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Assessment of the effect of velocity and residence time in CaSO4 precipitating flow reaction
Chemical Engineering Science 58 (2003) 3807 – 3816
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Assessment of the e"ect of velocity and residence time in CaSO4 precipitating (ow reaction P. Walker, R. Sheikholeslami∗ School of Chemical Engineering and Industrial Chemistry, The University of New South Wales, Sydney NSW 2052, Australia Received 11 June 2002; received in revised form 23 April 2003; accepted 21 May 2003
Abstract The e"ect of (ow hydrodynamics on kinetics of precipitation of sparingly soluble salts that cause fouling of process equipment has not been addressed previously. Precipitation of sparingly salts from solutions is usually considered to be due to crystallization on the surface. Precipitation in the bulk or in the boundary layer and its subsequent deposition has been traditionally neglected and to our knowledge, no previous research work has focused on this important area. Computational (uid dynamics was used as a tool to examine the e"ect of (ow on and the location within the (ow domain where calcium sulphate precipitation occurs. Initially, isothermal conditions were used to isolate the e"ect of (ow; non-isothermal conditions were examined for turbulent (ows usually encountered in heat exchange systems. Precipitation kinetics was modelled using a simple second-order reaction usually used in fouling of calcium sulphate. In laminar (ow, the characteristics of the velocity distribution led to the emergence of radial concentration gradients and as a result a radial di"usive (ux was induced. Similar behaviour was evident under turbulent conditions but not to the same extent. In turbulent (ow, particles were mainly produced in the viscous sub-layer rather than in the turbulent bulk. The reduced velocity within the laminar (ow or turbulent viscous sub-layer increased the residence time within these regions, enabling precipitation to take place and more particles to form. The generated particles could deposit by particulate fouling in addition to the fact that consumption of ionic species by precipitation in the boundary layer would induce ionic concentration gradient resulting in di"usion of ions to the walls and further crystallization fouling on the walls. Deposit layer formed by crystallization on the surface has a di"erent structure than that formed by particulate fouling. The e"ect of velocity and residence time on fouling, its mechanisms, and deposit structure should not be overlooked. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Fouling; Scaling; Crystallization; Precipitation; Kinetics; Particulate deposition
1. Introduction Fouling, the deposition of unwanted materials on process equipment, continues to be a major unresolved problem. The consequences of fouling include reduction in product purity, increased frequency of maintenance, increased energy consumption and reduction in heat or mass transfer (ux. To limit the consequences, one needs to advance the understanding of the fouling phenomenon. There has been extensive research into heat exchanger fouling and some research in membrane fouling; but the fouling problem is still largely unresolved. Most of the previous research has been conducted using experimental methods. The e"ect of (uid
∗ Corresponding author. Tel.: +61-2-9385-4343; fax: +61-2-9385-5966. E-mail address: [email protected] (R. Sheikholeslami).
0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0009-2509(03)00268-9
(ow on kinetics of precipitation and the exact location where the foulant forms, i.e., bulk, boundary layer or right at the surface, has never been systematically examined. Fouling has various mechanisms and stages, as depicted by Epstein’s 5 × 5 matrix (Epstein, 1983). The mechanisms describe the types of fouling. Two mechanisms of concern with sparingly soluble salts and as such related to this study are crystallization fouling and particulate fouling. Crystallization fouling involves the precipitation of soluble species directly onto the transfer surface. If any precipitant forms in the bulk or in the boundary layer, its deposition will be by means of the particulate fouling. Traditional research into fouling mainly assesses one mechanism of fouling in isolation. Such research considers that fouling, from an aqueous solution of a sparingly soluble salt, is only by the crystallization and neglects the formation of any precipitants within the bulk or the boundary layer and as such assumes no contribution by particulate fouling. However, it is possible for
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these mechanisms to coexist simultaneously in crystalline systems (Sheikholeslami, 2000). Examination of previous research into fouling prevails that for a given (uid, velocity and temperature are the two variables that have the greatest impact on fouling. Other variables that in(uence the severity of fouling in crystalline systems are the solution supersaturation, the physical conEguration of the unit and surface characteristics (Taberok, Ritter, & Palen, 1972). It is important to identify and quantify the complex e"ect that the interactions of these variables have on the fouling mechanisms. Temperature has a major in(uence on precipitation fouling since the solubility of normal salts increases with increasing temperature and for inverse soluble salts like calcium sulphate and many other sparingly soluble salts, solubility decreases with increasing temperature (Mukherjee, 1996). The attainment of supersaturation is a pre-requisite for precipitation to occur and once the supersaturated solution becomes unstable, the precipitant will form. Since the saturation concentration is dependant on temperature, the development of temperature gradients within a heat exchanger will induce the transportation of species within the solution into regions of di"ering degrees of supersaturation. This demonstrates that temperature has a dominant role on the precipitation process and as such there have been many investigations that address the temperature e"ects in heat exchangers. These studies are experimental and only look at the e"ect of bulk and surface temperature; the e"ect of temperature gradient on foulant formation cannot be examined experimentally. Velocity is usually considered to a"ect fouling in two di"erent ways. Firstly, higher velocities assist in the transportation of ions to the wall and crystallization at the wall. Secondly, removal rate is increased with increased velocities because of higher shear rate at the liquid–solid interface. There have also been some general remarks made regarding the in(uence that velocity may have on reactions and Bott (Bott, 1995) states that velocity a"ects chemical reaction mechanisms by in(uencing residence times but suggests that the impact is signiEcant if the reaction rate is relatively large. No observations regarding the e"ect of velocity on the bulk and boundary layer precipitation have ever been addressed. An issue of great interest in fouling in general is to locate the (ow region where the actual foulant forms. In case of sparingly soluble salts it is to locate the (ow region or regions where the precipitation occurs. This would be valuable in physical modelling of the fouling phenomena as well as modifying the operating or possibly design conditions to mitigate foulant formation. For example, the particles generated by precipitation in the bulk or the boundary layer can adhere to the surface by particulate fouling. In addition it can induce ionic concentration gradient and the di"usion of soluble species towards the solid surface which would promote crystallization at the surface. It should be noted that the structure, in terms of porosity and tenacity, of the deposits obtained from crystallization fouling can be very di"erent
from those in particulate fouling; therefore the behaviour of the precipitation process in the (ow domain and adjacent to the walls would impact fouling and its mechanisms significantly and as such would a"ect not only the fouling-cycle but as well the cleaning-cycle in operation of heat exchange and membrane units. Traditional experimental approach is to study fouling by examining and monitoring global temperature, (ux, and pressure changes within the system. Temperature values from the inlet and outlet streams, mass (uxes, or the pressure drops are used to calculate the formation of a fouled layer. In addition, an external window can be Etted to visually observe the deposition process (Bott, 1995). An external window is only suitable for some geometries and the window’s material may impact upon the deposition rate. To assess the relative magnitude of both crystallization and particulate fouling in-line Elters have been employed in experimental set-ups (Kho, 1998). This is a rather crude method as particles are present in solution either by generation within the system from solute or by detachment from the already deposited foulant layer. Also particles are Eltered out at the inlet or outlet of the unit, and therefore Eltration does not provide information on what takes place within the unit. 2. Scope In the introduction the previous studies and the impact of operating parameters on the fouling and some of the complexities in the crystalline systems were discussed. The focus of the current paper is to use computational (uid dynamics (CFD) as a tool to investigate the e"ect of (ow conditions on precipitation within the bulk and in the boundary layer. A tubular geometry which can be modiEed for future application to both tubular heat exchangers and tubular membranes have been used. The e"ect of geometry, tubular versus slit, on fouling mechanism has been addressed elsewhere (Walker & Sheikholeslami, 2002). An isothermal (ow, which enables to isolate and study the e"ect of velocity without interference from temperature, was used initially; it was later extended to non-isothermal (ow to assess the e"ect of temperature gradient. Information gained from the analysis is a fundamental study that will assist in assessing the possibility of particulate and crystallization fouling coexisting within the studied aqueous system. By using CFD details of the e"ect of hydrodynamics on kinetics of precipitation will be obtained that would be unattainable using traditional experimental investigations. CFD is a non-intrusive investigative technique, which can extract information that is diJcult to obtain experimentally and allows detailed examination of the (ow domain and enabling the intricate relationship between variables to be established. Hence, the advantage of CFD is that the in(uence that variables like temperature, velocity, and residence time have on fouling can be closely analysed. Also CFD
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
has the advantage of being able to isolate the phenomena so it can be understood individually before examining the process as a whole. The knowledge gained and future developments and reEnement of the approach can help to predict the location within the (ow domain where precipitation will take place and would assist in fundamental understanding of the process and in physical modelling of the fouling process and also in process modiEcation to mitigate fouling and limit the adverse consequences of fouling.
3809
ri ro
L
3. Approach Calcium sulphate is generated through the precipitation reaction from an ionic solution as shown by Eq. (1). Eq. (2) describes the second-order kinetic reaction that is usually used for calcium sulfate fouling (Liu & Nanacollas, 1970). kr SO2− 4(aq) CaSO4(s) ; kd
(1)
rppt = kr ([Ca2+ ][SO2− 4 ] − soln ksp ):
(2)
Ca2+ (aq)
+
The rate constant, kr , has an Arrhenius relationship (Liu & Nanacollas, 1970) and the solubility constant, ksp , is related to the saturation concentration and is a function of temperature (Raju & Atkinson, 1990). It should be noted that the e"ect of impurities and particle size distribution and population balance on precipitation could not be ignored. A recent study (Kostoglou & Karabelas, 1998) has addressed the e"ect of active species concentration and particle size distribution on precipitation in (uid (ow. However, the simpliEed kinetic model used here and which is usually used in physical modelling of fouling kinetics of sparingly soluble salts is suJcient to assess the e"ect of (ow velocity and domain which is the purpose of this paper. Some sparingly soluble salts such as calcium carbonate or calcium phosphate have complicated chemistry; as such calcium sulphate that is a relatively simple salt is used to allow one to focus on model development and the e"ects of operating parameters. The kinetic model will be reEned at later stages to incorporate other contributing factors. A commercial CFD package was used and sub-modules were developed and incorporated within it. To deEne the problem, the following discusses the (ow domain, the governing transport equations, the boundary conditions, and the veriEcation of the numerical solution. 3.1. The 1ow domain A simple 2D annular (ow domain is used in this investigation with the (uid entering and leaving perpendicular to the cross-section of the annulus. The simple geometry (Fig. 1) allows the physical phenomena to be observed. The e"ect of (ow geometry can be easily incorporated at later stages for various (ow channels and 3D conEgurations.
v in Fig. 1. The annular geometry in two dimensions.
3.2. Transport equations The transportation and interaction of species is modelled using the Eulerian approach. Each species from the expression in Eq. (1) has its own transport relationship and is described in terms of its mass fraction. The general transport equation for steady-state laminar (ow (which is usually encountered in membrane operations) is represented as ∇ · (soln v) = ∇ · ( ∇ ) + S :
(3)
The general transport property ( ) represents velocity, temperature and the mass fraction of each species involved in Eq. (1). Mass di"usivity of the species is used as the di"usion coeJcient ( ) in the equations representing transport of species. The solution density is calculated by summing the product of the density and the mass fraction of all the species plus the (uid. The source term for species transport is based on the generation of particulate matter or the consumption of ions in solution and is based on Eq. (2) and can be generalized as 2−
soln 2+
SO soln Ca 4 S i = MW i kr MW Ca2+ MW 2− SO
4
− ksp [soln (1 − ( CaSO4 + Ca2+ + SO2− ))]2 : 4
(4)
For turbulent (ow (which is more predominant for heat exchange units), Reynolds stresses are added to Eq. (3) to take into account the (uctuating nature of the turbulent (ow. The turbulent model used in this investigation is the low Reynolds number k–! turbulent model that is suitable for Reynolds numbers in range 5000 –30 000 and a more detailed discussion of the theory and operation of this model can be found in Wilcox (1998).
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816 0.0150
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0.0146
++
++
[Ca ] {mol/L}
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0.0144 0.0142
0.0148 0.0147 0.0146
Grid 1
Grid 2
Grid 3
Grid 4
0.6
0.8
0.0140
Re = 250 Re = 1500
0.0145 0.0
0.2
0.4
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 2. Grid analysis of the outlet radial calcium concentration Distribution, Re = 1000; T = 298 K.
3.3. Boundary conditions Uniform temperature, velocity and particulate free, equi-molar ionic concentrations were considered at the inlet. At the walls no slip velocity conditions apply. For non-isothermal conditions, the heat (ux is zero at the inner and outer walls and a constant heat (ux at the wall is applied for non-isothermal conditions; mass (ux of species at both inner and outer wall is assumed zero. The boundary conditions are varied to assess the relative in(uence of velocity under various operating conditions; both laminar and turbulent regions are examined. 3.4. Veri7cation The Erst step of the veriEcation is to perform grid analysis to ensure that an optimum grid is generated to adequately describe the process and to eJciently solve the problem. For this purpose, four structured, two-dimensional grids were developed with each consecutive grid having double the number of elements in the radial direction. The hybrid convective scheme was used to ensure stability. The radial concentration proEle of the soluble species and the generated calcium sulphate particles at the channel outlet were compared for grid analysis and a typical example is shown in Fig. 2. The major di"erence in each grid occurs at the walls. Grid 2 seems to be most suitable as it forms a similar shape to the Ener grids, maintaining this shape to the wall with relatively minimal change. In terms of the analysis, it is important to have a detailed description of this area because most of the activity appears to occur in this area. Hence, grid 2 with the dimensions of 20 × 670 is a sensible choice for the optimal grid. The second step of the veriEcation is to ensure that the key physical concepts of the process are correctly represented in the CFD model. Two methods were used to conErm this. Firstly, the precipitation rate constants obtained from CFD solutions were compared to those entered in the code of the source terms, which were obtained from literature (Raju & Atkinson, 1990). Secondly,
0.0
0.2
Re = 1000 Re = 2000 0.4
Re = 500
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 3. Radial calcium concentration at a given residence time for a fully developed (ow, T = 298 K.
the concentrations of soluble species were monitored as the degree of supersaturation declined due to the consumption of ions to ensure that it approached the solubility limit and then was maintained at that. Results from both methods veriEed that the model adequately reproduced the physical process. The third step of veriEcation was to construct the ionic concentration proEles for a fully developed laminar (ow at a given residence time but varying Reynolds numbers (Fig. 3) which demonstrates, as expected, that there is no di"erence between these proEles. A given mean residence time was chosen such that the (ow at the corresponding position along the geometry was fully developed for each Reynolds number.
4. Results and discussion As previously noted, this study aims to examine in detail the e"ect of velocity and residence time distribution in precipitating crystalline systems under both isothermal and non-isothermal conditions. The investigation demonstrates that velocity has a decisive in(uence on the behaviour of the precipitation process, especially through its relationship with (uid residence time that causes concentration gradients to emerge in the bulk and/or boundary layer depending on the (ow region. These concentration gradients are due to formation of particulate matter which can contribute to fouling by particulate deposition and in addition have a similar e"ect to those induced by temperature gradients in non-isothermal systems and in(uence di"usion of species towards the walls. Following discusses the results in laminar and fully developed turbulent (ow. For laminar (ow, the velocity e"ects have been examined at varying solution supersaturations and system temperatures to see whether or not the velocity e"ects would be exacerbated by the degree of supersaturations and system temperatures. For turbulent (ow that is mostly encountered in heat exchange systems, the e"ect of temperature gradient is assessed.
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
3811
0.016 0.014
++
[Ca ] {mol/L}
0.012 0.010
Ca, z = 0.5 Ca, z = 1.5
0.008
Part, z = 0.5 Part, z = 1.5
0.006 0.004 0.002 0.000 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 5. Radial concentration proEle at outlet for Re = 1000; T = 298 K.
0.08 0.07
Velocity {m/s}
0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 6. Developed velocity distribution for Re = 1000; T = 298 K.
Fig. 4. Calcium ion distribution, Re = 1000.
4.1. Precipitation in laminar 1ow 4.1.1. E8ect of velocity and residence time in laminar 1ow Fig. 4 shows the distribution of calcium ion mass fraction for a Reynolds number of 1000 at 25◦ C. It is observed that the inlet calcium ion mass fraction is uniform in the radial direction and as progress is made along the annulus the mass fraction of calcium ion in the bulk decreases. In some regions of the (ow, the mass fraction of calcium ions is 5.5% less than that at the inlet. For the sulphate ion the distribution and consumption is similar. In the corresponding distribution for particles the mass fraction increases along the annulus because the precipitation process is producing calcium sulphate particles. A distinct observation that can be made in Fig. 4 is that as progress is made along the annulus the mass fraction proEle in the radial direction does not remain constant. This suggests that the rate of ion consumption is varying not only in the axial direction but as well in the radial direction. The observed radial variation in precipitation rate and hence mass fraction is largest at the outlet. Fig. 5 shows the variation in the concentration proEles of species for calcium ion and particulate matter at two consecutive axial positions. At both of these positions the (ow
is fully developed and as such the di"erence is only due to the e"ect of residence time. Examination of Fig. 5 suggests that the precipitation is greater adjacent to the walls and more particles are produced closer to either wall than around the middle position. Since the process is isothermal and the solution supersaturation and hence concentration of species is uniform at the inlet, the variation in precipitation rate can only be due to the velocity gradient. Fig. 6 depicts a plot of the fully developed radial velocity proEle, which corresponds to both axial positions in Fig. 5, showing that the lower velocity regions correspond to the regions where more particles have been generated because at lower velocities the precipitation reaction has more time to proceed before the (uid exits the annulus. Fig. 5 also demonstrates that at two points in a fully developed (ow the overall concentration of particles and the concentration gradients for each species are larger downstream. The emergence of concentration gradients show the decisive in(uence that residence time has in production of particles and their possible contribution to fouling by mode of particulate deposition and also the ionic concentration gradient which would induce di"usion of ions to the wall and contribution to fouling by mode of crystallization. By changing the inlet velocity, whilst remaining within the laminar regime, the in(uence of velocity can be further examined (Fig. 7) indicating larger concentration gradients for lower Reynolds numbers. This of course is due to the residence time di"erence; as seen from Fig. 3 at a given
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816 0.0150
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0.020
[Ca ] {mol/L}
0.0140
++
++
[Ca ] {mol/L}
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0.0135 0.0130
0.015 0.010 0.005
Re =250
Re = 500
Re = 1000
Re = 1500
S.S. = 1.5
0.0125
S.S. = 3
S.S. = 4.5
0.000 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Position ([r-r i ]/[r o -r i ])
Fig. 7. At outlet radial calcium concentration distribution with varying Reynolds numbers. T = 298 K.
Fig. 9. The radial concentration proEle at the outlet for three di"erent inlet concentrations with equal molar feed of reactants: Re = 1000: T = 298 K.
0.01504
0.016
[Ca ] {mol/L}
0.01500
0.014 0.013
++
++
[Ca ] {mol/L}
0.015 0.01502
0.01498 Re = 250 Re = 1500
0.01496 0.0
0.2
Re = 500 Re = 2000 0.4
0.012 0.011
Re = 1000
T = 298 K 0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
T = 308 K
T = 318 K
0.010 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 8. The comparison of the radial calcium ion distribution for di"erent Reynolds number with the same residence time of 2 s in developing (ow. T = 298 K.
Fig. 10. E"ect of temperature on outlet calcium concentration for an inlet supersaturation of 3, Re = 1000.
residence time for a fully developed (ow, the concentration proEles are identical and the convective transport is zero. However, in the developing region, the convective (ow of momentum plays some role on transport of species. To more closely analyse the di"ering concentration proEles for different values of Reynolds numbers shown in Fig. 7 and to assess the relative e"ect of convective (ow of momentum in the developing region as opposed to residence time, the concentration proEles are plotted (Fig. 8) for a developing (ow for various Reynolds numbers at a given residence time. Fig. 8 demonstrates that the (ux is mainly di"usive and the degree of convection only a"ects the proEle next to the wall and there is no di"erence between these proEles within the bulk. As mentioned above, these observed characteristics have implications for both crystallization fouling and particulate fouling mechanisms. The generation of particles and the induced gradient in the ionic concentration would promote transfer of ions towards the surface and could promote the crystallization fouling in an actual heat or mass transfer unit. Particles produced could also adhere to the surface and promote particulate fouling. The location along the channel where the generation starts to take place would depend on the residence time and its interdependence on the velocity.
4.1.2. E8ect of velocity at varying inlet supersaturations The rate of reaction is a function of the degree of solution supersaturation. Fig. 9 was constructed to assess the relative in(uence of velocity at various inlet supersaturations. Whilst the radial trend described in the previous section exists in each proEle, it is obvious that the velocity e"ects become more signiEcant and pronounced at higher degrees of supersaturation. 4.1.3. E8ect of velocity at various system temperatures The e"ect of temperature on precipitation is well established. Comparative simulations have been carried out (Fig. 10) to show the relative e"ect of velocity on the generated concentration gradients at a given solution supersaturation but varying system temperatures (25 –45◦ C). The concentration gradient is much steeper at higher temperatures and further highlight the impact that velocity has on distribution of species at higher temperatures. Comparison of Figs. 9 and 10 indicates that temperature and its in(uence on the reaction rate constant has a more dominating impact on precipitation and the generated concentration proEle than the degree of solution supersaturation. In heat exchangers, the accumulated impact of velocity and the existing temperature gradients further
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
3813
0.015016
++
[Ca ] {mol/L}
0.015014 0.015012 0.015010 0.015008 Re = 10,000
Re = 20,000
0.015006 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 12. The comparison of the radial calcium ion distribution for the Reynolds number of 10 000 and 20 000 for a fully developed (ows at a given residence time. T = 298 K.
Fig. 11. Calcium ion distribution for isothermal turbulent (ow.
induces concentration gradients and would contribute to the magnitude of soluble species (ux towards the surface and also possibly the particulate fouling by generated particles; this will be assessed in turbulent (ows usually encountered in heat exchange systems. 4.2. Precipitation in fully turbulent 1ow (Re ¿ 10 000) 4.2.1. Isothermal fully turbulent 1ow The previous sections revealed that residence time has a signiEcant in(uence on precipitation in laminar (ow. Residence time also impacts the behaviour of precipitation in turbulent (ow. In speciEc, in the viscous sub-layer region of the turbulent (ow the behaviour of species is expected to be similar to that identiEed in laminar (ow because of the similar (ow characteristics. These characteristics were assessed and observed by carrying out CFD simulations for the low Reynolds number k–! turbulence model, which uses non-uniform grids to suJciently resolve the (ow behaviour adjacent to the walls. The bulk residence time is so short that the amount of particles produced in the turbulent bulk is considered to have a negligible e"ect on fouling. Fig. 11 demonstrates the concentration proEles for calcium ions along the annulus for
a Reynolds number of 10 000. In some regions of this turbulent (ow only a maximum of 0.15% of the calcium ions have been consumed which is much smaller than the corresponding 5.5% experienced in the laminar (ow of Fig. 4 indicating that the amount of particles produced in the bulk is insigniEcant. Within the viscous sub-layer the concentration gradients are similar to those that were developed in laminar (ow, which induce a radial, di"usive (ux of the species. The increased residence time adjacent to the walls allows more particles to form. Fig. 11 shows that the minimum calcium ion mass fraction appears adjacent to either walls of the annulus corresponding to a maximum amount of particles being produced near the walls and is explicable by the characteristics of the fully developed turbulent (ow having a fairly constant velocity in the bulk and laminar characteristics in the viscous sub-layer adjacent to the walls where viscous forces are more dominant than Reynolds stresses. Fig. 12 shows that though for both velocities the (ow is fully developed and the average residence times are the same, the di"erent degrees of turbulence does indeed a"ect the particle generation and the concentration proEles. 4.2.2. The e8ect of temperature gradients in fully turbulent 1ow Heat exchangers operate under various heat transfer conditions. Important to all is the impact that the resulting temperature gradient has on fouling. This section will examine the e"ect that temperature gradients have on precipitation in turbulent (ow, the (ow regime at which heat exchangers operate to minimize fouling. The temperature gradients are simulated by imposing a constant heat (ux as a boundary condition on the inner wall of the annulus. For a given run, the constant heat (ux corresponds to a constant temperature gradient adjacent the inner wall because the values of viscosity, heat capacity and thermal conductivity are assumed to be constant. Fig. 13 shows the drastic e"ect that a temperature gradient has on precipitation. As was noticed in Fig. 10, the varying temperature has a signiEcant e"ect on the rate of
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[Ca ] {mol/L}
0.01495 0.01490
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[Ca ] {mol/L}
3814
0.01485 q" = 14 kW/m2
0.01480
0.01495 0.01490 0.01485 Re = 10,000
0.01480
Re = 20,000
q" = 7 kW/m2 0.01475
0.01475 0.0
0.2
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0.2
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Position ([r-r i ]/[r o -r i ])
Position ([r-r i ]/[r o -r i ])
Fig. 13. Comparison of the radial ion distribution with di"erent heat (uxes at the same position for a Reynolds number of 20 000.
Fig. 14. Comparison of the radial ion distribution for the Reynolds number of 10 000 and 20 000 at the same residence time and the same heat (ux at ri Tbulk = 300 K.
precipitation and, hence, the amount precipitated. The position where most ionic species have been consumed corresponds to the position of highest temperature, at the inner surface. The temperature decreases toward the bulk and this corresponds to the decrease in the amount of ionic species consumed in Fig. 13. The resulting temperature gradient leads to the establishment of a concentration gradient adjacent to the heat transfer surface. Furthermore, a comparison of Figs. 12 and 13 suggest that temperature gradients have a more signiEcant impact on developing concentration gradients than the distribution in residence time; however, the e"ect of residence time distribution is not negligible though it is often neglected. Researchers have drawn general conclusions that higher surface temperatures, higher heat (uxes, lead to a greater amount of scale forming on the surface. The phenomenon that causes one to make this conclusion is clearly demonstrated and quantiEed in Fig. 13. It is observed that a higher heat (ux causes a more signiEcant concentration gradient to develop adjacent the heat transfer surface. As previously stated, these concentration gradients lead to a di"usive (ux of ionic species. In terms of fouling, these concentration gradients would have an in(uence on the transport mechanism. Hence, at a higher heat (ux there would be a superior di"usive (ux, which suggests that a greater amount of scale would form. Various researchers have compared results of di"erent (ows to speculate on whether transport or attachment is the rate-controlling step in the observed scale deposition. Similar conclusions cannot be drawn from these results because attachment is not examined. However, this study will provide an insight into how di"erent (ows e"ect the transport mechanism. Fig. 14 are the results for two Reynolds numbers for fully developed (ows at a given average residence time, heat (ux and bulk temperature. A larger concentration gradient has emerged for the lower Reynolds number. This implies that a greater amount ionic di"usive transport occurs for lower Reynolds numbers due to residence time distribution. The detail of the CFD results allows a better appreciation of the transition between the two possible controlling steps
of deposition and also di"erent mechanisms of fouling. It was previously observed that surface temperature relates to the temperature gradient and its impact on the concentration gradient. For these two runs (Fig. 14) that were at a given bulk temperature, given heat (ux, and a given average residence time, the lower (ow rate has a larger surface temperature. This is attributed to the lower Reynolds having a thicker boundary layer, a larger resistance to heat transfer. It is demonstrated by comparing the results in Fig. 14 that as the boundary layer is decreased so too will di"erence between bulk and surface temperature. This has the effect of reducing the size of the concentration gradient and, consequentially, the magnitude of di"use (ux. Therefore, it quantiEes how the controlling step of the deposition process alternates from di"usion controlled to surface reaction controlled as (ow increases. 5. Conclusion The CFD was used as a tool to highlight the vital role of velocity and residence time in detailing calcium sulphate precipitation in a (ow domain under both isothermal and non-isothermal conditions. The in(uence of velocity proEle on residence time distribution causes concentration gradients to emerge in both laminar and turbulent (ows. In addition to possible deposition of generated particles in the (ow domain on the transfer surface by particle deposition, the ionic concentration gradients induce a radial (ux of soluble species towards the surface and would further contribute to fouling by crystallization. The induced (ux is greatest in laminar (ow where velocities are lower, residence times are signiEcantly higher and considerable concentration gradients exists in all regions of the (ow. In turbulent (ow, similar concentration gradients appear only within the turbulent viscous sub-layer where laminar characteristics exist. In fouling by sparingly soluble salts, the mechanism of fouling is usually attributed to (ux of ionic species (caused by a temperature gradient in heat exchangers or concentration polarization in membranes) to the wall and precipitation at the wall. The Endings of this investigation infer that
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
velocity and residence time distribution also in(uence the behaviour of the precipitation process. Therefore, the combined e"ect of velocity, temperature or concentration polarization and their induced concentration gradients causes species to di"use towards the transfer surface and promotes deposit formation by both crystallization at the surface and particulate deposition by precipitate formed in the bulk and boundary layer. The e"ect of residence time distribution has been traditionally ignored. Kinetic e"ects are totally ignored in for example membrane units that have very short average residence times on the basis that there is no suJcient time for crystallization to take place. Fouling experiments base the optimum velocity on the balance between its e"ects on the removal rate with its e"ect on the convective mass transfer rate from the bulk. However, the velocity proEle and residence time distribution, in both laminar and turbulent (ow, causes concentration gradients to emerge in the (ow domain; these gradients become steeper at higher temperatures and higher supersaturations. Higher temperatures are encountered in heat exchangers and higher concentrations are encountered at Enal stages in the membrane units. Therefore, these additional e"ects of velocity on fouling, fouling mechanism and structure of the deposit should not be overlooked. Even though a simpliEed precipitation model was used, CFD has proven to be a useful tool in this investigation. Its ability to e"ectively simulate precipitation process has assisted in gaining improved understanding of velocity, temperature, and concentration gradients a"ecting fouling mechanism. At this stage the results are only to elucidate the fouling mechanism. However, the intention is to use the presented model as a basis for a comprehensive model of the fouling process in both heat exchange and membrane units. It is anticipated that this intended continuation in the development of the CFD model used in this investigation will result in better understanding of the fouling by sparingly soluble salts and of the fouling caused by solute polarization within membrane units. To achieve the planned result, a dynamic predictive model, would involve the inclusion of statistical methods like population distributions to improve the depth of the kinetic model, and of permeate (ux and concentration polarization in case of membrane units.
3815
solubility constant, mol=kg H2 O2 length of annulus, m molecular weight of species i , kg/mol r radial co-ordinate, m R universal gas constant, J/mol K Re Reynolds number ri inner radius of annulus, m ro outer radius of annulus, m precipitation reaction rate, kg=m3 s rppt ∗ r ri =ro S degree of solution supersaturation S source term for the equation, generation rate/volume ] sulphate ion concentration, mol=m3 [SO2− 4 solution T temperature, K v velocity vector, m/s
ksp L MW i
Greek letters
i k soln !
mass fraction of species i di"usion coeJcient of property rate of turbulent energy dissipation, m2 =s3 turbulent kinetic energy, m2 =s2 viscosity, Pa s solution density, kg=m3 general transport property, T; v; =k, s−1
Subscripts aq Ca2+ CaSO4 i in s SO2− 4
aqueous calcium ions calcium sulphate particle component i inlet solid sulphate ions
References Notation A1 ; A2 ; A3 ; A4 ; A5 [Ca2+ ] E kd kr
constants calcium ion concentration, mol=m3 solution activation energy for precipitation, J/mol dissolution rate constant, m3 solution/mol precipitate s precipitation reaction rate constant, m3 solution/mol precipitate s
Bott, T. R. (1995). Fouling of heat exchangers. Amsterdam: Elsevier. Epstein, N. (1983). Thinking about heat transfer fouling: A 5 × 5 matrix. Heat Transfer Engineering, 4(1), 43–56. Kho, T. C. S. (1998). E"ect of (ow distribution on scale formation in plate and frame heat exchangers. Department of Chemical and process engineering. University of Surrey, Guildford, Surrey UK. (p. 292). Kostoglou, M., & Karabelas, A. J. (1998). Comprehensive modeling of precipitation and fouling in turbulent pipe (ow. Industrial Engineering and Chemistry Research, 37, 1536–1550. Liu, S., & Nanacollas, G. H. (1970). The kinetics growth of crystal calcium sulphate dihydrate. Journal of Crystal Growth, 6, 281–289. Mukherjee, R. (1996). Conquer heat exchanger fouling. Hydrocarbon Processing, January, 121–127.
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Raju, K. U. G., & Atkinson, G. (1990). The thermodynamics of scale mineral solubilities 3. Calcium sulfate in aqueous NaCl. Journal of Chemical Engineering Data, 35, 361–367. Sheikholeslami, R. (2000). Calcium sulfate fouling—precipitation or particulate: A proposed composite model. Special Issue of Heat Transfer Engineering Journal, 21(2), 24–33. Taberok, J., Ritter, R. B., & Palen, J. W. (1972). Heat transfer; fouling: The major unresolved problem in heat transfer. Chemical Engineering Progress, 68(2), 59–67.
Walker, P., & Sheikholeslami, R. (2002). Preliminary numerical study of CaSO4 precipitation in laminar (ows in pipes and slits under isothermal conditions. The 9th APCChE congress and CHEMECA (pp. 612– 621). Wilcox, D. C. (1998). Turbulence modeling for CFD. DCW Industries Inc. La Canada, CA.
www.elsevier.com/locate/ces
Assessment of the e"ect of velocity and residence time in CaSO4 precipitating (ow reaction P. Walker, R. Sheikholeslami∗ School of Chemical Engineering and Industrial Chemistry, The University of New South Wales, Sydney NSW 2052, Australia Received 11 June 2002; received in revised form 23 April 2003; accepted 21 May 2003
Abstract The e"ect of (ow hydrodynamics on kinetics of precipitation of sparingly soluble salts that cause fouling of process equipment has not been addressed previously. Precipitation of sparingly salts from solutions is usually considered to be due to crystallization on the surface. Precipitation in the bulk or in the boundary layer and its subsequent deposition has been traditionally neglected and to our knowledge, no previous research work has focused on this important area. Computational (uid dynamics was used as a tool to examine the e"ect of (ow on and the location within the (ow domain where calcium sulphate precipitation occurs. Initially, isothermal conditions were used to isolate the e"ect of (ow; non-isothermal conditions were examined for turbulent (ows usually encountered in heat exchange systems. Precipitation kinetics was modelled using a simple second-order reaction usually used in fouling of calcium sulphate. In laminar (ow, the characteristics of the velocity distribution led to the emergence of radial concentration gradients and as a result a radial di"usive (ux was induced. Similar behaviour was evident under turbulent conditions but not to the same extent. In turbulent (ow, particles were mainly produced in the viscous sub-layer rather than in the turbulent bulk. The reduced velocity within the laminar (ow or turbulent viscous sub-layer increased the residence time within these regions, enabling precipitation to take place and more particles to form. The generated particles could deposit by particulate fouling in addition to the fact that consumption of ionic species by precipitation in the boundary layer would induce ionic concentration gradient resulting in di"usion of ions to the walls and further crystallization fouling on the walls. Deposit layer formed by crystallization on the surface has a di"erent structure than that formed by particulate fouling. The e"ect of velocity and residence time on fouling, its mechanisms, and deposit structure should not be overlooked. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Fouling; Scaling; Crystallization; Precipitation; Kinetics; Particulate deposition
1. Introduction Fouling, the deposition of unwanted materials on process equipment, continues to be a major unresolved problem. The consequences of fouling include reduction in product purity, increased frequency of maintenance, increased energy consumption and reduction in heat or mass transfer (ux. To limit the consequences, one needs to advance the understanding of the fouling phenomenon. There has been extensive research into heat exchanger fouling and some research in membrane fouling; but the fouling problem is still largely unresolved. Most of the previous research has been conducted using experimental methods. The e"ect of (uid
∗ Corresponding author. Tel.: +61-2-9385-4343; fax: +61-2-9385-5966. E-mail address: [email protected] (R. Sheikholeslami).
0009-2509/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0009-2509(03)00268-9
(ow on kinetics of precipitation and the exact location where the foulant forms, i.e., bulk, boundary layer or right at the surface, has never been systematically examined. Fouling has various mechanisms and stages, as depicted by Epstein’s 5 × 5 matrix (Epstein, 1983). The mechanisms describe the types of fouling. Two mechanisms of concern with sparingly soluble salts and as such related to this study are crystallization fouling and particulate fouling. Crystallization fouling involves the precipitation of soluble species directly onto the transfer surface. If any precipitant forms in the bulk or in the boundary layer, its deposition will be by means of the particulate fouling. Traditional research into fouling mainly assesses one mechanism of fouling in isolation. Such research considers that fouling, from an aqueous solution of a sparingly soluble salt, is only by the crystallization and neglects the formation of any precipitants within the bulk or the boundary layer and as such assumes no contribution by particulate fouling. However, it is possible for
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P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
these mechanisms to coexist simultaneously in crystalline systems (Sheikholeslami, 2000). Examination of previous research into fouling prevails that for a given (uid, velocity and temperature are the two variables that have the greatest impact on fouling. Other variables that in(uence the severity of fouling in crystalline systems are the solution supersaturation, the physical conEguration of the unit and surface characteristics (Taberok, Ritter, & Palen, 1972). It is important to identify and quantify the complex e"ect that the interactions of these variables have on the fouling mechanisms. Temperature has a major in(uence on precipitation fouling since the solubility of normal salts increases with increasing temperature and for inverse soluble salts like calcium sulphate and many other sparingly soluble salts, solubility decreases with increasing temperature (Mukherjee, 1996). The attainment of supersaturation is a pre-requisite for precipitation to occur and once the supersaturated solution becomes unstable, the precipitant will form. Since the saturation concentration is dependant on temperature, the development of temperature gradients within a heat exchanger will induce the transportation of species within the solution into regions of di"ering degrees of supersaturation. This demonstrates that temperature has a dominant role on the precipitation process and as such there have been many investigations that address the temperature e"ects in heat exchangers. These studies are experimental and only look at the e"ect of bulk and surface temperature; the e"ect of temperature gradient on foulant formation cannot be examined experimentally. Velocity is usually considered to a"ect fouling in two di"erent ways. Firstly, higher velocities assist in the transportation of ions to the wall and crystallization at the wall. Secondly, removal rate is increased with increased velocities because of higher shear rate at the liquid–solid interface. There have also been some general remarks made regarding the in(uence that velocity may have on reactions and Bott (Bott, 1995) states that velocity a"ects chemical reaction mechanisms by in(uencing residence times but suggests that the impact is signiEcant if the reaction rate is relatively large. No observations regarding the e"ect of velocity on the bulk and boundary layer precipitation have ever been addressed. An issue of great interest in fouling in general is to locate the (ow region where the actual foulant forms. In case of sparingly soluble salts it is to locate the (ow region or regions where the precipitation occurs. This would be valuable in physical modelling of the fouling phenomena as well as modifying the operating or possibly design conditions to mitigate foulant formation. For example, the particles generated by precipitation in the bulk or the boundary layer can adhere to the surface by particulate fouling. In addition it can induce ionic concentration gradient and the di"usion of soluble species towards the solid surface which would promote crystallization at the surface. It should be noted that the structure, in terms of porosity and tenacity, of the deposits obtained from crystallization fouling can be very di"erent
from those in particulate fouling; therefore the behaviour of the precipitation process in the (ow domain and adjacent to the walls would impact fouling and its mechanisms significantly and as such would a"ect not only the fouling-cycle but as well the cleaning-cycle in operation of heat exchange and membrane units. Traditional experimental approach is to study fouling by examining and monitoring global temperature, (ux, and pressure changes within the system. Temperature values from the inlet and outlet streams, mass (uxes, or the pressure drops are used to calculate the formation of a fouled layer. In addition, an external window can be Etted to visually observe the deposition process (Bott, 1995). An external window is only suitable for some geometries and the window’s material may impact upon the deposition rate. To assess the relative magnitude of both crystallization and particulate fouling in-line Elters have been employed in experimental set-ups (Kho, 1998). This is a rather crude method as particles are present in solution either by generation within the system from solute or by detachment from the already deposited foulant layer. Also particles are Eltered out at the inlet or outlet of the unit, and therefore Eltration does not provide information on what takes place within the unit. 2. Scope In the introduction the previous studies and the impact of operating parameters on the fouling and some of the complexities in the crystalline systems were discussed. The focus of the current paper is to use computational (uid dynamics (CFD) as a tool to investigate the e"ect of (ow conditions on precipitation within the bulk and in the boundary layer. A tubular geometry which can be modiEed for future application to both tubular heat exchangers and tubular membranes have been used. The e"ect of geometry, tubular versus slit, on fouling mechanism has been addressed elsewhere (Walker & Sheikholeslami, 2002). An isothermal (ow, which enables to isolate and study the e"ect of velocity without interference from temperature, was used initially; it was later extended to non-isothermal (ow to assess the e"ect of temperature gradient. Information gained from the analysis is a fundamental study that will assist in assessing the possibility of particulate and crystallization fouling coexisting within the studied aqueous system. By using CFD details of the e"ect of hydrodynamics on kinetics of precipitation will be obtained that would be unattainable using traditional experimental investigations. CFD is a non-intrusive investigative technique, which can extract information that is diJcult to obtain experimentally and allows detailed examination of the (ow domain and enabling the intricate relationship between variables to be established. Hence, the advantage of CFD is that the in(uence that variables like temperature, velocity, and residence time have on fouling can be closely analysed. Also CFD
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
has the advantage of being able to isolate the phenomena so it can be understood individually before examining the process as a whole. The knowledge gained and future developments and reEnement of the approach can help to predict the location within the (ow domain where precipitation will take place and would assist in fundamental understanding of the process and in physical modelling of the fouling process and also in process modiEcation to mitigate fouling and limit the adverse consequences of fouling.
3809
ri ro
L
3. Approach Calcium sulphate is generated through the precipitation reaction from an ionic solution as shown by Eq. (1). Eq. (2) describes the second-order kinetic reaction that is usually used for calcium sulfate fouling (Liu & Nanacollas, 1970). kr SO2− 4(aq) CaSO4(s) ; kd
(1)
rppt = kr ([Ca2+ ][SO2− 4 ] − soln ksp ):
(2)
Ca2+ (aq)
+
The rate constant, kr , has an Arrhenius relationship (Liu & Nanacollas, 1970) and the solubility constant, ksp , is related to the saturation concentration and is a function of temperature (Raju & Atkinson, 1990). It should be noted that the e"ect of impurities and particle size distribution and population balance on precipitation could not be ignored. A recent study (Kostoglou & Karabelas, 1998) has addressed the e"ect of active species concentration and particle size distribution on precipitation in (uid (ow. However, the simpliEed kinetic model used here and which is usually used in physical modelling of fouling kinetics of sparingly soluble salts is suJcient to assess the e"ect of (ow velocity and domain which is the purpose of this paper. Some sparingly soluble salts such as calcium carbonate or calcium phosphate have complicated chemistry; as such calcium sulphate that is a relatively simple salt is used to allow one to focus on model development and the e"ects of operating parameters. The kinetic model will be reEned at later stages to incorporate other contributing factors. A commercial CFD package was used and sub-modules were developed and incorporated within it. To deEne the problem, the following discusses the (ow domain, the governing transport equations, the boundary conditions, and the veriEcation of the numerical solution. 3.1. The 1ow domain A simple 2D annular (ow domain is used in this investigation with the (uid entering and leaving perpendicular to the cross-section of the annulus. The simple geometry (Fig. 1) allows the physical phenomena to be observed. The e"ect of (ow geometry can be easily incorporated at later stages for various (ow channels and 3D conEgurations.
v in Fig. 1. The annular geometry in two dimensions.
3.2. Transport equations The transportation and interaction of species is modelled using the Eulerian approach. Each species from the expression in Eq. (1) has its own transport relationship and is described in terms of its mass fraction. The general transport equation for steady-state laminar (ow (which is usually encountered in membrane operations) is represented as ∇ · (soln v) = ∇ · ( ∇ ) + S :
(3)
The general transport property ( ) represents velocity, temperature and the mass fraction of each species involved in Eq. (1). Mass di"usivity of the species is used as the di"usion coeJcient ( ) in the equations representing transport of species. The solution density is calculated by summing the product of the density and the mass fraction of all the species plus the (uid. The source term for species transport is based on the generation of particulate matter or the consumption of ions in solution and is based on Eq. (2) and can be generalized as 2−
soln 2+
SO soln Ca 4 S i = MW i kr MW Ca2+ MW 2− SO
4
− ksp [soln (1 − ( CaSO4 + Ca2+ + SO2− ))]2 : 4
(4)
For turbulent (ow (which is more predominant for heat exchange units), Reynolds stresses are added to Eq. (3) to take into account the (uctuating nature of the turbulent (ow. The turbulent model used in this investigation is the low Reynolds number k–! turbulent model that is suitable for Reynolds numbers in range 5000 –30 000 and a more detailed discussion of the theory and operation of this model can be found in Wilcox (1998).
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816 0.0150
0.0150
0.0148
0.0149
[Ca ] {mol/L}
0.0146
++
++
[Ca ] {mol/L}
3810
0.0144 0.0142
0.0148 0.0147 0.0146
Grid 1
Grid 2
Grid 3
Grid 4
0.6
0.8
0.0140
Re = 250 Re = 1500
0.0145 0.0
0.2
0.4
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 2. Grid analysis of the outlet radial calcium concentration Distribution, Re = 1000; T = 298 K.
3.3. Boundary conditions Uniform temperature, velocity and particulate free, equi-molar ionic concentrations were considered at the inlet. At the walls no slip velocity conditions apply. For non-isothermal conditions, the heat (ux is zero at the inner and outer walls and a constant heat (ux at the wall is applied for non-isothermal conditions; mass (ux of species at both inner and outer wall is assumed zero. The boundary conditions are varied to assess the relative in(uence of velocity under various operating conditions; both laminar and turbulent regions are examined. 3.4. Veri7cation The Erst step of the veriEcation is to perform grid analysis to ensure that an optimum grid is generated to adequately describe the process and to eJciently solve the problem. For this purpose, four structured, two-dimensional grids were developed with each consecutive grid having double the number of elements in the radial direction. The hybrid convective scheme was used to ensure stability. The radial concentration proEle of the soluble species and the generated calcium sulphate particles at the channel outlet were compared for grid analysis and a typical example is shown in Fig. 2. The major di"erence in each grid occurs at the walls. Grid 2 seems to be most suitable as it forms a similar shape to the Ener grids, maintaining this shape to the wall with relatively minimal change. In terms of the analysis, it is important to have a detailed description of this area because most of the activity appears to occur in this area. Hence, grid 2 with the dimensions of 20 × 670 is a sensible choice for the optimal grid. The second step of the veriEcation is to ensure that the key physical concepts of the process are correctly represented in the CFD model. Two methods were used to conErm this. Firstly, the precipitation rate constants obtained from CFD solutions were compared to those entered in the code of the source terms, which were obtained from literature (Raju & Atkinson, 1990). Secondly,
0.0
0.2
Re = 1000 Re = 2000 0.4
Re = 500
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 3. Radial calcium concentration at a given residence time for a fully developed (ow, T = 298 K.
the concentrations of soluble species were monitored as the degree of supersaturation declined due to the consumption of ions to ensure that it approached the solubility limit and then was maintained at that. Results from both methods veriEed that the model adequately reproduced the physical process. The third step of veriEcation was to construct the ionic concentration proEles for a fully developed laminar (ow at a given residence time but varying Reynolds numbers (Fig. 3) which demonstrates, as expected, that there is no di"erence between these proEles. A given mean residence time was chosen such that the (ow at the corresponding position along the geometry was fully developed for each Reynolds number.
4. Results and discussion As previously noted, this study aims to examine in detail the e"ect of velocity and residence time distribution in precipitating crystalline systems under both isothermal and non-isothermal conditions. The investigation demonstrates that velocity has a decisive in(uence on the behaviour of the precipitation process, especially through its relationship with (uid residence time that causes concentration gradients to emerge in the bulk and/or boundary layer depending on the (ow region. These concentration gradients are due to formation of particulate matter which can contribute to fouling by particulate deposition and in addition have a similar e"ect to those induced by temperature gradients in non-isothermal systems and in(uence di"usion of species towards the walls. Following discusses the results in laminar and fully developed turbulent (ow. For laminar (ow, the velocity e"ects have been examined at varying solution supersaturations and system temperatures to see whether or not the velocity e"ects would be exacerbated by the degree of supersaturations and system temperatures. For turbulent (ow that is mostly encountered in heat exchange systems, the e"ect of temperature gradient is assessed.
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
3811
0.016 0.014
++
[Ca ] {mol/L}
0.012 0.010
Ca, z = 0.5 Ca, z = 1.5
0.008
Part, z = 0.5 Part, z = 1.5
0.006 0.004 0.002 0.000 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 5. Radial concentration proEle at outlet for Re = 1000; T = 298 K.
0.08 0.07
Velocity {m/s}
0.06 0.05 0.04 0.03 0.02 0.01 0.00 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 6. Developed velocity distribution for Re = 1000; T = 298 K.
Fig. 4. Calcium ion distribution, Re = 1000.
4.1. Precipitation in laminar 1ow 4.1.1. E8ect of velocity and residence time in laminar 1ow Fig. 4 shows the distribution of calcium ion mass fraction for a Reynolds number of 1000 at 25◦ C. It is observed that the inlet calcium ion mass fraction is uniform in the radial direction and as progress is made along the annulus the mass fraction of calcium ion in the bulk decreases. In some regions of the (ow, the mass fraction of calcium ions is 5.5% less than that at the inlet. For the sulphate ion the distribution and consumption is similar. In the corresponding distribution for particles the mass fraction increases along the annulus because the precipitation process is producing calcium sulphate particles. A distinct observation that can be made in Fig. 4 is that as progress is made along the annulus the mass fraction proEle in the radial direction does not remain constant. This suggests that the rate of ion consumption is varying not only in the axial direction but as well in the radial direction. The observed radial variation in precipitation rate and hence mass fraction is largest at the outlet. Fig. 5 shows the variation in the concentration proEles of species for calcium ion and particulate matter at two consecutive axial positions. At both of these positions the (ow
is fully developed and as such the di"erence is only due to the e"ect of residence time. Examination of Fig. 5 suggests that the precipitation is greater adjacent to the walls and more particles are produced closer to either wall than around the middle position. Since the process is isothermal and the solution supersaturation and hence concentration of species is uniform at the inlet, the variation in precipitation rate can only be due to the velocity gradient. Fig. 6 depicts a plot of the fully developed radial velocity proEle, which corresponds to both axial positions in Fig. 5, showing that the lower velocity regions correspond to the regions where more particles have been generated because at lower velocities the precipitation reaction has more time to proceed before the (uid exits the annulus. Fig. 5 also demonstrates that at two points in a fully developed (ow the overall concentration of particles and the concentration gradients for each species are larger downstream. The emergence of concentration gradients show the decisive in(uence that residence time has in production of particles and their possible contribution to fouling by mode of particulate deposition and also the ionic concentration gradient which would induce di"usion of ions to the wall and contribution to fouling by mode of crystallization. By changing the inlet velocity, whilst remaining within the laminar regime, the in(uence of velocity can be further examined (Fig. 7) indicating larger concentration gradients for lower Reynolds numbers. This of course is due to the residence time di"erence; as seen from Fig. 3 at a given
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816 0.0150
0.025
0.0145
0.020
[Ca ] {mol/L}
0.0140
++
++
[Ca ] {mol/L}
3812
0.0135 0.0130
0.015 0.010 0.005
Re =250
Re = 500
Re = 1000
Re = 1500
S.S. = 1.5
0.0125
S.S. = 3
S.S. = 4.5
0.000 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Position ([r-r i ]/[r o -r i ])
Fig. 7. At outlet radial calcium concentration distribution with varying Reynolds numbers. T = 298 K.
Fig. 9. The radial concentration proEle at the outlet for three di"erent inlet concentrations with equal molar feed of reactants: Re = 1000: T = 298 K.
0.01504
0.016
[Ca ] {mol/L}
0.01500
0.014 0.013
++
++
[Ca ] {mol/L}
0.015 0.01502
0.01498 Re = 250 Re = 1500
0.01496 0.0
0.2
Re = 500 Re = 2000 0.4
0.012 0.011
Re = 1000
T = 298 K 0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
T = 308 K
T = 318 K
0.010 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 8. The comparison of the radial calcium ion distribution for di"erent Reynolds number with the same residence time of 2 s in developing (ow. T = 298 K.
Fig. 10. E"ect of temperature on outlet calcium concentration for an inlet supersaturation of 3, Re = 1000.
residence time for a fully developed (ow, the concentration proEles are identical and the convective transport is zero. However, in the developing region, the convective (ow of momentum plays some role on transport of species. To more closely analyse the di"ering concentration proEles for different values of Reynolds numbers shown in Fig. 7 and to assess the relative e"ect of convective (ow of momentum in the developing region as opposed to residence time, the concentration proEles are plotted (Fig. 8) for a developing (ow for various Reynolds numbers at a given residence time. Fig. 8 demonstrates that the (ux is mainly di"usive and the degree of convection only a"ects the proEle next to the wall and there is no di"erence between these proEles within the bulk. As mentioned above, these observed characteristics have implications for both crystallization fouling and particulate fouling mechanisms. The generation of particles and the induced gradient in the ionic concentration would promote transfer of ions towards the surface and could promote the crystallization fouling in an actual heat or mass transfer unit. Particles produced could also adhere to the surface and promote particulate fouling. The location along the channel where the generation starts to take place would depend on the residence time and its interdependence on the velocity.
4.1.2. E8ect of velocity at varying inlet supersaturations The rate of reaction is a function of the degree of solution supersaturation. Fig. 9 was constructed to assess the relative in(uence of velocity at various inlet supersaturations. Whilst the radial trend described in the previous section exists in each proEle, it is obvious that the velocity e"ects become more signiEcant and pronounced at higher degrees of supersaturation. 4.1.3. E8ect of velocity at various system temperatures The e"ect of temperature on precipitation is well established. Comparative simulations have been carried out (Fig. 10) to show the relative e"ect of velocity on the generated concentration gradients at a given solution supersaturation but varying system temperatures (25 –45◦ C). The concentration gradient is much steeper at higher temperatures and further highlight the impact that velocity has on distribution of species at higher temperatures. Comparison of Figs. 9 and 10 indicates that temperature and its in(uence on the reaction rate constant has a more dominating impact on precipitation and the generated concentration proEle than the degree of solution supersaturation. In heat exchangers, the accumulated impact of velocity and the existing temperature gradients further
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
3813
0.015016
++
[Ca ] {mol/L}
0.015014 0.015012 0.015010 0.015008 Re = 10,000
Re = 20,000
0.015006 0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Fig. 12. The comparison of the radial calcium ion distribution for the Reynolds number of 10 000 and 20 000 for a fully developed (ows at a given residence time. T = 298 K.
Fig. 11. Calcium ion distribution for isothermal turbulent (ow.
induces concentration gradients and would contribute to the magnitude of soluble species (ux towards the surface and also possibly the particulate fouling by generated particles; this will be assessed in turbulent (ows usually encountered in heat exchange systems. 4.2. Precipitation in fully turbulent 1ow (Re ¿ 10 000) 4.2.1. Isothermal fully turbulent 1ow The previous sections revealed that residence time has a signiEcant in(uence on precipitation in laminar (ow. Residence time also impacts the behaviour of precipitation in turbulent (ow. In speciEc, in the viscous sub-layer region of the turbulent (ow the behaviour of species is expected to be similar to that identiEed in laminar (ow because of the similar (ow characteristics. These characteristics were assessed and observed by carrying out CFD simulations for the low Reynolds number k–! turbulence model, which uses non-uniform grids to suJciently resolve the (ow behaviour adjacent to the walls. The bulk residence time is so short that the amount of particles produced in the turbulent bulk is considered to have a negligible e"ect on fouling. Fig. 11 demonstrates the concentration proEles for calcium ions along the annulus for
a Reynolds number of 10 000. In some regions of this turbulent (ow only a maximum of 0.15% of the calcium ions have been consumed which is much smaller than the corresponding 5.5% experienced in the laminar (ow of Fig. 4 indicating that the amount of particles produced in the bulk is insigniEcant. Within the viscous sub-layer the concentration gradients are similar to those that were developed in laminar (ow, which induce a radial, di"usive (ux of the species. The increased residence time adjacent to the walls allows more particles to form. Fig. 11 shows that the minimum calcium ion mass fraction appears adjacent to either walls of the annulus corresponding to a maximum amount of particles being produced near the walls and is explicable by the characteristics of the fully developed turbulent (ow having a fairly constant velocity in the bulk and laminar characteristics in the viscous sub-layer adjacent to the walls where viscous forces are more dominant than Reynolds stresses. Fig. 12 shows that though for both velocities the (ow is fully developed and the average residence times are the same, the di"erent degrees of turbulence does indeed a"ect the particle generation and the concentration proEles. 4.2.2. The e8ect of temperature gradients in fully turbulent 1ow Heat exchangers operate under various heat transfer conditions. Important to all is the impact that the resulting temperature gradient has on fouling. This section will examine the e"ect that temperature gradients have on precipitation in turbulent (ow, the (ow regime at which heat exchangers operate to minimize fouling. The temperature gradients are simulated by imposing a constant heat (ux as a boundary condition on the inner wall of the annulus. For a given run, the constant heat (ux corresponds to a constant temperature gradient adjacent the inner wall because the values of viscosity, heat capacity and thermal conductivity are assumed to be constant. Fig. 13 shows the drastic e"ect that a temperature gradient has on precipitation. As was noticed in Fig. 10, the varying temperature has a signiEcant e"ect on the rate of
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816 0.01505
0.01505
0.01500
0.01500
[Ca ] {mol/L}
0.01495 0.01490
++
++
[Ca ] {mol/L}
3814
0.01485 q" = 14 kW/m2
0.01480
0.01495 0.01490 0.01485 Re = 10,000
0.01480
Re = 20,000
q" = 7 kW/m2 0.01475
0.01475 0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Position ([r-r i ]/[r o -r i ])
Position ([r-r i ]/[r o -r i ])
Fig. 13. Comparison of the radial ion distribution with di"erent heat (uxes at the same position for a Reynolds number of 20 000.
Fig. 14. Comparison of the radial ion distribution for the Reynolds number of 10 000 and 20 000 at the same residence time and the same heat (ux at ri Tbulk = 300 K.
precipitation and, hence, the amount precipitated. The position where most ionic species have been consumed corresponds to the position of highest temperature, at the inner surface. The temperature decreases toward the bulk and this corresponds to the decrease in the amount of ionic species consumed in Fig. 13. The resulting temperature gradient leads to the establishment of a concentration gradient adjacent to the heat transfer surface. Furthermore, a comparison of Figs. 12 and 13 suggest that temperature gradients have a more signiEcant impact on developing concentration gradients than the distribution in residence time; however, the e"ect of residence time distribution is not negligible though it is often neglected. Researchers have drawn general conclusions that higher surface temperatures, higher heat (uxes, lead to a greater amount of scale forming on the surface. The phenomenon that causes one to make this conclusion is clearly demonstrated and quantiEed in Fig. 13. It is observed that a higher heat (ux causes a more signiEcant concentration gradient to develop adjacent the heat transfer surface. As previously stated, these concentration gradients lead to a di"usive (ux of ionic species. In terms of fouling, these concentration gradients would have an in(uence on the transport mechanism. Hence, at a higher heat (ux there would be a superior di"usive (ux, which suggests that a greater amount of scale would form. Various researchers have compared results of di"erent (ows to speculate on whether transport or attachment is the rate-controlling step in the observed scale deposition. Similar conclusions cannot be drawn from these results because attachment is not examined. However, this study will provide an insight into how di"erent (ows e"ect the transport mechanism. Fig. 14 are the results for two Reynolds numbers for fully developed (ows at a given average residence time, heat (ux and bulk temperature. A larger concentration gradient has emerged for the lower Reynolds number. This implies that a greater amount ionic di"usive transport occurs for lower Reynolds numbers due to residence time distribution. The detail of the CFD results allows a better appreciation of the transition between the two possible controlling steps
of deposition and also di"erent mechanisms of fouling. It was previously observed that surface temperature relates to the temperature gradient and its impact on the concentration gradient. For these two runs (Fig. 14) that were at a given bulk temperature, given heat (ux, and a given average residence time, the lower (ow rate has a larger surface temperature. This is attributed to the lower Reynolds having a thicker boundary layer, a larger resistance to heat transfer. It is demonstrated by comparing the results in Fig. 14 that as the boundary layer is decreased so too will di"erence between bulk and surface temperature. This has the effect of reducing the size of the concentration gradient and, consequentially, the magnitude of di"use (ux. Therefore, it quantiEes how the controlling step of the deposition process alternates from di"usion controlled to surface reaction controlled as (ow increases. 5. Conclusion The CFD was used as a tool to highlight the vital role of velocity and residence time in detailing calcium sulphate precipitation in a (ow domain under both isothermal and non-isothermal conditions. The in(uence of velocity proEle on residence time distribution causes concentration gradients to emerge in both laminar and turbulent (ows. In addition to possible deposition of generated particles in the (ow domain on the transfer surface by particle deposition, the ionic concentration gradients induce a radial (ux of soluble species towards the surface and would further contribute to fouling by crystallization. The induced (ux is greatest in laminar (ow where velocities are lower, residence times are signiEcantly higher and considerable concentration gradients exists in all regions of the (ow. In turbulent (ow, similar concentration gradients appear only within the turbulent viscous sub-layer where laminar characteristics exist. In fouling by sparingly soluble salts, the mechanism of fouling is usually attributed to (ux of ionic species (caused by a temperature gradient in heat exchangers or concentration polarization in membranes) to the wall and precipitation at the wall. The Endings of this investigation infer that
P. Walker, R. Sheikholeslami / Chemical Engineering Science 58 (2003) 3807 – 3816
velocity and residence time distribution also in(uence the behaviour of the precipitation process. Therefore, the combined e"ect of velocity, temperature or concentration polarization and their induced concentration gradients causes species to di"use towards the transfer surface and promotes deposit formation by both crystallization at the surface and particulate deposition by precipitate formed in the bulk and boundary layer. The e"ect of residence time distribution has been traditionally ignored. Kinetic e"ects are totally ignored in for example membrane units that have very short average residence times on the basis that there is no suJcient time for crystallization to take place. Fouling experiments base the optimum velocity on the balance between its e"ects on the removal rate with its e"ect on the convective mass transfer rate from the bulk. However, the velocity proEle and residence time distribution, in both laminar and turbulent (ow, causes concentration gradients to emerge in the (ow domain; these gradients become steeper at higher temperatures and higher supersaturations. Higher temperatures are encountered in heat exchangers and higher concentrations are encountered at Enal stages in the membrane units. Therefore, these additional e"ects of velocity on fouling, fouling mechanism and structure of the deposit should not be overlooked. Even though a simpliEed precipitation model was used, CFD has proven to be a useful tool in this investigation. Its ability to e"ectively simulate precipitation process has assisted in gaining improved understanding of velocity, temperature, and concentration gradients a"ecting fouling mechanism. At this stage the results are only to elucidate the fouling mechanism. However, the intention is to use the presented model as a basis for a comprehensive model of the fouling process in both heat exchange and membrane units. It is anticipated that this intended continuation in the development of the CFD model used in this investigation will result in better understanding of the fouling by sparingly soluble salts and of the fouling caused by solute polarization within membrane units. To achieve the planned result, a dynamic predictive model, would involve the inclusion of statistical methods like population distributions to improve the depth of the kinetic model, and of permeate (ux and concentration polarization in case of membrane units.
3815
solubility constant, mol=kg H2 O2 length of annulus, m molecular weight of species i , kg/mol r radial co-ordinate, m R universal gas constant, J/mol K Re Reynolds number ri inner radius of annulus, m ro outer radius of annulus, m precipitation reaction rate, kg=m3 s rppt ∗ r ri =ro S degree of solution supersaturation S source term for the equation, generation rate/volume ] sulphate ion concentration, mol=m3 [SO2− 4 solution T temperature, K v velocity vector, m/s
ksp L MW i
Greek letters
i k soln !
mass fraction of species i di"usion coeJcient of property rate of turbulent energy dissipation, m2 =s3 turbulent kinetic energy, m2 =s2 viscosity, Pa s solution density, kg=m3 general transport property, T; v; =k, s−1
Subscripts aq Ca2+ CaSO4 i in s SO2− 4
aqueous calcium ions calcium sulphate particle component i inlet solid sulphate ions
References Notation A1 ; A2 ; A3 ; A4 ; A5 [Ca2+ ] E kd kr
constants calcium ion concentration, mol=m3 solution activation energy for precipitation, J/mol dissolution rate constant, m3 solution/mol precipitate s precipitation reaction rate constant, m3 solution/mol precipitate s
Bott, T. R. (1995). Fouling of heat exchangers. Amsterdam: Elsevier. Epstein, N. (1983). Thinking about heat transfer fouling: A 5 × 5 matrix. Heat Transfer Engineering, 4(1), 43–56. Kho, T. C. S. (1998). E"ect of (ow distribution on scale formation in plate and frame heat exchangers. Department of Chemical and process engineering. University of Surrey, Guildford, Surrey UK. (p. 292). Kostoglou, M., & Karabelas, A. J. (1998). Comprehensive modeling of precipitation and fouling in turbulent pipe (ow. Industrial Engineering and Chemistry Research, 37, 1536–1550. Liu, S., & Nanacollas, G. H. (1970). The kinetics growth of crystal calcium sulphate dihydrate. Journal of Crystal Growth, 6, 281–289. Mukherjee, R. (1996). Conquer heat exchanger fouling. Hydrocarbon Processing, January, 121–127.
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Raju, K. U. G., & Atkinson, G. (1990). The thermodynamics of scale mineral solubilities 3. Calcium sulfate in aqueous NaCl. Journal of Chemical Engineering Data, 35, 361–367. Sheikholeslami, R. (2000). Calcium sulfate fouling—precipitation or particulate: A proposed composite model. Special Issue of Heat Transfer Engineering Journal, 21(2), 24–33. Taberok, J., Ritter, R. B., & Palen, J. W. (1972). Heat transfer; fouling: The major unresolved problem in heat transfer. Chemical Engineering Progress, 68(2), 59–67.
Walker, P., & Sheikholeslami, R. (2002). Preliminary numerical study of CaSO4 precipitation in laminar (ows in pipes and slits under isothermal conditions. The 9th APCChE congress and CHEMECA (pp. 612– 621). Wilcox, D. C. (1998). Turbulence modeling for CFD. DCW Industries Inc. La Canada, CA.