The Role Of Smartphones and Tablets In Numerical Problem Solving

 

       Numerical problem solving has been introduced into the Chemical Engineering education and practice in the early nineteen sixties (Shacham et al. 1996). At that time Fortran or other source code programming languages were used to implement the numerical solution algorithms on "mainframe" computers. Starting in the mid nineteen eighties the emphasis shifted to the use of general purpose mathematical software packages, such as MAPLE, MATHCAD, MATLAB, Mathematica and Polymath for numerical problem solving using the PC. PC based mathematical software packages are routinely used, nowadays, for numerical problem solving in engineering education and practice (Cutlip et al., 1998, Shacham and Cutlip, 1999) as they are easy to use and provide the most time efficient route for obtaining accurate solutions.

       The dominant position of the PC in the computing field has been challenged lately by the mobile devices such as smartphones and tablets. Furthermore, it is predicted that by 2017 the dominant operating system for all computing devices will be Google's Android (Wingfield, 2013). Relying on these trends and predictions we decided to check the possibility of the use of Android based smartphones and tablets for numerical problem solving in chemical engineering. For this aim, we have developed the PolyMathLite application, a slightly simplified version of the Polymath software package (Polymath is a trademark of Polymath Software http://www.polymath-software.com ).  PolyMathLite solves systems of linear and nonlinear (NLE) algebraic equations, systems of ordinary differential equations (ODE) and carries out polynomial, multiple linear and nonlinear regression of data.

       PolyMathLite was evaluated on both Android smartphones and tablets using the set of test problems listed in Table 1. Ten out of the eighteen problems are described in detail by Cutlip et al. (1998). These ten problems were solved by a group of educators on PC using the mathematical software packages Maple, Mathematica, MATLAB, MathCAD, Polymath and Excel. A comparison of the performance of the various packages in solving the set of 10 problems was reported by Shacham and Cutlip (1999). Six additional problems are described in the textbook of Cutlip and Shacham (2008). Sample problem 17 is from an NLE test problem library (Shacham et al., 2002) while problem 14 is described in the reference: Shacham et al., 2009.

       The test problems used represent a great variety with respect to the complexity of the mathematical model, the level of difficulty of the numerical solution and the associated ChE course. The various problems are applicable to required courses, such as: introduction to Ch. E., thermodynamics, fluid dynamics, heat transfer, mass transfer, separation processes, reaction engineering, process dynamics and control; and to the elective courses: mathematical methods, biotechnology and optimization. The simplest problems consist of a single NLE (Problems 2 and 4), while the most complex consist of 12 NLEs (Problem 18) and 9 ODEs. Some of the problems pose special challenges to the solution algorithms. In Prob. 11 a differential-algebraic system (DAE) need to be solved, in Prob. 12 a partial differential equation (PDE) is solved using the method of lines, and multiple linear regression of the data in Prob. 14 with a model that includes a free parameter yields an ill conditioned regression model while removing the free parameter provides a statistically valid model. Problems 10 and 13 require special stiff integration algorithms to find the correct solution. Problems 17 and 18 represent special challenges to the NLE solver algorithms as the values of some of the variables (mole fractions) are very close to zero at the solution, however these variables are not allowed to obtain negative values during the solution process as their logarithm or square root needs to be computed.

       In spite of the special challenges in terms of the problem size and algorithmic requirements, the solutions obtained using PolyMathLight on smartphones and tablets matched the solution obtained by Polymath 6.1 up to seven significant digits for all the problems included in the test problem set. The solutions were obtained in a very short time, even for the large scale, complex problems. On the negative side, the problem input via the touch screen has proven to be very cumbersome and time consuming.

       In the extended abstract and the presentation the testing of PolyMathLite will be described in more detail and the potential effects of the availability of numerical problem solving tools for smartphones and tablets on the CHE education and practice will be discussed.  

           

References

1.     Cutlip, M., Hwalek, J.J., Nuttall, H.E., Shacham, M., Brule, J., Widman, J., Han, T., Finlayson, B., Rosen, E. M. and Taylor, R., “A Collection of Ten Numerical Problems in Chemical Engineering Solved by Various Mathematical Software Packages.” Comput. Appl. Eng. Educ., 6(3), 169-180(1998)

2.     Cutlip, M. B. and Shacham, M.  Problem Solving In Chemical and Biochemical Engineering  with Polymath, Excel and MATLAB. Prentice-Hall, Upper Saddle River, New-Jersey, 2008.

3.     Shacham, M., M. B. Cutlip, and N. Brauner, "General Purpose Software for Equation Solving and Modeling of Data", pp. 73-84 in Carnahan, B. (Ed), “Computers in Chemical Engineering Education”, CACHE Corporation, Austin, TX, 1996.

4.     Shacham, M. and M.B. Cutlip, “A Comparison of Six Numerical Software Packages for Educational Use in the Chemical Engineering Curriculum”, Computers in Education Journal, IX(3), 9-15 (1999)

5.     Shacham, M., N. Brauner and M. B. Cutlip, “A Web-based Library for Testing Performance of Numerical Software for Solving Nonlinear Algebraic Equations”, Computers chem. Engng. 26(4-5), 547-554(2002)

6.     Shacham, M., Cutlip, M. B. and Elly, M., "Live Problem Solving via Computer in the Classroom to Avoid "Death by PowerPoint"", Computer Applications in Engineering Education Vol 17, No. 3, 285-294(2009)

7.     Wingfield, N., "PC Sales Still in a Slump, Despite New Offerings", The New York Times, April 10, 2013.

Table 1. List of Sample/Test Problems for PolyMathLite  

No.	COURSE	PROBLEM TITLE	MATHEMATICAL MODEL
1	Introduction to Ch. E.	Steady State Material Balances on a Separation Train	Simultaneous Linear Equations1
2	Introduction to Ch. E. and Thermodynamics	Molar Volume and Compressibility Factor from Van Der Waals Equation	Single NLE1
3	Thermodynamics and Separation Processes	Three Phase Equilibrium - Bubble Point	NLE System2
4	Fluid Dynamics	Terminal Velocity of Falling Particles	Single NLE1
5	Thermodynamics and Reaction Engineering	Reaction Equilibrium for Multiple Gas Phase Reactions	NLE System1
6	Mathematical Methods	Vapor Pressure Data Representation by a Polynomial and Equations	Polynomial Fitting, Linear and Nonlinear Regression1
7	Heat Transfer	Unsteady State Heat Exchange in a Series of Agitated Tanks	ODE's with known initial conditions1.
8	Mass Transfer & Reaction Engineering	Diffusion with Chemical Reaction in a One Dimensional Slab	ODE's with split boundary conditions1.
9	Reaction Engineering	Reversible, Exothermic, Gas Phase Reaction in a Catalytic Reactor	ODE System1
10	Process Dynamics and Control	Dynamics of a Heated Tank with PI Temperature Control	Stiff ODE System1
11	Separation Processes	Binary Batch Distillation	Differential-Algebraic System1
12	Heat Transfer	Unsteady state Heat Conduction in a Slab	Partial Differential equations2
13	Reaction Engineering	Rate Data Analysis for a Catalytic Reforming Reaction	Nonlinear Regression2
14	Mathematical Methods	Regression of Heat of Hardening Data of Cement	Multiple Linear Regression3
15	Biotechnology	Stoichiometric Calculations for Biological Reactors	Simultaneous Linear Equations2
16	Biotechnology, Reaction Engineering and Mathematical  Methods	Biological Reaction in a Batch Reactor	Stiff ODE System2
17	Reaction Engineering and Mathematical Methods	Combustion of Propane in Air	NLE system where some variables must be positive4
18	Reaction Engineering and Optimization	Complex Chemical Equilibrium	NLE system where some variables must be positive2

¹Reference: Cutlip et al., 1998

²Reference: Cutlip and Shacham, 2008

³Reference: Shacham et al., 2009

⁴Reference: Shacham et al., 2002