Patlak equation

Question

For solute flux across microvessel wall, these two equations are supposed to be equivalent:

$$ \begin{align} J_s &= J_v(1-\sigma_f)\frac{C_i - C_Le^{\mathrm{Pe}}}{1 - e^{\mathrm{Pe}}} \\ J_s &= J_v(1- \sigma_f) \left( C_L - \frac{\Delta C}{1 - e^{\mathrm{Pe}}}\right) \end{align} $$

where $ \Delta C = C_L - C_i $ and Pe is the Peclet number.

But how? It should be obvious, but I don't see it. I'm hoping it's just a mathematical argument that I'm missing, so somebody can enlighten me without the physical background.

Answer 1

$$\frac{C_i-C_Le^{Pe}}{1-e^{Pe}}=\frac{C_i-C_Le^{Pe}+C_L-C_L}{1-e^{Pe}}=\frac{C_L(1-e^{Pe})-(C_L-C_i)}{1-e^{Pe}}$$ $$=C_L-\frac{(C_L-C_i)}{1-e^{Pe}}=C_L-\frac{\Delta C}{1-e^{Pe}}$$

Answer 2

$C_L=\frac{C_L(1-e^{Pe})}{1-e^{Pe}}$