# Heat Exchanger (TL-MA)

Models heat exchange between a moist air network and a thermal liquid network

*Since R2020b*

**Libraries:**

Simscape /
Fluids /
Fluid Network Interfaces /
Heat Exchangers

## Description

The Heat Exchanger (TL-MA) block models a heat exchanger with one moist
air network, which flows between ports **A2** and
**B2**, and one thermal liquid network, which flows between ports
**A1** and **B1**. The fluid streams can be
aligned in parallel, counter, or cross-flow configurations.

A thermal liquid-moist air heat exchanger is not appropriate for refrigeration cooling systems. See Condenser Evaporator (2P-MA) or Condenser Evaporator (TL-2P) for heat exchangers that can be employed in refrigeration applications.

You can model the moist air side as flow within tubes, flow around thermal liquid
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 the latent heat that is released when water condenses on the
heat transfer surface. This liquid layer does not collect on the 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 thermal liquid 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 thermal liquid side as flow within tubes, flow around moist air tubing, or by an empirical, generic parameterization.

### Heat Exchanger Configuration

The heat exchanger effectiveness is based on the selected heat exchanger configuration, the fluid properties, 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 thermal liquid 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 Heat Exchanger (TL-MA) blocks in series or in parallel.

For example, to achieve a two-pass configuration on the moist air side and a single-pass configuration on the thermal liquid side, you can connect the moist air 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.

**Flow Geometry**

The **Flow geometry** parameter sets the flow arrangement of
the fluid of the respective dialog tab as either inside a tube or set of tubes,
or perpendicular to a tube bank. You can also specify an empirical, generic
configuration.

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

, use the
**Tube bank grid arrangement** parameter to define the tube
bank alignment of the other fluid as either `Inline`

or
`Staggered`

. The red, downward-pointing arrow in
the figure below indicates the direction of the fluid flowing external to the
tube bank. 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 fluid of the respective dialog tab, and *tube*
refers to the tubing of the other fluid. The **Length of each tube
segment in a tube row** parameter is indicated in the Staggered
figure.

Only one fluid can have **Flow geometry** set to
`Flow perpendicular to bank of circular tubes`

at a
time. The other fluid must be configured to either ```
Flow inside one
or more tubes
```

or `Generic`

. If
**Flow geometry** for both fluids is set to
`Flow perpendicular to bank of circular tubes`

, you
will receive an error.

**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
thermal liquid flows inside the tubes and the moist air flows outside of the
tubes, the **Total fin surface area** and **Fin
efficiency** parameters in the **Thermal Liquid
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 over the averaged properties of both fluids.

The heat transfer is calculated as:

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

where:

*C*_{Min}is the lesser of the heat capacity rates of the two fluids. 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,TL}is the inlet temperature of the thermal liquid.*T*_{In,MA}is the 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:

$$NTU=\frac{1}{{C}_{\text{Min}}R},$$

where *R* is the total thermal resistance between the two flows,
due to convection, conduction, and any fouling on the tube walls:

$$R=\frac{1}{{U}_{\text{TL}}{A}_{\text{Th,TL}}}+\frac{{F}_{\text{TL}}}{{A}_{\text{Th,TL}}}+{R}_{\text{W}}+\frac{{F}_{\text{MA}}}{{A}_{\text{Th,MA}}}+\frac{1}{{U}_{\text{MA}}{A}_{\text{Th,MA}}},$$

and where:

*U*is the convective heat transfer coefficient of the respective fluid. This coefficient is discussed in more detail in Heat Transfer Coefficients.*F*is the**Fouling factor**on the thermal liquid 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**.

**Effectiveness by Flow Arrangement**

The heat exchanger effectiveness varies according to its flow configuration and the mixing in each fluid.

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 & Moist Air 2
unmixed
```

or ```
Thermal Liquid 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 thermal liquid. 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 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.

### Heat Transfer Coefficients

The equations below apply to both the thermal liquid and moist air sides and use the respective fluid properties.

**Flow Inside Tubes**

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

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

where:

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

For turbulent flows, 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 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.

**Flow Across a Tube Bank**

When **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*is the direction of flow of the external fluid.*l*_{T}is the**Transverse tube pitch (perpendicular to flow direction)**, the distance between the centers of the tubing in one row of the other fluid.*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 measurements *l*_{L} and
*l*_{T} are shown in the tube bank
cross-section below. These distances are the same for both grid bank arrangement
types.

**Cross-section of Tubing with Pitch Measurements**

**Empirical Nusselt Number Formulation**

When the **Heat transfer coefficient model** parameter 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 defined in the **Coefficients [a, b, c] for
a*Re^b*Pr^c** parameter.

### Pressure Loss

The equations below apply to both the thermal liquid and moist air sides and use the respective fluid properties.

**Flows Inside Tubes**

The pressure loss due to viscous friction varies depending on flow regime and configuration.

For turbulent flows, when the Reynolds number is above the **Turbulent flow lower
Reynolds number limit** parameter, and when **Pressure loss
model** is set to ```
Correlations for flow inside
tubes
```

, the pressure loss due to friction is calculated in
terms of the Darcy friction factor.

For the thermal liquid side, the pressure differential between a 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 thermal liquid pipe**Internal surface absolute roughness**. The friction factor is dependent on the Reynolds number, and the block calculates this value at both ports for each liquid.*L*is the**Total length of each tube**on the thermal liquid side.*L*_{Add}is the thermal liquid side**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 **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, and when
**Pressure loss model** is set to ```
Correlations
for flow inside tubes
```

, 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, when **Pressure loss model** is set
to `Correlations for flow inside tubes`

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

**Empirical Formulation for Flows Inside Tubes**

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

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

, the pressure losses due to viscous
friction are calculated with an empirical pressure loss coefficient,
*ξ*. The same equations apply to both the moist air and
thermal liquid sides and use the respective fluid properties.

For the thermal liquid side, 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:

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

**Pressure Loss for Flow Across Tube Banks**

When **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 same equations
apply to both the moist air and thermal liquid sides and use the respective
fluid properties.

For the moist air side, 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*is the moist air fluid dynamic viscosity.*N*_{R}is the**Number of tube rows along flow direction**. When moist air is flowing external to a tube bank, this is the number of thermal liquid tube rows along the direction of the moist air flow.

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 Formulation for Flows Across Tubes**

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

**Thermal Liquid**

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

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

where:

*M*is the total mass of the thermal liquid._{TL}$$\dot{m}$$

_{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:

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

where:

*u*_{TL}is the thermal liquid 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 thermal liquid 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 Heat Exchanger
(TL-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
thermal liquid.

## 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 R2020b**