flow of supersaturated solutions in pipes. modeling ...

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FLOW OF SUPERSATURATED SOLUTIONS IN PIPES. MODELING BULK PRECIPITATION AND SCALE FORMATION a

a

M. KOSTOGLOU , N. ANDRITSOS & A. J. KARABELAS

a

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Chemical Process Engineering Research Institute and Department of Chemical Engineering, Aristotle University ofThessaloniki, P.O. Box 1517, Thessaloniki, GR 540 06, Greece Published online: 24 Apr 2007.

To cite this article: M. KOSTOGLOU , N. ANDRITSOS & A. J. KARABELAS (1995) FLOW OF SUPERSATURATED SOLUTIONS IN PIPES. MODELING BULK PRECIPITATION AND SCALE FORMATION, Chemical Engineering Communications, 133:1, 107-131, DOI: 10.1080/00986449508936313 To link to this article: http://dx.doi.org/10.1080/00986449508936313

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FLOW OF SUPERSATURATED SOLUTIONS IN PIPES. MODELING BULK PRECIPITATION AND SCALE FORMATION M. KOSTOGLOU, N. ANDRITSOS and A. 1. KARABELAS·

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Chemical Process Engineering Research Institute and Department of Chemical Engineering, Aristotle University of Thessaloniki, P.O. Box 1517. GR 54006 Thessaloniki, Greece (Received July 29.1994) Pipe flow of supersaturated solutions of sparingly soluble salts is simulated in this work. Models from the literature are selectedto represent individualprocesses,which may occur simultaneously.such as nucleation, particlegrowth, coagulation, and particulateand ionic deposition on the pipe wall. As regards formulation, a fairly comprehensive population balance equation is employed for the solid phase combined with mass balancefor the ionic species.A sufficiently accurate,yet computationally convenient,discretizationscheme is used in the simulation algorithm. Arbitrary parameters are avoided with the exception of surface reaction parameters which are determined from experimental data since no theoretical values are available. The PbS-water system (previously studied experimentally in this laboratory) serves as a test case. The PbS effectivesurfaceenergy is estimated fromexperimentaldata. Predictions are quite encouraging, reproducing statisfactorily the measured "bell't-shaped deposition versus pH curves.The predictedparticlesize distributions and their evolution are physically realistic, qualitatively and quantitatively. Despite necessary future improvements, the proposed simulation algorithm is already considered a useful tool for scale up in industrial and other applications. KEY WORDS

Supersaturated solutions

Pipe flow

Precipitation and scale formation

Lead sulfide.

INTRODUCTION In many industrial processes the change of a liquid property (e.g. pH, temperature, species concentration) can induce supersaturation of several (potentially precipitating) sparingly soluble compounds, thus promoting phenomena that can proceed concurrently. For pipe flow, particle nucleation, growth and coagulation can take place in the bulk, while deposition of ionic (or molecular) species and of particles may form scale on the pipe wall. Obviously, these phenomena result in variation of fluid medium properties along the flow path. Such very complex situations are encountered in many cases; e.g. in plants handling solutions of inorganic compounds (crystallization units, metallurgical plants), in heat exchangers, in condensation equipment etc. The particular problem motivating the present study is theevolution offluid and of scale properties along the flow path in geothermal installations. Geothermal fluids (brines) are usually rich in inorganic compounds (e.g. Gallup et al., 1990; Andritsos and Karabelas, 1991c) and supersaturation may be triggered by a pH • Author to whom all correspondence to be addressed. 107

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increase (due to CO 2 flashing) and, to a smaller degree, by temperature changes. Of particular interest here is the prediction of scaling rates and of scale properties which can vary significantly along the flow path. Scaling is the main operating problem in geothermal installations and considerable work has been carried out to understand the physico-chemical factors controlling these phenomena. However, there is no study reported in the literature aimed at simulating the entire flow system and obtaining a tool, which would be useful in engineering and other types of calculations. The objective of this study is to make a systematic effort in that direction. Various aspects of the problem treated here, as well as the general approach, have already been presented in the literature. A few typical examples of such efforts are mentioned here. A simplified form of population balances is employed in precipitation problems by Den Ouden and Thompson (1991). Population balance type of formulations are used for studying aerosol flow in pipes by Nadkarni and Mahalingam (1985) for laminar and turbulent flow, and by Xiong and Pratsinis (1991) for plug flow. The influence of the population balances on the deposition rate and on deposit morphology is studied by Okuyama et al. (1991) for chemical vapor deposition. Park and Rosner (1989) report on the effect of coagulation on deposition rate in boundary layer flow around bluff bodies. Wachi and Jones (1991) employ population balances to study precipitation effects on gas/liquid mass transfer. Relevant reviews on crystallization are those of Garside (1985) and Dirksen and Ring (1991). The present study deals with only one precipitating compound, namely lead sulfide (PbS), in water. A considerable amount of experimental data (scaling rates, scale and fluid properties) with this system have already been obtained in this Laboratory (Andritsos and Karabelas, 1991a and b) and are very useful in the modeling effort, for parameter estimation and model validation. Figure 1 shows various physical processes considered to occur simultaneously along the flow path, i.e. along a straight pipe. At pipe entrance (t = 0), the fluid becomes supersaturated with respect to PbS (or to other similar compounds dissolved in the geothermal brine). The driving force for all processes included in Figure I is the Gibbs free energy increase due to supersaturation of a specific species; i.e. PbS in this example. Ionic deposition (or crystallization) right on the pipe wall is a possible mechanism of scale formation, as already shown. Lead and sulfide ions can also be consumed in the bulk, by the processes of nucleation and further particle growth. Concurrently, nuclei and/or larger particles tend to coagulate upon collision promoted by Brownian diffusion or turbulent fluctuations. These processes result in a colloidal particle size distribution (PbS)' continuously evolving along the pipe. Particulate deposition from the bulk may also contribute to the total rate of scale formation. A related experimental study (Andritsos and Karabelas, 1991a) was carried out in a 13 mm J.D. pipeline with two test sections placed 10 m apart. The first test-section was located 20 em downstream from the point of mixing of the two reagent solutions (solutions of Na 2S x 9H 20 and Pb(N0 3)2) at the desired conditions (pH, liquid velocity, concentration). Each test section consisted offour flanged, 12-cm long, tubular segments and eight pairs of semiannular coupons-inserted therein. Crystalline mailer (PbS) was formed on the inner surface of the coupons made of stainless steel or Teflon"'. Its mass was measured with time after rinsing with distilled water and drying. The data showed that appreciable PbS deposit formation occurred only within a narrow pH

FLOW WITH PRECIPITATION AND SCALE FORMATION

109

Supersaturated

solution with respect to PbS. ConsumpUon of Pb·~

Wall/surface nucleation

I-

4-

+

+

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~

Particle growth

Ionicdeposition

-

Nucleation

Cosg".

+ Particle size distribution In bulk

~ SCALE FORMATION

Particulate

J

deposition

FIGURE 1 Physical processes occurring during pipe flow of a supersaturated solution. The case of lead sulfide.

range of about two pH units. A bell-shaped deposition curve was obtained at all conditions, while the maximum of the deposition curve depended upon the concentration of the reagents, the temperature and the liquid salinity. During the initial stage of deposit formation small isolated crystals-nuclei were observed on the substrate. These crystals grew with time forming a rather compact deposit layer when saline solutions were used and a less compact layer in the case of zero salinity solutions. In general, these observations are in accord with the description of chemical scale formation given by Siilinel and Garside (1992). At low supersaturation ratios (or for conditions corresponding to the increasing side of the deposition curve) only deposit formation apparently proceeds, while bulk precipitation does not take place. An important issue in the simulation presented here is the selection of an appropriate model to represent wall crystalIization. There are different views in the literature regarding the controlling mechanism for the precipitation of sparingly soluble salts. Almost all studies on bulk precipitation kinetics of sparingly soluble salts (e.g. Nancollas, 1983; Wiechers et al., 1975; Amjad, 1987) suggest that a surface controlled process is more likely to occur. This conclusion is based on the evidence that fluid dynamics (stirring rate) does not affect the crystallization rate, the rate is parabolic with respect to supersaturation, and the fact that considerably high values of activation energy are estimated. Additionally Sohnel and Garside (1992), reviewing the problem of scale formation recently, report that the rate of scale formation is usually controlled by a surface reaction mechanism. On the other hand, CaC0 3 deposition studies show that scale formation is controlled by convective diffusion (Hasson, 1981). For the PbS deposition experiments carried out in this Laboratory and for relatively large super-

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saturation ratios (S > 3) there is evidence that the process is controlled by convective diffusion of lattice ions towards the pipe surface. This conclusion is mainly based on the significant effect of fluid velocity on the initial deposition rate (Andritsos and Karabelas, 1992) and on the low value of an apparent activation energy, E = 18 kJ mol- 1, which is determined by plotting the logarithm of the deposition rate versus the inverse of temperature. For surface controlled precipitation the activation energy is typically greater than 40 kJ mol- 1, while for a diffusion controlled process it is expected to be less than 20 kJ mol - 1 (M ullin, 1993).In the present modeling effort both mechanisms are considered to influence the rate of deposit formation; i.e. the surface (polynuclear) step for low supersaturation values and the diffusion step for high values. In the following section the problem formulation is outlined. Particular emphasis is given to discretization of the system of integrodifferential equations, describing the physico-chemical processes considered here, and to the selection of an appropriate numerical method for solving the resulting system of discretized equations. Physical quantities and related expressions, for representing the various rate processes, are summarized in the next section. The results obtained with the proposed algorithm and comparisons with experimental data are presented in the last section.

2 PROBLEM FORMULATION 2./

Physicochemical Processes

The thermodynamic driving force for both bulk precipitation and wall crystallization is the change of the Gibbs free energy

6G= -RTInS

(1)

where R is the ideal gas constant, T is the absolute temperature and S is the saturation ratio, defined as S = [(Pb2+)(S2 -)JI /2

«;

(2)

Quantities in parentheses denote the activities of the respective species and K,p is the thermodynamic solubility product of PbS. Despite the fact that the driving force for the two basic processes (bulk and wall precipitation) is the same, it is possible that due to kinetic limitations one process may control, under certain conditions through the nucleation step. It is well known that bulk nucleation (leading to bulk precipitation) occurs following the lapse of an induction period, even in the presence of a foreign seed, and it is frequently correlated with a power of the inverse of the supersaturation ratio. Additionally, for bulk precipitation to occur a critical value of S, S" should be exceeded. On the other hand, nucleation on a pipe wall from a flowing supersaturated solution may occur at lower than S, values, or without the lapse of an induction period, in a similar way to the nucleation on an existing flat crystal surface (Dirksen and Ring, 1991). In the present modeling attempt the lead and sulfide ions are considered to be present in stoichiometric ratio, as in the related experiments. The consumption oflead ions will

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be followed in the analysis for the sake of simplicity, since sulfide may be present in three different forms. As already outlined, Figure I depicts the processes whereby lead ions are consumed, with dotted and full lines corresponding to ionic and particulate processes, respectively. For low supersaturation bulk precipitation is not observed (Andritsos and Karabelas, 1991a), at least for the short time period that the fluid is inside the test pipeline, and lead ions are consumed only through ionic transport and surface reaction right onto the tube wall. For larger supersaturation ratios, bulk nucleation appears and, consequently, the steps of particle growth and of coagulation eventually lead to a particle size distribution in the bulk. Under such dynamic conditions, scale formation on the pipe wall can take place by a combination of ionic transport to the surface (wall crystallization) and of particulate deposition. In modeling the above processes the following assumptions are made: .:- Steady state, plug pipe flow - Uniform radial concentration distribution, due to turbulent mixing - Particles of spherical shape - Macroscopic particle behavior independent of their size - Forces.between particles and particle-wall are neglected. The last assumption is justified on the basis of measured low zeta-potentials of bulk precipitated particles in the pH range of interest. It is also noted that particle breakage is not taken into account since the particle sizes involved in the present problem are too small and definitely smaller than the microscale of turbulence (Delichatsios and Probstein,1975). The supersaturation ratios and solubility data are computed by using the HYDRAQL code (Papelis et al., 1988), taking into account all possible sulfide and other complexes. All thermodynamic stability constants are from Smith and Martell (1976) except those of sulfide complexes which are taken from Barnes (1979). The code predicts solubility data which agree reasonably well with those shown in Figure 4 of Andritsos and Karabelas (1991 a). 2.2

Mathematical Formulation

Population balance (e.g. Ramkrishna, 1985) af(x t) 1 fX ----a/=2 B(y,x- Y)f(y,t)f(x~y,t)dy- f(x,t) f"" B(y,x)f(y,t)dy 0

-

0

aG(x, e)f(x, t) a - D(x)f(x, t) + H(c)b(x - a) X

(3)

M ass balance of lead ions de dt

roo 4 = -rp Jo G(x,c)f(x,t)dx-rH(c)pa-d;L(c)

(4)

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M. KOSTOGLOU et at.

Total mass of scale accumulated on the pipe surface over a spacetime period t

foo

dm 4Q -d =Qp xf(x,t)D(x)dx+-dL(c) t o r ,

(5)

Dimensionless quantities are introduced as follows _

x

X=a

_ af(x, t) F( x.t ) No

t

!=-

to

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- (x, y) = to N 0 B(:C, Y), B

D(x) = toD(x),

2.3

G(- x, c) -- toG(x,c) ,

H-(c) = toNH(C),

a

L(c) = 4L(c)t o d,co

0

C= ~ Co

Discretization of Equations

The numerical method employed here for simulations is based. on discretization of particle volume distribution in classes. The boundaries of class i are bi~ 1 == (3/4)2i - 1 and b, = (3/4)2i so that bl

ni =

l

F(x,t)dx

i= 1,2,... ,h

(6)

b'_l

Next, the discretization of each term in Eq. (3)is examined separately in order to assess systematically the suitability of the numerical technique. Coagulation

Equation (3) is transformed as follows for coagulation only:

where Bi ,) = B(2 i - ' , 2i - ' ). It will also be pointed out that for i = 1, only the 4th term is used, i = h, only terms 1st and 2nd are used.

This scheme, Eq. (7), was originally proposed by Hounslow et al. (1988)for crystallization. It conserves the total mass of the system, and belongs in a category of approximate methods usually referred to as "zero order methods". A rather thorough assessment of zero order methods is presented elsewhere (Kostoglou and Karabelas, 1994), showing that for coagulation problems the method by Hounslow et al. (1988) as applied here is the best choice. March et al. (1988) argue that in many cases the relatively simple "zero order" methods are better suited for simulations and comparisons with experimental data (of the type presented here) than methods of higher order, such as spline functions, collocation methods, etc.

FLOW WITH PRECIPITATION AND SCALE FORMATION

113

Particle Growth

For this phenomenon only, the population balance is of the form

aF(x, r)

aG(x)F(x, r) ax

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ar

(8)

Equation (8) is a first order P.D.E. of the hyperbolic type. As is well-known from the general literature (Lapidus and Pinder, 1982), as well as from the literature on .he problem treated here (Kim and Seinfeld, 1990), using finite differences in this case can lead to significant errors. Methods of polynomial representation of F(x, r] (Steemson and White, 1988),or the method of characteristics (Kim and Seinfeld, 1990)are suitable for solving Eq. (8), i.e. with no coagulation. However, the last two methods cannot be used efficiently in the full case treated here, where the grid (discretization) is determined by coagulation. The following version of the first order finite difference discretization of Eq. (8) is employed here

(9) where for i = I to h-l: 4 Ai=~ I

1-1

Ib'-

G(x)dx

bJ-l

and A o = A. = O. The main advantages of this method are the stability under all conditions and the conservation of total mass. The latter is a very important feature for this study. Numerical diffusion errors are considered a disadvantage of the method. These errors are reduced in higher order methods, such as second order finite differences and cubic spline polynomials. Even though these methods give very satisfactory results in some cases (when the distribution is smooth and the extent of size growth small), they are not recommended for general use since they exhibit dispersion errors. The latter grow with time, rendering eventually the numerical solution unstable. Furthermore, it is already known (Warren and Seinfeld, 1985) that higher order methods are unsuitable for problems of nucleation and fast size growth (as in precipitation). Figure 2 depicts results obtained for G(x) = 63 with first and second order finite difference methods. The accurate solution is ni = b(i -7). Diffusion errors of the numerical solution are evident. The first order finite difference method gives results closer to the true distribution, even though it is broader compared to the accurate one. Furthermore, it always provides correct positive concentrations whereas with second order finite differences it is possible to have oscillations around zero due to dispersion errors. The performance of the numerical method already selected is examined next, for the case of combined particle growth and coagulation. An initial distribution F(x,O) = exp( -x) is employed with a constant coagulation frequency B(x,y) = Band linear (with respect to particle volume) growth rate G(x) = Gx. The following analytical

114

M. KOSTOGLOU et al. 0.8 c:

C 0.6

--.. 0

~

c: 0.4 GI

U

c: 0

o

0.2

GI

U

.

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t:

a.

.,.0••. 'O~-().....o:_--'O.·o·

0.0 -0.2

a

15

10

5 Class

FIGURE 2 Comparison of numerical methods for solving the growth equation. Method (a): First order finite difference. Method (b): Second order finite difference.

solution is derived by integrating (for each class) the expression obtained via a Laplace transform (Ramabhadran et al., 1976): 2 M =-o

2+B

(10)

A comparison of distributions for G = B = 8 is made in Figure 3, implying a better agreement between numerical and analytical results than in the case of particle growth alone. This is attributed to the broadening of the initial monodisperse distribution caused by coagulation. To summarize, the accuracy of the selected numerical method appears to improve with the extent of coagulation. Discretization of the remaining terms in Eqs. (3), (4) and (5) is as follows: Eq. (3):

- D(2 i - 1 )n, + H(C)