Heat exchanger for systems with fluids susceptible to phase change

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

The Condenser Evaporator (2P-MA) block models a heat exchanger with
working fluid susceptible to phase change. The fluid, from the two-phase fluid domain,
enhances heat transfer by storing and releasing heat in both its sensible and latent
forms. In practice, it is often a refrigerant such as R-134a, with large heat of
vaporization to better absorb heat, and with boiling temperature near the desired
operating point. A gaseous mixture from the moist air domain is the common target of the
heat exchange. The two-phase fluid runs between ports **A1** and
**B1** and the moist air between ports **A2** and
**B2**.

The heat exchanger can be a condenser or an evaporator. The type modeled depends on
the phase change triggered—condensation or vaporization—which in turn varies with
position in the cooling or heating cycle. In a refrigerator, for example, the condenser
(**c** in the figure) sits past a compressor (**d**), where it cools pressurized refrigerant to liquid before
delivering it to an evaporator (**a**). The evaporator sits
past an expander (**b**), where it heats depressurized
refrigerant to vapor before returning it to the compressor for another cooling
cycle.

The two-phase fluid can change phase while inside—from subcooled liquid to
liquid-vapor mixture to superheated vapor, or from superheated vapor to liquid-vapor
mixture to subcooled liquid. The transitions create zones (**a**, **b**, and **c** in the figure) with distinct fluid properties and therefore also heat
transfer coefficients. The zones can shrink and grow, one at the expense of the others,
to occupy length fractions ranging from `0`

, if absent, to
`1`

, if present alone. Use port **Z**
to measure the zone length fractions, formatted as a vector, during simulation.

The moist air remains a gas mixture throughout. Its moisture contents, however, are
susceptible to condensation, normally as a film while circulating evaporator coils. The
film condensate forms on the coldest section of the wall, where humidity first reaches
saturation. Condensation consumes water vapor, creating a humidity gradient that drives
in more of the species for sustained condensation. The film drains from the flow and the
moist air, having lost mass to the liquid water and energy to latent heat, exits cooler
and drier. Use port **W** to measure the moisture
condensation rate.

Heat transfer occurs through a shared wall. The transfer is partly by convection, where fluid meets wall, and partly by conduction, in the thickness of the wall. Fouling can line the fluid boundaries, increasing thermal resistance and reducing heat transfer. The heat transfer rate is calculated using the Effectiveness-NTU, or E-NTU, method—NTU being the number of (heat) transfer units, a common measure of thermal size in heat exchangers. The E-NTU method obtains the heat transfer rate from the heat exchanger effectiveness, and the heat exchanger effectiveness from the number of transfer units.

The heat exchanger effectiveness depends not only on the states and properties of the fluids, but also on the disposition of the flows, the geometries of their channels, the mixing conditions in them, and the fins that in some heat exchangers serve to expand the thermal contact area.

The **Flow arrangement** parameter indicates how the flows
align. The flows can run in the same direction, in opposite directions, or in
perpendicular directions. These flow arrangements are available as
`Parallel flow`

, ```
Counter
flow
```

, and `Cross flow`

options. All
flow arrangements are single pass: the flows are assumed to meet once and run
lengthwise toward their respective outlets, without making the U turns
characteristic of shell-and-tube arrangements.

Note that the directions of the flows depend on the pressure gradients established during simulation between the ports. For accuracy, select a flow arrangement consistent with the expected flow conditions in the model.

The **Flow geometry** parameter indicates the type of channel
that is to carry the moist air. The moist air can flow inside a tube or tube
bundle, outside a tube or tube bundle (or *bank* in this
context), or through a channel of unspecified geometry. Unspecified geometries
use a generic parameterization, one less detailed in its calculations but better
suited for unconventional flow channels not otherwise captured in the block. As
the working fluid, two-phase fluid always flows inside a tube or tube
bundle.

Tubes can be circular, rectangular, or annular in cross section (**i**, **ii**, and **iii** in the figure). Circular tubes are parameterized
by diameter (*d*), rectangular tubes by width and height
(*w* and *h*), and annular tubes by inner
and outer diameters (*d*_{I} and
*d*_{O}). Tubes can also take less
conventional shapes, in which case the tube model is based on a (second) generic
parameterization. Use the **Tube cross section** parameter to
select a shape for tubes (active on the moist air side when configured with flow
inside tubes).

Tube bundles can be inline (**i** in the figure),
with tubes in one row located behind the tubes of the next row, or staggered
(**ii**), with tubes in one row located behind
the gaps of the next row. Inline and staggered bundles are characterized by the
same *longitudinal pitch*
(*l*_{L}) and *transverse
pitch* (*l*_{T}). The
longitudinal pitch is the distance between neighboring tube rows. The transverse
pitch is the distance between neighboring tubes within a row.

The flows can each be mixed (**i** in the figure)
or unmixed (**ii**). Mixing in this context is the
lateral movement of fluid in channels that have no internal barriers, normally
guides, baffles, fins, or walls. Such movement serves to even out temperature
variations in the transverse plane. Mixed flows have variable temperature in the
longitudinal direction alone. Unmixed flows have variable temperature in both
the transverse and longitudinal planes.

Flow in tube bundles is divided into branches and, unless the bundle is reduced to a single tube, is always unmixed. Flow around tube banks can be divided into similar branches—by wide plate fins, say—and it too can be unmixed. If no fins are used, or if the fins protrude only slightly, flow around tube banks is mixed. (Tube bundles and tube banks are both tube matrices. The difference in terms serves only to distinguish internal from external flow.)

The distinction between mixed and unmixed flows matters only in cross flow
arrangements. There, longitudinal temperature variation in one fluid produces
transverse temperature variation in the second fluid that mixing can even out.
In parallel and counter flow arrangements, longitudinal temperature variation in
one fluid produces longitudinal temperature variation in the second fluid.
Mixing, as it is of little effect here, is ignored. Use the **Cross flow
arrangement** parameter to mix each of the flows, one of the flows,
or none of the flows.

The flow channels can be plain or finned. Fins extend the heat transfer
surface, increasing the rate of heat transfer across it. They can protrude from
either side, but the moist air channel, being the typical bottleneck in heat
transfer, features them most often. The heat transfer boost due to the fins
depends on their total surface area—the sum over all the fins in the flow
channel—and on their efficiency, defined as the ratio of actual to ideal heat
transfer rates. The flow channels are plain if the fin surface area is specified
as `0`

and finned otherwise.

The fluids, being compressible and susceptible to phase change, can vary in density over time. Mass can then build (and dwindle) inside a flow channel. The rate at which it does depends on the flow rates across the channel bounds—the channel ports and, on the moist air side, the surface through which condensate must dribble away. The balance of mass flow rates, and therefore the mass accumulation rate, differs between the flow channels and is considered separately for each.

Mass flows through ports **A1** and **B1**
alone. The mass accumulation rate is:

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

where $$\dot{M}$$ is mass accumulation rate and$$\dot{m}$$ is mass flow rate. The subscripts denote fluid domain
(`2P`

for two-phase fluid) and domain ports
(**A1** and **B1**). Mass flow rate at a
port is positive when directed into the channel. The mass in the channel is the
product of the fluid volume (*V*) and the mean density in its
bounds (*ρ*):

$${M}_{\text{2P}}={\rho}_{\text{2P}}{V}_{\text{2P}},$$

Variations in density, as they are internal to the channel, reflect in the mass accumulation rate:

$${\dot{M}}_{\text{2P}}={\left[{\left(\frac{\partial \rho}{\partial p}\right)}_{u}\frac{dp}{dt}+{\left(\frac{\partial \rho}{\partial u}\right)}_{p}\frac{du}{dt}+{\rho}_{\text{L}}\frac{d{z}_{\text{L}}}{dt}+{\rho}_{\text{M}}\frac{d{z}_{\text{M}}}{dt}+{\rho}_{\text{V}}\frac{d{z}_{\text{V}}}{dt}\right]}_{\text{2P}}{V}_{\text{2P}},$$

where:

*p*is pressure.*u*is specific internal energy.*z*is zone length fraction—the length of a zone divided by the total length of the two-phase fluid channel. The subscripts denote subcooled liquid (`L`

), liquid-vapor mixture (`M`

), and superheated vapor (`V`

).

The first and second terms capture the compressibility of the fluid. The third through fifth terms capture the disposition of the fluid to phase change. Growth of a phase manifests in the length of its zone and therefore in the proportion of fluid possessing the density of that phase. The partial derivatives with respect to pressure and specific internal energy are calculated as sums over the fluid zones. For the first partial derivative:

$${\left(\frac{\partial \rho}{\partial p}\right)}_{u}={z}_{\text{L}}{\left(\frac{\partial \rho}{\partial p}\right)}_{u,\text{L}}+{z}_{\text{M}}{\left(\frac{\partial \rho}{\partial p}\right)}_{u,\text{M}}+{z}_{\text{V}}{\left(\frac{\partial \rho}{\partial p}\right)}_{u,\text{V}}.$$

For the second partial derivative:

$${\left(\frac{\partial \rho}{\partial u}\right)}_{p}={z}_{\text{L}}{\left(\frac{\partial \rho}{\partial u}\right)}_{p,\text{L}}+{z}_{\text{M}}{\left(\frac{\partial \rho}{\partial u}\right)}_{p,\text{M}}+{z}_{\text{V}}{\left(\frac{\partial \rho}{\partial u}\right)}_{p,\text{V}}.$$

Mass flows in part through ports **A2** and
**B2**. Moisture condensation is a sink, subtracting from
the moist air a part of its original mass. The mass accumulation rate is:

$${\dot{M}}_{\text{MA}}={\dot{m}}_{\text{A2}}+{\dot{m}}_{\text{B2}}-{\dot{m}}_{\text{Cond}},$$

The subscripts denote fluid domain (`MA`

for moist air),
domain ports (`A2`

and `B2`

), and phase change
type (`Cond`

for moisture condensation). Variations in density
due to compressibility of the fluid are internal to the channel and reflect in
the mass accumulation rate.

Mass conservation extends to the species in the flow, giving for moisture and trace gas each a mass balance expression. Moisture transits with the flow at the ports and with condensation both in the bulk flow and at the wall. Moisture accumulation reflects in its mass fraction in the fluid volume and in the total mass of that volume:

$${\dot{x}}_{\text{w}}{M}_{\text{MA}}+{x}_{\text{w}}{\dot{M}}_{\text{MA}}={\dot{m}}_{\text{w,A2}}+{\dot{m}}_{\text{g,B2}}-{\dot{m}}_{\text{Cond}},$$

where *x* denotes mass fraction and the
subscript `w`

denotes water vapor. The mass fraction of
moisture is also the specific humidity of the flow. Trace gas does not condense
and so is limited to flow through the ports:

$${\dot{x}}_{\text{g}}{M}_{\text{MA}}+{x}_{\text{g}}{\dot{M}}_{\text{MA}}={\dot{m}}_{\text{g,A2}}+{\dot{m}}_{\text{g,B2}},$$

where the subscript `g`

denotes trace
gas.

The pressure drop across a channel drives the flow between its ports. Viscous friction resists the flow, and quickly it reaches a steady flow rate, given by the balance of the opposing forces. The force, or momentum, balance is considered separately for each half volume.

The figure shows the half volumes of a tube bundle in parallel (left) and cross
flow arrangements (right). The circles indicate the nodes at which fluid states and
properties are defined. **A** and **B** are port
nodes and **I** is an internal node. Subscript `1`

corresponds to two-phase flow (referred to as side 1 in the block) and subscript
`2`

to moist air flow (side 2 in the block).

The pressure loss due to viscous friction depends on the flow geometry in the flow channel. Two-phase fluid is limited to flow inside tubes, but moist air can be configured in other flow geometries—inside tubes, across tube banks, or through channels with generic parameterizations. For flow inside tubes, the viscous friction calculation depends also on flow regime—laminar, transitional, or turbulent.

In turbulent flows, the viscous friction loss is proportional to the square of
mass flow rate. The proportionality is expressed in terms of the Darcy friction
factor. For the half volume nearest port **A**:

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

where *f*_{D} is the
Darcy friction factor, *L* is tube length,
*A*_{CS} is tube cross-sectional area,
and *D*_{H} is tube hydraulic diameter.
*L*_{Add} is the sum of the local
resistances expressed as a length. The hydraulic diameter is an effective
diameter, associated with cross sections both circular and otherwise, determined
from the tube cross-sectional area:

$${D}_{\text{H}}=\frac{4{A}_{\text{CS}}}{P},$$

where *P* is the perimeter of the cross
section. For the half volume nearest port **B**:

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

The Haaland correlation gives for the Darcy friction factor at port
**A**:

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

where *ε*_{R} is the
characteristic height of the microscopic protrusions that line the flow channel.
The **Tube internal absolute roughness** block parameter
specifies this height. Likewise at port **B**:

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

In laminar flows, the viscous friction loss is directly proportional to mass
flow rate. The proportionality is expressed in terms of the shape factor, an
empirical constant used to quantify the effect of tube shape on the friction
loss. For the control volume nearest port **A**:

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

where *ƛ* is the shape factor, specified in
the **Shape factor for laminar flow viscous friction** block
parameter. For the half volume nearest port **B**:

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

The flow is turbulent when the Reynolds number exceeds the **Turbulent
flow lower Reynolds number limit** block parameter. It is laminar
when the Reynolds number is below the **Laminar flow upper Reynolds
number limit** block parameter. In between, the flow is
transitional. The switch between flow regimes is smooth, with numerical blending
applied to remove discontinuities known to cause simulation problems.

The viscous friction loss is calculated from the Hagen number. The calculation
applies to laminar and turbulent flows alike. In the half volume nearest
**A**:

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

where *μ* is dynamic viscosity,
*N*_{R} is the number of tube rows in
the tube bank, and Hg is the Hagen number. The Hagen number is a function of the
Reynolds number and it depends on the tube bank geometry^{1}. In the half volume nearest port
**B**:

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

The viscous friction loss is based on the pressure loss coefficient, an
empirical measure of the pressure drop needed to sustain a mass flow rate. The
calculation applies to laminar and turbulent flows alike. In the half volume
nearest port **A**:

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

where *ξ* is the pressure loss coefficient.
In the half volume nearest port **B**:

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

Energy can build and dwindle inside. The energy accumulation rate depends on the energy flow rates across the channel bounds. Energy flows primarily by advection at the ports and by thermal convection at the wall. Thermal conduction in the fluid plays a role at the ports, but there advection dominates until flow slows near to a stop. Conduction is most often negligible. In moist air, condensate drains and so strips the flow of enthalpy, in effect acting as an energy sink—a departure from two-phase fluid, in which condensate (of what is often refrigerant) remains with the flow.

Energy flows partly by advection and conduction through ports
**A1** and **B1** and partly by convection
at the wall. The energy accumulation rate is:

$${\dot{E}}_{\text{2P}}={\varphi}_{\text{A1}}+{\varphi}_{\text{B1}}-Q,$$

where $$\dot{E}$$ is energy accumulation rate, $\phi$ is energy flow rate, and
*Q* is heat transfer rate. Advection and conduction both
factor into the energy flow rates at the ports. The heat transfer rate is
positive when directed from two-phase fluid to moist air. Heat lost from
two-phase fluid is heat gained in moist air. Energy accumulation reflects partly
in variations of specific internal energy and partly in variations of fluid mass:

$${\dot{E}}_{\text{2P}}={M}_{\text{2P}}{\dot{u}}_{\text{2P}}+{u}_{\text{2P}}{\dot{M}}_{\text{2P}},$$

where *u* is specific internal
energy.

Energy flows partly by advection and conduction through ports
**A2** and **B2** and partly by convection
at the wall. Moisture condensation is a sink, subtracting from the moist air a
portion of its original enthalpy content. The energy accumulation rate is:

$${\dot{E}}_{\text{MA}}={\varphi}_{\text{A2}}+{\varphi}_{\text{B2}}+Q-{\varphi}_{\text{Cond}},$$

where *Q* is the heat transfer rate
subtracted from the two-phase fluid channel, and the energy flow rate
*ϕ*_{Cond} is that due to moisture
condensation. The energy accumulation rate reflects in variations of specific
internal energy and total mass in the flow channel:

$${\dot{E}}_{\text{MA}}={M}_{\text{MA}}{\dot{u}}_{\text{MA}}+{u}_{\text{MA}}{\dot{M}}_{\text{MA}}.$$

Heat transfer is sensitive to phase and is considered piecewise by fluid zone. Liquid, mixture, and vapor zones are logical in two-phase fluid, but moist air, which flows as vapor only, allows for no such distinction. To carry out the heat transfer calculations then, zone boundaries are artificially mirrored on the moist air side, and two-phase fluid zones are each given a matching moist air zone. The zones in a pair are equal in length and are referred to by the name of the phase on the two-phase fluid side.

Heat transfer occurs solely between each zone pair. The total heat transfer rate between the fluids is the sum over the liquid, mixture, and vapor zones:

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

where the subscript `Z`

denotes (two-phase
fluid) zone: liquid (`L`

), mixture (`M`

), or vapor
(`V`

). The heat transfer calculations below apply separately to
each zone, but, for conciseness, the subscript is dropped.

With the exception of density in the mixture zone, two-phase fluid properties are zone averages. Moist air properties are channel averages—or averages over the combined length of the zones. Density in the mixture zone is modeled as a function of vapor quality:

$${\rho}_{\text{M}}=\frac{1}{{x}_{\text{Out}}-{x}_{\text{In}}}{\displaystyle {\int}_{{x}_{\text{In}}}^{{x}_{\text{Out}}}\frac{1}{\nu (x)}}dx,$$

where *x* is vapor quality and
*ν* is specific volume. The subscripts denote the entrance
(subscript `In`

) and exit (`Out`

) of the
mixture zone. The integral gives for density:

$${\rho}_{\text{M}}=\frac{1}{{\nu}_{\text{Out}}-{\nu}_{\text{In}}}\text{ln}\left(\frac{{\nu}_{\text{Out}}}{{\nu}_{\text{In}}}\right).$$

Fluid states vary in their treatment with fluid type and flow arrangement. For
two-phase fluid (side `1`

in the figure) and for moist air
(side `2`

) in a parallel or counter flow arrangement, zones
align lengthwise with respect to the flow. Mass flow rate ($$\dot{m}$$) is the same for all zones. Zone entrance temperature
(*T*_{In}) varies between zones, with
the outlet temperature of one giving the inlet temperature of the next.

For moist air in a cross flow arrangement, zones align crosswise with respect to the flow. Mass flow rate varies between zones and is a fraction of the total mass flow rate. That fraction is equal to the zone length fraction. Zone entrance temperature is the same for all zones.

The heat transfer rate in a zone follows from the Effectiveness-NTU method.
That method gives the actual heat transfer rate as a fraction of its maximum
theoretical value. The fraction is the heat exchanger effectiveness,
*ε*:

$$\u03f5=\frac{{Q}_{\text{Act}}}{{Q}_{\text{Max}}},$$

The subscripts `Act`

and `Max`

denote actual
and maximum values in a fluid zone. The maximum heat transfer rate occurs when
the temperature change in the flow least capable of absorbing heat is itself a
maximum. In terms of that temperature difference:

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

where *C* is heat capacity rate and
*T* is temperature—here at the entrances (subscript
_{In}) of the two-phase fluid (`2P`

)
and moist air (`MA`

) zones. The heat capacity rate measures the
ease with which a flow can absorb heat from its surroundings. The flow with the
smallest capacity rate limits, and therefore sets, the maximum heat transfer
rate possible between the fluids. The subscript `Min`

indicates
that heat capacity rate is the smallest of the two:

$${C}_{\text{Min}}=\text{min}\{{C}_{\text{2P}},{C}_{\text{MA}}\},$$

The heat capacity rates are each defined in terms of the respective fluid properties for the zone considered as:

$$C={\dot{m}}_{*}{c}_{\text{P}},$$

where *c*_{P} is
specific heat and $${\dot{m}}_{*}$$ is mass flow rate. redefined, for heat transfer calculations,
to be numerically smooth and always positive:

$${\dot{m}}_{*}=\sqrt{{\dot{m}}^{2}+{\dot{m}}_{\text{Th}}^{2}},$$

The square root ensures the smoothed variable does not drop below zero; $${\dot{m}}_{\text{Th}}$$—a threshold mass flow rate very near zero—ensures that $${\dot{m}}_{*}$$ does not reach true zero. Saturating the mass flow rate at a small threshold keeps the heat transfer rate from becoming undefined in stagnant or reversing flows.

The effectiveness and maximum heat transfer rate give for the actual heat transfer rate in a zone:

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

In the Effectiveness-NTU method of the block, the effectiveness is a function
of the heat capacity rate, *C*_{R}, and of
the number of transfer units, *NTU*:

$$\u03f5=f({C}_{\text{R}},NTU).$$

The capacity ratio is the fraction:

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

The subscript `Max`

indicates that the heat capacity rate is
the largest from among the fluids:

$${C}_{\text{Max}}=\text{max}\{{C}_{\text{2P}},{C}_{\text{MA}}\}.$$

The number of transfer units is:

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

*R* is the overall thermal resistance between the flows,
taken over the combined length of the fluid zones. The ratio *R*/*z* is the portion of that resistance encountered in a single
zone—the domain of the heat transfer calculations considered here.

The overall thermal resistance *R* is the sum of individual
resistances between the flows. Those resistances are due to convection on the
wet surfaces of the wall, conduction in the layers of fouling that over time
collect on those surfaces, and conduction in the thickness of the wall.
Convective and fouling resistances are specific to each of the flow channels.
The sum gives:

$$R=\frac{1}{{U}_{\text{2P}}{A}_{\text{Th,2P}}}+\frac{{F}_{\text{2P}}}{{A}_{\text{Th,2P}}}+{R}_{\text{W}}+\frac{{F}_{\text{MA}}}{{A}_{\text{Th,MA}}}+\frac{1}{{U}_{\text{MA}}{A}_{\text{Th,MA}}},$$

where *U* is the convective heat transfer
coefficient, *F* is the fouling factor, and
*A*_{Th} is the heat transfer surface
area, each for the flow indicated in the subscript.
*R*_{W} is the thermal resistance of
the wall. The heat transfer coefficients derive from empirical correlations
between Reynolds, Nusselt, and Prandtl numbers.

The heat transfer surface area increases with the use of fins. The increase is
determined in part by the thermal efficiency of the fins—a dimensionless number,
generally smaller than `1`

, defined as the ratio of actual to
ideal heat transfer rates. The heat transfer surface area is the sum of the
primary surface area, or that not covered by fins, and the effective surface
area of the fins:

$${A}_{\text{Th}}={A}_{\text{W}}+{\eta}_{\text{F}}{A}_{\text{F}}.$$

*A*_{W} is the primary
heat transfer surface area. *A*_{F} and
*η*_{F} are the fin surface area and
the fin efficiency. The effective heat transfer surface area is the product of
the two.

The effectiveness varies in its calculation with flow arrangement (parallel flow, counter flow, or cross flow) and with mixing condition (both flows unmixed, both flows mixed, or one flow mixed). The calculation is based on standard expressions from literature:

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

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

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

*Cross flow with flows each mixed*:$$\u03f5={\left[\frac{1}{1-\text{exp}\left(-NTU\right)}+\frac{{C}_{\text{R}}}{1-\text{exp}\left(-{C}_{\text{R}}NTU\right)}-\frac{1}{NTU}\right]}^{-1}$$

*Cross flow with just the flow of largest capacity rate mixed*:$$\u03f5={\left[\frac{1}{1-\text{exp}\left(-NTU\right)}+\frac{{C}_{\text{R}}}{1-\text{exp}\left(-{C}_{\text{R}}NTU\right)}-\frac{1}{NTU}\right]}^{-1}$$

*Cross flow with just the flow of lowest capacity rate mixed*:$$\u03f5=1-\text{exp}\left\{-\frac{1}{{C}_{\text{R}}}\left[1-\text{exp}\left(-{C}_{\text{R}}NTU\right)\right]\right\}$$

During phase change, the heat capacity ratio drops to zero, and the effectiveness expressions collapse to the same limiting form:

$$\u03f5=1-\text{exp}(-NTU).$$

The figure plots the effectiveness curves (*E*) against the
number of transfer units (*NTU*). The curves can differ sharply
between flow arrangements, with the difference becoming more pronounced as the
heat capacity ratio nears `1`

. Of the flow arrangements,
counter flow (**ii** in the figure) is the most effective,
followed by cross flow (**iii.a**–**iii.d**),
and then parallel flow (**i**).

The mixing condition has an impact, with unmixed flows
(**iii.a**) being the most effective and mixed flows
(**iii.b**) being the least. Mixing just the flow with the
smallest capacity rate (**iii.c**) tends to lower the
effectiveness more than mixing just the flow with the largest capacity rate
(**iii.d**). Curve **iv** is the limiting
form associated with a heat capacity ratio of `0`

.

Note that condensers and evaporators, in which phase change is generally
occurring, and for which the heat capacity ratio is therefore often close to
zero, have for much of the time the efficiency curve depicted in **iv.** Flow arrangement and mixing condition is of
little effect during phase change.

The heat transfer coefficient in a zone varies with the mean Nusselt number in that zone:

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

where Nu is Nusselt numbers and *k* is
thermal conductivity. In two-phase fluid, these each vary with phase, and so are
obtained separately for each zone. In moist air, which remains always a vapor
mixture, the same Nusselt number and thermal conductivity apply to all zones.

The Nusselt number derives from empirical correlations between Reynolds, Nusselt, and Prandtl numbers. Different correlations apply depending on the flow regime in effect (laminar or turbulent) and on the flow geometry of the channel (inside tubes, outside tubes, or through channels with generic parameterizations). Recalling that two-phase fluid always runs inside tubes:

**Inside tubes:**In turbulent moist air flow, and in turbulent two-phase flow in liquid and vapor zones, the Nusselt number is based on the Gnielinski correlation. The flow is turbulent when the Reynolds numbers exceeds the**Turbulent flow lower Reynolds number limit**parameter specified in the block. The Nusselt number is then:$$\text{Nu}=\frac{\frac{f}{8}(\text{Re}-1000)\text{Pr}}{1+12.7\sqrt{\frac{f}{8}({\text{Pr}}^{2/3}-1)}},$$

where Re is Reynolds number, Nu is Nusselt number, and Re is Reynolds number, each a mean for the zone considered in the calculation. The Darcy friction factor,

*f*, is the same used in pressure calculations.In the liquid-vapor mixture zone of a turbulent two-phase flow, the Nusselt number is based instead on the Cavallini-Zecchin correlation. The correlation is averaged over the change in vapor quality across the zone:

$$\text{Nu}=\frac{1}{{x}_{\text{Out}}-{x}_{\text{In}}}{\displaystyle {\int}_{{x}_{\text{In}}}^{{x}_{\text{Out}}}0}.05{\left(1-x+x\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}\right)}^{0.8}{\text{Re}}_{\text{SL}}^{0.8}{\text{Pr}}_{\text{SL}}^{0.33}dx$$

Or:

$$\text{Nu}=\frac{0.05{\text{Re}}_{\text{SL}}^{0.8}{\text{Pr}}_{\text{SL}}^{0.33}\left\{{\left[\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right){x}_{\text{Out}}+1\right]}^{1.8}-{\left[\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right){x}_{\text{In}}+1\right]}^{1.8}\right\}}{1.8\left(\sqrt{\frac{{\rho}_{\text{SL}}}{{\rho}_{\text{SV}}}}-1\right)({x}_{\text{Out}}-{x}_{\text{In}})}$$

where

*x*is the vapor quality at the entrance of the zone considered (subscript`In`

) or at the outlet (subscript`Out`

). The subscripts`SL`

and`SV`

indicate quantities measured in saturated liquid and saturated vapor, respectively.In laminar flow, for both fluids and for all zones, the Nusselt number is that specified in the

**Nusselt number for laminar flow heat transfer**block parameter for each fluid. The flow is laminar when the Reynolds number is below the**Turbulent flow lower Reynolds number limit**parameter specified in the block.Above the

**Laminar flow upper Reynolds number limit**parameter and below the**Turbulent flow lower Reynolds number limit**parameter, the flow is transitional. The switch between laminar and turbulent flows is not sudden but smooth. The smoothing results from a numerical blending of Reynolds numbers and ensures that simulation issues do not arise due to discontinuities.**Across tube banks:**Like the pressure drop for this flow geometry, the Nusselt number is calculated from the Hagen number. The calculation depends on the alignment of the tubes in the bank—`Inline`

or`Staggered`

—and on the proportion between tube spacing and tube diameter:$$\text{Nu}=\{\begin{array}{cc}0.404L{q}^{\text{1/3}}{\left(\frac{\text{Re}+1}{\text{Re}+1000}\right)}^{0.1},& Inline\\ 0.404L{q}^{1/3},& Staggered\end{array},$$

where (5):

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

*D*is tube diameter and*l*is tube spacing—longitudinal (subscript`L`

), transverse (subscript`T`

), or diagonal (subscript`D`

). The diagonal tube spacing is a function of the longitudinal spacing and the transverse spacing:$${l}_{\text{D}}=\sqrt{{\left(\frac{{l}_{\text{T}}}{2}\right)}^{2}+{l}_{\text{L}}^{2}}.$$

**In channels with generic flow parameterizations:**The Nusselt number follows from the Colburn equation. The equation applies to laminar and turbulent flows alike, and it correlates the Reynolds, Nusselt, and Prandtl numbers strictly in terms of empirical factors,*a*,*b*, and*c*. The factors can be tuned from experimental data, allowing for greater accuracy even where tube parameterizations suffice. From the Colburn equation:$$\text{Nu}=a{\text{Re}}^{b}{\text{Pr}}^{c},$$

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