SME
Separation Processes · Problem type
Minimum reflux ratio (Underwood)
Use this for “A saturated liquid feed holds 40% of the light component; the distillate must reach 95% and α = 2.5. What is the minimum reflux ratio?”
Diagram
Equilibrium
curve:
y
=
αx
1 + (α − 1)x
with
α
=
2.5;
the
45°
line
is
y
=
x.
Rectifying
line:
y
=
(
R
R + 1
)x
+
xD
R + 1
=
0.684x
+
0.3,
through
(xD,
xD).
q = 1 (saturated liquid feed): the q-line rises straight up from xF = 0.4.
At Rₘᵢₙ = 1.444 the rectifying line touches the curve where the q-line meets it (the pinch): stairs never get past it.
R = 2.167 clears it.
This is an example. Type your own values to start a new problem.
Rₘᵢₙ
=
1
−
1
×
xD
xF
−
×
1
−
1
−
R
=
1.5
×
Rₘᵢₙ
Formulas
Rₘᵢₙ
=
(
1
α
− 1
)
×
(
xD
xF
−
α
×
1 − xD
1 − xF
)
1.4444
=
(
1
2.5 − 1
)
×
(
0.95
0.4
−
2.5
×
1 − 0.95
1 − 0.4
)
R
=
1.5
×
Rₘᵢₙ
2.1667 = 1.5 × 1.4444
Assumptions
•
Constant relative volatility α and constant molar overflow.
•
Saturated liquid feed (q = 1): the pinch sits on the curve straight above xF.
•
The design reflux is taken as 1.5 times the minimum.
Step-by-step
WE KNOW
xF = 0.4, xD = 0.95, α = 2.5
FIND
minimum reflux ratio (Rₘᵢₙ), design reflux ratio (R)
Step 1 · Find minimum reflux ratio (Rₘᵢₙ)
Use Rₘᵢₙ = (1 ÷ (α − 1)) × (xD ÷ xF − α × (1 − xD) ÷ (1 − xF))
At minimum reflux the operating line from (xD, xD) just touches the equilibrium curve above xF; Underwood gives its slope.
Rₘᵢₙ
=
(
1
2.5 − 1
)
×
(
0.95
0.4
−
2.5
×
1 − 0.95
1 − 0.4
)
Rₘᵢₙ
=
1
1.5
×
(2.375
−
0.20833)
Rₘᵢₙ
=
2.1667
1.5
Rₘᵢₙ = 1.44444
Step 2 · Find design reflux ratio (R)
Use R = 1.5 × Rₘᵢₙ
A common design choice: run at 1.5 times the minimum reflux.
R
=
1.5
×
1.44444
R
=
2.1667
Check
Put the numbers back in. Both sides should match.
1.44444
=
(
1
2.5 − 1
)
×
(
0.95
0.4
−
2.5
×
1 − 0.95
1 − 0.4
)
✓
2.1667 = 1.5 × 1.44444 ✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Distillation
Main lesson
Rectifying operating line
Use this for “A column runs at reflux ratio 3 with a 90% distillate. What are the slope and intercept of the top operating line?”
Stages by McCabe–Thiele stepping
Use this for “α = 2.5, a saturated liquid feed at 40%, 95% distillate, 5% bottoms and R = 2.167. How many stages, and which is the feed stage?”