PROCESS CONTROL - EXAMPLE 6.1 : In a Laplace Transform Table, the Laplace transfer function of f(t) is F(s). When d(t) = f(t) then 1 = F(s). When x(t) = f(t) then X(s) = F(s). If d(t) is the impulse of a spring when d(t) = kx(t), then derive the equation for the impulse of a spring as X(s) in term of k. Next question : A controller has a transfer function a and the other controller has a transfer function b. The overall transfer function of both controllers is ab. What is the transfer function overall when both controllers have similar transfer function 1 / (Cs + k)?

Question 87 - According to Hardy-Weinberg Equation, p x p + 2 x p x q + q x q = 1 where p = dominant allele frequency and q = recessive allele frequency. Let p + q = 1. Fraction of population has 2 copies of the p gene = p x p. Fraction of population has 2 copies of the q gene = q x q. Fraction of population has a copy of p gene and a copy of q gene = 2 x p x q. In a small town, the allele frequency is q = 0.2 for a recessive gene, the delta-32 mutation, that gives human protection from HIV infection. (a) Find the allele frequency a dominant gene, p. (b) What percent of the population has at least a copy of the gene that cause the population either immune to HIV or less susceptible to the disease?

PROCESS CONTROL - EXAMPLE 6.3 : The differential equation is 3 dy / dt + 2y = 1 with y(0) = 1. (a) The Laplace transformation, L for given terms are : L (dy / dt) = sY(s) - y(0), L(y) = Y(s), L(1) = 1 / s. Use such transformation to find Y(s). (b) The initial value theorem states that : When t approaches 0 for a function of y(t), it is equal to a function of sY(s) when s approaches infinity. Use the initial value theorem as a check to the answer found in part (a).

HEAT TRANSFER - EXAMPLE 5.3 : In a cylinder with a hollow, let a is outside radius and b is the inside radius. In a steady state temperature distribution with no heat generation, the differential equation is (d / dr) (r dT / dr) = 0 where r is for radius and T is for temperature. (a) Integrate the heat equation above into T(r) in term of r. (b) At r = a, T = c; at r = b, T = d. Find the heat equation of T(r) in term of r, a, b, c, d.

Question 91 - In the application of Theory of Spectrometry in spectrophotometer, let n = N x C x V, V = A x t, e = a x N where n = number of molecules, N = Avogadro's number, V = volume of cuvette, A = area of cuvette, t = thickness of cuvette, C = concentration of fluid in the cuvette, e = extinction coefficient, a = effective area of molecule. (a) By using calculus in dI = -I x a x N x C x dt, prove that ln (I / Io) = -a x N x C x t, where dI is the small difference in I and dt is the small difference in t. I = intensity of light. Io = initial intensity of light. (b) Show by calculations that ln (Io / I) = e x C x t based on the answer in the previous question (a). (c) Find the equation of log (Io / I) as a function of e, C and t based on the answer in the previous question (b).

As we know multiplicity factor derivation equation is as below MF = main c t ratio / main p t ratio * meter c t ratio / meter p t ratio * meter dial constant Where in digital meters for HT kwh import export now a days meter constant are calibrated already which is 1 So my question is For HT 11kv 110 v line values are as below Main c t and p t ratio are 11000 v and 110 v and 20/5 means 20 amp and 5 amp Now meter values are 11000/√3/110√3 and primary and secondary amp are displayed as -/5 amp on kwh meter so what will be the MF for this meter ?

NEPC,

Question 94 - The resolution of separation, Rs for chromatography is given by the formula Rs = (difference in retention time) / (average width at the base). In a chromatogram, 3 peaks a, b and c are found. Average widths W at the bases of the solutes are : Wa = 20 s, Wb = 40 s, Wc = 30 s. Resolutions of separation, Rs for solutes b and c in comparison to a are 2 and 4 respectively. The differences in retention times T for b and c in comparison to a are (Tb - Ta) and (Tc - Ta), Ta = Tc - Tb : (a) Form 2 equations involving Rs as a function of Wa, Wb, Wc, Ta, Tb and Tc. (b) Find the values of Ta, Tb and Tc.

ENGINEERING MATHEMATICS - EXAMPLE 8.3 : Solve the first order differential equation : (Z + 1)(dy/dx) = xy in term of ln |y| = f(x). Z = (x)(x).

Question 98 - The Planck-Einstein relation connects the particulate photon energy E with its associated wave frequency f to produce E = hf. Let h to be the Planck constant. The frequency f, wavelength L and speed of light c are related by E = hc / L. With p denoting the linear momentum of a particle, the de Broglie wavelength L of the particle is given by L = h / p. (a) Find the equation of E as a function of p and c. (b) If E has a unit of electron-volt and f has a unit of 1 / second, then what is the unit of h?

What is the wave equation for the sound waves encountered in smart cell phones, specifically android?

ENGINEERING MATHEMATICS - EXAMPLE 8.4 : Let 1 ^ 1 = 1, 2 ^ 2 = 4, 3 ^ 3 = 27. By using the Excel computer programming - either by Solver or Goal Seek, find the value of v for the Van der Waals equation (P - a / v ^2) (v - b) = RT where a = 18.82, b = 0.1193, P = 2, R = 0.082, T = 5000 for benzene. Describe briefly how to use Solver and Goal Seek in Excel program of computer to find the solution quickly.

UNIT OPERATION - EXAMPLE 9.2 : A distillation column separates 10000 kg / hr of a mixture containing equal mass of benzene and toluene. The product D recovered from the condenser at the top of the column contains 95 % benzene, and the bottom W from the column contains 96 % toluene. The vapor V entering the condenser from the top of the column is 8000 kg / hr. A portion of the product from the condenser is returned to the column as reflux R, and the rest is withdrawn as the final product D. Assume that V, R, and D are identical in composition since V is condensed completely. Find the ratio of the amount refluxed R to the product withdrawn D. Hint : Solve the simultaneous equations as follow in order to find the answer (R / D) : 10000 = D + W; 10000 (0.5) = D (0.95) + W (0.04); 8000 = R + D.

UNIT OPERATION - EXAMPLE 9.3 : In the distillation of binary systems by Mc Cabe Thiele method, the equation for the line of top section is given by y = [ R / (R + 1) ] x + XD / (R + 1). 2 points on the line are (0.99, 0.99) and (0, 0.36). Find the reflux ratio of R and XD.

CHEMICAL ENERGY BALANCE - EXAMPLE 11.5 : According to Margules Equation, P = x(1) p(1) g(1) + x(2) p(2) g(2) for a two-component mixture where P is bubble pressure, x is mole fraction, p is saturation pressure, g is constant given by ln g(1) = x(2) A x(2). Find the value of A as a constant when P = 1.08 bar, p(1) = 0.82 bar, p(2) = 1.93 bar in a 50 : 50 mole fraction mixture. Estimate the pressure required to completely liquefy the 30 : 70 mixture using the same equation, by proving P = 1.39 bar. Take note that ln g(2) = x(1) A x(1), ln g(1) = x(2) A x(2).

ENGINEERING MATERIAL - EXAMPLE 12.1 : In crystal material, hexagonal crystal system could form 4-digit index in certain direction of solid. For [1(-1)0] direction in the hexagonal crystal systems of particular catalyst applied in fume removal of incinerator, what is the four-digit index for this direction? Hint : The transformation equations between the 3-digit [h'k'l'] and the 4-digit [hkil] indices are : h = (1/3) (2h'-k'); i = - (h + k); k = (1/3) (2k'-h'); l = l' A. [(-1)100] B. [1(-1)00] C. [(-1)000] D. [00(-1)(-1)] E. [(-1)0(-1)0]