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Modelo matematico de<= span style=3D'mso-spacerun:yes'>  viscosidad de soluciones tecnicas y puras  durante el procesamiento del azucar  de remolacha

Mathematical model of viscosity of technical and pure solutions during processing of  beet sugar

 

= Javier Ignacio Briones García. [1], = Keren Esther Vásquez Martínez. [2], Diana Nereida Villa Uvidia. [3= ] & Teresita Jackelin Mejía Reinoso. [4= ]

 

Recibido: 21-04-2019 / Revisado: 25-05-2019 /Aceptado: 24-06-2019/ Publicado: 15-07-2= 019

 

 Abstract.                                 DOI: https://doi.org/10.33262/cienciadigital.v= 3i3.1.697

 

This research = has as a main objective to study the variation of viscosity in the technical and pure solutions in the different stages of the process of sugar using a mathematical model. The optimization of the technological systems in the process of ebullition and crystallization of sugar will improve significant= ly the technical and economic profit of the plant, in order to carry out this task, it is necessary to control and determine a series of parameters such = as: viscosity, saturation, supersaturation, purity of the = intercrystalline solution and the molasses. It was carried out through the VisSim 7.0 program, which permitted the modulation of the isobaric evaporative crystallization process to obtain 60 tons of sugar, under the equations of = S.V. Shterman, the values of the total duration of t= he crystallization do not observe pronounced divergence, even though there is a discrepancy for the values of viscosity, which cannot be more than 1500 pas= cal, for the correct functioning of the vacuum apparatus and to avoid damage. By developing new and revolutionary technologies will per= mitid to obtain the normal purity of the intercrystalline solution in the vacuum devices and exclude of the technological line the polythermic crystallization process of saccharose, re= ducing the costs of operating and expenses in electricity, the use of cooling water consumption and simplify in a great quantity the technology to obtaining su= gar.

 

Keywords: Sugar, Computer Simulation, Mathematical Model, Optimization.

 

Resumen.

 

Esta investiga= ción tiene como objetivo estudiar la variación de la viscosidad en soluciones técnicas y puras en las diversas etapas del procesamiento del azúcar utiliz= ando un modelo matemático. La optimización del régimen tecnológico del proceso de ebullición y cristalización del azúcar mejorará significativamente el rendimiento técnico y económico de la planta, para realizar dicha  tarea, es necesario controlar y determi= nar una serie de parámetros como: viscosidad, saturación, sobresaturación, pure= za de la solución intercristalina y  m= elaza. Se llevó a cabo mediante el programa VisSim 7.0= , el cual permitió la modulación del proceso de cristalización evaporativa isobá= rica para la obtención de 60 toneladas  azúcar, bajo las ecuaciones de S.V. Shterman,  los valores de la duración total de la cristalización no observan divergencia pronunciada, aunque existe discrepancia  para los valores de viscosidad, los cuales no deben ser mayor a 1500 pascal, para el correcto funcionamiento del aparato de vacío y evitar dańos. Al desarrollar nuevas y revolucionarias tecnologías  permit= irán obtener la pureza normal de la solución intercristalina en los aparatos de vacío  y excluir de la línea tecnol= ógica el proceso de cristalización politérmica de la sacarosa, reduciendo  costos de operación, gastos en electricidad, consumo de agua refrigerante y simplific= aría en gran medida la tecnología de obtención de azúcar.

 

Palabras claves: Azúcar, Simulación Computarizada, Modelo Matemáti= co, Optimización.

Intro= ducción.

La remolacha azucarera es = la principal fuente de obtención de azúcar en muchos países, siendo el 90% de azúcar consumida en Europa a base de remolacha azucarera y el 30% a nivel mundial, gracias a su elevado contenido de sacarosa. En la producción de es= te tipo de azúcar, intervienen procesos físicos y químicos complejos,  donde la sacarosa es extraída de las cé= lulas en difusión  por acción  química y termo-física separando los az= úcares de los no-azúcares convirtiéndolo en un producto cristalino puro.

 

Cocción del jarabe – Es la etapa final en la producción del azúcar. El objetivo principal es la extracción de la sacarosa disuelta en el almíbar, en forma = de cristales. Durante el proceso de purificación llamado carbonatación – el  líquido resultante del proceso de difus= ión se mezcla con lechada de cal  [Ca(OH)_2 ]   y gas carbónico (CO_2 ), produciendo carbonato de calcio (CaCO_3 )  que se precipita, arrastrando la= mayor parte de las impurezas consigo, en el proceso se elimina un tercio de los n= o azucares.

 

Las sustancias restantes junto con la sacarosa son redirigidas al departamento = de producción, donde el jarabe pasa por el proceso de evaporación hasta que ll= egue a una concentración de 70-75% de contenido sólidos, después de lo cual está listo para su cristalización. El azúcar cristalino se obtiene mediante dos técnicas:

 

ˇ&nb= sp;        Cristalización politérmica;

ˇ&nb= sp;        Cristalización por evaporación isobárica.

 

La separación de la sacarosa es realizada en dos o tres etapas, teniendo en cu= enta la calidad de la materia prima. En el caso de materias primas de baja calid= ad, el azúcar es obtenida mediante  dos esquemas de cristalización (cristalización politérmica= ). Por lo contrario, si la materia prima es de alta calidad se utilizan tres esquemas (cristalización por evaporación isobárica).

Los investigadores centran cada vez más su atención en el desarrollo de nuevas tecnologías de cristalización, debido al gran contenido de impurezas acumul= adas en la solución madre, trayendo como resultado, mayor viscosidad y dificulta= d en el proceso de cristalización, disminuyendo el volumen final del producto. La reducción de la pureza de la melaza ha generado un mayor rendimiento productivo, pero ha alcanzado límites prácticos, llevando resultados indeseables.

 

Esto se debe a la elevada viscosidad de la solución intercristalina, lo que empe= ora las condiciones de nucleación, crecimiento cristalino y duración del proceso debido a una disminución en la velocidad de cristalización. Dado que la pur= eza de la melaza depende de la calidad de los productos que entran en ebullició= n, como del retorno y recirculación del licor madre, esto conduce a la gran acumulación de no-azucares, lo que afecta negativamente al rendimiento técn= ico y económico de la planta.

 

La modelación matemática en los últimos ańos ha sido un método efectivo y comúnmente utilizado en el desarrollo y estudio de procesos tecnológicos, proporcionando respuestas a muchas preguntas incluso en la etapa preliminar= de investigación, permitiendo determinar modos óptimos y esquemas apropiados en los procesos tecnológicos. Eliminando los gastos innecesarios de mano de ob= ra y recursos materiales en la construcción de esquemas irracionales y en la implementación de regímenes ineficientes.

 

El modelo matemático es una herramienta extremadamente flexible que permite reproducir cualquier situación, tanto real como hipotética, ya que no está sujeta a limitaciones físicas. Esto reduce la necesidad de sofisticados equ= ipos de laboratorio y pruebas operacionales de los procesos tecnológicos que se están creando.

 

Metodologia.

El análisis del material bibliográfico permitió concluir, que la viscosidad depende ampliamente de la densidad, temperatura, composición química y asociación de las moléculas en= las distintas soluciones. Cuanto más complicada sea la separación de la solución intercristalina de los cristales de sacarosa, mayor será la concentración de no-azúcares en el licor madre, requiriendo una mayor cantidad de  equipos para su procesamiento.

 

Numerosos estudios han demostrado que la viscosidad de las soluciones técnicas de azúcar, incluyen= do la melaza, no dependen de la velocidad de cizallamiento, es decir, estos fluidos son newtonianos. Sin embargo, estudios más detallados han demostrad= o el carácter no newtoniano de algunas melazas y que las propiedades de las soluciones de azúcar son similares a las soluciones de polímeros y de siste= mas dispersos.

 

Estas desviaciones se debe= n a la presencia de burbujas de gas, altos polímeros y sólidos suspendidos en l= as soluciones. Que en comparación con  soluciones  puras de sacaros= a, que contienen no-azucares, durante una misma concentración de sólidos tiene una viscosidad más baja. Pero esto sólo aplicaría a soluciones que no contengan pectinas, dextranos, polímeros cuya presencia aumenta considerablemente la viscosidad.

 

Diferentes profesionales d= e la industria azucarera han basado sus trabajos investigativos en el estudio de= la viscosidad en las diferentes etapas que presentan las soluciones, para la o= btención de los cristales de azúcar, dichas investigaciones han podido generar ecuaciones que describen la dependencia de esta, desde diferentes variables formando un grupo de 17 ecuaciones, las cuales se muestran en la tabla 1.

 

Sin embargo es imposible c= ompararlas entre sí, ya que la literatura no cuenta con valores de constantes para cada ecuación.

 

Tabla 1: Ecuaciones de viscosidad de soluciones técnicas y puras de la sacarosa

AUTORES

ECUACIÓN DE VISCOSIDAD

1

G. Waterman

<= /span>

2

J. Pidu

<= /span>

3

Y. Frankel

  <= /span>

4

R. Mishchuk, L. Belostotsky

5

G. Znamenskii

, μ=3D= (3.5-0.025*P)<= /span> <= /span>

6

D. Shliphake

<= /span>

7

K. Wagner

<= /span>

8

J. Jhonatelle

<= /span>

9

N. Lyusy<= /span>

<= /span>

10

V. Shterman

<= /span>

11

V. Shterman, A. Filippov

<= /span>

 

12

 

S. Petrov=

<= /span>

13

B. Maurandi

14

V. Shterman, G. Vovk

 

15

V. Shterman

                =

<= /span>

16

N. Silina=

<= /span>

17

A. Gromkovsky, V. Fursov

 

Elaborado: Por los autores

 

En base a los cambios en la viscosidad de soluciones técnicas y puras de la sacarosa en el proceso de ebullición y cristalización se tomo como refere= ncia las ecuaciones creadas por Sergey Shterman las = cuales coinciden con la línea investigativa del proceso planteado.

 

Primera ecuación de S. Shterman

 

            (1)<= /o:p>

        (2)

                                                =    (3)                             

                    (4)                

En donde:

 

mum =3D viscosidad  de la solución intercristalina; DBm3 =3D pureza  de la solución intercristalina; CVm3 =3D sólidos en la solución intercristalina; K =3D coeficiente de impurezas sobr= e la viscosidad de soluciones técnicas azucaradas; Ncv3 =3D fracción molar de  sólidos en la solución intercristalina;= muy =3D viscosidad de la masa cocida; gamy =3D fracción= de volumen de los cristales en la masa cocida; dkr= =3D tamańo de los cristales.

 

Viscosidad por la segunda ecuación de S. Shterman:

 

       (5)             

               (6)                          

            (7)    

 

Donde: <= /p>

 

mum1 =3D viscosidad de la = solución intercristalina; Ty3 - temperatura de la masa de cocio= n; Ncv3 y Nncx3 =3D fracción molar de las sustancias soli= das y los no azucares.

 

Por consiguiente, la viscosidad se describe mediante la ecuación:

 

     (8)

 

La in= vestigacion se implemento eficazmente utilizando el modelo matemático (VisSim 7.0), V= isSim – es  un lenguaje visual de program= ación, diseńado para modelar sistemas dinámicos, así como proyecciones basadas en modelos matemáticos.

 

VisSim combina un ensamblado de la interfaz intuitiva de arrastrar y colocar los diagramas de bloques a un motor de simulación de gr= an alcance. El lenguaje de programacion fue desarr= ollado por la compańía americana Visual Solutions, que= se encuentra en Westford (Massachusetts).

 

 

 

La ventaja de este modelo = es que con su ayuda es posible demostrar, desarrollar y analizar efectivamente nuevas tecnologías de ebullición y cristalización de masas cocidas, alcanza= ndo 60 toneladas de producto final y la concentracion requerida en la masa cocida, independientemente de los parametros tecnologicos, fisico-quimi= cos y termo-tecnicos del proceso.=

 

Está diseńado para estudia= r, fundamentar nuevas tecnologías de cristalización sobre la base del estudio y procesamiento de los resultados de un experimento informático durante el modelado matemático del proceso de cristalización industrial, por ejemplo, = la sacarosa

 

La si= mulacion se llevo a cabo tomando en cuentas los siguient= es parametros tecnologicos:<= o:p>

 

ˇ&nb= sp;        Modo típico de ebullición, dentro del cual se mant= uvo una sobresaturación constante de P =3D const y = el final del proceso fue determinado por la materia seca de la masa cocida. En este caso, la cantidad final de la masa cocida es siempre diferente, dependiendo= de los parámetros tecnológicos y de la composición de las impurezas.

ˇ&nb= sp;        Modo de soldadura con control automático. En este caso, la cantidad final de la masa cocida y las sustancias secas permanecen inalteradas, y en cualquier caso, hasta el final del proceso de ebullición,= los valores de volumen de producto final alcanzando 60 toneladas, con un coeficiente de sobresaturación dado.

ˇ&nb= sp;        El modo de cristalización cuando se alcanza la frecuencia normal, que es la base del modo típico.

ˇ&nb= sp;        El regimen de cristali= zación cuando se alcanza la pureza mínima.

ˇ&nb= sp;        El régimen de cristalización con la selección de cristales a sobresaturación constante es P =3D const.<= /span>

 =

&nbs= p;

Resultado= s.

La comparación de las ecuaciones de Sergey Shterman se ll= evó a cabo utilizando el modelo matemático antes mencionado, dando como resulta= do que las primeras ecuaciones descritas por el autor ruso corresponden a datos referenciales de parámetros finales para la melaza, pero la viscosidad es m= ucho mayor que en las ecuaciones mas recientemente creadas por este, afirmando q= ue estas poseen datos  mucho mas preci= sos (Shterman, 2006).

 

S.V. Shterman enfatiza tomar en cuenta el efecto de los no-azúcares sobre las principales características tecnológicas de los produ= ctos de ebullición y cristalización con la ayuda  de coeficientes m, b y k. Los dos primeros coeficientes reflejan la influencia de los no azucares sobre la solubilidad de la sacarosa, y la seg= unda sobre la viscosidad de las soluciones técnicas de azúcar.=

 =

 =

 =

Figura 1: Comparaci= ón de las ecuaciones 1, 4, 5, 8

 

<= v:shape id=3D"Imagen_x0020_8" o:spid=3D"_x0000_i1026" type=3D"#_x0000_t75" style= =3D'width:402.75pt; height:258pt;visibility:visible;mso-wrap-style:square'>

 =

Fuente: Los autore= s

 

Con el programa informático, se llevó a cabo una serie de experimentos, para estudiar el proceso de ebullición del licor madre por las ecuaciones de viscosidad de Shterman, para el ultimo producto con los siguientes datos:

 

Concentración de sólidos - 0,94 / 0,95 / 0,96, pureza - 0= ,76. El coeficiente "m" varía en el intervalo de 0,1 a 0,5; B =3D 1-m,= los coeficientes "c" y "k" se asumieron sin cambios, los da= tos se resumen en las tablas 2, 3 y 4. El programa proporcionó las siguientes funciones:

 

El modelado y control del proceso de cristalización por evaporación isobárica en la zona de crecimiento cristalino metaestable  manteniendo la relación calculada de la= masa de agua evaporada y el producto cristalizable a ańadir. Al mismo tiempo, el control del proceso se llevó a cabo de tal manera que el producto cristaliz= ado, como regla, necesariamente alcanza la concentración especificada y el volum= en

requerido del producto cristalizado, dependiendo del tipo= y diseńo del equipo utilizado. además, el programa permitió explorar a fondo = la cinética y la dinámica del proceso de cristalización, estudiar los patrones= de cambios en la tecnología, la ingeniería térmica, los indicadores físico-químicos y otros del proceso, sobre esta base justificar y desarroll= ar nuevas tecnologías para la cristalización de la sacarosa.=

 

 

 

 Figura 2: Modelo matemático por las ecuaciones de S. V. Shterman=

 

 

 =

Fuente: Los autor= es

 

 

 En las tablas 2,= 3 y 4 se  muestra como resultado que el m= enor tiempo de cristalización proporciona la ecuación 2 con parámetro b=3D0.5 y = m=3D0.5, el análisis de las mediciones en la viscosidad de la masa cocida y de la solución intercristalina durante el proceso de ebullición y cristalización = con el uso del modelo matemático ha demostrado que la regularidad de la medicio= n en la viscosidad con la composición de impurezas depende fuertemente de los valores integrales de los coeficientes m, b, k, los dos primeros coeficient= es reflejan la influencia de los no azucares sobre la solubilidad de la sacaro= sa, y la segunda sobre la viscosidad de las soluciones técnicas de azúcar.=


 

 

Tabla 2: Masa cocida de producto final con parámetros: Solidos =3D0,94; pureza=3D0,76; b= =3D0.9; 0.7; 0.5

 


Parámetros de investigación

m =3D 0.1

Cambios en  %

m =3D 0.3

Cambios en %

m =3D 0.5

Cambios en %

Ecuación 1

Ecuación 2

Ecuación  1

Ecuación 2

Ecuación 1

Ecuación 2

Tiempo total de cristalización t

7.82

7.9

+ 1%

5.75

5.65

- 1.7%

5.25

4.51

- 14%

Masa de jarabe ańadido mcd3

67692.2

67692.3

0%

66769.6

66766.7

-0.004%=

66769.2

66774.6

+ 0.008%

Masa general de agua evaporada MviOb

27694

27694.4

0%

26771.5

26772.7

+0.004%=

26770

26772.9

+ 0.01%

Cantidad total de azúcar cristalizada Mkr= ob3

26396.9

26403.7

+ 0.02%

16335.9

16345.4

+ 0.5%

8827.2

8834.12

+ 0.07%

Viscosi= dad de la masa cocida muy

325.5

5.56

- 98.3%

330.2

8.124

- 97.4%

1068

7.642

- 99.6%

Concent= ración final de solución intercristalina Cvm3

0.910758

0.910763

+ 0.005%

0.91759

0.917624

+ 0.002%

0.92966

0.929678

+ 0.001%

Pureza final de la solución intercristalina Dbm3

0.552993

0.552895

- 0.02%

0.66214

0.662061

- 0.01%

0.71547

0.715433

- 0.005%

Viscosi= dad de la solución intercristalina mum

25.14

3.561

- 86.5%

95.29

4.093

- 96.2%

594

4.736

- 99.8%

Tamańo final del cristal dkp

0.000376

0.000376

0%

0.00032

0.00032

0%

0.00026

0.000261

0%

 

Fuente: Los autores

 


 

Tabla 3: Masa cocida de producto final con parámetros: Solidos =3D0,95; pureza =3D0,76; b= =3D0.9; 0.7; 0.5

 

Parámetros de investigac= ión

m =3D 0.1

Cambios en  %

m =3D 0.3

Cambios en %<= /span>

m =3D 0.5

Cambios en %<= /span>

Ecuación 1

Ecuación 2

Ecuación  1

Ecuación 2

Ecu= acion 1

Ecu= acion 2

Tiempo total de cristalización t

7.82

7.9

+ 1%

6.32

6.09

- 3.6%=

6.35

4.92

- 22%<= /p>

Masa de jarabe ańadido mcd3

67692.2

67692.3

0%

67692.1

67689.5

- 0.003%

67692.3

67696.3

+ 0.005%

Masa general de agua evaporada MviOb<= /p>

27694<= /p>

27694.4

0%

27693.5

27697.9

+ 0.01%

27694<= /p>

27694<= /p>

0%

Cantidad total de azúcar cristalizada Mkrob3

26396.9

26403.7

+ 0.02%

18205<= /p>

18219<= /p>

+ 0.07%

10125.9

10131.9

+ 0.05%

Viscosidad de la masa cocida muy

325.5<= /p>

5.56

- 98.3%

1514

8.835<= /p>

- 99.8%

6191

8.872<= /p>

- 99.6%

Concentración final de solución intercristalinaCvm3

0.910758

0.910763

+ 0.005%

0.928249

0.928341

+ 0.009%

0.939866

0.939874

+ 0.008%

Pureza final de la solución intercristalinaDbm3

0.552993

0.552895

- 0.02%

0.647381

0.647249

- 0.2%=

0.708158

0.708123

- 0.004%

Viscosidad de la solución i= ntercristalinamum

25.14<= /p>

3.561<= /p>

- 86.5%

355.4<= /p>

4.452<= /p>

- 98.6%

3104

5.194<= /p>

- 99.3%

Tamańo final del cristal dkp

0.000376

0.000376

0%

0.000332

0.000332

0%

0.000273

0.000273

0%

 

 

Fuente: Los autores

 

 

 

Tabla 4: Masa cocida de producto final con parámetros: Solidos =3D0,95; pureza =3D0,76; b= =3D0.9; 0.7; 0.5

 

 

Parámetros de investigac= ión

m =3D 0.1

Cambios en  %

m =3D 0.3

Cambios en %<= /span>

m =3D 0.5

Cambios en %<= /span>

Ecuación 1

Ecuación 2

Ecuación  1

Ecuación 2

Ecu= acion 1

Ecu= acion 2

Tiempo total de cristalización t

8.69

8.73

+ 0.5%=

7.1

6.57

- 8%

8.55

5.38

- 37.4%

Masa de jarabe ańadido mcd3

68615.4

68615.8

0%

68615.4

68615<= /p>

0%

68615.4

68615.9

0%

Masa general de agua evaporada MviOb<= /p>

28617.5

28617<= /p>

0%

28617<= /p>

28617.5

0%

28616<= /p>

28616.5

0%

Cantidad total de azúcar cristalizada Mkrob3

28897.7

28891.7

- 0.02%

20083.2

20086.4

+ 0.01%

11422.5

11426.5

+ 0.03%

Viscosidad de la masa cocida muy

1565

5.26

- 99.3%

11631<= /p>

9.515<= /p>

- 99.7%

62784.5

10.36<= /p>

- 99.9%

Concentración final de solución intercristalinaCvm3

0.92289

0.92287

- 0.002%

0.93990

0.93992

+ 0.002%

0.95960

0.95060

0%

Pureza final de la solución intercristalinaDbm3

0.51837

0.51848

+ 0.02%

0.63153

0.63149

- 0.006%

0.70063

0.70061

- 0.002%

Viscosidad de la solución i= ntercristalinamum

79.34<= /p>

3.847<= /p>

- 95.4%

2169

4.91

- 99.8%

28195.9

5.77

- 99.9%

Tamańo final del cristal dkp

0.000387

0.000387

0%

0.000343

0.000343

0%

0.000284

0.000284

0%

 

 

Fuente: Los Autores


 

La simulación de procesos por computadora, ha permitido optimizar las diversas etapas de los diferentes esquemas tecnológicos utilizados por las industria= s, evitando gastos innecesarios en la implementación de regímenes y equipos ineficientes. Con el paso del tiempo la industria azucarera ha evolucionado, con la finalidad de aumentar su producción y disminuir perdidas del product= o, gracias a la implementación de equipos que aceleran la saturación de la mel= aza y permiten el control automático de su viscosidad.

 

 La cristalización del azúcar es consider= ada una de las etapas científicas-técnicas de mayor interés durante el procesamiento de la remolacha, porque allí ocurre la principal perdida de azúcar, alcanzando un 75% de  pérdi= da total dentro de la producción. La modelación de estos procesos permite determinar los parámetros óptimos para la obtención de una melaza normal sin gastos para la empresa.

 

En el marco de solución, se proponen y justifican = las siguientes variantes tecnologicas:

ˇ      =    Disminución de la viscosidad de la solución intercristalina disminuyendo la sobresaturación, comenzando desde el moment= o en que se alcanza la viscosidad máxima 50 ... 70 Pa.s.

ˇ      =    Disminución de la viscosidad de la masa cocida deb= ido a la selección de masa cristalina, a partir del momento en que se alcanza la viscosidad máxima 50 ... 70 Pa.s.

ˇ      =    Disminución de la viscosidad de la masa cocida deb= ido a un cambio en la pureza del producto ańadido.

ˇ      =    Disminución de la viscosidad de la masa cocida deb= ido a una combinación de variantes de los métodos anteriores.=

 

Conclusiones.

ˇ      =    El análisis bibliográfico arrojo 17 ecuaciones de diferentes autores referentes a la viscosidad en soluciones técnicas y puras durante el proceso de ebullición y cristalización de la sacarosa.

ˇ      =    Se construye un modelo matemático que proporciona = 60 toneladas de producto final con las ecuaciones de S. V. Shterman, tomando en cuenta que las soluciones en el aparato de vacío no pueden super= ar los 1500 pascal de viscosidad para evitar dańos y perdida del producto.

ˇ      =    Los valores de viscosidad según las ecuaciones de = Shterman mostraron diferencia, pero los resultados de= la modulación del proceso de cristalización evaporativa isobárica  en los valores de la duración total de = la cristalización no observan divergencia pronunciada.

ˇ      =    La validación de la modelación computacional reduc= iría significativamente los costos de operación de equipos voluminosos, los gast= o en electricidad, el consumo de agua refrigerante y simplificaría en gran medid= a la tecnología de obtención de azúcar.

 

 

 

Referencias bibliográficas= .

Sapronov, A. (2014). Tecnologia d= e la produccion de azucar. Kolos. 150-250.

Sherman, S. (2007). La composición y viscosidad óptimas del procesamiento del azucar= de romolacha y de la última cristalización en varias etapas de su producción. Sahar, 5. = 160.

Sherman, V. VOVK, G. (2006). Descripción analítica de la viscosidad de las soluciones técnicas de azúcar y melaza.Sahar,3. 250.

Slavyanckiy, A. (2006). Equipos tecnológicos de las fábricas de azúcar: clasificación, características técnicas, cálculos, diseńo. MGUPP. 60-100.

Slavyanckiy, A.Goldenberg, S. (2004). Cálculo de productos de un esquema tecnológico típ= ico y el uso de computadoras para mejorar la eficiencia de la producción de azú= car. MGUPP. 98.

Tuzhilkin, V. (2007). Cristalizacion del azucar. MGUPP. 336.

 Zaets, A. (2014). Regularizacion del azucar: problemas metodologicos de su creacion. Sahar, 5. 17-20.

 

 

 

 

 

 

 

 

 

 

 

 

 

PARA CITAR EL ARTÍCULO IND= EXADO.

 

 

Briones García, J., Vásquez Martínez, K., Villa Uvidia, D., & M= ejía Reinoso, T. (2019). Modelo matematico de viscos= idad de soluciones tecnicas y puras durante el procesamiento del azucar de remolacha. Cienc= ia Digital, 3(3.1), 229-243. https://doi.org= /10.33262/cienciadigital.v3i3.1.697

 

 


 

 

 

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

 

 

 

<= o:p> 



[1] Escuela Superior Politécnica= de Chimborazo, Sede Morona Santiago. Macas, Ecuador.    javier.briones@espoch.edu.ec

[2]     Unidad Educativa Particular Emanuel. = Macas, Ecuador. kvasquez@emanuel.edu.ec

[3] Escuela Superior Politécnica de Chimborazo, Sede Morona Santiago. Macas, Ecuador. diana.villa@espoch.edu.ec

[4] Escuela Superior Politécnica de Chimborazo, Sede Morona Santiago. Macas, Ecuador.    teresita.mejia@espoch.e= du.ec

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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////8AAAAAAAAAAP///wAAAP////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////wAAAAAAAAAA////AAAA//////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////AAAA//////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////AAAA////AAAA//////////////// ////////////////////////////////////////////////////////AAAA//////////////// ////////////////AAAA//////////////////////////////////////////////////////// ////////////////AAAA////////////////AAAA////////////////////////AAAA//////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////AAAAAAAAAAD///8AAAD///////////////////////////////////////////////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////8AAAAA AAAAAP///wAAAP////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////wAAAAAAAAAA////AAAA//////////////////////////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////wAAAAAAAAAA ////AAAA//////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////8AAAAAAAAAAP///wAAAP// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////////8Fj BcFjBf///////////////////////////////////////////////8FjBcFjBcFjBf////////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////8AAAAAAAAAAP///wAAAP////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////wAAAAAA AAAA////AAAA//////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////AAAAAAAAAAD///8AAAD///////////////////////////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////8AAAAAAAAAAP///wAAAP////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////wAAAAAAAAAA////AAAA//////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////AAAAAAAAAAD/ //8AAAD///////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////8AAAAAAAAAAP///wAAAP////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////wAAAAAAAAAA////AAAA //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////wAAAAAAAAAA////AAAA//////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////AAAAAAAAAAD///8AAAD///////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //8AAAAAAAAAAP///wAAAP////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// 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//////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// ////////////////////////AAAAAAAAAAD///8AAAD///////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////8AAAD///////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////8AAAAAAAAAAP///wAA AP////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /wAAAAAAAP////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////wAAAP////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////wAAAP////////////// //////////////////////////////////////////////////////////////////////////// /////wAAAP////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////wAAAAAAAAAA////AAAA//////////////////////////////////////// ////////////////////////////////////////////////////////////////AAAA////AAAA 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//////////////////////////////////////////////////////////////////////////// ////AAAAAAAAAAD///8AAAD///////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////8AAAAAAAAAAP///wAAAP////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// /////////////////////////////////////////////////////////////////////////wAA AAAAAAAA////AAAA//////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// //////////////////////////////////////////////////////////////////////////// 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