TK Solver’s Programming Power
Takes On Differential Equations

Lots of TK Solver users will tell you about its ease of use. If you want to hear about TK’s power as a programming environment, ask Dr. Kurt Lund.
Kurt Lund, Ph.D.  
Home: Del Mar, California  
TK User Since:1985  
Uses TK For: Differential equations in engineering consultancy  

“I’m an old hand at engineering from way back,” says Lund, principal of Applied Chemical and Engineering Systems, Del Mar, Calif. ( “I took my first baby steps in Fortran programming. I like to program.

“TK has programming capability with similarities to Fortran,” he says. ‘That’s an objection for some people, but, boy, not for me. And the Variable, Rule, and Function Sheets allow me to organize my calculations in the best of engineering tradition—numbers in one place, formulations in another.”

Lund uses TK to program differential equations, for projects that range over aerospace cooling systems, chemical processing, feedback control systems, and biomedical instrument development. Big projects in which he used TK included a job for TRW, now Northrup Grumman Space Technology: “This was an airborne cooling system that was on a very tight schedule. You could not build and try some design, and then try something else later; it had to work right the first time. With TK, I was able to model the system and predict performance, prior to fabrication.”

“I tend to model things theoretically from first principles,” Lund says. “I derive the differential equations that I want to solve and program those in TK Solver for numerical integration. I make frequent use of RK 4 (a TK Library procedure function for 4th-order Runge-Kutta method of numerical integration). In some cases I have to solve a two-point boundary value problem; for this the far boundary condition is the guess variable, and TK can iterate to solve for that.” A case in point is his recent article on thawing of bio-compounds i , where the fraction thawed, f, is described by:


By various transformations this nonlinear partial differential equation was reduced to

which was integrated numerically in TK Solver.
Lund began using TK while working as university teacher and researcher. His favorite features are backsolving and TK’s declarative structure—“the ability, on the Rule Sheet and in rule functions, to solve equations in any order, and with the unknowns to the right of the equal sign is a great benefit and simplification.”
“I use backsolving all the time, especially for design purposes,” he says: “I can get close to a solution with a set of equations—but if I know a variable can only be so much, I make it an input, change another to an output, and solve some more. That’s really unique and powerful. I don’t think any other software has that.”
Lund is happy to encourage a client to purchase TK Solver. Once he developed a model of the flow and heat transfer distribution in a nuclear reactor core. “The comparable program for this was a huge thing that only ran on big computers—such as at a national laboratory like Oak Ridge,” he says. “I developed a TK model for this client, and they bought five seats of TK Solver so they could use it.”
By the same logic—client need—Lund has begun to use Excel in design work, though he complains, “As an engineer I can’t understand why they prefer this accounting software to TK Solver.” He plans to make extensive use of the TK Solver-Excel Toolkit and the TK Packager/Player in working with clients who use Excel.
But Lund remains an enthusiastic TK booster. “TK is such a powerful tool for me that I’m glad to spend time talking about it,” he says. “Whenever I meet someone I recommend it. At first it may seem to be not as jazzy as some of the other packages on the market, but when I show them what it can do, they’re very impressed.”
 i Lund, K. O., “Thawing of Bio-Compounds in Frozen Microplates,” Proc. Int’l. Mech. Eng. Congress and Exposition, Anaheim, CA., Nov.13-19, 2004, paper No. IMECE2004-60110.