In the given problem, it involves the design of a vertical Thermosyphon reboiler, taken from the book Coulson and Richardson$$s Chemical Engineering Series, Chemical Engineering Design, Sixth Edition by Ray Sinnott and Gavin Towler. Example $12\mathrm{.}11($page $747)$

He calculated the static pressure by assuming that the fluid density in the tube is a linear function with the length of the tube. Explain why this assumption is necessary and prove the following equation in the example.

$P_s=g{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{dx}{v_i+x(v_o-v_i)}=\frac{gL}{(v_o-v_i)}Ln(\frac{v_o}{v_i})$