Chapter 3. Separation Equipment

Separations require a basis and a separating agent. The separation basis is the physical property being exploited. For example, when drying hair with a hair dryer, the difference in boiling points (or volatility) between water and hair is the separation basis. Distillation also exploits the difference in boiling points between components, as most students have seen in organic chemistry lab.

This illustrates the subtle differences in the nomenclature used for separations. Distillation refers to the separation where both components can vaporize at typical operating conditions. Evaporation refers to the separation where one component does not vaporize at typical operating conditions. In a chemical engineering context, a solid can be separated from a solvent by evaporating the solvent. Two components that have boiling points that differ by 20°C, for example, can be separated by preferentially boiling the component with the lower boiling point. Another difference between these two separations is that the solvent obtained through evaporation will be essentially pure; however, in distillation the lower boiling component will contain some of the higher boiling component.

The separating agent is employed to exploit the separation basis to effect the separation. In distillation and evaporation, the separating agent is energy. Most students are also familiar with extraction from organic chemistry lab, where a solute is transferred from one phase to another, immiscible, phase. In this case, the destination solvent is the separating agent, and the general category is called mass separating agents. Another familiar mass separating agent is involved in the brewing of real (not instant) coffee, in which hot water removes the flavor ingredients from the solid coffee bean, but the coffee bean does not dissolve in the water. This solid-liquid separation is called leaching.

Table 3.1 shows some typical separations used in chemical engineering, their basis, and the separating agent.

and the component mass balance for species i is

where the stream flowrates (F, L, V) are either in mass or mole units, and the fractions (x_i, y_i, z_i) are either mass fractions or mole fractions, respectively.

For a mass separating agent, such as extraction, as illustrated in Figure 3.1(b), wherein solute i is transferred from the liquid stream L to the vapor stream V or vice versa, the mass balances are

where, once again, if the flowrates are in mass units, the fractions are mass fractions, and if the flowrates are in mole units, the fractions are mole fractions.

The nomenclature used for these mass balances is in anticipation of liquid-vapor separations, hence the use of L and V. However, the mass balances are applicable to any separation modeled by either case in Figure 3.1.

where Q is the heat duty, in energy/mass or energy/mole, H is a vapor enthalpy, and h is a liquid enthalpy, both in energy/mass or energy/mole.

For the system illustrated in Figure 3.1(b), the energy balance is

For liquid-liquid separations such as extraction, all enthalpies are for liquids, and the energy balance is not usually needed, since little or no energy of solution is involved in transferring a solute between liquid phases, so the process is essentially isothermal. For some vapor-liquid separations, such as absorption and stripping, the energy balance may be involved, since dissolving a gas in a liquid is often accompanied by a heat of solution, which makes the process nonisothermal.

where, for vapor-liquid separations

where P_i is the partial pressure of component i; is the vapor pressure of component i; is the fugacity coefficient for component i at saturation; γ_i is the activity coefficient for component i; the exponential is known as the Poynting correction factor, with being the liquid molar volume; and is the fugacity coefficient for component i in the vapor phase. The Poynting correction factor only deviates from unity at very high pressures. The fugacity coefficients only deviate from unity at high pressures, and the activity coefficient only deviates from unity for nonideal systems. Therefore, for ideal systems at low pressures, Equation (3.8) reduces to

which is Raoult’s law. For liquid-liquid separations,

where the superscripts I and II refer to the two liquid phases. Usually, for liquid-liquid systems, experimental data are used, and if the data appear to be linear, a constant value for m can be determined. For gases dissolving, but not condensing, in liquids, m can be related to Henry’s law. In Henry’s law, the partial pressure in the vapor phase is related to the liquid mole fraction by p_A = H_Ax_A, where H_A has pressure units, so m in Equation (3.7) becomes H_A/P, where P is the total pressure.

where N_y is the flux of solute in mass or moles/interface area/time and S is the interfacial (mass transfer) area/flow cross-sectional area, which is assumed to be zero at z = 0 and increase proportionally with the coordinate z. Equation (3.11) reduces to

A similar development under the assumption that transfer is from the L phase to the V phase gives

The flux on the V side for this case is (Treybal, 1980)

where c_y is the total molar concentration, D_ABy is the diffusion coefficient of solute (A) in the V phase, δ_y is the film thickness in the V phase, y_A is the mole fraction in the V phase, and y_Ai is the mole fraction in the V phase at the interface. Similarly, for the L phase,

In this illustration, it is assumed that the direction of mass transfer is from the V phase to the L phase; however, the result is the same for transport in the opposite direction. At steady state, the flux from the V phase must be equal to the flux into the L phase. Equating Equations (3.14) and (3.15) yields

where F_y = (c_yD_ABy/δ_y) and F_x = (c_xD_ABx/δ_x). It is observed that F_x and F_y have the same units as N, moles/interfacial area/time. Since this model assumes a stagnant film, and in a real application there will also be convection, the parameters F_x and F_y are effectively mass transfer coefficients. The relationships for these mass transfer coefficients parallel those for heat transfer and are dependent on the actual flow geometry involved. Equation (3.16) can be used to calculate the values of y_Ai and x_Ai from the values far from the interface, based on the assumption that the interface is at equilibrium, so that y_Ai = m_Ax_Ai. It is important to understand that, when this model is used to describe a separation, equilibrium occurs only at the interface, whereas, in the well-mixed, equilibrium model described in Section 3.1.3, the phases are in equilibrium at all points in both phases, meaning that the outlet streams are in equilibrium. Equations (3.14), (3.15), and (3.16) can be simplified for special cases, such as dilute solutions, and these relationships are available (Treybal, 1980).

In the two-film model, the two mass transfer coefficients, F_x and F_y, can be combined into an overall mass transfer coefficient, just as in heat transfer. For an overall mass transfer coefficient based on the V phase, defined as K_y, this result is

and the overall mass transfer coefficient based on the L phase, defined as K_x, is

where the subscript A has been dropped. The result, presented here without derivation, is

In turbulent flow, the mass transfer coefficient is proportional to Re^0.8 or a power close to 0.8 (just as in heat transfer), so the approximation that V/F_y is constant is often made. In this case, Equation (3.20) becomes

This form has the advantage of having one term (V/F_y) that is dependent on the flow geometry and one term (the integral) that is dependent only on the compositions of the phases for any flow geometry. These two terms are usually given the definitions “height of a transfer unit” (H) and “number of transfer units” (N), respectively. The use of the word height is based on a typical application to vertical, packed columns, although, the word length might be a better term. The height of a transfer unit is a measure of 1/separation efficiency, and the number of transfer units is a measure of the difficulty of the separation. The larger the height of a transfer unit, the less efficient the separation, and the larger the number of transfer units, the more difficult the separation. Therefore,

and

A parallel derivation is possible for the L phase, and the results are

and

In principle, the size (interfacial area) of a continuous differential separation unit calculated from either Equation (3.24) or Equation (3.27) is identical. This is similar to the area of a heat exchanger being identical for the two cases of the overall heat transfer coefficient based on the internal surface area (U_i) and the overall heat transfer coefficient based on the external surface area (U_o). The decision on which method to use is based on computational issues; however, it is generally true that the transfer unit expression for the phase with the limiting resistance is the better method to use.

It is also possible to define transfer units based on the overall mass transfer coefficients in Equations (3.17) and (3.18). All cases are presented in Table 3.2.

Assuming that the interfaces are at equilibrium, and using the equilibrium expressions C_A1m = m₁C_A1i and C_A2m = m₂C_A2i, Equation (3.28) can be rearranged into a series resistance form:

It is observed that the denominator of Equation (3.29) resembles the series resistance form in Equation (2.24). The difference, other than geometry, is the different solubilities of the solute in the membrane and in the two external phases. Analogous parameters are not present in the heat transfer form, because the temperature criterion for equilibrium at an interface is equal temperatures. For mass transfer, the criterion for equilibrium is the ratio of the solubilities in the two phases, m_i, which is often called the partition coefficient.

In some cases, it is assumed that the external resistance is negligible, in which case Equation (3.29) reduces to

In the particular application discussed later in this chapter, the two external phases are gases, so it can be assumed that m₁ = m₂ = m, so Equation (3.30) further reduces to

where P is defined as the membrane permeability. (Note that some references define the permeability as P/t_m.) In addition, for applications involving gas permeation, the concentrations are often expressed as partial pressures using the ideal gas law.

Quite often, all four of these operations are called flash separations. Technically, only the reduction in pressure is a “flash” separation, specifically a flash vaporization.

The equations used to solve any of these flash separations are the material balances, Equations (3.1) and (3.2); the energy balance, Equation (3.5); and the equilibrium expression, Equation (3.8). If Equations (3.1), (3.2), and (3.8) are combined, Equations (3.32) and (3.33) can be obtained.

where the subscript i represents each component, and there are C components. In principle, either of these equations can be used to solve for one unknown, either V, temperature, or pressure (inside m_i) if the other two are specified. Then, the mole fractions can be found. It is also possible, in principle, to solve for two of V, T, or P, with any two outlet parameters, including component mole fractions and fractional recoveries. However, since Equations (3.32) and (3.33) are not always monotonic, numerical methods can fail. Additionally, the difference between Equations (3.32) and (3.33) is always monotonic, so it is a better choice for a numerical solution.

Equation (3.34) is often called the Rachford-Rice equation (Wankat, 2017). The energy balance, Equation (3.5), is only used to calculate the heat duty on the heat exchanger used for the partial condensation.

Examples 3.1 and 3.2 illustrate that, without consideration of energy requirements, Equation (3.34) can be used to solve for a single unknown with two specifications. The energy balance, Equation (3.5), can be used the get the heat duty. Problems that couple the energy balance with Equation (3.33) are also possible.

It is observed from the T-xy diagrams that the vapor phase is enriched in the more volatile component, while the liquid phase is enriched in the less volatile component. The horizontal line connecting the vapor and liquid phases in equilibrium is called a tie line. Any mixture brought to a point in the two-phase region separates into two phases connected by the tie line. It is further observed that neither phase is very pure in the enriched component. Now, suppose that the feed is vapor that is partially condensed, and the vapor phase, V₁, is partially condensed again, as illustrated in Figure 3.7, and the process is repeated several times. It is observed that the more volatile component can asymptotically approach purity. Similarly, Figure 3.7 also illustrates that the less volatile component can asymptotically approach purity by partially vaporizing the liquid phase. Each heat exchanger/drum combination is called a stage, and these are called equilibrium stages, because it is assumed that the two phases leaving the stage are in equilibrium. This sequence appears promising; however, as shown Figure 3.7, there are a significant number of waste streams, since only the top or bottom stream is the desired product. Furthermore, a significant number of heat exchangers is required. Suppose the top “waste” stream is recycled to the second-from-the-top stage. It can provide the necessary heat sink in place of one partial condenser. This is illustrated in Figure 3.8. Similarly, if the bottom waste stream is recycled from the second-from-the bottom stage, it can provide the energy for one partial vaporization. This is also illustrated in Figure 3.8. Every waste stream can be returned to the adjacent stage, and Figure 3.9 illustrates the entire process, and it is observed that there is one feed stream and two exit streams, both of which can be very pure in one of the components. Heat is added only at the bottom and energy is removed only at the top. This is how distillation works, although as will be seen later, the actual equipment is more compact.

It is important to realize that while the T-xy diagram provides a mental picture of how distillation works, it is not used for calculations, though it was before high-speed computing. The calculations are done exactly how the diagrams suggest, using a series of mass and energy balances combined with equilibrium expressions from stage to stage, as shown in Section 3.1.

If it is understood that energy is required to “unmix” components, which lowers the system entropy (since mixing is spontaneous), it is no surprise that energy must also be rejected to the surroundings. Overall, energy input is required to unmix the components, but energy must also be rejected to the surroundings, just as in a power cycle.

The material and energy balances, as described by Equations (3.3), (3.4), and (3.6), are written for every tray. An alternative method is to write these balances from the top to each tray above the feed and the bottom to each tray below the feed. One of these sets of balances, along with balances on the condenser, reboiler, and feed tray, are solved simultaneously. Therefore, the number of trays must be known, so that all balances can be written. This means that the outlet conditions can be predicted for a fixed number of trays, feed location, reflux ratio (L₀/D in Figure 3.10[a]), and feed conditions, which is a simulation, not a design. To design a column this way requires iterations, until the number of trays, reflux ratio, and feed location provide the desired outlet conditions. This is why process simulators require that the number of trays and feed location be provided for a rigorous distillation calculation.

There is a simplification that allows a graphical method to design a distillation column. It only works for binary distillations under certain circumstances. However, a complete understanding of this method provides a complete understanding of the operation of a distillation column. This is the approach taken here.

The assumption that simplifies binary distillation calculations is called constant molar overflow (also called constant molal overflow). The assumptions of constant molar overflow are

• Molar heat of vaporization (λ) is the same for each component.

• Column is adiabatic.

• Sensible heat is small compared to latent heat (C_pΔT < < λ).

As a consequence, the moles of vapor condensing on a tray equal the moles of liquid vaporizing on the tray. Therefore, the molar flowrates of liquid and vapor do not change from tray to tray in a given section of the column (a section is either above the feed or below the feed). Additionally, there is no need for an energy balance on any tray, because the energy need for vaporization equals the energy given up by condensation, and since no heat is lost, the heats of vaporization are identical, and they are much larger than any temperature changes between adjacent trays.

Material balances written from the top of the column to a tray (j) above the feed, as illustrated in the top portion of Figure 3.11, are

where the assumption of constant molar flowrates on every tray is used, so that L and V are constant and not indexed by tray number. (Note that trays are numbered from top to bottom.) Equation (3.36) is written on the more volatile component, A. Equations (3.35) and (3.36) can be rearranged to

where R = L/D. Equation (3.37) is now written as

where the subscript for the tray number has been dropped, since the equation is the same for a balance written from the top of the column to any tray. Equation (3.38) is the equation of a straight line with slope L/V and intercept x_AD/(R + 1). While it is possible to plot this line if L/V is known, an easier method is to observe that when y_A = x_A, x_A = x_AD. Therefore, two points are known, the intercept and (x_AD, x_AD). When this line is plotted on an equilibrium diagram, the line labeled 1 in Figure 3.12 is obtained. On Figure 3.12, the curve represents the vapor-liquid equilibrium, which can be obtained from the endpoints of the tie lines on a T-xy diagram such as shown in Figure 3.9. These data can be predicted from Raoult’s law or, in the most general case, Equation (3.8). The diagonal line is just a plot of y_A = x_A.

A similar analysis can be done by writing a material balance from the bottom of the column to any tray below the feed. The situation is illustrated in the lower portion of Figure 3.11. The equations are

where the overbar indicates molar flowrates below the feed. Equations (3.39) and (3.40) can be rearranged to yield, with removal of the index subscript

Equation (3.41) is the equation of a straight line with slope and intercept . If y_A = x_A, x_A = x_AB. Therefore, two points are known, the intercept and (x_AB, x_AB). This line is plotted on Figure 3.12 and labeled 2. It is observed that the intercept is negative because the slope .

The upper section of a distillation column, above the feed, is called the enriching section or the rectification section. The lower section, below the feed, is called the stripping section. The feed tray is the boundary between these two sections. In general, liquid in the feed goes down and vapor goes up; however, what exactly happens on the feed tray depends on whether the feed is saturated liquid, saturated vapor, a vapor-liquid mixture, superheated vapor, or subcooled liquid. It is important to understand that these terms are defined relative to the conditions on the feed tray. For example, if the feed is saturated liquid at 30°C, but the tray temperature is 50°C, then the feed is considered subcooled.

To complete the column model, the top and bottom sections must match at the feed. Figure 3.13 illustrates the feed tray.

The material and energy balances on the feed tray are

where the subscript f indicates the feed tray number, uppercase H indicates vapor enthalpy, lowercase h indicates liquid enthalpy, and h_F is the enthalpy of the feed regardless of its phase. If the feed stage is assumed to be at equilibrium, then the vapor and liquid enthalpies are for saturated conditions. From the constant molar overflow assumption, the enthalpies of each phase are constant across the trays, so, rearranging Equations (3.42) and (3.43), while dropping the subscripts involving f, yields an equation for the quality of the feed, q, defined as the fraction of the feed in the saturated liquid state (as opposed to the quality of steam, which is defined as the fraction of vapor in the steam):

If the feed is saturated liquid, q = 1, so the liquid flowrate below the feed is equal to the liquid flowrate above the feed plus the feed flowrate. The vapor flowrates above and below the feed are identical.

If the feed is saturated vapor, q = 0, so the vapor flowrate above the feed is equal to the vapor flowrate below the feed plus the feed flowrate. The liquid flowrates above and below the feed are identical.

If the feed is a vapor-liquid mixture, 0 < q < 1, so the liquid flowrate below the feed is equal to the liquid flowrate above the feed plus the liquid flowrate in the feed, L_F. The vapor flowrate above the feed is the vapor flowrate below the feed plus the vapor flowrate in the feed, V_F. Simply put, saturated liquid goes down, and saturated vapor goes up.

Subcooled liquid and superheated vapor are not as simple. For a subcooled feed, the feed enthalpy, h_F, is less than the saturated liquid enthalpy, h. Therefore, q > 1, which means that the liquid flowrate below the feed is greater than the liquid flowrate above the feed plus the feed flowrate. How is this possible? Since the denominator of the enthalpy expression in Equation (3.44) is the heat of vaporization (also called latent heat), more energy is required to bring the feed to equilibrium on the feed tray than can be supplied by the condensing vapor. In order for the subcooled liquid in the feed to come to the equilibrium conditions on the feed tray, enthalpy is required. This enthalpy comes from condensing some of the saturated vapor. Therefore, the liquid flowrate below the feed increases by more than the feed flowrate. Additionally, the vapor flowrate above the feed is lower than the vapor flowrate below the feed, since some vapor condenses on the feed tray.

A similar explanation exists for superheated vapor. Since h_F > H, q < 0. The superheated vapor must give up enthalpy to become saturated on the tray, which is obtained by vaporizing some liquid on the tray. Therefore, vapor flowrate increases above the feed tray and the liquid flowrate decreases below the feed tray.

It can now be seen that

where the subscript F indicates feed. Adding Equations (3.36) and (3.40) with Equation (3.45), and then applying the overall more volatile component balance

yields

Equation (3.47) is the equation of a line with slope q/(q – 1) and is called the q-line. If y_A = x_A, it can be seen that one point on the line is (z_AF, z_AF). Since the balances from the top and bottom sections were combined to give Equation (3.47), the top section, the bottom section, and the q-line must all intersect. This is illustrated in Figure 3.14 for a vapor-liquid mixed feed. Since the slope of the q-line is q/(q – 1), it can be seen that there are five possible q-lines, representing the different possible feeds, which are illustrated in Figure 3.15. Table 3.3 summarizes these results.

The ideal feed location is where the feed stream and the tray conditions match, which means matching composition, temperature, and pressure. Of course, the feed stream must be at a slightly higher pressure than the feed tray to allow flow into the column. So, a liquid feed would be brought close to saturation before entering the column. If the vapor effluent from a reactor were to be fed directly to a column, it would be cooled close to saturation. Two-phase feeds are also possible. If the column diameters above and below the feed are different enough to make it impossible to have a column of uniform diameter, the feed conditions could be adjusted to equalize the top and bottom diameters. Therefore, in extreme situations, subcooled liquid or superheated vapor feeds might be desirable.

The McCabe-Thiele method can be used to determine the number of equilibrium stages needed for a separation. If the desired top and bottom compositions are specified (or alternatively, fractional recoveries) and a reflux ratio chosen, the upper operating line can be drawn, and the lower operating line is drawn from the intersection of the upper operating line and the q-line. Then, as is illustrated in Figure 3.16, the number of stages can be stepped off starting at the top with alternating horizontal lines and vertical lines. A horizontal line to the equilibrium curve is the solution to the equilibrium on a tray, and the vertical line to the operating line is the material balance on that tray (or from the top to any tray above the feed, or from the bottom to any tray below the feed). Each step is an equilibrium stage, so the total number of steps is the number of equilibrium stages required. It is observed that the vertical lines change from the upper operating line to the lower operating line when the step straddles the intersection of the q-line and the operating lines. This represents the optimum feed location that minimizes the number of stages. In Figure 3.16, the feed is on Stage 4 (always count from the top), and there are >7 stages. In the design phase, fractional stages can exist. Before the column is constructed, the extra stage would be added, and it would be planned to operate at a slightly different reflux ratio to make the bottom step intersect the point (x_AB, x_AB).

A total condenser and a total reboiler are not equilibrium stages, since saturated vapor at the dew point is condensed to saturated liquid at the bubble point (total condenser), or vice versa (total reboiler). A partial condenser and a partial reboiler are equilibrium stages, since the condensation or vaporization is “partial” and there are equilibrium phases in equilibrium, just as in a flash operation. Partial condensers are used if a vapor product is needed or if there are noncondensable components. In appropriate applications, there can be a partial vapor-liquid condenser, in which noncondensables are removed as vapor, but a condensable, desired product is removed as a liquid. All of these configurations are illustrated in Figure 3.17. There is also a reflux drum present with a controlled liquid level to smooth out fluctuations. The reflux pump is necessary, because in an actual process, the condenser, reflux drum, and reflux pump are likely to be located close to the ground, so the pump provides pressure to overcome the head to return the reflux to the top of the column.

Despite the limitations of the McCabe-Thiele method, a complete understanding of the graphical representations provides an understanding of distillation, including distillation with more than two components and distillation systems for which the constant molar overflow assumption is not valid.

In a chemical plant, about the only variable that can be adjusted is a flowrate. In a distillation column, the flowrate that is adjusted is the reflux stream, which is typically described in terms of the reflux ratio, L/D. Figure 3.18 illustrates four different reflux ratios for a saturated liquid feed; however, the discussion that follows applies to any feed condition. Only the upper section of the column is shown. In Figure 3.18(a), the upper operating line intersects the equilibrium line and the q-line at the same point. If stages are stepped off, the intersection point is never reached. This is the minimum reflux ratio. The operating line in Figure 3.18(a) represents the minimum reflux ratio, which requires an infinite number of stages. However, if the reflux ratio is increased slightly, which lowers the y-intercept, as illustrated in Figure 3.18(b), a finite number of stages can be counted. As the reflux ratio increases, the intercept decreases, the steps get larger, and fewer equilibrium stages are needed. When the intercept equals zero, the reflux ratio is infinite, called total reflux, which is illustrated in Figure 3.18(d). At total reflux, there are no product streams and no feed. This is an operational limit, which is used to start up a distillation column and bring it to steady state. Therefore, the McCabe-Thiele diagram illustrates a key concept in distillation, that increasing the reflux ratio reduces the number of equilibrium stages required to obtain the same product specifications. A concept introduced in Chapter 1 is that just about the only way to adjust anything in a chemical process is to open or close a valve. In distillation, there would be valves included to adjust the ratio of L₀/D in Figure 3.17. This is how the reflux ratio is adjusted.

As the boiling points of the two components being distilled become closer, the equilibrium curve moves toward the diagonal line, since the equilibrium liquid and vapor mole fractions become closer together. Figure 3.19 illustrates that the minimum reflux ratio must be higher for a separation involving closer boilers, since, for the same feed, the y-intercept is smaller. Therefore, the actual reflux ratio must also increase for closer boilers.

The feed composition also affects the reflux ratio. The q-line moves to the left if the feed is dilute in the more volatile component. Figure 3.20 illustrates how a more dilute feed in the more volatile component requires a larger reflux ratio. Once again, only the minimum reflux ratio is shown.

The feed condition also affects the required reflux ratio, as is illustrated in Figure 3.21 for the minimum reflux ratio for each feed condition. Subcooled liquid requires the lowest reflux ratio, while superheated vapor requires the highest reflux ratio.

While it is difficult to illustrate, as the desired distillate mole fraction approaches 1.0 for a fixed number of stages, the slope of the upper operating line increases, so the intercept decreases, which means that a larger reflux ratio is needed.

As the reflux ratio increases, all of the internal flowrates increase. As will be seen later, there is an associated increase in column diameter, but, as already stated, fewer equilibrium stages are required. However, as the internal flowrates increase, the amount of liquid that must be condensed and the amount of vapor that must be reboiled both increase, thereby increasing the heat duties of the condenser and reboiler. For a total condenser, the energy balances is

Since h_L = h_D, because the enthalpy does not change when a stream splits, and since V = L + D, Equation (3.48) becomes

which is how the condenser heat duty can be calculated. The approximation in the last equality in Equation (3.49) includes the assumptions of ideal gas behavior and ideal liquid solutions. If these conditions are not valid, a standard thermodynamic method for calculating the enthalpies of the vapor and liquid mixtures would be used. Equation (3.49) also shows that as the flowrate of the vapor stream increases, the condenser heat duty increases. For a given separation, with fixed feed and outlet flowrates, the value of V increases as reflux ratio increases; therefore, the condenser heat duty also increases.

A similar development for the reboiler yields

where, once again, it is seen that as the internal flowrates increase, the reboiler heat duty also increases.

The energy balances clearly show that as the reflux ratio increases, the heat duties increase, which means that the operating cost (virtually all heating and cooling utilities) of a distillation column increases.

Table 3.4 summarizes the reasons for needing a high reflux ratio. Table 3.5 summarizes the effect of increasing the reflux ratio. Figure 3.22 illustrates the economics of a distillation column, and it is noted that Figure 3.22 is an illustration and not drawn to scale. As the reflux ratio increases, the number of trays decreases, so the cost of the column decreases. As the reflux ratio increases further, the diameter increases, so the cost of the column increases. As the reflux ratio increases, the cost of energy (utilities) increases, and the cost of the condenser and reboiler also increases. If the equipment cost is put on the same basis as the operating cost (utilities), which is done by assuming that the equipment cost is equivalent to a loan with periodic payments, the resulting curve suggests that there is an optimum reflux ratio. EAOC stands for Equivalent Annual Operating Cost, which is the sum of the actual operating costs (utilities) and the periodic payment on the initial equipment cost. The location of this optimum value for R/R_min is a strong function of the cost of energy. In the mid-20th century, when energy was inexpensive, optimum values in the 1.5 to 1.8 range were recommended. In the latter part of the 20th century, when energy prices increased, optimum values in the 1.2 to 1.5 range were recommended. In the early 21st century, energy prices have fluctuated significantly, so it is important to remember that the optimum value changes with energy prices.

The height can be obtained from the interfacial area/cross-sectional area, S, by defining a mass transfer area, a, that is specific to each type of packing, having units of mass transfer area/packed volume. The packed volume includes the void fraction, which is specific to the packing dimensions and shape. Therefore,

where Z is the height of the column.

If the mass transfer area, a, of a packing is known, then the height, Z, can be calculated from S in Equation (3.51). Manufacturers provide specifications on packings, and there is an extensive literature available for some of the most common packing materials. The interfacial area/cross-sectional area, S, can be calculated from the height and number of transfer units. For this explanation, vapor transfer units and heights are used, as described in Equation (3.24). The height of a transfer unit H_V = V/F_y in Equation (3.22) can be calculated if the internal flowrates are known, since there are correlations available for the mass transfer coefficient F_y. The question is how to evaluate the integral for the number of transfer units, N_V, in Equation (3.23). For this type of analysis, a McCabe-Thiele diagram is drawn, just as for staged separations; however, stages are not stepped off. The integral must be evaluated separately above the feed and below the feed, so one limit on each integral is the feed composition. As the values of mole fraction are varied between the feed and the distillate (top of column) and the feed and bottom (bottom of column), Equation (3.16) can be used with the material balance line (operating line) to determine pairs of corresponding values of y and y_i. The graphical representation of the simultaneous solution for this problem is illustrated in Figure 3.24. Consistent with the two-film model in Figure 3.3, points on the operating line represent passing stream mole fractions at every point in the column. Similarly, points on the equilibrium curve represent the interface composition, assumed to be in equilibrium, at every point in the column. This set of pairs of (x, y) and (x_i, y_i) allows the integral in Equation (3.23) to be evaluated numerically. The total height of the column is the sum of the heights of the upper and lower sections, and the feed height is clearly defined. An important concept to understand is that Figure 3.24 clearly shows that equilibrium occurs at the interface at every location in the column, but that the equilibrium values are different at different values of the column height.

The rigorous method is to solve Equations (3.3) and (3.4) simultaneously for every stage. For gas-liquid systems (absorption and stripping), energy balances might also be needed, since dissolving a gas into a liquid (absorption), HCl into water, for example, produces a heat of solution and a noticeable temperature change. For separations such as extraction and adsorption from a liquid, heat effects are usually negligible. If there are negligible heat effects and if the solvents are completely immiscible, the McCabe-Thiele method is applicable. If there is only one solute and two immiscible solvents and if the mole fractions are small enough, the total flowrates are approximately constant. Under these circumstances, Equation (3.4) can be rewritten as

where the subscript i has been dropped, since there is only one solute. This is the equation of a straight line on a plot of y versus x with slope L/V connecting the points representing the inlet and outlet mole fractions. This is illustrated in Figure 3.25, assuming countercurrent operation, with an arbitrary equilibrium curve. For multistage separations, countercurrent is the norm. If the two phases flow in the same direction, called cocurrent separations, once equilibrium is attained in the first stage, all subsequent stages would be useless, since the feed to those subsequent stages would already be at equilibrium.

Another difference between distillation and mass separating agents is that the operating line can be on either side of the equilibrium curve. As illustrated in Figure 3.25, if the solute transfers from the V phase to the L phase, x_out > x_in, and y_out < y_in, since each point on the operating line represents streams passing in the opposite direction. If the solute transfers from the L phase to the V phase, x_in > x_out, and y_in < y_out. If there are multiple stages, Equation (3.52) can be rewritten from one end of the cascade of stages to any intermediate stage

which is the equation of the operating line for the separation. Once this operating line is known, the stages can be stepped off, as illustrated in Figure 3.26.

Consider a separation involving transfer from the V phase to the L phase. The inlet mole fractions, y_in and x_in, would be known along with the inlet flowrate of the V phase, V. The purpose of the separation is to remove solute from the V phase, so the inlet mole fraction would be known. The inlet mole fraction of the L phase would presumably be known and be very small, if not zero. There would probably be a target amount of solute to be removed or a target outlet mole fraction, so y_out would also be known. Figure 3.27 shows that the flowrate L can have multiple values for a fixed value of V, but as L gets smaller, resulting in a smaller slope, the operating line gets closer to the equilibrium curve, resulting in more steps, meaning more stages are needed. This is analogous to the trade-off between reflux ratio and number of stages in distillation. The destination solvent rate replaces the reflux ratio. There is also a minimum solvent rate, which is determined either by the intersection of y_in with the equilibrium curve or by a tangent, depending on the curvature of the equilibrium curve.

For continuous differential separations, the operating line and equilibrium curves are similar to those in staged separations, since there is an equilibrium relationship and since the operating line represents the tray-to-tray mass balances. The method described in Section 3.2.2.1 would be used to evaluate the integral for the number of transfer units.

• Dilute solutions

• Total stream flowrates remain constant

• Equilibrium relationship is linear

• Isothermal operation

• No energy associated with transfer between phases

The results are presented without derivation. The result for transfer from the V phase to the L phase for staged separations is called the Kremser method (Kremser, 1930; Souders and Brown, 1932) and is

where A = L/mV and is called the absorption factor (because this method is traditionally applied to gas absorption into a liquid, but it is valid for any separation that is consistent with the assumptions), N is the number of stages, and m is the partition coefficient from the equilibrium expression y = mx. Equation (3.54) can be written explicitly for N as

There is no explicit relationship for A. For transfer from the L phase to the V phase, the equivalent relationship is shown in Table 3.6. When A = 1, Equations (3.54) and (3.55) are indeterminate. The results for such a case, obtained from application of L’Hôpital’s rule, are also presented in Table 3.6.

For continuous differential separations with transfer from the V phase to the L phase, the relationships are attributed to Colburn (1939) and are

and

where N_oV is defined in Table 3.2.

Table 3.6 summarizes the relationships presented for transfer from the V phase to the L phase and also presents the relationships for transfer from the L phase to the V phase. In the latter case, S is the stripping factor, which is the inverse of the absorption factor.

The Kremser and Colburn results are often presented in graphical form, and they are shown in Figures 3.28 and 3.29, respectively. Although it is easy to solve the equations, even for A, similar to the McCabe-Thiele method for distillation, a thorough understanding of these graphs provides a full conceptual understanding of separations involving mass separating agents.

Consider a separation with transfer from V to L. It is desired to remove solute from the V phase such that y_out/y_in = 0.1 using a pure solvent, so x_in = 0. Point a on Figure 3.30, which is a blown-up portion of Figure 3.28, shows one possible solution to the problem, N ≈ 5.0 and A = 1.2. Moving along the horizontal line in either direction shows that the same separation can be accomplished by increasing N and decreasing A, and vice versa. Since A ∝ L, this illustrates the trade-off between the number of stages and the solvent flowrate. If it is desired to improve the separation such that y_out/y_in = 0.005, there are two possible methods. One is to increase N at constant A, adding more stages, which makes sense. This is illustrated by following the A = 1.2 line down to Point b on Figure 3.30. Another possibility is to increase A at constant N, which is illustrated by the vertical line down to Point c on Figure 3.30. There are two ways to increase A, increase the solvent flowrate L, which should make sense. Another method is to decrease m. Since y = mx at equilibrium, decreasing m decreases the ratio y/x, which means that the L-phase mole fraction at equilibrium increases and/or the V-phase mole fraction decreases at equilibrium, favoring the L-phase. For the specific application of gas absorption, an expression for m might be m = P*/P. If the pressure increases, m decreases, which makes sense since increasing the pressure favors the liquid phase. If the temperature decreases, P* decreases, so m decreases, which makes sense since decreasing the temperature favors the liquid phase. For a given target separation, if a certain A value is needed, an unfavorable m value, which is a large value for absorption, can be overcome by increasing L as much as needed. While this works in principle, there could be operability issues.

To generalize, even though all separations involving mass separating agents do not follow the assumptions in the Kremser or Colburn methods, a thorough understanding of those graphs provides a thorough understanding of the trends in separations involving mass separating agents. To reiterate what is illustrated in Figure 3.30, Point a shows a target separation, as defined by a value on the y-axis (which is the same y-axis as in Figures 3.28 and 3.29). It is being accomplished with a certain number of stages (or transfer units) at a certain A (or S) value. The same level of separation, defined by the same value of the y-axis, can be achieved with a lower A value and more stages (Point d) and a larger A value and fewer stages (Point e). This illustrates a trade-off between the absorption (stripping) factor and the number of stages (transfer units). Within the absorption (stripping) factor, for the purposes of this example, it is assumed that the equilibrium constant cannot be changed and that the value of L (V) is fixed by upstream demands (meaning that the flowrate of material that must be processed in the separator is fixed). A higher A (S) value corresponds to a higher liquid flowrate (vapor flowrate). This means that the trade-off between absorption (stripping) factor and the number of stages (transfer units) is actually a trade-off between liquid (vapor) flowrate and the number of stages (transfer units). Since A (S) is the key parameter, this also demonstrates how to overcome an unfavorable equilibrium condition. For transfer from V to L, it would be desirable for m to be less than unity, meaning the mole fraction in the liquid phase is always less than that in the vapor phase. This means that the equilibrium favors the liquid phase. However, if m is greater than unity, which alone reduces the value of A (S), the value of A (S), the unfavorable equilibrium can be offset by increasing the value of the destination solvent, L (V). There are equipment limitations to the relative flowrates of the two phases that are dependent of the specific phases involved, and these will be discussed later in the chapter. Finally, at a constant number of stages (transfer units), if A (S) is increased, the value of the y-axis decreases, which means a better separation. This is illustrated by the vertical line through Points a and c in Figure 3.30.

If A < 1, where A is the absorption factor, L/(mV), there is a limiting behavior. It can be seen that for A < 1, increasing the number of stages does not improve the separation beyond an asymptotic limit. Normally, it would be expected that increasing the number of stages would improve the separation, but this is only true for A ≥ 1. The reason can be understood by looking at the McCabe-Thiele diagrams shown in Figure 3.31. If the equilibrium line is defined by y = mx, and the slope of the operating line is L/V, as shown in Section 3.2.2, then, if A < 1, L/V < m, so the operating line approaches the equilibrium line at the bottom of the column (see also Figure 3.25), which is the most concentrated part of the column. Therefore, since the solution is approaching saturation at the concentrated portion of the column, where solvent capacity is needed most, the ability to transfer solute is limited, causing the observed asymptote. If A ≥ 1, L/V > m, equilibrium is approached at the less concentrated portion of the column. For transfer in the opposite direction, while the operating line is below the equilibrium line, the argument is exactly the same.

For continuous differential separations, the discussion just presented is identical, except that the number of transfer units replaces the number of stages. Once again, it is important to remember that the number of transfer units is not numerically equal to a number of stages. However, as the number of transfer units increases, the required interfacial area increases (Equation [3.24]). For a gas-liquid separation, this means that the height of the packed column increases. Similarly, as the number of trays increases, the height of a tray column increases.

For other separations, such as liquid-liquid extraction, if the equilibrium is linear and the solutions dilute, the Kremser and Colburn methods can be used, and all of the same trends are true. Quite often, liquid-liquid equilibrium obtained experimentally is expressed in terms of mass fraction, so the flowrates become mass flowrates. In most textbooks, the letters used to represent these flowrates are different to conform with chemical engineering jargon. The output stream from the feed stream (F) is called the raffinate (R). The destination solvent input stream is denoted S, and its output stream is called the extract (E). To avoid confusion of letters, an alternative method is to retain the symbols L and V and use L for the heavier (more dense) phase (often water) and V for the lighter (less dense) phase (usually organic). In reality, as long as the equilibrium expression is written correctly, the choice of symbols used is irrelevant.

As discussed in Section 3.2.1, after a partial vaporization/condensation, the vapor-liquid mixture must be allowed to disengage so that separate vapor and liquid streams can be removed continuously. These drums, often called knockout drums, are usually vertically oriented with a typical height-to-diameter ratio between 2.5/1 and 5/1.

Distillation columns have reflux drums, as illustrated in Figure 3.17. The purpose of the reflux drum is to smooth out fluctuations, particularly in the reflux stream, by maintaining and controlling the liquid level in the drum. It is also important to understand that the reflux drum and reflux pump, while drawn at the top of the column, are typically located closer to the bottom of the column. It is usual to locate equipment such as pumps closer to the ground because it makes maintenance easier, and often the reflux drum will be located directly above the reflux pump, which is located at ground level. The height of the drum above ground level is often determined on the basis of the required net positive suction head (NPSH_R) of the pump (see Chapter 1, Section 1.5.2). Therefore, the reflux pump must be designed to supply the head necessary (plus a small frictional loss) for the reflux stream to flow from ground level to the top tray of the tower. Reflux drums are usually horizontally oriented, with a typical length-to-diameter ratio between 2.5/1 and 5/1.

While not specifically related to distillation equipment, drums are also used when mixing liquid streams, with the most common example being mixing recycled liquid and feed liquid. The liquid level in these drums is controlled by varying the fresh feed flowrate. These mixing drums serve a similar purpose as reflux drums; the downstream flowrate is maintained by controlling the liquid level in the drum. Mixing drums are usually vertically oriented, with a height-to-diameter ration between 2.5/1 and 5/1.

The holes in the trays can also have caps or valves. The former are bubble cap trays, and the latter are valve trays. Bubble cap trays were the earliest ones used, but sieve and valve trays have replaced them (Kister, 1992). Figure 3.33 illustrates various types of caps and one valve. Bubble caps make smaller bubbles than sieve trays, which can be seen by the design of the holes in the cap. However, they are more expensive than sieve trays. Valve trays have a cap on the hole without the extra holes for bubbles. Bubble caps and valves move up and down with the vapor flowrate, so they are more amenable than sieve trays to scale-down. Since, at low vapor flowrates, the bubble cap or valve covers the top of the hole, bubble cap and valve trays are less prone than sieve trays to a phenomenon called weeping. Weeping is where the liquid falls through the holes in the trays due to low vapor flowrate, and it is an undesirable situation. However, bubble cap trays are more expensive than valve and sieve trays, and they are more prone to fouling, because the small holes can collect solids, and the smaller bubbles mean a higher pressure drop. While sieve trays are more prone to weeping, they are easier to clean during plant shutdown and are recommended for services in which fouling or corrosion is anticipated. There are also fixed valve trays, in which the valves covering the holes do not move. They are not prone to sticking shut and erosion, and they are better suited for process upsets. More details are available elsewhere (Hebert and Sandford, 2016).

Figure 3.32 suggests a cross-flow pattern of liquid on alternative trays. This is the most common flow pattern, because of its simplicity. However, as the flow path of fluid becomes long (for large diameter trays) it may be desirable to reduce the flow path resulting in other flow patterns (double-pass trays, triple-pass trays, etc.). One reason for this is the difficulty in building a large-diameter column with perfectly horizontal trays, because very large-diameter trays are “field erected” in sections; they are not in one piece. Figure 3.34 shows one alternative, a two-pass tray, in which the flow is edge to center and back. One advantage of this flow pattern is that the flow is split, so that there is less liquid loading on a given tray. Another advantage is that variation in the height of liquid above the tray is more uniform and bubbles do not bypass the part of the column near the weir. The two-pass flow pattern is more common for large-diameter trays and high liquid rates. Its main disadvantages are the added complexity and cost.

As will be discussed later, the weir height, which is the main component of the level of liquid on the tray, affects the column pressure drop, since the vapor flowing upward must overcome the liquid head on each tray. While this suggests a small weir height is desirable, for a large-diameter column, it may be difficult to guarantee completely level trays, which may lead to puddles and dry spots on the tray causing poor liquid-vapor contact and low efficiencies. Therefore, weir heights less than 2 in are rare, although possible, when a low pressure drop is required. Another limitation is entrainment, which is a mist of liquid formed from the froth on the tray being carried up to the tray above. This results in remixing of liquid from the tray below to the tray above, counteracting any change in concentration that occurred in the liquid phase between trays. Clearly, this is counterproductive to the objective of the separation. If the weir height is kept low, entrainment is less likely. Typical weir heights are in the 2-in to 4-in range and are always less than one-half the distance (spacing) between the trays.

Entrainment is the situation where the upward-flowing vapor carries liquid from the tray below to the tray above. Effectively, this results in a mixing of liquids at different compositions, negating or reducing the separation that has occurred. Flooding can be caused by excessive entrainment.

The column diameter is based on flooding. Another way to think about flooding is that, if the upward vapor velocity is too large, the drag force on the liquid exceeds gravity, and the liquid does not fall through the column. A precursor to flooding is called loading, and this is manifested by a rapid increase in the pressure drop in the column. Based on the relationship between mass flowrate and velocity developed in Chapter 1, , for a given mass flowrate, the velocity can be reduced by increasing the cross sectional area of the column, that is, by increasing the column diameter. Typically, columns operate in the range of 75% to 80% of flooding; therefore, to determine the diameter of the column, the flooding velocity must be calculated.

Tray spacing also affects flooding. The typical tray spacing is between 18 and 24 in. This allows for easy access for maintenance. Tray spacings up to 36 in can be found, and, for good reason, smaller tray spacings are possible. The column height is the number of trays times the tray spacing plus additional height at the top and bottom of the column. In addition, the column is usually raised off of the ground by means of a column skirt. One reason is that the bottom product leaves the column as saturated liquid, and additional head is needed to avoid the NPSH issues discussed in Chapter 1 if a pump is needed to provide additional pressure to move the bottom product to its next location.

Flooding velocities are calculated from empirical correlations based on experimental results. The most commonly available result is for sieve trays. There are so many different kinds of bubble caps and valves that there is no single, universally applicable empirical correlation available for these types of trays. Sieve trays, on the other hand, are simple and generic. The correlation attributed to Fair and Matthews (1958) is one of the most commonly cited, and it is presented graphically in Figure 3.35. In Figure 3.35, the x-axis is a parameter usually called F_lv, which includes the ratio of mass flowrates through the column. The y-axis is a flooding capacity factor that allows calculation of the flooding velocity. Since it has been established that, in the most general case, these flowrates are different on every tray, process simulators perform the flooding calculation on every tray and provide a profile of diameters. This is called tray sizing, and the user input must include the tray spacing. There is also active area versus total area. The active area does not include the cross-sectional area for the downcomers and is the active area for the upward vapor flow. Typically, the active area is 85% to 90% of the total area, the column diameter is calculated on the basis of the total area, and this parameter is also input to a simulator. Typically, once the tray sizing profile is obtained from a simulator, a diameter is chosen, and a tray rating is performed for the given diameter, which results in a percentage-of-flooding profile for the column. As long as there are no trays that are flooding or are close to weeping (<40% of flooding for sieve trays), that diameter is appropriate. If a constant diameter column results in operation of some trays above the 80% flooding limit and some below the weeping limit, then the use of different column diameters above and below the feed may have to be considered. This situation is not uncommon for high vacuum operation.

To estimate the column diameter without a simulator, McCabe-Thiele approximate results are used. In this method, the flowrates are constant in each section of the column, so there are only two possible calculations, above and below the feed. An equation has been suggested to determine the limiting (larger diameter) section (Wankat, 2017). Intuitively, it may seem that the section with the larger molar flowrates will require a larger diameter. However, while the chemistry of phase equilibrium is based on molar flows and compositions, equipment sizing is based on mass flow-rates. Therefore, the molecular weight also affects the mass flowrate. Additionally, based on the relationship between mass flowrate and velocity, , the density affects the velocity, so a large temperature difference can be significant. Finally, especially in vacuum columns, the pressure drop can be a large fraction of the column pressure, which means that the vapor density can change significantly through the column. In general, the most desirable feed conditions are saturated liquid, saturated vapor, or a vapor-liquid mixture, because the feed should match column conditions (saturated). This suggests that the molar flowrates below the feed will be larger than those above the feed. Quite often, the higher boiler, which is at the bottom of the column, has the larger molecular weight. Thus far, it seems like designing for the bottom of the column is limiting. However, the pressure and temperature at the bottom of the column are both larger. For an ideal gas, the density ρ = PM/RT, where M is the molecular weight, so, depending on the exact numbers, the effects of temperature and pressure may offset. For vacuum columns, the pressure and therefore density will be significantly smaller above the feed. While it is usually most desirable to design a column with a uniform diameter, vacuum columns often have different diameters, larger above the feed and smaller below the feed, because the velocity above the feed is so much larger than the velocity below the feed. In a standard column, if the diameters above and below the feed are not compatible, that is, one section is so far below flooding to create weeping, the feed can be made into a vapor-liquid mixture to equalize the mass flowrates in both sections. In multicomponent distillation columns, flowrates change on every tray, and there are methods such as pump-arounds that can be used to avoid localized flooding (Wankat, 2017). At the level of the estimated diameter calculation, it is probably best to determine the diameter both above and below the feed.

The curves in Figure 3.35 have been fit to the equation (Wankat, 2017):

where the constants are given in Table 3.7 for the different tray spacings.

The parameter F_lv is defined as

where L and V are molar flowrates, and M_L and M_V are molecular weights used to convert the molar flowrates to mass flowrates.

The nomenclature in Equation (3.59) and Figure 3.35 must be understood. First of all, throughout the history of separations, the variables V and G have been used somewhat interchangeably. V stands for vapor, while G stands for gas. V is being used in this chapter for all gas-vapor phases, but the original correlation of Fair and Matthews (1958) used G. On the x-axis of Figure 3.35, it is assumed that L and G are in mass units; Equation (3.59) expresses this clearly. On the y-axis of Figure 3.35, the parameter C_sb,flood, which is abbreviated here as C_sb,f, is

where u_f is the flooding velocity in ft/sec, and σ is the surface tension of the liquid phase in dyne/cm². The units of Equation (3.60) are exactly as stated; there is no metric equivalent available. Since the surface tension term is raised to a small power, only a small amount of error is introduced by assuming the entire term is unity.

The algorithm to determine the column diameter required is as follows:

1. Determine all internal flowrates in molar units. For distillation, this involves knowing the reflux ratio and the feed condition. For mass separating agents, the flowrates are likely given. Convert these to mass units. For distillation, it is usually assumed that the molecular weights in the vapor and liquid phases are equal, because the phases are not that different in composition. For mass separating agents, especially in dilute solutions, the molecular weight is approximately that of the solvent. (This may not be true for non-dilute vapor phases.)

2. Determine the liquid and vapor densities. The liquid density must be found from a reference. The vapor density can be estimated using the ideal gas law or another equation of state. The top and bottom temperatures and pressures are used, depending on which section the calculation is based.

3. Calculate F_lv using Equation (3.59).

4. Use Equation (3.58) or Figure 3.35 to determine C_sb,f.

5. Calculate u_f (which will be in ft/sec) from Equation (3.60). If metric units are desired, convert it to m/s.

6. Calculate the actual velocity (u_actual) as a percentage of the flooding velocity.

7. Calculate the column active area from

where VM_V is the mass flowrate .

8. Determine the actual column area as actual area = active area/fraction active area.

9. Determine the column diameter from the actual area.

where the numerators represent the actual mole fraction change on a tray, and the denominators represent the mole fraction change on a tray if equilibrium is reached on the tray. The subscripts V and L represent a vapor-phase and a liquid-phase basis, respectively. While these definitions make intuitive sense, they are not very useful for a variety of reasons. One reason is that they differ for every tray, and experimental results would be needed for every tray before the design of a column.

Another method for estimation is to use empirical results gathered over many years of practice, and these results were presented by O’Connell (1946). The results are presented in Figure 3.36, which are for overall, average column efficiencies. There are different correlations for distillation columns and for absorbers/strippers. The parameter for distillation is αμ, where α is the relative volatility of the key components (only two components in binary distillation), and μ is the viscosity of the feed in cP (centipoise). The efficiency decreases as the relative volatility increases because more mass must be transferred to reach equilibrium, and the efficiency decreases as the viscosity increased because mixing is more difficult, which lowers the mass transfer rate. A curve fit of the points shown (in percent) has been suggested (Couper et al., 2012, page 471):

An alternative curve fit has also been suggested (Wankat, 2017):

For absorbers, the parameter is HP/μ, where H is Henry’s law constant (in lbmol/ft³/atm), P is the pressure (in atm), and μ is the viscosity (in cP).

Figure 3.37 is a schematic of the liquid flow across a tray. The vapor flowing upward must overcome the downcomer head (h_dc), which is the sum of all of the other heads shown in Figure 3.37,

where

h_weir = weir height, the dominant term

h_crest = crest over weir, the level of liquid above the top of the weir

h_grad = hydraulic gradient, difference in level from the downcomer side and the weir side

h_du = frictional loss of liquid flowing out of downcomer

h_dry = the “dry” head, the head to get vapor through holes in tray

If the weir height is assumed to be the dominant contribution to the pressure drop, and assuming that all weir heights are the same, the column pressure drop becomes

where the subscripts above and below refer to the location in the tower relative to the feed. The liquid densities are used as a conservative estimate, since the portion of the tray containing a froth (two-phase, vapor-liquid mixture) is unknown in an initial design. Since the estimation methods presented here are based on the properties of the more volatile component above the feed tray and the less volatile component below the feed tray, their liquid densities would be used for the appropriate section of the column.

For the reboiler, the energy balances and design equations are

where the subscript R denotes the reboiler, the subscript p denotes the process stream, and the subscript s denotes steam, which is a standard source of heat. The overbars indicate the bottom of the column. Built in to Equation (3.68) is the assumption that all phase changes are between saturated liquid and saturated vapor. The log-mean subscript is omitted from the temperature difference, because the temperature difference between the two streams is constant at all points, and thus ΔT_lm = ΔT.

Once the flowrate of the process stream is known, since the latent heat can be found in the literature, the heat duty, Q, is known. If the heat transfer coefficient and the available steam temperature are known, then there is a unique correspondence between T_p and A. So, a reboiler can be designed for any desired value of T_p. There is an exact pressure for this value of T_p, based on Antoine’s equation. Care must be taken to keep (T_s − T_p) from being too large to avoid film boiling, as discussed in Chapter 2.

The pressure in the condenser is obtained from the pressure in the reboiler minus the column pressure drop. The column pressure drop is not arbitrary but is related to the weir height, as discussed the previous section. From Antoine’s equation, the condensing temperature is then known. In the condenser, the energy balances and design equation are

where the subscript c denotes the condenser or condensation temperature, and the subscript cw denotes cooling water, which is the most desirable heat sink in a condenser. Therefore, if the heat transfer coefficient is known, there is only one condenser area that will satisfy Equation (3.69) for the calculated T_c.

When simulating a column, the pressure drop is an input variable. However, it is not known until the trays are designed, but the pressure drop is dependent on the number of trays, which is not known until the column is simulated. Therefore, column design is an iterative process, and this is not the only source of iterations. As a first approximation, a pressure drop could be assumed and the weir height calculated after determining the number of trays and feed location. If the weir height is reasonable, the result may be sufficient for a first step. If not, then the pressure drop must be changed to correspond to a reasonable weir height.

The condenser and reboiler also affect the pressure in the column. In a chemical plant, the least expensive source of cooling in the condenser is usually process cooling water. Air can be used, but the heat transfer coefficient from a gas is much smaller than from a liquid. While the following values may differ from plant to plant and seasonally, typically, process cooling water is available at 30°C and is returned at 40°C. (The returned cooling water can be cooled back to 30°C by evaporating about 2%, so, other than makeup for the 2% evaporated, it is a closed loop.) If a 10°C approach temperature between the cooling water and the condensation temperature is desired (Figure 3.38), then the top column pressure must be high enough for the boiling point of the top component to be 50°C (bubble point of mixture at top). Quite often, distillation of low-boiling components, such as ethane-ethylene or propane-propylene, are run at significant pressures to make the top temperature around 50°C.

Columns can also be run at vacuum by using either a steam ejector (covered in Chapter 5) or a vacuum pump at the top of the column. There are two possible reasons for running a column under vacuum. One is that a component in the distillation is temperature sensitive. For example, acrylic acid spontaneously polymerizes at 90°C. To keep the temperature below 90°C, the pressure must be 0.16 bar. So, in the purification of acrylic acid from acetic acid, the undesired product of a side reaction, the column is run at vacuum. (In this particular example, acrylic acid is the higher boiler, so it is the bottom pressure that is at 0.16 bar!) A second reason is the available steam temperature for high boilers. Typically, the highest pressure, saturated steam available for a reboiler is around 250°C. For example, phthalic anhydride’s normal boiling point is 295°C. Therefore, the column that purifies phthalic anhydride is run at around 0.4 bar, where phthalic anhydride boils at around 240°C.

There are several other differences between packed towers and tray towers. Packed towers flood, but entrainment and weeping do not occur. In general, packed towers have lower pressure drops than tray towers. One problem with packed towers is the potential for flow maldistribution, which is the uneven distribution of liquid across the cross section, which can create regions of uncoated, dry solid, reducing the expected mass transfer area. Since flow maldistribution manifests more often in larger-diameter towers, historically, tray towers were used for larger-diameter (i.e., larger-capacity) columns, such as distillation in oil refineries, and packed towers were used for smaller-diameter (i.e., smaller-capacity) columns, such as removing pollutants from exhaust streams. Flow maldistribution is also related to the column diameter and the packing material diameter. If column diameter/packing diameter >40, maldistribution is more likely. One method that can be used to mitigate flow maldistribution in larger diameter columns is to have the packing in sections with the liquid collected and redistributed across the cross section throughout the column. This is illustrated in Figure 3.39.

A more recent development is structured packings, which come in sections that slide into the cylindrical vessel. One key advantage of structured packings is a very-low pressure drop. Figure 3.41 illustrates a structured packing.

There is another method that can be used for calculating the height of a packed tower, the height equivalent to a theoretical plate (HETP). In this method, the number of trays (also called plates) is calculated as if designing a tray tower. If the HETP can be calculated, the tower height is

The expression for HETP is

where H_oV is defined in Table 3.2, and the flowrates L and V are molar flowrates. The mass transfer coefficient must still be known, since it is in H_oV, but the HETP expression replaces the tedious N_oV calculation described in Section 3.2.2.1. A possible heuristic is that HETP ≈ 0.5 m. Another heuristic might be HETP (ft) ≈ 1.5 (packing size, inches).

The diameter of a packed tower is based on a flooding calculation. The concept is identical to that for tray towers, but the empirical correlation is different. A graphical representation of the flooding limit is shown in Figure 3.42. The important line is the flooding line and a curve fit has been presented (Wankat, 2017) that corresponds to the equation

where F_lv is defined in Equation (3.59), G_f′ is the superficial mass flowrate of vapor at flooding (could be called V_f′) in lb/ft²/sec (area unit is cross-sectional area of tower), the densities are in lb/ft³, the viscosity (μ) is in cP, ψ is the inverse of the specific gravity of the liquid (ρ_water/ρ_L), F is a packing factor found in Table 3.8, and g_c is the unit conversion 32.2 ft lb_m/(lb_fsec²). The units must be exactly as stated in this empirical correlation, and they are not consistent with each other. The algorithm is as follows:

1. Determine all internal flowrates in molar units. For distillation, this involves knowing the reflux ratio and the feed condition. For mass separating agents, the flowrates are likely given. Convert these to mass units. For distillation, it is usually assumed that the molecular weights in the vapor and liquid phases are equal, because the phases are not that different in composition. For mass separating agents, especially in dilute solutions, the molecular weight is approximately that of the carrier solvent (non-absorbing component).

2. Determine the liquid and vapor densities. The liquid density must be found from a reference. The vapor density can be estimated using the ideal gas law or another equation of state. The top and bottom temperatures and pressures are used, depending on which section the calculation is based.

3. Calculate F_lv from Equation (3.59).

4. Use Equation (3.72) or Figure 3.42 to determine the parameter in the brackets.

5. Solve for G_f′, which has units lb/ft²/sec. This is the value of G′ at flooding.

6. Calculate the actual value of G′ as a desired percentage of the flooding velocity (typically 70%–80% of flooding for a packed tower).

7. Calculate the area from

where G is the mass flowrate of vapor in the column.

8. Determine the column diameter from the area.

Just as for tray towers, for a distillation column, this calculation should be done both above the feed and below the feed and a diameter chosen that satisfies both results. Since there is no lower percentage of flooding limit (no weeping), choosing an appropriate diameter is easier for a packed tower than for a tray tower.

An empirical correlation for the pressure drop has been presented (Wankat, 2017).

where α and β are listed in Table 3.8. L′ and G′ are superficial mass flowrates in lb/ft²/sec, the density is lb/ft³, and the resulting pressure drop is in inches H₂O/ft packed height.

Tray towers were traditionally used for larger-capacity (larger-diameter) operations. Trays are easier to clean than packing, so trays are more applicable to fouling operations. In a tray tower, access points are built in for maintenance (manways) and cleaning during shutdown. However, in a packed tower, the packing must be removed for the same access. In more advanced distillation operations, such as reactive distillation, adding heat transfer capability to remove heat produced by an exothermic reaction is easier in a tray tower by inserting a liquid take-off at a given tray and pumping the liquid through an external heat exchanger and returning the liquid to another tray.

Ultimately, the choice between a tray tower and a packed tower comes down to the economics. However, constraints for a specific application must also be considered.

If a distillation column must be scaled-up (scaled-down) at constant product purities, the material balance requires that inlet and outlet flowrates increase (decrease) by the same amount. The question is how the output changes based on equipment performance. Clearly, there is a limit to scale-up of a distillation column based on the flooding limit. If the inlet flowrate to a distillation column increases at constant composition, and if the outlet compositions are maintained, the reflux rate increases, since the reflux ratio (L/D) stays constant. Therefore, all internal flows increase, making flooding a real possibility. If, for example, a column was designed for 75% of flooding, a 10% to 15% scale-up might be possible, assuming that it is desired to keep the flooding percentage less than 85% to 90% to avoid loading. For scale-down, there might be more flexibility, since weeping does not typically occur until <40% of flooding, with the exact value dependent on the tray type.

However, the situation is not that simple. In Section 2.8, the performance of heat exchangers was discussed, and it was seen that changing the inlet flowrate to a heat exchanger affects the temperatures of both streams and the total heat load. Since it has been shown that the condenser and reboiler affect the pressure of the column, then scale-up or scale-down of a distillation column involves much more than flooding. Therefore, the performance of a distillation column cannot be analyzed without consideration of the performance of both the condenser and reboiler. This is illustrated with a case study.

Among the several possible operating strategies for accomplishing the necessary scale-down are the following:

1. Scale down all flows by 50%. This is possible only if the original operation is not near the lower velocity limit that initiates weeping or reduced tray efficiency. If 50% reduction is possible without weeping, reduced tray efficiency, or poor heat-exchanger performance, this is an attractive option.

2. Operate at the same boil-up rate. This is necessary if weeping or reduced tray efficiency is a problem. The reflux ratio must be increased in order to maintain the reflux necessary to maintain the same liquid and vapor flows in the column. In this case, weeping, lower tray efficiency, or poor heat-exchanger performance caused by reduced internal flows is not a problem. The downside of this alternative is that a purer product will be produced and unnecessary utilities will be used. Increasing the reflux ratio during scale-down increases the utility cost or the column per unit product.

In Example 3.9, the analysis will be done by assuming that it is possible to scale down all flows by 50% without weeping or reduced tray efficiency.

The pressure drop is assumed to remain constant after the scale-down because the weir height is not changed. In practice, there is an additional contribution to the pressure drop due to gas flow through the tray orifices. This would change if column flows changed, but the pressure drop through the orifices is small and should be a minor effect. Examples 3.10 and 3.11 illustrate how the performance of the reboiler and condenser affect the performance of the distillation column.

Example 3.10 shows that the reboiler operation requires that the pressure of the boil-up stream be increased. This is not the only possible alternative. All that is necessary is that the temperature difference be 21.7°C. This could be accomplished by reducing the temperature of the steam to 163.4°C. In most chemical plants, steam is available at discrete pressures, and the steam used here is typical of medium-pressure steam. Low-pressure steam might be at too low a temperature to work in the scaled-down column. However, the pressure and temperature of medium-pressure steam could be reduced if there were a throttling valve in the steam feed line to reduce the steam pressure. The resulting steam would be superheated, so desuperheating would also be necessary. This is usually accomplished by spraying water into the superheated steam. This system is called a desuperheater, and the benefit of a desuperheater is to permit flexibility in saturated steam conditions to facilitate process changes and process control. A change in a utility stream is almost always preferred to a change in a process stream, because it does not disrupt process operation at the design conditions.

The condenser is now analyzed in Example 3.11.

Observation indicates that in order to operate the distillation column at 50% scale-down with process flows reduced by 50%, a higher pressure is required and the cooling water will be returned at a significantly higher temperature than before scale-down. The operating pressure of the column is determined by the performance of the reboiler and condenser.

Examples 3.10 and 3.11 reveal a process bottleneck at the reboiler that must be resolved in order to accomplish the scale-down. In the process of changing operating conditions (capacity) in a plant, a point will be reached where the changes cannot be increased or reduced any further. This is called a bottleneck. A bottleneck usually results when a piece of equipment (usually a single piece of equipment) cannot handle additional change. In this problem, one bottleneck is the high cooling water return temperature, which would almost certainly cause excessive fouling in a short period of time. Another potential bottleneck is tray weeping due to the greatly reduced vapor velocity. In addition to the solution presented here, there are a variety of other possible adjustments, or debottlenecking strategies, which follow with a short explanation for each.

1. Replace the Heat Exchangers: Equation (E3.10b) shows that a new heat exchanger with half of the original area allows operation at the original temperature and pressure. The heat transfer area of the existing exchanger could be reduced by plugging some of the tubes, but this modification would require a process shutdown. This involves both equipment downtime and capital expense for a new exchanger. Your intuitive sense should question the need to get a new heat exchanger to process less material. You can be assured that your supervisor would question such a recommendation.

2. Keep the Boil-Up Rate Constant: This would maintain the same vapor velocity in the tower. If the operation occurs near the lower velocity limit that initiates weeping or lower tray efficiency, this is an attractive option. The constant boil-up increases the tower separation. This option results in much smaller temperature and pressure changes but is somewhat wasteful, as approximately the same amount of reboiler and condenser energy is being used to process only 50% of the original material.

3. Introduce Feed on a Different Plate: This strategy must be combined with Option 4. The plate should be selected to decrease the separation and increase the bottoms concentration of the lower boiling fraction. This lowers the process temperature and increases the ΔT for heat transfer and the reboiler duty. This may or may not be simple to accomplish depending on whether the tower is piped to have alternate feed plates. As in Option 2, the reflux and coolant inputs will change.

4. Recycle Bottoms Stream and Mix with Feed: This strategy must be combined with Option 3. By lowering the concentration of the feed, the concentration of the low boiler in the bottoms can be increased, the temperature lowered, and the reboiler duty increased. This represents a modification of process configuration and introduces a new (recycle) input stream into the process.

The next series of examples involves the performance of an absorber. The same methodology would be used for a stripper.

If there is a disturbance as in Example 3.12, the next question is how to compensate for the disturbance. In the next set of examples, some methods are suggested. Those that involve manipulation of temperature and pressure are based on the assumption that these parameters can be manipulated. The pressure can be manipulated by designing the upstream pressures to be higher than the column pressure so that valves can be used to set the inputs, or by adding pumps to lower pressure input streams. Heat exchangers can be designed upstream to adjust the inlet temperatures. It must be remembered that, unlike distillation columns, absorbers and stripper conditions are directly related to the feed conditions.

It is important to remember the qualitative trends illustrated in Examples 3.13 and 3.14. Even if the Colburn method is not applicable, the trends are identical, but the numbers would change.

A variation on a static column is a pulsed column, in which short pulses, up and down, are imposed on the flowing fluids. On the up-stroke, the light liquid is dispersed into the heavy phase, and on the down-stroke, the heavy phase in injected into the light phase. This is possible in both tray and packed columns. A subsequent variation on the pulsed column is the Karr extractor, where the trays themselves move up and down, a schematic of which is illustrated in Figure 3.45(d). The trays do not have downcomers and can have as little as 1-in tray spacing, which makes up for the very low efficiencies, which can be as low as the 5% to 10% range.

Additional information on extraction equipment is available (Koch and Shiveler, 2015; Koch Modular Processes, 2000–2016).

With the advent of technology that allows polymers to be spun into hollow fibers, membrane modules that have the appearance of a shell-and-tube heat exchanger were developed. A hollow-fiber module is illustrated in Figure 3.50. The hollow fibers are illustrated in Figure 3.51. Outside diameters are in the 200 to 1000 μm range, and wall thicknesses are in the 50 to 250 μm range. Most of these membranes are asymmetric, meaning that they have multiple layers with different properties. For example, many of these membranes have a support layer with a thin diffusing layer, with the thin diffusing layer on the order of 0.1 μm. For comparison, human hair is in the range of 30 to 100 μm. So, hollow fibers are about an order of magnitude larger than a human hair with a “hole” in the middle that allows the fiber to approximate a tube. Thousands of hollow fibers can be in a single module.

The advantage of hollow-fiber membranes is identical to shell-and-tube heat exchangers, which is that a large amount of surface area/unit volume of equipment is possible. While the module appears to mimic a shell-and-tube heat exchanger, the flow patterns do not. In a shell-and-tube heat exchanger, there are two inlet streams and two outlet streams. In a hollow-fiber module, there is one input and two outputs, with the two outputs being the permeate stream and the less permeable (called retentate) stream. In fact, the most analogous separation to gas permeation from an input-output perspective is a partial vaporization/partial condensation/flash separation (Section 3.2.1).

The subscripts p and r indicate permeate and retentate, respectively. The variables p and y indicate pressure and mole fraction, respectively, and F indicates molar flowrate. The upstream side is at a higher pressure than the permeate side, which increases the driving force across the membrane. For the simple model, a two-component system is assumed.

A total balance and a balance on the more permeable component are

Rearrangement of Equations (3.75) and (3.76) give

where θ = F_p/F_in, which is often called the stage cut. It is a similar term to V/F in the flash-type separations, one outlet flowrate divided by the feed flowrate.

The balances must be combined with rate expressions for transport across the membrane. The rate expressions, assuming ideal gas, are

where P_A and P_B are the membrane permeabilities of the more permeable component (A) and the less permeable component (B), respectively, and t_m is the membrane thickness. Tables of permeabilities are available (Geankoplis, 2003; Wankat, 2017). Taking the ratio of Equations (3.78) and (3.79) yields

where α_AB is the ratio of the permeabilities, P_A/P_B. Sometimes, the following rearranged form of Equation (3.80) makes calculations easier.

The area, A, can be obtained from either rate expression, Equation (3.78) or (3.79), for a given membrane and thickness, once all flows and mole fractions are known.

Clearly, given the geometry of hollow-fiber membrane modules, the well-mixed assumption is a crude model. Crossflow arrangements are possible, where the feed and retentate are in the middle of the hollow-fiber bundle. Depending on the exact geometry of the equipment, both countercurrent and cocurrent flows are possible, and it is possible to have both in the same piece of equipment. Figure 3.53 illustrates some of these possible flow configurations.

The mathematical analysis of these models involves differential equations, and they parallel both the development for shell-and-tube heat exchangers and continuous differential separators. Turbulent flow is usually assumed in modeling heat exchangers and packed bed, vapor-liquid separators. However, given the very small diameters involved in hollow-fiber membranes, laminar flow is possible, so the parabolic velocity profile may have to be included. Clearly, these models are far more complex than the well-mixed model, and numerical solutions of the differential equations are often required. A summary of some of these models is available (Geankoplis, 2003).

Colburn, A. P. 1939. “A Method of Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction.” Trans AIChE 29: 174.

Couper, J. R., W. R. Penney, J. R. Fair, and S. M. Walas. 2012. Chemical Process Equipment, Selection and Design, 3rd ed. Boston: Elsevier.

Eckert, J. S. 1970. “Selecting the Proper Distillation Column Packing.” Chem Eng Prog 66(3): 39–44.

Fair, J. R., and R. L. Matthews. 1958. “Better Estimate of Entrainment from Bubble-Cap Trays.” Petrol Refin 37(4): 153.

Geankoplis, C. 2003. Transport Processes and Separation Principles, 4th ed. Upper Saddle River, NJ: Prentice Hall.

Hebert, S., and N. Sandford. 2016. “Consider Moving to Fixed Valves.” Chem Engr Prog 112(5): 34–41.

Kister, H. 1992. Distillation Design. New York: McGraw-Hill.

Koch, J., and G. Shiveler. 2015. “Design Principles for Liquid-Liquid Extraction.” Chem Engr Prog 111(11): 22–30.

Koch Modular Processes. 2000–2016. Extraction Column Types. http://modularprocess.com/liquid-liquid-extraction/extraction-column-types.

Kremser, A. 1930. “Theoretical Analysis of Absorption Process.” Natl Petrol News 22(21): 42.

O’Connell, H. E. 1946. “Plate Efficiencies of Fractionating Columns and Absorbers.” Trans Amer Inst Chem Eng 42: 741.

Perry, R. H., and D. W. Green (Eds.). 1997. Perry’s Chemical Engineers’ Handbook, 7th ed. New York: McGraw-Hill, 2-61–2-75.

Pilling, M. 2006. “Design Considerations for High Liquid Rate Tray Applications.” Proceedings of 2006 AIChE Annual Meeting, Paper 264e.

Regents of the University of Michigan. 2014. Encyclopedia of Chemical Engineering Equipment. http://encyclopedia.che.engin.umich.edu/Pages/SeparationsChemical/Extractors/Extractors.html.

Seader, J. D., E. J. Henley, and D. K. Roper. 2011. Separation Process Principles. Chemical and Biochemical Applications, 3rd ed. New York: Wiley.

Souders, M., and G. G. Brown. 1932. “Fundamental Design of High Pressure Equipment Involving

Paraffin Hydrocarbons. IV. Fundamental Design of Absorbing and Stripping Columns for Complex Vapors.” Ind Eng Chem 24: 519–522.

Stichlmair, J. G., and J. R. Fair. 1998. Distillation. Principles and Practice. New York: Wiley.

Treybal, R. E. 1980. Mass Transfer Operations, 3rd ed. New York: McGraw-Hill.

Turton, R., R. C. Bailie, W. B. Whiting, J. A. Shaeiwitz, and D. Bhattacharyya. 2012. Analysis, Synthesis and Design of Chemical Processes, 4th ed. New York: Pearson.

Wankat, P. 2017. Separation Processes. Includes Mass Transfer Analysis, 4th ed. Upper Saddle River, NJ: Prentice Hall.

a. Illustrate a partial vaporization for a feed of 25 mol% of the more volatile component. Indicate the two phases in equilibrium, the bubble point, and the dew point.

b. Illustrate a partial condensation for a feed of 50 mol% of the more volatile component. Indicate the two phases in equilibrium, the bubble point, and the dew point.

2. Sketch a P-xy diagram for an ideal, binary mixture.

a. Illustrate a flash for a feed of 75 mol% of the more volatile component. Indicate the two phases in equilibrium, the bubble point, and the dew point.

b. Illustrate a partial condensation for a feed of 50 mol% of the more volatile component. Indicate the two phases in equilibrium, the bubble point, and the dew point.

3. What are the physical assumptions of constant molar overflow? What are the consequences? To what types of systems does constant molar overflow typically apply?

4. State three reasons for requiring a high reflux ratio. Illustrate these on a McCabe-Thiele diagram.

5. How does the feed condition (saturated, superheated, subcooled) affect the reflux ratio for a given feed mole fraction? Illustrate the answer on McCabe-Thiele diagrams.

6. What are the limiting cases for reflux ratio? Why are neither of these cases practical?

7. What is the best location for the distillation column feed?

8. What happens if the feed to a distillation column is not at the optimum location? Illustrate this on a McCabe-Thiele diagram.

9. A distillation column requires 18 equilibrium stages that are 40% efficient. It has a partial reboiler.

a. How many trays are in the column if there is a total condenser?

b. How many trays are in the column if there is a partial condenser?

10. Why are some distillation columns run at elevated pressures? State and explain as many reasons as you can.

11. Why are some distillation columns run at vacuum? State and explain as many reasons as you can.

12. The feed to a distillation column is significantly subcooled. What happens on the feed tray? What is the potential effect on column diameter? What is the effect on the reboiler and condenser duties?

13. The feed to a distillation column is significantly superheated. What happens on the feed tray? What is the potential effect on column diameter? What is the effect on the reboiler and condenser duties?

14. Explain the relationship between reflux ratio and the number of equilibrium stages in a distillation column.

15. What is flooding? What is loading? What is weeping?

16. Discuss the advantages and disadvantages of sieve trays versus bubble cap trays.

17. Why might vacuum columns be designed with a larger diameter above the feed compared to below the feed?

18. Does tray spacing have an effect on column diameter? Explain why or why not.

19. The feed to a column is to be saturated liquid. An initial design based on the column conditions below the feed shows that trays above the feed might be weeping. Suggest possible remedies.

20. Discuss the advantages and disadvantages of packed versus tray columns.

21. An existing distillation column is to be scaled up (increase throughput) without changing the distillate and bottom compositions. What happens to the reflux ratio? What problems might arise in the performance of this column?

22. For the distillation column in Problem 3.21, it is suggested to change the pressure of the column. Explain why this might work.

23. For the distillation column in Problems 3.21 and 3.22, how can the pressure in the column be changed?

24. For the distillation column in Problems 3.21, 3.22, and 3.23, someone suggests that the column pressure can be increased by increasing the feed pressure. What is your response? Explain the rationale for your response.

25. The preliminary design of a distillation column shows that it is flooding. If, for some reason, the diameter cannot be changed, suggest two other design modifications that could alleviate this problem, and explain why each works.

26. A colleague gives you a preliminary tray column design. It shows 12-in tray spacing with 8-in weirs. There are 30 actual trays, and the diameter is 5 ft. Comment on all aspects of this design and suggest changes, if necessary.

27. A colleague gives you a preliminary tray column design. It shows 24-in tray spacing with 6-in weirs. There are 60 actual trays and the diameter is 5 ft. Comment on all aspects of this design and suggest changes, if necessary.

28. What is a packing factor, and how does it affect packed column design?

29. What is the absorption (stripping) factor? How does it affect absorber (stripper) design?

30. State three operating conditions that favor absorption (i.e., make absorption easier). Provide a physical explanation for each condition.

31. State three operating conditions that favor stripping (i.e., make stripping easier). Provide a physical explanation for each condition.

32. An absorber is not performing as specified, meaning that the target outlet vapor composition is higher than desired. State as many possible reasons for this that you can identify, and explain why each reason makes physical sense.

33. A stripper is not performing as specified, meaning that the target outlet liquid composition is higher than desired. State as many possible reasons for this that you can identify, and explain why each reason makes physical sense.

34. Comment on the following statement: A transfer unit in a packed column is numerically equivalent to an equilibrium stage in a tray column.

35. What do height of a transfer unit and the number of transfer units indicate?

36. Explain the physical reason why, when the absorption or stripping factor is less than one, increasing the number of stages or increasing packed column height does not increase the recovery of solute.

37. What is the separation basis for distillation? What is the separating agent?

38. What is the separation basis for absorption? What is the separating agent?

39. What is the separation basis for extraction? What is the separating agent?

40. What is the separation basis for leaching? What is the separating agent?

41. What is the separation basis for gas permeation membranes? What is the separating agent?

a. A feed of 200 kmol/h is flashed at 5 bar and 50°C. What are the liquid and vapor flowrates and the mole fractions in the vapor and liquid exit streams?

A feed of 200 kmol/h is to be fractionated to give a distillate containing 98 mol% dimethyl ether and a bottoms product containing 98 mol% methanol. The feed is 20% vapor and 80% liquid. There is a total condenser and a partial reboiler.

b. Determine the molar flowrates of the distillate and bottoms.

c. Calculate all internal flows for operation at a reflux ratio of 0.1.

d. In order to use cooling water in the condenser, assume that the DME must condense no lower than 55°C. Estimate the top pressure for this column.

e. In order to use low-pressure steam in the reboiler, the bottom temperature must be no higher than 148°C. Estimate the bottom pressure for this column.

f. Determine the diameter of a sieve tray tower with 18-in tray spacing to operate at 75% of flooding. The active tray area is 85% of the column area. Determine the diameter both above and below the feed. There are six equilibrium stages with the feed on the third stage.

g. Assume that the tray efficiency is 42%. Estimate the maximum weir height for which trays can be designed to meet the pressure specifications.

h. Suppose you determine that the height to diameter ratio for this column exceeds 20, which is usually the maximum recommended value. Suggest at least one design change that would result in a height to diameter ratio smaller than 20. Explain the rationale for your suggestion and its consequences.

i. Determine the diameter of a packed tower to operate at 70% of flooding using 1.5-in, ceramic Berl saddles. Determine the diameter both above and below the feed.

j. Look up the behavior of the DME-methanol system. How would it impact the answers to this problem?

43. Cumene (isopropyl benzene) is manufactured from propylene and benzene. Most of the world’s supply of cumene is directly converted into phenol, which is a raw material for a variety of products such as plasticizers. Acetone is also manufactured as a by-product of phenol manufacture. The separation section of a cumene process consists of a flash, to remove the unreacted propylene, followed by a distillation column to separate the benzene and produce purified cumene as the bottom product.

a. For this problem, consider that the flash contains only propylene and benzene and operates at 3 atm and 100°C. The feed contains 80 mol% benzene and 20 mol% propylene. On a basis of 200 kmol/h feed, determine the flowrates of vapor and liquid leaving the flash.

The feed to the distillation column is 60% liquid at 200 kmol/h containing 48 mol% benzene. It is necessary to produce 99 mol% cumene in the bottom product and 98 mol% benzene in the top product for recycle. The distillation column contains a total condenser and a partial reboiler. The column operates at an average pressure of 3 atm.

b. Determine the flowrates of the distillate and the bottoms.

c. For a reflux ratio of 0.85, calculate all internal flows.

d. Estimate the heat loads (kJ/h) on the condenser and reboiler.

e. Estimate the column efficiency and determine the actual number of trays, the actual number of stages, and the actual feed location. There are 10 equilibrium stages, with feed on Stage 4.

f. Design a sieve tray column for 75% of flooding, with 85% active tray area and with 18-in tray spacing. Specify the column diameter and the actual column height. Ignore the surface tension correction. It is up to you to determine whether to design for the top section or for the bottom section. You should justify your choice.

g. Estimate the pressure drop (kPa) in the column if the trays have 6-in weirs.

h. Design a packed column for 75% of flooding with 2-in, ceramic Raschig rings. It is up to you to determine whether to design for the top section or for the bottom section. You should justify your choice.

i. There is an existing column with the correct number of trays with a 2 m diameter and 85% active area. Will this column work? Explain why or why not. Your answer must be supported with calculations. Could anything be done to make this column work? If so, what are the consequences?

j. The original column is now built and operational, and the feed is 60% liquid. Due to a reactor upset, it is now necessary to accept feed, still 60% liquid, at the same rate but containing 50 mol% benzene, with the remainder cumene. It is still necessary to produce 99 mol% cumene at the same rate from the bottom of this column. You assign your summer intern the job of determining new operating conditions. The answer you get back is to operate at a reflux ratio of 0.75. Will the column operate at these conditions?

Data:

For propylene,

For benzene,

Average column relative volatility = 6

Feed viscosity = 0.3 cP

Benzene density = 800 kg/m³

Cumene density = 700 kg/m³

Benzene heat of vaporization = 3.07 × 10⁴ kJ/kmol

Cumene heat of vaporization = 3.81 × 10⁴ kJ/kmol

Estimated top temperature = 56°C

Estimated bottom temperature = 178°C

Benzene viscosity = 0.4 cP

Cumene viscosity = 0.3 cP

44. A distillation column is fed 500 kmol/h of an equimolar mixture of pentane and hexane. The feed is saturated liquid. It is necessary to produce outlet streams containing 95 mol% pentane and 99 mol% hexane. The distillation column contains a partial reboiler and a total condenser.

a. Determine the exit flowrates D and B.

b. If the reflux ratio is one, determine all of the internal molar flowrates in the column.

c. Estimate the condenser and reboiler heat loads.

d. In the summer, cooling water is available at 30°C and is returned at 40°C. In the winter, cooling water is available at 25°C and is returned at 35°C. Assuming a 10°C approach in the condenser, based on the cooling water return temperature, determine the seasonal top column pressures.

e. If the column pressure drop is 15 kPa, determine the seasonal bottom column temperatures.

f. There are 15 equilibrium stages, with the feed on stage 5. Determine the diameter of a sieve tray column with 24-in tray spacing for 70% of flooding and an active tray area of 88%. Calculate the diameter for both the top and bottom sections. If the column is designed for the larger diameter, how will the other section perform?

g. Determine the diameter of a packed column with 2-in, ceramic Berl saddles at 70% of flooding. Calculate the diameter for both the top and bottom sections. If the column is designed for the larger diameter, how will the other section perform?

Data:

45. For this problem, the feed is 40 mol% heptane and 60 mol% ethylbenzene. A saturated liquid feed of 200 kmol/h is to be fractionated at atmospheric pressure to give a distillate containing 98 mol% heptane and a bottoms product containing 1 mol% heptane.

a. Determine the molar flowrates of distillate and bottoms.

b. Calculate all internal flows for operation at a reflux ratio of 1.5.

c. Estimate the condenser and reboiler duties, in kW.

d. Estimate the top and bottom temperatures, in °C.

e. Design a tray tower with 18-in tray spacing to operate at 75% of flooding. The active tray area is 75%. You should decide whether to design for the top or for the bottom of the column. There are 17 equilibrium stages with the feed on Stage 10.

f. Design a packed tower with 2-in, ceramic INTALOX™ saddles for 70% of flooding. You should decide whether to design for the top or for the bottom of the column.

Vapor pressure data:

For heptane,

For ethylbenzene,

Other data:

46. A feed of 500 kmol/h of methanol and ethanol are to be distilled at 2 bar. The feed is 60 mol% methanol, and the feed is 75% liquid and 25% vapor in equilibrium. The desired outlet compositions are 90 mol% methanol in the distillate and 95 mol% ethanol in the bottoms.

a. Determine the flowrates of the distillate and the bottoms (kmol/h).

b. Calculate all internal flows for a reflux ratio of 3.

c. Determine the diameter of a sieve tray tower with 24-in tray spacing at 70% of flooding with an active tray area of 89%. Design for the top and bottom sections. If the larger diameter is used for the entire column, how will the other section perform? There are 19 equilibrium stages with the feed on Stage 7.

Data:

For methanol,

For ethanol,

Methanol heat of vaporization = 40.5 kJ/mol

Ethanol heat of vaporization = 38.3 kJ/mol

Methanol specific gravity = 0.75

Ethanol specific gravity = 0.75

47. CO₂ is to be stripped from water at 20°C and 2 atm using a staged, countercurrent stripper. The liquid flowrate is 100 kmol/h of water with an initial mole fraction of CO₂ of 0.00005.

The inlet air stream contains no CO₂. A 98.4% removal of CO₂ from the water is desired. The Henry’s law constant for CO₂ in water at 20°C is 1420 atm.

a. Calculate the outlet mole fraction of CO₂ in water.

b. If there are seven equilibrium stages, calculate V and the outlet mole fraction of CO₂ in the air.

48. An absorber is designed to treat 40 kmol/h of air containing acetone with a mole fraction of 0.02 and reduce the mole fraction to 0.001. Pure water is the solvent, at a rate of 20 kmol/h. The column temperature is 26.7°C, and the pressure is 1 atm. Raoult’s law may be assumed to apply, and

a. How many equilibrium stages are needed?

b. Assume that a column with five equilibrium stages is built and operational. It is now observed that the outlet mole fraction of acetone in the air is 0.002. Describe and explain at least four possible causes of this situation. Quantify what is happening in the column for each case.

c. Suggest realistic ways to compensate for this problem. An example of a nonrealistic compensation method is to reduce the air feed, which is probably fixed upstream.

d. The column is operating normally, but you are told to expect a 10% increase in air to be treated. How can you change column operation to ensure the desired outlet mole fraction of acetone in air? Be quantitative.

e. The column is operating normally, but you are told to expect an inlet mole fraction of acetone of 0.025. How can you change column operation to ensure the desired outlet mole fraction of acetone in air? Be quantitative.

f. The column is operating normally, but you are told that the outlet mole fraction of acetone must be reduced to 0.00075. How can you change column operation to ensure the desired outlet mole fraction of acetone in air? Be quantitative.

49. Sulfur dioxide (SO₂) is removed from a smelter gas (which may be considered to have the properties of pure air) containing 0.80 mol% SO₂ by absorption into a solution with the properties of pure water in a countercurrent tower. The exit SO₂ concentration in the gas phase must be 0.04 mol%. The tower is designed to operate at exactly 30°C and at 4 bar. At these conditions, the equilibrium relationship is y = 0.1923x, where y is the mole fraction of SO₂ in the gas phase, and x is the mole fraction of SO₂ in the water phase. The feed rate of gas is 0.36 kmol/s and 1 mole of aqueous solution is used for every 4 moles of feed gas.

a. What is the ratio of the solvent rate to the minimum solvent rate?

b. For a tray tower, determine the number of equilibrium trays required for this separation. Include fractional trays in your answer.

c. Assume the number of equilibrium stages is 6.5. Due to a temporary upset upstream, it is now necessary to treat gas at the higher feed concentration of 1 mol% SO₂. If the temperature and pressure must remain constant at 30°C and 4 bar, what do you suggest to accommodate the higher concentration in the feed?

50. An air stream flowing at 1500 kmol/h containing benzene vapor is to be scrubbed using 25 kmol/h of hexadecane (C₁₆H₃₄) as the solvent. The mole fraction of benzene in the inlet gas is 0.010, and the required outlet mole fraction is 0.00050. The hexadecane is being returned to the scrubber from a stripper, so the mole fraction of benzene in the feed hexadecane is 0.00010. The column may be assumed to operate at 150 kPa and 35°C. Raoult’s law may be assumed to define the benzene equilibrium. Dilute solutions may be assumed.

a. How many equilibrium stages are needed? Keep the decimal places.

b. What is the outlet mole fraction of benzene in the hexadecane? What do you conclude about the assumptions made?

c. The column is operational, and the outlet mole fraction of benzene in the air is measured to be 0.0010. You send an operator to check the conditions in the column, and the temperature, pressure, air flowrate, and hexadecane flowrate are all as per design specifications. Suggest two possible causes for the process upset, and provide exact values for the off-spec conditions, assuming that only one condition at a time is off-spec.

d. Determine the diameter of a sieve tray column for an 18-in tray spacing for operation at 75% of flooding. The fraction of active tray area is 0.88.

e. Determine the diameter of a packed column at 75% of flooding with 2-in, ceramic Raschig rings.

51. A wastewater stream at 35,000 kmol/h containing benzene at a mole fraction of 0.001 is to be stripped with air in a column operating at 2 bar and 25°C. The outlet water must contain no more than 0.00004 mole fraction of benzene. The equilibrium between water and air for benzene at 2 bar and 25°C is y = 150x, where y is the mole fraction of benzene in air and x is the mole fraction of benzene in water. The air used is maintained in a closed loop. Benzene is recovered by condensation from the air stream; therefore, the air entering the stripper contains 0.005 mole fraction of benzene.

a. As part of the design evaluation for this process, it is necessary to determine whether it is feasible to use an existing tray tower for this process. If the tower contains 9 equilibrium stages, and an air rate of 325 kmol/h is the limit set by 80% of flooding, can the tower be used?

b. You observe that there is some flexibility in the operating conditions of the tower in Part (a). Suggest one set of conditions under which the existing tower could be used.

52. When coal is burned to form synthesis gas (syngas), which contains mostly CO, H₂, H₂S, and CO₂, the H₂S must be removed. The gas is called syngas because H₂ and CO are the building blocks for synthesis of hydrocarbons. One method for removal of H₂S is the Selexol™ solvent, in which the H₂S is highly soluble. The Selexol solvent is the dimethyl ether of polyethylene glycol and may be assumed to have a molecular weight of 300 kg/kmol, μ = 2.5 × 10⁻³ kg/m s, and ρ = 900 kg/m³.

Consider the removal of H₂S only from an otherwise inert mixture of gases; that is, none of the other gases are soluble in the Selexol solvent. The partition coefficient for H₂S between gas and solvent is given by the relationship

The problem at hand is the removal of H₂S from a gas stream with the properties of air, initially containing 2% H₂S, so that the exit gas only contains 0.05% H₂S. The H₂S-rich Selexol solvent is regenerated in a stripper using pure air. The absorber operates at 100 psig (6.8 atm gauge) and 70°F. The stripper operates at 1 atm absolute and 200°F. The stripper reduces the H₂S content of the Selexol solvent to 0.5%, which is the feed concentration to the absorber. The gas to be treated is flowing at 1000 kmol/h, and it may be assumed that the Selexol solvent circulates at 30,000 kg/h. In the packed-bed absorber, it is known that H_oV = 3.5 m. In the packed-bed stripper, it is known that H_oL = 0.25 m. It is proposed to use an existing stripper that is packed with a 5 m high section of 1.5-in, ceramic Raschig rings and has a diameter of 1.8 m.

a. Determine the required absorber height. You may assume that the Colburn method is valid.

b. Determine the exit H₂S mole fraction in the Selexol solvent. Comment on this value.

c. For the stripper, determine the amount of air needed.

d. At what percentage of flooding is the stripper operating? Do you recommend operating under these conditions?

e. Determine the pressure drop across the stripper, in kPa.

53. Consider the removal of H₂S only from air. The solvent has a molecular weight of 300, μ = 2.5 × 10⁻³ kg/m/s, σ = 30 dyne/cm², and SG = 0.90. The partition coefficient for H₂S between air and solvent is given by the relationship

The problem at hand is the removal of H₂S from an air stream initially containing 0.10% H₂S, so that the exit gas only contains 0.0050% H₂S. The absorber operates at an average pressure of 6 atm and an average temperature of 50°F. The inlet H₂S content of the solvent can be assumed to be zero. The gas to be treated is at 200 lbmol/h, and it may be assumed that the solvent circulates at 2100 lb/h.

a. How many equilibrium stages are needed for this separation?

b. There is a problem with the solvent pump, which must be taken off line soon, and the spare pump can only provide solvent at 5 atm at a flowrate no higher than 105% of the original flowrate. Therefore, the feed gas is to be throttled to 5 atm. Since the heat exchangers for this process already uses refrigerated water, 50°F is the lowest possible temperature. You ask two summer interns to evaluate the situation. One says that everything should be okay. The other says that the absorber will not work. Which intern would you hire?

c. What is the diameter of a sieve-tray column for this absorber with 24-in tray spacing for 70% of flooding with an active area of 90% of the total area?

d. If the trays are 20% efficient, is the height/diameter ratio within typical limits? Explain.

e. Estimate the pressure drop in the actual column if there are 6-in weirs.

f. What is the diameter of a packed column for this absorber with 1.5-in, ceramic Berl saddles at 75% of flooding?