Methodology for Calculating Evaporating Equipment with Direct Electric Heating
Abstract Generated abstract
This paper presents a calculation method for evaporation equipment using direct electric heating, in which heat is generated within the processed solution by current flow rather than transferred through a wall. It formulates the heat release and equipment power from the heat balance and Joule heating, then develops resistance calculations for the electrode, solution, and boundary layers, accounting for solution conductivity, electrode geometry, gas filling, and experimentally determined limiting resistances. The method gives design equations for plane parallel and cylindrical coaxial electrode systems, including current, voltage, power, and principal dimensions based on selected current density. Laboratory verification reported deviations of released power from calculated values of up to plus or minus 5 percent.
Full Text
Method for Calculating Evaporation Equipment with Direct Electric Heating
V. P. Pishchulin
(Presented by the scientific-methodological seminar of the Department of Processes, Machines, and Apparatuses of Chemical Production)
Electrical energy is the most effective and promising heat carrier in various processes and apparatuses of chemical technology. The development of a number of designs of apparatuses with direct electric heating requires a methodology for calculating the equipment. In the present communication, primary attention is devoted to the method for calculating evaporation equipment with direct electric heating, since with direct electric heating the heat is released directly in the material being heated itself as a result of the passage of electric current through the material, which serves as a resistance; that is, with direct electric heating, heat transfer through a wall is completely absent. At the same time, it is possible to release a large amount of energy in definite volumes, with a high rate of heating and, as a result, to obtain any desired temperatures. The coefficient of useful conversion of electrical energy into heat is 95–98%, heat supply is simplified and, naturally, so is the thermal calculation of the equipment. Electrothermal units are compact and simple in design.
The required amount of heat and the heater power for carrying out the process of heating or evaporating solutions are calculated from the heat balance of the apparatus. This heat is released when alternating electric current passes through the system electrode—solution—electrode according to the Joule–Lenz law
\[ Q = k I U \tau = k I^{2} R \tau, \tag{1} \]
where \(I\) is the current strength, A;
\(U\) is the operating voltage, V;
\(R\) is the resistance, ohm;
\(\tau\) is time;
\(k\) is the proportionality coefficient.
In the system electrode—solution—electrode, the electrodes possess electronic conductivity, while the heated or evaporated solution has ionic conductivity, and the resistance of the system will consist of the resistance of the electrodes, of the solution, and of two resistances at the electrode—solution boundary, determined by the resistance of the double electric layer and by chemical reactions on the surface of the electrodes. Since the resistance of the electrodes is a relatively small quantity, it may be considered that the resistance of the solution and the boundary resistances determine the total resistance of the system and the process of heat release.
To determine the resistance of the solution, it is necessary to know the specific…
the electrical conductivity of the solution, the distance between the electrodes, their shape and arrangement, and gas filling of the electrolyte.
\[ R_{\mathrm{p}}=K_{\mathrm{g}}\cdot K\cdot \frac{1}{\chi}\cdot \frac{\delta}{F}, \tag{2} \]
\[ K_{\mathrm{g}}=\frac{1}{1-1.78\Gamma+\Gamma^2}, \tag{3} \]
\[ \Gamma=\frac{V_{\mathrm{g}}}{V_{\mathrm{g}}+V_{\mathrm{p}}}, \tag{4} \]
where \(\chi\) is the specific electrical conductivity of the solution, \(\mathrm{ohm}^{-1}\cdot\mathrm{cm}^{-1}\);
\(\delta\) is the distance between the electrodes, cm;
\(F\) is the working surface area of the electrodes, \(\mathrm{cm}\);
\(K_{\mathrm{g}}\) is a coefficient taking into account gas filling of the electrolyte [1],
\(K\) is a coefficient taking into account the shape and arrangement of the electrodes;
\(\Gamma\) is gas filling;
\(V_{\mathrm{g}}\) is the volume of gas in the mixture;
\(V_{\mathrm{p}}\) is the volume of solution in the mixture.
The magnitude of the limiting resistance is determined by the specific limiting resistance and the working surface area of the electrodes. The specific limiting resistance depends on the electrode material, the composition and concentration of the solution, the process temperature, and the current density on the electrode surface, and is determined experimentally
\[ R_{\mathrm{gr}}=\frac{R_{\mathrm{ud.gr}}}{F}, \tag{5} \]
where \(R_{\mathrm{ud.gr}}\) is the specific limiting resistance, \(\mathrm{ohm}\cdot\mathrm{cm}^2\).
The current strength is determined by the current density and the working surface area of the electrode. The optimum current density is selected experimentally for each process, depending on the electrophysical and corrosion properties in the electrode—solution—electrode system. From the current density selected on the basis of experimental data and from the ratios of the geometric dimensions and configuration of the electrodes, the electrical resistance in the system, the linear dimensions, the operating voltage, and the current strength are determined for plane-parallel electrodes by jointly solving equations (6–12), and for cylindrical coaxial electrodes by equations (13–20).
\[ I=i\cdot F, \tag{6} \]
\[ R=K_{\mathrm{g}}\cdot K\cdot \frac{1}{\chi}\cdot \frac{\delta}{F} +2\frac{R_{\mathrm{ud.gr}}}{F} = \frac{K_{\mathrm{g}}\cdot K}{\chi\cdot n\cdot m\cdot \delta} + 2\frac{R_{\mathrm{ud.gr}}}{n\cdot m\cdot \delta^2}, \tag{7} \]
\[ U=IR=i\left(\frac{K_{\mathrm{g}}\cdot K\cdot \delta}{\chi}+2R_{\mathrm{ud.gr}}\right), \tag{8} \]
\[ N=IU=i^2\cdot F\left(\frac{\delta\cdot K_{\mathrm{g}}\cdot K}{\chi}+2R_{\mathrm{ud.gr}}\right) =i^2\cdot \delta^2 m\cdot n \left(\frac{\delta\cdot K_{\mathrm{g}}\cdot K}{\chi}+2R_{\mathrm{ud.gr}}\right), \tag{9} \]
\[ F=a\cdot b=n\cdot m\cdot \delta^2, \tag{10} \]
\[ a=n\cdot \delta, \tag{11} \]
\[ b=m\cdot \delta, \tag{12} \]
where \(i\) is the current density, \(\mathrm{A}/\mathrm{cm}^2\);
\(a\) is the electrode width, cm;
\(b\) is the electrode length, cm;
\(n\) is the coefficient of the ratio of the electrode width to the distance between the electrodes;
\(m\) is the coefficient of the ratio of the electrode length to the distance between the electrodes.
\[ I=i\cdot F_{\mathrm{avg}}=i\cdot 2\pi\cdot n\cdot r_{\mathrm{v}}\frac{r_{\mathrm{v}}(m-1)}{\ln m}, \tag{13} \]
\[ F_{\mathrm{avg}}=2\pi\cdot r_{\mathrm{avg}}H, \tag{14} \]
\[ r_{\mathrm{avg}}=\frac{r_{\mathrm{n}}-r_{\mathrm{v}}}{\ln \dfrac{r_{\mathrm{n}}}{r_{\mathrm{v}}}}, \tag{15} \]
\[ H=n\cdot r_{\mathrm{v}}, \tag{16} \]
\[ r_{\mathrm{n}}=m\cdot r_{\mathrm{v}}, \tag{17} \]
\[ R=R_{\mathrm{p}}+R_{\mathrm{gr.n}}+R_{\mathrm{gr.v}} =\frac{1}{2\pi n\cdot r_{\mathrm{v}}} \left( \frac{K_{\mathrm{n}}\cdot K}{\chi}\ln m +\frac{R_{\mathrm{gr.n}}}{m\cdot r_{\mathrm{v}}} +\frac{R_{\mathrm{gr.v}}}{r_{\mathrm{v}}} \right), \tag{18} \]
\[ U=I\cdot R =i\cdot r_{\mathrm{v}}\cdot\frac{m-1}{\ln m} \cdot \left( \frac{K_{\mathrm{g}}\cdot K}{\chi}\cdot \ln m +\frac{R_{\mathrm{gr.n}}}{m\cdot r_{\mathrm{v}}} +\frac{R_{\mathrm{gr.v}}}{r_{\mathrm{v}}} \right), \tag{19} \]
\[ N=I\cdot U =2\pi n\cdot i^{2}\cdot r_{\mathrm{v}}^{3} \left(\frac{m-1}{\ln m}\right)^{2} \cdot \left( \frac{K_{\mathrm{g}}\cdot K}{\chi}\ln m +\frac{R_{\mathrm{gr.n}}}{m\cdot r_{\mathrm{v}}} +\frac{R_{\mathrm{gr.v}}}{r_{\mathrm{v}}} \right), \tag{20} \]
where \(F_{\mathrm{avg}}\) is the average surface through which the current passes, \(\mathrm{cm}^{2}\);
\(r_{\mathrm{avg}}\) is the average radius of the interelectrode space, cm;
\(r_{\mathrm{n}}\) is the radius of the outer electrode, cm;
\(r_{\mathrm{v}}\) is the radius of the inner electrode, cm;
\(n\) is the coefficient of the ratio of the electrode length to the radius of the inner electrode;
\(m\) is the coefficient of the ratio of the radii of the outer and inner electrodes;
\(R_{\mathrm{gr.n}}\) is the specific boundary resistance of the outer electrode, \(\Omega\cdot\mathrm{cm}^{2}\);
\(R_{\mathrm{gr.v}}\) is the specific boundary resistance of the inner electrode, \(\Omega\cdot\mathrm{cm}^{2}\);
moreover, \(R_{\mathrm{gr.n}}\) and \(R_{\mathrm{gr.v}}\) are selected at the corresponding current density
\[ i_{\mathrm{n}}= \frac{r_{\mathrm{n}}-r_{\mathrm{v}}}{r_{\mathrm{n}}\cdot \ln \dfrac{r_{\mathrm{n}}}{r_{\mathrm{v}}}} \cdot i = \frac{m-1}{m\ln m}\cdot i, \tag{21} \]
\[ i_{\mathrm{v}}= \frac{r_{\mathrm{n}}-r_{\mathrm{v}}}{r_{\mathrm{v}}\cdot \ln \dfrac{r_{\mathrm{n}}}{r_{\mathrm{v}}}} \cdot i = \frac{m-1}{\ln m}\cdot i. \tag{22} \]
Verification of the calculation under laboratory conditions showed that the deviation of the power released in the apparatus during the investigation from the calculated value was up to \(\pm 5\%\).
REFERENCES
- N. P. Fedot’ev [et al.]. Applied Electrochemistry. “Khimiya,” 1967.
- Research in the Field of Industrial Electric Heating. Proceedings of VNIIETO. Issue 2. Edited by A. P. Al’tgauzen and L. E. Nikol’skii. “Energiya,” 1967.