A theoretical study of freezing fouling: limiting behaviour based on a heat and mass transfer analysis

Chemical Engineering and Processing 40 (2001) 335– 344 www.elsevier.com/locate/cep

A theoretical study of freezing fouling: limiting behaviour based on a heat and mass transfer analysis M.J. Fernandez-Torres a,1, A.M. Fitzgerald a, W.R. Paterson a, D.I. Wilson a,* a

Department of Chemical Engineering, Uni6ersity of Cambridge, New Museums Site, Pembroke Street, Cambridge, CB2 3RA, UK

Abstract An analysis is presented of freezing fouling for liquids in laminar flow through ducts. Solidification occurs on a cooled wall while the bulk liquid remains unsaturated. The approach assumes that intrinsically rapid heterogeneous nucleation occurs at the solid/liquid interface, so that solidification is controlled by heat and mass transfer rates rather than by the intrinsic rate of crystallisation. A simple one-dimensional model for a single crystallising solute predicts that a range of fouling behaviours can occur, ranging from linear fouling to asymptotic (Kern– Seaton) behaviour, depending on the operating conditions, without any need to invoke removal effects. Maximum fouling rates can be estimated and the occurrence of quasi-asymptotic fouling can be identified, so permitting an apt choice of parameters for the design and operation of systems subject to such fouling effects. The model is illustrated by a case study on ‘coring’ in food fat distribution pipelines. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Fouling; Crystallisation; Freezing; Fats; Modelling

1. Introduction Fouling refers to the formation of unwanted deposit layers on heat transfer surfaces, causing increased thermal resistance and other process penalties. Freezing fouling describes the phenomenon whereby a layer is formed on a heat transfer surface which is colder than the bulk fluid via crystallisation of dissolved species or of the solvent itself, and is therefore considered to be a particular case of crystallisation fouling [1,2]. The formation of solidified layers from moving fluids is not restricted to heat exchanger fouling, however; the formation of such layers represents a serious operating problem in undersea oil pipelines when wax fractions deposit, reducing the duct size and increasing pressure drops, and has prompted extensive research in this area [3,4]. Wall freezing is not always undesired; it is a fundamental part of various industrial processes, e.g. wax separation and ice cream manufacture in scrapedsurface heat exchangers. Keary and Bowen, further* Corresponding author. Tel.: + 44-1223-334777; fax: + 44-1223334796. E-mail address: ian – [email protected] (D.I. Wilson). 1 Present address: Departamento de Ingenieria Quimica, University of Alicante, Alicante, Spain.

more, have studied the case where freezing promoted by a subcooled wall can be usefully used to isolate a section of piping for inspection or maintenance [5]. Wiegand et al., reviewed the literature on freezing in forced convection flows inside ducts, but concentrated solely on pure liquids, usually water [6]. The related literature on wax formation (e.g. [7]) and continuous casting of metals (e.g. [8]) features related problems, complicated by aspects of mass transfer, namely concentration-dependent saturation temperatures and diffusive transport, including deposit ageing. Analogous problems arise in food processing plants, where mixtures of food fats cause deposition of semisolid layers on the walls of distribution lines; in that industry, the phenomenon is called ‘coring’. In the latter case, coring affects both the pressure drop and the rheology of the fat suspension due to subsequent difficulties in controlling its temperature. Fouling in such systems can be caused by (a) particulate fouling, where wax crystals present in the bulk liquid become attached to the wall; and (b) freezing fouling, where the presence of a cool wall causes local supersaturation and crystallisation there. The onset of particulate fouling, driven by crystallisation in the bulk liquid, has prompted extensive research into the thermodynamics of wax/hydrocarbon mixtures (e.g. [9]). This paper con-

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siders the latter case, where the bulk liquid is free of crystals. In practice, both mechanisms can occur simultaneously, or in sequence, as the bulk temperature changes along a pipeline. Singh et al., reported that surface shear prevented the attachment of particulates in their experiments, which were performed at low concentrations [7]. Experimental studies of wax solidification on heat transfer surfaces such as those reported by Bott and Gudmundsson [10] and Ghedamu et al. [11] showed strongly non-linear behaviour. In these studies the deposit thickness, lf, expressed as a fouling resistance Rf via Eq. (1), exhibited decreasing rate behaviour and even asymptotic fouling behaviour Eq. (2). Rf(t) =

1 1 lf − : U(t) U(0) uf

Rf(t) =R f (1− exp(− t/~))

(1) (2)

Eq. (2) was first reported by Kern and Seaton in an analysis of refinery fouling [12], and it has been found to describe fouling behaviour in a number of other instances. The basis of the Kern– Seaton fouling model is the difference between (i) a growth term and (ii) a removal term proportional to the deposit thickness, lf, viz. dRf 1 dlf 1 (m; −krlf) = 8 dt uf dt ufzf f

(3)

The experiments of both groups mentioned above, which featured Reynolds numbers in the quasi-turbulent region (Re \5000), exhibited deposit removal. Bott and Gudmundsson [10] found that the Reynolds number had a significant effect on the asymptotic (equilibrium) fouling resistance, which they related to the shear stress exerted by the fluid on the deposit surface. Their reported values of R increased with wax conf centration. They discussed their results in terms of wax crystals being formed in the cool viscous sublayer, and a deposit being subject to removal by surface shear. Similar results were reported by Ghedamu et al. [11] Crystallisation rates, like most chemical reaction rates, are very sensitive to temperature, due both to the temperature dependency of the reaction rate constant and of the degree of saturation. Both of the above-mentioned investigations were performed under constant temperature driving force conditions, so that the temperature at the deposit surface changed (increased) during an experiment, affecting (reducing) the solidification rate. Under such conditions, fouling will be auto-retarding, as discussed by Epstein [13,14], independent of any shear removal mechanism. The effects of shear therefore need to be decoupled from any auto-retardation effect. Modelling of crystallisation fouling also requires reliable crystallisation kinetics, which are not always available, particularly for mixtures of waxes or fats as described above.

This work describes an analysis of freezing fouling based on heat and mass transfer principles alone. Crystallisation fouling involves the diffusion of solute species to the wall and the evolution (and removal) of enthalpy of crystallisation. Heat and mass transport rates will therefore set limits on crystallisation rates and possible fouling behaviour: crystallisation kinetics and shear removal will give rise to behaviour within these limits. The flow conditions in oil pipelines and fat distribution systems are often laminar owing to the relatively high viscosity of the bulk fluid, so that the rates of convective heat and mass transfer are relatively low. This work investigates the types of fouling behaviour which could be expected for such systems without (i) knowledge of crystallisation kinetics — assumed to be intrinsically rapid, or (ii) shear removal effects — assumed to be negligible, i.e. it corresponds to a worst case scenario. It also provides a basis for assessing the true effects of these chemical and physical processes. The analysis is illustrated by a case study involving a binary mixture of a palm oil fat in a non-crystallising solvent.

2. Analysis Consider the laminar flow of a single-phase liquid mixture through a duct surrounded by a fluid at an ambient temperature Ta, less than the mean (mixingcup) temperature of the mixture, Tb. The following analysis considers the situation at a single (axial) co-ordinate; stream-wise integration as described by Ribeiro et al. [3] is not described here. The temperature profile over a cross section is shown schematically in Fig. 1, and features a cooled wall, at temperature Tw, which may cause the process fluid in contact with it to be saturated and thus to deposit crystals. For clarity, Fig. 1 and the subsequent analysis given in this section will be for the special case of transfer to a flat wall. The calculations reported, however, will be for the more practical case of the inner wall of a pipe: the corre-

Fig. 1. Schematic analysis of freezing fouling (near surface).

M.J. Fernandez-Torres et al. / Chemical Engineering and Processing 40 (2001) 335–344

sponding algebra for cylindrical geometry is summarised in the Appendix. Crystallisation will result in the formation of a layer of insulating solid with solidliquid surface temperature Ts; initially Tw =Ts. The temperature at the deposit surface can be calculated by assuming that the system is in pseudo-steady state, using film heat transfer coefficients to describe convective heat transfer fluxes. (Ts −Ta)



n

1 +Rf ho

−1

=[(Tb −Ts)hi +NDHm]

(4)

Here, N is the molar flux of crystallising solute; for a slab geometry, the fouling resistance Rf is given by lf/uf. Eq. (4) shows that crystallisation increases the amount of heat to be removed from the fluid. The external convective heat transfer coefficient, ho, includes any resistances due to the duct wall, insulation, radiation etc, i.e. it is the overall heat transfer coefficient from the inner pipe surface. We make the following assumptions concerning crystallisation of the solute: 1. The mixture exhibits ideal solution behaviour. The freezing point, Tf, of a solution containing mole fraction x of solidifying solute is given by DHm 1 1 ln(x)= − (5) R Tm Tf where Tm is the melting point of the pure solute. 2. The deposit consists of pure solute. In a multicomponent solid system, the deposit is treated as a mixture of crystals of pure solute, as opposed to a solid solution. This is the approach described by Firoozabadi [9] for wax deposition in hydrocarbons. The presence of entrained solvent within the crystal matrix is ignored, but could be incorporated via an appropriate void fraction. 3. Homogeneous crystallisation does not occur in the flowing liquid. 4. Intrinsically rapid heterogeneous crystallisation occurs at the wall. The achieved rate of crystallisation is determined by heat and mass transfer rates alone: neither crystallisation kinetics nor removal mechanisms are included in the model. Rapid crystallisation means that negligible supersaturation is required, so that crystallisation occurs if the mole fraction of solute at the deposit surface is equal to or greater than the mole fraction which would be in equilibrium with the surface at temperature Ts. This mole fraction is given by Eq. (5), and we simplify the calculation by setting the solute concentration at the surface to be in equilibrium with the solid at that temperature, i.e.









DHm 1 1 ln(xs )= − (6) R Tm Ts This equation, establishing the saturation conditions, would be augmented by crystallisation rate equations if such information were available.

337

Finally, we assume that the solution properties, such as diffusivity, are independent of composition so that diffusion of solute to the deposit surface can be described by the familiar ‘rich solution’ result for diffusion through a stagnant film: N= CTkm ln





1− xs 1− xb

(7)

There are evident analogies between this analysis and the case of condensation in the presence of inerts, where by Eq. (7) we are implicitly using the Stefan result for diffusion of A through ‘stagnant’ B. Eqs. (4), (6) and (7) are solved simultaneously to find N, xs and Ts for a given set of operating conditions, subject to the following bounds: (i) Composition 0 Bxs 5 xb

(8)

(ii) Mass transport 0B N5 Nmax

(9)

where Nmax − kmCT ln(1− xb): kmCTxb



(9a)



(iii) Heat transfer h Ta + Tb i + hiRf ho

U(Rf)5 Ts 5 Tb hi + hiRf + 1 ho

(10)

The lower bound on Ts in Eq. (10) is obtained from the heat transfer statement in Eq. (4) by setting N=0. Note that the fouling resistance used in Eq. (10) should be that based on the external surface area, such that the left hand side must be modified to include changes in surface area for curved geometries. The solidification rate can then be calculated from dlf N · M = dt zf

(11)

Porous layers, and the resulting diffusion-driven ageing described by Singh et al. [7] are not considered here. The changes in time and internal energy required for the temperature and concentration profiles within the system to adjust as the deposit grows are also neglected. Limits of the model regime can be identified by inspection of Eqs. (4), (6) and (7).

2.1. Crystallisation in the bulk liquid Crystallisation is deemed to occur in the bulk liquid when the mean temperature reaches the saturation temperature for the solute concentration, T*. This could occur via homogeneous or heterogeneous nucleation (e.g. dust, impurities), particularly for prolonged exposure at the cloud point. We therefore choose to focus

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boundaries need to be added to this diagram: (i) the region where the flow can no longer be described as laminar, at some higher value of Tb; (ii) the limit of applicability of the mass transfer term in Eq. (7); the predictions for higher concentrations are subject to uncertainty, as the diffusivity used was independent of concentration, an approximation which may not hold at large x.

3. Case study – coring in palm oil distribution lines

Fig. 2. Explanation of fouling regime map for fixed Ta, hi and ho.

on the temperature regime where Tb \T*, with T* calculated from Eq. (6), namely ln(xb)=



DHm 1 1 − R Tm T*



(12)

Thus we do not consider the case of a sub-cooled solution.

2.2. Zero solidification beha6iour Eq. (7) indicates that xs =xb when N =0, so that Ts = T* (from Eq. (6)). Eq. (10) therefore allows one to predict when asymptotic fouling, or no fouling, is likely to occur. If Ts, calculated from Eq. (10) with Rf =0, is greater than T*, then the ‘clean’ wall is too warm for solidification to occur. Heat transfer therefore sets an upper limit for the bulk temperature which will cause freezing fouling for given Ta and solution concentration xb (and T*). Fouling will cause Ts to increase over time when Tb and Ta are constant, and asymptotic fouling may occur when N =0 for finite Rf. Setting [ =T* allows one to calculate the asymptotic fouling resistance Rf for a given set of process conditions, and inspection of the corresponding fouling layer thickness will indicate whether this is a feasible result. An infeasible result for lf indicates that asymptotic fouling does not occur. For example, a fouling resistance which gave lf \ di /2 would be infeasible for solidification on the inside of a pipe. The next section presents the results of this analysis for a binary food fat system. The data are presented in two forms: as Rf – time profiles to illustrate fouling behaviour; and as regime maps such as the schematic shown in Fig. 2. The abscissa presents solution concentration (xb) in terms of the saturation temperature T*, given by the non-linear relationship in Eq. (12). The map shows that freezing fouling will occur in the region T* BTb B {Tb : Tw =T*}. We note that two further

Mixtures of palm oil fats are important feedstocks in the food industry, e.g. in the generation of plasticised fat and shortenings. Palm oil is a mixture of tri- and di-glyceride components with a range of melting points [15]. We here simplify the system by treating it as a binary system of one solidifying solute and a hydrocarbon solvent of lower melting point fractions. Table 1 summarises the operating parameters for the case study. The solute parameters are those for tri-palmitin (PPP), which is the most abundant high melting point component found in palm oil. The solvent and solution parameters were either taken from work in progress [16] or estimated from correlations. We consider a typical case where the fat solution is pumped along a pipe of di = 50 mm at constant mass flow rate w. The internal film heat transfer coefficient hi is obtained from Hausen and Kays’ correlation [17]: Nud = 3.66+

0.0668(ds/L)Red Pr 1+0.04[(ds/L)Red Pr]2/3

(13)

where L is taken to be 50 m and the duct dimension, ds = di –2lf , is updated as fouling occurs. The film mass transfer coefficient, km, is obtained from the analogous mass transfer correlation: Shd = 3.66+

0.0668(ds/L)RedSc 1+0.04[(ds/L)RedSc]2/3

(14)

The cylindrical geometry of the system is incorporated into the heat transfer statement, equation (4*), and the fouling resistance quoted is that based on the external surface area, Rf,o. It follows from Eq. (1) that this resistance is given by Rf,o =



   

do d d + o ln i dshi 2uf ds

−

do dihi,c

(15)

For convenience, the o subscript is dropped in the following discussion and presentation of results for a cylindrical geometry. For a given set of parameters (Ta, ho, xb) the limiting values of Tb were calculated, and fouling profiles were then predicted for a set of intermediate temperatures. Table 2 summarises the calculation procedure. Eqs. (4), (6) and (7) were solved simultaneously at a given value of lf using a modified Newton–Raphson method

M.J. Fernandez-Torres et al. / Chemical Engineering and Processing 40 (2001) 335–344

within Microsoft Excel, giving N, Ts and xs and therefore dRf/dt. The Rf – time profile was then obtained by integrating Eq. (11) stepwise in Rf. The effect of changing internal diameter on film transfer coefficients, and the film temperature dependency of physical properties, were incorporated in the model (Table 1). Eqs. (4), (6) and (7) were solved for increasing values of lf , until N reached zero, or the duct dimension reduced to an unreasonable value. The latter criterion was set by pressure drop in the pipe, which can be readily calculated assuming that the fouling layer is smooth. For laminar flow in a pipe, the ratio of pressure gradients is given by:

   





considered to be linear if the correlation coefficient, R 2, was greater than 0.99999. Fouling profiles and fouling regime maps were generated for different combinations of flow rates (w=0.1 and 0.7 kg/s) and ambient temperatures (Ta = 25, 15°C) over a range of concentrations (0.01B xb B 0.9) and bulk temperatures Tb. The values ho ranged from 1 W/m2 K, estimated for an insulated pipe, to 25 W/m2 K, typical of a very draughty location. Higher values of ho would arise if the pipe were surrounded by flowing cooling liquid [10,11].

4. Results and discussion

dP dz w d 4i = dP wclean (di −2lf)4 dz clean

(16)

A ratio of ten was used to terminate calculations. The alternative scenario, of constant pressure drop and varying mass flow rate, calculated from Eq. (16), was not considered here. The extent of auto-retardation depends on the operating parameters, with some profiles showing only small reductions in fouling rate over time. In such circumstances the Rf –t and lf – t profiles were almost linear, so a criterion based on regression was used to delineate linear from falling rate behaviour. The data were fitted to a linear regression model, i.e. Rf =a × t, and were

4.1. Fouling beha6iour The effect of bulk temperature is illustrated in Fig. 3(a) for a typical case {xb = 0.10 (T*= 324.5 K); ho =5 W/m2 K; w=0.7 kg/s; Ta = 25°C}. Linear fouling behaviour was predicted at low Tb, i.e. no auto-retardation, mass transfer control. The Figure shows the results obtained for Tb = 326 K, above which the Rf – time profile did not satisfy the linearity condition described above. The maximum pressure drop criterion placed a limit of 1.12 cm (Rf = 0.112 W/m2 K) on the deposit thickness in this scenario. Also plotted in Fig. 3(a) is the result for Tb = 327.35 K, which exhibits evident auto-retardation behaviour. The figure also

Table 1 Model parameters for palm oil case study Parameter Tripalmitin (PPP)

Value Tm ZHm zM D

uf RMM Solution

z CT vb ub Cp

Geometry

di do

Variables

Ta ho w

339

65.6°C 147.7 kJ/mol 1083 mol/m3 1.96×10−11 m2/s (325 K) 2.57×10−11 m2/s (330 K) 3.28×10−11 m2/s (335 K) 0.14 W/m K 807 g/mol

Source/Comment

Assumed constant Assumed constant Wilke and Chang correlation (1955) [18]

773 kg/m3 4500 mol/m3 0.037 Pa s (325 K) 0.029 Pa s (330 K) 0.023 Pa s (335 K) 0.15 W/m K 2109 J/kg K (325 K) 2121 J/kg K (330 K) 2132 J/kg K (335 K) 0.050 m 0.054 m 15, 25°C 1–50 W/m2 K 0.1–0.7 kg/s

Typical plant dimension Typical plant dimension

Typical plant value

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340

Table 2 Calculation sequence for generating fouling regime maps 1.

Select material and operating condition parameters, particularly Ta, ho, xb

2. 2.1 2.2

3. 3.1

3.2

3.3 4. 5.

Determine limiting values Foulant thickness given by maximum pressure drop criterion, lf,max Limiting values of Tb, given by (i) Tb =T* (ii) Hot wall condition; Tb [ Tw = T*, lf = 0 (Eq. (10)) Consider intermediate values of Tb : select Tb Check whether a true asymptote is reached Calculate Rf such that N= 0, i.e. [ (Eq. (10)) = T* Check whether the corresponding value of lf5lf,max If satisfied, behaviour is asymptotic If not satisfied, inspect behaviour as lf approaches lf,max by 3.2 Integrate to find Rf–t behaviour Discretise range of feasible foulant thicknesses: lf  {0, lf,max } For each lf, solve equations Eqs. (4), (6) and dlf (7) [ N [ (Eq. (11)) dt Integrate to get lf –t and thereby Rf–t profiles Inspect Rf–t profiles for (i) Linear behaviour: compare R 2 for fit to Rf = a+b.t (ii) Asymptotic behaviour; compare R 2 for fit to Eq. (2) Select next value of Tb Delineate asymptotic, falling rate and linear fouling regimes Select next value of xb (and T*)

shows the result from fitting a Kern– Seaton curve equation Eq. (2) to these data; the agreement is reasonable (R 2 = 0.998) and might even have been considered good if the model predictions were actually experimental data. The asymptotic behaviour is caused by the combination of temperature conditions and the strongly non-linear dependence of saturation on Ts. Between 326 and 327.35 K, the fouling profiles exhibited falling-rate fouling, with quasi-asymptotic behaviour, i.e. falling rate fouling, giving very low final fouling rates at Tb just less than 327.35 K. Asymptotic fouling was observed at all Tb \327.35 K, with the value of Rf decreasing with increasing Tb until the warm wall condition was reached, when Rf =0 (no solidification). Fig. 3(a) thus demonstrates that a range of fouling behaviours can be obtained for freezing fouling from a priori calculations without invoking any removal term; this analysis shows that the deposition term in Eq. (3) is strongly dependent on the extent of fouling. Introduction of a removal term incorporating shear forces would result in the onset of asymptotic behaviour at a lower value of Tb than that calculated using this analysis.

Fig. 3(b) shows the corresponding results for xb = 0.50. The same final Rf values appear because of the pressure-drop constraint used in the calculations, but the effect of xb on the time scales is noteworthy. Fig. 4 is a series of domain diagrams which summarise the results for a range of concentrations and fixed ho, Ta in a similar format to Fig. 2, with lines marking the transition from linear to asymptotic fouling. The upper boundary of the asymptotic region is given by the ‘warm wall’ criterion (N=0 at Rf =0). The asymptotic/falling rate boundary is given by inspection of the calculated Rf –t behaviour: if the data fitted an asymptotic model equation (Eq. (2)) with a regression coefficient (R 2)] 0.998, the data set was deemed to be ‘asymptotic’. The value of 0.998 was somewhat notional, but inspection of the data sets by eye suggested almost negligible final fouling rates. The boundary between the falling rate and linear fouling behaviours was set by regression to a linear fouling model, with a threshold regression coefficient value of 0.99999.

Fig. 3. Fouling behaviours predicted for PPP case study with ho = 5 W/m2 K; Ta =25°C; w =0.7 kg/s. Fouling resistances are based on external surface area, Rf,o. Linear fouling: circles, calculated points; solid lines, regressed model. Asymptotic fouling: crosses, calculated points; dashed lines, regressed model.

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Fig. 4. Domain maps for fouling behaviours in PPP case study. Ta =298 K; w=0.7 kg/s. Dotted line, limit of asymptotic behaviour; dashed line, transition to falling rate fouling; crossed line, limit of linear fouling behaviour; solid line, Tb =T*b (bulk crystallisation).

Fig. 4 shows that the type of fouling behaviour predicted is thus strongly dependent on the original operating conditions. Reading across the diagram, it can be seen that at a fixed value of Tb: (i) no solidification occurs for low xb (and T*); (ii) onset of fouling is observed at a critical concentration, beyond which its regime will change from asymptotic to falling rate to linear as xb increases, until (iii) the solution is saturated and solidification will occur from the bulk fluid. The solidification behaviour along a pipeline can be followed by reading downwards on a domain map (assuming that the effect of L on transport parameters is small). For example, Fig. 4(c) shows that for xb corresponding to 332 K, no solidification is observed until Tb is less than  343 K, when asymptotic fouling starts (at a low rate). Asymptotic behaviour will continue until ca. 337 K, when solidification (at increasing initial rate) no longer asymptotes, but continues with a steadily falling rate. At 334 K pseudo-linear behaviour will arise, increasing in rate until Tb =332 K, when the solution reaches saturation and bulk crystallisation is likely to occur.

Also marked on Fig. 4(c) is the locus for Re (initial)\ 1000, which is used here to indicate when laminar flow correlations are likely to give less reliable descriptions of heat and mass transfer in the system. It can be seen that most of the initial conditions used in this work lie within the laminar regime. Fig. 5 shows the effect of the flow rate and ambient temperature on the domain diagram for the palm oil case study in Fig. 4(c). It can be seen that the width of the fouling regions (expressed in terms of temperature) is reduced by a drop in Ta, but increased by a reduction in mass flow rate (and hence transport rates). Such diagrams can be used to understand the processes occurring in a distribution line or pipeline. The fat solution will initially be too warm to cause any solidification at the wall, but downstream, after some cooling, asymptotic fouling will arise. Even further downstream, assuming negligible change in solute concentration, linear fouling or bulk precipitation will occur. Solidification, however, will reduce the heat loss from the bulk solution so that at some later time, the temperature profile will be different and the fouling behaviour at a particular location is likely to have

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ond or higher order reaction behaviour. Such an interpretation would be false, as the results were generated entirely without reference to kinetics, being based purely on thermodynamic and transport criteria. The results in Fig. 6 also demonstrate how the model can generate bounds for fouling rates (and asymptotic levels of deposit) which could be used in estimating worst case behaviour, or time scales likely to be encountered in experimental investigations of these systems. The greatest source of uncertainty in the model lies in the thermodynamic relationships between equilibrium concentrations and temperatures at the solid/liquid interface, i.e. Eq. (1). Most systems of interest feature several crystallising species, such that the thermodynamics in solution and at the interface require careful elucidation via experimentation in order to be incorporated into a model of this form. Furthermore, no subcooling has been assumed to be necessary for growth, which may not be realistic in all applications. Comparison with experimental results is required in order to establish the applicability of this approach; in its current form, it does, nevertheless, provide a framework to explain the different behaviours which can be observed in these systems. Fig. 5. Effect of operating parameters on fouling behaviour. Domain maps for comparison with Fig. 4(c). (a) Ta = 288 K, ho = 10 W/m2 K, w = 0.7 kg/s. (b) Ta = 298 K, ho = 10 W/m2 K, w= 0.1 kg/s.

changed. These effects shed some light on the difficulties encountered in practice in assessing coring behaviour. The model can also provide estimates of (worst-case) fouling rates. Fig. 6 shows the variation of the initial fouling rate with PPP concentration at a constant bulk temperature of 64.9°C. The fouling rates are presented in the form of a dimensionless fouling Biot number, Bif =Rf × ho. The figure shows a non-linear increase in fouling rate with increasing concentration which, were the data experimental, might be taken to indicate sec-

Fig. 6. Variation of initial fouling rate with concentration. Tb = 64.9°C; Ta =25°C; ho = 25 W/m2 K; w= 0.7 kg/s.

5. Conclusions A model is presented for solidification, on a cooled wall, of fat from a binary fat/solvent solution in laminar flow through a pipe. The modelling is ‘parsimonious’; that is to say, in designing the model we have deliberately excluded various complications to investigate how wide a range of behaviour such a simple model can predict, when supplied with realistic parameter values. In particular, the key simplifications invoked concern: Crystallisation kinetics: rather than assume a rate law, an activation energy and so forth, we have simply assumed the intrinsic crystallisation rate to be high. Crystallisation location: all crystallisation is assumed to take place on the solid surface, either, initially, on the clean pipe wall, or later, on the exposed surface of the deposit. Streamwise variations: the model neglects these, except insofar as they are a function of local variables, e.g. local fluid radial-mean temperature, rather than upstream variables; it is a purely one-dimensional model which, by assumption, may be applied at any point along the pipe wall. Deposit properties: the solidifying component forms a pure layer of uniform thickness in the immediate neighbourhood of any point.

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Removal effects: these are omitted from the model. Whatever the shear stress on the surface of the deposit, the rate of removal of deposit is nil. Unsteady state temperature and composition profiles: these too are omitted; we have constructed a pseudo steady-state model. The key effects incorporated are: The reduction in solubility of the fat as temperature is reduced; That heat and mass transfer resistances can control the rate of solidification; The effect of release of heat of crystallisation; Heat transfer by conduction through the deposit; Deposit growth. The model proves capable of predicting different regimes of fouling, specifically: absence of fouling, linear fouling, reducing-rate fouling and asymptotic (Kern–Seaton) fouling. The predictions are presented in two forms: as time profiles of fouling thermal resistance, to illustrate these different modes of behaviour; and as regime maps. Maximum fouling rates are estimated and conditions for the occurrence of quasiasymptotic fouling are identified, so permitting an apt choice of parameters for the design and operation of systems subject to such fouling effects. A discussion is presented to demonstrate how such diagrams may be used to understand the processes occurring along a distribution line or pipeline subject to fouling.

where Ao is the external pipe surface area and ho is a lumped thermal resistance based on the external diameter. The thermal resistance of the pipe is negligible. Rearranging, Q





1 ln(di/ds) + = (Ts − Tw)+ (Tw − Ta)= (Ts −Ta) Aoho 2yLuf (A.2)

The rate of heat transfer to the inner surface by convection and crystallisation is given by: Q=





(Ts − Ta) = [hi(Tb − Ts)+ NDHm]As 1 ln(di/ds) + Aoho 2yLuf





(Ts − Ta)Ao = [hi(Tb − Ts)+ NDHm]As 1 + Rd ho

(4*)

where As, the area of the fouling layer/liquid interface, and ds decrease as solidification continues. This expression reduces to Eq. (4) for a slab geometry where Ao = As. Equation 10 Eq. (10)* is obtained by setting N= 0 in equation (4*), giving:

Ts =

MJFT wishes to acknowledge the provision of a fellowship from Generalitat Valenciana. A CASE Award for AMF from the BBSRC and United Biscuits, and discussions with Dr. Ian Smart, are also gratefully acknowledged.

(A.3)

yielding the analogous result to Eq. (4):





hi A + hi Rd s ho Ao hi A + hi Rd s + 1 ho Ao

Ta + Tb Acknowledgements

343





(10*)

Appendix B. Nomenclature

Appendix A The model equations are presented for a slab geometry; the corresponding relationships for a cylindrical configuration, as arise in the case study calculations, are presented here. Equation 4 Assuming that the system is in pseudo-steady state, (e.g. ignoring the time taken to establish the temperature profile through the fouling layer) the rate of heat transfer, Q, through the core layer by conduction equals that from the external surface by convection and radiation: Q=



2yLuf (Ts − Tw) =Aoho(Tw −Ta) di ln ds

(A.1)

Latin Ao,i surface area based on outer, inner pipe dimension (m2) CT total concentration (mol m−3) di internal (clean) diameter of pipe (m) do external diameter of pipe (m) ds Reduced diameter of pipe (di−2lf ) (m) D Diffusivity of solute in solvent (m2 s−1) hi Internal convective film heat transfer coefficient at time t (W m−2 K−1) hi,c Internal (clean) convective film heat transfer coefficient (W m−2 K−1) ho External film heat transfer coefficient (W m−2 K−1)

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ŠHm Molar enthalpy of melting (J mol km kr M m; f L N Nu P Q R R2 Re Rd Rf Rf,o R f Sc Sh Tb Ta Ts Tf Tm Tw T* U t w x

~

−1

)

Film convective mass transfer coefficient (m s −1 ) Removal rate constant (kg m−1 s−1) RMM of component (kg kmol−1) Mass ‘deposition’ rate (kg s−1) Axial length (m) Molar flux of solute towards surface (mol m−2 s−1) Nusselt number Pressure (Pa) Rate of heat transfer (W) Gas constant (J mol−1 K−1) Correlation coefficient (dimensionless) Reynolds number Thermal resistance of deposit (m2 K W−1) Fouling resistance (m2 K W−1) Fouling resistance based on external surface area (m2 K W−1) Asymptotic fouling resistance (m2 K W−1) Schmidt number Sherwood number Mean fluid temperature (K) Ambient temperature (K) Temperature at deposit–liquid interface (K) freezing point of solution of composition x (K) melting point of pure solute (K) temperature at heat transfer surface (wall) (K) saturation temperature for the solute concentration (K) overall heat transfer coefficient (W m−2 K−1) time (s) mass flow rate (kg s−1) mole fraction of solute (mol (mol solution−1))

Greek lf deposit thickness (m) [ surface temperature estimate, Eq. (10) (K) uf deposit thermal conductivity (W m−1 K−1) um pipe wall thermal conductivity (W m−1 K−1) ub solution thermal conductivity (W m−1 K−1) vb solution dynamic viscosity (Pa s) zf deposit density (kg m−3) zM deposit molar density (mol m−3)

.

fouling time constant (s)

References [1] T.R. Bott, Aspects of crystallization fouling, Exp. Thermal Fluid Sci. 14 (1997) 356 – 360. [2] N. Epstein, Thinking about heat transfer fouling: a 5 × 5 matrix, Heat Transfer Eng. 4 (1) (1981). [3] F.S. Ribeiro, P.R. Souza Mendes, S.L. Braga, Obstruction of pipelines due to paraffin deposition during the flow of crude oils, Intl. J. Heat Mass Transfer 40 (18) (1997) 4219 – 4328. [4] G.M. Elphingstone, K.L. Greenhill, J.J.C. Hsu, Modelling of multiphase wax deposition, J. Energy Resources Tech. 121 (1999) 81 – 85. [5] A.C. Keary, R.J. Bowen, On the prediction of local ice formation in pipes in the presence of natural convection, J. Heat Transfer 121 (4) (1999) 934 – 944. [6] B. Wiegand, J. Braun, S.O. Neumann, K.J. Rinck, Freezing in forced convection flows inside ducts: a review, Heat Mass Transfer 32 (1997) 341 – 351. [7] P. Singh, V. Venkatesan, H.S. Fogler, N. Nagarajan, Formation and aging of incipient thin film wax-oil gels, AIChEJ 46 (5) (2000) 1059 – 1074. [8] M.J.M. Krane, F.P. Incropera, Experimental validation of continuum mixture model for binary alloy solidification, J. Heat Transfer 119 (4) (1997) 783 – 791. [9] A. Faroozibadi, Thermodynamics of Hydrocarbon Reservoir Fluids, McGraw-Hill, New York, 1998. [10] T.R. Bott, J.S. Gudmunsson, Deposition of paraffin wax from kerosene in cooled heat exchanger tubes, Can. J. Chem. Eng. 55 (1977) 381 – 385. [11] M. Ghedamu, A.P. Watkinson, N. Epstein, Mitigation of wax oil buildup on cooled surfaces, in: C.B. Panchal (Ed.), Fouling Mitigation of Industrial Heat-Exchange Equipment, Bgell House, New York, 1997, pp. 473 – 489. [12] D.Q. Kern, R.E. Seaton, A theoretical analysis of thermal surface fouling, Brit. Chem. Eng. 14 (5) (1959) 258. [13] N. Epstein, Fouling of Heat Exchange Surfaces, VDI-GVC, Dusseldorf, Germany, 1990, pp. 1.1 – 1.17. [14] N. Epstein, Elements of particle deposition onto non-porous solid surfaces parallel to suspension flows, Exp. Thermal Fluid Sci. 14 (4) (1997) 323 – 334. [15] W.L. Ng, C.H. Oh, J. Am. Oil Chem. Soc. 71 (10) (1994) 1135 – 1139. [16] A.M. Fitzgerald, Modelling of Palm Oil Crystallisation in Food Fats, 1998 CPGS dissertation,University of Cambridge (unpublished data). [17] W.M. Kays, citing H.Z. Hausen, Trans. ASME, 77 (9) (1943) 1265. Ver. Dtsch. Ing. Beih. Verfahrenstech 4 (1955) 91. [18] C.R. Wilke, P. Chang, AIChEJ 1 (1955) 264.