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CHEN 4601 Lecture 12: 2015_9_24 Recitation

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mauvedinosaur421
30 Sep 2015
School
UMN
Department
Chemical Engineering
Course
CHEN 4601
Professor
Prodromos Daoutidious

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Document Summary

C(t = 0) = 0 nothing dissolved @ t = 0. Chen4601_processcontrol page 1 rate of stuff dissolving is proportional to the area so if we account for the dissolving, we would get a slower change. Chen4601_processcontrol page 2 we have constant flow in and out, constant density. What if ca,in is a step function or with if rearrange tau expression. Chen4601_processcontrol page 3 for very high mass transfer coefficient, static gain if k is high, we will have barely any a coming out since most reactsopposite if k is low. What if our system is 2nd order? if put in deviation terms and linearize it. Rsd in a stirred tank look at residence time distribution put some dye in a tank. Chen4601_processcontrol page 4 but in reality, can"t do delta function impulse instead, can do pulse. Chen4601_processcontrol page 5 to be closer to an impulse, we want h to be small h << v/q.

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Related Questions

Consider a dilute solution of glucose in water at 25 oC. The solution was in a horizontal tube of length 10 cm, at was prepared such that there was a linear gradation of glucose concentration from cglucose(0, 0) = 0.1M at x = 0 cm to cglucose(0, 10cm) = 0.05M at x = 10 cm. (A) What is the force (and direction) acting at the center of the tube (x = 5 cm)? Derive a formula for cglucose(0, x) for x ∈ [0, 10]cm. (B) We wish to solve the time time dependence of cglucose(t, x) for all times t > 0. The next steps of this question will lead you through how to solve such a problem. Consider Fick’s second law of diffusion in one dimension, ∂c ∂t = D ∂ 2 c ∂x2 (2) Show that a function of the form c(t, x) = exp(λt) × (Aeikx + Be−ikx) satisfies Eq. (2) when λ is related to k and D. Determine λ for such a solution. (C) For the problem like that considered in (A) for particles confined to a a region of length L where x ∈ (0, L) (in (A), L = 10 cm, for example) The flux at the end of the containers is equal to zero (no particles leave or enter the container). Mathematically, this imposes a boundary condition that J(x → 0−) = J(x → L+) = 0 (x → 0− means you approach x = 0 from the left and x → L+ means you approach x = L from the right). Show that these boundary conditions imply that A = B and k ≡ kn = nπ L for n = 0, 1, 2, ...∞. That is, show that solutions look like 2Aeλnt cos nπx L (where we now relate λn to kn). (D) Spatially normalize the solutions. Let’s denote the solutions to Eq. (2) with the above boundary conditions as cn(t, x) = A˜ ne λnt cos nπx d for n = 0, 1, 2, ...∞. We wish to verify that at t = 0, these solutions are orthonormal. What the means is we want to show: Z L 0 dxcn(0, x)cm(0, x) = 0 (3) for n 6= m and Z L 0 dxc0(0, x)c0(0, x) = L(A˜ 0) 2 = 1 Z L 0 dxcn6=0(0, x)cn6=0(0, x) = L 2 (A˜ n) 2 = 1 (4) 3 (Hint: it may be easier to verify the above integrals using the relationships that cos(θ) = e iθ+e−iθ 2 and sin(θ) = e iθ−e−iθ 2i ). From the above orthonormality conditions, determine what An for n = 0, 1, 2, ...∞. (E) Superposition principle. Verify that c(t, x) = P∞ n=0 bncn(t, x) satisfies Eq. (2) and also the boundary condition that ∂c(t,x) ∂x x→0− = ∂c(t,x) ∂x x→L+ = 0. The coefficients, bn are simple numbers telling you how much of cn(t, x) is part of your solution for c(t, x). (F) Imposing the initial condition at t = 0. Suppose that at t = 0, c(0, x) = F +Gx for x ∈ (0, L) and c(0, x) = 0 for x ≤ 0 and x ≥ L, i.e., there is a linear gradation to the concentration within the sample region. By the superposition principle, we can write c(t, x) as c(t, x) = X n=0 bncn(t, x) (5) At t = 0 we must have: c(0, x) = F + Gx = X n=0 bncn(0, x) (6) Using orthonormality conditions you derived above, show that bn = Z L 0 dxcn(0, x)(F + Gx) (7) Explicitly determine formulas for b0 and bn6=0. Using these expressions, finally write out your solution to c(t, x). (G) Finally, let’s go back to (A) and determine how the glucose concentration changes with time. The diffusion coefficient for glucose in water is D = 5.7 × 10−10 m2 s . Numerically calculate cglucose(t, x) for x ∈ (0, L) at t = 0.5 s, t = 1s, t = 10s, and t = 1000 s. You will need to approximate your formulas for c(t, x) by taking only a finite number of terms.

You are part of a team working on the development of a process in which a mineral species (labeled A for proprietary reasons) undergoes a reaction to form a new pigment to be used in house paints. In a series of experiments in a large well-mixed stirred tank reactor, you fill a tank with a known quantity of an inert liquid, bring the liquid up to a specified temperature, add a known quantity of A, and measure the concentration of A in the tank as a function of time. The reaction gives off heat as it progresses, but cooling water circulating in a jacket around the reactor keeps the temperature of the reacting mixture constant. The following data are recorded.

Concentration of A, CA(mol A/L)

t(min)

T = 94°C

T = 110°C

T = 127°C

T = 142°C

10

9.30×10−2

5.19×10−2

2.10×10−2

9.87×10−3

20

6.17×10−2

2.35×10−2

1.33×10−2

5.55×10−3

30

4.41×10−2

1.91×10−2

8.15×10−3

3.97×10−3

40

3.12×10−2

1.45×10−2

5.92×10−3

2.45×10−3

50

2.58×10−2

1.01×10−2

4.48×10−3

2.27×10−3

60

9.30×10−2

9.50×10−3

4.36×10−3

1.83×10−3

A research article indicates that concentration of A should vary with time according to the following expression:

CA(t)=11CA0+kt

(1)

where CA0 (mol/L) is the initial concentration of A in the reactor [CA(t=0)] and k is called the reaction rate constant. Despite being called a constant, k is a strong function of the absolute temperature in the reactor:

k(T)=k0 exp−EaRT(K)

(2)

In this equation (known as the Arrhenius equation after the Swedish chemist who proposed it), k0 is a constant, Ea(J/mol) is the reaction activation energy, and R=8.314 J/(mol·K) is the universal gas constant. Your task will be to verify that the expressions for CA(t) and k(T) fit the data, and if they do, to determine the parameters CA0 and k at each temperature and then the constants k0 and Ea.

(a)

What are the units of k, k0, and Ea if CA is in mol/L and t is in min?

(b)

Transform Equation 1 into an equation of the form y=at+b, so that if Equation 1 is valid, a plot of yvs. t would be a straight line. How would you determine CA0 and k from the slope and intercept of the line?

(c)

Create an Excel spreadsheet with the structure shown as Rows 1-10 in the figure at the end of this problem statement, and fill in the data table in Columns A-I. Then create the four scatterplots shown in Rows 15-21. In the “Trendline” options of each plot, check the boxes for “Display equation on chart” and “Display R-squared value on chart” but not the one for “Set intercept.” R2 is the coefficient of determination (it has several other names as well), and provides a measure of how well a straight line fits a set of data: the closer it is to 1, the better the fit. What can you conclude about Equation 1 from the four plots?

(d)

From the four trendline equations, calculate the values of k and CA0 for each of the four experimental temperatures and fill in Columns B-I in Rows 12 and 13 of the spreadsheet.

(e)

Enough A was initially added to the tank to make the initial concentration CA0=0.25 mol/L, yet you calculated four different values of CA0 in Part d. How do you explain this result?

i need answers of all the 12 questions. plz provide steps as well