P. M. Silin

Evaporation Temperatures in a Sugar Factory

When calculating the heating surface of the evaporation station of a sugar factory, among other factors—the amount of juice evaporated, the amount of exhaust steam [[unclear: sent to the station]], and so on—the temperatures of steam and juice in the various vessels of the evaporator are usually taken into account. There are no definite considerations for choosing these temperatures. They are specified as “customary,” but they are determined by several factors and indicate only, respectively, the heating surface on the evaporator vessels and the heating surface of the heaters, which depend on the temperatures in the evaporator and the heaters. Thus this introduces a certain arbitrariness into the whole calculation. However, the arbitrariness is limited somewhat by the results that are obtained. If “unsuitable” temperatures are taken, then an unreasonably large heating surface is obtained in the separate evaporator vessels and heaters. Gradually, empirically, the most suitable values of the temperatures were worked out.

But the temperatures assigned arbitrarily in the evaporator may be theoretically justified. They may be calculated. Such calculations are desirable and necessary if it is required to pass to a new, as yet untried type of evaporation station—for example, to install a turbine with high back pressure and to send the steam from it not to the 1st but to the 0th evaporator vessel. In such cases the “customary” temperatures may prove unsuitable.

The temperatures in the evaporator should be chosen with such a calculation that the sum of the costs of the heating surfaces—both of the evaporator and of the heaters and calorizators supplied with steam from the evaporator—be the least. More precisely, the sum of the products of the surfaces by the cost of 1 sq. meter of them must be the least (or, conversely, by the cost coefficient of increasing the heating surface of the apparatus by 1 sq. meter). Such a correction introduces only several coefficients. We shall disregard it. Considering the cost of 1 sq. m. of the surfaces of the various apparatuses the same, the temperature of the diluted juice may be considered known, as also the temperature of the exhaust steam of the steam engine, which depends on its pressure and on the construction of the engine. Consequently, we may take as given the pressure and the heating surface of the 0-vessel and of the apparatuses heated by steam from it, since the 0-vessel is heated by steam from the steam boiler, while the steam of the 0-vessel at pressure [[unclear: …]] heats the apparatuses with exhaust steam of the steam engine.

Let us adopt the following notation.

*) The average temperatures of the juice in the evaporators heated by steam from the corresponding body are taken: in the 1st body, generally, \(t_1-\Delta t_1\); the coefficients of heat transfer, generally, \(k_i^{(\alpha)}\), etc. For evaporators heated by steam from the \(i\)-th body these quantities have corresponding values [[unclear]].

The temperature of the feed to the 1st body is denoted by \(\theta\); \(\Delta t_1\) is the difference between the temperature of the heating steam and the boiling temperature of the juice of the 1st body; \(t_1+\Delta t_1\) is the surface of heating of the 1st body of the evaporator (for processing 100 kg of beets per minute):

\[ F_1=\frac{a_1 r_1}{(t_1-t_0-\Delta t_1)\,k_1} \]

The heating surface of the remaining bodies is equal to

\[ F_i=\frac{a_i r_i}{(t_i-t_{i-1}-\Delta t_i)\,k_i} \]

For the heating surface of evaporators heated by steam of this body with temperature \(t_i\), we have the general expression

\[ F_i^{(\alpha)}=\frac{[[unclear]]}{[[unclear]]} \]

For evaporators heated from the 1st body:

\[ F_i^{(1)}=\frac{[[unclear]]}{[[unclear]]} \]

For evaporators heated from the II body:

\[ F_i^{(2)}=\frac{[[unclear]]}{[[unclear]]} \]

The useful heat capacity of the apparatus actually employed for evaporation is thus:

\[ F_{\text{useful}}= F_1+F_2+F_3+F_4+ \sum F_i^{(1)}+ \sum F_i^{(2)}+ [[unclear]] \]

The total useful heating surface of the installation will be [[unclear]].

\[ [[unclear: long formula involving }F,\ a_i,\ r_i,\ t_i,\ \Delta t_i,\ k_i\text{]] \]

Here \(I, II, III\), and \(IV\) denote the bodies; \(R\) [[unclear]] the evaporators.

*) All quantities should be understood as referred to 100 kg of beets; thus \(a_i\) means the quantity of steam obtained in a given body from the juice, and the temperature is [[unclear]].

$$ \frac{a_1^{(n)} r_1}{(t_0-t_1^{(n)})\,k_1^{(n)}}+ \sum \frac{a_2^{(n)} r_2}{(t_2-t_2^{(n)})\,k_2^{(n)}}+ \sum \frac{a_3^{(n)} r_3}{(t_3-t_3^{(n)})\,k_3^{(n)}}=\cdots \tag{I} $$

In the literature \(t_0\) is known, as has already been clarified; \(t_4\) is also known (it depends on the degree of rarefaction produced by the air pump). It remains only to choose the most advantageous values of \(t_1\), \(t_2\), and \(t_3\). We shall regard \(t_0\) and \(t_4\) as constants. When the values of \(t_1\) change, the value of \(F\) will also change. It is necessary to choose such a value of \(t_1\) for which the surface \(F\) would be minimal. For these values, obviously, the first derivative of \(F\) with respect to \(t_1\) must be equal to zero, i.e.,

$$ \frac{\partial F}{\partial t_1} = \frac{a_1 r_1}{(t_0-t_1-\Delta t_1)^2\,k_1} - \frac{a_2 r_2}{(t_1-t_2-\Delta t_2)^2\,k_2} - \sum \frac{a_1^{(n)} r_1}{(t_1-t_1^{(n)})^2\,k_1^{(n)}} =0. \tag{II} $$

In the same way, taking \(t_2\), and then \(t_3\), as variables, we obtain two more equations:

$$ \frac{\partial F}{\partial t_2} = \frac{a_2 r_2}{(t_1-t_2-\Delta t_2)^2\,k_2} - \frac{a_3 r_3}{(t_2-t_3-\Delta t_3)^2\,k_3} - \sum \frac{a_2^{(n)} r_2}{(t_2-t_2^{(n)})^2\,k_2^{(n)}} =0, \tag{III} $$

$$ \frac{\partial F}{\partial t_3} = \frac{a_3 r_3}{(t_2-t_3-\Delta t_3)^2\,k_3} - \frac{a_4 r_4}{(t_3-t_4-\Delta t_4)^2\,k_4} - \sum \frac{a_3^{(n)} r_3}{(t_3-t_3^{(n)})\,k_3^{(n)}} =0. \tag{IV} $$

Thus, we have 3 equations (II, III, and IV) with three unknowns (\(t_1\), \(t_2\), and \(t_3\)), which can be solved. The following course of solution is possible. Eq. (II) gives, for any value of \(t_1\), for the three- or four-body apparatuses (for example, \(t_1=100\)—\(t_2=\) [[unclear: value]], \(t_1=108\)—\(t_2=98.7\)), a value of \(t_2\) which makes it possible to draw the curve representing the relation between \(t_2\) and \(t_1\). Eq. IV gives \(t_3\) as a function of \(t_2\). One can then calculate several values of \(t_2\) corresponding to certain values of \(t_3\), and draw the curve. The curves will now make it possible, for any value, for example, of \(t_1\), to find the corresponding values of \(t_2\) and \(t_3\). Of these conjugate values it remains only to choose those which would satisfy Eq. (III). A solution with accuracy sufficient for engineering purposes (up to \(1/2^\circ\text{C}\)) is easily carried out.

Let us give the results obtained by calculation for several particular cases, namely: 1) for the case of evaporation without utilization of the vapor to the side, 2) with utilization of the vapor from the I, II, and III bodies, 3) with utilization of the vapor only from the I and II bodies.

• If it is possible to indicate in advance approximately the temperatures in the evaporator, then one may specify the quantities \(r\); if, however, these temperatures cannot be indicated even approximately, then all \(r\)’s may be taken equal to one another; the error from such an assumption will be small.

Example I. We take the following data (see Prof. Zubov, “Heat Engineering in Sugar Production,” p. 111).

Temperature of the vapor heating the 1st body, \(t_0=112^\circ\).

There is no utilization of the vapor from evaporation to the side.

— angle II; the tube is … [III, IV, and II] with the fin … [[unclear]]

\[ [[unclear: displayed formula involving]]\quad t_1-t_0,\quad 0.5,\quad r,\quad 10.650,\quad 550 \]

On the right the angle is very easily found. It is obtained by an approximation; it is therefore sufficient to make one iteration. [[unclear]] $t_1=-104.1^\circ$; hence [[unclear]].

Example II. [[unclear: beginning of sentence]] from $1^\circ$ to $550^\circ$. The numerical data are as follows (see Prof. Zuev, Simplified calculation of superheaters (Journal of Soc. of Mech. Eng., 1913, No. 7)):

An approximate construction of the curve associated with a given task [[unclear]] the following data.

The coefficient of heat transfer for an undeformed calorifer will then be according to equation (3), [[unclear]]; for an economizer [[unclear]].

\[ [[unclear: displayed formula involving]] \quad (t_1-t_2-1.5),\quad 15,\quad \frac{10.650}{550} \quad =0 . \]

Hence we find: $t_1=106^\circ$, $t_2=54^\circ$; thus tubes III and II give the required temperature [[unclear]].

Example III. A tube with an enlarged [[unclear]] and a second row of tubes.

The tube [[unclear]].

[[unclear: final lines of the page, including text about the end of the curve and the amount of heat]]

From (11 and 12) we obtain

\[ \frac{15.685}{(t_1-0.5)^2\cdot 20} -\frac{22.550}{(t_1-t_2-\text{[[unclear]]})^2\cdot 15} -\frac{18.550}{(t_1-75)^2\cdot 3} +\frac{7.485}{(t_1-90)^2\cdot 8}=0 \]

\[ \frac{22.540}{(t_1-t_2-1.5)^2\cdot 15} -\frac{2.550}{(t_2-2.5)^2\cdot 10} +\frac{6.540}{(t_2-55)^2\cdot 5} +\frac{9.540}{(t_2-70)^2\cdot 8}=0 \]

\[ \frac{6.550}{(t_2-t_3-2)^2\cdot \text{[[unclear]]}} -\frac{7.565}{(t_3-60)^2\cdot 15}=0. \]

Hence: \(\text{[[unclear: value of }t_1\text{]]};\quad t_2=92;\quad t_3=78.\)

The resulting values are close to those that were obtained by direct construction on the diagram. This gives confidence in the correctness of the given method. This method, however, has the same limitations as diagrammatic construction: it presupposes knowledge, already from practice, of certain particularly common profiles; in cases of relatively uncommon tasks—under especially high requirements—it will be necessary to resort to other methods [[unclear: parenthetical reference]].