Numerical Methods in Chemical Engineering — Summer 2024

Differentiate between accuracy and precision with appropriate example.

Differentiate between accuracy and precision with appropriate example. 03 C R O1

Find the percentage error in the area of an ellipse where an error of 1% is made in measuring its major and minor axis.

Fit a second degree parabola to the following data. x 0 1 2 3 y 1 1.8 1.3 2 6.3

Fit a second degree parabola to the following data. 07 C A x 0 1 2 3 4 O5 y 1 1.8 1.3 2 6.3

Find the percentage error in the area of an ellipse where an error of 1% is made 04 C U in measuring its major and minor axis. O1

Use Descartes' rule of signs to find the number of positive, negative and 03 C U imaginary roots of the function: x6 − x5 − 10x + 7 = 0 O2

Solid particles having a diameter of 0.12 mm, shape factor Ф =0.88 and a 07 C A s O density of 1000 kg/m3 are to be fluidized using air at 2.0 atm and 25oC. The3 voidage at minimum fluidization is 0.42. The viscosity of air under these conditions is 1.845*10-5 kg/m.s. The molecular weight of air is 28.97 g/mol. The diameter of the particle is 1.2*10-4 m. Estimate the minimum fluidization velocity using newton Raphson method. The Ergun equation for packed bed is given below. [ 1.75 𝜌(1−e mf ) ]v2 +[ 150 𝜇(1−e mf )2 ]v - (1−e )( 𝜌 −𝜌)g = 0 Фsdp e m3 f mf Ф s 2d p 2 e m3 f mf mf p

Air at 25oC and 1 atm flows through a 4 mm diameter tube with an average 07 C A O velocity of 50 m/s. The roughness is e = 0.0015 mm. Density of air at 25 oC3 and 1 atm is 1.23 kg/m3. Calculate the friction factor using the Colebrook equation e 1 2.51 = −2.0log { D + } √f 3.7 Re√f Determine the pressure drop in 1 m section of the tube using the relation fLV2ꝭ ΔP = 2D4

Find root of the equation x3 - 2x - 5 = 0 using the bisection method correct 04 C U upto three decimal places. O2

Solid particles having a diameter of 0.12 mm, shape factor Ф =0.88 and a s density of 1000 kg/m3 are to be fluidized using air at 2.0 atm and 25oC. The voidage at minimum fluidization is 0.42. The viscosity of air under these conditions is 1.845*10-5 kg/m.s. The molecular weight of air is 28.97 g/mol. The diameter of the particle is 1.2*10-4 m. Estimate the minimum fluidization velocity using newton Raphson method. The Ergun equation for packed bed is given below. 1.75 𝜌(1−e ) 150 𝜇(1−e )2 [ mf ]v2 +[ mf ]v - (1−e )( 𝜌 −𝜌)g = 0 Фsdp e m3 f mf Ф s 2d p 2 e m3 f mf mf p1

Solve the following system of simultaneous equations by Gauss seidal method 04 C A 20x + y -2z = 17, 3x + 20y -z = -18, 2x - 3y + 20z = 25 O3

The function y= sinx is tabulated below 07 C U x 0 π/4 π/2 O y 0 0.70711 1.0 Using Lagrange’s interpolation formula find the value of sin(π/6)5

Define Eigen values and Eigen vectors.

Find numerically the largest Eigen value and corresponding Eigen Vector of the following matrix using power method.161 A = [1 2 0]003

Given the values x 5 7 11 13 17 f(x) 150 392 1452 2366 5202 Determine f(9).

Discuss the pitfalls of Gauss - Elimination method and techniques for improving solutions.

Solve the following system of simultaneous equations by Gauss seidal method 20x + y -2z = 17, 3x + 20y -z = -18, 2x - 3y + 20z = 25

Given that y = ln x, and x 4.0 4.2 4.4 4.6 4.8 5.0 5.2 y 1.3863 1.4351 1.4816 1.526 1.5686 1.6094 1.64871 5.2 Evaluate∫ lnx dx using simpsons 3/8 rule.4

Solve the following system of equations by Gauss Elimination method: 2x + y + z = 3x + 2y + 3z = 18 x + 4y + 9z = 16

State the formulas for Trapezoidal Rule, Simpsons 1/3rd rule, Simpsons 3/8th rule.

6 1 07 C A Evaluate ∫ dx taking h=1 using Simpson 1/3 rule. 0 1+x O4

Derive formula for Trapezoidal Rule of numerical integration. 03 C R O4

Given that y = ln x, and 07 C A O x 4.0 4.2 4.4 4.6 4.8 5.0 5.2 y 1.3863 1.4351 1.4816 1.5261 1.5686 1.6094 1.6487 5.2 Evaluate∫ lnx dx using simpsons 3/8 rule.4

Solve the following system of equations by Gauss Elimination method: 04 C A 2x + y + z = 10 O 3x + 2y + 3z = 18 x + 4y + 9z = 16

Using Runge Kutta method of fourth order, solve the following at x=0.2 and 07 C A 0.4. Take y(0)=1 O6 dy y2 −x2 = dx y2 +x2

Discuss in brief about initial and boundary value problems.

Prove the following

∆= E-1 (ii) ∇= 1-E-1

Using Euler’s method, find an approximate value of y corresponding to x = 0.1 for the following equation. dy y−x = dx y+x Take y(0)=1 and h=0.02 OR2

Explain Milne’s predictor corrector method

Given that x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 y 7.989 8.403 8.781 9.129 9.451 9.750 10.031 dy Find at x=1.1 dx