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engineering
numerical methods with chemical engineering applications

Numerical Methods With Chemical Engineering Applications 1st Edition Kevin D. Dorfman, Prodromos Daoutidis - Solutions

Consider the method of lines solution to the unsteady diffusion equationsubject to an initial condition c = 1 and first-order depletion reactions on each boundary, n · J = −kc, where n is an outward pointing normal vector and J = −D∇c is the flux. Determine the system of ordinary
We would like to use centered finite differences and the method of lines to solve the unsteady diffusion–reaction problemsubject to no-flux on the left boundaries, ∂c/∂x = 0 at x = 0, a constant concentration on the right boundary, c(1, t) = 1, and an initial concentration c(x, 0) = 1. We
We want to use the method of lines to solve the linear reaction–diffusion equationsubject to the boundary conditionat x = 0 and c(x = 1, t) = 1. The initial condition is c(x, 0) = 2. We will set D = 4 and k = 3. Using a grid with six nodes, determine the initial values for ci and the values of
Consider the solution to the diffusion equationon a 4 × 4 rectangular grid with equal spacing in the x and y directions. The boundary conditions for the problem areUsing the global index k and the ordering of equations and unknowns in order of the index k, determine the non-zero entries on the
Consider the solution to the steady state concentration profile ∇2c = 0 using centered finite differences. The domain is discretized into nine nodes with a spacing Δx and Δy. The concentration at the top and right sides of the slab is c = 2, and the concentration on the bottom and left sides of
Suppose you are interested in the steady state temperature profile T(x, y) of a square surface with a known spatially dependent thermal diffusivity α(x, y) subject to Dirichlet boundary conditions. The time-dependent PDE governing this system isand the boundary conditions are given by Fig.
Consider the solution of a 2D finite difference solution of the diffusion equation ∇2T = 0 where the boundary conditions correspond to fixed temperatures. What fraction of the entries in the matrix needed to solve this problem are non-zero if there are nx nodes in x and ny nodes in y? Assume that
Consider the case of unsteady diffusion of heat and mass in the presence of a first-order reaction. With a suitable choice of dimensionless variables, this system leads to the set of coupled partial differential equationswhere β, γ , and φ are the activation energy, Prater number, and Thiele
In heat transfer the heat transfer from a slab of thickness 2w initially at a uniform temperature T0 can sometimes be modeled by a lumped parameter analysiswhere ρ is the density of the slab, Cˆp is its heat capacity, h is the heat transfer coefficient, and T∞ is the temperature far from the
Write a MATLAB program that uses the method of lines, RK4, and finite differences to solve the diffusion equationsubject to an initial condition c(x, 0) = 1 and boundary conditions c(0, t) = 1 and c(1, t) = 0. Your program should automatically plot the solution at the times t = 0.0005, 0.005, 0.02,
We would like to solve the steady state heat transfer problem given by the drawing in Fig. 7.17. We have a composite solid material of area LxLy, with the left half having conductivity k1 and the right half having conductivity k2. The lower part of the material is in contact with a cold reservoir
Derive a centered finite difference formula for the first derivative that is accurate to O(Δx)4.
Write a MATLAB program that computes the solution to the (dimensionless) unsteady diffusion equationsubject to an initial condition c(x, 0) = 0 and the boundary conditions c(0, t) = 1 and ∂c/∂x = 0 at x = 1. In other words, this is the solution for the concentration (or temperature profile) in
Consider the partial differential equationsubject to c(0, t) = 0 andWe will consider three cases: c(x, 0) = 1, c(x, 0) = x and c(x, 0) = x2. Note that the second initial condition is a steady state solution to the PDE.(a) Use finite differences and the method of lines to recast this problem as a
This problem considers different aspects of the solution of the unsteady heat equation.(a) Use separation of variables to solve the PDEwith no flux boundary conditions at x = 0 and x = 1 and an initial condition T = 1 − x.(b) Write a MATLAB program that makes a plot of T(x, t) versus x for a
In this problem, we will look at the two-dimensional diffusion problem with a reactive surface at one of the boundaries. The steady state PDE in the domain isAlong all of the non-reactive boundaries, we will set the concentration to unityOn the upper boundary at y = 1, we will have a first-order
This problem involves the solution of the diffusion equationwith no flux boundary conditions at x = 0 and x = 1 and a concentration c = 1 at y = 1. The end goal is to determine the concentration profile when the boundary condition at y = 0 is zero for x ∈ [0, 0.25] and x ∈ [0.5, 0.75] and
This problem takes a closer look at flow inside a square duct. Read this entire problem statement first. You should try to construct your programs in parts (a) and (b) in a way that makes it easier for you to solve part (c).(a) For flow inside a square duct the dimensionless velocity profile is
Compute the following sums using base 10 and the indicated mantissa:(a) 12.4235 + 0.2134 with a 3 bitmantissa(b) 1.029439 + 4.3284 with a 2 bitmantissa(c) 4.532938 + 0.004938 with a 5 bit mantissa
Use the rules of operator precedence to compute the output for the following calculations: (2-4) ^-2+5 8+3+2^ (4-2) 3 9-3^3+ (2-3)^5 4 8/4+3+2+3/6 12 in 5 8/4+ (3+2)+*3/6
Determine the output of the following calculation: 1 2 3 4 5 x = 0; for i = 1:2:5 X = x+1; end X
Determine the output of the following calculation: 1 A = zeros (3,3); 2 for 3 4 5 6 7 end A i= = 1:3 for j = 1:3 end A(i, j) i-j;
Determine the output of the sum 10.3 + 0.442 using 3 bits of precision.
What are the entries of the matrix produced by the following code: 1 2 3 4 5 for i = 1:3 for end end j = 1:1 A(i, j) = i*j;
Determine the output of the following calculation: 1 stop = 0; j = 2; while j < 9 2 3 4 j = j^2; 5 6 end j
What is the value of k at the end of this function? function k = probleml_34 1 2 stop_loop = 2; 3 k = 0; 4 while stop_loop > 1 5 k k+1; 6 stop_loop = 7 end stop_loop 0.25;
Determine the output of the following calculation: 1 2 3 4 5 8 10 11 k = 5; if k < 3 12 13 14 15 16 17 18 19 20 q else end 3; if k > 2 end elseif k == 4 q = 2; 8¹ a = 4; elseif k < 1 q = 5; else a = 6; if k = 7 q = 1; else end a = 0;
It is possible to obtain the value of π through the infinite sumIn this problem, you will explore the convergence of this summation for a finite number of terms N,Write a MATLAB function that computes the value of this sum up to a desired value of the residualThe latter expression is only valid
The goal of your problem is to write an m-file that generates a table in a text file with the trace of A × A† for a matrix A with the entriesYou must use for loops to (i) create the matrix A, (ii) construct the transpose A†, (iii) compute the product A × A†, and (iv) compute the trace of A
In this problem, you are going to look into the error due to the addition of small numbers and the subtraction of very similar numbers. You should start with the large number x = 1010. We want to look at different ways to add 1 to this number and then subtract x from the result. If the calculation
Consider the functionfor the concentration profile as a function of time. In practice, we need to truncate the sum with a finite number k terms,(a) Some functions are very difficult to represent as a Fourier series. The initial condition for this problem is one such function. Write a MATLAB program
Consider the sumWrite a MATLAB function file that computes the sum and makes a semilog-x plot of the value of this sum as a function of k for k = 1, 2, . . . , 1000. Your numerical solution should suggest the value of this sum for k =∞. Derive the value of the infinite sum. k 1 s=Σ n(n + 1) n=1
This problem involves analyzing the solution ofsubject to the initial condition T(x, 0) = 1 and the boundary conditions T(0, t) = 0 and T(1, t) = 0. The Fourier series for the temperature is(a) Write a program that prints to the screen a formatted table of the values of the temperature at t = 0.1
For the following piecewise function:(a) Express y(x) as a Fourier series,(b) Write aMATLAB program that computes the value of the Fourier series for n = 10, n = 100 and n = 1000 using a grid in x that is linearly space from [−1, 1] using 101 points. The program should automatically plot these
What is the condition number for the matrixwhich has the inverse A 2 = [ 1₂ 1/2 ₂] -2 (2.3.44)
Write the determinant of the following 3×3 matrix A as a sum of the determinants of 2 × 2 matrices using co-factor expansion on the third row (i = 3): A = 6 3 5 215 23 2
Express the determinant of the following matrix as a sum of 2 × 2 determinants using co-factor expansion on the first row. You do not need to evaluate the 2 × 2 determinants. -2 4 13 3 1 -1 -1 −1 1
Solve the following system of equations by Gauss elimination: 1 1 23 -1 1 3 1 1 5 3 7 -2 1 1 -5 X2 X3 Х4 = 10 31 -2 18 (2.5.19)
What is the determinant of the following matrix? 4-3 0 1 0 0 0 0 4 2 -0.25 0 2 1 3 10
Solve the following system of equations by Gauss elimination with partial pivoting: 2 -2 0 4 2 40 12 1 2 -4 0 4 1 X1 X2 X3 Х4 H 13 2 -20 (2.6.30)
Write the determinant of the following 3×3 matrix A as a sum of the determinants of 2 × 2 matrices using co-factor expansion on the second row (i = 2): A = 635 215 23 2
What is the bandwidth of the matrix? 1 1 0 0 1000 1 1 1 0 0 1 1 1 10 0 1 1 1 1 1 1 1
Compute the determinant of the following matrix by co-factor expansion: A = 6 1 -2 3 2-2 4 1 -1 1 0-2 ㅜㅜ 2 1
Compute the determinant of the following matrix by co-factor expansion: A = 2 1 -5 2 8 3 -4 2 11 2 11 -3 = 1 2 -1 2
Solve the systemby Jacobi’s method with an initial guess (0, 0). 2x1 + x₂ = 2 x12x₂ = -2 (2.13.7)
Use co-factor expansion to reduce the determinant ofto the sum of the fewest number of 4 × 4 matrices. 35 -1 -1 8 -4 2 1 0 1 -3 2 1 4 0 1 0 3 -7 2 3 4 3 4 1 2
Repeat the solution of Example 2.8 using Gauss–Seidel.Example 2.8Solve the systemby Jacobi’s method with an initial guess (0, 0). 2x1 + x₂ = 2 x12x₂ = -2 (2.13.7)
Does the following system have a unique solution, no solution, or infinitely many solutions? 4 2 -1 1 00 21 00 24 00 08 X1 X2 X3 X4 0 1 4
Compute the convergence of successive relaxation for the problem in Eq. (2.13.7) as a function of the relaxation parameter ω. 2x1 + x₂ = 2 x1 - 2x₂ = -2 (2.13.7)
Use co-factor expansion to prove that the determinant of an n×n upper triangular matrix U is the product of the entries along its diagonal, n det U = Uij i=1
Use Cramer’s rule to find the solution to X 13 {][]-[] y 8 Z 20 21 3 12 21 2 3 4
Use the built-in MATLAB solver to find the solution to 2 4 X1 BAGHA TH 4 1 2 2 -2 -3 X3 1 3 -2 (2.15.1)
Use Cramer’s rule to compute the solution to the system of equations 2x+y=z=1 -x+ 2z = 4 4x - 2y + 3z0
Write a MATLAB program to do LU decomposition for the matrix in Example 2.12.Data from Example 2.12:Use the built-in MATLAB solver to find the solution to 2 4 X1 BAGHA TH 4 1 2 2 -2 -3 X3 1 3 -2 (2.15.1)
Use Cramer’s rule to solve the system of equations 3x + 2y + z = 5 x-y + 2z = 5 x3z-1
Use Gauss elimination to find the solution to 4 3 W 0 -3 ][-] y 4 Z 0 2 3 1 1 0 2 02 35 4-2 3 1
Determine the value of the matrix element a(2)4,3 during the naive Gauss elimination of the matrix A A = 2 2 3 3 7 5 3 5 3 6 3 5 4 2 1 5 3 28 2 3 2 1 1 2 4 2 1 8 8 2 2 8 3 2 1
Cramer’s rule grows like n! with the size n of the matrix. In this problem, you will construct the key steps of the proof of this scaling. The limiting step for Cramer’s rule is to compute the determinants of the matrices. So you want to show that the effort to compute the determinant of a
Use Gauss elimination to solve the system of equations 2x1 X] 3x1 + - 3x2 + Х2 X2 - - X3 2x3 X3 7 -2 0
Use simultaneous Gauss elimination to solve the system of equations Ax = c with A 5 2 4 1 -1 -2 3 -3 for C₁ = [3 -3 5] and c₂ = [2 -2 8]¹
Use simultaneous Gauss elimination to solve the system of equations Ax = c withfor c1 = [1 0 0]T, c2 = [0 1 0]T, and c3 = [0 0 1]T. Based on your results, what is A−1? A = 5 6 13 42 -3 1 -6
At the current step in Gauss elimination, show the result after partial pivoting for the augmented matrix 1 27 2 4 0 2 3 1 8 f53 0341 0 4 1 8 4
Which two rows (if any) should be swapped in the current step of Gauss elimination with the following matrix: 8 2 05 -5 00 0 24 00 0-1 00 0 1 2 1 4 00 2 12 00 0 11 00 3-2 5 5 1 4 1 2 4 21 4-1 2 12 24 0 -3 1 2 1 2 12325 2 3 -1 4
Use Gauss elimination with partial pivoting to find the solution to 4 2 22 1 13 0 022 3 -1 3 2 3 W X y N -2 [B] 2 -1
Use Gauss elimination with pivoting to solve -1 3 3 -2 -3 2 10 5 -5 -3 2 3 1 -3 2 7 y 1 1 1 1 1 ][ Z ۲۵ دن درا 0 4 U W X || -5 11 1 14 4
The code below takes the matrix A and creates the upper triangular matrix U.We have modified the code so that it does banded Gauss elimination p = q when the function is given the value of p as an input argument.Determine the entires for B through H and L through N. 1 2 3 4 5 6 7 8 for
Solve the systemusing Gauss elimination with pivoting. Determine the solution x and the determinant of the original matrix, A. 3 3 - 1 -1 -2 4 2 4 5 4 1 2 0 3-2 3 1 X1 0 X2 8 ][-] X3 2 X4 14
Consider some matrix A. After using Gauss elimination with three partial pivoting steps, you obtain the upper triangular matrixWhat is the determinant of the original matrix A? U= 16 3 5 4 2 0 1 3 5 6 8 0 0 2 4 2 1 000138 000029 00000 1
What is the bandwidth of the following matrix? 82 3 -2 5 512 05 -5 24-4 1 332 NIN 1 00 0 24 212 00 0-1 4 -1 2 00 0 1 2 1 4 00 2 12 12 00 0 11 00 0-3 1 2 5 24-1 2 1 4
Perform LU decomposition on the matrix 1 -2 1 -1 2 4 -5 1 2
Compute the bandwidth of the following matrix: A= 635 00 0 1 0 0 2 4 2 1 1 0 1 2 3 0 0 1 0 0 0 000042 0 0 0 0 2 1
In LU decomposition, determine the missing value of L for 2 4 2 6 13 9 2 8 17 1 = [₁ 2 1 L = 00 1 ;] - [ 0 L3,2 1 2 62 1 4 003 0 X = X1 X2 X3
Use Gauss elimination with pivoting to find the determinant of A = 2 3 4 1 1 1 464
What is the bandwidth of the following matrix? 0 4 2 2 1 0 0 0 0 2 1 3 0 0 0 0 2 2 3 1 0 0 0 0 1 3 2 0 23 40 0000532 0 0 0 1 4 2 3 000
Compute the LU decomposition of A 2 = [²43] 6 13

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