engr-25_programming-1.ppt 1 bruce mayer, pe engineering/math/physics 25: computational methods bruce...
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ENGR-25_Programming-1.ppt 3 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Prob Solve 1st Step → PLOT it Advice for Every Engineer and Applied Mathematician or Physicist: RRule-1: When in Doubt PLOT IT! RRule-2: If you don’t KNOW when to DOUBT, then PLOT EVERYTHINGTRANSCRIPT
[email protected] • ENGR-25_Programming-1.ppt1
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Bruce Mayer, PERegistered Electrical & Mechanical Engineer
Engr/Math/Physics 25
Prob 5.47, Prob 5.47, 5.575.57
TutorialTutorial
[email protected] • ENGR-25_Programming-1.ppt2
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Problem 5.47 Problem 5.47 Chemical Rcn Order Chemical Rcn Order 1st Order Rate Eqn is Expontnential
mxkt beyeCtC 0• By SemiLog Linearization we can “Discover”
parameters [m & b] [C(0) & −k 2nd Order Eqn can
be LINEARIZED as 011C
kttC
222 BmXY
• Thus ANOTHERLinearizable Fcn
bmxy 11
[email protected] • ENGR-25_Programming-1.ppt3
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Prob Solve 1st Step → PLOT itProb Solve 1st Step → PLOT it Advice for Every Engineer and Applied
Mathematician or Physicist:
Rule-1: When in Doubt PLOT IT!
Rule-2: If you don’t KNOW when to DOUBT, then PLOT EVERYTHING
[email protected] • ENGR-25_Programming-1.ppt4
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Prob 5.47Prob 5.47 When in Doubt PLOT
0 50 100 150 200 250 300-5.6
-5.4
-5.2
-5
-4.8
-4.6
t
1st O
rd =
> ln
(C)
0 50 100 150 200 250 300100
150
200
250
300
t
2nd
Ord
=>1
/C
Some CURVATURE
Straight• Better Model• t X• 1/C Y
[email protected] • ENGR-25_Programming-1.ppt5
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Linear Xform of 2Linear Xform of 2ndnd Order Reaction Order Reaction Now that the plot
has identified the Rcn as 2nd Order, Make Linear Xform
The 2nd Order Eqn
011C
kttC
222 BmXY
Use polyfit of order-1 to generate fitting parameters contained in vector k_1overC0
That is: k_1overC0 = [m, B2]; or• k_1overC0(1)
= m = k• k_1overC0(2)
= B2 = 1/C(0)
[email protected] • ENGR-25_Programming-1.ppt6
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
The 2The 2 ndnd O
rder Model
Order M
odel% Bruce Mayer, PE * 04Nov06% ENGR25 % file = Prob5_47_Chem_Concentration_0611.m% Find the Order of the chemical reaction%% CLEAR out any previous runsclear%% The Data Vectorst = [0,50,100,200,300];C = [0.01,0.0079,0.0065,0.0048,0.0038];%% WHEN IN DOUBT => PLOT%% in this plot vs t to reveal Rcn Order: ln(C) & 1/C%%% the Xformed DataVectors for RCN ORDERCfirst = log(C);Csecond = 1./C;%% Check which one gives straight linesubplot(2,1,1)plot(t,Cfirst,t,Cfirst,'*'), xlabel('t'), ylabel('1st Ord => ln(C)'), gridsubplot(2,1,2)plot(t,Csecond,t,Csecond,'o'), xlabel('t'), ylabel('2nd Ord =>1/C'), grid%% After Comparing two curves, 2nd order gives much straighter line%% use PolyFit to fit to 1/C(t)= k*t + 1/C0 => Y = mX + B%% Xform to Line => 1/C => Y, t => X, k => m, 1/C0 => B% Calc k & C0 showing in scientific notationformat short ek_1overC0 = polyfit(t,Csecond,1)k = k_1overC0(1)C0 = 1/k_1overC0(2)
[email protected] • ENGR-25_Programming-1.ppt7
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
P 5.47 AnswerP 5.47 Answer k_1overC0 = [m B2] = [k 1/C0] =
[5.4445e-001 9.9605e+001] k = 5.4445e-001
• k = 0.54445 C0 =
1/9.9605e+001 1.0040e-002
• C = 0.01004 0 50 100 150 200 250 30080
100
120
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200
220
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t
2nd
Ord
=>1
/C
1/C = 0.5445*t + 99.61
data Line PolyFitdata Pts
[email protected] • ENGR-25_Programming-1.ppt8
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
P5.57 GeometryP5.57 Geometry
x
y
r2r2r1 dy
dx1
dx2
Point at (x,y)
x-0.3
x-(-0.3) = x+0.3
2
2
1
1
041
rq
rqV
E-Field Governing Equation
[email protected] • ENGR-25_Programming-1.ppt9
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
The Distance CalcsThe Distance Calcs Using Pythagorean Theorem
222211 3.0 yxdydxr
22
2
222222
3.0
3.0
yxr
yxdydxr
[email protected] • ENGR-25_Programming-1.ppt10
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
The MeshG
rid PlotThe M
eshGrid Plot
% Bruce Mayer, PE * 04Nov06% ENGR25 % file = Prob5_57_Point_Charges_meshgrid_Plot_0611.m% Surface Plot eField from two Point Charges%% CLEAR out any previous runsclear% The Constant Parametersq1 = 2e-10; q2 = 4e-10; % in Coulombsepsilon = 8.854e-12; % in Farad/m%% Note the distances, r1 & r2, to any point(x,y) in the field by pythagorus% * r1 = sqrt((x-0.3)^2 + y^2)% * r2 = sqrt((x+0.3)^2 + y^2)%% Construct a 25x25 mesh[X Y] = meshgrid(-0.25:0.010:0.25);%% find r1 & r2 by pythagorus and array-opsr1 = sqrt((X-0.3).^2 +Y.^2); % note dots used with array operationr2 = sqrt((X-(-0.3)).^2 +Y.^2); % note dots used with array operation% use vectors r1 & r2, and array ops to find VV = (1/(4*pi*epsilon))*(q1./r1 + q2./r2);%% use %-Comment to toggle between SURF & MESHC plots% surf(X,Y,V), xlabel('X-distance'), ylabel('Y-distance'),... zlabel('Elect. Potential (V)'), title('2 Point-Charges Electical Field'),... grid onmeshc(X,Y,V), , xlabel('X-distance'), ylabel('Y-distance'),... zlabel('Elect. Potential (V)'), title('2 Point-Charges Electical Field'),... grid on
[email protected] • ENGR-25_Programming-1.ppt11
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
Meshc Plot MeshGridMeshc Plot MeshGrid
-0.4-0.2
00.2
0.4
-0.4
-0.2
0
0.2
0.40
20
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60
80
X-distance
2 Point-Charges Electical Field
Y-distance
Ele
ct. P
oten
tial (
V)
[email protected] • ENGR-25_Programming-1.ppt12
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
surf Plot by MeshGridsurf Plot by MeshGrid
-0.4-0.2
00.2
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0.410
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X-distance
2 Point-Charges Electical Field
Y-distance
Ele
ct. P
oten
tial (
V)
[email protected] • ENGR-25_Programming-1.ppt13
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
The Loop PlotThe Loop Plot
% Bruce Mayer, PE * 04Nov06 * ENGR25 % file = Prob5_57_Point_Charges_Loop_Plot_0611.m% Surface Plot eField from two Point ChargesClear % CLEAR out any previous runs% The Constant Parametersq1 = 2e-10; q2 = 4e-10; % in Coulombsepsilon = 8.854e-12; % in Farad/m% Note the distances, r1 & r2, to any point(x,y) in the field by pythagorus% * r1 = sqrt((x-0.3)^2 + y^2)% * r2 = sqrt((x+0.3)^2 + y^2)% Build up From Square XY Plane%% r1 goes to q1 at (0.3,0)%% r2 goes to q2 at (-0.3,0)x = linspace(-.25, .25, 50); %50 pts over x-range y = linspace(-.25, .25, 50); %50 pts over y-range for k = 1:length(x) for m = 1:length(y) % calc r1 & r2 using pythagorus r1 = sqrt((x(k)-0.3)^2 + y(m)^2); r2 = sqrt((x(k)-(-0.3))^2 + y(m)^2); % Find V based on r1 and r1 V(k,m) = (1/(4*pi*epsilon))*(q1/r1 +q2/r2); % Note that V is a 2D array using the x & y indices endendX = x;Y = y;% use %-Comment to toggle between SURF & MESHC plotssurf(X,Y,V), xlabel('X-distance'), ylabel('Y-distance'),... zlabel('Elect. Potential (V)'), title('2 Point-Charges Electical Field'),... grid on%meshc(X,Y,V), , xlabel('X-distance'), ylabel('Y-distance'),... zlabel('Elect. Potential (V)'), title('2 Point-Charges Electical Field'),... grid on
[email protected] • ENGR-25_Programming-1.ppt14
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
meshc Plot by Loopmeshc Plot by Loop
-0.4-0.2
00.2
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0.410
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X-distance
2 Point-Charges Electical Field
Y-distance
Ele
ct. P
oten
tial (
V)
Note that plot is TURNED
[email protected] • ENGR-25_Programming-1.ppt15
Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods
surf Plot by Loopsurf Plot by Loop
-0.4-0.2
00.2
0.4
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-0.2
0
0.2
0.410
20
30
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80
X-distance
2 Point-Charges Electical Field
Y-distance
Ele
ct. P
oten
tial (
V)
Note that plot is TURNED