Modeling of mass transfer process’ kinetics during dyeing and printing of textile fibers

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National Technical University of Ukraine "Ihor Sikorsky Kyiv Polytechnic Institute", Peremohy Ave., 37, Kyiv

G.E. Pukhov Institute for Modelling in Energy Engineering, Generala Naumova St., 15, Kyiv-164

Modeling of mass transfer process' kinetics during dyeing and printing of textile fibers

Furtat Irina Eduardivna

PhD (Technical Sciences), Associate Professor

Associate Professor of the Department of Thermal and

Alternative Energy of the Educational and

Scientific Institute of Energy Saving and Energy Management

Furtat Yurii Olehovych

PhD (Technical Sciences)

Leading Researcher of the Department of Modeling of Energy

Processes and Systems

Abstract

adsorbent molecule energy

We can single out two approaches currently used in the theory of adsorption in general. First of all, it is microscopic , based on the application of methods of statistical physics. Based on the idea of an averaged molecular field characterizing the interaction of a sorbing molecule with a sorbent surface, and then using the grand canonical ensemble method, it is possible in principle to determine the number of molecules adsorbed on the surface. The main difficulty here is the calculation of the interaction energy, since the method of successive approximations is inapplicable. In the framework of the so-called lattice model, attempts to calculate the interaction energy in the case of sorption of nonpolar adsorbent molecules were made, and it was shown that the dipole-dipole interaction was not taken into account, since the contribution of the dipole- quadrupole interaction is not small. An attempt to simulate the real structure of an active carbon grain, presenting it as a set of slitlike and cylindrical pores, and on this basis to calculate adsorption using the Lennard-Jones potential, was made in the work of Everett. A method was developed for calculating the interaction energy of polar sorbate molecules with an ionic cubic lattice. The comparability of the calculated and experimental data, therefore, largely depends on the adequacy of the chosen model to the real structure. Accounting for other types of interactions, which, generally speaking, take place during adsorption, is associated with significant mathematical difficulties.

Within the framework of this approach, the theory of sorption can also be constructed based on the kinetic consideration of the process, based on the use of the so-called chain of BBGKY equations, which considers the behavior of partial distribution functions in time that describe the correlation between molecules. The solution of such a system of equations is currently known only for simple cases. Significant difficulties that arise when describing the adsorption process in a statistical way force us to use a phenomenological approach that does not require detailed knowledge of the sorption mechanism. Based on general thermodynamic relationships, he establishes a relationship between the amount of sorbed substance and thermodynamic quantities: chemical affinity, heat of adsorption, free energy, etc. We will dwell on its application in describing adsorption from aqueous solutions in more detail, especially since the kinetics of the process in porous structures is also considered within the framework of this phenomenological approach.

Keywords: sorption, adsorption, energy of interaction, polar molecules, micropores.

Фуртат Ірина Едуардівна кандидат технічних наук, доцент, доцент кафедри теплової та альтернативної енергетики Навчально-наукового інституту енергозбереження та енергоменеджменту, Національний технічний університет України «Київський політехнічний інститут імені Ігоря Сікорського», Солом'янський район, пр-т Перемоги, 37, м. Київ, 03056, тел.: (063) 629-93-97, https://orcid.org/0000-0002-2197-8150

Фуртат Юрій Олегович кандидат технічних наук, провідний науковий співробітник відділу моделювання енергетичних процесів і систем, Інститут проблем моделювання в енергетиці ім. Г.Є. пухова НАНУ, вул. Генерала Наумова, 15, Київ-164, 03164, тел.: (063) 126-72-18, https://orcid.org/0000-0002-0775-5460

Моделювання кинетики процесу масопереносу при фарбуванні і друку текстильних волокон

Анотація

Можна виділити два підходи, що використовуються нині в теорії адсорбції взагалі. Насамперед, це мікроскопічний, заснований на застосуванні методів статистичної фізики. Виходячи з уявлення про усереднене молекулярне поле, що характеризує взаємодію сорбуючої молекули з поверхнею сорбенту, і використовуючи потім метод великого канонічного ансамблю, можна в принципі визначити кількість адсорбованих на поверхні молекул. Основну складність тут становить розрахунок енергії взаємодії, оскільки метод послідовних наближень не застосовується. У рамках так званої гратової моделі робилися спроби розрахувати енергію взаємодії у разі сорбції неполярних молекул адсорбенту, причому показана недостатність обліку диполь-дипольної взаємодії, оскільки внесок диполь-квадроупольної взаємодії не малий. Спроба змоделювати реальну структуру зерна активного вугілля, представивши її у вигляді сукупності щілинних і циліндричних пір, і на цій основі розрахувати адсорбцію з використанням потенціалу Леннарда- Джонса вжито в роботі Еверетта. Розроблено методику розрахунку енергії взаємодії полярних молекул сорбату з іонними кубічними гратами. Сумісність розрахункових та експериментальних даних, таким чином, значною мірою залежить від адекватності обраної моделі реальної структури. Облік інших типів взаємодії, що мають, взагалі кажучи, місце при адсорбції, пов'язаний із значними математичними труднощами.

В рамках цього підходу теорію сорбції можна побудувати, виходячи також і з кінетичного розгляду процесу, заснованого на використанні так званого ланцюжка рівнянь ББГКІ, що розглядає поведінку в часі часткових функцій розподілу, що описують кореляцію між молекулами. Рішення такої системи рівнянь нині відоме лише простих випадків. Значні труднощі, що виникають в описах процесу адсорбції статистичним чином, змушують використовувати феноменологічний підхід, який вимагає детального знання механізму сорбції. Виходячи із загальних термодинамічних співвідношень, він встановлює взаємозв'язок між кількістю речовини, що сорбується, і термодинамічними величинами: хімічною спорідненістю, теплотою адсорбції, вільною енергією тощо. Ми докладніше зупинимося на його застосуванні в описі адсорбції з водних розчинів, тим більше, що кінетика процесу в пористих структурах також розглядається в рамках цього феноменологічного підходу.

Ключові слова: сорбція, адсорбція, енергія взаємодії, полярні молекули, мікропори.

Problem statement

Since the process of mass transfer during dyeing and printing of textile fibers is presented as an inseparable unity of adsorption and diffusion stages, sufficiently theoretically substantiated data on each stage, both on its mechanism and on the temporal course, are needed.

The polymer-dye adsorption interaction is ultimately the basis of dyeing and printing processes. Not only the end result of the process as a whole, but also its kinetics are largely determined by the sorption isotherm.

According to modern concepts, even adsorption on a flat surface is a very complex theoretical problem.

There are two types of adsorption on the surface of a solid: physical and chemical.

Adsorption interaction means physical adsorption. In the case of adsorption of non-polar molecules on non-polar surfaces, its cause is dispersion interaction, the nature of which was elucidated by F. London. It is due to the mutual behavior of instantaneous dipoles caused by fluctuations in the electron density. It turns out that when two such molecules approach each other, their attraction to each other is more likely than repulsion.

The energy of the dipole-dipole interaction is inversely proportional to the sixth power of the distance and is on the order of several kJ/mol. Along with instantaneous dipoles, the appearance of instantaneous quadrupoles is also possible. The energy of their interaction with instantaneous dipoles, although it decreases much faster with distance, can, however, be comparable in order of magnitude to the energy of the dipole-dipole interaction.

If the sorption of non-polar molecules is happening on ionized surfaces, the induction forces of attraction of the dipole are added to the disperse interaction, which is created in the sorbate molecule due to the field of the solid surface. However, the contribution of this interaction is small and in some cases is only 5% of the fraction of dispersed forces. On the contrary, during the adsorption of polar molecules on a nonpolar sorbent, the induction forces of attraction have a great influence, so that the energy here is already on the order of several kJ/mol.

The orientational interaction is manifested by the adsorption of polar molecules on surfaces having electric charges, and thus is due to electrostatic forces between the dipole and the surface ion. The energy value can reach several kJ/mol.

Analysis of research and publications

Different aspects of the sorption process have been researched by Lennard-Jones S.E., Dent B.N., London F., Crank I. and many others.

But the model of the process' kinetics, that take into account different factors and conditions of the process, suitable for implementation in modern modeling systems, is yet to be fully developed.

Article goal is to analyze different factors of the mass transfer process during dyeing and printing of textile fibers, creating mathematical models of their influence suitable for further implementation in modeling computer systems.

Main material

First of all, we note that adsorption from solutions on solid sorbents fundamentally differs from the sorption of vapors and gases in that it has a displacement character and, thus, is carried out due to the redistribution of the solution at the interface. For this reason, the energy determined from the experiment should be considered as the difference between the energies of interaction with the adsorbent of both components of the solution, and the adsorption equilibrium constant is the ratio of the adsorption constants of the mixture components.

The absorbing capacity of a flat sorbent is usually characterized by the specific surface area, i.e. surface area per unit mass of the sorbent.

By its nature, adsorption in porous structures differs from that on a flat surface in that, firstly, porous sorbents are characterized by energy inhomogeneity, and, secondly, for * pores, the sizes of which are close to van - der Waals diameter of sorbate molecules , the fixation of the latter is probable under the influence of the general field of all atoms or molecules bounding the pore cavity. This implies that a noticeable effect of porosity on adsorption should be expected. This actually takes place during the adsorption of gases and vapors, which leads to the need to distinguish between a - sorption occurring in pores of different sizes.

In the processes of adsorption from solutions, the influence of porosity, on the contrary, turned out to be not so noticeable. In any case, adsorption isotherms of certain substances on porous carbon and non-porous soot are very little distinguishable, this circumstance can be explained, apparently, by the fact that adsorption is of a displacement nature and is largely determined by the difference in the interaction energies of both components with the adsorbent. The difference, however, is due to the steric factor.

Since sorption is ultimately determined by the type of interaction, the form of the adsorption isotherm of nonpolar substances on a nonpolar adsorbent should, generally speaking, differ from the adsorption isotherm of ions on a charged surface.

In the simplest case of sorption of non-polar molecules on a non-polar sorbent at not too high concentrations, when the solution and the state of molecules in the adsorbed phase can be considered as ideal subsystems, the adsorption value depending on the concentration of the sorbed substance in the solution is usually described by the Langmuir equation:

a = abc/1 + bc (1)

where a and am are the values of specific adsorption concentration c and limiting monolayer filling, b is a constant.

Strictly speaking, equation (1) in this form was applied when describing the process of gas sorption, while the constant b has the meaning of the adsorption equilibrium constant, defined as b = exp(-A^/RT),where A^is the chemical affinity. The application of (1) to the description of adsorption from solutions requires some substantiation already by virtue of the displacement nature of the latter.

Linear dependences usually describe the initial segment of the isotherm, when the number of active free sites is large compared to the number of sorbed molecules. Mathematically, they are expressed by Henry's equation:

a = kc,

where a is the concentration of the sorbant sorbed by the surface; c is the equilibrium concentration of the sorbant in the external solution.

For porous structures, equations of sorption isotherms were obtained, which are characterized by more complex dependencies than the Lengrum isotherm and, accordingly, differ from the latter in form. Later, a completely different approach to the problem of adsorption is proposed, which is based on the model of a "vacant" solution. In fact, a binary solution is considered, in which the role of the solvent is played by the adsorbent. A feature of a "vacant" solution is that it is always in a state of osmotic equilibrium. This makes it possible to use the osmotic theory of solutions. Interestingly, within the framework of this theory, a t does not depend on temperature, in contrast to the Langmuir theory and the theory of volume filling.

So far, we have considered ways of describing the adsorption of molecules. Let us pass to a brief description of the adsorption of ions . It is known that in a number of cases of ion adsorption at not too high surface potentials, it is quite satisfactorily described by a Langmuir- type isotherm or the Freundlich-type isotherm, proposed for the first time to describe the sorption of ions on a flat surface and having the form:

a = вп, (2)

where в, n are constants, and 0.2 < n < 0.9.

It is obvious, however, that the description of ion adsorption by the Langmuir isotherm does not mean, generally speaking, that a m or V a can have the same significance as in the sorption of neutral molecules, since the presence of electrostatic interaction can have a significant effect on the maximum density of their packing. Other things being equal, a m or V a for neutral molecules will obviously be the upper limit for the corresponding values during the sorption of ions on a surface simultaneously charged with them. This is confirmed by the fact that the addition of an electrolyte indifferent to the surface leads to an increase in adsorption values.

The main feature of ion adsorption in general is the formation of a double electric layer (DL) on the sorbing surface. Therefore, the value and distribution of the potential over the DW cross section has a very significant effect on the sorption value. The dependence of the potential and charge of the surface on the number of sorbed ions is, in the general case, very complex, and so far there is no rigorous theory explaining it. In addition, it is clear that both the kinetics and the equilibrium adsorption will also depend to a large extent on the DW thickness. If the double layers over the cross section of the pore space overlap very strongly, then with a small error we can assume that the grain volume is equipotential. The consideration can still be simplified if we additionally assume that the grain is a certain homogeneous phase and the value of the potential is not too large. The thermodynamics of such systems was developed by Guggenheim, and then the relationship between the concentrations inside the grain and outside it is expressed by the following formula:

cjcs =Xk

where Л is the Donnan distribution coefficient , defined as

Л = exp(-zcAp/RT),

where zc is the charge of the sorbed ion,Ap is the potential difference between two homogeneous phases: grain and external mortar. The value Л, and with it the value of the potential, can be determined using the conditions of electroneutrality.

Ion adsorption does not always proceed by the mechanism of charge accumulation. There is another ion-exchange type of sorption.

The question of the effect of porosity on the amount of ion adsorption is very important. It follows from general considerations that it must take place and is due to the effect of the DW thickness. Indeed, in macropores, the DW thickness at not too low concentrations is small compared to the pore size; xR ? 1, where xis the reciprocal of the Debye screening radius, R - pore radius. Therefore, ceteris paribus, compared with a plane, one should not expect a significant effect of grain porosity here. On the contrary, in micropores almost always xR < 1, so the porosity of the structure should affect the amount of adsorption.

In the simplest case, when the pore is assumed to be cylindrical and only its diffuse part is taken into account in the DS structure, the differential capacitance C, defined as C = dq, essentially depends on the value of xR, i.e.: dp '

c=Cf,W)=Cfl ^-xR) ' (3)

where Cji is the capacitance of a flat capacitor, I0 (xR)and I1 (xR) are the in modified. Bessel functions of zero and first order, respectively supermicropores and micropores, xR < 1, a significant decrease in the amount of sorbed ions should also be expected. For inorganic ions, this is indeed the case. Already in pores, for which xR < 10, adsorption per unit pore is much lower than on a flat surface.

The parameter ju(xR) with increasing xR ranging from xR ~ 1 to xR ~ 100,increases significantly from the value л = 0.005 to л

For large ions, such as surfactant ions, the filling of the adsorbed space is more complex. Indeed, the amount of adsorption of surfactant ions and molecules is largely determined by their orientation with respect to the sorbing surface.

The area occupied by one adsorbed cationic surfactant ion decreases with increasing pore size. In this case, adsorption is described by the Lengrum isotherm. Thus, in addition to the direct influence of the DW thickness on the adsorption value, the orientation factor plays a significant role in determining its value for molecules.

Such a complex behavior of surfactant adsorption, as well as its dependence on the number of functional groups on the surfactant molecule, the length of the hydrocarbon radical, makes it extremely difficult to study it. An even more complex dependence could be expected for dye adsorption by a fiber, which, as noted above, is a capillary-porous colloidal body with polydisperse pores, the size of which varies depending on temperature, solvent properties, etc.

At the same time, experimental data shows that the adsorption of dyes by fibrous materials is mainly characterized by isotherms described by the equations of Henry, Langmuir and Freundlich, with constant values that certainly differ from the corresponding values for ideal solutions. This situation can be observed in two cases: 1) sorption isotherm on the inner surface of the micro- and super p-micropores of the fiber corresponds to the isotherm on the outer surface of the fiber; 2) in the process of mass transfer of the dye from solution to the fiber in the volume of micro- and supermicropores , the concentration of the dye is established in a state of equilibrium in accordance with the average value of the thermodynamic potential in the surface layer and inside the micro- and supermicropores.

From our point of view, the second hypothesis more accurately corresponds to the physical essence of the phenomenon, which is the basis of the proposed model of the process. Let us now turn to the consideration of the diffusion stages of the process.

Starting with the work of Neil and Springfellow [1], it is believed that the kinetics of the staining process is largely due to the diffusion mechanism. They investigated the applicability of Fick `s law to describe the kinetics of dyeing and found that the diffusion coefficient is not a constant value, but depends on the concentration of the dye and the indifferent electrolyte. In the future, many experimental data, a review of which is contained in [2], this position was confirmed and developed.

Theories of dye diffusion in fibers can be classified into three groups: a) hydrodynamic theories; b) theories based on the porous matrix model; c) theories based on the free volume model.

Since, as shown above, in the presence of solvents, the transfer mechanism according to the free volume model is unlikely, it is not considered in this paper.

The hydrodynamic theory of diffusion [3] considers the system as a solid solution of a dye in a polymer , and the diffusion coefficient is determined from the expression:

(4)

where f c is the coefficient of friction of the dye molecule in the polymer matrix.

Thus, the hydrodynamic theory ignores the microheterogeneity of the polymer structure and models it as a homogeneous medium in which the dye diffusion flux is determined by the balance between the driving force and the friction resistance force. The solid solution model, in which formal kinetic parameters are introduced that do not explicitly reflect the characteristics of the polymer, does not make it possible to correctly predict the change in the fixation kinetics that occurs as a result of the transformation of the polymer matrix under the influence of external factors, and to take into account the effects of sorption saturation at high dye concentrations.

A significant contribution to understanding the essence of the kinetics of the dyeing process was made by the development of a model of a porous matrix, which proceeded from a more detailed understanding of the fiber as a structure penetrated by a system of pores filled with a solution, i.e. using the Damkeller-Wicke-Barrer porous body model [4]. Particularly successful was the use of the porous matrix model for solving problems of the kinetics of the dyeing process by Crank [5]. As a result, solutions for variable boundary conditions are obtained. In the theory of a porous matrix, the assumption is made that the entire dye sorbed by the fiber can be divided into two fractions: one in the internal solution and one adsorbed on the pore wall.

Crank, in contrast to earlier ideas about the coincidence of external and internal sorption isotherms, accepted that the internal isotherm differs from the external one due to the Donnan distribution of dye ions between internal and external solutions. This conclusion was drawn based on the equipotential volume model, according to which the dye ion must overcome the electrical potential barrier before entering the pore, but at the next stage of the transfer process, the electrical factor no longer has an effect. In fact, Crank did not use Donnan coefficients, but considered the Boltzmann distribution of charged ions near a charged surface. Thus, a clearer representation of the process mechanism should be considered an essential advantage of the porous matrix model, which made it possible to express the relationship between the true and apparent diffusion coefficients through the structural parameters of the systems, pore porosity and tortuosity. Within the framework of this model, the movement of a dye molecule or ion in a fiber can be represented as a sequence of diffusion jumps from one active center to another, during which the particle passes through a potential barrier separating one equilibrium position from another. Thus, activated diffusion in the pore is postulated. The value of the potential barrier is determined by two components: the adsorption interaction between dye molecules and polymer active centers and the restrictions imposed on the movement of dye molecules by the polymer structure.

In accordance with this mechanism, the expression for the temperature dependence of the diffusion coefficient is written similarly to the Arrhenius equation:

where E* - activation energy; B - pre-exponential multiplier.

It should be noted that experimental studies show that this dependence is valid in a very narrow temperature range, and E* changes according to an indefinite law: upon reaching a certain temperature E* decreases, then E* * increases sharply again. This behavior of E* contradicts the physical meaning of the activation energy, since, according to the definition, the activation energy can only decrease with increasing temperature. Therefore, this behavior of E*, in our opinion, allows us to make the assumption that E*, at least, is not uniquely determined by the activation energy.

It was natural to associate such an "anomalous" dependence of the diffusion coefficient on temperature, as well as on some other characteristics of the process, with the shortcomings of the porous matrix model, which consist in the complexity of determining the structural parameters; the mechanical nature of the model, ignoring the dynamics of the rearrangement of the polymer matrix with changes in temperature and other external factors; neglecting explicitly the properties of the solvent, etc. All this led to the fact that the model of the porous matrix in its current form, although it provides a certain accuracy in describing the kinetics of dye transfer at relatively long characteristic times of the process, does not allow one to analytically calculate the absolute values of the diffusion coefficient of the dye in the fiber, predict the change in t in a wide range of temperatures and etc.

An attempt to interpret the mass transfer of a dye in fibers as a "crawl" of its molecules along the outer and inner surfaces from one active center to another, also, in our opinion, did not provide a significant step forward compared to the traditional model of a porous matrix, because the physical properties of the solvent and changes in the structure of the material are still not explicitly taken into account. Moreover, the probability of dye "crawling " over the surface of the fiber, with certain reservations, exists only in those limited cases when the binding energy of the dye with the fiber in the presence of a solvent is higher than the binding energy of the solvent with the fiber and the dye.

When developing high-intensity methods for fixing dyes on textile materials in the presence of solvents, it seems justified to refine the models of the porous matrix by taking into account the properties of the solvent, the dynamics of the rearrangement of the polymer matrix, and the additional resistance of the surface layer of the fiber to the transition of the dye from the external solution into the internal volume.

Consideration of dye mass transfer processes in the system external solution - textile material is based mainly on a four-stage process scheme: dye diffusion in an external solution to the fiber surface, dye adsorption on the outer surface of the fiber, dye diffusion in the fiber, its adsorption on the inner surface of the fiber.

Such a scheme is quite valid for low-intensity processes, when the values of the diffusion coefficients of dyes in fibers are 10-9 - 10-20 m2 /s, i.e. two or more orders of magnitude smaller than their diffusion coefficients in solution ( melt). In this case, is the general kinetic problem, taking into account the mass transfer in the thread, is not considered on the basis that the dye relaxation time in the fiber much longer than its relaxation time with the space between the fibers.

Using a four-stage process scheme and methods of the theory of heat conduction, analytical solutions on the kinetics of isothermal and non-isothermal dyeing was obtained, taking into account the resistance of the boundary layer and the continuous model of dyeing.

It should be noted that the very existence of the so-called discontinuous and continuous dyeing models is a consequence of the imperfection of the fundamental equations of mass transfer, and can be eliminated by using methods that more accurately reflect the physical essence of the potential process.

In this article, we deliberately do not touch upon the chemical kinetics of fixation, since, on the one hand, the physicochemical aspects of the issue are detailed in the indicated sources, and on the other hand, we consider processes when the bonds of the solvent with fiber is higher than the affinity of the dye for the fiber and the probability of chemical interaction in the process of mass transfer is such that it cannot significantly affect its kinetics.

With high-intensity thermal dyeing methods, due to the rapid growth of DB and the relatively weak growth of Dh at material temperatures above a certain limit, different for different fiber-solvent combinations, tb < th , i.e. the duration of the mass transfer process and the uniformity of the distribution of the dye between the fibers is determined by the diffusion in the thread between the fibers. Therefore, for high- temperature methods, to solve the general kinetic problem, at least a 5-stage process scheme should be adopted: dye transfer in an external solution to the surface of the thread, diffusion in the space between the fibers, adsorption on the outer surface of the fiber, diffusion inside the fiber, adsorption on the inner surface of the fiber. This leads to the construction of a new, biporous, model of a sorbent (thread), represented as a set of microporous formations and gaps between them - transport pores. In the framework of the macroscopic theory, which proposes averaging equations over volumes containing a sufficiently large number of microporous inclusions, but small compared to the volume of the whole body, the system of equations describing the kinetics of adsorption in bodies with a biporous structure was studied.

This system of equations can be written as follows:

Here f\ (c) is adsorption on the walls of transport pores; characteristic relaxation time for the volume of microporous formation; A, quantities depending on the form of the microporous formation.

In this form, system (6) assumes that Db is constant and the sizes of all microformations are equal. With regard to threads modeled as a body consisting of many infinite capillary-porous colloidal cylinders with varying diffusion characteristics, no such solutions have been made. In addition, the question of the resistance at the entrance to the pore, as well as the case of nonisothermal mass transfer, remained unresolved.

Very fruitful for studying the system of equations (6) is the method of statistical moments, which makes it possible in principle to obtain two (or more) kinetic parameters of the system from one kinetic curve at once. In fact, on the basis of this method, methods were proposed for determining the diffusion coefficients, for the relationship between the characteristic diffusion times Ті and Tb for grains of various geometries and moments of the kinetic curve.

Conclusions

By representing more accurately the model of dye mass transfer in a textile material and solving the conjugate problem of mass transfer kinetics with the problem of heating the material, it is possible to determine the optimal modes of high-temperature fixation, in which uniform fixation of dyes over the textile material is achieved.

It should be noted that an increase in temperature is a significant factor in the intensification of the process in the case when the duration of the mass transfer process is determined by diffusion in the fiber. As will be shown below, in those cases where the process is determined by diffusion in the interfiber space, a further increase in the temperature of the material does not give the desired results either from the point of view of a significant intensification of the process, or from the point of view of the uniform fixation of the dye by individual fibers. To substantially intensify dye transfer between fibers, one can apply external transfer potentials and, in particular, electric fields.

A number of works are devoted to dyeing in electric fields. However, they dealt mainly with dyeing in microwave fields. When microwave fields are applied, most of the energy is converted into thermal energy; is equivalent to an increase in the heat flux density. Thus, in the case when the duration of the mass transfer process is determined by diffusion in the thread between the fibers, and the total thickness of the textile material is such that the temperature difference across the cross section can be neglected, the use of microwaves is hardly advisable. Another thing is products of considerable thickness (bobbins, ropes, etc.), when one of the main problems is to ensure temperature uniformity over the material section. To fix the dye on such materials, the use of microwaves is fully justified. If we add to this the low efficiency of modern microwave current generators, it becomes clear that the use of microwave currents to fix dyes on most types of fabrics is very problematic. It is necessary to apply an electric field to a textile material impregnated with a dye solution (suspension) that would facilitate the transfer of the dye between the fibers without significant conversion into thermal energy. The imposition of a direct electric current is not applicable due to the removal of the dye to the positive electrode and deposits on it. The required frequency of the electric current can be obtained, in a first approximation, depending on the electric field strength according to the Smoluchowski formula from the condition that in a half-cycle the dye particle penetrates half the thickness of the textile material. Calculations show that these conditions are met by ultra-low frequency currents of 4-90 Hz. However, since we are dealing with colloidal solutions (suspensions), it is necessary to consider in more detail the theory of particle transfer in an alternating electric current field, paying special attention to the issue of behavior, insufficiently studied in the theory of electrophoresis: charged particles in the near-electrode zone and their interaction in the presence of an indifferent electrolyte when an alternating electric field is applied in order to protect the electrodes from dye contamination and ensure the stability of dye solutions to coagulation.

References

1. Neale, S.M. & Stringfellow, W.A. (1933). The Mechanism of the Dyeing of Fibers. Transactions of the Faraday Society. Vol. 29 (№ 8). Pp. 1167-1180.

2. Vickerstaff, Т. (1956). Fizicheskaia himiia krasheniia [Physical Chemistry of Dyeing]. Moscow.: Gizlegprom. [In Russian].

3. Rogers. K. (1968). Rastvorimost i diffuziia [Solubility and Diffusion]. Problemy fiziki i himii tviordogo sostoianiia organicheskih soedinenii [Problems of solid state physics and chemistry of organic compounds] (p. 229). Moscow: Mir. 1968. [In Russian].

4. Barrer, R. (1948). Diffuziia v tviordyh telah [Diffusion on solid bodies]. Moscow: IL. [In Russian].

5. Crank, I. (1956). Mathematics of Diffusion. Oxford: Acad. Press.

Література

1. Neale S.M. The Mechanism of the Dyeing of Fibers / S.M. Neale, W.A. Stringfellow // Transactions of the Faraday Society. - 1933. - Vol. 29. - № 8. - Pp. 1167-1180.

2. Виккерстафф Т. Физическая химия крашения. / Т. Виккерстафф. - М.: Гизлегпром, 1956. - 575 с.

3. Роджерс К. Растворимость и диффузия. / К. Роджерс // Проблемы физики и химии твердого состояния органических соединений.- М.: Мир. 1968. - С. 229.

4. Баррер Р. Диффузия в твердых телах. / Р. Баррер. - М.: ИЛ, 1948. - 340 с.

5. Crank I. Mathematics of Diffusion. / I. Crank. - Oxford: Acad. Press, 1956. - 231 p.

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