Batch Distillation: Thermodynamic Efficiency

The batch distillation is a separation processes that requires great amounts of energy. Due to high energy costs, the study of energy consumption is of great interest in this process (Zavala et al. 2007). According to Luyben (1990), the energy consumption increases when operation occurs in a batch manner. Determining how efficient is the heat transfer under specific conditions and to modify them in order to find how efficient the heat is used, is an important task.


Introduction
The batch distillation is a separation processes that requires great amounts of energy. Due to high energy costs, the study of energy consumption is of great interest in this process (Zavala et al. 2007). According to Luyben (1990), the energy consumption increases when operation occurs in a batch manner. Determining how efficient is the heat transfer under specific conditions and to modify them in order to find how efficient the heat is used, is an important task.
The analysis of thermodynamic efficiency in a batch distillation column has been presented by Kim and Diwekar (2000), Zavala-Loría (2004), Zavala et al. (2007), Zavala & Coronado (2008) and Zavala et al. (2011). The first work only developed expressions to calculate thermodynamic efficiency while the rest of the works developed expressions to calculate thermodynamic efficiency and have applied them to a new problem of optimal control: Maximum thermodynamic efficiency.

Description of the process
For this study we used a conventional batch distillation column consisting of: Reflux drum  Receiver Figure 1 Shows a conventional batch column like the one used for this study.
Mass balances resulting from the process shown in Figure 1, allow us to obtain the mathematical model in For non-ideal mixtures, we can use the following equation: Where K (i) is the constant of the VLE, and y (i) is the mole fraction of the vapor phase, x (i) is the mole fraction of the liquid phase,  i represents the activity coefficient, P sat is the saturation pressure and  i  denotes the partial fugacity coefficient, all referring to component i, and P

Thermodynamic efficiency
By definition, the thermodynamic efficiency ( t ) of a process based on availability or exergy is defined as: where W is the work and LW is the total loss of work.
Since the minimum work is determined by changes in exergy, they can be determined using the First and Second Law of Thermodynamics. Figure 2 shows the control volume used to obtain the equations that represent thermodynamic efficiency in batch distillation. Figure 2 shows that the process can exchange energy with the environment but does not perform any mechanical work. The energy balance (enthalpy) given by the First Law of Thermodynamics, considering the reboiler, the column of trays and the condenser-reflux drum, is: According to the Second Law of Thermodynamics, the entropy balance is: where: I represents enthalpy, S is entropy, Q is the amount of heat, C represents the condenser, D represents the distillate, B represents the bottoms, m is the system mass, 0 is the reference state, t is the stage or tray and T is the temperature. The available work can be obtained if we combine equations (11) and (13). To do so, equation (13) must be multiplied by the temperature of the reference state T 0 (it is considered that the reference state is liquid at 25 °C and 1.0 atm):  (16) represent the difference between the exergy or the availability of flows entering and leaving the system. The term on the left hand side is the change of exergy in the system. Finding the values in Equation (16), the work loss is: can be calculated, if discretised, by multiplying Equation (12) by T 0 minus Equation (10): Another way to estimate the value of   sist dm dt A is by applying an exergy balance at the bottom of the column. Applying an exergy balance in the reboiler, we have: Introducing Equation (1) According to Kim and Diwekar (2000) and Zavala-Loría and Coronado-Velasco (2008), the total exergy can be calculated from its physical component ( phis A ) and its chemical where phis A considers the physical processes that involve thermal interactions with the surroundings and chem A considers the mass and heat transfer with the surroundings. The main contribution to this energy is due to mixing effects and can be estimated from the chemical potential at low pressures (Kim and Diwekar, 2000). phis A is relatively lower than chem A . Thus, the physical component can be regarded as constant for all chemical species and the derivative of this term is eliminated. The chemical component of exergy for an ideal mixture can be expressed as: whereas a non-ideal mixture can be calculated as: and the exergy exchange in the reboiler for an ideal mixture can be calculated as: for a non-ideal mixture: Taking the derivative of Equations (24) and (25) yields: The derivative of the term on the right hand side of Equations (26) and (27) Substituting Equation (28) on the right hand side of Equation (20) for an ideal mixture, and Equation (29) on the right hand side of Equation (20) for a non-ideal mixture yields: The exergy of the current production (dome) that will be used in Equation (6) for an ideal mixture can be calculated as: for a non-ideal mixture: The exergy transfer associated with the transmission of energy as heat in the process can be calculated by the energy balances in the reboiler and condenser. Considering that H vap is the same for each component and is not related to the temperature of the process, the Clausius-Clapeyron equation can be used to calculate H vap , and the first two terms on the right hand side of Equation (16) Non-ideal Mixture: where, K is the liquid-vapor equilibrium constant and Φ is defined as: Therefore, the term for exergy loss or work loss, LW in Equation (17) the minimum work can be calculated as: Equation (45) can be reduced if one considers that the changes of activity coefficients in the reboiler are so small that they can be neglected. Then: If the gas phase is ideal or close to ideality, then Equation (46) can be reduced even more: The solution of Equation (45) or its simplifications [Equation (46) and (47)

Results and discussion
Using the First and Second Law of Thermodynamics, this work has developed an expression for calculating the thermodynamic efficiency of a batch distillation process [Equations (44) and (45)]. To solve the mathematical model, the feeding was introduced at the top of the column at boiling temperature, neglecting the accumulation of vapor in each stage. The process was performed with total condenser, atmospheric pressure and adiabatic column.
To solve the model, two mixtures were considered. The first one is an ideal mixture of Hexane/Benzene/Chlorobenzene (HBC); the second one is a non-ideal mixture of Ethanol-Water (EW).

Case of study 1: Hexane/Benzene/Chlorobenzene mixture
The conditions and data used to solve the mathematical model of thermodynamic efficiency are given in Table 2 The first step for the resolution of the mathematical model was to reach the steady state to extract the product during one hour. Figure 3, shows the concentrations at the top of the column (including the steady state). The average thermodynamic efficiency is 17.58% for this production time while the average concentration of the most volatile component at the top of the column is 88.55% mol. Table 3, shows the variations observed when the reflux ratio is changed.  Table 3. Thermodynamic efficiency behavior based on variations in the reflux ratio. Figure 4 shows the thermodynamic efficiency behavior in the product obtaining. Such variations make evident the influence of the reflux ratio over the thermodynamic efficiency; in other words, the thermodynamic efficiency of the process is smaller when the reflux ratio is higher. We also could observe that the reflux ratio affects the product concentration. If the reflux ratio increases the product concentration increases as well.   Figure 5 shows the behavior of the concentrations at the top of the column, while the behavior of punctual thermodynamic efficiency is shown in Figure 6. Table 5 shows the variations observed when different reflux ratios are used. Fig. 6. Thermodynamic efficiency profile (obtaining product).  Table 5. Thermodynamic efficiency behavior with regards to the reflux ratio variation.

Reflux ratio
As in the last case, we observe that when the reflux ratio increases, the efficiency decreases; however, the concentration of the most volatile component presents also an increment. We can formulate the following heuristic rule: If the variables of the process are maintained, the reflux ratio will have an inverse effect on the thermodynamic efficiency of the process.

Concluding remarks
Using the First and Second Law of Thermodynamics (exergy concept), this work has developed an expression for calculating the thermodynamic efficiency of a batch distillation Time, h process. The resulting equation was used to find the batch distillation thermodynamic efficiency for an ideal mixture and a non-ideal mixture. The equation obtained is a generalization of the equation developed by Zavala-Loría et al. (2007), Zavala and Coronado (2008) and Zavala et al. (2011).
The results obtained by solving the Equations, allowed us to observe the relationship between reflux and thermodynamic efficiency of the process. Furthermore, variables such as the product purity are affected by the reflux ratio, in other words, the purity of the product requires a greater amount of reflux to obtain a higher concentration.