1. You have been asked to model the relationship between inlet flow-rate and product height in the conical hopper shown below:

The hopper is 10 meters tall and has a top diameter of 8 meters. The volume in the hopper is given by: and the flow-rate out of the hopper is given by (in m3/hr): The hopper is normally operated at a height of 9 meters. (a) Develop a first-principles model relating the hopper level and the inlet flow-rate. (b) Linearize your model and convert it to deviation variables (c) Convert your model to transfer function form (d) At some time tc, the inlet flow rate is reduced to 200 m3/hr, determine the time constant, τ, and the

new steady-state value for h. (30 Marks)

Solution: (a) We are interested in the level of the hopper; therefore the fundamental quantity of interest is mass.

A material balance for the hopper will be: dm / dt = min - mout

For the conical hopper, because r = 4 meters when h = 10 m, then we can get the following relationship

between r and h:

Then the volume of the hopper can be expressed as: Assuming constant density, and replacing the previous volume relationship in our mass balance:

CH E 446 – Process Dynamics and Control  Assignment 2 – Due at 4:00 pm, Tuesday January 31, 2012  Please write down the names of all the people you work with.  

Fo = 100√h

dρV

dt= ρFi − ρFo

(b) linearizing about h0 = 9 m:

Then,

Note that by definition the function f is equal to zero at the steady state, so that term cancels out, and introducing deviation variables: h’ = h – 9, and Fi’ = Fi – 300:

(c) Taking Laplace transform to the previous linearized equation: Rearranging in the standard form, we obtain the Transfer function:

(d) For a step response with α = - 100, the new steady state will be at α K = -100 * 0.06 = - 6, which is in

deviation variable, so h = h’ + 9 = - 6 + 9 = 3 m, the time constant is 2.44 hours, that is it will take 2.44 hrs to reach 63.2% of the new steady state value.

0.16π

3

d(h3)

dt= Fi − 100

√h

F 0i = F 0

o = 100√h0 = 100

√9 = 300

∂f

∂h

����(h0,F 0

i )

=−300 · 2 · 6.25

π93+

3 · 6.25 · 1002π95/2

= −312.5

243π

∂f

∂Fi

����(h0,F 0

i )

=1

0.16π92=

6.25

81π

f(h) =1

0.16πh2(Fi − 100

√h) ≈ f(h0, F 0

i ) +∂f

∂h

����(h0,F 0

i )

(h− h0) +∂f

∂Fi

����(h0,F 0

i )

(Fi − F 0i )

sH(s) =6.25

81πFi(s)−

312.5

243πH(s)

dh

dt≈ f(h0, F 0

i ) +6.25

81π(Fi − 300)− 312.5

243π(h− 9)

2. Problem 3.6. Using partial fraction expansion where required, find x(t) for: (a) (b) (c) (d) (e) (20 Marks)

Solution: (a)

α1 = 1; α2 = -6; α3 = 6;

(b)

α1 = -1/8; α2 = 2/13; α3 = -3/208; β3 = -11/208;

(c)

X(s) =s(s+ 1)

(s+ 2)(s+ 3)(s+ 4)

X(s) =s+ 1

(s+ 2)(s+ 3)(s2 + 4)

X(s) =s+ 4

(s+ 1)2

X(s) =1

s2 + s+ 1

X(s) =s+ 1

s(s+ 2)(s+ 3)e−0.5s

α1 = 1; α2 = 3;

(d)

(e)

The exponential term corresponds to a time delay, so we first ignore it:

with A = 1/6; B = ½; and C= -2/3; Then,

And introducing the time delay:

3. Problem 4.8. A surge tank is designed with a slotted weir so that the outflow rate, w, is proportional to

the liquid level to the 1.5 power; that is,

where R is a constant. If a single stream enters the tank with flow-rate wi, find the transfer function H(s)/Wi(s). Identify the gain and all time constants. Verify units. Consider that the cross-sectional area of the tank is A, and the liquid density ρ is constant.

(25 Marks)

Solution: Since we are required to find the Transfer function relating height (H) to the inlet mass flow rate (Wi), then we know that the fundamental quantity is mass and we start by doing a mass balance:

or,

Using Taylor series approximation to linearize the above equation: Transforming the above expression into deviation variables [h’ = h – h0 and wi’ = wi – wi

0], then: Then τ and K are: Then Taking Laplace Transform and re-arranging:

Note that τ has units of time as:

τ =ρA

1.5Rh0.50

H(s)

Wi(s)=

K

τs+ 1

4. Let us consider a cooking plate with mass m = 1 kg and specific heat capacity CP = 0.5kJ/kg K. The

cooking plate is heated by electric power and the supplied heat is Q1 = 2000 W. The heat loss from the cooking plate is UA(T - To), where T is the cooking plate’s temperature, To = 290 K is the temperature of the surroundings, A = 0.04 m2 and U is the overall heat transfer coefficient. If we leave the plate unattended, then we find that T = 1000 K when t goes to infinity. (a) Set up a proper energy balance equation to find the temperature of the plate as a function of time. (b) Find the overall heat coefficient. (Use the steady state information) (c) Assume that U and To are constant during the heating. Rewrite the above dynamic model in terms of deviation variables. (d) What is the time constant (value and units)? (25 Marks) Solution: (a) This is a closed system without mass flows and shaft work, and since the cooking plate is solid, we

can neglect energy related to pressure-volume changes. The energy balance around the cooking plate (the system) gives: Here, there are two contributions to the supplied heat Q, from electric power and from heat loss, that is: The enthalpy of the cooking plate is a function of temperature, that is:

(b) In order to determine the overall heat transfer coefficient U, we use the steady state temperature T0=1000K. At steady state, the energy balance is:

(c) The state variable is T and the input variable is Q1. The deviation variables are defined as: T’ = T – T0, and Q’ = Q – Q0

Then the original model, obtained in part (a), becomes:

(d) Rearranging the last equation into the standard form gives:

mCpdT �

dt= −UAT � +Q�

1

Replacing the corresponding numerical values give: