Процессы массопереноса с участием твердой фазы. Учебное пособие

supersaturation systems stability falls sharply and nucleation rate grows; respectively, supersaturation area is divided into two zones: a meta-stable and labile (Fig. 3.3). The border between labile and meta-stable regions is rather relative. This boundary is conventionally divides the region, where the probability of spontaneous crystallization within a certain time is low, and the region where this probability is high.

In general case, the emergence of stable crystalline nuclei is

where I is a number of equilibrium embryos, arising in a unit of time in a volume unit of solution; θ - coefficient of proportionality; - work of an embryo formation; k - Boltzmann constant; U - activation energy, T - temperature; µ - solvent viscosity; D - diffusion coefficient.

There is no any unified theory of the nuclei formation mechanism. Usually the process of nucleation is considered as a consequent connection of the third particle to two ions (molecules), etc., until the formation of an embryo of a critical size. Minimum critical embryo sizes are described by the expression:

where rcR is critical radius; - molecular mass of an embryo; ρ - embryo density; σ - interfacial surface tension, R - gas constant; - concentrations of supersaturated (ss) and saturated (s) solution respectively.

Crystals grow on germs, which have overcome critical dimensions. Growth of crystals occurs simultaneously in all of its faces. However, linear growth rates of different faces are usually not equal. Unified theory of crystal growth does not exist. There is no single theory, which explains in general the difference in faces growth rates, defects, stratification, the difference in speed of crystallization and dissolution.

Crystallization rate is not constant. It varies depending on changes in process conditions. Dependence of crystallization speed from supersaturation is described by the equation:

satiety (abundance); n – order of the process.

Rate of crystals` growth may be determined by the stage of delivery of molecules or ions of crystallizable substances to the surface of a crystal.

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In this case, parameter n in equation (3.4) is equal to the unity and it turns into a standard equation of mass delivery (1.49). However, the speed of crystal growth can be affected by the speed of molecules` embedding (ions) in the crystal lattice. In this case, the value β in (3.4) will represent some effective kinetic coefficient and the degree of n will differ from the unity.

In Fig. 3.4 is shown the evolution of crystal growth speed. Crystallization rate is not constant, it changes in time, depending on crystallization conditions in a wide range.

At the beginning the speed is equal to zero (period of induction), then it reaches short-term peak and again decreases to zero. If the extent of solution` supersaturation is large, then it is observed a sharp maximum of speed curve (1). In case of a low degree of supersaturation or if there are impurities, impeding crystallization, induction period is big enough and on a curve 2 is observed horizontal part t 2 − t 3 , i.e. the maximum speed for some time remains constant.

In addition, the speed of crystal growth, except supersaturation, is strongly influenced by temperature. The temperature influence is individual for each case, which is explained by positive or negative solubility of substances. Shape of crystals is determined by the nature of solidifying substances, presence of impurities in solution, of hydrodynamic conditions in the device, degree of supersaturation.

Fig. 3.4. Time dependence of crystallization speed for different solutions: 1 - with a relatively high degree of supersaturation;

2 - with a low degree of supersaturation.

Crystals` size, so much as crystals` shape, depends on the solution` degree of supersaturation, on hydrodynamic situation and presence of

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impurities (especially surface-active substances). For obtaining large crystals is necessary to introduce seed crystals in solution, which will replace embryos and become the centres of crystallization. Fractional composition of obtained crystals is regulated by solution` supersaturation, creation of a certain hydrodynamic situation or subsequent division of obtained crystals at fractions by mechanical methods. Degree of purity of obtained crystals depends from the cleanliness of initial solution.

3.4. Techniques of Crystallization

For crystallization process` realizing should be created the conditions for solution supersaturation. Supersaturation in a solution can be obtained by various procedures. Most widely used: cooling or, for some substances with a negative solubility, heating of the solution; partial removing of solvent by evaporation.

Technique, in which the solution temperature has to be changed, called isohydrational crystallization. In this case the amount of solvent remains constant. We may consider this process by using the solution cooling on solubility diagram (Fig. 3.5). Point A in the diagram corresponds to a solution at initial moment of time. At this point unsaturated solution has temperature T1 and concentration C1. Line AC characterizes the cooling of solution to the temperature T2 and intersects the solubility curve in point B.

Fig. 3.5. rystallization process` image in solubility diagram

If crystallization begins only at temperature T2 and at this temperature ends, crystallization process will be shown with CD line. Point D on solubility curve, corresponding to equilibrium concentration C2,

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corresponds to the end state of solution too. If the solution cannot be supersaturated, i.e. crystallization process begins at equilibrium concentration, then the process of cooling and crystallization can be noted by line D. In reality the process of cooling and crystallization may proceed in accordance with lines 'D'D or "D.

Technique, in which solution is transferred to supersaturated state by solvent removal, is called isothermal crystallization. In solubility diagram (Fig. 3.5) changing in solution` concentration from the beginning of the process (point A) will be noted by line AEG. Line AE, characterizing evaporating process up to the concentration of saturation, shows that with increasing of concentration rises the boiling point of solution. In most cases, crystallization by evaporation occurs at a constant concentration of solution, close to a state of saturation for a given

temperature T1'' .

There are also other methods of crystallization: super salting, super cooling, as a result of chemical reaction. In super salting a substance is added to a solution, which reduce solubility of extracted salt. In super cooling solution is cooled to the temperature of solvent` crystal formation (ice). It is possible to achieve supersaturation with the help of chemical reactions, taking place in solution, too.

The choice of a particular crystallization technique depends on the properties of solvent and extracted substances. For substances, which solubility sharply decreases with the lowering of temperature ( NO3, K2Cr2O3, NH4CL, etc), crystallization is expedient to conduct by isohydrational method. Otherwise (NaCL) is applied isothermal crystallization.

It follows from the above, that crystallization process occurs only when initial phase is in the state of supersaturation. Because supersaturated solutions are unstable, the excess amounts of dissolved substances exit from them, i.e. proceeds a process of crystallization. After cessation of crystals` extraction the solution becomes saturated. Such a solution is called mother`s (or matrix), it is separated from crystals by sedimentation, filtration, centrifugation, etc.

3.5. Design of Crystallizers

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Used method of crystallization dictates the choice of a rystallizer design. For example, machines can be of continuous or periodic action. Design of apparatus for carrying out isothermal crystallization does not essentially differs from evaporators.

Devices for carrying out isohydrational crystallization (with solution cooling) represent a heat-exchange apparatus of specific constructions. Specificity of heat-exchangers design is determined by the appearance of solid crystalline inclusions in solution.

In Fig. 3.6 is shown a crystallizer of periodic action with cooling shirt and mixer. At the beginning, body 1 grouts and mixer switches on. Then shirt filles with cold water (brine), whose temperature at 8 -10 degrees lower than solution temperature. After the end of crystallization got suspension unloads and divides on filters or centrifuges.

Fig. 3.6. Crystallizer with mixer and cooling shirt:

1 - body; 2 - cooling shirt; 3 - mixer; 4 – suspension` output.

Swinging crystallizer (Fig. 3.7) is a device of continuous action. It is a long shallow trough 1, fortified by bandages 2, based on rollers 3. The trough is installed with a slight slope along its longitudinal axis. By means of a special device (on figure is not shown) it is slowly swinging on the rollers. Solution is fed onto the top of the trough and slowly flows through it. On the way solution cools due to the loss of heat to atmosphere and partly due to evaporation. Because of slow cooling of the solution, the rate of nucleation is small, hence here form large crystals. Since crystals move slower than solution, then they are well washed by solution and acquire correct forms. Pendulum movement of trough prevents the inlay of its

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surface and shredding of crystals.

Fig. 3.7. Swinging Crystallizer:

1 - swinging trough; 2 - bandages; 3 – supporting rollers.

Swinging crystallizers disadvantage is low productivity due to slow cooling of solution because of low intensive heat exchange with the environment.

Fig. 3.8. Crystallizer with a ribbon mixer:1 - closed horizontal still trough; 2 - water-shirt; 3 - ribbon or screw mixer.

Screw crystallizers refer to the machines of continuous action. They consist of a horizontal still trough with water shirt, inside which rotates belt or screw mixer. It not only moves the produced crystals to the

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place of unloading, but also supports them in suspension` state retention, which facilitates free and uniform growth of crystals.

Fig. 3.8 shows the scheme of a crystallizer with a ribbon mixer. In the trough 1 with a water shirt 2 slowly rotates spiral mixer 3 with metal strips. Cooling water in a shirt moves counter-currently to solution. Screw crystallizers, unlike the ribbon one, have mixer in the form of endless screw-auger. In apparatuses of this type hot solution comes from one end of the trough, and moves, continuously cooling, together with the formed crystals, to output connection on the other end of the trough. Due to intensive cooling of solution crystals are small, but regular in shape and uniform in size. Disadvantage: possible the inlay of surfaces, contacting with solution.

Crystals can be obtained by evaporating of crystallizing solutions (isothermal crystallization). Isothermal method is implemented in evaporators devices, allowing to perform crystallization. There exist evaporators, which allow to perform thin (selective) crystallization process. With the purpose to obtain crystals of a certain size are used classifying crystallizers of a special ("Crystal") type.

Fig. 3.9. Crystallizers of "Crystal" type:

(a) crystallizer with evaporation; (b) vacuum-crystallizer

In this aqpparatus the supersaturation of solution by solvent` evaporation takes place in the circulating flow, where supersaturation is achieved in one part of the device and lost in the other. Supersaturated

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solution flows upwards through fluidized layer of crystals, allocating them by size.

Fig. 3.9 ("a" and" b") shows two types of "Crystal" devices, which differ primarily in the way of saturated state achieving.

In the first machine (Fig. 3.9, a) the supersaturation is achieved by circulating flow heating under sufficiently high static pressure (head). This, on one hand, prevents solution evaporation in circulation pipes, and, on another hand, contributes to solution evaporation in the upper part of the device A. Steam, produced in such a way, removes through tube U. Solution, supersaturated respectively to crystallization zone E, leaves evaporation zone A, passing through pipe B and filter partition P in zone E, where solution contacts with crystals, being over the filter partition. During the interaction with crystals the solution loses saturation, which leads to the crystals growth. Flow, moving from E zone to heater H, should be almost saturated. This solution mixes with the initial solution in pipe T. Crystals that have reached the desired size, fall down and get out from the machine through pipe M. Solution circulation is ensured by pump F. For solution heating by heat exchanger H is used a heating steam. Considered type of crystallizer is used in cases, when it is necessary to get supersaturation with the help of evaporation.

Fig. 3.9, b shows a vacuum-crystallizer, in which supersaturation is achieved by adiabatic reduce of the pressure over concentrated hot solution. Initial solution, coming in socket T, includes directly in the circulation flow, which is then becomes a mixture of initial and mother solutions. This mixture is vaporized in the evaporator A, in which exist some vacuum. Formed vapor drawn off through the pipe U. Supersaturated solution, passing through the tube B, interacts with the growing crystals in zone E. Mother fluid is drained through socket N, and crystalline mass - through socket M. Arising of crystallization centers in E zone may occur at the expense of crystals, existing in solution, or as a result of crystals` collisions either one with another, or with walls of a vessel. In case of continuous processing the rate of crystallization centers formation should correlate to the number of crystals, moved away in the form of final product.

3.6. Calculation of Crystallizers

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Procedure of calculation of the particular crystallizer depends on its design and the method of crystallization (isothermal, isohydrational). In any case, you can allocate the basic stages:

1.Making up of the equations of material and thermal balances of the process. With their help, and with the help of temperature dependence of the saturation concentration, you should define the final concentration of the solution, mass of obtained crystals and mother` solution, mass of deleted solvent and heat consumption for its evaporation during an isothermal procedure, or the cost of a cooling agent in case of an isohydrational technique.

2.Definition of the speed of solution movement and of the area of crosssection of a crystallizer in an apparatus of continuous action, depending on the design and required size of crystals.

3.Finding of the height (length) of a crystallizer on the basis of the process` conditions and its kinetics (3.4).

3.7. Dissolution

Under dissolution is understood the transition of a substance into solution from the surface of solid particles. The process of dissolution can be divided on two types - physical and chemical dissolution. In case of physical dissolution, the original substance can be re-obtained by crystallization from solution. Chemical dissolution is a heterogeneous chemical reaction, whose products are dissolved in a liquid phase. Return to the original substance by the way of crystallization in this case is impossible. Hereinafter we will consider only the physical dissolution, which can be represented as a process, reverse to crystallization, occurring when the concentration of solution becomes below the equilibrium.

Equilibrium during the dissolution, like during crystallization, is achieved in conditions of equality of chemical potentials of the substance in solution and solid state (1.81), which is determined by the equilibrium concentration (1.82) and as in the section 3.2.

Kinetics of dissolution is more simple in comparison with crystallization because of the absence of a stage, similar to the formation of embryos. Dissolution rate can be described by analogy with the speed of crystallization of the equation (3.4), in which = − , where C is the concentration of an unsaturated solution. It can be also divided into two stages: destruction of crystal lattice and transition of molecules (ions) into

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solution, as well as their withdrawal from the crystal border into the nuclear of solution.

«Devices» for carrying out the process of dissolution are called solvents, and their design is extremely diverse. The process can be intensified by the increase of coefficient of mass delivery in (3.4), and for this purpose machines of a periodic action are usually provided with mechanical mixers (figure 3.6), circulating pumps or devices of pneumatic mixing. In the machines of continuous action are also strive to obtain high velocities of the liquid phase` motion relatively to dissolving solid particles. For example, for this purposes can be used devices with a fluidized layer.

CHAPTER 4. MEMBRANE SEPARATION

4 .1. General Information. Classification of Membrane Separation

Techniques

Membrane separation is the process of gas or liquid mixtures separation with the help of membranes. Membranes – semi-tight partitions, selectively permeable to components of gas or liquid mixtures.

leachate

W, y

Fig. 4.1. Scheme of membrane separation process:

Lin , L fin ,W - mass consumptions and in , x fin , y - mass fractions of component B in initial mixture, concentrate and leachate;

in < x fin , y < in .

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