19888cictrl
A rectangular tank is fitted with a valve at the bottom and is used for storing a liquid. The area of crosssection of the tank is 10 m\(^2\) and the flow resistance of the valve (assumed constant) is 0.1 s/m\(^2\). The time constant of the tank will
be:
19888ciictrl
The transfer function of a system is given by, \(Y/X = 1/(s^{2} + 5s + 6)\). The roots of the characteristic equation are located,
19919ictrl
A certain thermocouple has a specific time constant of 2 s. If the process temperature changes abruptly from 800 to 900\(^\circ \)C, the temperature reading in an indicator attached to the thermocouple after 6 s will be approximately,
19929actrl
When a bare thermocouple is covered by a protective sheath, the response becomes:
19951uctrl
The response of two tanks of same size and resistance in series is
19951vctrl
The transfer function of a pure dead time system with dead time \(\tau _d\) is
19952sctrl
Identify an unbounded input from four inputs whose functions are given below:
1997124ctrl
The transfer function for a firstorder process with time delay is
1997126ctrl
For an input forcing function, \(X(t) = 2t^2\), the Laplace transform of this function is
1997219ctrl
The transfer function of a process is \(\displaystyle \frac {1}{16s^2 + 8s+4}\). If a step change is introduced into the system, then the response will be
1999126ctrl
A system with a double pole at the origin is unstable since the corresponding form in the time domain
1999128ctrl
A typical example of a physical system with underdamped characteristics is a
1999216ctrl
A control system has the following transfer function, \[ F(s) = \frac {(s  1)(s + 1)}{s(s  2)(s + 4)} \] The initial value of the corresponding time function is
2000125ctrl
2000126ctrl
The unit step response of the transfer function \( \dfrac{1}{s^2 + 2s + 3}\)
2000222ctrl
The initial value (\(t = 0^+\)) of the unit step response of the transfer function \(\dfrac{s + 1}{2s + 1}\) is
2001118ctrl
A process is initially at steady state with its output \(y = 1\) for an input \(u = 1\). The input is suddenly changed to 2 at \(t = 0\). The output response is \(y(t) = 1+2t\). The transfer function of the process is
2001220ctrl
A second order system can be obtained by connecting two first order systems \(1/(\tau_1s+1)\) and \(1/(\tau_2s+1)\) in series. The damping ratio of the resultant second order system for the case \(\tau_1 \ne \tau_2\) will be
200427ctrl
For the time domain function \(f(t) = t\), the Laplace transform of \(\displaystyle\int_0^t f(t) dt\) is given by
200478ctrl
Match first order system given in Group I with the appropriate time constant in Group II.
Group I 
Group II 

(P) Thermometer 
(1) \((mC_P)/(hA)\) 
(Q) Mixing 
(2) \(q/V\) 
(3) \(V/q\) 

(4) \((hA)/(mC_P)\) 
200479ctrl
The experimental response of a controlled variable \(y(t)\) for a step change of magnitude \(P\) in the manipulated variable \(x(t)\) is shown below.
The appropriate transfer function of the process is
20058ctrl
The unit step response of a first order system with time constant \(\tau\) and steady state gain \(K_p\) is given by
20059ctrl
An example of an openloop underdamped system is
200917ctrl
201410ctrl
201618ctrl
199219actrl
A thermometer follows firstorder dynamics with a time constant of 0.2 min. It is placed in a temperature bath at 100\(^\circ \)C and is allowed to reach steady state. It is suddenly transferred to another bath at 150\(^\circ \)C at time \(t\) = 0 and
is left there for 0.2 min. It is immediately returned to the original bath at 100\(^\circ \)C.
Calculate the readings (in \(^\circ \)C) at:
(i) \(t\) = 0.1 min
{#1}
(ii) \(t\) = 0.4 min
{#2}
199525ctrl
When a thermometer at 30\(^\circ \)C is placed in water bath at 90\(^\circ \)C, the initial rate of rise in thermometer temperature is found to be 2\(^\circ \)C/s.
(i) What is the time constant (in s) of the thermometer, assuming it is a first order
device with unity steady state gain?
{#1}
(ii) What will thermometer (in \(^\circ \)C) read after one minute?
{#2}
19919iiictrl
A system has the transfer function \(Y/X = 10/(s^{2} + 1.6s + 4)\). A step change of 4 units magnitude is introduced in this system. The percent overshoot is:
200376ctrl
Water is entering a storage tank at a temperature \(T_0\) and flow rate \(Q_0\) and leaving at a flow rate \(Q\) and temperature \(T\). There are negligible heat losses in the tank. The area of cross section of the tank is \(A_c\). The model that describes the dynamic variation of water temperature in the tank with time is given as
200657ctrl
200760ctrl
200761ctrl
200860ctrl
The unit impulse response of a first order process is given by \(2e^{0.5t}\). The gain and time constant of the process are, respectively,
200862ctrl
200863ctrl
200944ctrl
201344ctrl
201550ctrl
201653ctrl
For a unit step input, the response of a second order system is
\[ y(t) = K_p\left[1\frac{1}{\sqrt{1\zeta^2}}e^{\zeta t/\tau} \sin\left(\frac{\sqrt{1\zeta^2}}{\tau}t + \phi \right) \right] \]
where, \(K_p\) is the steady state gain, \(\zeta\) is the damping coefficient, \(\tau\) is the natural period of oscillation and \(\phi\) is the phase lag. The overshoot of the system is \(\displaystyle \exp\left(\frac{\pi \zeta}{1\zeta^2} \right)\). For a unit step input, the response of the system from an initial steady state condition at \(t = 0\) is shown in the figure below.
What is the natural period of oscillation (in seconds) of the system?
201831ctrl
201930ctrl
Consider two noninteracting tanksinseries as shown in figure. Water enters Tank 1 at \(q\) cm^{3}/s and drains down to Tank 2 by gravity at a rate \(k\sqrt{h_1}\) (cm^{3}/s). Similarly, water drains from Tank 2 by gravity at a rate of \(k\sqrt{h_2}\) (cm^{3}/s) where \(h_1\) and \(h_2\) represent levels of Tank 1 and Tank 2 respectively (see figure). Drain valve constant \(k=4\) cm^{2.5}/s and crosssectional areas of the two tanks are \(A_1=A_2=28\) cm^{2}.
At steady state operation, the water inlet flow rate is \(q_{s}=16\) cm^{3}/s. The transfer function relating the deviation variables \(H_2\) (cm) to flow rate \(Q\) (cm^{3}/s) is,
198918iictrl
A temperature alarm unit, a unity gain first order system with a time constant of two minutes, is subjected to a sudden 100 K rise because of fire. If an increase of 50 K is required to activate the alarm, what will be the delay (in minutes) in signaling
the temperature change?
199018ictrl
A thermometer of time constant 10 seconds, initially at 30\(^\circ \)C, is suddenly immersed into water at 100\(^\circ \)C. How long (in s) will it take for the thermometer reading to reach 90\(^\circ \)C?
200013ctrl
IN201532ctrl
199119iictrl
An aqueous solution (density = 1000 kg/m\(^3\), specific heat = 4 kJ/kg.\(^\circ \)C) at 300 K is continuously fed at a flow rate of 1 m\(^3\)/min to a continuous flow stirred tank of volume 1 m\(^3\) containing a heater having a heating capacity of 1000
kW. If the liquid in the tank is also at 300 K to start with, find the equation which predicts the exit temperature of the solution as a function of time after the heater is switched on.
199326actrl
A first order reaction \(A \rightarrow B\) with the rate constant \(k\) is taking place in CSTR fed with \(A\) at concentration \(C_{AF}\) which remains unchanged. There are likely to be some deviations in feed rate (\(F\)) of \(A\). Derive linearized
transfer function between concentration of \(A\) in the outlet and feed rate of \(A\) assuming that volume \(V\) of reacting mixture remains unchanged.
199426ctrl
Derive an analytical expression for a unit impulse response of a system whose transfer function is given by \[ \frac {Y(s)}{X(s)} = \frac {1.5}{s^2+3s+2} \]
199526ctrl
A first order system with transfer function \(\displaystyle G_p=\frac {Y(s)}{X(s)}=\frac {1}{s+1}\) is subjected to input \(x(t)=t\). Derive the expression for change in output \(y(t)\) as a function of time.
(i) What is the maximum and minimum difference
between input and output?
(ii) At what time does these difference occur?
Last Modified on: 02May2024
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