## N

Bulk liquid

Liquid to vapor transfer vv

Flgur» 4.3 Model of n Disequilibrium stage.

Stage liquid and vapor contacting zone nent along with the compositions in the bulk phase, the temperature in the bulk phase, and at the interface and the area of the interface, aj.

The correlations for the mass and heat transfer coefficients and in-terfacial also take into account packing or tray geometries for the actual column. The total mass and energy rates are calculated from intergrating the mass and energy fluxes across the total surface area, ar

The Taylor method. Krishnamurthy and Taylor (88, 89) present and test a nonequilibrium model which includes rate equations for mass transfer, and sometimes reaction, among the traditional MESH equations. These include individual mass and energy balances in the vapor and the liquid and across the interface. An equilibrium equation exists for the interface only. The solution method for these equations is the same as that of the block-banded matrices of the global Newton methods and the style of the method is similar to the Naphtali-Sandholm (Sec. 4.2.9).

The total mass transfer rates are added to an expanded set of the MESH equations called the MERQ equations. The new MERQ acronym stands for

Material balances for each component; one for the bulk vapor, one for the bulk liquid, and one across the interface.

Energy balance equations; one for the bulk vapor, one for the bulk liquid, and one across the interface.

Rate equations for mass transfer for all but one component; one from the interface to the bulk vapor and one from the bulk liquid to the interface, plus one energy transfer rate equation from the liquid to the vapor.

equilibrium equation at the interface only.

In component balances, feed and side product streams are considered in the bulk phases only. The component balance for the bulk vapor is

and the component balance for the bulk liquid is

Material transfer across the interface should cancel.

As in the component balances, a separate energy transfer equation is written for each bulk phase and all energy transfer between the two phases is at the interface. The energy balance for the bulk vapor is

Vj+JiJ+t - Vpj + {FjHf^) - (WjHf7) +(-) (QJ) + E)" = 0 (4.127) and the energy balance for the bulk liquid is

VAf-i " Ljhf + (Ff hfL) - (Wf- h]"') +(-)(Qf) - Ef = 0 (4.128)

Net energy gained or lost by each phase from the transfer between the two phases will cancel at the interface:

The mass transfer rate equations force values of the mass transfer coefficients such that the rates of each phase, N^V* and N^L*, are equal. The above equations for material and energy balance across the interface are used to form these rate equations and sire excluded from the final set of equations. Since the vapor and liquid are only at equilibrium at the interface, one equilibrium equation per component is drawn there:

There are (5C + 1) equations per section of packing or per tray. The n(5c + 1) equations of a complete column are arranged to have the block-banded form of a global Newton method and can be solved by the same numerical methods. The independent equations for a tray or a section of packing are

Ry Ry- ■ ■Rc-V R\j R%- • 'Rc-v Qv'' -Q'cj (4.131)

EjEfR'd

The independent variables for this global Newton method will be the bulk component vapor and liquid flow rates; compositions at the interface for each component less one; a mass transfer rate for each component and the temperatures of the bulk vapor, bulk liquid, and the interface:

X, = Vty ' 'Vq ly • 'Iq yJv ■ y'c-y 4 4- • -xh. v k n* • ■ -n"cj (4.132)

The number of equations, n(5c + 1), for a large number of trays and components, can be excessive. The global Newton method will suffer from the same problem of requiring initial values near the answer. This problem is aggravated with nonequilibrium models because of difficulties due to nonideal A'-values and enthalpies then compounded by the addition of mass transfer coefficients to the thermodynamic properties and by the large number of equations. Taylor et al. (80) found that the number of sections of packing does not have to be great to properly model the column, and so the number of equations can be reduced. Also, since a system is seldom mass-transfer-limited in the vapor phase, the rate equations for the vapor can be eliminated. To force a solution, a combination of this technique with a homotopy method may be required.

The methods based on the equilibrium stage model have existed for over 30 years and refinements continue, but serious development of nonequilibrium models has begun only recently. These methods are an alternative means to the stage model for predicting column performance. They are expected to make inroads, especially for systems for which stage efficiency prediction is very difficult, such as reactive distillation, chemical absorption, and three-phase distillation. However, their progress into systems where efficiency prediction is well-estab-lished is likely to be slower. Their complexity due to the restriction to a global Newton method can be overcome, but the ability to reliably predict mass transfer coefficients, interfacial areas, and diffusion coefficients must also be resolved.

For development of nonequilibrium methods to continue, the calculations for mass transfer coefficients and interfacial areas required by these models will have to be added to physical property packages. Krishnamurthy and Taylor (89) present methods and recommendations for calculating the mass and energy transfer coefficients and rates. Help may be available from published manuals or supplier literature.

There are commercially available programs that use nonequilibrium models or consider mass transfer rates when calculating revalues or efficiencies. A rate-based method can include both mass transfer and reaction rate equations. Programs with such a method are the Gasplant-Plus system of Taylor, Weiland and Associates of Potsdam, New York, and the RATEFRAC method in ASPENPlus from AspenTech of Cambridge, Massachusetts. Other programs that contain attributes of a rate-based system are TSWEET from Bryan Research and Engineering of Bryan, Texas, and the AMSYM program from D. W. Robinson and Associates of Edmonton, Alberta, which also can be installed in HYSIM from Hyprotech of Calgary, Alberta. The FraChem program of OLI Systems, Florham Park, New Jersey, supports both mass transfer and reaction equations in its global Newton method.

### 4.3 How to Use and Which to Use

Section 4.2.3 discusses the importance of problem setup, selecting specifications, and of initial profiles. Tools used in problem setup include x-y diagrams {Chap. 2) and shortcut methods such as Smoker and Fenske-Underwood-Gilliland {Chap. 3). These are useful for providing initial estimates, for troubleshooting failure to reach a converged solution, and for analyzing the rigorous solution. The global Newton methods are the most sensitive to the quality of initial estimates, but the guidelines in Sec. 4.2.3 also apply to other methods. The sections below provide additional guidelines on how to reach a good converged solution.

### 4.3.1 Hints for setting separation specifications

For the first four rigorous methods (Sees. 4.2.5 and 4.2.7 to 4.2.9) and most of their variations, there is only a limited set of compatible column specifications. Usually a reflux ratio (or reflux rate) specification and a product rate specification with possibly substituting the condenser and reboiler duties are compatible. These are inherent in the MESH equations, and any method begins with these specifications. While these are the easiest specifications to solve, designers often prefer to set purity specifications [including specifications such as a true boiling point (TBP), cut point, vapor pressure, or top tray temperature] or recovery of some component. Some of the above methods allow for their equations to be modified for these and other specifications.

For a product purity specification for some component C in the distillate, the purity specification equation is

In methods such as the 2N Newton or global Newton methods (Sees. 4.2.8 and 4.2.9), Eq. (4.133) can substitute for the energy balance of the top stage. A purity specification for the bottoms product replaces the energy balance of the bottom stage. On the other hand, a specification made on the reflux ratio, condenser duty, or reboiler duty does not require replacing the energy balance equation. Other purity specification equations, such as one based on the product temperature, will also replace the energy balance.

D. spec

For some methods, many specifications may not be used simultaneously and there is little freedom in setting specifications. In the 2N Newton or global Newton methods, there can usually be only one specification equation, such as Eqs. (4.133) or (4.134). The new equation ■rHI be sensitive to only a few of the independent variables, and if a second equation is added, any attempt to manipulate the Jacobian matrix will fail. For this reason, specifying both top and bottom purities often fails when using such methods.

When setting a column specification, some other variable must be allowed to change in order to meet the specification, and it and other variables should be particularly sensitive to the specification. For instance, in a complex column, the bottoms purity can be more sensitive to an interreboiler duty rather than that of the reboiler, especially if the interreboiler duty is large. In a simple column with a small condenser duty relative to the reboiler duty, the distillate purity will be more sensitive to reboiler duty.

Often these variables are something that can be manipulated in the rclumn operation and control to meet the specification such as reboiler duty (or steam), reflux rate, or a stripper steam rate. Both the speci fications and the variables should have some physical significance in column operation. The philosophy in matching specification to floating variable is the same as matching controlled parameter to manipulated variable in process control of a column.

If the specified component concentration is small (e.g., in the part-per-million range) there may not be any variables that are sensitive to Eq. (4.133). In this concentration range, temperature is insensitive to composition and a temperature specification is therefore unsuitable [Eq. (4.134)]. In these cases, another set of specifications and floating variables must be used.

In a simple column, with only two products and a condenser and a reboiler, there are only two degrees of freedom and there can only be two specifications and two floating variables. For complex columns, a degree of freedom is added with each intercondenser, interreboiler. side product, and so on.

How the MESH equations are arranged in the method should not have to restrict the number and combination of specifications and corresponding floating variables but can in the bubble-point (Sec. 4.2.5), sum-rates (Sec. 4.2.7), 2N Newton (Sec. 4.2.8), and global Newton (Sec. 4.2.9) methods. The best methods for invoking numerous column specifications are the inside-out methods (Sec. 4.2.10). There, the handling of specifications is inherent in the methods and they provide the greatest number and freedom in column specifications and floating variables. The inside-out methods allow a more natural way of mixing specifications and variables to where specifications and variables do not have to be paired at points in the column and can be well spread across the column.

Further discussion of separation specifications is in Sees. 4.1.1, 4.2.3, and 3.1.1.

### 4.3.2 Problems when setting simulation input

■ Feed stage location: Locating the feed stage far above or below the optimum will cause a composition pinch. The pinch represents a column that as specified cannot be solved. A pinch can best be detected with an x-y diagram (Sec. 2.4.1).

■ Feed temperature: Too cold or too hot a feed may disturb one or more stages and may also cause a composition pinch. The temperature of the lean oil to an absorber will affect the removal of the heavies from the vapor along with losses of the lean oil overhead. Convergence problems in the simulation may represent actual physical problems, suggesting that a feed exchanger is needed.

■ Reflux ratio or vapor boilup: These should be above the minimum for the separation. An x-y diagram (Sec. 2.4.1) is the most effective way to detect if the calculation is below the minimum. Also an Underwood minimum reflux calculation (Sec. 3.2.2) can show if the reflux or boilup is below the minimum. However, an Underwood calculation may be inaccurate if constant molal overflow (Sec. 2.2.2) cannot be assumed for the column.

It is common to design and operate reasonably close to the minimum reflux or minimum boilup (Sec. 3.1.4). A computer solution at such low reflux ratios can be unstable and fail. A solution may only be reached if very good initial values are available. The technique of sneaking up on an answer" is powerful in these cases. Initially, the column is solved at a higher reflux ratio. This solution is used as the initial value for the subsequent calculation, in which the reflux ratio is slightly lowered. This process is continued until the desired reflux ratio is reached. Other examples of how to use the solution of one simulation to initialize another simulation are described by Brierley and Smith (106). The "sneaking-up" technique is part of the basis of the homotopy methods (Sec. 4.2.12) and these and other forcing techniques may also be used.

■ More than one product purity or recovery specification: Not all methods will accept or solve. Replace one purity specification with some other such as reflux rate or a product rate. The methods best suited to solve multiple purity specifications are the inside-out methods (Sec. 4.3.1).

■ Two purities or recoveries for the same product: In a simple column, two purity specifications may be impossible. Once a purity is set, the concentrations of all other components become dependent variables. In a complex column, the degrees of freedom increase and just as there can be more than one purity specification across the column, there may be two specifications on the same product. This can be solved but only if the relative volatilities between the components of the specifications are high enough to make this feasible. The solution will also be difficult if the concentrations of both components are small and therefore insensitive to the floating variables.

■ Two purities or recoveries of the same component in different products: While this is a feasible separation, it is difficult for a method to solve. Directly specifying the purity of a component in different products will not give the program needed freedom in calculating the component balance.

■ Extreme purity or recovery specifications (e.g., 99.8 percent): The alternative is an impurity specification of all other components in the product or a loss specification of the component in the other product. Impurity specifications give the solution more freedom and are inherently easier to solve.

■ Combining purity or recovery and product flow specifications: These may clash, especially on binary systems. Free one and find something else to specify.

■ Specifying all product flow rates: This will not give the program freedom in establishing the overall material balance. At least one product flow rate must be allowed to float.

■ Specifying overhead vapor product in a system with noncondens-ables: Since the split in the condenser can be very sharp, there will be little freedom of movement. It may be better to specify a variable such as reflux rate, condenser temperature, or any specification on the liquid overhead product (if it exists).

■ Specifying stage temperatures: These are effective only where temperature is sensitive to composition, e.g., where composition changes significantly. They should not be used at a high-purity product location. Section 4.3.1 has additional discussion.

■ Product rates: A simple way to either specify or estimate the products is to examine the feed and using boiling point splits, decide what components will be in what products.

■ Specifying duties and reboiler boilup or reflux: Energy input and internal flows directly affect each other, and specifying both may not be possible.

■ Both the temperature and the product on a stage: There is only a narrow range of solution for this combination.

4.3.3 Recovering from failures and analyzing results

■ Negative flow rates: Some methods have safeguards that prevent negative flow rates or techniques to recover from them. A method that uses a Newton-Raphson procedure can reduce the step in the independent variables (Sec. 4.2.6). The inside-out methods apply a stripping factor to adjust the flow rates. Often the cause is poor estimates of the internal and product flows and the user may need to modify the estimates. In complex columns, negative flows can be caused by high (or excessive) interreboiler or intercondenser duties or side product flow rate specifications. The negative flows may be removed by increasing reflux, reducing interreboiler or intercondenser duties and side product flow rates, or by refining the initial estimates.

■ Oscillation in the column variables: This occurs where the temperature and flow rate profiles swing widely either side of what should be the final answer, often in the Newton-Raphson-based methods. Oscillation is caused by too large a step in the profiles from one column trial to the next. This oscillation is prevented by limiting the step or percentage change in the MESH variables to below the amount generated by the Newton-Raphson technique.

■ "Bad actor" components: Components such as water can seriously upset the VLE method and prevent solution without veiy good initial values. If the water is not a steam feed, use the sneaking-up technique by solving the column first without water in the feed to establish the initial profiles. Then slowly increase the water during the succeeding runs.

■ Systems with highly nonideal VLE suffer from requiring very good initial profiles: The sneaking-up technique can be used by first solving the column with a simple approximation of the VLE and then slowly introducing the nonideal VLE. This is described by Brierley and Smith (106) and is also the thermodynamic homotopy of Vickery and Taylor (81). As stated in Sees. 4.2.9 and 4.2.12, this can occur in the global Newton methods. The inside-out methods avoid these problems in their use of simple VLE models.

■ Initial estimates can also be generated using simpler solution methods: The 2N Newton (Sec. 4.2.8) and global Newton (Sec. 4.2.9) methods require good initial estimates and the more complex the method, the greater the need for good estimates. A relaxation method (Sec. 4.2.11) or a BP method (with a simple VLE method) t Sec. 4.2.5) are far less sensitive to initial estimates. These can be used to bring the profile close to the solution from where the preferred method can complete the calculation.

■ It is helpful for programs to be able to plot temperature, flow rate, composition, and stripping factor profiles plus x-y diagrams and key ratio plots: These are powerful in detecting composition pinches. They also help in analyzing a column's performance, such as when rating an operating column to understand why it fails to meet the required products. Section 3.4 discusses the use of these graphical techniques in analyzing simulation results.

Flat profiles or hardly any change in composition or temperature e ver a large range of stages can indicate the location of a pinch. If the range is above or below the feed, move the feed stage in the direction the range. If the range spans the feed stage, increase reflux or remove some stages. A reversed temperature profile (or temperature increasing up the column) may indicate a problem with the feed temperature. It may also be caused by a shift in the concentration of an inert component in the vapor phase.

Composition profile plots often show the buildup of some middle components in the column, forced there by lighter or heavier components. The profiles plots may not show the relative separation between components and there the x-y diagrams are better suited.

Using these plots in preliminary computer runs with simple specifications such as product rates and reflux or boilup will also help to determine if more or fewer stages are needed or if the feed stage is properly located. These simple runs will also help establish initial profiles and even whether the separation is feasible at all. Once such problems are conquered, the column can be simulated at the desired conditions and with the required specifications.

### 4.3.4 Which method to use

Friday and Smith (1) were the first to classify the methods and present recommendations on whieh method to use for different columns. A good recent source with recommendations on most of the types of methods is King (11). He presents a decision diagram to show which family of methods is best for a column. Figure 4.4 is an updated diagram. Table 4.1 summarizes the strengths and weaknesses of many common methods. The choice of method is based on the type of mixture and type of column. The diagram and table are intended to provide general guidelines only; favorable or unfavorable experience with a specific application may override these guidelines.

The inside-out methods (Sec. 4.2.10) can be used for most columns. The Russell method is simple to implement and does work well for a wide variety of refinery and hydrocarbon columns. The Boston method also works well for a wide range of columns and has been shown to work for superfractionators or tall, high-purity columns. Since the outer loops of the two methods are similar, they can be combined with a choice between the two methods, depending on the type of column, as part of the inside loop.

The global Newton method (Sec. 4.2.9) can be used for highly nonideal systems or reactive distillation systems with a homotopy forcing (Sec, 4.2.12) or relaxation technique (Sec. 4.2.11).

Nonequilibrium methods (Sec. 4.2.13) tend to be global Newton methods extended to solve mass-transfer-inhibited systems. Nonequilibrium methods are not yet completely extended to more common systems, but these methods should see the greatest amount of development in distillation modeling.

Alternates (marked as the entries in parentheses in Fig. 4.4) could

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