Midterms Problems

Question 1

Develop a transfer function relating the tank outlet temp to changes in the inlet temp. Determine the response to a step change of the inlet from 60 deg C to 80 deg C. What is the temp reading after 5 min?

V = 1000 L
volumetric flowrate (v dot) = 200 L/min

Answer 1

T2`(t) = 20 (1-e^(t/5))
T2(t) = 20 (1-e^(t/5)) + 80

Question 2

A tank having a time constant of 1 min and a resistance of 1/9 ft/cm is operating at steady state with an inlet flow of 10 ft3/min. At time t = 0, the flow is suddenly increased to 100 ft3/min for 0.1 min by adding an additional 9 ft3 of water to the tank uniformly over a period of 0.1 min. Plot the response in tank level and compare with the impulse response.

Answer 2

Q1 (t) = 90 (!-e^(-t))- 90 (1-e^(-t-0.1))u(t-0.1)

Question 3

A reactor was initially filled with 0.1 m3 of brine containing 50 g of salt. Fresh brine containing 5g of salt/ L enters the reactor at a rate of 1 L/s. Assuming the tank is well-stirred and the outflow runs at 1 L/s, what is the concentration of solution in the reactor at any given time? What is the concentration of brine in the reactor after 10 min?

Answer 3

y (t) = 4.5 (1-e^(-t/100)) + 0.5
y(10) = 4.989 g/L

Question 4

Water at Ti = 60 deg C and a rate of 200 kg/min enters a well-mixed tank with a cooling oil where it is cooled at a constant cooling rate to T = 25 deg C. The tank has a volume of 1000 L. At time t = 0, the inlet temp started to decrease at a rate of 0.5 deg C/min.

a. Determine the temp change of the water at the exit at any given time.
b. What is the temp of the exit water after 5 min?
c. When will the exit temp. fall beyond 15 deg C?

Answer 4

a. T2 (t) = -0.5t+2.5(1-e^(-t/5)) +25
b. T2(5) = 24.08 deg C
c. t= 24.966 min

Question 5

Two non-interacting tanks in series are as shown in the figure. If the inlet flowrate in Tank 1 was increased from 1 m3/min to 2.5 m3/min, determine the response of outlet flowrate in Tank 2 and what if magbase sa Tank 1.

R1= 2/3
R2 = 1
A1 = 3 m2
A2 = 1 m2

Answer 5

Q2(t) = 1.5-3e^(-t/2) +1.5 e^(-t)

What if magbase sa Tank 1?
Q1(t) = 1.5 (-e^(-t/2))

Question 6

Three identical tanks are operated in a non-interacting system. For each tank, R = 1, Tao = 1. If the deviation in flow rate of the 1st tank is an impulse function with a magnitude of 2, determine the response H3(t) of the 3rd tank. What is the maximum change and when?

Answer 6

H3(t) = t^2e^(-t)
tmax at 3rd tank = 2
H3 (tmax) = 0.541
tmax at 2nd tank = 1
H2 (tmax) = 0.73
H1 (tmax) = 2

Question 7

In the two interacting tanks mixing process, x varies from 0 lb salt/ft3 to 1 lb salt/ft3 according to a sleep function. At what time does the salt concentration in tank 2 rise by 0.6 lb salt/ft3? The hold-up volume of each tank is 6ft3.

q = 3ft3/min

Answer 7

C (t) = 1- e^(-1/2t)-1/2te^(-1/2t)
t=4.04 min

Question 8

Two interacting tanks in series are as shown in the figure. If the inlet flowrate in Tank 1 was increased from 1 m3/min to 2.5 m3/min, determine the response of outlet flowrate in tank 2.

A1=3 m2
R1= 2/3
A2= 1 m2
R2= 1

Answer 8

Q2(t) = 1.5-3/2cosh((sqrt7)/2)te^(-3/2t)-((9sqrt7)/14)e^(-3/2t)sinh ((sqrt7)/2)t

Question 9

Two non-interacting holding tanks in series both have an area of A = 9ft2 and resistance R = 1/9 ft/cm with a steady-state flow 10ft3/min. At time t = 0, the flow was increased to 20 ft3/min which only lasted for 5 min.

a. Determine the response equation of the exit flow rate and the height of the second tank.
b. What is the maximum change in the exit flowrate and height of the second tank?

Answer 9

Q2(t) = (10-10e^(-t)-10te^(-t))- (10-10e^(-(t-5))-10(t-5)e^(-(t-5))) u(t-5)
For H2(T),
Idivide lang ang Q2(t) ug 9

tmax= 5.03 min
Q2max(5.03) = 9.06 ft3/min
H2 (5.03) = 1.07 ft

Question 10

For the interacting tank system shown below, determine the change in height at t = 3min and 8 min of h1 and h2 for a change in inlet q from 2 m3/min to 4m3/min for 5 mins. (Naay pump sa second tank- q2)

A1 = 1 m2
R1= 2
A2= 1 m2
q2 = 3 m3/min

Answer 10

H1 (t) = 1 + t -e^(-t) - (1+(t-5)-e^-(t-5)) u (t-5)
H1 (3) = 3.95 m
H1 (8) = 5.049 m

H2 (t) = -1 + t + e^(-t) - (-1+(t-5)+e^-(t-5)) u (t-5)
H2(3) = 2.0498 m
H2 (8) = 4.951 m

Question 11

A steady-state non-interacting two-tank mixing process has an inlet stream of 2L/s carrying 0.5 kg/L of electrolytes. Both tanks have a holding volume of 10 L. An error in the operator side caused the inlet concentration to increase to 1 kg/L which lasted for 5 minutes before it was corrected back to the right concentration. Determine the concentration on both tanks at 3 and 6 minutes after the error occurred.

Answer 11

y1 (180 s) = 1 kg/L
y1 ( 360s) = 0.5 kg/L

y2 (180s) = 1 kg/L
y2 (360s) = 0.5 kg/L

Question 12

Two interacting tanks in series are shown. Initially, the inlet flow rate in Tank 1 is steady-state at qo= 1m3/min. Suddenly, the inlet flow increases to 10 m3/min. However, the error was brief enough that it can be described by an impulse function.

a. Determine the change in the liquid level in tanks 1 and 2 after 1 minute of disturbance.
b. What is the maximum change in the outlet flow q2?

Answer 12

Naa sa gc sa CHE4, pakilantaw nalang ko pls

Question 13

Water at Ti = 25 deg C and a rate of 200 kg/min enters a well-mixed heating tank where it is heated at a constant rate., q, to To = 60 deg C. The tank has a volume of 1000 L. At time t = 0, the inlet temperature increased up to 35 deg C.

a. Determine the transfer function.
b. Determine the temperature change of the water at the exit at any given time.
c. What is the temperature of the exit water after 5 min?
d. Determine a, b, c, for a scenario where the inlet temperature was unchanged at 25 deg C but the heating rate, q, was increased by 25%.

Answer 13

a. T(s)/To(s) = 1/(TAOs +1)
b. T`(t) = 10(1-e^-(t/5))
c. 66.32 deg C
d. qs = 29285 kJ/min
T’(t) = 8.75 (1-e^-(t/5))
T (5) = 65.531 deg C

Question 14

A steady-state 2-tank mixing process has an inlet stream of 2L/s carrying 1 kg/L of salt. Both tanks are identical with a volume of 15 L. At t=0, the inlet concentration was decreased to 0.2 kg/L which lasted for only 10 minutes before it was corrected back to the right concentration. Determine the concentration on both tanks at 5 and 15 minutes.

Answer 14

y1 (5) = 0.2 kg/L
y2 (5) = 0.2 kg/L
y1 (15) = 1 kg/L
y2 (15) = 1 kg/L

Question 15

Two interacting tanks in series are as shown in the figure. Initially, the inlet flow rate in Tank 1 is steady-state at qo = 1m3/min. Suddenly, the inlet flow qo increases to 5 m3/min. However, the error only lasted for 5 minutes.

a. Determine the change in the liquid level in tank 2 after 3 and 7 minutes of the disturbance.
b. What is the maximum change in the liquid level of tank 2?

A1 = 4 m2
R1 = 1/4
A2 = 2 m2
R2 = 1/2

Answer 15

Pakilantaw nalang kos gc please sa CHE4

a. H2 (3) = 1.035 m
H2 (7) = 0.930

b. tmax = 5.088 min
H2max (tmax) = 1.442 m

Question 16

Consider a well-mixed tank with a water steady flow rate F of 200 L/min. The volume of water in the tank is 1,000 L. The inlet water temperature is 60oC. The system is at steady state with heat input sufficient to heat the outlet water temperature to 80oC.

(a) Determine the response of the outlet tank temperature to a step change in the inlet temperature from 60oC to 70oC

(b) Determine the response of the outlet tank temperature to a step increase in the heat input of 42 kW.

(c) Determine the response of the outlet tank temperature to a simultaneous step change in the inlet temperature from 60oC to 70oC and a step increase in the heat input of 42 kW.

Answer 16

a. The actual tank outlet temperature is
T = Ts + T(t)
= 80 + 10(1 e-t/5)

b. T = Ts + T(t)
= 80 + 3(1 e-t/5)

c. T = Ts+ T(t)
= 80 + 13(1 e-t/5)

*

Question 17

Consider a well-mixed tank with a water steady flow rate F of 200 L/min. The volume of water in the tank is 1,000 L. The inlet water temperature is 60oC
. The system is at steady state with heat input sufficient to heat the outlet water temperature to 80oC. Suddenly the inlet temperature experiences a step change from 60oC to 70oC. The thermocouple measuring the tank temperature has a first order transfer function relating the measured temperature dTm to the actual temperature Td (s) in the tank according to

1/ (0.33S+1)

Plot the actual tank temperature and the measured temperature as a function of time.

Answer 17

Tmd(t) = 10 + 0.71exp( 3.03t) 10.71exp( 0.2t)

*

Question 18

A tank having a cross-sectional area of 2 ft* is operating at steady state with an
inlet flow rate of 2.0 cfm. The flow-head characteristics are shown in Fig. P6.3.

(a) Find the transfer function H(s)/Q(s).

(b) If the flow to the tank increases from 2.0 to 2.2 cfm according to a step change,
calculate the level h two minutes after the change occurs.

q0 from 1.0 ft3/min to 2.4 ft3/min
h from 0.3 ft to 1.0 ft

Answer 18

a. H(s)/Q(s) = 0.5/(s+1)

b. H (2) = 0.886 ft

Question 18

The liquid-level process shown in Fig. P6.9 is operating at steady state when the
following disturbance occurs: at time t = 0, 1 ft3 water is added suddenly (unit
impulse) to the tank; at t = 1, 2 ft3 of water is added suddenly to the tank. Sketch
the response of the level in the tank versus time and determine the level at t = 0.5,
1, and 1.5.

h = 10 cfm
tao = 1 min
R = 0.5

Answer 18

at t = 0.5 min
height = 5.303 ft

at t = 1 min
height = 6.1839 ft

at t = 1.5 min
height = 5.7181 ft

*

Question 19

A tank having a cross-sectional area of 2 ft2 and a linear resistance of R = 1 ft/cfm
is operating at steady state with a flow rate of 1 cfm. At time zero, the flow varies
as shown in Fig. P6.10.
(a) Determine Q(t) and Q(s) by combining simple functions. Note that Q is the
deviation in flow rate.
(b) Obtain an expression for H(t) where H is the deviation in level.
(c) Determine H(r) at r = 2 and t = INFINITY

Answer 19

Iask ang uban