REACTION ENGINEERING - EXAMPLE 13.2 : A batch reactor is designed for the system of the irreversible, elementary liquid-phase hydration of butylene oxide that produces butylene glycol. At the reaction temperature T = 323 K, the reaction rate constant is k = 0.00083 L / (mol - min). The initial concentration of butylene oxide is 0.25 mol / L = Ca. The reaction is conducted using water as the solvent, so that water is in large excess. (a) Let the molecular weight of water is 18 g / mol and the mass of 1 kg in 1 L of water, calculate the molar density of water, Cb in the unit of mol / L. (b) Determine the final conversion, X of butylene oxide in the batch reactor after t = 45 min of reaction time. Use the formula X = 1 - 1 / exp [ kt (Cb) ] derived from material balance. (c) Find the equation of t as a function of X.

REACTION ENGINEERING - EXAMPLE 13.3 : The half-life for first order reaction could be described in the differential equation dC / dt = -kC where k is a constant, C is concentration and t is time. (a) Find the equation of C as a function of t. (b) Find the half life for such reaction or the time required to reduce 50 % of the initial concentration, where k = 0.139 per minute. (c) When the initial concentration Co is 16 mol / cubic metre, how long does the reaction required to achieve the final concentration of 1 mol / cubic metre?

BIOPROCESS ENGINEERING - EXAMPLE 14.1 : In differential centrifugation of cells with diameter D in centimeter, the square of D is given by D x D = [18n ln (RF / RI) ] / [ (RP - RFF) Wt ] where n is the fluid viscosity (poise), RF is the final radius of rotation (cm), RI is the initial radius of rotation (cm), RP is cell density (g / ml), RFF is the fluid density (g/ml), W the square for the rotational velocity in (radians / s) (radians / s), t is the time required to sediment from RI to RF (s). Derive an equation for W as a function for D, n, RF, RI, RP, RFF and t, with the stated units above, in radian & degree.

BIOPROCESS ENGINEERING - EXAMPLE 14.3 : The kinetic behavior of an enzyme could be described using Michalis - Menten equation : Vo = Vmax [S] / (Km + [S]). Derive this equation from [ES] = [E]total [S] / (Km + [S]), Vmax = Kcat [E]total, Vo = Kcat [ES].

ENGINEERING MECHANIC - EXAMPLE 15.3 : A biochemical trolley of mass 15 kg is towing a trailer of mass 5 kg along a straight horizontal pathway. The trailer and the trolley are connected by a light inextensible tow-bar. The engine of the trolley exerts a driving force of magnitude 100 N. The trailer and the trolley experience resistances of magnitude 10 N and 30 N respectively. (a) Form 2 equations with unknowns T and a, that represents the equilibrium for the 2 systems of the trolley and trailer. (b) Solve the simultaneous equations from the 2 equations that are obtained in part (a) of this question. T is the tension of the tow-bar and a is the acceleration.

Question 105 - In a rigid rotor model in quantum chemistry, the moment of inertia I is given by an Equation E as I = Ma x La x La + Mc x Lc x Lc = m x L x L, where m = (Ma x Mc) / (Ma + Mc) and L = La + Lc, m is the reduced mass, Ma is the mass of a, Mc is the mass of c, La is the radius of a from point O, Lc is the radius of c from point O. Prove by simplest method that Equation E is wrong.

ELECTRICAL TECHNOLOGY - EXAMPLE 16.1 : According to Shockley equation, the I - V characteristic of a diode is approximated by I = IS [ exp (nVD / VT) - 1 ]. For silicon, let the reverse bias saturation current IS as 0.000000000001. If n is ideality factor with value of 1.5, VT as thermal voltage drop of 0.026 V at room temperature, what is the value of current I that passes through the silicon diode in the heater of evaporator when the forward voltage drop VD = 0.026 V? Please take note that exp is the exponential function with e(1) = 2.718, e(2) = 7.389.

COMPUTER PROGRAMMING FOR ENGINEERS - EXAMPLE 17.2 : There are 5 simultaneous equations with 5 unknowns as follow : 16 V - 4 X - Z = 36, 4 X - 4 W - Y + Z = 3, 8 V + 4 W - X + Y = 10, X - W = 1, V + W = 0. Find the values of V, W, X, Y and Z accurately within 15 minutes. State the computer program that you use to get the answers.

ENGINEERING NUMERICAL METHODS - EXAMPLE 19.1 : Solve the simultaneous equation using simple Excel program : r + s - t = 7, 2r - s = 0, 3r + 2t = 15. Explain the use of functions of MINVERSE and MMULT together with the keyboard C (Ctrl) + S (Shift) + E (Enter).

ENGINEERING NUMERICAL METHODS - EXAMPLE 19.2 : For a mixture of benzene (B), toluene (T) and xylene (X), the equation applies where x for B, T and X will sum up to 1. The equation of x for each component is x = (L / V + 1) (F) / (L / V + K). The data of F for each component are : 0.5 for B, 0.35 for T, 0.15 for X. The data of K for each component are : 1.98 for B, 0.76 for T, 0.24 for X. When x for B + x for T + x for X = 1, find the values of (a) L / V; (b) x for each component of B, T, X respectively. You may use Excel program - Data : What-If-Analysis for Goal Seek to perform the iterative calculations.

ENGINEERING NUMERICAL METHODS - EXAMPLE 19.3 : There are 2 simultaneous equations : (A1) x + (B1) y = D1 and (A2) x + (B2) y = D2. (a) By using Excel program, find the values of x and y when A1 = 80, A2 = 150, B1 = 52, B2 = 100, D1 = 3.5 and D2 = 2.3. (b) Write the expression of Excel in the form of =MMULT(MINVERSE(W:X),Y:Z) in order to get the values of x and y. W, X, Y and Z may be A1, A2, B1, B2, D1 and D2.

DIFFERENTIAL EQUATIONS - EXAMPLE 20.1 : By using Excel program either on laptop or desktop PC, solve the differential equation dy / dx = -2y + x + 4 with h = 0.005, initial values : x = 0, y = 1. The 4th order Runge-Kutta method provides : y(N + 1) = y(N) + (1/6) (k1 + 2k2 +2k3 + k4), k1 = h [ -2y(N) + x(N) + 4 ], k2 = h { -2 [ y(N) + k1 / 2 ] + x(N) + h / 2 + 4 }, k3 = h { -2 [ y(N) + k2 / 2 ] + x(N) + h / 2 + 4 }, k4 = h { -2 [ y(N) + k3 ] + x(N) + h + 4 }. What is the value of y at x = 0.5?

DIFFERENTIAL EQUATIONS - EXAMPLE 20.2 : During the landing process of an airplane, the velocity is constant at v. (a) If the displacement of the plane is x at time t, find the differential equation that relates t, x and v. (b) The plane has 2 parts of wheels - the front and the back, separated by a distance L. The front part of the wheel touches the land first, that allows the straight body of the plane to form an angle T with the horizontal land. If the vertical distance between the back part of the wheel and the horizontal land is y, find the equation of y as a function of L and T. (c) Find the differential equation that relates dy as a function of dt, v and sin T. (d) Find the differential equation that consist of dy as a function of y, L, v and dt. (e) Find the equation of y as a function of v, L, t and C where C is a constant. (f) When t = 0, prove that y = exp C as the initial value of y.

DIFFERENTIAL EQUATIONS - EXAMPLE 20.3 : A differential equation is given as y” + 5y’ + 6y = 0, y(0) = 2 and y’(0) = 3. By using Laplace transform, an engineer has correctly produced the equation L {y} = (2s + 13) / [(s + 2)(s + 3)] = A / (s + 2) + B (s + 3). (a) Find the values of A and B. (b) The inversed Laplace transform of 1 / (s + a) is given by exp (-at) where a is a constant. If the statement : L {y} = 9 L { exp (-2t) } - 7 L { exp (-3t) } is correct, find the equation of y as a function of t as a solution to the differential equation stated in the beginning of this question. When L {d} = 9 L {b} - 7 L {c}, then d = 9b - 7c with b, c and d are unknowns.

PROCESS DESIGN - EXAMPLE 21.1 : According to rules of thumb in chemical process design, consider the use of an expander for reducing the pressure of a gas when more than 20 horsepowers can be recovered. The theoretical adiabatic horsepower (THp) for expanding a gas could be estimated from the equation : THp = Q [ Ti / (8130a) ] [ 1 - (Po / Pi) ^ a ] where 3 ^ 3 is 3 power 3 or 27, Q is volumetric flowrate in standard cubic feet per minute, Ti is inlet temperature in degree Rankine, a = (k - 1) / k where k = Cp / Cv, Po and Pi are reference and systemic pressures respectively. (a) Assume Cp / Cv = 1.4, Po = 14.7 psia, (temperature in degree Rankine) = [ (temperature in degree Celsius) + 273.15 ] (9 / 5), nitrogen gas at Pi = 90 psia and 25 degree Celsius flowing at Q = 230 standard cubic feet per minute is to be vented to the atmosphere. According to rules of thumb, should an expander or a valve be used? (b) Find the outlet temperature To by using the equation To = Ti (Po / Pi) ^ a.