Professional Development Elements of Calculation Style James Anthony, Lockwood Greene Style helps readers to understand, and calculations to succeed. T he elements of style in calculations are the choices in composition that strengthen collaboration between writers and readers by helping them meet one another’s needs. The overriding need of both writers and readers is to not have to keep track of too much new, unfamiliar material at one time (1). Elements of style include separate sections for assumptions, data, calculations and summaries. Each section can be prepared and read with a minimum of in-depth thought, yet each section moves the solution forward and serves as a resource for use with the later sections. Elements of style at the formula level, which are even more helpful, include conventional symbols, reminders of variable definitions, reference names and pages, reminders of values of dependent variables, and equations that are visible, as shown in Figure 1. These elements let writers and readers understand and check formulas with a minimum of cognitive strain. Elements of style in calculations are elaborated below and illustrated with a sample calculation on pages 52–53. 1. Use calculation software Use software that displays the working formulas. The equations are the working parts of the calculation. A calculation is easier to use if its functions can be readily inferred by looking at 50 www.cepmagazine.org November 2001 CEP the features visible to the user. Calculation software packages, such as Mathcad and CalculationCenter, display a calculation’s working formulas and results together, and print them for convenient writing and reading (3, 4). Spreadsheets typically don’t display formulas, so only simple operations like totaling the numbers in a column can be readily inferred by looking at a spreadsheet. (Spreadsheets can be made to display formulas and results together through the use of named ranges (5) and user-defined functions, as shown in Figure 2). Programming languages, such as Visual Basic and FORTRAN, display and print the working formulas and results separate from each other, and often use coded syntax (when referencing data objects, for example), making calculations in these languages difficult to read and document. Use software that calculates units as well as numbers. Calculation software like Mathcad and CalculationCenter makes it possible to include dimensional units in values and formulas and have the software perform unit conversions automatically, bypassing an otherwise major source of errors. 2. Set up for easy viewing Make calculations read from top to bottom. Use the approach that experts use when solving easy problems — work forward from the known data to determine the unknown values that are needed (6). Enter data before calculations, and summarize inputs before summarizing results. In the same man- ■ Figure 1. Formula blocks help readers to understand and check formulas (2). ner, provide reference materials before referring to them. Place the table of contents before the actual contents, list references before they are used, and list variable names before they are mentioned. Use font, font size, and font style changes to help readers. Font changes can help readers distinguish equations from text. Font size changes can improve readability of equations. Both changes are performed automatically by the calculation software used to prepare Figure 1 and the sample calculation. A font style change to bold face for the headings can help readers rapidly scan through a calculation, as shown in the sample calculation. Use graphic lines mostly to convey information. When graphic lines are used sparingly, the lines that are used stand out better (7). Horizontal lines can be used as blanks for user-supplied data. Vertical lines can be used as revision bars. Blank space can serve the same function that graphics lines are often used for in forms, providing separation between unrelated items of information and helping readers read horizontally across rows of information in tables. Include equipment number and page number at the right on each page. Sets of calculations arranged by major equipment number including letter prefix can be leafed through easily to locate calculations that are of interest to the reader. Make formulas readable without comments. Help readers be able to review formulas independent of the explanatory comments. Place formulas on separate lines from comments. Center formulas on the page, or indent them. Provide punctuation and text to allow the resulting material to be read straight through more easily than the formulas alone could be read. 3. Provide supporting information Write clear sentences. Start by writing what you would say aloud. Then remove excessive words. Rearrange phrases to improve clarity or eliminate ambiguity. Add words wherever this will help readers understand without having to concentrate as hard and without having to reread (8). Reread the work yourself later and edit it again, repeating these steps. List the contents. Simple, descriptive headings provide enough useful help to readers to avoid the need for paragraphs of explanatory text. State the objective. Readers expect to find the most important information at the start, and if not there, then at the end. They spend more time reading the information at the start. When the key theme is identified up front, readers understand the subsequent material better as they proceed 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. 12 in = 1 ft. 6ε 6ε a 1 : = G1 and a 2 : = G2 : d vs1 d vs2 2 2 a 1 = 49.255 ft and a 2 = 49.648 ft . 3 3 ft ft through it, and they proceed through it more quickly. Sketch the system. Sketches with text help people understand problems more thoroughly and help people move further toward solutions (9). The more useful diagrams show spatial relationships, show key data at a glance, and place information near the associated objects so that symbolic labels are not needed (10). When solving a problem that requires the use of formulas to interpret physical information, experts tend to insert an intermediate step redescribing the problem qualitatively (11). Sketches can capture some of an expert’s understanding of the problem by emphasizing key considerations while leaving out secondary details. Unfortunately, people who have less trouble proceeding with a problem tend to draw fewer sketches. As a result, they miss out on opportunities to help novices develop the skill of going beyond the literal features clearly evident in problem statements to infer additional relationships that are important for constructing effective solutions, which is the skill that novices are usually most lacking (6). State the approach, noting the key methods used. Name the key method or methods used, and describe how they ■ Figure 2. Current spreadsheets, such as Microsoft Excel, can display formulas and results together. CEP November 2001 www.cepmagazine.org 51 S A M P L E Objectives Approach References Symbols Assumptions page 1 1 1 1 1 Constants and Conversions Hardware Data Property Data Operating Data 1 1 2 2 Calculations Results 2 2 Objective Estimate minimum recommended slurry velocities vs. pipe diameter. Approach Apply the Durand equation for the minimum transport velocity as recommended in Perry’s and Heywood, adding a reasonable margin to the velocity as recommended in Heywood. References Data Sheet General Info Heywood P & ID Perry’s World Minerals, “Harborlite 2000 Technical Data Sheet,” World Minerals, Lompoc, CA, 2000 (enclosed) World Minerals, “Perlite General Information,” World Minerals, Lompoc, CA, 2000 (enclosed) Heywood, N.I., “Stop Your Slurries from Stirring Up Trouble,” Chemical Engineering Progress, pp. 21-41, September 1999 (enclosed) WRC, “Filter Aid Storage/Delivery P&ID,” Dwg. No. PR-001, Rev. A, Confidential Client, 2000 (in project master file) 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 ft sec 2 - 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 ft sec ft sec lb 3 ft lb 3 ft Assumptions Assume pipe is Schedule 40S. Constants and Conversions The gravitational acceleration g is: g = 32.174 ft . sec 2 Hardware Data Consider several pipe diameters D (Perry’s pages 10-72 and 10-73) 1.610 D: = 2.067 in. 3.068 52 www.cepmagazine.org November 2001 CEP C A L C U L A T I O N Property Data The 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. Ws sgs Cs: = Ws 100% – Ws + sgs sgL 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 lb3 ; ρL = 62.45 lb3 . ρs := sgs ⋅ 62.45 lb3 ; ρs := 143.63 lb3 . ft ft ft ft s, the ratio of solid density ρs to liquid density ρL, is: ρs = 143.63 lb3 . ρL = 62.45 lb3 . ft ft ρL s := ρ ; s = 2.3. S 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 ft . sec 2 s = 2.3. 2.0 1.610 ft . For D = 2.067 in, given that 12 in = 1 ft and VM2 := FL ⋅ [2 ⋅ g ⋅ D ⋅ (s–1)]0.5, V M2 = 2.3 sec 2.8 3.068 The minimum slurry velocities V can be calculated given the minimum transport velocities VM2 vs. the pipe diameters D (Heywood page 28): 2.5 1.610 For D = 2.067 in, V := 125% ⋅ VM2 ; V = 2.8 ft . sec 3.5 3.068 Results Minimum slurry velocities V vs. pipe diameters D are: 1.610 For D = 2.067 3.068 2.5 in, V = 2.8 ft . sec 3.5 CEP November 2001 www.cepmagazine.org 53 Professional Development were located and chosen. Avoid describing the details of the calculation in text form, since the actual formulas will be displayed later when they are used and will be nearly self-explanatory, while text descriptions of the formulas would take extra effort and extra skill to write and would be less clear and less helpful. Name, list and enclose references. Identify convenient, clear references for each formula used and data value entered. Name each reference using a short descriptive name such as the lead author’s name. Give titles and page ranges. Provide readers copies of references, so that they can find out things for themselves right away, while they are most interested. Use conventional symbols. Match the conventional notation in the area of interest for ready recognition. Use the same main symbol for all variables of a given type, and use subscripts to differentiate the family members from one another. Greek letters and subscripts can be typed directly into a calculation when calculation software like Mathcad or CalculationCenter is used. List complete symbols, including subscripts, and provide complete descriptions and standard units. Descriptions that include subscripts can eliminate guesswork. Standard units provide added descriptions of the symbols. Promote alternative methods. Describe alternative approaches and possible outcomes. Present the alternatives as positive possibilities, so they will be considered more likely and will therefore more effectively counterbalance the base case that is being presented positively. Considering alternatives reduces overconfidence. This promotes progress on problems (12), improves decision-making, and may improve self-checking by writers and errorchecking by readers. 4. Include text comments and equation-style comments with the working formulas Provide comments that supplement formulas but do not describe them. The working formulas do the actual calculation. Writers and readers need to be helped to review the formulas carefully, and need to not be lulled into a false sense of security by comments that seem to tell a complete story, and as a result, encourage them to skip over the equations (13). Repeat the description of each symbol each time it is used in a formula. Provide the description and repeat the symbol, including any subscript. List a source for each formula. Identify a convenient source that states the formula clearly. List the source’s short, descriptive name from the reference list, and identify the page or pages where the formula is defined. Include the formula number from the source, where helpful. Repeat the value of each symbol used in standard units each time it is used. Help people learn the relative magnitudes of terms and check the values of input data and intermediate results at every opportunity they have to do so. 54 www.cepmagazine.org November 2001 CEP Provide conversion factors each time they are used. Conversion factors often help reassure readers of the reasonableness of calculations and occasionally help writers find mistakes. When calculation software is used, it takes little effort to call up predefined conversion factors. Check function definitions by calculating known values. Check function definitions for temperature-dependent properties, for instance, by evaluating the equations at temperatures where the property values are known. Show a formula’s comments together with the formula on the same page. Add a page break before an assembled formula block if needed to keep the block together on a single page. Self-contained formula definitions and evaluations that can be seen together at a glance are easier to review. 5. Provide assumptions, inputs and calculations Note assumptions. Assumptions can include notes on how the mathematical models that are used oversimplify the behavior that they describe. They can also include notes on how the experimental approach underlying a method differs from the particulars of the process that is being analyzed. The assumptions that can be the most difficult to recognize are the underlying beliefs shared by previous workers, the writer, and the readers when they all are from the same era and have similar backgrounds. Explaining assumptions early, especially assumptions about factors that cannot be changed by the writer or the readers, produces more realistic assessments about the reliability of results. Reducing overconfidence improves checking, which improves accuracy. Enter any assumed data values. Assumptions can also include reasonable guesses of data values that are not known for certain. Provide an equation block for each assumed data value, as shown in the sample calculation. Repeat the description and the symbol, including any subscripts. Display the value in standard units for reasonability checking. Enter the hardware data. Provide an equation block for each hardware data value. Repeat the description and the symbol, including any subscripts. Repeat the source’s name from the references and note the applicable page or pages. Enter the data in the dimensional units that were used in the source. Provide any conversion factors used, in units that are as familiar as possible. Display the data in standard units for reasonability checking. Enter the property data. Provide an equation block for each property data value. Repeat the description and the symbol including any subscripts, the source of the data, the data in the units used in the source, any conversion factors, and the data in standard units. If a property is a function of variables such as temperature, pressure or composition, enter the property as a function that can be evaluated later based on the values of the variables at that point in the calculation. Enter the operating data. Provide an equation block for each operating data value. Provide the description and the symbol including any subscripts, the source of the data, the data in the units used in the source, any conversion factors, and the data in standard units. Enter the calculation formulas. Provide a formula block, like that in Figure 1, for each calculation formula. Provide text descriptions of the dependent variables and the independent variables, and provide a reference source for the formula. Then provide the values of the independent variables in standard units, any conversion factors, the formula, and the results. 6. Provide summary information Repeat key assumptions. An assumption may be crucial, and well worth highlighting again by including it in the summary information at the end of a calculation. Summarize the key input parameter values. Summarizing key input parameters near the end of calculations highlights them for the writer as well as for the reader. Also, sometimes it is convenient to set up a file containing a single case and then change input parameters and save separate files for new cases. In such situations, it is critical to Literature Cited 1. Miller, G. A., “The Magical Number Seven, Plus or Minus Two: Some Limits on our Capacity for Processing Information,” The Psychological Review, 63 (2), pp. 81–97 (1956). 2. Anthony, J., “Chloroform Plan,” available via http://www.cepmagazine.org (2001). 3. Phillips, J. E., and J. D. Decicco, “Choose the Right Mathematical Software,” Chem. Eng. Prog., 95 (7), pp. 69-74 (July 1999). 4. Sandler, S. I., “Spreadsheets for Thermodynamics Instruction: Another Point of View,” Chem. Eng. Edu., 31 (1), pp. 18-20 (Winter 1997). 5. Lira, Carl T., “Advanced Spreadsheet Features for Chemical Engineering 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 Educational Research, 59 (1), pp. 43-64 (1989). 10. Larkin, J. H., and H. A. Simon, “Why a Diagram is (Sometimes) Worth Ten Thousand Words,” Cognitive Science, 11 (1), pp. 65-99 (1987). 11. Larkin, J. H., “Processing Information for Effective Problem Solving,” 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 Programming Style,” 2nd ed., McGraw-Hill, New York, pp. 141-152 (1978). 14. Allwood, C. M., “Error Detection Processes in Statistical Problem Solving,” Cognitive Science, 8 (4), pp. 413-437; see pp. 419 and 431 (1984). point out the values of the changed parameters to distinguish the cases from one another. Summarize the results. Conclude by returning to the big picture and recalling key intermediate results and final results for the benefit of casual readers and careful readers alike. 7. Get calculations checked Seek checking, whether by experts or by interested colleagues. Unless people get very detailed feedback on their performance, they tend to be overconfident in their own abilities. They do not perform nearly enough selfchecks on material they believe is correct (14). As a result, errors of omission are almost never identified and corrected by the person who made them. With checking, a fresh viewpoint enters the situation, and errors of omission can be corrected. Calculations will get easier The calculation approach shown here easily scales up to handle tougher problems. An example of a more difficult calculation 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 experimental data on a process are unavailable, data cannot be obtained cheaply and quickly, and readily available calculation methods do not cover the process in question. This list of uses barely hints at the broader roles that could quickly develop for approaches like this. Affordable calculation software already provides the capability to embed subprograms and the capability to define graphical symbols that have smart interconnections. Soon, such software could provide the capability to embed subprograms in smart symbols. Libraries of thermodynamic property calculation routines and unit operations could emerge easily from environments of friendly competition and sharing, in academic settings and in industry. It could ultimately be possible to simply connect components from reliable sources and produce accurate and reliable process simulations and other calculations. Even wider impacts are imaginable. Calculation approaches developed for process applications could easily be adapted to other uses in science and in education. Much can get easier when collaboration is improved by CEP style in calculations. 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. CEP November 2001 www.cepmagazine.org 55 Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 1 of 37 Prep, Check, Appr: J Anthony, _________, _________ LOCKWOOD GREENE Confidential Client Chloroform Effluent Plan page Objective Approach References Symbols Assumptions 1 1 2 3 6 Conversions and Constants Hardware Data Property Data Operating Data 7 8 8 13 Steam Flows Interfacial Areas Mass Transfer Coefficients Stripping and Dilution 14 19 27 29 Key Parameters Summary Results Summary 34 35 Objective Model the lime press effluent chloroform concentration after raffinate stripping, reslurrying using stripper effluent, and reslurrying using four chloroform-free streams. S team FO Lim e P ress S olids FO 1. S trip Liquor R affinate 2. 3. 4. 5. 1st S tage Lim e Liquor W ater W ater W ater 2nd S tage 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 LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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, Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 3 of 37 y Confidential Client, 11-8-99 (enclosed) g 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 a1 and a2 effective interfacial area per unit volume of vessels ft 2 ft ft 2 3 2 aT1 and aT2 total hole areas of first and second stage spargers ft Bw second virial coefficient of water vapor ft 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 water Dv1 and Dv2 dN1 and dN2 diameters of stripper first and second stage vessels ft sec in hole diameters of first and second stage spargers in 3 2 Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 4 of 37 inertial_force , of first and second stage vessels gravity_force Frgc1 and Frgc2 Froude numbers for transition to foaming flow Fr1 and Fr2 Froude numbers, lbmole GM1 and GM2 gas-phase molar fluxes based on vessel cross-sectional areas 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 - hr⋅ ft 2 in water in the first and second stage vessels KL1 and KL2 overall liquid-phase mass-transfer coefficients in vessels kL1 and kL2 individual liquid-phase mass-transfer coefficients in vessels kw heat capacity ratio of water vapor LM1 and LM2 liquid-phase molar fluxes based on vessel cross-sectional areas ft hr ft hr lbmole hr⋅ ft Lover1 and Lover2 level of clear liquid at overflow of first and second stage vessels % lb lbmole Mw and Mc molecular weight of water and of chloroform NA1 and NA2 molar fluxes based on interfacial areas in vessels 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 raffinate Rg universal gas constant r rc1 and rc2 pressure ratios of the orifices critical pressure ratios of restriction orifices lbmole hr⋅ ft - Sc1 and Sc2 inertial_force , of spargers viscous_force Schmidt numbers of first and second stage vessels - Sh1 and Sh2 Sherwood numbers of first and second stage vessels - T T0 absolute temperature absolute temperature upstream of the restriction orifices K K TF magnitude of the temperature in degrees F - Tref absolute temperature reference value K NOT CHECKED 2 gal min joule mole⋅ K - ReN1 and ReN2 Reynolds numbers, Chloroform Plan.mcd 2 LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 spargers ft sec V1 and V2 velocities in first and second stage restriction orifices ft sec rated capacities of first and second stage vessels gal Vcap1 and Vcap2 Voper1 and Voper2 operating volumes of first and second stage vessels W1 and W2 mass flows in first and second stage restriction orifices WT total mass flow of steam wcr weight fraction chloroform in raffinate gal lb hr lb hr - wcs weight fraction chloroform in stripped liquid - wcs_adj weight fraction chloroform in stripped liquid, adjusted - Weg1 and Weg2 Weber numbers, - x1v1 and y1v1 inertial_force , of spargers surface_tension_force liquid and vapor mole fractions chloroform - x2v1 and y2v1 at bottom - position 1 - of first stage vessel liquid and vapor mole fractions chloroform - x1v2 and y1v2 at top - position 2 - of first stage vessel liquid and vapor mole fractions chloroform - x2v2 and y2v2 at bottom - position 1 - of second stage vessel liquid and vapor mole fractions chloroform - β 1 and β 2 at top - position 2 - of second stage vessel diameter ratios of first and second stage restriction orifices - ε G1 and ε G2 gas holdups in first and second stage vessels - γc activity coefficient of chloroform at infinite dilution in water - µ lw liquid viscosity of water cP µ vw vapor viscosity of water lb ft⋅ sec ρ0 vapor density upstream of the restriction orifices ρ lw ρv liquid density of water vapor density in first and second stage vessels lb 3 ft lb 3 ft lb ft σw Chloroform Plan.mcd surface tension of water NOT CHECKED 3 newton m LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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): 3926 6 wcr := 10 + 3581 6 10 3 + 3760 6 10 : wcr = 3756 1 6 . 10 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 . Tv1 := 91 + 92 + 91 + 91 + 94 + 273.15 K and Tv2 := ( 100 + 273.15)K : 5 Tv1 = 364.95 K and Tv2 = 373.15 K . 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): 4.7 qr := Chloroform Plan.mcd gal gal + 3.67 min min gal ; qr = 4.185 . 2 min NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 := 10 −2 ⋅ poise . The millinewton mN is defined as follows: mN := 10 −3 ⋅ newton . The amount of substance kgmole is defined as follows: kg = 1000 gm , so kgmole := 1000mole . The amount of substance lbmole is defined as follows: lb = 453.592370 gm , 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): T ⋅ 1.8 − 459.69 . K Tref = 298.15 K : TF ( Tref ) = 76.980 . TF ( T) := The universal gas constant Rg is (Perry's page 1-19): 3 Rg := 8.3144 Chloroform Plan.mcd joule psi⋅ ft . Alternatively, Rg = 10.73 . mole⋅ K lbmole⋅ R NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 8 of 37 Hardware Data The restriction orifice diameters of the first and second stages, D1 and D2 , are (Baseline: JH): 16 19 ⋅ in and D2 := ⋅ in : 64 64 D1 = 0.250 in and D2 = 0.297 in . D1 := 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): dN1 := 11 13 in and dN2 := in . 12in = 1 ft . 64 64 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 := 18.015 Chloroform Plan.mcd kg gm lb ; Mw = 18.015 ; Mw = 18.015 . kgmole mole lbmole NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 0.0810 ... 1 + 1 − 647.13 0.3054 4.9669 + ( 333.15⋅ K < T) ⋅ ( T ≤ 403.15⋅ K) ⋅ 0.18740 T K 1 + 1 − 647.13 0.27788 4.3910 + ( 403.15⋅ K < T) ⋅ ( T ≤ 647.13⋅ K) ⋅ 0.25340 T K 1 + 1 − 647.13 0.24870 T K ⋅ kgmole ⋅ M ; M = 18.015 kg . w w 3 kgmole m ... Tref = 298.15 K : ρ lw ( Tref ) Mw ρ lw ( Tref ) = 995.122 kg 3 = 55.239 kgmole 3 . m . 1kg = 2.204623 lb . 1ft = 0.304800 m . m ρ lw ( Tref ) = 62.123 lb ft 3 . 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 73.649 + −7258.2 T T + −7.3037⋅ ln ... ⋅ Pa . K K 2.0000 T + 4.1653⋅ 10− 6⋅ K Tref = 298.15 K : Pvw ( Tref ) = 3170 Pa . 1Pa = 0.000145038 psi . Pvw ( Tref ) = 0.4598 psi . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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⋅ 10 ... 4 2610.5 T K 4 + 2.6790⋅ 10 ⋅ 2610.5 sinh T K 2 1169.0 T K + 8896.0⋅ 1169.0 cosh T K ⋅ joule . kgmole⋅ K 2 ... Tref = 298.15 K : CPgw ( Tref ) = 33578 joule kgmole⋅ K joule . 1 = 0.000238851 kgmole⋅ K BTU . CPgw ( Tref ) = 8.020 lbmole⋅ R Mw = 18.015 CPgw ( Tref ) Mw lb lbmole = 0.445193 BTU lbmole⋅ R . . BTU 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 = 298.15 K : CPgw ( Tref ) = 33578 joule . Rg = 8314 kgmole⋅ K kw ( Tref ) = 1.329106 . Chloroform Plan.mcd NOT CHECKED joule kgmole⋅ K . LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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) ⋅ 0.0222 + −26.38 7 + −1.675⋅ 10 T 3 T K K 19 21 + −3.894⋅ 10 + 3.133⋅ 10 8 9 T T K K 3 m . kgmole ... ⋅ Tref = 298.15 K : Bw ( Tref ) = −1.1536 3 m kgmole . 1m = 3.280840 ft . 1lbmole = 0.453592 kgmole . Bw ( Tref ) = −18.479 3 ft . lbmole The liquid viscosity of water, µ lw , as a function of the absolute temperature T is (DIPPR): µ lw ( T) := ( 273.16⋅ K ≤ T) ⋅ ( T ≤ 646.15⋅ K) ⋅ exp −52.843 + 3703.6 T T + 5.8660⋅ ln ... ⋅ Pa⋅ sec . K K 10.000 T + −5.8790⋅ 10− 29⋅ K Tref = 298.15 K : µ lw ( Tref ) = 0.0009 Pa⋅ sec . 1Pa⋅ sec = 999.978174 cP . µ lw ( Tref ) = 0.912511 cP . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 12 of 37 The vapor viscosity of water, µ vw , as a function of the absolute temperature T is (DIPPR): 6.1839⋅ 10 −7 µ vw ( T) := ( 273.16K ≤ T) ⋅ ( T ≤ 1073.2K) ⋅ 847.23 1+ T T ⋅ K 0.6778 4 + −7.3930⋅ 10 K T K ⋅ Pa⋅ sec . 2 Tref = 298.15 K : µ vw ( Tref ) = 9.7696 × 10 −6 Pa⋅ sec . 1Pa⋅ sec = 0.671969 µ vw ( Tref ) = 6.564835 × 10 −6 lb . ft sec lb . ft sec The surface tension of water, σ w , as a function of the absolute temperature T is (DIPPR): T K σ w ( T) := ( 273.16⋅ K ≤ T) ⋅ ( T ≤ 647.13⋅ K) ⋅ 0.1855⋅ 1 − 647.13 T T K K 2.7170+ − 3.5540⋅ + 2.0470⋅ 647.13 647.13 ⋅ Tref = 298.15 K : σ w ( Tref ) = 0.072825 newton m . 1newton = 1000 mN . σ w ( Tref ) = 72.825 σ w ( Tref ) = 0.072825 newton m mN m . . 1newton = 0.224814 lbf . 1ft = 0.304800 m . σ w ( Tref ) = 0.004990185 lbf . ft The molecular weight of chloroform, Mc , is (DIPPR): Mc := 119.38 Chloroform Plan.mcd kg kgmole ; Mc = 119.38 gm mole ; Mc = 119.38 NOT CHECKED 2 lb lbmole . newton . m LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 146.43 + −7792.3 T + −20.614⋅ ln ... ⋅ Pa . T K K 1.0000 T + 0.0246 ⋅ K Tref = 298.15 K : Pvc ( Tref ) = 26337 Pa . 1Pa = 0.000145038 psi . Pvc ( Tref ) = 3.8199 psi . The diffusivity of chloroform in infinite dilution in water, Dc0w , as a function of the absolute temperature T is (Yaws): − 1.4389+ Dc0w ( T) := ( 274⋅ K ≤ T) ⋅ ( T ≤ 394⋅ K) ⋅ 10 − 1051.706 T K Dc0w [ ( 25 + 273.15)K] = 1.08 × 10 −5 Dc0w [ ( 100 + 273.15)K] = 5.53 × 10 2 cm ⋅ sec . 2.54cm = 1 in . 2 cm . sec −5 2 cm . sec The activity coefficient of chloroform at infinite dilution in water, γ c , as a function of the absolute temperature T is (Wright, Mathcad Polynomial Regression): 20.0 + 273.15 818 20.0 + 273.15 818 γ c ( T) := interp regress 35.0 + 273.15 , 847 , 2 , 35.0 + 273.15 K , 847 , T ; 50.0 + 273.15 862 50.0 + 273.15 862 γ c [ ( 20.0 + 273.15)K] = 818 . γ c [ ( 35.0 + 273.15)K] = 847 . γ c [ ( 50.0 + 273.15)K] = 862 . γ c [ ( 100.0 + 273.15)K] = 811 . Operating Data The absolute pressure upstream of the orifices, P0 , is (P&ID): P0 := 30psi + 14.7psi : P0 = 44.7 psi . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 = 44.7 psi . Pv = 14.7 psi . Pv r := P0 : r = 0.328770 . The diameter ratios of the first and second stage restriction orifices, β 1 and β 2 , are calculated from the orifice diameters D1 and D2 and the orifice pipe inside diameters ID1 and ID2 (Perry's page 10-4): β 1 := D1 ID1 and β 2 := D2 ID2 : β 1 = 0.238322 and β 2 = 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 = 298.15 K , so T0 = 298.15 K . Given P0 = Pvw ( T0 ) , calculate T0 := Find ( T0 ) : T0 = 407.67 K . [ 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 β 1 and β 2 (Perry's page 10-14): T0 = 407.67 K . [ TF ( T0 ) = 274.11 .] kw ( T0 ) = 1.319786 . β 1 = 0.238322 and β 2 = 0.283008 . Start with rc1 := 1 and rc2 := 1 . 1 − kw( T0) Given ( rc1) kw( T0) 1 − kw( T0) (rc2) kw( T0) kw ( T0) − 1 + 2 2 ⋅ ( β 1) ⋅ ( rc1) 4 kw( T0) = 2 2 kw ( T0 ) + 1 kw( T0) kw ( T0) − 1 4 , = + ⋅ ( β 2) ⋅ ( rc2) 2 2 rc1 calculate := Find ( rc1 , rc2) : rc2 rc1 = 0.542571 and rc2 = 0.542962 . Chloroform Plan.mcd kw ( T 0 ) + 1 NOT CHECKED and LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 β 1 and β 2 alone (Benedict page 247 equation 22): 3 ( K1 := 1 − β 1 ) 2 2 + ( ) ( 1 2 ⋅ 1 − β 1 + 1.41⋅ 1 − β 1 2 ) 2 2 3 ( and K2 := 1 − β 2 ) 2 2 + ( ) ( ) 1 2 2 2 : ⋅ 1 − β 2 + 1.41⋅ 1 − β 2 2 K1 = 2.653 and K2 = 2.550 . The vapor density upstream of the restriction orifices, ρ 0 , 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 = 44.7 psi . Mw = 18.015 Rg = 10.73 psi⋅ ft 3 lbmole⋅ R . . K = 1.8 R . T0 = 407.67 K . [ TF ( T0 ) = 274.11 .] Bw ( T0 ) = −5.297421 Start with ρ 0 := 0.1 lb ft Given that P0 ρ0 Mw ⋅ Rg⋅ T0 = 1 + B w ( T0 ) ⋅ ρ0 Mw ρ 0 = 0.105532 3 . ( ) , calculate ρ 0 := Find ρ 0 : lb ft Chloroform Plan.mcd lb lbmole 3 . NOT CHECKED ft 3 lbmole . LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 ρ 0 (Benedict page 245 equation 1): P0 = 44.7 lbf 2 in . rc1 = 0.542571 and rc2 = 0.542962 . ft 1⋅ lbf = 32.174 lb⋅ 2 2 . 144in = 1 ft 2 . sec lb . K1 = 2.652828 and K2 = 2.550223 . ρ 0 = 0.105532 3 ft V1 := 2 (P0 − rc1⋅ P0)⋅ K ⋅ ρ 1 V1 = 822.655 and V2 := 2 (P0 − rc2⋅ P0)⋅ K ⋅ ρ 2 0 : 0 ft ft and V2 = 838.683 . sec 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 ρ 0 , the pipe diameters D1 and D2 , and the velocities in the orifices V1 and V2 (Benedict page 245 equation 3): ρ 0 = 0.105532 lb ft 3 . π = 3.141593 . D1 = 0.25 in . V1 = 822.655 D2 = 0.296875 in . V2 = 838.683 ft sec ft sec . 2 2 D1 D2 ⋅ V1 and W2 := ρ 0⋅ π ⋅ ⋅ V2 : 2 2 W1 := ρ 0 ⋅ π ⋅ W1 = 106.540 lb hr and W2 = 153.165 WT := W1 + W2 : WT = 260 Chloroform Plan.mcd NOT CHECKED lb hr . lb hr . 2 2 . 144in = 1 ft . LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 17 of 37 The vapor density at the vessels, ρ v , 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 = 14.696 psi . Mw = 18.015 Rg = 10.73 psi⋅ ft 3 lbmole⋅ R lb . lbmole . K = 1.8 R . Tv1 = 364.95 K . [ TF ( Tv2) = 211.98 .] Bw ( Tv2) = −7.242413 Start with ρ v := 0.1 lb ft Given that Pv ρv Mw = 1 + Bw ( Tv2) ⋅ ⋅ Rg⋅ Tv2 ρv Mw ρ v = 0.037288 lb ft 3 3 3 ft . lbmole . ( ) , calculate ρ v := Find ρ v : . 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 . 2 2 Dv1 Dv2 and A2 := π ⋅ : 2 2 A1 := π ⋅ 2 2 A1 = 2.182 ft and A2 = 3.142 ft . The superficial velocities in the first and second stage vessels, UG1 and UG2 , are calculated from the mass flows W1 and W2 , vapor density ρ v , and vessel cross-sectional areas A1 and A2 : lb lb and W2 = 153.165 . 3600sec = 1 hr . hr hr lb 2 2 . A1 = 2.182 ft and A2 = 3.142 ft . ρ v = 0.037288 3 ft W1 W2 and UG2 := : UG1 := ρ v⋅ A1 ρ v⋅ A2 W1 = 106.540 UG1 = 0.363794 ft sec and UG2 = 0.363195 UG1 = 0.110884 Chloroform Plan.mcd m sec ft sec . 1ft = 0.304800 m . and UG2 = 0.110702 NOT CHECKED m sec . LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 = 0.36 ft ft and UG2 = 0.36 . sec sec ft , "Quiescent" , "" and sec ft Vessel_Gas_Velocity2 := if UG2 ≤ 0.15 , "Quiescent" , "" ; sec ft ft Vessel_Gas_Velocity1 := if 0.15 < UG1 ⋅ UG1 < 0.20⋅ , "Quiescent or Turbulent" , Vessel_Gas_Velocity1 a sec sec Vessel_Gas_Velocity1 := if UG1 ≤ 0.15 ft ft < UG2 ⋅ UG2 < 0.20⋅ , "Quiescent or Turbulent" , Vessel_Gas_Velocity2 sec sec ft Vessel_Gas_Velocity1 := if 0.20⋅ ≤ UG1 , "Turbulent" , Vessel_Gas_Velocity1 and sec ft Vessel_Gas_Velocity2 := if 0.20⋅ ≤ UG2 , "Turbulent" , Vessel_Gas_Velocity2 : sec and Vessel_Gas_Velocity1 = "Turbulent" Vessel_Gas_Velocity2 = "Turbulent" . Vessel_Gas_Velocity2 := if 0.15 Vessel_Quiescence_vs_Max1 := UG1 0.15 Vessel_Quiescence_vs_Max2 := ft sec UG2 0.15 . 1 = 100 % . ft . 1 = 100 % : sec Vessel_Quiescence_vs_Max1 = 243 % and Vessel_Quiescence_vs_Max2 = 242 % . Vessel_Turbulence_vs_Min1 := UG1 0.20 Vessel_Turbulence_vs_Min2 := ft sec UG2 0.20 . 1 = 100 % . ft . 1 = 100 % : sec Vessel_Turbulence_vs_Min1 = 182 % and Vessel_Turbulence_vs_Min2 = 182 % . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 . 2 2 dN1 dN2 aT1 := π ⋅ ⋅ n1 and aT2 := π ⋅ ⋅ n2 : 2 2 2 2 aT1 = 0.003545 ft and aT2 = 0.004951 ft . 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 ρ v , and total hole areas aT1 and aT2 : W1 = 106.54 ρ v = 0.037288 lb ft 3 lb hr and W2 = 153.165 hr . 3600sec = 1 hr . 2 2 . aT1 = 0.003545 ft and aT2 = 0.004951 ft . uN1 := W1 ρ v⋅ aT1 uN1 = 223.907 Chloroform Plan.mcd lb ft sec and uN2 := W2 ρ v⋅ aT2 : and uN2 = 230.469 NOT CHECKED ft sec . LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 = 224 ft and sec ft . sec Vessel_Gas_Velocity1 = "Turbulent" uN2 = 230 and Vessel_Gas_Velocity2 = "Turbulent" . Sparger_Velocity1 := if uN1 ≤ 250 ft , "Not Excessive" , "" and sec Sparger_Velocity2 := if uN2 ≤ 250 Sparger_Velocity1 := if 250 ft < uN1 ⋅ uN1 < 300⋅ ft , "Not Excessive" , "" ; sec ft , "Possibly Excessive" , Sparger_Velocity1 and sec ft Sparger_Velocity2 := if 250 < uN2 ⋅ uN2 < 300⋅ , "Possibly Excessive" , Sparger_Velocity2 ; sec sec ft Sparger_Velocity1 := if 300⋅ ≤ uN1 , "Excessive" , Sparger_Velocity1 and sec ft Sparger_Velocity2 := if 300⋅ ≤ uN2 , "Excessive" , Sparger_Velocity2 : sec sec ft Sparger_Velocity1 = "Not Excessive" and Sparger_Velocity2 = "Not Excessive" . uN1 Sparger_Velocity_vs_Max_Okay1 := 250 ft sec uN2 Sparger_Velocity_vs_Max_Okay2 := 250 . 1 = 100 % . ft . 1 = 100 % : sec Sparger_Velocity_vs_Max_Okay1 = 90 % and Sparger_Velocity_vs_Max_Okay2 = 92 % . Sparger_Velocity_vs_Min_Excessive1 := Sparger_Velocity_vs_Min_Excessive2 := uN1 ft 300 sec uN2 . 1 = 100 % . . 1 = 100 % : ft 300 sec Sparger_Velocity_vs_Min_Excessive1 = 75 % and Sparger_Velocity_vs_Min_Excessive2 = 77 % . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 21 of 37 inertial_force (Perry's page 6-50), for the gas flows at the holes in the first viscous_force and second stage spargers, ReN1 and ReN2 , are calculated from the hole diameters dN1 and dN2 , hole The Reynolds numbers, superficial gas velocities uN1 and uN2 , vapor density ρ v , and vapor viscosity of water at the absolute temperature in the second stage vessel (as noted in the assumptions) µ vw ( Tv2) (Kumar page 508 Nomenclature section): ft ft lb and uN2 = 230.469 . ρ v = 0.037288 . 3 sec sec ft 12in = 1 ft . − 6 lb . Tv2 = 373.15 K . [ TF ( Tv2) = 211.98 .] µ vw ( Tv2) = 8.397656 × 10 ft sec dN1 = 0.171875 in and dN2 = 0.203125 in . uN1 = 223.907 ReN1 := dN1⋅ uN1⋅ ρ v µ vw ( Tv2) and ReN2 := dN2⋅ uN2⋅ ρ v µ vw ( 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" . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 22 of 37 The Weber numbers, inertial_force (Perry's page 6-50), for the gas flows at the holes in the surface_tension_force first and second stage spargers, Weg1 and Weg2 , are calculated from the vapor density ρ v , 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) σ w ( Tv1) and σ w ( Tv2) (Perry's page 14-71): ρ v = 0.037288 lb ft 3 . dN1 = 0.171875 in and dN2 = 0.203125 in . uN1 = 223.907 ft ft and uN2 = 230.469 . sec sec Tv1 = 364.95 K . [ TF ( Tv1) = 197.22 .] σ w ( Tv1) = 0.004098 lbf lbft . 1lbf = 32.174 . 2 ft sec Tv2 = 373.15 K . [ TF ( Tv2) = 211.98 .] σ w ( Tv2) = 0.003989 lbf Weg1 := ρ v⋅ dN1⋅ uN1 σ w ( Tv1) 2 and Weg2 := ft . 1lbf = 32.174 ρ v⋅ dN2⋅ uN2 lbft sec 2 . 2 σ w ( 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_We1 := Sparger_We_vs_Jet_Flow_We2 := Weg1 2 Weg2 2 . 1 = 100 % . . 1 = 100 % : Sparger_We_vs_Jet_Flow_We1 = 10153 % and Sparger_We_vs_Jet_Flow_We2 = 13060 % . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 σ w ( Tv1) , σ w ( Tv2) , ρ lw ( Tv1) , and ρ lw ( Tv2) (as noted in the assumptions), the vapor density in the spargers ρ v , 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 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] lbf lbf lbft and σ w ( Tv2) = 0.003989 . 1lbf = 32.174 . 2 ft ft sec lb lb lb ft and ρ lw ( Tv2) = 59.793 . ρ v = 0.037288 . g = 32.174 . ρ lw ( Tv1) = 60.155 3 3 3 2 ft ft ft sec σ w ( Tv1) = 0.004098 Define interpolation functions f1 ( ReN1) and f2 ( ReN2) . ( 0.425 log 0.32⋅ 2100 f1 ( ReN1) := 10 ( 0.425 log 0.32⋅ 2100 f2 ( ReN2) := 10 0.425 − 0.4 ) + log( 100⋅ 4000 ) − log( 0.32⋅ 2100 ) ⋅ ( log( ReN1) − log( 2100) ) log( 4000) − log( 2100) ) + log( 100⋅ 4000 ) − log( 0.32⋅ 2100 − 0.4 0.425 log( 4000) − log( 2100) ) ⋅ 0.425 = 8.262 . − 0.4 = 3.624 . f1 ( 2100) = 8.262 ; 0.32⋅ 2100 f1 ( 4000) = 3.624 ; 100⋅ 4000 and ( log( ReN2) − log( 2100) ) : f1 ( 3000) = 5.236 . 1 4 σ w ( Tv1) ⋅ dN12 . 1ft = 12 in . dvs1 := ( 1 < ReN1) ⋅ ( ReN1 ≤ 10) ⋅ 1.56⋅ ReN1 ... ⋅ ( ρ lw ( Tv1) − ρ v) ⋅ g 0.425 ... + ( 10 < ReN1) ⋅ ( ReN1 ≤ 2100) ⋅ 0.32⋅ ReN1 + ( 2100 < ReN1) ⋅ ( ReN1 < 4000) ⋅ f1 ( ReN1) ... − 0.4 + ( 4000 ≤ ReN1) ⋅ ( ReN1 < 70000) ⋅ 100⋅ ReN1 0.058 1 4 σ w ( Tv2) ⋅ dN22 ⋅ . 1ft = 12 in : dvs2 := ( 1 < ReN2) ⋅ ( ReN2 ≤ 10) ⋅ 1.56⋅ ReN2 ... ( ρ lw ( Tv2) − ρ v) ⋅ g 0.425 ... + ( 10 < ReN2) ⋅ ( ReN2 ≤ 2100) ⋅ 0.32⋅ ReN2 + ( 2100 < ReN2) ⋅ ( ReN2 < 4000) ⋅ f1 ( ReN2) ... − 0.4 + ( 4000 ≤ ReN2) ⋅ ( ReN2 < 70000) ⋅ 100⋅ ReN2 0.058 dvs1 = 0.285 in and dvs2 = 0.285 in ; dvs1 dN1 Chloroform Plan.mcd = 166 % and dvs2 dN2 = 140 % . NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 24 of 37 inertial_force (Perry's page 6-49), in the first and second stage vessels, Fr1 gravity_force and Fr2 , are calculated from the superficial velocities UG1 and UG2 , gravitational acceleration g , and The Froude numbers, vessel diameters Dv1 and Dv2 (Godbole pages 1235-1236): UG1 = 0.363794 g = 32.174 ft sec 2 ft ft . UG2 = 0.363195 . sec sec . Dv1 = 20 in . Dv2 = 24 in . 12in = 1 ft . Fr1 := UG1 g⋅ Dv1 . Fr2 := UG2 g⋅ Dv2 . Fr1 = 0.049680 . Fr2 = 0.045276 . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 25 of 37 The gas holdups in the first and second stage vessels, ε G1 and ε G2 , 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) , σ w ( Tv1) , σ w ( Tv2) , µ lw ( Tv1) and µ lw ( Tv2) , and the superficial velocities UG1 and UG2 (Akita page 78 equation 11): Tv1 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] g = 32.174 ft lb lb . Dv1 = 20 in and Dv2 = 24 in . 12in = 1 ft . ρ lw ( Tv1) = 60.155 and ρ lw ( Tv2) = 59.793 . 3 3 sec ft ft 2 lbf lbft − 3 lbf and σ w ( Tv2) = 3.989 × 10 . 1lbf = 32.174 . 2 ft ft sec − 6 lb − 6 lb and µ lw ( Tv2) = 187.815 × 10 . µ lw ( Tv1) = 206.177 × 10 ft sec ft sec σ w ( Tv1) = 4.098 × 10 −3 UG1 = 0.363794 ft sec and UG2 = 0.363195 ft sec . Start with ε G1 := 10% and ε G2 := 10% . Given that ε G1 (1 − ε G1)4 ε G2 (1 − ε G2)4 1 1 8 12 g⋅ Dv12⋅ ρ lw ( Tv1) g⋅ Dv13 UG1 1.0 = 0.20⋅ ⋅ ⋅ and 2 σ T ⋅ g D ( ) v1 w v1 µ lw ( Tv1) ρ lw ( Tv1) = 0.20⋅ 1 1 8 12 g⋅ Dv2 ⋅ ρ lw ( Tv2) g⋅ Dv23 UG2 1.0 ⋅ ⋅ , 2 σ w ( Tv2) µ lw ( Tv2) g⋅ Dv2 ρ lw ( Tv2) ε G1 calculate := Find ( ε G1 , ε G2) : ε G2 2 ε G1 = 19.468 % and ε G2 = 19.621 % . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 26 of 37 inertial_force (Perry's page 6-49), for transition to gravity_force The critical values of the Froude number, foaming flow in the first and second stage vessels, Frgc1 and Frgc2 , are calculated from the gas holdups ε G1 and ε G2 , 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): ε G1 = 0.195 and ε G2 = 0.196 . Frgc1 := 0.25⋅ ε G1 2 3 and Frgc2 := (1 − ε G1) 2 Frgc1 = 0.013110 0.25⋅ ε G2 2 3 . (1 − ε G2) 2 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_Nonfoaming1 := Vessel_Fr_vs_Min_Nonfoaming2 := Fr1 Frgc1 Fr2 Frgc2 . 1 = 100 % . . 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 ε G1 and ε G2 and the Sauter-mean bubble diameters or bubble volume-to-surface ratios dvs1 and dvs2 (Perry's pages 5-69 and 5-43): ε G1 = 0.194676 and ε G2 = 0.196213 . dvs1 = 0.284573 in and dvs2 = 0.284552 in . 12in = 1 ft . a1 := 6⋅ ε G1 dvs1 a1 = 49.255 ft ft Chloroform Plan.mcd 6⋅ ε G2 and a2 := and a2 = 49.648 2 3 : dvs2 NOT CHECKED ft ft 2 3 . LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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) , µ lw ( Tv1) , and µ lw ( Tv2) , the vapor density ρ v , and the gravitational acceleration g (Calderbank page 53): Tv1 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] lb lb ft . ρ v = 0.037288 . g = 32.174 . dvs1 = 0.284573 in . 12in = 1 ft . ρ lw ( Tv1) = 60.155 3 3 2 ft ft sec lb dvs2 = 0.284552 in . ρ lw ( Tv2) = 59.793 3 ft ft sec ( ) dvs1 ⋅ ρ lw ( Tv1) ⋅ ρ lw ( Tv1) − ρ v ⋅ g 3 Gr1 := lb µ lw ( Tv1) = 0.000206177 µ lw ( Tv1) 2 Gr1 = 36.503 × 10 6 and µ lw ( Tv2) = 0.000187815 . ( ) dvs2 ⋅ ρ lw ( Tv2) ⋅ ρ lw ( Tv2) − ρ v ⋅ g 3 and Gr2 := lb ft sec µ lw ( Tv2) and Gr2 = 43.452 × 10 6 2 : . 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) µ lw ( Tv1) , µ lw ( 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): Tv1 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] µ lw ( Tv1) = 0.000206177 ρ lw ( Tv1) = 60.155 lb ft 3 lb lb and µ lw ( Tv2) = 0.000187815 . ft sec ft sec and ρ lw ( Tv2) = 59.793 lb ft 3 . Dc0w ( Tv1) = 51.439 × 10 Dc0w ( Tv2) = 59.514 × 10 Sc1 := µ lw ( Tv1) ρ lw ( Tv1) ⋅ Dc0w ( Tv1) and Sc2 := −9 2 ft . sec µ lw ( Tv2) ρ lw ( Tv2) ⋅ Dc0w ( Tv2) Sc1 = 66.632 and Sc2 = 52.780 . Chloroform Plan.mcd NOT CHECKED : −9 2 ft and sec LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 × 10 6 and Gr2 = 43.452 × 10 Sc1 = 66.632 and Sc2 = 52.780 . 6 . dvs1 = 0.284573 in and dvs2 = 0.284552 in . 1in = 2.54 cm , so dvs1 = 0.722814 cm and dvs2 = 0.722762 cm . 1 1 Sh1 := ( dvs1 < 0.25cm) ⋅ 2 + 0.31⋅ Gr1 ⋅ Sc1 3 1 1 + ( 0.25cm ≤ dvs1) ⋅ 0.42⋅ Gr1 ⋅ Sc1 3 3 1 1 ... and Sh2 := ( dvs2 < 0.25cm) ⋅ 2 + 0.31⋅ Gr2 ⋅ Sc2 3 1 + ( 0.25cm ≤ dvs2) ⋅ 0.42⋅ Gr2 ⋅ Sc2 2 3 3 ... : 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): Tv1 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] Sh1 = 1554.180 . Dc0w ( Tv1) = 51.439 × 10 −9 Sh2 = 1481.527 . Dc0w ( Tv2) = 59.514 × 10 −9 ft 2 sec ft . 1hr = 3600 sec . 2 . 1hr = 3600 sec . sec dvs1 = 0.284573 in and dvs2 = 0.284552 in . 12in = 1 ft . kL1 := Sh1 ⋅ Dc0w ( Tv1) dvs1 kL1 = 12.136 and kL2 := Sh2 ⋅ Dc0w ( Tv2) dvs2 : ft ft and kL2 = 13.386 . hr 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): ft ft and kL2 = 13.386 . hr hr KL1 := kL1 and KL2 := kL2 : kL1 = 12.136 KL1 = 12.136 Chloroform Plan.mcd ft ft and KL2 = 13.386 . hr hr NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 ε G1 and ε G2 (Akita page 76 equation 1): Vcap1 = 32 gal and Vcap2 = 52 gal . Lover1 = 60 % and Lover2 = 60 % . ε G1 = 19.468 % and ε G2 = 19.621 % . 100% = 1 . Voper1 := Vcap1⋅ Lover1 100% − ε G1 and Voper2 := Vcap2⋅ Lover2 100% − ε G2 : Voper1 = 23.841 gal and Voper2 = 38.816 gal . 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 = 364.95 K . [ TF ( Tv1) = 197.22 .] qr = 4.185 gal min lb 3 . 1gal = 0.133681 ft . 1hr = 60 min . ρ lw ( Tv1) = 60.155 . 3 ft wcr = 0.003756 . Mw = 18.015 lb lbmole and Mc = 119.38 100lbwcr LM1 := qr⋅ ρ lw ( Tv1) ⋅ Chloroform Plan.mcd Mc + lb lbmole . A1 = 2.182 ft 2 100lb⋅ ( 1 − wcr) Mw 100lb NOT CHECKED ⋅ 1 lbmole : LM1 = 51.213 . 2 A1 hr⋅ ft LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 = 18.015 x2v1 := lb lb . Mc = 119.38 . lbmole lbmole 100lb⋅ wcr Mc 100lbwcr Mc + 100lb⋅ ( 1 − wcr) : x2v1 = 0.000569 . Mw 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 = 106.540 Mw = 18.015 GM1 := GM1 = 2.711 lb Mw⋅ A1 lbmole hr⋅ ft 2 hr . 2 lbmole W1 lb . A1 = 2.182 ft . and GM2 := W2 Mw⋅ A2 and GM2 = 2.706 : lbmole hr⋅ ft 2 . 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) , γ c ( Tv1) , and γ c ( Tv2) , and from the total pressure in the vessels Pv (Hwang page 1759 equation 1): Tv1 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] Pvc ( Tv1) = 36.595 psi and Pvc ( Tv2) = 45.276 psi . γ c ( Tv1) = 829.935 and γ c ( Tv2) = 810.889 . Pv = 14.696 psi . Kc1 := Pvc ( Tv1) ⋅ γ c ( Tv1) Chloroform Plan.mcd Pv and Kc2 := Pvc ( Tv2) ⋅ γ c ( Tv2) Pv NOT CHECKED : Kc1 = 2067 and Kc2 = 2498 . LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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): Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 32 of 37 x2v 1 y2v1 x2v 2 y2v2 x1v 1 y1v1 x1v 2 y1v2 Tv1 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] a1 = 49.255 GM1 = 2.711 ft 2 3 . Voper1 = 23.841 gal . LM1 = 51.213 ft lbmole hr⋅ ft 2 . KL1 = 12.136 a2 = 49.648 ft ft ft hr lbmole hr⋅ ft . A1 = 2.182 ft 2 2 . Voper2 = 38.816 gal . A2 = 3.142 ft . GM2 = 2.706 KL2 = 13.386 ft hr x2v1 = 0.000569 . lb lb . ρ lw ( Tv1) = 60.155 . Mw = 18.015 . Kc1 = 2067 . 3 lbmole ft 2 3 2 . ρ lw ( Tv2) = 59.793 lb ft 3 lbmole hr⋅ ft 2 . . Kc2 = 2498 Note: y1v1 := 0 . y1v2 := 0 . Let: x1v1 := 0 . y2v1 := 0 . NA1 := 0⋅ x1v2 := 0 . x2v2 := 0 . y2v2 := 0 . NA2 := 0 lbmole 2 hr⋅ ft lbmole hr⋅ ft 2 . . 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) ⋅ y2v1 y2v2 1 1 ⋅ x1v1 − and NA2 = KL2⋅ ρ lw ( Tv2) ⋅ ⋅ x1v2 − . Mw Kc1 Mw Kc2 LM2⋅ A2 = LM1⋅ A1 − NA1⋅ a1 ⋅ Voper1 . x1v1 = x2v2 . Calculate: Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 33 of 37 x 1v1 y 2v1 N A1 x 1v2 := Find ( x 1v1 , y 2v1 , N A1 , x 1v2 , x 2v2 , y 2v2 , N A2 , L M2) : x 2v2 y 2v2 N A2 L M2 x1v1 = 14.788 × 10 −6 . y2v1 = 10.462 × 10 x1v2 = 218.741 × 10 −9 −3 . x2v2 = 14.788 × 10 NA2 = 6.315 × 10 −6 lbmole hr⋅ ft 2 . NA1 = 394.136 × 10 −6 −6 lbmole . y2v2 = 191.357 × 10 . LM2 = 35.545 lbmole hr⋅ ft 2 hr⋅ ft −6 2 . . . 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 × 10 Mw = 18.015 wcs := lb lbmole −9 . . Mc = 119.38 100lbmole⋅ x1v2⋅ Mc lb lbmole 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 × 10 −6 . 125 + 125 + 115 + 115 + 70 5 −6 : w . wcs_adj := wcs⋅ cs_adj = 17.730 × 10 8.993 Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 × 10 wcf := wcs_adj⋅ −6 0.19 + 0.22 + 0.08 3 ( 125 + 125 + 115) + ( 115 + 70 + 75) + ( 50 + 85 + 100) 9 : wcf = 30.306 × 10 −9 . 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 = 4.185 gal min . The absolute pressure upstream of the orifices, P0 , is: P0 = 44.7 psi . The absolute pressures and temperatures in the vessels, Pv , Tv1 , and Tv2 , are: Pv = 14.7 psi . Tv1 = 364.95 K and Tv2 = 373.15 K . [ TF ( Tv1) = 197.22 and TF ( Tv2) = 211.98 .] 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 . 2 2 aT1 = 0.003545 ft and aT2 = 0.004951 ft . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 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 = 106.540 lb lb lb and W2 = 153.165 . WT = 260 . hr hr hr The superficial velocities in the first and second stage vessels, UG1 and UG2 , are: ft ft and UG2 = 0.363195 . sec sec m m and UG2 = 0.110702 . UG1 = 0.110884 sec sec UG1 = 0.363794 The vessel flow regimes indicated by the vessel superficial velocities Wi Dvi 2 are (Perry's ρ 4⋅ π ⋅ 2 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 ρ 0 ⋅ 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 % . Chloroform Plan.mcd NOT CHECKED LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 36 of 37 The sparger flow regimes indicated by the sparger hole Reynolds numbers dNi⋅ uNi⋅ ρ 4 are (Perry's µ vw ( T0 ) pages 14-70 and 14-71): Sparger_Re_Regime1 = "Possibly Jet" and Sparger_Re_Regime2 = "Possibly Jet" . 2 The sparger flow jet regimes indicated by the sparger hole Weber numbers ρ g⋅ dNi⋅ uNi σ w ( 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 dvs1 dvs2 stage spargers, dvs1 and dvs2 , and the ratios of the bubble diameters to the hole diameters, and , dN1 dN2 are: dvs1 = 0.285 in and dvs2 = 0.285 in . dvs1 dN1 = 166 % and dvs2 dN2 = 140 % . The gas volumetric holdups in the first and second stage vessels, ε G1 and ε G2 , are: ε G1 = 19.5 % and ε G2 = 19.6 % . The vessel foaming flow conditions indicated by the Froude numbers 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 % . Chloroform Plan.mcd NOT CHECKED UGi g⋅ Dvi are (Perry's page LOCKWOOD GREENE Confidential Client Job, Item: A7947 010704.01, CHCl3 Plan Date, Page: 7/22/2001, 37 of 37 The weight fraction chloroform in the stripped liquid, wcs , is: wcs = 1.450 × 10 −6 . The weight fraction chloroform in the stripped liquid, adjusted based on Process Qualification data, wcs_adj , is: wcs_adj = 17.730 × 10 −6 The weight fraction chloroform in the final rinsate, wcf , is: wcf = 30.306 × 10 Chloroform Plan.mcd −9 NOT CHECKED . .
