On the Relationship Between the Average Driving Forces of Cocurrent and Countercurrent Mass Transfer Processes
Abstract Generated abstract
This paper examines the relation between mean driving forces in cocurrent and countercurrent mass transfer processes. It extends a standard proof, which required both operating and equilibrium lines to be linear, by retaining linear operating lines while allowing the equilibrium relation between phase compositions to be any increasing nonlinear function satisfying the stated separation condition. Using expressions for the number of transfer units, the argument compares the corresponding integrals for the two flow arrangements and shows that the number of transfer units is greater for cocurrent flow. Since the mean driving force is inversely related to this quantity for the same composition change, the result establishes that the countercurrent mean driving force exceeds the cocurrent mean driving force under these assumptions.
Full Text
ON THE RELATIONSHIP BETWEEN THE MEAN DRIVING FORCES OF CO-CURRENT AND COUNTER-CURRENT MASS-TRANSFER PROCESSES
G. M. IZMAILOV
(Presented by the scientific seminar of the Department of Processes, Apparatuses, and Cybernetics of Chemical Production, KhTF)
In the proof of the superiority of the mean driving force in counter-current flow over the mean driving force in co-current flow, the requirement of linearity is imposed both on the operating lines and on the equilibrium line [1]. In the proof given below of the relationship between the mean driving forces, the requirement of linearity is retained only for the operating lines. A condition of a more general character is imposed on the relation between the equilibrium and phase compositions: it may also be a nonlinear increasing function.
Let the initial and final values of the composition for one of the phases be denoted by \(x_1\) and \(x_2\), respectively, and for the other phase by \(y_1\) and \(y_2\). The dependence between the equilibrium compositions of these phases is represented on the segment \([x_1, x_2]\) by the increasing function
\[ y^* = f(x), \tag{1} \]
and on this segment
\[ y^* < y_1; \tag{2} \]
here \(y^*\) is the composition of the given phase under equilibrium conditions and at the composition of the adjacent phase equal to \(x\).
The operating compositions of the phases for the processes compared by mean driving force are related by the following equations: for the counter-current process (line \(AB\) in Fig. 1)
\[ y = a(x - x_0) + y_0, \tag{3} \]
for the co-current process (line \(A_1B_1\))
\[ y = -a(x - x_0) + y_0; \tag{4} \]
here
\[ a = \frac{y_2 - y_1}{x_2 - x_1}, \tag{5} \]
\[ x_0 = \frac{x_1 + x_2}{2}, \tag{6} \]
\[ y_0 = \frac{y_1 + y_2}{2}. \tag{7} \]
The relationship between the mean driving forces will be determined below through the number of transfer units [2].
Fig. 1
On the basis of equations (3)—(7), the general expression for the number of transfer units
\[ m_y=\int_{y_1}^{y_2}\frac{dy}{y-y^*} \]
is represented in the following form:
for countercurrent flow
\[ m_y=a\int_{x_1}^{x_2}\frac{dx}{(y_0-y^*)+a(x-x_0)}, \tag{8} \]
for cocurrent flow
\[ \overline{m}_y=a\int_{x_1}^{x_2}\frac{dx}{(y_0-y^*)-a(x-x_0)}. \tag{9} \]
From (8) and (9) we find the difference between \(\overline{m}_y\) and \(m_y\), and then represent it in the form of the following sum of two integrals:
\[ \overline{m}_y-m_y = 2a^2\int_{x_1}^{x_0} \frac{x-x_0}{(y_0-y^*)^2-a^2(x-x_0)^2}\,dx + 2a^2\int_{x_0}^{x_2} \frac{x-x_0}{(y_0-y^*)^2-a^2(x-x_0)^2}\,dx. \tag{10} \]
On the segment \([x_1,x_0]\) introduce the variable \(z\) by the equation
\[ x=x_0-z, \tag{11} \]
where
\[ 0\le z\le z_1=\frac{x_2-x_1}{2}. \tag{12} \]
Denote
\[ \varphi(z)=f(x_0-z)=f(x)=y^*, \tag{13} \]
\[ F(z)=[y_0-\varphi(z)]^2-a^2z^2=[(y_0-y^*)^2-a^2(x-x_0)^2]. \tag{14} \]
On the segment \([x_0,x_2]\) introduce the variable \(z\) by the equation
\[ x=x_0+z; \tag{15} \]
Denote
\[ \varphi_1(z)=f(x_0+z)=f(x)=y^*, \tag{16} \]
\[ F_1(z)=\left[y_0-\varphi_1(z)\right]^2-a^2z^2=(y_0-y^*)^2-a^2(x-x_0)^2. \tag{17} \]
On the basis of equations (11)—(17), we give expression (10) the following form:
\[ \frac{\overline{m}_y-m_y}{2a^2} = \int_{z_1}^{0}\frac{z\,dz}{F(z)} + \int_{0}^{z_1}\frac{z\,dz}{F_1(z)}. \]
Hence we obtain
\[ \frac{\overline{m}_y-m_y}{2a^2} = \int_{0}^{z_1} \frac{F(z)-F_1(z)}{F(z)F_1(z)}\,z\,dz. \tag{18} \]
Let us find the relation between the functions \(F(z)\) and \(F_1(z)\).
From (2) and (16) it follows that
\[ \varphi_1(z)<y_1. \tag{19} \]
Using (5), (7), (12), (19), from (17) we obtain
\[ F_1(z)=\left[y_0-\varphi_1(z)\right]^2-a^2z^2>(y_0-y_1)^2-a^2z^2\geq (y_0-y_1)^2-a^2z_1^2=0. \]
Consequently,
\[ F_1(z)>0. \tag{20} \]
Owing to the increasing character of function (1), its values corresponding to the segment \([x_1,x_0]\) will be less than the values of this function determined on the segment \([x_0,x_2]\). Therefore, from (13) and (16) we obtain
\[ \varphi(z)<\varphi_1(z). \tag{21} \]
From (14), (17), and (21) it follows that
\[ F(z)-F_1(z)=\left[y_0-\varphi(z)\right]^2-\left[y_0-\varphi_1(z)\right]^2>0. \tag{22} \]
Now from (20) and (22) we obtain
\[ F(z)>F_1(z)>0. \tag{23} \]
From (23) it is evident that in (18) the integrand is positive. Consequently, in (18) the integral and, together with it, the difference between \(\overline{m}_y\) and \(m_y\) are positive quantities [3]. It follows from this that
\[ \overline{m}_y>m_y. \]
The mean driving force has the following expressions:
for co-current flow
\[ \overline{\Delta y}_{\mathrm{cp}}=\frac{y_2-y_1}{\overline{m}_y}, \]
for counter-current flow
\[ \Delta \overline{y}_{\mathrm{cp}}=\frac{y_2-y_1}{m_y}. \]
Applying the last inequality to the given expressions for the mean driving force, we obtain the desired relation
\[ \Delta y_{\mathrm{cp}}>\overline{\Delta y}_{\mathrm{cp}}. \]
REFERENCES
-
B. N. Devyatov, Proceedings of the Ural Scientific Research Chemical Institute (UNIKhIM). Works on the Technology of Inorganic Substances and Automatic Control. Leningrad, Goskhimizdat, 1958, issue VII, 147—149.
-
Chemist’s Handbook, vol. V. “Khimiya,” 1966.
-
V. A. Il’in, E. G. Poznyak, Foundations of Mathematical Analysis. Moscow, “Nauka,” 1967.