Analysis, Simulation, and Optimization of the Hemihydrate Process for

Analysis, Simulation, and Optimization of the Hemihydrate Process for the Production of Phosphoric Acid from CaIcareous Phosphorites Francesco Gioia,’ Giampaolo Mura, and Antonio Viola lstituto di Chimica Applicata e Metallurgia, Facolta di lngegneria, The University of Cagliari, Cagliari, ltaly

In this paper it is shown how the methods used in chemical engineering for the analysis and simulation of processes may be applied to explore the possibilities of an economical utilization of low grade calcareous phosphate rocks for the production of phosphoric acid. Attention has been focused on the hemihydrate process for which the necessary fundamental experimental data, although fragmentary, are available in literature. The results of the analysis have shown that a reasonable description of the process at hand is possible by the proposed method. However, the basic phenomena which take place must be understood and a knowledge of a few specific parameters is essential. The mathematical model resulting from the analysis has enabled fixed and operating costs of the process and their dependence on the variables of interest to be established. The characterization of the “optimal configuration” of the plant is then dealt with. The results of the optimization may be used, among other things, for pricing low grade calcareous phosphorites.

Introduction In the chemical industry one often comes across different processes having the same final product. In choosing between the alternatives, economics play the leading role. If a wise choice is to be made, the capital and operating costs of the plant must be known for each process and then compared with each other. Furthermore, even for a single scheme of the overall process, the costs will change according to the values of some input and internal variables and a set of these values may exist which characterizes an optimum configuration for the scheme. In order to single out this set of values, the influence of the variables on the performance of the process must be well known. One way of finding this out is to build a pilot plant and effect changes in the variables while observing how the process functions. Of course if the understanding of the basic chemical and physical phenomena that make up the process is poor, the size of the pilot plant must be close to that of the actual process. In this last case, such a technique is not only time consuming and expensive, but actually may be impossible to carry out. Furthermore, such a replica of the actual process could never be so versatile as to permit the investigation of the many possible schemes of the process. A much more convenient and economical method of analysis which is largely used in chemical engineering can be employed. This consists of making a mathematical modelof the process in order to simulate the influence of the variables of interest on its performance and cost. Of course, contrary to the pilot plant method, in this case a good knowledge of the basic chemical and physical phenomena that take place is necessary. The better this knowledge, the more reliably the mathematical model simulates the process. Moreover, since the model resorts to first principles, the parameters it includes will be “fundamental” in the sense that they are related to these principles. Therefore, another feature of the method is that the experimental data necessary to evaluate those of these parameters which are specific for the particular process under analysis, may be obtained cheaply (as compared to a pilot plant construction) in the laboratory. The purpose of this paper is to apply, as an illustration, this approach to the “hemi-hydrate” process for the production of phosphoric acid from phosphorites. A literature survey has shown that this has not been done for this particular process nor for the other several kinds of 390

Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977

the processes (Slack, 1968; Scott et al., 1974) producing the same product. Only a few authors-among them Hatfield e t al. (1959), Davenport et al. (1965), Patterson et al. (1967),and Libhaber et al. (1974)-have tackled the problem of studying the influence of some variables on the performance of both conventional and unconventional wet processes, but they have generally utilized experimental data from pilot plants. Therefore, their results are difficult to extrapolate to different situations. The analysis presented in this paper will be based, as much as possible, on the general results and methods of transport phenomena, chemical kinetics, and population balances. Because of the complexity of the process, of the limited basic experimental data reported in literature, and the limitations of mathematics, the model will be somewhat idealized and will give a reliable representation of the influence of only a few of the variables that affect the performance of the process. However, it will be demonstrated that even with the above limitations, a reasonable description of its performance is possible. The model will then be used to find the optimal design and operating conditions as the most significant variables are varied. Particular attention will be devoted to the existing possibilities of utilizing low grade calcareous phosphate rocks. In fact, with increasing use of complex and single phosphatic fertilizers, the requirements of phosphoric acid are rising rapidly and the possibility of utilizing such low grade rocks begins to be taken into consideration. Analysis of the Process The process consists of an attack on phosphorites by sulfuric acid producing a solution mainly composed of calcium sulfate and phosphoric acid. The stoichiometry of the overall dissolution reaction depends on the chemical composition of the phosphate rock. The calcium sulfate separates by crystallizing as CaSO&H20 if the values of the temperature and composition (H3P04and HZS04) existing in the solution are in a well defined range (Slack, 1968). Filtration and a distillation to concentrate H3P04 follow. For further details see Slack (1968). According to the general strategy of analysis of complex systems, the process will be divided into subsystems. From Figure 1the following subsystems can be determined: (i) re-

i

N

Q

Q Calcium

Sulfate disposal

I Rock

I

dissolut cristalliz

preliminary treat ments

f iltral

phosphoric acid

ion

concenfr

1

i

Figure 2. Flow sheet of the overall process.

Figure 1. Flow sheet of the reactor-filtration subsystem.

actors (for dissolution and crystallization); (ii) filtration; (iii) distillation; (iv) sulfuric acid production; and (v) calcium sulfate disposal. The last two points will not be dealt with in this work. Reactors-Crystallizers. They represent the core of the plant where many phenomena (chemical reactions, crystallization etc.) take place. Unfortunately, for the process at hand, these phenomena have been little studied or, at least, few results have been reported in literature. However, the characterization of the basic chemical and physical principles that lie behind them, together with few results available in literature and with data strictly valid for other similar chemical systems, will furnish a reasonable description of this part of the plant. The processes taking place in the j t h reactor are dissolution of phosphate rock and CaSO&H20 crystallization. The reactant fluid is thus made up of a suspension of solid particles in a liquid (slurry). Therefore the use of continuous stirred tank reactors (CSTR) is most appropriate. In the following analysis we will postulate a flow pattern characterized by perfect macromixing and segregation for solid particles. Perfect macromixing implies an exit age distribution function given by

E ( t ) = 1 e-t/T t where t is the mean residence time defined as

t = m/W,

(1)

The use of the above ideal flow pattern is in general a good approximation of real mixed reactors. However, for a different flow pattern, the following analysis may be easily modified by using a more appropriate E ( t )function. Let us assume that the plant is composed of N CSTRs in series with recycle. By referring to Figure 2 we will start writing the mass balance equations. Total mass rate:

WMN

(3)

water:

The sum of eq 8,9,10,11, and 12 gives the slurry overall mass balance: = W,j

+ QM,;-~ % (6 - 6 * ) M h F ; M g

(13)

The liquid solution "state equation" which relates liquid density to composition and temperature can be found in Slack (1968) and is reported further on, a is:

Input quantities to reactor 1 are:

Ws,o = WMF + Q c ~ ( p + c ~P N ~ ) where: ( = -QR

M

M,

(8)

calcium sulfate:

H3P04 mass rate: 2 a(WMF -

- xj)= W M j

Mass balances on components in liquid phase, neglecting the COZ that absorbs: phosphoric acid:

Ws,;-1

(2)

WMF + P ~ F Q ~=F W e p + Q a P P N

-

wMj-1(1

WMN)wg

=

QaPCa,N

Q ~ F

(4)

('

Qo=Qc~

HzSO4 mass rate:

cF

+ PNE )

CaO mass rate:

Mass Balances f o r Reactor j ( F i g u r e 2). Definition of Ws,j

Mineral mass balance:

where: Ind. Eng. Chem., Process Des. Dev., Vol. 16,No. 3, 1977

391

CdF

= PcF

- CcF

The time T for the complete dissolution of a particle is:

Furthermore

(19)

ZO(L) = 0 $0” = 0 The input particle size distribution function fo(R)may be approximated by a Dawson integral, i.e.:

fo(R) = e - R P

LR

etBdt

The integration of the above equations must be accomplished by using eq 47 for k L ” , where d is substituted by 2 r . The resulting equation then enables the conversion X (for a particle) defined as:

(16)

(20)

rnl”

in particular, for the mineral examined, it was /3 = 3, R,,, = 1.5 x 10-5. In order to solve the above set of mass balance equations, we must develop expressions for the two quantities and B, which are related to the rock dissolution rate and to the crystallization rate, respectively. Therefore we must postulate the elementary steps that make up the dissolution and the crystallization mechanisms. We will consider, following Gilbert and Moreno (1965), the following two cases determined by the values of the parameter G: (i) 1 < G < 2.5, the rock dissolves and CaS04.1/2H20crystallizes on nuclei and crystals existing in the bulk of the liquid; (ii) G L 2.5, the rock dissolves and CaS04J/2H20 crystallizes directly on the external surface of the dissolving particles. In the first case it is possible to analyze separately the dissolution and crystallization mechanisms. 1 < G < 2.5-Dissolution of a Single Particle. Let us assume that the rock particle is spherical. Shape factors will be utilized to correct for the actual geometry. We can visualize the following steps occurring during dissolution. (i) At the solid/liquid interface the salts that are contained the mineral dissolve and dissociate. We will postulate a t this interface thermodynamic equilibrium whose conditions are regulated by the solubility products. (ii) The reactant H+, deriving from H2S04 dissociation, diffuses from the core of the liquid toward the liquidholid interface. (iii) H+ reacts with the main constituents of the rock. The reactions taking place in liquid phase are:

x,

+ 3H+

PO43-

FC0a2-

+ 2H+

+ Hi+

-

+

H2C03

1-

HF -+

H20

+ COn

= hL°Cc


x=

The time for the complete dissolution is:

(17)

where h ~ ’is the “physical” mass transfer coefficient given by eq 47 and c, is the H2SO4 concentration in the bulk of the liquid. Thus the time required for dissolution of a particle from its initial radius R to radius r is given by:

392

and the conversion for such a particle is:

HzP04

and other minor reactions that will not be considered here. (iv) All reaction products diffuse back into the main body of the liquid. The above reactions occur by proton transfer mechanism and therefore can be assumed to be “instantaneous” with respect to diffusion. The overall dissolution process is thus controlled by the diffusion of reactants toward a reaction plane. The situation is analogous to that encountered in the process of “gas absorption with instantaneous chemical reaction” (Roper et al., 1962; Astarita and Gioia, 1965).On the basis of this analogy, since the “physical” solubility of rock (in water) is much smaller than the H2S04 concentration, the results reported in the quoted works permit the following expression to be written for the dissolution rate, expressed as grams of H2SOJ consumed per unit of time and unit of surface of rock particle: VM

to be related to the time of reaction t . A test of the postulated mechanism can be made by utilizing the experimental results of Gilbert and Moreno (1965). They report experimental dissolution times for particles of phosphate rock which are in good agreement with those obtainable by applying eq 19. G L 2.5-Dissolution of a Single Particle. For these values of G , as shown by Gilbert and Moreno (1965), covering of rock particles by CaS04a1/2H20 takes place. Thus, an additional elementary step exists in the dissolution mechanism: diffusion of reactants and products through the layer of CaSO&H20 (solid) to the interface of the unreacted core (the reaction surface). It may be easily verified that in this case this last diffusional step controls the overall dissolution process. Therefore a t any time t the concentration profiles are those pertaining to the “unreacted core model controlled by diffusion” (Wen, 1968). If we assume a pseudosteady state (Wen, 1968),as is reasonable in this case, the relationship between time and the position of the moving boundary r of the unreacted core for a particle of external radius R and initial unreacted core ri is:

(23)

Equations 21 and 23 for ri = R reduce to the equations reported by Wen (1968). The value of the diffusion coefficient De, failing direct experimental data, can be assumed to be of the same order of magnitude as that reported by Cordell (1968), Le., De = 7 X 10-8. This value was obtained experimentally for a system quite similar to the one under discussion. A test of the postulated mechanism can again be made by utilizing the results of Gilbert and Moreno (1968). The application of eq 21 and 22 for ri = R to the experimental conversion data reported by the above authors for cases in which deposition of CaS04.1/nH20 took place, produces times that can be considered in good agreement with the experimental values. Evaluation of for 1 < G < 2.5. The dissolution eq 18 obtained in the previous paragraph allows us to calculate conversion in reactor j (for entering particles of uniform radius R ) by the equation:

zj

The integration is extended to the physical values of time 7(R) furnished by eq. (19). Actually the mineral feed to reactor j consists of a mixture of different sized particles. The size distribution of this feed can be represented by a continuous particle size distribution function f,-i(R); namely, f;-1(R) dR is the fraction of particles in the size range R to R dR.Therefore the actual conversion in reactor j is given by the equation:

+

+

Finally, in order to apply eq 25 to reactor j 1 we must develop expressions for the function f J ( R ) . Equation 18, which for brevity we write as: t = g(R,r)

In conclusion, they find the following functional dependence between nucleation rate and supersaturation:

(30) the value of the k , constant is not given in the cited work. The process of crystal growth has been studied extensively and many mechanisms have been postulated. Basically all these mechanisms involve the following two steps in series: (i) diffusion of crystallizing compound from the bulk solution to the crystal surface; (ii) integration of the above compound in the crystal lattice. Both steps depend on the value of the supersaturation s. In particular Amin and Larson (1968) find from experiments that for calcium sulfate crystallization, the following rate equation holds:

gives the reaction time necessary for a particle of radius R to attain a final radius r. Then:

2

fJ-i(R)Ejk(R,r)\

d r dR

(26)

is the weight ratio of particles entering with initial common radius R and having radius between r and r dr in the exit stream. Then the required exit particle size distribution function is given by:

+

v, = kes

(31)

but unfortunately information suitable to calculate the constant k , is omitted from the paper. However, eq 31 implies that both steps (i) and (ii) are first order in supersaturation, and if so it must be:

Furthermore from Amin and Larson's (1968) results one can postulate for T > 70 "C, k R >> k L " , and therefore write: Evaluation of zj f o r G z 2.5. For this case it may be assumed that fJ-l(R) = f,-z(R) = . . . = fo(R). However, due to the kinetic eq 21 and to the residence time distribution E ( t ) , the particles entering the reactor j show different sized unreacted core radii ri. Let us say FJ-l(R,ri)dri dR is the weight fraction of particles entering reactor j having unreacted core radius between ri and r, dri and external radius between R and R dR. Then, writing for brevity eq 21 as:

+

(33)

= kLoS

V,

where k L o can be calculated by eq 47 (where LQ; is substituted to d) and v, is expressed as weight of Cas04 that crystallizes for unit of time and unit of crystal surface. The linear growth rate dL/dt = V L is then given by: (34)

+

the conversion in reactor j is

X

dG

- dR dr dri (28) dr

A check for the above model can be made by applying eq 34 to Amin and Larson's (1968) experimental V L data. This enables a supersaturation to be estimated (not reported by the above) that correspond to a G value in the right range (12.5). Extension of Crystallization Model t o R e a c t o r j . A population balance on the j t h reactor in steady-state operation with perfect macromixing can be written as (Randolph and Larson, 1962; Randolph, 1965):

The exit particle size distribution function is given by:

Crystallization Mechanism for 1 < G < 2.5. The crystallization process is usually divided into nucleation and crystal-growth mechanism (Schoen, 1961; Moyers and Randolph, 1973). The actual process of nucleation is still uncertain; many mechanisms have been postulated according to the physicochemical steps involved in the nuclei formation. Roughly, one can assume both a heterogeneous and a homogeneous mechanism for nuclei formation. The relative importance of these nuclei sources is strongly dependent on the system under examination and must be determined experimentally for each case. The experimental work of Amin and Larson (1968) concerning just the dissolution of phosphorites and CaSOC$H20 crystallization gives information on nucleation rate for this system. These authors indicate that the predominant source of nuclei is homogeneous nucleation and the nuclei generation rate is not dependent on solid present, but only on supersaturation.

(35) Assuming that the mass transfer coefficient is constant and equal to the value corresponding to the "dominant crystal size" (on a weight basis) defined as:

then V L is independent of L and can be taken out of differentiation and eq 35 gives: r ~ =. + j - i + ( + j o - icj-1) exp L (37) J

(- z)

rCj0 is the nuclei population density defined as: $0

dn = 1'im L-o

dL

then dn dt

- = +OVL Ind. Eng. Chem.. Process Des. Dev., Vol. 16, No. 3, 1977

393

Ch*

and by eq 30 and 34

where (39)

+ 0.024) - 3.46 x 10-'C,

= ~ ( 7 . 2 7x 10-'T

The degree of supersaturation G, is defined (Gilbert and Moreno, 1965) as the ratio between the product of concentration of calcium and sulfate ions in the supersaturated solution and in the equilibrium solution (for the same values of P20,5and H?S04 concentrations), namely:

According to Amin and Larson's (196813) data, for the system under consideration we can postulate: $0

= 2.15

x 1019(itL's)1,6

(40)

The quantity B appearing in eq ( 7 ) ,(11)and (12) can now be calculated:

(48)

(49) by assuming calcium ion concentration equal to sulfate ion concentration, eq 49 becomes: (50) where s = C h - C h * is the supersaturation. For hemihydrate precipitation it must be (Gilbert and Moreno, 1965): 1< G

The unreacted cores leaving reactor N are lost in the filtrate. This loss is equal to W M , N . Parameters Mass Transfer Coefficient. The correlation of Brian and Hales (1969) (Satterfield, 1970) for mass transfer to spheres suspended in an agitated liquid, relates the mass transfer coefficient k L o to the agitator power input "p" expressed as power per unit mass of slurry. For the system under examination the above correlation may be written as: kL"d pl/3d4/3p In = 0.359 In (43) D(SC)'.~~ Y valid for: pl/3d4/3p 10-2 < < 10 (44) ~

~

~

Y

this range includes all possible operating conditions for our system. Rushton et al. (1950),Foust et al. (1960),and O'Connell and Mack (1950) provide useful correlations of agitator total power P as a function of vessel geometry and stirrer speeds for several types of agitators. In particular, for an impeller "flat paddle 2 blades" with six baffles, the correlation is: (45) valid for

where: R ~ D I ~ Re = Y

This range of Reynolds number corresponds to operation of agitators in practice. Assuming a height of liquid equal to the diameter of tank, eq 43 with eq 45 gives the correlation that has been used for mass transfer coefficient in this work. Namely: kL"d In -= 0.479 In 0.359 In Re - 0.533 (47) D ( Sc)o,33 Solubility of a-Hemihydrate Calcium Sulfate. From the data of Taperova and Shulgina (1945) (reported also by Slack, 1968) it is possible to obtain the following correlation for the solubility of CaS04JhH20:

+

394


< 2.5

Density of Liquid Solution. The liquid density p has been evaluated as a function of temperature and liquid composition by using the relationship reported by Slack (1968). p = 1284 - 0.56(T -20)

+ (11- 2 X 10-'(T + (9 - 2 X 10-'(T

- 20))

- 20))

(0 . 9 8 +~ ~0.71ch 100- 2)

(51)

Viscosity of Liquid Solution (Slack, 1968). = 3.6 [100.4i9-0.0lO'T + ~0-l.'s:i+'."xlo-:'T + C a U . 2 4 - O.Ol:iT)//>]( 5 2 ) Results of the Analysis for t h e Reactor Subsystem Before proceeding further with the analysis, the above system of equations has been solved with an IBM 370/135 computer, for the case of major interest in practice (1 < G < 2.51, in order to analyze the influence of the main variables on the reactor's configuration. This has enabled some preliminary conclusions to be drawn. These conclusions, although of interest by themselves, will help in fixing the operating limits of the other subsystems that compose the plant. One of the most significant results of these calculations is that the limiting step of the process is the crystallization. In fact, this step requires a residence time longer than that necessary for the complete mineral dissolution. However, even so, the overall process is quite fast. Examination ofTables I. 11, and 111,where a few of these results are reported, shows that one reactor with half an hour residence time is sufficient for the complete transformation of the mineral with crystals of a reasonable size (and size distribution, as shown by the other results not reported here). Smaller residence times would result in the crystals being too small. Of course, this fast crystallization rate is due to the high temperature that makes k R >> h ~ and " the high value of h,?'.Therefore, the main feature of the hemihydrate process, as roughly indicated also in the literature (Slack, 1968) is a short residence time as compared with other processes. Moreover, still from the results reported in the above tables, it is possible to infer the role played by the recycle ratio on the performance of the process. In brief, 5 controls the supersaturation in reactors and then the parameter G, namely, the possibility of the covering of the mineral particles by CaSOd.1/2Hn0,the growth rate of crystals and the nucleation rate are strongly influenced by 0) both for 1 < G < 2.5; tau, WMF,w g , and the consequent 6 will be assumed as input data. Their values are reported in Tables VI and VII.

-

Minimization of t h e Objective Function-Discussion of Results Many mathematical methods of minimizing a function of several variables have been proposed. In our case, because the

Table IV. Ci, Annual Installed Costs of Process Equipment for Ten Year Life Equipment description Reactor-crystallizers height is 1.25Dt

Cost, $/year 103ZDto,’’

Electrical motors for reactors. Installed power is 1.5 that required

1.1x 10-41

Two rotary drum filters. Each for total service

3.11

Two recycle pumps. Each for total service. im

1.27 x 1041

Equation

Chilton (1960)

Re3p3

(---)

0.5

D 2

(

Lit. citation

pN1,48

45 46

Aries and Newton (1955)

60


58

Chilton (1960)

58


62


= 50

Two electrical motors for recycle pumps irn= 50

Steam boiler, 200 psi. Assume h independent of T Three-stage vacuum evaporator-crystallizers. Forced circulation Total annual installed cost of process equipment Other items forming the “physical plant cost”: Piping Instrumentation Insulation Electrical Additional utilities Buildings Annual “physical plant cost”

(

4.11 Q a p p ~(1 21.51(p~Q,p)~,’~

Chilton (1960)

zCl Aries and Newton (1955) 0.25ZC, 0.03ZC1 0.06 Z C, O.072CL

0.15ZC, 0.25ZC1 1.81ZC,


Table V. Annual (8700 h) ODerating Costs Cost, $/year

Process description

2.68 X lo-’

Reactors’ agitation

Equation

Re3p3 -

45,46

DIPj

58 Recycle” 61 Filtration” 0.5 atm vacuum 80%pump efficiency 200 psi steam boiler efficiency 70% assume X 62 independent of T Mineral cost; assume 198.7 $/ton of P205 1.73 x 1 0 3 ~ ~ ~ ~ ~ 623.4Wc Sulfuric acid cost, 65.0 $/ton anhydrous H2S04 1.26ZCi Maintenance: 7% of “total physical plant costs” The cost of electric power has been assumed equal to 0.02 $/kW-h. * The oil cost for steam production has been assumed equal to 0.081 $/kg.

objective function depends on the variables through the computer program previously discussed, a nonlinear optimization algorithm based on a nongradient method (Rosenbrock, 1960; Powell, 1964; Zellnik e t al., 1962) should he used. However, it is possible t o avoid this time-consuming computer procedure by solving the objective function for a few values of the variables. The results will contain sufficient information to locate the minimum. This has been done for two minerals having compositions wg = 0.35 and wg = 0.25, respectively, as reported in Tables VI and VII. From these tables the following consideration can be made. (i) As already mentioned, the economy of the process is dominated by the raw material costs. The sost per unit of weight of P205 produced changes with C; and t , only on the third digit. (ii) The cost of concentration is, for each mineral composition and for a given product concentration, constant, because for all practical values of residence t and recycle ratio F , the reactor shows a total conversion. Therefore only reactor, recycle, and filtration costs change with [ and t. (iii) As a first approximation the objective function may he considered only a function of C;. Its

constancy with t is due to the feet that the decreasing reactor costs with decreasing values of t are roughly compensated by the increasing filtration costs as the crystal size decreases with

t. In conclusion, the minimum of the objective function depends essentially on C; and occurs for that value of the recycle ratio where G = 2.5. Of course, the above conclusion is valid in the limit that the Cas04 content in the final phosphoric acid does not influence its trade value. In fact, discounting data in the literature on this aspect of the problem, we have not accounted for any variation of P205value with its purity. On the other hand, this is not a strong limitation for fertilizer production purposes because as G ranges between 1 and 2.5 the PzOj purity (on a basis of PzO5 CaS04)ranges between 95.5 and 93.4 and part of the CaS04 is eliminated in the concentration process. A comparison between the results of Table VI and those of Table VI1 shows that the cost of unit of weight of Pz05produced is higher for a poorer mineral. This is because we have assumed, as the market shows, that the mineral cost is pro-

+


397

Table VI"

t

E

Values of variables D, ( P z O ~ % ) N L D X lo6

+

G

Annual partial costs (phys. operat.): $/yr Mineral: 4.32 X lo6; H2S04:5.99 X 106 Reactors Recycle Filtrn Concn

10.0 6.08 43.14 187.8 1.026 2.73 X lo4 6.0 5.30 203.9 1.032 1.75 X lo4 43.24 2.0 4.19 43.26 1.040 9.82 X lo3 230.0 2.0 3.81 43.24 211.8 1.050 8.28 X lo3 1.5 1 2.0 3.33 43.24 185.6 1.060 6.81 X lo3 2.0 2.64 43.27 151.5 1.090 5.25 X lo3 0.5 " WMF = 10 000; c,, = 1.86 (54%P205);wg = 0.35; uc = 0.8. 2 2 2

1.32 X 1.20 X 1.04 X 9.78 X 9.03 X 7.88 X

lo4

lo3

1.32 X

lo3 lo3 lo2 lo2 lo2

1.11 X lo4 6.01 X lo3 6.32 X lo3 6.82 X lo3 7.69 X lo3

6.29 X 6.28 X 6.22 X 6.23 X 6.23 X 6.23 X

lo4 lo4 lo4 lo4 lo4 lo4

cost of product, $/ton

CaN c a N + CbN

402.72 401.89 400.97 400.92 400.88 400.83

98.03 98.03 98.03 98.02 98.00 97.98

cost of product, $/ton

C.N c a N + CbN

485.88 484.78 483.57 483.51 483.44 483.38

96.82 96.82 96.81 96.80 96.78 96.74

Table VII"

t

Values of variables Dt ( P 2 0 5 % ) ~LD X lo6

+

G

Annual partial costs (phys. operat.): $/yr Mineral: 4.32 X lo6; H2SO4 = 5.99 X IO6 Reactors Recycle Filtrn Concn

10.0 6.10 188.0 1.021 2.58 X lo4 35.00 6.0 5.32 35.02 199.9 1.024 1.68 X lo4 2.0 4.21 35.02 2 231.4 1.030 9.70 X lo3 1.038 8.22 X lo3 2.0 3.82 35.02 213.5 1.5 187.6 1.048 6.78 X lo3 1 2.0 3.34 35.02 35.02 156.8 1.080 5.25 X lo3 2.0 2.65 0.5 " WMF= 10 000; caU = 1.86 (54%P205);ag = 0.25; wC = 0.8. 2 2

portional to up-while (for a given W M Fthe ) sulfuric acid requirement and the plant size are practically constant with wg. This result gives a hint for a discussion on the selling price of low grade calcareous phosphate rocks. As is known, a few years ago, mainly rich phosphate minerals (about 35% P205) were utilized to produce phosphoric acid. However, the increase of the use of phosphatic fertilizers and consequently the abrupt increase in their price has made convenient the utilization also of low grade rocks from mines mainly located in the third world countries. Problems arise for these countries when they must establish the selling price of these low grade rocks. In fact, data on production costs of phosphoric acid are available in the literature only for acid produced from the above standard rich minerals. In the absence of data regarding rocks of different composition the tendency has been to establish the selling price by assuming a value of the unit of weight of P205 contained in the mineral (for an assigned composition range) equal to that of 35% mineral and irrespective of its inert content. On the contrary, the results given in this paper have shown that this is not a correct approach. In fact, for an assigned value of the unit of weight of PzOs produced as phosphoric acid, the unit of weight of P2O5 contained in a poor mineral has less value than that contained in a richer mineral. In particular, by utilizing the data for example in Tables VI and VI1 it would be possible to find the relationship between mineral cost and its P2O5 content which makes the value of the unit of weight of P2O5 (as phosphoric acid) independent of rock composition. Then, it would also be possible to determine the value of w g which gives a null value to the mineral. Finally a few words must be said about another wet process which has not been analyzed in this paper. As it is industrial practice to improve the filtrability of calcium sulfate salt produced while keeping the phosphoric acid concentration high, the hemihydrate-dihydrate process is preferred (Slack, 1968). Briefly, the wet process consists of adding few more tanks where by dropping the temperature to the range of dihydrate stability (50-60 "C), the reaction: 398 Ind. Eng. Chem., Process Des. Dev., Vol. 16, No. 3, 1977

1.30 X 1.19 X 1.01x 9.60 X 8.86 X 7.73 X

lo3 lo3 103 lo2 lo2 lo2

1.54 X 1.12 X 5.95 x 6.25 X 6.74 X 7.48 X

lo4 lo4

103 lo3 lo3

lo3

CaS0442H20

6.19 X 6.19 X 6.20 x 6.22 X 6.22 X 6.23 X

lo4 lo4

104 lo4 lo4 lo4

+ -3 H20

-

CaS04.2H20

2 takes place. The main features of this wet process are: (i) higher phosphoric acid concentrations as compared with those obtainable from the direct dihydrate process; (ii) production of large, well formed crystals of CaSO4-2Hz0and, therefore better filtrability; (iii) the eventual phosphate substituted in the CaS04. $$HZ0 lattice is released and dissolved during recrystallization; and (iv) the hydration of CaSOCHH20 on filters, which is one of the drawbacks of the hemihydrate process, is avoided. The analysis made in this paper could have been easily extended to the hemihydrate-dihydrate process. Unfortunately, this has not been possible for the complete lack of experimental data suitable to estimate the rate of the above reaction. Conclusions The use of the hemihydrate process for the production of phosphoric acid has been examined as an illustration of application of first principles to the analysis of a "real process". It has been shown that by this approach the knowledge of very few fundamental parameters is sufficient in order to obtain a reasonable description of the process. Information of industrial importance, regarding, among other things, the optimal configuration and operating conditions of the process as well as the trade value of low grade phosphorites was obtained. Throughout the work, where possible, we have resorted to simple but efficient procedures and reasonings, instead of complex mathematical procedures. Nomenclature Af = filtering area, m2 A t = total filtering area, m2 B = mass of CaSO&H20 crystals per unit mass of slurry, k g k c,, = phosphoric acid concentration out of evaporators, kg/m3

= anhydrous sulfuric acid concentration in Qcp, kg/m:I cl; = liquid phase concentration of compound i in reactor j , kg/m:j C, = annual installed cost of process equipment i, $/year d = diameter of particles, m D = diffusion coefficient in liquid, m2/h D e = diffusion coefficient in CaS04.l/zH?O, m2/h D I = impeller diameter, m D , = pipe diameter, m D, = reactor diameter, m E ( t ) = residence time distribution function, l/h f = Fanning factor?dimensionless f, ( R ) = particle size distribution function of the phosphate rock, out of reactor j , kg/kg m G = degree of supersaturation, see eq 49, dimensionless k , = overall constant in crystal growth rate equation, m/h h ~ , ' = constant in the linear crystal growth rate, m4/h kg hl," = physical mass transfer coefficient, m/h h , = constant of the nucleation rate, eq 30, (number/ h kg)(m:'/kg)?.G k,,' = see eq 39, (number/m kg)(h/m)l.G k~ = constant of integration rate in crystal growth mechanism, m/h I = ratio between M and S indexes, dimensionless m = mass of slurry in reactor, kg M = molecular weight, kg/mol L = crystal size, m L I , = dominant crystal size. m L,, = total length of recycle line, m n = number of nuclei, number/kg N = number of reactors, dimensionless p = power for unit mass of slurry, m'/h3 P = power of agitator, kg m2/h:j Pi.= power requirements for filtration, kg m2/h3 PR = power requirements for recycle, kg m2/h3 Q = volumetric liquid phase flow rate, m3/h r = particle radius. m R = radius of particles entering the reactor, m s = supersaturation = ct) - c b * , kg/m:j Sc = Schmidt number, dimensionless t = time, h f = mean residence time, h T = temperature, " C V = volume of slurry in reactor, m3 W = mass rate, kg/h 2,( L 1 = cumulative weight ratio of crystals out of reactor j , kg/kg X = conversion degree of phosphate rock for a single particle, -W k g X = conversion degree of particles entering the reactor with =common radius R , kg/kg X = conversion degree in reactor; mass of the rock converted per unit mass of the rock feed to the reactor, kg/kg Greek L e t t e r s ( \ = part of anhydrous sulfuric acid per part of rock; for stoichiometric acidulation, kg/kg 6 = see eq 16, dimensionless 6 = mole ratio CaOiPpOS of the rock. For a mineral composed of CaCOx and Ca:j(PO& it is: 6 = ( 4 3 M h - MR) + M,l/((Mf + M h ) ~ dimensionless ~ l 6* = value of 6 for Ca:j(PO,),; 6* = 3, dimensionless AH = heat required for concentration, kcal/kg AP = pressure drop in recycle line, kg/m h* A P v = pressure drop across filter cake, kg/m h2 h = latent heat of vaporization, kcal/kg p = viscosity of liquid phase, kg/m h u, = crystal growth rate, kg/h m 2 VL = linear crystal growth rate, m/h r'x~ = dissolution rate of phosphate rock per unit of particle surface, kg/h m2 ( = recycle ratio, eq 15, dimensionless p = density, kg/mz 7 = time for complete dissolution of a rock particle, h C,F

= crystal shape factor, dimensionless = particle shape factor, dimensionless $ = crystal population density (per unit mass of suspension), @M

number/kg m $" = nuclei population density (per unit mass of suspension),

number/kg m anhydrous sulfuric acid content of Q c ~kg/kg , phosphorus pentoxide content of the rock, kg/kg of rock R = runs of the impeller per unit of time, l / h { I = concentration, kg/m" Subscripts a = H3P04 (also H3P04 solution) b = Cas04 c = H2S04 (also I42S04 solution) d = H20 e = CaSOd.f/2H20 f = CaO g = P20j h = CO2 F = feed j = out of reactor j or into reactor j 1 m = maximum v d u e * M = mineral P = product R = recycle s = suspension oc = og =

+

Literature Cited Abramowitz, M., Segun, I. A., "Handbook of Mathematical Functions", 5th ed, p 887,Dover Publications, New York, N.Y., 1965. Amin, A. B., Larson, M. A., Ind. Eng. Chem., Process Des. Dev., 7, 133

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Received for revieu

M a y 13, 1976 Accepted M a r c h 24, 1977

This work has been supported by a grant from t h e Consiglio Nazionale delle Ricerche, I t a l y . It was presented in p a r t a t t h e XI I n t e r n a t i o n a l M i n e r a l Processing Congress-Seminar o n B e n e f i c i a t i o n of L e a n Phosphates w i t h Carbonate Gangue.


399

Analysis, Simulation, and Optimization of the Hemihydrate Process for

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