State whether the following is true or false.
A washout of a well-mixed tank removes the last of the solute in a finite time.
Answer: C
The concentration falls exponentially and approaches zero without ever reaching it, so a specification must state an acceptable residual. Draining and refilling is the practical alternative when a true zero is required.
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Match each situation with the form of its unsteady balance.
List I
A. Tank filling with no outlet
B. Well-mixed washout
C. Batch reactor
D. Semi-batch reactor
List II
1. accumulation equals the generation term
2. accumulation equals the inlet flow
3. accumulation equals inlet plus generation
4. accumulation equals minus the outlet term
Answer: D
Each case keeps a different subset of the general balance, and identifying which terms survive is the whole of setting the problem up. The integration that follows is then a matter of technique. Identifying the surviving terms is the real work.
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Arrange the steps of solving an unsteady state balance in the usual order.
1. state the initial condition
2. choose the system and write the general balance
3. integrate to obtain the time dependence
4. delete the terms that are zero for this case
Answer: A
The initial condition must be stated before integrating, since it fixes the constant that the integration introduces. Deleting terms first keeps the equation as simple as the problem allows. An unnecessary term makes the integration harder.
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A batch reactor has no inlet or outlet, so its species balance reduces to
Answer: B
Correct answer: accumulation equalling generation
A batch vessel is closed, so both flow terms vanish and whatever the reaction generates simply accumulates: \[ \frac{\mathrm{d}N_i}{\mathrm{d}t} = r_iV. \] The equation has time as its independent variable rather than position, which is what makes a batch reactor an unsteady state problem by nature.
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All of the following are unsteady state situations EXCEPT
Answer: C
The column at design conditions has settled and nothing changes with time. The other three all have a non-zero accumulation term and require a differential treatment. Design conditions are by definition a steady state.
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A semi-batch operation is one in which
Answer: D
Feeding one reactant slowly into a charge of another is the common industrial case, used to control a strongly exothermic reaction. The balance keeps both an inlet and a generation term. Only the outlet term is missing from the general form.
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Assertion (A): A strongly exothermic reaction is often run semi-batch rather than as a full batch.
Reason (R): Feeding one reactant slowly limits the rate at which heat is released.
Answer: A
Correct answer: Both A and R are true, and R is the correct explanation of A
Both are true and the reason gives the mechanism. With one reactant fed gradually only a little is present at any instant, so the reaction rate - and with it the heat release - is set by the feed rate rather than by the kinetics. The operator controls the exotherm with a valve, and closing it stops the heat almost at once, which a full batch charge does not allow.
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A tank initially holds 50 kilograms of salt in 500 litres. Pure water enters at 5 litres per minute and the well-mixed solution leaves at the same rate. What is the salt content after 100 minutes?
Answer: B
Correct answer: about 18.4 kilograms
With pure water entering and well-mixed liquid leaving at the same rate, the salt decays exponentially: \[ m = m_0e^{-t/\tau}, \qquad \tau = \frac{V}{\dot{V}} = \frac{500}{5} = 100\ \mathrm{min}. \] At \( t=\tau \) that gives \( 50e^{-1} = 18.4 \) kg. The 31.6 offered is the amount *removed* rather than the amount left, and the volume stays at 500 litres throughout because the flows balance.
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Batch heating of a stirred tank by a coil gives a temperature that
Answer: C
Correct answer: approaches the coil temperature
The driving force for heat transfer is the difference between the coil and the contents, so as the contents warm the difference shrinks and the heating slows. The approach is exponential, and the contents can never exceed the coil temperature - which is why a steam coil at a given pressure sets a ceiling on what the batch can reach, however long it is heated.
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Three of the following require an unsteady state analysis. Which is the odd one out?
Answer: D
Correct answer: a heat exchanger at constant duty
A heat exchanger at constant duty is at steady state: nothing accumulates, and its balance is input equals output with no time derivative. The other three all change with time - a column being started up, a tank being filled, a batch reaction proceeding - so each needs the accumulation term and an integration over time.
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