where Q is the still heat duty, U is the overall heat transfer coefficient, A is the heat transfer area, Δtmean is the mean temperature difference, fj is jacket fluid flow and Hj is jacket fluid specific heat.
Δtmeanwith a still bottoms temperature tbotand jacket-fluid inlet and outlet temperatures t1 and t2 is approximated by:
Δtmean = (t2 – t1)/2 – tbot(25)
The overall heat transfer coefficient is the sum of the individual resistances, i.e.:
1/U = 1/hi + 1/hfi + 1/(kw/x) + 1/ho + 1/hfo (26)
where hi is the inside film coefficient, hfi is the inside fouling coefficient, kw is the thermal conductivity of the still wall, x is the vessel wall thickness, ho is the outside film coefficient and hfo is the outside fouling coefficient.
For glass-lined equipment, the wall thickness consists of the metal thickness, xm, and the glass thickness, xg. The reactor-wall thermal conductivity includes the contributions of both materials:
kw = (km + xg)/(xg/kg + km/km)(27)
where km and kg are the thermal conductivities of the metal and glass, respectively.
The inside and outside fouling coefficients are determined by practical experience; extensive literature exists on this subject. The combined fouling coefficient is given by:
hf = hfihfo/(hfi + hfo) (28)
The stirred-batch-reactor inside film coefficient can be predicted using the Sieder-Tate equation:
Nu = KRe0.667Prn( μb /μw)q(29)
where Nu is the Nusselt number, Re is the Reynolds number, Pr is the Prandt number, K, n and q are empirical constants provided by the agitator manufacturer, μb is the process-side bulk viscosity and μw is the process-side viscosity at the wall. In low viscosity applications, μb/μw approaches unity and can be ignored.
The Nusselt number is:
Nu = hid/kpf (30)
where d is the reactor inside diameter and kpf is the process fluid thermal conductivity.
The Reynolds number is:
Re = ρrI2/μb (31)
whereρ is the density of the process fluid,r is the impeller rotation rate andI is the impeller diameter.
The Prandtl number is:
Pr = Hpfμb/khf(32)
Table 1 provides some typical agitator constants.
The outside film coefficient depends on the type of heat transfer surface, which can be external jacket(s) with or without spiral baffles or mixing nozzles, external half coil(s) or internal coil(s).
For spiral baffled jackets without mixing nozzles, coils and half coils, under turbulent flow conditions ho is determined via:
Nu = hodequ/khf = 0.027Re0.8Pr0.33(33)
where Re is found using the equivalent diameter, dequ, given by:
dequ = 4s/W (34)
wheres is flow cross-sectional area, andW is the wetted perimeter for heat transfer (which is equal to the spiral baffle pitch or inside pipe diameter for external half-coil construction).
ho is evaluated under actual process conditions in the jacket.
For mixing nozzle applications, manufacturers’ proprietary methods are used to calculateRe, raised to a modified coefficient, depending on turbulent or laminar flow conditions.
The batch still duty is calculated via:
Q = UA(t1 – tbot)(35)
As the distillation proceeds, the still bottoms temperature tbotis increasing and the heat transfer areaA is reducing, so the temperature difference must be raised to maintain an acceptable heat duty and boil-up rate. Figure 3 shows typical heat-transfer coefficients that can serve for initial estimates of the still heat duty.
A heat balance allows prediction of the heat-up and cool-down cycle times:
Q = UA(t1 – tbot) = 60(dt/dθ)(wHstill + MHbot + JHj) (36)
where dt/dθ is the still bottoms temperature change expressed as rate per minute (Q and U units based on hours), w is the still equipment weight, Hstill is the specific heat of the still material of construction, M is the mass of the still bottoms fluid, Hbot is the specific heat of the still bottoms fluid, J is the jacket fluid mass and Hj is the specific heat of the jacket fluid.
The rate of temperature change during heat-up is calculated via:
dt/dθ = Q/[60(wHstill + MHbotj)] (37)
The time θ in minutes for temperature change from t1bot to t2bot can be calculated using mean values for the specific heat andU in the temperature range under consideration:
θ = [(wHstill + MHbot + JHj)/UA] ln (t1 – t1bot)/(t1 – t2bot) (38)
Applying batch distillation simulation software can lead to significant savings in process development and production costs. Achieving a successful simulation requires an understanding of the relevant thermodynamics for the batch distillation process.
As demonstrated, the mass and energy balances control the interdependence of process parameters. Setting one process parameter fixes all dependent parameters; setting two independent process parameters defines the batch distillation operating state.