# Condenser Evaporator (2P-MA)

Models heat exchange between a moist air network and a two-phase fluid network

**Libraries:**

Simscape /
Fluids /
Heat Exchangers /
Two-Phase Fluid - Moist Air

## Description

The Condenser Evaporator (2P-MA) block models a heat exchanger with one
moist air network, which flows between ports **A2** and
**B2**, and one two-phase fluid network, which flows between ports
**A1** and **B1**. 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 moist air side as flow within tubes, flow around the two-phase fluid
tubing, or by an empirical, generic parameterization. The moist air side comprises air,
trace gas, and water vapor that may condense throughout the heat exchange cycle. The
block model accounts for energy transfer from the air to the liquid water condensation
layer. This liquid layer does not collect on the heat transfer surface and is assumed to
be completely removed from the downstream moist air flow. The moisture condensation rate
is returned as a physical signal at port **W**.

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 moist air 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].

### Heat Exchanger Configuration

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.

**Flow Arrangement**

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.

**Thermal Mixing**

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

, use the **Cross flow arrangement**
parameter to indicate whether the two-phase fluid or moist air 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 (2P-MA) 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 moist air side, you can connect the two-phase fluid sides in series and the moist air 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.

**Flow Geometry**

The **Flow geometry** parameter sets the moist air 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 moist air flow. Also indicated in the Inline figure
are 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 moist air 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.

**Fins**

The heat exchanger configuration does not have fins when the **Total
fin surface area** parameter is `0 m^2`

. Fins
introduce additional surface area for heat transfer. Each fluid side has a
separate fin area and parameters that describe the fins in that fluid. For
example, when the two-phase fluid flows inside the tubes and the moist air flows
outside of the tubes, the **Total fin surface area** and
**Fin efficiency** parameters in the **Two-Phase
Fluid 1** section describe the fins protruding into the tube. The
**Total fin surface area** and **Fin
efficiency** parameters in the **Moist Air 2**
section describe the fins outside of the tubes.

### Effectiveness-NTU Heat Transfer

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:

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

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,MA}is the zone inlet temperature of the moist air.*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:$$R=\frac{1}{{U}_{\text{2P}}{A}_{\text{Th,2P}}}+\frac{{F}_{\text{2P}}}{{A}_{\text{Th,2P}}}+{R}_{\text{W}}+\frac{{F}_{\text{MA}}}{{A}_{\text{Th,MA}}}+\frac{1}{{U}_{\text{MA}}{A}_{\text{Th,MA}}},$$

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 Moist Air Correlations.*F*is the**Fouling factor**on the two-phase fluid or moist air 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*), and superheated vapor (

_{M}*Q*):

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

**Effectiveness by Flow Arrangement**

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 ```
Two-Phase Fluid 1 mixed & Moist Air 2
unmixed
```

or ```
Two-Phase Fluid 1 unmixed
& Moist Air 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:

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

### Condensation

On the moist air side, a layer of condensation may form on the heat transfer
surface. This liquid layer can influence the amount of heat transferred between the
moist air and two-phase fluid. The equations for E-NTU heat transfer above are given
for *dry* heat transfer. To correct for the influence of
condensation, the E-NTU equations are additionally calculated with the wet
parameters listed below. Whichever of the two calculated heat flow rates results in
a larger amount of moist air side cooling is used in heat calculations for each zone
[1]. To use this method, the Lewis number is assumed to be close to 1 [1], which is
true for moist air.

**E-NTU Quantities Used for Heat Transfer Rate Calculations**

Dry calculation | Wet calculation | |
---|---|---|

Moist air zone inlet temperature | T_{in,MA} | T_{in,wb,MA} |

Heat capacity rate | $${\overline{\dot{m}}}_{MA}{\overline{c}}_{p,MA}$$ | $${\overline{\dot{m}}}_{MA}{\overline{c}}_{eq,MA}$$ |

Heat transfer coefficient | U_{MA} | $${U}_{MA}\frac{{\overline{c}}_{eq,MA}}{{\overline{c}}_{p,MA}}$$ |

where:

*T*is the moist air zone inlet temperature._{in,MA}*T*is the moist air wet-bulb temperature associated with_{in,wb,MA}*T*._{in,MA}$${\overline{\dot{m}}}_{MA}$$ is the dry air mass flow rate.

$${\overline{c}}_{p,MA}$$ is the moist air heat capacity per unit mass of dry air.

$${\overline{c}}_{eq,MA}$$ is the equivalent heat capacity. The

*equivalent heat capacity*is the change in the moist air specific enthalpy (per unit of dry air), $${\overline{h}}_{MA}$$, with respect to temperature at saturated moist air conditions:$${\overline{c}}_{eq,MA}={\left(\frac{\partial {\overline{h}}_{MA}}{\partial {T}_{MA}}\right)}_{s}.$$

The mass flow rate of the condensed water vapor leaving the moist air mass flow depends on the relative humidity between the moist air inlet and the channel wall and the heat exchanger NTUs:

$${\dot{m}}_{cond}=-{\overline{\dot{m}}}_{MA}\left({W}_{wall,MA}-{W}_{in,MA}\right)\left(1-{e}^{-NT{U}_{MA}}\right),$$

where:

*W*_{wall,MA}is the humidity ratio at the heat transfer surface.*W*_{in,MA}is the humidity ratio at the moist air flow inlet.*NTU*_{MA}is the number of transfer units on the moist air side, calculated as:$$NT{U}_{MA}=\frac{{U}_{MA}\frac{{\overline{c}}_{eq,MA}}{{\overline{c}}_{p,MA}}{A}_{Th,MA}}{{\overline{\dot{m}}}_{MA}{\overline{c}}_{eq,MA}}.$$

The energy flow associated with water vapor condensation is based on
the difference between the vapor specific enthalpy,
*h*_{water, wall}, and the specific
enthalpy of vaporization, *h*_{fg}, for water:

$${\varphi}_{Cond}={\dot{m}}_{cond}\left({h}_{water,wall}-{h}_{fg}\right).$$

The condensate is assumed to not accumulate on the heat transfer surface, and does not influence geometric parameters such as tube diameter. The condensed water is assumed to be completely removed from the downstream moist air flow.

### Two-Phase Fluid Correlations

**Heat Transfer Coefficient**

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.

**Empirical Nusselt Number Formulation**

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

, the block calculates the
Nusselt number for the subcooled liquid and superheated vapor zones by using the
empirical Colburn equation

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

where *a*, *b*, and
*c* are values 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 block calculates the Nusselt number for liquid-vapor mixture zones by
using the Cavallini-Zecchin equation with the variables in the
**Coefficients [a, b, c] for a*Re^b*Pr^c in mixture zone**
parameter.

**Pressure Loss**

The pressure loss due to viscous friction varies depending on flow regime and
configuration. The pressure loss calculation uses the overall density, which is
the total fluid mass divided by the total fluid volume. When the
**Pressure loss model** parameter is
`Correlation for flow inside tubes`

, the pressure
loss due to friction depends on the Darcy friction factor.

For turbulent flows, when the Reynolds number is above the **Turbulent
flow lower Reynolds number limit** parameter, the block calculates
the pressure loss due to friction in terms of a calculated Darcy friction
factor. The pressure differential between port **A1** and the
internal node I1 is:

$${p}_{\text{A1}}-{p}_{\text{I1}}=\frac{{f}_{\text{D,A}}{\dot{m}}_{\text{A1}}\left|{\dot{m}}_{\text{A1}}\right|}{2\rho {D}_{\text{H}}{A}_{\text{CS}}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$$

where:

$$\dot{m}$$

_{A1}is the total flow rate through port**A1**.*f*is the Darcy friction factor according to the Haaland correlation. When the_{D,A}**Local resistance specification**parameter is`Aggregate equivalent length`

,$${f}_{\text{D,A1}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{\text{A1}}}+{\left(\frac{{\u03f5}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}},$$

where

*ε*_{R}is the value of the**Internal surface absolute roughness**parameter. The friction factor is dependent on the Reynolds number, and the block calculates this value at both ports for each liquid.*L*is the value of the**Total length of each tube**parameter.*L*_{Add}is the value of the**Aggregate equivalent length of local resistances**parameter, 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*is the tube cross-sectional area._{CS}

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

$${p}_{\text{B1}}-{p}_{\text{I1}}=\frac{{f}_{\text{D,B}}{\dot{m}}_{\text{B1}}\left|{\dot{m}}_{\text{B1}}\right|}{2\rho {D}_{\text{H}}{A}_{\text{CS}}^{2}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$$

where $$\dot{m}$$_{B1} is the total flow rate through port
**B1**.

When the **Local resistance specification** parameter is
`Aggregate equivalent length`

, the Darcy friction
factor at port **B1** is:

$${f}_{\text{D,B1}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{\text{B1}}}+{\left(\frac{{\u03f5}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}}.$$

When the **Local resistance specification** parameter is
`Local loss coefficient`

, the Darcy friction
factors at ports **A1** and **B1** are

$$\begin{array}{l}{f}_{\text{D,A1}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{A1}}+{\left(\frac{{\u03f5}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}}+{C}_{localloss}\left(\frac{{D}_{H}}{L}\right)\\ {f}_{\text{D,B1}}={\left\{-1.8{\text{log}}_{\text{10}}\left[\frac{6.9}{{\text{Re}}_{B1}}+{\left(\frac{{\u03f5}_{\text{R}}}{3.7{D}_{\text{H}}}\right)}^{1.11}\right]\right\}}^{\text{-2}}+{C}_{localloss}\left(\frac{{D}_{H}}{L}\right)\end{array}$$

where *C _{localloss}*
is the value of the

**Total local loss coefficient**parameter.

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

. Otherwise, the block calculates the
value internally. The pressure differential between port **A1**
and internal node I1 is:

$${p}_{\text{A1}}-{p}_{\text{I1}}=\frac{\lambda \mu {\dot{m}}_{\text{A1}}}{2\rho {D}_{\text{H}}^{2}{A}_{CS}}\left(\frac{L+{L}_{\text{Add}}}{2}\right),$$

where *μ* is the fluid dynamic viscosity.
The pressure differential between port **B1** and internal node
I1 is:

$${p}_{\text{B1}}-{p}_{\text{I1}}=\frac{\lambda \mu {\dot{m}}_{\text{B1}}}{2\rho {D}_{\text{H}}^{2}{A}_{CS}}\left(\frac{L+{L}_{\text{Add}}}{2}\right).$$

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

**Empirical Pressure Loss Formulation**

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 **A1** and internal
node I1 is:

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

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

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

### Moist Air Correlations

**Heat Transfer Coefficient for Flows Inside One or More Tubes**

When the moist air **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 supercooled liquid or superheated vapor. See Heat Transfer Coefficient for more
information.

**Heat Transfer Coefficient for Flows Across a Tube Bank**

When the moist air **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 moist air 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 [6].

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**

**Empirical Nusselt Number Formulation**

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 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.

**Pressure Loss for Flow Inside Tubes**

When the moist air **Flow geometry** is set to
`Flow inside one or more tubes`

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

**Pressure Loss for Flow Across Tube Banks**

When the moist air **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 **A2** and internal node I2 is:

$${p}_{\text{A2}}-{p}_{\text{I2}}=\frac{1}{2}\frac{{\mu}^{2}{N}_{\text{R}}}{\rho {D}^{2}}\text{Hg}(\text{Re}),$$

where:

*μ*_{MA}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 moist air flow direction.

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

$${p}_{\text{B2}}-{p}_{\text{I2}}=\frac{1}{2}\frac{{\mu}^{2}{N}_{\text{R}}}{\rho {D}^{2}}\text{Hg}(\text{Re}).$$

**Empirical Pressure Loss Formulation**

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 **A2** and internal
node I2 is:

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

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

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

### Conservation Equations

**Two-Phase Fluid**

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

$$\frac{d{M}_{\text{2P}}}{dt}={\dot{m}}_{\text{A1}}+{\dot{m}}_{\text{B1}},$$

where:

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

_{A1}is the mass flow rate of the fluid at port**A1**.$$\dot{m}$$

_{B1}is the mass flow rate of the fluid at port**B1**.

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:

$${M}_{2P}\frac{d{u}_{2P}}{dt}+{u}_{2P}\left({\dot{m}}_{A1}+{\dot{m}}_{B1}\right)={\varphi}_{\text{A1}}+{\varphi}_{\text{B1}}-Q,$$

where:

*u*_{2P}is the two-phase fluid specific internal energy.*φ*_{A1}is the energy flow rate at port**A1**.*φ*_{B1}is the energy flow rate at port**B1**.*Q*is heat transfer rate, which is positive when leaving the two-phase fluid volume.

**Moist Air**

There are three equations for mass conservation on the moist air side: one for the moist air mixture, one for condensed water vapor, and one for the trace gas.

**Note**

If **Trace gas model** is set to
`None`

in the Moist Air
Properties (MA) block, the trace gas is not modeled
in blocks in the moist air network. In the Condenser
Evaporator (2P-MA) block, this means that the
conservation equation for trace gas is set to 0.

The moist air mixture mass accumulation rate accounts for the changes of the entire moist air mass flow through the exchanger ports and the condensation mass flow rate:

$$\frac{d{M}_{\text{MA}}}{dt}={\dot{m}}_{\text{A2}}+{\dot{m}}_{\text{B2}}-{\dot{m}}_{\text{Cond}}.$$

The mass conservation equation for water vapor accounts for the water vapor transit through the moist air side and condensation formation:

$$\frac{d{x}_{w}}{dt}{M}_{\text{MA}}+{x}_{\text{w}}\left({\dot{m}}_{\text{A2}}+{\dot{m}}_{\text{B2}}-{\dot{m}}_{\text{Cond}}\right)={\dot{m}}_{\text{w,A2}}+{\dot{m}}_{\text{w,B2}}-{\dot{m}}_{\text{Cond}},$$

where:

*x*_{w}is the mass fraction of the vapor. $$\frac{d{x}_{w}}{dt}$$ is the rate of change of this fraction.$${\dot{m}}_{\text{w,A2}}$$ is the water vapor mass flow rate at port

**A2**.$${\dot{m}}_{\text{w,B2}}$$ is the water vapor mass flow rate at port

**B2**.$${\dot{m}}_{Cond}$$ is the rate of condensation.

The trace gas mass balance is:

$$\frac{d{x}_{\text{g}}}{dt}{M}_{\text{MA}}+{x}_{\text{g}}\left({\dot{m}}_{\text{A2}}+{\dot{m}}_{\text{B2}}-{\dot{m}}_{\text{Cond}}\right)={\dot{m}}_{\text{g,A2}}+{\dot{m}}_{\text{g,B2}},$$

where:

*x*_{g}is the mass fraction of the trace gas. $$\frac{d{x}_{g}}{dt}$$ is the rate of change of this fraction.$${\dot{m}}_{\text{g,A2}}$$ is the trace gas mass flow rate at port

**A2**.$${\dot{m}}_{\text{g,B2}}$$ is the trace gas mass flow rate at port

**B2**.

Energy conservation on the moist air side accounts for the change in specific internal energy due to heat transfer and water vapor condensing out of the moist air mass:

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

where:

*ϕ*_{A2}is the energy flow rate at port**A2**.*ϕ*_{B2}is the energy flow rate at port**B2**.*ϕ*_{Cond}is the energy flow rate due to condensation.

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

## Ports

### Conserving

### Output

## Parameters

## References

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

[2] Braun, J. E., S. A. Klein, and
J. W. Mitchell. "Effectiveness Models for Cooling Towers and Cooling Coils."
*ASHRAE Transactions* 95, no. 2, (June 1989):
164–174.

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

[4] Ding, X., Eppe J.P., Lebrun,
J., Wasacz, M. "Cooling Coil Model to be Used in Transient and/or Wet Regimes.
Theoretical Analysis and Experimental Validation." *Proceedings of the Third
International Conference on System Simulation in Buildings* (1990):
405-411.

[5] Mitchell, John W., and James
E. Braun. *Principles of Heating, Ventilation, and Air
Conditioning in Buildings*. Wiley, 2013.

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

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

## Extended Capabilities

## Version History

**Introduced in R2019a**