, the hydraulic diameter for heat transfer calculations is

$${D}_{h,heat}=\frac{4\left(\pi /4\right)\left({D}_{o}^{2}-{D}_{i}^{2}\right)}{\pi {D}_{i}}=\frac{{D}_{o}^{2}-{D}_{i}^{2}}{{D}_{i}},$$

while the hydraulic diameter for pressure calculations is

$${D}_{h,p}=\frac{4\left(\pi /4\right)\left({D}_{o}^{2}-{D}_{i}^{2}\right)}{\pi \left({D}_{i}+{D}_{o}\right)}={D}_{o}-{D}_{i},$$

where:

*D*_{o}is the outer annulus diameter.*D*_{i}is the inner annulus diameter.

**Annulus Schematic**

## Ports

### Output

### Conserving

## Parameters

## Extended Capabilities

## Version History

**Introduced in R2016a**