PRACTICE PROBLEMS 43 Chapter IVP Practice Problems Use Excel and VBA to solve the following problems. Document your solutions using the Expert Problem Solving steps outlined in Table... Find an approximate solution for the range 0 x subject to the initial condition y = 0 at x = 0. Plot y versus x in a scatter plot. x yexp y. Find an approximate solution for the range 0 x subject to the initial condition y = 0 and / = 0.5 at x = 0. Plot y versus x in a scatter plot. d y x y 3. For a set of first-order consecutive reactions A B C, the concentrations of A, B, and C at any time in a batch reactor are governed by the differential equations. A B C kc kc B kc B A kc The initial concentrations are C A0 = 0. M, C B0 = 0, and C C0 = 0. The rate constants are k = 0.0 s - and k = 0.005 s -. Plot the concentrations in the range 0 t 300 s. 4. Solve for the range 0 t 0 for the values K = 9, 6,, 0, -. The initial conditions are y = 0 and / = 0. d y K 9y The initial concentrations are C A0 = 0. M, C B0 = 0, and C C0 = 0. The rate constants are k = 0.0 s - and k = 0.005 s -. Plot the concentrations in the range 0 t 300 s. 5. An unstea-state mass balance for a chemical in a well-mixed reactor with a second-order reaction is V F QC VkC where V is the working volume ( m 3 ), C is the chemical concentration (g/m 3 ), F is the feed rate (00 g/min), Q is the volumetric flow rate ( m 3 /min), and k is the second order rate constant (0.0 m 3 /gmin). The initial concentration is zero. Solve for the time to reach 99% of stea-state concentration. Use a trapezoidal method in Excel. A Answer: Time =. min 6. A non-isothermal, well-mixed, batch reactor can be described by the following equations involving an exothermic, first-order reaction: 0 exp C T 73 dt 0 000exp C0T 0 T 73
44 PRACTICAL NUMERICAL METHODS where C is the concentration of the reactant and T is the temperature of the reactor. Initially the reactor is at 5 o C with a concentration of reactant C =.0 gmol/l. Use Euler s method in Excel to find the concentration and temperature of the reactor versus time. Plot your results in a two-y axis yx plot. Explain your results. 7. Repeat problem 6 using the macro RK45ODE in Excel. Compare the results using RK45ODE and Euler s methods. 8. The irreversible first-order reaction: A B takes place in two identical well-mixed reactors in series (as shown in the schematic). The unstea-state mass balances for each chemical species in both reactors are: C A0 C A C B C A C B A CA0 CA kca B CB kca A CACA kca B CBCB kca where C A0 is the feed concentration to the first reactor, C A and C B are the exit concentrations from the first reactor and feed to the second reactor. C A and C B are the exit concentrations from the second reactor, is the residence time in a reactor, and k is the reaction rate constant. Use Excel to find the concentrations of A and B in each of the reactors during the first 30 minutes of operation if C A0 is 0, = 5 min, and k = 0./min. The initial concentrations of all chemical species are 0 in both reactors. Plot your results and determine the stea-state concentrations in both reactors. What is the approximate time to reach stea-state operation? Answer: C A = 6.7, C B = 3.3, C A = 4.4, C B = 5.6 9. Repeat problem 4 using the macro RK45VS in Excel. 0. Integrate the following system of differential equations for 0 < t < 50. 0.3x.5xy 0.036xy 0.y At t 0, x, y 0.05 Use 0 print steps. Plot x vs. y. What are x and y when t = 0, 0, 30, 40, 50? 7. For the following differential equation and boundary conditions: dt 7 4. 0 T 73 550 T 0
PRACTICE PROBLEMS 45 At x 0, T 00 At x 0.5, T 00 Calculate dt at each boundary, x = 0 and x = 0.5. 8. Solve the following system of differential equations and initial conditions for x and y as a function of time, t, for the range 0 < t < 3. Use the explicit trapezoidal method in Excel. Use 0 integration steps. Plot the results for x and y vs. t. What are the stea-state values for x and y (t)? Show your work. 0.05y0.5x x 000 at t 0 0.5x y y 00 at t 0 9. Plot the solutions for the following system of equations for 0 < x <. Solve the equations using the trapezoidal method in Excel with a step size of 0.. y y3 x at x 0, y 3x at x 0, y 3 y exp( x) at x 0, y3 0. Solve the following pair of differential equations using Excel over the range x = 0 to with y(0) = and z(0) = 4. Plot the results for y and z vs. x.. Solve the following system of equations using the trapezoidal method in Excel, using 0 integration steps. The initial conditions at x = 0 are y = 0, y =. Plot the results for y and y over the domain 0 x 3y 9 y 4 y y xy. Solve the following system of initial value equations for the range 0 t by the implicit trapezoidal method in an Excel worksheet (do not use any PNMSuite macros). Use a step size of Δt = 0.. At t = 0, x =, x =, x 3 = -. Plot all x versus t. Solve the system of equations and plot the results using Excel. 3xx x3 x x x3t 3 xx x3 e 3. Solve for y and y at x =. The initial conditions at x = 0 are y = and y =. Use VBA user functions for the derivatives. Plot the values for y and y over the range 0 x. Label both axis on the plots and include legends. t
46 PRACTICAL NUMERICAL METHODS.7 yy 0.67 y 0.67 y a. Use Euler s method with 0 steps in an Excel worksheet. b. Use the trapezoidal method with 0 steps in an Excel worksheet. 4. A stone is thrown vertically upward in air at an initial velocity V of 50 m/s. A frictional drag AV acts on the stone in a direction opposite to that of the motion. The stone rises until the velocity becomes zero and then accelerates downwards to the ground. Its velocity is governed by the following equations, while going upward and while coming down, respectively dv gav dv g AV where g = 9.8 m/s is the acceleration due to gravity, and A = 0-3 m -. Use Euler s method in Excel. (a) Solve the first equation to calculate the time when the upward velocity of the stone is zero. (b) Use the second equation to find the velocity of the falling stone after falling for the amount of time taken for the upward motion. (c) Plot the complete velocity profile in Excel for the total time. Label the axis. 5. A tank is filled with water and simultaneously drained from the bottom of the tank through a valve. The change in water level (h) in the tank is described by the following differential equation: dh 4 in F C h d Initial level of liquid in the tank at t = 0 h 0 ft Inlet liquid flow rate F in 000 lb/hr Valve constant C 500 lb/(ft / hr) Water density ρ 6.5 lb/ft 3 Tank diameter d 6 ft (a) Create a VBA user function for the differential equation. Use your function for the rest of this problem. (b) At stea-state, dh/ = 0, or the rate of water in equals the rate of water out. Use Goal Seek to find the stea-state water level in the tank. (c) Use Simpson s /3 rule in an Excel worksheet with 0 intervals to calculate the time to reach 90% of the stea-state water level found in part (b). (d) Repeat part c using the VBA macro ROMBERGT. (e) Solve the differential equation to achieve 90% of the stea-state level using Euler s method in an Excel worksheet. Use 0 time steps. (f) Repeat part e using the VBA macro RK4ODE (use 000 integration steps and 0 print steps). (g) Plot the water level versus time from part f (h on y-axis, t on x-axis). Answers: (b) 4 ft (c) 5.9 hr (d) 5.9 hr (e) h=3.69 ft at t = 5.9 hr (f) h = 3.6 ft at t = 5.9 hr 6. Cells are grown in a fermentation bioreactor. The kinetics of cell growth and substrate utilization are described by a system of first-order differential equations: SX Km S ds 0SX K S dx Y 0 bx m
PRACTICE PROBLEMS 47 Initial substrate concentration (t=0) S 0 8500 g/l Initial cell biomass concentration (t=0) X 0 40 g/l Maximum growth rate μ 0 6.5 Yield coefficient Y 0.045 Monod constant K m 3.4 g/l Decay coefficient b 0.03 (a) Create VBA user functions for the differential equations. Use your functions to evaluate the derivatives for the rest of the parts in the problem. (b) Solve the differential equations using the implicit trapezoidal method and the macro RK45VS. (c) Plot the results in a scatter plot with X on the primary axis and S on the secondary axis. 7. A reversible chemical reaction is used in a batch reactor: A B. The unstea-state mole balances for the reactant and product are: A B kc A kc B kc A kc B Initial concentration of A C A0 mol/l Initial concentration of B C B0 0 Rate constant k min - Rate constant k 0.5 min - (a) Create VBA user-functions for the derivative expressions. Use your functions for the rest of the problem. (b) Use the Trapezoidal method in an Excel worksheet to solve the differential equations for 0 t min with a step size of 0. min. (c) Plot the results from part b for C A and C B versus t in a scatter plot. Use the primary y-axis for C A and a secondary-axis for C B. (d) Repeat part b using the VBA macro RK45VS with 0 print steps. Answers: (b) t = min, C A = 0.366 mol/l, C B = 0.633 mol/l (d) t = min, C A = 0.36653, C B = 0.63348 8. A copper sphere of diameter 5 cm is initially at temperature 00C. It cools in air by convection and radiation. The temperature T of the sphere is governed by the equation dt 4 4 cv p AT Ts ht Ts with the following properties for the sphere and heat transfer: Density = 9000 kg/m 3 Specific heat c p = 400 J/kg K Radius r = 0.05 m Area A = 4r Volume V = 4r 3 /3 Emissivity ε = 0.5 Stephan-Boltzmann constant σ = 5.67x0-8 W/m K 4 Surrounding temperature T s = 5C Heat transfer coefficient h = 5 W/m K Note that the temperature in the differential equation must be in degree K. Solve for the time to cool the sphere to 50C. Use the trapezoidal method in Excel. Plot the time (horizontal axis) versus temperature (vertical axis). Label each axis with appropriate variable symbol and units. 9. Solve the membrane problem from Section..3. The mole balances for N and O in the permeate section of the membrane. Plot the results and compare with Figure. and Figure.3. Feed Conditions: y nf = 0.79, y of = 0., P f = 0 atm, k n = 0.3, k o = 8 k k
48 PRACTICAL NUMERICAL METHODS Closed System (0 < t < 5) dp dpo Open System (0 < t < ) n dp n n nf f n k y P P P n = 0.79 at t = 0 o of f o k y P P P o = 0. at t = 0 n nf f n n k y P P QP P n = 7.9 at t = 0 dpo koyofpf Po QPo P o =. at t = 0