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Modeling and Simulation of Vertical Continuous Cooling Crystallizers for the Sugar Industry Mario Llano-Restrepo* School of Chemical Engineering, Universidad del Valle, Apartado 25360, Cali, Colombia
Vertical continuous cooling crystallizers (VCCC) are commonly used in the sugar industry for the economically important final recovery of sucrose from the mother liquor present in the massecuite obtained at the end of the vacuum evaporative crystallization stages. In this work, a steady-state one-dimensional distributed model is formulated to simulate well-designed VCCC in which the massecuite follows approximately an ideal plug-flow pattern. The model consists of seven ordinary differential equations comprising mass, energy, and crystal population balances. The model describes the changes (along the crystallizer height) of the massecuite weight fraction of crystals, mother liquor dry substance content, mother liquor purity, crystal average size, crystal size coefficient of variation, massecuite temperature, and cooling water temperature. Simulation of the model yields a very accurate description of the reported performance of a VCCC [with vertically oscillating horizontal cooling coils (OVCC system)] from a beet sugar factory in Germany. 1. Introduction 1.1. Background. Sugar production is one of the largest industries in the world, and crystallization is extremely important in that business. Both sugar beets and sugar cane are sucrose sources. In sugar factories, the main crystallization of sucrose occurs over a series of three vacuum evaporative crystallizers (known as pans), where water is evaporated to an extent such that the solution (known as mother liquor) becomes supersaturated, and sucrose crystals may form from a suitable seed magma charged to the pan. The three evaporation stages are known as boilings.1,2 At the end of the first and second boilings, the massecuite (i.e, the mixture of crystals and mother liquor) is dropped from the pan, and the crystals are separated from the mother liquor in a centrifugal. The mother liquor recovered is charged to the next boiling. Sucrose crystals obtained from the first boiling are regarded as raw sugar, and those obtained from the second boiling are recycled as seed magma to the first boiling. After the third boiling, the purity (i.e., the sucrose content) of the mother liquor has decreased to a level such that it is not practical to crystallize more sucrose by evaporation. Cooling crystallization is then an economically important last resource to recover sucrose before it is lost in the final mother liquor. The low-purity massecuite obtained in the third boiling is thereby transferred to a cooling crystallizer for the exhaustion of the mother liquor, after which point the crystals and the exhausted mother liquor (known as final molasses) are separated in a centrifugal. Crystals are recycled as seed magma to the second boiling. Final molasses can be used to produce ethanol by fermentation. Even though cooling crystallization is an absolute requirement for the effective exhaustion of the mother liquor in low-purity massecuites from the third boiling, it is sometimes also desirable and advantageous for the * To whom correspondence should be addressed. Tel: +57-2-3312935. Fax: +57-2-3392335. E-mail: mllano@ univalle.edu.co.
exhaustion of the mother liquor in higher purity massecuites from first or second boilings.1,3-5 In the past, exhaustion of the mother liquor in sugar massecuites was carried out in horizontal cooling crystallizers.3-6 However, the increasing quantities of mother liquor in the massecuites produced in many sugar factories could not be sufficiently exhausted in the existing horizontal cooling crystallizers, because of the insufficient cooling surface associated with their conventional design. In the decade of 1975-1985, the vertical continuous cooling crystallizer (VCCC) emerged as a solution for the increasing cooling capacity that was required for an adequate exhaustion of the mother liquor in sugar massecuites. A VCCC can be used for the exhaustion of the mother liquor in high-grade or low-grade massecuites. A VCCC is built as a cylindrical tower-type container, which is often erected out in the open, outside the sugar house, thereby requiring no supporting steelwork. For the most efficient operation, all designs are provided with a counter-flow arrangement of the massecuite and the cooling water (See Figure 1). In typical industrial designs, the massecuite containing the mother liquor to be exhausted is continuously fed at the top of the tower, it is gradually cooled as it travels down, and it is withdrawn from the bottom of the tower by the action of a positive displacement pump. In turn, cooling water flows upward from the bottom to the top of the tower through a set of cooling tubes (banks or coils), which can be either static or mobile, horizontal or vertical, in parallel or series flow, depending on the specific design. The geometrical arrangement of the cooling tubes inside the container (See Figure 2) is also one of the features that makes some designs different from one another. The Toury crystallizer6-9 was the first type of vertical crystallizer used in the sugar industry. In the Toury crystallizer, and also in the first version of the BMA vertical crystallizer,10-13 a certain number of partitioning floors were provided along the height of the container for a compulsory guidance of the massecuite flow. Cooling tube banks were static and horizontal, and
10.1021/ie0504046 CCC: $30.25 © 2005 American Chemical Society Published on Web 10/20/2005
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Figure 1. Schematic drawing of a vertical continuous cooling crystallizer (VCCC) with vertically oscillating horizontal cooling coils (MA ) massecuite, CW ) cooling water), according to the description given by Selwig & Lange.26
Figure 2. Schematic drawing of a VCCC’s horizontal cooling coil with its set of concentric tubular hexagonal rings, according to the description given by Selwig & Lange.26
arranged between two consecutive partitioning floors. Stirring arms, fastened to a hydraulically driven central tubular shaft, were mounted horizontally between the cooling coils. Because of drag forces, the stirring arms kept the massecuite in a slow horizontal rotary motion against the static cooling tube banks. In the first version of the BMA crystallizer, the stirring arms had adjustable blades, to scrape closely the cooling surface.10,12 A schematic drawing of the Toury crystallizer can be found in the paper by Austmeyer.6 Detailed drawings of the
first version of the BMA crystallizer are shown in the paper by Cronewitz10 and the handbook by Chen and Meade.13 The basic features of the first BMA design can also be found with some modifications or improvements in the Buckau-Wolf,14,15 Silver,13,16 Sudeco,17 TongaatHulett,18 Fives Cail,19 Honiron,20 and Fletcher-Smith21 vertical crystallizers. Based on residence time distributions obtained from LiCl tracer tests conducted at the Zeil sugar factory14 in Germany, it was concluded that the massecuite approximately follows an ideal plug-flow pattern in the Buckau-Wolf vertical crystallizer. Plug flow of the massecuite has also been claimed by the manufacturers of the Fives Cail19 and the FletcherSmith21 vertical crystallizers. In contrast, on the basis of residence time distributions obtained from zinc tracer tests, it was observed in a recent study22 that, possibly due to short-circuiting, the flow of the massecuite seems to deviate severely from an ideal plug-flow pattern in the Silver13,16 and Honiron20 vertical crystallizers that were examined. In the DDS vertical crystallizer,10,23-25 the cooling tubes are static and vertical, arranged in two axial sections that are separated by a central rotating shaft with a stirring wing that oscillates in the compartment located between the two sections to displace the massecuite along the outer surface of each cooling tube. Based on a residence time distribution obtained from LiCl tracer tests conducted at the Assens sugar factory25 in Denmark, it was concluded that the flow of the massecuite in the DDS vertical crystallizers almost follows an ideal plug-flow pattern. A detailed drawing of the DDS vertical crystallizer can be found in the paper by Klein.25 In the improved Selwig & Lange vertical crystallizer6,26,27 and in the current BMA oscillating vertical cooling crystallizer (OVCC system),28,29 the cooling system consists of a certain number of standardized blocks that are installed at different heights along the crystallizer. The cooling blocks are mounted on six actuating rods (up-and-down motion mechanism). As schematically shown in Figure 1, each cooling block typically consists of four horizontal levels of coils,6,26,27 and each coil is comprised of a set of concentric tubular hexagonal rings, as shown in Figure 2. The rings in each coil, the coils in each block, and the blocks themselves are altogether connected in series for the flow of the cooling water upward from the bottom to the top of the crystallizer. The entire cooling system slowly oscillates in the vertical direction, which is due to the motion of the actuating rods that are driven by six cylindrical pistons symmetrically arranged on the top cover of the crystallizer. The self-cleaning effect of the vertically moving coils ensures a good heat transfer between the massecuite and the cooling water. In contrast to the previous designs, no partitioning floors are required to guide the flow of the massecuite through the crystallizer. A staggered arrangement of the tubular rings is used, i.e., there is a gap between the tubular rings in one horizontal level and the rings in the level above or below it, such that the massecuite flows through the slotlike openings between the tubes, slowly moving downward. As shown in the detailed drawings of the relevant patent,26 a low-speed engine-driven rotary distributor, located above the uppermost cooling block, evenly spreads the entering massecuite over the cross section of the crystallizer, and a conical bottom discharges the exiting massecuite from the crystallizer into
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the suction line of a positive displacement pump. The combination of all these features ensures a uniform distribution of the massecuite velocity over the entire cross section of the crystallizer, and, in consequence, an optimum residence time distribution (i.e., a nearly ideal plug-flow) of the massecuite in the crystallizer,26-29 as shown by Dunker27 from LiCl tracer tests for the Selwig & Lange VCCC at the Su¨derdithmarschen beet sugar factory in Germany. This type of vertical crystallizer has no rotary stirring arms; therefore, it needs no antifriction bearings or stuffing boxes and it has a low specific drive power associated with its operation. In summary, the main purpose of a well-designed VCCC is to provide cooling of the sugar massecuite, gradual in the axial direction, and uniform in the radial direction, such that the massecuite follows an almostideal plug-flow pattern, with a sufficient residence time to allow the maximum possible exhaustion of the mother liquor by a steady growth of the entering sucrose crystals without the formation of new nuclei. 1.2. Prior Work. There are very few published studies on the modeling and simulation of cooling crystallizers for the exhaustion of the mother liquor in sugar massecuites. Batch cooling crystallizers were simulated by Maudarbocus and White,30 and horizontal continuous cooling crystallizers were modeled and simulated by Rouillard4,5 and Rein.31 Harris and co-workers32-36 formulated a transportphenomena model that consisted of a set of equations that govern the conservation of mass, momentum, and energy in an axis-symmetrical coordinate system and numerically solved it for the velocity and temperature distributions in a VCCC formerly used for the exhaustion of the mother liquor in low-grade cane sugar massecuites at the Victoria sugar factory12 in Australia, and corresponding to the old version of the BMA crystallizer already described in Section 1.1.10,11,13 Using computational fluid dynamics, it was shown that some major modifications to the design of the studied crystallizer were necessary to avoid a highly nonuniform cooling of the massecuite due to a significant shortcircuiting of flow between the entrance and the exit of the crystallizer. The proposed design modifications were determined to be successful in improving the actual performance of the crystallizer by narrowing the crystal residence time distribution and increasing the purity drop of the mother liquor. An attempt to model and simulate VCCC for the beet sugar industry has recently been made by Gros et al.37 Their steady-state model consisted of integral mass balances on the mother liquor and the sucrose crystals at each section into which the modeled crystallizer is divided, a crystal growth rate expression that includes surface reaction and volume diffusion mass-transfer coefficients, crystal population densities based on the Rosin-Rammler distribution, and a modified Stokes equation for hindered settling of particles in a viscous medium. For the formulation of their model, Gros et al.37 did not include energy balances either on the massecuite or the cooling water. Therefore, a constant supersaturation value was assumed along the crystallizer tower for the purpose of computing a temperature profile for the massecuite. This model was simulated for two cases differing for the entering massecuite weight fraction of crystals and the entering crystal average size. In both cases, Gros et al.37 found that the total linear velocity of the massecuite in the tower was much larger than
the settling velocities of all crystal sizes, leading to a practically uniform crystal residence time distribution. This result favors the assumption of an ideal plug-flow pattern in a well-designed VCCC. Even though profiles along the crystallizer height were obtained by Gros et al.37 for some quantities such as the massecuite temperature, the mother liquor dry substance content, and the massecuite weight fraction of crystals, simulation results were not compared to actual sugar factory data. A previous attempt to formulate a model of steadystate continuous cooling crystallizers for the cane sugar industry was made by Llano-Restrepo et al.38 The model included mass and energy balances, and empirical expressions for the average crystal linear growth rate and the massecuite-film local heat-transfer coefficient. 1.3. Overview. The purpose of this work is to formulate a steady-state one-dimensional distributed plug-flow model of well-designed VCCCs for the sugar industry and to provide a comparison of simulation results with a beet sugar factory data set available from the open literature. In Section 2, the mathematical model of the crystallizer is presented in detail. The model consists of seven ordinary differential equations (ODEs) that describe the changes (along the crystallizer height) of the following variables: massecuite weight fraction of crystals, mother liquor dry substance content, mother liquor purity, crystal average size, crystal size coefficient of variation, massecuite temperature, and cooling water temperature. These equations are derived, respectively, from a mass balance on crystallized sucrose, a mass balance on water in mother liquor, a mass balance on dissolved sucrose, the crystal population balances, an energy balance on the massecuite, and an energy balance on the cooling water. In this work, a rigorous combined kinetic approach39-41 is applied for the computation of the growth rate of the sucrose crystals, as needed for the solution of the crystal population balances. The correlations for the thermophysical and transport properties that are required for the implementation of the crystallizer model are given in Appendix A. The correlations to compute all the quantities that are involved in the crystal growth rate model are given in Appendix B, and those required to calculate the massecuite and cooling water local heattransfer coefficients are given in Appendix C. In Section 3, the method for the computational solution of the model is explained. In Section 4, as an illustrative example, the model is numerically solved for a VCCC (with vertically oscillating horizontal cooling coils (OVCC system)) of a beet sugar factory27 in Germany. Simulation results are compared with the actual reported performance of that VCCC, with regard to the total mother-liquor purity drop, the exiting massecuite temperature and weight fraction of crystals, the exiting mother liquor dry substance content, the overall mass rate of crystallization, the cooling water heat duty, the logarithmic mean temperature difference between the massecuite and cooling water, and the overall heattransfer coefficient in the crystallizer. The steady-state one-dimensional distributed plug-flow model formulated in the present work was observed to yield a very accurate description of the reported performance of such a well-designed VCCC. 2. Formulation of the Model From a consideration of the vertical crystallizer design and operation features that were reviewed in Section
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1, the model proposed in this work for the simulation of well-designed VCCCs used in the sugar industry is based on the following assumptions: (a) The crystallizer is in steady-state operation; that is, massecuite properties at each section of the crystallizer are time-independent. (b) All crystals have the same residence time in the crystallizer; that is, an ideal plug-flow pattern is assumed for the flow of the massecuite through the crystallizer. Therefore, massecuite properties change only along the crystallizer height. (c) No water evaporation or addition occurs in the crystallizer; that is, the amount of water in the mother liquor does not change along the crystallizer height. (d) The crystallizer is operated at a level of supersaturation well below the nucleation threshold; that is, nuclei (i.e., false grain) formation does not occur in the crystallizer. (e) The sucrose crystal population may be represented by a crystal average size (i.e., a characteristic linear dimension equivalent to the mean aperture from sieve analysis), and a crystal size coefficient of variation. The average crystal growth rate corresponds to the crystal average size. Crystal growth rate dispersion due to random fluctuations is modeled by the incorporation of a crystal growth rate diffusivity into the population balance equations. (f) Crystal volume and surface shape factors remain constant along the crystallizer height. (g) Impurities present in the mother liquor do not crystallize; that is, impurities in the mother liquor remain in solution and their amount does not change along the crystallizer height. 2.1. Mass Balance on Sucrose Crystals. Let z be the vertical distance from the top of the tower (where the massecuite enters the crystallizer) downward. Let (V˙ mc)z and (Fmc)z be the volumetric flow and the mass density of the massecuite, respectively, and let (ωc)z be the weight fraction of crystals in the massecuite at the distance z. A mass balance on sucrose crystals in a crystallizer element of infinitesimal height ∆z, is given by
(V˙ mcFmcωc)z + Jcrys AT∆z ) (V˙ mcFmcωc)z+∆z
(1)
where AT ) πDT2/4 is the cross section area of the crystallizer, with DT as the tower diameter, and Jcrys is the mass rate of sucrose crystallization per unit volume of crystallizer. Taking the limit of eq 1 as ∆z f 0 yields
d(V˙ mcFmcωc) ) AT Jcrys dz
(2)
Because of the conservation of the massecuite total mass in the crystallizer, the mass flow rate m ˘ mc ) V˙ mcFmc does not change along the tower height. As a consequence, eq 2 simplifies to the expression
dωc AT Jcrys ) dz m ˘ mc
at the distance z, defined as the following weight fraction:
Bsol )
mS + mI mS + mI + mW
(4)
where mS is the mass of dissolved sucrose, mI the mass of dissolved impurities, and mW the mass of water in solution. Because of assumption (c) in the model, a mass balance on water in the mother liquor in a crystallizer element of infinitesimal height ∆z is given by
[V˙ mcFmc(1 - ωc)(1 - Bsol)]z ) [V˙ mcFmc(1 - ωc)(1 - Bsol)]z+∆z (5) Taking the limit of eq 5 as ∆z f 0 yields
d [V˙ F (1 - ωc)(1 - Bsol)] ) 0 dz mc mc
(6)
Taking into account that the massecuite mass flow rate V˙ mcFmc is constant along the tower height, and by developing the derivative of the indicated product, eq 6 is simplified to
(1 - Bsol) dωc dBsol )dz (1 - ωc) dz
(7)
Substitution of eq 3 into eq 7 leads to
AT Jcrys (1 - Bsol) dBsol )dz m ˘ mc (1 - ωc)
(8)
which is the differential equation of change for the mother liquor dry substance content. 2.3. Mass Balance on Dissolved Sucrose. Let (Psol)z be the sucrose content (i.e., the purity) of the mother liquor at the distance z, defined as the following weight fraction:
Psol )
mS mS + mI
(9)
where mS is the mass of dissolved sucrose and mI is the mass of impurities in solution. Because of assumption (g) in the model, the mass balance on dissolved sucrose in a crystallizer element of infinitesimal height ∆z is given by
[V˙ mcFmc(1 - ωc)BsolPsol]z - Jcrys AT∆z ) [V˙ mcFmc(1 - ωc)BsolPsol]z+∆z (10) Taking the limit of eq 10 as ∆z f 0 yields
(3)
which is the differential equation of change for the massecuite weight fraction of crystals. 2.2. Mass Balance on Water in the Mother Liquor. Let (Bsol)z be the dry substance content (or refractometric Brix) of the solution (i.e., mother liquor)
d [V˙ F (1 - ωc)BsolPsol] ) - AT Jcrys dz mc mc
(11)
Taking into account that the massecuite mass flow rate m ˘ mc ) V˙ mcFmc is constant along the tower height, and by developing the derivative of the indicated product, eq 11 can be written in the form
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Bsol(1 - ωc)
dPsol dBsol AT Jcrys - Psol(1 - ωc) )+ dz m ˘ mc dz dωc (12) Bsol Psol dz
Substitution of eqs 3 and 8 into eq 12 finally yields
AT Jcrys (1 - Psol) dPsol )dz m ˘ mc Bsol(1 - ωc)
(13)
which is the differential equation of change for the mother liquor purity. Using the definitions for Bsol and Psol given in eqs 4 and 9, the following expression can be obtained for the nonsucrose-to-water ratio in the mother liquor (qNS/W,sol ) mI/mW):
qNS/W,sol )
Bsol(1 - Psol) 1 - Bsol
(14)
Because of assumptions (c) and (g) in the model, this ratio remains constant at the value of the right-hand side of eq 14 computed for the mother liquor entering the crystallizer (i.e., at z ) 0). Use of this expression as a relationship between Bsol and Psol would make simultaneous integration of eqs 8 and 13 unnecessary: integration of just one of them would suffice. However, for the present work, both equations are integrated simultaneously and the conservation of qNS/W,sol via eq 14 is used as a tool for checking the consistency of simulation results. 2.4. Crystal Population Balance Equations. The moment transformation of the crystal population balances for a steady-state continuous flow crystallizer without nucleation [assumption (d) in the model] yields the following set of equations of change for the moments of the crystal size distribution:35,42,43
vz
dµj ) jG h µj-1 + j ( j - 1)DG µj-2 dz
(for j g1) (15)
where vz is the crystal velocity in the downward direction along the crystallizer axial coordinate z, G h is an average crystal linear growth rate (corresponding to the average crystal size), DG is the crystal growth rate diffusivity, and µj, µj-1, µj-2 are the moments of order j, j - 1, and j - 2, respectively, where the jth moment is defined as
µj )
∫
∞ j L f (L) 0
dL
(16)
In eq 16, L is the crystal size (characteristic linear dimension) and f(L) is the crystal size distribution, which is normalized to unity (i.e., the condition µ0 ) 1 is fulfilled). Because of assumption (b) in the model and the results from the work of Gros et al.37 (see Section 1.2 above), the crystal velocity vz practically equals the massecuite velocity vmc for all crystal sizes:
vmc )
m ˘ mc ATFmc
(17)
A knowledge of the complete crystal size distribution (i.e., the entire set of distribution moments) is unneces-
sary in many systems of engineering interest.43 Because of assumption (e) in the model, the first- and secondorder moments µ1 and µ2 will suffice here to describe the crystal size distribution. From eq 16, it follows that
µ1 ) L h
(18)
where L h is the average crystal size. From eq 16 and the definition of the variance σL2 of the crystal size distribution, it can be shown that
h )2(cV2 + 1) µ2 ) (L
(19)
where cV ) σL/L h is the coefficient of variation of the crystal size distribution, with σL as the standard deviation of the distribution. Substitution of the normalization condition µ0 ) 1 and eqs 17 and 18 into the first-order moment population balance (eq 15 for j ) 1) yields
( )
ATFmc dL h ) G h dz m ˘ mc
(20)
which is the differential equation of change for the average crystal size. Substitution of eqs 17-20 into the second-order moment population balance (eq 15 for j ) 2) leads to
( )[
]
G h cV ATFmc DG dcV ) 2 dz m ˘ mc (L h ) cV L h
(21)
which is the differential equation of change for the crystal size coefficient of variation. The total mass of sucrose crystals in the massecuite (mc) is given by the expression44
h )3 mc ) ncFckV(L
(22)
where nc is the number of sucrose crystals, Fc is the mass density of sucrose, kV is the volume shape factor of sucrose crystals, and L h is the average crystal size, equivalent to the mean aperture from sieve analysis. From eq 22, it follows that the mass flow rate of crystals m ˘ mc(ωc)z at the distance z is related to the number flow of crystals (n˘ c)z by the expression
m ˘ mc(ωc)z ) (n˘ c)zFckV(L h )3
(23)
Because of assumption (d) in the model, the number flow of crystals remains constant along the crystallizer height, so that
(n˘ c)z )
m ˘ mcωc | | ) n˘ c,0 | FckV(L h )3 |z)0
(24)
where n˘ c,0 is the number flow of crystals in the entering massecuite. Taking the derivative of eq 23, with respect to the distance z, and assuming that the change in the sucrose crystal mass density by cooling is much smaller than the change in crystal size by growth, the following expression is obtained:
[
]
3n˘ c,0FckV(L h )2 dL dωc h ) dz m ˘ mc dz
(25)
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Substitution of eq 20 into eq 25 leads to
dωc 3n˘ c,0FckV(L h )2ATFmcG h ) 2 dz (m ˘ )
Combination of eqs 29 and 30 finally yields
(26)
G h )
mc
A comparison of eqs 3 and 26 yields
Jcrys )
[
]
3n˘ c,0FckV(L h )2Fmc G h m ˘ mc
(27)
which is the expression for the mass rate of crystallization per unit volume of crystallizer, in terms of the average crystal linear growth rate. 2.5. Crystal Growth Rate Model. A rigorous combined kinetic approach to the computation of the crystal growth rate in pure and impure sucrose solutions was formulated and validated by Ekelhof and Schliephake39-41 from a comprehensive experimental study in pilot plant crystallizers. This kinetic growth rate model takes into account three sequential steps for crystallization: (i) diffusion of the sucrose molecules from the solution through the flow boundary layer to the crystal surface; (ii) adsorption, desorption, and diffusion of the sucrose molecules on the crystal surface; and (iii) incorporation of the sucrose molecules into the crystal lattice. Based on the condition that the molecular flows are identical in each sequential step, the individual kinetic differential equations were combined to yield the following expression for the mass rate of crystallization (dmc/dt) per unit surface area of crystals (Ac), designated here as GM:
GM )
∆ceff (1/kVD) + [(1/kSD)(csol,sat/∆ceff)] + (1/kLI)
(28)
where kVD is the rate coefficient for volume diffusion [step (i)], kSD is the rate coefficient for surface diffusion [step (ii)], kLI is the rate coefficient for incorporation into the lattice [step (iii)], csol,sat is the sucrose mass concentration in the saturated solution, and ∆ceff (the so-called effective oversaturation) is the excess of sucrose mass concentration of the actual supersaturated solution, relative to the final value of solution concentration after a very long crystallization time. The expressions to compute the rate coefficients kVD, kSD, and kLI, the concentration csol,sat, and the effective oversaturation ∆ceff are given in detail in Appendix B. According to Ekelhof,41 this growth rate model is valid in the temperature range of 40 °C < T < 80 °C and the nonsucroseto-water ratio range of 1.5 < qNS/W,sol < 5. The total surface area Ac of the sucrose crystals in the massecuite is given by the expression44
h )2 Ac ) nckA(L
(29)
where nc is the number of sucrose crystals, kA is the surface area shape factor of sucrose crystals, and L h is the average crystal size. Taking the time derivative of eq 22, the following expression is obtained:
dmc ) 3ncFckV(L h )2G h dt
(30)
where G h ) dL h /dt is the average crystal linear growth rate.
( )
kA G 3FckV M
(31)
as the relationship between G h and the mass rate of crystallization per unit area GM computed from eq 28. In principle, because of the volume diffusion resistance (first term in the denominator of eq 28), the crystal linear growth rate is size-dependent. However, as will be noted from the simulation results presented in Section 4.2, the surface diffusion resistance (second term in the denominator of eq 28), which is size-independent, has a tendency to dominate the kinetics of sucrose crystallization by cooling. Thus, the linear growth rate G has a tendency to be weakly dependent on the crystal size L, and, in consequence, use of G h in the right-hand side of the moment population balance (eq 15) seems to be well-justified. 2.6. Energy Balance on the Massecuite. Let (Hmc)z be the specific enthalpy of the massecuite at the distance z, let q˘ cw be the heat transfer rate from the massecuite to the cooling water, let q˘ sa be the heat rate lost by natural convection to the surrounding air through the crystallizer outside wall, let w˘ s be the mechanical power supplied to the massecuite by the rotary stirring mechanism (in the traditional VCCC designs) or by the vertically oscillating cooling coils (in the most current VCCC design), and let q˘ crys be the heat rate evolved in the massecuite because of the sucrose crystallization, where the mechanical power and all heat rates are given in units of energy rate per unit height of crystallizer. An energy balance on the massecuite in a crystallizer element of infinitesimal height ∆z is given by
(V˙ mcFmcHmc)z - q˘ cw∆z - q˘ sa∆z + w˘ s∆z + q˘ crys∆z ) (V˙ mcFmcHmc)z+∆z (32) Taking into account that the massecuite mass flow rate (m ˘ mc ) V˙ mcFmc) is constant along the crystallizer height, and taking the limit of eq 32 as ∆z f 0, yields
- q˘ cw - q˘ sa + w˘ s + q˘ crys dHmc ) dz m ˘ mc
(33)
The specific isobaric heat capacity (CP,mc) of the massecuite is defined as
CP,mc )
( ) ∂Hmc ∂T
p,x b
(34)
where p is the pressure and b x ) (ωc, Bsol, Psol). Integration of eq 34 from an arbitrary reference temperature T0 (e.g., the temperature of the massecuite entering the crystallizer) (at which the massecuite specific enthalpy is set to zero) to the massecuite temperature Tmc at the distance z yields
Hmc )
∫TT
mc
0
CP,mc dT
(35)
Taking the derivative of eq 35 with respect to distance, the following expression is obtained:
dTmc dHmc ) CP,mc dz dz A comparison of eqs 33 and 36 yields
(36)
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dTmc - q˘ cw - q˘ sa + w˘ s + q˘ crys ) dz m ˘ mcCP,mc
(37)
which is the differential equation of change for the massecuite temperature. The heat-transfer rate q˘ cw from the massecuite to the cooling water at the distance z can be written in the form
q˘ cw ) htot
( )
Acw (Tmc - Tw) ZT
(38)
where Acw is the total cooling tube external surface area, ZT the total vertical distance traveled by the massecuite, and Tw the cooling water temperature at the distance z. The variable htot is the total local (i.e, at the distance z) heat-transfer coefficient between the massecuite and the cooling water, given by the expression
htot )
1 (39) (1/hmc) + [(do/2λt) ln(do/di)] + [do/(dihw)]
In eq 39, hmc is the massecuite-film local heat-transfer coefficient, do the outside diameter of the cooling tubes, λt the thermal conductivity of the tube wall, di the inside diameter of the cooling tubes, and hw the cooling waterfilm local heat-transfer coefficient. The correlations used to compute both of the local heat-transfer coefficients are given in detail in Appendix C. In the case of VCCC without external thermal insulation, the heat rate q˘ sa lost through the crystallizer outside wall (by natural convection to the external surrounding air) is given by the expression
q˘ sa ) hsa
()
AL (T - Tsa) ZT mc
(40)
where hsa is a total local (i.e, at the distance z) heattransfer coefficient between the massecuite and the external surrounding air, AL the lateral external surface area of the tower (AL ) πDTZT), and Tsa the temperature of the external surrounding air. In practice, this heat loss can be responsible for a significant portion (20%30%) of the overall massecuite cooling in the case of VCCC without external thermal insulation.34 The heat rate q˘ crys evolved in the massecuite due to crystallization can be written as the product of the differential change (along the crystallizer height) in the mass of crystallized sucrose and the specific heat (change of enthalpy) of crystallization ∆hcrys:
(
˘ mc q˘ crys ) m
)
dωc ∆hcrys dz
(41)
The use of eq 3 into eq 41 yields
q˘ crys ) AT Jcrys∆hcrys
(42)
2.7. Energy Balance on the Cooling Water. Let (V˙ w)z, (Fw)z, and (Hw)z be the volumetric flow, the mass density, and the specific enthalpy of the cooling water, respectively, at the distance z. An energy balance on the cooling water (that flows from the bottom to the top of the tower) in a crystallizer element of infinitesimal height ∆z is given by
(V˙ wFwHw)z+∆z + q˘ cw∆z ) (V˙ wFwHw)z
(43)
Taking into account that the mass flow rate of the cooling water (m ˘ w ) V˙ wFw) remains constant along the tower height, and taking the limit of eq 43 as ∆z f 0, yields
dHw q˘ cw )dz m ˘w
(44)
The specific isobaric heat capacity of the cooling water (CP,w) is defined as
CP,w )
( ) ∂Hw ∂T
p
(45)
Integration of eq 45 from an arbitrary reference temperature T0 (e.g., the temperature of the massecuite entering the crystallizer) (at which the water specific enthalpy is set to zero) to the cooling water temperature Tw at the distance z yields
Hw )
∫TT CP,w dT w
0
(46)
Taking the derivative of eq 46 with respect to distance, the following expression is obtained:
dHw dTw ) CP,w dz dz
(47)
A comparison of eqs 44 and 47 finally yields
q˘ cw dTw )dz m ˘ wCP,w
(48)
which is the differential equation of change for the temperature of the cooling water. 3. Solution of the Model The mathematical model formulated in Section 2 consists of seven ODEs of change for the same number of variables: eq 3 for the massecuite weight fraction of crystals, eq 8 for the mother liquor dry substance content, eq 13 for the mother liquor purity, eq 20 for the average crystal size, eq 21 for the crystal size coefficient of variation, eq 37 for the massecuite temperature, and eq 48 for the cooling water temperature. This set of equations must be solved along the vertical distance z from the top of the crystallizer tower (i.e., at z ) 0) to the bottom (i.e., at z ) ZT) with the help of seven subsidiary expressions: eq 24 for the number flow of crystals, eq 27 for the mass rate of crystallization per unit volume of crystallizer, eq 28 for the mass rate of crystallization per unit surface area of crystals, eq 31 for the average crystal linear growth rate, eqs 38 and 39 for the heat transfer rate from the massecuite to the cooling water, eq 40 for the heat rate lost through the crystallizer outside wall, and eq 42 for the heat rate evolved in the massecuite due to crystallization, together with the entire set of correlations for the thermophysical and transport properties (see Appendix A), for the quantities involved in the crystal growth rate model (see Appendix B), and for the heat-transfer coefficients (see Appendix C). The model is numerically solved by use of the fourth-order Runge-Kutta method,45 implementing the technique of step doubling to estimate the local error of each integrated variable. The model and its solution algorithm were translated into a
Ind. Eng. Chem. Res., Vol. 44, No. 24, 2005 9251
computer program coded in the FORTRAN 95 programming language. To start the integration of all seven ODEs, it is necessary to specify the values of the massecuite weight fraction of crystals ωc, the mother liquor dry substance content Bsol, the mother liquor h , the crystal size purity Psol, the crystal average size L coefficient of variation cV, and the massecuite temperature Tmc at z ) 0 (i.e, where the massecuite enters the crystallizer); in addition, because of the counter-flow arrangement of the massecuite and the cooling water, it is also necessary to assume a trial value for the unknown temperature Tw (at z ) 0) of the exiting cooling water. The correct value of that temperature is determined by performing several trial runs of the computer program until the known temperature of the entering cooling water (at z ) ZT) is matched at the end of the integration process. 4. Testing of the Model 4.1. Input Data. Operating and performance data for VCCCs are very scarce in the open literature. The only sufficiently comprehensive experimental report available was given by Dunker,27 for a Selwig & Lange6,26,27 VCCC (see Section 1.1) with vertically oscillating horizontal cooling coils (OVCC system), for the 1980 campaign at the Su¨derdithmarschen beet sugar factory in St. Michaelisdonn, Germany. This VCCC was designed with seven cooling blocks, evenly spaced along the tower height, with each cooling block consisting of four horizontal levels of coils, and each coil being comprised of a set of concentric tubular hexagonal rings (see Figures 1 and 2). This VCCC was initially built without thermal insulation for the outside wall. An almost-ideal plugflow pattern for the massecuite in this VCCC was shown from LiCl tracer field tests. The factory data given in Dunker’s report27 (hereafter designated as DR-27) were used for the testing of the model formulated in the present work. The input data required for the numerical solution of the model are given in Table 1. The total vertical distance traveled by the massecuite (ZT), the diameter of the crystallizer tower (DT), the effective volume occupied by the massecuite (VT), the total cooling tube surface area (Acw), and the outside diameter of the cooling tubes (do) were taken from Section 2.2 of DR27. The entering massecuite temperature (Tmc,0) was taken from Table 1 of DR-27. The temperature of the external surrounding air (Tsa) was set to the typical average value reported by Spoelstra.46 The purity of the entering mother liquor Psol,0 was taken from Table 2 of DR-27. The weight fraction of crystals in the entering massecuite ωc,0 was taken from Figure 8 of DR-27. The mechanical power supplied to the massecuite by the vertically oscillating coils (W ˙ s) was taken from Section 3.4 of DR-27 and corresponds to the power value required for the travel of the hydraulically driven piston in the downward direction. The other input data shown in Table 1 were not explicitly given in DR-27 and had to be determined from other reported data. The inside diameter of the cooling tubes (di) was obtained from the reported value of the outside diameter do and a reasonable value of the tube wall thickness. The longitudinal and transversal pitch ratios were determined from the drawings of the relevant patent.26 The vertical velocity of the oscillating cooling coils (vOVCC) was obtained by dividing the reported travel (0.58 m) of the hydraulically driven piston by the reported duration (43.2 s) of that
Table 1. Input Data for the Simulation of the Su 1 derdithmarschen Beet Sugar Factory’s Vertical Continuous Cooling Crystallizer input data
symbol
total vertical distance traveled by the massecuite diameter of the crystallizer tower effective volume occupied by the massecuite total cooling tube surface area outside diameter of the cooling tubes inside diameter of the cooling tubes longitudinal tube pitch ratio transversal tube pitch ratio vertical velocity of the oscillating cooling coils mechanical power supplied to the massecuite mass flow rate of the massecuite weight percentage of crystals in the entering massecuite dry substance content of the entering mother liquor purity of the entering mother liquor temperature of the entering massecuite temperature of the external surrounding air average size of the entering crystals size coefficient of variation of the entering crystals mass flow rate of cooling water temperature of the exiting cooling water
value
ZT
13 m
DT VT
4.5 m 170 m3
Acw do di pL/do pT/do vovcc
315 m2 0.102 m 0.094 m 2.6 3.2 1.34 cm/s
W ˙s
4.77 kW
m ˘ mc 100ωc,0
8705 kg/h 38.90%
100Bsol,0
90.08%
100Psol,0 Tmc,0 Tsa
59.72% 70.4 °C 30.0 °C
L h0 100cV,0
250 µm 25.0%
m ˘w Tw,f
6503 kg/h 51.17 °C
stroke in the downward direction. Using the value of the massecuite residence time (τmc ) 29.5 h) reported in Section 2.2 of DR-27, the mass flow rate of the massecuite m ˘ mc was obtained from the following definition for τmc:
Fmc,0VT m ˘ mc
τmc )
(49)
where VT is given in Table 1 and Fmc,0 is the mass density of the entering massecuite, calculated from eq A17 (from Appendix A) under the inlet conditions specified in Table 1 (i.e., ωc,0, Bsol,0, Psol,0, and Tmc,0). The dry substance content of the entering mother liquor Bsol,0 was not explicitly given in DR-27 and had to be obtained as follows. The weight fraction of crystals ωc, dry substance content Bmc, and purity Pmc of the massecuite are defined as the following ratios:
ωc )
mc mc + mS + mI + mW
(50)
mc + mS + mI mc + mS + mI + mW
(51)
mc + mS mc + mS + mI
(52)
Bmc )
Pmc )
where mc is the mass of sucrose crystals, mS the mass of dissolved sucrose, mI the mass of dissolved impurities, and mW the mass of water in solution. It can be shown that the combination of eq 9 for Psol with eqs 50-52 leads to the expression
Bmc ) ωc
(
)
1 - Psol Pmc - Psol
(53)
9252
Ind. Eng. Chem. Res., Vol. 44, No. 24, 2005
Because of the conservation of mass, the values of Bmc and Pmc must remain constant along the crystallizer. Therefore, by writing eq 53 at z ) 0 (i.e., for ωc,0 and Psol,0) and at z ) ZT (i.e, for ωc,f and Psol,f), and equating the two expressions, the following result for Pmc is obtained:
Pmc )
ωc,f (1 - Psol,f)Psol,0 - ωc,0(1 - Psol,0)Psol,f ωc,f (1 - Psol,f) - ωc,0(1 - Psol,0)
(54)
Substitution of the entering (Psol,0 ) 0.5972) and exiting (Psol,f ) 0.5414) mother liquor purities (given in Table 2 of DR-27) and the entering (ωc,0 ) 0.389) and exiting (ωc,f ) 0.456) massecuite weight fractions of crystals (taken from Figure 8 of DR-27) into eq 54 yields the massecuite purity: Pmc ) 0.7640. Substitution of either the entering or exiting values into eq 53 yields the massecuite dry substance content: Bmc ) 0.9394. It can be shown that the combination of eq 4 with eqs 50 and 51 leads to the expression
Bsol )
Bmc - ωc 1 - ωc
(55)
Substitution of Bmc ) 0.9394 and the DR-27 values ωc,0 ) 0.389 and ωc,f ) 0.456 into eq 55 finally yields the corresponding entering (Bsol,0 ) 0.9008) and exiting (Bsol,f ) 0.8886) factory values of the mother liquor dry substance content. The crystal average size and its coefficient of variation were not given in DR-27. Because the European beet sugar factories typically work with sucrose crystals in the average size range of 250-300 µm, the average size of the entering crystals (L h 0) was set to 250 µm. The coefficient of variation cV,0 of the entering crystal size distribution was set to a value of 0.25, which is the midpoint of the 0.20-0.30 range typically obtained in sugar factories. The coefficients a and b of the Wiklund relationship for the saturation coefficient ysat (see Section B.6 in the Appendix) were not given by Dunker27 in his report; therefore, these coefficients had to be regarded as adjustable parameters for the present work. The values a ) 0.17 and b ) 0.80 were determined to be required in the attempt to reproduce the reported performance27 of the Su¨derdithmarschen beet sugar factory’s VCCC as best as possible from simulation. These values of a and b turn out to be quite reasonable, considering the values already reported by Cronewitz10 for other beet molasses from Germany. The mass flow rate of cooling water (m ˘ w) was obtained from the following expression:
∫TT
m ˘w
w,f
w,0
CP,w dT ) Q˙ cw
(56)
where Tw,0 and Tw,f are the entering (at z ) ZT) and exiting (at z ) 0) cooling water temperatures, respectively, and
Q˙ cw ) UAcw∆Tlog
Table 2. Main Output Data from the Simulation of the Su 1 derdithmarschen Beet Sugar Factory’s Vertical Continuous Cooling Crystallizer output data
symbol
value
weight percentage of crystals in the exiting massecuite dry substance content of the exiting mother liquor purity of the exiting mother liquor temperature of the exiting massecuite average size of the exiting crystals size coefficient of variation of the exiting crystals temperature of the entering cooling water
100ωc,f
45.60%
100Bsol,f
88.86%
100Psol,f Tmc,f L hf 100cV,f
54.13% 40.87 °C 264 µm 25.1%
Tw,0
36.2 °C
performance data U ) 34.7 W m-2 K-1 and ∆Tlog ) 10.4 °C (given in Table 1 of DR-27) into eq 57 yields Q˙ cw ) 113.7 kW. Substitution of eq A37 (from Appendix A) and the values Tw,0 ) 36.2 °C and Tw,f ) 51.2 °C (given in Table 1 of DR-27) into eq 56 finally yields the value of m ˘ w shown in Table 1 of the present work. As explained in Section 3, the exiting cooling water temperature Tw,f (given in Table 1) was obtained by performing several runs of the computer program until the value of the entering cooling water temperature, which was measured at the factory (Tw,0 ) 36.2 °C), was matched at the end of the integration process. 4.2. Output Data. Corresponding to the input data given in Section 4.1, the output data (obtained by running the simulation program coded for this work) are reported in Tables 2-4. With an integration step size of ∆z ) 0.01 m, implementation of the technique of step doubling for the fourth-order Runge-Kutta method (see Section 3) yielded a maximum local integration error of the order of 10-15. Consistency of the simulation results was checked via eq 14 by monitoring the conservation of the nonsucrose-to-water ratio (qNS/W,sol) at the entering value of 3.66. Even though the crystal average size L h shows a small increase of 14 µm from inlet to outlet, the massecuite weight percentage of crystals increases from 38.9% to 45.6%, yielding an overall mass rate of crystallization of 584 kg/h. The crystal size coefficient of variation cV undergoes an almost imperceptible increase from the inlet to the outlet (from 25.0% to 25.1%), indicating a practically negligible broadening of the crystal size distribution throughout the crystallizer. The simulation results reported for the VCCC are not sensitive to a change of the assumed value of cV for the entering massecuite when that value is set in the range typically obtained at sugar factories (from 20% to 30%). Assumption (d) in the model (i.e., nucleation does not occur in the crystallizer) can be tested from the condition given by Kelly and Drinnen47 for the minimum crystal surface area required to keep crystallization below the nucleation threshold. In terms of the crystal average size, that condition can be written as follows:
L h
