other physical examples of first-order system_lec 5

7/27/2019 Other Physical Examples of First-Order System_LEC 5

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Other Physical Examples of First-

Order System

CHE 516


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MERCURY THERMOMETER

A = surface area of bulb for heat transfer,

ft2

C = heat capacity of mercury, Btu/(lbm-

0F)

m = mass of mercury in bulb, lbm

t= time, h

h = film coefficient of heat transfer,

Btu/(ft2 - h- 0F)Cross-sectional view of

thermometer


3/12

Energy Balance: Input Rate Output Rate = Rate ofAccumulation

hA (x-y) 0 = mCdy/dt equation 1

At steady state: hA (xs ys) = mCdys/dt equation 2

Subtracting: hA[(x-xs) (y-ys)] = mCd(y-ys)/dt

In deviation variables: hA (X Y) =mCdY/dtnote: mC/hA = lbm * Btu/(lbm-

0F) = hour =

[Btu/(ft2-h-0F)]*ft2

Thus,X-Y = mCdY/hAdt

X-Y = dY/dt

Transforming: X(s) Y(s) = sY(s)

Rearranging to std. form: Y(s) = 1

X(s) s + 1


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Mixing Process

Assume: Density is constant.

q = constant volumetric flow rate

V = constant holdup volume

x = concentration of salt in the entering stream, mass of

salt/volume

y = outlet concentration


5/12

Mass Balance:

Flow rate of salt in Flow rate of salt out = Rate ofaccumulation of salt in the tank

qx qy = Vdy/dt

At steady state: qxs qys = Vdys/dt

Subtracting: q[(x-xs) (y-ys)] = Vd(y-ys)/dt

Deviation Variables: q(X-Y) = VdY/dt

Note: V/q =

Thus, X-Y = dy/dt

Transforming: X(s) Y(s) = sY(s)

Rearranging to std. form: Y(s) = 1

X(s) s + 1


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Heating Process

Ti= inlet stream temperature

T = temperature of the exiting stream

w = constant flowrate, lb/hour

Tref= reference temperature

C = heat capacity of the fluid


7/12

Energy Balance:

Rate of energy - Rate of energy + Rate of energy flow in = Rate of

flow into tank flow out of tank from heater accumulation

of energy in tank

wC(Ti-Tref) - wC(T-Tref) + q = VCdT/dt

At steady state: wC(Tis - Ts) + qs = VCdTs/dt

Subtracting: wC(Ti - Tis ) - wC(T - Ts ) + (q-qs) = VCd(T-Ts)/dt

Assume Tiis constant (Ti=Tis)

Deviation Variables: -wCT + Q = VCdT/dt

Transforming: -wCT (s) + Q(s) = VCsT(s) Rearranging to std. form: T(s) = 1/wC = Kp

Q(s) (V/w)s + 1 s + 1


8/12

A thermometer having a time constant of 0.2 min is placed in

a temperature bath, and after the thermometer comes toequilibrium with the bath, the temperature of the bath is

increased linearly with time at a rate of 1/min. Find the

difference between the indicated temperature and the bath

temperature.

( a ) 0.1 min after the change in temperature begins

( b ) 1.0 min after the change in temperature begins

( c ) What is the maximum deviation between

indicated temperature and bath temperature,

and when does it occur?


9/12

A thermometer having a time constant of 1

min is initially at 50C. It is immersed in a

bath maintained at 100C at t = 0. Determine

the temperature reading at t = 1.2 min.


10/12

Consider the stirred-tank reactor shown. The reaction occurring is AB and it

proceeds at a rate r = kCo.

where

r = (moles A reacting)/(volume)(time)

k = reaction rate constant

Co(t) = concentration of A in reactor at any

time t (mol A /volume)

V = volume of mixture in reactor

F = constant feed rate ,volume/time

Ci(t) = concentration of A in feedstream, moles/volume

Assuming constant density and constant volume V, derive the transfer function relating the

concentration in the reactor to the feed-stream concentration. Sketch the response of the

reactor to a unit-step change in Ci.


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The two-tank mixing process shown in contains a recirculation loop that transfers

solution from tank 2 to tank 1 at a flow rate ofqo .

(a) Develop a transfer function that relates the concentration c2 in tank 2 to the

concentration x in the feed, that is, C2(s)/X(s) where C2 and X are deviation

variables. Forconvenience, assume that the initial concentration c2 = 0.

(b) If a unit-step change in X occurs, determine the time needed for c2 to reach 0.60

for the case where =0.


12/12

Consider the mixed tank heater shown. Develop a transfer function relating the

tank outlet temperature to changes in the inlet temperature.

Determine the response of the outlet temperature of the tank to a step change in

the inlet temperature from 60 to 700C. Before we proceed, intuitively what would

we expect to happen? If the inlet temperature rises by 100C, we expect the outlet

temperature to eventually rise by 100C if nothing else changes (q =qs

).

other physical examples of first-order system_lec 5

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