Simulation, Parameter Estimation and Optimization of an Industrial-Scale Evaporation System - Simulation and Modeling Methodologies, Technologies and Application - page 71

4 Simulation Results

Figure 2 shows that the trajectories of the smooth model with smoothing parame-
ter τ =0 . 002 bar and of the original hybrid model deviate only marginally from
each other. When the pressure meets the transition condition p = p c (see inset of
Figure 2(a)) the evaporator switches from the non-evaporating mode to the evaporating
mode. As a consequence, the mass fractions of the volatile components B and C jump
according to Eq. 5 from 0 in the non-evaporating mode (no vapor is present) to the finite
values given by the phase equilibrium (see Figure 2(c)) and vapor starts to escape from
the tank (Figure 2(d)). The evaporation of the volatile components B and C leads to a
decrease of their mass fractions w B ,w C in the liquid. The decrease of w C (ethanol)
is more pronounced due to the higher vapor pressure and thus the higher outflow of C.
Consequently, the vapor mass fractions cross each other near t = 1150 s (Figure 2(c)).
Since the pressure in the evaporator and also the vapor outflow (Figure 2(a) and 2(d))
depend on the (temperature-dependent) vapor pressure and the mass fractions of all liq-
uid components, both first increase due to the increasing temperature (Figure 2(b)) and
later decrease due to the reduced mass fractions w B ,w C in the liquid.
In the transition region, the dynamics is given by the linear combination (Eq. 7) of
both operating modes involved. Figure 3 demonstrates that the smooth model approxi-
mates the hybrid model the better the smaller the smoothing parameter τ is chosen: The
slope of the state trajectory ξ C increases and the transition region narrows. The increas-
ing slope of ξ C is to lead back to the increasing slope of the smoothing function ϕ ( x )
(Eq. 6). The width of the transition region can be estimated by looking at the variation
of the smoothing function

Δϕ
Δt ≈

∂ϕ
∂ψ

∂ψ
∂x

∂x
∂t

(8)

linearized around the exact transition point ψ ( x ∗ )= ψ ( x ( t ∗ )) = 0 . With Δϕ =1 ,the
estimate of the width of the transition region will be

4 τ

∂x x ∗ ∂t t ∗

Δt = . (9)

∂ψ

∂x

For the transition between the non-evaporating and the evaporating mode, this takes the
form

4 τ
∂p

∂t t ∗

Δt = (10)

with ∂p/∂t =0 . 002 bar/s (see Figure 2(a)). These estimates are shown in Figure 3. The
actual transition width obey rather well the predicted ratio Δt 1 : Δt 2 : Δt 3 =5:2:
1 . Closely related to the above estimation of the width of the transition region is the
argument put forward by us in [18] that the approximate model matches the original
model satisfactorily for practical use if

ψ≈ 0 ψ≈ 0

dϕ
dt

dx i
dt

∀i, i =1 ...n .

(11)