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Implement Mathematical Operations - Computational Methods - Lecture Slides, Slides of Calculus for Engineers

National Institute of Industrial Engineering Calculus for Engineers

These are the Lecture Slides of Computational Methods which includes Thévenin’s Equivalent Circuit, Circuit Simplification, Analysis of Power Transfer, Voltage Division, Analytical Game Plan, Array Operation, Element Operations, Number of Allowable Values etc.Key important points are: Implement Mathematical Operations, Simulations of Dynamic Control, Integrator Parameters, Atmospheric Pressures, Cascading Tanks, Valve Resistance, Square-Root Relation, Fluid Density, Complicated Integrand

Typology: Slides

2012/2013

Uploaded on 03/26/2013

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Learning Goals

•Implement Mathematical Operations in

MATLAB using SimuLink Functional Blocks

•Employ FeedBack in the SimuLink

Environment to numerically Solve ODEs

•Create Simulations of Dynamic Control

Systems using SimuLink Block Models

–Export Simulation result to MATLAB WorkSpace

for Further Analysis

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Discover Slides of Calculus for Engineers National Institute of Industrial Engineering

Partial preview of the text

Download Implement Mathematical Operations - Computational Methods - Lecture Slides and more Slides Calculus for Engineers in PDF only on Docsity!

Learning Goals

Implement Mathematical Operations in

MATLAB using SimuLink Functional Blocks

Employ FeedBack in the SimuLink

Environment to numerically Solve ODEs

Create Simulations of Dynamic Control

Systems using SimuLink Block Models

Export Simulation result to MATLAB WorkSpace

for Further Analysis

Problem 10.30 (1)

Make A subsystem

Block for

 It has been found

that for many Valves

the Flow Thru the

valve is Related to

the Pressure Drop

 Inputs

q (kg/s)
P (^) l & P (^) r (Pa)
R (^) l & R (^) r ([√ΔP]/[kg/s])

q ∝ Phi − Plo = ∆ P

 Using the Industry

Constant of

Proportionality, Cv

q = Cv ∆ P

Problem 10.30 (3)

To Account for

potential BACK Flow

under NEGATIVE ΔP

Conditions use the

Signed

Square-Root Relation;

the “SSR”

 Where the SSR Fcn

 Back to the Tank;

ID the In-Flows

Assuming Pr & Pl

are Less than Pbot

i.e., There is

OUTFLOW at the

Left & Right

SSR ( P )

R

q

If 0

P P

SSR P

SSR Digression

The SSR fcn is BUILT into

SimuLink

An quick Example
- The Result
- For This Problem We’ll

Build our OWN

SSR

|u|

Signed Sqrt

P10_30_SSR_demo_1111.mdl

Yssr To Workspace

Yin To Workspace

Sine Wave

|u| Signed Sqrt

Scope

-10 0 1 2 3 4 5 6 7 8 9 10

9sin(t); SSR(10sin(t))

sint SSR

plot(tout,

(^) Yin,

(^) tout,

(^) Yssr,

(^) 'LineWidth',

(^) 3),

(^) xlabel('t'),

ylabel('9sin(t);

(^) SSR(10sin(t))'),

(^) grid,

(^) legend('sint',

(^) 'SSR')

Problem 10.30 (5)

Now the OutFlows in

Terms of the Fluid-

Flow Resistances

p ≡ Pressure at the

BOTTOM of the Tank

(Pa)

 And From Fluid

Mechanics ^ A NONlinear

ODE in h(t)

( (^) l r )

l r

l r SSR p p R

q (^) ,

1 = −

p = ρ gh

 Next, Sub into the

dmT/dt eqn

q ( q (^) l qr )

dt

dh

ρ A = − +



  

 = − − + − r r

l l

SSR p p R

q dt

dh A

1 1 ρ



  

 = − − + − r r

l l

SSR gh p R

q dt

dh ρ A ρ ρ

1 1

Problem 10.30 (6)

Put the ODE into Integrable Form

= − − + − r r

l l

SSR gh t p R

q dt A

( )

∫ ∫ (^ ( ) )^ (^ ( ) )

= 

z t

z r r

l l

SSR gh z p dz R

SSR gh z p R

q A

dy 0 0

( )

∫ ∫

= 

z t

z r

h t l

dz R

gh z p

gh z p q A

dy 0 0

1 ρ^ ρ

 If the Pressure in the Tank is greater than

Outside the ODE simplifies to

Problem 10.30 (7)

Now Make a SimuLink

Model To Determine

h(t)

In This Case The

Parameters will be

VARIABLES with values

Taken from the

WORKSPACE

The Parameter List:
- A ≡ Tank Cross- Section Area - Assumed Circular (Cylindrical Tank)
- R (^) l,r ≡ Hydraulic

Resistances of the

LEFT & RIGHT

Valves

ρ ≡ Liquid Density
q ≡ Liquid InFlow
h(0) ≡ Liquid Height

at t = 0

Problem 10.30 (8)

Design a SimuLink Model to Solve for h(t)

Given

ql q r

( ) ( ) (^) ∫ ( ) ( )

= 

z t

z r r

l l

SSR gh p dz R

SSR gh p R

q A

h t h 0

( ) ( ) (^) ∫ ( ( ) ) ( ( ) )

= 

z t

z r r

l l

SSR p h p dz R

SSR p h p R

q A

h t h 0

q l (^) qr

Final Model

q_mi

h1, h Scope

Left P

Right P

InFlow , q

Liquid Height

Bottom P

Tank

Left P

Right P

InFlow , q

Liquid Height

Bottom P

Tank 1

f(u) SSR

0 Pr2 at Atmos

Pr1 at Atmos

0 Pl2 = 0

0 Pl1 = 0

1/R

Result for 1hr Simulation

Prob 10.30 (1)

INPORT Block for Rt &

Lt Pressure Values

Inport ≡ Create an

input port for a

subsystem or an external input

Library → Ports &

Subsystems, Sources

Chg Label, No

Parameters

Summing Bloks for Rt &

Lt ΔP’s = P-Pl,r

Sum ≡ Add or subtract

inputs

Library → Math

Operations

Painful RePosition of

“+” & “-” connection

Locations

Top Node = |-+
Bot Node = +-|

Prob 10.30 (2)

Fcn Blok for SSR
- Fcn ≡ Apply a specified expression to the input
- Library → User-

Defined Functions

Need to Implement

for u = ΔP

 Parameters for Fcn

( ) 

 



− <

= If 0

If 0

u u

u u SSR u

Prob 10.30 (4)

Now (^)  INport Blok for Inflow
- Click on Block, and

Use FORMAT to Flip &

Twist Block

( (^) l r )

l r

l r SSR p p R

q (^) ,

1 = −

 Sum the OUTflow =

q l + q r

f(u) SSR

1/R_r Right R

1/R_l Left R

2 Right P

1 Left P

q l

q r

+|+

|−+

+−|

Prob 10.30 (5)

Then the NET INflow =

q - (ql + qr)

 Parameters for

Scaling Gain-Blok

q l

f(u) q^ r SSR

f(u) SSR

1/R_r Right R

1/R_l Left R

(^3) In

2 Right P

1 Left P q

 Now Scale Net

InFlow by 1/ρA

ρ & A values set in

WorkSpace

Implement Mathematical Operations - Computational Methods - Lecture Slides, Slides of Calculus for Engineers

Related documents

Partial preview of the text

Download Implement Mathematical Operations - Computational Methods - Lecture Slides and more Slides Calculus for Engineers in PDF only on Docsity!

Learning Goals

MATLAB using SimuLink Functional Blocks

Environment to numerically Solve ODEs

Systems using SimuLink Block Models

for Further Analysis

Problem 10.30 (1)

Block for

 It has been found

that for many Valves

the Flow Thru the

valve is Related to

the Pressure Drop

 Inputs

q ∝ Phi − Plo = ∆ P

 Using the Industry

Constant of

Proportionality, Cv

q = Cv ∆ P

Problem 10.30 (3)

potential BACK Flow

under NEGATIVE ΔP

Conditions use the

Signed

Square-Root Relation;

the “SSR”

 Where the SSR Fcn

 Back to the Tank;

ID the In-Flows

Assuming Pr & Pl

are Less than Pbot

SSR ( P )

R

q

If 0

If 0

P P

P P

SSR P

SSR Digression

SimuLink

Build our OWN

SSR

Problem 10.30 (5)

Terms of the Fluid-

Flow Resistances

 And From Fluid

Mechanics ^ A NONlinear

ODE in h(t)

 Next, Sub into the

dmT/dt eqn

dt

dh

Problem 10.30 (6)

1 ρ^ ρ

 If the Pressure in the Tank is greater than

Outside the ODE simplifies to

Problem 10.30 (7)

Model To Determine

h(t)

Parameters will be

VARIABLES with values

Taken from the

WORKSPACE

Problem 10.30 (8)

Given

Final Model

Result for 1hr Simulation

Prob 10.30 (1)

Lt Pressure Values

Parameters

Lt ΔP’s = P-Pl,r

“+” & “-” connection

Locations

Prob 10.30 (2)

for u = ΔP

 Parameters for Fcn