ENGR-25_Programming-1.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Registered Electrical

[email protected] • ENGR-25_Programming-1.ppt1

Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods

Bruce Mayer, PERegistered Electrical & Mechanical Engineer

[email protected]

Engr/Math/Physics 25

Prob 5.47, Prob 5.47, 5.575.57

TutorialTutorial



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



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



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



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)



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)



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

140

160

180

200

220

240

260

280

t

2nd

Ord

=>1

/C

1/C = 0.5445*t + 99.61

data Line PolyFitdata Pts



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



The Distance CalcsThe Distance Calcs Using Pythagorean Theorem

222211 3.0 yxdydxr

22

2

222222

3.0

3.0

yxr

yxdydxr



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



Meshc Plot MeshGridMeshc Plot MeshGrid

-0.4-0.2

00.2

0.4

-0.4

-0.2

0

0.2

0.40

20

40

60

80

X-distance

2 Point-Charges Electical Field

Y-distance

Ele

ct. P

oten

tial (

V)



surf Plot by MeshGridsurf Plot by MeshGrid

-0.4-0.2

00.2

0.4

-0.4

-0.2

0

0.2

0.410

20

30

40

50

60

70

80

X-distance


Y-distance

Ele

ct. P

oten

tial (

V)



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



meshc Plot by Loopmeshc Plot by Loop

-0.4-0.2

00.2

0.4

-0.4

-0.2

0

0.2

0.410

20

30

40

50

60

70

80

X-distance


Y-distance

Ele

ct. P

oten

tial (

V)

Note that plot is TURNED



surf Plot by Loopsurf Plot by Loop

-0.4-0.2

00.2

0.4

-0.4

-0.2

0

0.2

0.410

20

30

40

50

60

70

80

X-distance


Y-distance

Ele

ct. P

oten

tial (

V)

Note that plot is TURNED

Documents

ENGR-25_Programming-1.ppt 1 Bruce Mayer, PE Engineering/Math/Physics 25: Computational Methods Bruce Mayer, PE Registered Electrical