# Mechanistic Modelling of Chemical Processes

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## Question 1

A well mixed tank is fed with a stream with total flowrate *F
kmols/h* containing *x* mol fraction of a solute.
The outflow is arranged so that the tank holdup will remain constant at
a value of * M kmol*. The mol fraction of solute in the
tank is equal to *y* - which is unknown.

- Set up the equations for the system.

- Determine the time constant and steady state gain associated with
*y* when the feed rate is 10 *kmol/h* and the
tank holds 20 *kmols*.

- Sketch the form of the response of
*y* to a sudden
increase, or *step change* in *x* from an initial
steady state.

- If the solute reacts isothermally in a first order with rate
constant 0.5
*/hr* without change of volume in the tank,
what now will be the time constant for the system?

## Question 2

A thermometer whose bulb has mass *m*, heat capacity
*Cp* and surface area *A* indicates a
temperature *T*. The temperature of its surroundings is
*Te* and heat is transferred convectively with overall heat
transfer coefficient *U*. Assume that the thermometer bulb
is at a uniform temperature. Choose realistic values for the parameters
and estimate how fast a typical thermometer indication might respond to
changes in the temperature being measured.

## Question 3

A tank with input flowrate *F kg/s* has variable holdup
*M(t) kg*. The outlet flowrate *L kg/s* is a
function of holdup -

- Set up a model of the system.

- The tank is initially at steady state and contains 100
*kg* of liquid. *Estimate* the steady state gain for
changes in holdup w.r.t inlet flow and the time constant for the tank holdup.

- Suppose that the tank capacity is 500
*kg* and the
maximum feed flowrate is 2 *kg/s*. Given that there is 100
*kg* in the tank when there is a 50 % increase in feed
flowrate, evaluate the dimensionless gain.

- Sketch the response of the tank holdup to this sudden increase.

**Optional** - perform a numerical simulation and compare the
results with the sketched response. Comment on how and why these differ.

## Question 4

A well mixed vessel is fitted with a heating jacket whose
temperature is uniform and has value *Tw*. Fluid enters the
vessel at *F kg/s* and leaves at the same rate. The feed
temperature is * Tf*. The vessel holdup is *M
kg* and its temperature is *T*.

- Set up a model for the system.

- Derive expressions for the steady state gains for
*T* w.r.t

A pressure vessel discharges to atmosphere through a relief valve
with square root characteristic.

- Set up a model for the system.

- Derive an approximate expression for its pressure time constant.

The vessel has a volume of 2 *m3* and is initially at 10
*bar*. The relief valve has a diameter of 20 *mm*.

*Estimate* how long it will take for the pressure to fall to 5
*bar* after the valve has opened.

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