CME220-Long Quiz 1

This page contains 5 questions, worth a total of 15 points. Each question is separate a Your answer should be placed in the yellow highlighted cell. Random (Write Excel formula in B11) X(guess) F(x) 1 0 2 3 4 5 a.) b.) Iteration step 3. [3 pts] A function f(x) is plotted below. If Excel’s Solver is used to minimize f(x), what will be the approximate estimate for x that minimizes the function starting with an initial guess of x = (a) 2.5 or (b) 0.5? 1. [3 points] What function will return a random number between the values of 0 and 1? 2. [3 points] Using the Direct Substitution Method (F(x) = x), find a root for the following equation 𝑥^3+4 ^ 𝑥 2−33𝑥=22 use an initial guess of 0, using only 5 iteration steps

4. [3 pts] using an excel function, report the tangent of 33.3 Degrees Angle 33.3 degrees Tangent X Y sin(3X) X^2 X^3 0 0 0 0 0 0.1 1.495101 0.29552 0.01 0.001 0.2 2.883212 0.564642 0.04 0.008 A 0.3 4.029135 0.783327 0.09 0.027 B 0.4 4.820195 0.932039 0.16 0.064 C 0.5 5.174975 0.997495 0.25 0.125 0.6 5.049238 0.973848 0.36 0.216 0.7 4.438547 0.863209 0.49 0.343 0.8 3.377316 0.675463 0.64 0.512 0.9 1.934399 0.42738 0.81 0.729 1 0.2056 0.14112 1 1 1.1 -1.696228 -0.157746 1.21 1.331 1.2 -3.652602 -0.44252 1.44 1.728 1.3 -5.551331 -0.687766 1.69 2.197 1.4 -7.297879 -0.871576 1.96 2.744 1.5 -8.825151 -0.97753 2.25 3.375 1.6 -10.10082 -0.996165 2.56 4.096 1.7 -11.13157 -0.925815 2.89 4.913 1.8 -11.96382 -0.772764 3.24 5.832 1.9 -12.68093 -0.550686 3.61 6.859 2 -13.39708 -0.279415 4 8 2.1 -14.24843 0.016814 4.41 9.261 2.2 -15.38229 0.311541 4.84 10.648 2.3 -16.9453 0.57844 5.29 12.167 2.4 -19.07166 0.793668 5.76 13.824 2.5 -21.8725 0.938 6.25 15.625 2.6 -25.42728 0.998543 6.76 17.576 2.7 -29.77805 0.96989 7.29 19.683 5. [3 points] Using the Regression Tool and the table given below, report the coefficients that best fits the foll equation

lowing

Problem 1 (15 points total) Part a. 5 points Initial guess Xd 0.95 Part B 5 points for setting up table correctly 5 points for using cor

Problem 2 (30pts total) 7 points for setting up tables correctly, t(min) Mt/Minf 0 0.000 3 points f 0.5 0.040 1 0.052 2 0.073 4 0.111 6 0.126 2 points fo 8 0.157 12 0.185 radius 24 0.242 36 0.306 45 0.353 55 0.354 65 0.388 75 0.441 85 0.451 95 0.459 105 0.511 115 0.553 3 points 125 0.550 SSE 135 0.591 SST 155 0.600 175 0.611 195 0.676 215 0.698 235 0.704 255 0.757 275 0.739 295 0.806 315 0.827 Find the R 2 R 2 Drug loaded microspheres are often used a means of controlling the release rate of active compounds into the body. To determine the rate of release, passive diffusion of the drug molecule through the sphere can be described by the following simplified solution of Fick’s Second Law for a sphere. 𝑴 𝒕 𝑴 ∞ = 𝟒 ξ𝝅 ൬ 𝑫𝒕 𝒓 𝟐 ൰ 𝟏/ 𝟐 − 𝑫𝒕 𝒓 𝟐 − 𝟏 𝟑ξ𝝅 ൬ 𝑫𝒕 𝒓 𝟐 ൰ 𝟑/ 𝟐 Where, M t /M inf is the normalized amount of drug released at a given time, t. r is the radius of the microsphere, and D is the diffusivity of the drug in the sphere’s matrix. In this instance, microspheres were produced to be 100µm in diameter. However, from drug release studies (see data below), 100µm does not fit the actual data. Using non-linear regression, determine the actual size of sphere that fits the data assuming the diffusivity of the drug is 7.5x10 -10 cm 2 /s. (note be sure to keep track of units). Report your answer in µms. Use the nominal microsphere radiusas your initial guess.

335 0.821 355 0.854 375 0.856 395 0.885 415 0.910 435 0.914 455 0.881 475 0.984 495 0.993 515 0.907

Problem 3 (15 points total) a. Write [A] and [b], where [x] is shown below (5pts) [A] [b] m1 m2 m3 m4 b. Is the set of linear equations solvable? (5 pts) Clearly label units. c. Determine m 1 to m4 using matrix functions in Excel. (5pts) A two-stage separation cascade designed to purify a product (A) is given the figure t right. The following series of material balances can be used to find the mole flow ra the streams. Using matrix algebra, solve this set of linear equations.

[x]= m 1 m 2 m 3 m 4 to the ates of