Two methods of design of condensers for multicomponent mixtures are used in industrial practice: the equilibrium method of Silver (1947) and the rate model of Colburn et al. (1934) and (1937). They are both film-based models in which the resistances of a series of layers are added, but the former is essentially a heat transfer model in which the mass transfer rate is approximated, whereas the latter provides an accurate description of heat and mass transfer processes.

Mass transfer is important in many condensation processes and the fundamentals are given in the entries on Diffusion and the Maxwell Stefan Equations. In a mixture of n components, the *Diffusion Law* gives the n independent fluxes of condensation

It is the which are needed in design and the above equation shows a diffusive and a convective contribution to these fluxes. The theories of diffusion allow the evaluation of J_{g}, the diffusive fluxes, and therefore the prediction of mass transfer effects in condensation. In most cases of condensation,

Convection, the condensation flux towards the interface, induced by removal of energy, dominates the transfer process and the diffusion is a small effect riding on top. However, there are many condensation processes, e.g., reflux condensers, where the diffusional transfer rate becomes important. It is worth emphasizing that there are n independent fluxes, , but only (n – 1) independent diffusive fluxes, J_{i}. Condensation is not fully defined by diffusional processes and the additional determinacy relation required is the total condensation rate.

There are three starting points for the description of multicomponent mass transfer. The following form of Fick’s Law describes multicomponent mass transfer in a dilute mixture of vapors in a noncondensing gas, where it is a form of *Effective Diffusivity Method*. It is exact for binary mixtures.

The second is the Generalized Fick’s Law equation. The *Linearized Theory* uses this equation as starting point.

Thirdly, the Maxwell-Stefan Equations may be used. Their advantage is that there is an *Exact Matrix Solution* of the M-S equations for a film model.

Equation [2a] differs from the other two equations in that independent diffusion is implied. Each component transfers under the action of its own composition driving force alone. This is acceptable in most cases, but sometimes the latter two interactive methods give better predictions. No significant difference between these two arises in film methods.

A one-dimensional, steady film of thickness sf is considered adjacent to the condensate surface. The transfer of mass follows the diffusion law and all changes in concentration occur across this film, from a vapor which is completely mixed to a condition of equilibrium at the condensate surface. The thickness of the hypothetical film is not required, but is incorporated within the binary mass transfer coefficient, β_{ik}. (See Figure 1.)

The solution of Eq. (1) for the J_{i} has been achieved with each of the three above methods of determining the diffusive fluxes to give a solution of the same form,

Here, braces and square brackets show (n – 1) column and square matrices, and the usual rules of matrix multiplication apply. In this equation, [B] is a matrix of mass transfer coefficients. In general, it will have off-diagonal elements of finite size so that it will show an interaction between species, where the transfer of any component is influenced by all the independent driving forces in the mixture.

The matrix [Φ] is present in the equation to show the effect of a finite rate of transfer within the film. It is akin to the *Ackermann correction factor*, which arises when mass transfer takes place in the direction of a temperature gradient. With [M(m)] defined as a function of by

Table 1 shows how to evaluate matrices [B_{g}] and [Φ] in Eq. (3) by the methods of Eqs. (2a), (2b), and (2c).

In the effective diffusivity method, the matrices are diagonal, shown by , and each species transfers independently. More detailed descriptions of the developments leading to the above equation are given in Webb (1980) and Taylor and Krishna, (1993).

Figure 2 shows the region near the interface and defines nomenclature when condensation is treated by a film model.

The energy equation, with terms which describe steady-state conduction and convection, is quoted and integrated across the film to give the conductive flux across the interface, .

where

The heat transfer coefficient, α_{g}, is that for conductive heat transfer alone because in the limit of zero mass transfer rate,

For simplicity, liquid subcooling effects are not included here, but are treated later. It is assumed that continuity of energy across the interface may be written as . An overall liquid side heat transfer coefficient, α_{0}, is defined including condensate, wall, coolant film and dirt,

The rate of cooling of the gas phase, , may be obtained. The two heat fluxes, and , differ by the sensible heat change over the gas film, so that the rate of gas cooling is less than the heat flux at the condensate surface.

The term ε/(e^{z}−1) is the *Ackermann Correction Factor*. It is the factor by which the heat transfer is modified to account for mass transfer. A similar factor, ø/(e^{ø}−1) occurs in Eq. (3). Comparison of and shows that the higher the rate of condensation, the slower the rate of gas cooling.

Standard correlations may be used for the geometry in question. The correlations define the unknown thicknesses, s_{t} and s_{f}, of the thermal and mass transfer films.

With an analogy, only a single correlation is needed. The analogy effectively links s_{f} and s_{t}. The *Chilton-Colburn analogy* (1934) may be written as follows:

Vapor liquid equilibrium must be satisfied at the interface. The behavior of the condensate lies between two limiting conditions of liquid mixed and unmixed. These are compared in Figure 3.

At any location in a condenser, gas temperature, T_{g}, gas composition, and cooling fluid temperature, T_{o}, are known. The calculation of the local rates of mass transfer, , and energy transfer, , involves the estimation of interfacial conditions, T_{s}, and , a total of (2n + 1) unknowns. Solving the (2n + 1) equations allows the solution of the local problem. The equation of continuity of energy, Eq. (6), may be rewritten by defining a dimensionless interfacial temperature, χ=(T_{s}−T_{o})/(T_{g}-T_{o}), between 0 and 1.

The physical properties , and ε are flux dependent because the components contribute through their condensation rate. Moreover, the equation is implicit in that ε depends on . Neither is a problem in that the equation is usually solved within an iterative sequence to determine the fluxes, and the implicit dependence on in ε is weak. With diffusive fluxes,

it is seen that Eq. (9) is a determinacy condition allowing the determination of the n independent , from the J_{i}. The interfacial vapor composition is defined by the assumption of equilibrium with the liquid.

The (2n + 1) equations required are:

Continuity of energy (1);

Mixed or unmixed liquid conditions (n); and

Vapor-liquid equilibrium at the interface (n).

Computer algorithms for determining the equilibrium state of an n component mixture usually involve two iteration loops and this is the structure of methods to solve these equations.

Factors that affect the solution of the local problem are:

The liquid mixing condition, Mixed or Unmixed;

Presence of noncondensing gas;

Miscible or immiscible condensate.

Liquid mixing has a strong effect and changes the structure of the algorithm required to calculate the transfer rates. A noncondensing gas is usually a component above its critical temperature. However, in the case of vapors of immiscible liquids with one condensate phase present, but above the azeotropic temperature, all components that will form the second immiscible phase behave as noncondensing species.

The case of mixed condensate is easier because the interfacial liquid condition is defined by prior condensation. In the absence of noncondensing gas, irrespective of whether one- or two-liquid phases are present, a bubble point calculation defines the interfacial state. The above equations are then explicit in transfer rates, iteration being required only to determine the flux dependent quantities in Eqs. (1), (3) and (9), [B] [Φ], and properties and .

When noncondensing gases are present, and these may be vapors having the potential to form a second condensate phase, the calculation of interfacial temperature becomes iterative. For each noncondensing gas, the liquid composition is , and the vapor composition can be obtained directly from the Maxwell-Stefan equations as:

where is the number of species present in the liquid.

Condenser design for the case of an unmixed condensate is similar to the vapor-liquid phase evaluation calculation and can be done in the same way, despite being rate-controlled. It involves finding an interfacial temperature χ (between 0 and 1), and compositions and for specified vapor conditions, T_{g} and , and coolant temperature, T_{o}.

In comparison, the phase evaluation seeks the vapor fraction, θ (between 0 and 1), and compositions and for specified overall composition, , temperature and pressure. A phase evaluation uses two nested iterations (see Cooling Curve):

Outer loop to converge θ,

Inner loop on composition.

Condenser design algorithms may be devised with similar structure. The unmixed liquid condition, Eq. (8), is combined with Eqs. (1) and (3) to give

The term f(χ) is a function only of the interfacial temperature.

The following shows an algorithm for the case of an unmixed, single condensate with or without a noncondensing gas: In such an algorithm, the following must be taken into account:

The guess of χ in the outer loop is made as follows. If Σ>1.0 increase χ and vice versa.

Properties and are flux-averaged. Their values are updated in the inner loop as fluxes are defined.

There are only (n – 1) independent, , in Step 5. The last composition is obtained by difference from 1.0.

If the value of in the inner loop, K

_{i}is obtained using the normalized estimate, .In the first iteration, evaluate the mass transfer coefficients and rate factors using known quantities,

The correct dependence, as shown in Table 1, can be introduced as the dependent values become available.

Design or rating a condenser involves integration of the ordinary differential equations expressing the downstream development of the independent variables. Here, we consider T_{g} - Gas Temperature, T_{o} - Coolant Temperature, - Gas Flowrates.

A full treatment should also include pressure, if pressure drop causes a significant loss of saturation temperature, and condensate temperature to allow for subcooling (see later). Flowrate, , and temperatures, T_{g} and T_{o}, must be known at the vapor inlet to define an initial value problem, Figure 4.

Design involves the calculation of the area to achieve a given degree of condensation, while rating involves the calculation of the degree of condensation produced in a given area. Design is usually achieved as a series of rating calculations of guessed condensers until a satisfactory performance is achieved. Rating calculations are presented below.

Mass and energy balances are carried out over an element of differential area, δA, as shown in Figure 5.

Mass balances over the area δA give for any component, for the mixture as a whole and for a noncondensing gas

Energy balance over the same increment of area gives

where

The rate of gas cooling in the presence of mass transfer is suppressed by the above factor, relative to heat transfer in the absence of mass transfer. For the coolant in co-(+) or countercurrent (−) flow

The integration of the above equations as a means of condenser design has been described in many works, e.g., Schrodt (1973), Price and Bell (1974) and Webb and McNaught (1980).

Colburn and Edison (1941) showed that, based on Eqs. (13) and (14), it is possible to calculate the condensation path as the rate of change of partial pressure with gas temperature,

This allows verifying the likelihood of fogging by comparing the vapor pressure of the component with the condensation path predicted by the above, Figure 6.

An excursion into the supersaturation zone does not, of itself, ensure fogging. Steinmeyer (1972) defines Supersaturation Ratio, S, for clean systems that must be exceeded for fogging to occur; but dust or fine spray may hasten the onset,

where

C=1.76×10^{7} for water, alcohols, etc., |

C=1.38×10^{7} for nonpolars. |

The Colburn-Edison equation also shows the conditions that must be satisfied by a vapor if it is to remain saturated during a condensation process. These are small driving forces and Le = Sc/Pr = 1, rarely satisfied in real processes. This implies that the equilibrium approach can only approximate condenser behavior.

The methods initiated by these authors are special cases of the above. For binary mixtures, the condensation rate can be expressed, following Colburn and Drew (1937), as

When considering the special case of an unmixed condensate, zi becomes the liquid composition. Alternatively, if component 2 is a noncondensing gas, where z_{1}=1 and = , the Colburn-Hougen (1934) equation is obtained. In each of these special cases, a determinacy condition has been added and, as above, this is the specification of total flux. It is this binary form of the equations that has found the widest application in industrial design practice.

In this approach, the detailed mass transfer equations are not used. However, a gas side resistance is included by assuming that the sensible heat change of the saturated gas mixture must be conducted across the gas film. It has been stated by Bell that this assumption is usually conservative. The overall heat transfer coefficient is then calculated by:

where Z is the ratio of the heat flux of the gas side to the total heat flux in an interval.Consider Figure 7; by definition,

hence

The appropriate heat transfer coefficient, α_{g,eff}, in Eq. (19) remains unspecified. In early work, it has been taken as the gas film heat transfer coefficient, α_{g}, for the geometry in question. However, McNaught (1979) has shown that better results are given when it is corrected by the Ackermann factor.

Care should be taken with superheated vapors. The early practice was to provide a desuperheating zone which was designed as a gas cooler, but this was too conservative because it ignored wet wall desuperheating. McNaught (1981) has suggested the following equation to allow the use of the equilibrium model in cases of wet wall desuperheating.

Caution is also needed to distinguish the gas and saturated gas temperatures, T_{g} and T*_{g}, respectively. The gas temperature T_{g} can only be followed using the equations of the film model. The application of the equilibrium method requires the calculation of the appropriate mean temperature driving force and this is considered in the section on Cooling Curve.

The equilibrium model is applied very widely in condenser design practice. Examination of the Colburn-Edison equation shows that it is only exact in the limit of low driving forces and Le = 1. In fact, it is only conservative if the Lewis Number is less than 1 and its accuracy for Le > 1 is not published.

A second case that has not been treated satisfactorily in the literature is the condensation of one liquid alone from vapors that form immiscible liquids. This is similar to the wet wall desuperheating case and the method may be very conservative.

The presence of multicomponent vapors does not cause special problems unless immiscible condensates are formed. In this case, the following methods have been applied.

Bernhardt et al. (1972), considered the two condensates to form separate layers around the surface, and recommended the following combination of the Nusselt coefficients of the two condensates, α_{c1} and α_{c2}:

with v_{i} = Volume fraction of the phases.

Akers and Turner (1962) proposed a model which essentially considers a homogeneous condensate layer with the viscosity of the surface wetting layer,

where ρ_{aν} is the mass fraction average density, η_{c} the viscosity of the surface wetting fluid and λ_{aν} is the volume average thermal conductivity. These simple equations seem to fit most data within about ±30%. The influence of turbulence, vapor shear and inundation on immiscible condensates are not known.

Condensate subcooling has not been included in the above film model treatment. It is implicit in the Silver method, when the integral cooling curve is applied, because the entire mixture is assumed to be cooled to the equilibrium temperature.

With the assumption of laminar flow and a linear temperature profile in the condensate, the heat flux, , at the condensate surface is related to , passing to the coolant (see Figure 2), by

Two terms are added to the previous treatment where it was assumed that . They describe subcooling of new condensate to the mean temperature, T_{c}=T_{s}-3(T_{s}-T_{w})/8, and subcooling of all prior condensate, respectively.

A new method is presented. It is assumed that all heat of subcooling is conducted across the full condensate layer, which is conservative. Both terms above are included, with the second estimated by Eqs. (14) and (15) to predict downstream change, assuming constant relative film heat transfer resistances. Equation (9) holds and all previous algorithms apply if modified quantities , and are defined,

The Silver approach is preferred for industrial design of multicomponent condensers. With unspecified mixtures, where only a cooling curve is provided, it must be used. Its advantage is that no further vapor-liquid equilibrium calculations are needed, irrespective of geometry. The more physically realistic film model has been used to correct the equilibrium method; but the equilibrium method remains less reliable. These are: Where condensate composition must be predicted in a partial condenser (reflux condensers); where pressure drop is a large fraction of pressure; where vapors of immiscible liquids condense above the azeotropic temperature; and where the Lewis Number exceeds unity.

#### REFERENCES

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