Prof. S. V. Lebedev and Eng. V. N. Stabnikov

On the Question of the Choice of the Type of Dryer for Beet Cossettes

—and—

The Basic Principles of Calculation

Tomsk

1928

Prof. S. V. Lebedev and Eng. V. N. Stabnikov.

On the Choice of the Type of Dryer for Beet Cossettes and the Basic Principles of Calculation

I. Conditions for Drying Sugar Beets.

The problem of drying beets in recent years has been subjected to a rather lively discussion in our specialized periodical press. A considerable part of the works that have appeared concerns the clarification of the question of sugar losses during drying and the finding of such drying conditions under which sugar losses would be minimal. Alongside original works there is a whole series of abstracts of works and official reports concerning, chiefly, the operation of Italian factories processing dried beets by the de Vecchi method.

These works allow the question of the most favorable conditions for drying sugar beets to be considered, to a significant degree, resolved.

Researchers who worked on the question of drying beets before de Vecchi had no success, mainly because the temperatures they used were limited to 50–60°C. De Vecchi gave a new impetus to the advancement of the problem of drying beets in the direction of its technical realization. He found that, if the drying conditions he had established were observed, then, when temperatures above 100°C were used, no sugar losses were observed either as a result of invert formation or as a result of caramelization. Researchers who examined the operation according to the de Vecchi method directly at the Italian sugar factory in Loreo came (11) to the following conclusions:

1. No formation of invert sugar or loss of sucrose is observed during the entire process of drying beet cossettes.

2. The yield of crystalline sugar is quite normal. Other researchers (Prof. Mezzadroli, Dr. Frigerio) showed in their works that the indicated conclusions are correct.

In the report (11) of the English Ministry of Agriculture on the results of the operation, by the de Vecchi method, of the experimental installation at Eynsham, the authors of the report come to the conclusion: “when heated to 110°C, no formation of invert sugar or signs of caramelization were detected in the dried beet; it was found that the drying rate had to be increased very substantially so that the moist material would not be subjected to more or less significant heating. When the drying process lasted up to one hour or less, significantly better results were obtained.” In the same report the danger of raising the drying temperature above 110°C is noted. Laboratory work by Eng. Benin on the question of drying beets (10) established the following with respect to the temperature and duration of drying:

1. “All other conditions being equal, the loss of sugar during drying is the greater, the longer the drying process.”

1. “Drying can be carried out at temperatures above 100°C, but in this case it is necessary to monitor carefully the condition of the cossettes in the drying apparatus.”

Thus, on the basis of the foregoing, it may be accepted that the question of using high temperatures (above 100°C), under appropriate drying conditions, is resolved in the affirmative.

The drying process in the operating dryer at Loreo, as has been noted more than once in print, (6, 8, 11, 12) is essentially as follows. The dryer is a rectangular chamber, with external dimensions of 12.2 m in length, 3 m in width, and 7 m in height. The drying chamber is divided vertically by an air duct into two parts. In each part there are three horizontal belt conveyors made of a material that readily allows air to pass through. The fresh cossettes enter the upper conveyor, which they traverse in 20 minutes. The next five conveyors are traversed by the cossettes at half that speed, i.e., in 40 minutes each.

The cossettes are removed at 90°C from the lower part of the dryer. Heating and drying of the cossettes are carried out with air at a temperature from 90° to 130°C. The heating sequence is as follows: air heated to 90°C, in an amount of 142 cubic meters per minute, enters the lower part of the dryer and moves from bottom to top, counter to the cossettes. The air then enters the lower air chamber, whence it is drawn off by a fan. To this air a further 368 cubic meters of air from outside are added, and all this air, heated to 130°C, is blown into the upper part of the dryer (6). The air then descends, following the movement of the cossettes, and enters the upper air chamber, whence it leaves to the outside at 110°C. Heating of the cossettes and removal from them of 75% of the water take place in the upper part of the dryer, while in its lower part final drying proceeds, with only 2.5% of water, by weight of raw beets, being removed. The special furnaces in which the air was heated were fired with oil, the consumption of which amounted to 8.54% by weight of beets. The commission whose report (12) supplied these data notes that the dryer in question was not designed altogether successfully. The authors of the report express the hope that fuel consumption will be reduced when the drying method is rationalized.

II. Prof. Ramzin’s Diagram.

The question of the most economical variant of the drying process is of considerable interest. Regardless of the construction of the dryer, provided that the latter is technically sound, the application of one or another principle of heat utilization may ensure a greater or lesser degree of economy in drying.

A theoretical analysis of various variants of the drying process was given by Prof. Ramzin (1) in the article “Calculation of Dryers and the J-d Diagram.” In that work three variants of a normal drying process were examined. The same work also describes the J-d diagram proposed by Prof. Ramzin, which considerably simplifies calculations connected with dryers operating with heated air. In addition, the J-d diagram makes it possible to represent the drying process very clearly. Assuming that the work mentioned is still little known among broad circles, brief information on the J-d diagram is given below, within the limits needed for the subsequent exposition. On Prof. Ramzin’s diagram

as coordinate axes are taken: the moisture content \(d\), in grams per kilogram of dry air (horizontal axis), and the heat content \(J\) of moist air, in calories per 1 kg of dry air (vertical). The calculations were made for a barometric pressure of 745 mm of mercury. This pressure was taken as close to the mean annual pressure of the Central Industrial Region of the U.S.S.R.

The angle between the lines \(J=\mathrm{const.}\) and \(d=\mathrm{const.}\) is taken equal to \(135^\circ\) (Drawing 1). In these coordinates are constructed isotherms, representing lines close to straight lines, and curves of constant degree of saturation. The curve at the bottom of the diagram gives the partial pressure of water vapor in the mixture, as a function of the moisture content. The method of using the diagram is explained by the example given below.

Vertical axis: Heat content \(J\), cal. per 1 kg of dry air.
Horizontal axis: Moisture content \(d\), g per 1 kg of dry air.
Label: \(J\)-\(d\) diagram, Prof. L. K. Ramzin.
Drawing 1.

Example: Find the consumption of air and heat for the evaporation of one kilogram of water, if the air is heated in the heater to \(95.0^\circ\mathrm{C}\), its initial temperature is \(20^\circ\mathrm{C}\) and humidity \(60\%\), and the final temperature is \(35^\circ\mathrm{C}\). Let us assume that drying proceeds without useless losses, which are denoted by \(\Delta\), i.e., let us assume that \(\Delta=0\). On the diagram we find the point corresponding to the state of the outside air. This will be point A, where the lines \(\varphi=60\%\) and the isotherm for \(20^\circ\mathrm{C}\) intersect. When this air is heated in the heater, before entering the drying chamber, its moisture content \(d\) (which we find by dropping a perpendicular from point A to the horizontal axis) does not change. Only the temperature changes. Consequently, the heating process will be represented by the line AB, parallel to the vertical axis.

The state of the air at the outlet from the heater will be represented on the diagram by point B, the intersection of line AB with the isotherm for \(95.0^\circ\mathrm{C}\). The heated air enters the dryer. For a dryer in which \(\Delta=0\), the amount of heat in the air remains constant throughout the entire drying time. Only a redistribution of heat occurs. The heat content of the air coming from the heater decreases, and as a result water evaporates. The total heat content of the mixture remains constant and equal to the initial heat content of the air coming from the heater. On the diagram this process of evaporation of moisture at the expense of cooling of the air coming from the heater will be represented by the line

BC, for which \(J = \mathrm{Const}\). The final point of the process will be represented on the diagram by the intersection of the line BC and the isotherm for \(35^\circ\mathrm{C}\). We find the moisture content of one kilogram of dry air by drawing from point C a perpendicular to the horizontal axis. The relative humidity corresponds, as is seen from the diagram, to \(90\%\). The final point of the process can also be obtained in the case where the relative humidity of the outgoing air were specified, while its temperature were unknown. Then point C would be found at the intersection of the line BC with the curve corresponding to the specified relative-humidity condition. Thus, the process of heating and drying is represented by the line ABC. Let us draw the auxiliary line CD, which will be the perpendicular from point C to the line AB.

The air consumption for the evaporation of one kilogram (1000 g) of water, denoted by \(l_0\), is

\[ l_0 = \frac{1000}{d_2 - d_0}, \]

where \(d_2\) is the moisture content of the air (the amount of water in grams per 1 kg of dry air) at the outlet from the dryer, and \(d_0\) is that at the outlet from the calorifier. On the diagram this difference is represented, on a certain scale, by the line DC. Let us denote the scale of moisture contents for the diagram of the given size by \(M_d\). Then the air consumption is equal to

\[ \frac{1000}{\mathrm{DC} \times M_d}, \]

where DC is measured in millimeters. The heat consumption for drying without losses, i.e. for \(\Delta = 0\), is found from the equation \(q_0 = l_0 (J_1 - J_0)\), where \(J_1\) is the heat content of the air per 1 kg of dry air at the outlet from the calorifier, and \(J_0\) is the heat content of the air entering the calorifier. Obviously, on the diagram the difference \((J_1 - J_0)\) corresponds to the line AB. Measuring it in mm and multiplying by the heat-content scale, we obtain \((J_1 - J_0)\) in calories. Consequently, \(q_0 = l_0(\mathrm{AB} \times M_i)\), where \(M_i\) is the heat-content scale. Hence

\[ q_0 = \frac{1000}{\mathrm{DC} \times M_d} \times \mathrm{AB} \times M_i = \frac{1000 M_i}{M_d} \times \frac{\mathrm{AB}}{\mathrm{DC}} \quad \text{or} \quad q_0 = m \frac{\mathrm{AB}}{\mathrm{DC}}, \]

where

\[ m = \frac{1000 M_i}{M_d}. \]

Using these relations, it is very simple to find the consumption of air and heat per 1 kg of water evaporated in the dryer.

The calculation of a dryer where \(\Delta\) cannot be taken equal to zero becomes somewhat more complicated. Let us agree that \(\Delta\) is the amount of heat lost or imparted to the steam-air mixture in the dryer, referred to 1 kg of moisture evaporated in the dryer. Let us solve the example considered above, complicating its conditions by assuming that \(\Delta > 0\), i.e. that in the dryer a certain amount of heat is imparted to the steam-air mixture for each kg of moisture evaporated. Obviously, the ray (Fig. 1) on which the final point of the drying process will lie will, under the indicated condition, occupy another position; moreover, since the heat content of the mixture will now be higher than in the first case, when \(\Delta = 0\), the new ray will pass above the ray BC. The rule for finding, according to Ramzin, the position of the new ray \(BC_1\) is as follows: (see Fig. 1) “On the ray BC, going from point B parallel to the line \(J = \mathrm{const}\), we choose an arbitrary point \(e\); from \(e\) we draw \(ef\) perpendicular to the horizontal axis of ordinates and lay off upward the segment \(eE\), equal to \(ek \frac{\Delta}{n}\) mm. [(1), p. 26], where \(ek\) is the length of the perpendicular from \(e\) to AB in mm, and \(n\) is a coefficient depending on the scale of the diagram.” “By joining the points B and E, we obtain the desired ray, on which lies the required point C of the end of the process.” If ...

\(\Delta\) were less than zero, i.e., if the losses in the dryer exceeded the additionally introduced heat, the construction of the new ray would be the same, with only the difference that in this case the segment is laid off not upward but downward and, consequently, the new ray will take a position below ray \(BC\) (Fig. 1).

Let us note one more property of the diagram: the heat consumption for drying without losses \((\Delta = 0)\) depends on the angle of inclination of line \(AC\), drawn through the initial and final points of the process, to the horizontal axis. The heat consumption increases with an increase in this angle. The angle of inclination of line \(AC\), in turn, is determined by the position of points \(A\) and \(C\).

Consequently, the economy of the drying process is determined by the mutual arrangement of the final and initial points of the process. Processes having a common initial and final point are equally economical.

Restricting ourselves to these brief data on the \(J\)-\(d\) diagram, in what follows, using it, we shall consider various variants of the drying process as applied to those conditions which are imposed by the properties of beet cossettes.

III. Variants of the Drying Process.

Normal process. By a normal drying process Prof. Ramzin (1) means a process in which the air is heated once and works in the dryer without additional heat being supplied to it.

For the normal drying process Prof. Ramzin (1) established the following dependences of the consumption of air and heat on the temperature to which the air is heated in the calorifer.

1. With an increase in the temperature—\(t_1\)—of the air entering the dryer, its consumption falls.

2. The law of change of heat consumption as a function of \(t_1\) is determined by the ratio of the values of the relative humidity of the air entering the calorifer \(\varphi_0\) and the humidity of the air leaving the dryer \(\varphi_2\). Under ordinary conditions the beets will be dried in autumn, when \(\varphi_0\) has a large value. Therefore it may be assumed that \(\varphi_0 > \varphi_2\). For this same case the heat consumption grows continuously as the temperature is lowered. The dependence of the consumption of heat and air on the magnitude of the humidity of the exhaust air is formulated by Prof. Ramzin (1) as follows: “With a decrease in \(\varphi_2\), the consumption of air \(l_0\) and heat \(q_0\) increases rapidly.” Hence it is clear that, in order to carry out under such conditions \((\varphi_0 > \varphi_2)\) the most economical normal drying process, it is necessary to take the maximum temperature permissible for the given material and the maximum humidity of the exhaust air. The maximum temperature permissible in drying beets, as indicated above, may be taken as \(130^\circ\text{C}\).* As for the maximum relative humidity, it may be taken equal to \(70\%\), since at a higher value of it, with a local drop in temperature in some corner of the dryer, “steaming” of the product is possible (2). Under these conditions the consumption of air and heat per 1 kg of evaporated moisture is

\[ l_0 = 29 \text{ kg}, \]

\[ q_0 = 900 \text{ cal.}, \]

and for 1 kg of beets, provided that 75% of water by weight of these beets is evaporated from them:

\[ l_1 = 21.8 \text{ kg}. \]

\[ q_1 = 675 \text{ cal.} \]

Thus, the limiting economy of the normal drying process for a beet dryer corresponds to a heat consumption per 1 kg of beets of 675 cal. for a dryer in which \(\Delta = 0\).

* In the last stages of drying the temperature should be lowered, but for simplicity we shall assume that \(t_1 = \text{const.}\) and equal to \(130^\circ\).

By raising the temperature of the air heating (if the material being dried permitted this), we could obtain a more economical process.

DEPENDENCE
of the heat consumption
for evaporating 1 kg of water on the value of \(t_1\)

Drawing 2

AIR CONSUMPTION
kg per 1 kg of evaporated water as a function of \(t_1\).

Drawing 3

In the diagrams of Drawings 2 and 3 are given the changes in heat consumption and air consumption per 1 kg of water evaporated in the dryer without losses, depending on the temperature of the air entering the dryer. The relative humidity of the air leaving the dryer was taken in the calculation as equal to 70%, and that entering the calorifier as 90%. The temperature

air entering the calorifier is equal to \(0^\circ\)C. From the diagrams it is evident that, as the temperature rises, the quantities of evaporated water, heat, and air expended per 1 kg decrease at first very intensively. With the transition into the region of temperatures above \(450^\circ\)C, the rate of change in the consumption of heat and air slows down; in its further development the curve approaches, in character, a straight line slightly inclined to the horizon. Raising the temperature beyond \(450—600^\circ\)C, even considerably, no longer produces a large effect with respect to reducing the consumption of heat and air. However, these temperatures are also inadmissible in drying sugar beets, owing to their destructive action on the sugar. Thus, the problem of economical drying of beets is reduced to this: without raising the temperature beyond the limits permissible for beets, i.e. above \(130^\circ\)C, to carry out the drying process at least as economically as a process with a temperature equal to \(450^\circ\)C. Obviously, on the basis of the preceding, such a process, irrespective of the character of its course, must have the same initial and final points as in the process with a temperature of \(450^\circ\)C. Modern drying technology has at its disposal several methods for increasing the economy of the drying process while maintaining low drying temperatures.

The following variants of the normal drying process belong here:

1. A dryer with continuous heating of the air in the drying chamber.
2. A dryer with periodic heating of the air either in the drying chamber or outside it.
3. A dryer with return of the spent air.

Let us consider all three variants in succession.

Variant 1.—Let us suppose that we are assigned the task of carrying out, by various means, without raising \(t_1\) above \(130^\circ\), a process equal in economy to the process with \(t_1 = 450^\circ\)C. This latter is represented by the line ABC (Fig. 11). Its final point is C, its initial point A. We saw above that, by introducing additional heat into the dryer, one can change the direction of the ray BC (Fig. 1), and the new ray \(BC_1\) will then occupy a position above BC. Proceeding from this, in order to achieve the stated aim, one may proceed as follows: heat the air in the calorifier to \(130^\circ\)C and admit it into the drying chamber. The process will be represented by the line \(AB_1C_1\) (Fig. 11). By introducing additional heat into the dryer, we change the direction of the ray \(B_1C_1\). In doing so, with a successive increase in the heat introduced, the ray \(B_1C_1\) rotates about the point \(B_1\) in the counterclockwise direction. At a certain definite quantity of heat supplied to the dryer, the ray \(B_1C_1\) will coincide with the ray \(B_1C\), i.e. the process will have common initial and final points with the process for which \(t_1 = 450^\circ\)C. This variant of the drying process is represented on the diagram by the line \(AB_1C\). As is evident, it proceeds at temperatures not above \(130^\circ\)C.

Variant 2.—We could arrive at the same point C (Fig. 11) by conducting the process as shown by the line \(AUFSRPONML \ldots C\), i.e. by heating the air in the first calorifier to a temperature not exceeding \(130^\circ\)C and admitting it into the working space of the dryer. The air cooled in it, owing to the evaporation of water, is again heated and again admitted into the working space. The air consumption, identical for all stages, as before, is equal to

\[ l_0 = \frac{Md}{DC}\ \text{kg}. \]

The heat consumption in the individual calorifiers is

\[ \frac{Au}{DC}m;\ \frac{FS}{DC}m;\ \frac{PR}{DC}m\ \text{etc.},\quad \text{where } m = \frac{Md}{Mi}. \]

Intermediate heating of the air may be carried out either outside the dryer, in special heaters, or within the dryer itself, by suitably arranged heating surfaces, as has been done in the dryers with multiple air circulation proposed by Prof. Grum-Grzhimailo (2,3). As is evident from the diagrams, with the same slope of the line \(AC\), as also for the process with a temperature of \(450^\circ C\), and consequently with the same expenditure of heat, the maximum temperature of the process is \(130^\circ C\). We can carry out the process in the most varied ways, varying the temperature of the intermediate heatings and the degree of saturation of the air in the various chambers of the dryer. If we assume that the temperature of the air admitted after heating into the working space is always equal to \(t.\max.\), and that the temperature of the air entering each intermediate heater is equal to \(t.\min.\), then, for determining the number of intermediate heatings, the formula of Prof. Grum-Grzhimailo (2,3) is applicable:

\[ Q=\frac{a}{b}\left[\left(1+\frac{b}{\lambda \min}\right)^n-1\right]\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (1) \]

where
\(\quad n\) — number of heatings,
\(\quad a\) — \((t.\max.-t.\min.)\times 0.2375\) cal.,
\(\quad\) — \(0.2375\), the heat capacity of air,
\(\quad b=(\lambda \max-\lambda \min)\)
\(\quad \lambda \max.\) — total heat content of the vapor at \(t.\max.\)
\(\quad \lambda \min.\) — total heat content of the vapor at \(t.\min.\)
\(\quad Q\) — content of water vapor in 1 kg of spent air at the adopted value of the relative humidity and the temperature \((t.\min.)\) of the outgoing air.

Equation (1), after taking logarithms and the corresponding transformations, assumes the form:

\[ n=\frac{\lg(1+1.284\,Q)}{\lg\frac{\lambda \max}{\lambda \min}}\ .\ .\ .\ .\ .\ .\ .\ .\ .\ (2) \]

Let us specify the temperatures \(t\max.\) and \(t\min.\), and find the corresponding \(\lambda \max.\) and \(\lambda \min.\). Substituting them in equation (2), we easily find the number of heatings necessary to attain a water content of \(Q\) kg in 1 kg of air leaving the dryer.

Variant 3.—A dryer with return of the spent air. In this case part of the spent air is discharged into the atmosphere, while the other part is mixed with fresh air and, after being heated, is introduced again into the dryer. Analyzing the various cases represented by this variant, Prof. Ramzin (1) comes to the conclusion: “The return of the latter (air) in most cases gives an insignificant saving, for a large saving is obtained only at an unusually low relative humidity of the air leaving the dryer.” However, the return of spent air has found wide application in drying technology in those cases where the air admitted into the dryer must be humidified. Since the experience of drying beet cossettes has shown that drying them proceeds well without humidifying the air, this variant of the drying process will not be considered by us.

IV. Survey of dryers.

The dryer described above at Loret operates by carrying out a normal drying process almost in its pure form.

Indeed, the main mass of air—368 cubic meters, supplied per minute—is heated only once. The remaining 142 cubic meters are heated twice; that is, only one third of the air operates in the pro-

process with one intermediate heating. Thus, in the dryer installed in Loreo, not a single one of the possibilities is being used to the proper extent which would make it possible to increase the economy of the drying process without raising the temperature of the air heating above the limit permissible for drying sugar beets. On the basis of the preceding, it is quite clear in what direction design investigations must be carried out in order to create an economical drying process.

For these purposes it is necessary to use, for drying beet cossettes, dryers with air reheating.

In this work we shall not consider dryers in which the air is heated continuously. We shall dwell on dryers with periodic, repeated heating of the air; moreover, from the entire variety of dryer designs we shall consider only two types that have become widespread in industrial drying on a large scale of such agricultural products as vegetables and starch, namely:

1. corridor or channel dryers,
2. belt dryers.

Both of the indicated types permit the use of periodic (intermediate) air reheating. Below we shall consider the design schemes of such dryers.

Belt dryers. We have not encountered belt dryers with intermediate heating, except for the very primitive dryers in this respect in Loreo. The operating scheme of a dryer built on the indicated principle is presented in Drawing 4.

The air, heated in the calorifer, travels along the belt in the direction of its motion or against it; at the point where the material being dried passes from one belt to another, the air must be separated and directed into the heating chamber.

The heating devices are placed outside the drying chambers, as shown in the scheme. But they may also be arranged as is done in dryers with multiple air circulation by Prof. Grum-Grzhimailo (see below: corridor dryers).

The capacity of a belt dryer may vary, depending on the dimensions of the belts and on the loading of the drying surface, i.e., the output of a unit area per unit time. The dryer in Loret had a capacity of 12.5 centners per hour with a drying time of 3 h. 40 min.

Diagram of a corridor dryer with intermediate heating

Fig. 5

Corridor dryers.—In engineer Lur’e’s report, “A Survey of Modern Dryers” (Proceedings of the Second Thermal-Engineering Congress, 1928), several structural types of corridor dryers with intermediate heating in special heaters are given. A diagram of this type is presented in Drawing 5.

The air passes successively through a heater, a chamber, then again a heater and a chamber, etc.; as may be seen, intermediate heating in dryers of this kind is carried out very simply.

A special place is occupied by the corridor-type dryer with intermediate heating, with the heating surface placed in the drying chambers themselves. Drawing 6 shows a transverse section of such a dryer (2).

Diagram of a corridor dryer of Prof. Grum-Grzhimailo

Such a dryer was proposed by Prof. Grum-Grzhimailo in accordance with his theoretical investigations in this field. Heating surfaces in the form of tubes are installed vertically along the walls of the dryer, on the sides of the train of cars. The air cooled in the dryer descends to the dryer floor, is heated near the tubes, and rises upward owing to the difference in weight between warm and cold air. Descending again, passing through the grating floors and the material being dried, it is heated again, and so on.

The capacity of a channel dryer may be very varied; however, it is possible approximately, proceeding from the structural

considerations and from practical data, determine the average value of their productivity. The principal element of the drying surface in a channel dryer is a sieve; a series of sieves is placed one above another on a trolley moving along the channel. The dimensions of the sieve for various dryers are given in Table 1 (4,5).

Table 1.

On the basis of the data of Table 1, the area of the sieve surface may be taken as equal to 2 sq. m. This value will correspond to the maximum of the cited surfaces of a single sieve.

If it is assumed that each trolley carries 10 sieves, then under these conditions one trolley will contain 20 sq. m. of surface on which the material to be dried is placed. Let us take the number of trolleys as 10. Then the total drying surface is \(20 \times 10 = 200\) sq. m. If the drying time is 1 h 30 min. (the time the cossettes remain in the upper chamber of the Loreau dryer), then the drying surface will be used during a day \(24 : 1\tfrac{2}{3} = 14.4\) times. Let us take the load per 1 sq. m. of surface to be the same as for the Loreau dryer, i.e. 14.5 kg of raw beet. Then the daily productivity of a dryer of the indicated dimensions will be \(14.4 \times 200 \times 14.5 \simeq 42000\) kg, or 420 mt. centners, which corresponds to 17.5 mt. centners per hour, calculated as fresh beet. The corridor dryer, as we saw above, makes it possible, very simply from the constructional point of view, to apply the principle of intermediate heating; therefore it, on a par with the belt dryer, deserves full attention.

The dryers considered above, in terms of their productivity, are suitable for the industrial drying of beet cossettes. Both, with the proper design and the application of the above-mentioned principle of reheating, can give a cheap and good-quality product. Lacking operating figures, it is difficult to say which type should be given preference. However, proceeding from considerations of simplicity of construction, it may be noted that corridor dryers have a certain advantage.

Below we shall dwell on the method of calculating a dryer of either type with reheating outside the drying chamber in special calorifers.

The scheme of a dryer with intermediate heating of the operating air is presented in Drawing 7. As is seen from the drawing, the air is first heated in the calorifer and enters the first drying chamber, then is heated again, etc.

Drawing 7.
Visible labels in the diagram: “drying chambers”; “calorifer”; “calorifers”; “path of air”; chamber numbers 1, 2, 3, 4; calorifer numbers 2, 3, 4; symbols \(q_0, t, d\), \(q_1, t, d_1\), \(q_2, t_2, d_2\), \(q_3, t_3, d_3\), \(q_4, t_4, d_4\).

The drawing gives the characteristics of the air at the outlet from each chamber and calorifier. The designations are as follows:

$\varphi$—relative humidity of the air.
$d$—its absolute humidity (the quantity of water per 1 kg of dry air).
$t$—air temperature.

As we have seen above, determining the air consumption and the heat balance of the dryer is very simple if we are able to represent it on the $J$—$d$ diagram. To represent it, however, it is necessary to know: 1) the initial state of the air; 2) the temperature to which the air is heated in each calorifier; 3) the temperature or humidity of the air leaving the chamber. The conditions usually known in the practical solution of the problem are the initial state of the air and the heating temperatures. Consequently, what remains is to find a method for determining the third condition, namely—the temperature or humidity of the air leaving each chamber.

The state of the air leaving any chamber in a theoretical dryer, where $\Delta = O$, is determined by the quantity of evaporated water. The latter depends on the rate of drying in the given chamber. Therefore, in order to plot the drying process with sufficient accuracy on the $J$—$d$ diagram, it is necessary to determine the nature of the course of the drying process of sugar beets as a function of time; in other words, this means dealing with the question of the dynamics of drying sugar beets.

V. Dynamics of drying sugar beets.

The question of the dynamics of drying various water-containing materials represents an almost untouched field (21, 22). The question of the dynamics of drying sugar beets likewise has so far scarcely been investigated. The scanty material available in the literature concerns only the total duration of drying, without taking into account the significance of the individual factors affecting the drying time. There is an experimental work in which an attempt was made to differentiate the process of drying beets (13). This work, carried out in the laboratory of food-substance technology of the Siberian Technological Institute by Prof. S. V. Lebedev and N. G. Zabegalov, concerns the rate of drying of beet cossettes and contains several drying experiments performed at various temperatures.

Wishing to find the law of variation of the drying rate of beet cossettes, we approached this question below theoretically and obtained a certain dependence expressed by a certain curve. Comparing the latter with the curve constructed on the basis of experimental data, under the same conditions, we were able to verify the degree of suitability of the method used for theoretically determining the drying-rate curve for practical purposes and to introduce the necessary corrections.

Let us assume that beet cossettes in the amount of one kilogram are arranged in a thin layer on the surface of a sieve placed in a drying chamber and are washed by air at temperature $t^\circ$, at a partial pressure of water vapor $l_b$. The general equation for the drying rate has the following form [see: (9), p. 78]:

$$ W = f \times O \times (1 + \alpha t) \times \sqrt{v} \times (l_w - l_b) \ldots \ldots (3) $$

where $W$—the quantity of water evaporated per unit time.
$O$—the surface of the material being dried.
$(1 + \alpha t)$—the temperature binomial of expansion.
$V$—the velocity of air flow.

\(l_w\)—the pressure of water vapor at the surface of the body being dried.
\(f\)—the coefficient of the drying rate, depending on the character of the material being dried.

At a constant drying temperature, with the rate and the character of the material being dried unchanged, the term \(f(1+\alpha t)\sqrt{v}\) may be taken as constant in magnitude. Denote it by \(C\)

\[ C=f(1+\alpha t)\sqrt{v}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (4) \]

Then the amount of water removed per unit time is determined by the expression:

\[ W=C\times O\times(l_w-l_b)\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (5) \]

The amount of water \(Q\), removed during the time \(Z\), will be expressed as follows:

\[ Q=Z\times W=Z\times C\times O\times(l_w-l_b)\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (6) \]

Let us divide the drying process into a number of periods, which we shall take of such duration that in each of them 5% of the initial weight of the material being dried will be evaporated. In our case 5% amounts to 50 grams of water. Then equation (6) will take, for a number of successive periods, the form:

\[ \begin{aligned} \text{for the 1st}\quad &50=C\times Z_1\times O_1\times(l_{w_1}-l_{b_1})\\ \text{“ the 2nd}\quad &50=C\times Z_2\times O_2\times(l_{w_2}-l_{b_2})\\ \text{“ the n-th}\quad &50=C\times Z_n\times O_n\times(l_{w_n}-l_{b_n})\ . \ . \ . \ . \ . \ . \ . \ . \ 7. \end{aligned} \]

Solving this equation with respect to \(Z_n\), we find

\[ Z_n=\frac{50}{C\times O_n(l_{w_n}-l_{b_n})}\ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ . \ (8) \]

where \(Z_n\) is the time during which the \(n\)-th 50 grams of water are evaporated. It is obvious that the time \(Z_n\) changes depending on the magnitude of the surface \((O)\) of the beet cossettes being dried and on the difference \((l_w-l_b)\), which have unequal values for different drying periods. If it is assumed that the mean humidity of the air is constant throughout the entire drying time, then the partial pressure of water vapor may be considered constant. In equation (8), then, only \(O\) and \(l_w\) will be variables; both these quantities decrease during the drying process. The surface area of the material being dried \((O)\) continuously decreases as water is removed. The pressure of water vapor \((l_w)\), with removal of water from the juice and with an increase in the concentration of sugar in it, tends toward a certain minimum value corresponding to the vapor pressure saturated at the given temperature of the solution. Hence it is clear that during the course of the drying process its rate must continuously decrease.

Let us compute the duration of the individual periods of time \(Z_1, Z_2, Z_3 \ldots Z_n\), required for the evaporation of successive 50 grams of water from one kilogram of beet cossettes.

In calculating \(Z\) we shall take, for the values of \(l_w\) and \(O\), their mean value for the given drying period. First let us find how the surface area of the cossettes changes during drying.

We shall make the following assumptions:

1. The sections of the cossette by a plane perpendicular to its axis, in the different periods from the beginning of drying, give similar figures.

2. The volume of the cossettes decreases in each period by the amount of the volume of the water evaporated during that period.

Let us denote the length of the cossette by \(l\), the area of its section by \(p\), and the perimeter of the section by \(k\). The volume of the cossette \((M)\), with a prismatic form, is determined as follows: \(M=pl\) cu. cm.

The surface area of the cossette—\(S\), if the end areas are not taken into account, will be expressed as follows: \(S=kl\) sq. cm.

Let us take the ratio of the volume of beet cossettes to their surface area; it is as follows:

\[ M : S = pl : kl = p : k \]

The ratio of the area of a figure to its perimeter can be easily determined for every case. For similar figures, however, this ratio will remain the same under all quantitative changes. Nevertheless, when calculating the ratio of the area of the cossettes to their perimeter, it is necessary to take into account the circumstance that the cossettes will be obtained with different cross-sections depending on the different mutual arrangement of the knives in the slicer. Professor Zuev, in vol. I of his Encyclopedia of the Sugar Industry (p. 135), cites Goller’s calculations, who found that, when slicing with Chizek knives (cossette thickness 1.5 mm), there are 1.6 mm of perimeter per 1 sq. mm of area. For Goller knives the same ratio is 2.35 mm. Expressing areas in sq. cm and lengths in cm, we find for Chizek knives \(M : S = 1 : 16\) cm. For Goller knives—\(M : S = 1 : 23.5\) cm. Hence it is not difficult, knowing the volume of the cossettes, to find their surface. We assumed that the volume of cossettes in each period decreases by the volume of water evaporated during that period. If the specific weight of the cossettes is taken as 1.00, then the area of one kilo of cossettes cut by Chizek knives will be: \(S = M \times 16\ \text{cm}^2 = 1000 \times 16\ \text{cm}^2 = 1.6\ \text{sq. m}\), and for Goller knives 2.35 sq. m. This, of course, is true only for a definite thickness (1.5 mm) of cossettes. In the general case we have \(S_0 = M \times b\), where \(b\) is the ratio \(k : p\).

In the first period, during which 50 grams of water, or 5% by weight of the raw beet, are evaporated, the volume decreases by 5% according to the assumption made. Correspondingly, the area will also decrease by 5%. Under these conditions we have: the area of the cossettes at the beginning of the first period—\(S_0\), at the end \(0.950 S_0\), on average—\(0.975 S_0\). At the beginning of the 2nd period \(0.950 \times S_0\), at the end \(0.900 \times S_0\), and on average—\(0.925 \times S_0\), etc. With a constant setting of the knives, \(S_0\) always has the same value; in that case, in equation (4) it may be assumed that \(C_0 = S_0(1 + \alpha t) f \sqrt{v}\). Equation (5) will take the form:

\[ W = C_0 U (l_w - l_b) \ldots \ldots \ldots \ldots \ldots \ldots (9) \]

where \(U\) is the numerical coefficient at \(S_0\), depending on the stage of drying; equation (8) will now be expressed as follows:

\[ Z_n = \frac{50}{C_0 \times U_n \times (l_{w_n} - l_{b_n})} \ldots \ldots \ldots (10) \]

The values of \(U_n\) for a series of successive periods, during which 50 grams each are evaporated, as we saw above, constitute an arithmetic progression:

\[ 0.975,\ 0.925,\ 0.875,\ 0.825,\ \text{etc.} \]

where the \(n\)-th term of this series is equal to

\[ U = 0.975 - (n - 1)0.05. \]

Then the general expression for the duration of drying of the \(n\)-th portion of 50 grams will be expressed as follows:

\[ Z_n = \frac{50}{C_0[0.975 - (n - 1)0.05](l_{w_n} - l_{b_n})} \]

or

\[ Z_n = \frac{50}{C_0(1.025 - 0.05\,n)(l_{w_n} - l_{b_n})} \ldots \ldots (11) \]

Let us now find the value of \(l_w\)—the pressure of water vapor at the surface of the solution—for successive periods of drying, during which 50 grams of water each are evaporated. Let us take the composition of normal beet juice as follows:

Bx. — 20.0, Sax — 17.5, Db. — 87.5,

non-sugars — 2.5, water — 80.0%, juice coefficient 90%.

Determining the value of \(l_w\) presents considerable difficulties. To determine the pressure of water vapor above a liquid, it is necessary to know the temperature of the liquid. The work of the drying laboratory of the Heat-Engineering Institute has established (21) that “the temperature of a liquid and of a material, during evaporation of its free moisture, placed in a stream of air of definite temperature and humidity and deprived of heat losses to the environment, is, in the established state, equal to the temperature of the wet-bulb thermometer—psychrometer—placed there.” But when the material begins to evaporate bound moisture, its temperature increases, with a parallel decrease in the pressure of water vapor at its surface.

In the case of drying beet, we do not have water that could be treated as completely free. As is known, beet water consists of the water of the normal juice and the water of the pulp, or so-called imbibition water. The water of the normal juice, with the above-indicated composition, amounts, calculated on the weight of the beet, to \((0.80 \times 0.90) \times 100 = 72\%\). The water of the pulp usually amounts to about 5% by weight of the beet. The total quantity of water in sugar beet, therefore, is 77% by weight of the beet. Obviously, from the very beginning of drying, the water of the beet is in a bound state, and the temperature of the material is higher than the temperature of the wet-bulb thermometer—psychrometer, while the vapor pressure is lower than the vapor pressure over the surface of water. We shall calculate the vapor pressure of water above the surface of the juice according to Raoult’s equation:

\[ l_w = l_{w0}\frac{N}{N+n} \]

where: \(l_w\) — the pressure of water vapor above the surface of the solution,

\(l_{w0}\) — the pressure of vapor of pure water at the temperature of the material.

\(N\) — the number of moles of solvent.

\(n\) — the number of moles of dissolved substance.

According to the foregoing, the temperature of the material will be lower than the temperature of the air and will lie between the temperature of the wet bulb of the psychrometer placed in this air and the temperature of the air itself. Moreover, as the drying process proceeds, the temperature of the material approaches the temperature of the air. In view of the lack of study of this question for beet, we shall take \(l_{w0}\) at the temperature of the air flowing in the dryer. This, of course, introduces some error into the calculations.

According to the given composition of the beet, 1 kg of it in the fresh state contains 770 grams of water and 180 grams of sugar and dissolved non-sugars; we shall assume that the non-sugar contained in the solution has a molecular weight equal to that of sugar, which will only slightly change the results of the calculations.

Then the number of moles of solvent per 1 kg of beet is \(770 : 18 = 43\), where: 18 is the molecular weight of water. The number of moles of dissolved substance is

\[ \frac{180}{342} = 0.526 \]

moles (342 is the molecular weight of sucrose). Let us take the temperature of the air flowing in the dryer as \(85^\circ\mathrm{C}\). The initial pressure of water vapor at the surface of the solution, according to the assumptions made, will be

\[ l_w = \frac{43}{43 + 0.525} \times 433^*) = 428 \]

mm of mercury.

\[ \text{*) where: } 433 \text{ expresses the vapor pressure of pure water at } 85^\circ. \]

At the end of the first period the amount of solvent decreases by 5%, or 2.8 mol. The pressure \(l_{w_2}\) at the end of the first period is found from the equation

\[ l_{w_2}=\frac{40.2}{40.2+0.525}\times 433; \]

obviously, the average pressure for the first period is equal to

\[ \frac{l_{w_1}+l_{w_2}}{2}. \]

In the same way we shall find the pressure for all the other periods.

Thus, with a known approximation, we have found it possible to determine the values of the variable quantities in equation (11). In Table II below are presented the results of calculations made for a temperature of 85°C. Everywhere the numerical values of the calculations have been rounded.

Table II.

The average partial pressure of water vapor in the air is taken as equal to 90 mm, which approximately corresponds to air at 85°, having a relative humidity of 20%.

The temperature of 85°C was chosen so that the results of the calculations could be compared with the results of the experimental work (13). From the latter, only two drying experiments at 85°C were taken, since in the others, conducted at lower temperatures, the conditions of the experiment were complicated by side circumstances depending on the design of the drying apparatus. Drying in the cited work was carried out in an electric drying cabinet, on the lower shelf of the latter, to which fresh heated air flowed.

The figures in column 7 of Table II are calculated as follows: we find the expression for the total duration of drying \(n\times 50\) g of water, where \(n\) determines the number of periods during each of which 50 g of water evaporates. We had (see equation 11).

\[ Z_1=\frac{50}{C_0\times U_1\times (I_{w_1}-I_{b_1})} \]

\[ Z_2=\frac{50}{C_0\times U_2\times (I_{w_2}-I_{b_2})} \]

\[ Z_n=\frac{50}{C_0\times U_n\times (I_{w_n}-I_{b_n})} \]

whence the sum of the first \(n\) periods of drying time is equal to

\[ \sum_{1}^{n} Z=\sum_{1}^{n}\frac{50}{C_0\times U\times (I_w-I_b)} =\frac{50}{C_0}\sum_{1}^{n}\frac{1}{U(I_w-I_b)} \ldots\ldots (12). \]

We have two experiments carried out under the drying conditions adopted above. Both experiments lasted 420 minutes; during this time, in each experiment, 75% of water by beet weight was removed. Consequently, for this case, corresponding to 15 periods with evaporation in each of them of 50 g of water, we have

\[ \sum_{1}^{15} Z = 420\ \text{minutes}. \]

On the basis of equation (13) and column 6 of Table II, we have:

\[ \sum_{1}^{15} Z=420=\frac{50}{C} \left( \frac{1}{330}+\frac{1}{312}+\frac{1}{295}+\ldots+\frac{1}{67.5} \right). \]

From this it is not difficult to determine that \(C_0=0.0105\).

Substituting the value of \(C_0\) into equation (10), we shall find the duration of each period, given in column 7 of Table II.

In Table III the data of the theoretical calculations and of experimental run 13 are compared. The same is presented graphically in the diagrams of drawings 8 and 9.

Curve
of drying rate
at \(t=85^\circ\mathrm{C}\). \(\sum z=420\) min.
Experiment 1.

Vertical axis: Evaporated water, as % of initial weight
Horizontal axis: Time, hours

Legend:
— · — theoretical curve
—— curve obtained experimentally

Drawing 8.

Experiment 2.

Curve
of the drying rate
at \(t = 85^\circ\mathrm{C}\). \(\sum z = 420\) min.

[Graph: percentage of water evaporated vs. time in hours. Legend: dash-dot line—“Theoretical curve”; solid line—“Curve found experimentally.”]

Drawing 9

From Table III and the diagrams it is evident that, despite a certain inaccuracy and a large number of assumptions, the theoretically obtained results agree fairly well with those obtained in practice. This permits us to suppose that the method we have adopted for constructing the drying-rate curve, when refined, can yield results that are quite satisfactory in accuracy.

Table III.

As can be seen from drawings 8 and 9, the curve obtained in practice passes somewhat above the theoretical curve. This is explained by the fact that actually, in the first hours, a larger quantity of water evaporates than is given by the theoretical calculations. The small number of experiments does not make it possible to establish any definite law of corrections. However, it may be said that up to the evaporation of 50% of the water by weight of the beets, the corrections almost always have a positive value, while after evaporation of 50% of the water they are always negative. The time which, on the theoretical curve, corresponds to 50% evaporated water, on the experimentally obtained curve corresponds to 53–55%. The incompleteness of the factual material compels us to postpone the final resolution of the question; for the time being, for our further constructions, we shall adopt the following rule for finding the drying-rate curve.

1. Having specified the drying conditions, and using equations (11) and (12), we find, by the method indicated above, the theoretical drying curve.

2. We find the time \(Zq\) corresponding to evaporation of 50% of the water by weight of the beets.

3. We draw a new curve having the same beginning and end as the theoretically found one, but on which the time \(Zq\) corresponds not to 50% of the water, but to \(50 + 3 = 53\%\) of the water.

Using the curve constructed in this way, we shall subsequently try to connect the static calculation of the dryer with the dynamics of the drying process.

VI. Calculation of the Dryer.

As we saw above, the drying time \(\sum_1^n Z\), required for the removal of

\(n \times 50\) g of water from 1 kilogram of beets, is determined with sufficient accuracy for practice in the case where \(C_o\) is known from the equation

\[ \sum_1^n Z = \frac{50}{C_o}\sum_1^n \frac{1}{(l_w - l_b)u} \quad \cdots \cdots (12), \]

where all quantities, for the specified drying conditions, are known.

In this equation \(C_o\) represents the product

\[ (1+\alpha t)\times V^v \times f \times S_o. \]

If \(S_o\) is a known quantity, then the same equation can be written as follows:

\[ \sum_1^n Z = \frac{50}{C}\sum_1^n \frac{1}{(l_w - l_b)O} \quad \cdots \cdots (13), \]

where \(O\), according to the preceding, is equal to: \(S_o \times U\), and

\[ C = (1 \pm \alpha t)\times V^v \times f \quad \cdots (4) \]

In equation (4) the undetermined quantity is the coefficient \(f\).

If experimental determinations of the coefficient \(f\) showed that it is a constant quantity, then, finding from equation (4) the quantity \(C\), assigning values to \(V\) and \(t\), and substituting \(C\) in equations (11) and (13), it would be possible easily to find the total duration of drying and the character of the drying process. However, we do not have at our disposal material for judging the magnitude of \(f\). In view of this, we shall make use of the data on the duration of drying of beets in the Loreau dryer. At the plant in Loreau the duration of drying

3 h. 40 minutes. In the first 100 minutes 75% of the water is removed, and in the following 120 minutes only 2.5%. The air velocity during the first 100 minutes, judging from the descriptions of the dryer, is about 0.25–0.5 m/sec. Therefore, in what follows, for our calculation we shall take the time for evaporation of 75% of the water by weight of the raw beet, or, in other words, \(\sum_{1}^{15} z = 100\) minutes. The question of the limit of drying at which the dried beet can be stored without danger of spoilage requires special study. As we see, the drying out of the last 2.5% of water requires 2 hours, whereas the drying out of the first 75% requires only 1 h. 40 minutes. Naturally, by raising the limit of drying by 2.5%, we greatly simplify the design of the dryers. On this subject Scott Carret and Riley (12) write: “It is very probable that dried cossettes will keep stably even with a moisture content above 10–12%*) provided that the proteins contained in them are converted into a ‘hornlike state,’ i.e., if they lose their hygroscopic capacity and if the material is sufficiently sterilized.”

It is doubtful, however, that the process of hornification occurred in the last stage of drying, which in Loreau took place at a temperature of 90°C, whereas Kroeker’s experiments [(16) p. 29] established that beet pulp loses its capacity for imbibition during drying only at 110°C. In any case, until the question of the rational limit of drying has been resolved experimentally, one should stop at the limit adopted by Loreau. Therefore, in the further calculation we shall assume that the beet, after 75% of the water has been removed from it, is dried further in a special “after-drying chamber.”

Let us proceed to finding the drying curve for \(\sum_{1}^{15} z = 100\) minutes. We choose the mean temperature of the material being dried equal to 100°C.**) In accordance with the preceding, this same temperature will be the mean temperature of the air in each chamber (see p. 15).

We shall assume that the dryer operates according to the countercurrent principle. The air successively passes through the after-drying chamber and all the drying chambers, moving in the direction opposite to the movement of the cossettes being dried. At the same time the partial pressure of the water vapor in it increases at the expense of the evaporating moisture. In order to take into account the increase in the partial pressure of the water vapor, it is necessary to know the quantity of air introduced into the dryer. As is not difficult to see on the \(J\)—\(d\) diagram, the partial pressure at a given barometric pressure depends on the moisture content. For a given moisture content, expressed in grams per kilogram of dry air, regardless of temperature, the partial pressure will be the same. Thus, by assigning a definite weight of air per one kg of evaporated moisture, and taking into account the increment of its moisture content, we can find the partial pressure of the water vapor in it at any moment. (1)

Above we pointed to the drying process with a temperature of 450°C as the limit up to which raising the drying temperature affords considerable economic advantages. Therefore we shall set ourselves the goal of obtaining a process economically equivalent to it and proceeding within the lim-

*) 10–12% by weight of dry substance is the limit to which beet is dried by the de Vecchi method.

**) It is assumed that, owing to circulation, a temperature is maintained in the chamber equal to the mean between the temperatures of the entering and leaving air.

at temperatures allowable for beets. The corresponding consumption of air will be 6.8 kg per 1 kg of evaporated moisture. The partial pressure of water vapor in the air can be found from the equation

\[ l_b=b\frac{d}{653+d}\ldots\ldots\ldots\ldots\ldots\ldots\ldots (14), \]

where:
\(l_b\) — partial pressure,
\(b\) — barometric pressure,
\(d\) — moisture content in grams per 1 kg of dry air (see I, p. 10).

Let us divide, as we did in the preceding chapter, the drying process into a series of periods during which 50 grams of water per 1 kg of dry beets evaporate, and calculate for each period the mean value of the variables. The method for calculating the variables—the surface of the beet cossettes—\(O\), and the pressure of the water vapors on the surface of the solution, \(l_w\), was given above. Here we shall show how the value of \(l_b\) is found.

We take the initial moisture content of the air entering the final-drying chamber as equal to 3.0 grams, which corresponds approximately to 80% relative humidity and a temperature of 0.0°C. Let us suppose that in the final-drying chamber 20 grams of water per 1 kg of beets evaporates, i.e. 2%. Under the adopted conditions, the air consumption per 1 kg of evaporated water is 6.8 kg, and per 1 kg of beets:

\[ \frac{77}{100}\times 6.8=5.25\ \text{kg}, \]

where \(77=75+2\) is the percentage of water evaporated from the beets.

The air entering the dryer has a humidity of

\[ 3+\frac{20}{5.25}=3+3.7=6.7 \]

grams per 1 kg of dry air. The partial pressure of the water vapors in it is found from equation (14)

\[ l_b=b\frac{6.7}{623+6.7}. \]

For

\[ b=745\ \text{mm},\quad l_b=745\times\frac{6.7}{629.7}=7.8\ \text{mm}. \]

After the evaporation of 50 grams of water in the first drying period, the water content per 1 kg of dry air is

\[ 6.7+\frac{50}{5.25}=6.7+9.55=16.25\ \text{g}, \]

and the pressure of the water vapors, according to equation (14), is

\[ l_b=745\times\frac{16.25}{639.25}=18.9\ \text{mm}. \]

The mean pressure for the first drying period is

\[ \frac{18.9+7.8}{2}=\frac{26.7}{2}=13.35\ \text{mm}\sim 13\ \text{mm}. \]

Similarly we find the moisture content and the partial pressure for any period.

Table IV gives the results of these calculations; it is assumed that the cossettes are cut by knives of the Chizhek system and that their thickness is equal to 1.5 mm.

Table IV.

The coefficient \(C\) is found from equation (13)

\[ \sum_{1}^{15} Z = 100\ \text{min.}; \qquad \sum_{1}^{15} Z = \frac{15}{C}\sum_{1}^{15}\frac{1}{O \times (l_w-l_b)}; \]

whence \(C = 0{,}0134\). Substituting \(C\) in equation (8)

\[ Z_n = \frac{50}{C \times (l_{wn}-l_{bn}) \times O_n}, \]

we find the values of \(Z_n\), given in column 10.

On the basis of the data in the table a curve (drawing 10) has been constructed for the drying rate at

\[ \sum_{1}^{15} Z = 100 \]

and \(t = 100^\circ \mathrm{C}\). In the same drawing there is given the corrected drying-rate curve, in accordance with the indications given on p. 19. As is evident from Table IV, the same curve gives the relation between the duration of drying and the increase in the absolute humidity of the air operating in the dryer (Table IV, columns 9 and 10). On the diagram in drawing 10, therefore, the moisture content in the air per 1 kg of dry air is plotted on the second vertical axis. Using this curve, one can easily plot on the \(J—d\) diagram the drying process with intermediate heating of the working air. The basis—

we shall carry out this construction for a corridor dryer. For this purpose, we shall first determine the duration of the cossettes’ stay in the chambers. The time during which the cossettes remain in each chamber is determined by the last chamber, from which the spent air is removed from the dryer and into which fresh cossettes enter. We have taken the temperature of the removed air to be \(70^\circ\text{C}\). According to the condition, the mean air temperature must be equal to \(100^\circ\text{C}\). Consequently, the temperature to which the air is heated before being admitted into the last chamber is \(130^\circ\text{C}\). The drying process in the last chamber is represented on the \(J\)—\(d\) diagram (see Fig. 11) by the line CD. The amount of water absorbed by the air in this chamber per 1 kg of dry air is found by dropping perpendiculars from points C and D to the horizontal axis. This amount, represented in the drawing by the segment \(ba\), is equal to 29 grams. We find on the drying-rate diagram (Fig. 10) that for the absorption by the air

CURVE

of the drying rate
at \(t = 100^\circ\text{C}\). \(\sum \tau = 100\) min.

Fig. 10

29 grams will require 11 minutes. Thus, the time during which the wagon remains in the first chamber must be equal to 11 minutes. In the other chambers, obviously, the wagon may remain either 11 minutes, or for a time that is a multiple of 11 minutes. Hence the number of wagons is equal to \(100:11 \simeq 9\).

Let us divide, on the drying-rate diagram (Fig. 10), the time into intervals of 11 minutes each (segments \(a_1b_1;\ b_1c_1;\ c_1d_1;\ d_1e_1;\ e_1f_1;\ f_1g_1;\ g_1h_1;\ h_1k_1;\ k_1l_1\)). The quantity of water absorbed by 1 kg of dry air during each interval is easily found from the diagram.

\(J\)-\(d\) diagram
Prof. V. Ramzin

Fig. 11

(Fig. 10). This is what is given by the segments ab, bc, cd, de, ef, fg, gh, hk, kl. By setting the residence time of the wagon in the various chambers, one can find the amount of water evaporated in them. Let us take, for example, the residence time of the wagon in the first 5 chambers as 11 minutes each. Consequently, 5 wagons will be accommodated in them. The remaining 4 we shall place in two chambers, two in each.

Consequently, the residence time of the wagon in chambers 6 and 7 will be equal to 22 minutes.

Table V.

The amount of water evaporated in each chamber per 1 kg of dry air, found from the drying-rate diagram, is given in Table V.

Let us now return to the J—d diagram. On the horizontal axis of the J—d diagram (Fig. 11), on a known scale, the moisture contents in grams per one kg of dry air are plotted. On this axis let us successively lay off the segments representing the amount of water entering the dryer in grams per 1 kg of dry air—(on), the amount of water absorbed by the air in the “finishing-drying chamber”—(nl), and the amounts of water absorbed by the air in the successive chambers (hl, fh, ef, de, cd, bc, ab). Drawing through these points lines perpendicular to the horizontal axis and, consequently, satisfying the equation \(d = \mathrm{const}\), we shall obtain the lines of air heating in the calorifers between neighboring chambers. On these lines will lie the initial and final points of the processes taking place in the individual chambers. This construction, obviously, assumes that the drying process proceeds in each chamber in exactly the same way both at the moment the wagon enters and at the moment it leaves it. The amount of moisture absorbed in an individual chamber that we lay off represents a certain average value for that chamber. Obviously, in reality the “drying process,” during the residence time of the beets in an individual chamber, changes the more appreciably the longer the beets remain in it. Therefore our construction is not entirely exact. However, owing to the short time during which the wagon with cossettes is in an individual chamber, the drying rate during this time changes only slightly and, carrying out the calculation with the average magnitude of moisture absorption, we do not make a large error.

Thus, we have obtained the lines of air heating in the intermediate calorifiers. Having the lines of air heating in the intermediate calorifiers, it is not difficult to construct the lines representing the work of the air in the individual chambers. The following conditions are available for their construction.

1. The initial and final points of these lines lie on the heating lines.

2. For a theoretical dryer these lines are directed parallel to the line \(J = \mathrm{const}\).

3. The mean temperature of the process represented is, by specification, equal to \(100^\circ\mathrm{C}\).

For each chamber let us choose a line satisfying these conditions. In this way, on the \(J\)—\(d\) diagram we obtain the broken line CDEFGHKLMNOPRSFUA, representing the drying process. This line pertains to a dryer without losses, i.e. where \(\Delta = 0\). Having determined the losses, it is not difficult to plot on the diagram the process taking place in an actual dryer, in accordance with the rules set forth in Chapter II.

Thus, using the \(J\)—\(d\) diagram and the drying-rate curve, we have obtained a representation of the drying process for a corridor dryer divided into a number of isolated chambers. The calculation of a dryer divided into chambers with materials moved in them by band conveyors (belt dryers, see drawing 4) can be modified somewhat, since belt dryers are not bound by the condition that the residence time of the beet in the individual chambers be a multiple of a common value. We shall proceed as follows. First we plot on the \(J\)—\(d\) diagram the drying process desired by us, subject only to the conditions that, as in the case analyzed above, for a corridor dryer \(t_{\max} \leq 130^\circ\), \(t_{\mathrm{mean}} = 100^\circ\mathrm{C}\). On the \(J\)—\(d\) diagram we find the amount of water evaporated in each chamber per 1 kg of dry air. Plotting these segments on the drying-rate diagram, we shall find the residence times of the beet cossettes in each chamber. The speed of the individual conveyors must be such that, in the time found for the given chamber, the cossettes move by the amount of the chamber length. The method of construction is very simple and therefore is not explained by a special drawing.

The present work represents one of the first attempts to link the dynamics of the drying process with a static calculation. It is perfectly clear that, as a first approximation to the solution of the problem, the calculation presented contains a large number of assumptions, which in the present case is unavoidable. The very construction of the drying-rate curve, for lack of the corresponding experimental material, can for the present be carried out with a certain number of assumptions. Nevertheless, even in its present form the proposed work can, to a certain extent, help to elucidate the process of drying sugar beets and to design dryers for beet cossettes.

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