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Dynamic Analysis of First Order Instruments

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## Dynamic Analysis of First Order Instruments

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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**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.**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.**First‐order Instrument Step Response**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**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**Ts(t)**Ttf(t) Rs Rcond Rtf Change in energy of thermometer:**Step Response of Thermometers**Time constant**Response of Thermometers: Periodic Loading**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.**U-tube Manometer : A Second Order System**• 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**Dynamic response of a U tube manometer**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.**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**Second Order System**The essential parameters The static sensitivity: The dimensionless damping ratio: The Natural Frequency:**The transfer function is parameterized in terms of 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)**y(t)**t**General Response of A Second Order System**t z=0 y(t) t z=0.5**z=0.707**z=1.0 t