The Manual Calculation Using A Linear Model Engineering Essay

This study is presented in two parts. Part1 involves the manual computation utilizing a additive theoretical account ( regression analysis ) . Part 2 involves the usage of existent acquired informations and using the non-linear theoretical account fitting process utilizing LabView VI. In Part2, the study shows an experimental analysis of the transeunt response of a type K thermocouple to a sudden measure rise in temperature by sudden submergence of the thermocouple into hot H2O, the scene and constellation of different parametric quantities, and besides gives a distinct account of the front panel and the block diagrams contained in the LabView VI. A graph is plotted to demo the transient response, while accent was placed on the assorted variables used in plotting the curve ( swerve suiting analysis ) .

Table OF CONTENTS

1. Introduction 3

2. Separate 1 4

2.1 AIM… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … ..4

2.2 CALCULATION AND RESULT… … … … … … … … … … … … … … … … … … … … … … … … … … … .4

3. Separate 2 7

3.1 AIM… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … ..7

3.2 Theory… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 7

3.2.1 CURVE Adjustment… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … .7

3.2.2 NON-LINEAR REGRESSION… … … … … … … … … … … … … … … … … … … … … … … … … .8

3.3 EXPERIMENTAL SET-UP AND PROCEDURE… … … … … … … … … … … … … … … … … … 10

3.3.1 SET-UP AND CONFIGURATION… … … … … … … … … … … … … … … … … … … … … … .10

3.3.2 LABVIEW VI PROGRAM… … … … … … … … … … … … … … … … … … … … … … … … … … ..12

3.3.3 PROCEDURE… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … ..14

3.4 Consequence… … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … 15

4. Decision 16

Mentions

Appendix

1. Introduction

The most popular transducer for mensurating temperature is the thermocouple. It is one of the simplest of all detectors ; it is an cheap, rugged device that can run over a really broad scope of temperatures. The thermocouple besides has alone signal conditioning demands.

Thermocouples operate on the rule that the junction of two dissimilar metals generates a electromotive force that varies with temperature. The end product is a little electromotive force measured between the two wires ( National Instruments, 2010 )

Figure1: Thermocouple ( culled from www.capgo.com/Resources/Temperature/Thermocouple/thermocouple.html )

The thermocouple Acts of the Apostless as a temperature detector and it produces a comparatively little end product electromotive force. To show a more feasible consequence from electromotive force, signal conditioning is required by either linearizing or amplifying. The thermocouple used is a type K which consists of chromel and alumel.

In order to input the information from the thermocouple into a computing machine, vitual instrumentality is used to treat the signal by utilizing suited package which generates studies and consequences. The thermocouple ( which is a transducer ) converts temperature to voltage and a information acquisition board is so used to change over the parallel signals into digital signals which are so fed to the computing machine. The DAQ acquisition package ( Labview VI ) converts the digital signals into graphical indexs utilizing the front panel and block diagrams. Display charts are so placed to reexamine the signal acquired.

The study is split into two parts, Part 1 involve a manual computation utilizing a additive theoretical account implemented on a spreadsheet ( Microsoft Excel ) , while Part 2 involves the acquisition of existent informations followed by a non-linear theoretical account fitting process utilizing LabView.

2. Separate 1

Part 1 involves a manual computation between Track A and Track B. Track A involves a set of recorded informations covering with a thermal resistor discussed in the Measurement theory and Devices faculty, while Track B involves decay transient with an exponential map of clip. Path B was chosen for the study.

2.1 AIM

The purpose is to bring forth a manual computation utilizing a additive theoretical account implemented on a spreadsheet for specific informations which represents a decay transient which can be modelled as an exponential map of clip, and bring forth a ensuing normal equation of. The information provided is shown below.

Time ( X )

Response V

0.01

0.812392

0.02

0.618284

0.03

0.425669

0.04

0.328861

0.05

0.260562

0.06

0.18126

0.07

0.150454

0.08

0.11254

0.09

0.060903

0.1

0.070437

Table 1: Table informations for Part 1

With ; , which can be made additive in two coefficients, ( ln ) and 1/I„ , as shown: .

2.1 CALCULATION AND RESULT

Ten

Yttrium

clip ( sec )

response ( V )

Ln ( V )

0.01

0.812939

-0.20777

0.02

0.618284

-0.48081

0.03

0.425669

-0.85409

0.04

0.328861

-1.11212

0.05

0.260562

-1.34491

0.06

0.181260

-1.70782

0.07

0.150454

-1.8941

0.08

0.112540

-2.18445

0.09

0.060903

-2.79847

0.10

0.070437

-2.65304

Table 1: Sample Data for a Thermimocouple

[ Y ] =

-0.20777

-0.48081

-0.85409

-1.11212

-1.34491

-1.70782

-1.8941

-2.18445

-2.79847

-2.65304

[ Z ] =

1

0.01

1

0.02

1

0.03

1

0.04

1

0.05

1

0.06

1

0.07

1

0.08

1

0.09

1

0.10

[ Z ] T =

1

1

1

1

1

1

1

1

1

1

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

We say, allow R = [ Z ] T [ Z ] , and P= [ Z ] T [ Y ] , so:

R = [ Z ] T [ Z ] =

10

0.55

0.55

0.0385

P= [ Z ] T [ Y ] =

-15.2369

-1.07602

From [ R ] [ A ] = [ P ] , [ A ] = [ R ] -1 [ P ]

[ R ] -1 =

0.466667

-6.66667

-6.66667

121.2121

[ A ] =

0.06263

-28.8434

3. Separate 2

3.1 AIM

The purpose of Part 2 is to make a LabView VI to get transient informations from a Type K thermocouple, exposing the transient informations, fitted theoretical account response and deliberate parametric quantities.

3.2 Theory

3.2.1 CURVE Adjustment

Nonlinear curve adjustment is utile in suiting an equation ( additive and nonlinear ) to a set of informations points to take measuring noise, fill in losing information points, interpolate values between informations points, extrapolate hereafter or old values, integrate and differentiate digital informations ( Chugani et al, 1998 ) . The intent of curve adjustment is to happen a map degree Fahrenheit ( ten ) in a map category I¦ for the information ( xi, Lolo ) where i=0, 1, 2, aˆ¦ , n-1. The map degree Fahrenheit ( x ) minimizes the residuary under the weight W. The remainder is the distance between the information samples and degree Fahrenheit ( ten ) . A smaller residuary means a better tantrum. In geometry, curve adjustment is a curve y=f ( ten ) that fits the information ( xi, Lolo ) where i=0, 1, 2, aˆ¦ , n-1 ( National Instruments, 2010 ) . The average square mistake shows how accurate the fitted curve is in regard of the information points, and the closer to zero, the better the adjustment.

The LabView VI has a curve adjustment map which helps to cipher the best tantrum of a given set of informations. The set of informations given in this study is that of a Type K thermocouple, by obtaining the thermocouple response to a sudden rise in temperature. We therefore can state that due to the rapid rise and alteration in temperature an exponential growing is hence present, where ;

( 3.1 )

This in the LabView VI curve puting map is set as ;

( 3.2 )

Where Y = , x = T, a = , B = I„ , and hundred = , which is for a non-linear arrested development curve.

A graphical representation is shown below.

Figure 2: Exponential growing curve.

3.2.2 NON-LINEAR Arrested development

Nonlinear arrested development is a signifier of arrested development analysis in which experimental informations are modelled by a map which is a nonlinear combination of the theoretical account parametric quantities and depends on one or more independent variables. The informations are fitted by a method of consecutive estimates ( Wikipedia, 2010 ) .

The basic thought of nonlinear arrested development is the same as that of additive arrested development, i.e. to associate a response Yttrium to a vector of forecaster variables x = . Nonlinear arrested development is characterized by the fact that the anticipation equation depends nonlinearly on one or more unknown parametric quantities. A nonlinear arrested development theoretical account has the signifier

( 3.3 )

Where the Yi are responses, degree Fahrenheit is a known map of the covariate vector = and the parametric quantity vector I? =and Iµi are random mistakes. The Iµi are normally assumed to be uncorrelated with average nothing and changeless discrepancy ( Smyth, 2002 ) .

The Gauss Newton method uses Taylor ‘s series enlargement to show the original nonlinear look in about additive signifier. From the rule of Taylor ‘s series enlargement, at a point near to f ( ten ) defined as degree Fahrenheit ( x+m ) can be predicted by,

( 3.4 )

For a general instance of two parametric quantities ;

( 3.5 )

Where T is the independent variable, are the nonlinear coefficients and inferior one represent ith loop. The Taylor series enlargement of above is,

( 3.6 )

Where J is the initial conjecture and j+1 is the anticipation. Giving ;

( 3.7 )

( 3.8 )

Where D is a matrix of, is the matrix of partial differential coefficients and matrix of nonlinear coefficients.

3.3 EXPERIMENTAL SET-UP AND PROCEDURE

Figure 3: Block diagram for Experiment with hardware filter.

3.3.1 SET-UP AND CONFIGURATION

For the circuit connexion, a type K thermocouple and a pre-calibrated signal conditioner AD595 were supplied. The circuit for thermocouple conditioning was built utilizing a veroboard.

Figure 4: Thermocouple Signal Conditioning Circuit ( Culled from Analogue Devices Monolithic Thermocouple Amplifiers with Cold Junction Compensation AD594/AD595 Datasheet ) .

The Pin 8 is the positive end product terminus which is connected to the linear channel on the DAQ, and provides the unfiltered signal. The common or signal land is connected to the linear channel land. Besides, pin 11 with an end product of +5V was connected to the DAQ, to feed a electromotive force beginning to the system. The connexion manner used is the referenced individual ended ( RSE ) because the full negative terminuss are common ( i.e. grounded ) . The LabView VI package plan is used to analysis and show the transient response. The plan responds to a electromotive force trigger on the parallel input channel to get down the gaining control of informations.

The DAQ constellation is shown below ;

Figure 5: DAQ Configuration

Figure 6: DAQ Trigger puting

Figure 5 shows the DAQ constellation, while Figure 6 shows the trigger set at 280mv ( 28A°C ) .

3.3.2 LABVIEW VI PROGRAM

A practical instrument plan is made up both the front panel and the block diagram. The front panel acts as the user interface to the measuring systems and it contains controls and indexs. A VI was created to analyze and expose temperature signal. Graphic indexs are required to see and analyze the information. The figure below shows the VI for analysis.

Figure 7: Block diagram of signal for transient analysis

Figure 8: Front panel demoing transeunt analysis for thermocouple

Figure 7 shows the block diagram consists of the DAQ aid, curve suiting VI, show and storage subVIs. The signal informations end product from the DAQ is sent into the curve adjustment for analysis. Figure 8 shows the front panel for the transeunt analysis with a graphical show.

The scene for the curve adjustment is shown below ;

Figure 9: Configuration of curve adjustment.

In order to configure the curve adjustment maps, a non-linear map is selected and harmonizing to the non-linear theoretical account ( in equation 3.1 ) initial conjectures are set for the theoretical account parametric quantities with given values. Where, a = 0.15V, B = 1sec, c = 0.25V. The coefficients are sent to its end product as a twine of dynamic informations and to obtain each single value, it was into an array to expose the consequence. The curve adjustment besides displays an index for the mean squared mistake, which shows how true the values are. The best tantrum is so analyzed by a graph index, and the informations are stored for mention.

The scene for the write to register measuring is shown below ;

Figure 10: Configuration of the write to register measuring

3.3.3 Procedure

Harmonizing to thermocouple thermodynamics, the initial temperature of the thermocouple detector in air is given by T0 merely earlier t = 0. Then the temperature is all of a sudden raised by plunging the thermocouple in hot H2O at temperature Tf. The heat balance equations below govern this physical phenomenon ( Bentley, 1995 ) ,

This process is started by running the VI and rapidly dunking the thermocouple into the hot H2O. As temperature of the thermocouple rises to the start trigger electromotive force degree scene, the DAQ starts the acquisition of 300 samples after which it completes the curve adjustment procedure and displays the consequence.

3.4 Consequence

Figure 11: Transeunt Analysis of Type K Thermocouple

Initial Guess

Best Fit Coefficients

Remark

15.0000

17.4296

A a?†V

1.0000

0.03205

A Time invariable ( I„ )

25.0000

28.8872

A

A

0.046025

Mean Squared Error

Table 3: Non-Linear Coeffiencts

4. Decision

Datas for the transient analysis and curve adjustment for the Type K thermocouple where obtained. This was possible by puting the DAQ to trip at an false voltage/temperature ( 280mv/28A°C ) . When temperature rises above the triggered point, readings are taken at an false clip of 1second, at 300 sample and 1 KHz rate. It could be seen that by utilizing the non-linear arrested development methods, the clip invariable was determined to be 0.03205 seconds to a grades of uncertainness of A±3.205 % .

Mentions

Bentley J. P. 1995, “ Dynamic Characteristics of Measurement Systems ” Principles of Measurement Systems 3rd edition Singapore: Pearson Education Asia, ISBN 0-582-23779-3, pp. 43.

Capgo, 2010. Capgo Thermocouples. [ on-line ] Available at: & lt ; hypertext transfer protocol: //www.capgo.com/Resources/Temperature/Thermocouple/Thermocouple.html & gt ; [ Accessed 05 October 2010 ]

Chugani M. L. , Samant A. R. , Cerna M. , 1998 “ Curve Fitting ” LabVIEW Signal Processing, USA: Prentice Hall PTR, ISBN 0-13-972449-4, pp. 260.

Gordon K. Smyth, 2002. Nonlinear Regression [ Pdf ] . Encyclopedia of Environmentrics ( ISBN 0471899976 ) Volume 3, pp1405-1411 Available at: & lt ; hypertext transfer protocol: //people.brandeis.edu/~moshep/Resources/nlr.pdf & gt ; [ Accessed 05 November 2010 ]

National Instruments, 2010. National Instruments. [ on-line ] Available at: & lt ; hypertext transfer protocol: //zone.ni.com/devzone/cda/tut/p/id/4084 & gt ; [ Accessed 05 November 2010 ]

National Instruments, 2010. Overview of Curve Fitting Models and Methods in LabVIEW. [ on-line ] Available at: & lt ; hypertext transfer protocol: //zone.ni.com/devzone/cda/tut/p/id/6954 & gt ; [ Accessed 05 November 2010 ]

Wikipedia, 2010. Non-Linear Regression. [ on-line ] Available at: & lt ; hypertext transfer protocol: //en.wikipedia.org/wiki/Nonlinear_regression & gt ; [ Accessed 05 November 2010 ]