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Gas side of heat exchanger

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

The Heat Exchanger Interface (G) block models thermal transfer by a gaseous flow within a heat exchanger. Use a second heat exchanger block to model the fluid pair. The interfaces can be in different fluid domains, such as one in liquid and one in gas. Use an E-NTU Heat Transfer block to couple the interfaces and capture the heat exchange between the fluids.

The fixed-volume construction of the block allows you to capture variations in fluid mass flow rates due to compressibility. The overall mass accumulation rate is equal to the sum of the mass flow rates through the ports:

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

where $$\dot{M}$$ is the mass accumulation rate and $$\dot{m}$$ is the mass flow rate. The subscripts denote ports
**A** and **B**. The mass flow rate is
positive when it is directed into the gas channel. Variations in density are
reflected in the mass accumulation rate:

$$\dot{M}=\left[{\left(\frac{\partial \rho}{\partial p}\right)}_{u}\frac{dp}{dt}+{\left(\frac{\partial \rho}{\partial u}\right)}_{p}\frac{du}{dt}\right]V,$$

where:

*ρ*is density.*p*is pressure.*u*is specific internal energy.*V*is volume.

Balancing momentum between the inlet and outlet ports of the heat exchanger dictates the flow direction and speed within the exchanger. Changes in momentum are due primarily due to friction losses from pipe turns, which translate to changes in pressure. Local resistances, such as bends, elbows, and tees can result in flow separation that leads to minor additional pressure losses. For steady flows, the mass flow rate remains constant.

A momentum balance is applied to each segment of the gas (pipe) volume. This
figure shows a tube bank divided into two volumes and three nodes. The nodes
correspond to ports **A** , **B**, and the fluid
volume, `I`

. Fluid states, such as pressure and temperature, and
fluid properties, such as density and viscosity, are defined at these nodes.

Note that flow inertia is negligible and the flow is considered to be
quasi-steady state. The translation of transients to mass flow rates can be
offset: due to coupling between density, pressure, and temperature,
propagation of changes throughout the system is not instantaneous. Other
sources and sinks of momentum, such as differences in head between ports or
radial deformations of the channel wall, are not considered. The momentum
balance for the half volume at port **A** is:

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

where *p* is the pressure at the node
indicated in the subscript. *Δp*_{f,A}
is the total pressure loss between the port node and the internal node due
to friction. The total pressure loss includes both major and minor losses.
For the half volume at port **B**, the momentum balance is:

$${p}_{\text{B}}-{p}_{\text{I}}=\Delta {p}_{\text{f,B}}.$$

Pressure changes due to friction vary with the square of the mass flow rate for
turbulent flows and with the magnitude of the mass flow rate for laminar flows. This
pressure change is characterized by three dimensionless parameters: the Darcy
friction factor, the pressure loss coefficient, and the Euler number. These numbers
are calculated from empirical correlations or are estimated from lookup tables,
depending on the **Pressure loss parameterization**
parameter.

Classification of "laminar" or "turbulent" flow is based on the Reynolds number.
When the Reynolds number is above the **Turbulent flow lower Reynolds number
limit** parameter, the flow is fully turbulent. Below the
**Laminar flow upper Reynolds number limit** parameter, the
flow is fully laminar. Reynolds numbers in between these values indicate
transitional flow. Transitional flows show characteristics of both laminar and
turbulent flows. In the Simscape™
Fluids™ language, numerical blending is applied between these bounding
values.

`Correlations for tubes`

For tubes, the Darcy friction factor,
*f*_{D}, is used. In the half volume at
port **A**, the momentum balance is:

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

where *L* is the tube length and
*L*_{Add} is the added tube length that
would reproduce the minor viscous losses if used in place of elbows, tees,
unions, or other local resistances. *A* is the tube
cross-sectional area; in the event of a non-uniform cross-sectional area,
*A*_{min} should be used.
*D*_{H} is the tube hydraulic diameter,
or the diameter of a circle equal in area to the tube cross section:

$${D}_{\text{H}}=\sqrt{\frac{4{A}_{\text{Min}}}{\pi}}.$$

If the tube has a circular cross section, the hydraulic diameter and the tube diameter are the same.

For the half volume at port **B**, the momentum balance is:

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

For turbulent flows, the Darcy friction factor is calculated with the Haaland correlation. The Reynolds number is established at the bounding port:

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

where *ε*_{R} is wall
roughness, taken as a characteristic height. This parameter is specified in the
**Internal surface absolute roughness** parameter.

For laminar flows, the friction factor depends on the tube shape and is calculated with the tube shape factor:

$${f}_{\text{D}}=\frac{\lambda}{\text{Re}},$$

where λ is the shape factor. The Reynolds number is calculated at the bounding port as:

$$\text{Re}=\frac{{D}_{\text{H}}\dot{m}}{\mu {A}_{\text{Min}}}.$$

Substituting Re into the pressure loss equation at port
**A**, the momentum balance is reformulated as:

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

Likewise, for the half volume at port **B**, the momentum
balance is:

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

For channels other than tubes, use the pressure loss coefficient,
*ξ*. For turbulent flows in the half volume at port
**A**, the momentum balance is:

$${p}_{\text{A}}-{p}_{\text{I}}=\frac{1}{2}\xi \frac{{\dot{m}}_{\text{A}}\left|{\dot{m}}_{\text{A}}\right|}{2{\rho}_{\text{A}}{A}_{\text{Min}}^{2}},$$

For turbulent flows in the half volume at port
**B**, the momentum balance is:

$${p}_{\text{B}}-{p}_{\text{I}}=\frac{1}{2}\xi \frac{{\dot{m}}_{\text{B}}\left|{\dot{m}}_{\text{B}}\right|}{2{\rho}_{\text{B}}{A}_{\text{Min}}^{2}},$$

For laminar flows in the half volume at port **A**, the
momentum balance is:

$${p}_{\text{A}}-{p}_{\text{I}}=\frac{1}{2}\xi {\text{Re}}_{\text{L}}\frac{{\dot{m}}_{\text{A}}{\mu}_{\text{A}}}{2{D}_{\text{H}}{\rho}_{\text{A}}{A}_{\text{Min}}},$$

where Re_{L} is the **Laminar
flow upper Reynolds number limit** block parameter. For laminar
flows in the half volume at port **B**, the momentum balance is:

$${p}_{\text{B}}-{p}_{\text{I}}=\frac{1}{2}\xi {\text{Re}}_{\text{L}}\frac{{\dot{m}}_{\text{B}}{\mu}_{\text{B}}}{2{D}_{\text{H}}{\rho}_{\text{B}}{A}_{\text{Min}}}$$

```
Tabulated data - Darcy friction factor vs. Reynolds
number
```

You can use tabulated data to determine the Darcy friction factor based on
the Reynolds number for tube flows. For the half volume at port
**A**, the momentum balance is:

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

For the half volume at port **B**, the momentum balance is:

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

For the turbulent regime, the friction factor is determined from a tabulated function of the Reynolds number:

$${f}_{\text{D}}={f}_{\text{D}}(\text{Re}).$$

The breakpoints of the tabulated function derive from the vector block
parameters. The **Reynolds number vector for Darcy friction
factor** parameter specifies the independent variable and the
**Darcy friction factor vector** parameter specifies the
dependent variable. Linear interpolation is applied between breakpoints. Outside
of the tabulated data range, the nearest breakpoint determines the friction
factor.

In the laminar regime, the friction factor is calculated from the shape factor, λ:

$${f}_{\text{D}}=\frac{\lambda}{\text{Re}}.$$

```
Tabulated data - Euler number vs. Reynolds
number
```

You can use tabulated data to determine the Euler number based on the Reynolds number. This calculation is dependent on flow regime, and the Euler number is formulated as a tabulated function of the Reynolds number:

$$\text{Eu}=\text{Eu}(\text{Re}).$$

The breakpoints in ```
Tabulated data - Euler number vs. Reynolds
number
```

are specified by Reynolds number and Euler number
vectors. The **Reynolds number vector for Euler number**
parameter specifies the independent variables, the Reynolds numbers, and the
**Euler number vector** parameter specifies the dependent
variable, the Euler number, at each Reynolds number. Linear interpolation is
used to determine values between breakpoints. Outside of the tabulated data
range, the value at the nearest breakpoint is used.

For turbulent flows, the momentum balance for the half volume at port
**A** is:

$${p}_{\text{A}}-{p}_{\text{I}}={\text{Eu}}_{\text{A}}\frac{{\dot{m}}_{\text{A}}\left|{\dot{m}}_{\text{A}}\right|}{{\rho}_{\text{A}}{A}_{\text{Min}}^{2}},$$

where Eu is the Euler number at port **A**.
For turbulent flows, the momentum balance for the half volume at port
**B** is:

$${p}_{\text{B}}-{p}_{\text{I}}={\text{Eu}}_{\text{A}}\frac{{\dot{m}}_{\text{B}}\left|{\dot{m}}_{\text{B}}\right|}{{\rho}_{\text{B}}{A}_{\text{Min}}^{2}}.$$

For laminar flow in the half volume at port **A**, the
momentum balance is:

$${p}_{\text{A}}-{p}_{\text{I}}={\text{Eu}}_{\text{L}}{\text{Re}}_{\text{L}}\frac{{\dot{m}}_{\text{A}}{\mu}_{\text{A}}}{4{D}_{\text{H}}{\rho}_{\text{A}}{A}_{\text{Min}}},$$

where Re_{L} is the **Laminar flow upper Reynolds
number limit** parameter and Eu_{L} is the
Euler number evaluated from tabulated data at that Reynolds number. For laminar
flow in the half volume at port **B**, the momentum balance is:

$${p}_{\text{B}}-{p}_{\text{I}}={\text{Eu}}_{\text{L}}{\text{Re}}_{\text{L}}\frac{{\dot{m}}_{\text{B}}{\mu}_{\text{B}}}{4{D}_{\text{H}}{\rho}_{\text{B}}{A}_{\text{Min}}},$$

The energy balance within the gas volume is the sum of its flow rates across channel boundaries and the associated heat transfer. Energy can be transferred by advection at the ports and by convection at the wall. While conduction contributes to the energy balance at the ports, it is often negligible in comparison to advection. However, conduction is non-negligible in near-stationary fluids, such as when fluids are stagnant or changing direction. The energy balance equation is:

$${\dot{E}}_{\text{2P}}={\varphi}_{\text{A}}+{\varphi}_{\text{B}}+Q,$$

where:

$$\frac{\partial U}{\partial p}$$ is the partial derivative of the internal energy of the gas volume with respect to pressure at constant temperature and volume.

*p*_{I}is the pressure of the gas volume.$$\frac{\partial U}{\partial T}$$ is the partial derivative of the internal energy of the gas volume with respect to temperature at constant pressure and volume.

*T*_{I}is the temperature of the gas volume.*Φ*_{A}and*Φ*_{B}are energy flow rates at ports**A**and**B**, respectively.*Q*is the heat transfer rate.

Advection and conduction are accounted for in *Φ*, and convection
is accounted for in *Q*. The heat transfer rate is positive when
directed into the gas volume.

Heat transfer between the two heat exchanger fluids occurs in multiple ways: through convection at the fluid interfaces, conduction through layers of built-up fouling, and conduction through the thickness of the wall.

Heat transfer extends beyond the gas channel and therefore requires other blocks to model the entire heat exchanger system. A second heat exchanger interface block models the second flow channel while an E-NTU Heat Transfer block models the heat flow across the wall. Heat transfer parameters that are specific to the gas channel, but required by the E-NTU Heat Transfer block, are available through the physical signal ports:

Port

**C**outputs the heat capacity rate, which is a measure of the gas' ability to absorb heat and is required for calculating the number of heat transfer units (NTU). The heat capacity rate is calculated as:$${C}_{\text{R}}={c}_{\text{p}}\dot{m},$$

where

*C*_{R}is the heat capacity rate and*c*_{p}is the specific heat.Port

**HC**outputs the heat transfer coefficient,*U*.If the heat transfer coefficient is treated as a constant, its value is uniform across the flow channel. If the heat transfer coefficient is variable, it is calculated at each port from the expression:

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

where Nu is Nusselt number,

*k*is thermal conductivity, and*D*_{H,Q}is a hydraulic diameter for heat transfer. The hydraulic diameter*D*_{H,Q}is calculated as:$${D}_{\text{H,Q}}=\frac{4{A}_{\text{Min}}{L}_{\text{Q}}}{{S}_{\text{Q}}},$$

where

*S*_{Q}is the**Heat transfer surface area**parameter and*L*_{Q}is the**Length of flow path for heat transfer**parameter.The lower bound of the mean heat transfer coefficient is the

**Minimum gas-wall heat transfer coefficient**parameter.

The Nusselt number is derived from empirical correlations with the Reynolds and
Prandtl numbers. Use the **Heat transfer parametrization**
parameter to select the most appropriate formulation for your simulation.

The simplest parameterization, ```
Constant heat transfer
coefficient
```

, obtains the heat transfer coefficient directly from
the value of the **Gas-wall heat transfer coefficient ** parameter.
`Correlations for tubes`

uses analytical expressions
with constant or calculated parameters to capture Nusselt number dependence on the
flow regime for tube flows.

The remaining parameterizations are tabulated functions of the Reynolds number. These are useful for varying Nusselt numbers, or heat transfer coefficients, across flow regimes. The functions are generated from experimental data relating the Reynolds number to the Colburn factor or the Reynolds and Prandtl numbers to the Nusselt number.

`Constant heat transfer coefficient`

Using `Constant heat transfer coefficient`

, specified
in the **Gas-wall heat transfer coefficient** parameter, sets
the heat transfer coefficient as a constant, and does not use the Nusselt number
in calculations. Use this parameterization as a simple approximation for gas
flows confined to the laminar regime.

`Correlation for tubes`

The Nusselt number depends on flow regime when using ```
Correlation
for tubes
```

. For turbulent flows, its value changes in
proportion to the Reynolds number and is calculated from the Gnielinski correlation:

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

where Re is the Reynolds number, Nu is the Nusselt number,
and Pr is the Prandtl number. The friction factor, *f*, is the
same as the factor used in tube pressure loss calculations. For laminar flows,
the Nusselt number is a constant. Its value is obtained from the
**Nusselt number for laminar flow heat transfer**
parameter, Nu_{L}:

$$\text{Nu}={\text{Nu}}_{\text{L}}.$$

```
Tabulated data - Colburn factor vs. Reynolds
number
```

You can use tabulated data to determine the Colburn factor based on the
Reynolds number. The Colburn equation is used to determine the Nusselt number,
which varies in proportion to the Reynolds number. The Colburn
*j*-factor is a measure of proportionality between the
Reynolds, Prandtl, and Nusselt numbers:

$$\text{Nu}=j({\text{Re}}_{\text{Q}}){\text{Re}}_{\text{Q}}{\text{Pr}}^{1/3}.$$

Re_{Q} is the Reynolds number based on the hydraulic
diameter for heat transfer, *D*_{H, Q}, and
from the minimum free-flow area of the channel,
*A*_{Min}:

$${\text{Re}}_{\text{Q}}=\frac{\dot{m}{D}_{\text{H,Q}}}{{A}_{\text{Min}}\mu}$$

```
Tabulated data - Nusselt number vs. Reynolds number &
Prandtl number
```

You can use a tabulated function to determine the Nusselt number from the
Prandtl and Reynolds numbers. Linear interpolation is used to determine values
between breakpoints. The Nusselt number is a function of both Re and Pr, and
therefore the **Reynolds number vector for Nusselt number**,
**Prandtl number vector for Nusselt number**, and
**Nusselt number table, Nu(Re,Pr)** parameters define the
table breakpoints:

$$\text{Nu}=\text{Nu}({\text{Re}}_{\text{Q}},\text{Pr}).$$

The tabulated Reynolds number must be calculated using the hydraulic diameter
for heat transfer, *D*_{H,Q}.