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Models heat exchange between a thermal liquid network and a network that can undergo phase change

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

The Condenser Evaporator (TL-MA) block models a heat exchanger with one
thermal liquid network, which flows between ports **A1** and
**B1**, and one two-phase fluid network, which flows between ports
**A2** and **B2**. The heat exchanger can act as a
condenser or as an evaporator. The fluid streams can be aligned in parallel, counter, or
cross-flow configurations.

**Example Heat Exchanger for Refrigeration Applications**

You can model the thermal liquid side as flow within tubes, flow around the two-phase fluid tubing, or by an empirical, generic parameterization.

The block uses the Effectiveness-NTU (E-NTU) method to model heat transfer through the shared wall. Fouling on the exchanger walls, which increases thermal resistance and reduces the heat exchange between the two fluids, is also modeled. You can also optionally model fins on both the thermal liquid and two-phase fluid sides. Pressure loss due to viscous friction on both sides of the exchanger can be modeled analytically or by generic parameterization, which you can use to tune to your own data.

You can model the two-phase fluid side as flow within a tube or a set of tubes. The
two-phase fluid tubes use a boundary-following model to track the sub-cooled liquid (L),
vapor-liquid mixture (M), and super-heated vapor (V) in three zones. The relative amount
of space a zone occupies in the system is called a *zone length
fraction* within the system.

**Zone Length Fractions in the Two-Phase Fluid Piping**

The sum of the zone length fractions in the two-phase fluid tubing equals
`1`

. Port **Z** returns the zone
length fractions as a vector of physical signals for each of the three phases: [L, M,
V].

The heat exchanger effectiveness is based on the selected heat exchanger configuration, the fluid properties in each phase, the tube geometry and flow configuration on each side of the exchanger, and the usage and size of fins.

The **Flow arrangement** parameter assigns the relative flow
paths between the two sides:

`Parallel flow`

indicates the fluids are moving in the same direction.`Counter flow`

indicates the fluids are moving in parallel, but opposite directions.`Cross flow`

indicates the fluids are moving perpendicular to each other.

When **Flow arrangement** is set to ```
Cross
flow
```

, use the **Cross flow arrangement**
parameter to indicate whether the two-phase fluid or thermal liquid flows are
separated into multiple paths by baffles or walls. Without these separations,
the flow can mix freely and is considered *mixed*. Both
fluids, one fluid, or neither fluid can be mixed in the cross-flow arrangement.
Mixing homogenizes the fluid temperature along the direction of flow of the
second fluid, and varies perpendicular to the second fluid flow.

Unmixed flows vary in temperature both along and perpendicular to the flow path of the second fluid.

**Sample Cross-Flow Configurations**

Note that the flow direction during simulation does not impact the selected flow arrangement setting. The ports on the block do not reflect the physical positions of the ports in the physical heat exchange system.

All flow arrangements are single-pass, which means that the fluids do not make multiple turns in the exchanger for additional points of heat transfer. To model a multi-pass heat exchanger, you can arrange multiple Condenser Evaporator (TL-2P) blocks in series or in parallel.

For example, to achieve a two-pass configuration on the two-phase fluid side and a single-pass configuration on the thermal liquid side, you can connect the two-phase fluid sides in series and the thermal liquid sides to the same input in parallel (such as two Mass Flow Rate Source blocks with half of the total mass flow rate), as shown below.

The **Flow geometry** parameter sets the thermal liquid flow
arrangement as either inside a tube or set of tubes, or perpendicular to a tube
bank. You can also specify an empirical, generic configuration. The two-phase
fluid always flows inside a tube or set of tubes.

When **Flow geometry** is set to ```
Flow
perpendicular to bank of circular tubes
```

, use the
**Tube bank grid arrangement** parameter to define the
two-phase fluid tube bank alignment as either `Inline`

or `Staggered`

. The red, downward-pointing arrow
indicates the direction of thermal liquid flow. The Inline figure also shows the
**Number of tube rows along flow direction** and the
**Number of tube segments in each tube row** parameters.
Here, *flow direction* refers to the thermal liquid flow, and
*tube* refers to the two-phase fluid tubing. The
**Length of each tube segment in a tube row** parameter is
indicated in the Staggered figure.

The heat exchanger configuration is without fins when the **Total fin
surface area** parameter is set to `0 m^2`

. Fins
introduce additional surface area for additional heat transfer. Each fluid side
has a separate fin area.

The heat transfer rate is calculated for each fluid phase. In accordance with the three fluid zones that occur on the two-phase fluid side of the heat exchanger, the heat transfer rate is calculated in three sections.

The heat transfer in a zone is calculated as:

$${Q}_{zone}=\u03f5{C}_{\text{Min}}({T}_{\text{In,2P}}-{T}_{\text{In,TL}}),$$

where:

*C*_{Min}is the lesser of the heat capacity rates of the two fluids in that zone. The heat capacity rate is the product of the fluid specific heat,*c*_{p}, and the fluid mass flow rate.*C*_{Min}is always positive.*T*_{In,2P}is the zone inlet temperature of the two-phase fluid.*T*_{In,TL}is the zone inlet temperature of the thermal liquid.*ε*is the heat exchanger effectiveness.

Effectiveness is a function of the heat capacity rate and the number
of transfer units, *NTU*, and also varies based on the heat
exchanger flow arrangement, which is discussed in more detail in Effectiveness by Flow Arrangement. The
*NTU* is calculated as:

$$NTU=\frac{z}{{C}_{\text{Min}}R},$$

where:

*z*is the individual zone length fraction.*R*is the total thermal resistance between the two flows, due to convection, conduction, and any fouling on the tube walls:$$R=\frac{1}{{U}_{\text{2P}}{A}_{\text{Th,2P}}}+\frac{{F}_{\text{2P}}}{{A}_{\text{Th,2P}}}+{R}_{\text{W}}+\frac{{F}_{\text{TL}}}{{A}_{\text{Th,TL}}}+\frac{1}{{U}_{\text{TL}}{A}_{\text{Th,TL}}},$$

where:

*U*is the convective heat transfer coefficient of the respective fluid. This coefficient is discussed in more detail in Two-Phase Fluid Correlations and Thermal Liquid Correlations.*F*is the**Fouling factor**on the two-phase fluid or thermal liquid side, respectively.*R*_{W}is the**Thermal resistance through heat transfer surface**.*A*_{Th}is the heat transfer surface area of the respective side of the exchanger.*A*_{Th}is the sum of the wall surface area,*A*_{W}, and the**Total fin surface area**,*A*_{F}:$${A}_{\text{Th}}={A}_{\text{W}}+{\eta}_{\text{F}}{A}_{\text{F}},$$

where

*η*_{F}is the**Fin efficiency**.

The total heat transfer rate between the fluids is the sum of the heat transferred
in the three zones by the subcooled liquid
(*Q _{L}*), liquid-vapor mixture
(

$$Q={\displaystyle \sum {Q}_{\text{Z}}}={Q}_{\text{L}}+{Q}_{\text{M}}+{Q}_{\text{V}}.$$

The heat exchanger effectiveness varies according to its flow configuration and the mixing in each fluid. Below are the formulations for effectiveness calculated in the liquid and vapor zones for each configuration. The effectiveness is $$\epsilon =1-\mathrm{exp}(-NTU)$$ for all configurations in the mixture zone.

When

**Flow arrangement**is set to`Parallel flow`

:$$\u03f5=\frac{1-\text{exp}[-NTU(1+{C}_{\text{R}})]}{1+{C}_{\text{R}}}$$

When

**Flow arrangement**is set to`Counter flow`

:$$\u03f5=\frac{1-\text{exp}[-NTU(1-{C}_{\text{R}})]}{1-{C}_{\text{R}}\text{exp}[-NTU(1-{C}_{\text{R}})]}$$

When

**Flow arrangement**is set to`Cross flow`

and**Cross flow arrangement**is set to`Both fluids unmixed`

:$$\u03f5=1-\text{exp}\left\{\frac{NT{U}^{\text{0}\text{.22}}}{{C}_{\text{R}}}\left[\text{exp}\left(-{C}_{\text{R}}NT{U}^{\text{0}\text{.78}}\right)-1\right]\right\}$$

When

**Flow arrangement**is set to`Cross flow`

and**Cross flow arrangement**is set to`Both fluids mixed`

:$$\u03f5={\left[\frac{1}{1-\text{exp}\left(-NTU\right)}+\frac{{C}_{\text{R}}}{1-\text{exp}\left(-{C}_{\text{R}}NTU\right)}-\frac{1}{NTU}\right]}^{-1}$$

When one fluid is mixed and the other unmixed, the equation for
effectiveness depends on the relative heat capacity rates of the fluids.
When **Flow arrangement** is set to ```
Cross
flow
```

and **Cross flow arrangement**
is set to either ```
Thermal Liquid 1 mixed & Two-Phase
Fluid 2 unmixed
```

or ```
Thermal Liquid 1
unmixed & Two-Phase Fluid 2 mixed
```

:

When the fluid with C

_{max}is mixed and the fluid with C_{min}is unmixed:$$\u03f5=\frac{1}{{C}_{\text{R}}}\left(1-\text{exp}\left\{-{C}_{R}\left\{1-\mathrm{exp}\left(-NTU\right)\right\}\right\}\right)$$

When the fluid with C

_{min}is mixed and the fluid with C_{max}is unmixed:$$\u03f5=1-\text{exp}\left\{-\frac{1}{{C}_{\text{R}}}\left[1-\text{exp}\left(-{C}_{\text{R}}NTU\right)\right]\right\}$$

*C*_{R} denotes the ratio
between the heat capacity rates of the two fluids:

$${C}_{\text{R}}=\frac{{C}_{\text{Min}}}{{C}_{\text{Max}}}.$$

The convective heat transfer coefficient varies according to the fluid Nusselt number:

$$U=\frac{\text{Nu}k}{{D}_{\text{H}}},$$

where:

*Nu*is the zone mean Nusselt number, which depends on the flow regime.*k*is the fluid phase thermal conductivity.*D*_{H}is tube hydraulic diameter.

For turbulent flows in the subcooled liquid or superheated vapor zones, the Nusselt number is calculated with the Gnielinski correlation:

$$\text{Nu}=\frac{\frac{{f}_{D}}{8}(\text{Re}-1000)\text{Pr}}{1+12.7\sqrt{\frac{f}{8}}({\text{Pr}}^{2/3}-1)},$$

where:

*Re*is the fluid Reynolds number.*Pr*is the fluid Prandtl number.

For turbulent flows in the liquid-vapor mixture zone, the Nusselt number is calculated with the Cavallini-Zecchin correlation:

$$\text{Nu}=\frac{{\text{aRe}}_{\text{SL}}^{b}{\text{Pr}}_{\text{SL}}^{c}\left\{{\left[\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right){x}_{\text{Out}}+1\right]}^{1+b}-{\left[\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right){x}_{\text{In}}+1\right]}^{1+b}\right\}}{\left(1+b\right)\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right)({x}_{\text{Out}}-{x}_{\text{In}})}.$$

where:

*Re*_{SL}is the Reynolds number of the saturated liquid.*Pr*_{SL}is the Prandtl number of the saturated liquid.*ρ*_{SL}is the density of the saturated liquid.*ρ*_{SV}is the density of the saturated vapor.*a*= 0.05,*b*= 0.8, and*c*= 0.33.

For laminar flows, the Nusselt number is set by the **Laminar flow
Nusselt number** parameter.

For transitional flows, the Nusselt number is a blend between the laminar and turbulent Nusselt numbers.

When the **Heat transfer coefficient model** parameter is set
to `Colburn equation`

, the Nusselt number for the
subcooled liquid and superheated vapor zones is calculated by the empirical the
Colburn equation:

$$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c},$$

where *a*, *b*, and
*c* are defined in the **Coefficients [a, b, c] for
a*Re^b*Pr^c in liquid zone** and **Coefficients [a, b, c]
for a*Re^b*Pr^c in vapor zone** parameters.

The Nusselt number for liquid-vapor mixture zones is calculated with the
Cavallini-Zecchin equation, with the coefficients specified in the
**Coefficients [a, b, c] for a*Re^b*Pr^c in mixture zone**
parameter.

The pressure loss due to viscous friction varies depending on flow regime and configuration. The calculation uses the overall density, which is the total two-phase fluid mass divided by the total two-phase fluid volume.

For turbulent flows, when the Reynolds number is above the **Turbulent
flow lower Reynolds number limit**, the pressure loss due to
friction is calculated in terms of the Darcy friction factor. The pressure
differential between port **A2** and the internal node I2 is:

$${p}_{\text{A2}}-{p}_{\text{I2}}=\frac{{f}_{\text{D,A}}{\dot{m}}_{\text{A2}}\left|{\dot{m}}_{\text{A2}}\right|}{2\rho {D}_{\text{H}}{A}_{\text{CS}}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$$

where:

$$\dot{m}$$

_{A2}is the total flow rate through port**A2**.*f*_{D,A}is the Darcy friction factor, according to the Haaland correlation:$${f}_{\text{D,A2}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{\text{A2}}}+{\left(\frac{{\u03f5}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}},$$

where

*ε*_{R}is the two-phase fluid pipe**Internal surface absolute roughness**. Note that the friction factor is dependent on the Reynolds number, and is calculated at both ports for each liquid.*L*is the**Total length of each tube**on the two-phase fluid side.*L*_{Add}is the two-phase fluid**Aggregate equivalent length of local resistances**, which is the equivalent length of a tube that introduces the same amount of loss as the sum of the losses due to other local resistances in the tube.*A*_{CS}is the total tube cross-sectional area.

The pressure differential between port **B2**
and internal node I2 is:

$${p}_{\text{B2}}-{p}_{\text{I2}}=\frac{{f}_{\text{D,B}}{\dot{m}}_{\text{B2}}\left|{\dot{m}}_{\text{B2}}\right|}{2\rho {D}_{\text{H}}{A}_{\text{CS}}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$$

where $$\dot{m}$$_{B2} is the total flow rate through port
**B2**.

The Darcy friction factor at port **B2** is:

$${f}_{\text{D,B2}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{\text{B2}}}+{\left(\frac{{\u03f5}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}}.$$

For laminar flows, when the Reynolds number is below the **Laminar
flow upper Reynolds number limit**, the pressure loss due to
friction is calculated in terms of the **Laminar friction constant for
Darcy friction factor**, *λ*. *λ*
is a user-defined parameter when **Tube cross-section** is set
to `Generic`

, otherwise, the value is calculated
internally. The pressure differential between port **A2** and
internal node I2 is:

$${p}_{\text{A2}}-{p}_{\text{I2}}=\frac{\lambda \mu {\dot{m}}_{\text{A2}}}{2\rho {D}_{\text{H}}^{2}{A}_{CS}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$$

where μ is the two-phase fluid dynamic viscosity. The
pressure differential between port **B2** and internal node I2 is:

$${p}_{\text{B2}}-{p}_{\text{I2}}=\frac{\lambda \mu {\dot{m}}_{\text{B2}}}{2\rho {D}_{\text{H}}^{2}{A}_{CS}}\left(\frac{L+{L}_{\text{Add}}}{2}\right).$$

For transitional flows, the pressure differential due to viscous friction is a smoothed blend between the values for laminar and turbulent pressure losses.

When **Pressure loss model** is set to ```
Pressure
loss coefficient
```

, the pressure losses due to viscous friction
are calculated with an empirical pressure loss coefficient,
*ξ*.

The pressure differential between port **A2** and internal
node I2 is:

$${p}_{\text{A2}}-{p}_{\text{I2}}=\frac{1}{2}\xi \frac{{\dot{m}}_{\text{A2}}\left|{\dot{m}}_{\text{A2}}\right|}{2\rho {A}_{\text{CS}}^{2}}.$$

The pressure differential between port
**B2** and internal node I2 is:

$${p}_{\text{B2}}-{p}_{\text{I2}}=\frac{1}{2}\xi \frac{{\dot{m}}_{\text{B2}}\left|{\dot{m}}_{\text{B2}}\right|}{2\rho {A}_{\text{CS}}^{2}}.$$

When the thermal liquid **Flow geometry** is set to
`Flow inside one or more tubes`

, the Nusselt number
is calculated according to the Gnielinski correlation in the same manner as
two-phase subcooled liquid or superheated vapor. See Heat Transfer Coefficient for more
information.

When the thermal liquid **Flow geometry** is set to
`Flow perpendicular to bank of circular tubes`

, the
Nusselt number is calculated based on the Hagen number, Hg, and depends on the
**Tube bank grid arrangement** setting:

$$\text{Nu}=\{\begin{array}{cc}0.404L{q}^{\text{1/3}}{\left(\frac{\text{Re}+1}{\text{Re}+1000}\right)}^{0.1},& Inline\\ 0.404L{q}^{1/3},& Staggered\end{array}$$

where:

$$Lq=\{\begin{array}{cc}1.18\text{Pr}\left(\frac{4{l}_{\text{T}}/\pi -D}{{l}_{\text{L}}}\right)\text{Hg}(\text{Re}),& Inline\\ 0.92\text{Pr}\left(\frac{4{l}_{\text{T}}/\pi -D}{{l}_{\text{D}}}\right)\text{Hg}(\text{Re}),& Staggeredwith{l}_{L}\ge D\\ 0.92\text{Pr}\left(\frac{4{l}_{\text{T}}{l}_{\text{L}}/\pi -{D}^{2}}{{l}_{\text{L}}{l}_{\text{D}}}\right)\text{Hg}(\text{Re}),& Staggeredwith{l}_{L}<D\end{array}$$

*D*is the**Tube outer diameter**.*l*_{L}is the**Longitudinal tube pitch (along flow direction)**, the distance between the tube centers along the flow direction.*Flow direction*refers to the thermal liquid flow.*l*_{T}is the**Transverse tube pitch (perpendicular to flow direction)**, shown in the figure below. The transverse pitch is the distance between the centers of the two-phase fluid tubing in one row.*l*_{D}is the diagonal tube spacing, calculated as $${l}_{\text{D}}=\sqrt{{\left(\frac{{l}_{\text{T}}}{2}\right)}^{2}+{l}_{\text{L}}^{2}}.$$

For more information on calculating the Hagen number, see [3].

The longitudinal and transverse pitch distances are the same for both grid bank arrangement types.

**Cross-Section of Two-Phase Fluid Tubing with Pitch Measurements**

When the **Heat transfer coefficient model** is set to
`Colburn equation`

or when **Flow
geometry** is set to `Generic`

, the
Nusselt number is calculated by the empirical the Colburn equation:

$$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c},$$

where *a*, *b*, and
*c* are the values defined in the** Coefficients
[a, b, c] for a*Re^b*Pr^c** parameter.

When the thermal liquid **Flow geometry** is set to
`Flow inside one or more tubes`

, the pressure loss
is calculated in the same manner as for two-phase flows, with the respective
Darcy friction factor, density, mass flow rates, and pipe lengths of the thermal
liquid side. See Pressure Loss for more
information.

When the thermal liquid **Flow geometry** is set to
`Flow perpendicular to bank of circular tubes`

, the
Hagen number is used to calculate the pressure loss due to viscous friction. The
pressure differential between port **A1** and internal node I1 is:

$${p}_{\text{A1}}-{p}_{\text{I1}}=\frac{1}{2}\frac{{\mu}^{2}{N}_{\text{R}}}{\rho {D}^{2}}\text{Hg}(\text{Re}),$$

where:

*μ*_{TL}is the fluid dynamic viscosity.*N*_{R}is the**Number of tube rows along flow direction**. This is the number of two-phase fluid tube rows along the thermal liquid flow direction.

The pressure differential between port **B1**
and internal node I1 is:

$${p}_{\text{B1}}-{p}_{\text{I1}}=\frac{1}{2}\frac{{\mu}^{2}{N}_{\text{R}}}{\rho {D}^{2}}\text{Hg}(\text{Re}).$$

When the **Pressure loss model** is set to ```
Euler
number per tube row
```

or when **Flow geometry**
is set to `Generic`

, the pressure loss due to viscous
friction is calculated with a pressure loss coefficient, in terms of the Euler
number, *Eu*:

$$\text{Eu}=\frac{\xi}{{N}_{R}},$$

where *ξ* is the empirical pressure loss
coefficient.

The pressure differential between port **A1** and internal
node I1 is:

$${p}_{\text{A1}}-{p}_{\text{I1}}=\frac{1}{2}{N}_{R}Eu\frac{{\dot{m}}_{\text{A1}}\left|{\dot{m}}_{\text{A1}}\right|}{2\rho {A}_{\text{CS}}^{2}}.$$

The pressure differential between port
**B1** and internal node I1 is:

$${p}_{\text{B1}}-{p}_{\text{I1}}=\frac{1}{2}{N}_{R}Eu\frac{{\dot{m}}_{\text{B1}}\left|{\dot{m}}_{\text{B1}}\right|}{2\rho {A}_{\text{CS}}^{2}}.$$

The total mass accumulation rate in the two-phase fluid is defined as:

$$\frac{d{M}_{\text{2P}}}{dt}={\dot{m}}_{\text{A2}}+{\dot{m}}_{\text{B2}},$$

where:

*M*is the total mass of the two-phase fluid._{2P}$$\dot{m}$$

_{A2}is the mass flow rate of the fluid at port**A2**.$$\dot{m}$$

_{B2}is the mass flow rate of the fluid at port**B2**.

The flow is positive when flowing into the block through the port.

The energy conservation equation relates the change in specific internal energy to the heat transfer by the fluid:

$${M}_{2P}\frac{d{u}_{2P}}{dt}+{u}_{2P}\left({\dot{m}}_{A2}+{\dot{m}}_{B2}\right)={\varphi}_{\text{A2}}+{\varphi}_{\text{B2}}-Q,$$

where:

*u*_{2P}is the two-phase fluid specific internal energy.*φ*_{A2}is the energy flow rate at port**A2**.*φ*_{B2}is the energy flow rate at port**B2**.*Q*is heat transfer rate, which is positive when leaving the two-phase fluid volume.

The total mass accumulation rate in the thermal liquid is defined as:

$$\frac{d{M}_{\text{TL}}}{dt}={\dot{m}}_{\text{A1}}+{\dot{m}}_{\text{B1}}.$$

The energy conservation equation is:

$${M}_{TL}\frac{d{u}_{TL}}{dt}+{u}_{TL}\left({\dot{m}}_{A1}+{\dot{m}}_{B1}\right)={\varphi}_{\text{A1}}+{\varphi}_{\text{B1}}+Q,$$

where:

*ϕ*_{A1}is the energy flow rate at port**A1**.*ϕ*_{B1}is the energy flow rate at port**B1**.

The heat transferred to or from the thermal liquid,
*Q*, is equal to the heat transferred from or to the
two-phase fluid.

[1] *2013
ASHRAE Handbook - Fundamentals.* American Society of Heating,
Refrigerating and Air-Conditioning Engineers, Inc., 2013.

[2] Çengel, Yunus A. *Heat and Mass Transfer: A Practical Approach*. 3rd ed,
McGraw-Hill, 2007.

[3] Shah, R. K., and Dušan P.
Sekulić. *Fundamentals of Heat Exchanger Design*. John
Wiley & Sons, 2003.

[4] White, Frank M. *Fluid Mechanics*. 6th ed, McGraw-Hill, 2009.

Condenser Evaporator (2P-MA) | E-NTU Heat Transfer | Heat Exchanger (G-TL) | Heat Exchanger (TL-MA) | Heat Exchanger (TL-TL) | Thermostatic Expansion Valve (2P)