First Order Dynamics Bulb Thermometer

8/10/2019 First Order Dynamics Bulb Thermometer

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FIRST ORDER DYNAMICS: BULB

THERMOMETER

AIM: To determine time constant and study the frst order dynamics or

mercury bulb thermometer

APPARATUS:

A ask to heat the liquid.

Ethylene glycol solution.

A mercury thermometer to determine the temperature at regular

point o time.

A hot-plate to heat the ask containing solution.

A stop-watch to note the time.

PROCEDURE:

• Fill the liquid in a ask completely and heat it below the boiling point

up to around !"#o$.

• %ote down the thermometer reading& which is the room

temperature.

• 'nsert the thermometer into the liquid bath till mercury shows the

highest le(el.• Take the thermometer out& wipe it and let the temperature all down

to !##o$

• )tart the stop watch and take reading o time or e(ery * +$ all in

temperature.

• ,epeat the eperiment or about -/ times.

THEORY:

0ynamics o a frst order system can written in the orm as gi(en below

ζdy

dt + y= Ku

1ere&

2 3 Time constant or the system Tau4

y 3 response o the system

u 3 input o the system.

5 3 gain o the system



0ynamics o a frst order system can be studied only by knowing one parameter

i.e 2

The time constant determines how ast the system reaches the steady state.

)ome o the eamples or the frst order system are

• A thermal system

• A resistance-capacitor circuit

• 6iquid le(el in a o(er-head tank

Amount o heat liberated rom the thermometer 3 mcp 7 dT8dt

Amount o heat gained by the surroundings 3 hA To 9T4

1ence at steady state

mcpdT8dt 3 hA To 9T4

mcp8hA 7 dT8dt 3 To 9T4 mcp8hA 7 dT8dt : T 3 To

From the abo(e equation& we can see that time constant 24 3 mcp8hA

1ere&

mcp 3 $apacitance to store energy

!8hA 3 ,esistance to heat transer

1ence time constant 2 3 $apacitance to store energy47,esistance to heat

transer4

'n general in one time constant the thermometer response decreases to

• ;."< or one 2

• =*< or "2

• ==.* or * 2

'n a frst order system response is independent o the input step si>e.

%ow in order to calculate time constant& i we integrate the equation in(ol(ing

frst order dynamics we get

0−¿Y f Y ¿¿

Y ( t )=Y 0−¿

)ol(ing this ormula& we get& time constant ? as@

τ = −t

ln ( Y ( t )−Y f

Y 0−Y f )



Observation Table:

Tempo$4 t! t" t t/ ta(g T - Ts

lnT - Ts4

!## # # # # # */.//";

*!

=* /.B *./B *.!" /.! *.#/* #/."#

"B

=# =.;" !#.=B !#.!* =."B!#.##"

* B*/.!B/

* !/.=B !;.* !*.; !.* !*." B#/."//

=*

# "#.B "".* "#.B !=.!" "#.B* ;* /.!B/B

B* "B."! ".=; ";.* "/./;";.B"

* ;#/.#=/

/*

B# /. ;.;; ".== "=.=/.*=B

* **/.##B

;* /!.*= /*.# /!." *.* /#.B *#.=!"#

"

;# /=.=" */. /=./ /".!*/=.!BB

* /*.#;;

;"

** *=.=* ;B.B* *.=* *!.B"*=.*="

* /#.;

B=

*# B#.B! #.; ;=.!* ;#./B B#."/ *.***

/

/* "." =/.; #.!* B!."* ".!!* #./#!!

=B

/# =*.B*!!!.

/ =".B! ."==*.=B

* "*."!

B;

*!!.B

!"=.

!#.

=/.!B!!!.;;

* "#".==*B

"

# !/.=!*;.

=!";.B

=!!".#

"!".*"

* !*".B##

*

"*

!*.;

!

!B;.*

* !*!./

!#.

!

!*/./

"* !#

".#"*

*



Observation Carts:

# "# /# ;# # !## !"# !/# !;#

#!#

"#

#

/#

*#

;#

B#

#

=#

!##4 3 - #.== : ==.=B

,C 3 !

Te!" #s t

Ti!e $se%&

Te!"erat're $oC&

# "# /# ;# # !## !"# !/# !;#

"."

".B

."

.B

/."4 3 - #.#! : /./;

,C 3 !

ln$T(Ts& #s t

Ti!e $se%&

ln$T(Ts&



Conclusion:

The value of time constant was found to be equal to 86.08s.

Since this is an example of first order dynamics, when the value was plotted it comes as

exponential decaying function.

The time constant is calculated using initial slope method, by the first order equation and

steady state heat balance. rom all these methods we got almost same result.

The error in results is mainly because of initial readings, initially the rate is too fast to

measure using stopwatch, so more reading were ta!en and average was ta!en for better result.

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First Order Dynamics Bulb Thermometer