Design of solids processes: production of potash - Industrial

Xiaoyan Ji, Xin Feng, Xiaohua Lu, Luzheng Zhang, Yanru Wang, and Jun Shi, Yunda Liu. Industrial & Engineering Chemistry Research 2002 41 (8), 2040-204...
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Ind. Eng. Chem. Res. 1988,27, 2071-2078

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Design of Solids Processes: Production of Potash Shankar Rajagopal, Ka M.

Ng,*and James M. Douglas*

Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003

Various issues encountered in the design of solids processes are examined through the analysis of a concrete example-the production of potash. Economic trade-offs from a systems point of view are identified, and the sensitivity of the total process cost to each key design variable is quantified. It is shown that the dissolvei. and crystallizer temperatures cannot be set at the optimum conditions unless pressurization and refrigeration are employed a t the respective units. The economic impact of the degree of saturation in the dissolver is found to be significant, because of its coupling t o the recycle flow. Furthermore, the sensitivity of the total process cost to the uncertainty in various model parameters is evaluated. In addition to the potential benefits to the design and operation of a potash plant, this study serves as the first step in the development of a comprehensive hierarchical procedure for the design of solids processes. During the development of a design for a new process, there are two main issues that need to be considered: 1. When we scale-up the process from the laboratory recipe of a chemist (or microbiologist) to a commercial production rate, how can we be certain that the process will actually operate at the design conditions? 2. How much profit will the commercial process generate? In current industrial practice both of these problems are considered simultaneously, and the greatest effort often is allocated to the operability problem, because it is the most difficult to predict. However, we propose that it is more efficient to examine the economics first. That is, if we use short-cut design procedures and process synthesis ideas to find the best three or four flow sheets and the range of the optimum design conditions for these alternatives based on our best data available, then it is a fairly simple task to estimate the sensitivity of the processing costs to uncertain data, to uncertainties in the models, and to many uncertain operability problems. With this approach, we avoid any consideration of operability problems unless the design under consideration is profitable, and we can use the results of the sensitivity studies to help plan additional laboratory and pilot-plant experiments that are needed to gather better data and to resolve operability questions. As discussed below, this approach is of particular importance in the design of solids processes. We use the term solids processes to refer to plants that involve extensive solids processing in the main process train, regardless of whether the solids are feeds, intermediates, or final products. This includes the production of chemicals from minerals, petrochemicals, inorganics, polymers, and biomaterials. Excluded from this definition are plants that involve primarily crushing or milling of solids and vapor-liquid processes in which the presence of solids, such as solid catalysts, is incidental. Despite the economic significance of solids processes, relatively little is available in the extensive literature concerning process synthesis and process retrofitting, on plants that contain solids processing steps. And systems problems in solids processes such as the identification of key design variables and evaluation of their impact on plant economics have not received sufficient attention. However, there has been a growing interest in the integration between crystallization and sedimentation/ filtration in the production of inorganic compounds (Sohneland Matejckova, 1981; Jones, 1985) and between feed preparation, reaction, and solid-liquid separation for coal li-

* To whom correspondence should be addressed.

Table I. Hierarchical Decision Procedure for Synthesis of Solids Processes level 0 input information level 1 batch versus continuous process level 2 input-output structure of the flow sheet level 3 recycle structure of the flow sheet level 4 separation system level 4a general structure level 4b vapor recovery system level 4c liquid separation system level 4d solids recovery system level 5 energy integration evaluation of process alternatives level 6

quefaction processes (Leu and Tiller, 1984). Recently, Rossiter and Douglas (1986a,b) proposed a hierarchical procedure for the synthesis of solids processes; an abbreviated and modified version is presented in Table I. Rossiter (1986) applied this decision procedure to an IC1 salt plant, and he found that the process was not operating at the optimum operating conditions. In addition to systems issues, it is generally agreed among process engineers that considerable operating problems are frequently experienced in solids processes. In a recent survey of the performance of 37 solids plants (Merrow, 1985)) it was found that on the average they operate at an abysmal 64% design capacity. The cause of difficulty ranges from abrasion problems and fouling within equipment to failures in transferring solids from one location to another. To provide a tool for the design and improved operation of solids processes, we propose to begin with the generalization and refinement of the procedure presented in Table I by adding the necessary details. Then, this will be followed by the identification and amelioration/elimination of potential operating problems. We do not include the feed preparation steps for solids such as crushing or any solids products finishing operations such as bagging because we expect that the costs for these steps are the same for most of the process alternatives. As the first step toward such a comprehensive design procedure, this study examines in detail the economic trade-offs encountered in a potash plant. The sensitivity of total process cost to uncertainty in model parameters is evaluated. Furthermore, it is demonstrated that efforts are needed to understand the physics of some solids processing units, to develop design heuristics, and to identify process alternatives in the design of solids processes.

Process Synthesis and Conceptual Design The synthesis/analysis to be presented below embodies a nascent, systematic procedure for the synthesis of solids

0888-5885/88/2621-2011~0~.50/00 1988 American Chemical Society

2072 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988

processes. To set the stage, a brief review of process synthesis and conceptual design is essential. Most of the research on process synthesis has been focused on heatexchanger networks and trains of distillation columns. This work is too voluminous to review here, but an excellent review has been published by Nishida et al. (1981). The number of studies on the synthesis of complete plants is much smaller, and these are discussed below. There were two early attempts to develop computeraided design programs that would synthesize process flow sheets, AIDES (Siirola et al., 1971; Powers, 1972; Siirola and Rudd, 1971) and BALTAZAR (Mahalec and Motard, 1977a,b). These codes consider both single-product and multiproduct plants, but neither calculates the equipment sizes or costs so that no estimate of the optimum design conditions can be obtained. AIDES invents a flow sheet by making a series of decisions; Le., (a) which raw materials should be used and what new byproducts will be produced when a desired product or set of products and a set of reactions has been specified?; (b) what separation tasks are required to isolate the products, byproducts, and recycle streams?; (c) what equipment is required to accomplish the separations?; and (d) how should the process be energy integrated? BALTAZAR develops a flow sheet one product at a time, starting with a product, then looking for reactions that produce this product, and next looking for raw materials that will supply the reaction requirements. As the program proceeds through this sequence of decisions, the program identifies the need for separations, the need for heating and cooling, etc. A look-back strategy is used to eliminate redundant components and parts of the flow sheet, and an evolutionary search is used at the end to improve the structure of the flow sheet. An alternate approach has been proposed by Papoulias and Grossmann (1983) and by Duran and Grossmann (1986). With this approach, a superstructure is created first that includes all of the process alternatives of interest. Then, a mixed-integer nonlinear programming code is used to find both the flow-sheet equipment and the values of the design variables that optimize the process. Hence, this structural optimization approach treats the synthesis problem as strictly an optimization problem. This approach seems to be ideally suited for situations where a relatively small number of alternatives have been identified. A systematic procedure for the conceptual design of a limited class of petrochemical processes has been proposed by Douglas (1985), the genesis of which is based on the following scenario. When new products are being developed, the primary consideration is to establish an early position in the market. However, experience indicates that less than 1% of the ideas for new designs at the research stage of a project development ever become commercialized. Hence, it is common practice to develop a new design through a series of levels where more detail and more accuracy are added to the design at each level in the hierarchy. The goal in the first of these levels, often called a conceptual design, is to determine if any flow sheet is profitable and, if so, to find the best three or four flow sheets. Hence, for a conceptual design we consider the following: (a) the selection of the process units; (b) the selection of the interconnection between these units (including the best heat-exchanger network); (c) the identification of the dominant design variables; (d) a procedure for obtaining a quick estimate of the optimum design conditions for the alternative under consideration; (e) the identification of

other process alternatives that should be considered; (f) a procedure for quickly screening the alternatives in order to determine the best three or four flow sheets for further consideration. Douglas’ final procedure is limited to a vapor-liquid system and is a subset of Table I, but has all the necessary details. A similar procedure, such as a detailed version of Table I, is particularly useful for solids processes, and we begin by focusing on items c, d, and e mentioned above in this study. It is well-known that the models for solids processing operations are less reliable than those for vapor-liquid processes. How detailed and accurate should a model be for solids processing operations? We suggest that we begin with short-cut models which are easy to use, but with all the pertinent physics. In conceptual design, a model is adequate if it can be used for identifying the dominant design variables and it provides reasonably accurate estimates of the capital and operating costs. The results of a conceptual design study based on these models can then be used to identify the sensitivity of the total process cost to the solids processing units, and then more rigorous models can be developed for the sensitive units. We emphasize that a unit with significant influence on total cost is not necessarily expensive itself. In this way, a hierarchy of models can be developed, but the modeling effort is carried out in a context where it can be economically justified. As discussed below, a simple dissolver model is used in the analysis of a potash plant, which illustrates some of these issues in a more concrete manner.

Process Description and Analysis Potash, a basic fertilizer, has a yearly consumption rate of about 6 X lo6 tons in the US alone at a price of $70 per ton (Brown, 1987). We consider here a plant with a production rate of 1 X lo6 tonslyear and a process in which potash (KC1) is produced by mining sylvinite (a KC1-NaC1 mixture), followed by a solution recovery step. We assume that the ore has a composition of 40% KCl and 60% NaCl by weight, although the composition varies from mine to mine and a small amount of water and impurities is generally present in the feed. Separation of the two solids relies on the fact that the joint solubility of KC1, i.e., the solubility in an aqueous solution saturated with both KC1 and NaC1, increases with temperature while the joint solubility of NaCl decreases with temperature. The dependence of the joint solubilities on temperature can be represented by the following linear relationships (Krull, 1956): S , ( T ) = C1 + C2T S , ( T ) = C3

+ C4T

The values of C,-C,, as well as some other input parameters for this study, are summarized in Table 11. The flow-sheet structure is essentially a dissolver-filter-crystallizer-filter-dryer train (Figure 1). Obviously, there are different types of filters, crystallizers, and dryers, but the results to be presented below are based on that of rotary vacuum filter, vacuum crystallizer, and rotary drum dryer. Other equipment types can be easily included in our analysis, but they are not expected to change our conclusions qualitatively. Water leaves the system from the crystallizer at a rate of W , through vaporization and as residual liquid within the filter cakes of the two rotary vacuum filters. Makeup water is added to the dissolution tank at a rate of W,. The vacuum in the crystallizer is maintained with a condenser, and a steam ejector allows

Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2073 Table 11. Values of Some Input Parameters for the Potash Plant 1 x lo6 tons/year production rate joint solubilities C1 = 0.085, Cz = 0.0027 OC-' KCl C3 = 0.299, C4 = -0.0005 OC-' NaCl densities KC1 2000 kg/m3 NaCl 2160 kg/m3 heat capacities KC1 1318 J/(kg "C) air 1046 J/(kg "C) water 4180 J/(kg "C) latent heat of vaporization for 2.36 X lo6 J/kg water particle diameter in the feed (KC1 1600 pm and NaC1) temp/price of steam for dissolver 148 "C/(0.62 cents/kg) preheater 186 "C/(0.75 centsjkg) energy input for mixing in the 1.4 X J/(kg s) dissolver per unit mass of liquid (see Happel and Jordan (1975)) dissolver mass-transfer coeff, k, 2.01 X m/s rotation speed of filters 10 rpm submergence of filters 145O 35000 N/m2 vacuum level in filters porosity of filter cakes 0.4 residual liquid saturation after 0.155 dewatering in the KC1 filter cake (see Wakeman (1979)) i = 4.99, j = 0.14 crystallization kinetics for KCl (see Garside and Shah (1980)) k, = 3 x 1042


dissolver) in the second filter, dewatered, and then dried in the rotary drum dryer. The condition that none of the NaCl in the feed dissolves in the dissolver and all the KC1 is dissolved is achieved by adjusting the recycle, W,, and makeup water flow rates, i.e., (wr

+ w , ) s p ( T d ) = w r s p ( T c ) + Mp


+ wm)ss(Td)





Equation 3 expresses the demand that the capacity for KC1 in the dissolver must be equal to the amount in the recycle and that of the feed. Equation 4 shows that the capacity for NaCl in the dissolver just equals the amount in the recycle, so none of the feed NaCl is dissolved. Solution of eq 3 and 4 gives (5)

The subscript min indicates that these are the minimum flow rates for the process. As discussed in the following section, we have to use more water to ensure process operability. Equation 5 shows that because of the linearity of the solubility curves the makeup water flow rate is independent of the choice of dissolver and crystallizer temperatures.

Dissolver, Makeup, and Recycle Flow Rates Design equations for rotary vacuum filter, MSMPR crystallizer, rotary drum dryer, preheater, steam ejector, etc., are available in many standard texts (e.g., Peters and Timmerhaus (1980) and Treybal(l980)) and the Chemical Engineers' Handbook (Perry and Chilton, 1973),and they are not repeated in this paper. The only exception is the dissolver, for which a model does not seem to exist. A design equation for the dissolver volume is derived in the Appendix as



Pp'p0W2 h&2[wsp(Td) - w,sp(Tc)- Mpl



w = w,+ w, Figure 1. Scheme of potash plant.

the removal of noncondensible gases. The temperature of the dissolver, Td, is higher than that of the crystallizer, Tc. Cooling of the mother liquor in the crystallizer is achieved primarily through the latent heat of vaporization of w,. The mixture of KC1 and NaCl particles enters the plant at the dissolution tank at mass flow rates of Mp and M,, respectively. The plant is operated in such a way that the solution within the dissolver is saturated with NaC1. Consequently, the NaCl particles in the feed remain undissolved and leave the process at the first filter. On the other hand, the KC1 particles in the feed are completely dissolved. They remain dissolved until reaching the crystallizer where crystallization of KC1 takes place because of the lowering in temperature. Note that NaCl does not crystallize because its joint solubility increases with decreasing temperature. The KC1 crystals are subsequently separated from the mother liquor (which is recycled to the


Note that the terms in the square brackets correspond to eq 3. It equals zero if Wm,min and Wr,minare used and V tends to infinity. In physical terms, the amount of water is just enough to dissolve all the KC1 in the feed under this condition. The concentration difference between the surface of a dissolving KC1 particle and the bulk solution within the dissolver (see eq A3), i.e., the driving force for dissolution, becomes zero and an infinitely large dissolver is needed. We cannot increase W, and/or W, arbitrarily to avoid this problem, however. Dividing eq 3 and 4 throughout by W,, we get, respectively, (1 + R)Sp(Td) =


+ Mp/w,


and (1 + R)Ss(Td) =




2074 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 800 1 1


7 Dissolver


L --1





R,+ Figure 2. Dependence of the dissolver volume on dilution ratio.

The appropriate strategy is to increase both W , and W , while keeping R constant. Obviously, eq 10 is identical with eq 4 and none of the NaCl in the feed is dissolved. For W, > Wr,mh,the left-hand side of eq 9 becomes larger than the right-hand side, i.e. the capacity for KC1 in the dissolver is greater than the amounts in the recycle and the feed combined. Thus, the KCl concentration within the dissolver is below its saturation value, and there is a finite driving force for dissolution. As a result, the dissolver volume decreases as W is increased above W,, (Figure 2). Beyond a dilution ratio (R, W / Wmh) value of about 2, however, the dissolver volume begins to increase. At that point, the KC1 concentration within the dissolver is essentially zero. The time required for the dissolution of a KCl particle reaches its asymptotic minimum value (see eq A4 by setting Coutequal to zero). Increasing W necessitates an increase in dissolver volume to provide the necessary residence time for particle dissolution. Before proceeding further, it is important to examine the effect of dissolver and crystallizer temperatures on the recycle flow rate. It can be seen in eq 6 that Wr,,in decreases as Td increases, while keeping T, constant. The numerator of the quotient within the square brackets is the joint solubility of NaCl (see eq 2), which decreases with increasing temperature. Note that C4 is negative and Td is larger than T,; the denominator clearly increases with increasing Td. Alternatively, this observation can be easily understood on physical grounds in terms of the KC1 solubility (eq 1). As Td increases, the differences between the KC1 solubility in the dissolver, Sp(Td),and that in the crystallizer, Sp(T,),increases. Thus, less recycle water is needed to dissolve the given amount of KC1 in the feed, Mp. Similar arguments can be used to conclude that WrF increases with increasing crystallizer temperature, while keeping Td fixed. Economic Trade-offs The influence of various key design variables on the economics of the potash plant is examined below. For the purpose of preliminary design, it is sufficient to include only the following capital and operating costs: (1)dissolver, (2) steam for dissolver, (3) heat exchanger for dissolver, (4)rotary vacuum filter no. 1, (5) vacuum crystallizer, (6) rotary vacuum filter no. 2, (7) rotary drum dryer including motor and blower, (8) steam for the dryer, and (9) preheater. Thus, the total annualized cost is given by TAC = fl + fi + ... + fs (12) A capital charge factor of year is used to annualize capital costs. The steam ejector, condenser, and dewatering operation of the KC1 filter cake in the second filter are not included in the total cost since our preliminary estimates indicate that these costs are relatively small. The various cost models can be found in Peters and Timmerhaus (1980) and Happel and Jordan (1975), and current

120 Td



9 OC

Figure 3. Dependence of the total annualized cost, dissolver cost, and steam cost for the dissolver on dissolver temperature.



0-01 -20




T, 9OC Figure 4. Dependence of the total annualized cost, steam cost for the preheater, and dissolver cost on crystallizer temperature.

costs are estimated with the Marshall and Swift index. The important design variables identified during the flow-sheet development stage are dissolver temperature, crystallizer temperature, dilution ratio at the dissolver (R,), dominant particle size of potash crystals (LD),air temperature at the dryer inlet (Ta,h),and the approach tem- Tp,,J. To demonstrate the impact perature (Tapp= T%out of a single design variable on the total annualized cost of the plant, each variable is changed one at a time from the following values: Td = 100 "C, T, = 30 "C, R, = 1.15, LD = 600 pm, Ta,in= 176 "C, and Tapp= 20 "C, which correspond to that of a local optimum in TAC. Figure 3 shows the dependence of the TAC on dissolver temperature. As mentioned, the recycle water flow rate decreases with increasing dissolver temperature. This results in lower costs for the dissolver, both filters and the crystallizer, with the dissolver cost dropping the most, and thus the TAC also decreases. Note that in this figure as well as other figures below only the items with the largest increasing or decreasing cost are shown. As the dissolver temperature is further increased, the increasing cost of the steam required for heating the recycle mother liquor from T, = 30 "C to Td begins to overshadow the savings from other equipment, causing an increase in the TAC. The minimum TAC at about 140 "C is probably not physically realizable because boiling might have occurred. However, we can easily determine the economic incentive for removing this constraint by considering a process alternative where we pressurize the dissolver. In Figure 4 the effect of crystallizer temperature on the TAC is illustrated. Similar to the previous case, decreasing crystallizer temperature leads to a smaller recycle water flow rate and lower equipment costs. Below a temperature of about 0 "C, the increasing cost of the steam required for heating the KCl crystals from T , to Tp,out, which we assume to be fixed at 60 "C, begins to dominate the TAC. The degree of cooling necessary to reach the minimum TAC is probably not physically realizable with the vacuum Crystallizer at hand, and refrigeration should be considered as a process alternative. Of course, the freezing point of

Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2075 L

2o 1

---h w-3


E. m








'1.0 0



. 10.0






15 1 141


l6 5





* E

/ '





E E 1







8 3






u 9C


1 Dissolver 2 Steaml 3 Heal Exchan. 4 Filter1 5 Crystalliser

6 Filter2

7 Dryer 8 steam2

9 Preheat.




1 2 3 4 5 6 7 8 9

Figure 6. Dependence of the total annualized cost, crystallizer cost, and its downstream filter cost on dominant crystal size.


. * 8

6 11-

104 0


Figure 8. Dependence of the total annualized cost, cost of steam required for the preheater, and dryer cost on approach temperature.





Figure 5. Dependence of the total annualized cost, steam cost for the dissolver, and dissolver cost on dilution ratio.

. *




Figure 9. Major annual capital and operating costs for the case corresponding to a local minimum in TAC.

the steam cost for the air preheater and the dryer cost, and vice versa. Thus, there is an optimal air inlet temperature. Note that drying operating conditions do not affect the rest of the process. Similarly, if we increase the approach temperature (T,out - TP& at the cocurrent flow dryer (again, Tp,out, is fixed at 60 "C), the dryer cost decreases but more air must be supplied and the steam cost for the air preheater increases (Figure 8). Thus, there is an optimal approach temperature. Finally, the relative costs of fi to fs for the base case are presented in Figure 9. The dominant operating cost is clearly the steam required for the dissolver. Although it is beyond the scope of this paper to carry out a complete heat integration analysis, the economic incentive is obvious. For instance, we can use a train of vacuum crystallizers with decreasing temperatures so that the vapor from the upstream crystallizers can be used to heat up the recycle mother liquor.

i 3 L





12-. 11100 130


E 5

Steam2 -


L I8 Preheater 150 Ta,in,




Figure 7. Dependence of the total annualized cost, preheater cost, and the cost of steam required for the preheater on the air temperature a t the dryer inlet.

water adjusted by the freezing point depression is the lowest temperature possible. Figure 5 demonstrates the importance of the water flow rate to the dissolver, and thus the dilution ratio, on the total annualized cost. The cost of the steam required to heat up the recycle stream increases with water flow rate, while the cost of the dissolver increases rapidly when the water flow rate is close to the saturation requirement, i.e., at R, = 1. The minimum TAC occurs at an R, value of around 1.15. It is expected that more case studies would allow the development of a design heuristic for choosing the dilution ratio for dissolvers. The influence of the dominant crystal size on the TAC of the potash plant is shown in Figure 6. Larger crystals are easier to filter as the filter cake resistance becomes smaller but require a larger crystallizer to make. Thus, with increasing crystal size, the filter cost decreases while the crystallizer cost increases, and vice versa. In practice, solids products have to be within a certain particle size range. However, we want to estimate the optimum size so as to minimize the cost while matching the customer requirement in particle size. Figure 7 shows the dependence of the TAC on the air temperature at the dryer inlet. An increase in the temperature increases the air preheater cost while decreasing

Proximity and Rank-Order Parameters As mentioned, the chosen values for R,, LD,Ta,in,and Tappare relatively close to their local minimum in TAC, while Tdand T, are set close to the constraints mentioned. To quantify the closeness of a design variable, x j , to its optimal value, the proximity parameter

can be useful. It approaches zero at xj,opt.andapproaches unity as x j moves far away from the optimum (Fisher et al., 1985). As can be seen in Table 111, the proximity parameters for R,, LD, Tbh, and Tappare very close to zero for our chosen values. We can also determine the relative importance of the design variables on process economics with the rank-order parameter

2076 Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 Table 111. Proximity and Rank-Order Parameters for the Potash Plant with the Design Variable Values CorresDondina to a Local Minimum in TAC ($1000) afi/aRw (1) dissolver -6819 (2) steam 1 4823 (3) heat exchanger 35 (4) filter 1 175 (5) crystallizer 481 414 (6) filter 2 -0.1 (7) dryer -0.2 (8) steam 2 -0.04 (9) preheater 0.07 PI 100 rj fi



0 0 0 0 1.6 -1.6 -0.001 -0.001 -0.0002 0.01 12.4

0 0 0 0 0 0 -2.1 -3.7 5.4 0.04 7.3


($1000) afi/aRw (1) dissolver -2565 4823 (2) steam 1 34.2 (3) heat exchanger 167 (4) filter 1 828 (5) crystallizer 283 (6) filter 2 -0.1 (7) dryer -0.2 (8) steam 2 (9) preheater -0.03 0.41 Pj 100


0 0 0 0

1.7 -0.7 -0.0002 -0.0004 -0.0001 0.42 22.2



0 0 0 0 0 -11.9 9.9







0.002 3.1



0 0 0 0 0 0 -1.7 -5.0 10.8

0 0 0 0 0 0 -7.7 12.2 2.3 0.31

0.24 16.1



Table IV. Proximity and Rank-Order Parameters for the Potash Plant for a Case Away from the Local Optimal: R, = 1.25, Ln = 1000 gm, T..,. = 180 "C, and T ,. = 30 OC fi



Here, Axj," is the maximum expected range of x i . The larger the rank-order parameter, the more impact the corresponding design variable has on the TAC (Fisher et al., 1985). Clearly, the dilution ratio, R,, is the most important optimization variable (Table 111). The partial derivatives in each of the four columns in Table I11 can be used to identify the dominant increasing or decreasing cost in varying a design variable. Thus, the dissolver and steam 1 costs are the most significant in varying R,, in agreement with Figure 5. The crystallizer and filter 2 costs are the most significant in varying L D , in agreement with Figure 6, and so on. Table IV illustrates the change in proximity and rank-order parameters for a case away from the chosen optimum values. The proximity parameters are larger than that in Table 111, but the order of the rank-order parameters remains the same.

Process Alternatives We pointed out earlier, with reference to the dissolver and crystallizer temperatures (Figures 3 and 4), that pressurization and refrigeration, respectively, should be considered. Of course, consideration of equipment types other than rotary vacuum filter, vacuum crystallizer, and rotary drum dryers also generates process alternatives. Here, we want to point out another process alternative. Instead of removing NaCl in the first filter and KC1 after the crystallizer as in the original scheme (Figure l),we can remove KC1 in the first filter and NaCl after the crystallizer. This is achieved by reversing the temperatures of the dissolver and crystallizer (Figure 10). For the alternate scheme, we replace eq 3 and 4 with

and Td is less than T,. For Td = 30 "C and T,= 100 "C, both the makeup and recycle flow rates in the alternate process are larger than that in the original scheme (Figure





Figure 10. Comparison of the makeup and recycle water flow rates in kilograms/hour in the original and the alternate schemes. Table V. Sensitivity of TAC to Uncertainty in Model Parameters percentage TAC, $ X lo6 change, % model parameter +20% -20% +20% -20% porosity 13.55 13.66 0.76 1.58 13.47 13.42 0.16 -0.19 Kr i 12.75 32.58 -5.18 142.31 j 13.46 13.43 0.13 -0.13 k, 13.19 13.81 -1.87 2.70 2.11 -2.14 residual filter cake saturation 13.73 13.61

lo), hence higher equipment costs. This observation concurs with a common design heuristic "separate the most plentiful (NaC1 in this case) first". Sensitivity of TAC to Uncertainty in Model Parameters Model parameters, particularly those for solids processing units, generally are not exact. We propose that we can determine the acceptable degree of uncertainty in a model parameter in conceptual design by evaluating the impact on TAC. To illustrate this approach, we change the filter cake porosity, crystallization kinetics parameters (Kr,i, j ) , dissolver mass-transfer coefficient, and residual liquid saturation in the KC1 filter cake by f20% of the values reported in Table 11, one parameter at a time. The impact is rather small for most of the model parameters considered and depends on whether the parameter value is increased or decreased (Table V). The only exception is the crystallization parameter i; an increase in its value has a significant impact on TAC. Clearly, this analysis helps us to identify which model parameter demands a more accurate value. Concluding Remarks We have carried out a systematic study to identify the economic trade-offs in a potash plant. This should not be viewed as an isolated case study. Rather, it serves as a concrete example in our search of the generic issues encountered in the design of solids processes. Following are some of the salient points in this study not previously emphasized in the process synthesis literature. (1) In a dissolver-filter-crystallizer-filter train, the dissolver and crystallizer temperatures can have a profound impact on process economics, as well as the operation of

Ind. Eng. Chem. Res., Vol. 27, No. 11, 1988 2077 the plant. They determine which solid comes out at the first filter. Pressurization of the dissolver and refrigeration for the crystallizer are identified as potential candidates in retrofitting to reduce the cost of the plant. (2) The design variables-dissolver and crystallizer temperatures, crystal particle size, dilution ratio in the dissolver, air temperature at the dryer inlet, approach temperature-are unique to solids processes. (3) A model which captures the essential physics of a dissolution tank is derived. It is based on the assumption that the solid particles pass through the system in a plug-flow manner while the liquid is perfectly mixed. Although this dissolver model is rather simple, it helps to show that the dissolver has significant influence on the total cost as it is coupled to the recycle and makeup water flows through the dilution ratio in the tank. This underscores the importance of the hierarchy of models concept alluded to earlier. The focus on economics helps to point out the incentive of a more detailed model, preferably with experimental verification, than the one derived in the Appendix. This study is limited to specific equipment types. Also, there is a model mismatch between the crystallizer and the second filter, in that the crystals have an exponential particle size distribution, while the equation for flow through a filter cake in the literature is limited to a single particle size. This is why only the dominant crystal size is considered in this analysis. Efforts to relax these limitations are under way. Nomenclature C = concentration, kg/kg of water C1, C3 = constants in eq 1 and 2, dimensionless C2, C4 = constants in eq 1 and 2, OC-’ f l ,...,f9 = cost functions, $ i = crystallization kinetics parameter, dimensionless j = crystallization kinetics parameter, dimensionless k, = mass-transfer coefficient, m/s k, = rate constant for crystallization, no./(m3s (kg/m3r’(mls)’) LD = dominant crystal size, pm M = mass flow rate of solids, kg/s p = proximity parameter, eq 13, dimensionless R = ratio of W, to W,, eq 11 Rw = ratio of W to Wmin r = rank-order parameter, $, eq 14; particle radius, m, eq A3 S = solubility functions, kg/kg of water, eq 1 and 2 T = temperature, “C Tapp= approach temperature (=Ta,out- Tp,out),“C T,, T d = crystallizer and dissolver temperature, respectively, OC V = volume of dissolution tank, m3 W = W, 4- W,, kg of water/s, eq 8 W,, W,, W, = water flow rate out of the crystallizer, in the makeup water stream and recycle, respectively, kg of water/s x = design variable

Greek Symbols density, kg/m3 T = time at which a KC1 particle dissolves completely, s

p =

Subscripts a = air i = dummy index for a cost function in = inlet j = dummy index for a design variable o = original opt = optimum out = outlet p = potassium chloride s = sodium chloride w = water

Appendix. Derivation of the Dissolver Model, Equation 7 Consider a stirred tank of volume V. Water enters the tank at a rate of W (= W , + W,) and KCl at Mp. Since the amount of KC1 in the recycle is W,Sp(T,),the inlet concentration is given by W,Sp(T,) (AI) W A mass balance of the solute yields the outlet concentration: Cin =

Cout = Cin + Mp/ W


Now consider the dissolution of a single KC1 particle of radius rp inside the tank. The rate of dissolution is given by

where k,, the mass-transfer coefficient, can be evaluated with a graphical correlation in Sherwood et al. (1975). With an initial size of rpo,the time at which the KC1 dissolves completely can be obtained by integrating eq A3: r =

PPrPo kcPw(sp(Td)

- Cout)


Equating r to the mean residence time of the liquid (=V/(W/p,)),we get for the volume of the stirred tank,

Substitution of eq A1 and A2 into eq A5 gives eq 7. Registry No. KC1, 7447-40-7.

Literature Cited Brown, J. M. “Price Ruling on Potash from Canada Divides the Fertilizer Industry in US.”.Wall St. J. 1987,Aug 25, 4. Douglas, J. M. “A Hierarchical Decision Procedure for Process Synthesis”. AZChE J. 1985,31, 353. Duran, M. A.; Grossmann, I. E. “A Mixed-Integer Nonlinear Programming Algorithm for Process Systems Synthesis”. AZChE J. 1986,32, 592. Fisher, W. R.; Doherty, M. F.; Douglas, J. M. “Evaluating Significant Economic Trade-offs for Process Design and Steady State Control Optimization Problems”. AZChE J. 1985,31, 1538. Garside, J.; Shah, M. B. “Crystallization Kinetics from MSMPR Crystallizers”. Znd. Eng. Chem. Process Des. Deu. 1980,19,509. Happel, J.; Jordan, D. G. Chemical Process Economics; MarcelDekker: New York, 1975. Jones, A. G. “Crystallization and Downstream Processing Interactions”. Znst. Chem. Eng. Symp. Ser. 1985,91. Krull, 0. “Influence of the Magnesium Chloride Content of Dissolving Liquids on the Potassium Chloride Content of Condenser Salts in the Manufacture of Sylvite”. Chem. Tech. (Leipzig) 1956, 8, 212. Leu, W.; Tiller, F. M. “An Overview of Solid-Liquid Separation in Coal Liquefaction Processes”. Powder Technol. 1984,40, 65. Mahalec, V.; Motard, R. L. “Procedures for the Initial Design of Chemical Processing Systems”. Comput. Chem. Erg. 1977a,1,57. Mahalec, V.;Motard, R. L. “Evolutionary Search for an Optimal Limiting Process Flowsheet”. Comput. Chem. Eng. 1977b,1, 149. Merrow, E. W. “Linking R&D to Problems Experienced in Solids Processing”. Chem. Eng. Prog. 1985,May, 14. Nishida, N.; Stephanopoulos, G.; Westerberg, A. W. “A Review of Process Synthesis”. AIChE J. 1981,27,321. Papoulias, S. A.; Grossman, I. E. “A Structural Optimization Approach in Process Synthesis. Part 111: Total Processing Systems”. Comput. Chem. Eng. 1983,7,723.

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Perry, R. H., Chilton, C. H., Eds. Chemical Engineers’ Handbook, 5th ed.; McGraw-Hill: New York, 1973. Peters, M. S.; Timmerhaus, K. D. Plant Design and Economics for Chemical Engineers, 3rd ed.; McGraw-Hill, New York, 1980. Powers, G. J. “Heuristic Synthesis in Process Development”. Chem. Eng. Prog. 1972, 68, 88. Rossiter, A. P. “Design and Optimization of Solids Processes Part 3-Optimization of a Crystalline Salt Plant Using a Novel Procedure”. Chem. Eng. Res. Des. 1986, 64, 191. Rossiter, A. P.; Douglas, J. M. “Design and Optimization of Solids Processes Part 1-A Hierarchical Decision Procedure for Process Synthesis of Solids Systems”. Chem. Eng. Res. Des. 1986a, 64, 175. Rossiter, A. P.; Douglas, J. M. “Design and Optimization of Solids Processes Part 2-Optimization of Crystallizer, Centrifuge and Dryer Systems”. Chem. Eng. Res. Des. 1986b, 64, 184. Sherwood, T. K.; Pigford, R. L.; Wilke, C. R. Mass Transfer; McGraw-Hill: New York, 1975; p 223.

Siirola, J. J.; Rudd, D. F. “Computer-Aided Synthesis of Chemical Process Designs”. Ind. Eng. Chem. Fundam. 1971,10,353. Siirola, J. J.; Powers, G. J.; Rudd, D. F. “Synthesis of System Designs 111: Toward a Process Concept Generator”. AIChE J . 1971, 17, 677. Sohnel, 0.;Matejckova, E. “Batch Precipitation of Alkaline Earth Carbonates, Effect of Reaction Conditions on the Filterability of Resulting Suspensions”. Ind. Eng. Chem. Process Des. Dev. 1981, 20, 525. Treybal, R. E. Mass Transfer Operations, 3rd ed.; McGraw-Hill, New York, 1980. Wakeman, R. J. “The Performance of Filtration PostTreatment Process 1. The Prediction and Calculation of Cake Dewatering Characteristics”. Filtr. Sep. 1979, NovlDec, 655.

Received f o r review February 10, 1988 Revised manuscript received June 30, 1988 Accepted July 31, 1988

SEPARATIONS Adsorptive Drying of Toluene Sudhir Joshi and James R. Fair* Department of Chemical Engineering, T h e University of Texas at Austin, Austin, T e x a s 78712

The adsorption of water from toluene was studied using the following desiccants: activated alumina, 3A molecular sieves, and 4A molecular sieves. Water isotherms were measured, mass-transfer rates were studied by means of batch-type kinetics experiments, and breakthrough data were determined by means of flow experiments using fixed beds of the desiccants. A linear driving force mass-transfer model was found to fit the breakthrough results, with equilibrium and mass-transfer parameters determined from the isotherm and kinetics experiments. The results showed that intraparticle diffusion offers the controlling resistance to the transport of water. In the process industries, it is often necessary to dry fluids before they can be processed further. Such concerns as hydrate formation, catalyst poisoning, and corrosion require that the fluid streams contain exceedingly small amounts of water. Adsorptive drying is a logical method, though not the only method of water removal. Liquid drying by adsorption is an important unit operation and despite its scant coverage in the literature is practiced extensively in industry, often empirically. The objective of this work was to study the adsorptive removal of water from hydrocarbons with limited water solubility. This study was to be experimental as well as mechanistic. A mathematical model utilizing the mechanistic characteristics of the process was needed which could be useful in designing commercial drying systems. Toluene was chosen as a model compound, representative of hydrocarbon liquids requiring water removal in processing operations.

Previous Work Burfield and co-workers (Burfield et al., 1977,1978,1981, 1984; Burfield and Smithers, 1978, 1980, 1982, 1983) studied static and dynamic drying of laboratory solvents and reagents using several adsorbents activated aluminas, silica gels, molecular sieves, and ion-exchange resins, together with several chemical agents. Temperature of activation was shown t~ have a strong impact on the moisture

holding capacity of molecular sieves and ion-exchange resins. The 4A molecular sieves, when regenerated at 130 “C, had 7.3% w/w water holding capacity from p-dioxane. This capacity increased to 18.5% w / w when regenerated at 250 “C (Burfield et al., 1978). Most organic molecules have effective diameters in excess of 4 A. For toluene, the effective diameter is about 12 A based on a Lennard-Jones potential constant. This effectively excludes most of the hydrocarbons from the main adsorption space in the micropores of 3A and 4A molecular sieves. In such cases, if water is the only adsorbate, Basmadjian (1984) concluded that moisture uptake isotherms theoretically might be independent of the nature of the solvent and thus yield one common isotherm from all nonadsorbable solvents on a particular molecular sieve. This isotherm might then be related thermodynamically to the vapor adsorption isotherm by equating the fugacities of all three phases. Thus, the calculated vapor isotherm should represent the generalized moisture uptake isotherm on a particular molecular sieve from all nonadsorbable solvents. Actually, many of the reported moisture uptake isotherms from liquid solvents differ substantially from the calculated vapor isotherm (Goto et al., 1972; Selin et al., 1964; Stuchkov, 1975; Varga and Beyer, 1967). The data reported by these authors on 4A molecular sieves were compiled and analyzed by Basmadjian (1984). A diversity

0888-588518812627-2078$01.50/0 0 1988 American Chemical Society