Dynamic modeling and control of soybean meal drying in a direct rotary dryer

food and bioproducts processing 8 8 ( 2 0 1 0 ) 90–98

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Dynamic modeling and control of soybean meal drying in a direct rotary dryer Gianini Regina Luz, Wagner André dos Santos Conceic¸ão, Luiz Mário de Matos Jorge, Paulo Roberto Paraíso, Cid Marcos Gonc¸alves Andrade ∗ Department of Chemical Engineering, State University of Maringá, P.O. Box 87020-900, Maringá, PR, Brazil

a b s t r a c t This paper presents the analysis of the dynamic model and the application of Proportional, Integral and Derivative (PID) control in the soybean meal drying in a industrial direct rotary dryer by computational tests. Therefore, load disturbances as step, pseudorandom, and impulse were applied in the inlet speed and in the inlet moisture of soybean meal in the dryer. Later, the output responses for perturbations were investigated with control systems. The application of feedback PID control showed satisfactory results, returning expected output moisture. According to the Integral Square Error (ISE) the manipulated variables that showed better controllability were the inlet speed, when the perturbation was in the soybean meal inlet moisture; and the inlet temperature of drying air, when the perturbation was in the inlet speed of soybean meal. These results are coherent with literature and conclude that, the tuning feedback PID control keeps the output moisture of soybean meal inside the specifications with fast results. © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Soybean meal; Rotary drying; Dynamic model; Feedback PID control

1.

Introduction

Soybean meal is an important product of the soybean processing industry, playing a significant role worldwide. It is used as main source of protein for animal feed and as supplementary food for humans. Soybean meal is obtained by a set of operations including drying which consumes a great amount of energy, reaching at 80 kg of steam per ton of processed soybean, according to Jongeneelen (1976). To maintain the soybean meal quality at a lower cost it is necessary to establish strategies to optimize the drying process. Currently industrial drying is done in an indirect rotary dryer. This process presents deficiency in the heat and mass transfer and great variability in the quality of the final product. These facts show the necessity of investigating other alternatives for the industrial drying of soybean meal. The chosen alternative for the study was the drying of soybean meal in the direct rotary dryer. This alternative provides better gradients of temperature and product moisture inside the dryer. Furthermore, this dryer shows better facility in controlling the input variables of the process and consequently, the final quality of soybean meal. A dynamic investigation of

∗

the process is essential for a complete analysis of behavior of the variables during the drying process. In addition, the investigation contributes to implant forms of control to minimize the process disturbance and maximize the production. For the successful completion of these the limits of moisture and temperature of the product must be respected (Courtois and Trystram, 1994). The investigation of the process dynamics requires the modeling in transient regime based on mass and energy balances and constitutive equations, creating a nonlinear system. From this, a series of simulations are done to verify the behavior of the involved variables in this process. Conforming to Åström and Hägglund (2006), the analysis of dynamics of process can be performed by the application of disturbances on some input variables of the model. This action permits the observance of the deviations of output variables. From the dynamic study realized, the control system is the initial part. According to Yoannou and Sun (1996), this system must ensure a good performance of the process from model. This is necessary due to the model corresponding to an approximate behavior of the plant, and may show discrep-

Corresponding author. Tel.: +55 44 3261 4752; fax: +55 44 3261 4792. E-mail address: [email protected] (C.M.G. Andrade). Received 27 May 2009; Received in revised form 12 December 2009; Accepted 20 January 2010 0960-3085/$ – see front matter © 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.fbp.2010.01.008

food and bioproducts processing 8 8 ( 2 0 1 0 ) 90–98

Nomenclature A, B, D

coefficients matrix of the controlled variables, input variables and output variables a11 , a12 , a21 , a22 components of the matrix A b11 , b12 , b13 , b21 , b22 , b23 components of the matrix B available cross-sectional area of the soybean Adryer meal in the dryer (m2 ) C(s) load variable Ca specific heat of the dry air (J/kg ◦ C) specific heat of the soybean meal (J/kg ◦ C) Cs specific heat of water steam (J/kg ◦ C) Cv specific heat of water (J/kg ◦ C) Cw e(s) error Ga mass flow of the air (kg/min) gm margin of gain Gs mass flow of soybean meal (kg/min) Gp (s) transfer function of process Grt (s) transfer function of the delay for transport transfer function of the sensor/transmitter Gt (s) G1 (s) transfer function of the manipulated variable transfer function of the manipulated variable G1 (s) with delay for transport G2 (s) transfer function of the load variable G2 (s) transfer function of the load variable with delay for transport average mass transfer coefficient of the soyKm bean meal-air (s−1 ) kc, ku gain process L total length of the dryer (m) M(s) manipulated variable Ma dry air inlet mass (kg/s) Ms dry soybean meal inlet mass (kg/s) P atmospheric pressure (kPa) P(s) variable signal pm margin of phase pu period of oscillation S(s) output signal of control t time (s) tr time delay in transporting the soybean meal (s) average temperature of the air (◦ C) Tam Taini inlet temperature of the air (◦ C) Tbsm average dry bulb temperature of the air (◦ C) average temperature of the soybean meal (◦ C) Tsm Tsini inlet temperature of soybean meal (◦ C) average wet bulb temperature of the air (◦ C) Tum Uv volumetric heat transfer coefficient (kW/m3 ◦ C) u input variables urm average relative humidity of air volume of the dryer (m3 ) Vsec inlet speed of the soybean meal (m/s) vs va inlet speed of the air (m/s) average equilibrium moisture of the soybean Xem meal (d.b.) x controlled variables x˙ differential equations to mass and energy balances Xsm average outlet moisture of the soybean meal (d.b.) Xsini inlet moisture of the soybean meal (d.b.) z length of the control volume of the dryer (m)

Ym y Yini Y(s) Yr (s) w, wcp,  m  i d

91

average air moisture (d.b.) output variables air inlet moisture (d.b.) controlled variable desired value (set point) wcg frequencies difference between the values in transient regime and permanent regime average latent heat of vaporization of water at the temperature of the soybean meal (J/kg) time delay for transport (s) integral time derivative time

ancies. As such, the control system requires computational tests before their real application. The design of control is based on choice of manipulated and led variables, transformation of the mathematical model in state equations, dynamics of process, combination of these variables and in the selection of the best configuration (Mujumdar, 2006). Cao and Langrish (2000) reported that the rotary dryers show challenging characteristics in the project development and implementation of control systems. These facts occur due to physical properties of the product and the design and operational conditions of the dryer. According to Pérez-Correa et al. (1998) the input temperature, the gas flow rate, the solid feed rate and the rotation speed of the dryer can be manipulated in a direct rotary dryer. The controlled variables may be the inlet and the outlet gas temperature, the gas flow rate, temperature and moisture of the solid and output gas moisture (Lipták, 1998; Savaresi et al., 2001). Recent researches are being made in the attempt to improve the drying of solid materials. The work of Yliniemi (1999) shows the application of step load disturbances in the drying air temperature, moisture content and inlet speed of calcite in the direct rotary dryer. This study aimed to implement various control strategies such as feedforward-feedback, fuzzy logic and neural network. Jover and Alastruey (2006) applied step load disturbances in the initial moisture of a material in order to manipulate the initial temperature, airflow, rotation speed and flow of product in a direct rotary dryer. Conforming to Åström and Hägglund (2006), currently the Proportional, Integral and Derivative (PID) control is used in around 90% of industrial plants, being the first form used experimentally for feedback control. Willis (1999) describes the feedback PID control as a robust algorithm, easily understood and its performance depending on the dynamic process of the plant. However, the feedback PID control can show unsatisfactory performance in some cases due to poor tuning (Arruda et al., 2008). The aim of this work was to evaluate the dynamics of industrial drying of soybean meal and the application of feedback PID control in a direct rotary dryer by computational tests. Therefore, step, pseudorandom and impulse load disturbances were applied to the inlet speed and inlet moisture of the soybean meal in the dryer.

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Fig. 1 – Scheme of the direct rotary dryer.

2.

Direct rotary dryer

Fig. 1 shows the configuration of a typical direct rotary dryer. It consists of a long cylindrical shell, slightly inclined to the horizontal mounted on large steel rings, termed riding rings, or tires that are supported on fixed trunnion roller assemblies (Devki Energy Consultancy Pvt. Ltd., 2006). The wet product is fed into the upper extremity of the cylinder, dries and outputs in the lower extremity. The product passes through the cylinder by lifts fixed on its wall longitudinally arranged to promote loss of moisture of the cascading solid in contact with heated air. This air can circulate in concurrent or against the solid flux. The removal of wet air is done by exhaustion (Treybal, 1980). For computational tests in this work, it was assumed that the direct rotary dryer is adiabatic and shows concurrent flow; has a drum length L of 17 m; diameter of 1.84 m; slope of 0.087 m/m, rotates at 7 rpm, has an area Adryer of 2.68 m2 and volume Vdryer of 46 m3 . These properties correspond to a real industrial indirect rotary dryer of soybean meal projected by L.C. Masiero Ltda. (São Paulo - Brazil) to an industry in the northwestern region of the State of Paraná in Brazil. The mathematical model of the dryer was developed from mass and energy balances considering the solid phase composed of soybean meal and gas phase composed by heated air as shown in Fig. 2. The inlet and outlet data of the dryer operating on permanent regime were specified in Table 1 assuming that the soybean meal and air speed, specific heat of the soybean meal, water and air and the soybean meal and air mass remain constant throughout time.

3.

Methodology

direct drying of soybean meal and the application of feedback PID control, by computer, in this mathematical model.

3.1. Dynamic mathematical model of soybean meal drying in the direct rotary dryer The developed mathematical model for direct rotary dryer is based on the following equations of mass and energy balances.

3.1.1.

Mass balance

Mass balance equation for moisture in the soybean meal is represented by Eq. (1): dXs vs = − (Xs − Xsini ) − K(Xs − Xe ) dt L

(1)

Mass balance equation for moisture in the air is dY Ms va K(Xs − Xe ) = − (Y − Yini ) + dt L Ma

(2)

Mass transfer coefficient, K, varied with time and was calculated in function of Xs and of air inlet temperature, Ta , according to Eq. (3) (Luz et al., 2006c): K = (−0.33 × 10−7 Ta + 4.60 × 10−5 )Xs 2 + (7 × 10−6 Ta + 1.42 × 10−3 )Xs + 8.44 × 10−3

(3)

However, for the soybean meal moisture in fractional unit it is necessary to use K according to Eq. (4): K = (−0.33 × 10−11 Ta + 4.60 × 10−9 )Xs 2 + (7 × 10−8 Ta + 1.42 × 10−5 )Xs + 8.44 × 10−3

The methodology used in this work consists of two steps: the development of the dynamic mathematical model for the

(4)

Equilibrium moisture, Xe , varied along the time and was calculated by Eq. (5) according to Luz et al. (2006a): Xe = Fig. 2 – Scheme of a control volume of the dryer for the mass and energy balances.

0.834 (1 + 0.036(Ts + 273.15) ln(1/ur))

(5)

Relative humidity of the air, ur, was obtained by Eq. (6) according to Marin et al. (2001) and it was considered constant

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food and bioproducts processing 8 8 ( 2 0 1 0 ) 90–98

Table 1 – Inlet and outlet date of the direct rotary dryer. Operational data Property

Symbol

Mass flow of the dry soybean meal (kg/min) Mass flow of the dry air (kg/min) Dry soybean meal inlet mass (kg) Dry air inlet mass (kg) Inlet moisture content of soybean meal (d.b.) Outlet moisture content of soybean meal (d.b.) Air inlet moisture (d.b.) Air outlet moisture (d.b.) Inlet temperature of the soybean meal (◦ C) Outlet temperature of the soybean meal (◦ C) Air inlet temperature (◦ C) Air outlet temperature (◦ C) Inlet speed of the soybean meal (m/s) Inlet speed of the air (m/s) Specific heat of the dry soybean meal (kJ/kg ◦ C) Specific heat of water liquid (kJ/kg ◦ C) Specific heat of water vapor (kJ/kg ◦ C) Specific heat of the dry air (kJ/kg ◦ C)

Data

Gs

5933.5

Ga Ms Ma Xsini

74 3634 29,074 0.22

Xf

0.14

Yini Y Tsini

0.13 0.1322 85

Ts

89

Taini Ta vs va Cs

89 ◦ C 90 ◦ C 0.082 0.46 1.76

Cw Cs Ca

4.19 1.89 1.01

Volumetric heat transfer coefficient between air and product, Uv , was calculated according to Eq. (9) was obtained experimentally in kcal/◦ C min m3 by Lisboa et al. (2004). Uv was considered constant along the time: G0.68 Uv = 10.87G0.19 s a

(9)

Considering that transient model of the indirect rotary dryer can be validated, for analysis and tuning of the control system, from of permanent model validated by Luz et al. (2009) according to Vogel (1992), and the transient model of the direct rotary dryer only shows the heat transfer coefficient as different term among these transient models terms (Eq. (9) validated by Lisboa et al., 2004), it was concluded that the proposed model of direct rotary dryer can be used within an acceptable error range.

3.1.3.

Dynamic model

The dynamic model obtained from mass and energy balances equations was linearized using Taylor’s series in the form of Eqs. (10) and (11): x˙ = Ax + Bu

(10)

y = Dx

(11)

along the time: 0.6108 × 107.5Tu /(237.3+Tu ) − 0.00067P(Tbs − Tu ) ur = 0.6108 × 107.5Tu /(237.3+Tu )

(6)

Being P local atmospheric pressure, Tu wet bulb temperature and Tbs dry bulb temperature. The additional equations necessary for the solution of the differential equations are the same ones shown in Luz et al. (2006b), Luz (2006) and Table 1.

3.1.2.

Energy balance

Energy balance equation for moisture in the soybean meal is represented by Eq. (7): dTs vs = − (Ts − Tsini ) dt L 1 + (Cs + Xsini Cw )



Uv Vdryer (Ta − Ts ) Ms

 − K(Xs − Xe )

(7)

Mass balance equation for moisture in the air is dTa va = − (Ta − Taini ) dt L +

1 Ca + Yini Cv

 −

Uv Vdryer (Ta − Ts ) Ma



+

Ms K(Xs − Xe ) Ma

(8)

Being x˙ the differential equations of mass and energy balances, x the controlled variables, u the input variables, y the output variables and A, B, D are the coefficients of the respective variable. The choice of possible variables to be manipulated was made with simulations of soybean meal drying in a direct rotary dryer. From this, it was observed that the variables that influence most the variation of soybean meal outlet moisture, Xs , and the soybean meal outlet temperature, Ts , were the soybean meal inlet speed in the dryer, vs , the heating air temperature of the product, Ta , and soybean meal inlet moisture, Xsini (Luz, 2008). From these results, the cases shown in Table 2 were suggested to define the best strategy of control for the soybean meal drying. The study of Xsini and vs as perturbed variables in Table 2 due to changes in the soybean meal output moisture in the desolventizating/toastering that is the previous process to drying and to imposition of short and or long time of contact among the involved phases that resulting in dry less or over drying, respectively. ˙ x, u, A, B and D can be written by Eqs. (12)–(19): Thus, x,

⎡ dX

sm

x˙ = ⎣

dt dTsm dt

⎤ ⎦

(12)

Table 2 – Variables of the feedback loop. Cases I II III

Disturbed variable Xsini vs Xsini

Manipulated variable vs Taini Taini

Controlled variable Xs Xs Xs

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Fig. 3 – Block diagram of a feedback PID control system.

 x=

Xsm

(13)

Tsm

a22 = −



Being for case I:



Xsini

u=

vs + L

×



1 Cs + Xsini Cw

Uv Vdryer Ms

− m

0.0834



Being for case II:



vs

u=

+Km Cv



(14)

vs







−0.0834 1+0.036(Tsm +273.15)(log(1/urm ))



0.036(log(1/urm )) (1 + 0.036(Tsm + 273.15)(log(1/urm )))

− Cv Xsm





2

(23)

(15)

Taini

Being for case I: Being for case III:



Xsini

u=



A=

 B=

vs L

(24)

b22 =

(Tsini− Tsm ) L

(25)

b12 =

(Xsini− Xsm ) L

(26)

(16)

Taini



b11 =

a11

a12

a21

a22

b11

b12

b21

b22

(17)

(18)

b21 =

1 1

× (2(−0.33 × 1011 Tam + 4.6 × 10−9 )Xsm + 7 × 10−8 Tam



a21 =



−0.0834

1 Cs +Xsini Cw

(1 + 0.036(Tsm + 273.15)(log(1/urm )))



b21 = (20)

b22 =



0.036(log(1/urm )) 2

Ms

 − m Km (Xem− Xsm )

(Xsini− Xsm ) L

+ 2(7 × 10−8 Tam + 1.42 × 10−5 )Xsm + 8.44 × 10−3 )

(26 )

(27)

(Tsini− Tsm ) L



1 Ca + Yini Cv



(21)

×

Uv Vdryer Ms

(28) (29)



− m (Xem− Xsm )(0.33 × 10−11 Tam Xsm 2



+ 7 × 10−8 Xsm )

2 (−m (3(−0.33 × 10−11 Tam + 4.6 × 10−9 )Xsm )

(30)

Being for case III:

+ Xem (2(−0.33 × 10−11 Tam + 4.6 × 10−9 )Xsm + 7 × 10−8 Tam + 1.42 × 10−5 )

Uv Vdryer (Tam − Tsm )

b12 = (Xem− Xsm )(0.33 × 10−11 Tam Xsm 2 + 7 × 10−8 Xsm )

+ 2(7 × 10−8 Tam + 1.42 × 10−5 )Xsm + 8.44 × 10−3 )Xem

+ 1.42 × 10−5 )



Being for case II: b11 =

vs 2 = − − (3(−0.33 × 1011 Tam + 4.6 × 10−9 )Xsm L

a12 =Km

×

(19)

Being the elements of the matrixes A and B defined by Eqs. (20)–(34): a11

−Cw Cs + Xini Cw



 D=



(22)

b11 =

vs L

(31)

food and bioproducts processing 8 8 ( 2 0 1 0 ) 90–98

b12 = (Xem− Xsm )(0.33 × 10−11 Tam Xsm 2 + 7 × 10−8 Xsm )

b21 =



−Cw Cs + Xini Cw



×

95

(32)



Uv Vdryer (Tam − Tsm ) Ms

 − m Km (Xem− Xsm )

(33) Fig. 4 – Rate of drying curves to air temperature at 50, 74, 85 and 97 ◦ C.

b22 =



1 Ca + Yini Cv



×

Uv Vdryer Ms

+ 7 × 10

−8



according to Eq. (29): Xs (s) = G1 (s)M + G2 (s)C

− m (Xem− Xsm )(0.33 × 10−11 Tam Xsm 2

 Xsm )

(34)

The application of a disturbance in C annuls M and the transfer function G2 (s) can be calculated by Eq. (37): G2 (s) =

Later, the step, impulse and pseudorandom disturbances were applied at Xsini of u (case I – Eq. (14), case II – Eq. (16)) and at vs of u (case I – Eq. (15)). The step was at 0.07 in the first and second types of disturbance. Later all the respective response curves were checked. The solution of the system in each kind of disturbance was obtained numerically and graphically using the MATLAB® software.

3.2.

Application of feedback PID control

The block diagram shown in Fig. 3 represents the feedback PID control loop. This one shows a sensor/transmitter, a control and a process. The transfer functions that represent this diagram are given in Eq. (28) (Lipták, 1998):

(36)

Xs −1 = D(sI − A) B C

(37)

Being I the identity matrix. Similarly, a disturbance in M annuls C and the transfer function G1 (s) can be calculated by Eq. (38): G1 (s) =

Xs −1 = D(sI − A) B M

(38)

At first, the system was considered as concentrate parameters, according to Eqs. (1), (2), (7) and (8). However, the system behaves as distributed parameters; to add this characteristic to the model, a function of delay for transportation in the dryer was added, Grt (s) according to Eq. (39): Grt (s) = exp(− · s)

(39)

Being  given by Eq. (40): Ys (s) =

Gc (s)G1 (s)Gp (s) M(s) 1 + Gc (s)G1 (s)Gp (s)Gt (s) +

G2 (s)Gp (s) C(s) 1 + Gc (s)G1 (s)Gp (s)Gt (s)

= (35)

Each of these functions the tuning parameters of feedback PID control were determined by using the MATLAB® software and the system was divided in sections shown as follows. The transfer functions G1 (s) and G2 (s) were determined for each case of Table 2 initially considering the control without system (open loop) as shows in Fig. 4. In Fig. 4, M is the manipulated variable, C the disturbed variable and Xs the controlled variable. From this, the variation of soybean meal outlet moisture, Xs , to each case of Table 2 was obtained by the linear systems in the dominion of Laplace

L vs

(40)

After, the transfer functions G1 (s) and G2 (s) were multiplied by Eq. (39) and became G1 (s) and G2 (s) as show by Eqs. (41) and (42) respectively. In the next step, the transfer functions G1 (s) and G2 (s) were added to system by suggestion from Coughanowr (1991): G2 (s) =

Xs −1 = (D(sI − A) B) exp(− · s) C

(41)

G1 (s) =

Xs −1 = (D(sI − A) B) exp(− · s) M

(42)

Then, they were implemented in a system with control as shown in Fig. 5.

Fig. 5 – Block diagram of a system with control.

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Table 3 – Adjusted parameters of the feedback PID control. Cases

Process gain, kc

Integral time,  i

Derivative time,  d

10.8985 22.3441 10.8985

36.2839 36.2825 36.2839

9.0710 9.0706 9.0710

I II III

Fig. 7 – Response curves of the output moisture with a disturbance in vs and a manipulation in Ta .

Fig. 6 – Response curves of the output moisture with a disturbance in Xsini and a manipulation in vs .

Luyben, 1997): In Fig. 5, a block diagram can be observed of a system with control in the dryer that has controlled variable Xs (s). The transfer functions that represent this diagram may be conferred by Eq. (43):

Xs (s) =

Gc (s)G1 (s)

1 + Gc (s)G1 (s)Gt (s) +

G2 (s)

ku 1.7

(47)

i =

pu 2

(48)

d =

pu 8

(49)

M(s)

1 + Gc (s)G1 (s)Gt (s)

C(s)

(43)

In this system, transfer functions G1 (s) and G2 (s) were defined by Eqs. (41) and (42). Gt (s) and Gc (s) are presented below. Transfer function Gt (s) for the time delay in transporting the soybean meal between the wake of the dryer outlet and the temperature reading sensor, tr , at 20 s, is defined by Eq. (44). The value of tr was estimated from an industrial situation: Gt (s) = exp(−tr · s)

kc =

(44)

Using the values obtained by Eqs. (47)–(49) was possible to calculate the transfer function of the PID control, Gc (s), shown by Eq. (50): Gc (s) =

kc (  s + i s + 1) i s i d

Thus, after determining G1 (s), G2 (s), Gt (s) and Gc (s) the new value of the control variable, Xs (s), was calculated by Eq. (43). The performance of the PID control for each case of Table 2 was evaluated according to the Integral Square Error (ISE), as

Transfer function Gc (s) was obtained by multiplying the transfer function G2 (s) and Gt (s) to find the magnitude, mag, phase in step, phase, and frequency, w, by bode method. These properties were used to calculate the margin of gain, gm, and of phase, pm, and the frequencies, wcg and wcp (for SISO). After, these new properties were used to obtain the constants of process gain, ku, and the period of oscillation, pu, according to Eqs. (45) and (46): ku = gm pu =

2 wcg

(45)

(46)

From ku and pu were estimated the constants of the process gain, kc, integral time,  i , and derivative time,  d , according to Ziegler-Nichols method shown by Eqs. (47)–(49) (Luyben and

(50)

Fig. 8 – Response curves of the output moisture with a disturbance in Xsini and a manipulation in Ta .

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Table 4 – ISE calculated for disturbances. Cases

Step disturbance

Pseudorandom disturbance

−7

−9

3.8375 × 10 2.9949 × 10−14 6.4728 × 10−16

I II III

3.5722 × 10 1.5496 × 10−17 2.6714 × 10−18

shown in Eq. (51):



∞

e2 (t) dt

ISE =

(51)

0

4.

Results and analysis

The analysis of performance for the feedback PID control on the direct rotary dryer of soybean meal was made by comparison of the obtained results between the cases with and without control. The adjusted parameters to the feedback PID control calculated according to Ziegler-Nichols method for each case of the Table 2 can be conferred in Table 3. The results in Table 3 show small variations in the gain process and in the integral time among the examined cases. The obtained response curves by variation of product output moisture with the time without a control system (item 3.1) and with a PID control system (item 3.2) for cases proposed I, II and III are shown below. Figs. 6–8 can be observed that the response curves have a time delay of 207 s for the dryer without and with control. After this time, occurs a gradual increase of soybean meal moisture in the without control system until this one reaches the steady state. In Figs. 6 and 8 the moisture stabilizes in 490 s with 0.36 (d.b.) and in Fig. 7 in 536 s with 0.18 (d.b.). Considering these results, the variation of soybean meal inlet moisture is the variable that influences most the soybean meal outlet moisture as shown in Figs. 6 and 8 and by Yliniemi (1999) in the calcite drying. In each controlled system can be observed that after passing the time delay the response curves have a little oscillation in function of the kind of disturbance in Xsini or vs . For the step disturbance, the soybean meal moisture presents a variation followed of reach of the steady state. In Fig. 6 this variation happens up to 475 s with 3.64 × 10−3 , −2.63 × 10−4 , and 1.96 × 10−5 (d.b.), in Fig. 7 up to 460 s with 1.01 × 10−6 , −7.34 × 10−8 , and 5.47 × 10−9 (d.b.), and in Fig. 8 up to 460 s with 1.48 × 10−7 (d.b.), −1.08 × 10−8 and 8.04 × 10−10 (d.b.). In these cases, the period of oscillation is of 72.5 s, of rise time is of 295 s, the time to reach the first peak is of 233 s and the decay ratio is of 5.4 × 10−3 . This behavior has already been expected to be characteristic of controlled systems by feedback PID according to Åström and Hägglund (2006) and Alastruey and Jover (2006). For pseudorandom disturbances, can be verified that several alterations of the soybean meal outlet moisture are present with magnitude at 10−4 (d.b.) for Fig. 6, and at 10−8 (d.b.) for Figs. 7 and 8. This occurs due to the control system responds the constant changes of soybean meal inlet moisture. For the impulse disturbance, the soybean meal outlet moisture shows small oscillations but they swiftly achieve the steady state. In Fig. 6 these variations are of 2.13 × 10−4 , −1.05 × 10−4 and 5.25 × 10−6 (d.b.) up to 283 s, in Fig. 7 of 5.69 × 10−8 , −2.86 × 10−8 , of 1.47 × 10−9 and −1.11 × 10−10 (d.b.) up to 468 s, and in Fig. 8 of 8.36 × 10−9 , −4.20 × 10−9 , and

Impulse disturbance 8.9887 × 10−10 6.5354 × 10−17 1.4126 × 10−18

2.16 × 10−10 (d.b.) up to 410 s. The time to reach the first peak is of 215 s in Fig. 6 and of 218 s in Figs. 7 and 8. At the end of obtaining the response curves, the value of the Integral Square Error, ISE (Eq. (51)) was determined for cases I–III to evaluate the performance of feedback PID control. The respective values are shown in Table 4. Table 4 shows that the load disturbance in vs or Xsini and manipulation in Ta (cases II and III) results similar performance. Moreover it can be observed that the three cases provide satisfactory results in the application of the feedback PID control.

5.

Conclusions

The dynamic modeling of the soybean meal direct rotary dryer was developed and showed consistent results with the literature. Furthermore, this model also showed good capacity to correct the variations of soybean meal outlet moisture when submitted to disturbances in the soybean meal inlet speed and soybean meal inlet moisture. The application of the feedback PID control on the variables that suffered disturbances also showed satisfactory results returning the drying process to desired value of the soybean meal outlet moisture. Comparing the response curves among the controlled systems with disturbance on soybean meal inlet moisture among cases I–III, similar changes in the soybean meal outlet moisture were obtained. According to the criterion for control tuning of the Integral Square Error, ISE, the manipulated variables that showed better controllability were the inlet speed when the perturbation was in the soybean meal inlet moisture and the inlet temperature of drying air when the perturbation was in the inlet speed.

Acknowledgment The authors thank gratefully acknowledgement CAPES for financial support.

References Åström, K.J. and Hägglund, T., (2006). Advanced PID Control. (ISA-Instrumentation, Systems and Automation Society, Research Triangle Park, NC 27709). Arruda, L.V.R., Swiech, M.C.S., Neves, F., Jr. and Delgado, M.R., 2008, Um método revolucionário para sintonia de controladores PI/PID em processos multivariáveis. Controle & Automac¸ão, 19(1): 1–15. Cao, W.F. and Langrish, T.A.G., 2000, The development and validation of a system model for a countercurrent cascading rotary dryer. Dryer Technol, 1–2(18): 99–115. Coughanowr, D.R., (1991). Process Systems Analysis and Control. (McGraw-Hill, New York). Courtois, F. and Trystram, G., 1994, Study and control of the dynamics of drying processes. ACoFoP, 289–296.

98

food and bioproducts processing 8 8 ( 2 0 1 0 ) 90–98

Devki Energy Consultancy Pvt. Ltd., 2006, Best Practice Manual Dryers, http://www.energymanagertraining.com/ CodesandManualsCD-5Dec%2006/BEST%20PRACTICE% 20MANUAL%20-%20DRYERS.pdf (accessed March 2007). Jongeneelen, H.P.J., 1976, Energy conservation in solvent extraction plants. J Am Oil Chem Soc, 53: 291. Jover, C. and Alastruey, C.F., 2006, Multivariable control for an industrial rotary dryer. Food Control, 17: 653–659. Lipták, B., (1998). Optimization Industrial Unit Processes. (URC Press, Florida), pp. 227–252 Lisboa, M.H., Alves, M.C., Vitorino, D.S., Delaiba, W.B., Finzer, J.R.D. and Barrozo, M.A.S., 2004, Study of the performance of the rotary dryer with fluidization, In Proceedings of the 14th International “Drying Symposium” (IDS 2004) , pp. 1668–1675. Luyben, W.L. and Luyben, M.L., (1997). Essentials of Process Control. (McGraw-Hill, New York). Luz, G.R., 2006, Modelagem Matemática e Análise do Secador Rotativo de Farelo de Soja, Dissertac¸ão (M.Sc.), Universidade Estadual de Maringá, Maringá, PR, Brasil. Luz, G.R., Sousa, L.H.C.D., Jorge, L.M.M. and Paraíso, P.R., 2006, Estudo das Isotermas de Equilíbrio do Farelo de Soja. Ciência e Tecnologia de Alimentos, 26(2): 408–413. Luz, G.R., Andrade, C.M.G., Jorge, L.M.M. and Paraíso, P.R., 2006, Coeficiente de Transferência de Massa na Secagem de Farelo de Soja, In XXXII Congresso Brasileiro de Sistemas Particulados Maringá-PR, Luz, G.R., Andrade, C.M.G., Jorge, L.M.M. and Paraíso, P.R., 2006, Análise energética da secagem de farelo de soja em secador rotativo indireto. Acta Sci Technol, 28(July/December(2)): 173–180.

Luz, G.R., 2008, Avaliac¸ão das estratégias de operac¸ão e de controle da secagem industrial de farelo de soja, Tese de Qualificac¸ão (D.Sc.), Universidade Estadual de Maringá, Maringá, PR, Brasil. Luz, G.R., Paraíso, P.R., Jorge, L.M.M. and Andrade, C.M.G., 2009, Modeling and energetic analysis of soybean meal drying in the indirect rotary dryer. CPPM, 4(1). Article 8 Marin, F.B., Angelocci, L.B., Coelho Filho, M.A. and Villa Nova, N.A., 2001, Construc¸ão e avaliac¸ão de psicrômetro aspirado de termopar. Sci Agric, 58(4): 839–844. Mujumdar, A.S., (2006). Handbook of Industrial Drying. (Taylor & Francis, Philadelphia), pp. 1161–1179 Pérez-Correa, J.R., Cubillos, F., Zavala, E., Shene, C. and Álvarez, P.I., 1998, Dynamic simulation and control of direct rotary dryers. Food Control, 9(4): 195–203. Savaresi, S., Bitmead, R. and Peirce, R., 2001, On modeling and control of a rotary dryer. Contr Eng Pract, 9: 249–266. Treybal, R.E., (1980). Mass Transfer Operations (3rd ed.). (Mc Graw-Hill Book Co. Inc, Tokyo), pp. 655–716 Vogel, E.F., 1992, Plantwide process control simulation, in Practical Distillation Control, Luyben, W.L. (ed). (Van Nostrand Reinold, New York), p. 92 Willis, M.J., 1999, Proportional – Integral – Derivative Control, June 12, 2006, http://lorien.ncl.ac.uk/ming/PID/PID.pdf. Yliniemi, 1999, Advanced control of a rotary dryer, PhD thesis, University of Oulu, Oulu, Finland, October 17, 2005, http://herkules.oulu.fi/isbn9514252810. Yoannou, P. and Sun, J., 1996, Robust Adaptive Control (Prentice Hall, Inc.), December 05, 2006, http://wwwrcf.usc.edu/∼ioannou/Robust Adaptive Control.htm.