pure water 4gpm 4gpm 4gpm a b - cpp

pure water

4GPM4GPM 4GPM

A B

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #1

Note: Your answers must be correct to 3 significant figures and have the appropriate units.

I. 2

7

8 12s s s =

A

s +

1

B

s r +

2

C

s r

In this equation, r1 > r2

(1) B = ____________

A

s +

1

B

s r +

2

C

s r=

A

s +

6

B

s +

2

C

s

7 = A(s2 + 8s + 12) + Bs(s + 2) + Cs(s + 6)

s = −6 B = 7/(6)/( 6 + 2) = 0.2917

(2) C = ____________

s = −2 C = 7/(2)/( 2 + 6) = −0.875

II. (3) In tank A are 200 gal of brine containing 80 lbs of

dissolved salts. Solution from this tank runs at a rate of 4

GPM into a second tank, B, which contains initially 100 gal

of brine with a concentration of 0.2 lb/gal of solution.

Similarly, solution runs from tank B at the same rate.

Determine the concentration of salt in tank A after 20

minutes.

A A( )d V

dt

= 4A A = 0.4exp( 0.02t) = 0.4exp( 0.02×20) = 0.2681 lb/gal

III. (4) Consider a heat exchanger where the process outlet temperature is controlled by varying

the steam flow rate to the exchanger. __________

A. The controller compares the setpoint to the actual process outlet temperature and send

out an appropriate signal to the control valve.

B. Temperature sensor with longer time constant is more desirable.

a. A and B are true b. Only A is true c. Only B is true d. A and B are false (A)

IV. (5) Given Y(s) =

5( 2)

1 2

s

s s s

. Determine y(t ∞) =

_________

y(t ∞) = 0

lim ( )s

sY s

= 0

5( 2)lim

1 2s

s

s s

= − 5

V. (6) Two consecutive, first order reactions take place in a perfectly mixed, isothermal

continuous reactor (CSTR).

A B Ck

1 2k

Volumetric flow rates (F) and density are constant. The reactor operates at steady state. The inlet

stream to the reactor contains only A with CA,in = 10 kmol/m3. If k1 = 2 min-1, k2 = 3 min-1, F =

0.1 m3/min, V = 2 m

3, and CB =

1 21+ 1+

K

k k where ( = V/F), determine the numerical value

(with correct unit) of K.

_________

0 = CA,inF k1CAV CAF CA,in k1CA CA = 0 CA = A,in

11+

C

k ,

0 = (k1CA k2CB)V CBF (k1CA k2CB) CB = 0 CB = 1 A

21+

k C

k

K = k1CA,in = 2×(2/.1)×10 = 400 kmol/m3

VI.(7) Water flows steadily through a 2-in.-inside diameter pipe at the rate of 170 gal/min (1 ft3

= 7.48 gal). The 2-in. pipe branches into two 1-in.- inside diameter pipes. If the average velocity

in one of the 1-in. pipes is 25 ft/s, the average velocity in the other 1-in. pipe is

__________

170/(7.48×60) = ×25/(4×144) + ×V/(4×144)

V = [4×144×170/(7.48×60) − ×25]/ = 44.4 ft/s

VII. (8) A two-lane highway carries cars traveling at an average speed of 60 mph. In a

construction zone, where the cars have merged into one lane, the average speed is 20 mph and

the average distance between front bumpers of successive cars is 25 ft. (1 mile = 5280 ft)

The average distance between front bumpers in each lane of the two-lane section is __________

D = 25×2×60/20 = 150 ft

VIII. (9) Find the Laplace transform of e-5t

cos 3t

____________

We have L{cos 3t} = 0

cos3ste tdt

= 2 23

s

s

Replacing s by s + 5 gives

L{e-5t

cos 3t} = 2

5

( 5) 9

s

s

IX. (10) F(s) =

22

104

1 4 10s s s =

1

A

s + 2 4

Bs C

s

+

2

2 4

Ds E

s

+

10

F

s

F = ____________

104 = A(s + 10)( s2 + 4)

2 + (Bs + C)(s + 1)( s

2 + 4)(s + 10) + (Bs + C)(s + 1)(s + 10) + F(s + 1)( s

2 + 4)

2

s = −10 F =

2

104

100 4 10 1 = − 1/(104×9) = − 0.0010684

pure water

4GPM4GPM 4GPM

A B

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #1


I. 2

6

8 15s s s =

A

s +

1

B

s r +

2

C

s r

In this equation, r1 > r2

(1) B = ____________

A

s +

1

B

s r +

2

C

s r=

A

s +

5

B

s +

3

C

s

6 = A(s + 5)(s + 3) + Bs(s + 3) + Cs(s + 5)

s = − 5 B = 6/(5)/( 5 + 3) = 0.6

(2) C = ____________

s = − 3 C = 6/(3)/( −3 + 5) = −1

II. In tank A are 200 gal of brine containing 60 lbs of

dissolved salts. Solution from this tank runs at a rate of 4

GPM into a second tank, B, which contains initially 100 gal

of brine with a concentration of 0.2 lb/gal of solution.

Similarly, solution runs from tank B at the same rate. The

differential equation describing the salt concentration in

tank B (B) is given by: Bd

dt

+ aB = bexp( 0.02t).

Determine the numerical values (with correct units) for a and b.

(3) a = ____________ and (4) b = _____________

A A( )d V

dt

= 4A A = 0.3exp( 0.02t)

B B( )d V

dt

= 4A 4B = 1.2exp( 0.02t) 4B

100 Bd

dt

+ 4B = 1.2exp( 0.02t)

Bd

dt

+ 0.04B = 0.012exp( 0.02t) a = 0.04 /min, b = 0.012 lb/(galmin)

III. (5) A tank containing 5 m3 of 20% (by volume) NaOH solution is to be purged by adding

pure water at a rate of 2 m3/h. If the solution leaves the tank at a rate of 4 m3/h, determine the

time necessary to purge 99% of the NaOH by mass from the tank. Assume perfect mixing.

Specific gravity of pure NaOH is 1.22.

t(h) = ________

Adm

dt = Am

VF A

A

dm

m =

4

5 2tdt ln

A,i

A,i

0.01m

m

= 2ln(1 0.4t)

ln(0.01) = ln(1 0.4t)2 t = (1 0.01

1/2)/.4 = 2.25 hr

IV (6) F(s) =

22

25

1 4 5s s s =

1

A

s + 2 4

Bs C

s

+

2

2 4

Ds E

s

+

5

F

s

A = ____________

25 = A(s + 5)( s2 + 4)

2 + (Bs + C)(s + 1)( s

2 + 4)(s + 5) + (Bs + C)(s + 1)(s + 5) + F(s + 1)( s

2 + 4)

2

s = −1 A =

2

25

1 4 1 5 = 25/(25×4) = 0.25

V. (7) _________

A. Feedback and feedforward control both require a measured variable..

B. The process variable to be controlled is measured in feedback control.

a. A and B are true (A) b. Only A is true c. Only B is true d. A and B are false

VI. (8) In tank A are 300 gal of brine containing 400 lbs of dissolved salts. Solution from this

tank runs at a rate of 3 GPM into a second tank, B, which contains initially 500 gal of pure water.

Similarly, solution runs from tank B at the same rate (3 GPM). Assume perfect mixing,

A B

The steady state concentration of salt A(lb/gal) in tank B is _________

400/800 = 0.5 lb/gal

VII. An isothermal CSTR with a first-order irreversible reaction A —> B and

rA = 0.5CA mol/(ft3min)

has a constant flow rate of 8 ft3/min. The reactor volume is 100 ft

3. The inlet concentration CAi

is 6 moles/ft3.

9) The steady state outlet concentration CA is _______________

FCAi 0.5 CAV FCA = 0 CA = CAi/(0.5V/F + 1) = 6/(0.5×100/8 + 1) = 0.828 mol/ft3

10) At t = 0 the initial outlet concentration CA is 3.0 mol/ft3. Determine the outlet concentration

CA after 2 minute.

___________

AdC

dt= FCAi/V 0.2CA FCA/V = 0.08×6 0.5CA 0.08CA = 0.48 0.58CA

ln0.48 0.58

0.48 0.58 3

AC

= 0.58t 0.48 − 0.58CA = (0.48 − 1.74)exp(0.58t)

CA = [0.48 + 1.26exp(0.58×2)]/0.58 = 1.508 mol/ft3

0 1 2 3 4 5 60

0.5

1

1.5

2

t

f

TT TC

FT

HeatexchangerCooler

90 Fo

Hot oil

Cooled oil

70 Fo

Refrigerant

50 Fo

+

-Set point

(Reference input)

Gc G Gp

GsFeedback Signal

ErrorSignal

Actuating Signal

Manipulated Variable

CV

Disturbance

X(s)

Y(s)

R(s)

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #2 (50 minutes)


I. 1) Plot the following function from t = 0 to t = 6:

f(t) = 0.5u(t − 2) − (0.5t − 3)u(t − 3) + (0.5t − 3)u(t − 4)

t = 2, f(t) = 0.5

t = 3, f(t) = 0.5 − 0.5t + 3 = 3.5 − 1.5 = 2

t 4, f(t) = 3.5 − 2 = 1.5

t > 4, f(t) = 0.5 − 0.5t + 3 + (0.5t − 3) = 0.5

II. Consider the given feed back loop with the

following transfer functions:

1 2 3, 2, ,and 1

2 1p s c

sG G G G

s s

2) Determine the numerical value of the transfer function Y(s)/ X(s) when s = 0.5 ________

( )

( ) 1

p

p s c

GY s

X s G G G G

=

1

2 12 1(2 3)

12 1

ss

s s

=

1

2 0.5 12 2 0.5 3

12 0.5 1 0.5

= 0.05556

3) Determine the steady state value of ( )y t when X(s) = 0 and R(s) = 1/s. ________

Y(s) = 1

c p

p s c

RG GG

G G G G=

1 1(2 3)

1 2 12 1(2 3)

12 1

s

s sss

s s

=1 2 3

(2 1) 2(2 3)

s

s s s s

( )y t = lim

0s sY(s) =

lim

0s

2 3

(2 1) 2(2 3)

s

s s s

= 0.5

III. 4) Consider the given control system with all

instrumentation in electronic (4 to 20 mA). If the range of

the orifice-differential pressure flow transmitter on the

water line is 0-1000 gpm, the value from the flow

transmitter is for a water flow rate of 300 gpm is: _______

4 + 16

2300

1000

= 5.44 mA

IV. Consider the continuous stirred tank (CST) thermal mixer shown below:

The process parameters and variables are defined as:

F1: mass flow rate of stream 1 (initially 4 kg/s); F2: mass flow rate of stream 2 (4 kg/s)

M: mass of liquid in the mixer (100 kg) = constant (perfect level control)

T1: temperature of stream 1 (30oC); T2: temperature of stream 2 (90

oC)

Ts: the time constant for the temperature sensor on the product stream (0.5 s) , the

dynamic model for the sensor is sdT

dt=

1

Ts(T Ts) where T is the liquid temperature in the tank

and Ts is the sensor temperature. Reference temperature is 0oC. At time equal to 0 seconds, the

liquid temperature in the tank is 60oC, a step change in the flow rate for stream 1 is made from 4

kg/s to 8 kg/s. The temperature sensor can be obtained by solving the following ODE where time

is in second:

sdT

dt+ ATs = B + Cexp(Dt). Determine the numerical values with correct units for B, C, and D

5) B = __________ 6) C = __________ 7) D = __________

__________

MdT

dt= F1T1 + F2T2 (F1 + F2)T 100

dT

dt= (8)(30) + (4)(90) 12T = 600 12T

10060

600 12

TdT

T = t ln600 12

600 12 60

T

= .12t T = 50 + 10exp(0.12t)

sdT

dt=

1

0.5(T Ts) sdT

dt+ 2Ts = 100 + 20exp(0.12t)

B = 100oC/s C = 20

oC/s D = 0.12/s

F1

T1

T

TT

F2

T2

FT

FC(F

1)spec

V. The temperature of a CSTR is controlled by an electronic (4 to 20 mA) feedback control

system containing (1) a 100 to 300oF temperature transmitter, (2) a PI controller with integral

time set at 2 minutes and proportional band at 20, and (3) a control valve with linear trim, air-to-

open action, and Cv = 40 gpm/psi0.5

through which cooling water flows. The pressure drop across

the valve is a constant 20 psi.

8) If the steady state controller output, CO, is 14 mA, how much cooling water is going through

the valve?

____________

F = Cv f(x)SG

Pv = Cv f(x) vP

At CO = 14 mA, stem position x = 14 4

20 4

= 0.625. For linear trim f(x) = 0.625. The steady

cooling water through the valve is

F = (40)(0.625) 20 = 111.8 gpm

9) If a sudden disturbance increases reactor temperature by 10oF, what will be the immediate

change on the controller output?

CO = _________

Tm = 10 oF

o

16 mA

200 F

= 0.8 mA

CO = KcTm = 100

20

(0.8) = 4.0 mA

VI. (10) A tank containing 4 m3 of 20% (by volume) NaOH solution is to be purged by adding

pure water at a rate of 2 m3/h. The solution leaves the tank at a rate of 3 m3/h. Assume perfect

mixing. Specific gravity of pure NaOH is 1.22. Determine the time necessary to purge 60% of

the NaOH by mass from the tank.

t(h) = ________

Adm

dt = Am

VF A

A

dm

m =

3

4 tdt ln

A,i

A,i

0.4m

m

= 3ln(1 0.25t)

ln(0.4) = ln(1 0.25t)3 t = (1 0.4

1/3)/.25 = 1.0528 hr

+

-Set point

(Reference input)

Gc G Gp

GsFeedback Signal

ErrorSignal

Actuating Signal


CV

Disturbance

X(s)

Y(s)

R(s)

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #2 (50 minutes)


I. Consider the given feed back loop with the


1 2 3, 2, ,and 1

2 1p s c

sG G G G

s s

1) Determine the numerical value of the transfer function Y(s)/ R(s) when s = 0.5 ________

( )

( ) 1

c p

p s c

G GGY s

R s G G G G

=

1 1(2 3)

2 12 1(2 3)

12 1

s

s ss

s s

=2 3

(2 1) 2(2 3)

s

s s s

= 0.444

2) Determine the steady state value of ( )y t when X(s) = 1/s and R(s) = 0. ________

Y(s) =( )

1

p

p s c

X s G

G G G G=

1

1 2 12 1(2 3)

12 1

sss

s s

=1

(2 1) 2(2 3)

s

s s s s

( )y t = lim

0s sY(s) =

lim

0s (2 1) 2(2 3)

s

s s s = 0

II. Consider the water supply control system in a drum-boiler

that generated saturated steam.

3) If the range of the orifice-differential pressure flow

transmitter on the saturated steam line is 100-300 gpm, the

value from the flow transmitter in mA for a steam flow rate

of 200 gpm is:

__________

4 + 16

2200 100

300 100

= 8 mA

4) At the design conditions, the water insider the boiler is saturated at 100oC, the inlet and outlet gases to

the boiler are 500oC and 200

oC respectively. The area for heat transfer is 70 m

2 and the overall heat

transfer coefficient is 1000 W/m2K. Determine the feed water flow rate in kg/s if the heat of

evaporation is 1941 kJ/kg for saturated water at 100oC.

____________

Tlm = 400 100

400ln

100

= 216.4 m = lm

evap

UA T

H

=

1000 70 216.4

1,941,000

= 7.8 kg/s.

III. An isothermal CSTR with a first-order irreversible reaction A —> B and

rA = − 0.2 CA mol/(ft3 - min)

has a constant flow rate of 8 ft3/min. The reactor volume is 40 ft

3 . If the inlet concentration CAi

changes from 2.2 to 6 moles/ft3 (a step change) determine

5) The initial steady state concentration in the tank _________

0 = Fv (CAi CA) Vr k CA 0 = CAi CAs k CAs

CAs = CAi/(1 + k) = 2.2/(1 + 5×0.2) = 2.2/2 = 1.1 moles/ft3

6) The process time constant is (numerical value with unit) _________

In terms of the deviation variables:

VrAdC

dt= Fv (CAi CA) Vr k CA sCA(s) = CAi(s) CA(s) k CA(s)

CA(s)/CAi(s) = 1/(s + k + 1) =1/(5s + 2)

Process time constant = 5/2 = 2. 5 min

7) The steady state gain is (numerical value with unit) __________

Steady state gain = 1/2 = 0.5

VI. . Consider the continuous stirred tank (CST) thermal mixer shown below:

The process parameters and variables are defined as:

F1: mass flow rate of stream 1 (initially 4 kg/s); F2: mass flow rate of stream 2 (4 kg/s)

M: mass of liquid in the mixer (100 kg) = constant (perfect level control)

T1: temperature of stream 1 (30oC); T2: temperature of stream 2 (90

oC)

Ts: the time constant for the temperature sensor on the product stream (0.5 s) , the

dynamic model for the sensor is sdT

dt=

1

Ts(T Ts) where T is the liquid temperature in the tank

and Ts is the sensor temperature. Reference temperature is 0oC. At time equal to 0 seconds, the

liquid temperature in the tank is 60oC, a step change in the flow rate for stream 1 is made from 4

kg/s to 8 kg/s. In term of the deviation variables and unit of time in second, the temperature

sensor in Laplace domain is given by (Note: B < C):

Ts(s) = 1

( )( )

A

s s B s C

8) A = __________ 9) B = __________ 10) C = __________

MdT

dt= F1T1 + F2T2 (F1 + F2)T 100

dT

dt= (8)(30) + (4)(90) 12T = 600 12T

dT

dt= 6 0.12T sT(s) = 6/s 0.12T(s) T(s) =

6 1

( 0.12)s s

sdT

dt=

1

0.5(T Ts) Ts(s) =

2

( 2)s T(s) =

6 1

( 0.12)s s

2

( 2)s =

1 12

( 0.12)( 2)s s s

A = 12 B = 0.12 C = 2

F1

T1

T

TT

F2

T2

FT

FC(F

1)spec

Feed

Product

AT

FT

FC

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #3


I. Consider the following CSTR with first order reaction and the process parameters and

variables are given by

CA: reactant concentration (initially

0.25 mol/L)

CA0: feed concentration (1.0 mol/L)

Cp: heat capacity of the reactor feed

and product (1000 cal/kgK) = Cv

E/R: normalized activation energy (21,000 K)

F: mass feed rate and product rate (10 kg/s)

k0: rate constant (1.97×1024

s-1

)

Q: heat addition rate (initially 700,000 cal/s)

T: reactor temperature (initially 350 K)

T0: feed temperature (400 K)

t: time (s)

Vr: reactor volume (100 L)

: the constant density of the reactor feed and product (1 kg/L)

Hrxn: heat of reaction (120,000 cal/mol)

1) Determine the reaction rate constant inside the reactor at t = 0. ________

k = k0exp( E/RT) = 1.97×1024

exp( 21000/350) = 0.01725 s-1

For question (2) & (3) use k = 1 s-1

2) Determine the initial 0

A

t

dC

dt

________

VrAdC

dt= Fv (CA0 CA) Vr k CA

0

A

t

dC

dt

= [10(1 0.25)/100 1×0.25] = 0.175 mol/Ls

3) Determine the initial 0t

dT

dt

________

CpVrdT

dt= FCp(T0 T) Vr HrxnkCA + Q

0t

dT

dt

= [10×1000(400 350) 100×120,000×1×0.25 + 700,000]/(1000×100) = 18 K/s

+

+ +

-

2

4/(s+1) 1/(s+2)X

5

II) A second order response is given by

Y(t) = 1 1.2exp( 0.25t) sin(0.75t + 1.2), where t is in minute.

4) Determine the rise time with proper unit __________

sin(0.75t + 1.20) = 0 0.75t + 1.20 = tr = ( 1.2)/0.75 = 2.589 min.

5) Determine the period of oscillation with proper unit __________

= 0.75 rad/min f = 0.75/(2) = 0.11937 cycle/min period = 8.378 min

III) Consider the given block diagram:

6) Determine the numerical value of the transfer

function Y(s)/ X(s) when s = 0.5

________

4 1

( ) 1 24 1 4( )

1 5 21 2 1

Y s s s

X s

s s s

= 4

( 1)( 2) 20 8(s 2)s s

=4

1.5 2.5 20 8 2.5 = 1.0667

7) Determine the steady state value of ( )y t when X(s) = 1/s. ________

( )y t = lim

0s sY(s) =

lim

0s =

4

( 1)( 2) 20 8(s 2)s s = 2/3 = 0.667

IV. (8) A thermometer having first-order dynamics with a time constant of 2 min is at 100oF. The

thermometer is suddenly placed in a bath at 180oF at t = 0. Calculate the thermometer reading at t

= 2 min.

_________

X(t) = 80u(t) X(s) = 80

s

( )

( )

Y s

X s =

1

1s =

1

2 1s Y(s) =

80

s

1

2 1s = 80

1 2

2 1s s

Y(t) = 80[1 exp(0.5t)]

At t = 2 min Y(t) = 80[1 exp( 1)] = 50.57 150.57oF

V. At t = 0, an error is introduced into a system (t) = sin(0.5t) where t is in minute and is in

mA.

(9) If the bias value of a PI controller is 12 ma, KC = 2, I = 5 minutes , determine the controller

output at 4 minutes

__________

CO = bias + KCsin(0.5t) + (KC/I) 4

0sin(0.5 )t dt = 12 + 2sin(0.5×4) − (2/2.5)[cos(2) − 1]

CO = 14.95 mA.

(10) If the bias value of a PD controller is 12 ma, KC = 2, d = 4 minutes , determine the

controller output at 4 minutes

CO = bias + KCsin(0.5t) + (KCd) d

dt

= 12 + 2sin(0.5×4) + (2×4×0.5)cos(2)

CO = 12.15 mA.

Reflux drum PT PC

I/P

PM

SP

CO

PV

Control valveCoolingwater Condenser

Vapor

+

-Set point

(Reference input)

Gc G Gp

GsFeedback Signal

ErrorSignal

Actuating Signal


CV

Disturbance

X(s)

Y(s)

R(s)

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #3


I. Consider the given feed back loop with the


21 2 3

, 2, ,and 12 1

p s c

s sG G G G

s s

1) Determine the numerical value of the transfer function Y(s)/ R(s) when s = 1 ________

( )

( ) 1

c p

p s c

G GGY s

R s G G G G

=

2

2

1 1(s 2 3)

2 1

2 1(s 2 3)1

2 1

s

s s

s

s s

=

2

2

2 3

(2 1) 2(s 2 3)

s s

s s s

= 0.40

2) Determine the steady state value of ( )y t when X(s) = 1/s and R(s) = 0. ________

Y(s) =( )

1

p

p s c

X s G

G G G G=

2

1

1 2 1

2 1(s 2 3)1

2 1

s

ss

s s

=2

1

(2 1) 2(s 2 3)

s

s s s s

( )y t = lim

0s sY(s) =

lim

0s 2(2 1) 2(s 2 3)

s

s s s = 0

II. The overhead vapor from a

depropanizer distillation column is

totally condensed in a water-cooled

condenser at 120oF and 250 psig.

Cooling water inlet and outlet

temperatures are 75 and 95oF,

respectively. The condenser heat transfer

area is 1000 ft2

and the overall heat

transfer coefficient is 120 Btu/hroFft

2.

The process pressure is measured by an electronic (4-20 mA) pressure transmitter whose range is

200-350 psig.

3) Calculate the cooling water flow rate (gpm) at design conditions. Water density is 62.3 lb/ft3

and 1 ft3 = 7.48 gal. Cp of water is 1 Btu/lb

oF.

____________

Q = 1000×120[(120 75) (120 95)]/ln(45/25) = 4.0831×106 Btu/hr

Water flow rate = (4.0831×106)(7.48)/(20×1×62.3×60) = 408.5 gpm

TT TC

Steam jacket

PMSP

Steam

Condensate

w, Ti

w, T

Water

4) If the cooling water flow rate is 300 gpm at design conditions, calculate the value of the signal

PM at design condition:

PM = _________

PM = 4 + 16 250 - 200

350 - 200

= 9.33 mA

III. Consider a kettle through which water flows

at a variable rate w [lb/min]. The entering water is

at temperature Ti [oF] which may vary with time.

The well-mixed water is heated by steam

condensing in the jacket at temperature Ts.

5) Assume negligible heat loss to the atmosphere, the energy equation for the water is given by

MC dT

dt = wC(Ti T) + UA(Ts T) where M, C, U, and A are constant.

A. w is the mass flow rate of water into the tank.

B. U is the overall heat transfer coefficient for heat transfer between water and steam.

a. A and B are true ✓ b. Only A is true c. Only B is true d. A and B are false

6) A. The ordinary differential equation in (5) is a linear differential equation

B. Both the water temperature and the steam jacket temperature are functions of time.


7) A. The control valve on the steam line is fails shut.

B. The bias value for the current leaving the temperature controller is normally 4 mA.

a. A and B are true A b. Only A is true ✓ c. Only B is true d. A and B are false

+

-Set point

(Reference input)

Gc G Gp

GsFeedback Signal

ErrorSignal

Actuating Signal


CV

Disturbance

X(s)

Y(s)

R(s)

IV. 8) Consider the given feed back loop with the


1 2 3, 2, ,and 1

2 1p s c

sG G G G

s s

Determine the offset when X(s) = 0 and R(s) = 1/s.

( )

( ) 1

c p

p s c

G GGY s

R s G G G G

=

1 1(2 3)

2 12 1(2 3)

12 1

s

s ss

s s

=2 3

(2 1) 2(2 3)

s

s s s

Y(s) =

1 2 3

(2 1) 2(2 3)

s

s s s s

R(∞) Y(∞) = lim

0s s [R(∞) Y(∞)] =

lim

0s 1 −

2 3

(2 1) 2(2 3)

s

s s s

= 1 − 3/6 = 0.5

IV. A liquid storage tank is shown with a diameter of

10 ft. The tank has a valve which acts as a linear

resistance to flow with the flow-head relation of q =

6h, where q is in ft3/min and h is in ft. Suppose that

each tank is initially at steady state with h = 6 ft and

qi = 36 ft3/min and that at time t = 0 the inlet flow

rate suddenly changes from 36 to 50 ft3/min.

Note: The transfer function must be in term of s and number only, for example: 2

5 1s

9) Determine the transfer function H(s)/Qi(s): ____________

In term of the deviation variable

A dH

dt = Qi 6H AsH(s) = Qi(s) 6H(s) H(s)/Qi(s) =

1

6As =

1

78.54 6s

10) The liquid level h in the tank after 5 minute is ____________

A dh

dt = qi 9h = 50 6h A

50 6

dh

h = dt (A/6)ln

50 6

50 36

h

= 5

50 6h = 14 exp(30/78.54) h = (50 9.555)/6 = 6.74 ft

qi

q

h

Tank 1

+

+ +

-

2

4/(s+1) 1/(s+2)X

5

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #4


I) Consider the given block diagram:

1) Determine the numerical value of the transfer

function Y(s)/ X(s) when s = 1

________

4 1

( ) 1 24 1 4( )

1 5 21 2 1

Y s s s

X s

s s s

= 4

( 1)( 2) 20 8(s 2)s s

=4

2 3 20 8 3 = 2

2) Determine the steady state value of ( )y t when X(s) = 1.5/s. ________

( )y t = lim

0s sY(s) =

lim

0s =

1.5 4

( 1)( 2) 20 8(s 2)s s

= 1

II) A PI controller has the characteristic equation: 6 Is2 + (1 + 12KC) Is + 12KC = 0, with KC = 2

and I = 4. Put this equation into standard second-order form and determine the numerical values

for the time constant and damping coefficient .

3) = _____________ 4) = _____________

6 Is2 + (1 + 12KC) Is + 12KC = 0 24s

2 + 100s + 24 = 0

s2 + 4.1667s + 1 = 0 compared with the standard form: 2

s2 + 2s + 1 = 0

= 1,

2 = 4.1667 = 2.083

III. At t = 0, and error is introduced into a system (t) = 2sin(1.2t) where t is in minute and is in

mA.

(5) If the bias value of a PI controller is 12 ma, KC = 3, I = 4 minutes , determine the controller

output at 4 minutes

__________

CO = bias + KC2sin(1.2t) + (KC/I) 4

02sin(1.2 )t dt

= 12 + 3×2sin(1.2×4) − (3×2/4.8)[cos(4.8) − 1] = 7.16 mA.

Controller

(Set point )

V, T

C , k1

1 1

V, T

C , k2

2 2

Compositionmeasuringelement

Sample stream

Product stream

F, C0

Heating coil

Pure A

F+m/A

6) If the bias value of a PD controller is 12 ma, KC = 3, d = 4 minutes , determine the controller

output at 4 minutes

CO = bias + KC2sin(1.2t) + (KCd) d

dt

= 12 + 3×2sin(1.2×4) + (3×4×1.2×2)cos(4.8)

CO = 8.54 mA.

IV)

m

Consider a Chemical-Reactor control system where a liquid stream enters tank 1 at a volumetric

flow rate F in cfm and contains reactant A at a concentration of C0 [mol A/ft3]. Reactant A

decomposes in the tanks according to the irreversible chemical reaction: A B. The reaction is

first order with reaction rate constant k1 and k2 for tank 1 and tank 2 respectively. Data at initial

steady state: MwA = 100, A = 1 lbmol/ft3, C0s = 0.5 lbmol/ft

3, F = 100 cfm, ms = 2.5 lbmol/min,

k1 = 0.5 min-1

, k2 = 0.75 min-1

, V = 200 ft3. Neglect the volumetric flow rate m/A compared to F.

7) Determine the initial steady state concentration C1s: _________

(F + k1V)C1s = FC0s + ms C1s = (FC0s + ms)/(F + k1V)

C1s = 100 0.5 2.5

100 0.5 200

= 0.2628 lbmol/ft

3

8) Let C1d = C1 C1s, C0

d = C0 C0s, and M

d = m ms. Determine the steady state gain

(numerical value with units) for the transfer function C1d(s)/M

d (s)

___________

V 1dC

dt + (F + k1V)C1 = FC0 + m

1

V

F k V

1dC

dt + C1 =

1

F

F k VC0 +

1

1

F k Vm

Steady state gain = 1

1/

1

F

k =

1/100

1 0.5 2 = 0.005 min/ft

3

V) The five feedback control loops are shown for the distillation column below:

9) For the flow control loop FC-1:

A. Flow rate of the feed stream is the controlled variable.

B. Flow rate of the feed stream is the manipulated variable..


10) For the level control loop LC-1:

A. The level of drum D-111 is the manipulated variable.

B. The flow rate of reflux stream to T-111 is the controlled variable.

a. A and B are true b. Only A is true c. Only B is true d. A and B are false ✓

Feed

Product

AT

FT

FC

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #4





0.25 mol/L)




E/R: normalized activation energy (21,000 K)


k = 2/s (rate constant at 350 K)

Q: heat addition rate (initially 1,200,000 cal/s)





Hrxn: heat of reaction (120,000 cal/mol)


A

t

dC

dt

________

VrAdC


0

A

t

dC

dt

= [10(1 0.25)/100 2×0.25] = 0.425 mol/Ls


dT

dt

________

CpVrdT


0t

dT

dt

= [10×1000(400 350) 100×120,000×2×0.25 + 1,200,000]/(1000×100) = 43 K/s

II. 3) For the control system of the CSTR

A. Flow rate of the feed stream is the controlled variable.

B. Flow rate of the feed stream is the manipulated variable..


III) A second order response is given by

Y(t) = 1 1.2exp( 0.25t) sin(0.5t + 1.6), where t is in minute.

4) Determine the rise time with proper unit __________

sin(0.5t + 1.60) = 0 0.5t + 1.60 = tr = ( 1.6)/0.5 = 3.08 min.

5) Determine the period of oscillation with proper unit __________

= 0.5 rad/min f = 0.5/(2) = 0.079577 cycle/min period = 12.57 min

IV. 6) Given Y(s) = 2

1

3 9s s determine y(t)

Y(s) = 2

1

( 1.5) 9 2.25s =

2

1

( 1.5) 9 2.25s =

2 2

1

( 1.5) 2.5981s

= 2 2

1 2.5981

2.5981 ( 1.5) 2.5981s

y(t) = 1

2.5981exp(−1.5t)sin(2.5981t)

V) 7) Determine the temperature reading corresponding to a 12.4 mA analog signal from a temperature

transmitter that has a span of 200oF and a zero of 30

oF.

_______

T = 30 + 20012.4 4

20 4

= 135 o

F

VI) 8) The liquid phase reactions (A B ) and (A + B C) are

carried out in a jacketed, semibatch stirred tank reactor. Reaction 1

(A B ) is first order while the second reaction is first order in both

reactants. The reactor is cylindrical in shape with inner diameter D.

The reactor wall is relatively thin.

The number of conservations equations that would be required to

completely describe the dynamic processes of the reactor and jacket

is

_______

1) Jacket energy balance, 2 reactor energy balance, 3 reactor mole balance on species A, 4

reactor mole balance on species B, 5 reactor mole balance on species C

Controller

(Set point )

V, T

C , k1

1 1

V, T

C , k2

2 2


Sample stream

Product stream

F, C0

Heating coil

Pure A

F+m/A

VII.






3, F = 100 cfm, ms = 2.5 lbmol/min,

k1 = 0.5 min-1

, k2 = 0.75 min-1


9) Let C1d = C1 C1s, C0

d = C0 C0s, and M

d = m ms. Determine the time constant (numerical

value with units) for the transfer function C1d(s)/C0

d(s)

_________

V 1dC

dt + (F + k1V)C1 = FC0 + m

1

V

F k V

1dC

dt + C1 =

1

F

F k VC0 +

1

1

F k Vm

Time constant = 1

V

F k V=

200

100 200 0.5 = 1.0 min

10) Determine the steady state gain (numerical value with units) for the transfer function

C1d(s)/C0

d(s)

___________

11dC

dt + C1 =

1

1

1 k C0 +

1

1/

1

F

k m where = V/F and 1 =

1

V

F k V

= 200/100 = 2 min

Steady state gain = 1

1

1 k = 1/(1+ 0.5×2) = 0.500

Controller

(Set point )

V, T

C , k1

1 1

V, T

C , k2

2 2


Sample stream

Product stream

F, C0

Heating coil

Pure A

F+m/A

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #5


I.

m






3, F = 60 cfm, ms = 2.5 lbmol/min, k1

= 0.75 min-1

, k2 = 0.5 min-1


1) Determine the initial steady state concentration C1s: _________

(F + k1V)C1s = FC0s + ms C1s = (FC0s + ms)/(F + k1V)

C1s = 60 0.5 2.5

60 200 0.75

= 0.155 lbmol/ft

3

2) Let C1d = C1 C1s, C0

d = C0 C0s, and M

d = m ms. Determine the time constant (numerical

value with units) for the transfer function C1d(s)/C0

d(s)

_________

V 1dC

dt + (F + k1V)C1 = FC0 + m

1

V

F k V

1dC

dt + C1 =

1

F

F k VC0 +

1

1

F k Vm

Time constant = 1

V

F k V=

200

60 200 0.75 = 0.952 min

R1

R2

A1

A2

h1

h2

Q1

Q2

20 ft /min3 10 ft

3

3) The block diagram for the Chemical-Reactor control system can be obtained from the

combination of the transfer function of each component.

The output of the measuring element varies from 4 to 20 mA as the concentration of A varies

from 0.01 to 0.81 lbmol/ft3.

Determine Km (numerical value with units) _________

Km = 20 4

0.81 0.01

= 20

3

mA

lbmol/ft

II. The frequency response of a system with first order transfer function G(s) = ( )

( )

Y s

X s=

2

1s to a

sinusoidal input function X(t) = 3sin(3t) at large time is Y(t)|s = Asin(3t + ) where

(4) A = ______________. (5) = ______________.

Let s = i = 3i G(i) = 2

3 1i =

2 3 1

3 1 3 1

i

i i

=

2

10

6

10

i= 0.2 0.6i

R = magnitude = 2 2a b = (0.22 + 0.6

2)0.5

= 0.6325 A = 3×0.6325 = 1.897

= angle = tan-1

b

a= tan

-1(3) = 1.249 rad

III. The two-tank system shown on the right is operating at

steady state. At time t = 0, 10 ft3 of water is quickly added

to the first tank. Data: A1 = A2 = 10 ft2, R1 = 0.2 ft/cfm, R2 =

0.5 ft/cfm.

Note: Q1 = h1/R1. cfm = ft3/min.

The transfer function for the system is given by 2

0

d

d

Q

Q =

2

1

1as bs

Determine the numerical value with appropriate unit if exists for

6) a = __________ 7) b = __________

G1G2

+++

_

C 'RCR

mA

P

mAKC KT

PT

psigKV A

Mlbmol/min

1/F

M/FC0

Km exp(- s)d

B

mA

Km

Q1d = 1

1

dh

R 1( )dd h

dt = R1

1( )dd Q

dt

1 1( )dd A h

dt = Q0

d Q1

d A1R1

1( )dd Q

dt = Q0

d Q1

d

A1R1sQ1d = Q0

d(s) Q1

d(s) 1

0

d

d

Q

Q =

1 1

1

1R A s

2

1

d

d

Q

Q =

2 2

1

1R A s 2

0

d

d

Q

Q =

1 1

1

1R A s 2 2

1

1R A s =

2

1 1 2 2 1 1 2 2

1

( ) 1R A R A s R A R A s

a = A1R1 A2R2 = (0.2)(10)(0.5)(10) = 10 min2

b = A1R1 + A2R2 = (0.2)(10) + (0.5)(10) = 7 min

8) If H2d(t) =

2

3exp(t/4)

2

3exp(t),

determine the maximum deviation in level (feet) in tank 2 ___________

d

2dH

dt=

1

6exp(t/4) +

2

3exp(t) = 0 exp(t/4 + t) = 4 t = 4ln(4)/3 = 1.8484 min

H2d(t) =

2

3exp(t/4)

2

3exp(t) = 0.315 ft

IV. 9) a) In general, the time constant is the ratio of capacitance over conductance

b) The conductance is a measured of the ability of the process to accumulate the

quantity conserved (mass or energy).

a. A and B are true b. Only A is true ✓ c. Only B is true d. A and B are false

V. 10) For systems with loop interactions

a) The change in the sign of the gain occurs when the interaction change is less than

the original change.

b) Positive interaction can be a more severe problem than negative interaction

a. A and B are true b. Only A is true c. Only B is true d. A and B are false ✓

Feed

Product

AT

FT

FC

CHE 426 (Spring 2019) __________________

LAST NAME, FIRST

Quiz #5





0.50 mol/L)





Q: heat addition rate (initially 700,000 cal/s)



t: time (s)



Hrxn: heat of reaction (−160,000 cal/mol)

The reaction rate constant, k, at 350 K is 2.5 s-1


A

t

dC

dt

(numerical value with units) ________

VrAdC


0

A

t

dC

dt

= [10(1 0.5)/50 1.2×0.5] = 1.15 mol/Ls


dT

dt

(numerical value with units) ________

CpVr

dT


0t

dT

dt

= [10×1000(400 350) + 50×160,000×2.5×0.5 + 700,000]/(1000×50) = 224 K/s

II. A liquid storage tank is shown with a diameter of 10 ft. The

tank has a valve which acts as a linear resistance to flow with

the flow-head relation of q = 7h, where q is in ft3/min and h is

in ft. Suppose that each tank is initially at steady state with h =

8 ft and qi = 56 ft3/min and that 20 ft

3 is quickly added to the

tank at time t = 0. The transfer function H(s)/Qi(s) is a first

order transfer function.

3) The steady state gain of H(s)/Qi(s) (numerical value with unit) is ____________

In term of deviation variables:

A dH

dt = Qi 7H AsH(s) = Qi(s) 7H(s) H(s)/Qi(s) =

1

7As Kp = 1/7 = 0.143 min/ft

2

4) The change in liquid level h in the tank after 2 minute is ____________

Qi(s) = 20 H(s) = 20

78.54 7s =

20 / 78.54

7 / 78.54s H(t) = 0.2564exp(0.08913t)

At 2 minute, H(t) = 0.213 ft

III. An electrical kettle at has a mass of 0.5 kg is used to boil 0.75 kg of water at 20°C. Assume

that the kettle is always at the water temperature. The average specific heat for the kettle and the

water is 2.5 kJ/kgK. The power of the kettle is 1200 W.

(5) If heat is lost with the ambient temperature being 20°C, the convection heat transfer

coefficient is 20 W/m2K, and the kettle’s external surface area is 0.050 m

2, determine the time in

seconds it take to boil the water. ___________

MCdT

dt = Ah(T 20) + 1200 1250×2.5

dT

dt = 0.05×20(T 20) + 1200 = T + 1220

t = 3125 ln 1220 100

1220 20

= 215.6 s

(6) When the water boils, at what rate is steam produced if the heat of evaporation is 2257 kJ/kg?

__________

[1200 0.05×20(100 20)]/2257 = 0.496 g/s

qi

q

h

Tank 1

IV. The frequency response of a system with first order transfer function G(s) = ( )

( )

Y s

X s=

3

4 1s to

a sinusoidal input function X(t) = 2sin(3t) at large time is Y(t)|s = Asin(3t + ) where

(7) A = ______________. (8) = ______________.

Let s = i = 3i G(i) = 3

12 1i =

3 12 1

12 1 12 1

i

i i

=

3

145

36

145

i

R = magnitude = 2 2a b =

1/22 2

2 2

3 36

145 145

= 0.2491 A = 2×0.2491 = 0.498

= angle = tan-1

b

a= tan

-1(12) = 1.488 rad

V. (9) Determine the integral time for a PI controller using C-C settings (I = 30 3 /

9 20 /

dd

d

T TT

T T

).

The process reaction curve for the process is Gp(s) = 2exp 1.5

0.6 1

s

s

, time is in minute.

I = 1.530 3 1.5 / .6

9 20 1.5 / .6

= 0.953 minutes

(10) Determine the integral time for a PI controller using C-C settings. The tangent line to the

process reaction curve at the inflection point is B = 2.4 + 1.2t. The ultimate value Bu of the

measured variable B is 0.6. Time is in minute.

Td = 2.4/1.2 = 2 minutes, T = uB

S = 0.6/1.2 = 0.5 minute __________

I = 230 3 2 / .5

9 20 2 / .5

= 0.944 minutes

pure water 4gpm 4gpm 4gpm a b - cpp

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