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Flash Evaporation: Modelling and Constraint Formulation
0263–8762/03/$23.50+0.00 # Institution of Chemical Engineers Trans IChemE, Vol 81, Part A, October 2003
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FLASH EVAPORATION: MODELLING AND CONSTRAINT FORMULATION A. BOUCHAMA, P. SE´BASTIAN and J.-P. NADEAU Laboratoire Energe´tique et Phe´nome`nes de Transfert L.E.P.T.-ENSAM, CNRS UMR 8508, Esplanade des Arts et Me´tiers, Talence, France
T
he ash evaporation process has received growing recent interest in the agro-industry, particularly for the post-harvest treatment of grapes. This paper is devoted to the design and the optimization of a two-stage ash evaporator. An embodiment design approach of this process based on constraint satisfaction problem-solving techniques is presented. The aim is to perform the identi cation and the constraint formulation of the process model, especially the coupling between the two-stage evaporation chamber and the high and low pressures condensers. As an application, the functioning of a ash evaporator is studied using ow rates and temperatures de ned on real number intervals. The problem is treated by a CSP solver involving the global physical model of the two-stage ash evaporator and the functional requirements. The analysis has been validated by several experimentations performed on a ash evaporator for the treatment of vintage and leads to a decision support system. Keywords: embodiment design; constraints satisfaction problem; condensation; condenser; vacuum; evaporation.
INTRODUCTION Several works concerned by the treatment of citrus fruits by ash evaporation have been developed (Ageron et al., 1995; Escudier et al., 1998) to increase the extraction of avours from certain exotic fruits. The ash evaporation process is used in the wine industries to concentrate and to improve wine quality. The vintage is treated after being harvested and heated at a temperature ranging between 70 and 90¯ C. Entering the ash evaporator, the vintage is suddenly cooled by a violent ash evaporation causing the simultaneous extraction of the colour and the tannins. The vacuum effect favours the destruction of the alcoholoxydases, which are responsible for the fruit juices tanning effect. This paper is focused on the thermodynamic aspects of the process functioning using a cascade evaporation. Some previous works (Sebastian and Nadeau, 2000, 2002) have been performed on the thermodynamic analysis of a mono-staged pilot performance. Based on these works, a new two-stage pilot plant has been developed aiming to reduce the apparatus volume, to increase his production capacity and to improve the control of the cooling temperature of the vintage. This new pilot includes a oat that controls the product ow between the two stages of the separation chamber (Nadeau et al., 2001). This paper is focused on the constraint formulation step (Fischer et al., 2002; Pahl and Beitz, 1996) in the embodiment design phase (see Figure 1). The text mainly concerns the constraints formulation of the thermodynamic behaviour
of the coupling between the two-stage ash chamber and the high and low pressure condensers.
PROCESS PRESENTATION The experimental pilot plant (see Figure 2) includes two subsets: (1) The rst subset ( 1 in Figure 2) is the product treatment zone of the installation, including a two-stage separation chamber and a oat that regulates the vintage ow rate between the high and low pressure stages. The product enters this part of the system at temperatures between 70 and 90 C and at a pressure of approximately 30 mbar. Owing to the vacuum effect, the product is aspired in the high pressure stage and some of the liquid phase is abruptly vaporized ( ash evaporation), releasing some droplets; the pressure reaches approximately 100 mbar. The level of the product increases in the high pressure stage and actuates the oat. A similar scenario occurs in the low pressure stage. (2) The second subset ( 2 in Figure 2) is devoted to the vapour treatment and includes mist eliminators on the vapour outlets of the separation chamber (which enables recovery of droplets generated during the ash evaporation), a condenser at each stage and a vacuum setting device composed of a liquid-ring vacuum pump and an
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model precision is necessarily low. Owing to the coupling of several physical phenomena (condensation, evaporation, droplet releasing, vapour ow mode) and the uncertainties linked to the physical parameters, the experimental model must take into account important uncertainties (about 15–20%). Uncertainties appear through the variable domains of the model, which are de ned on real number intervals. The CSP solver used in the numerical treatment of the model is based on interval analysis and naturally takes into account this kind of variable. Global Model Reduction
Figure 1. Design approach.
ejector (which lowers the pressure in the system below the critical pressure of the vacuum pump).
Heat and mass transfer balance in the separation chamber Previous studies (Sebastian and Nadeau, 2000, 2002) based on the investigation of a mono-stage pilot behaviour have showed that the ash evaporation phenomenon may be modelled through a simple heat and mass transfer balance in the separation chamber (see Figure 4). The product is almost instantaneously cooled at the saturation temperature inside the separation chamber. High pressure stage: qv1 Dhv1 ˆ qpi (CplpiTpi ¡ Cpl1 Tv1 )
(1)
Tp12 ˆ Tv1
(3)
qv2 Dhv2 ˆ qp12(Cpl1 Tv1 ¡ Cpl2 Tv2 )
(4)
Tpo ˆ Tv2
(6)
qpi ¡ qv1 ¡ qp12 ˆ 0
(2)
Low pressure stage:
MODELLING This paragraph aims to assess and model the physical phenomena that determine the evaporator performances (see Figure 3). Two models are de ned and compared: a theoretical and an experimental model. The real behaviour of the pilot plant components appears to be somewhat different from the behaviour predicted by theoretical models in the literature (Sebastian and Nadeau, 2002; Rohsenow et al., 1985; Chen, 1961; Berman, 1968). Therefore, whereas the theoretical model is precise and inaccurate (the model resolution leads to precise numerical results but rather far from the experimental observations), the experimental
qp12 ¡ qv2 ¡ qpo ˆ 0
Heat and mass transfer balance inside the condensers The shell and tube condensers used in the experimental pilot were arranged according to a rotated square (45¯ ) tube layout pattern (130 tubes). The NTU-e reduced model [see
Figure 2. Installation diagram, sharing sections of the plant.
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Figure 3. Design approach.
equation (7)] resulted from the heat and mass transfer balance formulation in a condenser (see Figure 5) with the following assumptions: (i) the vapour is saturated (ii) the vapour pressure drop is neglected; and (iii) the energy loss through the shell is neglected. ³ ´ 1 ˆ NUTcl ln (7) 1 ¡ ecl
with Tclo ¡ Tcli Tv:sat ¡ Tcli kcd ¢ Ae,int NUTcl ˆ qcl ¢ Cpcl 1 1 1 dint ˆ ‡ kcd hcl hv:cd dext
ecl ˆ
(8) (9) (10)
Equation (10) assumes that the thermal resistances of the wall and the fouling phenomena are negligible. Identi cation of the Condensation Transfer Coef cient In the literature, most of the condensation transfer coef cients are derived from the Nusselt coef cient. Nusselt (Rohsenow et al., 1985; Kakac¸ et al., 1981) determined an
Figure 4. Separation chamber.
Figure 5. Condenser.
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average heat transfer coef cient in a stagnant vapour around a horizontal tube analysing, on a differential element of the condensate lm, the equilibrium between gravity forces (decreased by buoyancy forces) and forces due to shear stresses. In this study two expressions are taken into account: the Nusselt formulation coupled with the Kern model (1958) leads to the de nition of an average heat transfer coef cient in a stagnant vapour for a tube bundle; the Chen (1961) formulation is a modi cation of the Nusselt formulation considering a low velocity vapour. According to Chen, the condensation transfer coef cient is: Á !0:25 g ¢ (rvliq ¡ rv:sat ) ¢ rvliq ¢ l3vliq ¢ Dh0v hv,cd ˆ B ¢ mvliq ¢ xcondenser ¢ (Tv:sat ¡ Tw )
(11)
xcondenser ˆ Ntubes vert dext (12) ³ ´ T ¡ Tw B ˆ 0:725 ¢ 1 ‡ 0:2 ¡ v:sat ¢ (Ntubes vert ¡ 1) Dhv (13) The correction factor, B, of the condensation transfer coef cient is due to the accumulation of condensate on the tubes and is usually attributed to phenomena such as external vibrations causing ripples at the liquid surface or liquid splashing on the tubes. Berman Analysis Berman (1968) compared numerous experimental values of condensation transfer coef cients and computed values using the Nusselt formulation on a horizontal smooth tube in a stagnant vapour. It results from this study that, for a temperature difference (Tv.sat 7 Tw) lower than 15¯ C, the ratio hv.cd experimental=hv.cd Nusselt varies between 0.5 and 1.8. For temperature differences higher than 15¯ C, the ratio remains close to 1. A similar analysis was performed to compare experimental results with theoretical values. The results of this study are given in the following section. EXPERIMENTAL ANALYSIS
Figure 6. Energy released during condensation (Energy.v) and energy transmitted to the cooling liquid (Energy.cl) vs. the inlet temperatures of the product.
Figure 6 displays the vapour condensation energy of and the cooling liquid heating energy vs. the inlet temperature of the product. According to this gure, the difference between these energies appears to be negligible and the energy losses are lower than the measurement uncertainties. Above, by writing the conservation energy law in the condenser, the vapour was been supposed to be saturated inside the system. In order to validate this assumption through experimentation, the saturated temperatures were calculated using pressure measurements (carried out at the inlet and the outlet side of the condenser) and compared with the temperature measurements. Figure 7 shows the temperature measurements (Tvi, Tvo) and the saturated temperatures [Tv.sat(Pi), Tv.sat(Po)] vs. the inlet temperature of the product. The vapour appears to be saturated. Models Comparison Identi cation of the experimental condensation transfer coef cients The heating effectiveness of the cooling liquid was calculated from the sensor temperature measurements and the global condensation coef cient (kcd) was derived from the NTU-e model. By introducing the heat transfer
Several series of tests have been performed on the experimental pilot plant in order to identify each component behaviour and to validate the assumptions related to the reduced global physical model. The experimental results being presented in this part only concern one condenser and thereafter are used for the high pressure and low pressure stages. Each point of the graphs presented in this paragraph corresponds to an experiment. Every pressure, temperature and ow rate measurement is performed once the system reaches the stationary operating conditions. Global Model Validation By writing heat and mass transfer balances in condensers, we assume that all the energy released during condensation is transmitted to the cooling liquid without energy loss through the shell (the energy loss is about 5% of the condensation energy). Based on experimental measurements, Trans IChemE, Vol 81, Part A, October 2003
Figure 7. Temperature measurements (Tvi, Tvo) and saturation temperatures vs. the inlet temperature of the product.
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coef cient of the cooling liquid (calculated by the Sieder–Tate correlation (Kuppan, 2000; Bu¨yu¨kalaca and Jackson, 1998), we de ned the experimental condensation transfer coef cient. Analysis Figures 8 and 9 display respectively the theoretical and experimental condensation transfer coef cients, vs. the temperature difference (Tv.sat 7 Tw). Theoretical coef cients are more precise than the experimental results but appear to be rather inaccurate. In Figure 9, the lines surrounding the experimental measurements correspond to the uncertainties interval limits on the experimental condensation transfer coef cient. This interval is easily taken into account by the model as this model is treated using a constraint satisfaction problem solver. To take into account the complexity of the condensate ow pattern [in a tube bundle arranged in a rotated square (45¯ )] and the interference between two close vertical tubes rows, an average vertical tube number of two staggered rows was used; the number of tubes was 22 in our case. Equation (14) may be used to calculate the tube number using the inner diameter of shell, the inner diameter of tube, the tube pitch and staggered row con guration: r 2 pD2int Ntubes vert ˆ (14) p 4 Condensation transfer coef cients estimated using ‘Nusselt with stagnant vapour’ and ‘Nusselt for low vapour velocities’ have been compared to our experimental values. Figure 10 highlights that: The results obtained using the Nusselt model for low vapour velocities are closer to the condenser experimental behaviour than those obtained considering a stagnant vapour. For small temperature differences (Tv.sat 7 Tw) the ratio hv.cd experimental=hv.cd Nusselt is close to 1. Indeed, due to our process characteristics, the vapour ow rate is thoroughly linked to the temperatures Tv.sat and Tw. More to the point the temperature Tv,sat depends on the heated product temperature Tpi as the condenser and the separation chamber are linked. When the inlet product temperature
Figure 8. Theoretical heat transfer coef cients [Equation (11)] for low vapour velocities vs. (Tv.sat 7 Tw).
Figure 9. Experimental condensation transfer coef cient vs.(Tv.sat 7 Tw).
reaches 80 or 90 C, the vapour release in the separation chamber occurs at high temperature and the vapour velocity at the condenser inlet is high. For lower heated vintage temperatures, and thus low vapour velocities, the transfer coef cient ratios are close to 1. Our study supplements Berman’s (1968) publication as low vapour velocities for low temperatures differences (Tv.sat 7 Tw) are considered in this paper. A model based on the low velocity vapour Nusselt model and on our experimental results was investigated. This model takes into account our experimental measurements by deriving a parameter B as a linear function of the wall tube temperature. This parameter is de ned on the interval [Bmin (Tw), Bmax (Tw)]. hv:cd
Á !0:25 g(rvliq ¡ rvsat )rvliq l3vliq Dh0v ˆ B(Tw ) mvliq xcondenser(Tv:sat ¡ Tw )
(15)
Wall temperature: Tw ˆ Tv:sat ¡
hcl ¢ (Tv:sat ¡ Tcl ) hv:cd ‡ hcl
(16)
Figure 10. Comparison between experimental values of the heat transfer coef cient and computed values using the Nusselt model for a stagnant vapour and the Nusselt model for a low vapour velocity on smooth horizontal tube bundles.
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FLASH EVAPORATION Experimental uncertainty is taken into account by the parameter B(Tw) Bmax (Tw )iB(Tw )iBmin (Tw )
(17)
width Bmin (Tw ) ˆ ¡0:000577Tw ‡ 0:113
(18)
Bmax (Tw ) ˆ ¡0:0032Tw ‡ 0:626
(19)
and
CONSTRAINT SATISFACTION PROBLEM SOLVING Numerical Resolution Tool A numeric constraint satisfaction problem (CSP) is de ned by a set of variables, domains and constraints taking into account discrete or continuous domains and relations involving equalities, inequalities or logical rules (Lhomme, 1993). A CSP solver nds the problem solutions by determining values for each variable domain satisfying all the constraints. CSP solver performance is limited by the number of variables and constraints they are able to treat. Based on the NTU-e method, the physical model being developed in this paper includes few variables even though it is close to the real behaviour of our experimental pilot plant. A CSP solver called a Constraint explorer (CE) was used, which was developed within the framework of the RNTL project ‘CO2’ (www.industrie.gouv.fr=rntl=AAP2001= Fiches_Resume=CO2; Re´seau National des Technologies Logicielles ‘COnception par COntraintes’). This software was developed by the companies Dassault Aviation and CRIL Technology, the computer science laboratory of Paris VI (LIP6, UMR CNRS 7606), the research institute in computer science of Nantes (IRIN, Upres-ea 2157), the laboratory of engineering processes and industrial services (LIPSI-estia) and our team within the energy and transfer phenomena laboratory (LEPT-ensam, UMR CNRS 8508). Here we are interested in the software structuring and the embodiment design for mechanical and process engineering problems.
CE is a numeric CSP solver based on interval analysis (Jaulin et al., 2001; Benhamou and Older, 1997; Granvilliers and Monfroy, 2000) and treating variables de ned on integer or real number spaces by gradually restricting their value domains in coherence with the problem constraints. The mathematical model bases underlying the algorithm involved in the solver functioning have been presented by Lhomme (1993). The propagation procedure (HC3-like algorithm) is discussed by Cleary (1987). The solutions provided by the constraint solver are an over-approximation of the solver solution space since the resolution algorithm is based on inner approximation. This means that the solver determines sub domains for the values of the problem variables in which a real design may be found. The design problem discussed in this paper does not involve any criteria optimization. The numerical simulation results displayed in this paper were computed using a personal computer (Pentium 4, 1.6 GHz, 256 Mo Ram), and required about 10 s of the CPU time.
Functional Requirements An industrial pilot plant adapted to the treatment of vintage in the area of Bordeaux (France) was developed in collaboration with Les Vignobles Andre´ Lurton Company. Owing to the dispersion of the harvest zones of grapes in this area, the pilot plant must be transportable and must operate on several sites. As wine growers do not use the same equipment and do not have the same production objectives, the inlet, outlet and functional parameters of the pilot plant (see Figure 11) are variables of the design problem. Therefore, the variable domains of the evaporator model were de ned on sets. In particular, most of the continuous variables were de ned on real number intervals. The evaporator design problem was treated by solving simultaneously the process model and the functional requirements (see Table 1) using a CSP solver. The solver exhaustively determined the whole problem solution, which is discussed in the following section.
Figure 11. Functional parameters diagram.
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Table 1. Inlet, outlet and functional parameters. Inlet parameters Functional parameters Outlet parameters Geometrical parameters
Tpi ˆ {70, 72, . . . , 88} qpi ˆ {0:2, 0:3, . . . , 0:9} qcli1 ˆ qcli2 ˆ 0:64 Tcli1 ˆ Tcli2 ˆ 18 Tpo ˆ [25, 30] Ae ˆ 10
CSP Solving Results Figure 12 presents the vapour temperature values Tv1 and Tv2 at the steady state of the process functioning. These temperatures correspond respectively to the high and low pressure stages of the separation chamber. They depend on the inlet product temperature. As the numerical solutions delivered by the CSP solver are de ned on real number intervals, the points plotted on the gure correspond to the middles of the solution intervals. It may be pointed out that the temperature Tv2 is also the outlet temperature of the product. We observe that: contrary to the temperature Tv2, Tv1 is not constrained by the functional requirements and varies inside a broader domain; Tv2 is always higher than 30 C, which means the solver did not nd solutions involving a mono staged evaporator; the product mass ow rate must be lower than 0.4 kg s 1 considering the functional requirements of the design problem; for such a mass ow rate, the product inlet temperature must be lower than 81 C; as the condenser ef ciency is xed by the cooling liquid characteristics (mass ow rate and temperature), the pilot is likely to cool the product too much; thus, when the product mass ow rate is 0.3 kg s 1 (the lower value de ned by the functional requirements), the product inlet temperature must be higher than 75 C; the CSP solver used to treat the evaporator design problem is based on inner-approximation solving techniques. This means in particular that the found solutions correspond to
Inlet product temperature (¯ C) Product ow rate (kg s¡1) Cooling liquid mass ow rate (kg s¡1) Cooling liquid temperature (¯ C) The cooling liquid is water Vapour and product temperature in the low pressure stage (¯ C) Total heat transfer area (m2) Shell and tubes condenser Co-current
sub-domains of some continuous variables de ned on intervals such as the transfer coef cients.
CONCLUSION A constraint formulation approach for the modelling of coupling phenomena between two stages of a separation chamberand the two condensersof a cascade ash evaporation process has been developed and implemented. An analysis of the global transfer coef cients in the condensers has been performed and an experimental model has been compared with several models resulting from the literature. The behaviour model of the process has been validated through an experimental approach jointly with the transfer coef cients. Using CSP solving techniques, a simple design problem has been treated taking into account the global physical model of the evaporator and operational requirements based on variable functioning conditions. The solutions put forward correspond to a small installation that is able to treat about one ton per hour of product. This design problem is a rst phase towards the realization of a global design model of a cascade ash evaporation process. APPENDIX This appendix summarizes the model constraints involved in the Constraint Satisfaction Problem. These constraints correspond to the physical laws describing the physical phenomena, the geometric parameters and the operating conditions of the design problem.
Figure 12. Vapour temperatures in the high and low pressure stages vs. the inlet temperature of the product. The plotted points correspond to the middle of the solution domains.
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FLASH EVAPORATION
Physical Laws High pressure stage chamber: balance energy qv1 Dhv1 ˆ qpi (Cplpi Tpi ¡ Cpl1 Tv1 )
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Nusselt number Nu ˆ
Reynolds number
mass conservation qpi ¡ qv1 ¡ qp12 ˆ 0 assumption on the evaporation temperature Tp12 ˆ Tv1 Low pressure stage chamber:
hcl dint lcl
Re ˆ
rcl vcl dint mcl
Prandtl number Pr ˆ
mcl Cpcl lcl
balance energy qv2 Dhv2 ˆ qp12(Cpl1 Tv1 ¡ Cpl2 Tv2 ) mass conservation qp12 ¡ qv2 ¡ qpo ˆ 0 assumption on the evaporation temperature Tpo ˆ Tv2
wall tube temperature Tw ˆ Tv:sat ¡
Condensers:
balance energy ³ ´ 1 ˆ NUTcl ln 1 ¡ ecl effectiveness of heating of the liquid cooling T ¡ Tcli ecl ˆ clo Tv:sat ¡ Tcli Units Transfer Number of the cooling liquid NUTcl ˆ
number of the tubes in vertical bank r 2 pD2int Ntubes vert ˆ p 4
kcd ¢ Ae,int qcl ¢ Cpcl
total coef cient transfer 1 1 1 dint ˆ ‡ kcd hcl hv:cd dext coef cients transfer in vapour condensation Á !0:25 g(rvliq ¡ rvsat )rvliql3vliq Dh0v hv,cd ˆ B(Tw ) mvliq Ntubes vert (Tv:sat ¡ Tw)
hcl ¢ (Tv:sat ¡ Tcl ) hv:cd ‡ hcl
parameter taking into account the experimental uncertainties Bmax (Tw )iB(Tw )iBmin (Tw ) Bmin (Tw ) ˆ ¡0:000577Tw ‡ 0:113 Bmax (Tw ) ˆ ¡0:0032Tw ‡ 0:626
Geometrical Parameters Shell diameter, 0.4 m Tube length, 1.485 m Number of tubes, 130 Tube diameter internal, 0.0149 m Tube diameter external, 0.0181 m Angle of tubes arrangement, 45
correction of the latent heat due to the condensates cooling Dh0v ˆ Dhv ‡ 0:68Cpvliq (Tv:sat ¡ Tw ) Sieder state correlation Á !0:14 ³ ´0:5 dint m cl Nu ˆ 1:86 ¢ Re ¢ Pr ¢ ¢Pr1=3 ¢ L mvliq Trans IChemE, Vol 81, Part A, October 2003
Figure A1. Staggered rows con guration.
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Functional Requirements Inlet parameters: Tpi ˆ {70, 72, . . . , 88},
qpi ˆ {0:2, 0:3, . . . , 0:9},
¯
C kg s
qcli1 ˆ qcli2 ˆ 0:64, kg s¡1 Tcli1 ˆ Tcli2 ˆ 18, ¯ C Outlet parameters: ¯
C
NOMENCLATURE Cp d D Dh P L p Q q T U k h Ae B N x NTU Pr Re Nu
heat capacity, J kg 1 K 1 tube diameter, m shell diameter, m latent heat, J kg 1 pressure, Pa tube length, m tube pitch, m volumic ow rate, m3 s 1 mass ow rate, kg s 1 temperature, K velocity, m s 1 global heat transfer coef cient, W m 2 K 1 local heat transfer coef cient, W m 2 K 1 total heat transfer area, m2 correction factor number equivalent diameter number of transfer units Prandtl number Reynolds number Nusselt number
Greek symbols e heating effectiveness of the cooling liquid l conduction coef cient, W m 1 K 1 m dynamic viscosity, kg m 1 s 1 r density, kg m 3 Subscripts v cl p 1 2 sat i o cdst l vliq cd int ext
vapour cooling liquid product high pressure stage low pressure stage saturation inlet outlet condensates liquid liquid vapour condensation interior exterior
condenser tube wall vertical
REFERENCES ¡1
Functional parameters:
Tpo ˆ [25, 30],
w vert
Ageron, D., Escudier, J.L., Abbal, Ph. and Moutonet M., 1995, Pre´traitement des raisins par ash de´tente sous vide pousse´, Rev Franc¸ Oenol, 153: 50–54. Benhamou, F. and Older, W., 1997, Applying interval arithmetic to real integer and boolean constraints, Logic Program, 32(1): 1–24. Berman, L.D., 1968, Heat transfer during lm condensation of vapour on horizontal tubes in traverse ow, AEC-tr-6877. Bu¨yu¨lkalaca, O. and Jackson, D., 1998, The correction to take account of variable property effects on turbulent forced convection to water in a pipe, Int J Heat Mass Transfer, 41(4–5): 665–669. Chen, M.M., 1961, J Heat Transfer, 83: 55. Cleary, J.G., 1987, Logical arithmetic, Fut Comput Syst, 2(2): 125–149. Escudier, J.L., Mikolajczak, M. and Moutounet M., 1998, Pre´-traitement des raisins par ash de´tente sous vide et caracte´ ristiques des vins, J Int Sci Vigne Vin, Trait Phys Mouˆts Vin, 105–110. Fischer, X., Nadeau, J.P. and Sebastian, P., 2002, Decision support in integrated mechanical design through qualitative constraint, IDMME 2000, Selected Paper Book (Kluwer Academic, Dordrecht). Granvilliers, L. and Monfroy, E., 2000, Declarative modelling of constraint propagation Strategies, in Proceedings of the First Biennial International Conference on Advances in Information Systems (ADVIS’2000), LNCS, Izmir (Springer, Berlin). Jaulin, L., Kieffer, M., Didrit, O. and Walter, E., 2001, Applied Interval Analysis (Springer, London). Kakac¸, S., Bergles, A.E. and Mayinger, F., 1981, Heat Exchangers, ThermoHydraulic Fundamentals and Design (McGraw-Hill, New York, USA). Kern, D.Q., 1958, Mathematical development of tube loading in horizontal condensers, AIChE J, 4(2): 157–160. Kuppan, T., 2000, Heat Exchanger Design Handbook (Marcel Dekker, New York, USA). Lhomme, O., 1993, Consistency techniques for numeric CSPs, in Proceedings of IJCA193, Chambery, pp 232–238. Nadeau, J.P., Sebastian, P., Cadiot, D., Callede, D. and Gaillard, M., 2001, Installation de refroidissement par vaporisation partielle a´ basse pression d’un jus chauffe´, CNRS=Les Vignobles Andre´ Lurton patent, Brevet FR 01 07177. Pahl, G. and Beitz W., 1996, Engineering Design, a Systematic Approach (Springer, Berlin, Germany). Rohsenow, W.M., Hartnett, J.P. and Ganic, E.N., 1985, Handbook of Heat Transfer Fundamentals (McGraw-Hill, New York, USA). Sebastian, P. and Nadeau, J.P., 2000, Etude expe´rimentale et nume´rique d’un syste´me de ash de´tente pour le refroidissement de la vendange, Proceedings of Congre´s franc¸ais de Thermique (SFT 2000), Lyon (Elsevier, Oxford, UK), pp 41–46. Sebastian, P. and Nadeau, J.P., 2002, Experiments and modeling of falling jet ash evaporators for vintage treatment, Int J Therm Sci, 41: 269–280.
ADDRESS Correspondence concerning this paper should be addressed to Dr A. Bouchama, Laboratoire Energe´tique et phe´nome`nes de Transfert L.E.P.T.-ENSAM, CNRS UMR 8508, Esplanade des Arts et Me´tiers, 33405 Talence Cedex, France. E-mail: [email protected] The paper was presented at the 9th Congress of the French Society of Chemical Engineering held in Saint-Nazaire, France, 9–11 September 2003. The manuscript was received 10 February 2003 and accepted for publication after revision 28 August 2003.
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www.ingentaselect.com=titles=02638762.htm
FLASH EVAPORATION: MODELLING AND CONSTRAINT FORMULATION A. BOUCHAMA, P. SE´BASTIAN and J.-P. NADEAU Laboratoire Energe´tique et Phe´nome`nes de Transfert L.E.P.T.-ENSAM, CNRS UMR 8508, Esplanade des Arts et Me´tiers, Talence, France
T
he ash evaporation process has received growing recent interest in the agro-industry, particularly for the post-harvest treatment of grapes. This paper is devoted to the design and the optimization of a two-stage ash evaporator. An embodiment design approach of this process based on constraint satisfaction problem-solving techniques is presented. The aim is to perform the identi cation and the constraint formulation of the process model, especially the coupling between the two-stage evaporation chamber and the high and low pressures condensers. As an application, the functioning of a ash evaporator is studied using ow rates and temperatures de ned on real number intervals. The problem is treated by a CSP solver involving the global physical model of the two-stage ash evaporator and the functional requirements. The analysis has been validated by several experimentations performed on a ash evaporator for the treatment of vintage and leads to a decision support system. Keywords: embodiment design; constraints satisfaction problem; condensation; condenser; vacuum; evaporation.
INTRODUCTION Several works concerned by the treatment of citrus fruits by ash evaporation have been developed (Ageron et al., 1995; Escudier et al., 1998) to increase the extraction of avours from certain exotic fruits. The ash evaporation process is used in the wine industries to concentrate and to improve wine quality. The vintage is treated after being harvested and heated at a temperature ranging between 70 and 90¯ C. Entering the ash evaporator, the vintage is suddenly cooled by a violent ash evaporation causing the simultaneous extraction of the colour and the tannins. The vacuum effect favours the destruction of the alcoholoxydases, which are responsible for the fruit juices tanning effect. This paper is focused on the thermodynamic aspects of the process functioning using a cascade evaporation. Some previous works (Sebastian and Nadeau, 2000, 2002) have been performed on the thermodynamic analysis of a mono-staged pilot performance. Based on these works, a new two-stage pilot plant has been developed aiming to reduce the apparatus volume, to increase his production capacity and to improve the control of the cooling temperature of the vintage. This new pilot includes a oat that controls the product ow between the two stages of the separation chamber (Nadeau et al., 2001). This paper is focused on the constraint formulation step (Fischer et al., 2002; Pahl and Beitz, 1996) in the embodiment design phase (see Figure 1). The text mainly concerns the constraints formulation of the thermodynamic behaviour
of the coupling between the two-stage ash chamber and the high and low pressure condensers.
PROCESS PRESENTATION The experimental pilot plant (see Figure 2) includes two subsets: (1) The rst subset ( 1 in Figure 2) is the product treatment zone of the installation, including a two-stage separation chamber and a oat that regulates the vintage ow rate between the high and low pressure stages. The product enters this part of the system at temperatures between 70 and 90 C and at a pressure of approximately 30 mbar. Owing to the vacuum effect, the product is aspired in the high pressure stage and some of the liquid phase is abruptly vaporized ( ash evaporation), releasing some droplets; the pressure reaches approximately 100 mbar. The level of the product increases in the high pressure stage and actuates the oat. A similar scenario occurs in the low pressure stage. (2) The second subset ( 2 in Figure 2) is devoted to the vapour treatment and includes mist eliminators on the vapour outlets of the separation chamber (which enables recovery of droplets generated during the ash evaporation), a condenser at each stage and a vacuum setting device composed of a liquid-ring vacuum pump and an
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model precision is necessarily low. Owing to the coupling of several physical phenomena (condensation, evaporation, droplet releasing, vapour ow mode) and the uncertainties linked to the physical parameters, the experimental model must take into account important uncertainties (about 15–20%). Uncertainties appear through the variable domains of the model, which are de ned on real number intervals. The CSP solver used in the numerical treatment of the model is based on interval analysis and naturally takes into account this kind of variable. Global Model Reduction
Figure 1. Design approach.
ejector (which lowers the pressure in the system below the critical pressure of the vacuum pump).
Heat and mass transfer balance in the separation chamber Previous studies (Sebastian and Nadeau, 2000, 2002) based on the investigation of a mono-stage pilot behaviour have showed that the ash evaporation phenomenon may be modelled through a simple heat and mass transfer balance in the separation chamber (see Figure 4). The product is almost instantaneously cooled at the saturation temperature inside the separation chamber. High pressure stage: qv1 Dhv1 ˆ qpi (CplpiTpi ¡ Cpl1 Tv1 )
(1)
Tp12 ˆ Tv1
(3)
qv2 Dhv2 ˆ qp12(Cpl1 Tv1 ¡ Cpl2 Tv2 )
(4)
Tpo ˆ Tv2
(6)
qpi ¡ qv1 ¡ qp12 ˆ 0
(2)
Low pressure stage:
MODELLING This paragraph aims to assess and model the physical phenomena that determine the evaporator performances (see Figure 3). Two models are de ned and compared: a theoretical and an experimental model. The real behaviour of the pilot plant components appears to be somewhat different from the behaviour predicted by theoretical models in the literature (Sebastian and Nadeau, 2002; Rohsenow et al., 1985; Chen, 1961; Berman, 1968). Therefore, whereas the theoretical model is precise and inaccurate (the model resolution leads to precise numerical results but rather far from the experimental observations), the experimental
qp12 ¡ qv2 ¡ qpo ˆ 0
Heat and mass transfer balance inside the condensers The shell and tube condensers used in the experimental pilot were arranged according to a rotated square (45¯ ) tube layout pattern (130 tubes). The NTU-e reduced model [see
Figure 2. Installation diagram, sharing sections of the plant.
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Figure 3. Design approach.
equation (7)] resulted from the heat and mass transfer balance formulation in a condenser (see Figure 5) with the following assumptions: (i) the vapour is saturated (ii) the vapour pressure drop is neglected; and (iii) the energy loss through the shell is neglected. ³ ´ 1 ˆ NUTcl ln (7) 1 ¡ ecl
with Tclo ¡ Tcli Tv:sat ¡ Tcli kcd ¢ Ae,int NUTcl ˆ qcl ¢ Cpcl 1 1 1 dint ˆ ‡ kcd hcl hv:cd dext
ecl ˆ
(8) (9) (10)
Equation (10) assumes that the thermal resistances of the wall and the fouling phenomena are negligible. Identi cation of the Condensation Transfer Coef cient In the literature, most of the condensation transfer coef cients are derived from the Nusselt coef cient. Nusselt (Rohsenow et al., 1985; Kakac¸ et al., 1981) determined an
Figure 4. Separation chamber.
Figure 5. Condenser.
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average heat transfer coef cient in a stagnant vapour around a horizontal tube analysing, on a differential element of the condensate lm, the equilibrium between gravity forces (decreased by buoyancy forces) and forces due to shear stresses. In this study two expressions are taken into account: the Nusselt formulation coupled with the Kern model (1958) leads to the de nition of an average heat transfer coef cient in a stagnant vapour for a tube bundle; the Chen (1961) formulation is a modi cation of the Nusselt formulation considering a low velocity vapour. According to Chen, the condensation transfer coef cient is: Á !0:25 g ¢ (rvliq ¡ rv:sat ) ¢ rvliq ¢ l3vliq ¢ Dh0v hv,cd ˆ B ¢ mvliq ¢ xcondenser ¢ (Tv:sat ¡ Tw )
(11)
xcondenser ˆ Ntubes vert dext (12) ³ ´ T ¡ Tw B ˆ 0:725 ¢ 1 ‡ 0:2 ¡ v:sat ¢ (Ntubes vert ¡ 1) Dhv (13) The correction factor, B, of the condensation transfer coef cient is due to the accumulation of condensate on the tubes and is usually attributed to phenomena such as external vibrations causing ripples at the liquid surface or liquid splashing on the tubes. Berman Analysis Berman (1968) compared numerous experimental values of condensation transfer coef cients and computed values using the Nusselt formulation on a horizontal smooth tube in a stagnant vapour. It results from this study that, for a temperature difference (Tv.sat 7 Tw) lower than 15¯ C, the ratio hv.cd experimental=hv.cd Nusselt varies between 0.5 and 1.8. For temperature differences higher than 15¯ C, the ratio remains close to 1. A similar analysis was performed to compare experimental results with theoretical values. The results of this study are given in the following section. EXPERIMENTAL ANALYSIS
Figure 6. Energy released during condensation (Energy.v) and energy transmitted to the cooling liquid (Energy.cl) vs. the inlet temperatures of the product.
Figure 6 displays the vapour condensation energy of and the cooling liquid heating energy vs. the inlet temperature of the product. According to this gure, the difference between these energies appears to be negligible and the energy losses are lower than the measurement uncertainties. Above, by writing the conservation energy law in the condenser, the vapour was been supposed to be saturated inside the system. In order to validate this assumption through experimentation, the saturated temperatures were calculated using pressure measurements (carried out at the inlet and the outlet side of the condenser) and compared with the temperature measurements. Figure 7 shows the temperature measurements (Tvi, Tvo) and the saturated temperatures [Tv.sat(Pi), Tv.sat(Po)] vs. the inlet temperature of the product. The vapour appears to be saturated. Models Comparison Identi cation of the experimental condensation transfer coef cients The heating effectiveness of the cooling liquid was calculated from the sensor temperature measurements and the global condensation coef cient (kcd) was derived from the NTU-e model. By introducing the heat transfer
Several series of tests have been performed on the experimental pilot plant in order to identify each component behaviour and to validate the assumptions related to the reduced global physical model. The experimental results being presented in this part only concern one condenser and thereafter are used for the high pressure and low pressure stages. Each point of the graphs presented in this paragraph corresponds to an experiment. Every pressure, temperature and ow rate measurement is performed once the system reaches the stationary operating conditions. Global Model Validation By writing heat and mass transfer balances in condensers, we assume that all the energy released during condensation is transmitted to the cooling liquid without energy loss through the shell (the energy loss is about 5% of the condensation energy). Based on experimental measurements, Trans IChemE, Vol 81, Part A, October 2003
Figure 7. Temperature measurements (Tvi, Tvo) and saturation temperatures vs. the inlet temperature of the product.
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coef cient of the cooling liquid (calculated by the Sieder–Tate correlation (Kuppan, 2000; Bu¨yu¨kalaca and Jackson, 1998), we de ned the experimental condensation transfer coef cient. Analysis Figures 8 and 9 display respectively the theoretical and experimental condensation transfer coef cients, vs. the temperature difference (Tv.sat 7 Tw). Theoretical coef cients are more precise than the experimental results but appear to be rather inaccurate. In Figure 9, the lines surrounding the experimental measurements correspond to the uncertainties interval limits on the experimental condensation transfer coef cient. This interval is easily taken into account by the model as this model is treated using a constraint satisfaction problem solver. To take into account the complexity of the condensate ow pattern [in a tube bundle arranged in a rotated square (45¯ )] and the interference between two close vertical tubes rows, an average vertical tube number of two staggered rows was used; the number of tubes was 22 in our case. Equation (14) may be used to calculate the tube number using the inner diameter of shell, the inner diameter of tube, the tube pitch and staggered row con guration: r 2 pD2int Ntubes vert ˆ (14) p 4 Condensation transfer coef cients estimated using ‘Nusselt with stagnant vapour’ and ‘Nusselt for low vapour velocities’ have been compared to our experimental values. Figure 10 highlights that: The results obtained using the Nusselt model for low vapour velocities are closer to the condenser experimental behaviour than those obtained considering a stagnant vapour. For small temperature differences (Tv.sat 7 Tw) the ratio hv.cd experimental=hv.cd Nusselt is close to 1. Indeed, due to our process characteristics, the vapour ow rate is thoroughly linked to the temperatures Tv.sat and Tw. More to the point the temperature Tv,sat depends on the heated product temperature Tpi as the condenser and the separation chamber are linked. When the inlet product temperature
Figure 8. Theoretical heat transfer coef cients [Equation (11)] for low vapour velocities vs. (Tv.sat 7 Tw).
Figure 9. Experimental condensation transfer coef cient vs.(Tv.sat 7 Tw).
reaches 80 or 90 C, the vapour release in the separation chamber occurs at high temperature and the vapour velocity at the condenser inlet is high. For lower heated vintage temperatures, and thus low vapour velocities, the transfer coef cient ratios are close to 1. Our study supplements Berman’s (1968) publication as low vapour velocities for low temperatures differences (Tv.sat 7 Tw) are considered in this paper. A model based on the low velocity vapour Nusselt model and on our experimental results was investigated. This model takes into account our experimental measurements by deriving a parameter B as a linear function of the wall tube temperature. This parameter is de ned on the interval [Bmin (Tw), Bmax (Tw)]. hv:cd
Á !0:25 g(rvliq ¡ rvsat )rvliq l3vliq Dh0v ˆ B(Tw ) mvliq xcondenser(Tv:sat ¡ Tw )
(15)
Wall temperature: Tw ˆ Tv:sat ¡
hcl ¢ (Tv:sat ¡ Tcl ) hv:cd ‡ hcl
(16)
Figure 10. Comparison between experimental values of the heat transfer coef cient and computed values using the Nusselt model for a stagnant vapour and the Nusselt model for a low vapour velocity on smooth horizontal tube bundles.
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(17)
width Bmin (Tw ) ˆ ¡0:000577Tw ‡ 0:113
(18)
Bmax (Tw ) ˆ ¡0:0032Tw ‡ 0:626
(19)
and
CONSTRAINT SATISFACTION PROBLEM SOLVING Numerical Resolution Tool A numeric constraint satisfaction problem (CSP) is de ned by a set of variables, domains and constraints taking into account discrete or continuous domains and relations involving equalities, inequalities or logical rules (Lhomme, 1993). A CSP solver nds the problem solutions by determining values for each variable domain satisfying all the constraints. CSP solver performance is limited by the number of variables and constraints they are able to treat. Based on the NTU-e method, the physical model being developed in this paper includes few variables even though it is close to the real behaviour of our experimental pilot plant. A CSP solver called a Constraint explorer (CE) was used, which was developed within the framework of the RNTL project ‘CO2’ (www.industrie.gouv.fr=rntl=AAP2001= Fiches_Resume=CO2; Re´seau National des Technologies Logicielles ‘COnception par COntraintes’). This software was developed by the companies Dassault Aviation and CRIL Technology, the computer science laboratory of Paris VI (LIP6, UMR CNRS 7606), the research institute in computer science of Nantes (IRIN, Upres-ea 2157), the laboratory of engineering processes and industrial services (LIPSI-estia) and our team within the energy and transfer phenomena laboratory (LEPT-ensam, UMR CNRS 8508). Here we are interested in the software structuring and the embodiment design for mechanical and process engineering problems.
CE is a numeric CSP solver based on interval analysis (Jaulin et al., 2001; Benhamou and Older, 1997; Granvilliers and Monfroy, 2000) and treating variables de ned on integer or real number spaces by gradually restricting their value domains in coherence with the problem constraints. The mathematical model bases underlying the algorithm involved in the solver functioning have been presented by Lhomme (1993). The propagation procedure (HC3-like algorithm) is discussed by Cleary (1987). The solutions provided by the constraint solver are an over-approximation of the solver solution space since the resolution algorithm is based on inner approximation. This means that the solver determines sub domains for the values of the problem variables in which a real design may be found. The design problem discussed in this paper does not involve any criteria optimization. The numerical simulation results displayed in this paper were computed using a personal computer (Pentium 4, 1.6 GHz, 256 Mo Ram), and required about 10 s of the CPU time.
Functional Requirements An industrial pilot plant adapted to the treatment of vintage in the area of Bordeaux (France) was developed in collaboration with Les Vignobles Andre´ Lurton Company. Owing to the dispersion of the harvest zones of grapes in this area, the pilot plant must be transportable and must operate on several sites. As wine growers do not use the same equipment and do not have the same production objectives, the inlet, outlet and functional parameters of the pilot plant (see Figure 11) are variables of the design problem. Therefore, the variable domains of the evaporator model were de ned on sets. In particular, most of the continuous variables were de ned on real number intervals. The evaporator design problem was treated by solving simultaneously the process model and the functional requirements (see Table 1) using a CSP solver. The solver exhaustively determined the whole problem solution, which is discussed in the following section.
Figure 11. Functional parameters diagram.
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Table 1. Inlet, outlet and functional parameters. Inlet parameters Functional parameters Outlet parameters Geometrical parameters
Tpi ˆ {70, 72, . . . , 88} qpi ˆ {0:2, 0:3, . . . , 0:9} qcli1 ˆ qcli2 ˆ 0:64 Tcli1 ˆ Tcli2 ˆ 18 Tpo ˆ [25, 30] Ae ˆ 10
CSP Solving Results Figure 12 presents the vapour temperature values Tv1 and Tv2 at the steady state of the process functioning. These temperatures correspond respectively to the high and low pressure stages of the separation chamber. They depend on the inlet product temperature. As the numerical solutions delivered by the CSP solver are de ned on real number intervals, the points plotted on the gure correspond to the middles of the solution intervals. It may be pointed out that the temperature Tv2 is also the outlet temperature of the product. We observe that: contrary to the temperature Tv2, Tv1 is not constrained by the functional requirements and varies inside a broader domain; Tv2 is always higher than 30 C, which means the solver did not nd solutions involving a mono staged evaporator; the product mass ow rate must be lower than 0.4 kg s 1 considering the functional requirements of the design problem; for such a mass ow rate, the product inlet temperature must be lower than 81 C; as the condenser ef ciency is xed by the cooling liquid characteristics (mass ow rate and temperature), the pilot is likely to cool the product too much; thus, when the product mass ow rate is 0.3 kg s 1 (the lower value de ned by the functional requirements), the product inlet temperature must be higher than 75 C; the CSP solver used to treat the evaporator design problem is based on inner-approximation solving techniques. This means in particular that the found solutions correspond to
Inlet product temperature (¯ C) Product ow rate (kg s¡1) Cooling liquid mass ow rate (kg s¡1) Cooling liquid temperature (¯ C) The cooling liquid is water Vapour and product temperature in the low pressure stage (¯ C) Total heat transfer area (m2) Shell and tubes condenser Co-current
sub-domains of some continuous variables de ned on intervals such as the transfer coef cients.
CONCLUSION A constraint formulation approach for the modelling of coupling phenomena between two stages of a separation chamberand the two condensersof a cascade ash evaporation process has been developed and implemented. An analysis of the global transfer coef cients in the condensers has been performed and an experimental model has been compared with several models resulting from the literature. The behaviour model of the process has been validated through an experimental approach jointly with the transfer coef cients. Using CSP solving techniques, a simple design problem has been treated taking into account the global physical model of the evaporator and operational requirements based on variable functioning conditions. The solutions put forward correspond to a small installation that is able to treat about one ton per hour of product. This design problem is a rst phase towards the realization of a global design model of a cascade ash evaporation process. APPENDIX This appendix summarizes the model constraints involved in the Constraint Satisfaction Problem. These constraints correspond to the physical laws describing the physical phenomena, the geometric parameters and the operating conditions of the design problem.
Figure 12. Vapour temperatures in the high and low pressure stages vs. the inlet temperature of the product. The plotted points correspond to the middle of the solution domains.
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Physical Laws High pressure stage chamber: balance energy qv1 Dhv1 ˆ qpi (Cplpi Tpi ¡ Cpl1 Tv1 )
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Nusselt number Nu ˆ
Reynolds number
mass conservation qpi ¡ qv1 ¡ qp12 ˆ 0 assumption on the evaporation temperature Tp12 ˆ Tv1 Low pressure stage chamber:
hcl dint lcl
Re ˆ
rcl vcl dint mcl
Prandtl number Pr ˆ
mcl Cpcl lcl
balance energy qv2 Dhv2 ˆ qp12(Cpl1 Tv1 ¡ Cpl2 Tv2 ) mass conservation qp12 ¡ qv2 ¡ qpo ˆ 0 assumption on the evaporation temperature Tpo ˆ Tv2
wall tube temperature Tw ˆ Tv:sat ¡
Condensers:
balance energy ³ ´ 1 ˆ NUTcl ln 1 ¡ ecl effectiveness of heating of the liquid cooling T ¡ Tcli ecl ˆ clo Tv:sat ¡ Tcli Units Transfer Number of the cooling liquid NUTcl ˆ
number of the tubes in vertical bank r 2 pD2int Ntubes vert ˆ p 4
kcd ¢ Ae,int qcl ¢ Cpcl
total coef cient transfer 1 1 1 dint ˆ ‡ kcd hcl hv:cd dext coef cients transfer in vapour condensation Á !0:25 g(rvliq ¡ rvsat )rvliql3vliq Dh0v hv,cd ˆ B(Tw ) mvliq Ntubes vert (Tv:sat ¡ Tw)
hcl ¢ (Tv:sat ¡ Tcl ) hv:cd ‡ hcl
parameter taking into account the experimental uncertainties Bmax (Tw )iB(Tw )iBmin (Tw ) Bmin (Tw ) ˆ ¡0:000577Tw ‡ 0:113 Bmax (Tw ) ˆ ¡0:0032Tw ‡ 0:626
Geometrical Parameters Shell diameter, 0.4 m Tube length, 1.485 m Number of tubes, 130 Tube diameter internal, 0.0149 m Tube diameter external, 0.0181 m Angle of tubes arrangement, 45
correction of the latent heat due to the condensates cooling Dh0v ˆ Dhv ‡ 0:68Cpvliq (Tv:sat ¡ Tw ) Sieder state correlation Á !0:14 ³ ´0:5 dint m cl Nu ˆ 1:86 ¢ Re ¢ Pr ¢ ¢Pr1=3 ¢ L mvliq Trans IChemE, Vol 81, Part A, October 2003
Figure A1. Staggered rows con guration.
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Functional Requirements Inlet parameters: Tpi ˆ {70, 72, . . . , 88},
qpi ˆ {0:2, 0:3, . . . , 0:9},
¯
C kg s
qcli1 ˆ qcli2 ˆ 0:64, kg s¡1 Tcli1 ˆ Tcli2 ˆ 18, ¯ C Outlet parameters: ¯
C
NOMENCLATURE Cp d D Dh P L p Q q T U k h Ae B N x NTU Pr Re Nu
heat capacity, J kg 1 K 1 tube diameter, m shell diameter, m latent heat, J kg 1 pressure, Pa tube length, m tube pitch, m volumic ow rate, m3 s 1 mass ow rate, kg s 1 temperature, K velocity, m s 1 global heat transfer coef cient, W m 2 K 1 local heat transfer coef cient, W m 2 K 1 total heat transfer area, m2 correction factor number equivalent diameter number of transfer units Prandtl number Reynolds number Nusselt number
Greek symbols e heating effectiveness of the cooling liquid l conduction coef cient, W m 1 K 1 m dynamic viscosity, kg m 1 s 1 r density, kg m 3 Subscripts v cl p 1 2 sat i o cdst l vliq cd int ext
vapour cooling liquid product high pressure stage low pressure stage saturation inlet outlet condensates liquid liquid vapour condensation interior exterior
condenser tube wall vertical
REFERENCES ¡1
Functional parameters:
Tpo ˆ [25, 30],
w vert
Ageron, D., Escudier, J.L., Abbal, Ph. and Moutonet M., 1995, Pre´traitement des raisins par ash de´tente sous vide pousse´, Rev Franc¸ Oenol, 153: 50–54. Benhamou, F. and Older, W., 1997, Applying interval arithmetic to real integer and boolean constraints, Logic Program, 32(1): 1–24. Berman, L.D., 1968, Heat transfer during lm condensation of vapour on horizontal tubes in traverse ow, AEC-tr-6877. Bu¨yu¨lkalaca, O. and Jackson, D., 1998, The correction to take account of variable property effects on turbulent forced convection to water in a pipe, Int J Heat Mass Transfer, 41(4–5): 665–669. Chen, M.M., 1961, J Heat Transfer, 83: 55. Cleary, J.G., 1987, Logical arithmetic, Fut Comput Syst, 2(2): 125–149. Escudier, J.L., Mikolajczak, M. and Moutounet M., 1998, Pre´-traitement des raisins par ash de´tente sous vide et caracte´ ristiques des vins, J Int Sci Vigne Vin, Trait Phys Mouˆts Vin, 105–110. Fischer, X., Nadeau, J.P. and Sebastian, P., 2002, Decision support in integrated mechanical design through qualitative constraint, IDMME 2000, Selected Paper Book (Kluwer Academic, Dordrecht). Granvilliers, L. and Monfroy, E., 2000, Declarative modelling of constraint propagation Strategies, in Proceedings of the First Biennial International Conference on Advances in Information Systems (ADVIS’2000), LNCS, Izmir (Springer, Berlin). Jaulin, L., Kieffer, M., Didrit, O. and Walter, E., 2001, Applied Interval Analysis (Springer, London). Kakac¸, S., Bergles, A.E. and Mayinger, F., 1981, Heat Exchangers, ThermoHydraulic Fundamentals and Design (McGraw-Hill, New York, USA). Kern, D.Q., 1958, Mathematical development of tube loading in horizontal condensers, AIChE J, 4(2): 157–160. Kuppan, T., 2000, Heat Exchanger Design Handbook (Marcel Dekker, New York, USA). Lhomme, O., 1993, Consistency techniques for numeric CSPs, in Proceedings of IJCA193, Chambery, pp 232–238. Nadeau, J.P., Sebastian, P., Cadiot, D., Callede, D. and Gaillard, M., 2001, Installation de refroidissement par vaporisation partielle a´ basse pression d’un jus chauffe´, CNRS=Les Vignobles Andre´ Lurton patent, Brevet FR 01 07177. Pahl, G. and Beitz W., 1996, Engineering Design, a Systematic Approach (Springer, Berlin, Germany). Rohsenow, W.M., Hartnett, J.P. and Ganic, E.N., 1985, Handbook of Heat Transfer Fundamentals (McGraw-Hill, New York, USA). Sebastian, P. and Nadeau, J.P., 2000, Etude expe´rimentale et nume´rique d’un syste´me de ash de´tente pour le refroidissement de la vendange, Proceedings of Congre´s franc¸ais de Thermique (SFT 2000), Lyon (Elsevier, Oxford, UK), pp 41–46. Sebastian, P. and Nadeau, J.P., 2002, Experiments and modeling of falling jet ash evaporators for vintage treatment, Int J Therm Sci, 41: 269–280.
ADDRESS Correspondence concerning this paper should be addressed to Dr A. Bouchama, Laboratoire Energe´tique et phe´nome`nes de Transfert L.E.P.T.-ENSAM, CNRS UMR 8508, Esplanade des Arts et Me´tiers, 33405 Talence Cedex, France. E-mail: [email protected] The paper was presented at the 9th Congress of the French Society of Chemical Engineering held in Saint-Nazaire, France, 9–11 September 2003. The manuscript was received 10 February 2003 and accepted for publication after revision 28 August 2003.
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