2.3.2. Solution Procedure Using Model Equations

Step 1: Assume *C*<sup>p</sup> = *C*pA (*initial guess C*pA = 0) Step 2: Calculate {*H*+} <sup>=</sup> <sup>10</sup>−*pH* Step 3: Calculate ∆*P*<sup>i</sup> = *P*<sup>i</sup> − *P*<sup>p</sup> Step 4: Calculate *A*<sup>w</sup> and *B*<sup>S</sup> (Equations (27) and (28)) Step 5: Calculate *φ* (Equation (18)) Step 6: Calculate *F*<sup>o</sup> (Equation (15)) Step 7: Calculate *P*<sup>o</sup> (Equation (16)) Step 8: Calculate ∆*P*<sup>o</sup> = *P*<sup>o</sup> − *P*<sup>p</sup> Step 9: Calculate *J*wi = 2·*A*w∆*P*<sup>i</sup> 1+ *A*w*iγ B*S *TC*p and *J*wo = 2·*A*w∆*P*<sup>o</sup> 1+ *A*w*iγ B*S *TC*p Step 10: Calculate *v*<sup>i</sup> = *F*i *A*f and *v*<sup>o</sup> = *F*o *A*f Step 11: Calculate *C*<sup>o</sup> = *C*<sup>p</sup> + *<sup>F</sup>*i(*C*i−*C*p) *F*o Step 12: Calculate *Re*f,i, *Re*p,i, *Sc*<sup>i</sup> and *Re*f,o, *Re*p,o, *Sc*<sup>o</sup> Step 13: Calculate *Sh*<sup>i</sup> and *Sh*<sup>o</sup> (Equation (14)) Step 14: Calculate *k*<sup>i</sup> = *Sh*<sup>i</sup> ·*D*<sup>i</sup> *d*e and *k*<sup>o</sup> = *Sh*o·*D*<sup>o</sup> *d*e Step 15: Calculate *C*pi = <sup>h</sup> *C*i 1+ (*J*Wi/*B*<sup>s</sup> ) *exp*(*J*Wi/*k*i) <sup>i</sup> and *C*po = <sup>h</sup> *C*o 1+ (*J*Wo/*B*<sup>s</sup> ) *exp*(*J*Wo/*k*<sup>o</sup> ) i Step 16: Calculate *C*pA = *C*pi+*C*po 2

Step 17: Calculate *C*iw = *C*pA + *J*wi·*C*pA *B*S Step 18: Calculate *K*a1 (Equation (32), where *S* = *C*iw·*MW*NaCl) Step 19: Calculate *B*<sup>B</sup> (Equations (29)–(31)) Step 20: Calculate *C*Bp = <sup>h</sup> *C*Bi 1+ (*J*Wi/*B*<sup>B</sup> ) *exp*(*J*Wi/*k*i) i

Step 21: If *C*<sup>p</sup> − *C*pA  > tolerance → Go to step 5, otherwise stop

In step 21 a tolerance of <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> was implemented to give a reasonable convergence of the calculated concentration. Additionally, the dimensionless numbers were calculated using the following correlations [22,31]:

$$Re\_{\rm f} = \frac{\rho \, d\_{\rm e} \, v}{\mu} \tag{35}$$

$$Re\_{\rm P} = \frac{\rho \, d\_{\rm e} \, J\_{\rm w}}{\mu} \tag{36}$$

$$\text{Sc} = \frac{\mu}{\rho \, D} \tag{37}$$

where *d*<sup>e</sup> = *t*f/2 is the equivalent diameter [31]. In addition, the density and viscosity can be estimated through the following correlations of Koroneos et al. [39]:

$$
\rho = 498.4m + \sqrt{248400m^2 + 752.4 \text{ m S}} \tag{38}
$$

$$m = 1.0069 - 2.757 \times 10^{-4} \cdot (T - 273.15) \tag{39}$$

$$\mu = 1.234 \times 10^{-6} \cdot \exp\left[0.0212 \cdot S + \frac{1965}{T}\right] \tag{40}$$

$$D = 6.725 \times 10^{-6} \times \exp\left[0.1546 \times 10^{-3} \text{S} - \frac{2513}{T}\right] \tag{41}$$
