- Email: [email protected]
Modelling local heat and mass transfer in food slabs due to air jet impingement
Journal of Food Engineering 78 (2007) 230–237 www.elsevier.com/locate/jfoodeng
Modelling local heat and mass transfer in food slabs due to air jet impingement M.V. De Bonis, G. Ruocco
*
CFDfood, DITEC, Universita´ degli studi della Basilicata, Campus Macchia Romana, 85100 Potenza, Italy Received 24 March 2005; accepted 26 September 2005 Available online 17 November 2005
Abstract Adequate design and verification of drying by a forced convection enhanced technique (gaseous jet impingement) can be carried out by numerical analysis, but customary transport calculations need to be integrated to account for complex (simultaneous) energy and mass transfer. In this paper the available procedures are reviewed and applied to food substrates: temperature, mass concentration and velocity fields are computed even for non-linear couplings (i.e. when local species concentration depends on temperature) using a specific solution strategy. Validity and limitations of the adopted notation and related integration into a proprietary software are discussed. A comparison is also brought forth with the available literature data. 2005 Elsevier Ltd. All rights reserved. Keywords: Jet impingement heat and mass transfer; Transient CFD; Food dehydration; Local water activity; Evaporation kinetics
1. Introduction Among the available forced convection processes, the gaseous jet impingement (JI) is frequently used for its excellent heat and mass transfer characteristics, where localized, controlled and rapid surface transfer is desirable. Studies on JI have been performed extensively over the past five decades, nevertheless the coupling and interdependence between simultaneous mass/heat transfer and fluid dynamics still needs to be fully analyzed, with special reference to local distribution of transfer rates on substrates of different shape (see for example Olsson, Ahrne´, & Tra¨ga˚rdh, 2004; Sarghini & Ruocco, 2004). Additional difficulties arise when in the subject impinged solid a multi-phase transport is allowed. JI can be successfully employed in drying or dehydration of foods by forced air convection, a most energy-intensive process, which is
*
Corresponding author. E-mail address: [email protected] (G. Ruocco).
0260-8774/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2005.09.032
commonly used in food engineering to extend food shelflife. Here, the majority of the unbound water normally present in a food is removed by applying heat under controlled conditions. The reduction in relative humidity (water activity) inhibits microbial growth and enzyme kinetics, also resulting in transport and storage costs reduction. In turn, drying may cause deterioration of both eating quality and nutritional value of the food. In food engineering, the design and operation of drying equipment aim to minimize these changes by selection of appropriate conditions. Dehydration involves a rather complex combination of application of heat and removal of moisture from a food medium (Barbosa-Ca`novas & Vega-Mercado, 1996; Fellows, 2000). In addition to air temperature and relative humidity, the rate of moisture removal is controlled by the air velocity. When hot air is locally blown over a moist food, water vapor diffuses through the boundary layer and is carried away (Fig. 1). A water vapor pressure gradient is therefore established from the moist interior to the external food surfaces. The boundary layer acts as a barrier to both heat transfer and water vapor removal during drying.
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
231
Nomenclature aw c cp d D Ea k K K0 l p r R S Sct t T u v x
water activity (dimensionless) concentration (g solid/g water) specific heat at constant pressure (kJ/kg K) jet diameter (m) mass diffusivity (m2/s) activation energy (kJ/mol) turbulent kinetic energy (J/kg) rate of production of water vapor mass per unit volume (1/s) reference rate constant (1/s) length (m) pressure (Pa) radial coordinate (m) universal gas constant (kJ/mol K) sucrose equivalent conversion factor (dimensionless) turbulent Schmidt number (dimensionless) time (s) temperature (K) x-component velocity (m/s) r-component velocity (m/s) axial coordinate (m)
Fig. 1. A drying air jet onto a food slab: flow of moisture during process.
Localized forced convection patterns, such as in JI, contribute to boundary layer destruction, hence in increase of moisture removal, while equipment costs are kept to a minimum. Nonetheless, local conditions have to be carefully monitored to ensure product uniformity. Porous and multi-phase media drying has been long speculated, and a large number of studies are available, following the seminal works be De Vries (1958) and Whitaker (1977), that carry complete descriptions of physics, as in Stanish, Schajer, and Kayihan (1986), Ben Nasrallah and Perre (1988) and Chen and Pei (1989). Boukadida and Ben Nasrallah (1995) first described a two-dimensional
Greek / k m q x
mass fraction (dimensionless) thermal conductivity (W/m K) kinematic viscosity (m2/s) density (kg/m3) specific dissipation rate (1/s)
Superscripts f fluid side, across the interface s solid side, across the interface Subscripts 0 initial a air i, m i-species in the mixture j jet s solid sr solid, along r sx solid, along x t turbulent v water vapor w liquid water 1r undisturbed, along r
progress of bulk convective drying of clay, but no contributions have been found in the available literature, with reference to localized convective drying in extended (i.e. at least 2D) porous media. A first contribution of JI drying of a moist, porous solid was presented by Francis and Wepfer (1996) with a thorough transient physical analysis, yet limited as one-dimensional. Furthermore, this model does not allow for a truly coupled transfer mechanism, as the surface transfer rates are externally implied. This limitations are also found in more recent works by Moreira (2001) and Braud et al. (2001), who first applied JI to food drying. Within this framework, the numerical analysis by a computational fluid dynamics (CFD) approach can gain importance as it leads to complete multi-dimensional and transient process description, yet ancillary calculation procedures are still needed to account for fully coupled energy and mass transfer. The present work has been performed with the specific aim to merge an in-house computation routine into a proprietary software (FLUENT 6.1 UserÕs Guide, 2003) in order to incorporate the multi-dimensional, transient calculation of an evaporation process due a gaseous impinging, heating jet.
2. Problem formulation A drying process of a thin food substrate is devised by using JI: hot, fully turbulent air is discharged through a
232
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
Fig. 2. Characteristic regions in submerged, confined JI. Fig. 3. Geometry and nomenclature.
nozzle with a given velocity distribution (upon recognition of the nozzle internal geometry). Upon impact of the free jet on the substrate, the characteristic stagnation and wall jet regions are first formed, then a secondary pattern can be identified in the lateral regions (Fig. 2). If the impinged food is water saturated, liquid can be converted into vapor depending on the heat perturbation front within the substrate, which can then be modelled as a multi-phase medium (a mixture of bulk solid, liquid water and water vapor). Liquid water normally moves from the interior of the substrate to the surface (1) by capillary forces, and (2) by diffusion caused by differences in the concentration of solutes at the surface and in the interior, while water vapor moves (3) by diffusion in air spaces within the substrate caused by vapor pressure gradients. In this work, for the sake of simplicity, Fickian diffusion is assumed only for both liquid and water vapor, nevertheless a highly nonuniform drying still results within the substrate, and a given process time can lead to local overheating and/or incomplete liquid conversion. 2.1. Driving assumptions In Fig. 3 the subject geometry in cylindrical coordinates is reported (only half section of the domain is considered due to the geometry and transfer symmetries). The domain under scrutiny consists in two interfaced fluid-and-substrate multi-species sub-domains, sharing the biomaterialÕs exposed surface. The fluid is a binary system comprising of 1. water vapor (v) 2. air (a) while the substrate is a ternary system comprising of 1. water vapor (v) 2. liquid water (w) 3. solid matter proper (s).
The following additional assumptions are adopted: 1. The flow is axisymmetric, with constant properties and incompressible (negligible pressure work and kinetic energy). 2. The viscous heat dissipation is neglected. 3. Due to the adopted flow regime, no body force is accounted for. 4. No-slip is enforced at every solid surface. 5. Due to the nature of the interacting species, no diffusion fluxes are accounted for in the energy equation. 6. The dilute-mixture assumption is appropriate in each sub-domain (the velocity components, temperature and pressure of each species are related to bulk mass in each governing equation). 7. As the turbulence-chemistry interaction is neglected, the production of water vapor in the substrate is determined by an Arrhenius expressions (laminar-finite rate model).
2.2. Governing equations With reference to the previous statements, the standard governing RANS and energy equations are enforced, to yield for velocity components, temperature, pressure mass fractions in both sub–domains (Bird, Stewart, & Lightfoot, 2002) Continuity, for each fluid species o/ o/ o/ v þu þv þ ot ox or r mt 1 o o/ o2 / r þ ¼ Di;m þ þ KðT Þ ð1Þ r or or ox2 Sct where • Di,m (from the FickÕs law) is the diffusivity of the water vapor–air system in the fluid, and the water vapor– liquid water system in the substrate sub-domain;
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
• in the substrate, K is the rate of production of water vapor mass per unit volume (K = 0 in the fluid sub-domain); • in the fluidPsub-domain, the overall mass fraction conservation / ¼ 1 apply, whereas in the solid sub– domain this condition is not required during the drying P process ( / < 1: voids formation). Momentum in the axial direction, in the fluid sub– domain ou ou ou op o ou þu þv ¼ þ 2ðm þ mt Þ ot ox or ox ox ox 1 o ou ov ðm þ mt Þr þ þ ð2Þ r or or ox Momentum in the radial direction, in the fluid sub–domain ov ov ov op o ou ov þu þv ¼ þ ðm þ mt Þ þ ot ox or or ox or ox 1 o ov v 2ðm þ mt Þr þ ð3Þ ðm þ mt Þ 2 r or or r
q
Fluid, at undisturbed distance (outlet) (r = l1r, 0 < x < lsx + lj) o/a;v ov oT ¼ 0; ¼0 ð8Þ ¼ 0; u ¼ 0; or or or Upper confinement plate (d/2 < r 6 l1r, x = lsx + lj) o/v;a ¼ 0; u ¼ 0; v ¼ 0; T ¼ T 0 ð9Þ ox Lower confinement plate (0 < r 6 l1r, x = 0) o/a;v o/w;v ¼ 0 or ¼ 0; where applicable; ox ox u ¼ 0; v ¼ 0; T ¼ T 0
o X ðcp /ÞT ot o X o X ðcp /ÞT þ v ðcp /ÞT þq u ox or 2 X o 1 o o X r ðcp /ÞT ¼ ðk þ kt Þ 2 ðcp /ÞT þ r or or ox
ð10Þ
Finally, denoting with the superscripts f and s respectively the fluid and the substrate side across the interface, along the horizontal interface (for x = lsx and 0 6 r 6 lsr) /fv ¼ /sv ;
o/a;w;s ¼ 0; ox
u ¼ 0;
v ¼ 0;
Tf ¼ Ts
ð11Þ
and along the vertical interface (for r = lsr and 0 6 x 6 lsx) /fv ¼ /sv ;
Energy
233
o/a;w;s ¼ 0; or
u ¼ 0;
v ¼ 0;
Tf ¼ Ts
ð12Þ
2.4. Turbulence treatment
In the substrate, the usual constraints of u = v = 0 apply in Eqs. (1)–(4), whereas mt and kt are zero (laminar mass and thermal diffusions only).
In Eq. (1) Sct is constant and held to 0.7, while the closure relationships for mt and kt in Eqs. (2)–(4), respectively, can be assumed from the chosen turbulence model. For the present study, the k x shear stress transport has been adopted, having determined elsewhere (Angioletti, Nino, & Ruocco, 2005) its relative merit for the given flow configuration. Being its treatment beyond the scope of the present work, the Reader is referred to FLUENT 6.1 UserÕs Guide (2003) for its complete formulation.
2.3. Initial and boundary conditions
2.5. Rate of production of vapor
The food is initially in thermal equilibrium (T = T0) with the quiescent ambient (u0 = v0 = 0), and saturated with liquid water (/w = /w0, /v0 = 0); the mass fraction of solid is constant throughout the treatment (/s = /s0 = const; /w0 + /s0 = 1). Water vapor is allowed, during treatment, to flow through the interface, while no liquid water is allowed in the fluid sub–domain. With reference to Fig. 3, the mass, momentum and thermal boundary conditions are as follows: Jet inlet (0 6 r 6 d/2, x = lsx + lj)
In this paper a model of evaporation of unbound water has been adopted, based on a first-order irreversible kinetics (Roberts & Tong, 2003). Liquid water to water vapor conversion can be generally taken as an Arrhenius firstorder reaction, with a rate dependent on temperature
/v ¼ 0;
2.6. Numerical method and additional considerations
ð4Þ
/a ¼ 1;
u ¼ uj ;
v ¼ 0;
T ¼ Tj
ð5Þ
Substrate symmetry axis (r = 0, 0 6 x 6 lsx) o/v;w ¼ 0; or
oT ¼0 or
ð6Þ
Fluid symmetry axis (r = 0, lsx < x 6 lj) o/v;a ¼ 0; or
ou ¼ 0; or
v ¼ 0;
oT ¼0 or
ð7Þ
KðT Þ ¼ K 0 eEa =RT
ð13Þ
where the reference rate constant K0 is 4.96 · 106 1/s and the activation energy Ea is 48.7 kJ/mol.
The effect of the different values of domain radial length has first been monitored, to enforce the boundary condition of undisturbed flow. A value of 20 nozzle diameters was finally chosen along r. A triangular pave grid of approx. 9800 cells has been employed (Fig. 4), highly stretched to resolve the species mass fraction, velocity, temperature and pressure gradients in the boundary layer and within
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M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
Fig. 4. Computational grid.
the food, induced by the heating, evaporation, impingement and redirection of flow. A grid independence positive check has also been performed on a 15,000 cells grid. A finite-volume segregated solver with 2nd order unsteady implicit formulation has been employed throughout, with the SIMPLEC pressure–velocity coupling, and QUICK stencils for all other variables. The residuals were kept to 1 · 103 for all variables, except for temperature, 1 · 105. The time step has been kept small to 1 · 103 s to ensure stability; at each time step, the temperature– dependance was updated in Eq. (1). Execution time for t = 300 s elapsed time was approx. 60 h on a Pentium Xeon with two processor elements in serial mode (WindowsXP OS, 3.0 GHz, 2 GB RAM). A specific in-house code was devised to deal with water vapor transfer and interpolation across the food interface, and incorporated in the software.
3.2. Flow and temperature field
3. Results
The non-homogeneous heat transfer at the interface favors a simultaneous vapor transfer across the interface to the air flow, and the mass diffusion of vapor or liquid water in the impinged substrate. The peculiarity of the dehydration by JI is emphasized by the compared examination of liquid water and water vapor mass fractions in Fig. 6. For t = 30 s, /w is function of r only, Fig. 6a, due to the limited thickness of the substrate. The evaporation and depletion of liquid water is evident in the stagnation region, directly under the jet (at r = 0), where the temperature is
3.1. Configuration and material A comparison with the available yet scarce literature data (Braud, Moreira, & Castell-Perez, 2001; Moreira, 2001) has been tried: a baking process of a thin (lsx = 1.85 mm) food substrate (corn tortilla) was configured, with Tj = 418 K, T0 = 298 K and uj = 40 m/s, at a lsr = lj = 0.1 height. A corn starch–water mixture with /s0 = 0.77 and /w0 = 0.23 was adopted.
From examination of Fig. 5a it is first seen that after t = 300 s the heating forms a gradual thermal gradient in the entire substrate, varying from T = 358 K on the exposed surface directly under the jet (at r = 0, x = lsx), to T = 298 K at the tortillaÕs bottom (at r = 0, x = 0), to T = 303 K at the tortillaÕs end side (at r = lsr). The heat/ mass transfer rates are therefore highly non-uniform along the exposed surface, and the conduction in the substrate contributes to lateral heat transfer. The correspondent velocity distribution in Fig. 5b is fully developed, showing all regions already denoted in Fig. 1. 3.3. Moisture content
Fig. 5. (a) Qualitative temperature field and (b) qualitative velocity field at t = 300 s.
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
235
Fig. 6. Quantitative plots of (a) /w and (b) /v isolines at t = 30 s.
highest. This conversion favors in part the diffusion of vapor in the food, Fig. 6b, below the stagnation region, and where the wall jet region starts to form (for approx. r>5 mm) the surface vapor is blown away by the air flow, inducing more diffusion within the food. The /v isolines are densely packed along x, due to the high surface mass transfer rate. Far from stagnation, the mass fraction of vapor decreases, and for r > 10.0 cm a small vapor cloud appears. The effect of JI heating leads to complete /w drying after t = 300 s. A residual /v transport in the wall jet region is well evidenced in Fig. 7, while in the close-up of Fig. 8 a non-monotone behavior is detected along r, due to the joint effect of jet on the exposed surface, the internal diffusion of vapor along r and the removal from the side. From the comparison with the available literature reference (Braud et al., 2001), a discrepancy has been found on the process duration. It must be recalled that (Braud et al., 2001) do not solve for the flow field (an average heat trans-
fer mechanism is attributed at the surface), therefore the present relatively high uj value has been reconstructed using the cited bibliography through a local Nusselt number calculation based on jet diameter and height. In the present configuration, after only 5 min the thin tortilla has lost almost completely all liquid water, whereas a duration of 20 min was reported in Braud et al. (2001). This difference is attributed to the strongly different models compared here. 3.4. Evaluation of water activity The importance of water activity aw in food processing has been recalled earlier. aw is evaluated as the ratio between the vapor pressure in the substrate and the vapor pressure of pure water at the same temperature. The vapor pressure in a given product varies with the affinity of water with food constituents: the greater the affinity, the lower is the vapor pressure as few water molecules are available to be released upon processing. Solutes depress the vapor
Fig. 7. Quantitative plots of /v isolines at t = 300 s.
Fig. 8. /v map at t = 300 s.
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M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
Fig. 9. Water activity aw maps at (a) t = 20 s and (b) t = 30 s.
pressure of the solvent: this depression is usually given by RaoultÕs law but in foods there are substantial deviations from the ideal relationship. A number of approaches have been taken insofar to estimate the water activity of a mixture, in order to predict it using few characteristic parameters. But water activity has never been calculated on a local basis: the knowledge of the local distribution of aw is essential to food structure, consistence, perishability, and to yield for product/process optimization. One of the empirical equations for calculating water activity is the method of Grover (Barbosa-Ca`novas & Vega-Mercado, 1996). With this method, different ingredients are assigned a ‘‘sucrose equivalent conversion factor’’ S. This factor was based on experimental vapor pressures measured in such ingredient solutions, hence in part it incorporates corrections to RaoultÕs law. Water activity is assessed by the concentration ci of each ingredient, multiplied by S factor for the specific ingredient, as in Eq. (14) aw ¼ 1:04 0:1
X
X 2 S i ci þ 0:0045 S i ci
The local depletion near the lateral edge of the tortilla can be attributed to the particular flow field, which relatively favors vapor transport at the end-side, as seen in Figs. 6b and 7. 4. Conclusions Dehydration in a food slab has been accomplished by an impinging heated air jet. Time–dependent governing equations have been integrated to predict local moisture, temperature and velocity distributions. The evaporation kinetics has been tackled by a simple Arrhenius notation. Coupled moisture and temperature gradients have been shown to determine a strong process non-uniformity. A local pseudo-water activity has also been computed, for a thin corn tortilla. The non-monotone radial progress of aw is attributed to the peculiar surface heat/mass transfer mechanism. The model shows how the integration of transport and biochemical notations in foods can be employed to pursue process optimization.
ð14Þ
For the present case, S = 0.8 for corn starch. It must be observed, though, that as aw assessment is strongly dependent on temperature, it should be carried out in equilibrium conditions: only pseudo-aw transient values can be calculated for the drying process. For the case at hand, after few seconds a rapid decrement is already detected, from an initial average value of 0.800 0.760 in the stagnation region (not shown). At t = 20 s (Fig. 9a) the aw distribution is clearly influenced by the jet-offset: the lowest value of 0.700 is detected under the stagnation region (uniform along x due to the x-wise limited extension). Its progress is monotone with r, increasing up to 2 cm from the edge (maximum value 0.729), then decreasing again very slightly. The same progress is found for a later time t = 30 s (Fig. 9b). It is evident that even such a short duration increase contributes to decisive dehydration, as the lowest and highest values are 0.589 and 0.625 in this case.
Acknowledgements This work was funded by MIUR Italian Ministry of Scientific Research, grant no. 2004090750003 entitled ‘‘Analysis of transport phenomena due to jet impingement on substrates in industrial applications’’. References Angioletti, M., Nino, E., & Ruocco, G. (2005). International Journal of Thermal Science, 44(4), 349. Barbosa-Ca`novas, G. V., & Vega-Mercado, H. (1996). Dehydration of foods. New York: Chapman & Hall. Ben Nasrallah, S., & Perre, P. (1988). International Journal of Heat and Mass Transfer, 31(5), 957. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport phenomena. New York: John Wiley & Sons. Boukadida, N., & Ben Nasrallah, S. (1995). Drying Technology, 13(3), 661. Braud, L. M., Moreira, R. G., & Castell-Perez, M. E. (2001). Journal of Food Engineering, 50, 121.
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237 Chen, P., & Pei, D. C. T. (1989). International Journal of Heat and Mass Transfer, 32(2), 297. De Vries, D. A. (1958). Transaction, American Geophysical Union, 39(5), 909. Fellows, P. J. (2000). Food processing technology. Boca Raton: CRC Press. FLUENT 6.1 UserÕs Guide, Fluent Inc., 2003. Francis, N. D., & Wepfer, W. J. (1996). International Journal of Heat and Mass Transfer, 39(9), 1911. Moreira, R. G. (2001). Journal of Food Engineering, 49, 291.
237
Olsson, E. E. M., Ahrne´, L. M., & Tra¨ga˚rdh, A. C. (2004). Journal of Food Engineering, 63(4), 393. Roberts, J. S., & Tong, C. H. (2003). International Journal of Food Properties, 6(3), 355. Sarghini, F., & Ruocco, G. (2004). International Journal of Heat and Mass Transfer, 47(8-9), 1711. Stanish, M. A., Schajer, G. S., & Kayihan, F. (1986). AlChE Journal, 32(8), 1301. Whitaker, S. (1977). Advances in Heat Transfer, 13, 119.
Modelling local heat and mass transfer in food slabs due to air jet impingement M.V. De Bonis, G. Ruocco
*
CFDfood, DITEC, Universita´ degli studi della Basilicata, Campus Macchia Romana, 85100 Potenza, Italy Received 24 March 2005; accepted 26 September 2005 Available online 17 November 2005
Abstract Adequate design and verification of drying by a forced convection enhanced technique (gaseous jet impingement) can be carried out by numerical analysis, but customary transport calculations need to be integrated to account for complex (simultaneous) energy and mass transfer. In this paper the available procedures are reviewed and applied to food substrates: temperature, mass concentration and velocity fields are computed even for non-linear couplings (i.e. when local species concentration depends on temperature) using a specific solution strategy. Validity and limitations of the adopted notation and related integration into a proprietary software are discussed. A comparison is also brought forth with the available literature data. 2005 Elsevier Ltd. All rights reserved. Keywords: Jet impingement heat and mass transfer; Transient CFD; Food dehydration; Local water activity; Evaporation kinetics
1. Introduction Among the available forced convection processes, the gaseous jet impingement (JI) is frequently used for its excellent heat and mass transfer characteristics, where localized, controlled and rapid surface transfer is desirable. Studies on JI have been performed extensively over the past five decades, nevertheless the coupling and interdependence between simultaneous mass/heat transfer and fluid dynamics still needs to be fully analyzed, with special reference to local distribution of transfer rates on substrates of different shape (see for example Olsson, Ahrne´, & Tra¨ga˚rdh, 2004; Sarghini & Ruocco, 2004). Additional difficulties arise when in the subject impinged solid a multi-phase transport is allowed. JI can be successfully employed in drying or dehydration of foods by forced air convection, a most energy-intensive process, which is
*
Corresponding author. E-mail address: [email protected] (G. Ruocco).
0260-8774/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2005.09.032
commonly used in food engineering to extend food shelflife. Here, the majority of the unbound water normally present in a food is removed by applying heat under controlled conditions. The reduction in relative humidity (water activity) inhibits microbial growth and enzyme kinetics, also resulting in transport and storage costs reduction. In turn, drying may cause deterioration of both eating quality and nutritional value of the food. In food engineering, the design and operation of drying equipment aim to minimize these changes by selection of appropriate conditions. Dehydration involves a rather complex combination of application of heat and removal of moisture from a food medium (Barbosa-Ca`novas & Vega-Mercado, 1996; Fellows, 2000). In addition to air temperature and relative humidity, the rate of moisture removal is controlled by the air velocity. When hot air is locally blown over a moist food, water vapor diffuses through the boundary layer and is carried away (Fig. 1). A water vapor pressure gradient is therefore established from the moist interior to the external food surfaces. The boundary layer acts as a barrier to both heat transfer and water vapor removal during drying.
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
231
Nomenclature aw c cp d D Ea k K K0 l p r R S Sct t T u v x
water activity (dimensionless) concentration (g solid/g water) specific heat at constant pressure (kJ/kg K) jet diameter (m) mass diffusivity (m2/s) activation energy (kJ/mol) turbulent kinetic energy (J/kg) rate of production of water vapor mass per unit volume (1/s) reference rate constant (1/s) length (m) pressure (Pa) radial coordinate (m) universal gas constant (kJ/mol K) sucrose equivalent conversion factor (dimensionless) turbulent Schmidt number (dimensionless) time (s) temperature (K) x-component velocity (m/s) r-component velocity (m/s) axial coordinate (m)
Fig. 1. A drying air jet onto a food slab: flow of moisture during process.
Localized forced convection patterns, such as in JI, contribute to boundary layer destruction, hence in increase of moisture removal, while equipment costs are kept to a minimum. Nonetheless, local conditions have to be carefully monitored to ensure product uniformity. Porous and multi-phase media drying has been long speculated, and a large number of studies are available, following the seminal works be De Vries (1958) and Whitaker (1977), that carry complete descriptions of physics, as in Stanish, Schajer, and Kayihan (1986), Ben Nasrallah and Perre (1988) and Chen and Pei (1989). Boukadida and Ben Nasrallah (1995) first described a two-dimensional
Greek / k m q x
mass fraction (dimensionless) thermal conductivity (W/m K) kinematic viscosity (m2/s) density (kg/m3) specific dissipation rate (1/s)
Superscripts f fluid side, across the interface s solid side, across the interface Subscripts 0 initial a air i, m i-species in the mixture j jet s solid sr solid, along r sx solid, along x t turbulent v water vapor w liquid water 1r undisturbed, along r
progress of bulk convective drying of clay, but no contributions have been found in the available literature, with reference to localized convective drying in extended (i.e. at least 2D) porous media. A first contribution of JI drying of a moist, porous solid was presented by Francis and Wepfer (1996) with a thorough transient physical analysis, yet limited as one-dimensional. Furthermore, this model does not allow for a truly coupled transfer mechanism, as the surface transfer rates are externally implied. This limitations are also found in more recent works by Moreira (2001) and Braud et al. (2001), who first applied JI to food drying. Within this framework, the numerical analysis by a computational fluid dynamics (CFD) approach can gain importance as it leads to complete multi-dimensional and transient process description, yet ancillary calculation procedures are still needed to account for fully coupled energy and mass transfer. The present work has been performed with the specific aim to merge an in-house computation routine into a proprietary software (FLUENT 6.1 UserÕs Guide, 2003) in order to incorporate the multi-dimensional, transient calculation of an evaporation process due a gaseous impinging, heating jet.
2. Problem formulation A drying process of a thin food substrate is devised by using JI: hot, fully turbulent air is discharged through a
232
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
Fig. 2. Characteristic regions in submerged, confined JI. Fig. 3. Geometry and nomenclature.
nozzle with a given velocity distribution (upon recognition of the nozzle internal geometry). Upon impact of the free jet on the substrate, the characteristic stagnation and wall jet regions are first formed, then a secondary pattern can be identified in the lateral regions (Fig. 2). If the impinged food is water saturated, liquid can be converted into vapor depending on the heat perturbation front within the substrate, which can then be modelled as a multi-phase medium (a mixture of bulk solid, liquid water and water vapor). Liquid water normally moves from the interior of the substrate to the surface (1) by capillary forces, and (2) by diffusion caused by differences in the concentration of solutes at the surface and in the interior, while water vapor moves (3) by diffusion in air spaces within the substrate caused by vapor pressure gradients. In this work, for the sake of simplicity, Fickian diffusion is assumed only for both liquid and water vapor, nevertheless a highly nonuniform drying still results within the substrate, and a given process time can lead to local overheating and/or incomplete liquid conversion. 2.1. Driving assumptions In Fig. 3 the subject geometry in cylindrical coordinates is reported (only half section of the domain is considered due to the geometry and transfer symmetries). The domain under scrutiny consists in two interfaced fluid-and-substrate multi-species sub-domains, sharing the biomaterialÕs exposed surface. The fluid is a binary system comprising of 1. water vapor (v) 2. air (a) while the substrate is a ternary system comprising of 1. water vapor (v) 2. liquid water (w) 3. solid matter proper (s).
The following additional assumptions are adopted: 1. The flow is axisymmetric, with constant properties and incompressible (negligible pressure work and kinetic energy). 2. The viscous heat dissipation is neglected. 3. Due to the adopted flow regime, no body force is accounted for. 4. No-slip is enforced at every solid surface. 5. Due to the nature of the interacting species, no diffusion fluxes are accounted for in the energy equation. 6. The dilute-mixture assumption is appropriate in each sub-domain (the velocity components, temperature and pressure of each species are related to bulk mass in each governing equation). 7. As the turbulence-chemistry interaction is neglected, the production of water vapor in the substrate is determined by an Arrhenius expressions (laminar-finite rate model).
2.2. Governing equations With reference to the previous statements, the standard governing RANS and energy equations are enforced, to yield for velocity components, temperature, pressure mass fractions in both sub–domains (Bird, Stewart, & Lightfoot, 2002) Continuity, for each fluid species o/ o/ o/ v þu þv þ ot ox or r mt 1 o o/ o2 / r þ ¼ Di;m þ þ KðT Þ ð1Þ r or or ox2 Sct where • Di,m (from the FickÕs law) is the diffusivity of the water vapor–air system in the fluid, and the water vapor– liquid water system in the substrate sub-domain;
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• in the substrate, K is the rate of production of water vapor mass per unit volume (K = 0 in the fluid sub-domain); • in the fluidPsub-domain, the overall mass fraction conservation / ¼ 1 apply, whereas in the solid sub– domain this condition is not required during the drying P process ( / < 1: voids formation). Momentum in the axial direction, in the fluid sub– domain ou ou ou op o ou þu þv ¼ þ 2ðm þ mt Þ ot ox or ox ox ox 1 o ou ov ðm þ mt Þr þ þ ð2Þ r or or ox Momentum in the radial direction, in the fluid sub–domain ov ov ov op o ou ov þu þv ¼ þ ðm þ mt Þ þ ot ox or or ox or ox 1 o ov v 2ðm þ mt Þr þ ð3Þ ðm þ mt Þ 2 r or or r
q
Fluid, at undisturbed distance (outlet) (r = l1r, 0 < x < lsx + lj) o/a;v ov oT ¼ 0; ¼0 ð8Þ ¼ 0; u ¼ 0; or or or Upper confinement plate (d/2 < r 6 l1r, x = lsx + lj) o/v;a ¼ 0; u ¼ 0; v ¼ 0; T ¼ T 0 ð9Þ ox Lower confinement plate (0 < r 6 l1r, x = 0) o/a;v o/w;v ¼ 0 or ¼ 0; where applicable; ox ox u ¼ 0; v ¼ 0; T ¼ T 0
o X ðcp /ÞT ot o X o X ðcp /ÞT þ v ðcp /ÞT þq u ox or 2 X o 1 o o X r ðcp /ÞT ¼ ðk þ kt Þ 2 ðcp /ÞT þ r or or ox
ð10Þ
Finally, denoting with the superscripts f and s respectively the fluid and the substrate side across the interface, along the horizontal interface (for x = lsx and 0 6 r 6 lsr) /fv ¼ /sv ;
o/a;w;s ¼ 0; ox
u ¼ 0;
v ¼ 0;
Tf ¼ Ts
ð11Þ
and along the vertical interface (for r = lsr and 0 6 x 6 lsx) /fv ¼ /sv ;
Energy
233
o/a;w;s ¼ 0; or
u ¼ 0;
v ¼ 0;
Tf ¼ Ts
ð12Þ
2.4. Turbulence treatment
In the substrate, the usual constraints of u = v = 0 apply in Eqs. (1)–(4), whereas mt and kt are zero (laminar mass and thermal diffusions only).
In Eq. (1) Sct is constant and held to 0.7, while the closure relationships for mt and kt in Eqs. (2)–(4), respectively, can be assumed from the chosen turbulence model. For the present study, the k x shear stress transport has been adopted, having determined elsewhere (Angioletti, Nino, & Ruocco, 2005) its relative merit for the given flow configuration. Being its treatment beyond the scope of the present work, the Reader is referred to FLUENT 6.1 UserÕs Guide (2003) for its complete formulation.
2.3. Initial and boundary conditions
2.5. Rate of production of vapor
The food is initially in thermal equilibrium (T = T0) with the quiescent ambient (u0 = v0 = 0), and saturated with liquid water (/w = /w0, /v0 = 0); the mass fraction of solid is constant throughout the treatment (/s = /s0 = const; /w0 + /s0 = 1). Water vapor is allowed, during treatment, to flow through the interface, while no liquid water is allowed in the fluid sub–domain. With reference to Fig. 3, the mass, momentum and thermal boundary conditions are as follows: Jet inlet (0 6 r 6 d/2, x = lsx + lj)
In this paper a model of evaporation of unbound water has been adopted, based on a first-order irreversible kinetics (Roberts & Tong, 2003). Liquid water to water vapor conversion can be generally taken as an Arrhenius firstorder reaction, with a rate dependent on temperature
/v ¼ 0;
2.6. Numerical method and additional considerations
ð4Þ
/a ¼ 1;
u ¼ uj ;
v ¼ 0;
T ¼ Tj
ð5Þ
Substrate symmetry axis (r = 0, 0 6 x 6 lsx) o/v;w ¼ 0; or
oT ¼0 or
ð6Þ
Fluid symmetry axis (r = 0, lsx < x 6 lj) o/v;a ¼ 0; or
ou ¼ 0; or
v ¼ 0;
oT ¼0 or
ð7Þ
KðT Þ ¼ K 0 eEa =RT
ð13Þ
where the reference rate constant K0 is 4.96 · 106 1/s and the activation energy Ea is 48.7 kJ/mol.
The effect of the different values of domain radial length has first been monitored, to enforce the boundary condition of undisturbed flow. A value of 20 nozzle diameters was finally chosen along r. A triangular pave grid of approx. 9800 cells has been employed (Fig. 4), highly stretched to resolve the species mass fraction, velocity, temperature and pressure gradients in the boundary layer and within
234
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237
Fig. 4. Computational grid.
the food, induced by the heating, evaporation, impingement and redirection of flow. A grid independence positive check has also been performed on a 15,000 cells grid. A finite-volume segregated solver with 2nd order unsteady implicit formulation has been employed throughout, with the SIMPLEC pressure–velocity coupling, and QUICK stencils for all other variables. The residuals were kept to 1 · 103 for all variables, except for temperature, 1 · 105. The time step has been kept small to 1 · 103 s to ensure stability; at each time step, the temperature– dependance was updated in Eq. (1). Execution time for t = 300 s elapsed time was approx. 60 h on a Pentium Xeon with two processor elements in serial mode (WindowsXP OS, 3.0 GHz, 2 GB RAM). A specific in-house code was devised to deal with water vapor transfer and interpolation across the food interface, and incorporated in the software.
3.2. Flow and temperature field
3. Results
The non-homogeneous heat transfer at the interface favors a simultaneous vapor transfer across the interface to the air flow, and the mass diffusion of vapor or liquid water in the impinged substrate. The peculiarity of the dehydration by JI is emphasized by the compared examination of liquid water and water vapor mass fractions in Fig. 6. For t = 30 s, /w is function of r only, Fig. 6a, due to the limited thickness of the substrate. The evaporation and depletion of liquid water is evident in the stagnation region, directly under the jet (at r = 0), where the temperature is
3.1. Configuration and material A comparison with the available yet scarce literature data (Braud, Moreira, & Castell-Perez, 2001; Moreira, 2001) has been tried: a baking process of a thin (lsx = 1.85 mm) food substrate (corn tortilla) was configured, with Tj = 418 K, T0 = 298 K and uj = 40 m/s, at a lsr = lj = 0.1 height. A corn starch–water mixture with /s0 = 0.77 and /w0 = 0.23 was adopted.
From examination of Fig. 5a it is first seen that after t = 300 s the heating forms a gradual thermal gradient in the entire substrate, varying from T = 358 K on the exposed surface directly under the jet (at r = 0, x = lsx), to T = 298 K at the tortillaÕs bottom (at r = 0, x = 0), to T = 303 K at the tortillaÕs end side (at r = lsr). The heat/ mass transfer rates are therefore highly non-uniform along the exposed surface, and the conduction in the substrate contributes to lateral heat transfer. The correspondent velocity distribution in Fig. 5b is fully developed, showing all regions already denoted in Fig. 1. 3.3. Moisture content
Fig. 5. (a) Qualitative temperature field and (b) qualitative velocity field at t = 300 s.
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235
Fig. 6. Quantitative plots of (a) /w and (b) /v isolines at t = 30 s.
highest. This conversion favors in part the diffusion of vapor in the food, Fig. 6b, below the stagnation region, and where the wall jet region starts to form (for approx. r>5 mm) the surface vapor is blown away by the air flow, inducing more diffusion within the food. The /v isolines are densely packed along x, due to the high surface mass transfer rate. Far from stagnation, the mass fraction of vapor decreases, and for r > 10.0 cm a small vapor cloud appears. The effect of JI heating leads to complete /w drying after t = 300 s. A residual /v transport in the wall jet region is well evidenced in Fig. 7, while in the close-up of Fig. 8 a non-monotone behavior is detected along r, due to the joint effect of jet on the exposed surface, the internal diffusion of vapor along r and the removal from the side. From the comparison with the available literature reference (Braud et al., 2001), a discrepancy has been found on the process duration. It must be recalled that (Braud et al., 2001) do not solve for the flow field (an average heat trans-
fer mechanism is attributed at the surface), therefore the present relatively high uj value has been reconstructed using the cited bibliography through a local Nusselt number calculation based on jet diameter and height. In the present configuration, after only 5 min the thin tortilla has lost almost completely all liquid water, whereas a duration of 20 min was reported in Braud et al. (2001). This difference is attributed to the strongly different models compared here. 3.4. Evaluation of water activity The importance of water activity aw in food processing has been recalled earlier. aw is evaluated as the ratio between the vapor pressure in the substrate and the vapor pressure of pure water at the same temperature. The vapor pressure in a given product varies with the affinity of water with food constituents: the greater the affinity, the lower is the vapor pressure as few water molecules are available to be released upon processing. Solutes depress the vapor
Fig. 7. Quantitative plots of /v isolines at t = 300 s.
Fig. 8. /v map at t = 300 s.
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Fig. 9. Water activity aw maps at (a) t = 20 s and (b) t = 30 s.
pressure of the solvent: this depression is usually given by RaoultÕs law but in foods there are substantial deviations from the ideal relationship. A number of approaches have been taken insofar to estimate the water activity of a mixture, in order to predict it using few characteristic parameters. But water activity has never been calculated on a local basis: the knowledge of the local distribution of aw is essential to food structure, consistence, perishability, and to yield for product/process optimization. One of the empirical equations for calculating water activity is the method of Grover (Barbosa-Ca`novas & Vega-Mercado, 1996). With this method, different ingredients are assigned a ‘‘sucrose equivalent conversion factor’’ S. This factor was based on experimental vapor pressures measured in such ingredient solutions, hence in part it incorporates corrections to RaoultÕs law. Water activity is assessed by the concentration ci of each ingredient, multiplied by S factor for the specific ingredient, as in Eq. (14) aw ¼ 1:04 0:1
X
X 2 S i ci þ 0:0045 S i ci
The local depletion near the lateral edge of the tortilla can be attributed to the particular flow field, which relatively favors vapor transport at the end-side, as seen in Figs. 6b and 7. 4. Conclusions Dehydration in a food slab has been accomplished by an impinging heated air jet. Time–dependent governing equations have been integrated to predict local moisture, temperature and velocity distributions. The evaporation kinetics has been tackled by a simple Arrhenius notation. Coupled moisture and temperature gradients have been shown to determine a strong process non-uniformity. A local pseudo-water activity has also been computed, for a thin corn tortilla. The non-monotone radial progress of aw is attributed to the peculiar surface heat/mass transfer mechanism. The model shows how the integration of transport and biochemical notations in foods can be employed to pursue process optimization.
ð14Þ
For the present case, S = 0.8 for corn starch. It must be observed, though, that as aw assessment is strongly dependent on temperature, it should be carried out in equilibrium conditions: only pseudo-aw transient values can be calculated for the drying process. For the case at hand, after few seconds a rapid decrement is already detected, from an initial average value of 0.800 0.760 in the stagnation region (not shown). At t = 20 s (Fig. 9a) the aw distribution is clearly influenced by the jet-offset: the lowest value of 0.700 is detected under the stagnation region (uniform along x due to the x-wise limited extension). Its progress is monotone with r, increasing up to 2 cm from the edge (maximum value 0.729), then decreasing again very slightly. The same progress is found for a later time t = 30 s (Fig. 9b). It is evident that even such a short duration increase contributes to decisive dehydration, as the lowest and highest values are 0.589 and 0.625 in this case.
Acknowledgements This work was funded by MIUR Italian Ministry of Scientific Research, grant no. 2004090750003 entitled ‘‘Analysis of transport phenomena due to jet impingement on substrates in industrial applications’’. References Angioletti, M., Nino, E., & Ruocco, G. (2005). International Journal of Thermal Science, 44(4), 349. Barbosa-Ca`novas, G. V., & Vega-Mercado, H. (1996). Dehydration of foods. New York: Chapman & Hall. Ben Nasrallah, S., & Perre, P. (1988). International Journal of Heat and Mass Transfer, 31(5), 957. Bird, R. B., Stewart, W. E., & Lightfoot, E. N. (2002). Transport phenomena. New York: John Wiley & Sons. Boukadida, N., & Ben Nasrallah, S. (1995). Drying Technology, 13(3), 661. Braud, L. M., Moreira, R. G., & Castell-Perez, M. E. (2001). Journal of Food Engineering, 50, 121.
M.V. De Bonis, G. Ruocco / Journal of Food Engineering 78 (2007) 230–237 Chen, P., & Pei, D. C. T. (1989). International Journal of Heat and Mass Transfer, 32(2), 297. De Vries, D. A. (1958). Transaction, American Geophysical Union, 39(5), 909. Fellows, P. J. (2000). Food processing technology. Boca Raton: CRC Press. FLUENT 6.1 UserÕs Guide, Fluent Inc., 2003. Francis, N. D., & Wepfer, W. J. (1996). International Journal of Heat and Mass Transfer, 39(9), 1911. Moreira, R. G. (2001). Journal of Food Engineering, 49, 291.
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Olsson, E. E. M., Ahrne´, L. M., & Tra¨ga˚rdh, A. C. (2004). Journal of Food Engineering, 63(4), 393. Roberts, J. S., & Tong, C. H. (2003). International Journal of Food Properties, 6(3), 355. Sarghini, F., & Ruocco, G. (2004). International Journal of Heat and Mass Transfer, 47(8-9), 1711. Stanish, M. A., Schajer, G. S., & Kayihan, F. (1986). AlChE Journal, 32(8), 1301. Whitaker, S. (1977). Advances in Heat Transfer, 13, 119.