Case Study: Rapid Conversion of a Technical Paper to a Mathematical Model
Introduction
A technical paper, in PDF form, containing
an algorithm with over sixty linear, nonlinear and integral equations is
converted to a mathematical model in TK Solver and subsequently linked to Excel
and a web application. The math model
was completed in one day and the links and refinements to the interface were
completed on day two.
The Model Building Process
The technical paper was ESDU Data Item
85007, entitled “Contact Phenomena III:
Calculation of Individual Stress Components in Concentrated Elastic
Contacts Under Combined Normal and Tangential Loading.” It was made available in PDF form.
The PDF file was briefly reviewed and it
was noted that two complete examples were provided within the document, either
of which could be used to verify a computerized version of the formulas and
calculations.
The formulas were entered into TK
Solver. Here is a sample of some of the
formulas, in TK’s MathLook View including the equation numbers from the paper:
There are 57 such equations in the math model, in addition to
several functions which were added to perform tasks such as generate
plots. These will be discussed in more
detail later.
The variables in the equations are
automatically summarized on TK’s variable sheet, including descriptions and
units. Here is a portion of the variable
sheet, with the inputs grouped at the top.
The model also includes many auxiliary
variables used by the ESDU algorithm. Those variables are grouped together on the
variable sheet. During the testing of
the model, it was very useful to be able to verify that each intermediate value
was computed as specified by the paper.
Here is a portion of that listing.
The calculations required three numerical
integrations. Fortunately, TK Solver
includes built-in functions for those calculations. Here are the integrands.
The limits of the integrals were determined from the solution
of a sixth order polynomial. It was
necessary to find all the roots and then use the largest real root in the upper
limit of integration. TK Solver includes
a root finder that was perfect for the task, based on Bairstow’s algorithm for
finding real and complex roots of nth-order polynomials. A loop was added to select the best one for
this application. TK’s unique capability
to mix rule-based equation solving with calls to procedural subroutines was a
time-saver.
Here is the procedure that sets up the polynomial, solves it with a call to the Bairstow function and then picks the best solution for the application.
The model takes a fraction of a second to
solve and the solutions verify the values from the example in the paper.
Interfacing with Excel
The TK file was linked to an Excel
spreadsheet, as shown below. There are
several advantages to this. The
underlying formulas can be hidden. The
interface can be customized. The TK
model can be used as a function within a larger spreadsheet application. The spreadsheet and model can be packaged together
and shared with anyone.
The spreadsheet dynamically updates as the
inputs are changed. The plots were added
by taking advantage of TK’s powerful “List Solve” feature to repeatedly solve
the equations over a range of inputs for one of the variables.
Interfacing with the Web
The TK Solver model was also processed
using a UTS tool called Galaxy, which automatically builds a browser interface
for a web-based application. Again, the
variables displayed in the browser interface are dynamically linked to the
model.
Odds and Ends
There are a number of additional features
of TK Solver that make it an excellent tool for such applications.
Units management is one TK’s strengths. In
this application, unit conversions are available for all the inputs and
outputs. You can globally swap between
US and SI units or mix and match as you like.
This feature carries through to the Excel and Web interfaces as well.
Table and plot generation is another asset. Any
mathematical model built in TK Solver can be enhanced without any additional
programming required. As shown, the
effects of varying the friction angle are plotted and automatically updated as
other variables change. Even the
locations of the plot annotations dynamically change.
Backsolving
is perhaps TK Solver’s most valuable feature.
Backsolving allows users to input a desired value for one or more
outputs and solve for the required inputs.
One example for this model would be to input a stress value and have TK
determine the required friction angle or contact dimension. Here is one such example. Note that fy, the component stress with the
maximum absolute value, has been input and the contact length, a, has been solved.
In this way, a single TK Solver model
actually solves much more than the initial intent of the technical paper, with
no additional programming required.
In summary, a technical paper containing
dozens of pages of equations is now accompanied by a mathematical model that
the reader can immediately use to solve any related problems. With much more functionality than FORTRAN or
BASIC translations of the formulas could provide, the TK Solver model gives the
reader a dynamic way of turning a complex document into a handful of variables
that are easily understand and manipulated.
Best of all, the entire process only required a couple days.
TK Solver for Chemical Engineers
Overview
TK Solver is frequently used for
calculations involving mass balances, fluid flow, distillation,
agitation/mixing, combustion, heat transfer, dust/mist collection, and reactor
design. The program includes built-in
thermodynamic and transport property functions based on the NIST Standard
Reference Database 23. Additional material
property tables and curves can be integrated into TK models. TK’s unique backsolving capability allows
engineers to optimize designs according to their process/performance
requirements. The versatile TK library
of mathematical and statistical tools handles critical number crunching tasks
such as nonlinear curve-fitting, simulation, and solutions of differential
equations.
Chemical engineering educators also find TK
useful because of its structured approach to model building and its powerful
equation solving and unit conversion abilities.
Students can progressively build on previously learned concepts. TK’s List Solve feature empowers students to
explore the sensitivity of the variables in their models, generating tables and
plots of important relationships -- even those involving large sets of
nonlinear equations. Engineering
textbooks are available which integrate TK into the teaching of chemical
engineering.
Glimpses
of the History of TK Solver in Chemical Engineering
Since its initial release in 1983, TK
Solver has proven to be a very powerful and practical tool for chemical
engineers. It has also been favorably
reviewed many times from a variety of perspectives. Here are some examples:
Mark Jennings, Digital Kinetics -- Mr.
Jennings used TK Solver to develop and publish PCOST, an extensive collection
of process and pollution control equipment costing models based on a series of
articles published in Chemical Engineering magazine.
Vijay Singh, Schering-Plough, for Chemical
Engineering Magazine -- Mr. Singh presented material balance and flash
calculation examples, concluding that “TK reduces the time spent on problems by
allowing the engineer to concentrate on defining the solution rather than
working out the mechanics of the computer program required. It allows for the formulation of design
models in a natural equation form... It
greatly facilitates the exploration of “what-if” situations.”
D.W. Pepper and A.H. Weber, DuPont, for the
Journal of the Air Pollution Control Association -- Pepper and Weber discussed
calculations involving accidental atmospheric dispersions and reported that TK
Solver is easy to program and, because it can backsolve problems, can yield
quick, reasonable estimate of the source term.
Allan L. Smith, Drexel University
Department of Chemistry -- “Every student said that TK Solver was easier to use
than the programming languages they knew.”
Professor Smith has used TK Solver to model the composition of “acid
rain”.
Gerald Rich for Pollution Engineering
Magazine -- Mr. Rich reviewed TK Solver and summarized, “This is a simple,
fun-to-use program for solving equations.”
Edward M. Rosen, Monsanto Co. -- Mr. Rosen
investigated TK Solver’s equation solving capabilities as applied to four
chemical engineering problems:
fractional conversion in a chemical reactor; energy and material
balances; plug flow reactor equilibrium conversion and temperature;
kinetics. Successfully solving all four
problems with TK Solver, he concuded that TK is a useful tool for chemical
engineers.
Curt Arnold, M.W. Kellogg, quoted in
Digital News -- “Although we have promoted TK inside the company primarily for
relatively simple calculations, we have some very large models with 800 to
1,000 rules. These systems handle
complicated problems such as modeling steam systems for process plants, very
difficult and very important calculations.”
Ali Cinar, Illinois Institute of Technology
-- Volume II of the HAWK MANUAL, published by IIT to assist engineering
students in learning to use computers, features 22 engineering case studies and
corresponding TK Solver models. Mr. Cinar’s
case study models an ammonia reactor operation and the effects of inerts
molefraction level on recycle streams.
Amoco Production Company and the American
Petroleum Institute teamed up with UTS to develop TK Solver applications for
the API Gas Measurement Manual.
Z.Z. Zhang, Donald L. Sparks, and Noel
Scrivner; Characterization and Modeling
of the Al-Oxide/Aqueous Solution Interface, published in the Journal of
Colloid and Interface Science -- The authors, from DuPont and the University of
Delaware, successfully used TK Solver to build their mathematical model and
study the sensitivity of the variables involved.
Dan Deitz, Mechanical Engineering
Magazine -- To determine the
specifications for a combustor in a high-speed civil transport plane, Pratt
& Whitney engineers are using TK Solver to solve hundreds of complex
equations. The article focuses on TK’s
many unique features put to use in solving heat transfer problems.
UTS released Heat and Mass Transfer on TK,
a collection of TK Solver models co-authored with Professor David DeWitt of
Purdue University.
UTS and BHR Group Ltd. produced an on-line
electronic information system called Intermix, based on the FMP and HILINE
Design Guides for fluid mixing. The
following topics are covered -- motionless mixers as liquid-phase and
gas-liquid reactors, ejectors as gas-liquid reactors, rotor stator mixers as
liquid phase reactors, and centrifugal pumps as liquid phase reactors.
A Case Study:
DuPont’s Chemical Process Design Standards are an excellent example of web-based smart standards. DuPont engineers have accumulated four volumes of chemical process design standards. There are well over one thousand pages of formulas, figures, and descriptions. DuPont worked with Universal Technical Systems, Inc. to produce the web-based version of these standards.
Many topics are covered:
· fluid flow through pipes and orifices
· heat transfer
· agitation and mixing
· distillation
· dust and mist collection
· physical properties of materials
Many of the paper standards required the users to refer to figures for use in intermediate calculations. Many of these figures required users to visually interpolate within families of curves. The TK Solver software was used to process the data within these curves numerically, interpolating automatically where necessary.
Unit conversions were also handled using TK Solver. The equations were all entered using the established nomenclature for each variable. Conversion factors are automatically applied if the user wishes to view a variable in a different unit.
Perhaps the most important use of TK Solver is for backsolving problems. Once the equations are set up in TK, they can be solved in any direction depending on which variables are known and unknown. For example, a standard which computes the pressure drop from turbulent flow through a pipe can also be used to size a pipe for an allowable pressure drop. A single standard easily solves many combinations of problems.
Another major enhancement to the standards was to incorporate a database of physical properties into the calculations. DuPont has a license for the DIPPR physical properties database. That information is now used during the calculations from the standards. Users specify the substances they are working with and properties such as viscosity, density, and vapor pressure are applied at various temperatures. The integration with TK Solver means that these material property calculations can also backsolved. The result is that users can backsolve for temperatures at which to operate various processes for desired performance. Users gain a better understanding of the relationships between variables.
A Summary of TK’s Competitive Advantages for Chemical Engineers
TK Solver has many great features but the following ones really make TK stand out as the preferred tool.
• TK is easy to program. -- TK’s rule-based approach to model building allows engineers to enter equations just as they know them, without the need for algebra. Other programs require assignment statements where an unknown must be isolated.
• TK can backsolve. -- TK can solve problems for many combinations of inputs and outputs without any additional programming or rearranging of variables or equations. Backsolving facilitates better designs.
• TK can list solve. -- TK can solve problems repeatedly over a range of inputs to generate data for tables and plots. This feature works for models involving simultaneous nonlinear equations as easily as for simpler problems. This feature can easily be used for filtering random numbers through math models for Monte Carlo simulations.
• TK can optimize -- TK can solve problems repeatedly within given constraints to determine the optimal configuration of inputs.
• TK does table look-ups. -- TK can access material property tables and charts during model resolution and can interpolate within them as well. TK also backsolves table look-ups to find necessary inputs producing the desired properties.
• TK is uniquely programmable. -- TK combines both rule-based and procedural languages for the best of both worlds. Users can implement their own algorithms and use the resulting functions within TK as they would any of the built-in functions. Even backsolve them! It’s easy to save these functions and merge them into other models as necessary.
• The TK Library. -- Engineers with math and statistical requirements such as solving differential equations, fitting curves to experimental data, or even a combination of these two, will find a treasury of pre-programmed tools in the TK Library, all at no extra cost.
TK Solver Library Highlights:
List/array manipulation
Matrix manipulation
Root finders
Numerical differentiation
Numerical integration
Differential equation integrators
Optimization algorithms
Geometry and CAD functions
Functions of complex variables
Data analysis and statistical testing
Linear and nonlinear curve-fitting
Curve-fitting of differential equations
Probability distributions
Maximum likelihood estimation
Monte Carlo simulation analysis
FOR IMMEDIATE RELEASE
December 9, 2009
Advanced Spring Design 7 Release is “Unbelievable”
New Features and Spring Types Make ASD7 an Essential Upgrade.
A joint project between
Universal Technical Systems, Inc. (UTS) and the
Spring Manufacturers Institute (SMI) since its first launch in 2003, Advanced Spring Design Release 6 has always lived up to its name, advancing spring design to the next level, saving spring designers time and reducing the margin for human error. After a number of intermediate updates, a totally new ASD 7 is now released making the newest product a must have for every spring designer. ASD7 meets all of your spring design needs in a single program, combining engineering expertise with customized calculations created in TK Solver’s collaborative environment to help you design quality springs.
Dick Carter, an Engineering Manager from Michigan Spring & Stamping has been using ASD6 since its beta version in 2003 and has done several thousand designs. He notes “with the upgrade to ASD7, spring design becomes increasingly robust over a larger variety of spring types.”
Some of the new spring types included in ASD7 are:- Systems of compression springs in series and in parallel
- Internal and external garter springs
- Constant force spiral springs
- Stacked Belleville washers
- Internal and External snap rings
Carter continues, “ASD7 is a powerful tool which cuts design time by eliminating the need for running multiple designs. Your response to the customer can be almost instantaneous and with a professional in-depth detailed datasheet complete with DXF and 3D drawings, as well as plots. ASD7 contains the combined science from throughout the spring industry pertaining to spring design theory in a simple to use software.”
Dick Carter isn’t the only user impressed by ASD7, Tom Lusinski, a Plant Manager, Spring Division, and Spring Estimator for Stanley Spring & Stamping has been using ASD6 for the past few years for all of his spring design needs on a day-to-day basis. Describing ASD6 as “fantastic,” Lusinski was blown away by ASD7. “It is unbelievable… the new features and help module offered in ASD7 are state-of-the-art.”
New Features in ASD7 Include:
- Non-linear rates for conical compression springs
- Ability to calculate torsion at arm and arm length calculations
- Variable pitch and diameter on single springs
- Context-sensitive help linked to the SMI Encyclopedia
- Built-in curve-fitting routine for tensile strength data
- Life plots with default or user-defined S-N curves
- Improved management of saved designs
- 3D images produced for each design
- Notes can be saved with each design
- “Enterprise” option for sharing designs across networks
“The 3-D viewer is tremendous. I could go on and on. Anyone that was familiar with how ASD6 operated will be very pleased to work with ASD7”, says Lusinski.
ASD7 contains a flexible database of materials with a new estimation routine for minimum tensile strength, making it easier to add your own materials. Other time-saving features include the ability to combine cylindrical and conical compression springs in a series (stacked or continuous) and in parallel (nested). Because it runs on TK Solver, ASD7 lets you “backsolve” your designs, allowing you to solve your problem in any direction, and gives you a means of designing a spring with variable pitch, rate, size, and other characteristics.
“I never thought that spring design software could do any more than ASD6 was doing, but I was terribly wrong,” says Lusinski. Other ASD7 users are bound to agree.
Since 1933, the
Spring Manufacturers Institute (SMI) has served North American manufacturers of springs and their associate suppliers.
Universal Technical Systems, Inc. has been a leading provider of high-productivity problem-solving software products and custom developed solutions, including the highly-regarded TK Solver, since 1984. Visit
http://www.uts.us.com/ or call 815-963-2220 to schedule your free trial and demonstration of ASD 7.
Contact: Todd Piefer,
Todd.piefer@uts.us.com (815) 963-2220
###
Additonal Information:ASD7 Product PageBrochure and Fact Sheet
PTS - Document Management Module
Every activity in a Project Plan involves documentation, which may be shared across teams or among team members. PTS provides upload, filtering and search facilities for viewing information pertaining to Project activities. These may include documents that are internally generated and then uploaded for sharing with other Project or team members.
Documents can be readily viewed. Supported formats include Microsoft Office, PDF, TXT and RTF and DWG files. Support for specific file extensions can be added as needed.
This post is the ninth in a ten-part series about UTS's Project Tracking System. Check back next week for the final installment, which will discuss the core Administrative Module.
Labels: Project Tracking System
