## L

«

Ljf + Wip

lip -Lphp

ngure 4.1 (Continued) Stage and column models, (c) feed soage model; (d) sidestream product withdrawal stage model.

ngure 4.1 (Continued) Stage and column models, (c) feed soage model; (d) sidestream product withdrawal stage model.

ike feed at the feed stage pressure but using the enthalpy of the feed, Hf, before the feed enters the column.

If a feed is subcooled liquid, it condenses some of the vapor ascending from the stage below the feed stage. If the feed is superheated, it vaporizes some of the liquid descending from the stage above the feed stage. For the bottom stage of a rectifier, the vapor feed substitutes the vapor entering from the reboiler. Likewise, for the top stage of a nripper, the liquid feed substitutes the reflux.

Side product stage model. For a stage with a side product, the side product, WP, is subtracted from the material leaving the stage (i.e., WP is drawn from the vapor leaving the stage to the stage above when WP is a vapor product and is drawn from the liquid leaving the stage to the stage below if WP is a liquid product). This stage is illustrated in Fig. 4.1c?.

4.1.2 Basic (MESH) equations of rigorous distillation

The basic equations below fully describe a distillation column. They must be satisfied in any solution technique. These equations define the overall column total material balances, energy balances, and product compositions. Internal to the column, they describe equilibrium conditions, internal (stage-to-stage) component and total material balances, and internal energy balances. The independent variables of a column are the product rates and compositions, internal vapor and liquid rates and compositions, and stage temperatures. Equilibrium constants, iC-values, and mixture enthalpies are dependent variables. Each stage is assumed to be at equilibrium (a theoretical stage), though an efficiency can be applied in the equations.

The rigorous methods thus convert a column to a group of variables and equations. The equations were first referred „o as the MESH equations by Wang and Henke (24). The MESH variables are often referred to as state variables. These are

■ Stage temperatures, TJs

■ Internal total vapor and liquid rates, Vj's and L/s

■ Stage compositions, y^s and x^'s, or instead, component vapor and liquid rates, I'^'s and Z„'s.

The summation equation. The summation equation or composition constraint simply states that the sum of the mole fractions on each stage is equal to unity. For the liquid phase, c c c

2^-1 = 0 or ^ vo/Vj "1 = 0 or £ KijXij - 1 = 0 (4.2:

The equilibrium equation. The equilibrium equation is

The equilibrium constant, K¿j, can be complex function itself, dependent on %ij and y^

as described in Chap. 1. The dependence of K^ on x^ and y^ often appears in the MESH equations. The component rates of Eq. (4.3) can also be expressed in the terms of each other, giving vv = ll} (Kv Vj/Lj) = l,} Sy (4.5)

K.j VJLj is termed the stripping factor, S^, while Lj/Ky Vj is termed :he absorption factor, A^. The stripping and absorption factors have been used since the earliest absorber and distillation methods.

The component balances. A great effort in all rigorous methods is in solving the component balances. Proper construction of the equations and choice of numerical method is important. For a simple column single feed, no side products), the overall component balance equation is fi-di- bt « 0 (4.7)

The component balance for the simple stage (no feed or side product), of Fig. 4.16, is

Only the liquid portion of a feed, appears in the component balance for feed stage, f, of Fig. 4.1c, that is, vl/+1 + Vi + UF-Vif-l>f= 0 (4.9)

The vapor portion, viF, appears in the component balance for stage - 1. For stage p of Fig. 4.1c?, from which material is withdrawn, the component material balance is

By convention, material leaving a tray has a negative value and material entering a tray has a positive value.

The total material balances. The total material balances are organized in the same manner as the component balances. The total material balance for the simple stage of Fig. 4.16 is

The same convention applies to feed and product trays where the total flow rate of a feed, Ff, is positive and the product, W^, is negative.

The bubble- and dew-point equations. The equilibrium equation (Eq. 4.3) and the composition constraint [Eq. (4.1)] are combined to get the bubble-point equation,

The bubble-point and dew-point equations are used in some of the solution methods to help determine the stage temperature.

The energy balance equations. The energy balance equations are required in any rigorous method. In narrow-boiling mixtures, they influence the internal total flow rates. In wide-boiling mixtures and in columns where there are great heat effects (e.g., oil refinery fraction-ators), they also strongly influence stage temperatures. The overall energy balance for a column with one feed and side product is

The enthalpy terms, H and h, are per mole of mixture. Note that the enthalpies of the top and side products are written so that a vapor or liquid enthalpy can be substituted, depending on the phase of the product. The energy balance for the simple stage of Fig. 4.16 is

The enthalpies (energy per mole) for each phase are functions of temperature, pressure, and composition

For feed stages, sidestream product stages, and stages with inter-condensers or interreboilers, additional terms are included in the energy balance equations [Eqs. (4.14) and (4.15)]. These terms are the same as those used in the overall column energy balance. The energy balance for the reboiler is

and for a partial condenser with both vapor and liquid products:

Subcooling is accounted for in h1 (the enthalpy of the reflux Lj and the liquid distillate d).

The energy balances are not solved in the same manner as the component or total material balances. With some solution methods, they are simultaneously solved with other MESH equations to get the independent cc'umn variables; in others they are used in a more limited manner to get a new set of total flow rates or stage temperatures.

Tray efficiencies. To characterize the deviation from ideality, stage efficiencies are often used (Sec. 2.1.1). Most computer simulations work with ideal stages. Once the number of ideal stages is established, the number of actual trays or packing height is calculated using stage efficiencies. Tray and packing efficiencies are discussed in detail in Chaps. 7 and 9. Some rigorous procedures incorporate tray efficiencies in the method. Commonly, a Murphree vapor efficiency (Sec. 7.1.1) is used for each component, given as yu - y ij* i y*a - yo*i

where y'0 is what the vapor composition would be if the vapor were in equilibrium with the actual liquid on the stage and and ytj _ x are actual vapor compositions. If the absorption factor is used [Eq. (4.6)], the vapor efficiency can be expressed in terms of variables already presented:

The Murphree efficiency can be rearranged to form a modified equilibrium equation

and can now appear in the absorption and stripping factors of the component balances and in the bubble-point and dew-point equations.

A vaporization efficiency Ev based on the Murphree efficiency was defined by Holland and Liapis (10)

This can be used in the MESH equations to account for stage non-ideality. This vaporization efficiency is applied to the equilibrium constant K{j and appears as the product E^ify- The vaporization efficiency does solve a computational problem in placing an efficiency in the MESH equations. As shown by Lockett (105), a major disadvantage of the vaporization efficiency is that it does vary with composition. Near the top of a high-purity column, as y^ + l and xy approach unity, Eu also approaches unity, and so a vaporization efficiency does not truly reflect stage nonidealities.

Caution then should be used in any choice of efficiency. Some limitations are outlined above; others are discussed in Chap. 7 and by Lockett (105). Both the author of the book and the author of this chapter feel that usually it is best to perform the rigorous calculation using ideal stages and then apply an overall column efficiency to account for stage nonideality,

4.2 Rigorous Computational Methods

4.2.1 The basic classification of the methods

Friday and Smith (1) and King (11) have divided the rigorous methods into four basic classes. These are

1. The bubble-point methods (BP)

2. The sum-rates methods (SR)

3. The 2JV Newton methods

4. The global Newton or simultaneous correction (SC) methods

The original method of Wang and Henke (24) gave the name to the BP methods because the stage temperatures are found by directly solving the bubble-point equation. The SR methods use the energy balances to update the stage temperatures. The 2N Newton's methods calculate temperatures and total flow rates together but compositions are calculated in a separate, dependent step. These three classes are referred to as equation tearing or decoupling methods because the

MESH equations are divided and grouped or partitioned and paired with MESH variables to be solved in a series of steps. The SC methods attempt to solve all of the MESH equations and variables together. Additional classes are

5. Inside-out methods

6. Relaxation methods

7. Homotopy-continuation methods ». Nonequilibrium models

The relaxation, inside-out, and homotopy-continuation methods are extensions of whole or part of the first four methods in order to solve difficult systems or columns. The nonequilibrium models are rate- or transport phenomena—based methods that altogether do away with the ideal-stage concept and eliminate any use of efficiencies. They are best suited for columns where a theoretical stage is difficult to define and efficiencies are difficult to predict or apply.

### 4.2.2 Precomputer methods

The Thiele-Geddes (15) and Lewis-Matheson (16) methods are rigorous methods referred to as stage-to-stage methods. Both preceded the computer and suit manual calculations.

The Thiele-Geddes method. This method is one of the first (1933) rigorous methods for distillation and is the basis of most modern rigorous methods. A key point of the Thiele-Geddes method is the use of the absorption and stripping factors. These will appear in the computer methods. It is known as the rating method because it requires the specification of all feed conditions, feed-stage locations, reflux rate, total product rates, and number of stages. Product compositions and exchanger duties are calculated.

The original form of the Thiele-Geddes method uses the stripping and absorption factors to calculate liquid and vapor compositions. The compositions for the rectifying section are calculated stage to stage by successive substitution, beginning with the distillate product, until the feed stage is reached.

y„ = XDi {Ai} Ai + Ali.i---All + - - + A,2An + An + 1) (4.24)

Compositions for the stripping section are found by the same procedure, but beginning with the reboiler and calculating up to the feed. With these compositions, the temperatures are updated, usually by a trial-and-error bubble-point technique. With the new temperatures, the total flow rates are calculated from the energy balances. A test is then made of the overall total and component balances. If the column is out of balance, the top and bottom compositions are adjusted and the calculation is repeated.

The Lewis-Matheson method. This method, published in 1932, is the converse of the Thiele-Geddes method. Here, the product light-key and heavy-key component distributions are specified, while the number of stages above or below the feed stage are changed. Stage-to-stage calculations are made from the top stage down to the feed stage and from the reboiler up to the feed stage. A comparison is made between the feed stage compositions from the top-down and from the bottom-up calculations. If the two do not match, stages are added to or removed from either the rectifying or stripping sections and the procedure is repeated. Temperatures and total flow rates can be found as in the Thiele-Geddes method. Because the number of stages is adjusted, the Lewis-Matheson method is called the design method.

The Lewis-Matheson method does not work well for wide-boiling mixtures because of the required component distribution. There is also the difficulty of deciding what is a good match at the feed stage.

The Lewis-Matheson method can be modified so that the number of stages is fixed and instead the product compositions or the reflux ratio is adjusted. The criterion of a match at the feed stage is preserved.

4.2.3 The strategy of solution using a rigorous method

A rigorous method proceeds in the following steps:

1. Problem setup

2. Initialization of the MESH variables

3. Actual calculation

4. Solution test

5. Output and engineer's evaluation

Problem setup. Section 4.1.1 describes the variables that are usually specified. All of these, except the separation specification, are straightforward.

Section 3.1.1 states that in a process design, a separation is specified in terms of purities and product flows. For a simple column, two specifications are made and at least one must be a purity. Section 3.1.1 also states that the purity specification can be substituted by a physical property which is a function of the purity or composition, while a product flow can be substituted by a recovery specification.

In a computer simulation, the number of stages and the feed point location are fixed (Sec. 4.1.1) for a given solution. Once these are fixed, distillate {or bottom) composition becomes a function of reflux rate, and the purity specification is met through a reflux rate specification. Since the reflux is only one of the internal flows in the column, a similar argument will apply to any other internal stage flow. It follows that the purity specification can be met by specifying any internal stage flow, either vapor or liquid. Similarly, other variables that can be specified include reflux-to-distillate ratio, boilup-to-bottoms ratio, reboiler heat duty, and temperature of a given stage.

The option of specifying an internal flow instead of a purity specification is particularly useful with some of the methods described below that do not permit direct purity specifications. However, the larger number of options available to the user also increases the pitfalls of inconsistent specifications. These pitfalls are discussed in Sees. 4.3.1 and 4.3.2.

For a column with side products, the number of specified variables increases with each side product. In most methods, the product flow rate is specified for each side product, but sometimes it is possible to specify the purity of a side product. For columns with interreboilers or Lntercondensers, the number of specified variables increases by the number of these exchangers. Usually, the interreboilers or intercon-denser duties are specified, but in some methods, these duties may be allowed to vary to meet a product specification. With these complex columns, inconsistent specifications are a major pitfall and simpler specifications are preferred.

Initial values. Before a trial is begun, stage temperatures, Tjs, and total flow rates, V/s and L/s, have to be given initial values. The stage component rates, v;j s and l^'s, do not have to be estimated since these can be calculated from the component balances. The component balances are dependent on the if-values and for the first component balances, composition-independent Jf-values must be used. A composition-independent Jf-value can be found from the pure component fugacities calculated from an equation of state:

Ki} (composition independent) = 7yX(pure)//yv(pure) (4.25)

For the rest of the first column trial and for succeeding trials, the composition-dependent if-values are used.

The initial total flow rate and temperature profiles can make the difference between success and failure of a rigorous method. Usually for distillation columns, the condenser and reboiler temperatures are estimated and a calculation that assumes constant molal overflow Sec. 2.2.2) is used to initialize the internal vapor and liquid flow rates. These carry down through the column to any feed or side product stage. The problem at these stages is that there is an abrupt change in the internal vapor and/or liquid rates when the feed is added or the product subtracted. At the feed stage, the feed is flashed at the stage pressure to give its distribution (Sec. 4.1,1). Also, a product rate can be defined as a ratio to the remaining vapor or liquid to keep from starving a stage.

Any program should permit the user to initialize the internal flow rates and temperatures. The best way is to have the user initialize a few internal flow rates and temperatures and have the program calculate the others by linear interpolation. User-provided internal flows and temperatures are especially important for

■ Wide-boiling mixtures and energy-coupled systems where the profiles may shift drastically from linear.

■ Where constant molar overflow does not work well, such as with nonideal systems or where there is a drastic difference between internal vapor and liquid rates. Section 2.2.2 discusses the applicability of constant molal overflow.

■ Complex columns—e.g., multiple feeds, side products, interreboilers and intercondensers, pumparounds and side strippers—at or near the stages where these appear. See Sec. 4.1.1 for other examples.

■ The stage below a subcooled receiver where the temperature difference between the two may be great,

■ The overhead vapor and overhead liquid products of a partial condenser. In a column with a high amount of noncondensables, the split between these two products is very sharp and wrong estimates can prevent a solution.

■ Reflux in high-purity or narrow-boiling columns.

For some systems, the initial values have to be near the expected solution results. For superfractionators, and columns with purity specifications and highly nonideal systems, the initial temperature profile should be near the expected results. For narrow-boiling systems, an accurate reflux estimate is necessary.

Solution criteria. With any rigorous method it is imperative to set criteria to determine when a solution is reached. Each rigorous method has its own unique criteria, but they all must conform to some physical criteria. Section 4.1.1 states that at solution, overall mass, component, and energy balances are achieved. There are other criteria a method should meet. If there are specifications such as product flow rates, heat duties, or product compositions, these must be matched within some tolerance. At the solution, the temperature and total flow rate profiles should not change between iterations; i.e., the accumulated fractional change in stage temperatures and in stage vapor rates between trials k and k + 1 should be very small:

Errors in the MESH equations of Sec. 4.1.2 should be small, including the stage energy; total material and component balances and summation equation should be small. The physical solution criteria above should take precedence over any mathematical criteria, such as having Newton-Raphson functions approach zero (Sec. 4.2.6).

4.2.4 Trldiagonal matrix method for the material balances

Calculating the component (low rates. The tridiagonal matrix method introduced by Wang and Henke (24), is a fast and accurate technique for calculating the component and total flow rates. This method for calculating the component flow rates is used in most of the following rigorous methods.

The component liquid rates are eliminated from Eq. (4.10) using the absorption factors [Eq. (4.6)] to leave only the component vapor rates. Any feed rates are moved to the right of the equal sign. The terms that multiply the component vapor rates are the factors in the diagonals of the tridiagonal matrix:

This equation expresses both the component liquid and vapor rates for the stage in terms of vapor rates only. In matrix terms, the system of equations is expressed as

where the transposed vapor rate vector is v, = (d„ vn, v,.2,- ■ -, viN.u vlNf

and the transposed feed rate vector is:

The feed to a stage is split into the two phases of an adiabatic flash as viF for the vapor portion and liF for the liquid portion. The coefficients of the component vapor rates in Eq. (4.29) form the elements of the tridiagonal matrix m,! 1 0 An m,o 1 0 0 Ai2 mi3 0 .. 0 Ai3

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