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Modela= ción matemática para el control de temperatura de salida en tanques de almacenamiento térmico de la Industria láctea

Mathematical modeling for outlet temperature control in thermal stor= age tanks in the dairy industry

= ¹

Danielita Fernanda Borja= Mayorga

https://orcid.org/0000-0002-8438-064X

Escuela Superior Politécnica de Chimborazo

danielita.borja@espoch.edu.ec=

= ²

Mónica Lilián Andrade Av= alos

https://orcid.org/0000-0001-5736-5607

Escuela Superior Politécnica de Chimborazo

moandrade@es= poch.edu.ec=

= ³

Edgar Gualberto Salazar Alvarez

https://orcid.org/0000-0003-0988-0641=

Escuela Superior Politéc= nica de Chimborazo

edgar.salazar@espoch.edu.ec

Artículo de Investigación Científica y Tecnológica

Enviado: 05/03/2022

Revisado: 06/04/2022

Aceptado: 28/05/2022

Publicado:10/06/2022

DOI: https://doi.org/10.33262/concienciadigi= tal.v5i2.2235 =

Cítese: <= /b>

&= nbsp;

Borja Mayorga, D. F., Andrade Avalos, M. L., & Salazar Alvarez, E. G. (2022= ). Modelación matemática para el control de temperatura de salida en tanques= de almacenamiento térmico de la Industria láctea. ConcienciaDigital, 5(2), 251-269. https://doi.org/10.33262/concienciadigital.v5i2.2235=

&= nbsp;

CONCIENCIA DIGITAL, es una Revista Multidisciplinar= , Trimestral, que se publicará en soporte electrónico tiene como misión contribu= ir a la   formación de profesionales competentes con visión humaníst= ica y crítica que sean capaces de exponer sus resultados investigativos y científicos en la misma medida que se promueva mediante su intervención cambios positivos en la sociedad. https://concienciadigital.org

La rev= ista es editada por la Editorial Ciencia Digital (Editorial de prestigio regis= trada en la Cámara Ecuatoriana de Libro con No de Afiliación 663) www.celibro.org.ec<= span lang=3DES style=3D'font-size:8.0pt;line-height:115%;font-family:"Times Ne= w Roman",serif; mso-fareast-font-family:"Times New Roman"'>

Esta revista está protegida ba= jo una licencia Creative Commons Attribution Non Commercial No Derivatives 4= .0 International. Copia de la licencia: http://creativec= ommons.org/licenses/by-nc-nd/4.0/

Palabr= as claves: modelado matemático; tanque térmico; balance = de energía; validación.

&= nbsp;

Resumen =

Introducción. En el Ecuador = la industria Láctea a nivel general representa el 6,1 % de la actividad en agricultura y ganadería del país. En cuanto al consumo de leche fresca sin elaborar se observa que alrededor del 43,4% de la producción se destina al consumo final, el 56% restante al consumo intermedio, dentro del cual destacan la elaboración de productos lácteos (89% del total consumido). <= b>Objetivos. Desarrollar el modelado matemático de los tanques térmicos de almacenamie= nto de leche de la Estación Experimental de Tunshi de la Escuela Superior Politécnica de Chimborazo en el control de la temperatura de salida en el proceso de refrigeración y calentamiento. Metodología. Se realizó = la determinación de la temperatura inicial de ingreso del fluido al centro de acopio con un valor de 19,6 °C y de 8°C para la refrigeración, la tempera= tura de calentamiento se estableció hasta los 80°C, el proceso fue modelado a partir del balance de energía y la simulación mediante el programa Matlab (Simulink) se trabajó con la capacidad de 875 litros de un tanque tipo vertical en un tiempo total de 4800 segundos. Resultados. Para la validación se determinaron tres diferentes comportamientos de curvas tanto para el balance de energía del tanque térmico, su función de transferenci= a en el calentamiento de la leche y del comportamiento tanto de recepción, enfriamiento y calentamiento hasta la temperatura de 80 °C. Conclusiones. Se evidenció el comportamiento de las curvas y se verifico de manera experimental dando validez a la modelación establecida de los dos diferentes procesos.<= /o:p>

Keywords:

mathematical modeling; thermal tank; energy balance; validation..<= /span>

Abstract= <= /o:p>

Introduction. In Ecuador, the dairy industry at a general lev= el represents 6.1% of the activity in agriculture and livestock in the count= ry. Regarding the consumption of fresh unprocessed milk, it is observed that around 43.4% of production is destined for final consumption, the remaini= ng 56% for intermediate consumption, within which the production of dairy products stands out (89% of the total consumed). Objectives. Develop the mathematical modeling of the thermal milk storage tanks of the Tunshi Experimental Sta= tion of the Polytechnic Higher School of Chimborazo in the control of the outl= et temperature in the cooling and heating process. Methodology. The determination of the initial temperature of entry of the fluid to the collection center was made with a value of 19.6= °C and 8 °C for refrigeration, the heating temperature was established up to= 80 °C, the process was modeled Based on the energy balance and the simulation using the Matlab program (Simulink), worked with the 875-liter capacity o= f a vertical-type tank in a total time of 4800 seconds. Results. For the validation, three different behaviors of cur= ves were determined both for the energy balance of the thermal tank, its tran= sfer function in heating the milk and the behavior of both reception, cooling = and heating up to a temperature of 80 °C. Conclusion. The behavior of the curves was evidenced and verified experimentally, validating the established modeling of the two different processes.<= /o:p>

Introducción=

= La leche obtenida tras el ordeńo y que posteriorme= nte es refrigerada es a lo que se denomina leche cruda, a la que no ha sufrido cambios químicos, y no se le ha aplicado ningún tratamiento térmico por cal= or. Esta leche cruda es una excelente materia prima debido a la gran variedad de productos que se pueden elaborar a partir de ella, tales como, leche de consumo, quesos, yogures, helados, mantequilla, nata, cuajada y postres lác= teos (García, 2015)

El almacenamiento en refrigeración inhibe el desarrollo y la proliferación de los microorganismos en la leche, aunque si este almacenamiento es prolongado se convierte en un medio ideal para el desarrollo de los microorganismos psicrófilos (Alonso, 1996).

Efecto de la aplicación= de frío sobre las propiedades de la leche

A nivel de los componen= tes de la leche, con las temperaturas de refrigeración aumenta la estabilidad de las micelas de caseína, sin embargo, si bajan estas temperaturas y se produ= ce la congelación parcial de la leche los glóbulos de grasa se cristalizarán produciendo fisuras en la membrana del glóbulo, liberando de este modo los triglicéridos que serán atacados por las enzimas lipasas ocasionando los fenómenos de enranciamiento (Rosado & Rosado, 2015).

Efecto de la aplicación= de calor sobre las propiedades de la leche

En la termización, la aplicación de temperaturas bajas seguida de una rápida refrigeración alrede= dor a los 4-8 °C no represente un efecto importante sobre los componentes de la leche y sus propiedades, por tanto, no afecta al valor nutritivo (Callejo, 2013).

Para conseguir el efecto deseado de los tratamientos térmicos es tan importante alcanzar la temperat= ura como respetar el tiempo de tratamiento. El tratamiento es eficaz por la combinación de temperatura-tiempo. Cuando el calentamiento es excesivo puede tener consecuencias en los componentes de la leche como degradación de lact= osa, reacciones entre la lactosa y las proteínas (García, 2015).

Según Alais (2018), las temperaturas estables de conservación de leche se estable= cen como:

·&nb= sp;        A 40 ° C, la conservación de la leche en depósi= tos de refrigeración en la granja durante un tiempo a dos días tiene riesgos considerables. La barrera del millón de gérmenes no se alcanza generalmente hasta transcurridos 3 días.

·&nb= sp;        A 0-1 °C, la leche de buena calidad bacteriológ= ica inicial puede recogerse dos veces por semana solamente (conservación 3 y 4 días) sin que aparezcan defectos.

·      =    La refrigeración instantánea después del ordeńo es una práctica muy recomendab= le, sobre todo cuanto la leche se destina a fabricaciones especiales, como puede ser las de alimentos infantiles.

Equipos de enfriamiento para leche

Un tanque de leche o enfriador de leche consiste en una tina interior y otra exterior, realizada= s en acero inoxidable de calidad alimenticia. El tanque de expansión directa, soldado en interior, tiene un sistema (evaporador) de placas y tubos en los= que circula gas refrigerante (R22). Este gas absorbe el calor del líquido conte= nido en la tina (la leche). Los tanques de expansión directa se entregan con un compresor y una grilla de condensación en la que también circula gas refrigerante. El mismo principio que para un refrigerador/heladera (refrigeración por comprensión) (Pendini, 2012).

Tanques de enfriamiento vertical o silo

Acorde a Milkplan (2021= ), fabricadores de tanques abiertos de refrigeración en sus catálogos de produ= ctos establece dos tipos de tanques tipo silo o verticales para almacenamiento de leche:

a)      Tanque vertical de 50-300 litros

b)      Tanque vertical de 400-2500 litros <= /span>

Control de Temperatura = de Tanques Verticales

El equipo para el contr= ol de la temperatura de la leche deberá operar satisfactoriamente con cualquier volumen entre 10 % y 100 % del volumen nominal a temperaturas de la leche d= e 0 a 35 °C. También deberá ser capaz de soportar sin pérdida de calibración, temperaturas dentro del recipiente de 10 a 70 °C y temperaturas de funcionamiento (International Standard [ISO], 1983.)

Metodología

Material y Métodos

Para llevar a cabo el desarrollo del presente trabajo de investigación se realizó el estudio previo de las propiedades fisicoquímicas de recepción de la leche previo al almacenamiento en los tan= ques térmicos (Madrid, 1996), se realizó una experimentación con la capacidad má= xima de proceso de refrigeración y calentamiento se procedió a la modelación de = la función respuesta de transferencia con el software Matlab (Simulink) en su versión R2021a. (Unidad States.The MathWorks, Inc. [MATLAB], 20= 21).

Figura 1

Diagrama de bloqu= es de la propuesta del modelado del tanque de refrigeración<= /i>

Nota: = Esquema gráfico del planteamiento del modela= do al tanque de refrigeración para el almacenamiento de la leche en la industr= ia láctea

Para el desarrollo previo se analizó las con= diciones preliminares del tanque de almacenamiento como sus propiedades físicas y térmicas para la conservación de calor, para este estudio se realizó previó= el análisis físico químico de la leche en la recepción y el planteamiento del balance de energía en su estado estable (Cuevas & Fonseca, 2016).

Las condiciones se operación se estableciero= n y se dio el planteamiento gráfico del sistema simulado dando respuesta a la temperatura de salida del proceso real.

Propiedades de la leche de almacenamiento en el tanque =

El estudio de las propiedades fisicoquímicas de recepción de la leche se los tomó a partir del instrumento analítico Milk analyzer Ecomilk 120 (Milkplan, 2021), previamen= te calibrado a partir de soluciones buffer estándar, obteniendo los siguientes= valores:

Tabla 1=

Propied= ades físico-químicas de la leche a recepción

Condición

Valor

Unidades

Contenido de grasa=

4.14

Sólidos no grasos<= /span>

5.93

Densidad

1.028

Proteína

2.28

Temperatura

19.6

Lactosa

3.22

Conductividad

4.36

pH

5.81

-

Fuente: Manual Ekomilk Ultra <= /span>

Condiciones de operación del tanque vertical

Se estableció las condiciones del tanque vertical predeterminado de capacidad = de 875 litros sus propiedades de diseńo y construcción.

De igual manera se analizó y se estableció los parámetros de diseńo de la pared del tanque vertical en su espesor como lámina establecida en su interior.

Tabla 2=

Parámet= ros de diseńo del tanque vertical

Condición

Valor

Unidades

Capacidad del tanque

875

L

Presión atmosférica (P_atm)

1

atm

Longitud del tanque

0.97

M

Diámetro del tanque

0.66

M

Espesor del tanque=

0.10

M

Cálculos del tanq= ue de almacenamiento vertical

Para la determina= ción de la geometría del tanque y dimensión, se trabajó con el tipo cilíndrico, = con el material acorde al Instituto Ecuatoriano de Normalización (acero inoxida= ble AISI 304).

Tabla 3=

Dimensionamiento del tanque vertical

Determinación

Fórmula

Valor

Unidades

Volumen del tanque=

V: volumen del tanque, (l) =

p: producción anual de leche, (l) =

N: duración anual de producción

n: número de días de almacenamiento de la lec= he

c: coeficiente de relleno (constante)=

730.16

L

Esfuerzo Tangencial

σ: Esfuerzo, (N/m²⁾

P: Presión hidrostática, (N/m2)

D: Diámetro, (m)

t: Espesor, (m)

N/m²

32330.1

Cálculo de la masa de leche

m: masa, (kg)

δ: densidad de la leche, (kg/m<= sup>3)

V: capacidad del recipiente, (m^=
3)

901.25

Kg

Cálculo del calor de refrigeración

Q: calor, (KJ)

m: masa, (kg)

D<= span lang=3DES-MX style=3D'font-size:10.0pt;line-height:150%;font-family:"Time= s New Roman",serif; color:black;mso-ansi-language:ES-MX'>T: variación de temperatura, (°C)

Cp: calor específico de la leche, (KJ/kg*°C)

-41.08

KJ

Cálculo del calor absorbido en el calentamiento

Q: calor, (KJ)

m: masa, (kg)

Tfinal: 80, (°C) <= /span>

Tinicial: 8, (°C)<= /span>

Cp: calor específico de la leche, (KJ/kg*°C)

255

KJ

Tabla 3=

Dimensionamiento del tanque vertical (contin= uación)

Determinación

Fórmula

Valor

Unidades

Flujo de entrada al tanque

q: flujo volumétrico, (m³/s)

V: volumen del recipiente, (m³)

t: tiempo transcurrido, (s)

0.000426

m³/s

=

Resultados y Discusión

Planteamiento del modelo matemático al tanque de refrigeración vertical

Para el planteami= ento correcto de la modelación matemática se tomó a interés la función de transferencia de la temperatura de salida 𝑇𝑓= ;(t), respecto a los cambios de su temperatur= a de entrada 𝑇𝑖(t). Tomando el contenido del tanque como el volumen de control, el balance de energía se lo estableció en estado estacionario (Smith & Corripio, 2014, p.108).<= /p>
Tabla 4<= /p>
Condici= ones de operación para el modelo y simulación

Condición

Valor

Unidades

Flujo de entrada

0.000426

Densidad del fluido de entrada<= /o:p>

0.001032

Densidad del fluido de salida

0.001035

Capacidad calorífica inicial de la l= eche

0.0039

Capacidad calorífica a volumen const= ante

0.034

Volumen líquido del tanque

0.875

Temperatura de entrada del fluido

19.6

Dinámica del proceso

800

Ecuaciones del Balance de Energía

El balance de energía es efectuado con la finalidad de conocer la energ= ía requerida y desechada en cada operación del proceso (Vega, 2010).

𝑞<= /span>1𝜌𝑖h1𝑇𝑖(𝑡)−𝑞<= /span>1𝜌𝑓h2𝑇𝑓(𝑡)=3D 𝑑𝑈= ;/𝑑𝑡 𝐸𝑐.1

Donde:

𝑞1 =3D flujo volumétrico de entrada, (𝑚3/𝑠) =

𝜌𝑖<= /span>, 𝜌𝑓 =3D densidad del lí= quido de entrada y salida, (kg/𝑚3)

h<= span style=3D'mso-bookmark:_Hlk81985521'>1=3D entalpía del líquido de entrada, (J/kg) <= /p>
h<= span style=3D'mso-bookmark:_Hlk81985521'>2 =3D entalpía del líquido de salida, (J/kg) <= /p>
𝑇𝑖<= /span> =3D Temperatura de entrada del fluido, (°C) <= /p>
𝑇𝑓<= /span> =3D Temperatura de salida del fluido, (°C)
U =3D Energía int= erna del líquido en el tanque, (J/kg)

En términos de energía interna del sistema:

𝑞<= /span>1𝜌𝑖𝐶𝑝𝑖w= 879;𝑖(𝑡)−𝑞<= /span>1𝜌𝑓𝐶𝑝𝑓w= 879;𝑓(𝑡)=3D 𝑑<= /span>[𝑉𝜌𝐶𝑣= (𝑡)]/= 𝑑𝑡= ;   𝐸𝑐.2

Donde:

𝐶𝑝𝑖= ; =3D capacidad calorífica de entrada a presión constante del líquido, (= J/kg * °C)

𝐶𝑝𝑓= ; =3D capacidad calorífica de salida a presión constante del líquido, (J= /kg * °C)

𝐶𝑣 =3D capacidad calorífica del líquido a volu= men constante, (J/kg * °C)

V =3D Volumen líq= uido en el tanque, (𝑚3)

La densidad y la capacidad calorífica se suponen constantes e iguales en el rango de temperaturas de la operación a refrigeración

𝑞<= /span>1𝜌𝑖𝐶𝑝𝑖w= 879;𝑖(𝑡)− <= /span>𝑞<= /span>1𝜌𝑓𝐶𝑝𝑓w= 879;𝑓(𝑡)=3D𝑉𝜌= ;𝐶𝑣𝑑𝑇𝑓(𝑡)/<= span style=3D'mso-bookmark:_Hlk81985521'>𝑑𝑡= ;     𝐸𝑐.3

La expresión matemática anterior se expone como el modelo a simulación para ello la resolución mediante la programación en Simulink indicará la respuesta de la temperatura de salida como una función del tiempo al calentamiento de la le= che teniendo

𝜌𝑖= ;𝐶𝑝𝑖𝑇𝑖(𝑡)𝑞<= /span>1(𝑡)− <= /span>𝜌𝑓= ;𝐶𝑝𝑓𝑇𝑓(𝑡)𝑞<= /span>1(𝑡)=3D𝑉𝜌= ;𝐶𝑣𝑑𝑇𝑓(𝑡)/<= span style=3D'mso-bookmark:_Hlk81985521'>𝑑𝑡= ;   𝐸𝑐.4

Balance de energí= a en estado estacionario

𝜌𝑖= ;𝐶𝑝𝑖𝑇𝑖̅𝑞<= /span>1− 𝜌𝑓= ;𝐶𝑝𝑓𝑇𝑓̅𝑞<= /span>1=3D0    𝐸𝑐.5

Donde:

𝑇𝑖<= /span>̅ =3D valor inicial de estado estacionario de entrada, (°C) <= /o:p>

𝑇𝑓̅ =3D valor inicial de estado estaciona= rio de salida, (°C)

𝑞<= /span>1𝜌𝐶𝑝[𝑇𝑖(𝑡)−𝑇𝑖= ;̅]−𝑞<= /span>1𝜌𝐶𝑝[𝑇𝑓(𝑡)−𝑇𝑓= ;̅]=3D𝑉𝜌= ;𝐶𝑣𝑑[𝑇𝑓(𝑡)−𝑇𝑓= ;̅]/𝑑𝑡= ;

𝑉𝜌= ;𝐶𝑝𝑑𝑇̅2/𝑑𝑡= ;=3D 𝑞1𝜌𝐶= ;𝑝𝑇1̅−𝑞<= /span>1𝜌𝐶𝑝𝑇= ̅2

𝑑𝑇= ;̅2/𝑑𝑡= ;=3D 𝑞1/<= span style=3D'mso-bookmark:_Hlk81985521'>𝑣<= /span>1*𝑇1̅−𝑞<= /span>1/𝑣1*<= span style=3D'mso-bookmark:_Hlk81985521'>𝑇<= /span>̅2

𝑑𝑇̅2/𝑑𝑡= ;=3D 𝑞1/<= span style=3D'mso-bookmark:_Hlk81985521'>𝑣<= /span>1*(𝑇̅1−𝑇<= /span>̅2)

Se definió las variables de desviación al proceso obteniendo:

𝛤<= /span>(𝑡)=3D𝑇𝑓= ;(𝑡)−𝑇<= /span>̅𝑓<= /span>

𝛤𝑖= ;(𝑡)=3D𝑇𝑖= ;(𝑡)− <= /span>𝑇<= /span>̅𝑖<= /span>

Donde:

𝛤 =3D variable de desviación de la temperatura de salida, (°C)

𝛤𝑖 =3D variable de desviación de la temperatur= a de entrada, (°C)

𝑑𝛤= ;2(𝑡)/<= span style=3D'mso-bookmark:_Hlk81985521'>𝑑𝑡= ;=3D 𝑞1/<= span style=3D'mso-bookmark:_Hlk81985521'>𝑉<= /span>1*(𝛤1(<= span style=3D'mso-bookmark:_Hlk81985521'>𝑡<= /span>)−𝛤<= /span>2(𝑡))

Transformadas de = La place

𝑠𝛤= ;2(𝑠)=3D 𝑞<= /span>1/𝑉1*(= 𝛤<= /span>1(𝑠)−𝛤<= /span>2(𝑠))

𝛤2(<= span style=3D'mso-bookmark:_Hlk81985521'>𝑠<= /span>) (𝑠+ <= span style=3D'mso-bookmark:_Hlk81985521'>𝑞<= /span>1𝑉1) =3D 1∗𝛤<= /span>1(𝑠)/<= span style=3D'mso-bookmark:_Hlk81985521'>𝜏<= /span>1

𝛤<= /span>2(𝑠)=3D (𝛤<= /span>1(𝑠)/<= span style=3D'mso-bookmark:_Hlk81985521'>𝜏<= /span>1) / (𝑠<= /span>+ 1/𝜏<= /span>1) =3D (𝛤<= /span>1(𝑠)/<= span style=3D'mso-bookmark:_Hlk81985521'>𝜏<= /span>1) /(𝜏<= /span>1𝑠+1/= 𝜏<= /span>1)

Función de transf= erencia del proceso

𝛤<= /span>2(𝑠)=3D 1/(𝜏<= /span>1𝑠+1) *𝛤<= /span>1(𝑠) <= span style=3D'mso-bookmark:_Hlk81985521'>𝐸𝑐= ;.6

Figura 2

Simulación del modelado mediante Matlab (Similunk)

Nota: = Mediante el balance de energía representado = en un sistema de bloques por la herramienta Simulink y factor integrador para determinar la incógnita de la ecuación diferencial se pudo determinar una respuesta de temperatura de salida en un lapso de 800 segundos de 20,51 °C.=

Curva del comportamiento dinámico=

Figura 3

Curva del comportamiento dinámico

Nota: <= span lang=3DES-MX style=3D'font-size:10.0pt;line-height:115%;font-family:"Times = New Roman",serif; color:black;mso-themecolor:text1;mso-ansi-language:ES-MX'>La curva del comportamiento dinámico se determinó de forma creciente, completando su estabilidad en un tramo de 800 segundos, representado por el comando Scope = de Simulink.

Se establece la función= de transferencia del proceso real mediante la ganancia del proceso y la dinámi= ca para ello:

𝑘=3D h𝑓&#= 119892;𝜌/𝐶𝑝𝑞1   𝐸w= 888;.7

Donde:

𝑘 =3D ganancia del proceso, (°C*s/kg)

h𝑓&#= 119892; =3D entalpia de vaporización, (kJ/kg)

𝜌 =3D densidad del fluido, (kg/𝑚3)

𝐶𝑝 =3D capacidad calorífica de la leche, (J/kg*°C= )

𝑞1 =3D caudal de entrada, (𝑚3/𝑠)

Para el cálculo de la entalpía del proceso mediante las tablas termodinámicas de (Cengel & Bo= les, 2012).

Tabla 5

Condiciones para el cálculo de la ganancia de proceso

Condición

Valor

Unidades

Entalpía de proceso a 19,6°C

2454000

Densidad de la leche

0.00132

Capacidad calorífica de la leche

3.930

Caudal de entrada<= /span>

0.000426

Masa de vapor

0.216

=

Con los datos obtenidos de la tabla se obtuvo:

𝑘=3D1.11 °𝐶∗𝑠/𝑘𝑔

=

=

Tabla 6

Determinación de = la variación de la ganancia del proceso

Determinación

Ecuación

Valor

Variación de la temperatura

∆ =3D ∗ <= /span>

k =3D ganancia de proceso del proceso dinámico, (°C*s/kg)

𝑚𝑣

0.24 °C

Temperatura máxima del proceso<= /o:p>

= 𝑇. = 𝑚á= 𝑥𝑖𝑚𝑎 = 𝑑𝑒 = 𝑝𝑟𝑜𝑐𝑒𝑠𝑜

=3D 80 °𝐶 + ∆= = 𝑇

80.24 °C

Ganancia total del proceso

= 𝑘 =3D
∆= 𝑠𝑎𝑙𝑖𝑑𝑎/∆= 𝑒𝑛𝑡𝑟𝑎𝑑𝑎

1,64

Dinámiaveca del proceso

𝜏1 <= span lang=3DES-MX style=3D'font-family:"Times New Roman",serif;mso-fareast-fon= t-family: "Cambria Math";mso-ansi-language:ES-MX'>=3D
𝑉𝑟𝑒𝑐𝑖𝑝𝑖𝑒𝑛= 𝑡𝑒 / 1

2053.99 s

La dinámica de proceso se recomienda= que debe ser 10 veces mayor al proceso real

Figura 4

Temperatura de salida al proceso mediante factor integrador

Nota: Se reemplaza la función de transferencia del siste= ma al tanque y se determinó la temperatura del proceso ideal dinámico en un la= pso igual a 800 segundos.

Figura 5<= /o:p>

Comportamiento de la curva del proceso con función de transferencia

Nota: <= /span>Mediante la función de transferen= cia se determinó la temperatura del proceso ideal con un total de 21,66 °C del proceso de calentami= ento del fluido en un lapso de 800 segundos. El proceso óptimo se lo pl= anteó hasta los 80 °C.

Se estableció las condiciones de trabajo para el proceso en un lapso de 0 a 19.6 °C estableciendo que no existe cambios entre las propiedades de la leche como su estructura y cambio de calor latente de fusión.

0 < 𝑇 = < 19.6

Para el desarrollo de la ecuación= de refrigeración y su función de transferencia se partió de la ley de enfriamiento de Newton indicando:

𝑇(𝑡) = =3D (𝑇𝑂 − 𝑇𝑠) = 𝑒⁻^𝑘w=
905; = + 𝑇𝑠        𝐸𝑐.8

Donde:

𝑇(𝑡) =3D temperatura luego de transcur= rir t horas

𝑇𝑠 =3D temperatura constante del ambiente

𝑇o =3D temperatura inicial=

k =3D tasa relativa

t =3D tiempo

Se aplica la linealización matemá= tica para obtener la función de transferencia de refrigeración=

Se determinó el script de los parámetros de la segunda función de transferenci= a de enfriamiento y calentamiento del proceso industrial en la elaboración de leche a partir de lote de producció= n a partir de los datos experimentales obtenidos y la capacidad del tanque de los 875 litros.

Figura 6

Script de los parámetros de la simulación del proceso

Nota: El sist= ema representado por Simulink en el proceso tanto de enfriamiento y calentamien= to se lo estableció mediante bloq= ues condicionales y las funciones de transferencia de cada uno de los procesos en el desarrollo experimental.

&n= bsp;

Figura 7

Simulación del modelado de refrigeración y calentamiento del tanque de 500 L

&n= bsp;

Nota: Se determinó el script de los parámetros de la segunda función de transferenci= a de enfriamiento y calentamiento del proceso industrial en la elaboración de leche a partir de lote de producció= n del departamento de lácteos de la Base Experimental.

Figura 8

Comportamiento de las funciones de transferencia del proceso

&n= bsp;

Nota: Al desarrollo del proceso se determinó la curva del proceso bajo las condicion= e y función de transferencia en un lapso de desarrollo experimental= de 4800 segundos.

Validación del modelado del sistema del tanque y refrigeración

Acorde a Kishurim (2013), el solucionar el problema valida el modelo mediante la comprobación de los resultados matem= áticos y su interpretación como razonables y compatibles en términos de la información recolectada dado por el probl= ema original. Cuando el proceso de = validación arroje resultados no satisfactorios todo el proceso debe repetirse con una modificación, si se logra la<= span style=3D'letter-spacing:-.15pt'> solución del problema original real = se da como valido al modelado (p.156).<= /span>

Tabla 7

Datos Experimentales = y Referencial al Modelado

Datos experimentales y de referencia al modelado

Condición

Modelado=

Experimental

Validación

Temperatura de salida = al proceso de calentami= ento

21.6

21.6

El proceso mediante su función de transferencia se lo estableció en un lapso a los 800 segundos garantizand= o lo experimental con el modelado correspondiente.

Temperatura de salida al proceso de refrigerac= ión

8 °C

8 °C

Acorde al setpoint generado de los 30 °C= y su refrigeración hasta los 8 = °C se determinó la temperatura de salida del fluido y la generación del modelado<= span style=3D'letter-spacing:.05pt'> hasta el límite de la operación considerando= la capacidad térmica del banco h= ielo.

Conclusiones

·      =    = La modelación matemática empleada para el estudio de la temperatura de salida mediante su simulación en Simulink se demostró en el proceso tanto experime= ntal y representado los valores del balance de energía como en función de ecuaci= ón de transferencia el proceso parametrizado el cumplimiento con lo establecid= o en el lapso total de operación, garantizando estable el modelado con el desarr= ollo experimental.

·         = El comportamiento de la curva de transferencia del proceso real se lo determin= ó en dos diferentes condiciones a partir de la ecuación diferencial de las varia= bles de entrada y su función de transferencia del modelado , demostrando un crecimiento gradual creciente en el proceso de calentamiento similar a una ecuación de segundo grado o parabólica , la representación final tanto de la refrigeración y calentamiento generó la curva total de un proceso real indicando los puntos generales del cambio del temperatura bajo condicionales como el punto desde los 19.6 °C y su refrigeración gradual decreciente el aumento de la temperatura hasta su proceso estable de lo planteó hasta los = 80 °C indicando el crecimiento final de temperatura. =

Referencias Bibliográficas

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Cengel, Y. A., & Boles, M. A. (2012). Termodinámica. (7.Ş ed.). Editoria= l Mc Graw Hill Education. http://joinville.ifsc.edu.br/~evandro.dario/Termodin%C= 3%A2mica/Material%20Did%C3%A1tico/Livro%20-%20Cengel/Termodinamica%20-%20Ce= ngel%207th%20-%20espanhol.pdf

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Vega, L. (2010). Diseńo y Construcción de Equipo de Refrigeración de Leche Cruda en Sitio de Producción. [Tesis de maestría, Universidad Nacional de Colombia sede Amazonía] Repositorio institucional Virtualpro.   https:/= /www.virtu= alpro.co/biblioteca/diseno-y-construccion-de-equipo-de-refrigeracion-de- leche-cruda-en-sitio-de-produccion

El artículo que se publica es de exclusiva responsabilidad de los autores y no necesariamente reflejan el pensamiento = de la Revista Conciencia Digital.

El artículo queda en propiedad de la revista y, por tanto, su publicación parcial y/o total en otro medio tiene que ser autoriz= ado por el director de la Revista Conc= iencia Digital.

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ISSN: <= b>2600-5859

Vol. 5 = No 2, pp. 251 – 269 , abril-junio= 2022

www= .concienciadigital.org

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                                     Ge= stión y Procesos                Página 2511=

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= ¹	Danielita Fernanda Borja= Mayorga	https://orcid.org/0000-0002-8438-064X
	Escuela Superior Politécnica de Chimborazo danielita.borja@espoch.edu.ec=
= ²	Mónica Lilián Andrade Av= alos	https://orcid.org/0000-0001-5736-5607
	Escuela Superior Politécnica de Chimborazo moandrade@es= poch.edu.ec=
= ³	Edgar Gualberto Salazar Alvarez	https://orcid.org/0000-0003-0988-0641=
	Escuela Superior Politéc= nica de Chimborazo edgar.salazar@espoch.edu.ec

	Artículo de Investigación Científica y Tecnológica Enviado: 05/03/2022 Revisado: 06/04/2022 Aceptado: 28/05/2022 Publicado:10/06/2022 DOI: https://doi.org/10.33262/concienciadigi= tal.v5i2.2235 =

Cítese: <= /b>		&= nbsp; Borja Mayorga, D. F., Andrade Avalos, M. L., & Salazar Alvarez, E. G. (2022= ). Modelación matemática para el control de temperatura de salida en tanques= de almacenamiento térmico de la Industria láctea. ConcienciaDigital, 5(2), 251-269. https://doi.org/10.33262/concienciadigital.v5i2.2235= &= nbsp;
		*CONCIENCIA DIGITAL,* es una Revista Multidisciplinar= , Trimestral, que se publicará en soporte electrónico tiene como misión contribu= ir a la formación de profesionales competentes con visión humaníst= ica y crítica que sean capaces de exponer sus resultados investigativos y científicos en la misma medida que se promueva mediante su intervención cambios positivos en la sociedad. https://concienciadigital.org La rev= ista es editada por la Editorial Ciencia Digital (Editorial de prestigio regis= trada en la Cámara Ecuatoriana de Libro con No de Afiliación 663) www.celibro.org.ec<= span lang=3DES style=3D'font-size:8.0pt;line-height:115%;font-family:"Times Ne= w Roman",serif; mso-fareast-font-family:"Times New Roman"'>
		Esta revista está protegida ba= jo una licencia Creative Commons Attribution Non Commercial No Derivatives 4= .0 International. Copia de la licencia: http://creativec= ommons.org/licenses/by-nc-nd/4.0/
Palabr= as claves: modelado matemático; tanque térmico; balance = de energía; validación.	&= nbsp;	Resumen = Introducción. En el Ecuador = la industria Láctea a nivel general representa el 6,1 % de la actividad en agricultura y ganadería del país. En cuanto al consumo de leche fresca sin elaborar se observa que alrededor del 43,4% de la producción se destina al consumo final, el 56% restante al consumo intermedio, dentro del cual destacan la elaboración de productos lácteos (89% del total consumido). <= b>Objetivos. Desarrollar el modelado matemático de los tanques térmicos de almacenamie= nto de leche de la Estación Experimental de Tunshi de la Escuela Superior Politécnica de Chimborazo en el control de la temperatura de salida en el proceso de refrigeración y calentamiento. Metodología. Se realizó = la determinación de la temperatura inicial de ingreso del fluido al centro de acopio con un valor de 19,6 °C y de 8°C para la refrigeración, la tempera= tura de calentamiento se estableció hasta los 80°C, el proceso fue modelado a partir del balance de energía y la simulación mediante el programa Matlab (Simulink) se trabajó con la capacidad de 875 litros de un tanque tipo vertical en un tiempo total de 4800 segundos. Resultados. Para la validación se determinaron tres diferentes comportamientos de curvas tanto para el balance de energía del tanque térmico, su función de transferenci= a en el calentamiento de la leche y del comportamiento tanto de recepción, enfriamiento y calentamiento hasta la temperatura de 80 °C. Conclusiones. Se evidenció el comportamiento de las curvas y se verifico de manera experimental dando validez a la modelación establecida de los dos diferentes procesos.<= /o:p>
Keywords: mathematical modeling; thermal tank; energy balance; validation..<= /span>		Abstract= <= /o:p> Introduction. In Ecuador, the dairy industry at a general lev= el represents 6.1% of the activity in agriculture and livestock in the count= ry. Regarding the consumption of fresh unprocessed milk, it is observed that around 43.4% of production is destined for final consumption, the remaini= ng 56% for intermediate consumption, within which the production of dairy products stands out (89% of the total consumed). Objectives. Develop the mathematical modeling of the thermal milk storage tanks of the Tunshi Experimental Sta= tion of the Polytechnic Higher School of Chimborazo in the control of the outl= et temperature in the cooling and heating process. Methodology. The determination of the initial temperature of entry of the fluid to the collection center was made with a value of 19.6= °C and 8 °C for refrigeration, the heating temperature was established up to= 80 °C, the process was modeled Based on the energy balance and the simulation using the Matlab program (Simulink), worked with the 875-liter capacity o= f a vertical-type tank in a total time of 4800 seconds. Results. For the validation, three different behaviors of cur= ves were determined both for the energy balance of the thermal tank, its tran= sfer function in heating the milk and the behavior of both reception, cooling = and heating up to a temperature of 80 °C. Conclusion. The behavior of the curves was evidenced and verified experimentally, validating the established modeling of the two different processes.<= /o:p>

Condición	Valor	Unidades
Contenido de grasa=	4.14
Sólidos no grasos<= /span>	5.93
Densidad	1.028
Proteína	2.28
Temperatura	19.6
Lactosa	3.22
Conductividad	4.36
pH	5.81	-

Determinación	Fórmula	Valor	Unidades
Volumen del tanque=	V: volumen del tanque, (l) = p: producción anual de leche, (l) = N: duración anual de producción n: número de días de almacenamiento de la lec= he c: coeficiente de relleno (constante)=	730.16	L
Esfuerzo Tangencial	σ: Esfuerzo, (N/m²⁾ P: Presión hidrostática, (N/m2) D: Diámetro, (m) t: Espesor, (m)	N/m²	32330.1
Cálculo de la masa de leche	m: masa, (kg) δ: densidad de la leche, (kg/m<= sup>3) V: capacidad del recipiente, (m^= 3)	901.25	Kg
Cálculo del calor de refrigeración	Q: calor, (KJ) m: masa, (kg) D<= span lang=3DES-MX style=3D'font-size:10.0pt;line-height:150%;font-family:"Time= s New Roman",serif; color:black;mso-ansi-language:ES-MX'>T: variación de temperatura, (°C) Cp: calor específico de la leche, (KJ/kg*°C)	-41.08	KJ
Cálculo del calor absorbido en el calentamiento	Q: calor, (KJ) m: masa, (kg) Tfinal: 80, (°C) <= /span> Tinicial: 8, (°C)<= /span> Cp: calor específico de la leche, (KJ/kg*°C)	255	KJ

Determinación	Ecuación	Valor
Variación de la temperatura	∆ =3D ∗ <= /span> k =3D ganancia de proceso del proceso dinámico, (°C*s/kg) 𝑚𝑣	0.24 °C
Temperatura máxima del proceso<= /o:p>	= 𝑇. = 𝑚á= 𝑥𝑖𝑚𝑎 = 𝑑𝑒 = 𝑝𝑟𝑜𝑐𝑒𝑠𝑜 =3D 80 °𝐶 + ∆= = 𝑇	80.24 °C
Ganancia total del proceso	= 𝑘 =3D ∆= 𝑠𝑎𝑙𝑖𝑑𝑎/∆= 𝑒𝑛𝑡𝑟𝑎𝑑𝑎	1,64
Dinámiaveca del proceso	𝜏1 <= span lang=3DES-MX style=3D'font-family:"Times New Roman",serif;mso-fareast-fon= t-family: "Cambria Math";mso-ansi-language:ES-MX'>=3D 𝑉𝑟𝑒𝑐𝑖𝑝𝑖𝑒𝑛= 𝑡𝑒 / 1	2053.99 s
La dinámica de proceso se recomienda= que debe ser 10 veces mayor al proceso real

Datos experimentales y de referencia al modelado
Condición	Modelado=	Experimental	Validación
Temperatura de salida = al proceso de calentami= ento	21.6	21.6	El proceso mediante su función de transferencia se lo estableció en un lapso a los 800 segundos garantizand= o lo experimental con el modelado correspondiente.
Temperatura de salida al proceso de refrigerac= ión	8 °C	8 °C	Acorde al setpoint generado de los 30 °C= y su refrigeración hasta los 8 = °C se determinó la temperatura de salida del fluido y la generación del modelado<= span style=3D'letter-spacing:.05pt'> hasta el límite de la operación considerando= la capacidad térmica del banco h= ielo.