Batches often are concentrated by applying heat from steam in a jacket (or similar means). Common references such as Kern’s “Process Heat Transfer”  and “Perry’s Chemical Engineers’ Handbook”  cover transient heating and cooling, as does an earlier paper of mine . However, their unsteady-state analyses treat wetted area as a constant, which clearly isn’t the case when a batch is boiled down.
Equally unhelpful, batch distillation literature primarily focuses on predicting component splits via vapor/liquid equilibrium and the Rayleigh Equation. No publication appears to offer a quick way to predict the time needed to distill or evaporate a batch to a desired concentration. This article introduces just such an equation.
DERIVING THE EQUATION
For simplification, let’s assume the liquid is in a jacketed tank and remains on the straight side of the vessel as it is concentrated (Figure 1). For purposes of integration, let’s also assume the heat of vaporization (ΔH), liquid density (ρ), overall heat transfer coefficient (U) and the temperature differential between the steam or other heat-transfer medium in the jacket and the liquid in the tank (ΔT) remain constant.
First, let’s consider the geometry of the batch as it remains on the tank straight side:
The wetted area is:
AS = πDh (1)
The volume relationship is:
VS = πD2h/4 (2)
Dividing Eq. 1 by Eq. 2 provides the area-to-volume relationship:
AS/VS = 4/D (3)
So, the total wetted area is:
AT = AH + 4(VT-VH)/D (4)
Differentiating Eq. 4 with respect to time gives:
dAT/dt = (4/D)dVT/dt (5)
Next, let’s write an alternative expression for dVT/dt using the heat transfer rate and fluid properties:
dVT/dt = -Q/(ρΔH) (6)
Substituting the standard expression for heat transfer yields:
dVT/dt = -UATΔT/ρΔH (7)
Using Eq. 5 to eliminate volume in the relationship leads to:
(D/4)dAT/dt = -UATΔT/ρΔH (8)
Solving for dAT/dt gives:
dAT/dt = -4UATΔT/ρDΔH (9)
At this point, it is convenient to define a characteristic time constant:
Θ = ρDΔH/4UΔT (10)
Substituting Eq. 10 into Eq. 9 gives:
dAT/dt = -AT/Θ (11)
dAT/AT = -dt/Θ (12)
We then can solve Eq.12 per standard procedure:
dln(AT ) = -1/Θdt (13)
Integrating to the indefinite solution gives:
ln(AT) = -t/Θ (14)
Then, substituting the integration limits (At at time t, A0 at time 0), we obtain:
ln(At/A0) = <-t/Θ (15)
Rewriting Eq. 15 provides the following simple equation:
At/A0 = e–t/Θ (16)
where Θ is given by Eq. 10.
We want to concentrate a batch from 735 gal to 617 gal in a 5-ft-diameter tank. The bottom head holds 74 gal and provides 23 ft2 of wetted area, in addition to the jacket. The solvent has a heat of vaporization of 1,036 Btu/lb and a density of 62.3 lb/ft3. We can assume the overall heat transfer coefficient is 50 Btu/(hr∙ft2∙°F). What is the time to evaporate the batch if the ΔT across the jacket is maintained at 165°F?
First, we must convert volumes to cubic feet:
V0 = 735/7.481 = 98.25 ft3
Vt = 617/7.481 = 82.48 ft3
VH = 74/7.481 = 9.89 ft3
Then, we calculate wetted areas per Eq. 4:
A0 = 23 + 4(98.25 - 9.89)/5 = 93.7 ft2
At = 23 + 4(82.48 - 9.89)/5 = 81.1 ft2
From Eq. 10, we determine characteristic time:
Θ = (62.3 × 5 × 1,036)/(4 × 50 × 165) = 9.78 hr
Using a rearrangement of Eq. 15 we get:
t = -Θ ln(At/A0) = - 9.78 ln(81.1/93.7) = -9.78 ln(0.8655) = 1.4 hr
So, the time needed to concentrate the batch from 735 gallons to 617 gallons is 1.4 hr.
MICHAEL J. GENTILCORE is Hazelwood, Mo.-based chemical process engineering manager for Mallinckrodt, a business unit of Covidien that will become a standalone company in 2013. E-mail him at Mike.Gentilcore@Covidien.com
A Area, ft2
D Tank diameter, ft
h Liquid height above head, ft
ΔH Heat of vaporization, Btu/lb
Q Heat transfer rate, Btu/hr
ΔT Temperature difference between jacket and process, °F
t Time, hr
U Overall heat transfer coefficient, Btu/(hr∙ft2∙°F)
V Volume, ft3
ρ Liquid density, lb/ft3
Θ Time constant (per Eq. 10), hr
0 Time zero
H Bottom head
S Straight side of vessel
t Time t
1. Kern, D. Q., “Process Heat Transfer,” p. 624, McGraw-Hill, New York (1950) [still available in McGraw-Hill Classic Textbook Reissue Series].
2. Green, D. W. and Perry, R. H., “Perry’s Chemical Engineers’ Handbook,” 6th Ed., p. 10-38, McGraw Hill, New York (1997).
3. Gentilcore, M. J., “Estimate Heating and Cooling Times for Batch Reactors,” p. 41, Chem. Eng. Progress (March 2000).