2.8. Detailed equipment equations

A distillation column can be described using several material and energy balance equations. Figure 4 shows a typical distillation column.

Tray distillation column.

The overall material balance can be found as follows:

where F, D, and W are the feed, distillate, and residue rates, respectively (kg/h).

The component material balance can be found as follows:

where x_f, x_d, and x_w are the distillate compositions in the feed, distillate, and residue, respectively.

The overall heat balance can be found as follows:

where Q_r and Q_c are the reboiler duty and condenser duty, respectively, and h_f, h_d, and h_w are the specific enthalpies (J/kg) for the feed, distillate, and residue, respectively.

Here, c_p is the sensible heat (J/kg °C).

Here, V and L are the rates of the vapor at the top plate and liquid flow returning to the top plate (kg/h), respectively.

Here, λ_mix and λ _steam are the latent heat of vaporization (J/kg) values for the overhead mixture and reboiler steam, respectively.

The material balance equations were presented in Eq. (10). The general equation for the heat transfer across a surface is as follows:

where Q, U, A, and ΔT_m are the heat transfer per unit time (W), overall heat transfer coefficient (W/m² °C), area of heat transfer (m²), and log mean temperature difference (°C), respectively. The log mean temperature difference was calculated using the following equation:

where T₁ is the hot-fluid inlet temperature, T₂ is the hot-fluid outlet temperature, t₁ is the cold fluid inlet temperature, and t₂ is the cold fluid outlet temperature.

The total energy required can be calculated from the following equation:

where W is the work done (J/kg), Δz is the elevation difference between z₁ and z₂ (m); ΔP is the system pressure difference between P₁ and P₂ (N/m²); ΔPf is the pressure drop due to friction, including miscellaneous losses and equipment losses (N/m2); ρ is the liquid density (kg/m3); and g is the acceleration due to gravity (m/s²) [24]. Figure 5 shows the piping system.

Piping system.