P. M. Silin

Theory of Operation of a Diffusion Battery

The factors—the most important factors influencing the operation of a diffusion battery—are:

1. the physicochemical properties of the beet (normal, frozen, thawed, more or less sugary, etc.);
2. the design of the diffusers;
3. the quantity of juice drawn off;
4. the temperature;
5. the speed of operation;
6. the size of the diffusers and their shape;
7. the dimensions of the heaters, the capacity of the communications.

Of the first factors—the properties of the beet, its ability to diffuse—of course play a very large role, but they cannot be taken into account. Therefore, in what follows we shall take this factor to be constant (the factory works with beet of approximately the same quality; the capacity for diffusion does not change).

To take account of the influence of temperature [[unclear: is]] possible; therefore the temperature regime must be established by experiments. Rapid heating after charging has, [[unclear: among other things]], the advantage that the temperature rises to approximately [[unclear: 55–60]]° C. The further rise in temperature should not go above 85°, since higher temperatures destroy the cell walls, which retard diffusion¹. In the last diffusers, before discharge, the temperature should be lowered below the normal diffusion temperature (40°); this presents the danger of pectin substances passing into solution. At present the temperatures in different diffusers do not differ so greatly from one another; they fluctuate only within the limits 80–60, [[unclear: sometimes]] (a difference of 20–15°). All the same, in the last diffusers the temperature is several degrees lower than in the first. This should reduce the diffusion constant, but this decrease is partly compensated by the fact that in the last diffusers the concentration of sugar is much lower than in the first, owing to which the concentration increases and becomes a constant. Thus, one may consider the diffusion constant to be approximately the same for all diffusers. (Experimental confirmation—according to factory and laboratory data—will be given below.)

The form of the diffuser must be such that leaching proceeds uniformly, with no jets or dead spaces being produced. Evidently this condition is fulfilled in the oval form of diffusers, for which the surface of diffusion is almost perfect.

The volume (not the heating surface) of the calorimeters and the communications, as will be evident from what follows, plays no major role and is quite mistakenly sometimes treated as one of the chief factors in diffusion operation.

The number of diffusors is not taken into account; it, too, plays only a secondary role (provided, of course, that the sum of their volumes remains constant).

There remain the degree of filling, the magnitude of the juice withdrawal, and the rate of operation—these are the factors whose influence on the operation of the battery we shall clarify below.

Review of theories of the operation of the diffusion battery. In attempts to establish a relation between the various quantities characterizing the operation of a diffusion battery, the process occurring in it is usually considered as stepwise: a diffusor with shavings containing normal juice of a definite concentration is filled with juice from the preceding diffusor; this juice remains in the diffusor until, as a result of diffusion, its concentration becomes equal to the concentration of the normal juice of the shavings. Such is the starting point of G. Biryukov in his theory of diffusion.^1) This principle is applicable and is applied to such purely stepwise processes as, for example, discontinuous decantation by Dorr’s method—in metallurgy; but it cannot be applied to the operation of a diffusion battery of a sugar factory. With assumptions such as are usually made, it is impossible to take into account the influence of the rate of operation. Equalization of concentrations in reality never occurs. On the contrary, there is always a difference in the concentrations of the normal and diffusion juice in the diffusor, by virtue of which the diffusion process itself becomes possible. At the time when Biryukov wrote his theory, it had some justification: previously, when working with diffusion with peripheral circulation, the condition of continuity of the process was absent; since in the circulation the first, more concentrated portions of juice pumped from the diffusor were mixed with the subsequent, less concentrated ones. After heating, this mixture entered the next diffusor. In diffusion, however, the process already approaches a continuous one: the juice enters the most leached part of the diffusor charge, then passes to layers richer in sugar, gradually increasing its concentration in doing so. Biryukov’s theory leads to the completely paradoxical conclusion that the volume of the diffusors plays no essential role; what matters is only their number, without regard to what it occupies.

Meanwhile, factory practice gives us examples of diffusion batteries operating equally successfully and of equal productivity, but consisting of from 6 to 16 diffusors (the total volume of them differing little).

G. Rosenberg^2) in 1886 likewise gives a mutual dependence of the quantities characterizing the operation of a diffusion battery. His assumptions are the same as Biryukov’s, only their mathematical treatment is different. Time is not taken into account. Rosenberg apparently considers his equations suitable for any beet and any operating conditions. Smolensky^3) objected to this, pointing out a whole series of factors not taken into account. Rosenberg, among other things, comes to

^1) Zapiski K. O. I. T. O. po sv. sakh. pr. 1875, vol. V, 281, and 1876, vol. VI, 1. There is also a separate edition.

^2) Zap. sv. sakh. pr. 1886, 320—On the calculation of the diffusor charge.

^3) Zap. sv. sakh. pr. 1886, 471 and 1887, 118.

[[unclear: beginning of paragraph]] that, with an infinite number of diffusers, the % of sugar in the diffusion juice = the % of sugar in the beet. Smolensky makes the correction: % of sugar in the diffusion juice = % of sugar in the normal juice. Smolensky has absolutely no idea of the continuity of the diffusion process.

In 1892, in the article¹ “On the Least Withdrawal of Juice,” Rosenberg changes his point of view regarding the concentration of the diffusion juice: he believes that all the juice from the space in the diffuser and the communication, not occupied by cossettes, must be withdrawn. Hence the conclusion: the greater the filling, the smaller the withdrawal of juice.

The same is the point of view of Baudry, who, proceeding from similar considerations, even compiles a table² of juice withdrawals for various cossette densities. (These considerations are a consequence of an incorrect conception of the diffusion process as stepwise: if the displacement had occurred a large number of times, the difference in concentrations would not be so important. Rosenberg says this quite consciously in his answer to Smolensky’s objections, and even explains: the diffusion surface is so large that, with a minimal difference in concentrations, diffusion proceeds very rapidly³.) The confidence that greater filling does not require greater withdrawal of juice leads to the conclusion that, with greater filling, the diffusion juice is obtained more concentrated; that good filling is very advantageous. From these same views it follows that communications, heaters, etc., are “harmful spaces”—they require greater withdrawal.

Vivien⁴, in the article “Diffusion Method of Extracting Juice,” on the contrary, states that the heater must be equal in volume to the volume in the diffuser occupied by the cossettes. A large heater, filled with diffusion juice of intermediate concentration. Thus Vivien sees the advantages of a continuous change in the concentration of the juice, strives toward it, but does not notice that such a continuous change is achieved of itself in any diffusion battery, and without large heaters, which can only hinder the uninterrupted course of the process (owing to the mixing of juices of different concentrations).

Levitsky⁵, in 1912, in the article “How Should Diffusers Be Filled?” comes, by experimental means, to the conclusion that firm filling does not increase the concentration of the diffusion juice, and in general gives no advantages; consequently, it is irrational. This article was met, apparently, with the mistrust that is justified by Levitsky’s strange and confused conclusions (an increase in unspecified losses with greater filling, the absence of automatic scales for beets at the factory where the experiments were carried out). Meanwhile, putting aside the balances cited by G. Levitsky, the basic idea of his article follows from entirely correct observations.

Grille⁶, in 1914, establishes, by a purely empirical method, the continuity of the diffusion process, but gives no theoretical illumination of this phenomenon.

The contradictory conclusions of various investigators compel us to subject anew the theory of the operation of the diffusion battery. Up to now the process of diffusion has not been at all clarified: some say that the intermediate volume is harmful; others—that it is useful; according to some it is useful, according to others—harmful. Smolensky asserts that the concentration of the diffusion juice may approach indefinitely the concentration of the normal juice; Rosenberg gives quite a different limit; some say that there is in the diffusers some difference between the concentration of the normal and the diffusion juice; others say that it is too small and may be neglected.

Fundamentals. The operation of a sugar-factory diffusion battery is very closely approximated by the operation of an ideal continuously operating diffusion apparatus, which from one end is continuously and uniformly fed with water, while into it from the other end fresh cossettes enter, continuously and uniformly, moving countercurrent to the water. Where water enters, exhausted cossettes are removed; and where fresh cossettes are introduced, sugar-containing diffusion juice is drawn off. Each diffuser is only a part of one continuously operating apparatus.

The process in a diffusion battery may be represented schematically as follows. A cossette, located in a given place, has a certain surface \(S\). Let us unfold its surface into a practically straight line \(AB\) (Fig. 1). The entering water moves countercurrent to the cossette, but one may imagine that the diffusion surface \(S\) remains in place (for it does not in fact change in magnitude). On one side of it there moves normal juice of variable concentration \(C\); on the other side, counter to it, along the surface \(S\), from \(A\) to \(B\), moves diffusion juice, gradually increasing in concentration \(c\) as a result of diffusion.

In 1 min, through the area \(ds\), a quantity of sugar will diffuse

\[ dq = k(C - c)\,ds \ldots (1), \]

where \(k\) is the coefficient of proportionality. Let, in 1 min, \(N\) liters of normal juice and \(n\) liters of diffusion juice pass through any transverse section of the battery. The concentrations on either side of \(ds\), as a result of diffusion, change from point \(s\) to point \(s + ds\) by \(dC\) and \(dc\) (they increase). The amount of sugar given up by the normal juice is equal to the amount received by the diffusion juice. Consequently,

\[ dq = N\,dC = n\,dc \ldots (2). \]

In (1) and (2) we have 3 equations with 4 variables—\(C\), \(c\), \(s\), \(q\). Each of them can be expressed as a function of any of the others. It is also of interest to be able, for any place in the battery, to calculate the difference in concentrations \(C - c\), which we shall denote by \(\Delta c\).

Let us find certain relations, introducing the following notation:

\(C_2\)—concentration (volume %) of sugar in the normal juice of the charged cossette.

\(c_2\)—concentration of sugar in the diffusion juice.

\(C_1\)—concentration of sugar in the exhausted cossette (pulp).

\(c_1\)—concentration of sugar in the extracting liquid (if this is water, then \(c_1 = 0\)).

1. From (2):

\[ \left. \begin{aligned} dC &= \frac{1}{N}\,dq,\\ dc &= \frac{1}{n}\,dq \end{aligned} \right\} \quad \text{whence} \quad d(C - c) = \left(\frac{1}{N} - \frac{1}{n}\right)dq \quad \text{or} \quad d\Delta c = \left(\frac{1}{N} - \frac{1}{n}\right)dq. \]

Together with

\[ \Delta c = [[\unclear: formula involving \Delta N,\ a,\ b,\ t,\ \lambda]] \]

we obtain

\[ \frac{\Delta c}{\Delta c_0} = [[\unclear: expression involving logarithms and exponential factors]] \]

The maximum gain occurs for

\[ [[\unclear: equation for the condition of a maximum]] \]

or

\[ [[\unclear: transformed maximum condition]] \]

By introducing the notation

\[ \frac{\Delta c}{\Delta c_0}=\eta, \]

one can express the preceding equation in the following form:

\[ \eta_{\mathrm{opt}}(1-\eta_{\mathrm{opt}}) = [[\unclear: right-hand side involving } \Delta c_0,\ b,\ a,\ \lambda,\ t [[\unclear]] \tag{10} \]

or, for small values of the argument,

\[ \eta_{\mathrm{opt}} = [[\unclear: expression involving } \ln [[\unclear]] \text{ and } \Delta c_0 [[\unclear]] \]

This formula determines the dependence of the optimum value of the useful effect on the initial concentration difference and on the dimensions of the diffusion battery.

V. Formula Relating the Efficiency to the Concentration Difference

Let us determine the dependence of the efficiency on the concentration difference at the terminals of the battery. We shall make use of the result obtained above, namely,

\[ \Delta c = [[\unclear: expression involving } \Delta c_0,\ \Delta c_1,\ \lambda,\ t,\ a,\ b [[\unclear]] \]

whence

\[ \frac{\Delta c_1}{\Delta c_0} = [[\unclear: fraction containing } \Delta c,\ \Delta c_0 [[\unclear]] \]

and therefore

\[ \eta = [[\unclear: expression for efficiency as a function of } \Delta c/\Delta c_0 [[\unclear]] \]

or, after transformation,

\[ \eta = 1- \frac{[[\unclear]]}{[[\unclear]]} \left(\frac{\Delta c}{\Delta c_0}\right)^{[[\unclear]]} \tag{11} \]

This gives the required relation between the coefficient of useful action of the battery and the decrease of concentration in the external circuit.

The value of \(\Delta c\), at which the efficiency is a maximum, is obtained from the condition

\[ \frac{d\eta}{d(\Delta c)}=0. \]

This condition gives

\[ [[\unclear: equation involving } \Delta c,\ \Delta c_0,\ \ln [[\unclear]] \]

or

\[ [[\unclear: equivalent logarithmic equation]] \tag{12} \]

The latter formula may also be written in the form

\[ [[\unclear: formula containing } \eta_{\max},\ \Delta c/\Delta c_0 [[\unclear]] \]

Thus the optimum regime is determined by the ratio of the actual concentration difference to the initial one; this ratio depends on the geometrical dimensions of the battery and on the time during which diffusion proceeds.

In the limiting case, when the concentration difference is small, the preceding formula simplifies and gives

\[ \eta = [[\unclear: approximate expression]] \]

and, consequently,

\[ \eta_{\max} = [[\unclear: approximate maximum-efficiency expression]] \]

The comparison of these formulae shows that the efficiency increases with increasing initial concentration difference, but only up to a certain limiting value. Further increase of the concentration difference gives no corresponding increase in useful effect, since the losses due to diffusion through the separating layer increase at the same time.

For a battery of given dimensions the most advantageous operating condition is therefore determined by the equation

\[ [[\unclear: final displayed condition involving } \Delta c,\ \Delta c_0,\ a,\ b,\ \lambda,\ t [[\unclear]] \]

which must be satisfied by the concentration difference maintained at the terminals of the battery.

\[ c=\frac{(c_2-c_1)\,\Delta c_1}{\Delta c_2-\Delta c_1} \left(\frac{\Delta c_2}{\Delta c_1}\right)^{s/s_0} -\frac{c_2\Delta c_1-c_1\Delta c_2}{\Delta c_2-\Delta c_1}\ldots(6) \]

Let us note that, in this way, \(\log\left(c+\dfrac{c_2\Delta c_1-c_1\Delta c_2}{\Delta c_2-\Delta c_1}\right)\) is in a rectilinear dependence on \(S\).

1. From (2): \(NdC=ndc\); \(\theta=\dfrac{n}{N}\), and \(\theta=\mathrm{const}\). But at \(c=c_1\), \(C=C_1\), i.e.

\[ C_1=\frac{n}{N}c_1+\mathrm{const}. \]

Hence,

\[ C-C_1=\frac{n}{N}(c-c_1)\ldots(8). \]

At \(c=c_2\),

\[ C_2-C_1=\frac{n}{N}(c_2-c_1), \]

whence from (8) follows:

\[ C-C_1=\frac{C_2-C_1}{c_2-c_1}(c-c_1)\ldots(10). \]

When leaching with pure water (therefore, without return of the diffusion waters), which we shall subsequently speak of exclusively, i.e. when \(c_1=0\), the formulas take the following form (in this case, \(\Delta c=c\)):

\[ dq=k(C-c)\,ds\ldots(1'),\qquad dq=N\,dc \ [[unclear]]. \]

\[ d\Delta c=\left(\frac{1}{N}-\frac{1}{n}\right)dq\ \text{whence}\quad s=\frac{g}{s_0}\ln\frac{\Delta c_2}{\Delta c_1}+\ln\Delta c_1\ldots(4') \]

\[ \Delta c=\Delta c_1\left(\frac{\Delta c_2}{\Delta c_1}\right)^{s/s_0}\ldots(6') \]

\[ q_0=k\,s_0\,\Delta c_2\, \frac{1-\dfrac{\Delta c_1}{\Delta c_2}}{\ln\dfrac{\Delta c_2}{\Delta c_1}} \ldots(8') \]

or

\[ q_1=k\,s_0\,\Delta c_1\, \frac{1-\dfrac{\Delta c_1}{\Delta c_2}}{\ln\dfrac{\Delta c_2}{\Delta c_1}} \ [[unclear]] \]

and

\[ c=\frac{c_2\Delta c_1}{\Delta c_2-\Delta c_1} \left(\frac{\Delta c_2}{\Delta c_1}\right)^{s/s_0} -\frac{c_1\Delta c_1}{\Delta c_2-\Delta c_1} \ldots(6') \]

\[ C-C_1=\frac{n}{N}c\ldots(8'),\qquad C-C_1=\frac{C_2-C_1}{c_2}\,c\ldots(10'). \]

The application and operation of semidiffusers. In applying the formula, \(C\) denotes the concentration of sugar in the leached chips, taken continuously at that place in the battery where the leaching water continuously enters. In the operation of an ordinary diffusion battery the composition of such a product changes continuously: the feed is carried out periodically, with part of the total charge being discarded each time. Thus, in the section being transferred, the more leached chips are replaced by newly leached ones. Analysis in the laboratory gives us the % of sugar, \(C'-C_1'\), remaining in the spent chips; \(C_1'\), of course, is greater than \(C_1\) (the identity being fulfilled only under continuous operation).

in the intervals between the discharged pulp there is not water \((c_1 = 0)\), but a weak sugar-containing solution—“molasses,” or diffusion water—of concentration \(c_1'\), which is also known from the data of laboratory analysis.

In order to apply everything said about the continuous process to the actual factory diffusion process, it is necessary to be able to calculate, from \(C_1'\) and \(c_1'\), the “ideal” sugar content of the press cake—\(C_1\), which can then be substituted into the formulas given. The calculation is very easy to carry out using formula (10), substituting in it, instead of \(C\) and \(c\), \(C_1'\) and \(c_1'\):

\[ C_1' - C_1 = \frac{C_2 - C_1}{c_2}\cdot c_1' : C_1 \left(1-\frac{C_1 c_1'}{c_2}\right) = C_1' - \frac{C_2 \cdot c_1'}{c_2}; \]

but the quantity

\[ \frac{C_1 c_1'}{c_2} \]

is very small (as the product of the small quantities \(C_1 - c_1'\)); it may be discarded. Thus,

\[ C_1 = C_1' - \frac{C_2 c_1'}{c_2}. \tag{11} \]

The inverse calculation—if it is required, from the changed value \(C_1\), to find new values for \(C_1'\) and \(c_1'\)—may be carried out by regarding \(C_1'\) and \(c_1'\) as approximately proportional to \(C_1\).

One may also proceed much more simply: take

\[ C_1 = C_1' - c_1'. \tag{11'} \]

All possible coefficients and constants in the calculations change somewhat because of this, but the final results obtained are the same. Examples of the application of formulas (11) and (11′) will be given below.

Experimental confirmation. For a theoretical discussion such as the one set forth above to correspond to reality, it is necessary that the quantities calculated by formula (8) and \((8')\), namely \(\log \Delta c\) and

\[ \log \left(c+\frac{c_2 \Delta c_1}{\Delta c_2-\Delta c_1}\right), \]

against \(S\) should be expressible graphically by a straight line. Knowing \(c\) for the various diffusers (samples taken simultaneously), we can easily calculate \(C\) by formula \((10')\), and consequently also \(\Delta c\). The application of formula \((8')\) is quite simple.

There turned out to be few experimental data suitable for verification in the literature, namely: 1) Stammer’s—2 series of figures; 2) in Herzfeld’s experiments, with a laboratory battery of the Inst. f. Zuck.-Ind. Ver. d. D. Z. Ind.,² carried out by Sachs—2 series of figures; 3) Juten’s experiments³ of 1910—at a factory under modern conditions—also 2 series of figures.

The figures were subjected to the following treatment. The weight percentage of sugar was given for the diffusion juice of the various diffusers. We convert it to mass percent (multiply by sp. gr.) and thus find the quantity \(c\) for all diffusers. We likewise convert to volumetric percent the sugar in the normal juice of the beet being processed—we obtain \(C_2\). By formula (11) we find \(C_1\)—the percent sugar of the “ideal” press cake.

Next, by formula \((10')\),

\[ C - C_1 = \frac{C_2 - C_1}{c_2}\,c, \]

we find the concentration of the normal juice \(C\) for any diffuser. Thus the concentration difference \(C-c\) and its \(\log\) for any diffuser will also be known. Plotting the found values of \(\log(C-c)\) on a diagram—as ordinates, and

... arriving at the abscissa corresponding to the number of the diffuser, we must obtain a straight line. This may serve as a criterion of the correctness of the theory.

Just as exactly, and still more simply, from formula (8) one may find the quantity

\[ \log\left[c+\frac{c_{2}\Delta c_{1}}{\Delta c_{2}-\Delta c_{1}}\right]. \]

The dependence of its log on the number of the diffuser should also be represented by a straight line. For Stammer’s experiments we calculate \(\lg \Delta c\), and for the others—

\[ \log\left[c+\frac{c_{2}\Delta c_{1}}{\Delta c_{2}-\Delta c_{1}}\right]. \]

A straight line can be obtained only for diffusers beginning with the second. For the first diffuser the condensation drops will be different. The fall in concentration for it must evidently be expected to be greater: here only heating, moistening of the gas, and diffusion, which has not yet begun, take place.

The results of the calculations are given in the form of tables and drawings.

Stammer’s data

\[ c_1=c_1'-\frac{a_2c_1'}{a_1}=0.07-\frac{[[unclear]]}{[[unclear]]}=0.0[[unclear]] \]

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[[unclear: body text between tables; several inline mathematical expressions and numbers are visible but not reliably decipherable. Fragments include “95”, “88.8”, “1.07”, and “0.85”, but the surrounding text and formulas are unreadable.]]

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The curves obtained for $\lg \Delta C$ from Stammer’s data deviate from the ideal straight line (the curve for the second row already comes very close to a straight line). Beginning with the fourth diffuser, they (also for the first row) are indeed arranged almost along a straight line, but within the limits of the third and fourth diffusers the curve is inclined more steeply. This is understandable for the former working conditions (1872), when it was considered necessary to heat only the first diffusers. Thus the deviations are explained by the fact that in the third and fourth diffusers the temperature was probably higher than in the others. In the fifth–thirteenth diffusers it was more uniform; therefore the result is a straight line. For the working conditions of 1872, the agreement with the theory may be regarded as quite satisfactory.

The drawing relating to the experiments of Inst. f. Z.-Ind. gives curves that are rather close to the ideal straight lines. The greater deviation from the straight line for the second battery is quite understandable, since the experiment in the second battery was carried out with shavings that were too thick (5 mm). It must be taken into account that the last diffusers in these experiments were not heated, and the water for diffusion had a temperature of only 20° C. Moreover, even in the heated diffusers of a small laboratory battery (diffuser capacity—2 ltr.) it was, of course, difficult to keep the temperature uniform. Taking this into account, the agreement of the experiment with the theory must be considered sufficient.

The most consistent with the theory are Juten’s data—comparatively recent (1910) and obtained not in a laboratory but at a factory.

The general character of the deviations from the theoretical straight lines in all the cited experiments is as follows: 1) In the first diffuser almost no diffusion is observed (the experimental curves are inclined only very slightly to the abscissa axis). 2) In the immediately following diffusers the drop in concentration is somewhat weaker than according to the theory (the slope of the curve is less than the slope of the theoretical straight line): here not all the cells have yet been warmed through. 3) In the middle diffusers of the batteries there is accelerated diffusion, since the temperature is still high and all the cells have already been warmed through (the slope of the empirical curve is greater than the slope of the theoretical straight line). 4) Further, in the last diffusers before discharge, the slope of the curve decreases and becomes much less than the theoretical one, since diffusion slows down owing to the lowering of the temperature.

The deviations are not so large, because the accelerating influence of the high temperature in the first diffusers is compensated by the circumstance that the shavings have not yet been warmed through (in addition, see the explanation on p. 1).

From the comparisons made, it appears possible to draw the following conclusion: the process taking place in the diffusers must differ, and indeed does differ, from the theoretical process, but this difference is not so great as to make the application of the theory impossible (the theoretical diagram of the operation of a steam engine also differs from the diagram given by the indicator). Probably, in the future, by setting up experiments at a factory, it would be possible to introduce corrections into the formulas of the theory (as, for example, is practiced in calculating air blowers). At present, having at hand only scant experimental material, it would be premature to introduce corrections. We shall take into account only one circumstance—the absence of diffusion in the first diffuser—and shall consider that the first diffuser, as it were, does not exist, and that the period of leaching is the time during which the shavings remain in the other diffusers, except the first.

Fundamental equations. In addition to the above, let us introduce the following notation:

\(C_2\) is the content of sugar in 1 hl of the diffuser capacity.

\(C_1\) is the content of sugar in 1 hl of the diffuser capacity, measured from the moment when this diffuser has become first and up to its unloading.

The quantity of sugar extracted from one hl of diffuser capacity per one revolution of the battery \((t)\) is equal to

\[ (C_2-C_1)\frac{a}{100}\ \text{kg, or in }1' \quad q_0=\frac{C_2-C_1}{t}\cdot\frac{a}{100}. \]

But \(q_0\) may also be determined by formula (7):

\[ q_0=k\,Q\,\Delta c_9\,\alpha, \qquad \alpha=\frac{1-\dfrac{\Delta c_1}{\Delta c_2}}{\ln\dfrac{\Delta c_2}{\Delta c_1}}. \]

Equating the first expression for \(q_0\), we obtain

\[ \frac{C_2-C_1}{t}\cdot\frac{a}{100} = \alpha kQ\Delta c_9. \]

The magnitude of the surface of the cossettes in 1 hl of diffuser capacity depends on the degree of filling of the diffuser and on the thickness of the cossettes, \(d\), namely: \((Q\) is proportional to \(a\), and proportional to \(\frac{1}{d}\); the surface of the elementary plate increases proportionally to \(d^2\), but the number of such plates in a given weight is, evidently, proportional to \(\frac{1}{d^3}\).)

Thus,

\[ Q=k_1\frac{a}{d}. \]

Consequently, the quantity of sugar extracted is proportional to

\[ C_2-C_1=\frac{k\,k_1}{100}\cdot\frac{a t}{d}\,\Delta c_9. \]

Replacing \(100\,k\,k_1\) by \(k\), we have

\[ C_2-C_1=k\,\frac{at}{d}\,\Delta c_9. \tag{12} \]

Since the diffusion juice contains sugar from the normal juice \((C_2)\) without the sugar of the juice \((C_1)\), but diluted in proportion to \(a_1:a\), then

\[ c_2=(C_2-C_1)\frac{a}{a_1}. \tag{13} \]

For simplicity we introduce the following designations:

\[ \frac{kt}{d}=A \tag{14} \]

— the coefficient of time, and

\[ \frac{a}{a_1}=B \tag{15} \]

— the coefficient of juice withdrawal.

Then the equations derived earlier take the form:

\[ C_2-C_1=A\,a\,\Delta c_9, \tag{16} \]

\[ c_2=B(C_2-C_1), \tag{17} \]

where

\[ \alpha=\frac{1-\dfrac{\Delta c_1}{\Delta c_9}}{\ln\dfrac{\Delta c_9}{\Delta c_1}}. \]

Here we also add

\[ \Delta c_9=C_0-c_2=C_0(1-B)+B\,C_1. \tag{18} \]

For practical application, the five equations (14–18) cited here are sufficient. From them one can determine the values \(C_1, C_2, \Delta c_9\) for any given values \(C_0, A\), and \(B\). If \(C_1\) has already been determined, then \(C_2\) is found very easily from (17). Therefore we shall show how to use equations (14–18) to determine \(C_1\).

Substituting \(\Delta c_2\) from (18), after the necessary corrections, gives the found value of \(a\cdot \Delta c_2\). Substituting this value into the first formula (17), we find:

\[ \ln \frac{C_2(1-B)+B C_1}{C_1} = \frac{A a \Delta c_2}{C_1} \left( 1-\frac{C_1}{C_2(1-B)+B C_1} \right). \]

After simplification and reduction:

\[ \ln \frac{C_2(1-B)+B C_1}{C_1} = A(1-B). \tag{19} \]

Having found \(\ln\), we obtain from the table of logarithms the corresponding number

\[ \frac{C_2(1-B)+B C_1}{C_1}=M, \]

whence

\[ C_2=\frac{C_1(1-B)}{M-B}. \]

When using (18), the calculation is very simple. Since a table of logarithms is available, one may also use decimal logarithms (as will be shown below). This will be reflected only in the numerical value of the coefficient \(A\) and \(a\).

Applying the formulas derived above (14–19), one may, knowing how the diffusion battery works under certain definite conditions, answer the question of what will be obtained when conditions are changed, namely (more precisely stated, approximately, within known limits). For illustration we shall solve several practical examples.

1. a) With a charge to the diffuser of \(a=54\ \dfrac{\mathrm{kg}}{\mathrm{hl}}\) and with a normal juice sugar content of \(15.7\%\), a diffusion juice of \(12.1\%\) sugar was obtained, while the pulp contained \(0.42\), and the diffusion water \(0.18\%\) sugar. Determine the quantities \(A\) and \(B\) characterizing the operation.

If the juice contains \(98\%\), then in \(1\ \mathrm{hl}\) of tank volume there will be contained normal juice,

\[ a=54\cdot \frac{0.93}{1.08}=46.5\ \mathrm{hl} \]

of normal juice; \(C_1=15.8\times 1.08=17.0\); \(C_2=12.1\times 1.06=12.8\). From formula (11),

\[ c_2=0.40-\frac{17.0}{12.8}\cdot 0.18=0.17; \quad \Delta c_2=17.0-12.8=4.2; \quad q= \frac{1-\frac{0.17}{4.2}}{\frac{12.8}{0.17}} =0.6889. \]

By (18):

\[ A=\frac{C_2-C_1}{a\Delta c_2} = \frac{17.0-0.17}{0.6889\cdot 4.2} = 5.82; \]

by (17):

\[ B=\frac{C_2}{A a+C_1} = \frac{12.8}{16.83} = 0.761. \]

Now we can decide how much sugar there will be in the juice and in the diffusion water if the working conditions are changed. Let us give examples of such calculations.

b) What will the juice be like if the charge is increased by \(5\%\), while the other conditions remain unchanged?

\[ A=5.82; \]

since evidently \(A=\frac{\alpha}{d}\) does not depend on the degree of charge \(a\), but

\[ B=\frac{a_1}{a} \]

is proportional to \(a\); consequently, now

\[ B=0.761\cdot 1.05=0.799. \]

Using (19):

\[ \lg \frac{C_2(1-B)+BC_1}{C_1} = A(1-B) = 5{,}82\cdot 0{,}201 = 1{,}1698. \]

The corresponding number \(M=14{,}8\);

\[ C_1=\frac{C_2(M-B)}{M-B} =\frac{17{,}0\cdot 0{,}201}{14{,}8-0{,}799} =0{,}244, \]

instead of the former \(C_1=0{,}17\). Proportionally increasing also

\[ C'_1=0{,}4\frac{0{,}244}{0{,}17}=0{,}57\%, \qquad c'_1=0{,}18\frac{0{,}244}{0{,}17}=0{,}26\%. \]

Thus, with an increase in the fill, the elevation of the sugar loss in the bagasse is possible; this is found in accordance with an increase of \(B\). In order that the quantity of sugar in the juice not change, it is sufficient simultaneously with the fill \((a)\) to increase proportionally to it the extraction of juice \((a_1)\). Then

\[ B=\frac{a}{a_1} \]

will remain unchanged, but at the same time the strong juice will disappear; the concentration of the diffusion juice will remain unchanged.

c) If the cutting does not keep pace with the work of the diffusion battery, then with an increase of the fill both the duration of diffusion and the extraction will be obtained, since the cutting, in order to give a larger fill, must work longer. Let us calculate the \(\%\) of sugar in the bagasse for such a case.

What will the bagasse be if the fill is increased by \(5\%\) and \(t\)—the duration of diffusion—is also increased by \(5\%\)?

For these conditions—

\[ A=5{,}82\cdot 1{,}05=6{,}11; \qquad B=0{,}761\cdot 1{,}05=0{,}799. \]

\[ \lg \frac{C_2(1-B)+BC_1}{C_1} = 6{,}11\cdot 0{,}201 = 1{,}2281; \qquad M=16{,}9; \qquad C_1=\frac{17{,}0\cdot 0{,}201}{16{,}9-0{,}799} =0{,}212. \]

\[ C'_1=0{,}4\frac{0{,}212}{0{,}17}=0{,}50\%; \qquad c'_1=0{,}18\frac{0{,}212}{0{,}17}=0{,}22\%. \]

The loss in the refuse has nevertheless increased, although not to the same extent as in the preceding example. In any case, with an increase in the fill one should expect a decrease in the \(\%\) of sugar in the juice; an increased extraction with an increased fill is impossible. Baudry and Rosenberg’s considerations regarding the dependence of extraction on the degree of filling are erroneous, since they neglect the leveling of the concentration difference inside and outside the beet slices. In reality, with an increase of the fill the surface increases (under otherwise equal conditions), while at the same time the volume of liquid in all the diffusers decreases; consequently the limiting concentration difference \(\Delta c\), which is obtained as a result of this, also decreases proportionally.

d) What will the bagasse be if the extraction of juice is increased by \(5\%\) (instead of \(100\)—\(105\))?

\[ A=5{,}82 \quad \text{(does not change)}; \qquad B=\frac{0{,}761}{1{,}05}=0{,}725. \]

\[ \lg \frac{C_2(A-B)+BC_1}{C_1} =5{,}82\cdot 0{,}276 =1{,}606; \qquad M=39{,}6. \]

\[ C_1=\frac{17{,}0\cdot 0{,}276}{39{,}6-0{,}725} =0{,}12\%. \]

\[ C'_1=0{,}4\frac{0{,}12}{0{,}18}=0{,}26\%; \qquad c'_1=0{,}18\frac{0{,}12}{0{,}17}=0{,}18\%. \]

e). What will the juice be if the extraction is increased by 5% (instead of 100—95)

\[ A=5.82;\quad B=\frac{0.761}{0.95}=0.801;\quad \lg\frac{C_2(1-B)+BC_1}{C_1}=5.82\cdot0.199=1.158;\quad M=14.4. \]

\[ C_1=\frac{17.0\cdot0.199}{14.4-0.801}=0.25;\quad C'_1=0.40\cdot\frac{0.25}{0.17}=0.60\%;\quad C''_1=0.18\cdot\frac{0.25}{0.17}=0.27\%. \]

Thus, a change in the amount of extraction has a very considerable effect on the amount of sugar in the juice.

f). What will be the composition of the arriving-to-the-battery pressed-out diffusion juice, assuming that there are only 10 diffusers, of which 9 are simultaneously working?

We must count only 9 diffusers in action, since in the freshly filled diffuser the chips are only beginning to diffuse. Thus, there were 9 diffusers; on addition of one more, there are 10 of them. The duration of diffusion \((t)\) increased in the ratio \(10/9=1.11\).

Magnitude

\[ A=5.82\cdot1.11=6.46;\quad B=0.761\quad \text{(unchanged)}. \]

\[ \lg\frac{C_2(1-B)+BC_1}{C_1}=6.46\cdot0.238=1.544;\quad M=35.0; \]

\[ C_1=\frac{17.0\cdot0.238}{35.0-0.761}=0.12; \]

\[ C'_1=0.40\cdot\frac{0.12}{0.17}=0.28\%;\quad C''_1=0.18\cdot\frac{0.12}{0.17}=0.13\%. \]

g). What will be the juice if the thickness of the chips is increased by \(1\frac{1}{2}\) times?

\[ B=0.761\quad \text{(unchanged)};\quad A=\frac{5.82}{1.5}=3.88 \quad \left(\text{since } A=\frac{kt}{d},\ \text{where } d \text{ is the thickness of the chips}\right). \]

\[ \lg\frac{C_2(1-B)+BC_1}{C_1}=3.88\cdot0.238=0.924;\quad M=8.46. \]

\[ C_1=\frac{17.0\cdot0.238}{8.46-0.76}=0.52;\quad C'_1=0.40\cdot\frac{0.52}{0.17}=1.21\%;\quad C''_1=0.18\cdot\frac{0.52}{0.17}=0.57\%. \]

Thus, the thickness of the chips apparently greatly decreases the comparatively slight diffusion; however, it should be noted that the calculation of the resistance coefficient of the juice moving inside the chips is very complicated and has not yet been performed; therefore it is difficult to predict how it will change when the thickness of the chips changes.

II. The experiments of Mr. Levitsky¹ led him to a conclusion surprising even to himself, concerning the uselessness of strong circulation. From the standpoint of the old theory of diffusion work, the conclusion is indeed strange. Let us see whether it is surprising from the standpoint of our theory.

a). The charge of the diffuser in the 1st experiment of Mr. Levitsky was 102 p. (capacity—260 ved.). Extraction—139.67 ved. The diffusion juice obtained contained 14.26% sugar; the juice with 0.412%, and the diffusion water—0.151% sugar. The rate of work: in 6 hours 81 diffusers were worked.

Converting these liters into kg, we obtain extraction of water:

\[ 52.3\ \frac{\mathrm{kg}}{\mathrm{hl}} . \]

Of normal juice in 1 h:

\[ 52.3 \cdot 0.95 = 49.7\ \mathrm{kg} \]

(\(0.95\) is the weight coefficient). The number of liters of normal juice in 1 hl is

\[ a=\frac{49.7}{1.08}=46.0\ \mathrm{litr.} \]

\[ (1.08\text{—sp. wt.});\qquad C_1=\frac{[[unclear]]}{[[unclear]]}=15.26\ \frac{\mathrm{litr.}}{\mathrm{hl}};\qquad B=\frac{46.0}{53.8}=0.855; \]

\[ C_2=14.26 \times 1.07 - 15.27;\qquad C_2=\frac{[[unclear]]}{0.855}+0.41=18.26\quad(\text{formula 17}); \]

\[ C_1=0.41\ ? - 0.15 = \]

\[ [[unclear]]\qquad C_2=18.26,\qquad 15.26=3.00\cdot a= \frac{1-\frac{0.26}{3.0}}{1-\frac{3.0}{0.26}}=0.860. \]

Then

\[ A=\frac{C_2-C_1}{C_2\cdot \Delta\rho} =\frac{18.26-0.26}{0.860\cdot 3.00}=6.98 . \]

b) In the following 6 hours work was carried out on the same beet, but with a different extraction—108 poods per diffuser in 260 buckets. Extraction—143.61%; in the diffusion juice, 14.24% sugar; in the pulp—0.495%; in the diffusion water—0.195%. In 6 hours, 83 diffusers were worked.

Compared with the first experiment the extraction increased in the ratio

\[ \frac{143.61}{139.87}, \]

and the charge in the ratio

\[ \frac{108}{106}; \]

therefore the dilution increased in the ratio

\[ \frac{143.61\cdot 108}{139.87\cdot 106}=1.045 \]

and, consequently,

\[ B = [[unclear]] = \frac{35\cdot 106}{31\cdot 108}=1.085. \]

Therefore, for the second experiment,

\[ A=\frac{6.98}{1.085}=6.43;\qquad C_2=\frac{14.24\cdot 1.07}{0.881}+0.495=17.79. \]

\[ \frac{C_2(1-B)+BC_1}{C_1} =6.43\cdot 0.119^2=0.7652;\qquad C_1=\frac{17.79\cdot 0.119}{0.7652-0.881} =[[unclear]] . \]

Of the experiment \(C_1=0.495-0.195=0.300\), which is less than was obtained theoretically. Thus, there is nothing surprising in the fact that in the first experiment the percentage of sugar in the pulp is not lower than in the second. The difference of 0.4% between the theoretical and the estimated sugar content in the pulp is wholly admissible with the given arrangement of the experiment, since it is a matter of small phenomena. (For example, in experiment II there proved to be 0.77% irrecoverable losses, taking into account our data; probably these irrecoverable losses include this discrepancy.)

Thus, the proposed theory gives a more correct conception of the process of diffusion in countercurrent apparatuses; the active factor is the difference in concentrations—in contrast to the former theories based on the principle of repeated mixing.

1. The theory set forth makes it possible, knowing the results of the operation of a diffusion battery under certain given conditions, to calculate the results of its operation under changed conditions (apart from the conditions of temperature and the quality of the beet).

1. With an increase in imbibition, without changing the other operating conditions, the amount of sugar in the bagasse increases; consequently, the withdrawal must be increased. Thus, the advantages of strong imbibition are not so great.

2. The influence of the amount of juice withdrawn is the reverse of the influence of imbibition: when the withdrawal is increased, the % of sugar in the bagasse decreases.

3. The speed of operation in diffusion, which previous theories did not take into account at all, is taken into account by the theory set forth; moreover, it turns out to have almost as great an importance as imbibition and withdrawal.

4. The number of diffusers, when new ones are added to the existing diffusers, has the effect both of increasing the total volume of the battery and, consequently, of increasing the duration of diffusion. If, however, without changing the total volume, the number of diffusers is changed, the influence will not be so significant (too small a number of diffusers is disadvantageous, since with the diffuser being discharged a considerable part of the total charge of the battery is carried away; the process is greatly removed from the continuous one; therefore, if the total volume is not increased, the loss of sugar will be greater).

5. The capacity of the calorisators and of the communications is of no substantial importance. It is important only that in the transfer spaces the juices not be mixed. From this point of view it is useful to increase the velocity of the juice and to construct long calorisators with tubes of small diameter.

Further development of the theory set forth may be useful both in the field of sugar production and in application to analogous processes of other chemical technologies.

In the field of sugar production, an experimental development of the theory at the factory is necessary; it would be of interest to determine, for various conditions, the value \(k\) (equation 14), which may indicate a way of taking account of the influence of temperature and of measuring the ability of beet chips to diffuse. From deviations from the theoretical curves it will probably be possible to draw conclusions about desirable changes in the course of operation, and to direct it in the same way as an indicator diagram of a steam engine indicates defects in its operation and makes it possible to correct them.

Continuous countercurrent processes, similar to the process in the diffusion battery of a sugar factory, are widely distributed in all branches of chemical technology. They occur everywhere where it is necessary to carry out heating, cooling, leaching, and drying. The theory of operation of a diffusion battery, without any changes, could be applied to such heating processes as take place, for example, in the operation of all kinds of shaft, chamber, channel, and roasting furnaces (one need only, in the zone of greatest heating, take into account the thermal effect that usually occurs there from chemical reaction). The operation of a diffusion battery is entirely analogous to the operation of other continuous countercurrent leaching apparatuses, for example, Shanks’s apparatus. The operation of countercurrent dryers could also be considered from a similar point of view, introducing certain modifications (experimental development is necessary).

Theory of Operation of a Diffusion Battery

Fig. 4.

Labels in the diagram:

• \(S\)
• \(c\)
• \(ds\)
• Fresh jet
• Diffusion trail

Graph labels and annotations:

• Stanton’s data
• \( \log \Delta c \)
• Diffusers
• I, II, III
• I–XIII

Graph labels and annotations:

• Neufeld’s data
  (methylene blue)
• \( \log \left(\dfrac{c_k}{c_0}\right) \) [[unclear: remainder of vertical-axis formula]]
• Diffusers
• I, II
• I–XIII

Graph labels and annotations:

• Zsigmondy’s data
• \( \log \left(\tau + \dfrac{c_k - c_0}{c_k - c_1}\right) \) [[unclear: formula partially illegible]]
• Diffusers
• I–XII

1. Zap. Sv.-Sakh. Prom., 1912, 103. ↩↩↩

2. Z. d. Ver. f. Rüb.-Zuck.-Ind. 1889, XXIX, 337. ↩↩

3. Bull. Ch. Sucr. et Dist. t. XXVIII, 1910, 137. ↩↩

4. Zap. sv. sakh. pr. 1886, 106—107. ↩

5. Zap. sv. sakh. pr. 1912, 103. ↩

6. Vestn. sakh. pr. 1914 I 457; Centr.—Bl. 1914 No. 24, p. 870. ↩