Surprise Quiz 1
Q1. Consider a counter-current gas absorption column with 3 ideal equilibrium stages to
remove a polutant from a gas stream. Given: Gas flow V = 1 mol/s, liquid flow L = 1
mol/s, equilibrium relation yi = Kxi with K = 0.8, inlet solvent x0 = 0.1, inlet gas to
last stage y4 = 0.
(a) Write the stage balances and equilibrium equations for stages i = 1, 2, 3.
L
.
(b) Rearrange these equations using the stripping factor, S = KV
(c) Arrange the equations in matrix form T x = b with x = (x1 , x2 , x3 )⊤ . (No need to
solve.)
(d) what is the direction of mass transfer – from x to y or from y to x?
(e) In addition to y4 = 0, if x3 = 0.005 is also specified, and you have to determine V
that is needed for this stripping problem, can you still formulate the problem as a
linear algebra problem? If not how should one proceed with solving this problem?
Q2. A perfectly mixed tank of volume VT contains a solute with concentration C(t). The
system has the following characteristics:
• Flow in = Flow out = F (constant)
1
• Inlet concentration:
(
Cin,1 , 0 ≤ t < ts ,
Cin (t) =
Cin,2 , t ≥ ts
• First-order reaction kinetics: r = kC with k > 0
• Initial condition: C(0) = C0
Define τ = VT /F and λ = k + 1/τ .
(a) Write the differential balance for C(t).
(b) For the step input above, derive the piecewise solution C(t) for 0 ≤ t < ts and
t ≥ ts , ensuring continuity at ts .
(c) What will be the steady state outlet concentration?
(d) If the feed rate is increased, how will the outlet concentration change (increase or
decrease)?
2
kC
(e) If the reaction rate expression is replaced by r = (1+αC
2 ) , comment on the possibility
to get an analytical expression for outlet concentration? If that is not possible, how
will you approach to get a numerical solution for C(t)?
Q3 Let λ1 , λ2 denote the eigenvalues of the Jacobian J at the origin (they may be real or
a complex-conjugate pair). For each of the following fixed-point types at the origin, state
the condition on the eigenvalues explicitly in terms of whether they are real or complex
and the sign of their real parts:
• stable node
• unstable node
• saddle point
• stable spiral (focus)
• unstable spiral (focus)
• centre
Q4. Consider the nonlinear ODE
ẋ = x − x3 .
(a) Find all fixed points and classify their stability.
(b) Sketch the phase portrait on the real line (arrow directions in each region).
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(c) For x(0) = 0.5, describe qualitatively how x(t) evolves as t → ∞.
4
0
