Elementsof Calculation Style

50 www.cepmagazine.org November 2001 CEP

Professional Development

he elements of style in calcu-lations are the choices in com-position that strengthen collab-oration between writers andreaders by helping them meetone another’s needs. The over-

riding need of both writers and readers is to nothave to keep track of too much new, unfamiliarmaterial at one time (1).

Elements of style include separate sections forassumptions, data, calculations and summaries.Each section can be prepared and read with a min-imum of in-depth thought, yet each section movesthe solution forward and serves as a resource foruse with the later sections.

Elements of style at the formula level, whichare even more helpful, include conventional sym-bols, reminders of variable definitions, referencenames and pages, reminders of values of depen-dent variables, and equations that are visible, asshown in Figure 1. These elements let writers andreaders understand and check formulas with aminimum of cognitive strain.

Elements of style in calculations are elaboratedbelow and illustrated with a sample calculation onpages 52–53.

1. Use calculation softwareUse software that displays the working formu-

las. The equations are the working parts of thecalculation. A calculation is easier to use if itsfunctions can be readily inferred by looking at

the features visible to the user. Calculation soft-ware packages, such as Mathcad and Calculation-Center, display a calculation’s working formulasand results together, and print them for conve-nient writing and reading (3, 4). Spreadsheetstypically don’t display formulas, so only simpleoperations like totaling the numbers in a columncan be readily inferred by looking at a spread-sheet. (Spreadsheets can be made to display for-mulas and results together through the use ofnamed ranges (5) and user-defined functions, asshown in Figure 2). Programming languages,such as Visual Basic and FORTRAN, display andprint the working formulas and results separatefrom each other, and often use coded syntax(when referencing data objects, for example),making calculations in these languages difficultto read and document.

Use software that calculates units as well asnumbers. Calculation software like Mathcad andCalculationCenter makes it possible to include di-mensional units in values and formulas and havethe software perform unit conversions automatical-ly, bypassing an otherwise major source of errors.

2. Set up for easy viewingMake calculations read from top to bottom. Use

the approach that experts use when solving easyproblems — work forward from the known data todetermine the unknown values that are needed (6).Enter data before calculations, and summarize in-puts before summarizing results. In the same man-

Style helps readers to understand,and calculations to succeed.

Elements of Calculation Style

T

James Anthony,Lockwood Greene

CEP November 2001 www.cepmagazine.org 51

ner, provide reference materials before referring to them.Place the table of contents before the actual contents, listreferences before they are used, and list variable names be-fore they are mentioned.

Use font, font size, and font style changes to help read-ers. Font changes can help readers distinguish equationsfrom text. Font size changes can improve readability ofequations. Both changes are performed automatically bythe calculation software used to prepare Figure 1 and thesample calculation. A font style change to bold face for theheadings can help readers rapidly scan through a calcula-tion, as shown in the sample calculation.

Use graphic lines mostly to convey information. Whengraphic lines are used sparingly, the lines that are usedstand out better (7). Horizontal lines can be used as blanksfor user-supplied data. Vertical lines can be used as revisionbars. Blank space can serve the same function that graphicslines are often used for in forms, providing separation be-tween unrelated items of information and helping readersread horizontally across rows of information in tables.

Include equipment number and page number at theright on each page. Sets of calculations arranged by majorequipment number including letter prefix can be leafedthrough easily to locate calculations that are of interest tothe reader.

Make formulas readable without comments. Help read-ers be able to review formulas independent of the explana-tory comments. Place formulas on separate lines fromcomments. Center formulas on the page, or indent them.Provide punctuation and text to allow theresulting material to be read straightthrough more easily than the formulasalone could be read.

3. Provide supporting informationWrite clear sentences. Start by writing

what you would say aloud. Then removeexcessive words. Rearrange phrases to im-prove clarity or eliminate ambiguity. Addwords wherever this will help readers un-derstand without having to concentrate ashard and without having to reread (8).Reread the work yourself later and edit itagain, repeating these steps.

List the contents. Simple, descriptiveheadings provide enough useful help toreaders to avoid the need for paragraphs ofexplanatory text.

State the objective. Readers expect tofind the most important information at thestart, and if not there, then at the end.They spend more time reading the infor-mation at the start. When the key theme isidentified up front, readers understand thesubsequent material better as they proceed

through it, and they proceed through it more quickly. Sketch the system. Sketches with text help people un-

derstand problems more thoroughly and help people movefurther toward solutions (9). The more useful diagramsshow spatial relationships, show key data at a glance, andplace information near the associated objects so that sym-bolic labels are not needed (10). When solving a problemthat requires the use of formulas to interpret physical in-formation, experts tend to insert an intermediate step re-describing the problem qualitatively (11). Sketches cancapture some of an expert’s understanding of the problemby emphasizing key considerations while leaving out sec-ondary details. Unfortunately, people who have less trou-ble proceeding with a problem tend to draw fewer sketch-es. As a result, they miss out on opportunities to helpnovices develop the skill of going beyond the literal fea-tures clearly evident in problem statements to infer addi-tional relationships that are important for constructing ef-fective solutions, which is the skill that novices are usual-ly most lacking (6).

State the approach, noting the key methods used. Namethe key method or methods used, and describe how they

■ Figure 1. Formula blockshelp readers to understand and

check formulas (2).

The effective interfacial areas per unit volume in the first and sec-

ond stage vessels, a1 and a2, are calculated from the gas holdups

εG1 and εG2 and the Sauter-mean bubble diameters or bubble vol-

ume-to-surface ratios dvs1 and dvs2 (Perry’s pages 5-69 and 5-43):

εG1 = 0.194676 and εG2 = 0.196213

dvs1 = 0.284573 in and dvs2 = 0.284552 in. 12 in = 1 ft.

■ Figure 2. Current spreadsheets, such as Microsoft Excel, can display formulas and results together.

a1 : =6 εG1

dvs1

and a2 : =6 εG2

dvs2

:

a1 = 49.255ft

2

ft3

and a2 = 49.648ft

2

ft3.


pageObjectives 1Approach 1References 1Symbols 1Assumptions 1

Constants and Conversions 1Hardware Data 1Property Data 2Operating Data 2

Calculations 2Results 2

ObjectiveEstimate minimum recommended slurry velocities vs. pipe diameter.

ApproachApply the Durand equation for the minimum transport velocity as recommended in Perry’s and Heywood, adding areasonable margin to the velocity as recommended in Heywood.

ReferencesData Sheet World Minerals, “Harborlite 2000 Technical Data Sheet,” World Minerals, Lompoc, CA, 2000 (enclosed)General Info World Minerals, “Perlite General Information,” World Minerals, Lompoc, CA, 2000 (enclosed)Heywood Heywood, N.I., “Stop Your Slurries from Stirring Up Trouble,” Chemical Engineering Progress,

pp. 21-41, September 1999 (enclosed)P & ID WRC, “Filter Aid Storage/Delivery P&ID,” Dwg. No. PR-001, Rev. A, Confidential Client, 2000

(in project master file)Perry’s Green, Don W., editor, “Perry’s Chemical Engineers’ Handbook - 7th Ed.,” McGraw-Hill, New York,

pages 6-30, 6-31, 10-72, and 10-73, 1997 (enclosed)Symbols

Cs maximum volume fraction solids -

D pipe diameters in

d particle diameter mm

FL Durand factor for minimum suspension velocity -

g gravitational acceleration

s ratio of solid density to liquid density -

sgL specific gravity of liquid -

sgs specific gravity of solid -

V minimum recommended slurry velocities

VM2 minimum transport velocities

Ws maximum weight fraction solids -

ρL density of liquid

ρs density of solid

AssumptionsAssume pipe is Schedule 40S.

Constants and ConversionsThe gravitational acceleration g is:

g = 32.174 .

Hardware DataConsider several pipe diameters D (Perry’s pages 10-72 and 10-73)

ft

sec 2

lb

ft3

lb

ft3

D: =

1.610

2.067

3.068

in.

ftsec

ftsec

ft

sec 2

S A M P L E

Property DataThe specific gravity of the liquid, sgL, is (P&ID):

sgL := 1.0.

The specific gravity sgs and particle size d of the solid are (Data Sheet):

sgs := 2.3. d := 0.0431 mm.Operating Data

The maximum weight fraction solids, Ws, is (General Info page 19):

Ws := 10%.Calculations

The maximum volume fraction solids, Cs, follows from the maximum weight solids fraction Ws and the specific gravities of the liquid and solid, sgL and sgs:

Ws = 10%, sgL = 1.0, and sgs = 2.3.

Cs = 4.6%. Since 100% = 1, Cs = 0.046.

The Durand factor FL can be read from a chart given the particle size d and the maximum volume fraction solid, Cs

(Perry’s page 6-31 Figure 6-33):For d = 0.043 mm and Cs = 0.05, FL := 0.6.

The densities of the liquid and solid, ρL and ρs, follow from specific gravities sgL and sgs, within engineering accuracy:sgL = 1.0 and sgs = 2.3.

ρL := sgL ⋅ 62.45 ; ρL = 62.45 . ρs := sgs ⋅ 62.45 ; ρs := 143.63 .

s, the ratio of solid density ρs to liquid density ρL, is:

ρs = 143.63 . ρL = 62.45 .

s := ; s = 2.3.

The minimum transport velocities VM2 as a function of the Durand factor FL, gravitational acceleration g, pipe diameters, D, and ratio of liquid density to solid density, s (Perry’s ):

FL = 0.6. g = 32.174 . s = 2.3.

For in, given that 12 in = 1 ft and VM2 := FL ⋅ [2 ⋅ g ⋅ D ⋅ (s–1)]0.5,

The minimum slurry velocities V can be calculated given the minimum transport velocities VM2 vs. the pipe diametersD (Heywood page 28):

For in, V := 125% ⋅ VM2 ;

ResultsMinimum slurry velocities V vs. pipe diameters D are:

For in, V =

2.5

2.8

3.5

ftsec

.D =

1.610

2.067

3.068

V =

2.5

2.8

3.5

ftsec

.D =

1.610

2.067

3.068

VM2 =

2.0

2.3

2.8

ftsec

.D =

1.610

2.067

3.068

ft

sec 2

ρLρS

lb

ft3

lb

ft3

lb

ft3

lb

ft3

lb

ft3

lb

ft3

Cs: =

Ws

sgs

Ws

sgs

+100% – Ws

sgL


C A L C U L A T I O N

were located and chosen. Avoid describing the details ofthe calculation in text form, since the actual formulas willbe displayed later when they are used and will be nearlyself-explanatory, while text descriptions of the formulaswould take extra effort and extra skill to write and wouldbe less clear and less helpful.

Name, list and enclose references. Identify convenient,clear references for each formula used and data value en-tered. Name each reference using a short descriptive namesuch as the lead author’s name. Give titles and pageranges. Provide readers copies of references, so that theycan find out things for themselves right away, while theyare most interested.

Use conventional symbols. Match the conventional no-tation in the area of interest for ready recognition. Use thesame main symbol for all variables of a given type, and usesubscripts to differentiate the family members from oneanother. Greek letters and subscripts can be typed directlyinto a calculation when calculation software like Mathcador CalculationCenter is used.

List complete symbols, including subscripts, and pro-vide complete descriptions and standard units. Descrip-tions that include subscripts can eliminate guesswork.Standard units provide added descriptions of the symbols.

Promote alternative methods. Describe alternative ap-proaches and possible outcomes. Present the alternativesas positive possibilities, so they will be considered morelikely and will therefore more effectively counterbalancethe base case that is being presented positively. Consider-ing alternatives reduces overconfidence. This promotesprogress on problems (12), improves decision-making,and may improve self-checking by writers and error-checking by readers.

4. Include text comments and equation-stylecomments with the working formulas

Provide comments that supplement formulas but do notdescribe them. The working formulas do the actual calcu-lation. Writers and readers need to be helped to reviewthe formulas carefully, and need to not be lulled into afalse sense of security by comments that seem to tell acomplete story, and as a result, encourage them to skipover the equations (13).

Repeat the description of each symbol each time it isused in a formula. Provide the description and repeat thesymbol, including any subscript.

List a source for each formula. Identify a convenientsource that states the formula clearly. List the source’sshort, descriptive name from the reference list, and identifythe page or pages where the formula is defined. Include theformula number from the source, where helpful.

Repeat the value of each symbol used in standard unitseach time it is used. Help people learn the relative magni-tudes of terms and check the values of input data and inter-mediate results at every opportunity they have to do so.

Provide conversion factors each time they are used.Conversion factors often help reassure readers of the rea-sonableness of calculations and occasionally help writersfind mistakes. When calculation software is used, it takeslittle effort to call up predefined conversion factors.

Check function definitions by calculating known values.Check function definitions for temperature-dependentproperties, for instance, by evaluating the equations at tem-peratures where the property values are known.

Show a formula’s comments together with the formulaon the same page. Add a page break before an assembledformula block if needed to keep the block together on a sin-gle page. Self-contained formula definitions and evaluationsthat can be seen together at a glance are easier to review.

5. Provide assumptions, inputs and calculationsNote assumptions. Assumptions can include notes on

how the mathematical models that are used oversimplifythe behavior that they describe. They can also includenotes on how the experimental approach underlying amethod differs from the particulars of the process that isbeing analyzed. The assumptions that can be the most diffi-cult to recognize are the underlying beliefs shared by pre-vious workers, the writer, and the readers when they all arefrom the same era and have similar backgrounds. Explain-ing assumptions early, especially assumptions about factorsthat cannot be changed by the writer or the readers, pro-duces more realistic assessments about the reliability of re-sults. Reducing overconfidence improves checking, whichimproves accuracy.

Enter any assumed data values. Assumptions can also in-clude reasonable guesses of data values that are not knownfor certain. Provide an equation block for each assumed datavalue, as shown in the sample calculation. Repeat the de-scription and the symbol, including any subscripts. Displaythe value in standard units for reasonability checking.

Enter the hardware data. Provide an equation block foreach hardware data value. Repeat the description and thesymbol, including any subscripts. Repeat the source’sname from the references and note the applicable page orpages. Enter the data in the dimensional units that wereused in the source. Provide any conversion factors used, inunits that are as familiar as possible. Display the data instandard units for reasonability checking.

Enter the property data. Provide an equation block foreach property data value. Repeat the description and thesymbol including any subscripts, the source of the data,the data in the units used in the source, any conversionfactors, and the data in standard units. If a property is afunction of variables such as temperature, pressure orcomposition, enter the property as a function that can beevaluated later based on the values of the variables at thatpoint in the calculation.

Enter the operating data. Provide an equation block foreach operating data value. Provide the description and the

Professional Development


symbol including any subscripts, the source of the data, thedata in the units used in the source, any conversion factors,and the data in standard units.

Enter the calculation formulas. Provide a formulablock, like that in Figure 1, for each calculation formula.Provide text descriptions of the dependent variables andthe independent variables, and provide a reference sourcefor the formula. Then provide the values of the indepen-dent variables in standard units, any conversion factors, theformula, and the results.

6. Provide summary informationRepeat key assumptions. An assumption may be crucial,

and well worth highlighting again by including it in thesummary information at the end of a calculation.

Summarize the key input parameter values. Summariz-ing key input parameters near the end of calculations high-lights them for the writer as well as for the reader. Also,sometimes it is convenient to set up a file containing a sin-gle case and then change input parameters and save sepa-rate files for new cases. In such situations, it is critical to

point out the values of the changed parameters to distin-guish the cases from one another.

Summarize the results. Conclude by returning to the bigpicture and recalling key intermediate results and final resultsfor the benefit of casual readers and careful readers alike.

7. Get calculations checkedSeek checking, whether by experts or by interested col-

leagues. Unless people get very detailed feedback ontheir performance, they tend to be overconfident in theirown abilities. They do not perform nearly enough self-checks on material they believe is correct (14). As a re-sult, errors of omission are almost never identified andcorrected by the person who made them. With checking, afresh viewpoint enters the situation, and errors of omis-sion can be corrected.

Calculations will get easierThe calculation approach shown here easily scales up to

handle tougher problems. An example of a more difficultcalculation is available at www.cepmagazine.org (2).

This approach produces accurate results, is easy to read,and is easy to reuse. It is particularly helpful when experi-mental data on a process are unavailable, data cannot beobtained cheaply and quickly, and readily available calcu-lation methods do not cover the process in question.

This list of uses barely hints at the broader roles thatcould quickly develop for approaches like this. Affordablecalculation software already provides the capability toembed subprograms and the capability to define graphicalsymbols that have smart interconnections. Soon, such soft-ware could provide the capability to embed subprograms insmart symbols. Libraries of thermodynamic property cal-culation routines and unit operations could emerge easilyfrom environments of friendly competition and sharing, inacademic settings and in industry. It could ultimately bepossible to simply connect components from reliablesources and produce accurate and reliable process simula-tions and other calculations.

Even wider impacts are imaginable. Calculation ap-proaches developed for process applications could easilybe adapted to other uses in science and in education.

Much can get easier when collaboration is improved bystyle in calculations. CEP


JAMES ANTHONY is a process engineer with Lockwood Greene, St. Louis,

MO (Phone: (314) 919-3208; Fax: (314) 919-3201; E-mail:

[email protected]). He has process design experience with chemical,

pharmaceutical, and beverage applications, which have included the

manufacture of iodine products, abrasives, inorganic salts, alkyds,

polyesters, polyurethanes, synthetic pharmaceuticals, soy sauce, and

tea. He also has aerospace design experience developing jet engine air

inlets, piston-propeller systems, sensors, adhesive-bonded structures

and molded plastic parts. He has a BS in chemical engineering from the

Univ. of Missouri – Rolla and an MS in mechanical engineering from

Washington Univ. He is a registered professional engineer.

Literature Cited1. Miller, G. A., “The Magical Number Seven, Plus or Minus Two:

Some Limits on our Capacity for Processing Information,” The Psy-chological Review, 63 (2), pp. 81–97 (1956).

2. Anthony, J., “Chloroform Plan,” available via http://www.cep-magazine.org (2001).

3. Phillips, J. E., and J. D. Decicco, “Choose the Right MathematicalSoftware,” Chem. Eng. Prog., 95 (7), pp. 69-74 (July 1999).

4. Sandler, S. I., “Spreadsheets for Thermodynamics Instruction: An-other Point of View,” Chem. Eng. Edu., 31 (1), pp. 18-20 (Winter1997).

5. Lira, Carl T., “Advanced Spreadsheet Features for Chemical Engi-neering Calculations,” submitted to Chem. Eng. Edu.,http://www.egr.msu.edu/~lira/spreadsheets.pdf (2000).

6. Chi, M. T. H., et al., “Expertise in Problem Solving,” in Sternberg,R. J., “Advances in the Psychology of Human Intelligence,” Vol. 1,Lawrence Erlbaum Associates, Publishers, Hillsdale, NJ, pp. 7-75;see pp. 18, 19, 35, and 71 (1982).

7. Tufte, E. R., “The Visual Display of Quantitative Information,”Graphics Press, Cheshire, CT, p. 96 (1983).

8. Cook, C. K., “Line by Line: How to Improve Your Own Writing,”Houghton Mifflin, Boston (1985).

9. Mayer, R. E., “Models for Understanding,” Review of EducationalResearch, 59 (1), pp. 43-64 (1989).

10. Larkin, J. H., and H. A. Simon, “Why a Diagram is (Sometimes) WorthTen Thousand Words,” Cognitive Science, 11 (1), pp. 65-99 (1987).

11. Larkin, J. H., “Processing Information for Effective Problem Solv-ing,” Engineering Education, 70 (3), pp. 285-288 (December 1979).

12. Platt, J. R., “Strong Inference,” Science, 146 (3642), pp. 347-353(1964).

13. Kernighan, D. W., and P. J. Plauger, “The Elements of Program-ming Style,” 2nd ed., McGraw-Hill, New York, pp. 141-152 (1978).

14. Allwood, C. M., “Error Detection Processes in Statistical Problem Solv-ing,” Cognitive Science, 8 (4), pp. 413-437; see pp. 419 and 431 (1984).

LOCKWOOD GREENEConfidential Client

Chloroform Effluent Plan

Job, Item:Date, Page:

Prep, Check, Appr:

A7947 010704.01, CHCl3 Plan 7/22/2001, 1 of 37 J Anthony, _________, _________

page

Objective 1Approach 1References 2Symbols 3Assumptions 6

Conversions and Constants 7Hardware Data 8Property Data 8Operating Data 13

Steam Flows 14Interfacial Areas 19Mass Transfer Coefficients 27Stripping and Dilution 29

Key Parameters Summary 34Results Summary 35

Objective

Model the lime press effluent chloroform concentration after raffinate stripping, reslurrying using stripper effluent, and reslurrying using four chloroform-free streams.

2nd S tage1st S tage

Steam

R affinate

2. L im e Liquor3. W ater4. W ater5. W ater

1. S trip L iquor

L im e Press SolidsFOFO

Approach

Calculate restriction orifice flows using Benedict's general equation as suggested on Perry's page 10-16. Calculate sparger mass transfer per Perry's page 5-69, using bubble diameters calculated as referenced on Perry's page 14-71 and interfacial areas calculated as referenced on Treybal page 144. Calculate dilution using available data.

Chloroform Plan.mcd NOT CHECKED



A7947 010704.01, CHCl3 Plan 7/22/2001, 2 of 37

References

Akita Akita, K., and F. Yoshida, "Gas Holdup and Volumetric Mass Transfer Coefficient in Bubble Columns", Industrial and Engineering Chemistry Process Design and Development, 12(1), pp. 76-80, 1973 (enclosed)

Baseline Anthony, J., "Confidential Client Lime Press Chloroform Effluent Baseline", [ Chloroform Baseline].mcd, Lockwood Greene, St. Louis, MO, 2000 (enclosed)

Benedict Benedict, R. P., "Loss Coefficients for Fluid Meters", Journal of Fluids Engineering, 99(1), pp. 245-248, 1977 (enclosed)

Calderbank Calderbank, P. H., and M. B. Moo-Young, "The Continuous Phase Heat and Mass-Transfer Properties of Dispersions", Chemical Engineering Science, 16, pp. 39-54, 1961 (enclosed);

Calderbank, P. H., in Uhl, V. and J. Grey, Editors, "Mixing", Volume 2, Chapter 6, Academic Press, New York, 1967 (source of small bubble-size mass transfer correlation listed in Perry's page 5-69, per Perry's page 5-8 reference 109, Kirwan 1987) (not enclosed; not available at Washington University until 9/18 or later)

Cussler Cussler, E. L., "Diffusion, Mass Transfer in Fluid Systems", page 251, Cambridge University Press, 1984 (enclosed)

DIPPR Daubert, T. E., R. P. Danner, H. M. Sibul, and C. C. Stebbins, "Physical and Thermodynamic Properties of Pure Chemicals: Data Compilation", chloroform's fixed properties, vapor pressure, and surface tension; water's fixed properties, vapor pressure, ideal gas heat capacity, second virial coefficient, vapor viscosity, and surface tension, Taylor & Francis, Bristol, PA, extant 1994 (enclosed)

Geankoplis Geankoplis, C. J., "Transport Processes and Unit Operations", 3rd Ed., pp. 450-453, PTR Prentice Hall, Englewood Cliffs, NJ, 1993 (enclosed)

Godbole Godbole, S. P., and Y. T. Shah, "Design and Operation of Bubble Column Reactors", in Cheremisinoff, N. P., "Encyclopedia of Fluid Mechanics" Vol. 3, pp. 1216-1239, Gulf Publishing, Houston, TX, 1986 (enclosed)

Grace Grace, H. P., and C. E. Lapple, "Discharge Coefficients of Small-Diameter Orifices and Flow Nozzles", Transactions of the ASME, 73, pp. 639-647, 1951 (enclosed)

JH Anthony, J., Notes from phone call, 6/22/00 (enclosed)

Hwang Hwang, Y.-L., J. D. Olson, and G. E. Keller II, "Steam Stripping for Removal of Organic Pollutants from Water. 2. Vapor-Liquid Equilibrium Data", Industrial and Engineering Chemistry Research, 31, pp. 1759-1768, 1992 (enclosed)

Kumar Kumar, A., T. E. Degaleesan, G. S. Laddha, and H. E. Hoelscher, "Bubble Swarm Characteristics in Bubble Columns", The Canadian Journal of Chemical Engineering, 54, pp. 503-508, December 1976 (enclosed)

Manual SS, Confidential Company, "Confidential Client Raffinate Stripping System Project A-7819 Operating Manual", Sparging Hole Requirements, Confidential Client, 11/11/99 (enclosed)

Mathcad Help Mathsoft, Mathcad 2000 Solve Block help, Mathsoft, Cambridge, MA, 2000; Mathsoft, Resource Center: Polynomial Regression, Mathsoft, Cambridge, MA,

2000 (enclosed)

P&ID SS, Confidential Company, "Raffinate Stripper P&ID", Dwg. 0000-102-001, Rev. 8,




A7947 010704.01, CHCl3 Plan 7/22/2001, 3 of 37

y gConfidential Client, 11-8-99 (enclosed)

Perry's Green, Don W., editor, "Perry's Chemical Engineers' Handbook", 7th Ed., pages 1-18, 1-19, 2-355, 4-7, 5-7, 5-8, 5-56, 5-69, 6-49, 6-50, 10-4, 10-14, 10-16, 10-72, 10-140, 14-70, 14-71, and 14-74, McGraw-Hill, New York, 1997 (enclosed)

Pfaudler Pfaudler, "30 Gal. (20 I.D. x 24 3/4 Dp.) POWCT VR-30 Gl. Stl. "P" Tank", Dwg. CE279-0863-65, Pfaudler, Rochester, NY, 11/27/79;

Pfaudler, "50 Gal. (24" I.D. x 28 3/4" Dp.) JOWCT Glasteel Vacuum Receiver", Dwg. CE279-0860, Pfaudler, Rochester, NY, 12/3/79;

Pfaudler, "Chemstor/Storage Tank", Pfaudler, Rochester, NY, 4/97 (enclosed)

Pipe Spec SS, Confidential Company, "Raffinate Stripper Piping Material Specifications", Dwg. 0007-702-004, Rev. 0, Confidential Client, 5/28/99 (enclosed)

PQ Data SS, Confidential Company, "Chloroform Concentration in Raffinate Stripper", Confidential Client, 11/01/00 (enclosed)

PQ Report Keeler, R., "Amendment to Process Qualification Protocol of the Raffinate Stripper", Confidential Client, 1/5/00 (enclosed)

SS Anthony, J., Notes from meeting, 7/26/00 (enclosed)

Treybal Treybal, R. E., "Mass-Transfer Operations", 3rd Ed., pp. 143-144 and 211-217, McGraw-Hill, New York, 1980 (enclosed)

Wilkinson Wilkinson, P. M., and L. L. van Dierendonck, "A Theoretical Model for the Influence of Gas Properties and Pressure on Single-Bubble Formation at an Orifice", Chemical Engineering Science, 49(9), pp. 1429-1438, 1994 (enclosed)

Wright Wright, D. A., S. I. Sandler, and D. DeVoll, "Infinite Dilution Activity Coefficients and Solubilities of Halogenated Hydrocarbons in Water at Ambient Temperatures", Environmental Science and Technology, 26(9), pp. 1828-1831, 1992 (enclosed)

Yaws Yaws, C. L., "Handbook of Transport Property Data", pp. 141-145, Gulf Publishing, Houston, TX, 1995 (enclosed)

Symbols

A1 and A2 cross-sectional areas of first and second stage vessels ft2

a1 and a2 effective interfacial area per unit volume of vesselsft

2

ft3

aT1 and aT2 total hole areas of first and second stage spargers ft2

Bw second virial coefficient of water vaporft

3

lbmole

CPgw ideal gas heat capacity of water vapor joule

kgmole K⋅D1 and D2 diameters of first and second stage restriction orifices in

Dc0w diffusivity of chloroform in infinite dilution in waterft

2

sec

Dv1 and Dv2 diameters of stripper first and second stage vessels in

dN1 and dN2 hole diameters of first and second stage spargers in




A7947 010704.01, CHCl3 Plan 7/22/2001, 4 of 37

Fr1 and Fr2 Froude numbers, inertial_force

gravity_force, of first and second stage vessels -

Frgc1 and Frgc2 Froude numbers for transition to foaming flow -

GM1 and GM2 gas-phase molar fluxes based on vessel cross-sectional areas lbmole

hr ft2⋅

Gr1 and Gr2 Grashof numbers of first and second stage vessels -

ID1 and ID2 piping inside diameters of restriction orifices in

K1 and K2 loss coefficients of first and second stage restriction orifices -

Kc1 and Kc2 vapor-liquid equilibrium ratios of chloroform at infinite dilution -

in water in the first and second stage vessels

KL1 and KL2 overall liquid-phase mass-transfer coefficients in vesselsft

hr

kL1 and kL2 individual liquid-phase mass-transfer coefficients in vesselsft

hr

kw heat capacity ratio of water vapor -

LM1 and LM2 liquid-phase molar fluxes based on vessel cross-sectional areaslbmole

hr ft2⋅

Lover1 and Lover2

level of clear liquid at overflow of first and second stage vessels %

Mw and Mc molecular weight of water and of chloroformlb

lbmole

NA1 and NA2 molar fluxes based on interfacial areas in vesselslbmole

hr ft2⋅

n1 and n2 hole counts of first and second stage spargers -

P0 absolute pressure upstream of the restriction orifices psi

Pv absolute pressure in first and second stage vessels psi

Pvc vapor pressure of chloroform psi

Pvw vapor pressure of water psi

qr volumetric flow of raffinategal

min

Rg universal gas constantjoule

mole K⋅r pressure ratios of the orifices -

rc1 and rc2 critical pressure ratios of restriction orifices -

ReN1 and ReN2 Reynolds numbers, inertial_force

viscous_force, of spargers -

Sc1 and Sc2 Schmidt numbers of first and second stage vessels -

Sh1 and Sh2 Sherwood numbers of first and second stage vessels -

T absolute temperature K

T0 absolute temperature upstream of the restriction orifices K

TF magnitude of the temperature in degrees F -

Tref absolute temperature reference value K




A7947 010704.01, CHCl3 Plan 7/22/2001, 5 of 37

Tv1 and Tv2 absolute temperatures in first and second stage vessels K

uN1 and uN2 superficial velocities based on hole areas of spargersft

sec

V1 and V2 velocities in first and second stage restriction orificesft

sec

Vcap1 and Vcap2

rated capacities of first and second stage vessels gal

Voper1 and Voper2

operating volumes of first and second stage vessels gal

W1 and W2 mass flows in first and second stage restriction orificeslb

hr

WT total mass flow of steamlb

hr

wcr weight fraction chloroform in raffinate -

wcs weight fraction chloroform in stripped liquid -

wcs_adj weight fraction chloroform in stripped liquid, adjusted -

Weg1 and Weg2 Weber numbers, inertial_force

surface_tension_force, of spargers -

x1v1 and y1v1 liquid and vapor mole fractions chloroform -

at bottom - position 1 - of first stage vesselx2v1 and y2v1 liquid and vapor mole fractions chloroform -

at top - position 2 - of first stage vesselx1v2 and y1v2 liquid and vapor mole fractions chloroform -

at bottom - position 1 - of second stage vesselx2v2 and y2v2 liquid and vapor mole fractions chloroform -

at top - position 2 - of second stage vesselβ1 and β2 diameter ratios of first and second stage restriction orifices -

εG1 and εG2 gas holdups in first and second stage vessels -

γc activity coefficient of chloroform at infinite dilution in water -

µlw liquid viscosity of water cP

µvw vapor viscosity of waterlb

ft sec⋅

ρ0 vapor density upstream of the restriction orificeslb

ft3

ρ lw liquid density of waterlb

ft3

ρv vapor density in first and second stage vesselslb

ft3

σw surface tension of waternewton

m




A7947 010704.01, CHCl3 Plan 7/22/2001, 6 of 37

Assumptions

The levels of clear liquid at overflow of the first and second stage vessels, Lover1 and Lover2 , are

taken to be (initial design, 100% : SS):

Lover1 60%:= and Lover2 60%:= .

Vessel appurtenances - sparger pipes, level elements, etc. - are neglected in calculating vessel cross-sectional areas.

The flows upstream of the restriction orifices are taken to be saturated vapor.

The absolute pressures downstream of the restriction orifices are taken to equal those of the first and second stage vessels. For critical flow through thick-plate orifices, the flow increases only about 2% as the pressure ratio is decreased to well below the critical pressure ratio (Grace pages 645 and 640).

The weight fraction chloroform in the raffinate, wcr , is taken to be (PQ Data from 7/7/99):

wcr

3926

106

3581

106

+3760

106

+

3:= : wcr 3756

1

106

= .

The absolute pressures and temperatures in the first and second stage spargers are taken to be those of the vessels, not those of the restriction orifices. Bubbles reach pressure equilibrium with the vessel before they disengage from the sparger, which is reasonable since the flow through the sparger orifices is subsonic. Bubbles reach thermal equilibrium with the vessel before they disengage from the sparger, which is reasonable given the experimental observation that the gas-phase mass transfer resistance is negligible (Perry's page 5-69 Table 5-25 Condition Z).

The absolute pressures and temperatures in the first and second stage vessels, Pv , Tv1 , and Tv2 ,

are taken to be the following values at all points in the vessels, neglecting the small liquid heads at the sparger holes and the small pressure drops in the vapor discharge piping and condenser (PQ Data from 10/5/99):

Pv 14.696psi:= .

Tv191 92+ 91+ 91+ 94+

5273.15+

K:= and Tv2 100 273.15+( )K:= :

Tv1 364.95K= and Tv2 373.15K= .

The vapor densities and vapor viscosities in the first and second stage vessels are taken to be

those of water at the conditions in the second stage vessel. The liquid densities and surface tensions in the first and second stage vessels are taken to be those of water at the conditions in the vessels. Changes in properties due to chloroform and tar present are neglected.

The volumetric flow of raffinate, qr , is taken to be (PQ Data from 9/17/99):

qr

4.7gal

min3.67

gal

min+

2:= ; qr 4.185

gal

min= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 7 of 37

The bubble diameter correlation (Kumar) is for spargers mounted horizontally, while the spargers are mounted vertically, which may increase bubble coalescence, reducing interfacial area and mass transfer.

The gas holdup correlation (Akita) and mass transfer correlation (Perry's page 5-69, Calderbank) is for bubbles uniformly distributed throughout the cross-section of the vessel. The spargers, on the other hand, are vertical pipes with holes every 90 degrees around (Manual), and the spargers are mounted on off-center nozzles. The nonuniform bubble distributions may reduce mass transfer.

Conversions and Constants

The centipoise cP is defined as follows:

cP 102−

poise⋅:= .

The millinewton mN is defined as follows:

mN 103−

newton⋅:= .

The amount of substance kgmole is defined as follows:

kg 1000gm= , so kgmole 1000mole:= .

The amount of substance lbmole is defined as follows:

lb 453.592370gm= , so lbmole 453.592370mole:= .

The reference temperature, Tref , where the temperature-dependent property values listed in the

DIPPR fixed properties list are calculated, is:

Tref 298.15K:= .

The magnitude of the temperature in degrees F, TF , as a function of the absolute temperature T is

(Perry's page 1-19):

TF T( )T

K1.8⋅ 459.69−:= .

Tref 298.15K= : TF Tref( ) 76.980= .

The universal gas constant Rg is (Perry's page 1-19):

Rg 8.3144joule

mole K⋅:= . Alternatively, Rg 10.73

psi ft3⋅

lbmole R⋅= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 8 of 37

Hardware Data

The restriction orifice diameters of the first and second stages, D1 and D2 , are (Baseline: JH):

D116

64in⋅:= and D2

19

64in⋅:= :

D1 0.250 in= and D2 0.297 in= .

The pipe inside diameters of the first and second stage restriction orifices, ID1 and ID2 , are (P&ID,

Pipe Spec, Perry's page 10-72):

ID1 1.049in:= and ID2 ID1:= .

The sparger hole diameters of the first and second stages, dN1 and dN2 , and hole counts n1 and

n2 are (Baseline: Manual Sparging Hole Requirements):

dN111

64in:= and dN2

13

64in:= . 12in 1 ft= .

n1 16 4+ 2+:= and n2 n1:= .

The vessel diameters of the first and second stages, Dv1 and Dv2 , are (Pfaudler):

Dv1 20in:= and Dv2 24in:= .

The rated capacities of the first and second stage vessels, Vcap1 and Vcap2 , are (Pfaudler

Chemstor/Storage Tank):

Vcap1 32gal:= and Vcap2 52gal:= .

These equal the straight side capacity plus the bottom head capacity, based on a check using the standard formula for the capacity of an ellipsoidal head with a given height, using data for a Pfaudler VR-50 (Perry's page 10-140 and Pfaudler).

Property Data

The molecular weight of water, Mw , is (DIPPR):

Mw 18.015kg

kgmole:= ; Mw 18.015

gm

mole= ; Mw 18.015

lb

lbmole= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 9 of 37

The liquid density of water, ρ lw , as a function of the absolute temperature T is defined by the liquid

molar density as a function of the absolute temperature and the molecular weight of water Mw (DIPPR):

ρ lw T( ) 273.16 K⋅ T≤( ) T 333.15 K⋅≤( )⋅5.4590

0.30541 1

T

K

647.13

−

0.0810

+

⋅

333.15 K⋅ T<( ) T 403.15 K⋅≤( )⋅4.9669

0.277881 1

T

K

647.13

−

0.18740

+

⋅+

...

403.15 K⋅ T<( ) T 647.13 K⋅≤( )⋅4.3910

0.248701 1

T

K

647.13

−

0.25340

+

⋅+

...

kgmole

m3

⋅ Mw⋅:= ; Mw 18.015kg

kgmole= .

Tref 298.15K= :

ρ lw Tref( )Mw

55.239kgmole

m3

= .

ρ lw Tref( ) 995.122kg

m3

= . 1kg 2.204623 lb= . 1ft 0.304800m= .

ρ lw Tref( ) 62.123lb

ft3

= .

The vapor pressure of water, Pvw , as a function of the absolute temperature T is (DIPPR):

Pvw T( ) 273.16 K⋅ T≤( ) T 647.13 K⋅≤( )⋅ exp 73.6497258.2−

T

K

+ 7.3037− lnT

K

⋅+

4.1653 106−⋅

T

K

2.0000

⋅+

...

⋅ Pa⋅:= .

Tref 298.15K= :

Pvw Tref( ) 3170Pa= . 1Pa 0.000145038psi= .

Pvw Tref( ) 0.4598psi= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 10 of 37

The ideal gas heat capacity of water, CPgw , as a function of the absolute temperature T is (DIPPR):

CPgw T( ) 100.00K T≤( ) T 2273.1K≤( )⋅ 3.3363 104⋅

2.6790 104⋅

2610.5

T

K

sinh2610.5

T

K

2

⋅+

...

8896.0

1169.0

T

K

cosh1169.0

T

K

2

⋅+

...

⋅joule

kgmole K⋅⋅:= .

Tref 298.15K= :

CPgw Tref( ) 33578joule

kgmole K⋅= . 1

joule

kgmole K⋅0.000238851

BTU

lbmole R⋅= .

CPgw Tref( ) 8.020BTU

lbmole R⋅= .

Mw 18.015lb

lbmole= .

CPgw Tref( )Mw

0.445193BTU

lb R⋅= .

The heat capacity ratio of water vapor, kw , as a function of the ideal gas heat capacity of water

vapor Cpw T( ) and the universal gas constant Rg is (Perry's page 4-7):

kw T( )CPgw T( )

CPgw T( ) Rg−:= .

Tref 298.15K= :

CPgw Tref( ) 33578joule

kgmole K⋅= . Rg 8314

joule

kgmole K⋅= .

kw Tref( ) 1.329106= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 11 of 37

The second virial coefficient of water vapor, Bw , as a function of the absolute temperature T is

(DIPPR):

Bw T( ) 273.15 K⋅ T≤( ) T 2273.1 K⋅≤( )⋅ 0.022226.38−

T

K

+1.675− 10

7⋅

T

K

3+

3.894− 1019⋅

T

K

8

3.133 1021⋅

T

K

9++

...

⋅m

3

kgmole⋅:= .

Tref 298.15K= :

Bw Tref( ) 1.1536−m

3

kgmole= . 1m 3.280840 ft= . 1lbmole 0.453592kgmole= .

Bw Tref( ) 18.479−ft

3

lbmole= .

The liquid viscosity of water, µlw , as a function of the absolute temperature T is (DIPPR):

µlw T( ) 273.16 K⋅ T≤( ) T 646.15 K⋅≤( )⋅ exp 52.843−3703.6

T

K

+ 5.8660 lnT

K

⋅+

5.8790− 1029−⋅

T

K

10.000

⋅+

...

⋅ Pa⋅ sec⋅:= .

Tref 298.15K= :

µlw Tref( ) 0.0009Pa sec⋅= . 1Pa sec⋅ 999.978174cP= .

µlw Tref( ) 0.912511cP= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 12 of 37

The vapor viscosity of water, µvw , as a function of the absolute temperature T is (DIPPR):

µvw T( ) 273.16K T≤( ) T 1073.2K≤( )⋅6.1839 10

7−⋅T

K

0.6778

⋅

1847.23

T

K

+7.3930− 10

4⋅

T

K

2+

⋅ Pa⋅ sec⋅:= .

Tref 298.15K= :

µvw Tref( ) 9.7696 106−× Pa sec⋅= . 1Pa sec⋅ 0.671969

lb

ft sec= .

µvw Tref( ) 6.564835 106−×

lb

ft sec= .

The surface tension of water, σw , as a function of the absolute temperature T is (DIPPR):

σw T( ) 273.16 K⋅ T≤( ) T 647.13 K⋅≤( )⋅ 0.1855⋅ 1

T

K

647.13−

2.7170 3.5540−

T

K

647.13⋅+ 2.0470

T

K

647.13

2

⋅+

⋅newton

m⋅:= .

Tref 298.15K= :

σw Tref( ) 0.072825newton

m= . 1newton 1000mN= .

σw Tref( ) 72.825mN

m= .

σw Tref( ) 0.072825newton

m= . 1newton 0.224814 lbf= . 1ft 0.304800m= .

σw Tref( ) 0.004990185lbf

ft= .

The molecular weight of chloroform, Mc , is (DIPPR):

Mc 119.38kg

kgmole:= ; Mc 119.38

gm

mole= ; Mc 119.38

lb

lbmole= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 13 of 37

The vapor pressure of chloroform, Pvc , as a function of the absolute temperature T is (DIPPR):

Pvc T( ) 207.15 K⋅ T≤( ) T 536.40 K⋅≤( )⋅ exp 146.437792.3−

T

K

+ 20.614− lnT

K

⋅+

0.0246T

K

1.0000

⋅+

...

⋅ Pa⋅:= .

Tref 298.15K= :

Pvc Tref( ) 26337Pa= . 1Pa 0.000145038psi= .

Pvc Tref( ) 3.8199psi= .

The diffusivity of chloroform in infinite dilution in water, Dc0w , as a function of the absolute

temperature T is (Yaws):

Dc0w T( ) 274 K⋅ T≤( ) T 394 K⋅≤( )⋅ 10

1.4389−1051.706−

T

K

+

⋅cm

2

sec⋅:= . 2.54cm 1in= .

Dc0w 25 273.15+( )K[ ] 1.08 105−×

cm2

sec= .

Dc0w 100 273.15+( )K[ ] 5.53 105−×

cm2

sec= .

The activity coefficient of chloroform at infinite dilution in water, γc , as a function of the absolute

temperature T is (Wright, Mathcad Polynomial Regression):

γc T( ) interp regress

20.0 273.15+

35.0 273.15+

50.0 273.15+

818

847

862

, 2,

20.0 273.15+

35.0 273.15+

50.0 273.15+

K,

818

847

862

, T,

:= ;

γc 20.0 273.15+( )K[ ] 818= .

γc 35.0 273.15+( )K[ ] 847= .

γc 50.0 273.15+( )K[ ] 862= .

γc 100.0 273.15+( )K[ ] 811= .

Operating Data

The absolute pressure upstream of the orifices, P0 , is (P&ID):

P0 30psi 14.7psi+:= : P0 44.7psi= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 14 of 37

Steam Flows

The pressure ratio of the restriction orifices, r , depends on the absolute pressure upstream P0 and

the absolute pressure downstream Pv :

P0 44.7psi= . Pv 14.7psi= .

rPv

P0:= : r 0.328770= .

The diameter ratios of the first and second stage restriction orifices, β1 and β2 , are calculated from

the orifice diameters D1 and D2 and the orifice pipe inside diameters ID1 and ID2 (Perry's page 10-4):

β1D1

ID1:= and β2

D2

ID2:= :

β1 0.238322= and β2 0.283008= .

The absolute temperature upstream of the restriction orifices, T0 , is calculated iteratively from the

absolute pressure upstream P0 using the water vapor pressure as a function of the absolute temperature,

Pvw T( ) (Mathcad Help for Solve Block):

Start with T0 Tref:= . Tref 298.15K= , so T0 298.15K= .

Given P0 Pvw T0( )= , calculate T0 Find T0( ):= :

T0 407.67K= . [ TF T0( ) 274.11= .]

The critical pressure ratios of the first and second stage restriction orifices, rc1 and rc2 , are

calculated iteratively from the heat capacity ratio of the water vapor at the upstream absolute temperature, kw T0( ) , and the diameter ratios of the orifices β1 and β2 (Perry's page 10-14):

T0 407.67K= . [ TF T0( ) 274.11= .] kw T0( ) 1.319786= .

β1 0.238322= and β2 0.283008= .

Start with rc1 1:= and rc2 1:= .

Given rc1( )1 kw T0( )−

kw T0( ) kw T0( ) 1−

2

β1( )4⋅ rc1( )

2

kw T0( )⋅+

kw T0( ) 1+

2= and

rc2( )1 kw T0( )−

kw T0( ) kw T0( ) 1−

2

β2( )4⋅ rc2( )

2

kw T0( )⋅+

kw T0( ) 1+

2= ,

calculate rc1

rc2

Find rc1 rc2,( ):= :

rc1 0.542571= and rc2 0.542962= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 15 of 37

The restriction orifices are in critical flow if their pressure ratios r are smaller than their critical

pressure ratios rc1 and rc2 (Perry's page 10-14):

r 0.329= . rc1 0.543= and rc2 0.543= .

flow1 if r rc1≤ "critical", "not critical",( ):= and flow2 if r rc2≤ "critical", "not critical",( ):= :

flow1 "critical"= and flow2 "critical"= .

The loss coefficients of the first and second stage restriction orifices, K1 and K2 , can be estimated

from the orifice diameter ratios β1 and β2 alone (Benedict page 247 equation 22):

K1 1 β12

−( )2 1

21 β1

2−( )⋅+ 1.41 1 β1

2−( )

3

2⋅+:= and K2 1 β2

2−( )2 1

21 β2

2−( )⋅+ 1.41 1 β2

2−( )

3

2⋅+:= :

K1 2.653= and K2 2.550= .

The vapor density upstream of the restriction orifices, ρ0 , is calculated iteratively from the absolute

pressure upstream P0 , universal gas constant Rg , molecular weight of water vapor Mw , absolute

temperature upstream T0 , and second virial coefficient of water upstream Bw T0( ) (Perry's page 2-355):

P0 44.7psi= . Mw 18.015lb

lbmole= .

Rg 10.73psi ft

3⋅lbmole R⋅

= . K 1.8R= . T0 407.67K= . [ TF T0( ) 274.11= .] Bw T0( ) 5.297421−ft

3

lbmole= .

Start with ρ0 0.1lb

ft3

:= .

Given that P0

ρ0

MwRg⋅ T0⋅

1 Bw T0( )ρ0

Mw⋅+= , calculate ρ0 Find ρ0( ):= :

ρ0 0.105532lb

ft3

= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 16 of 37

The velocities in the first and second stage restriction orifices, V1 and V2 , are calculated from the

absolute pressure upstream P0 , the critical pressure ratios of the orifices rc1 and rc2 , the loss coefficients

of the orifices K1 and K2 , and the vapor density upstream ρ0 (Benedict page 245 equation 1):

P0 44.7lbf

in2

= . rc1 0.542571= and rc2 0.542962= . 1 lbf⋅ 32.174 lbft

sec2

⋅= . 144in2

1 ft2= .

K1 2.652828= and K2 2.550223= . ρ0 0.105532lb

ft3

= .

V1 P0 rc1 P0⋅−( ) 2

K1 ρ0⋅⋅:= and V2 P0 rc2 P0⋅−( ) 2

K2 ρ0⋅⋅:= :

V1 822.655ft

sec= and V2 838.683

ft

sec= .

The mass flows in the first and second stage restriction orifices, W1 and W2 , and the total mass

flow of steam WT are calculated from the vapor density upstream ρ0 , the pipe diameters D1 and D2 , and

the velocities in the orifices V1 and V2 (Benedict page 245 equation 3):

ρ0 0.105532lb

ft3

= . π 3.141593= . D1 0.25 in= . V1 822.655ft

sec= . 144in

21 ft

2= .

D2 0.296875 in= . V2 838.683ft

sec= .

W1 ρ0 π⋅D1

2

2

⋅ V1⋅:= and W2 ρ0 π⋅D2

2

2

⋅ V2⋅:= :

W1 106.540lb

hr= and W2 153.165

lb

hr= .

WT W1 W2+:= : WT 260lb

hr= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 17 of 37

The vapor density at the vessels, ρv , is calculated iteratively from the absolute pressure at the

spargers Pv , the universal gas constant Rg , the molecular weight of water vapor Mw , the absolute

temperature of the second stage vessel Tv2 (as noted in the assumptions), and the second virial coefficient

of water Bw Tv2( ) (Perry's page 2-355):

Pv 14.696psi= . Mw 18.015lb

lbmole= .

Rg 10.73psi ft

3⋅lbmole R⋅

= . K 1.8R= . Tv1 364.95K= . [ TF Tv2( ) 211.98= .] Bw Tv2( ) 7.242413−ft

3

lbmole= .

Start with ρv 0.1lb

ft3

:= .

Given that Pv

ρv

MwRg⋅ Tv2⋅

1 Bw Tv2( )ρv

Mw⋅+= , calculate ρv Find ρv( ):= :

ρv 0.037288lb

ft3

= .

The cross-sectional areas of the first and second stage vessels, A1 and A2 are calculated from the

vessel diameters Dv1 and Dv2 (neglecting vessel appurtenances as noted in the assumptions):

π 3.141593= . Dv1 20 in= and Dv2 24 in= . 12in 1 ft= .

A1 πDv1

2

2

⋅:= and A2 πDv2

2

2

⋅:= :

A1 2.182 ft2= and A2 3.142 ft

2= .

The superficial velocities in the first and second stage vessels, UG1 and UG2 , are calculated from

the mass flows W1 and W2 , vapor density ρv , and vessel cross-sectional areas A1 and A2 :

W1 106.540lb

hr= and W2 153.165

lb

hr= . 3600sec 1hr= .

ρv 0.037288lb

ft3

= . A1 2.182 ft2= and A2 3.142 ft

2= .

UG1W1

ρv A1⋅:= and UG2

W2

ρv A2⋅:= :

UG1 0.363794ft

sec= and UG2 0.363195

ft

sec= . 1ft 0.304800m= .

UG1 0.110884m

sec= and UG2 0.110702

m

sec= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 18 of 37

The vessel velocity regimes indicated by the vessel superficial velocities UG1 and UG2 are (Perry's

page 14-74):

UG1 0.36ft

sec= and UG2 0.36

ft

sec= .

Vessel_Gas_Velocity1 if UG1 0.15ft

sec≤ "Quiescent", "",

:= and

Vessel_Gas_Velocity2 if UG2 0.15ft

sec≤ "Quiescent", "",

:= ;

Vessel_Gas_Velocity1 if 0.15ft

secUG1<

UG1 0.20ft

sec⋅<

⋅ "Quiescent or Turbulent", Vessel_Gas_Velocity1,

:= a


secUG2<

UG2 0.20ft

sec⋅<

⋅ "Quiescent or Turbulent", Vessel_Gas_Velocity2,

:=


sec⋅ UG1≤ "Turbulent", Vessel_Gas_Velocity1,

:= and


sec⋅ UG2≤ "Turbulent", Vessel_Gas_Velocity2,

:= :

Vessel_Gas_Velocity1 "Turbulent"= and Vessel_Gas_Velocity2 "Turbulent"= .

Vessel_Quiescence_vs_Max1UG1

0.15ft

sec

:= . 1 100%= .

Vessel_Quiescence_vs_Max2UG2

0.15ft

sec

:= . 1 100%= :

Vessel_Quiescence_vs_Max1 243%= and

Vessel_Quiescence_vs_Max2 242%= .

Vessel_Turbulence_vs_Min1UG1

0.20ft

sec

:= . 1 100%= .

Vessel_Turbulence_vs_Min2UG2

0.20ft

sec

:= . 1 100%= :

Vessel_Turbulence_vs_Min1 182%= and

Vessel_Turbulence_vs_Min2 182%= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 19 of 37

Interfacial Areas

The total hole areas of the first and second stage spargers, aT1 and aT2 , are calculated from the

hole diameters dN1 and dN2 and hole counts n1 and n2 :

dN1 0.171875 in= and dN2 0.203125 in= . 12in 1 ft= . n1 22= and n2 22= .

aT1 πdN1

2

2

⋅ n1⋅:= and aT2 πdN2

2

2

⋅ n2⋅:= :

aT1 0.003545 ft2= and aT2 0.004951 ft

2= .

The superficial gas velocities based on total hole area of the first and second stage spargers, uN1

and uN2 , are calculated from the mass flows W1 and W2 , gas density ρv , and total hole areas aT1 and aT2 :

W1 106.54lb

hr= and W2 153.165

lb

hr= . 3600sec 1hr= .

ρv 0.037288lb

ft3

= . aT1 0.003545 ft2= and aT2 0.004951 ft

2= .

uN1W1

ρv aT1⋅:= and uN2

W2

ρv aT2⋅:= :

uN1 223.907ft

sec= and uN2 230.469

ft

sec= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 20 of 37

The sparger hole superficial velocities uN1 and uN2 (for open-end pipe, perforated plate, or ring- or

cross-style perforated-pipe spargers in quiescent vessels) are considered (Perry's page 14-74):

uN1 224ft

sec= and

uN2 230ft

sec= .

Vessel_Gas_Velocity1 "Turbulent"= and

Vessel_Gas_Velocity2 "Turbulent"= .

Sparger_Velocity1 if uN1 250ft

sec≤ "Not Excessive", "",

:= and

Sparger_Velocity2 if uN2 250ft

sec≤ "Not Excessive", "",

:= ;

Sparger_Velocity1 if 250ft

secuN1<

uN1 300ft

sec⋅<

⋅ "Possibly Excessive", Sparger_Velocity1,

:= and


secuN2<

uN2 300ft

sec⋅<

⋅ "Possibly Excessive", Sparger_Velocity2,

:= ;


sec⋅ uN1≤ "Excessive", Sparger_Velocity1,

:= and


sec⋅ uN2≤ "Excessive", Sparger_Velocity2,

:= :

Sparger_Velocity1 "Not Excessive"= and Sparger_Velocity2 "Not Excessive"= .

Sparger_Velocity_vs_Max_Okay1uN1

250ft

sec

:= . 1 100%= .

Sparger_Velocity_vs_Max_Okay2uN2

250ft

sec

:= . 1 100%= :

Sparger_Velocity_vs_Max_Okay1 90%= and

Sparger_Velocity_vs_Max_Okay2 92%= .

Sparger_Velocity_vs_Min_Excessive1uN1

300ft

sec

:= . 1 100%= .

Sparger_Velocity_vs_Min_Excessive2uN2

300ft

sec

:= . 1 100%= :

Sparger_Velocity_vs_Min_Excessive1 75%= and

Sparger_Velocity_vs_Min_Excessive2 77%= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 21 of 37

The Reynolds numbers, inertial_force

viscous_force(Perry's page 6-50), for the gas flows at the holes in the first

and second stage spargers, ReN1 and ReN2 , are calculated from the hole diameters dN1 and dN2 , hole

superficial gas velocities uN1 and uN2 , vapor density ρv , and vapor viscosity of water at the absolute

temperature in the second stage vessel (as noted in the assumptions) µvw Tv2( ) (Kumar page 508

Nomenclature section):

dN1 0.171875 in= and dN2 0.203125 in= . uN1 223.907ft


ft

sec= . ρv 0.037288

lb

ft3

= .

12in 1 ft= .

Tv2 373.15K= . [ TF Tv2( ) 211.98= .] µvw Tv2( ) 8.397656 106−×

lb

ft sec= .

ReN1dN1 uN1⋅ ρv⋅

µvw Tv2( ):= and ReN2dN2 uN2⋅ ρv⋅

µvw Tv2( ):= ;

ReN1 14240= and ReN2 17322= .

The sparger flow regimes indicated by the sparger hole Reynolds numbers ReN1 and ReN2 are

(Perry's pages 14-70 and 14-71):

ReN1 14240= and

ReN2 17322= .

Sparger_Re_Regime1 if ReN1 200< "Single-Bubble", "",( ):= and

Sparger_Re_Regime2 if ReN2 200< "Single-Bubble", "",( ):= ;

Sparger_Re_Regime1 if 200 ReN1≤( ) ReN1 2100≤( )⋅ "Intermediate", Sparger_Re_Regime1, := and

Sparger_Re_Regime2 if 200 ReN2≤( ) ReN2 2100≤( )⋅ "Intermediate", Sparger_Re_Regime2, := ;

Sparger_Re_Regime1 if 2100 ReN1< "Possibly Jet", Sparger_Re_Regime1,( ):= and

Sparger_Re_Regime2 if 2100 ReN2< "Possibly Jet", Sparger_Re_Regime2,( ):= :

Sparger_Re_Regime1 "Possibly Jet"= and

Sparger_Re_Regime2 "Possibly Jet"= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 22 of 37

The Weber numbers, inertial_force

surface_tension_force (Perry's page 6-50), for the gas flows at the holes in the

first and second stage spargers, Weg1 and Weg2 , are calculated from the vapor density ρv , the hole

diameters dN1 and dN2 , the hole superficial gas velocities uN1 and uN2 , and the surface tensions of water

at the absolute temperatures in the vessels (as noted in the assumptions) σw Tv1( ) and σw Tv2( ) (Perry's

page 14-71):

ρv 0.037288lb

ft3

= . dN1 0.171875 in= and dN2 0.203125 in= . uN1 223.907ft


ft

sec= .

Tv1 364.95K= . [TF Tv1( ) 197.22= .] σw Tv1( ) 0.004098lbf

ft= . 1lbf 32.174

lbft

sec2

= .

Tv2 373.15K= . [ TF Tv2( ) 211.98= .] σw Tv2( ) 0.003989lbf

ft= . 1lbf 32.174

lbft

sec2

= .

Weg1ρv dN1⋅ uN1

2⋅

σw Tv1( ):= and Weg2ρv dN2⋅ uN2

2⋅

σw Tv2( ):= :

Weg1 203.062= and Weg2 261.198= .

The sparger flow jet regimes indicated by the sparger hole Weber numbers Weg1 and Weg2 are

(Perry's pages 14-70 and 14-71, and Wilkinson page 1433):

Weg1 203.062= and

Weg2 261.198= .

Sparger_We_Regime1 if Weg1 2> "Jet", "Not Jet",( ):= and

Sparger_We_Regime2 if Weg2 2> "Jet", "Not Jet",( ):= :

Sparger_We_Regime1 "Jet"= and

Sparger_We_Regime2 "Jet"= .

Sparger_We_vs_Jet_Flow_We1Weg1

2:= . 1 100%= .

Sparger_We_vs_Jet_Flow_We2Weg2

2:= . 1 100%= :

Sparger_We_vs_Jet_Flow_We1 10153%= and

Sparger_We_vs_Jet_Flow_We2 13060%= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 23 of 37

The Sauter-mean bubble diameters or bubble volume-to-surface ratios of the first and second stage spargers, dvs1 and dvs2 , are calculated from the Reynolds numbers for the gas flows at the holes

ReN1 and ReN2 , the hole diameters dN1 and dN2 , the surface tensions and liquid densities of water at the

absolute temperatures in the vessels σw Tv1( ) , σw Tv2( ) , ρ lw Tv1( ) , and ρ lw Tv2( ) (as noted in the

assumptions), the vapor density in the spargers ρv , and the gravitational acceleration g (Kumar page 504

equations 4 to 6):

ReN1 14240= and ReN2 17322= .

dN1 0.171875 in= and dN2 0.203125 in= . 12in 1 ft= .

Tv1 364.95K= and Tv2 373.15K= . [ TF Tv1( ) 197.22= and TF Tv2( ) 211.98= .]

σw Tv1( ) 0.004098lbf

ft= and σw Tv2( ) 0.003989

lbf

ft= . 1lbf 32.174

lbft

sec2

= .

ρ lw Tv1( ) 60.155lb

ft3

= and ρ lw Tv2( ) 59.793lb

ft3

= . ρv 0.037288lb

ft3

= . g 32.174ft

sec2

= .

Define interpolation functions f1 ReN1( ) and f2 ReN2( ) .

f1 ReN1( ) 10log 0.32 21000.425⋅( ) log 100 4000 0.4−⋅( ) log 0.32 21000.425⋅( )−

log 4000( ) log 2100( )−

log ReN1( ) log 2100( )−( )⋅+:= and

f2 ReN2( ) 10log 0.32 21000.425⋅( ) log 100 4000 0.4−⋅( ) log 0.32 21000.425⋅( )−

log 4000( ) log 2100( )−

log ReN2( ) log 2100( )−( )⋅+:= :

f1 2100( ) 8.262= ; 0.32 21000.425⋅ 8.262= .

f1 4000( ) 3.624= ; 100 40000.4−⋅ 3.624= .

f1 3000( ) 5.236= .

dvs1 1 ReN1<( ) ReN1 10≤( )⋅ 1.56⋅ ReN10.058⋅

10 ReN1<( ) ReN1 2100≤( )⋅ 0.32⋅ ReN10.425⋅+

...

2100 ReN1<( ) ReN1 4000<( )⋅ f1 ReN1( )⋅+...

4000 ReN1≤( ) ReN1 70000<( )⋅ 100⋅ ReN10.4−⋅+

...

σw Tv1( ) dN12⋅

ρ lw Tv1( ) ρv−( ) g⋅

1

4

⋅:= . 1ft 12 in= .

dvs2 1 ReN2<( ) ReN2 10≤( )⋅ 1.56⋅ ReN20.058⋅

10 ReN2<( ) ReN2 2100≤( )⋅ 0.32⋅ ReN20.425⋅+

...

2100 ReN2<( ) ReN2 4000<( )⋅ f1 ReN2( )⋅+...

4000 ReN2≤( ) ReN2 70000<( )⋅ 100⋅ ReN20.4−⋅+

...

σw Tv2( ) dN22⋅

ρ lw Tv2( ) ρv−( ) g⋅

1

4

⋅:= . 1ft 12 in= :

dvs1 0.285 in= and dvs2 0.285 in= ;

dvs1

dN1166%= and

dvs2

dN2140%= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 24 of 37

The Froude numbers, inertial_force

gravity_force (Perry's page 6-49), in the first and second stage vessels, Fr1

and Fr2 , are calculated from the superficial velocities UG1 and UG2 , gravitational acceleration g , and

vessel diameters Dv1 and Dv2 (Godbole pages 1235-1236):

UG1 0.363794ft

sec= . UG2 0.363195

ft

sec= .

g 32.174ft

sec2

= . Dv1 20 in= . Dv2 24 in= . 12in 1 ft= .

Fr1UG1

g Dv1⋅:= . Fr2

UG2

g Dv2⋅:= .

Fr1 0.049680= . Fr2 0.045276= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 25 of 37

The gas holdups in the first and second stage vessels, εG1 and εG2 , are calculated iteratively from

the gravitational acceleration g , the vessel diameters Dv1 and Dv2 , the liquid densities, surface tensions,

and liquid viscosities of water at the absolute temperatures in the vessels (as noted in the assumptions) ρ lw Tv1( ) , ρ lw Tv2( ) , σw Tv1( ) , σw Tv2( ) , µlw Tv1( ) and µlw Tv2( ) , and the superficial velocities UG1 and

UG2 (Akita page 78 equation 11):


g 32.174ft

sec2

= . Dv1 20 in= and Dv2 24 in= . 12in 1 ft= . ρ lw Tv1( ) 60.155lb

ft3

= and ρ lw Tv2( ) 59.793lb

ft3

= .

σw Tv1( ) 4.098 103−×

lbf

ft= and σw Tv2( ) 3.989 10

3−×lbf

ft= . 1lbf 32.174

lbft

sec2

= .

µlw Tv1( ) 206.177 106−×

lb

ft sec= and µlw Tv2( ) 187.815 10

6−×lb

ft sec= .

UG1 0.363794ft

sec= and UG2 0.363195

ft

sec= .

Start with εG1 10%:= and εG2 10%:= .

Given that εG1

1 εG1−( )40.20

g Dv12⋅ ρ lw Tv1( )⋅

σw Tv1( )

1

8

⋅g Dv1

3⋅

µlw Tv1( )ρ lw Tv1( )

2

1

12

⋅UG1

g Dv1⋅

1.0

⋅= and

εG2

1 εG2−( )40.20

g Dv22⋅ ρ lw Tv2( )⋅

σw Tv2( )

1

8

⋅g Dv2

3⋅

µlw Tv2( )ρ lw Tv2( )

2

1

12

⋅UG2

g Dv2⋅

1.0

⋅= ,

calculate εG1

εG2

Find εG1 εG2,( ):= :

εG1 19.468%= and εG2 19.621%= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 26 of 37

The critical values of the Froude number, inertial_force

gravity_force (Perry's page 6-49), for transition to

foaming flow in the first and second stage vessels, Frgc1 and Frgc2 , are calculated from the gas holdups εG1

and εG2 , and the vessel foaming flow conditions are determined by comparing the calculated critical values

of the Froude number to the actual values Fr1 and Fr2 (Godbole page 1221):

εG1 0.195= and εG2 0.196= .

Frgc10.25 εG1

2⋅

1 εG1−( )3

2

:= and Frgc20.25 εG2

2⋅

1 εG2−( )3

2

:= .

Frgc1 0.013110= and Frgc2 0.013356= .

Fr1 0.049680= . Frgc1 0.013110= . Fr2 0.045276= . Frgc2 0.013356= .

Vessel_Fr_No1 if Fr1 Frgc1< "Foaming Flow", "Not Foaming Flow",( ):= and

Vessel_Fr_No2 if Fr2 Frgc2< "Foaming Flow", "Not Foaming Flow",( ):= :

Vessel_Fr_No1 "Not Foaming Flow"= and

Vessel_Fr_No1 "Not Foaming Flow"= .

Vessel_Fr_vs_Min_Nonfoaming1Fr1

Frgc1:= . 1 100%= .

Vessel_Fr_vs_Min_Nonfoaming2Fr2

Frgc2:= . 1 100%= :

Vessel_Fr_vs_Min_Nonfoaming1 379%= and

Vessel_Fr_vs_Min_Nonfoaming2 339%= .

The effective interfacial areas per unit volume in the first and second stage vessels, a1 and a2 , are

calculated from the gas holdups εG1 and εG2 and the Sauter-mean bubble diameters or bubble

volume-to-surface ratios dvs1 and dvs2 (Perry's pages 5-69 and 5-43):

εG1 0.194676= and εG2 0.196213= .

dvs1 0.284573 in= and dvs2 0.284552 in= . 12in 1 ft= .

a16 εG1⋅

dvs1:= and a2

6 εG2⋅

dvs2:= :

a1 49.255ft

2

ft3

= and a2 49.648ft

2

ft3

= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 27 of 37

Mass Transfer Coefficients

The Grashof numbers for the first and second stage vessels, Gr1 and Gr2 , are calculated from the

Sauter-mean bubble diameters or bubble volume-to-surface ratios dvs1 and dvs2 , the liquid densities and

liquid viscosities of water at the absolute temperatures in the vessels (as noted in the assumptions) ρ lw Tv1( ), ρ lw Tv2( ) , µlw Tv1( ) , and µlw Tv2( ) , the vapor density ρv , and the gravitational acceleration g (Calderbank

page 53):


dvs1 0.284573 in= . 12in 1 ft= . ρ lw Tv1( ) 60.155lb

ft3

= . ρv 0.037288lb

ft3

= . g 32.174ft

sec2

= .

dvs2 0.284552 in= . ρ lw Tv2( ) 59.793lb

ft3

=

µlw Tv1( ) 0.000206177lb

ft sec= and µlw Tv2( ) 0.000187815

lb

ft sec= .

Gr1dvs1

3 ρ lw Tv1( )⋅ ρ lw Tv1( ) ρv−( )⋅ g⋅

µlw Tv1( )2:= and Gr2

dvs23 ρ lw Tv2( )⋅ ρ lw Tv2( ) ρv−( )⋅ g⋅

µlw Tv2( )2:= :

Gr1 36.503 106×= and Gr2 43.452 10

6×= .

The Schmidt numbers for the first and second stage vessels, Sc1 and Sc2 , are calculated from the

liquid viscosities and liquid densities of water at the absolute temperatures in the vessels (as noted in the assumptions) µlw Tv1( ) , µlw Tv2( ) , ρ lw Tv1( ) , and ρ lw Tv2( ) , and the diffusivities of chloroform at infinite

dilution in water at the absolute temperatures in the vessels Dc0w Tv1( ) and Dc0w Tv2( ) (Calderbank page 53):


µlw Tv1( ) 0.000206177lb

ft sec= and µlw Tv2( ) 0.000187815

lb

ft sec= .

ρ lw Tv1( ) 60.155lb

ft3

= and ρ lw Tv2( ) 59.793lb

ft3

= . Dc0w Tv1( ) 51.439 109−×

ft2

sec= and

Dc0w Tv2( ) 59.514 109−×

ft2

sec= .

Sc1µlw Tv1( )

ρ lw Tv1( ) Dc0w Tv1( )⋅:= and Sc2

µlw Tv2( )ρ lw Tv2( ) Dc0w Tv2( )⋅

:= :

Sc1 66.632= and Sc2 52.780= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 28 of 37

The Sherwood numbers for the first and second stage vessels, Sh1 and Sh2 , are calculated from

the Grashof numbers Gr1 and Gr2 , Schmidt numbers Sc1 and Sc2 , and Sauter-mean bubble diameters or

bubble volume-to-surface ratios dvs1 and dvs2 (Perry's page 5-69):

Gr1 36.503 106×= and Gr2 43.452 10

6×= .

Sc1 66.632= and Sc2 52.780= .

dvs1 0.284573 in= and dvs2 0.284552 in= . 1in 2.54cm= , so

dvs1 0.722814cm= and dvs2 0.722762cm= .

Sh1 dvs1 0.25cm<( ) 2⋅ 0.31 Gr1

1

3⋅ Sc1

1

3⋅+

0.25cm dvs1≤( ) 0.42⋅ Gr1

1

3⋅ Sc1

1

2⋅+

...:= and Sh2 dvs2 0.25cm<( ) 2⋅ 0.31 Gr2

1

3⋅ Sc2

1

3⋅+

0.25cm dvs2≤( ) 0.42⋅ Gr2

1

3⋅ Sc2

1

2⋅+

...:= :

Sh1 1554.180= and Sh2 1481.527= .

The individual liquid-phase mass-transfer coefficients in the first and second stage vessels, kL1

and kL2 , are calculated from the Sherwood numbers Sh1 and Sh2 , the diffusivities of chloroform at infinite

dilution in water at the absolute temperatures in the vessels Dc0w Tv1( ) and Dc0w Tv2( ) , and the

Sauter-mean bubble diameters or bubble volume-to-surface ratios dvs1 and dvs2 (Calderbank page 53):


Sh1 1554.180= . Dc0w Tv1( ) 51.439 109−×

ft2

sec= . 1hr 3600sec= .

Sh2 1481.527= . Dc0w Tv2( ) 59.514 109−×

ft2

sec= . 1hr 3600sec= .

dvs1 0.284573 in= and dvs2 0.284552 in= . 12in 1 ft= .

kL1Sh1 Dc0w Tv1( )⋅

dvs1:= and kL2

Sh2 Dc0w Tv2( )⋅

dvs2:= :

kL1 12.136ft

hr= and kL2 13.386

ft

hr= .

The overall liquid-phase mass-transfer coefficients KL1 and KL2 are taken to equal the individual

liquid-phase mass-transfer coefficients kL1 and kL2 since the resistance is entirely in the liquid phase for

most gas-liquid mass transfer (Perry's page 5-69 Table 5-25 Condition Z, and page 5-56 equation 5-257):

kL1 12.136ft

hr= and kL2 13.386

ft

hr= .

KL1 kL1:= and KL2 kL2:= :

KL1 12.136ft

hr= and KL2 13.386

ft

hr= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 29 of 37

Stripping and Dilution

The operating volumes of the first and second stage vessels, Voper1 and Voper2 , are calculated

from the rated capacities Vcap1 and Vcap2 , the levels of clear liquid at overflow Lover1 and Lover2 , and the

gas holdups εG1 and εG2 (Akita page 76 equation 1):

Vcap1 32gal= and Vcap2 52gal= .

Lover1 60%= and Lover2 60%= .

εG1 19.468%= and εG2 19.621%= .

100% 1= .

Voper1 Vcap1Lover1

100% εG1−⋅:= and Voper2 Vcap2

Lover2

100% εG2−⋅:= :

Voper1 23.841gal= and Voper2 38.816gal= .

The liquid-phase molar flux in the first stage vessel, LM1 , is calculated from the volumetric flow of

raffinate, qr , the liquid density of water as a function of the absolute temperature in the vessel (neglecting

the chloroform present, as noted in the assumptions) ρ lw Tv1( ) , the weight fraction chloroform in the

raffinate wcr , the molecular weights of water and chloroform Mw and Mc , and the cross-sectional area of

the vessel A1 (Cussler page 251):

Tv1 364.95K= . [ TF Tv1( ) 197.22= .]

qr 4.185gal

min= . 1gal 0.133681 ft

3= . 1hr 60min= . ρ lw Tv1( ) 60.155lb

ft3

= .

wcr 0.003756= .

Mw 18.015lb

lbmole= and Mc 119.38

lb

lbmole= . A1 2.182 ft

2=

LM1 qr ρ lw Tv1( )⋅

100lbwcr

Mc

100lb 1 wcr−( )⋅

Mw+

100lb⋅

1

A1⋅:= : LM1 51.213

lbmole

hr ft2⋅

= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 30 of 37

The liquid mole fraction chloroform at the top of the first stage vessel, x2v1 , is calculated from the

weight fraction chloroform in the raffinate wcr and the molecular weights of water and chloroform Mw and

Mc :

Basis: 100lb of raffinate.

wcr 0.003756= .

Mw 18.015lb

lbmole= . Mc 119.38

lb

lbmole= .

x2v1

100lb wcr⋅

Mc

100lbwcr

Mc

100lb 1 wcr−( )⋅

Mw+

:= : x2v1 0.000569= .

The gas-phase molar fluxes in the first and second stage vessels, GM1 and GM2 , are calculated

from the restriction orifice mass flows W1 and W2 , the molecular weight of water Mw , and the vessel

cross-sectional areas A1 and A2 (Cussler page 251):

W1 106.540lb

hr= .

Mw 18.015lb

lbmole= . A1 2.182 ft

2= .

GM1W1

Mw A1⋅:= and GM2

W2

Mw A2⋅:= :

GM1 2.711lbmole

hr ft2⋅

= and GM2 2.706lbmole

hr ft2⋅

= .

The vapor-liquid equilibrium ratios of chloroform at infinite dilution in water in the first and second stage vessels, Kc1 and Kc2 , are calculated from the vapor pressures of chloroform and the activity

coefficients of chloroform at infinite dilution in water, both evaluated at the absolute temperatures in the vessels, Pvc Tv1( ) , Pvc Tv2( ) , γc Tv1( ) , and γc Tv2( ) , and from the total pressure in the vessels Pv (Hwang

page 1759 equation 1):


Pvc Tv1( ) 36.595psi= and Pvc Tv2( ) 45.276psi= . γc Tv1( ) 829.935= and γc Tv2( ) 810.889= .

Pv 14.696psi= .

Kc1Pvc Tv1( ) γc Tv1( )⋅

Pv:= and Kc2

Pvc Tv2( ) γc Tv2( )⋅

Pv:= : Kc1 2067= and Kc2 2498= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 31 of 37

The remaining concentrations, molar fluxes based on interfacial areas, and molar fluxes based on vessel cross-sectional areas, x1v1 , y2v1 , NA1 , x1v2 , x2v2 , y2v2 , NA2 , and LM2 , are calculated iteratively

from the interfacial areas per unit volume a1 and a2 , operating volumes Voper1 and Voper2 , liquid-phase

molar flux based on cross-sectional area of the first stage vessel LM1 , gas-phase molar fluxes based on

cross-sectional areas GM1 and GM2 , overall liquid-phase mass transfer coefficients KL1 and KL2 , liquid

densities of water at the vessel absolute temperatures (as noted in the assumptions) ρ lw Tv1( ) and ρ lw Tv2( ), molecular weight of water Mw , and vapor-liquid equilibrium ratios of chloroform at infinite dilution in water in

the vessels Kc1 and Kc2 (Geankoplis page 451 Example 7.4-1):




A7947 010704.01, CHCl3 Plan 7/22/2001, 32 of 37

x2v 1 y2v 1

x1v 1 y1v 1

x2v 2 y2v 2

x1v 2 y1v 2


a1 49.255ft

2

ft3

= . Voper1 23.841gal= . LM1 51.213lbmole

hr ft2⋅

= . A1 2.182 ft2= x2v1 0.000569= .

GM1 2.711lbmole

hr ft2⋅

= . KL1 12.136ft

hr= . ρ lw Tv1( ) 60.155

lb

ft3

= . Mw 18.015lb

lbmole= . Kc1 2067= .

a2 49.648ft

2

ft3

= . Voper2 38.816gal= . A2 3.142 ft2= . GM2 2.706

lbmole

hr ft2⋅

= .

KL2 13.386ft

hr= . ρ lw Tv2( ) 59.793

lb

ft3

= . Kc2 2498=

Note:y1v1 0:= . y1v2 0:= .

Let:

x1v1 0:= . y2v1 0:= . NA1 0lbmole

hr ft2⋅

⋅:= .

x1v2 0:= . x2v2 0:= . y2v2 0:= . NA2 0lbmole

hr ft2⋅

:= . LM2 LM1:= .

Given

NA1 a1⋅ Voper1⋅ LM1 A1⋅ x2v1 x1v1−( )⋅= and NA2 a2⋅ Voper2⋅ LM2 A2⋅ x2v2 x1v2−( )⋅= .

NA1 a1⋅ Voper1⋅ GM1 A1⋅ y2v1 y1v1−( )⋅= and NA2 a2⋅ Voper2⋅ GM2 A2⋅ y2v2 y1v2−( )⋅= .

NA1 KL1 ρ lw Tv1( )⋅1

Mw⋅ x1v1

y2v1

Kc1−

⋅= and NA2 KL2 ρ lw Tv2( )⋅1

Mw⋅ x1v2

y2v2

Kc2−

⋅= .

LM2 A2⋅ LM1 A1⋅ NA1 a1⋅ Voper1⋅−= . x1v1 x2v2= .

Calculate:




A7947 010704.01, CHCl3 Plan 7/22/2001, 33 of 37

x 1v1

y 2v1

N A1

x 1v2

x 2v2

y 2v2

N A2

L M2

Find x 1v1 y 2v1, N A1, x 1v2, x 2v2, y 2v2, N A2, L M2,( ):= :

x1v1 14.788 106−×= . y2v1 10.462 10

3−×= . NA1 394.136 106−×

lbmole

hr ft2⋅

= .

x1v2 218.741 109−×= . x2v2 14.788 10

6−×= . y2v2 191.357 106−×= .

NA2 6.315 106−×

lbmole

hr ft2⋅

= . LM2 35.545lbmole

hr ft2⋅

= .

The weight fraction chloroform in the stripped liquid, wcs , is calculated from the mole fraction

chloroform in the stripped liquid x1v2 and the molecular weights of water and chloroform Mw and Mc :

Basis: 100lbmole of stripped liquid.

x1v2 218.741 109−×= .

Mw 18.015lb

lbmole= . Mc 119.38

lb

lbmole= .

wcs100lbmole x1v2⋅ Mc⋅

100lbmole x1v2⋅ Mc⋅ 100lbmole 1 x1v2−( )⋅ Mw⋅+:= : wcs 1.450 10

6−×= .

The weight fraction chloroform in the stripped liquid, adjusted based on Process Qualification data, wcs_adj , is calculated from the weight fraction chloroform in the stripped liquid wcs based on PQ data

(Baseline calculation wcs , PQ Data from 10/5/99):

wcs 1.450 106−×= .

wcs_adj wcs

125 125+ 115+ 115+ 70+5

8.993⋅:= : wcs_adj 17.730 10

6−×= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 34 of 37

The weight fraction chloroform in the final rinsate, wcf , is calculated from the adjusted weight

fraction chloroform in the stripped liquid wcs_adj based on PQ data (PQ Report):

wcs_adj 17.730 106−×=

wcf wcs_adj

0.19 0.22+ 0.08+3

125 125+ 115+( ) 115 70+ 75+( )+ 50 85+ 100+( )+9

⋅:= : wcf 30.306 109−×= .

Key Parameters Summary

The levels of clear liquid at overflow of the first and second stage vessels, Lover1 and Lover2 , are

taken to be:

Lover1 60%= and Lover2 60%= .

The volumetric flow of raffinate qr is taken to be:

qr 4.185gal

min= .

The absolute pressure upstream of the orifices, P0 , is:

P0 44.7psi= .

The absolute pressures and temperatures in the vessels, Pv , Tv1 , and Tv2 , are:

Pv 14.7psi= .


The restriction orifice diameters of the first and second stages, D1 and D2 , are:

D1 0.250 in= and D2 0.297 in= .

The sparger total hole areas, aT1 and aT2 , are:

dN1 0.172 in= and dN2 0.203 in= .

n1 22= and n2 22= .

aT1 0.003545 ft2= and aT2 0.004951 ft

2= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 35 of 37

Results Summary

The mass flows in the first and second stage restriction orifices, W1 and W2 , and the total mass

flow of steam WT are:

W1 106.540lb

hr= and W2 153.165

lb

hr= . WT 260

lb

hr= .

The superficial velocities in the first and second stage vessels, UG1 and UG2 , are:

UG1 0.363794ft

sec= and UG2 0.363195

ft

sec= .

UG1 0.110884m

sec= and UG2 0.110702

m

sec= .

The vessel flow regimes indicated by the vessel superficial velocities Wi

ρ4 π⋅Dvi

2

2

⋅

are (Perry's

page 14-74):

Vessel_Gas_Velocity1 "Turbulent"= and

Vessel_Gas_Velocity2 "Turbulent"= .

Vessel_Quiescence_vs_Max1 243%= and

Vessel_Quiescence_vs_Max2 242%= .

Vessel_Turbulence_vs_Min1 182%= and

Vessel_Turbulence_vs_Min2 182%= .

The sparger hole superficial velocities Wi

ρ0 aTi⋅ (for open-end pipe, perforated plate, or ring- or

cross-style perforated-pipe spargers in quiescent vessels) are considered (Perry's page 14-74:

Sparger_Velocity1 "Not Excessive"= and

Sparger_Velocity2 "Not Excessive"= .

Sparger_Velocity_vs_Max_Okay1 90%= and

Sparger_Velocity_vs_Max_Okay2 92%= .

Sparger_Velocity_vs_Min_Excessive1 75%= and

Sparger_Velocity_vs_Min_Excessive2 77%= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 36 of 37

The sparger flow regimes indicated by the sparger hole Reynolds numbers dNi uNi⋅ ρ4⋅

µvw T0( ) are (Perry's

pages 14-70 and 14-71):

Sparger_Re_Regime1 "Possibly Jet"= and

Sparger_Re_Regime2 "Possibly Jet"= .

The sparger flow jet regimes indicated by the sparger hole Weber numbers ρg dNi⋅ uNi

2⋅

σw Ts( ) are

(Perry's pages 14-70 and 14-71, and Wilkinson page 1433):

Sparger_We_Regime1 "Jet"= and

Sparger_We_Regime2 "Jet"= .

Sparger_We_vs_Jet_Flow_We1 10153%= and

Sparger_We_vs_Jet_Flow_We2 13060%= .

The Sauter-mean bubble diameters or bubble volume-to-surface ratios of the first and second

stage spargers, dvs1 and dvs2 , and the ratios of the bubble diameters to the hole diameters, dvs1

dN1 and

dvs2

dN2,

are:

dvs1 0.285 in= and dvs2 0.285 in= .

dvs1

dN1166%= and

dvs2

dN2140%= .

The gas volumetric holdups in the first and second stage vessels, εG1 and εG2 , are:

εG1 19.5%= and εG2 19.6%= .

The vessel foaming flow conditions indicated by the Froude numbers UGi

g Dvi⋅ are (Perry's page

6-47, and Godbole page 1221):

Vessel_Fr_No1 "Not Foaming Flow"= and

Vessel_Fr_No1 "Not Foaming Flow"= .

Vessel_Fr_vs_Min_Nonfoaming1 379%= and

Vessel_Fr_vs_Min_Nonfoaming2 339%= .




A7947 010704.01, CHCl3 Plan 7/22/2001, 37 of 37

The weight fraction chloroform in the stripped liquid, wcs , is:

wcs 1.450 106−×= .

The weight fraction chloroform in the stripped liquid, adjusted based on Process Qualification data, wcs_adj , is:

wcs_adj 17.730 106−×= .

The weight fraction chloroform in the final rinsate, wcf , is:

wcf 30.306 109−×= .


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