# Batch Distillation

• Print
This chapter is from the book

## 9.7 Multicomponent Simple Batch Distillation

In the derivation of the Rayleigh equation in section 9.1, we noted that the equations are valid for multicomponent mixtures. However, since the general equilibrium expression

depends on the concentration of all components, a numerical solution is required. We previously considered multicomponent simple batch distillation in section 8.5.2 when we discussed residue curves. Although the notations and the procedures are different, the developments in Sections 8.5.2 and 9.2.1 are equivalent (see Problem 9.C4). Thus, the residue curve gives us the path of the xi during the batch distillation, and from the required equilibrium calculation, it has also calculated the yi values that correspond to the xi, although the vapor values may not be reported. The profiles of xi and yi values are sufficient information to numerically integrate the Rayleigh equation.

The procedure is easiest to illustrate if the relative volatility is constant. For batch distillation the equilibrium expression, Eq. (5-28) given in Problem 5.C1, is

The Rayleigh equation for the light component A becomes

To numerically integrate the integral in Eq. (9-27), we start with the spreadsheet calculation of a residue curve (see Section 8.5.2 and Figure 8-B1). For each increment in k, the incremental area is

Then the calculated value of the integral is

where ΔxA,k = xA,k – xA,k–1, and (1/[yA – xA])avg = 0.5[(1/[yA – xA])k-1 + (1/[yA – xA])k].

For example, if we calculate the area for k = 10 for the spreadsheet in Figure 8-B1, we have

|ΔxA,k| = |0.989960081 – 0.990021592| = 0.000061511, and

1/(yA – xA)avg = 0.5[1/(0.99617266 – 0.990021592) + 1/(0.996149968 – 0.989960081)] = 162.06.

Incremental area = (0.000061511)(162.06) = 0.009969.

If the values of h and kfinal from a residue curve calculation using the Euler method (Section 8.5.2) starting with the initial conditions of the batch distillation are available, then an estimate of the area of the integral in Eq. (9-27) is

where kfinal is the step for which xA = xA,final. This equation is useful for checking the result of a numerical integration. The rationale behind Eq. (9-28c) is explored in Problem 9.C5.

#### EXAMPLE 9-3. Multicomponent simple batch distillation

A simple batch distillation of the ideal system benzene (B), toluene (T), and cumene (C) is run. Charge is 1.0 kmole, and the feed is 70.0 mol% benzene, 20.0 mol% toluene, and 10.0 mol% cumene. Determine Wfinal, Dtotal, xT,fin, xC,fin, xD,B,avg, xD,T,avg, and xD,C,avg if xB,fin = 0.4. Pressure is 1.0 atm, and the relative volatilities are αBT = 2.4, αTT = 1.0, and αCT = 0.21.

Solution

1. Define. The apparatus is shown in Figure 9-1. F = 1.0 kmol, xW,B,initial = 0.70, xW,T,initial = 0.20, xW,C,initial = 0.1, and xB,fin = 0.40. Find Wfinal, Dtotal, xT,fin, xC,fin, xdist,B,avg, xdist,T,avg, and xdist,C,avg.

2. Explore. We need to find the area of the integral in Eq. (9-27) so that we can calculate Wfinal. Once the total area has been found, Wfinal is given by Eq. (9-11b). The values of xT,fin and xC,fin can be determined from the residue curve. We can then find Dtotal from the overall mass balance, and the xdist,i,avg values are determined from component mass balances.

3. Plan. To find the integral of the area in Eq. (9-27) we can use a spreadsheet to generate a residue curve starting with the initial conditions. The values of xT,fin and xC,fin are on the residue curve at the value of xB,fin. The more accurate method to calculate the area is Eq. (9-28b). Thus, we will calculate the y values from Eq. (9-28b) for every step k in the calculation of the residue curve. Then 1/(y – x)avg can be calculated for each step k, and the area of each step can be determined.

4. Do It. The spreadsheet and VBA program for the calculations is shown in this chapter’s appendix with h = 0.001. The values in the spreadsheet at xB,fin = 0.4 are found by inspecting the output manually and are shown in Table 9.1.

The cumulative area at xB,fin = 0.40 is approximately 1.1919.

From these results, xT,fin = 0.3175, and xC,fin = 0.2825.

Wfinal = F exp(–Area) = 1.0 exp(–1.1919) = 0.3036 kmol

Dtotal = F – Wfinal = 1 – 0.3036 = 0.6964 kmol

xB,dist,avg = [FxB,init – WfinalxB,fin]/Dtotal = [0.7 – (0.3036)(0.4)]/0.6964 = 0.8308

xT,dist,avg = [FxT,init – WfinalxT,fin]/Dtotal = [0.2 – (0.3036)(0.3175)]/0.6964 = 0.1488

xC,dist,avg = [FxC,init – WfinalxC,fin]/Dtotal = [0.1 – (0.3036)(0.2825)]/0.6964 = 0.0204

#### TABLE 9-1. Results from spreadsheet to determine xB,fin = 0.4 and area

 k xB xT xC yB yT yC 1/(yB-xB) Cum. Area 1193 0.400051 0.3175 0.2824 0.71815 0.2375 0.0444 3.144 1.1917 1194 0.39973 0.3176 0.2827 0.71792 0.2377 0.0444 3.143 1.1927
5. Check. ∑ xi,dist,avg = 1.0000 which is OK.

In addition, the approximate solution for the area, Eq. (9-28c), gives

Area = (k – 1)h = (1193 – 1)(0.001) = 1.192

which agrees with the area calculation from the spreadsheet.

6. Generalize. This method can be applied to any number of components if the relative volatilities are constant.

If the relative volatilities are not constant, we have two choices. First, we can divide the range of the batch operation into a series of intervals and use an average value for each interval. A reasonable definition of is the geometric average for each interval:

The recursion Eq. (8-30) uses values that correspond to the interval in which they appear.

The second procedure, which is more accurate, is to do a bubble-point calculation at every time step. The use of bubble-point calculations is explored in section 5.3 in the context of a steady-state, multistage analysis. For batch distillation the bubble-point calculation would be done at each step k. In essence, the resulting spreadsheet is a combination of the spreadsheets in Appendices 5A and 9A.

Fractions (Section 9.2.2) are often collected in multicomponent batch distillation. With only a single-stage, simple batch distillation does not give pure compounds unless the relative volatilities are very large. The calculation method is quite similar to the method shown in Example 9.3. For example, if a second fraction was collected for Example 9.3 when xW,B,2 = 0.3, we would look at the results of the spreadsheet in Appendix A to find the mole fractions and the cumulative area at this concentration. The values bracketing xW,B,2 = 0.3 are shown in Table 9.2.

#### TABLE 9-2. Calculations continuing Example 9.3 for fraction at xW,B,2 = 0.3

 k xB xT xC yB yT yC 1/(yB-xB) Cum. Area 1497 0.30018 0.3364 0.3635 0.63579 0.2969 0.0674 2.9796 1.495699 1498 0.29984 0.3364 0.3637 0.63547 0.2971 0.0675 2.9795 1.496699

The cumulative area at xW,B,2 = 0.3 is approximately 1.4962. If we subtract this value from the cumulative area = 1.1919 at xW,B = 0.4, we determine the area for the fraction from xW,B = 0.4 to xW,B = 0.3 is 0.3043. The initial charge for this fraction was Wfinal = 0.3036 calculated in Example 9.3. Then W(at xW,B = 0.3) = (0.3036)exp(–0.3043) = 0.2239 kmole. (See Problem 9.D27a for an equivalent calculation and Problem 9.D27b for the completion of the problem.)

Obviously, multistage batch columns are also used for multicomponent separations. One fairly simple approach is to assume that the changes in the column are quite slow and use a steady-state analysis similar to that employed in Section 9.6 for binary separations. Examples are the Fenske-Underwood-Gilliland shortcut method or stage-by-stage calculation with bubble-point calculation on each stage. Since these analyses ignore holdup on the stages and dynamics of the stages, they are considered simple models. More complicated models include stage holdup and dynamics and may include a mass transfer analysis. Readers interested in these more advanced topics should consult Sorensen (2014) for references and Diwekar (2012) and Mujtaba (2004) for calculation details.

• 🔖 Save To Your Account

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.