Minimizing the Cost of a Distillation Column

The previous sections showed the number of stages required for one particular set of operating parameters. Four of these parameters could be adjusted to reduce the operating costs of the separation. By far the most important is the reflux ratio. We picked 20% higher than the minimum, but this was arbitrary. Here are a couple of other cases:

>>ptop(1.05*Rmn,110,0,0.8009,0.9997,0.05,100)
ans =
   48.9401
>>ptop(1.5*Rmn,110,0,0.8009,0.9997,0.05,100)
ans =
   25.8435
 

As the reflux ratio increases the number of stages required to meet our specifications goes down. In economic terms, the larger number of stages (at reflux ratios close to the minimum) requires large capital expenditures for the distillation column. As the reflux ratio is increased however, we find that utility costs go up to pat for the increased fluid flow rates in the column. Since the vapor through the column requires heat in the reboiler and the liquid reflux requires cooling, both add to the utilities cost.

The program enbal1 uses the mass and energy balance modules to find the energy requirements in the condenser and reboiler. It also executes ptop to find the number of stages. Here are several executions of the program to help construct a plot showing how the reflux ratio influences the column size and utility usage.

>>help enbal1
 
  function [N, RN, QC, QR, TC, TR]=enbal1(F,Rormin,p,q,xf,xd,xb,npx)
  Returns:
   N  theoretical plates
  RN  min R
   QC condenser heat load kJ/time
   QR reboiler heat load  kJ/time
   TC temperature in the condenser units of Tdeg
   TR temperature in the reboiler units of Tdeg
  Parameters:
   F flow rate in mols/time
   Rormin: R/Rmin
   p pressure kPa
   q quality of feed: 1=sat vapor, 0=sat liquid
   xf, xd, xb mol fraction of 1st compound in feed, 
      distillate, bottom
  npx max number of plates; default is 50.
 

In order to estimate the cost of the column we also need to know its diameter. The example in Peters & Timmerhaus use an estimate based on the allowed velocity of the vapor in the column. This approximation is used in the program: VapFlow:

>>help VapFlow
 
  [D,vol]=VapFlow(F,Rormin,p,q,xf,xd,xb)
  Returns
   D    column diameter in m
   vol  Volumetric flow rate of Vapor m3/s
  Parameters:
   F flow rate in mols/hr
   Rormin: R/Rmin
   p pressure kPa
   q quality of feed: 1=sat vapor, 0=sat liquid
   xf, xd, xb mol fraction of 1st compound in feed, 
      distillate, bottom
  Tdeg should be K
 

The two programs were used in a loop to find the variation of the main design parameters with reflux ratio:

>>for k=1:length(RORs)
  [N, RN, QC, QR, TC, TR]=enbal1(170000,RORs(k),110,0,0.8009,0.9997,0.05,100);
  Ns=[Ns,N];
  QCs=[QCs,QC];
  QRs=[QRs,QR];
  Ds=[Ds,VapFlow(170000,RORs(k),110,0,0.8009,0.9997,0.05)];
end
 

Here are plots of the results from the executions of enbal1 and VapFlow.

>>plot(RORs,Ns)
>>grid
>>title('Ideal Stages for Benzene-Toluene Separation')
>>xlabel('Reflux/Min Reflux')
>>ylabel('Stages')
>>gtext('P=110kPa, q=0.0, xf=.8009, xd=.9997, xb=.05')
 
 

 
>>plot(RORs,QCs, RORs,QRs)
>>QCs(8)<-- Making sure we know which is the condenser
 
ans =
   2.0794e+07
>>grid
>>title('Energy Required for Benzene-Toluene Separation')
>>gtext('P=110kPa, q=0.0, xf=.8009, xd=.9997, xb=.05')
>>xlabel('Reflux/Min Reflux')
>>ylabel('Heat load kJ/hr')
>>gtext('Feed rate = 170 kg mol/hr')
>>gtext('Reboiler')
>>gtext('Condenser')
 
 

 
 
>>plot(RORs,Ds)
>>grid
>>title('Column Diameter for Benzene-Toluene Separation')
>>xlabel('Reflux/Min Reflux')
>>ylabel('Diameter meters')
>>gtext('P=110kPa, q=0.0, xf=.8009, xd=.9997, xb=.05')
>>gtext('Maximum Velocity = 0.76 m/s')
>>gtext('Feed Rate = 170 kg mol/hr')
 

A costing function was written to estimate various costs for the column and associated heat exchangers. Help applied to this function gives most of the information about what it does. The equipment and utility costs were taken from the Peters and Timmerhaus example.

>>help cost1
 
  [TotC, STC, CWC, FC, COL, COND, REB]=cost1...
   (N,D,p,TC,QC,TR,QR)
  Argument  Gives
     N      Number of Ideal Stages
     D      Column diameter  m
     p      pressure in kPa
     TC     Condenser Temperature K
     QC     Condenser heat load kJ/hr
     TR     Reboiler Temperature
     QR     Reboiler heat load kJ/hr
  Returned    Gives 
  TotC     Total yearly cost
  STC      yearly cost of steam
  CWC      Cooling Water yearly cost
  FC       annual fixed costs
  COL      Column cost
  COND     Condenser cost
  REB      Reboiler cost
Other Notes:
  QC is in kJ/hr.: dH=116.18kJ/kg from steamprop.
    so is the density giving 437.7 kJ/gal and the cost factor
    is: 1.028e-07 $/kJ or to get from QC in kJ/hr to CWC in 
    $/yr we need to multiply by: 8.75e-4.
 

We can see how some of the major costs vary with reflux by:

>>for k=1:length(RORs)
  [TotC,STC,CWC,FC,COL,COND,REB]=cost1(Ns(k),Ds(k),...
   110,TC,QCs(k),TR,QRs(k));
   TCs=[TCs,TotC];
   Sts=[Sts,STC];
   Cws=[Cws,CWC];
   FCs=[FCs,FC];
   COLs=[COLs,COL];
   CONs=[CONs,COND];
   REBs=[REBs,REB];
 end
 
 
 

 
 
>>plot(RORs,TCs)
>>grid
>>title('Total Cost for Benzene-Toluene Separation')
>>xlabel('Reflux/Min. Reflux')
>>ylabel('Total Cost $/yr')
>>gtext('P=110kPa, q=0.0, xf=.8009, xd=.9997, xb=.05')
>>gtext('Feed Rate = 170 kg mol/hr')
 

We can see that the minimum total cost occurs very close to the minimum reflux ratio. The main reason for this is the high cost of steam compared to the other items shown in the next figure.

>>plot(RORs,[Sts;Cws;FCs])
>>title('Steam, Cooling Water & Fixed Costs')
>>xlabel('Reflux/Min Reflux')
>>ylabel('Cost $/yr')
>>gtext('P=110kPa, q=0.0, xf=.8009, xd=.9997, xb=.05')
>>grid
>>gtext('Feed Rate = 170 kg mol/hr')
>>gtext('Steam')
>>gtext('Cooling Water')
>>gtext('Fixed Equipment')
 

Return to Matlab Distillation Table of Contents.