View source for Multiple reactions - 2013

{{ClassSidebarYouTube
| date = 06 March to 14 March
| dates_alt_text = 
| vimeoID1 = yPH7JvEi1Jw
| vimeoID2 = SpLniff-kcw
| vimeoID3 = fX8v4189Riw
| vimeoID4 = 0W48zitUYVw
| vimeoID5 = MvRWk6R0oSc
| course_notes_PDF = 
| course_notes_alt = Course notes
| overheads_PDF = 
| assignment_instructions = 
| assignment_solutions = 
| video_download_link_MP4 = http://learnche.mcmaster.ca/media/3K4-2013-Class-08B-2.mp4
| video_download_link_MP4_size = 142 M
| video_notes1 =
| video_download_link2_MP4 = http://learnche.mcmaster.ca/media/3K4-2013-Class-08C.mp4
| video_download_link2_MP4_size = 367 M
| video_notes2 =
| video_download_link3_MP4 = http://learnche.mcmaster.ca/media/3K4-2013-Class-09A.mp4
| video_download_link3_MP4_size = 211 M
| video_notes3 =
| video_download_link4_MP4 = http://learnche.mcmaster.ca/media/3K4-2013-Class-09B.mp4
| video_download_link4_MP4_size = 331 M
| video_notes4 =
| video_download_link4_MP4 = http://learnche.mcmaster.ca/media/3K4-2013-Class-09C.mp4
| video_download_link5_MP4_size = 243 M
| video_notes5 =
}}__NOTOC__

== Textbook references ==

* F2011: Chapter 8
* F2006: Chapter 6

== Suggested problems ==

{| class="wikitable"
|-
! F2011
! F2006
|-
| Problem 8-12
| Problem 6-12
|-
| Problem 8-14 (covered in class)
| Problem 6-15 (covered in class)
|-
| Problem 8-18 (set up equations) 
| Problem 6-21 (set up equations) 
|}

==Class materials ==

=== 06 March 2013 (08B-2) ===

* [http://learnche.mcmaster.ca/media/3K4-2013-Class-08B-2.mp3 Audio] and [http://learnche.mcmaster.ca/media/3K4-2013-Class-08B-2.mp4 video] recording of the class.

=== 07 March 2013 (08C) ===

* [http://learnche.mcmaster.ca/media/3K4-2013-Class-08C.mp3 Audio] and [http://learnche.mcmaster.ca/media/3K4-2013-Class-08C.mp4 video] recording of the class.

Polymath code for example in class. Make sure you plot the instantaneous selectivity, overall selectivity and yield over time. Compare these 3 plots during the batch to understand what each of these 3 variables mean.
<syntaxhighlight lang="text">
# ODEs
d(CA) / d(t) = -k1*CA 
d(CB) / d(t) = k1*CA - k2*CB
d(CC) / d(t) = k2*CB

# Initial conditions
CA(0) = 2 # mol/L
CB(0) = 0 # mol/L
CC(0) = 0 # mol/L

# Algebraic equations
k1 = 0.5 # 1/hr
k2 = 0.2 # 1/hr

# The 3 important algebraic variables: plot these 3 against time and interpret them.
S_DU = if (t>0.001) then (k1*CA - k2*CB) / (k2*CB)  else 0
Overall_SDU = if (t>0.001) then CB/CC else 0
Yield = if (t>0.001) then CB / (2 - CA) else 0

# Independent variable details
t(0) = 0
t(f) = 3.1  # hours
</syntaxhighlight>

=== 11 March 2013 (09A) ===

* [http://learnche.mcmaster.ca/media/3K4-2013-Class-09A.mp3 Audio] and [http://learnche.mcmaster.ca/media/3K4-2013-Class-09A.mp4 video] recording of the class.

=== 13 March 2013 (09B) ===

* [http://learnche.mcmaster.ca/media/3K4-2013-Class-09B.mp3 Audio] and [http://learnche.mcmaster.ca/media/3K4-2013-Class-09B.mp4 video] recording of the class.
Code for the CSTR example:
<syntaxhighlight lang="matlab">
tau = 0:0.05:10;

CA0 = 2;  % mol/L
k1 = 0.5; % 1/hr
k2 = 0.2; % 1/hr

CA = CA0 ./ (1 + k1 .* tau);
CB = tau .* k1 .* CA ./ (1 + k2 .* tau);
CC = tau .* k2 .* CB;

instant_selectivity = (k1.*CA - k2.*CB) ./ (k2.*CB);
overall_selectivity = CB ./ CC;
overall_yield = CB ./ (CA0 - CA);
conversion = (CA0 - CA)./CA0;

plot(tau, CA, tau, CB, tau, CC)
grid on
xlabel('\tau')
ylabel('Concentrations [mol/L]')

figure 
plot(tau, overall_selectivity)
xlabel('\tau')
ylabel('Overall Selectivity')
grid on

figure 
plot(tau, overall_yield)
xlabel('\tau')
ylabel('Overall Yield')
grid on

figure
plot(tau, conversion)
xlabel('\tau')
ylabel('Conversion')
hold on
grid on
</syntaxhighlight>

=== 14 March 2013 (09C) ===
* [http://learnche.mcmaster.ca/media/3K4-2013-Class-09C.mp3 Audio] and [http://learnche.mcmaster.ca/media/3K4-2013-Class-09C.mp4 video] recording of the class.

Despite the fact that Polymath code is shorter to write, ''I still recommend you use MATLAB or Python''. For example, comparing two simulations and generating plots is so much easier in MATLAB than Polymath.
{| class="wikitable"
|-
! MATLAB
! Polymath
|-
| 
'''<tt>pfr.m</tt>'''
<syntaxhighlight lang="matlab">
function d_depnt__d_indep = pfr(indep, depnt)
% Dynamic balance for the reactor
% 
%    indep: the independent ODE variable, such as time or length
%    depnt: a vector of dependent variables
% 
%    Returns d(depnt)/d(indep) = a vector of ODEs
 
% Assign some variables for convenience of notation
FA = depnt(1);
FB = depnt(2);
FC = depnt(3);
FD = depnt(4);
FE = depnt(5);
FG = depnt(6);
FW = depnt(7);
y  = depnt(8);
 
% Constant(s)
k1 = 0.014;    % L^{0.5} / mol^{0.5} / s
k2 = 0.007;    % L/(mol.s)
k3 = 0.14;     % 1/s
k4 = 0.45;     % L/(mol.s)
alpha = 0.002; % 1/L
CT0 = 1.0;     % mol/L
FA0 = 10;      % mol/s
FB0 = 5.0;     % mol/s
FT0 = FA0 + FB0;

FT = FA + FB + FC + FD + FE + FW + FG;

CA = CT0 * FA/FT * y;
CB = CT0 * FB/FT * y;
CC = CT0 * FC/FT * y;
CD = CT0 * FD/FT * y;
CE = CT0 * FE/FT * y;
CG = CT0 * FG/FT * y;
CW = CT0 * FW/FT * y;

% Reaction 1: A + 0.5B -> C
r1A = -k1*(CA)*(CB)^(0.5);
r1B = 0.5*r1A;
r1C = -r1A;

%# Reaction 2: 2A -> D
r2A = -k2*(CA)^2;
r2D = -r2A/2;

% Reaction 3: C -> E + W
r3C = -k3*(CC);
r3E = -r3C;
r3W = -r3C;

% Reaction 4: D + W -> G + C
r4D = -k4*(CD)*(CW);
r4W = r4D;
r4G = -r4D;
r4C = -r4D;

% Output from this ODE function must be a COLUMN vector, with n rows
n = numel(depnt);
d_depnt__d_indep = zeros(n,1);
d_depnt__d_indep(1) = r1A + r2A;
d_depnt__d_indep(2) = r1B;
d_depnt__d_indep(3) = r1C + r3C + r4C;
d_depnt__d_indep(4) = r2D + r4D;
d_depnt__d_indep(5) = r3E;
d_depnt__d_indep(6) = r3W + r4W;
d_depnt__d_indep(7) = r4G;
d_depnt__d_indep(8) = -alpha/(2*y) * (FT / FT0);
</syntaxhighlight>
'''<tt>ODE_driver.m</tt>'''
<syntaxhighlight lang="matlab">
% Integrate the ODE
% -----------------
 
% The independent variable: requires an initial and final value:
indep_start = 0.0;   % kg
indep_final = 500.0; % kg
 
% Set initial condition(s) for dependent variables
FA_depnt_zero = 10.0;   % i.e. FA(W=0) = 10.0
FB_depnt_zero = 5.0;    % i.e. FB(W=0) = 10.0
FC_depnt_zero = 0.0;    % i.e. FC(W=0) = 10.0
FD_depnt_zero = 0.0;    % etc
FE_depnt_zero = 0.0;
FG_depnt_zero = 0.0;
FW_depnt_zero = 0.0;
y_depnt_zero = 1.0;     % i.e. y(W=0) = 1.0
 
IC = [FA_depnt_zero, FB_depnt_zero, FC_depnt_zero, FD_depnt_zero, ...
      FE_depnt_zero FG_depnt_zero, FW_depnt_zero, y_depnt_zero];
  
% Integrate the ODE(s):
[indep, depnt] = ode45(@pfr, [indep_start, indep_final], IC);
 
% Calculate Yields and Selectivities
FA = depnt(:,1);
FC = depnt(:,3);
FD = depnt(:,4);
FE = depnt(:,5);

Yield_C = FC ./ (FA_depnt_zero - FA);
S_CE = FC./FE;
S_CD = FC./FD;

% Plot the results:
clf;
plot(indep, depnt(:,1), indep, depnt(:,2), indep, depnt(:,3), ...
     indep, depnt(:,4), indep, depnt(:,5), indep, depnt(:,6), ...
     indep, depnt(:,7), indep, depnt(:,8))
grid('on')
hold('on')
plot(indep, depnt(:,2), 'g')
xlabel('Catalyst weight, W [kg]')
ylabel('Concentrations and pressure drop')
legend('FA', 'FB', 'FC', 'FD', 'FE', 'FG', 'FW', 'y')
</syntaxhighlight>
| valign="top"|
<syntaxhighlight lang="text">
k1 = 0.014    # L^{0.5} / mol^{0.5} / s
k2 = 0.007    # L/(mol.s)
k3 = 0.14     # 1/s
k4 = 0.45     # L/(mol.s)
alpha = 0.002 # 1/L
CT0 = 1.0     # mol/L
FA0 = 10      # mol/s
FB0 = 5.0     # mol/s
FT0 = FA0 + FB0

# Concentration functions (isothermal conditions)
CA = CT0 * FA/FT * y
CB = CT0 * FB/FT * y
CC = CT0 * FC/FT * y
CD = CT0 * FD/FT * y
CE = CT0 * FE/FT * y
CG = CT0 * FG/FT * y
CW = CT0 * FW/FT * y

FT = FA + FB + FC + FD + FE + FW + FG

# Reaction 1: A + 0.5B -> C
r1A = -k1*(CA)*(CB)^(0.5)
r1B = 0.5*r1A
r1C = -r1A

# Reaction 2: 2A -> D
r2A = -k2*(CA)^2
r2D = -r2A/2

# Reaction 3: C -> E + W
r3C = -k3*(CC)
r3E = -r3C
r3W = -r3C

# Reaction 4: D + W -> G + C
r4D = -k4*(CD)*(CW)
r4W = r4D
r4G = -r4D
r4C = -r4D

# ODE's
d(FA) / d(W) = r1A + r2A
FA(0) = 10

d(FB) / d(W) = r1B
FB(0) = 5

d(FC) / d(W) = r1C + r3C + r4C
FC(0) = 0

d(FD) / d(W) = r2D + r4D
FD(0) = 0

d(FE) / d(W) = r3E
FE(0) = 0

d(FW) / d(W) = r3W + r4W
FW(0) = 0

d(FG) / d(W) = r4G
FG(0) = 0

W(0) = 0    # kg
W(f) = 500 # kg

Yield_C = if (W>0.001) then (FC / (FA0 - FA)) else (0)
S_CE = if (W>0.001) then (FC/FE) else (0)
S_CD = if (W>0.001) then (FC/FD) else (0)

d(y) / d(W) = -alpha/(2*y) * (FT / FT0)
y(0) = 1.0
</syntaxhighlight>
|}