Mass Flow Dynamic Modeling and Residence Time Control of a

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Mass Flow Dynamic Modeling and Residence Time Control of a Continuous Tubular Reactor for Biomass Pretreatment Ismael Jaramillo, and Arturo Sanchez ACS Sustainable Chem. Eng., Just Accepted Manuscript • DOI: 10.1021/ acssuschemeng.8b00882 • Publication Date (Web): 30 May 2018 Downloaded from http://pubs.acs.org on May 30, 2018

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ACS Sustainable Chemistry & Engineering

Mass Flow Dynamic Modeling and Residence Time Control of a Continuous Tubular Reactor for Biomass Pretreatment Ismael Jaramillo and Arturo Sanchez∗ Laboratorio de Futuros en Bioenergía Centro de Investigación y de Estudios Avanzados del IPN Unidad Guadalajara (CINVESTAV) Av. del Bosque #1145, Zapopan, CP 45019, Jalisco, México E-mail: [email protected]

Abstract

This work presents a dynamic discrete model that describes the mass ow and residence time behavior in a pretreatment continuous tubular reactor using lignocellulosic biomass as feedstock. The model consists of a set of linear dierence equations that accurately describe the output ow as a function of the operation conditions. Model parameters account for mechanical and rheological properties of dierent lignocellulosic biomass, as well as complex backow phenomena commonly encountered in these reactors. The proposed dynamic model was experimentally validated for wheat straw and corn stover. Additionally, a control strategy was proposed for the residence time and was correctly validated through simulations. Keywords:

dynamic discrete modeling, pretreatment reactor, lignocellulosic biomass,

residence time control, autohydrolysis, parameter estimation, experimental validation

1

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Introduction Lignocellulosic biomass (LB) is an agricultural residue with the potential to replace fossil materials for producing fuels and special products (e.g, plastics, pharma precursors, etc.) 1 in biochemical platform bioreneries 2 . One of the main processing challenges in this type of biorenery is LB feedstock pretreatment 3 . As a rst step, the pretreatment process aims to weaken the cellulose and lignin matrix, as seen in Figure 1, in order to recover the hemicellulose in oligo and monomeric forms, as well as to reduce cellulose crystallinity and increase the fraction of amorphous cellulose for further enzymatic attack in downstream biorening stages.

Figure 1: Pretreatment eect over the LB structure 4 . Pretreatment technologies are usually designed taking the LB and reactor characteristics into consideration 5 . Continuous technologies are being hailed as an important step toward improving the economics of biorening processes. Therefore, the use of continuous tubular reactors (CTR) to carry out the autohydrolysis pretreatment process has been widely studied 69 in dierent reactors in order to achieve maximal product yields and sugar recoveries. Geometric and mechanical characteristics of the CTR must be considered for calculating the mass ow output 10 . The LB's rheological properties, combined with the operation conditions, produce a non-ideal mass ow characterized by phenomena such as backmixing 2


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ow, LB compression and its adherence to the metallic surface of the CTR 11 . Several mass ow estimation approaches and techniques have been published in the literature. Grattoni et al. 12 presented dierent methods for solving the Langmuir type partial dierential equation (PDE), describing the convection, dispersion and adsorption phenomena of owing material. However, this becomes impractical due to the diculty of measuring the required model parameters. Another approach, presented by Wan and Hanley 13 , shows a model for predicting mass ow patterns inside a tubular reactor. Although, the required rheological properties of the LB and other parameters (permeability, mean particle diameter, void volume, etc.) are not reported. In another study, Owen and Cleary 14 simulated a screw conveyor for transporting idealized particles using the discrete element method (DEM) and predicted the screw performance for dierent particle speeds, mass ow rates and operation conditions (inclination degree, volumetric ll level, etc.), providing a good insight into the inuence of these variables on the mass ow rate. Nevertheless, the simulation conditions and characteristics dier dramatically from the real LB particle properties, resulting in a non-accurate analysis. Regarding the residence time (RT) measurements, Sievers et al. 15 presented a technique for carrying out an on-line measurement of the RT using specially designed sensors to measure the bulk electric conductivity. Based on this measurement technique, Sievers and Stickel 16 identied several factors that aect the RT mean and standard deviation and proposed a semiempirical model for the RT distribution (RTD). The signal measurement presented a noisy behavior and had to be tted to a gamma distribution curve. This RT estimation may not be suitable for realistic operation conditions (semi-random input ow signal and variable reactor lling levels). In another perspective, Xi et al. 17 executed a classical RT measurement using impulse response experiments and analyzed the eects of operational parameters, inclination angle and design parameters over the mean RT. As expected, dierent materials with similar shapes (cubic, spherical or irregular) presented similar RTD under the same 3



experimental conditions. Therefore, a dynamic model of the output mass ow of a pretreatment CTR suitable for system analysis and control is not available in the open literature. This work proposes a dynamic model that accurately describes the output mass ow and RTD curve based on the model parameters. A control scheme is also proposed for this latter variable. The model equations are based on experimental data, and thus considering non-ideal mass ow dynamics and non-modeled phenomena. The pilot-scale pretreatment CTR located at the Energy Futures Laboratory of CINVESTAV, Guadalajara served as a platform for carrying out the validation experiments using two biomass types with dierent physical properties (i.e, mechanical and rheological): wheat straw (WS) and corn stover (CS).

Equipment and Experiment description Figure 2 shows a schematic diagram of the pilot-scale CTR in question. The CTR is composed by areas A (i.e. extruder) and B (i.e. CTR body) to perform three process phases (ISA S88 18 ): extrusion ( 1 , 2 , 3 ), autohydrolysis ( 4 , 5 , 6 ) and steam explosion ( 7 , 8 ). Circled numbers represent the dierent parts of each area (see Table 1).

Figure 2: Structure diagram of the CTR used in this study.

4


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Table 1: Process components in Figure 2. ID Device Description 1 Extruder motor 11 RPM (xed), 10 cm pitch and 4 ights screw 2 Hopper LB feeding 3 Extruder body 2:1 reduction ratio 4 Conveyor motor 9.2 RPM max, 10 cm pitch and 14 ights screw 5 CTR body 1.6 m long, 10 cm diameter 6 Saturated steam input 180 °C, 1.034 MPa 7 Discharge valves Explosion every 1 min 8 Discharge box Receives exploded LB For a better mechanical mass transport performance, LBs were milled and soaked with water previous to the experiments. The extrusion starts with the LB being fed (by hand) through 2 and entering the extruder body. The mass ow input data U is acquired by measuring the time required to feed a xed amount of LB (500g containers of feedstock were hand-metered into the reactor and start/stop times for each container were used to calculate input mass owrate). Once inside, the screw 1 pushes and compacts the LB inside the extruder body 3 . The time between the LB input and its exit from the extruder is termed cone lling delay (CFD). The goal of the extrusion phase is to form a compact and dense biomass body in the CTR input to prevent steam from escaping through the extruder. Extruder mass output yE enters the CTR body 5 and once inside, it is transported by the rotation of 4 at a variable motor speed ω2 (normalized). During transportation, the LB is subjected to high pressure and temperature produced by saturated steam entering through 6 . The total delay time between LB input and output TR is determined by ω2 . If ω2 changes its value from ω2o to ω2f , there is a transition period with a duration of Φ minutes

while the output ow changes. After this period, the output mass ow returns to its steady state. Let kc be the instant in which ω2 is modied from ω2o to ω2f (i.e, ∆ω2 = ω2o − ω2f ). If ∆ω2 > 0 (decreasing motor speed), the LB extruded after time kc will take longer to travel the length of the CTR body than LB fed at any k < kc , reducing the output rate and increasing the internal mass accumulation. Despite the reduction of mass ow output 5



for a period of time, a ow will still exist, indicating that the inner accumulated mass Ai is being expelled from the CTR body. If ∆ω2 < 0 (increasing motor speed), the LB extruded after kc will take less time to reach the reactor exit, pushing part of the internal accumulated mass out of the CTR body, thus increasing the output rate and decreasing the internal mass accumulation during the transient period Φ. Finally, LB reaches the discharge point, where two valves 7 open and close in a sequence with a xed time interval such that the pressure dierence between the interior of the CTR body and the environment expels the LB at high speed (steam explosion) to the receiver box 8 . Output samples are collected, weighed and registered every minute (sample time Ts ). Input and output LB moisture was measured with a thermogravimetric balance.

CTR modeling Figure 2 shows the division of the CTR utilized for modeling purposes. The areas labelled A and B contain the elements involved in the extruder and the CTR body mass ow output models, respectively. The identication procedure of the output ow of the extruder and CTR body areas is presented in the following subsections.

Identication procedure Identication experiments were carried out separately for the extruder section and the complete CTR and consisted of registering the system response to an impulse tracer feed (temperature and pressure resistant polymer). The tracer was weighed and sectioned in particles of 1 cm2 before entering the reactor. Each discharge of pretreated biomass in the output collection box was manually inspected to recover the tracer particles and weighed to register their TR delay. For the extruder experiments, the screw speed was set to maximum and output samples were collected every minute. Experimental results from the extruder iden6


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tication for CS and WS, shown in Figure 3, exhibit some dierences for each LB response to the impulse input due to their rheological properties. For the CTR experiments, four conveyor speeds were tested (ω2 =1.00, 0.75, 0.50 and 0.33). These variations were applied up to two times during each experimental run to identify the inuence of screw speed upon the mass ow output and RTD. CTR body identication focused on three issues: 1) the time (delay) required for the tracer impulse to travel the length of the CTR body (subtracting the extrusion phase mean RT from the complete process delay), 2) the time required for the impulse to be fully expelled from the CTR (output duration) and 3) how much of the tracer exits the CTR at each sample. Table 2 shows the delay and output duration registered in the experiments carried out at dierent ω2 and Figure 4 shows the tracer mass recovered in every sample. 100

100

80

80

mass (g)

mass (g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


60 40 20 0 0

60 40 20

5

10

15

0 0

20

5

10

time (min)

time (min)

(a) CS.

(b) WS.

15

20

Figure 3: Tracer recovery from the extruder identication experiments. Table 2: Experimental ω2 relation with delay and output duration at four speed conditions. ω2

1.00 0.75 0.50 0.33

Delay (min) Output duration (min) 16 5 26 7 33 9 52 11

7


12

12

10

10

mass (g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

mass (g)


8 6 4 2 0 0

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8 6 4 2

10

20

30

40

50

60

70

0 0

80

10

20

time (min)

30

40

50

60

70

80

time (min)

(a) CTR with ω2 = 1.00.

(b) CTR with ω2 = 0.33.

Figure 4: Tracer recovery from the WS complete pretreatment identication experiments for two conditions of ω2 .

Extruder model The LB mass impulse response shown in Figure 3 can be described by a second order overdamped transfer function. Overdamping leads to a mass ow output yE ≥ 0 if the mass input U ≥ 0 (i.e, real roots), which is experimentally consistent. Additionally, the CFD phenomenon is taken into consideration by including a sigmoid-shaped gain K1 for the calculation of exit ow. K1 is an adimensional gain, which is a function of the total LB fed to the extruder

Pk 0

U (k), the extruder mass capacity Cc and a rising parameter m to t the

experimental behavior, as seen in Eq. (1), for experiments with both LBs m = 0.2.

K1 (k) = 1/(1 + e−m(

Pk 0

U (k)−Cc )

)

(1)

The extruder model is shown in Eqs. (2)-(3).

xE (k + 1) = θ1 xE (k) − θ2 xE (k − 1) + θ3 U (k) − θ4 U (k − 1)

(2) (3)

yE (k) = K1 (k)xE (k)

θ1 , θ2 , θ3 , θ4 parameters are functions of the damping parameter ζ , the natural frequency

8


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of the system ωn and the sampling time Ts . Their expressions are given in Appendix A.

CTR body model The CTR body considered in this work was modeled as a series of n continuous stirred tank reactors (CSTR) 19 with n set to 14 due to the geometrical characteristics of the device, as shown in Figure 5. This proposal reduces the theoretical 115 tanks in series calculated from the RT moments of distribution. Therefore, the extruder output yE is the rst CSTR (i.e, subreactor) input and the mass ow output of the CSTR xRi−1 is the mass ow input of the CSTR xRi . The output of the CSTR xR14 is the system output yR .

Figure 5: CTR body divided into 14 regions of analysis. Each subreactor mass ow dynamics are proposed as a rst order discrete transfer function dierence equation, as seen in Eqs. set (4). Also, the total TR is divided into 14 equal δ2 delays, as shown in Eq. (6).

x1R (k + 1) = θ5 x1R (k) + θ6 υˆ1

(4a)

x2R (k + 1) = θ5 x2R (k) + θ6 υˆ2

(4b)

.. . x14R (k + 1) = θ5 x14R (k) + θ6 υˆ14

(4c)

yR (k) = x14R (k)

(5)

TR (k) = δ2 (k − 14) + δ2 (k − 13) + · · · + δ2 (k − 1)

(6)

9



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θ5 and θ6 parameters are functions of the system's time constant τ2 and sampling time Ts . Their expressions are given in Appendix A. Delay TR and output duration (Table 2 and

Figure 4) are functions of ω2 and are considered in the model within the parameters δ2 and τ2 , respectively. Additionally, step changes of ω2 aect each subreactor input variable υî ;

when ω2 remains constant (∆ω2 = 0), υî is a function of the mass ow input fed to the i-th subreactor νi (i.e, steady state mass ow). If ω2 changes (∆ω2 6= 0), there is a period of Φ minutes during which υî is a function of the accumulated mass Ai inside the corresponding subreactor (i.e, transient mass ow). Once the transient period has concluded, υî returns to its steady state mass ow form. Eqs. set (7) shows the steady state and transient forms of υî .

   yE (k − δ2 ) , if υˆ1 (k) =   A1R (k)K2 (k) , if    xR1 (k − δ2 ) , if υˆ2 (k) =   A2R (k)K2 (k) , if

∆ω2 = 0

(7a)

∆ω2 6= 0 ∆ω2 = 0

(7b)

∆ω2 6= 0

.. .    xR13 (k − δ2 ) , if ∆ω2 = 0 υˆ14 (k) =   A14R (k)K2 (k) , if ∆ω2 = 6 0

With: AiR (k + 1) = AiR (k) + νi (k) − xiR (k)    ω2f + ω2o −ω2f , if ∆ω2 > 0 k−kc +1 K2 (k) =   ω2f , if ∆ω2 < 0   1+δ2  , if ∆ω2 > 0    ω2f Φ= δ2 , if ∆ω2 < 0      0 , if ∆ω2 = 0

10


(7c)

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Accumulated mass Ai is calculated as a simple mass balance. Gain K2 and period Φ were experimentally proposed to determine whether the input increases or decreases and the length of the transient behavior, respectively.

Parameter Estimation The parameter estimation was carried out using a PSO algorithm 20 (100 particles and 1000 iterations) for all parameters in Eqs. (2) and (4) using the mean square error between experimental and model output as the objective function. For the extruder model, the PSO algorithm was utilized to nd the values of ωn corresponding to a xed ζ = 1.001 (overdamped) in order to calculate the parameters in Eq. (2) for both LBs. The relation between ζ and ωn determines the pole placement of the transfer function to produce an output (Eq. (3)) that reproduces the experimental results. Figures 6a and 6b show the prediction results of tracer experiments. Estimated parameter values are shown in Table 3. Table 3: PSO parameter estimation results for the extruder model. LB ζ ωn CS 1.001 1.23E −3 WS 1.001 1.95E −3

Experimental output Model output

80 60 40

100

mass (g)

100

mass (g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


20 0 0


80 60 40 20

5

10

15

0 0

20

5

10

time (min)

time (min)

(a) CS.

(b) WS.

15

20

Figure 6: Comparison between extruder mass ow output experimental and model data using the estimated parameter ωn for the two LB tracer experiments. For the CTR experiments, similar delay and output durations (Table 2) were observed for both biomass types. The procedure of estimating the parameter values for the CTR 11



body mass ow output Eqs. (4) was divided into two stages: rst, the behavior shown in Figure 4 was modeled with a discrete second order overdamped transfer function (i.e, approximation function xˆR ) shown in Eq. (8), whose dynamics depend on parameters ζ2 and ωn2 to reproduce the experimental RTD. Second, parameters δ2 and τ2 in Eqs. (4) were estimated with the PSO algorithm to t the xˆR behavior.

(8)

xˆR (k + 1) = f (ˆ xR (k), xˆR (k − 1), yE (k), yE (k − 1), ζ2 , ωn2 )

Again, ζ2 = 1.001 was proposed as a xed value in order to estimate ωn2 . The resulting values are presented in Table 4 and shown in Figure 7 for the dierent ω2 conditions. Table 4: ω2 relation with damping and natural frequency parameters of xˆR . ω2

ζ2

1 0.75 0.50 0.33

12

12

Experimental output Approximation function

10

6 4 2 0 0

0.0088 0.0072 0.0056 0.0045

mass (g)

8

ωn2

1.001 1.001 1.001 1.001

Experimental output Approximation function

10

mass (g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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8 6 4 2

10

20

30

40

50

60

70

0 0

80

10

time (min)

20

30

40

50

60

70

80

time (min)

(a) CTR with ω2 = 1.00.

(b) CTR with ω2 = 0.33.

Figure 7: Comparison between WS mass ow output experimental and approximation function xˆR data for two conditions of ω2 . Parameters τ2 and δ2 in Eqs. (4) were estimated for Eq. (5) to t xˆR behavior (RTD curve). PSO results are presented in Table 5 and shown in Figure 8 compared with xˆR for dierent ω2 conditions. Expressions for δ2 and τ2 calculation are given in Appendix B. 12


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Table 5: ω2 relation with each subreactor delay δ2 , transport delay TR and time constant τ2 . 1.00 0.75 0.50 0.33

δ2 (min)

0.8 1.43 1.82 3.05

TR =14δ2 (min)

11.2 20.1 25.6 42.7

10

34.1 41.5 53.2 65.5

Approximation function Model output ω 2 = 0.50

15

ω 2 = 0.75

20

ω 2 = 1.00

25

τ2 (s)

ω 2 = 0.33

ω2

mass (g)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


5 0 0

20

40 time (min)

60

80

Figure 8: Comparison between yR and xˆR using the estimated parameters for dierent values of ω2 based on WS identication experiments. The complete mass ow model is formulated by joining Eqs. (2)-(6) in a cascade arrangement.

Validation Validation was carried out according to Zekki et al. 21 (mass balance, output error, correlation coecient and error distribution shown in Table 6). Figures 9-10 show (a) the model input, (b) the model output compared to experimental data (dierent from identication data) and (c) the accurate TR prediction, for two typical runs.

13



Table 6: Validation data. Analysis WS experiment CS experiment Required Input (kg) 26.1 24 > output Real/Model Output (kg) 25.3/25 20.9/20.4 < input Output Error 0.051 0.025 ≈0 Correlation Coecient 0.86 0.77 ≈1 Error Mean (g) 3 7 ≈0 Error Standar Deviation (g) 32 26 Reasonable

14


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200

Input flow

150 100 50 0 0

100

200 time (min)

300

400

mass flow rate (g/min)

(a) WS U .

300 200 100 0 0


100

200 time (min)

300

400

300

400

(b) WS yR .

Transport delay (min)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



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60 ω2=1.00

ω2=0.33

40 20 0 0

100

200 time (min) (c) WS TR .

Figure 9: WS experiment registered variables: a) mass ow input, b) experimental output vs model output and c) estimated TR . 15


200

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Input flow

150 100 50 0 0

100

200 300 time (min)

400

500

(a) CS U .


300 200 100 0 0


100

200 300 time (min)

400

500

(b) CS yR .


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60



60 ω2=1.00

ω2=0.50

ω2=0.33

40 20 0 0

100

200 300 time (min)

400

500

(c) CS TR .

Figure 10: CS experiment registered variables: a) mass ow input, b) experimental output vs model output and c) estimated TR .

16


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Design of RT control A PI controller is proposed for the TR regulation. The block diagram of the control scheme is shown in Figure 11. The controller tuning was carried out using the transfer function shown in Eq. (9), where KP > qKI , (q > 10) 22 .

Figure 11: Block diagram of the control scheme (TR to ω2 block expression presented in Appendix B).

C(s) = KP +

KI s

(9)

The controller was tested using simulation conditions as similar as possible to the experimental conditions. Figure 12 shows the dierent variables obtained from a control simulation to reach three dierent TR references. Figure 12a shows the controller action over ω2 to achieve the TR regulation for the reference step changes. Figure 12b shows the correct TR regulation reaching the 12, 26 and 41 minute references. Figures 12c and 12d show the

eect of the ω2 variations over mass ow output with two input cases: ideal and non-ideal, respectively. It is clearly seen that the mass ow input signal determines the mass ow output behavior. Also, the transient periods of time when ω2 is modied are consistent with the experimental data. The control scheme was validated by analyzing the controller's correct 17



1

60

Reference ω2

0.8 ω2


behavior.

0.6

Estimated TR

40

TR reference

20

0.2 0 0

50 30

0.4

10

100

200 time (min)

300

400

0 0

100

80 60

Simulated input Model output

0 0

100

200

300

time (min)

400


100

20

300

400

(b) TR regulation.

120

40

200

time (min)

(a) ω2 controller output. mass flow rate (g/min)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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(c) Ideal input simulation.

120 100 80 60 40

Simulated input Model output

20 0 0

100

200

300

400

time (min)

(d) Random input simulation.

Figure 12: Control simulations.

Conclusion The experiments showed that LB mass ow in a pretreatment CTR is a complex phenomenon. The main variables involved in characterizing the output mass ow behavior are the LB input mass ow and the screw conveyor speed, which produce the erratic material ow behavior and the ow dynamics (TR and τ2 which dene the RTD curve), respectively. A meticulous acquisition of LB input mass ow provided reliable data to accurately reproduce the mass ow dynamics with the proposed model. Both LBs presented behavior dierences at the extrusion phase output, but similarities at the CTR body output, probably caused by the high temperature and pressure conditions of the autohydrolysis process. The model's output was successfully validated by means of a statistic method and simulations. Additionally, a PI controller produced a correct regulation of TR and consistent mass ow output results.

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Acknowledgements Financial support is kindly acknowledged from the Energy Sustainability Fund 2014-05 (CONACYT- SENER, Mexico) Grants 245750 and 249564 (Mexican Bioenergy Innovation Center, Bioalcohols Cluster). I. J. acknowledges nancial support from CONACYT, Mexico, in the form of MSc scholarship number 589640.

Appendix A θ1 , θ2 , θ3 , θ4 , θ5 , θ6 included in Eqs. (2) and (4) are constant parameters calculated from

the Z-transform table when discretizing a continuous transfer function (ZOH method) 23 . Expressions for these parameters are shown in Eqs. (10) - (19).

θ1 = eTs r1 + eTs r2

(10)

θ2 = eTs (r1 +r2 )

(11)

θ3 = (r3 /r1 ) 1 − eTs r1 + (r4 /r2 ) 1 − eTs r2 θ4 = (r3 /r1 ) 1 − eTs r1 eTs r2 + (r4 /r2 ) 1 − eTs r2 eTs r1

(12)

θ5 = e−Ts /τ2

(14)

θ6 = 1 − e−Ts /τ2

(15)

(13)

ζ2 − 1 p r2 = −ζωn − ωn ζ 2 − 1

(16)

r4 = ωn2 /(r1 − r2 )

(18)

r3 = −r4

(19)

r1 = −ζωn + ωn

p

(17)

19



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Appendix B The time constants and delay from Table 5 are explicit functions of ω2 . These parameter values are calculated with an interpolation polynomial for any ω2 ∈ [0.33, 1] as seen in Eqs. (20)-(21). Similarly, the TR to ω2 block in Figure 11 used a numerically calculated polynomial to approximate ω2 from TR (Eq. (22)) based on the information shown in Table 5. Coecients for these polynomials are shown in Table 7.

δ2 (k + 1)(ω2 (k)) = a2 ω2 (k)2 + a1 ω2 (k) + a0

(20)

τ2 (k + 1)(ω2 (k)) = b2 ω2 (k)2 + b1 ω2 (k) + b0

(21)

ω2 (k + 1)(TR (k)) = c2 TR (k)2 + c1 TR (k) + c0

(22)

Table 7: δ2 (ω2 ), τ2 (ω2 ) and ω2 (TR ) polynomial coecients. Coecient a2 a1 a0 Value 3.948 -8.343 5.26 Coecient b2 b1 b0 Value 44.453 -105.533 95.291 Coecient c2 c1 c0 Value 0.000626 -0.055663 1.560155

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Appendix C Table 8: List of symbols (in order of appearance). Symbol Description U Extruder model input mass ow ω1 Extruder motor speed (constant) yE Extruder output mass ow ω2 Screw conveyor motor speed yR CTR body output mass ow TR CTR body total transport delay ω2o Screw conveyor speed before a step change ω2f Screw conveyor speed after a step change Φ CTR body mass ow transition period duration after a step change in the screw conveyor speed kc Instant of speed step change ∆ω2 Screw conveyor speed dierential (ω2f -ω2o ) k Instant of time Ai Accummulated mass inside the i-th subreactor Ts System sampling time K1 Extruder model output gain Cc Extruder cone mass capacity m Raising parameter for gain K1 xE Extruder model state θi Model parameters ζ Damping coecient of the extruder model ωn Natural frequency of the extruder model

Symbol Description n Number of subreactors to represent the CTR xiR CTR body model state δ2 CTR body mass transport delay presented in each subreactor υî CTR body model input variable τ2 Subreactor time constant νi Input ow to each subreactor K2 Transient period gain xˆR Approximation function for CTR body identication ζ2 Damping coecient for approximation function ωn2 Natural frequency for approximation function KP PI controller proportional gain q PI controller gain scale KI PI controller integral gain C(s) PI controller transfer function TRref Transport delay reference TRerror Transport delay error ω2error Screw conveyor speed error ω2ref Screw conveyor speed reference ai Coecients for delay polynomial approximation bi Coecients for time constant polynomial approximation ci Coecients for TR to ω2 block from Figure 11 polynomial approximation

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Graphical TOC Entry

Synopsis

An explicit dynamic model was developed to predict the mass ow output and to control the residence time in a CTR for the lignocellulosic biomass pretreatment.

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Mass Flow Dynamic Modeling and Residence Time Control of a

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