Drying Dynamics of Sugar Beet Pulp
Abstract Generated abstract
This paper develops a method for constructing drying rate curves for sugar beet pulp under varied drying conditions, motivated by the need to calculate multi-chamber dryers with intermediate air heating. Using experiments on beet plates, it examines how surface area changes during drying as a function of water removed, cut location and direction, plate thickness, and drying intensity, and incorporates these relations into an empirical drying rate equation. The authors derive a practical formula for estimating the time required to evaporate successive quantities of water and compare calculated intervals with experimental results, finding generally close agreement sufficient for technical dryer calculations.
Full Text
Prof. S. V. LEBEDEV and Engineer V. N. STABNIKOV
DYNAMICS OF DRYING
SUGAR BEET PULP
Tomsk — 1929
Prof. S. V. Lebedev and Eng. V. N. Stabnikov
Dynamics of Drying Sugar Beet Pulp
In the article “On the question of selecting a type of dryer for beet cossettes and the basic principles of calculation” (1), the authors of the present work noted the possibility of plotting, on Prof. Ramzin’s \(J—d\) diagram, the process of drying beets taking place in a multi-chamber dryer with intermediate heating of the air. A necessary condition, however, is a preliminary representation of the process of drying sugar beets in the coordinates: time—quantity of water removed. This auxiliary curve we have called the drying-rate curve. Lacking experimental material, in the work mentioned we were compelled to forgo differentiation of the drying process at the various stages of drying and to adopt uniform temperature conditions of drying for all chambers of the dryer. However, in this lies a very substantial inconsistency with the very character and purpose of intermediate air heating in the dryer, which makes it possible to differentiate the drying process by creating specific conditions for each stage. For differentiating the drying process in calculating the dryer, as was indicated in the article mentioned, it is necessary to find certain empirical coefficients that have not hitherto been determined. The task we pursue in the present case reduces to the following: to find a method for constructing the drying-rate curve under any conditions. One might limit oneself to obtaining such curves experimentally for various conditions, but in that case the task would be quite impracticable to the necessary extent, since there may be very many combinations of the various factors affecting the drying rate. Undoubtedly such a path may be used for solving a separate particular problem, but in general the solution must be based on applying to the process under study some equation for the drying rate and on finding the empirical coefficients entering into it. The ranges of variation of the values of these coefficients will at the same time indicate the applicability of the equation adopted.
1. Equations of the Drying Rate
From the theoretical standpoint, the phenomenon of drying represents a little-investigated field. Some authors (p. 59) directly note the impossibility of a theoretical accounting of the phenomenon, in view of its complex character. Nevertheless, equations do exist that relate the evaporation rate to the drying conditions and to the quality of the material being dried. Of such equations we shall note two. The first of them is cited by Prof. Deshevyi (III, p. 10). This equation is derived from purely physical considerations concerning the quantity of heat transmitted through a definite area under definite conditions. It has the form:
\[ h=\frac{S \times p \times d}{100 \times K \times T \times O}\ldots\ldots\ldots\ldots\ldots\ldots (1) \]
here:
\(h\) — drying time in hours,
\(S\) — the quantity of heat required for heating the water and converting it into steam at the known temperature,
\(d\) — percentage moisture of the material being dried,
\(p\) — weight of the material being dried,
\(K\) — coefficient of heat transfer from the air to the inner layers of the material,
\(O\) — surface of the material,
In this equation the unknown empirical coefficient is the heat-transfer coefficient \(K\), expressed by the equation:
\[ \frac{1}{K}=\frac{1}{a}+\frac{0.5\delta}{\lambda}+\frac{0.5\delta}{\beta}\ldots\ldots\ldots\ldots\ldots\ldots (2) \]
where \(a=2+10\sqrt{v}\); in this expression \(v\) is the velocity of the flowing air in meters per second, \(\delta\) is the thickness of the material being dried in mm., \(\lambda\) and \(\beta\) are numerical coefficients found empirically. Obviously, using equation (1), we can find only the coefficient \(K\). The coefficient \(K\), as is seen from equation (2), is a function of the velocity of air motion \(v\), of the thickness of the plate \(\delta\), and of the coefficients \(\lambda\) and \(\beta\). Therefore, in the present case the problem reduces to finding these coefficients \(\lambda\) and \(\beta\), which presents rather great difficulties. A simpler problem is to find the coefficient entering into the second equation known to us for the rate of drying. This equation (II p. 78) may be represented as follows:
\[ \Delta w=f\times(1+\alpha t_1)\times O\times\Delta P\times\sqrt{v}\times\Delta z\ldots\ldots (3) \]
here:
\(\Delta w\) — the quantity of water in grams evaporated during the time \(\Delta z\) min., referred to a unit of weight.
\(f\) — the sought coefficient.
\((1+\alpha t_1)\) — the binomial of gas expansion.
\(O\) — surface of the material per unit weight.
\(\Delta P\) — difference between the vapor pressure of water on the surface of the material being dried and the partial pressure of water vapor in the air, in mm of mercury.
\(v\) — velocity of air motion (in m per min.).
The value of the coefficient \(f\) may be determined in the following way.
Let us assume that:
- The expansion binomial \((1+\alpha t_1)=1\), i.e. \(t_1=0\).
- The area \(O=1\) sq. m per 1 kg of material.
- \(\Delta P=1\) mm.
- \(\sqrt{v}=1\) m per min., i.e. \(v=1\) m per min.
- \(\Delta z=1\) min.
then \(f=w\)
i.e. \(f\) is equal to the quantity of water in grams evaporated in one minute from one kilogram of material being dried at a temperature of \(0^\circ\) C, differ-
at a pressure difference \(\Delta P = 1\) mm, an area of one kilogram \(0 = 1\) sq. m, and an air velocity of 1 m per min. Thus, it may be assumed that the coefficient \(f\), obtained from equation (3), has a definite value and, if expression (3) is applicable to the drying process, must be numerically stable.
To calculate the coefficient \(f\), it is necessary to know the values of \(0\), \(\Delta P\), \(t_1\), and \(v\). Of these quantities, during the drying process \(0\) and \(\Delta P\) change their magnitude. Therefore, it is first necessary to turn to the study of the change in surface and pressure during the drying process.
II. Change in the Surface Area of Beet Pulp During the Drying Process.
During the drying process, the surface area of sugar beet is considerably reduced as a result of the decrease in the amount of water in it. The character of the relationship between the amount of water removed and the decrease in surface area is unknown. The purpose of the experiments presented below was to determine the relation between the amount of water removed and the decrease in the surface area of the beet pulp being dried. To make the results comparable in the calculations, the change in weight, or, in other words, the amount of water removed, was taken as a percentage of the initial weight of the beet. The change in surface was expressed as a percentage of the initial surface area of the beet.
The choice of the first quantity was to some extent arbitrary. It would seem that, with greater justification, one could take the percentage decrease in weight not from the initial weight of the beet, but from the water content in the original product. However, as will be shown below, there is a very close relationship between the quantities we have taken, quite satisfactory for our purposes. Thus, the problem was reduced to finding the character of the dependence between the amount of water removed, in percent of the initial weight, plotted on the \(X\)-coordinate, and the decrease in the surface area of beet slicing, in percent of the initial surface area, plotted on the \(Y\)-coordinate.
In studying the character of the change, it would naturally be best to take for the investigation beet in the same form in which it is taken for practical purposes. However, studying the change in the surface of beet in the form of normal pulp presents an extremely difficult problem, since its cross-section has a complex shape, which moreover changes during the drying process. Therefore, for studying the change in surface it is better to take not the pulp as a whole, but its constituent elements, i.e., plates bounded by mutually parallel planes. In essence, beet pulp usually consists of two plates joined at an angle. Thus, it will not be an error to extend the regularities found for the change in the area of plates to pulp representing the joining of two such plates. In connection with this, beet plates were investigated subsequently.
The drying of beet was carried out in one of two drying cabinets, depending on the required velocity of the flowing air. One of them had electric heating and air supplied from below,
and with the exhaust of spent air from above by means of a pipe (Fig. 1); the second (Fig. 2)—with heating of the air in a calorifer and with its delivery by a fan into the upper part of the drying chamber. For the removal of air there is a round opening in the lower bottom of the chamber, with a diameter equal to the diameter of the anemometer inserted for determining the air velocity in the outlet opening. The air temperature was recorded by two thermometers located at a distance of 13.5 cm from one another along the height of the chamber. The tested plates of the beet being dried, in both cabinets, were suspended on a wire in the drying space without touching one another. Such an arrangement guaranteed that the entire surface of the plate took part in giving off moisture. The plates were weighed with an accuracy of 0.01 gram. The area was determined—
Fig. 1.
Fig. 2. Drying apparatus with air blowing.
—as follows: the tested plate was placed on a sheet of millimeter paper and traced with a finely sharpened pencil. By counting the squares inside the contour, the area of the plate was found. It was assumed that the surface area of the whole plate changes proportionally to one of its planes, which, as will be seen below, is close to the truth. By carrying out a series of weighings and determinations of area during the drying process (for which the plates were taken out of the dryer), and plotting the results of the counts as a graph, we obtained a line giving the relationship between the change in surface and the amount of water removed.
It could be assumed that the following factors may influence the character of the change in area:
- The place from which the tested plate is cut (head, tail, ...) from the beet root,
- Direction of the cut (transverse, longitudinal, oblique).
- Thickness of the plate.
- Drying rate.
The experiments described below had as their purpose to answer the question of the character of the influence of the factors noted on the change in the surface of the beet plate being dried. Let us consider the significance of each factor separately.
Factor 1.—Place of the cut. In Fig. 3 are given the drying curves, under identical conditions \((t = 90^\circ \mathrm{C},\ v = 2.5\ \mathrm{m/min})\), of plates of identical thickness, cut in the longitudinal direction from one root. One plate was cut from the upper part of the beet, the other from the tail. The curves show that, if there is any difference between the character of the process of decrease in area of these two cuts, it is very slight, namely: the decrease in area proceeds more intensively for the plate cut from the upper part, where the vascular-fibrous bundles run almost parallel to the axis of the beet. Taking this circumstance into account, in the subsequent experiments plates cut only from the upper part were taken, since precisely this part constitutes the main mass of the beet and therefore is of greatest practical interest.
Fig. 3. Dependence of the change in area on the place of the cut.
With an insignificant divergence of the results of changes for the head and tail parts, the conclusions valid for the first may, without great error, be extended also to the second.
Factor 2.—Direction of the cut. The curves in Figs. 4, 5, 6, and 7 give a representation of the change in the surface area of plates differing only in the direction of the cut, under identical drying conditions. Three directions were taken: transverse, longitudinal (tangential), and oblique at an angle of \(45^\circ\) to the axis of the root. As can be seen from the drawings, a more intensive change of the surface is observed for the longitudinal—
Fig. 4. Dependence of the change in area on the direction of the cut.
directions of cut. This is especially sharply manifested for plates of small thickness (about 1 mm). For plates 3–4 mm thick no substantial difference is observed and, if such a difference is noticeable, then only in the last periods of drying. Therefore one may conclude: the character of the change in the surface of the beet plate being dried, when its thickness is 3–4 mm, depends little on the direction of cut.
Fig. 5. Dependence of the change in area on the direction of cut.
On the basis of the foregoing, in what follows we shall take as the object of study one definite direction, having the greatest practical importance—namely, the longitudinal one. However, it will not be an error to extend the conclusions made for longitudinal cuts to all other directions of cuts.
Factors 3 and 4.—Plate thickness and drying rate. To determine the influence of these factors, experiments were set up with longitudinal cuts from the upper part of the beet. As noted above, precisely these cuts are of greatest interest for practical purposes; at the same time, the conclusions made for them may, without great error, be extended also to all otherwise cut plates.
Two series of experiments, A and B, were carried out under the following conditions:
Fig. 6. Dependence of the change in area on the direction of cut.
• Series A.—\(t = 90^\circ\mathrm{C}\), \(v = 2\) m/min.; drying was carried out in a cabinet with electric heating.
Series A included 3 groups of experiments:
\[
A_{2—2.5}
\]
with plate thickness from 2 to 2.5 mm.
\[
A_{3—3.5}
\]
” ” ” ” 3 to 3.5 ”
\[
A_{4—4.5}
\]
” ” ” ” 4 to 4.5 ”
Series B.—\(t = 105^\circ\)C, \(\nu = 12\) m/min; drying was carried out in a dryer with a calorifer.
Series B included 4 groups of experiments:
\[ \begin{aligned} \mathrm{B}_{2-2.5} &\text{ with slices } 2\text{--}2.5 \text{ mm thick,}\\ \mathrm{B}_{3-3.5} &\quad „ \quad „ \quad 3\text{--}3.5 \quad „\\ \mathrm{B}_{4-4.5} &\quad „ \quad „ \quad 4\text{--}4.5 \quad „\\ \mathrm{B}_{5-5.5} &\quad „ \quad „ \quad 5\text{--}5.5 \quad „ \end{aligned} \]
Each group of experiments included from 10 to 16 experiments with slices from different roots.
It seemed doubtful whether the results of the experiments could be attributed to the conditions under which the investigation was carried out, since in the course of drying the beet slices were repeatedly removed from the drying chamber in order to weigh them and determine their surface area. It may be assumed that this introduced a certain change into the drying process. In order to determine to what extent this circumstance distorts the character of the change in area, the following two experiments were set up (Figs. 8 and 9).
Fig. 7. Dependence of the change in area on the direction of the cut.
In each of the experiments, several slices identical in all respects were dried under the same conditions. At the same time, one of them was dried in the usual manner, being periodically weighed and measured, while the other 3–4 were weighed and measured only once (not counting the determination of weight and area before drying), each of them giving only one point on the diagram.
Fig. 8. Influence on the process of change in area of periodic cooling of the slices.
The distance at which these individual points are located from the curve given by observations on the first slice gives the magnitude of the distortion introduced by periodically removing the first slice from the chamber. As can be seen from Figs. 8 and 9,
these points lie very close to the curve, i.e. brief periodic cooling of the plate during drying has almost no effect on the character of the change in the surface of the plate.
Fig. 9. Effect of periodic cooling of the plates on the process of surface change.
The processing of the results of experiments establishing the influence of plate thickness and drying rate on the character of the change in area was carried out as follows: each individual experiment gave a line of dependence of the change in surface on the amount of water removed: the first quantity—the change in surface—was plotted on the \(Y\)-axis, the second—the amount of water removed—on the \(X\)-axis. Further processing was carried out separately for each group that included experiments with plates of the same thickness, under identical drying conditions. We describe the processing of the experiments of series B with \(d = 2\text{–}2.5\) mm.
Thirteen curves were obtained giving the dependence of \(X\) on \(Y\). Each curve corresponds to one experiment. On the basis of these curves, a curve is constructed of the mean values of the quantity \(Y\) (change in area) corresponding to a definite value of \(X\) (amount of water removed). (Fig. 10). Table 1 gives the values of the mean quantities \(Y\) corresponding to definite values of \(X\), with \(X\) taken at intervals of 5% of evaporated water. The same table gives the values of the standard deviation*) of the quantity \(Y\), which provide a measure
Fig. 10. Graph of the change in area for the experiments of group \(B_{2-2.5}\).
) “The standard deviation of a series of values of some quantity is called the square root of the sum of the products of the squares of the deviations of these values from their arithmetic mean by the relative frequencies corresponding to these values.” Prof. Romanovsky, Elementary Exposition of the Theory of Means, p. 9; see also A. A. Chuprov, Fundamental Problems of the Theory of Correlation*, p. 28.
of the scatter of the values of \(Y\). From Table 1 it is seen that the magnitude of the mean square deviation for all series, except the first and the last, is approximately equal to 3.5% of the initial area.
Table 1.
For the experiments of group \(B_{2-2.5}\).
| % of evaporated water | Mean value \(Y\) | Mean square deviation | Magnitude \(M\), absolute value | Magnitude \(M\), % of mean \(Y\) | Limits of the most probable values of \(Y\) |
|---|---|---|---|---|---|
| 5 | 3.5% | 1.61% | 0.5 | 14.3 | 4—3 |
| 10 | 7.0% | 3.28% | 1.0 | 14.3 | 8—6 |
| 15 | 10.0% | 3.44% | 1.0 | 10.0 | 11—9 |
| 20 | 14.0% | 3.33% | 1.0 | 7.2 | 15—13 |
| 25 | 17.0% | 3.08% | 0.9 | 5.5 | 18—16 |
| 30 | 20.0% | 3.72% | 1.1 | 5.5 | 21—19 |
| 35 | 24.0% | 3.68% | 1.1 | 4.6 | 25—23 |
| 40 | 27.0% | 3.77% | 1.1 | 4.1 | 29—26 |
| 45 | 29.0% | 3.72% | 1.1 | 3.8 | 30—28 |
| 50 | 33.0% | 3.08% | 0.9 | 2.7 | 34—32 |
| 55 | 36.0% | 3.37% | 1.0 | 2.8 | 37—35 |
| 60 | 39.0% | 3.37% | 1.0 | 2.6 | 40—38 |
| 65 | 42.5% | 3.72% | 1.1 | 2.6 | 44—41 |
| 70 | 45.0% | 4.8% | 1.4 | 3.1 | 46—44 |
For the first series these quantities constitute a very large fraction of the mean value of \(Y\), since the value of the latter is subject to large fluctuations. This fluctuation, however, can in no case be attributed solely to the changing qualities of the beet. The main role here is played by observational error. Indeed, in determining the area of the plate we used the method of transferring the outline of the plate onto millimeter paper and counting the squares falling within the outline. In tracing, there is always the possibility of an accidental displacement of the pencil point by as much as 0.25 mm to one side or the other. Therefore the error along the outline when counting squares may fluctuate around \(0.25 \times l\), where \(l\) is the length of the perimeter of the outline.
The value of \(l\), for a plate area of 1500 mm, is approximately 150 mm. Thus, the possible error along the outline is equal to
\[ \frac{150 \times 0.25}{1500}\,100 = 2.5\% \]
of the initial area. In addition to this error, one must also take into account a similar error depending on the thickness of the pencil, amounting to about 1% of the initial area, and random errors in counting the area lying inside the outline. Altogether, the error in determining the area may exceed
2.5–1 = 3.5% of the initial area, and consequently the deviations from the mean value \(Y\) obtained by us do not exceed the error of the experiment. Thus, one may say with confidence that the deviations depend chiefly on errors of computation, and that the limits of fluctuation of the true value of \(Y\) are much smaller than those obtained.
It is highly probable to assume that for normal (not dried out, not spoiled) beets from a single plot of sowing, which was subjected to investigation, the value of \(Y\) corresponding to a definite value of \(X\) is a constant quantity.
Under these conditions it is possible to find the most probable values of the quantities between which the true quantity \(Y\) lies. The magnitude of these quantities is determined on the basis of calculating the mean error of the arithmetic mean, which we shall denote by \(M^*\). Table 1 gives the values of \(M\) for each series of observations in the experiments of group \(B_{2-2.5}\), in absolute magnitudes. The same table gives the values of the mean error of the arithmetic mean as percentages of the corresponding mean value of \(Y\). As is seen from the table, for most series the latter does not exceed 5%.
In the last column of Table 1 are given the limits within which the most probable values of \(Y\)—the amount of decrease in area—lie. These limits will be \(Y+M\) and \(Y-M\). Table 2 gives the magnitudes of the mean error of the arithmetic mean for various groups of experiments, as percentages of the corresponding mean value of \(Y\).
Table 2.
| % of evaporated water | \(B_{2-2.5}\) | \(B_{3-3.5}\) | \(B_{4-4.5}\) | \(B_{5-5.5}\) | \(A_{2-2.5}\) | \(A_{3-3.5}\) | \(A_{4-4.5}\) |
|---|---|---|---|---|---|---|---|
| 5 | 14.3 | 20.0 | 26.6 | 23.4 | 20.0 | 6.8 | 15.0 |
| 10 | 14.3 | 15.0 | 11.0 | 15.4 | 14.3 | 6.7 | 20.0 |
| 15 | 10.0 | 11.4 | 12.0 | 11.1 | 10.0 | 6.1 | 11.4 |
| 20 | 7.2 | 7.0 | 9.6 | 7.0 | 7.0 | 6.3 | 8.5 |
| 25 | 5.5 | 4.7 | 10.0 | 9.4 | 6.1 | 5.1 | 7.1 |
| 30 | 5.5 | 3.7 | 7.2 | 4.3 | 5.4 | 4.6 | 6.5 |
| 35 | 4.6 | 2.7 | 5.3 | 3.5 | 5.4 | 3.6 | 5.3 |
| 40 | 4.1 | 2.3 | 3.3 | 3.3 | 5.5 | 2.9 | 4.4 |
| 45 | 3.8 | 2.8 | 3.2 | 2.6 | 5.0 | 2.5 | 4.0 |
| 50 | 2.7 | 2.7 | 4.7 | 2.1 | 4.1 | 2.2 | 4.1 |
| 55 | 2.8 | 2.2 | 3.3 | 1.9 | 3.6 | 2.4 | 4.5 |
| 60 | 2.6 | 2.3 | 2.9 | 2.0 | 2.4 | 2.5 | 4.1 |
| 65 | 2.6 | 2.6 | 2.2 | 2.6 | 2.2 | 3.1 | 3.7 |
| 70 | 3.1 | 2.0 | — | 2.4 | 2.0 | 2.6 | 4.0 |
*) “The mean square deviation (error) of the arithmetic mean is equal to their mean square deviation divided by the square root of their number.” Prof. Romanovskii, Elements of the Exposition of the Theory of Means, p. 27; see also Avrinskii, Course in the Theory of Probability, p. 84.
In general, the magnitudes of the errors, and consequently also of the limits within which the sought quantity lies, are fairly close for all experiments. The error is the greater, the smaller the value of \(X\). For \(X\) greater than 20, the magnitude of the error in most experiments does not exceed 5%. From Table 2 it may be concluded that the results of the separate groups of experiments are equivalent, or almost equivalent, with respect to accuracy, and that the mean values of \(Y\) obtained in the different groups of experiments may be compared. The lines expressing the desired dependence are very well represented, for all experiments, by the equation of a straight line passing through the origin—\(y = X \times \operatorname{tg} a\).
Table 3 gives the values of \(\operatorname{tg} a\) for the different groups of experiments.
Table 3.
| Series A — 90° C | Series A — 90° C | Series B — 105° C | Series B — 105° C |
|---|---|---|---|
| Thickness | \(\operatorname{tg} a\) | Thickness | \(\operatorname{tg} a\) |
| 2—2,5 mm | 0,56 | 2—2,5 mm | 0,66 |
| 3—3,5 " | 0,57 | 3—3,5 " | 0,64 |
| 4—4,5 " | 0,62 | 4—4,5 " | 0,62 |
| 5—5,5 " | 0,58 |
Table 4 gives, for comparison, the values of \(Y\) in different groups of experiments.
Considering Tables 4, 5, and 6, we arrive at the following conclusions:
- At an air velocity in the dryer of 10 m/min and an air temperature of 105° C, as the thickness of the plate increases, the magnitude of the change in surface decreases.
Table 4.
| % evaporated water | 2—2,5 mm, A | 2—2,5 mm, B | 3—3,5 mm, A | 3—3,5 mm, B | 4—4,5 mm, A | 4—4,5 mm, B |
|---|---|---|---|---|---|---|
| 5 | 1,5 | 3,7 | 2,4 | 2,0 | 2,2 | 1,3 |
| 10 | 3,4 | 6,9 | 4,5 | 4,1 | 4,2 | 4,8 |
| 15 | 6,3 | 10,3 | 7,1 | 7,2 | 7,3 | 7,5 |
| 20 | 8,5 | 13,7 | 10,0 | 9,9 | 10,5 | 11,4 |
| 25 | 10,6 | 17,2 | 13,4 | 15,0 | 13,8 | 15,0 |
| 30 | 14,9 | 20,2 | 16,5 | 18,6 | 17,1 | 17,8 |
| 35 | 18,4 | 23,6 | 19,8 | 22,1 | 20,7 | 21,2 |
| 40 | 22,0 | 26,7 | 23,0 | 25,7 | 24,6 | 24,2 |
| 45 | 26,0 | 29,3 | 26,5 | 29,2 | 29,7 | 27,9 |
| 50 | 29,4 | 33,1 | 30,0 | 32,8 | 32,3 | 31,8 |
| 55 | 33,1 | 36,2 | 33,3 | 36,2 | 36,1 | 34,1 |
| 60 | 36,6 | 39,3 | 36,4 | 39,4 | 39,1 | 38,0 |
| 65 | 40,0 | 42,4 | 39,2 | 42,5 | 42,6 | 42,0 |
| 70 | 43,7 | 44,7 | 41,0 | 44,6 | 45,6 | 45,0 |
-
At an air velocity of 2.5 m/min and a temperature of 90° C, the opposite is observed, although in the latter case to a lesser extent than in the former.
-
When drying plates of the same thickness under different conditions, the following is observed:
a) \(d = 4—4.5\) mm: no significant difference is observed in the nature of the change in surface.
b) \(d = 3—3.5\) mm. At a higher temperature and air velocity, i.e., at a greater intensity of drying, a more considerable change in the surface is observed.
c) \(d = 2—2.5\) mm. The same as in the preceding case, but to an even greater degree.
- The tangent of the angle formed by the straight line expressing the dependence of the change in the surface area of the plate on the amount of water removed fluctuates within the limits \(0.56—0.66\).
For technical reasons it did not prove possible to follow the course of the process of surface reduction at velocities greater than 12 m/min, but, judging from the results of comparing the experiments of series A and B, with a chip thickness above 3 mm, no great difference in the course of the process will be observed.
The results of the experiments performed make it possible, with greater reliability than was done by us in the work “On the Question of Choosing the Type of Dryer for Beetroot” (1), to speak of the surface area of the beet in one or another period of drying, provided that the initial magnitude of the surface of the cossettes being dried is known.
If we denote the initial surface by \(O_0\), then after the evaporation of \(n\%\) of water the area will be equal to:
\[ O_n = O_0\left(1 - \frac{n \times \tg z}{100}\right), \quad \text{since the change in area is equal to } O_0 \frac{n \times \tg z}{100}. \]
Denoting \(\dfrac{\tg \alpha}{100}\) by \(\tg z_0\), we have
\[ O_n = O_0(1 - n \tg z_0)\ldots \tag{4} \]
4. Determination of the pressure \(P_1\) of water vapor at the surface of the material being dried.
To determine the magnitude \(\Delta P = (P_1 - P)\)—the difference between the pressure of water vapor at the surface of the material being dried and in the air—it is necessary to know the magnitude \(P_1\) in each period of drying. Below an attempt is made to solve this question theoretically.
On the basis of data in the literature (IV, p. 11 and VI, p. 26), it is known that the temperature of the material being dried during the evaporation of free moisture is equal to the temperature of a wet-bulb thermometer placed in the drying chamber. Using this proposition, one can find the pressure of water vapor at the surface of a sugar solution, knowing the temperature of the air flowing over it, its humidity, and the concentration of the solution being dried.
Note: In this, the influence of the structure of the beet on the magnitude of the vapor pressure of the solution is not taken into account.
In the case of evaporation of pure water, the pressure of its vapors would correspond to the pressure of saturated water vapor at the temperature of the wet-bulb thermometer. Let us call the temperature of the wet-bulb thermometer \(t_0\), and the saturated-vapor pressure corresponding to this temperature \(P_0\). If it were not water but a sucrose solution that were subjected to evaporation, then, obviously, the pressure of water vapor at the surface of the solution would be lower.
The decrease can be calculated from the equation:
\[ P_1 = P_0 \frac{N}{N+n} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (5) \]
where: \(N\)—the number of molecules of the solvent,
\(n\)—“ ” of the dissolved substance.
Substituting the found value \(P_1\) into the psychrometric equation, one can approximately calculate the temperature of the evaporating layer of the solution.
When drying sugar beet, however, difficulties arise in calculating the pressure \(P_1\) by equation 5, since beet juice contains not only sucrose but also certain amounts of substances of different composition and molecular weight. Moreover, it can only be assumed conditionally that the evaporation of water proceeds uniformly in all cells of the beet; most probably, evaporation of water occurs more rapidly in the peripheral cells than in the deeper ones. Therefore, in order to calculate \(P_1\), it is necessary to make the assumption that concentration of the juice proceeds perfectly uniformly throughout the entire mass of the beet. On the basis of this assumption, let us calculate the change in pressure \(P_1\) for a specific case.
The beet investigated by us had, on average, the following juice composition, expressed as a percentage by weight of the beet.
Dry substances . . . . . . . . 20.8%
Sugar . . . . . . . . . . . . 18.1%
Non-sugars . . . . . . . . . . 2.7%
Let us assume conditionally that the composition of the non-sugars is such that their molecular weight corresponds to \(1/3\) of the molecular weight of sugar. Consequently, in terms of depression, 3 parts by weight of sucrose are equivalent to one part of non-sugars. Under this condition, expressing the dry substances of the juice in sucrose equivalents, we have per 1000 g of juice
\[ \frac{18.1 + 3 \times 2.7}{342} = 0.77 \text{ mol. of dry substances.} \]
According to a number of determinations, the beet studied contained pulp on average \(5\%\), or 50 grams per 1000 grams of beet. Sugar, according to the preceding, was 181 grams; non-sugars, 27 grams. The total amount of dissolved dry substances and pulp is \(27 + 181 + 50 = 258\) g. Water, therefore, is 742 grams, or in moles \(742 : 18 = 41.2\) moles. Let us calculate the value of the coefficient
\[ \frac{N}{N+n}, \]
which we shall denote by \(r\), for different moments of the process of water evaporation. We shall take intervals of 50 grams of water and calculate the pressure at the end of successive periods, during each of which 50 grams of water are removed. Table 5 gives the results of the calculations. Table 5, in addition, gives the average values of the coeffi-
pressure coefficients for the different periods. The beet that we studied was, in composition, of average quality.
Table 5.
| Period | % water removed | $\dfrac{N}{N+n}=r$ | $r$ av. |
|---|---|---|---|
| 1 | 0% | 0.982 | 0.981 |
| 2 | 5% | 0.981 | 0.980 |
| 3 | 10% | 0.980 | 0.979 |
| 4 | 15% | 0.978 | 0.977 |
| 5 | 20% | 0.975 | 0.974 |
| 6 | 25% | 0.973 | 0.972 |
| 7 | 30% | 0.970 | 0.968 |
| 8 | 35% | 0.965 | 0.963 |
| 9 | 40% | 0.960 | 0.958 |
| 10 | 45% | 0.955 | 0.951 |
| 11 | 50% | 0.947 | 0.940 |
| 12 | 55% | 0.932 | 0.922 |
| 13 | 60% | 0.912 | 0.891 |
| 14 | 65% | 0.870 | 0.813 |
| 15 | 70% | 0.756 | 0.661 |
| 75% | 0.565 |
Taking into account the fact that, as calculations for beets of another composition have shown, variations in the latter have little effect on the magnitude of the indicated coefficient, one may take the values of the quantity
$$ \frac{N}{N+n}, $$
calculated above, as constants. These coefficients, when multiplied by the value $P_0$, will give the vapor pressure of the sugar solution in the beet being dried. As is clear from the preceding, we differentiate the process by taking fairly large intervals, but it is very easy to give an equation expressing the vapor pressure of the solution under a continuous change in the concentration of the solution, as a function of the amount of water removed. Let us denote the initial water content in 1000 grams of beet, in moles, by $N_0$, and the dry substances, calculated as sugar, by $n_0$ moles; then, when $n\%$ of water is removed by weight of beet, there is removed
$$ 10\times n \text{ g of water, or in moles } \frac{10n}{18}=0.555\,n \text{ mol.}, $$
whence
$$ P_1=P_0\frac{N_0-0.555\times n}{(N_0-0.55\times n)+n_0}\ldots\ldots\ldots\ldots\ldots\ldots (6) $$
substituting into equation (6) the percentage of water removed, we find, knowing $P_0$, the pressure $P_1$ corresponding to the given moment.
Having determined \(P_1\) and the surface—\(O\) of the beet being dried, as a function of the amount of water removed, we proceed to finding the coefficient \(f\)—the drying-rate coefficient in equation (3).
5. Determination of the drying-rate coefficient—\(f\).
The experimental part of determining the coefficient \(f\) in equation (3) amounted to drying beet plates of a definite weight, thickness, area, and direction of cut under conditions of constant temperature of the air entering from the heater and of its precisely determined velocity. The apparatus in which the drying was carried out was the drying apparatus described above (Fig. 2), with a vertical drying chamber. The temperature of the air entering from the heater during the experiment deviated from the mean value by no more than \(2^\circ\)C and was determined by thermometer \(A\) (see Fig. 11). The vapor pressure of the water in the air was determined with the aid of a psychrometer installed at the level of the suction opening of the blower.
The air velocity was measured with an anemometer at intervals of 20–30 min. The beet plates \(a—a\), weighed to an accuracy of 0.01 gram with a total weight of about 20 grams on the wire \(b—b\), were suspended in the drying chamber (Fig. 11). At intervals of 5–15 minutes, which increased as the drying of the beet proceeded, the plates were removed from the dryer and weighed as quickly as possible. Usually the weighing lasted about 1 min. Thus, if there was any disturbance of the drying regime, it was quite insignificant. The air temperature on both thermometers of the dryer, and also on the thermometers of the psychrometer, was determined at the moments preceding the weighing of the plates.
Fig. 11. Diagram of the drying chamber.
On the basis of the experimental data, curves of the drying rate were constructed. One of them, relating to an experiment with an air temperature of \(105^\circ\)C and a velocity of 8 m/min, is given in Fig. 12. On the drying-rate curve, the dependence is given of the amount of water removed, in percent of the initial weight of the beet, on the time \(z\).
The drying-rate coefficient was calculated in the following way: from the experimental data, the mean values of the quantities entering into equation (3) were determined for successive periods during which 50 g of water evaporates per 1 kg of beet;* these values were substituted into equation (3), which was solved with respect to \(f\).
Equation 3 for the case under consideration takes the form:
\[ f=\frac{50}{\Delta z \times O \times \Delta P \times Vv(1+\alpha t_1)} \cdots\cdots\cdots (7) \]
Let us show successively the method of finding the values of the quantities entering into equation (7). We begin with \(\Delta z\)—the duration of the period under consideration.
* In other words, 5% of the weight of the plates taken.
On the \(Y\)-axis of the drying-rate diagram (Fig. 11) we lay off segments corresponding in magnitude to 5% by weight of the beets. From the points thus obtained we draw lines parallel to the \(X\)-axis. From the points of intersection of the parallels with the drying-rate curve we drop perpendiculars to the \(X\)-axis. The distance between two perpendiculars gives, on the scale of the diagram, the duration of removal of the corresponding 50 grams of water per 1 kg of beets.
Fig. 12. Drying-rate curve.
The surface area of 1 kg of beets at the moment drying began was determined by direct measurement of the plates and recalculation of their area to 1 kg. The area during the successive drying periods was determined from equation (4), with the value of \(\operatorname{tg}\alpha\) being taken, depending on the experimental conditions and the thickness of the plates, from Table 3.
The air velocity \(v\) was determined, as indicated above, with an anemometer.
As for the quantities \(\Delta P = (P_1 - p)\) and \(t_1\), their determination is somewhat more complicated. The quantity \(P_1\), as indicated above, is determined for individual periods by multiplying \(P_0\)—the vapor pressure at the surface of pure water—by the corresponding coefficient, found
from Table 5. Consequently, the problem reduces to determining \(P_0\). The pressure \(P_0\) itself can be found from the psychrometric equation*)
\[ p = P_0 - \alpha (t_1 - t_0) \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (8) \]
where
\[ \begin{aligned} p &\text{— the partial pressure of water vapor in the air,}\\ t_0 &\text{— the temperature of the wet-bulb thermometer, or, what is the same, the temperature of the material during the evaporation of pure water,}\\ P_0 &\text{— the desired pressure of water vapor, equal to the pressure of saturated vapor at temperature } t_0,\\ \alpha &\text{— the psychrometric coefficient multiplied by the barometric pressure corresponding to the moment of the experiment,}\\ t_1 &\text{— the air temperature, or, what is the same, the dry-bulb temperature.} \end{aligned} \]
From equation (8) we have:
\[ P_0 + \alpha t_0 = p + \alpha t_1 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots (9) \]
We find \(p\) and \(t_1\) experimentally, by the method indicated below. Then in equation 9 the unknowns will be \(t_0\)—the temperature of the evaporating layer, or, what is the same, the temperature of the wet-bulb thermometer placed in the dryer, and \(P_0\)—the pressure of saturated water vapor at this temperature. The dependence of the saturated-vapor pressure on temperature is given by the corresponding tables; using the tables, we select such mutually corresponding values of vapor pressure and its temperature that, after substituting these values in equation 9, the left-hand side becomes equal to the right-hand side. These values of vapor pressure and temperature will be the desired \(P_0\) and \(t_0\).
Let us now find \(p\) and \(t_1\).
The partial pressure of water vapor was determined with the aid of a psychrometer placed outside the dryer at the level of the air-intake opening of the blower. Since the moisture content of the air did not change in the calorifer, the partial pressure of water vapor—\(p_0\)—of the air entering the dryer, depending only on the moisture content, will be the same as outside the dryer. However, in the dryer the partial pressure does not remain constant, since the moisture content of the air increases owing to the evaporation of water from the beet, and accordingly the partial pressure of water vapor also increases.
Let us calculate the average increase in the pressure of water vapor in the air working in the dryer. For this purpose let us consider the thermal phenomena occurring in our drying chamber. In Fig. 11 a diagram of the dryer is given. The diagram shows: \(A\) and \(B\), thermometers, \(a\)—\(a\), beet plates suspended on wire \(b\)—\(b\).
The distance between the thermometers is 13.5 cm. The distance from the upper thermometer to the upper edge of the beet plate is 2.5 cm. Let us denote the temperature drop due to heat transfer from the walls of the drying chamber over the distance from the upper thermometer to the lower one by \(\Delta t_1\); then, if we assume that the temperature drop is uniform over the whole length of the dryer, the temperature drop per 1 cm is equal to \(\frac{\Delta t_1}{13.5}\). We are interested in the mean temperature of the drying chamber and the mean partial pressure of the air in it, or otherwise—the mean temperature and partial pressure of the air working in the dryer.
) \(P_0\) can also be found graphically with the aid of Prof. L. K. Ramzin’s diagram (see the article by engineer Lurie, “How to Determine Graphically the Relative Humidity of Air from a Psychrometer and a Condensation Hygrometer by Means of Prof. L. K. Ramzin’s \(J\)—\(d\) Diagram,” No. 6 (29), Izvest. Tepl. I-ta*, 1927).
It may be assumed that this mean temperature and pressure are established at the level of the middle of the slices being dried. Therefore all further investigations will be directed toward finding the temperature and the partial pressure of the air at the level of the middle of the slices.
Let the length of the slices be equal to 1 cm. Then the fall in temperature to its middle is equal to \(\dfrac{\Delta t_1}{13.5}\times\dfrac{1}{2}\); the total fall in \(t\) from the upper thermometer to the middle of the slice is
\[ \frac{\Delta t_1 \times 2.5}{13.5}+\frac{\Delta t_1 \times 1}{13.5 \times 2}, \]
where 2.5 is the distance from the upper thermometer to the edges of the slices.
Usually the slices were taken 6 cm long. At this length we have a fall in temperature to the middle of the slice of approximately \(0.4\Delta t_1\). In the evaporation of moisture from beets the phenomenon is complicated by cooling due to evaporation, owing to which the difference in temperatures indicated by thermometers A and B increases.
Let us denote the fall in air temperature due to evaporation by \(\Delta t_2\). Evaporation of water takes place along the entire length of the slice. We shall assume that evaporation proceeds uniformly along the entire length and that the air, on reaching its middle, loses \(\dfrac{\Delta t_2}{2}\) due to evaporation. One kilogram of air, cooling by \(\dfrac{\Delta t_2}{2}\), gives off heat
\[ \frac{\Delta t_2}{2}\times 0.24 \text{ cal}, \]
where 0.24 is the heat capacity of air. At the expense of this heat there may be evaporated water, assuming that the water has the evaporation temperature,
\[ \Delta d=\frac{\Delta t_2 \times 0.24}{2 \times 0.540}\ \text{grams}. \]
Here it is assumed that the latent heat of evaporation of one gram of water, in large calories, is on average equal to 0.540. According to the equation of Prof. Ramzin (see p. 10), which gives the relation between the moisture content of the air and the partial pressure of water vapor in it, we have
\[ \Delta p=b\frac{\Delta d}{623+\Delta d}, \]
where \(b\) is the barometric pressure. Adding the \(\Delta p\) found in this way to the psychrometrically determined \(p_0\), we find the mean partial pressure of water vapor in the dryer. It is equal to \(p_0+\Delta p=p\).
From the preceding it is not difficult to determine \(t_1\)—the mean temperature of the air in the dryer at each given moment. Obviously it is equal to \(t_0\)—the temperature of the air coming from the calorifier, as indicated by thermometer A—reduced by the amount \(0.4\Delta t_1+0.5\Delta t_2\).
Thus, both sought quantities, \(p\) and \(t_1\), are determined through observations of thermometers A and B.
As is clear from the foregoing, the magnitude of the temperature fall \(\Delta t_1\), which depends on the heat transfer of the chamber walls, will be constant when the thermal state of the system has become steady.
The magnitude \(\Delta t_2\), however, which depends on the evaporation of moisture from the beets, will change, decreasing as water evaporates from the beets and as the rate of drying slows.
The magnitude \(\Delta t_1\) was determined from a preliminary experiment by observation of the dryer before beets were loaded into it, under a steady thermal state.
The quantity \(\Delta t_2\) was determined during the drying of the plates as the difference between the actual drop in temperature over the interval from A to B and the previously found drop in temperature due to heat transfer.
Knowing \(\Delta t_1\) and \(\Delta t_2\), we find, from the preceding, \(p\) and \(t_1\), and, substituting them into equation 9 and solving it as indicated above, with the aid of a table for the pressure of water vapor, we find \(P_0\) for each drying period.
The solution of equation 9, however, shows that the changes in \(p\) and \(t_1\) during the drying process are such that the value of \(P_0\), calculated from equation 9, varies very little in magnitude. Therefore, in order to calculate the vapor pressure of the solution at the surface of the plates being dried, it is sufficient to determine \(P_0\) for the initial moment of drying, taking it as constant throughout the entire drying.
The magnitude of the fluctuations of \(p\) during the drying process is shown in Diagram 12, by a graph on which the values of \(p\) are plotted for the different drying periods. As can be seen, the fluctuations of the partial pressure are insignificant, and the difference between the beginning and the end of drying is only 3 mm of mercury. Using this graph, it is very convenient to find the mean partial pressure during a given period. To do this, it is only necessary to measure the magnitude of the ordinate corresponding to the middle of the period.
Thus, it is possible either by direct measurement or indirectly to determine the mean value of all quantities entering into equation (7) for a number of successive drying periods, during each of which we regard these quantities as constant. Solving equation 7, we find \(f\) for each period.
Two groups of experiments were carried out. The first group had the temperature of the air issuing from the calorifier equal to \(105^\circ\text{C}\); the second group had an air temperature of \(90^\circ\text{C}\). The first group included 7 experiments; the second, 8 experiments. The air velocity in the individual experiments fluctuated around 10 m/min. The thickness of the plates was taken to be about 3 mm.
Tables 6 and 7 give the mean values of \(f\) for 14 successive drying intervals, during which a total of 70% of the water by weight of the beet evaporates.
Table 6.
1st group; \(t_0 = 105^\circ\text{C}\).
| Period | Mean value of \(f\) | Mean quadratic deviation, absolute value | Mean quadratic deviation, % of \(f\) |
|---|---|---|---|
| 1 | 0.151 | 0.021 | 14.0% |
| 2 | 0.153 | 0.021 | 13.7% |
| 3 | 0.144 | 0.021 | 14.6% |
| 4 | 0.136 | 0.023 | 16.9% |
| 5 | 0.133 | 0.012 | 9.1% |
| 6 | 0.132 | 0.023 | 17.5% |
| 7 | 0.128 | 0.012 | 9.4% |
| 8 | 0.132 | 0.011 | 8.35% |
| 9 | 0.121 | 0.021 | 17.4% |
| 10 | 0.114 | 0.013 | 11.4% |
| 11 | 0.107 | 0.019 | 17.7% |
| 12 | 0.094 | 0.021 | 20.0% |
| 13 | 0.081 | 0.013 | 16.0% |
| 14 | 0.056 | 0.024 | 43.0% |
Table 7.
2nd group; \(t_0 = 90^\circ\mathrm{C}\).
| Period | Mean value \(f\) | Mean square deviation, absolute value | Mean square deviation, in % of \(f\) |
|---|---|---|---|
| 1 | 0.137 | 0.027 | 19.7 |
| 2 | 0.140 | 0.014 | 10.0 |
| 3 | 0.138 | 0.015 | 10.9 |
| 4 | 0.137 | 0.015 | 11.0 |
| 5 | 0.137 | 0.016 | 11.7 |
| 6 | 0.136 | 0.015 | 11.0 |
| 7 | 0.133 | 0.016 | 12.2 |
| 8 | 0.130 | 0.016 | 12.8 |
| 9 | 0.127 | 0.019 | 14.9 |
| 10 | 0.124 | 0.026 | 20.1 |
| 11 | 0.121 | 0.025 | 20.6 |
| 12 | 0.105 | 0.014 | 17.7 |
| 13 | 0.090 | 0.011 | 14.6 |
| 14 | 0.062 | 0.013 | 21.4 |
In the last two columns of Tables 6 and 7 are given the values of the mean square deviation of the coefficient \(f\), in absolute numbers and in percentages of the mean value of the coefficient. The fluctuations in the value of the coefficient in our determinations for the majority of the series do not exceed 10—20% of its mean value. The coefficient \(f\) has its most stable values during periods 2 through 9.
The fluctuations increase considerably in the last periods of drying, presumably because formula (4), by which the area of the beet was determined, is not entirely accurate for these periods.
Comparing the results of calculating the coefficient \(f\) for both groups of experiments (Tables 6 and 7), we find that the values of the coefficient \(f\) obtained from both experiments are very close to one another. Moreover, in experiments with a temperature of \(105^\circ\mathrm{C}\), the coefficient \(f\) in most cases has a smaller value than in experiments with a temperature of \(90^\circ\mathrm{C}\). This circumstance suggests simplifying expression (7) by removing from it the term \((1 + at)\); then equation (7) takes the form:
\[ \Delta w = \frac{50}{\Delta z \times \theta \times \Delta P \times ]v}\ldots\ldots\ldots (10) \]
In accordance with this, the coefficients calculated above take the following values, presented in Table 8.
Table 8.
| Period | Coefficient \(f\) at \(t = 90^\circ\mathrm{C}\) | Coefficient \(f\) at \(t = 105^\circ\mathrm{C}\) |
|---|---|---|
| 1 | 0.183 | 0.216 |
| 2 | 0.187 | 0.218 |
| 3 | 0.184 | 0.206 |
| 4 | 0.183 | 0.194 |
| 5 | 0.183 | 0.190 |
| 6 | 0.182 | 0.189 |
| 7 | 0.178 | 0.183 |
| 8 | 0.173 | 0.178 |
| 9 | 0.170 | 0.173 |
| 10 | 0.166 | 0.163 |
| 11 | 0.162 | 0.153 |
| 12 | 0.140 | 0.135 |
| 13 | 0.120 | 0.116 |
| 14 | 0.083 | 0.080 |
Comparing the results of calculating the coefficients by equations 7 and 10 (Tables 6, 7, and 8), we conclude that the discrepancy between the results of the calculations for identical periods of both groups of experiments is equally large both when using formula 7 and when using formula (10). Formula 10, being simpler, has an advantage over formula 7 as more convenient to use.
Comparing the values of the coefficient \(f\) given in Tables 6, 7, and 8, we arrive at the following conclusions regarding the coefficient of the rate of drying of beets:
-
The coefficient of the drying rate in equation 3 is not a constant quantity for different moments of drying.
-
Its value is maximal in the initial stage of drying and minimal in the final stage.
-
The magnitude of the coefficient \(f\), calculated for individual periods of drying, fluctuates for each definite period, independently of the drying conditions, within rather narrow limits.
-
This latter circumstance makes it possible to regard the value of the coefficient \(f\) as a function of the amount of water removed.
-
In equation 3, not only \(O\)—the surface of the beet—and \(P\)—the pressure difference—turn out to be variable, but also \(f\); moreover, all these quantities are functions of the amount of water removed, continuously decreasing in the drying process.
6. Derivation of the drying-rate equation for beet.
As was indicated earlier in Chapter 4, the vapor-pressure coefficients \(r\) for the various periods may, for beets of average composition, be taken as constant and equal to the coefficients given in Table 5. Since, in the calculation of the coefficient \(r\), owing to the uncertainty of the composition of the beet, a convention is introduced by taking the molecular weight of non-sugar to be equal to \(1/3\) of the molecular weight of sugar, there is no point in refining this part of the calculations by computing the vapor-pressure coefficients anew each time. We shall therefore take, conventionally, the vapor-pressure coefficients given in Table 5 to be constant for any healthy beet that has not undergone any spoilage. As was indicated, the drying-rate coefficient \(f\) may also be taken as constant for a given period, and consequently the product of \(f\) by \(r\) will be constant in each period. Table 9 gives the mean value of the coefficient \(f\) in the temperature interval \(90\)—\(105^\circ\mathrm{C}\), found as the mean of its values in Table 8.
In the same Table 9 is given the product of \(r\) by \(f\), denoted by \(D\).
Table 9.
| Period | \(f\) | \(r\) | \(D = f \times r\) |
|---|---|---|---|
| 1 | 0.200 | 0.981 | 0.196 |
| 2 | 0.202 | 0.980 | 0.198 |
| 3 | 0.200 | 0.979 | 0.196 |
| 4 | 0.188 | 0.977 | 0.184 |
| 5 | 0.186 | 0.974 | 0.181 |
| 6 | 0.185 | 0.972 | 0.180 |
| 7 | 0.180 | 0.968 | 0.174 |
| 8 | 0.175 | 0.963 | 0.169 |
| 9 | 0.171 | 0.958 | 0.164 |
| 10 | 0.165 | 0.951 | 0.157 |
| 11 | 0.157 | 0.940 | 0.148 |
| 12 | 0.138 | 0.922 | 0.127 |
| 13 | 0.118 | 0.891 | 0.105 |
| 14 | 0.081 | 0.813 | 0.066 |
The coefficients \(f\) and \(r\) are functions of the amount of water removed; obviously, their product may likewise be represented as a function of the amount of water removed. In Fig. 13 the dependence of \(D\) on the degree of drying is shown graphically, the amount of evaporated water being given in % of the initial weight of the beet, and it is assumed that the value of \(D\) given in Table 9 corresponds to the mean of the corresponding
period. From Figure 13 it is seen that the broken line obtained can be represented by two straight lines forming an obtuse angle.
The first straight line gives the dependence of \(D\) on the amount of water removed within the limits \(0\)—\(52.5\%\), the second—within the limits \(52.5\)—\(70\%\). The equation of the first may be represented in the form:
\[ D = 0.209 - 0.00105\,n \ldots \ldots \ldots \ldots (11'), \]
the equation of the second—
\[ D = 0.448 - 0.0056\,n \ldots \ldots \ldots \ldots (11''), \]
where \(n\) is the number of percent, by weight of beets, of water removed.
Fig. 13. Dependence of coefficient \(D\) on the amount of water removed.
The general form of the dependence of \(D\) on \(n\) will be as follows—
\[ D = b - \tg \varphi\, n \ldots \ldots \ldots \ldots (11) \]
where
\[ \begin{aligned} b_1 &= 0.209 \\ b_2 &= 0.448 \\ \tg \varphi_1 &= 0.00105 \\ \tg \varphi_2 &= 0.0056. \end{aligned} \]
Let us write equation (3), omitting the term \((1 \div at)\), as follows:
\[ \Delta w=(fP-fp)\times O\times V^{v}\times \Delta z \ldots \ldots \ldots (12) \]
According to the preceding, \(P=rP_0\), where \(P_0\) is a constant quantity and \(r\) is a variable coefficient. In addition, \(D=fr\).
Hence \(fP=frP_0\) and \(fP=DP_0\). The product \(f\times p\) may, without great error, be replaced by \(Dp\) (see Table 9), since \(p\) is a small quantity and, when multiplied by it, the error from replacing \(f\) by \(D\) increases only slightly. In that case equation 11 will take the form
\[ \Delta w=D(P_0-p)\times O\times V^{v}\times \Delta z= \]
\[ =D\times \Delta P\times O\times V^{v}\times \Delta z,\quad \text{where } P_0-p=\Delta P \]
Let us express in the last equation the variables \(D\) and \(O\) through \(n\) from equations (11) and (4).
We obtain:
\[ \Delta w=V^{v}\times \Delta z\times P\Delta_0\times (b-\operatorname{tg}\varphi\, n)(O_0-O_0\operatorname{tg}\alpha_0\, n); \]
Denoting \(V^{v}\times \Delta P\) by \(\gamma\), we may write:
\[ \frac{\Delta w}{\Delta z} = \gamma(b-\operatorname{tg}\varphi\, n)(O_0-O_0\operatorname{tg}\alpha_0\, n) \]
Carrying out the multiplications on the right-hand side, reducing like terms, and replacing the coefficients obtained at the various powers of \(n\) by \(A\), \(B\), and \(C\), we obtain
\[ \frac{\Delta w}{\Delta z}=An^2-Bn+C \ldots \ldots \ldots \ldots \ldots \ldots (13), \]
where
\[ A=\gamma\times O_0\times \operatorname{tg}\alpha_0\times \operatorname{tg}\varphi \]
\[ B=\gamma\times O_0\times (\operatorname{tg}\varphi+\operatorname{tg}\alpha_0\, b) \]
\[ C=b\times O_0\times \gamma \]
Equation 13 makes it possible to determine \(\Delta z\)
\[ \Delta z=\frac{\Delta w}{An^2-Bn+C}\ldots \ldots \ldots \ldots \ldots \ldots (14). \]
Equations 13 and 14 will be the more exact, the smaller the quantity \(\Delta w\) taken. This equation will be perfectly exact (within the limits of the established accuracy of the experimental part and of the assumptions made) for an infinitely small value of \(\Delta w\). However, although equation 14, after replacing \(\Delta w\) by \(\Delta n\), assumes, for infinitely small \(\Delta n\) and \(\Delta z\), a form very convenient for integration, the expression obtained by integration is extremely complicated and is unsuitable for technical purposes. Therefore we leave the equation for the drying rate in the form in which it is given in expression 14, and shall consider the durations \(\Delta z\) of periods, during each of which 50 grams per 1 kg of beet are evaporated. In this case equation 14 assumes the following form: for the first period \((0—52.5\%)\)
\[ \Delta z=\frac{50}{A_1n^2-B_1n+C_1}\ldots \ldots \ldots \ldots \ldots (14'), \]
for the second period (52.5—70%)
\[ \Delta z=\frac{50}{A_2 n^2-B_2 n+C_2}\ldots\ldots\ldots\ldots\ldots\ldots (14'') \]
Substituting into these equations the mean \(n\) for the period under consideration, we shall find the duration of the period. The solution of the problem of finding the theoretical curve of the drying rate is thus reduced to the following:
1) Finding the coefficients \(A_1, B_1, C_1\) and \(A_2, B_2,\) and \(C_2\).
2) Calculating the time \(\Delta z\) from equation 14 by substituting into it these coefficients and the mean value of \(n\) for the period under consideration.
For example, let us solve the following problem: the drying conditions are given
\[ \frac{l}{v}=2.7\ \text{m t. per min.} \]
\[ (P_0-p)=42\ \text{mm.} \]
\[ \theta_0=0.81\ \text{sq. m per 1 kg.} \]
\[ d=3\ \text{mm.} \]
\[ t_0=105^\circ C. \]
From the values of \(d\) and \(t_1\) we find the values of \(\operatorname{tg}\alpha_0\) from Table 3: \(\operatorname{tg}\alpha_0=0.0064\). From the preceding,
\[ \operatorname{tg}\varphi_1=0.00105;\quad b_1=0.209 \]
\[ \operatorname{tg}\varphi_2=0.0056;\quad b_2=0.448. \]
\[ \lambda=2.7\times42=113.4 \]
On the basis of these data we find the values of \(A_1, B_1, C_1\) and \(A_2, B_2, C_2\)
\[ \begin{aligned} A_1&=0.00062 & A_2&=0.0036\\ B_1&=0.22 & B_2&=0.78\\ C_1&=19.2 & C_2&=41.2 \end{aligned} \]
We compose equations of the type \(14'\) and \(14''\).
\[ 1)=\frac{50}{0.00062\,n^2-0.22\,n+19.2} \]
\[ 2)=\frac{50}{0.0036\,n^2-0.78\,n+41.2} \]
Using these equations, we shall calculate the time intervals \(\Delta z\) and compare them with \(\Delta z\) obtained empirically under analogous conditions.
The results of the calculations, as well as the \(\Delta z\) obtained empirically for comparison, are given in Table 10.
Table 10.
| Period | \(\Delta z\) by calculation | \(\Delta z\) from experiment | Period | \(\Delta z\) by calculation | \(\Delta z\) from experiment |
|---|---|---|---|---|---|
| 1 | 2.68 min. | 2.5 min. | 9 | 4.6 min. | 3.6 min. |
| 2 | 2.84 “ | 2.5 “ | 10 | 4.93 “ | 4.35 “ |
| 3 | 3.02 “ | 2.6 “ | 11 | 5.38 “ | 4.7 “ |
| 4 | 3.22 “ | 3.16 “ | 12 | 6.07 “ | 6.0 “ |
| 5 | 3.44 “ | 3.34 “ | 13 | 7.0 “ | 8.35 “ |
| 6 | 3.67 “ | 3.54 “ | 14 | 10 “ | 12.7 “ |
| 7 | 3.94 “ | 3.34 “ | |||
| 8 | 4.24 “ | 3.6 “ | Sum. | 65.81 min. | 64.28 min. |
Table 10 shows that the difference between the empirically obtained values of \(\Delta z\) and the \(\Delta z\) found from equation 14 is not large. For most periods it does not exceed 5–10% of the determined value.
The obtained equation (14) makes it possible to solve, quite simply, the question of the drying rate in a multi-chamber dryer with different drying conditions in the individual chambers.
To solve the problem, in this case, for each of the drying zones differing in conditions, it is necessary to find the values of the coefficients \(A\), \(B\), and \(C\) and to compose equations of type (14), which presents no great difficulty.
Thus, the drying-rate curve may, by this method—in contrast to the method proposed earlier by the authors in the article “On the Question of Choosing the Type of Dryer for Sugar-Beet Cossettes”—be obtained for any drying process differentiated according to conditions.
10 February 1929
Tomsk.
Bibliography.
I. Prof. Lebedev and Eng. Stabnikov. On the question of choosing the type of dryer for beet cossettes and the basic methods of calculation. Proceedings of the Siberian Technological Institute, 1928.
II. Roitman and Makovetskii. Drying with air and flue gases.
III. Prof. M. A. Deshevoi. Calculation of drying installations for wood. Publ. Sh. 1928.
IV. Eng. M. Yu. Lur’e. Dryer for printing ink. Proceedings of the Heat-Engineering Institute, No. 10 (33), 1927.
V. Prof. L. K. Ramzin. Calculation of dryers and the \(J\)--\(d\) diagram. Proceedings of the Heat-Engineering Institute, No. 1 (24), 1927.
VI. G. P. Ivantsov. Calculation of natural circulation in multiple-saturation dryers. 1928. Transactions of the State Bureau of Metallurgical and Heat-Engineering Structures.