ANS:D - tray to tray calculations.

Tray-to-tray calculations, also known as stage-by-stage calculations, are a rigorous method used in the design and analysis of distillation columns. In tray-to-tray calculations, mass and energy balances are performed for each individual stage or tray of the distillation column to determine the composition and temperature profiles along the column height. Here's a basic overview of how tray-to-tray calculations are conducted:

1. Assumptions: Before starting the calculations, certain assumptions are made regarding the behavior of the system. These assumptions may include ideal stage behavior, constant molar overflow on trays, constant molar heat capacities, and others depending on the complexity of the system.
2. Component Mass Balances: For each stage of the distillation column, component mass balances are performed to determine the compositions of the liquid and vapor phases leaving each stage. These balances account for the inflow and outflow of each component on the tray, considering factors such as vapor-liquid equilibrium and tray efficiency.
3. Energy Balances: Energy balances are conducted for each stage to determine the temperature profiles along the column height. These balances take into account the heat input from the reboiler at the bottom of the column and the heat removal from the condenser at the top, as well as heat transfer between the liquid and vapor phases on each tray.
4. Equilibrium Relations: Equilibrium relations are used to calculate the compositions of the vapor and liquid phases in equilibrium on each tray. These relations are typically represented by the equilibrium curve or equations specific to the mixture being distilled.
5. Iteration: Tray-to-tray calculations often involve iterative procedures to converge on the correct tray temperatures and compositions. This may involve adjusting parameters such as reflux ratio, reboiler duty, or tray efficiencies until convergence is achieved.
6. Tray Efficiency: Tray efficiency factors may be incorporated into the calculations to account for non-idealities such as liquid entrainment, vapor bypassing, and imperfect mixing on the trays.
7. Sensitivity Analysis: After obtaining the tray compositions and temperatures, sensitivity analysis may be performed to assess the effects of changes in operating conditions or column parameters on the column performance.

Tray-to-tray calculations provide detailed insights into the behavior of distillation columns and are particularly useful for complex systems with multiple components, non-ideal behavior, and stringent separation requirements. However, they can be computationally intensive and require careful consideration of various factors and assumptions.