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Refractories and Industrial Ceramics
Vol. 39, Nos. 5 - 6, 1998
HEAT ENGINEERING UDC 666.041-936
CALCULATING THE SATURATION CONCENTRATION IN CIRCULATING MEDIA OF INDUSTRIAL POWER INSTALLATIONS N. A. Tyutin1 Translated from Ogneupory i Tekhnicheskaya Keramika, No. 6, pp. 33 - 35, June, 1998. Original article submitted January 12, 1998.
Expressions for calculating the saturation concentration of the substance liberated from the processed material and saturating the circulating medium are derived, namely, C = Cj and C = 1 -X
, where Cr is the
V(I - Y )
concentration of the liberated substance, Am is the mass of the liberated substance, Vis the consumption of the circulating medium, and Xis the recirculation coefficient of the medium.
The variety of industrial power devices that use circulating working media in a gaseous or liquid state can be divided into systems of three kinds of systems, namely, open (o), semi-closed (s), and closed (c). The fundamental difference consists in the degree of recirculation of the worked-out circulating medium. In an open system the worked-out medium does not return to the system, in a semi-closed system it returns partially, and in a closed system it returns fully. The degree of recirculation of the medium in these systems is characterized by a recirculation coefficient X understood as the ratio of the consumption AV of the recirculating medium (returned to the system) to the consumption V of the circulating medium, i.e., X = AV/V. For an open system X = 0, for a semi-closed system 0 < X < 1, and for a closed system X = 1. Diagrams of these systems are presented in Fig. 1. The essence of the processes that occur in each system in treating the material consists in the following. The treated material and the circulating medium with the parameters M~ and T~, respectively, enter the reactor. In the reactor the circulating medium is involved in mass and energy exchange with the material and is saturated with the substance liberated from the material. The worked-out material and the workedout medium exit the reactor with the parameters M 2 and T2, respectively. The worked-out medium in a semi-closed system returns partially (in a closed system fully) to the system through a generator; the circulation of the medium is provided by a blower.
In designing a power engineering device the parameter of the circulating medium denoted by the generalized symbols Tl and T2 and being the concentration of the substance liberated into the circulating medium from the treated material in the reactor is of primary importance. For this reason, it is
M2 I a
l ' ~ - ~ - " 1 T1 t
Fig. 1. Diagrams of open (a), semi-closed (b), and closed (c) systems with a circulating medium: R ) reactor; G ) generator; B ) blower; the other notation is given in the text.
Eastern Institute of Refractories, Ekaterinburg, Russia.
expedient to analyze each of the systems from the standpoint of saturation of the medium circulating in them with the treated material in order to determine dependences for calculating the concentration of the substance in the circulating medium at the outlet from the reactor, i.e., in the worked-out medium. In analyzing the systems we will consider a continuous sequence of individual cycles that occur in individual time intervals. The time of the process T, the number of its cycles (1 + n ), and the time of a cycle Az are related as T=(1 + n)Az,
the mass of the liberated substance Am and the consumption of the circulating medium V per cycle are constant in time, and the concentration of the substance of the first cycle in the circulating medium at the inlet to the reactor is equal to zero. The concentration of the substance of the first cycle in the circulating medium at the outlet from the reactor in each system Cl (the concentration of the liberated substance) will be determined as the ratio of the mass of the liberated substance to the consumption of the circulating medium, i.e., Am Cl=---ff-.
In an open system the concentration of the substance in the circulating medium at the outlet from the reactor for any cycle CO will be determined as in the first cycle. It will be equal to the concentration of the liberated substance C l and will be the saturation concentration at the same time, i.e.,
Co = q .
In a semi-closed system the concentration of the substance in the circulating medium at the outlet from the reactor Csi will be composed of the sum of two concentrations, beginning from the second cycle, namely, C l and the recirculation concentration of the cycle C r ~:
The recirculation concentration for any cycle Cr ~, beginning with the second one, is equal to the product of the concentration of the substance of the previous cycle in the circulating medium at the outlet from the reactor Cs(i_ ~) by the recirculation coefficient of the medium X, i.e.,
C. = Cs(i_ t)X.
Thus, the concentration of the substance in the circulating medium at the outlet from the reactor for any cycle, beginning with the second one, will be equal to
Csi = q + Cs(i_l)X in correspondence with Eqs. (4) and (5).
Taking into account the considerations presented above, we can derive equations for the concentration of the substance Cs that saturates the medium in any circulation cycle in the reactor: cycle 1 (7)
Csl = C 1 ,
Cs2 m CI + Csl Xm CI + CIX,
cycle 3 Cs3 = CI + Cs2X= G + CIX+ G X2,
cycle (1 + n) Cs(l+n) = C l + e s n X - = C s + q x T q x 2
The first term in Eqs. (7) - (10) characterizes the concentration of the substance liberated from the treated material in the circulating medium, and the other terms characterize the recirculation concentration in the medium at the outlet from the reactor. In the steady-state regime, i.e., at an infinitely large number of cycles of the circulating medium in the reactor (1 + n), the substance concentration Cs at the output obeys the equation Cs = Cl(l + X + X 2 +...+ X n ).
The polynomial I + X + X 2 + . . . + X " in Eq. (11) is a power series that converges at infinitely large n and an absolute value of X less than unity, i.e., at n --->ov and [XJ < 1. The domain of convergence of the series is (- 1, + 1) with the exception of X = - 1 and X= 1. The sum of the series 1 + X + X 2 +...+ X" in the domain of convergence is equal to 1 1-X . Substituting the value of the sum of the series, equal to 1 1 - X ' into Eq. (11), we will obtain an equation for calculating the saturation concentration of the circulating medium at the outlet from the reactor in the form
1- x "
In a closed system the concentration of the substance in the circulating medium at the outlet from the reactor Cci , beginning with the second cycle, will be expressed by an equation similar to that derived for the semi-closed system, i.e., Cci = Cl + Cri 9
Since a closed system differs from a semi-closed one by different values of X, we can repeat our reasoning and obtain an equation for the concentration of the substance in the me-
Calculating the Saturation Concentration in Circulating Media
three systems and equal to the concentration of the liberated substance CI. Beginning with the second cycle, the concentrations in each system differ. In an open system the concentration CO remains constant during the entire process and equals the concentration of the liberated substance Cj. In a semi-closed system the concentration Cs increases with time but has a limit
Co = C 1
Cs- 1-X' I
Fig. 2. Curves of the variation in the concentrations.
dium at the outlet from the reactor for an infinitely large number of recirculation cycles of the medium, which will be similarto Eq. (I1), i.e.,
Cc = Cl(1
The polynomial 1 + X + X 2 + . . . + X n is a power series and does not have a limit at n ~ oo and X = 1; its sum tends to infinity. After substituting X = 1 into Eq. (14), the equation for calculating Cc acquires the form Cc=Cl(1 + n ) .
In a closed system the concentration Cc increases with time without bound and has no limit, i.e., C c = C I (1 + n ) ~ oo. Equations (3) and (15) are particular solutions of Eq. (12) at X = 0 and X = 1, respectively. Therefore, the saturation concentration in each system can be calculated from Eq. (12). The final formulas for calculating the saturation concentration in circulating media of each of the systems after replacement of the notation C S in (12) by its generalized symbol C and with allowance for relation (2) will have the form C-
and Am C - )X---------~ . V ( l_
Functional dependences of the concentration on the number of cycles (1 + n ) or the time z, because they are related by Eq. (1), for each of the systems are presented in Fig. 2. After the first cycle the saturation concentrations are the same in all
REFERENCES 1. M. Ya. Vygodskii, A R e f e r e n c e B o o k o n H i g h e r M a t h e m a t i c s [in Russian], Gos. Izd. Tekh.-Teoret. Liter., Moscow (1956), p. 572.