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Dynamic Analysis of First Order Instruments. P M V Subbarao Professor Mechanical Engineering Department. Capability to Reach Exact Equilibrium ……. First Order Instruments.

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Dynamic analysis of first order instruments l.jpg

Dynamic Analysis of First Order Instruments

P M V Subbarao

Professor

Mechanical Engineering Department

Capability to Reach Exact Equilibrium ……


First order instruments l.jpg
First Order Instruments

A first order linear instrument has an output which is given by a non-homogeneous first order linear differential equation

  • In these instruments there is a time delay in their response to changes of input.

  • The time constant t is a measure of the time delay.

  • Thermometers for measuring temperature are first-order instruments.


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  • The time constant of a measurement of temperature is determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

  • A cup anemometer for measuring wind speed is also a first order instrument.

  • The time constant depends on the anemometer's moment of inertia.


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First‐order Instrument Step Response determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

b0

The complex function F(s) must be decomposed into partial fractions in order to use the tables of correspondences. This gives


Thermometer a first order instrument l.jpg
Thermometer: A First Order Instrument determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

Conservation of Energy during a time dt

Heat in – heat out = Change in energy

of thermometer

Assume no losses from the stem.

Heat in = Change in energy

of thermometer


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T determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.s(t)

Ttf(t)

Rs

Rcond

Rtf

Change in energy of thermometer:


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Step Response of Thermometers determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

Time constant


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Response of Thermometers: Periodic Loading determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

If the input is a sine-wave, the output response is quite different;

but again, it will be found that there is a general solution for all situations of this kind.


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T determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.s,max- Ttf,max

f


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U-tube Manometer : A Second Order System determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

  • The pressure to be measured is that of a system that involves a fluid (liquid or a gas) different from the manometer liquid.

  • Let the density of the fluid whose pressure being measured be ρf and that of the manometer liquid be ρm.

  • Equilibrium of the manometer liquid requires that there be the same force in the two limbs across the plane AA.

  • We then have

p

patm

This may be rearranged to read


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Dynamic response of a U tube manometer determined by the thermal capacity of the thermometer and the thermal contact between the thermometer and the body whose temperature is being measured.

h

  • The manometer liquid is assumed to be incompressible the total length of the liquid column remains fixed at L.

  • Assume that the manometer is initially in the equilibrium position.

  • The pressure difference Δp is suddenly applied across it.

  • The liquid column will move during time t > 0.


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The forces that are acting on the length L of the manometer liquid are:

Force disturbing the equilibrium

Inertial Force

Forces opposing the change:

a. Weight of column of liquid

b. Fluid friction due to viscosity of the liquid :

  • The velocity of the liquid column is expected to be small and the laminar assumption is thus valid.

  • The viscous force opposing the motion is calculated based on the assumption of fully developed Hagen-Poiseuelle flow.

The fricitional pressure drop



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Second Order System liquid are:

The essential parameters

The static sensitivity:

The dimensionless damping ratio:

The Natural Frequency:



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  • The transfer function is parameterized in terms of liquid are:ζ and ωn.

  • The value of ωn doesn’t qualitatively change the system response.

  • There are three important cases—withqualitatively different system behavior—as ζ varies.

  • The three cases are called:

  • Over Damped System (ζ >1)

  • Critically Damped System (ζ =1)

  • Under Damped System (ζ <1)


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y(t) liquid are:

t


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General Response of A Second Order System liquid are:

t

z=0

y(t)

t

z=0.5


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z=0.707 liquid are:

z=1.0

t