**2. Process description**

Figure 2 show the input and output streams in a vertical generic effect evaporator with long tubes. The solution to be concentrated circulates inside the tubes, while the steam, used to heat the solution, circulates inside the shell around the tubes.

Predictive Control for the Grape Juice Concentration Process 93

distributed parameters model must be developed in order to be used as a real plant to test

In this work, it is used the mathematical model of the evaporator developed by Ortiz *et al.*  (2006), which is constituted by mass and energy balances in each effect. The assumptions are: the main variables in the gas phase have a very fast dynamical behavior, therefore the corresponding energy and mass balances are not considered. Heat losses to surroundings are neglected and the flow regime inside each effect is considered as

> *i si i dW W WW*

> > *i i ii*

*i i i i si si i i si i*

*i i si i*

*UA T T <sup>W</sup> H h* 1 1 ( )

*si ci*

For each effect, the enthalpy can be estimated as a function of temperatures and

*dt* 1 1 <sup>1</sup> ( ) (3)

in this equations *W i <sup>i</sup>* , 1,...,4 are the solution mass flow rates leaving the effects 1 to 4, respectively. *W*<sup>0</sup> is the input mass flow rate that is fed to the equipment. *W i si* , 1,...,4 are the vapor mass flow rates coming from effects 1 to 4, respectively. *dMi dt i* / , 1,...,4

> *d WX W X WX dt* 1 1

where, *X i <sup>i</sup>* , 1,...,4 are the concentrations of the solutions that leave the effects 1 to 4,

*dW h W h Wh W H AU T T*

where, *<sup>i</sup> h i*, 1,...,4 are the liquid stream enthalpies that leave the corresponding effects, h0 is the feed solution enthalpy, and *H i si* , 1,...,4 are the vapor stream enthalpies that leave the corresponding effects and, *Ai* represents the heat transfer area in each effect. The model also includes algeb raic equations. The vapor flow rates for each effect are calculated neglecting the following terms: energy accumulation and the heat conduction across the

*dt* 1 (1)

( ) (2)

(4)

*H T si si* 2509.2888 1.6747 (5)

*ci si h T* 4.1868 (6)

*i*

*i i*

*si*

represent the solution mass variation with the time for each effect.

respectively. is the concentration of the fed solution.

*i i*

advance control strategies by simulation.

a. Global mass balances in each effect:

b. Solute mass balances for each effect:

completely mixed.

c. Energy balances:

, *Xo*

tubes. Therefore:

concentrations (Perry, 1997). Them:

The evaporator operates in co-current. The solution to be concentrated and the steam are fed to the first effect by the bottom and by the upper section of the shell, respectively. Later on, the concentrated solution from the first effect is pumped to the bottom of the second effect, and so on until the fourth effect. On the other hand, the vapor from each effect serves as heater in the next one. Finally, the solution leaving the fourth effect attains the desired concentration.

Each effect has a baffle in the upper section that serves as a drops splitter for the solution dragged by the vapor. The vapor from the fourth effect is sent to a condenser and leaves the process as a liquid. The concentrated solution coming from the fourth effect is sent to a storage tank.

Fig. 2. Photo of evaporator and scheme of effect i in the four-stage evaporator flow sheet. ݅ ൌ ͳǡ ڮ ǡͶǤ
