Chapter 14. Washing, Leaching, and Supercritical Extraction

A number of other separation processes can be analyzed, at least under limiting conditions, using McCabe-Thiele diagrams or the Kremser equation with an approach very similar to that used for absorption and stripping. These processes all require addition of a mass-separating agent. If flows are not constant, triangular diagram methods developed for extraction can often be extended to these separations. The triangular diagram method is developed for leaching (or solid-liquid extraction [SLE]) in Section 14.4.

1. Explain in general terms how McCabe-Thiele and Kremser analyses can be applied to other separation schemes, and delineate when these procedures are applicable

2. Explain what washing is, and apply McCabe-Thiele and Kremser procedures to washing problems

3. Explain what leaching is, and apply McCabe-Thiele and Kremser methods to leaching problems

4. Apply the triangular diagram analysis to leaching problems with variable flow rates

5. Explain how supercritical extraction works, and discuss its advantages and disadvantages

All McCabe-Thiele graphs are plots of concentration in one phase versus concentration in a second phase. There is a single equilibrium curve, and there is one operating line for each column section. It is desirable for each operating line to be straight. In addition, although it is not evident on the graph, we satisfy the energy balance and mass balances for all other species.

To obtain a single equilibrium curve, enough variables need to be specified so that only one degree of freedom remains. For binary distillation specifying constant pressure is sufficient. For absorption, stripping, and extraction we specify that pressure and temperature are constant, and if there are several solutes, we assume that they are independent. In general, we specify that pressure and/or temperature are constant, and for multisolute systems we assume that the solutes are independent.

To have a straight operating line for binary distillation we assume constant molal overflow (CMO) is valid, which means in each section total flow rates are constant. For absorption, stripping, and extraction we assume total flow rates are constant for dilute systems. The more dilute the solute, the more likely its total flow rates are constant. For more concentrated systems we assume there is one chemical species in each phase that does not transfer into the other phase; then flow rate of this species (carrier gas, solvent, or diluent) is constant. In general, we have to assume either total flow rates are constant or flow rates of nontransferred species are constant.

The flow rate assumptions control the concentration units used to plot McCabe-Thiele diagrams. If total flow rates are constant, solute mass balances are written in terms of fractions, and fractions are plotted on the McCabe-Thiele diagram. If flow rates of nontransferred species are constant, ratio units must be used, and ratios are plotted on the McCabe-Thiele diagram.

The McCabe-Thiele operating line satisfies the more volatile component (MVC) or solute mass balance. In binary distillation CMO forces total vapor and liquid flow rates to be constant; therefore, the overall mass balance is satisfied. In absorption, when constant carrier and solvent flows are assumed, mass balances for these two chemicals are automatically satisfied. In general, if overall flow rates are assumed constant, we are satisfying the overall mass balance. If flow rates of nontransferred species are constant, we are satisfying balances for these species.

The energy balance is automatically satisfied in distillation when CMO is valid. In absorption, stripping, and extraction, energy balances are satisfied by assuming constant temperature and negligible heat of absorption, stripping, or mixing. In general, for separations that require a mass separating agent we assume constant temperature and negligible heat involved in contacting two phases.

The Kremser equation was used for absorption, stripping, and extraction. When total flows, pressure, and temperature are constant, heat of contacting the phases is negligible, and equilibrium is linear; we can use the Kremser equation. When similar assumptions are valid, the Kremser equation can be used for other separations.

Of course, assumptions required to use McCabe-Thiele analysis or the Kremser equation may not be valid for a given separation. If assumptions are not valid, results of analyses could be garbage. To determine validity of assumptions, the engineer has to examine each specific case.

In the remainder of this chapter these principles are applied to generalize the McCabe-Thiele approach and the Kremser equation for a variety of unit operations. A listing of various applications is given in Table 14-1.

At washing equilibrium the solute concentration in the underflow liquid equals the solute concentration in the overflow liquid. This statement does not say anything about solid, which changes the relative underflow and overflow flow rates but does not affect concentrations. Thus, equilibrium equation is

where y = the mass fraction of solute in the overflow liquid, and x = the mass fraction of solute in the underflow liquid.

For the mass balance envelope shown in Figure 14-1B, the steady-state mass balance is

where O_j and U_j are total overflow and underflow liquid flow rates in kg/h leaving stage j. Units for Eq. (13-30) are kg solute/h. To develop the operating equation, solve Eq. (14-2) for y_j+1:

If underflow liquid and overflow liquid flow rates are constant, Eq. (14-3) will plot as a straight line.

Washers are often specified by required dry solids or wet solids flow rates. The underflow liquid flow rate can be calculated from the volume of liquid entrained with the solids. Let ε be the porosity (void fraction) of the solids in the underflow. That is,

and then

Once ε is known, we can calculate the underflow liquid flow rate, U_j. Suppose we are given the flow rate of dry solids. Then the volume of solids per hour is

and the total volume of underflow is

Then the volume of underflow liquid in m³ liquid per hour is

and finally the kg/h of underflow liquid is

In these equations ρ_f is the fluid density and ρ_s is the density of dry solids, both in kg/m³.

If the solids rate, ε, ρ_f, and ρ_s are all constant, then from Eq. (14-5d) U_j = U = constant. If U is constant, then an overall mass balance shows that the overflow rate, O_j, must also be constant. Thus, to have constant flow rates we assume:

1. No solids are in the overflow, and the solids do not dissolve. This ensures that the solids flow rate will be constant.

2. ρ_f and ρ_s are constant. Constant ρ_f implies that the solute has little effect on fluid density or that the solution is dilute.

3. Porosity ε is constant. Thus, the liquid volume entrained from stage to stage is constant.

When these assumptions are valid, O and U are constant, and the operating equation simplifies to

which plots as a straight line on a McCabe-Thiele diagram (Figure 14-2). This equation is similar to the other McCabe-Thiele operating equations we have developed; only the nomenclature has changed.

An alternative way of stating the problem is to specify volume of wet solids processed per hour. Then the underflow volume is

and

If densities and ε are constant, volumetric flow rates have to be constant, and washing problems can be solved using volumetric flow rates and concentrations in kg solute/m³.

We could have assumed that overflow and underflow rates are constant and derived Eq. (14-6). However, it is more informative to show the three assumptions required to make overflow and underflow rates constant. These assumptions show the McCabe-Thiele analysis of washing is likely to be invalid if the settlers are not removing all solids, if for some reason the amount of liquid entrained changes, and if the fluid density changes markedly. The first two problems do not occur in well-designed systems. The third is easy to check with density data.

The McCabe-Thiele diagram in Figure 14-2 differs from other McCabe-Thiele diagrams since temperature and pressure do not affect washing equilibrium. However, temperature does affect the rate of attaining equilibrium and hence efficiency because at low temperatures more viscous solutions are difficult to wash off the solid.

The analysis for washing can be extended to a variety of modifications. These include simulation problems, use of efficiencies, calculation of maximum U/O ratios, and calculations for crossflow systems. The Kremser equation can also be applied to countercurrent washing with no additional assumptions. This adaptation is a straightforward translation of nomenclature and is illustrated in Example 14-1. Brian (1972) discusses application of the Kremser equation to washing in considerable detail.

The washing analysis presented here is for a steady-state, completely mixed system in which wash water and water entrained by the solid matrix are in equilibrium. When filter cakes are washed, the operation is batch, the system is not well mixed because flow is close to plug flow, and the operation is not at equilibrium since the entrained fluid has to diffuse into the wash liquid. This case is analyzed by Harrison et al. (2015) and Mullin (2001).

Equipment and operation of washing and leaching systems are often very similar. In both cases a solid and a liquid must be contacted, allowed to equilibrate, and then separated from each other. Thus, mixer-settlers (Figure 14-1) are also commonly used for leaching easy-to-handle solids. A variety of specialized equipment has been developed to move solid and liquid countercurrently during leaching. Coulson et al. (1978), Lydersen (1983), Prabhudesai (1997), and Schwartzberg (1980) present good introductions to leaching equipment.

In leaching, the solute is initially a part of the solid matrix; thus, when the solute dissolves into the liquid the solid is partially dissolving. In washing, which can be considered a special case of leaching, the solute is initially retained in between solid particles and, if the solid has permanent pores, in the pores. The solid does not dissolve in washing. In leaching, equilibrium is usually not y = x, and total solids flow rate is usually not constant. Since diffusion rates in a solid are low, mass transfer rates are low. Thus, equilibrium may take days to leach excess salt from large pickles and years for in-situ leaching of copper ores (Lydersen, 1983). A rigorous analysis of leaching requires that the changing solid and liquid flow rates be included. This situation is very similar to partially miscible extraction and is analyzed in Section 14.4. In this section we look at simple cases in which a modified McCabe-Thiele analysis or the Kremser equation can be used.

A countercurrent cascade for leaching is shown in Figure 14-3A. Consider an idealized case in which entrainment of the liquid with the solid underflow can be ignored. The assumptions are:

1. The system is isothermal.

2. The system is isobaric.

3. No solvent dissolves into the solid.

4. No solvent is entrained with the solid.

5. There is an insoluble solid backbone or matrix.

6. Heat of mixing of the solute in the solvent is negligible.

7. Stages are equilibrium stages.

8. No solid is carried with the overflow liquid.

With these assumptions energy balances are automatically satisfied. The mass balance envelope shown in Figure 14-3A is often convenient since solute concentrations in entering solid and exiting solvent stream are often specified. To obtain a straight operating line we need to write the mass balance in terms of flow rates that are constant (solvent, not extract, and insoluble solid backbone, not total solid) and use weight ratios that correspond to these flow rates. Appropriate weight ratios and flow rates are

the operating equation is

This equation represents a straight line as plotted in Figure 14-3B. The equilibrium curve is now the equilibrium of solute between the solvent and solid phases. Equilibrium data must be measured experimentally. If the equilibrium line is straight, which is unusual, the Kremser equation can be applied.

Unfortunately, the analysis is oversimplified because assumptions 4 and 7 are almost always faulty. There is always entrainment of liquid in the underflow (for the same reason that there is an underflow liquid in washing). Since diffusion in solids is very slow, equilibrium is rarely attained in real processes. The combined effects of liquid in underflow and nonequilibrium stages are often included by determining an “effective linear equilibrium constant, m_E” so that equilibrium is y = m_Ex (Schwartzberg, 1980). The constant m_E depends on flow conditions and residence times and is valid only for the conditions at which it was measured. In other words, the effective equilibrium constant is not a fundamental quantity. However, it is easy to measure and use. The McCabe-Thiele diagram will look similar to Figure 14-3B, except equilibrium will be linear, and the operating line may be curved. The Kremser equation can be used if you assume that overall underflow and overflow rates are constant (see Problems 14.D8, D9, D16, and D17).

A schematic of a countercurrent leaching system is shown in Figure 14-4 with appropriate nomenclature. Since leaching is quite similar to LLE, the same nomenclature is used (see Table 13-2). Even if flow rates E and R vary, the differences in total and component flow rates for passing streams are constant, which defines a difference point [see Eqs. (13-42) and (13-43) for the LLE cascade in Figure 13-20]. Since the cascades are the same (compare Figures 13-20 and 14-4), the results for leaching are the same as for LLE.

The calculation steps for countercurrent leaching operations are exactly the same as for LLE:

1. Plot the equilibrium data.

2. Plot the locations of known points.

3. Find mixing point M.

4. Locate E_N.

5. Find the Δ point.

6. Step off stages.

This procedure is illustrated in Example 14-2.

Equilibrium data for leaching must be obtained experimentally, since the equilibrium data will depend on the exact nature of the solids, which may change from source to source. If there is no entrainment, the overflow (extract) stream will often contain no inert solids (diluent). However, the raffinate stream will contain solvent. Test data for extraction of oil from meal with benzene are given in Table 14-2 (Prabhudesai, 1997). In these data, inert solids are not extracted into benzene. Data are plotted on a triangular diagram in Figure 14-5 (see Example 14-2). The conjugate line is constructed in the same way as for extraction.

A technique that is very similar to triangular diagrams uses ratio units on a Ponchon-Savarit diagram (Lydersen, 1983; McCabe et al., 2005; Prabhudesai, 1997; Treybal, 1980). McCabe-Thiele diagrams can also be used for leaching if flow rates are close to constant. If flow rates are not constant, curved operating lines can be constructed on the McCabe-Thiele diagram with triangular or Ponchon-Savarit diagrams.

Leaching can be a very difficult separation system to design and operate in an efficient manner. Many solids have terrible flow properties, and a semblance of countercurrent flow requires mechanical movement (e.g., in compartments of a separator). Entrainment of fines is not uncommon, and filtration or centrifugation may be required to obtain a clear overflow liquid. In addition, mass transfer rates are often very low, and equilibrium can take months or years to achieve. Since leaching is often done with natural products, raw materials can vary significantly. For example, soybeans of a particular type from one field may have significantly different properties than soybeans of another type from a different location. Because of these difficulties, a new leaching process requires significant pilot plant studies before it is scaled up.

First, what is an SCF? Figure 14-6A shows a typical pressure-temperature diagram for a single component. Above critical temperature T_c, it is impossible to liquefy the compound. Critical pressure p_c is the pressure required to liquefy the compound at the critical temperature. Critical temperatures and pressures for a large number of compounds have been determined (Paulaitis et al., 1983; Poling et al., 2001). SCFs of interest include primarily CO₂ (p_c = 72.8 atm, T_c = 31.0°C, ρ = 0.47 g/mL) but also propane (p_c = 41.9 atm, T_c = 97.0°C, ρ = 0.22 g/mL), and water (p_c = 217.7 atm, T_c = 374°C, ρ = 0.32 g/mL). An SCF behaves like a gas in that it expands to fill the confines of a container.

As can be seen from this very short list, SCFs have densities much greater than those of typical gases and less than those of liquids by roughly a factor of 2 to 3. Viscosities of SCFs are about one-tenth those of liquids. This leads to low pressure drops. High diffusivities, roughly 10 times those of liquids and lack of a phase boundary lead to very high mass transfer rates and low HETP values in packed beds. SCFs can often dissolve almost as much solute as a good liquid solvent. Extraction can often be carried out at low temperatures, particularly when CO₂ is used. This is particularly advantageous for extraction of foods and pharmaceuticals. Supercritical CO₂ is completely natural, or “green,” and is totally acceptable for processing foods and pharmaceuticals if no additives are used (Allen and Shonnard, 2002). However, CO₂ should not be vented to the atmosphere.

Solubility of a solute in an SCF is a complex function of temperature and pressure. Figure 14-6B shows solubility of naphthalene in CO₂ (Hoyer, 1985; Paulaitis et al., 1983). As pressure is increased solubility first decreases and then increases. At both high and low pressures naphthalene is more soluble at high temperatures than at low temperatures. This is the expected behavior because vapor pressure of naphthalene increases with increasing temperature. Immediately above the critical pressure solute is more soluble at the lower temperature. This is a retrograde phenomenon; however, if naphthalene solubility is plotted versus CO₂ density, retrograde behavior is not evident (Gupta and Johnston, 2008). In addition to having high solubilities, SCF should be selective for the desired solutes. Solute-solvent interactions can affect solubility and selectivity of SCFs, and therefore entrainers (additives) are often added to increase solubility and selectivity. However, addition of additives may change the nontoxic, natural characteristics of CO₂.

Figure 14-6B also shows how solute can be recovered from CO₂. If pressure is dropped, naphthalene solubility will plummet, and naphthalene will drop out as finely divided solid. A typical process using pressure reduction is shown in Figure 14-7. Batch processes are usually used with solids because feeding and withdrawing solids at the high pressures of supercritical extraction is difficult. Regeneration can also be achieved by changing temperature, distilling the CO₂-solute mixture at high pressure, or absorbing solute in water (McHugh and Krukonis, 1994). In many cases one of these processes is preferable because it decreases compression costs.

Several SCF applications have been widely publicized. Kerr-McGee developed its Residuum Oil Supercritical Extraction (ROSE) process in the 1950s (Gupta and Johnston, 2008; Humphrey and Keller, 1997; Johnston and Lemert, 1997; McHugh and Krukonis, 1994). When oil prices went up, the process attracted considerable attention, since it has lower operating costs than competing processes. The ROSE process uses an SCF such as propane to extract useful hydrocarbons from residue left after distillation. The high temperatures and pressures employed for residuum treatment lead naturally to SCF extraction.

Food and pharmaceutical industries are interested in SCF because of the desire to have completely natural processes, which cannot contain any residual hydrocarbon or chlorinated solvents (Humphrey and Keller, 1997). Supercritical carbon dioxide has been the SCF of choice because it is natural, nontoxic, cheap, and completely acceptable as a food or pharmaceutical ingredient and often has good selectivity and capacity. Currently, supercritical CO₂ is used to extract caffeine from green coffee beans to make decaffeinated coffee. Supercritical CO₂ is also used to extract flavor compounds from hops to make a hop extract that is used in beer production. The leaching processes that were replaced were adequate in all ways except that they used solvents that were undesirable in the final product.

A variety of other SCF extraction processes have been explored (Gupta and Johnston, 2008; Hoyer, 1985; McHugh and Krukonis, 1994; Paulaitis et al., 1983). These include extraction of oils from seeds such as soybeans, removal of excess oil from potato chips, fruit juice extraction, extraction of oxygenated organics such as ethanol from water, dry cleaning, removal of lignite from wood, desorption of solutes from activated carbon, and treatment of hazardous wastes. Not all of these applications were successful, and many that were technically successful are not economical.

The main challenge in applying SCF extraction on a large scale has been scaling up for the high pressure required. High-pressure equipment becomes quite heavy and expensive. In addition, methods for charging and discharging solids continuously at high pressure are not well developed. Another problem has been lack of design data for supercritical extraction. Supercritical extraction is not expected to be a cheap process. Thus, most likely applications are extractions for which existing separation methods have at least one serious drawback and for which SCF extraction does not have major processing disadvantages.

McCabe-Thiele and Kremser methods have also been applied to analyze less common separation methods. Modifications of the McCabe-Thiele method have been applied to cyclic adsorption and ion exchange processes including parametric pumping (Grevillot and Tondeur, 1977) and cycling zone adsorption (Wankat, 1986). McCabe-Thiele and Kremser methods can also be used to analyze three-phase separations (Wankat, 1980).

Brian, P. L. T., Staged Cascades in Chemical Processes, Prentice Hall, Upper Saddle River, NJ, 1972.

Brown, G. G., et al., Unit Operations, Wiley, New York, 1950.

Coulson, J. M., J. F. Richardson, J. R. Backhurst, and J. H. Harker, Chemical Engineering, Vol. 2, 3rd ed., Pergamon Press, Oxford, 1978, Chapter 10.

Grevillot, G., and D. Tondeur, “Equilibrium Staged Parametric Pumping. II. Multiple Transfer Steps per Half Cycle and Reservoir Staging,” AIChE J., 23, 840 (1977).

Gupta, R. B., and K. P. Johnston, “Supercritical Fluid Separation Processes,” in D. W. Green and R. H. Perry (Eds.), Perry’s Chemical Engineers’ Handbook, 8th ed., McGraw-Hill, New York, 2008, pp. 20-14 to 20-19.

Harrison, R. G., P. Todd, S. R. Rudge, and D. P. Petrides, Bioseparations Science and Engineering, 2nd ed., Oxford University Press, New York, 2015.

Hoyer, G. G., “Extraction with Supercritical Fluids: Why, How, and So What,” Chemtech, 440 (July 1985).

Humphrey, J. L., and G. E. Keller, II, Separation Process Technology, McGraw-Hill, New York, 1997.

Johnston, K. P., and R. M. Lemert, “Supercritical Fluid Separation Processes,” in R. H. Perry and D. W. Green (Eds.), Perry’s Chemical Engineers’ Handbook, 7th ed., McGraw-Hill, New York, 1997, pp. 22-14 to 22-19.

Lydersen, A. L., Mass Transfer in Engineering Practice, Chichester, UK, Wiley, 1983.

McCabe, W. L., J. C. Smith, and P. Harriott, Unit Operations of Chemical Engineering, 7th ed., McGraw-Hill, New York, 2005.

McHugh, M. A., and V. J. Krukonis, Supercritical Fluid Extraction: Principles and Practice, 2nd ed., Butterworth-Heinemann, Boston, 1994.

Mullin, J. W., Crystallization, 4th ed., Oxford and Butterworth-Heinemann, Boston, 2001.

Paulaitis, M. E., J. M. L. Penniger, R. D. Gray, Jr, and P. Davidson, Chemical Engineering at Supercritical Fluid Conditions, Butterworths, Boston, 1983.

Perry, R. H., and D. Green (Eds.), Perry’s Chemical Engineer’s Handbook, 6th ed., McGraw-Hill, New York, 1984.

Poling, B. E., J. M. Prausnitz, and J. P. O’Connell, The Properties of Gases and Liquids, 5th ed., McGraw-Hill, New York, 2001.

Prabhudesai, R. K., “Leaching,” in P. A. Schweitzer (Ed.), Handbook of Separation Techniques for Chemical Engineers, McGraw-Hill, New York, 1997, Section 5.1.

Schwartzberg, H. G., “Continuous Countercurrent Extraction in the Food Industry,” Chem. Eng. Progress, 76 (4), 67 (April 1980).

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

Wankat, P. C., “Calculations for Separations with Three Phases: 1. Staged Systems,” IEC Fundam. 19, 358–363 (1980).

Wankat, P. C., Large-Scale Adsorption and Chromatography, CRC Press, Boca Raton, FL, 1986.

Wankat, P. C., Mass-Transfer Limited Separations, Kluwer, Amsterdam, 1990.

A1. Develop your key relations chart for this chapter. Remember that a key relations chart is not a core dump but is selective.

A2. How do the ideas of a general McCabe-Thiele procedure and the concept of unit operation relate to each other?

A3. In the unit operation called Washing (select the best answer),

a. solute is physically held to the solid particles.

b. at equilibrium the underflow liquid has same solute concentration as overflow liquid.

c. underflow liquid is liquid entrained with the particles.

d. all of the above.

e. a and b.

f. a and c.

g. b and c.

h. None of the above.

A4. In leaching the final saturated raffinate often contains a significant amount of solvent. This occurs in Example 14-3 and for Problems 14.E1 to 14.E3. How do you recover this solvent?

A5. How does solid enter into washing calculations? Where does solids flow rate implicitly appear in Figure 14-1?

A6. Explain why pilot plant studies are needed before designing and scaling up a new leaching process. Why are laboratory-scale experiments useful but not sufficient? Give a specific example.

A7. Show how Figure 14-7 could be modified to use a temperature swing instead of a pressure swing. What might be advantages and disadvantages of doing this?

A8. What are some of the properties you would look for in a good solvent for extraction, leaching, and supercritical extraction?

A9. Would you expect stage efficiencies to be higher or lower in leaching than in supercritical extraction? Explain.

A10. With any model (McCabe-Thiele, Kremser, triangular diagram, computer simulation, mass transfer analysis) you should ask how well the model captures key aspects of the physical situation. For staged separations key aspects include equilibrium, flow ratios, cascading, and mass transfer. Critique the model used in Problem 14.D11 based on these aspects.

A11. In washing, the equilibrium between the overflow and the underflow is written as y = mx. The value of the equilibrium constant m (select the best answer)

a. depends on temperature.

b. must be determined from experiment.

c. depends on pressure.

d. a, b, and c.

e. equals 1.0.

f. equals gas constant R.

g. equals 0.0.

A12. Washing clothes is most similar to large-scale washing when (select the best answer)

a. the clothes are washed by hand.

b. the clothes are washed with detergent but no soap.

c. the clothes are washed with soap but no detergent.

d. the clothes are rinsed.

C. Derivations

C1. Derive Eq. (14-10) and the form of this equation if the system is very dilute.

C2. Adapt the Kremser equation to leaching.

C3. Derive Eq. (14-11) and the corresponding equation for x_D,M.

C4. Develop procedures for single-stage and crossflow systems for leaching using a triangular diagram. Hint: Study Section 13.9.

C5. Batch washing is similar to batch extraction, except in continuous-addition batch washing (Figure 13-11) there is no need to presaturate the wash liquid with the solid, and a filter is used instead of a settler. If the terms are translated, the same equations can be used for batch washing.

a. Convert Eqs. (13-24b) and (13-21) to equations for batch washing.

b. Convert Eq. (13-28) to batch washing with continuous addition of wash liquid.

D. Problems

*Answers to problems with an asterisk are at the back of the book.

D1. We plan to wash dilute sulfuric and hydrochloric acids from crushed rock in a countercurrent system. Operation is at 25.0°C and 1.0 atm. 100.0 m³/day of wet rock are to be washed. After settling, porosity is constant at 0.40. Thus, the underflow rate is 40.0 m³/day. Initial concentration of the underflow liquid is 1.0 kg sulfuric acid/m³ and 0.75 kg hydrochloric acid/m³. The outlet underflow sulfuric acid concentration is 0.09 kg/m³. The wash liquid (overflow) rate is 50.0 m³/day, and the inlet wash water is pure. For these dilute solutions assume that the solution densities are constant and the same as pure water, 1000.0 kg/m³. Find

a. Number of equilibrium stages required.

b. Outlet concentration (kg/m³) of hydrochloric acid in the underflow liquid.

D2.* You are working on a new glass factory near the ocean. The sand is to be mined wet from the beach. However, wet sand carries with it seawater entrained between sand grains. Several studies have shown that 40.0% by volume seawater is consistently carried with sand. Seawater is 3.5 wt% salt, which must be removed by a washing process.

Densities: Water, 1.0 g/cm³ (assume constant); Dry sand, 1.8 g/cm³ (including air in voids); Dry sand without air, 1.8/0.6 = 3.0 g/cm³.

a. We desire a final wet-sand product in which entrained water has 0.2 wt% salt. For each 1000.0 cm³ of wet sand fed we will use 0.5 kg of pure wash water. In a countercurrent washing process, how many stages are required? What is the outlet concentration of the wash water?

b. In a crossflow process we wish to use seven stages with 0.2 kg of pure wash water added to each stage for each 1000.0 cm³ of wet sand fed. What is the outlet concentration of the water entrained with sand?

D3. Alumina solids are being washed to remove NaOH from liquid entrained with solids. The wet feed is a mixture of 20.0 vol% solids (dry basis) and 80.0 vol% liquid. Underflow streams from settlers maintain the same vol% of solids and liquid. The feed flow rate (dry basis) is 1500.0 kg/h of solids, and NaOH concentration in the entrained liquid is 4.5 wt%. We want to remove 85.0% of NaOH. Find the flow rate of the required pure wash water.

Data: ρ_water = 1.0 kg/L, ρ_solids = 2.5 kg/L (dry, crushed).

a. One equilibrium stage.

b. Three-stage crossflow system with equal amounts of wash water in each stage.

c. Three-stage countercurrent system.

D4.* Wash alumina solids to remove NaOH from the entrained liquid. Underflow from the settler tank is 20.0 vol% solid and 80.0 vol% liquid. Two feeds to the system are also 20.0 vol% solids. In one of these feeds, the NaOH concentration in the liquid is 5.0 wt%. This feed’s solid flow rate (on a dry basis) is 1000.0 kg/h. The second feed has a NaOH concentration in the liquid of 2.0 wt%, and its solids flow rate (on a dry basis) is 2000.0 kg/h. We desire the final NaOH concentration in underflow liquid to be 0.6 wt% NaOH. A countercurrent operation is used. Inlet washing water is pure and flows at 4000.0 kg/h. Find the optimum feed location for the intermediate feed and the number of equilibrium stages required.

Data: ρ_w = 1.0 kg/liter (constant), ρ_alumina = 2.5 kg/liter (dry crushed).

D5. Your boss thinks it will be just as good to combine the two feeds in Problem 14.D4 rather than keeping them separate. Calculate the number of equilibrium stages required to achieve the same outlet concentrations with the same flow rates if two feeds are combined before being fed to the washing cascade. Compare with your answer to Problem 14.D4.

D6. Barium sulfide (BaS) is recovered from insoluble solids (see Problem 14.D14 for data) in a batch washing process. The initial charge is 10.0 kg of wet solids (each kg dry insoluble solids carries with it 1.5 kg of underflow liquid). The initial underflow liquid is an aqueous solution with 5.0 wt% BaS. The wet solids are mixed with 10.0 kg of pure water, solids are allowed to settle, and overflow liquid is removed. This process is then repeated twice using 10.0 kg of pure wash water each time. Assume that liquid density is constant and that the underflow liquid held by the solids has the same concentration of BaS as the overflow liquid.

a. Find the mass fractions of BaS in the overflow and underflow liquids after the first, second, and third washing steps.

b. If operation is done continuously with 10.0 kg/h of a wet solids feed in a three-stage crossflow system using 10.0 kg/h of pure wash water in each stage, find the mass fractions of BaS in the overflow and underflow liquids from each of the three stages.

c. If a continuous countercurrent three-stage process is used with a total of 30.0 kg/h of wash water and 10.0 kg of wet solids/h, find the mass fractions of BaS in outlet underflow and overflow.

D7.* You are designing a new glass factory near the ocean. Sand is to be mined wet from the beach. However, wet sand carries with it seawater entrained between sand grains. Data are given in Problem 14.D2. Salt must be removed by a washing process. A crossflow process will be employed with 0.2 kg of wash water added to each stage for each 1000.0 cm³ of wet sand fed. Wash water outlet from last stage will be used as wash water inlet for stage 3. Wash water outlet from stage N-1 will be used as wash inlet for stage 2, and wash water outlet from stage N-2 as wash water inlet to stage 1. All other stages have pure wash water inlet (see figure). Outlet concentration ≤ 0.002 weight fraction salt in the entrained liquid. How many stages are required? Note: This problem is not guess and check.

D8.* In leaching sugar from sugarcane, water is used as solvent. Typically about 11 stages are used in a countercurrent Rotocel or other leaching system. On a volumetric basis liquid flow rate/solid flow rate = 0.95. The effective equilibrium constant is m_E = 1.18, where m_E = (concentration, g/liter, in liquid)/(concentration, g/liter, in solid) (Schwartzberg, 1980). If pure water is used as inlet solvent, predict the recovery of sugar in the solvent.

D9. Experimental data for leaching sugar from sugarcane with water shows that a reasonable value for effective equilibrium constant y/x = m_E is 1.18 where y and x are the solute concentration in g/(L liquid) and g/(L solid), respectively (Schwartzberg, 1980). Batch leaching is similar to batch extraction, and if the terms are translated, the same equations can be used. We are doing a single-stage batch leaching of sugarcane with water to recover sugar. The feed to the leaching process is 1.0 kg of solids (on a dry basis). In this feed, cane contains 5.5 wt% sugar. The inlet solvent is pure water. Assume the overflow and underflow amounts are constant.

a. If 3.0 kg of wash liquid are used in a normal batch leaching operation, what are the outlet overflow liquid weight fraction of sugar (y) and the weight fraction of sugar in the solid (x)?

b. If we want the outlet overflow liquid to be 0.4 wt% sugar, how many kg of wash water are required in a normal batch leaching operation?

D10.* Use of slurry adsorbents has received some industrial attention because it allows for countercurrent movement of the solid and fluid phases. Your manager wants you to design a slurry adsorbent system for removing methane from a hydrogen gas stream. The actual separation process is a complex combination of adsorption and absorption, but total equilibrium can be represented by a simple equation. At 5.0°C, equilibrium can be represented as

Weight fraction CH₄ in gas = 1.2 × (weight fraction CH₄ in slurry)

At 5.0°C, no hydrogen could be detected in the slurry, and heat of sorption was negligible. We wish to separate a gas feed at 5.0°C that contains 100.0 lb/h of hydrogen and 30.0 lb/h of methane. An outlet gas concentration of 5.0 wt% methane is desired. Entering slurry contains no methane and flows at a rate of 120.0 lb/h. Find the number of equilibrium stages required for this separation and the mass fraction of methane leaving in the slurry.

D11. To provide a simplified calculation method for the variable flow rate leaching problem solved in Example 14-2, your boss asks you to force-fit the problem so that a Kremser equation solution can be obtained. To do this define a constant flow rate of the meal, F_M = 800.0 kg meal/h, and a constant solvent flow rate, F_S = F_S,E kg solvent (benzene)/h, which is an effective solvent rate. Partial miscibility will be ignored, and its effect will be included in F_S,E and in the effective equilibrium constant m_E, Y = m_E X where Y = kg oil/kg solvent and X = kg oil/kg meal. From the results of Example 14-2, find the values of F_S,avg and m_E that satisfy both the Kremser equation (written in terms of ratios Y and X) and the external oil mass balance. Note that F_S,in = 662 > F_S,E. Assume extra solvent is used to saturate the meal and F_S,E is solvent available to do the separation. Do Problem 14.A10 immediately after solving 14.D11.

D12.* A countercurrent leaching system is recovering oil from soybeans with five stages. On a volumetric basis, liquid flow rate/solids flow rate = 1.36. Recovery of oil in the solvent is 97.5%. The solvent is pure. Determine the effective equilibrium constant, m_E, where m_E is (kg/m³ of solute in solvent)/(kg/m³ of solute in solid) and is given by equation y = m_Ex. Assume the overall volumetric flow rates are constant.

D13. Batch leaching is similar to batch extraction, and the equations developed in Section 13.6 can be adapted when the solution is dilute or there is an insoluble solid matrix. We have 12.5 l of pure water that we will use to leach 10.0 l of wet sugarcane solids. Equilibrium data are in Problem 14.D8.

a. Find the fractional recovery of sugar in the water if the water and wet sugarcane solids are mixed together and after settling, the water layer is removed.

b. Find the fractional recovery of sugar in the water if a continuous solvent addition batch leaching system analogous to Figure 13-11 is used.

D14. BaS is produced by reacting barium sulfate ore with coal. The result is barium black ash, which is BaS plus insoluble solids. Since BaS is soluble in water, it can be leached out with water. In thickeners, insoluble solids in underflow typically carry with them 1.5 kg of liquid per kg of insoluble solids. At equilibrium overflow and underflow liquids have the same BaS concentrations (Treybal, 1980). We want to process 350.0 kg/h of insoluble solids plus its associated underflow liquid containing 20.0 wt% BaS. Use a countercurrent system with 2075 kg/h of water as solvent. The entering water is pure. The outlet underflow liquid is 0.001 wt% BaS. Find:

a. The BaS mass fraction in the exiting overflow liquid.

b. The number of equilibrium stages required.

D15. We are removing BaS from insoluble solids (see Problem 14.D14 for data). We are processing 1000.0 kg/h of dry insoluble solids plus underflow liquid carried with the solids. The entering underflow liquid is 15.0 wt% BaS. A 99.0% recovery of BaS is desired. The entering solvent (water) is pure. Use a countercurrent process.

a. Find the minimum overflow rate (minimum entering water rate) (Ov)_min.

b. If Ov = 1.2(Ov)_min, find the number of stages required.

c. If separation actually requires 15 stages at Ov = 2000.0 kg/h, find the overall stage efficiency and alternatively, the effective equilibrium constant m_E. (m_E includes both equilibrium and efficiency. It represents the value of the equilibrium constant that will result in stages having 100.0% efficiency.)

D16. Repeat Problem 14.D13 for following conditions:

a. If we want 80.0% recovery of sugar in the water, how much water should we use in a batch leaching system to leach 10.0 liter of wet sugarcane solids?

b. If we want 80.0% recovery of sugar in water, how much water should we use in a continuous solvent addition batch leaching system to leach 10.0 liter of wet sugarcane solids?

Equilibrium data are in Problem 14.D8.

D17. Continuous water addition batch washing is similar to continuous solvent addition batch extraction, and if the terms are translated, the same equations can be used (see Problem 14.C5b). We are doing a single-stage, continuous water addition batch washing of alumina solids to remove NaOH from the entrained liquid. The feed to the washing process is 1.0 kg of solids (on a dry basis). Each kg of dry solids entrains 2.0 kg of underflow liquid. In this feed, the entrained liquid contains 6.0 wt% NaOH. The inlet washing liquid is pure water. Assume the overflow and underflow amounts are constant.

a. If 2.0 kg of wash liquid are used in a continuous water addition batch washing operation, what is the outlet underflow liquid weight fraction of NaOH?

b. If we want the outlet underflow liquid to be 0.5 wt% NaOH, how many kg of wash water are required in a continuous water addition batch washing operation?

c. Compare the results to the results of Problem 14.D16.

D18.* Repeat Example 14-2 except for a single-stage system and unknown underflow product concentration.

D19.* Repeat Example 14-2 except for a three-stage countercurrent system and unknown underflow product concentration.

D20.* Repeat Example 14-2 except for a three-stage crossflow system, with pure solvent at a rate of 421 kg/h added to each stage and unknown underflow product concentration.

D21. A slurry of pure NaCl crystals, NaCl in solution, NaOH in solution, and water is sent to a system of thickener(s) at a rate of 100.0 kg/min. The feed slurry is 45.0 wt% crystals. The mass fractions of the entire feed (including crystals and solution) are: x_NaCl = 0.5193, x_NaOH = 0.099, and x_water = 0.3187. Remove NaOH by washing with a saturated NaCl solution. Thickener(s) are operated so that underflow is 80.0% solids (crystals) and 20.0% liquid. There are no solids in the overflow. Solubility of NaCl in caustic solutions (y value) is listed in following table (Brown et al., 1950). The first two rows of x values (weight fraction in underflow including both crystals and solution) have been calculated so that you can check your calculation procedure. The x values are specific for this operation in which the underflow is 80.0% solids, and the crystals are pure NaCl. [x_NaCl = 0.8(1.0) + 0.2 y_NaCl, x_NaOH = 0.8(0) + 0.2 y_NaOH]

a. Complete the table, plot the saturated extract (y) and saturated raffinate (x) curves, and construct a conjugate line.

b. If the feed is mixed with 20.0 kg/min of a saturated NaCl solution (y_NaCl = 0.27, y_NaOH = 0.0) in a single thickener, find the flow rates of the underflow and the overflow, the compositions of these streams, and the weight of the crystals in the underflow. * Partial answer in part d.

c. If a countercurrent cascade of thickeners is used with S = 20.0 kg/min of a saturated NaCl solution (y_NaCl = 0.27, y_NaOH = 0.0) and we desire a raffinate product that has x_NaOH = 0.01, determine the flow rates of R₁ and E_N and the number of stages required.

d. Raffinate from part b, R₁ = 56.63 kg/min, x_R1,NaCl = 0.833, is mixed in a second stage with fresh solvent, which makes this a crossflow system. The goal is to have raffinate from the second stage have x_R2,NaOH = 1.0 wt%. Find the flow rate of the solvent that has to be added, the flow rates of R₂ and E₂, and the compositions of R₂ and E₂.

Note: Although this is a washing problem, there is not a constant flow rate of solids. Thus, it needs to be solved like the leaching problems in this chapter.

D22. We are testing a leaching system with five stages to leach sugar from sugarcane. On a volumetric basis, liquid flow rate/solid flow rate = 1.25. When pure water is used as the inlet solvent, recovery of sugar in the solvent is 95.0%. Determine the effective equilibrium constant, m_E = (concentration, g/liter, in liquid)/(concentration, g/liter, in solid).

D23. We plan to remove dilute sulphuric acid and dilute HCl from crushed rock by washing it with water in a continuous countercurrent process with five equilibrium stages. The feed is 1000.0 kg/h (dry, crushed) rock that entrains underflow liquid with it. The porosity of the settled rock is ε = 0.70. The initial weight fraction of sulphuric acid in the underflow liquid is x_S,0 = 0.013, and the initial weight fraction of hydrochloric acid is x_HCl,0 = 0.018. We want to reduce the sulphuric acid final weight fraction to x_S,N = 0.001 or less, and we want to reduce the HCl weight fraction to x_HCl,N = 0.002 or less. (One of the solutes should be exactly at the specified weight fraction, and the other solute should be at a lower weight fraction.) Find the required overflow flow rate in kg/h and the value of x_N for the solute that is below its maximum weight fraction. The entering wash water is pure.

Data: Density of dry crushed rock ρ_s = 2.6 kg/L. Density of overflow fluid ρ_f = 1.0 kg/L (assume constant).

D24. In your plant the effective equilibrium constant m_E for recovering oil from soybeans is 1.3 for the equation C_{oil_in_solvent} = m_E C_{oil_in_solid}. C_{oil_in_solvent} is defined as (kg oil in solvent)/(m³ overflow), and C_{oil_in_solid} is defined as (kg oil in solid/m³ underflow). The ratio of the volumetric flow rate of overflow/volumetric flow rate of underflow = 1.5. The entering overflow is pure solvent. Assume that the solids and fluid densities are constant. If there are four stages, what is the fractional recovery of the oil in the solvent?

E. More Complex Problems

E1. You are processing halibut livers that contain approximately 25.2 wt% fish oil and 74.8 wt% insoluble solids. The following data for leaching fish oil from halibut livers using diethyl ether solvent is given by Brown et al. (1950):

where y_oil is the mass fraction oil in the overflow, which was found to contain no insoluble solids, and Z is the pounds of solution per pound of oil-free solids in the underflow.

a. Convert the data to the mass fraction of oil and solvent in the overflow and to mass fraction of oil, solvent, and solids in the underflow. Plot the data as a saturated extract curve and a saturated raffinate curve with the appropriate tie lines on a graph with y_oil or x_oil as the ordinate (y axis) and y_solids or x_solids as the abscissa (x axis). Develop the conjugate line.

b. We have a total of 1000.0 lb of fish (oil + solids). We mix the fish with 500.0 lb of pure diethyl ether, let the mixture settle, and draw off the solvent layer. Calculate the weight fraction of oil in the extract and raffinate layers and the amounts of extract and raffinate.

c. The raffinate from step b is now mixed with 500.0 lb of pure diethyl ether. After settling, calculate the weight fraction of oil in the extract and raffinate layers and the amounts of extract and raffinate.

d. We have a total of 1000.0 lb of fish (oil + solids)/h. We continuously mix the fish with 500.0 lb of pure diethyl ether/h in a thickener and draw off the solvent layer. Calculate the weight fraction of oil in the extract and raffinate layers and the flow rates of extract and raffinate.

e. The raffinate from step d is now mixed with 500.0 lb/h of pure diethyl ether in a second thickener. Calculate the weight fraction of oil in the extract and raffinate layers and the flow rates of extract and raffinate.

f. 1000.0 lb of fish (oil + solids)/h are continuously processed in a countercurrent cascade of thickeners. Use 500.0 lb/h of pure diethyl ether. We want the outlet raffinate to contain 2.0 wt% fish oil. Find the number of stages required.

E2. Your company continues to have the stinky job of processing halibut livers detailed in Problem 14.E1. Since halibut arrive in batches on the fishing boats, you decide to evaluate a continuous solvent addition batch operation. The system is similar to the extraction system shown in Figure 13-11, except a filter is used instead of a settler. Derivation of Eqs. (13-28a) to (13-29a) is valid, but since equilibrium is not linear, Eq. (13-29b) is not valid. The feed is the same 25.2 wt% fish oil, 74.8 wt% insoluble solids feed as in Problem 14.E1. 1000.0 kg of this feed are to be processed using pure diethyl ether as solvent. Operation proceeds by first adding enough solvent so that the mixture is in a two-phase region. Then pure oil is skimmed off at the same rate as the solvent is added.

a. How much solvent must be added so that the mixture is in a two-phase region? The two-phase region starts when the mixing line crosses the saturated curve. Since there can be no removal of extract until this amount of solvent is added, Eq. (13-29a) is not valid until this amount of solvent is added.

b. Continuous solvent addition batch operation continues adding solvent until the weight fraction of oil in raffinate remaining in the tank is 0.02. Determine the total kg of solvent required, the weight of the raffinate remaining in tank, and the concentration of the raffinate. Also find the weight of the extract collected and the average weight fraction of the oil and the solvent in the extract. Note: Since Eq. (13-29a) assumes that R_t is constant, approximate constant R_t by using the average value of R_t in Eq. (13-29a) over the period when the extract is removed.

E3. Solve Problem 14.E2, part b, but do not assume R_t is constant. Derive an alternative for Eq. (13-29a) for nonconstant R_t but with a constant amount of insoluble solids.