Using the Software Package Mathcad As a Tool to Teach Soil Physics
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1. Without MathCAD the course would have been difficult because of the math In their comments they expanded on their responses Although I have several university courses in calculus my working knowledge of math was not to the level that would have made the physics course easy MathCAD really helped Although it helped in plotting analytical solutions and vary ing variables it did not help with partial differential equations Difficult to say perhaps I did not realize the amount of time I would have spent without MathCAD To understand MathCAD s output you must understand the input They agreed they would continue to use MathCAD even if it were not an integral part of a course Typical statements were I m using it to analyze my thesis data and I ll continue to use MathCAD because it has a wide range of math functions and operations In response to the question In general how do you think MathCAD helped you learn soil physics concepts in this course one responded It helped me under stand the relative importance impact of changes in vari ous input parameters on the output parameter i e salt concentration Other responses indicated that MathCAD had not hindered them in learning soil phys ics concepts in the course Balkovick et al 1985 found that students have difficulty translating an abstract mathematical formula tion into an understan2. Communications of the Association for Computing Machinery 28 1214 1224 Campbell G S 1985 Soil physics with BASIC Transport models for soil plant systems Elsevier New York Lapidus L and N R Amundson 1952 Mathematics of adsorption beds VI The effect of longitudinal diffusion in ion exchange and chromatographic columns J Phys Chem 56 984 988 Lindstrom F T R Haque V H Freed and L Boersma 1967 Theory on the movement of some herbicides in soils Linear diffusion and convection of chemicals in soils Environ Sci Technol 1 561 565 Taylor G 1 1953 Dispersion of soluble matter flowing slowly through a tube Proc R Soc London A 219 186 203 Van Genuchten M Th and J C Parker 1984 Boundary conditions for displacement experiments through short laboratory soil columns Soil Sci Soc Am J 48 703 708 Van Genuchten M Th and P J Wierenga 1986 Solute dispersion coefficients and retardation factors p 1025 1054 In A Klute ed Methods of soil analysis Part 1 2nd ed Agron Monogr 9 ASA and SSSA Madison WI
3. in the book or available on a disk from the pub lisher In addition to using the computer program provid ed students are encouraged to explore various solutions to the equations using different initial and boundary con ditions and to vary the magnitudes of variables in the equations This activity results in a deeper understand ing and appreciation of the physics of the process and its sensitivity to the magnitudes of various parameters A major advantage of this approach is that the student does not have to spend an undue amount of time mak ing numerical calculations or having to write computer programs Commercial computer software has become more sophisticated during the past several years Specialized toolbox software programs are now available that allow users to solve mathematical problems ranging from simple algebraic operations to complex ones involving for example integration differentiation and nonlinear curve fitting The availability of this software creates another opportunity for instructors to refine the manner in which soil physics is taught The purpose of this article is i to briefly describe some of the features of the mathematical software pack age MathCAD and ii to show several examples of how this package was used in teaching a graduate course in water and solute transport MathCAD software was selected for teaching the course because of the authors familiarity with it WH
4. AT IS MathCAD MathCAD is a software package for personal com puters MathCAD allows the user to write equations in standard two dimensional format to use standard mathematical symbols and to solve them In addition graphs in two or three dimensions can be generated MathCAD uses standard mathematical symbols for oper ators and variables including Greek symbols To create simple expressions such as addition or multiplication the user enters the numbers and operators for example for addition and for multiplication using the keys on a standard keyboard To create more complicated con structs such as plots or definite integrals the appropri ate built in function is selected using dedicated keys on the keyboard The user simply fills in the designated blanks to complete the construct For example once the proper key has been pressed to initiate the integral con struct the user must key in four entries to fill in the fol lowing four blanks associated with the integral i the expression to be integrated ii the upper limit of integra tion iii the lower limit of integration and iv the in tegrating variable Once variables are defined and the equations or constructs are written the program is complete and MathCAD finds the numerical solution The defined vari ables and equations of the program become part of a document that can be printed or saved in a computer file Documents in pieces or in entirety ca
5. Peclet number p 1i i cl 0 5 a i 1l p i Taylor 1953 2 p i 1 p i 2 jp i C2 Cl 0 5 exp B erf Je Lapidus and Amundson 1952 i i Lindstrom et al 1967 iB 1 p T E ot c3 Cl jp exp 0 5 P B p B exp B f1 erf jE i i in 4p i i 2 P i Example 2 Plotting Functions Plotting the functional relationships between several variables is a powerful learning tool because the student can observe visually how a change in one or more in dependent variables affects the dependent variable The concept of the breakthrough or effluent concentration curve is important in understanding solute transport in soils Van Genuchten and Wierenga 1986 For this problem the breakthrough curve is defined as the con centration of nonreacting solute flowing out of a soil sys tem e g out the base of a soil profile plotted vs p the number of pore volumes of fluid flowing through the soil system Problem Statement Plot the breakthrough curves for one dimensional solute transport of a nonreacting chem ical The first solution C1 for brevity the equations are not stated here but are shown in Fig 2 assumes an in finite soil column Taylor 1953 Solutions C2 Lapidus and Amundson 1952 and C3 Lindstrom et al 1967 assume first type concentration and third type flux boundary conditions respectively at the inlet end of a semi infinite soil column For a comparison of these and oth
6. Using the Software Package MathCAD as a Tool to Teach Soil Physics D K Cassel and D E Elrick ABSTRACT Many students avoid taking a course in soil physics because they are apprehensive about higher mathematics The computer software MathCAD one of several toolbox software pro grams for solving mathematics problems was used extensively in teaching a graduate course on water and solute transport Students gained proficiency using the software as homework assignments gradually became more complex Students solved simple arithmetic problems such as the computation of bulk density and water content problems of intermediate difficulty such as graphing functions to describe the concentration of a nonreactive chemical species in the effluent of a soil system and complex problems such as nonlinear curve fitting to con struct a smooth curve through experimental data A survey 7 mo after the course was completed indicated that students found MathCAD easy to learn the software allowed the student to focus on physics rather than mathematics and some of the stu dents planned to continue using MathCAD to solve problems in other courses and in their research OIL PHYSICS is perceived by many students to be a very difficult subject Indeed most practicing agron omists and soil scientists as well as graduate and ad vanced undergraduate students in these disciplines equate soil physics with advanced mathemat
7. actual copies of the display on the monitor Example 1 Simple Calculation Problem Statement A cylindrical core of wet soil hav ing a radius of 3 7 cm and a height of 7 6 cm weighs 573 g and loses 110 g upon oven drying Calculate bulk den sity and water content on a weight basis The easiest approach to solve this simple problem is to use the keyboard in the same manner as one uses a calculator After the numbers and operators defining the equation have been keyed in the numerical answer ap pears on the screen immediately after entering the equal sign Approach A Fig 1 This approach is rapid and the information on the screen can be either saved in a data file printed or discarded Problem 1 Calculation of BULK DENSITY and WATER CONTENT A First approach Db OOOC 1 416 B Second approach Given information Mw 110 g Mass of soil water Mt 573 g Total mass of wet soil r 3 7 cm Radius of soil core L 7 6 cm Length of soil core 2 V rnr L Volume of soil core Mt Mw 3 Db 1 416 g cm V Mw Ow 0 238 Mt Mw Mw Ow 23 758 Mt Mw Fig 1 Document showing bulk density and water content calculations and output using MathCAD J Nat Resour Life Sci Educ Vol 21 no 1 1992 75 Using Approach B to solve this problem is more satis fying Fig 1 The append command allows the user to select one of three scientif
8. agree agree neither agree nor dis agree disagree and strongly disagree They agreed the user manual made it possible for them to quickly use MPA 5000 MPA is the best guess or estimate Given of the amount of solute added per unit area SSE o u4 MPA 0 Il l MINERR o u MPA al oval uval and Mval are the Fea values using the nonlinear fittin pval 2 562991 procedure oval 0 374983 3 Mval 5 066342 10 The just calculated values oval uval and MvaI are now used to plot the curve shown below MPA 9 Mval i 1 100 g oval Boi pval MPA 1 inf t zna o inf L ft r i T READPRN ian4ot LWA ARE Fig 3 Document showing MathCAD program for nonlinear curve fitting J Nat Resour Life Sci Educ Vol 21 no 1 1992 77 MathCAD They also agreed the plot function was easy to learn They agreed using MathCAD made it possible to focus on the physical properties of soils without spend ing too much time on the math Comments included It allowed me to look at the physics of the question without having to review my calculus and The physics aspects of soil properties and their mathematical descriptions are supplementary to each other therefore without one the other cannot be understood properly Students agreed that using MathCAD helped them clar ify the functional relationships between variables They neither agreed nor disagreed with the statement
9. ding of the phenomena that they can use Because of computational difficulty faculty may confine examples to problems that can be solved on the 78 J Nat Resour Life Sci Educ Vol 21 no 1 1992 blackboard This may limit assignments With the com puter the students can be asked to pose a variety of What if questions about problems so sufficiently rich that not even the qualitative answers may be obvi ous Balkovick et al 1985 p 1215 Instructor Response We conclude that the various mathematical problem solving procedures in MathCAD represent a powerful tool for students as well as researchers to solve compli cated problems Advantages of using MathCAD in an ad vanced soil physics course include the following i relatively easy to learn to use ii capable of solving com plex equations and iii relatively inexpensive We are not promoting the idea that MathCAD or any other computer software package be substituted for learning mathemat ics However we do believe MathCAD and similar soft ware greatly enhance the learning process by allowing students and faculty to explore and experiment with complex mathematical relationships in a time efficient manner The favorable student response to MathCAD supports our enthusiasm for using it as an aid to under standing soil physics REFERENCES Balkovick E S Lerman and R P Parmelee 1985 Computing in higher education The ATHENA experience
10. er results see Van Genuchten and Parker 1984 The complete program to generate 25 pairs of num bers to plot the theoretical breakthrough curves for so lutions C1 C2 and C3 are shown in Fig 2 After assigning the index variable values for 1 to 25 the pore volume variable p which occurs in all three equations cl i E E 2 269814 10 0 000003 0 000398 9 00621 0 033945 0 102447 _ 0 214598 __ 0 354694 0 5 0 631979 _ 9 740697 __ i 0 884001 0 92554 9 353234 0 971161 0 99878 0 999301 L_9 999602 The following curve for B 25 was obtained by sirply retyping 5 by 25 ed in the first line of the program Only eeconds are requir Fig 2 Document showing MathCAD program and output for plotting complex functions 76 J Nat Resour Life Sci Educ Vol 21 no 1 1992 is defined in terms of i The Brenner number B also called the column Peclet number is a dimensionless parameter which represents the ratio of the convective to the dispersive processes After typing the three equa tions as shown in Fig 2 the compute command is given and the four columns of data displayed in Fig 2 are generated The plot function is activated with a pre assigned key allowing the three curves to be plotted A wide range of plot formats can be formulated using the guidance of a pulldown menu The average length of time required by a student learning MathCAD
11. gt ir SE ala 1 1 i Guess values o 5 These are the best guess values of o the variance and p the mean of the lognormal distribution Presented in Fig 3 are A the data files and defined constants B the program to solve for the numerical values of three parameters p and MPA subject to minimizing the least square mean error C the comput ed values of the three fitting parameters and D a plot of the curve along with the input data Although the pro gram looks complex the user manual provides a step by step description of the process Reasonable guesses of the three unknown parameter values are required to be gin the iterative curve fitting process The plot proce dure discussed in Example 2 was used to create the graph DISCUSSION Student Response Questionnaires to assess their views about MathCAD were sent to all nine M S and Ph D students 7 mo after they had completed the advanced soil physics course In their responses seven students said they used the com puter very often and one said often On a five point scale of very easy easy neither easy nor difficult difficult or very difficult five students found learning MathCAD was easy and three found it neither easy nor difficult One commented Once you know what the function keys do it is easy However it was somewhat frustrat ing at first Additional statements were evaluated based on a five point scale of strongly
12. ic unit files resident in MathCAD For example appending the MKS SI unit file allows the user to specify the units of each variable in the equation in SI units The answer is then automati cally displayed in SI units To begin solving this problem the numerical value for each symbol appearing in the equations to follow is defined Each equation is then writ ten in terms of the previously defined symbols and im mediately upon keying in the equal sign for each equation its numerical answer and units are given No units exist for the computer water content because the units cancel By pressing the percent key after the unit less answer the equation is automatically copied and the answer given in percent The equations are essentially templates and can be used to solve the same problem repeatedly for different sets of input data simply by changing the appropriate numerical value s of the vari ables defined in the Given information section in Fig 1 This example was chosen to illustrate several MathCAD features In practice it would be easier and faster for a student to solve the problem on a hand held calculator The following two examples illustrate the types of problems solved by graduate students using MathCAD in an Advanced Soil Physics class at the University of Guelph during the 1990 spring semester None of the stu dents had previous experience using MathCAD before the course began B is the Brenner or column
13. ics This sit uation arises because mathematics is a necessary tool for stating the relationships among variables to describe a particular physical state or process Problems involving D K Cassel Dep of Soil Science Box 7619 North Carolina State Univ Raleigh NC 27695 7619 and D E Elrick Dep of Land Resource Science Univ of Guelph Guelph Ontario NIG 2W1 Canada Received 19 Mar 1991 Corresponding author Published in J Nat Resour Life Sci Educ 21 74 78 1992 74 J Nat Resour Life Sci Educ Vol 21 no 1 1992 water transport for example may require the solution of a partial differential equation or in the case of drainage the use of complex variables It is our belief that students worry less about the physics than they do about the mathematics in a soil physics course Many students avoid soil physics courses for this very reason Progress in overcoming students apprehension of soil physics courses was made possible with the introduction of the book Soil Physics with BASIC by Campbell 1985 This book was designed for students who have mastered differential and integral calculus but who have not taken a course in differential equations Using Campbell s text book the student first learns the physical concepts con cerning a particular process and then solves related problems using microcomputers This is accomplished using computer programs coded in BASIC which are printed
14. n be import ed into other documents The numerical algorithms for evaluating integrals matrix inversion and equation solv ing are standard predictable and robust Commands are given from pull down menus by name or by dedicated keys An editor in MathCAD allows the user to rearrange the information on the screen and to add text and labels if desired to create a self explanatory document when sent to a printer capable of printing graphics The printed document is identical in every detail to the information displayed on the monitor Software Specifications MathCAD 2 5 requires at a minimum an IBM Apple or compatible computer with 512 RAM at least one 5 25 or 3 50 inch disk drive hard disks are supported and MS DOS or PC DOS 2 0 or higher It requires a graph ics printer or plotter and operates on either a color or The use of trade names in this publication does not imply endorse ment by the North Carolina Agric Exp Stn or the University of Guelph of the products named nor criticism of similar ones not mentioned monochrome monitor No math coprocessor is required but machine operations are much faster with one MathCAD is licensed by MathSoft Inc MathCAD 2 5 sells for about 180 1991 price the student version MathCAD 2 0 sells for 40 EXAMPLES The following examples were used in the soil physics course and are presented to illustrate several of the kinds of problems solved The accompanying figures are
15. to gener ate these plots was less than h Once the program is run the influence of the Brenner number on the breakthrough curves is easily obtained by typing in a new value of B The time required to run the entire program for one value of B was 5 s using a 286 computer with math coprocessor Example 3 Nonlinear Curve Fitting Problem Statement Chloride concentration vs time at the 20 cm depth was obtained by Ian van Wesenbeeck Ph D student at the Delhi Research Station in Ontario ona sandy soil For brevity the data are shown only in the output section of Fig 3 Chloride was applied to the soil surface at the rate of 80 g of chloride per square meter of soil surface The average water content in the trans mission zone was 0 30 m m the average bulk density was 1 59 g cm and the average infiltration rate was 3 5 cm h Use nonlinear curve fitting to construct a smooth curve to fit the data File called LANNLSU Q ORIGIN t t READPRN ian40t Ce READPRN iand0c 1 1 5 1 75 2 2 25 t 2 5 Input of experimental 2 75 data 3 3 25 3 n length t n 21 i 1 n I is the cumulative amount of input pulse in cm inf is the infiltration rate in cm hr t is the time duration of the input pulse L is the calibration depth z is the depth of calculation 2 t inf L t 1 ln mm B MPA 1 1 z C t o u MPA exp OOOO dr inf t t r 2 2 0 20a 2 SSE o u MPA
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