Numerical Methods For Engineers, 7 Ed
Numerical Methods For Engineers, 7 Ed
7th Edition
ISBN: 9789352602131
Author: Canale Chapra
Publisher: MCGRAW-HILL HIGHER EDUCATION
bartleby

Videos

Question
Book Icon
Chapter 28, Problem 15P
To determine

To calculate: The concentration of each reactant as the function of distance by using the finite difference approach, and apply centred finite-difference approximations with Δx=0.05 m.

Expert Solution & Answer
Check Mark

Answer to Problem 15P

Solution:

The concentration of each reactant as the function of distance is,

Numerical Methods For Engineers, 7 Ed, Chapter 28, Problem 15P , additional homework tip  1

The below plot shows the distance versus reactant.

Explanation of Solution

Given Information:

The series of first order, liquid phase reactions is,

Ak1Bk2C

The second order ODEs by using the steady-state mass balance.

Dd2cadx2Udcadxk1ca=0Dd2cbdx2Udcbdx+k1cak2cb=0Dd2ccdx2Udccdx+k2cb=0

Here, c is the concentration (mol/L), x is the distance (m), D is the dispersion coefficient (0.1 m2/min), U is the velocity (1 m/min), k1 is the reaction rate (3/min), k2 is the reaction rate (1/min).

Refer to the Prob 28.14, the Danckwerts boundary conditions is,

Ucin=Uc(x=0)Ddcdx(x=0)

dcdx(x=L)=0

Here, ca,in is the concentration in the inflow (10 mol/L), and L is the length of the reactor (0.5 m).

Δx=0.05 m

Formula used:

The finite divided difference formula is,

f(x0)=ci12ci+ci+1Δx2

Calculation:

Recall the ordinary differential equations,

Dd2cadx2Udcadxk1ca=0Dd2cbdx2Udcbdx+k1cak2cb=0Dd2ccdx2Udccdx+k2cb=0

Substitute the finite divided difference formula in the above differential equations.

D(ca,i12ca,i+ca,i+1Δx2)U(ca,i+1ca,i12Δx)k1ca,i=0D(cb,i12cb,i+cb,i+1Δx2)U(cb,i+1cb,i12Δx)+k1ca,ik2cb,i=0D(cc,i12cc,i+cc,i+1Δx2)U(cc,i+1cc,i12Δx)+k2cb,i=0

Substitute (0.1 m2/min) for D, (1 m/min) for U, 0.05 m for Δx, (3/min) for k1, and (1/min) for k2 yield unit.

(0.1)(ca,i12ca,i+ca,i+1(0.05)2)(1)(ca,i+1ca,i12(0.05))(3)ca,i=0(0.1)(cb,i12cb,i+cb,i+1(0.05)2)(1)(cb,i+1cb,i12(0.05))+(3)ca,i(1)cb,i=0(0.1)(cc,i12cc,i+cc,i+1(0.05)2)(1)(cc,i+1cc,i12(0.05))+(1)cb,i=0

Solve further,

50ca,i1+83ca,i30ca,i+1=050cb,i1+81cb,i30cb,i+1=3ca,i50cc,i1+80cc,i30cc,i+1=cb,i

Now solve for inlet node i=1, use a finite difference approximation for the first derivative.

Here use the second order version from the Table 19.3 for the interior nodes,

Uca,in=Uca,1Dca,3+4ca,23ca,12Δx

Ucb,in=Ucb,1Dcb,3+4cb,23cb,12Δx

Ucc,in=Ucc,1Dcc,3+4cc,23cc,12Δx

Can be solved for,

(3D2Δx2+UΔx)ca,1(2DΔx2)ca,2+(D2Δx2)ca,3=UΔxca,in

(3D2Δx2+UΔx)cb,1(2DΔx2)cb,2+(D2Δx2)cb,3=UΔxcb,in

(3D2Δx2+UΔx)cc,1(2DΔx2)cc,2+(D2Δx2)cc,3=UΔxcc,in

Substitute (0.1 m2/min) for D, (1 m/min) for U, 0.05 m for Δx, and 10 mol/L for ca,in yield unit.

80ca,180ca,2+20ca,3=200

80cb,180cb,2+20cb,3=0

80cc,180cc,2+20cc,3=0

Solve for the outer node (i=10), the zero derivative condition which implies that c11=c9,

(2DΔx2)c9+(2DΔx2+k)c10=0

The similar equations can be written for the other nodes, because the condition does not include reaction rates Substitute all the parameter gives,

80ca,9+83ca,10=080cb,9+81cb,10=3ca,10

80cc,9+80cc,10=cb,10

Rearrange the all equations in matrix form for each reactant separately, because the reactions are in series.

Write for the reactant A.

[808020000000005083300000000005083300000000005083300000000005083300000000005083300000000005083300000000005083300000000005083300000000005083300000000008380][ca,1ca,2ca,3ca,4ca,5ca,6ca,7ca,8ca,9ca,10ca,11]=[2000000000000]

Write the following code in MATLAB.

A=[80 -80 20 0 0 0 0 0 0 0 0

-50 83 -30 0 0 0 0 0 0 0 0

0 -50 83 -30 0 0 0 0 0 0 0

0 0 -50 83 -30 0 0 0 0 0 0

0 0 0 -50 83 -30 0 0 0 0 0

0 0 0 0 -50 83 -30 0 0 0 0

0 0 0 0 0 -50 83 -30 0 0 0

0 0 0 0 0 0 -50 83 -30 0 0

0 0 0 0 0 0 0 -50 83 -30 0

0 0 0 0 0 0 0 0 -50 83 -30

0 0 0 0 0 0 0 0 0 83 -80];

b=[200;0;0;0;0;0;0;0;0;0;0];

c=A\b;

c'

The output is,

ans =

Columns 1 through 5

8.0655 7.1509 6.3416 5.6270 4.9988

Columns 6 through 10

4.4516 3.9847 3.6050 3.3328 3.2123

Column 11

3.3328

Write the all the above equations in matrix form for the reactant B.

[808020000000005081300000000005081300000000005081300000000005081300000000005081300000000005081300000000005081300000000005081300000000005081300000000008180][cb,1cb,2cb,3cb,4cb,5cb,6cb,7cb,8cb,9cb,10cb,11]=[021.452719.024816.881014.996413.354811.954110.81509.99849.63699.6369]

Write the following code in MATLAB.

A=[80 -80 20 0 0 0 0 0 0 0 0

-50 81 -30 0 0 0 0 0 0 0 0

0 -50 81 -30 0 0 0 0 0 0 0

0 0 -50 81 -30 0 0 0 0 0 0

0 0 0 -50 81 -30 0 0 0 0 0

0 0 0 0 -50 81 -30 0 0 0 0

0 0 0 0 0 -50 81 -30 0 0 0

0 0 0 0 0 0 -50 81 -30 0 0

0 0 0 0 0 0 0 -50 81 -30 0

0 0 0 0 0 0 0 0 -50 81 -30

0 0 0 0 0 0 0 0 0 81 -80];

b=[0;21.4527;19.0248;16.881;14.9964;13.3548;

11.9541;10.815;9.9984;9.6369;9.6369];

c=A\b;

c'

The output is,

ans =

1.6486 2.4090 3.0415 3.5629 3.9880 4.3295 4.5977 4.7997 4.9358

4.9938 4.9358

Write the all the above equations in matrix form for the reactant C.

[808020000000005080300000000005080300000000005080300000000005080300000000005080300000000005080300000000005080300000000005080300000000005080300000000008080][cc,1cc,2cc,3cc,4cc,5cc,6cc,7cc,8cc,9cc,10cc,11]=[02.40903.04153.56293.98804.32954.59774.79974.93584.99384.9358]

Write the following code in MATLAB.

A=[80 -80 20 0 0 0 0 0 0 0 0

-50 80 -30 0 0 0 0 0 0 0 0

0 -50 80 -30 0 0 0 0 0 0 0

0 0 -50 80 -30 0 0 0 0 0 0

0 0 0 -50 80 -30 0 0 0 0 0

0 0 0 0 -50 80 -30 0 0 0 0

0 0 0 0 0 -50 80 -30 0 0 0

0 0 0 0 0 0 -50 80 -30 0 0

0 0 0 0 0 0 0 -50 80 -30 0

0 0 0 0 0 0 0 0 -50 80 -30

0 0 0 0 0 0 0 0 0 80 -80];

b=[0;2.4090;3.0415;3.5629;3.9880;4.3295;

4.5977;4.7997;4.9358;4.9938;4.9358];

c=A\b;

c'

The output is,

Columns 1 through 7

0.2859 0.4402 0.6169 0.8101 1.0133 1.2191 1.4178

Columns 8 through 11

1.5957 1.7321 1.7949 1.7332

The reaction is in series, thus the system for each reactant is,

Numerical Methods For Engineers, 7 Ed, Chapter 28, Problem 15P , additional homework tip  2

The below plot shows the distance versus reactant.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
1. Give a subset that satisfies all the following properties simultaneously: Subspace Convex set Affine set Balanced set Symmetric set Hyperspace Hyperplane 2. Give a subset that satisfies some of the conditions mentioned in (1) but not all, with examples. 3. Provide a mathematical example (not just an explanation) of the union of two balanced sets that is not balanced. 4. What is the precise mathematical condition for the union of two hyperspaces to also be a hyperspace? Provide a proof. edited 9:11
2. You manage a chemical company with 2 warehouses. The following quantities of Important Chemical A have arrived from an international supplier at 3 different ports: Chemical Available (L) Port 1. 400 Port 2 110 Port 3 100 The following amounts of Important Chemical A are required at your warehouses: Warehouse 1 Warehouse 2 Chemical Required (L) 380 230 The cost in £ to ship 1L of chemical from each port to each warehouse is as follows: Warehouse 1 Warehouse 2 Port 1 £10 £45 Port 2 £20 £28 Port 3 £13 £11 (a) You want to know how to send these shipments as cheaply as possible. For- mulate this as a linear program (you do not need to formulate it in standard inequality form). (b) Suppose now that all is as in the previous question but that only 320L of Important Chemical A are now required at Warehouse 1. Any excess chemical can be transported to either Warehouse 1 or 2 for storage, in which case the company must pay only the relevant transportation costs, or can be disposed of at the…
choose true options in these from given question a) always full and always crossing. b) always full and sometimes crossing. c) always full and never crossing. d) sometimes full and always crossing. e) sometimes full and sometimes crossing. f) sometimes full and never crossing. g) never full and always crossing. h) never full and sometimes crossing. i) never full and never crossing.

Chapter 28 Solutions

Numerical Methods For Engineers, 7 Ed

Ch. 28 - An on is other malbatchre actor can be described...Ch. 28 - The following system is a classic example of stiff...Ch. 28 - 28.13 A biofilm with a thickness grows on the...Ch. 28 - 28.14 The following differential equation...Ch. 28 - Prob. 15PCh. 28 - 28.16 Bacteria growing in a batch reactor utilize...Ch. 28 - 28.17 Perform the same computation for the...Ch. 28 - Perform the same computation for the Lorenz...Ch. 28 - The following equation can be used to model the...Ch. 28 - Perform the same computation as in Prob. 28.19,...Ch. 28 - 28.21 An environmental engineer is interested in...Ch. 28 - 28.22 Population-growth dynamics are important in...Ch. 28 - 28.23 Although the model in Prob. 28.22 works...Ch. 28 - 28.25 A cable is hanging from two supports at A...Ch. 28 - 28.26 The basic differential equation of the...Ch. 28 - 28.27 The basic differential equation of the...Ch. 28 - A pond drains through a pipe, as shown in Fig....Ch. 28 - 28.29 Engineers and scientists use mass-spring...Ch. 28 - Under a number of simplifying assumptions, the...Ch. 28 - 28.31 In Prob. 28.30, a linearized groundwater...Ch. 28 - The Lotka-Volterra equations described in Sec....Ch. 28 - The growth of floating, unicellular algae below a...Ch. 28 - 28.34 The following ODEs have been proposed as a...Ch. 28 - 28.35 Perform the same computation as in the first...Ch. 28 - Solve the ODE in the first part of Sec. 8.3 from...Ch. 28 - 28.37 For a simple RL circuit, Kirchhoff’s voltage...Ch. 28 - In contrast to Prob. 28.37, real resistors may not...Ch. 28 - 28.39 Develop an eigenvalue problem for an LC...Ch. 28 - 28.40 Just as Fourier’s law and the heat balance...Ch. 28 - 28.41 Perform the same computation as in Sec....Ch. 28 - 28.42 The rate of cooling of a body can be...Ch. 28 - The rate of heat flow (conduction) between two...Ch. 28 - Repeat the falling parachutist problem (Example...Ch. 28 - 28.45 Suppose that, after falling for 13 s, the...Ch. 28 - 28.46 The following ordinary differential equation...Ch. 28 - 28.47 A forced damped spring-mass system (Fig....Ch. 28 - 28.48 The temperature distribution in a tapered...Ch. 28 - 28.49 The dynamics of a forced spring-mass-damper...Ch. 28 - The differential equation for the velocity of a...Ch. 28 - 28.51 Two masses are attached to a wall by linear...
Knowledge Booster
Background pattern image
Advanced Math
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Advanced Engineering Mathematics
Advanced Math
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Wiley, John & Sons, Incorporated
Text book image
Numerical Methods for Engineers
Advanced Math
ISBN:9780073397924
Author:Steven C. Chapra Dr., Raymond P. Canale
Publisher:McGraw-Hill Education
Text book image
Introductory Mathematics for Engineering Applicat...
Advanced Math
ISBN:9781118141809
Author:Nathan Klingbeil
Publisher:WILEY
Text book image
Mathematics For Machine Technology
Advanced Math
ISBN:9781337798310
Author:Peterson, John.
Publisher:Cengage Learning,
Text book image
Basic Technical Mathematics
Advanced Math
ISBN:9780134437705
Author:Washington
Publisher:PEARSON
Text book image
Topology
Advanced Math
ISBN:9780134689517
Author:Munkres, James R.
Publisher:Pearson,
Orthogonality in Inner Product Spaces; Author: Study Force;https://www.youtube.com/watch?v=RzIx_rRo9m0;License: Standard YouTube License, CC-BY
Abstract Algebra: The definition of a Group; Author: Socratica;https://www.youtube.com/watch?v=QudbrUcVPxk;License: Standard Youtube License