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differing dielectric constants (Equation 5.33) between domains I and II to be able to get both positive and negative values for ∆φ ̃s. 5.2.1 A Note on Evaluation Stability Since L ̃ can get very large in equations 5.22 – 5.26, both the numerator and denomi- nator can get very large, to the point at which calculation tools will set them equal to infinity, which is of course problematic for evaluation of these integration constants necessary for calculation of ∆φ ̃s. To avoid problems with calculating these fractions, they had to be rearranged. First, we define a ratio of hyperbolic sines that will help us in evaluation. sinh(b2L ̃) e(b2−b1)L ̃ − e−(b1+b2)L ̃ r1 = sinh(b1L ̃) = b1b2 = δ1 1 − e−2b1L ̃ (5.35) (5.36) (5.37) Further, we note that We also defined a modified c6, since the term it appears in is always the same. q ̃b3r−b3−ρ ̃b (b2−b2)sinh(bL ̃) ̃ s211q ̃sδc12 2 cˆ6 = c6ξ exp (−rλL) = (b2−b2) b3 cosh(b L ̃)r −b3 cosh(b L ̃)− 1 2 sinh(b L ̃) 21112ξδc 2 c For the KCl electrolyte, this was all the modification that was required to be able to reliably calculate all constants. However, for the other two electrolytes, further mod- ification to remove all hyperbolic functions was required. By replacing the hyperbolic functions with their exponential equivalents and manipulating these, we obtain cˆ6 = 3 ( b − b ) L ̃ (b −b ) 21 2 2 (5.38) q ̃ 2b3r e−b2L ̃ −2b3e−b2L ̃ − ρ ̃b (b2 −b2)1−e−2b2L ̃ s21 1 q ̃sδc12 b2r1 e1 2 − ( b + b ) L ̃ +e 1 2 3 − 2 b L ̃ −b1 1+e 2 − ξδc − 2 b L ̃ 1−e 2 123PDF Image | Shock Electrodialysis for Water Purification and Electrostatic Correlations
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