MASS AND ENERGY BALANCES
The primary objective in a batch distillation is to minimize both batch cycle time and heat input by optimizing the number of stages and reflux ratio to achieve the required separation [3, 5]. The process variables are interdependent as determined by the mass and energy balances and the mode of operation.
The reflux ratio
R = L/D (6)
relates to the slope of the operating line,m, via:
m = L/V = R/(R+ 1) (7)
The overall mass balance at the top of the column is:
D/V= 1/(R+ 1) (8)
The mass balance demonstrates that the top composition is established by the D/V ratio, which depends upon the reflux ratio. If the D/V ratio is high, separation will be low and withdrawal of distillate must be stopped while a relatively high value of the mvc mole fraction, XD, remains, i.e., light ends recovery will be poor. If theD/Vratio is reduced to enhance recovery, the distillation may consume an uneconomic amount of time and energy.
A mass balance on the mvc yields the following relationship, known as the Rayleigh equation:
(dS)XC = d(SXS) (9)
ln (S0/S1) = (10)
The overall mass balance for the system gives:
S0 – S1 = C(11)
A mass balance on the mvc yields:
S0XS0 – S1XS1 = CXC (12)
and by transposing:
C/S0 = (XS0 – XS1)/(XC – XS1)(13)
S1 = S0(XS0 – XC)/(XS1 – XC) (14)
The Fenske equation uses a separation factor, F, to establish the minimum number of theoretical stages, NMIN, required at total reflux to achieve a specified separation of a binary mixture with near-ideal behavior:
NMIN ln α = ln F(15)
F = [XD/(1 – XD)][(1 – XS)/XS] (16)
where XD and XS are the mole fractions of the mvc in the distillate and still compositions, respectively, and α is the relative volatility of the two components.
If a given column can achieve the required separation at total reflux, the next step is to determine the minimum reflux ratio, Rmin, using Underwood’s equation for a binary system:
Rmin = [1/(α - 1)][(XD/XS) – α(1 – XD)/(1 – XS)](17)
When the distillate must contain the mvc at high purity, i.e., XD >0.995 mole fraction, Eq. 17 simplifies to:
Rmin = 1/(α - 1)XS(18)
For a high separation factor, a minimum relative volatility of 1.5 is considered reasonable, thus setting a top limit of Rmin at 2/XS. Batch distillations should start with Rmin equal to that required for a continuous split; Rmin increases as the amount of the mvc in the still decreases.
The total reboiler heat input to reduce the reactor contents from S0 to S1 moles for a variable top composition, achieved by setting a fixed reflux ratio, is:
Q = λ(S0 – S1)(R + 1) (19)
where λ is the latent heat of vaporization.
The reboiler heat input for a fixed top composition, achieved by varying the reflux ratio to maintain a fixed top temperature at constant pressure, is:
Q = λ(S0 – S1) (20)
Both relationships indicate that the reflux ratio must be kept to a minimum, subject to satisfying the requirements for the desired separation specification, to minimize the heat input.
The boil-up rate,V, without reflux is:
V = Q/λ (21)
At total reflux, the reboiler heat duty is:
Q = V(λ + Ht (22)
where H is the liquid specific heat andtsub is condenser subcool in °C.
The batch time, θbatch, at constant reflux ratio is given by:
θbatch = [(R +1)/V](S0 – S1) = λ/V (23)
BATCH STILL HEAT TRANSFER
The boil-up rate achievable with stirred jacketed reactors depends upon many factors, including operational temperature difference, jacket heating media and heat transfer considerations [2, 6].
The fundamental equation for heat transfer is:
Q = UAΔtmean = fjHj(t2 – t1)(24)