Computational Aptitude of Handheld Calculator

Computational Aptitude of Handheld Calculator

J. K. Adewole & A. S. Osunleke, 2011

USE OF HANDHELD CALCULATORS AS SUPPORTIVE TOOLS INENGINEERING EDUCATION

BYADEWOLE J. K1. and OSUNLEKE A. S2.

Department of Chemical Engineering1King Fahd University of Petroleum & Minerals

2Obafemi Awolowo University, Ile Ife, Nigeria

ABSTRACTThe computational capability of a handheld non – programmable calculator as a supportive tool in effective teaching of soft-computing skill is demonstrated in this work. Seven engineering problems were solved using some of its essential in-built scientific functions and accessories. The results obtained compared favourably well with those from sophisticated algebraic software packages such as MATLAB and MS Excel in terms of numerical accuracy. The percentage error obtained for all the seven problems are 0%. Acquiring skills in the use of this calculator will therefore not only enhance the learning of modern computing software, thereby preparing students and engineers for solving real-time world problems but will also serve as an affordable alternative to the available algebraic software packages in situations where these software are out reach. It is hopedthat this new effort will pave way for a more pragmatic approach of teaching and assessing computing skills in our higher institutions.

Keywords: sustainable development; engineering education; computing skills; simulation software; technical competencies.

1.0 INTRODUCTION

Effective teaching in engineering education is an important tool for sustainable industrial development. The act of teaching the principles of engineering to produce well baked products has been described as the most innovative and continually evolving challenges (1). With the emergence of many different technologies, students need to be taught the basic knowledge required for them to be relevant in this jet age. Globally, engineering environment today demands both technical competencies and excellent computing skills. Engineers and scientists are expected to be skilful in simulation software. Gone are the days when computer applications were consigned to the ranks of senior staff. The emerging proliferation of personal computers has greatly transformed application software into common tools for problem solving.Numerous sophisticated mathematics processing software packages are now available. It has been observed by Foley(2) that students in engineering and science have been using this software but often without good results. The author disagreed with teaching analysis in one place (course) and computing in another place (another course), when the two can be taught concurrently. In an attempt to

solve this problem, Hill and Rajeev et al (3, 4) wrote on fundamentals for getting accurate simulation results.In most schools (especially in the developing countries where the MDGs are expected to be met by 2015), the opportunities for hands-on practice with process simulators may have been nonexistent or underutilised. This is due to the fact that adequate method of teaching computing skills has not been put in place. Students population combined with the cost of sophisticated software have contributed to this problem. Millions of dollars is spent on acquiring licences for software with little or nothing to show for it. It is quite clear that once people are oblivious of the basics underlying the use of this software, then it becomes difficult to use them. Also, it has been observed that students are not interested in learning whatever will not come out in their examinations. The end result of this is that if a lecturer attempts to teach them software skills, the first question they will ask is whether it will come in the examinations or not. Of course students know quite well that it is very difficult for them to be asked questions that will entail real time use of computer in the examination halls. With this in mind, the only alternative for the lecturers is to give assignment which majority will not do personally. All these factors coupled with other social distractions have really



contributed to lack of student’s interest and hence little or no understanding of the basic knowledge of technical computing. In essence, there is the need to inculcate the use of affordable, accessible and efficient methods that will give room for real time use of computer in the examination halls. Accessibility in the sense that it is not necessary to use thousands of dollars to procure the tools needed to implement this method. We need not to havestudents to queue up to practise what they have learnt. The tool should be efficient such that moderate problems can be solved and analysed using these tools and results obtained are comparable with those from expensive software packages. While the use of calculator can never be a substitute for the available mathematical software, it can be used, at the initial stage of learning computing skills, to boost student morale and make computing more interesting. This paper is written to demonstrate the use of handheld calculator in solving engineering problems. It discusses itsapplication as a supportive tool in teaching computing skills. In justifying the use of handheld calculator, engineering problems were solved using 1 Casio fx –991MMS calculator and results obtained compared with those using software like 2 Matlab and MS Excel. The paper will present a step by step procedure of using the available built in Functions and Modes in the handheld calculator. The choice of this calculator is due to itsabundance and the fact that it is not programmable. It will therefore not give room for examination malpractice. Deep familiarization with such a calculator will give student good background skills in using simulation software applications.

2.1 Computer and Handheld Calculators

The history of computer can be traced back to abacus which was developed as a calculating device in the ancient kingdom of Babylonia in SW Asia and widely used in early Greek and Roman times (5). The first general purpose electronic digital calculator that was the prototype of most computers used today was invented by J. P. Eckert and co-invented with J. W. Mauchly in 1946. In about four years later, Eckert and Mauchly produced a Binary Automatic Computer (BINAC) which is capable

1 Casio is a registered trademark of the Casio Computer Co., Ltd2 Matlab is a registered trademark of the Math Works Inc. and MS Excel is a registered trademark of Microsoft Inc

of storing information on magnetic tape rather the than earlier punched cards. An all transistor calculator was introduced in 1965 by Sharp in Japan. Further research on calculator ushered in Scientific Calculator produced by Hewlett-Packard (HP) in 1972. Comparison between computer and calculator can further be illustrated graphically as shown in Fig 2.1. Details relating the common features regarding the design and development of the software can be found in any standard Computer Science and Engineering Texts.

3.0 APPLICATIONS OF AVAILABLE FUNCTIONS TO ENGINEERING PROBLEMS

3.1 CALC Function

Problem: One of the mathematical models proposed by Taiwo and Adewole (6) for predicting dynamic liquid hold up can be expressed as:P1 = 0.0323Q – 0.0003XF + 0.0249 ed – 0.0116 (3.1)

for 0 d 0.12, where XF is the feed composition, Q(KJ/s) is the heat flow rate and d (in m) is the distance of the point of measurement from the top. This can be implemented using the CALC Function

Representation: Let Q, XF, d and P1 be represented as A, B, C and D respectively.

Display Screen

Key Board

Fig. 2.1: Common Features of a Computer and a Handheld Calculator



Algorithm: ALPHA D ALPHA = 0.0323 x ALPHA A –0.0003 x B + 0.0249 x SHIFT ex (C) - 0.0116 CALCFor the first calculations press 0.195 = 0.1 = 0 = and theanswer will be displayed as 0.01957. The complete results are as shown below in Table 3.1

Table 3.1: Results obtained using CALC FunctionQ(A)

XF

(B)d (C)

P1 (CALC)(D)

P1 (MS EXCEL)

% ERROR

0.195 0.1 0 0.01957 0.01957 0

0.235 0.1 0.02 0.02136 0.02136 0

0.195 0.2 0.04 0.02055 0.02055 0

0.235 0.2 0.06 0.02237 0.02237 0

0.195 0.4 0.08 0.02155 0.02155 0

0.235 0.4 0.1 0.02339 0.02339 0

0.195 0.6 0.12 0.02259 0.02259 0

3.2 SOLVE Function

The SOLVE Function employs the Newton’s method to find approximations to more complex equations. It can be used to find the roots of non – linear equations such as

f (x) = 2x – x2

and

5.0)])sin())[cos(exp(1()( tttty where , and are constants.

Problem 1: Colebrook equation relating the friction factor ( f ) for a turbulent flown of an incompressible fluid in a

pipe with the roughness ( ) and the diameter (D) of the pipe is given by the non – linear expression

051.2

7.3ln86.0

1

Re

fND

f

(3.2)

This equation can be solved for f using the CALC

function.

Representation: Given that 0001.0D and

100000Re N , the equation can be rewritten as

0100000

51.2

7.3

0001.0ln86.0

1

ff.

Let f = F.

Algorithm: (1 ÷ ALPHA F + 0.86 × ln (0.0001 ÷ 3.7

+ 2.51 ÷ (100000 × ALPHA F))) ALPHA CALC 0

SHIFT CALC SHIFT CALC

Results: F = 0.01885.

Table 3.2: Results Obtained using SOLVE Function to Solve Colebrook Equation

f (CALCULATOR)

(F)

f (Text) % ERROR

0.0189 0.0189 0

The same answer was obtained as in ref 7 using a ten page Matlab code. However the matlab program used was an interactive one with graphical display making it more robust. The above algorithm can also be made interactive

for various values of D and ReN by defining them as

variables (say X and Y).

Problem 2: The boiling of an equimolar mixture of benzene and toluene at 101.3KN/m2 (760mm Hg) can also be computed using the SOLVE Function. Given Antoine constant for Benzene as

K1A= 6.90565, K2A = 1211.033 K3A = 220.79

Antoine constant for Toluene is

K1B=6.95334, K2B = 1343.943 K3B = 219.377

Representation: At the boiling point

1 BA yy (3.3)

For equimolar mixture 5.0 BA xx

1 oB

oA

A PPP

x(3.4)



From Antoine equation

3

21

10kT

kk

oP

(3.5)

Substituting (3.5) into (3.4), with A representing Benzene and B Toluene, we have

110103

21

3

21

BkTBk

Bk

AkTAk

Ak

P

xA

Substituting for the constants and let T = X, we have

15201010377.219

134395334.6

7.220

033.121190565.6

XX

Algorithm:

( SHIFT log ( 6.90565 – (1211.033 ÷ ( ALPHA X + 220.7 ) ) ) + SHIFT log ( 6.95334 – ( 1343 ÷ (ALPHA X + 219.377 ) ) ) ALPHA CALC 1520 SHIFT CALC SHIFT CALC

The result obtained from the above is 92.1058oC(365.1058K). This result compares favourably with that in ref (8) which uses trial and error to determine at which temperature PB +PT =101.3 KN/m2 (Table 3.3).

Table 3.3: Results from Trial and Error

T (K) oBP o

TP BP TP TPPB

373

353

363

365

365.1

180.006

100.988

136.087

144.125

144.534

74.152

38.815

24.213

57.810

57.996

90.003

50.494

68.044

772.062

72.267

37.076

77.631

27.106

28.905

28.998

127.079

128.125

95.150

100.967

101.265

3.3 Differential Calculation Function

This solver can be used to obtain a numerical derivative of explicit expressions such as )cos()( xxxf . To solve

such problem, three inputs are needed

i. The function of the variableii. The point at which the differentiation is

calculatediii. The change in Δx or step size

The general expression for solving such problem is

SHIFT dx

dexpression , a , b , Δx )

Problem 1: Determine the first derivative of the expression )exp()( xxxf at x = 0 using a step size

of h = 0.1

Representation: Let x = X

Algorithm: SHIFT dxd ALPHA X x SHIFT ln ( -

ALPHA X ) , 0 , 0.1 )

Results: X = 1

3.4 Integration Calculation Function

The integration function is used to solve moderate integration problems. To use this Function, four inputs are required

The function with variable x The boundary a and b which define the

integration range of the definite integral The number of partition n.

The general expression for solving such problem is

dx expression, a , b , n )

Problem 1: Solve for I(x) given that

2

02sin25.01

sin)(

dxx

xxI (3.3)

Algorithm: dx (( sin ALPHA X ) ÷ ( ( 1 – 0.25 x

(sin ALPHA X ) 2x ) ) , 0 , ÷ 2 , 4)

Table 3.4: Results obtained using Integration Function and Analytical Method

)(xI (CALCULATOR) )(xI (Analytical) % ERROR

1.09861 1.098612 0



The analytical solution was obtained from (9). It should be noted that the calculator must be in radian mode to solve this problem. The function can also be used to solve integrals such as

(i) 150

1800

CpdT given that

]0473.0exp[235.0 21

TCp

(ii) dxx

xxI

1

02

23

)1cos(ln(2

)exp()1,0(

(iii)

1

021

1)1,0( dx

xI (

(iv)

2

024

1)2,0( dx

xI

(v)

1

1

)]exp[sin()1,1( dxxI

(vi) 150

20

4 )1042.9855.0()( dTTxKgKJU

for 5n

[ANS (i) -1731.12cal/g (ii) 0.04585 (iii) 0.785398 (iv) 0.8813736 (v) 2.532132 (vi) 121.5591 KJ/Kg]

3.5 Matrix Calculations

In MAT Mode, the calculator can be used to create maximum of three matrices each with up to three rows and three columns. Matrices operations such as addition, subtraction, multiplication, transpose, inverse, and determinant can be performed on the matrices created.

Problem: The annual manufacturing cost for a specialty chemical is

)( 321 DKKnDKC and DQn

Where K1, K2, K3 and Q are constants(i) Obtain an expression for optimum D

(ii) Estimate the optimum value of C(take Q = 100, 000 and use Table 3.5)

Table 3.5: Variation of Diameter with the Manufacturing Costs

Representation: Given

)( 321 DKKnDKC (3.6)

DQn (3.7)

Substitution (3.7) into (3.6) we have,

QKD

QKDKC 3

21 (3.8)

Differentiating (3.6) and equating to zero, the optimum diameter can be expressed as

1

2

K

QKDopt (3.9)

The simplified form of (3.8) is given as

32

1 100000100000 KD

KDKC (3.10)

Using the values from Table 3.4, a matrix structure can be obtained as follows

3

2

1

33

22

11

3

2

1

100000100000

100000100000

100000100000

K

K

K

DD

DD

DD

C

C

C

(3.11)

Rearranging (3.11) gives

D 20,000 40,000 60,000

C 470,000 465,000 477,000



3

2

1

1

33

22

11

3

21

1

100000100000

100000100000

100000100000

C

C

C

DD

DD

DD

K

K

K

(3.12)

By using values from Table 3.4, K1, K2 and K3 can be obtained as follows:

477000

465000

470000

10000067.160000

1000005.240000

1000005200001

3

2

1

K

K

K

For the ease of computing K1, K2 and K3, let

BA

K

K

K1

3

2

1

Algorithm:MODE MODE MODE 2 SHIFT 4 1 1 3 = 3 = 20000 5 100000 40000 2.5 100000 60000 1.67 100000 SHIFT 4 1 2 3 = 1 = 470000 465000 477000 SHIFT 4 3 1 X-1 xSHIFT 4 3 2 =

The results are shown in Table 3.6.Table 3.6:Comparison of results obtained using ExcelLink in Matlab and Matrix Function in Calculator

CALCULATORMS Excel Link/Matlab

% ERROR

K1 1.022 1.022 0

K2 10179.641 10179.641 0

K3 3.987 3.987 0

4

10000

0.1

987.3

641.10179

022.1

3

2

1

K

K

K

78.31622optD and 55.463245optC

3.6 Regression Calculations

The REG Mode can be used to perform linear, logarithmic, exponential, power, inverse and quadratic regressions. This mode is very useful in data analysis, mathematical modelling and in solving optimization problems.

Problem: Table 3.7 shows experimental data in which the independent variable x is the mole percentage of a reactant and the dependent variable y is the yield (in percent). Determine the value of x that maximises the yield.

Table 3.7: Experimental Data on Percentage Reactant and Yield

X 20 20 30 40 40 50 50 50 60 70

Y 73 78 85 90 91 87 86 91 75 65

Representation: The data is first fit with a quadratic model of the form

2CxBxAy (3.13)

Algorithm: MODE MODE 2 ► 3 20,73 M+ 20,78 M+ 30,85 M+ 40,90 M+ 40,91 M+ 50,87 M+ 50,86 M+ 50,91 M+ 60,75 M+ 70,65 M+ SHIFT 2 ► ► 1 SHIFT 2 ► ► 2 = SHIFT 2 ► ► 3

The figure that shows on the screen after pressing M+ is the number of data that has been entered. It is strongly recommended that the calculator memory is cleared before initiating a new set of calculations. This can be done by pressing SHIFT MODE 3 = =

Results:Table 3.8: Comparison of results obtained using MS Excel and REG Mode in Calculator

CALCULATOR MS Excel %ERROR

A 35.66 35.66 0

B 2.63 2.63 0

C -0.032 -0.032 0

The values for the MS Excel ware obtained from (10)

3.7 Equation Calculations



The EQN Mode can be used to solve equations up to three degrees and simultaneous linear equations with up to three unknowns. The Mode can be applied in various field of engineering including but not limited to Thermodynamics, Mass Transfer, Optimization and Process Control.

Problem: Consider the function

36602165365)( 3456 xxxxxf

Determine the minimum point

Representation: The first derivative of this equation is

018033018030)( 2345' xxxxxf (3.14)

The optimum points can be obtained from (3.14) as follows

0)6116(30 232 xxxx

Thus 0x and 06116 23 xxx

Algorithm: MODE MODE MODE 1 ► 3 1 = -6 = 11 = -6 =

Continuous pressing of the = key will display the values of X1 = 3, X2 = 1 and X3 = 2Other functions and Modes in this calculator are Complex Number, Base n, Statistical Calculations, Vector Calculations, Metric Conversions, and Scientific Constants.

Conclusion and Recommendation

In this work, we have demonstrated the capability of using handheld calculator to perform some seemingly difficult scientific computations with same level of numerical accuracy as with commonly known sophisticated algebraic software packages. From the outcome of this work, we have shown that the hitherto familiar handheld calculators have so far been under-utilized because of lack of knowledge of its functional capabilities. It can be concluded that calculator can be used as a supportive tool in effective teaching of soft computing skills. It is therefore recommended that the use of calculator beintroduced in courses related to some high levelcomputing. Students can later build on this basic knowledge with the use of computer as they climb higher

the ladder of their academic pursuit. With the better understanding of the use of this important tool, theassessment of teaching of computing skills can now easily be done even in the examination hall so as to inject the spirit of speed and accuracy into the students.

Dedication: To my mother “Obinrin rere” and other lovely mothers around the world. “Gbogbo yin le o jere omo o” (ameen).

Acknowledgement: My profound appreciation to my favourite secondary school maths teacher, Engr Atoyebi and other individuals (too numerous to mention) for their contributions to this manuscript.

REFERENCES(1) Ziemlewski, J., “Designing the New Global

Chemical Engineer.” Chem Eng Progress, 104 (2), pp.6-9, (2009)

(2) Foley, H. C., “An Introduction to Chemical Engineering Analysis using Mathematica,”Academic Press, California, (2002).

(3) Hill, D., “Process Simulation From the Ground Up,”Chem Eng Progress, 104 (3), pp. 50 – 53, (2009)

(4) Rajeev, A., Yau-Kun, L., Oscar, S., Marco, A. S., and V. Andrew, “Uncovering the Reality of Simulation,” Chem Eng Progress, 97 (5 and 6), pp.42 -52 and 64 – 72, (2001).

(5) Shaw, C. M., “Engineering Probelm Solving: A Classical Perspective,” Noyes Publication, New York,(2001).

(6) Taiwo, E. A., and Adewole, J. K., “Mathematical Model for Predicting Dynamic Liquid Hold Up in Packed Distillation Colum,” Nigerian Society of Chemical Engineers, Enugu, (2007).

(7) Constantinides, A. and Mostoufi, N., “Numerical Methods for Chemical Engineers with MATLAB Applications,” Prentice Hall, New Jersey, (1999).

(8) Richardson, F. J., Harker, H. J. & Backhurst, R. J.,“Particle Technology and Separation Processes,”Butherworth Heinemann, Oxford, (2005).

(9) Osunleke, A. S., “Engineering Analysis II Solution to Tutorial Set,” Department of Chemical Engineering, Obafemi Awolowo University, Ile Ife, (2000).

(10) Edgar, F. T., Himmeblau, M. D., and Ladson, L.,“Optimization of Chemical Processes,” 2nd ed.,

McGraw-Hill Chemical Engineering Series, New York, (2001).

Education

Computational Aptitude of Handheld Calculator