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# Condenser Evaporator (2P-MA)

Heat exchanger for systems with fluids susceptible to phase change

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## Description

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.

### Flow Arrangements, Geometries, Mixing, and Fins

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.

#### Flow Arrangements

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.

#### Flow Geometries

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 (dI and dO). 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 (lL) and transverse pitch (lT). The longitudinal pitch is the distance between neighboring tube rows. The transverse pitch is the distance between neighboring tubes within a row.

#### Mixing

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.

#### Fins

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.

### Mass Balance

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.

#### Two-Phase Fluid

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

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

where $\stackrel{˙}{M}$ is mass accumulation rate and$\stackrel{˙}{m}$ 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:

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

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:

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

#### Moist Air

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:

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

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:

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

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:

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

where the subscript g denotes trace gas.

### Momentum Balance

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).

### Viscous Friction

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.

#### Inside Tubes

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:

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

where fD is the Darcy friction factor, L is tube length, ACS is tube cross-sectional area, and DH is tube hydraulic diameter. LAdd 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:

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

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

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

where εR 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:

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

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:

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

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:

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

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.

#### Across Tube Banks

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:

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

where μ is dynamic viscosity, NR 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 geometry1. In the half volume nearest port B:

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

#### In Channels with Generic Parameterizations

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:

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

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

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

### Energy Balance

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.

#### Two-Phase Fluid

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

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

where $\stackrel{˙}{E}$ 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:

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

where u is specific internal energy.

#### Moist Air

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:

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

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

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

### Heat Transfer Rate

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:

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

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.

#### Zone Properties and States

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:

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

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 ($\stackrel{˙}{m}$) is the same for all zones. Zone entrance temperature (TIn) 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.

#### Dry Heat Transfer

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:

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

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:

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

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:

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

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

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

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

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

The square root ensures the smoothed variable does not drop below zero; ${\stackrel{˙}{m}}_{\text{Th}}$—a threshold mass flow rate very near zero—ensures that ${\stackrel{˙}{m}}_{*}$ 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:

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

#### Heat Exchanger Effectiveness

In the Effectiveness-NTU method of the block, the effectiveness is a function of the heat capacity rate, CR, and of the number of transfer units, NTU:

$ϵ=f\left({C}_{\text{R}},NTU\right).$

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:

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

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 ATh is the heat transfer surface area, each for the flow indicated in the subscript. RW 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:

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

AW is the primary heat transfer surface area. AF and ηF are the fin surface area and the fin efficiency. The effective heat transfer surface area is the product of the two.

#### Effectiveness Curves

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:

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

• Counter flow:

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

• Cross flow with flows each unmixed:

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

• Cross flow with flows each mixed:

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

• Cross flow with just the flow of largest capacity rate mixed:

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

• Cross flow with just the flow of lowest capacity rate mixed:

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

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

$ϵ=1-\text{exp}\left(-NTU\right).$

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.aiii.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.

#### Heat Transfer Coefficients

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:

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

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:

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

Or:

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

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:

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

where (5):

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

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:

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

## Ports

### Output

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Vector with the instantaneous values of the zone length fractions for subcooled liquid, two-phase mixture, and superheated vapor in the two-phase fluid channel.

Instantaneous value of the water condensation rate in moist air flow. The condensate is assumed to drain away as it forms.

### Conserving

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Opening for moist air to enter and exit its side of the heat exchanger.

Opening for moist air to enter and exit its side of the heat exchanger.

Opening for two-phase fluid to enter and exit its side of the heat exchanger.

Opening for two-phase fluid to enter and exit its side of the heat exchanger.

## Parameters

expand all

#### Common Tab

Manner in which the flows align in the heat exchanger. The flows can run in the same direction, counter directions, or perpendicular directions relative to each other.

Mixing condition in each of the flow channels. Mixing in this context is the lateral movement of fluid as it proceeds along its flow channel toward the outlet. The flows remain separate from each other. Unmixed flows are common in channels with plates, baffles, or fins. This setting reflects in the effectiveness of the heat exchanger, with unmixed flows being most effective and mixed flows being least.

#### Dependencies

This parameter applies solely to the Flow arrangement setting of Cross flow.

Resistance of the wall to heat flow by thermal conduction. Wall resistance adds to convective and fouling resistances to determine the overall heat transfer coefficient between the flows.

Area normal to the direction of flow at port A1.

Area normal to the direction of flow at port B1.

Area normal to the direction of flow at port A2.

Area normal to the direction of flow at port B2.

#### Two-Phase Fluid 1 Tab

Cross-sectional shape of a tube. Circular tubes are most common but rectangular and annular tubes are standard in some applications. For tubes of still other shapes, a generic parameterization is available. If the channel comprises a tube bundle, the cross section is that of just one tube.

Internal diameter of the cross section of a tube. If the channel is a tube bundle, the diameter is that of just one tube. The cross section is uniform along a tube and so the diameter is constant throughout. The diameter factors into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Circular.

Internal width of the cross section of a tube. If the channel is a tube bundle, the width is that of just one tube. The cross section is uniform along a tube and so the width is constant throughout. The width and height together factor into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Rectangular.

Internal height of the cross section of a tube. If the channel is a tube bundle, the height is that of just one tube. The cross section is uniform along a tube and so the height is constant throughout. The width and height together factor into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Rectangular.

Smaller diameter of the annular cross section of a tube. Heat transfer occurs solely through the inner surface of the annulus. If the channel is a tube bundle, the inner diameter is that of just one tube. The cross section is uniform along a tube and so the inner diameter is constant throughout. The inner diameter factors into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Annular.

Larger diameter of the annular cross section of a tube. If the channel is a tube bundle, the outer diameter is that of just one tube. The cross section is uniform along a tube and so the outer diameter is constant throughout. The outer diameter factors into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Annular.

Internal area normal to the direction of flow in a single tube. The total area of the channel is the sum over the tubes that comprise it when a tube bundle. The cross section is uniform along a tube and so the area is constant throughout. The area factors in the pressure loss calculation.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Generic.

Perimeter of the tube cross section for pressure loss calculations. If the tube is in a bundle, the perimeter is that of a single tube. The cross section is uniform along a tube and so the perimeter is constant throughout.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Generic. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Perimeter of the tube cross section for heat transfer calculations. If the tube is in a bundle, the perimeter is that of a single tube. The cross section is uniform along a tube and so the perimeter is constant throughout.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Generic.

Distance traversed between the ports of a tube. This distance is not generally the shortest between the ports. If the tube is in a bundle, the length is that of a single tube. The tubes are identical to each other and so the length of one is the length of all.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow inside tubes.

Number of tubes through which to pass the flow between the ports. The greater the count, the greater the length subjected to viscous friction but the greater the surface area available for heat transfer.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow inside tubes.

Aggregate minor pressure loss expressed as a length. This length is that which all local resistances, such as elbows, tees, and unions, would add to the flow path if in their place was a simple wall extension. The larger the equivalent length, the steeper the minor pressure loss due to the local resistances.

Mean height of the surface protrusions from which wall friction arises. Higher protrusions mean a rougher wall for more friction and so a steeper pressure loss. Surface roughness features in the Haaland correlation from which the Darcy friction factor derives and on which the pressure loss calculation depends.

Start of transition between laminar and turbulent zones. Above this number, inertial forces take hold and the flow grows progressively turbulent. The default value is characteristic of circular pipes and tubes with smooth surfaces.

End of transition between laminar and turbulent zones. Below this number, viscous forces take hold and the flow grows progressively laminar. The default value is characteristic of circular pipes and tubes with smooth surfaces.

Measure of thermal resistance due to fouling deposits which over time tend to build on the exposed surfaces of the wall. The deposits, as they impose between the fluid and wall a new solid layer through which heat must traverse, add to the heat transfer path an extra thermal resistance. Fouling deposits grow slowly and the resistance due to them is accordingly assumed constant during simulation.

Constant value assumed for Nusselt number in laminar flow. The Nusselt number factors in the calculation of the heat transfer coefficient between fluid and wall, on which the heat transfer rate depends. Typical values include 3.66 for tubes with circular cross sections, 2.98 for those with square cross sections, and 3.99 for those with rectangular cross sections with aspect ratio of 2:1. (4)

#### Dependencies

This parameter applies solely to the Tube cross section setting of Generic.

Secondary heat transfer surface area provided by fins. Its product with the fin efficiency gives the effective heat transfer surface area provided by the fins. The total heat transfer surface area is the sum of the effective fin surface area with the primary surface area—that not covered by fins—calculated from channel geometry.

Ratio of the actual heat transfer rate to the theoretical maximum predicted for a fin held uniformly at its base temperature. The product of fin efficiency with fin surface area gives the effective heat transfer surface area provided by the fins. The total heat transfer surface area is the sum of the effective fin surface area with the primary surface area—that not covered by fins—calculated from channel geometry.

Measure of fluid energy whose initial value to specify.

Mean pressure in the two-phase fluid channel, specified relative to absolute zero, at the start of simulation. This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

Mean temperature in the two-phase fluid channel, specified relative to absolute zero, at the start of simulation. This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial fluid energy specification option is set to Temperature.

Mass fraction of vapor in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial fluid energy specification option is set to Vapor quality.

Volume fraction of vapor in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial fluid energy specification option is set to Vapor void fraction.

Enthalpy per unit mass in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial fluid energy specification option is set to Specific enthalpy.

Internal energy per unit mass in the two-phase fluid channel at the start of simulation. This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial fluid energy specification option is set to Specific internal energy.

#### Moist Air 2 Tab

Type of channel that is to carry the moist air. The flow can run across tubes in a bank—external to the tubes and perpendicular to them—or inside tubes in a bundle. A tube bank can span multiple rows of tubes, each row with multiple tubes. For other flow geometries, a generic parameterization is available.

Cross-sectional shape of a tube. Circular tubes are most common but rectangular and annular tubes are standard in some applications. For tubes of still other shapes, a generic parameterization is available. If the channel comprises a tube bundle, the cross section is that of just one tube.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow inside tubes.

Internal diameter of the cross section of a tube. If the channel is a tube bundle, the diameter is that of just one tube. The cross section is uniform along a tube and so the diameter is constant throughout. The diameter factors into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Circular. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Internal width of the cross section of a tube. If the channel is a tube bundle, the width is that of just one tube. The cross section is uniform along a tube and so the width is constant throughout. The width and height together factor into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Rectangular. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Internal height of the cross section of a tube. If the channel is a tube bundle, the height is that of just one tube. The cross section is uniform along a tube and so the height is constant throughout. The width and height together factor into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Rectangular. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Smaller diameter of the annular cross section of a tube. Heat transfer occurs solely through the inner surface of the annulus. If the channel is a tube bundle, the inner diameter is that of just one tube. The cross section is uniform along a tube and so the inner diameter is constant throughout. The inner diameter factors into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Annular. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Larger diameter of the annular cross section of a tube. If the channel is a tube bundle, the outer diameter is that of just one tube. The cross section is uniform along a tube and so the outer diameter is constant throughout. The outer diameter factors into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Annular. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Internal area normal to the direction of flow in a single tube. The total area of the channel is the sum over the tubes that comprise it when a tube bundle. The cross section is uniform along a tube and so the area is constant throughout. The area factors in the pressure loss calculation.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Generic. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Perimeter of the tube cross section for pressure loss calculations. If the tube is in a bundle, the perimeter is that of a single tube. The cross section is uniform along a tube and so the perimeter is constant throughout.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Generic. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Perimeter of the tube cross section for heat transfer calculations. If the tube is in a bundle, the perimeter is that of a single tube. The cross section is uniform along a tube and so the perimeter is constant throughout.

#### Dependencies

This parameter applies solely to the Tube cross section setting of Generic. The Tube cross section parameter in turn is active only for the Flow geometry parameterization of Flow inside tubes.

Distance traversed between the ports of a tube. This distance is not generally the shortest between the ports. If the tube is in a bundle, the length is that of a single tube. The tubes are identical to each other and so the length of one is the length of all.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow inside tubes.

Number of tubes through which to pass the flow between the ports. The greater the count, the greater the length subjected to viscous friction but the greater the surface area available for heat transfer.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow inside tubes.

Geometrical placement of a tube row against its neighbors. A row can have its tubes in line with those of its neighbors or staggered against them. This setting determines the expression to use for the Nusselt number and it impacts the heat transfer rate between the flows.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow across tube banks.

External diameter of the cross section of a tube. If the channel is a tube bundle, the diameter is that of just one tube. The cross section is uniform along a tube and so the diameter is constant throughout. The diameter factors into the cross-sectional area for pressure loss calculations and into the wall surface area for heat transfer calculations.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow across tube banks.

Offset between the rows of the tube bank in the direction of the moist air flow.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow across tube banks.

Offset between the tubes of a row in the bank perpendicular to the direction of flow.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow across tube banks.

Length of each tube from inlet to outlet. The tubes each of the same length.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow across tube banks.

Number of tube rows in the tube bank. The rows are distributed evenly in a direction perpendicular to the tube length and to the moist air flow. Set the number of tube rows and the number of tubes per tube row to 1 to model flow across a single tube.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow across tube banks.

Number of tubes in each row of the tube bank. The tubes are distributed evenly for each row in the direction of the moist air flow. Set the number of tube rows and the number of tubes per tube row to 1 to model flow across a single tube.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Flow across tube banks.

Mathematical model for pressure loss by viscous friction. This setting determines which expressions to use for calculation and which block parameters to specify as input. The pressure loss models available depend on the Flow geometry setting. Only two are available for each setting.

#### Dependencies

This parameter applies solely to the Flow geometry settings of Flow inside tubes and Flow across tube banks. The pressure loss models available and the choice of default differ between the flow geometry settings.

Aggregate minor pressure loss expressed as a length. This length is that which all local resistances, such as elbows, tees, and unions, would add to the flow path if in their place was a simple wall extension. The larger the equivalent length, the steeper the minor pressure loss due to the local resistances.

#### Dependencies

This parameter applies solely to the Pressure loss model setting of Correlations - flow in tubes.

Mean height of the surface protrusions from which wall friction arises. Higher protrusions mean a rougher wall for more friction and so a steeper pressure loss. Surface roughness features in the Haaland correlation from which the Darcy friction factor derives and on which the pressure loss calculation depends.

#### Dependencies

This parameter applies solely to the Pressure loss model setting of Correlations - flow in tubes.

Start of transition between laminar and turbulent zones. Above this number, inertial forces take hold and the flow grows progressively turbulent. The default value is characteristic of circular pipes and tubes with smooth surfaces.

End of transition between laminar and turbulent zones. Below this number, viscous forces take hold and the flow grows progressively laminar. The default value is characteristic of circular pipes and tubes with smooth surfaces.

Aggregate loss coefficient for all flow resistances in the flow channel—including the wall friction responsible for major loss and the local resistances, due to bends, elbows, and other geometry changes, responsible for minor loss.

The loss coefficient is a dimensionless empirical parameter commonly used to express the pressure loss incurred during flow due to viscous friction. It can be calculated from a nominal operating condition or be tuned to fit experimental data. It is defined as:

$\xi =\frac{\Delta p}{\frac{1}{2}\rho {v}^{2}},$

where ξ is the pressure loss coefficient, ρ is the fluid density, and v is the flow velocity.

#### Dependencies

This parameter applies only to Flow geometry settings of Generic and Flow inside tubes, the latter only for a Pressure loss model setting of Pressure loss coefficient.

Pressure loss correction for flow cross section in laminar flow conditions. This parameter is commonly referred to as the shape factor. Its ratio to the Reynolds number gives the Darcy friction factor for the pressure loss calculation in the laminar zone. The default value belongs to cylindrical pipes and tubes.

The shape factor derives for certain shapes from the solution of the Navier-Stokes equations. A square duct has a shape factor of 56, a rectangular duct with aspect ratio of 2:1 has a shape factor of 62, and an annular tube has a shape factor of 96, as does a slender conduit between parallel plates (8).

#### Dependencies

This parameter applies solely to the Pressure loss parameterization setting of Correlations for tubes.

Euler number for one row of the tube bank. Rows are the tube arrays aligned perpendicular both to the two-phase flow through them and to the moist air flow around them.

The Euler number is a dimensionless empirical number much like the pressure loss coefficient, a measure of the pressure loss incurred during flow due to viscous friction. It can be calculated from a nominal operating condition or be tuned to fit experimental data. It is defined for a tube row as:

$\text{Eu}=\frac{\Delta p}{N\frac{1}{2}\rho {v}^{2}},$

where Eu is the Euler number per tube row, N is the number of tube rows in the tube bank, ρ is the fluid density, and v is the flow velocity.

#### Dependencies

This parameter applies solely to the Pressure loss parameterization setting of Euler number per tube row.

Total area of the flow cross section measured where the channel is its narrowest and the flow its fastest. If the channel is a collection of ducts, tubes, slots, or grooves, the area is the sum over the collection (minus any occlusion due to walls, ridges, plates, and other barriers).

#### Dependencies

This parameter applies solely to the Flow geometry setting of Generic.

Effective surface area used in heat transfer between fluid and wall. The effective surface area is the sum of primary and secondary surface areas, or those of the wall, where it is exposed to fluid, and of the fins, if any are used. Fin surface area is normally scaled by a fin efficiency factor.

Total volume of moist air in the flow channel.

#### Dependencies

This parameter applies solely to the Flow geometry setting of Generic.

Mathematical model for heat transfer between fluid and wall. The choice of model determines which expressions to apply and which parameters to specify for heat transfer calculation. See the block description for the heat transfer calculations.

Three-element vector with the empirical factors for the general form of the Colburn equation. The factors determine the Nusselt number given the Reynolds and Prandtl numbers. The Nusselt number in turn determines the heat transfer coefficient between the fluid and the wall. The general form of the Colburn equation is:

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

where Nu is the Nusselt number, Re the Reynolds number, and Pr the Prandtl number. The coefficients can be calculated from a nominal operating condition or tuned to fit experimental data. The default values depend on the Flow geometry setting. For all but the Flow across tube banks setting, the defaults give the exact form of the Colburn equation:

$\text{Nu}=0.23{\text{Re}}^{0.8}{\text{Pr}}^{1/3}$

For the Flow across tube banks setting, the defaults give the alternate expression:

$\text{Nu}=0.27{\text{Re}}^{0.63}{\text{Pr}}^{0.36}$

#### Dependencies

This parameter applies to all Flow geometry settings, but in the Flow inside tubes and Flow across tube banks settings only for a Heat transfer coefficient model setting of Colburn equation.

Measure of thermal resistance due to fouling deposits which over time tend to build on the exposed surfaces of the wall. The deposits, as they impose between the fluid and wall a new solid layer through which heat must traverse, add to the heat transfer path an extra thermal resistance. Fouling deposits grow slowly and the resistance due to them is accordingly assumed constant during simulation.

Constant value assumed for Nusselt number in laminar flow. The Nusselt number factors in the calculation of the heat transfer coefficient between fluid and wall, on which the heat transfer rate depends. Typical values include 3.66 for tubes with circular cross sections, 2.98 for those with square cross sections, and 3.99 for those with rectangular cross sections with aspect ratio of 2:1 (4).

#### Dependencies

This parameter is active only when the Flow geometry setting is Flow inside tubes, the Tube cross section setting is Generic, and the Heat transfer parameterization setting is Correlation - flow in tubes.

Secondary heat transfer surface area provided by fins. Its product with the fin efficiency gives the effective heat transfer surface area provided by the fins. The total heat transfer surface area is the sum of the effective fin surface area with the primary surface area—that not covered by fins—calculated from channel geometry.

Ratio of the actual heat transfer rate to the theoretical maximum predicted for a fin held uniformly at its base temperature. The product of fin efficiency with fin surface area gives the effective heat transfer surface area provided by the fins. The total heat transfer surface area is the sum of the effective fin surface area with the primary surface area—that not covered by fins—calculated from channel geometry.

Pressure in the moist air channel at the start of simulation. The pressure is read against absolute zero. It can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

Temperature in the moist air channel at the start of simulation. The temperature is read against absolute zero. It can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

Measure of water vapor level whose initial value to specify.

Relative humidity in the moist air channel at the start of simulation. The relative humidity is the ratio of the specific humidity to its saturated value. The specific humidity is the mass fraction of water vapor to the total mass of moist air—that of water vapor, trace species, and dry air combined.

This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial moisture specification option is set to Relative humidity.

Specific humidity in the moist air channel at the start of simulation. The specific humidity is the mass fraction of water vapor to the total mass of moist air—the sum over the water vapor, trace species, and dry bulk in the channel.

This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial moisture specification option is set to Specific humidity.

Mole fraction of water vapor in the moist air channel at the start of simulation. The mole fraction is relative to the molar quantity of water vapor, trace species, and dry bulk combined.

This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial moisture specification option is set to Mole fraction.

Humidity ratio in the moist air channel at the start of simulation. The humidity ratio is the mass fraction of water vapor to the sum of water vapor and dry bulk alone—without considering the trace species that normally accompanies moist air flows.

This parameter can be a scalar or a two-element vector. As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

#### Dependencies

This parameter is active when the Initial moisture specification option is set to Humidity ratio.

Measure of trace gas level whose initial value to specify.

Mass fraction of trace gas in the moist air channel at the start of simulation. The mass fraction is relative to the total mass of moist air—the sum over water vapor, trace gas, and dry bulk. It can be a scalar or a two-element vector.

As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

This parameter is ignored if the Trace gas model parameter in the Moist Air Properties (MA) block is set to None.

#### Dependencies

This parameter is active when the Initial trace gas specification option is set to Mass fraction.

Mole fraction of trace gas in the moist air channel at the start of simulation. The mole fraction is relative to the total quantity of moist air—the sum over water vapor, trace gas, and dry bulk contents. It can be a scalar or a two-element vector.

As a scalar, it gives the mean value in the channel. As a two-element vector, it gives a linear gradient between the ports. The first element then describes the inlet and the second the outlet. Whether a port is the inlet or the outlet depends on the initial direction of the flow.

This parameter is ignored if the Trace gas model parameter in the Moist Air Properties (MA) block is set to None.

#### Dependencies

This parameter is active when the Initial trace gas specification option is set to Mole fraction.

Relative humidity at which water vapor begins to condense. Raise its value above 1 to allow for supersaturation of water vapor—a state in which moist air holds more water vapor than is stable at its thermodynamic conditions.

## References

[1] ASHRAE Standard Committee. 2013 ASHRAE handbook: fundamentals. 2013.

[2] Braun, J. E., S. A. Klein, and J. W. Mitchell. "Effectiveness models for cooling towers and cooling coils." In ASHRAE Transactions, vol. 95, no. 2, 164–174, 1989.

[3] Çengel, J. Heat and mass transfer: a practical approach. Boston, MA: McGraw-Hill, 2007.

[4] Lebrun, J., Xin Ding, J.P. Eppe, and M. Wasacz. "Cooling coil models to be used in transient and/or wet regimes-theoretical analysis and experimental validation." Proceedings of SSB, 405-411, 1990.

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[7] White, F. M. Fluid Mechanics. Boston, MA: McGraw-Hill, 1999.

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