Revista de Ciencias Tecnológicas (RECIT). Universidad Autónoma de Baja California ISSN 2594-1925
Volumen 3 (2):120-135. Abril-Junio 2020 https://doi.org/10.37636/recit.v32120135.
120
Dynamics model for the thermal performance from a
lyophilization process, based on a complete transfer
functions matrix
Modelo dinámico para el rendimiento térmico de un proceso de
liofilización, basado en una matriz de funciones de transferencia
completa
Rodríguez-Ibarra María Elizabeth
1,2
, Rodríguez-Vázquez Eloy Edmundo
1,3
, Arteaga-
Martínez Ana Marell
1,2
, Narváez-Granados Samantha Lilia
1
, Zúñiga-Osorio Helen
Janeth
3
, Villasana-Velázquez Víctor Miguel
4
1
Engineering Center for Industrial Development (CIDESI), National Laboratory for Cooling
Technology Research (LaNITeF), Av. Pie de la Cuesta 207, Desarrollo San Pablo, Querétaro, 76250
México.
2
Universidad TecMilenio, Fudamental Science Department, Camino Real a Humilpan, Corregidora,
Querétaro.
3
Universidad Anáhuac Querétaro, School of Engineering, Quantitative Methods and Fundamental
Science. Circuito Universidades, El Marques, Querétaro
4
Universidad Politécnica de Querétaro, School of Engineering, Carretera a los Cues, El Marques,
Querétaro
Corresponding author: Dr. Eloy Edmundo Rodríguez Vázquez, Engineering Center for Industrial
Development (CIDESI), National Laboratory for Cooling Technology Research (LaNITeF), Av. Pie de
la Cuesta 207, Desarrollo San Pablo, Querétaro, 76250 México. E-mail: eloy.rodriguez@cidesi.edu.mx.
Recibido: 30 de Junio del 2019 Aceptado: 15 de Mayo del 2020 Publicado: 30 de Junio del 2020
Abstract. – During the beginning of the XX century lyophilization was developed as an
alternative technology to extend the storage time for fruit and vegetables or other kind of food;
however, the energetic consumption of this technology makes it not an option for common food
producers, less over for those one that work by the open field cultivation technique. The main
energy consumption in a lyophilization systems are the motors from the vacuum pump and from
the refrigerant compressors; due to the temperature range needs the lyophilization systems use
to have more than one cooling thermodynamic system based on vapor compression. This paper
describes an experimental methodology to get a complete state transfer functions matrix, based
on the graphical analysis of the concerned transfer functions magnitude spectra. This
experimental data came from a set of test performed at the National Laboratory for Cooling
Technology Research (LaNITeF) at the Engineering Center for Industrial Development
(CIDESI). The intention of this transfer functions matrix is to be applied in a control strategy to
then optimize the energetic performance of the concerned lyophilization system. This function
transfer matrix is considered complete because there is not a dynamic order reduction
considering its degrees of freedom. The transfer functions matrix describes the dynamic
relationship between both the inputs variables that describe the energetic consumption of the
lyophilization system, and the ambient conditions, as well as the output variables that represent
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the dynamical states vector with the variables of interest from the concerned process. The
simulation from an experimental scenario worked as the graphical validation of the transfer
functions matrix characterized experimentally, so the main conclusion of this scientific work is
that this transfer functions matrix can be used as dynamic model to implement control and
optimization algorithms.
Keywords: Transfer functions matrix; Thermal performance; Lyophilization process.
Resumen. - Durante el comienzo del siglo XX, la liofilización se desarrolló como una
tecnología alternativa para extender el tiempo de almacenamiento de frutas y verduras u otro
tipo de alimentos; Sin embargo, el consumo energético de esta tecnología hace que no sea una
opción para los productores de alimentos comunes, menos para aquellos que trabajan con la
técnica de cultivo en campo abierto. El principal consumo de energía en un sistema de
liofilización son los motores de la bomba de vacío y de los compresores de refrigerante; Debido
al rango de temperatura que necesitan los sistemas de liofilización para tener más de un sistema
termodinámico de enfriamiento basado en la compresión de vapor. Este artículo describe una
metodología experimental para obtener una matriz completa de funciones de transferencia de
estado, basada en el análisis gráfico de los espectros de magnitud de las funciones de
transferencia en cuestión. Estos datos experimentales provienen de un conjunto de pruebas
realizadas en el Laboratorio Nacional de Investigación de Tecnología de Refrigeración
(LaNITeF) en el Centro de Ingeniería para el Desarrollo Industrial (CIDESI). La intención de
esta matriz de funciones de transferencia es aplicarla en una estrategia de control para luego
optimizar el rendimiento energético del sistema de liofilización en cuestión. Esta matriz de
transferencia de funciones se considera completa porque no hay una reducción de orden
dinámico considerando sus grados de libertad. La matriz de funciones de transferencia describe
la relación dinámica entre las variables de entrada que describen el consumo energético del
sistema de liofilización y las condiciones ambientales, así como las variables de salida que
representan el vector de estados dinámicos con las variables de interés del proceso en cuestión.
La simulación de un escenario experimental funcionó como la validación gráfica de la matriz
de funciones de transferencia caracterizada experimentalmente, por lo que la conclusión
principal de este trabajo científico es que esta matriz de funciones de transferencia puede usarse
como modelo dinámico para implementar algoritmos de control y optimización.
Palabras clave: Matriz de funciones de transferencia; Rendimiento térmico; Proceso de
liofilización.
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1. Introduction
Because of the cooling speed and load
capacity, traditional refrigeration and
freezing process based on the vapor
compression technologies are the most used
alternatives for the perishables products
conservation (production, storage,
transportation, distribution and exhibition)
[3 – 5].
It is estimated that just in Mexico the 30 %
of the food production is wasted because of
several issues with the cold chain and
refrigeration [1]; and it occurs almost the
same with the final availability for the
perishable medicine in rest of Latin
America [2].
But the wastes from the production of food
and medicine is not the only opportunity for
the refrigeration process; because,
nowadays the temporal cultivation of fruits
and vegetable through open field
techniques, does not represent a convent
opportunities for communities that have this
economical access naturally. And the due to
the inefficiency of the conventional cooling
technology, the producers do not have the
opportunity to apply it to extend their
market presence by transporting or
exporting their products and then increasing
their utility [6 INEGI].
It is well known that the natural cooling
process into the vapor compression
technologies affects the electrochemical
properties of food and medicine [6, 7]; so,
that, since the beginning of the XX century
the lyophilization process appears [8] by
being an alternative for the food and
medicine preservation process, because its
preserves their structural composition by a
fast cooling process into a vacuum ambient
for the water extraction [9].
Both lyophilization sub-stages mainly
defined as the fast cooling and the vacuum
ambient are high energy expending process;
due to the motors power that move the
refrigerant compressors and the vacuum
pumps. Therefore, a mathematical model
for the dynamic behavior of the thermal and
energetic variables from a lyophilization
device are proposed, with the intention to
use it in future works for the tuning of
control algorithms that can optimize this
lyophilization process energetic
consumption.
The mathematical model for the specific
lyophilization process described in this
document comes from a state space model,
and consists in a transfer function matrix
characterized by the spectral response of the
thermodynamic variables involved on the
concerned process.
All experimental effort and analysis was
developed at the National Laboratory for
Cooling Technology Research (LaNITeF)
at the Engineering Center for Industrial
Development (CIDESI), whom is part of the
Public Research Centers and the National
Laboratories Network form the National
Council for Science and Technology
(CONACYT).
The lyophilization process modeling is
described in this document by starting with
the lyophilization process description and
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its variables identification in the second
chapter. Then the experimental testing of
the lyophilization process is documented
into the third chapter, and just after it, on
section forth the experimental results
analysis is. This results analysis works as
reference for the conclusions synthesis in
the last chapter.
2. Lyophilization
Mainly, the lyophilization process consist in
three sub-process:
1. “Conduction” to remove most of the
food water.
2. “Diffusion stage 1” to consolidate the
food electrochemical properties.
3. “Diffusion stage 2” to consolidate the
food physical properties.
In general, the food lyophilization consist
on its gases solidification by passing
through the liquid phase just with the water
concentration needed to preserves its
original molecular structure [9 - 11].
To clarify the temperature behavior of a
lyophilization process, Fig. 1 illustrates the
temperature of the cooling chamber from
the lyophilization device used on this paper,
working with 87.37 grams of fruit load.
Fig. 2 shows the schematic diagram of the
lyophilization systems used in this
researching work, where it is possible to see
that this device consists of two vapor
compression refrigeration systems, one for
the R-23 refrigerant gas (system 1) and the
second one for the R-507a (system 2)
refrigerant.
Figure 1. Cooling chamber temperature.
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Figure 2. Schematic diagram of the lyophilization device.
Table 1 lists the variables involved in the
dynamics model for the lyophilization
process, considering the thermodynamic
states of the refrigerant gases as the model
outputs and as inputs the variables related
with both the energetic consumption as well
as the ambient conditions.
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Table 1. Lyophilization process model variables.
Variable
Description
Units
High Temperature at High Pressure on System 1
°C
Low Temperature at High Pressure on System 1
°C
Low Temperature at Low Pressure on Systems 1
°C
Common High Temperature at Low Pressure
°C
High Temperature at High Pressure on System 2
°C
Low Temperature at High Pressure on System 2
°C
Low Temperature at Low Pressure on System 2
°C
Ambient Temperature
°C
Atmospheric Pressure
Bar
Compressor Energy Consumption System 1
W
Compressor Energy Consumption System 2
W
Vacuum Pump Energy Consumption
W
Heat Source Energy Consumption
W
3. Lyophilization dynamics
As it has been clarified the intention of this
mathematical model is to be the base of an
energetic optimization strategy, by
modifying the dynamic response of the
described lyophilization system. Then the
dynamic model proposed as state space, was
defined as:
, (1)
where A and B are the dynamics matrix and
the input matrix respectively, then applying
the Fourier transfer and by solving for the
state vector, we have:
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, (2)
where the matrix in the between of the state
vector and the input vector is known as the
transfer function matrix, which can be re-
written as:
, (3)
where:
is the polynomial conformed by the
natural frequencies (Eigen values), and
represents the zeros polinomiun
from the concerned spectral relationship
between n-output and m-input variables.
4. Experimental characterization
With the intention to know both poles
(eigenvalues) and zeros, a set of
experiments where performed in the
lyophilization device, where all model
variables listed on table 1 were measured
and processed to know the spectral response
of each them. Figs 3 to 8 and 9 to 15 plot the
inputs and outputs variables behavior
respectively.
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Figure 3. Dynamic behavior of the compressor
energy consumption from system 1
Figure 4. Dynamic behavior of the compressor
energy consumption from system 2
Figure 5. Dynamic behavior of the vacuum pump
energy consumption
Figure 6. Dynamic behavior of the heat source
energy consumption.
Figure 7. Dynamic behavior of the ambient
temperature.
Figure 8. Dynamic behavior of the atmospheric
pressure.
0
50
100
150
200
250
300
350
400
450
0 50000
Power (W)
Time (s)
0
50
100
150
200
250
300
350
400
450
0 50000
Power (W)
Time (s)
0
200
400
600
800
1000
1200
1400
0 50000
Power (W)
Time (s)
0
50
100
150
200
250
0 50000
Power (W)
Time (s)
0
5
10
15
20
25
30
35
40
0 50000
Temperature (
°C)
Time (s)
0.8220
0.8230
0.8240
0.8250
0.8260
0.8270
0.8280
0 50000
Pressure (bar)
Time (s)
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Figure 9. Dynamic behavior of the high temperature
at high pressure on system 1.
Figure 10. Dynamic behavior of the low temperature
at high pressure on system 1.
Figure 11. Dynamic behavior of the low temperature
at low pressure on system 1.
Figure 12. Dynamic behavior of the common high
temperature at low pressure.
Figure 13. Dynamic behavior of the high
temperature at high pressure on system 2.
Figure 14. Dynamic behavior of the low
temperature at high pressure on system 2.
20
22
24
26
28
30
32
34
0 50000
Temperature (
°C)
Time (s)
20
25
30
35
40
45
0 50000
Temperature (
°C)
Time (s)
20
25
30
35
40
45
50
0 50000
Temperature (ªC)
Time (s)
20
22
24
26
28
30
32
34
0 50000
Temperature (
°C)
Time (s)
20
22
24
26
28
30
32
34
0 50000
Temperature (
°C)
Time (s)
-20
-10
0
10
20
30
40
0 50000
Temperature (ªC)
Time (s)
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Figure 15. Dynamic behavior of the low temperature at low pressure on system 2.
Then by considering the transfer functions
from each relationships of these
experimental results, their magnitude
spectra was estimated by a Fast Fourier
Transform algorithm, and by analyze them
graphically the poles (maximums) and zeros
(crosses) were estimated as well as Figs. 16
and 17 show it. Therefore the resulted poles
polynomial is
(4)
Figure 16. Graphical analysis from a transfer function spectra to get the poles location.
20
22
24
26
28
30
32
0 50000
Temperature (ªC)
Time (s)
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Figure 17. Graphical analysis from a transfer function spectra to get the zeros location.
And the concerned zeros for each variables relationship are listed on table 2.
Table 2. Model zeros list.
Z1 [Hz]
Z2 [Hz]
Z3 [Hz]
Z4 [Hz]
Z5 [Hz]
Z6 [Hz]
Z
1,1
1.1990E-05
1.6181E-05
3.0303E-05
0.00010417
Z
1,2
1.1820E-05
1.5015E-05
2.9240E-05
5.7471E-05
0.00033333
Z
1,3
1.3123E-05
2.1368E-05
6.4103E-05
0.00055556
Z
1,4
1.1655E-05
1.5432E-05
1.9841E-05
2.4155E-05
0.00166667
Z
1,5
1.4749E-05
3.9683E-05
9.8039E-05
0.000555556
Z
1,6
1.1494E-05
1.5873E-05
3.4722E-05
9.8039E-05
0.000416667
Z
2,1
1.1261E-05
1.5015E-05
3.6232E-05
0.00033333
Z
2,2
1.1038E-05
1.4493E-05
1.8315E-05
2.8736E-05
3.7879E-05
0.00055556
Z
2,3
1.4124E-05
2.1368E-05
7.5758E-05
0.00055556
Z
2,4
1.1038E-05
1.5576E-05
2.6042E-05
0.00041667
Z
2,5
1.1037E-05
1.5723E-05
3.3333E-05
0.000111111
0.000166667
0.000208333
Z
2,6
1.2073E-05
1.5576E-05
3.3333E-05
0.000128205
0.000208333
0.000416667
Z
3,1
1.2531E-05
1.7921E-05
3.4722E-05
0.00033333
Z
3,2
1.1111E-05
1.4124E-05
1.7361E-05
2.9240E-05
5.9524E-05
0.00041667
Z
3,3
1.3550E-05
1.9841E-05
0.00011905
0.00055556
Z
3,4
1.1111E-05
1.1820E-05
1.4368E-05
1.9608E-05
2.4876E-05
0.00041667
Z
3,5
1.0964E-05
0.001666667
3.7788E-05
0.00023810
0.00033333
0.00041667
Z
3,6
1.1737E-05
3.8760E-05
0.0002381
0.0003333
0.0005556
Z
4,1
1.1338E-05
1.3550E-05
1.9608E-05
2.4876E-05
9.2593E-05
0.00041667
Z
4,2
1.1038E-05
1.4124E-05
1.8727E-05
2.6882E-05
4.3860E-05
0.00055556
Z
4,3
1.5015E-05
2.0833E-05
5.9524E-05
0.00055556
Z
4,4
1.1038E-05
1.2438E-05
1.6835E-05
1.8315E-05
2.4155E-05
0.00055556
Z
4,5
1.263E-05
1.9380E-05
2.4876E-05
5.9524E-05
0.000238095
0.000333333
Z
4,6
1.1905E-05
1.9380E-05
2.5641E-05
8.7719E-05
0.000333333
0.001666667
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Z
5,1
1.1905E-05
1.2438E-05
1.5723E-05
1.9157E-05
2.5253E-05
0.00041667
Z
5,2
1.1111E-05
1.3661E-05
1.7921E-05
2.8249E-05
3.7879E-05
0.00041667
Z
5,3
1.2438E-05
1.8939E-05
2.0576E-05
6.1728E-05
0.00055556
Z
5,4
1.3228E-05
1.7921E-05
1.9608E-05
2.4510E-05
0.00033333
Z
5,5
0.0000142
0.0000192
0.0000249
0.0000476
0.0002083
0.0003333
Z
5,6
1.1494E-05
1.5873E-05
3.4722E-05
9.8039E-05
0.000416667
Z
6,1
1.1261E-05
1.3550E-05
2.0325E-05
2.4876E-05
8.3333E-05
0.00055556
Z
6,2
1.1111E-05
1.4493E-05
1.7007E-05
2.7778E-05
3.7037E-05
Z
6,3
1.5576E-05
2.1097E-05
5.5556E-05
0.00055556
Z
6,4
1.1111E-05
1.8315E-05
2.4876E-05
0.00012821
0.00055556
0.00166667
Z
6,5
1.1494E-05
1.4881E-05
1.9380E-05
2.4510E-05
0.000119
0.000417
Z
6,6
1.1111E-05
1.6340E-05
3.1447E-05
9.8039E-05
0.000333333
5.556E-04
Z
7,1
1.4124E-05
2.6042E-05
0.00016667
0.00083333
Z
7,2
1.1261E-05
1.4006E-05
1.7361E-05
2.8736E-05
3.7037E-05
0.00033333
Z
7,3
1.4881E-05
2.1368E-05
6.4103E-05
0.00055556
Z
7,4
1.1038E-05
1.3774E-05
3.2051E-05
0.00166667
Z
7,5
1.1737E-05
1.4881E-05
3.2680E-05
0.000111111
0.000333333
0.000416667
Z
7,6
1.1111E-05
1.6340E-05
3.1447E-05
9.8039E-05
0.000333333
0.000555556
Due to the sample time of the data acquisition system the all frequency measurements have
±0.0016 Hz of uncertainty. Fig. 18 shows the experimental setup of the lyophilization device
used in this work.
Figure 18. Experimental setup of the lyophilization device.
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5. Results analysis
Based on the transfer function matrix from
the complete dynamical system of the
analyzed lyophilization process, the model
simulation results (output variables
performance) for the same experimental
scenario (temporal behavior of the set of
input variables) was predicted. Figs. 19 and
20 show the temporal confrontation
between the experimental results and the
model simulation, for the internal
temperature and pressure from the
concerned lyophilization chamber.
Conclusion
Due to the zeros location listed on Table 2,
it is true that electrical inertial moments are
not significant for the spectral of interest in
this study case. As the poles location
dictates, the dynamic link between ambient
temperature and pressure is strong and
cannot be decoupling, statement that
reinforces the thermodynamics in the
between.
Figure 19. Results of the temperature prediction from the lyophilization chamber.
Figure 20. Results of the pressure prediction from the lyophilization chamber.
It is shown the poles location does not
depend of the three sub-process of the
lyophilization process, because they are
define based on the transfer function and
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do not on the activation time of the input
variables which define the lyophilization
device energetic performance.
The strongest dynamics links between
input and output variables are the ones
which present almost the same zeros
location, and they came from the variables
that represents the same energetic stage of
the refrigerant gas in both vapor
compression systems.
The results analysis statement listed
before, has been reviewed by a
refrigeration specialist board conformed
by Technologists from the several
Companies as BOHN, MABE, Guentner
and by Scientifics from LaNITeF; then the
most important results of this work is that
this model can work as the baseline for the
future development of a control algorithm
and an energy optimization strategy
consequently, because results are
according with the time response expected
from the lyophilization system.
Acknowledgment
Authors thank CONACYT for the support
through the Scholarships No. 906807 and
903766 and by the National Laboratories
Program Project No. 299090 from the
National Laboratory for Cooling
Technology Research (LaNITeF).
Thanks, are extended to the School of
Engineering of the Anahuac University of
Querétaro and to the LaNITeF from the
Engineering Center of Industrial
Development (CIDESI).
Part of the result reported in this paper
comes from the project
VINCULACIÓN/2018/03 developed by
CIDESI and WALWORTH México with
the support of COMECYT by the
PROGRAMA PARA LA
VINCULACIÓN DE EMPRESAS DEL
ESTADO DE MÉXICO CON
INSTITUCIONES DE EDUCACIÓN
SUPERIOR Y CENTROS DE
INVESTIGACIÓN.
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