1. (a) Consider the following series of chemical reactions A kdy B k2 c. respectively Letna(t),...
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
K1=10, k2=7. Please do both
3) Using Laplace transforms, solve the following differential equations with the initial conditions indicated. Sketch the resulting functions of time dy 0, with y(O) - k2 bk2)+ kik20, with z(0) - 0, t( 2 dt2
7. Consider e following mass-spring systemn f (t) winFwin k2 r(t) y(t) The spring near the wall has spring constant ki, and the one between the objects has constant k2. Assume there is no friction in the system. The position of mass m at time t is given by x(t), and the position of mass M is y(t). The rest position of the two masses are, respectively, xo and yo- Starting from time t = 0 a horizontal force f(t)...
h 1 (25 Pts) Consider the system shown below C2. C1 ki k2 ky ka kı = 8 N/m, k,-100 N/m, k3-k,-50 N/m and c,-c2-16Ns/m. a) Determine the equation of motion for the system b) Compute the time constant and natural frequency of oscillation tain the free response for the initial conditions x(0)-1 and (0)-1
2. (30 pts) CSTR Reactor Multiple Reactions: Consider the following set of series reactions which occurs in a CSTR reactor, with the given intrinsic reaction rate equations and kinetic constants Rxn 1: 2A → B; r1=kCA; ki = 0.5 (L/mole)-min Rxn 2: 2B → C; r = k Ce?; k2 = 0.2 (/mol)-min! a. Algebraic equation for each species: Write out the applicable algebraic equations for CA C using the space time parameter t, for these reactions occurring in a...
The conceptual model for decay of Pu-241 to Am-241 to Np-237 is illustrated below. Nh Nh N 2 , In this notation, Ni, N2, and Nz are the number of atoms of Pu-241, Am-241, and Np 237, respectively, and 11, 12, and 13 are the decay constants for Pu-241, Am-241, and Np-237, respectively. (a) Write the differential equations that describe the number of atoms of Pu-241, Am-241, and Np-237 at time t. The differential equations should be in the form...
3) Compute the Z-Transforms of the following time series: (a) x(n)k2"u(n) (b) + 1) x(n) = u(-n (c) x(n) -k2"u-1) (d) x(n) 0.5%1(n) + 3"11(-n) (e) x(n) = 4-nu(n) + 5-nu(n + 1) In the above, u(n) stand for the unit step signal in the discrete time domain. Also, if you can in each case determine the region of convergence of the Z-Transform you obtain.
l c The amounts x1 (t) and x2 (t) of salt in two brine tanks satisfy the differential equations below where ki = for i= 1. 2. The volumes are V,-50 (gal) and v2-25 (gal). First solve for x1(t) and x2(t) assuming that r#20 (gai in). x1(0): 25(b), and x2(0)-Ο Then find the maximum amount of salt ever in tank 2 Finally, construct a figure showing the graphs of x,(t) and x2t) dx1 dx2
l c The amounts x1 (t)...
7. Solve the following differential equations. dy 2 y= 5x, x>0. + a) dx dx 1+2x 4e', t>0 b) t dt
7. Solve the following differential equations. dy 2 y= 5x, x>0. + a) dx dx 1+2x 4e', t>0 b) t dt
LOGISTI We know that if the number of individuals, N, in a population at time t follows an exponential law of growth, then N-N, exr where k >0 and No is the population when t -o. es that at time, t, the rate of growth, N, of the population is proportional to dt dN the number of individuals in the population. That is, kN Under exponential growth, a population would get infinitely large as time goes on. In reality, when...