# Optimize Batch Distillation

## Proper design depends upon an understanding of key relationships.

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, *X _{D}*, 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)X _{C} = d(SX_{S}) (*9)

*ln (S _{0}/S_{1}) = (*10)

The overall mass balance for the system gives:

*S _{0} – S_{1} = C*(11)

A mass balance on the mvc yields:

*S _{0}X_{S}0 – S_{1}X_{S}1 = CX_{C}* (12)

and by transposing:

*C/S _{0} = (X_{S}0 – X_{S}1)/(X_{C} – X_{S}1*)(13)

*S _{1} = S_{0}(X_{S}0 – X_{C})/(X_{S}1 – X_{C})* (14)

The Fenske equation uses a separation factor, F, to establish the minimum number of theoretical stages,* N*_{MIN}, required at total reflux to achieve a specified separation of a binary mixture with near-ideal behavior:

*N _{MIN} ln α = ln F(*15)

*F = [X _{D}/(1 – X_{D})][(1 – X_{S})/X_{S}]* (16)

where *X*_{D} and *X _{S}* 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, *R*_{min}, using Underwood’s equation for a binary system:

*R _{min} = [1/(α - 1)][(X_{D}/X_{S}) – α(1 – X_{D})/(1 – X_{S})](*17)

When the distillate must contain the *mvc* at high purity, i.e., *X _{D} >0.995* mole fraction, Eq. 17 simplifies to:

*R _{min} = 1/(α - 1)X_{S}(*18)

For a high separation factor, a minimum relative volatility of 1.5 is considered reasonable, thus setting a top limit of *R*_{min} at 2/X_{S}. Batch distillations should start with* R _{min}* equal to that required for a continuous split;

*R*

_{min}increases as the amount of the

*mvc*in the still decreases.

The total reboiler heat input to reduce the reactor contents from *S*_{0} to *S*_{1} moles for a variable top composition, achieved by setting a fixed reflux ratio, is:

*Q = λ(S _{0} – S_{1})(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(S _{0} – S_{1})* (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,* θ*_{batch}, at constant reflux ratio is given by:

*θ _{batch} = [(R +1)/V](S_{0} – S_{1}) = 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 = UAΔt _{mean} = fjHj(t_{2} – t_{1})*(24)

where Q is the still heat duty, U is the overall heat transfer coefficient, A is the heat transfer area, *Δt _{mean}* is the mean temperature difference,

*f*

_{j}is jacket fluid flow and

*H*

_{j}is jacket fluid specific heat.

*Δt _{mean}*with a still bottoms temperature

*t*and jacket-fluid inlet and outlet temperatures

_{bot}*t*

_{1}and

*t*is approximated by:

_{2}*Δt _{mean} = (t_{2} – t_{1})/2 – t_{bot}*(25)

The overall heat transfer coefficient is the sum of the individual resistances, i.e.:

*1/U = 1/h _{i} + 1/h_{fi} + 1/(k_{w}/x) + 1/ho + 1/hfo* (26)

where h_{i} is the inside film coefficient, h_{fi} is the inside fouling coefficient, k_{w} is the thermal conductivity of the still wall, x is the vessel wall thickness, h_{o} is the outside film coefficient and h_{fo} is the outside fouling coefficient.

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