ENCH 445: Lecture 8 --  Multicomponent Distillation

Consider the case of the distillation of a multicomponent mixture in a multistage distillation column.  As before, the number of degrees of freedom is determined by the discription rule (i.e., D.O.F. = number of variables set during construction or controlled during operation by independent means).  In particular, when a partial condenser and reboiler are used, we have D.O.F. = N + 7, where N is the number of components.  Generally, all the variables associated with the feed, such as its composition, flow rate and enthaply are set, as is the column pressure, which leaves four degrees of freedom and two basic types of problems (design and simulation) as was the case for a binary distillation.

For a design problem, the goal is to determine the number of plates needed and the location of the feed plate, and the following is generally specified

Separation variable #1 (e.g., the recovery of the light key component in the top (distillate) product.

Separation variable #2 (e.g., the recovery of the heavy key component in the bottom product.

The fact that the optimal feed plate is used that minimizes the total number of plates.

The reflux ratio.

For a simulation problem the goal is to determine the compositions of the top and bottom products and the following is generally specified:

The number of plates above the feed plate.

The number of plates below the feed plate

One external flow, such as the flowrate of the top product.

One interal flow, such as the reflux flow from the condenser (or the reflux ratio).

Generally, in a multicomponent distillation column, only two components will exist in significant quantities in both the bottom and top products.  These are the two key components.  The heavy non-key components will essentially all end up in the bottom product while the light non-key components will essentially all end up in the top product.  This is why the separation variables described above are given in terms of the key components.

Solving either a design or sumulation problem involves solving a large number of coupled algebraic equations, roughly two for each component on each plate.  In principle, these equations can be all solved simultaneously using a multivariable Newton's method, especially with todays powerful computers.  However, a full simultaneous solution method is often prone to convergence problems, and may need considerable computer time.  The simplified methods described next involve the optimal ordering of the equations so they can be solved one equation at time sequentially and, in most cases, analytically.  This has the great advantage of simplfying the computer programming needed.  In addition, there is no numerical method that will guarantee a solution for several algebraic equations solved simultaneously, so solving smaller equation subsets, preferrably just one equation at a time, may be advantageous.   

Consider first the design problem for the special (and very common) case where there are no light non-keys or no heavy non-keys.  For the first case, an overall material balance can be used to determine the bottom composition to a high degree of precision since the distribution of the keys is determined by the product specification and all the heavy nonkeys can be assumed to end up in the bottom product.  Conversely, for the second case, an overall material balance can be used to determine the top composition to a high degree of precision since the distribution of the keys is again determined by the product specification and all the light nonkeys can be assumed to end up in the top product.

Using the product composition calculated as just described as the starting point, the material and equilibrium relations can be solved for stage by stage until the opposite end of the column is reached.  This procedure, termed the Lewis Matheson method, is a multicomponent analog of the McCabe Thiele method described earlier, although it must be performed analytically, not graphically.    In particular,  the internal total molar flow are first determined using the reflux information in the problem specification as was the case for the binary problem.  Then, starting from the bottom product, the following equations are used sequentially:

1.  Determine the composition of the vapor rising from the reboiler (or current stage) using a bubble point

calculation, or more simply, if the relative volatility is known, using:

      y_i_p  =   alpha_i  * x_i_p / [sum_j (alpha_j * x_i_p) ]

where i and j are the component indices, p is the current plate number, sum_j denotes the sum over all j, and alpha_j is the relative volatility of component j with respect to an arbitrary reference component, usually chosen to the the least volatile component so that alpha_j is unity or greater.  Vapor flow rates can then be determined using 

      v_i_p = V * y_i_p

2.  Determine the composition of the liquid stream passing the vapor stream just calculation using the following equation if the current location is below the the feed plate:

    l_i_p = v_i_p-1 + x_i_b b

Or the following equation if the current location is above the feed plate:

  v_i_p = l_i_p+1 + x_i_D D

3.  Return to step 1 with p incremented by unity and repeat until the compositions of the key components are those specified for the top product.

The optimal feed plate can be determined by varying the location of the feed plate until the minimum number of total plates is obtained.

The above calculation procedure can be easily implemented using a spreadsheet, as described elsewhere on this webiste. 

The minimum reflux ratio can be determined by (in the case where the calculaction starts at the reboiler) by setting the feed plate high in the column, such as at the 50th plate.  Then, initially try a small reflux ratio.  Next, step up to the feed plate using the Lewis Matheson method.  It should be obvious that a pinch exists below the feed plate, i.e., there will be a small change in compositions from plate to plate below the feed plate.  Then step across the feed plate and determine the compositions on the next few plates above the feed plate.  If there is a large change in composition from plate to plate above the feed plate, and expecially if the compositions become negative or more than unity, then the relux ratio being used is less than the minimum reflux ratio.  The objective is to increase the reflux ratio until the pinch conditions apply to the regios above and below the feed plate.  This will then be the minimum reflux ratio.  

The above method to determine the minimum reflux ratio can be automated by having the solver in Excel determine the flux ratio where there is a pinch above and below the feed plate. 

As in the case of binary distillation, the optimal design that minimizes the combination of capital and operating costs is often when the operating reflux ratio is 1.5 times the minimum reflux ratio. 

For the simulation problem, two options are possible as discussed in class.  First, a procedure called the Thiele-Geddes method can be used which involves guessing a temperature profile in the column, then performing a procedure much like the stage-to-stage method described above, except that ratios of flow rates (such as v_i_p/b_i) are determined initially and then and a matching procedure at the feed plates permits the determination of compositions everywhere in the column.  Then, an improved temperature profile is estimated from the liquid phase compositions just calculated and the procedure repeated to achieve an improved solution.  Convergence is achieved when the temperature profile does not change from iteration to iteration. 

Alternatively, the dynamic behavior (i.e., time dependent behavior) of the column can be simulated from some arbitrary starting point to the final steady state.  This second method, often called a relaxation method, is easily implemented on a spreadsheet using Euler's method as described elsewhere on this course website.