psod lecture 2. matrices and vectors in chemical & process engineering appear in calculations...

PSODPSOD

Lecture 2Lecture 2

MatricesMatrices and vectors and vectors in Chemical in Chemical & Process& Process Engineering Engineering

Appear in calculations when process is Appear in calculations when process is described by the system of equations:described by the system of equations:– Piping systemPiping system– Cascade ofCascade of

» ReactorsReactors

» Heat exchangersHeat exchangers

» Mixers Mixers

– System of System of apparatus and streams inapparatus and streams in chemical chemical plantplant

concentrations give 4-elements vector concentrations give 4-elements vector cc To find solution we need system of 4 To find solution we need system of 4

equationsequations Equation parameters creates square matrixEquation parameters creates square matrix

Matrices in ChemicalMatrices in Chemical & Process & Process EngineeringEngineering

L, c1

L, cs

L, c2

L, c3

L, c4

V

V

V

V

V, c

1

V, c

s

V, c

2

V, c

3

V, c

4


)(reactionsourceoutputinput

kVcLcLcs 11

221 kVcLcLc

332 kVcLcLc

443 kVcLcLc

sLcccckVLc 4321 000

000 4321 ccckVLLc

000 4321 cckVLLcc

000 4321 ckVLLccc

V, c

1

V, c

s

V, c

2

V, c

3

V, c

4


kVLL

kVLL

kVLL

kVL

00

00

00

000

4

3

2

1

c

c

c

c

0

0

0sLc

MathCAD – vectors and matrixMathCAD – vectors and matrix

MathCAD – vectors and matrixMathCAD – vectors and matrix Matrix operationsMatrix operations

– Multiply by constantMultiply by constant– Matrix transpose [ctrl]+[1]Matrix transpose [ctrl]+[1]– Inverse [^][-][1]Inverse [^][-][1]– Matrix multiplyingMatrix multiplying– DeterminantDeterminant

To read the matrix elements ATo read the matrix elements Ar, kr, k: key [[] r- : key [[] r-

row row nrnr, k – column , k – column nrnr– e.g. element Ae.g. element A1,11,1 keystrokes: [A] keystrokes: [A][[][[][1][,][1][=][1][,][1][=]

To chose matrix columnTo chose matrix column:: MM<col.nr><col.nr>

– First column A( AFirst column A( A<0><0>):):keys [A][ctrl]+[6][0]keys [A][ctrl]+[6][0]

Default first columnDefault first column&row&row number is 0, number is 0, – (to change : Math/Options/Array Or(to change : Math/Options/Array Oriigin)gin)



Calculations of dot product and cross Calculations of dot product and cross product of vectorsproduct of vectors

Special definition of matrix elements as a Special definition of matrix elements as a function of row-column number function of row-column number MMi,ji,j==ff((i,ji,j))

– E.g. Value of element is equal to product of E.g. Value of element is equal to product of column and row numbercolumn and row number


Constrain: function arguments have to be integer

MathCAD 3D graphsMathCAD 3D graphs 3D graphs3D graphs of function of function on the base of matrix : on the base of matrix : [ctrl][ctrl]

+[2]+[2] [M] [M] – M – matrix defined earlierM – matrix defined earlier

3D Graphs of function of real type 3D Graphs of function of real type argumentsarguments

– Using procedure: CreateMesh(function, Using procedure: CreateMesh(function, lb_v1, ub_v1, lb_v2, ub_v2lb_v1, ub_v1, lb_v2, ub_v2, v1grid, v2grid, v1grid, v2grid))

– Assign result to variableAssign result to variable– Plot of the variable Plot of the variable is is similarsimilar to to plot of plot of

matrix ([ctrl]+[2])matrix ([ctrl]+[2])

MathCAD 3D graphsMathCAD 3D graphs

Boundaries can be the real numbers. (def. –5,5)

Grids have to be integer numbers (def. 20)

MathCAD 3D graphsMathCAD 3D graphs

MathCAD 3D graphs - formatingMathCAD 3D graphs - formating

MathCAD 3D graphs – formatting: fill optionsMathCAD 3D graphs – formatting: fill options

MathCAD 3D graphs – formatting: fill optionsMathCAD 3D graphs – formatting: fill options

Contours colour filled

MathCAD 3D graphs – formatting: MathCAD 3D graphs – formatting: lineline options options

MathCAD 3D graphs – formatting: MathCAD 3D graphs – formatting: LightingLighting

MathCAD 3D graphs – formatting: Fog and MathCAD 3D graphs – formatting: Fog and perspectiveperspective

MathCAD 3D graphs – formatting: MathCAD 3D graphs – formatting: BBackplane and ackplane and Grids Grids

MathCAD 3DMathCAD 3D scatter scatter graphs graphs

Data given as three Data given as three vectors of each point vectors of each point coordinates coordinates – Equal vector sizeEqual vector size– Button on Graph Button on Graph

toolbar: 3D Scatter toolbar: 3D Scatter PlotPlot

– In the placeholder In the placeholder type in brackets the type in brackets the vectors names vectors names separated by comasseparated by comas

Predefined constantsPredefined constants

e = 2,718 – natural logarithm basee = 2,718 – natural logarithm base g = 9,81 m/sg = 9,81 m/s22 – acceleration of gravity – acceleration of gravity = 3,142 – circle perimeter/diameter ratio= 3,142 – circle perimeter/diameter ratio

Solving of algebraic equation Solving of algebraic equation

When equation is implicitWhen equation is implicit When we don’t want to separate variablesWhen we don’t want to separate variables

MathCAD equation solvMathCAD equation solversers

Single equation (one unknown value)Single equation (one unknown value)1.1. Given-Find methodGiven-Find method

» Input start point of variableInput start point of variable» Type "Given"Type "Given"» Type equation with using [Type equation with using [==]] ([ctrl]+[=]) ([ctrl]+[=])» Type Find(variable)=Type Find(variable)=

MathCAD MathCAD equation solvingequation solving

Given-Find – solving methods Given-Find – solving methods – Linear (function of type y=cLinear (function of type y=c00x + cx + c11) –starting ) –starting

point choice do not affects on results.point choice do not affects on results.– Nonlinear – according to nonlinear equation. Nonlinear – according to nonlinear equation.

Obtained result could depend on starting Obtained result could depend on starting point. Available methods: point. Available methods:

» Conjugate Gradient Conjugate Gradient » Quasi – NewtonQuasi – Newton» Levenberg-MarquardtLevenberg-Marquardt» Quadratic Quadratic

The choice of method is automatic by The choice of method is automatic by default. User can choose method from the default. User can choose method from the pop-up menu over word Find.pop-up menu over word Find.

Single equation (one unknown value)Single equation (one unknown value)2.2. Root procedure:Root procedure:

Root(function, variable, low_Root(function, variable, low_limitlimit, up_, up_limitlimit)=)=– Values of function at the bounds must have different signsValues of function at the bounds must have different signs

or

MathCAD equation solvingMathCAD equation solving

Single equation (one unknown value)Single equation (one unknown value)2.2. Root procedureRoot procedure

methods:methods:1.1. Secant methodSecant method2.2. Mueller method Mueller method (2(2ndnd order polynomial) order polynomial)


x3x2

y3x1

y1

y2

x4

x5

32

32224 yy

xxyxx

ii

iiiii yy

xxyxx

1

1111

Single equation (one unknown value)Single equation (one unknown value)3.3. Special procedure: polyroots for the Special procedure: polyroots for the

polynomialspolynomials.. Argument of procedure is a Argument of procedure is a vector of polynomial coefficients (avector of polynomial coefficients (a00, a, a11...). ...).

The result is a vector too.The result is a vector too.


Methods:1. Laguerre's method2. companion matrix

Laguerre's method

k

k

xp

xpG

Polynomial p(x) of degree n. Starting from assumed xk.

k

k

xp

xpGH

2

21 GnHnG

na

axx kk 1

The system of linear equationsThe system of linear equations– Solving on the base of matrix Solving on the base of matrix toolbartoolbar::

» Prepare square matrix of equations coefficients Prepare square matrix of equations coefficients (A) and vector of free terms (B)(A) and vector of free terms (B)

» Do the operation x:=ADo the operation x:=A-1-1BB and show result: x= and show result: x=

OrOr

» Use the procedure LSOLVE: lsolve(A,B)=Use the procedure LSOLVE: lsolve(A,B)=

MathCAD, the system of equations solvingMathCAD, the system of equations solving

The system of nonlinear equationThe system of nonlinear equation– Can be solved using given-find methodCan be solved using given-find method

» Assign starting values to variablesAssign starting values to variables

» Type GivenType Given

» Type the equationType the equationss using using == sign sign ((boldbold) )

» Type Find(var1, var2,...)=Type Find(var1, var2,...)=


Differential eq. Solvers in Differential eq. Solvers in MathCADMathCAD

Ordinary differential equations Ordinary differential equations solvingsolving

Numerical methods:Numerical methods:– Gives only values not functionGives only values not function– Engineer usually needs values Engineer usually needs values – There is no need to make complicated There is no need to make complicated

transformations (e.g. variables separation)transformations (e.g. variables separation)– Basic method implemented in MathCAD is Basic method implemented in MathCAD is

Runge-Kutta 4Runge-Kutta 4thth order method. order method.


Numerical methods principleNumerical methods principle– Calculation involve bounded Calculation involve bounded rangerange of of

independent variable onlyindependent variable only– Every point is being calculated on the base of Every point is being calculated on the base of

one or few points calculated before or givenone or few points calculated before or given starting points.starting points.

– Independent variable is calculated using step:Independent variable is calculated using step:

xxi+i+11 = x = x i i + h = x + h = xii++xx– Dependent value is calculated according to the Dependent value is calculated according to the

methodmethodyyi+i+11 = = yy i i + +yy= = yy i i + +KKii


Runge-Kutta 4Runge-Kutta 4thth order method principles: order method principles:– New point of the integral is calculated on the New point of the integral is calculated on the

base of one point (given/calculated earlier) and base of one point (given/calculated earlier) and 4 intermediate values 4 intermediate values

51

4321

34

23

12

1

26

1

,

2

1,

2

1

2

1,

2

1

,

hOKyy

kkkkK

kyhxhFk

kyhxhFk

kyhxhFk

yxhFk

ii

ii

ii

ii

ii

MathCAD differential equationsMathCAD differential equations

Single, first order differential equationSingle, first order differential equation

1.1. Assign the initial value of dependent variable Assign the initial value of dependent variable (optionally(optionally))

2.2. Define the derivative functionDefine the derivative function

3.3. Assign to the new variable the integrating function Assign to the new variable the integrating function rkfixed:rkfixed:

R:=rkfixed(init_v, low_bound, up_bound, num_seg, function)R:=rkfixed(init_v, low_bound, up_bound, num_seg, function)

),( yxfdx

dy 00 0

, yyxx xx Initial

condition

4.4. Result is matrix (table) of two columns: first Result is matrix (table) of two columns: first contain independent values second dependent onescontain independent values second dependent ones

5.5. To show result as a plot: RTo show result as a plot: R<1><1>@R@R<0><0>

NN y

y

y

y

x

x

x

x

R

,1

2,1

1,1

0,1

2

1

0

......

MathCADMathCAD,, differential equations differential equations

System of first order differential equationsSystem of first order differential equations

1.1. Assign the vector of initial conditions of dependent Assign the vector of initial conditions of dependent variables (starting vector)variables (starting vector)

2.2. Define the Define the vectorvector function of derivatives (right-hand function of derivatives (right-hand sides of equations)sides of equations)

3.3. Assign to the variable function rkfixed:Assign to the variable function rkfixed:R:=rkfixed(init_vect, low_bound, up_bound, num_seg, function)R:=rkfixed(init_vect, low_bound, up_bound, num_seg, function)

101

100

,,

,,

yyxfdx

dy

yyxfdx

dy

011

000

0

0

0

yy

yy

xx

xx

xx


4.4. Result is matrix (table) of three columns: Result is matrix (table) of three columns: first contain independent values, 2first contain independent values, 2ndnd column column containscontains first dependent first dependent variable variable values, values, third second ones :third second ones :

5.5. Results as a plot: RResults as a plot: R<1><1>,R,R<2><2>@ R@ R<0><0>

NNN y

y

y

y

y

y

y

y

x

x

x

x

R

,2

2,2

1,2

0,2

,1

2,1

1,1

0,1

2

1

0

.........


011

000

0

0

0

yy

yy

xx

xx

xx


Single second order equationSingle second order equation

1.1. Transform the second order equation to the Transform the second order equation to the system of two first order equations:system of two first order equations:

dx

dyyxf

dx

yd,,

2

2

0

00

0

0,

ydx

dy

yyxx

xx

xx

Initial

condition

dx

dz

dx

ydz

dx

dz

dx

dyzy 1

2

2

10

0 , ,

101

10

,, zzxfdx

dz

zdx

dz

011

000

0

0

0

zz

zz

xx

xx

xx



Example:Example:Solve the second order differential equation Solve the second order differential equation

(calculate(calculate:: values of function values of function and itsand its first first derivatives) given by equation:derivatives) given by equation:

While While yy=10 and =10 and yy’=-1 for ’=-1 for xx=0=0In the range of In the range of xx=<0,1>=<0,1>

yyxdx

yd 322

2


System of equationsStarting vector Vectoral function

psod lecture 2. matrices and vectors in chemical & process engineering appear in calculations...

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