Equations used during Reflux Distillation Calculations

Knowing the about the wash & column, we can calculate things like recommended min & max column diameters, the purity we expect off the top column, and how long it will take to heat up to temperature. Then, we can look at the column performance over time - how much we expect to collect from it, and how this changes (as does the column performance) over the course of the run.

I've collated all the required Physical constants, etc for water & ethanol on a data page

Min & Max Diameter

(call this "y") vs (call this "x")

Plotting the flooding line, I found : y = =1.4427 x² -1.0271 x + 0.2312 where

U_t = superficial gas velocity (m/s)
a_p = total area of packing (m²/m³ bed)
e = fractional voids in dry packing
g = gravitational constant = 9.8067 m/s²
G' = gas mass rate (kg/s.m²)
L' = liquid mass rate (kg/s.m²)
r_g and r_l = gas and liquid densities (kg/m³)
m_l = liquid viscosity, m Pa.s (cP)

Through the top section we have

V = vapour flowrate (kg/s)
D = distillate (kept) flowrate (kg/s), and
L = liquid flowrate (kg/s)

Based on the composition of the wash, and the power input (Q _distill), you can estimate the amount of vapour you are generating (V in kg/s)

Q _distill = (V * %ethanol * H_{vap ethanol}) + (V * %water * H_{vap water})

Then G' = V / column area ( pD²)

The amount of liquid (L) being refluxed will depend on the Reflux ratio; since D=V-L, and the Reflux ratio is L/D, then

So then L' = L / column area

So... calculate out x =, solve for y, then work out U_t

Since U_t = G' / area of column, and knowing area = p D² you can solve for D _Flood

This is the flooding diameter - eg smaller than this will cause the vapour flowrate to be too fast to allow the liquid to drain down past it. It is the absolute minimum diameter.

So the column needs to be a little wider. Standard recommendations are to use approx 65% the flooding velocity, so scale up the diameter we should use by

Likewise, if the column is too wide, there won't be enough interaction between the liquid and the vapour. I can't find any recommendations for this, but I'm guessing about 25% would be about it, so :

Now the flowrates of Liquid & Gas will be different at different heights of the column... so once you've been through the whole exercise, redo this calculation a couple of times, at a couple of different heights (flowrates). You find out that the differences aren't really worth worrying about.

HETP (Height Equivalent to a Theoretical Plate)

Calculate the effective area (a_w) using :

where:

a_w = effective interfacial area of packing per unit volume (m²/m³)
a = actual area of packing per unit volume (m²/m³)
s_c = critical surface tension for the particular packing material (see table below)
s_L = liquid surface tension mN/m
L^*_w = mass flowrate per unit cross-sectional area, kg/m²s = L'

Then calculate the liquid and gas mass transfer coefficients (k_L and k_G) using :

where:

K₅ = 5.23 for packing sizes >15mm and 2.0 for sizes <15mm
V^*_w = gas mass flowrate per cross-sectional area [kg/m²s] = G'
d_p = packing size [m]
k_L = gas film mass transfer coefficient,[kmol/m²s atm or kmol/m²s bar]
k_G = liquid film mass transfer coefficient [kmol/m²s or kmol/m³ = m/s depending on whether use R=0.08206 atm.m³/kmol K or R=0.08314 bar m³/mol K]

From these you can then calculate the film transfer heights:

where:

P = column operating presure, [atm or bar]
C_t = total concentration, kmol/m³ = r _L / molecular weight solvent
G_m = molar gas flowrate per unit cross-sectional area, kmol/m²s
L_m = molar liquid flowrate per unit cross-sectional area, kmol/m²s

From these you can then estimate the height of the overall gas-phase transfer unit:

m = slope of the equilibrium line (I reckon its about 0.49)
Gm/Lm = slope of the operating line

Now... for a section of the packed column in which the operating and equilibrium lines can be considered straight (hey- that's almost us !), theoretical stages can be converted to numbers of transfer units by :

and then

Z_p = packed bed height
N_t = number of theoretical stages

Mass Balance

Estimate the purity at the top of the column by stepping off the correct number of theoretical stages on the Equilibrium diagram

y = -31.065 x⁶ + 116.08 x⁵ - 169.95 x⁴ + 123.99 x³ - 47.195 x² + 9.1398 x

So... pick a suitable time-step, then calculate the volume removed (flowrate * time). You know the purity of it, so you can then update the total removed, the aggregate purity, and the purity and volume of the wash left in the pot.

For each time-step you can then go back to the start and redo it all again (HETP etc) if you really want to....but its not really worth the effort, as these don't change much.

 Just keep a good track of the various units you're using, cos there's a bit of switching between moles, kgs, and grams along the way (let alone dabbling in Imperial units...).