A. Heat transfer and pressure drop for the shell side of heat exchangers with twisted tubes in longitudinal flow

The pressure loss for the shell-side flow of a variable density fluid (i.e. a gas) can by calculated from the momentum equation for a section of finite length:

\[\label{eq1} \Delta p = f \frac{l_{c}}{d_{eq}}\left(\dfrac{\dot{M}}{F_{f}}\right)\dfrac{1}{2 \rho_{mean}}+\left(\frac{\dot{M}}{F_{f}}\right)^{\!2}\left[\left(\frac{1}{\rho_{2}}-\frac{1}{\rho_{1}}\right)\right], \tag{1}\]

where F_f is the cross sectional flow area and where

\[\rho_{mean} = \frac{p_{mean}}{RT_{mean}}, \quad T_{mean}=\dfrac{T_{1}+T_{2}}{2}, \quad p_{mean}=\dfrac{p_{1}+p_{2}}{2}.\]