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Experimental investigation and numerical modeling of pilot-scale fluidized-bed drying of yeast: Part A – Drying model development and validation
Journal Pre-proof Experimental investigation and numerical modeling of pilot-scale fluidized-bed drying of yeast: Part A - drying model development and validation Behdad Soltani Dale D. McClure Farshad Oveissi Timothy A.G. Langrish John M. Kavanagh
PII:
S0960-3085(19)30335-9
DOI:
https://doi.org/doi:10.1016/j.fbp.2019.11.008
Reference:
FBP 1177
To appear in:
Food and Bioproducts Processing
Received Date:
16 April 2019
Revised Date:
11 November 2019
Accepted Date:
12 November 2019
Please cite this article as: Soltani, B., McClure, D.D., Oveissi, F., Langrish, T.A.G., Kavanagh, J.M.,Experimental investigation and numerical modeling of pilot-scale fluidized-bed drying of yeast: Part A - drying model development and validation, Food and Bioproducts Processing (2019), doi: https://doi.org/10.1016/j.fbp.2019.11.008
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Experimental investigation and numerical modeling of pilot-scale fluidizedbed drying of yeast: Part A - drying model development and validation Behdad Soltani, Dale D. McClure, Farshad Oveissi, Timothy A.G. Langrish§, John M. Kavanagh§*
*
: Equal senior authors
: Corresponding Author:
John Kavanagh (Email: [email protected])
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§
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The University of Sydney, School of Chemical and Biomolecular Engineering, Sydney, 2006, Australia
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Abstract
In this work we have developed an extensive experimental data-set for the fluidized-bed drying
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of yeast. A range of experimental conditions including inlet air temperatures from 40 to 80 °C, bed temperatures from 10 to 60 °C and bed masses of 400-600 g were used. This data was used to validate a mathematical model of the drying process. The model developed uses the reaction
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engineering approach to estimate the drying kinetics and has a strong physical basis with the
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only fitted parameters being those used for the GAB isotherm. Agreement between model predictions and experimental data was excellent, the maximum root-mean-square errors were 3.1
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°C, 1.8 g of water per kg of dry air, and 2.3% in the bed temperature, air humidity and the final moisture content on a wet-basis, respectively. Such results demonstrate that the model developed can be used to predict the behavior of the drying process across a broad range of operating conditions. Results from this work can be applied to the drying of yeast and other starter cultures used in the food industry.
Keywords: fluidised-bed dryer, yeast, lumped modeling, drying, reaction engineering approach
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1
Introduction
Drying is one of the oldest techniques used to preserve the quality of food. It is also used to prolong the shelf life of many food starter cultures, including yeast used for baking and those used in the dairy industry (Goderska 2012). The principle of drying is based on reducing the water content in the product to minimize microbial growth and enzymatic activity. Drying also has the advantage of reducing the mass of the product, thereby lowering transport costs (Mujumdar 2014).
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In yeast drying, fluidized-beds using warm dry air are the most common, whilst for dairy starter cultures low temperature air is used (Mujumdar 2014). Fluidized-bed dryers have the advantages of providing good mixing of the solids as well as having large interfacial areas which leads to
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relatively high rates of heat and mass transfer (Strumiłło and Kudra 1986). Drying of microorganisms is particularly challenging as it is necessary to remove sufficient water to
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improve the shelf life of the product (dried yeast typically has a final moisture content of the order 4-6 wt%) while avoiding damage to the cells. It is known that both the temperature and
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duration of drying are key factors in this regard (Beker and Rapoport 1987), with temperatures above 40 °C being reported to be damaging for baker’s yeast (Bayrock and Ingledew 1997). An ideal drying process would remove sufficient moisture to ensure the shelf life of the product
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while minimizing its exposure to damaging temperatures. Such considerations are particularly
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important in the drying of microorganisms as damaging conditions reduces the viability of the cells which is essential to their use in baking or as starter cultures.
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The drying rate (J) is defined as the rate of moisture removal from the solid per unit time. During a typical drying process, there are three periods: a warming up period, an unhindered (constant rate) drying period and an hindered (falling rate) drying period (Porter et al. 1984). In the unhindered drying period, the rate of drying is limited by the rate of mass transfer from the wetted film on the particle to the air. If the external conditions (i.e. the gas temperature, velocity and humidity) are constant drying will occur at a fixed rate and the temperature of the bed will be approximately constant. As the solids dry a point is reached where the rate of moisture transfer from the interior of the particle to the surface is less than the gas-particle mass transfer rate. It is at this point unhindered drying stops and the hindered drying phase commences. This part of the drying curve continues, with the moisture content approaching the equilibrium value. The bed 2 Page 2 of 35
temperature can rapidly increase during the hindered stage, such increases can be difficult to control. As previously noted high temperatures can be damaging to the viability of starter cultures and hence should ideally be avoided to maintain the product quality. Being able to predict such increases in the temperature is obviously desirable from the perspective of both controlling and optimizing drying processes for microorganisms including yeast. It is for this reason a number of studies into the modeling of yeast drying have been carried out in the last decade (Debaste et al. 2008; Spreutels et al. 2013; Spreutels et al. 2014;
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Yuzgec, Turker, and Becerikli 2004; Türker et al. 2006; Yüzgeç, Türker, and Becerikli 2008; Van Engeland et al. 2019). Köni et al. (2009) developed a black box model using an artificial neural network and adaptive neural network-based fuzzy inference system structures. Their
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model gave good predictions of the moisture content (d.b) as a function of time. However, given that these were black box models, the physical process understanding incorporated in the model
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is unclear.
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The majority of authors (Debaste et al. 2008; Türker et al. 2006; Spreutels et al. 2014) have developed models using mass and energy balance equations. A range of approaches have been used to calculate the drying rate, Türker et al., (2006) used a kinetic approach. It was found that
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this modelling approach gave good agreement for small (0.6 mm diameter) yeast particles. However, agreement was less good for larger diameter (1.2 mm) particles. This difference was
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attributed to the fact that diffusive transport of water within the particles became significant as
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the particle diameter increased, hence it was necessary to account for this in the model. Spreutels et al. (2014) modelled the flux of two different ‘types’ of water; corresponding to extracellular and intracellular moisture. A kinetic model was used to model the drying rate for intracellular water, while the drying rate of extracellular water was modelled using a reaction engineering type approach where diffusion of moisture inside the yeast particles was also accounted for. Debaste et al. (2008) used a similar approach, using the GAB isotherm to calculate the relative humidity. It was found this model gave good agreement in the constant rate period, but poor agreement during the falling rate portion of the drying. To address this shortcoming, they replaced the GAB isotherm with a new model which accounts for the diffusion of moisture through the yeast particle during the falling rate period.
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Based on the previous work it may be necessary to model the diffusion of moisture within the yeast particles to accurately predict drying during the falling rate period. Ideally, any model should have a minimum number of adjustable parameters that are specific to a particular experimental configuration and the model should also be validated against a broad range of experimental data. The aim of this study was to develop and validate a quantitative model capable of accurately predicting the behavior (temperature and moisture content) as a function of time during the
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fluidized bed drying of yeast. Such a model can be used to design and optimize industrial drying processes for starter cultures. A key focus in the development of the model was minimizing the number of configuration specific fitted parameters, in order to ensure that the model would have
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a broad range of application. Results from this paper will be used in the second part of the work to develop a model where the drying conditions are related to the viability of the yeast (Soltani et
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al. 2019). Conclusions from this work can be applied to a broad range of drying processes in the
2 2.1
Materials and methods Yeast preparation
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food industry.
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Cream yeast (79-82% moisture content, wet-basis) was supplied by an industrial source. The
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yeast suspension was stored at 4 °C prior to use. An homogenized dispersion of Sorbitan monostearate (SMS) was added to the yeast suspension at the industrially recommended rate of 1
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wt% (calculated on the mass of dry matter). The mixture of yeast and SMS was then filtered under a vacuum of 50 kPa for one hour to concentrate the cream yeast to 70% moisture content, on a wet-basis. The yeast cake was pressed for 30 min and extruded through ∼0.6 mm diameter holes at a pressure of 500-600 kPa using a pneumatic press. The prepared samples were weighed before being placed into the dryer. In this paper a yeast particle is defined as the cylindrical pellets produced by the extrusion of the yeast cake. 2.2
Pilot-scale dryer
All experiments were performed in a pilot-scale fluidized-bed dryer as shown schematically in Figure 1. This unit was made of polycarbonate (4.5 mm wall thickness) and has a height of
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670 mm with a diameter of 100 mm at the base, increasing to 250 mm at the top of the conical section. Cool air (4-6 °C) was heated using a heat exchanger before being passed through a Munters desiccant dehumidifier to achieve a relative humidity of 2 to 3% and a temperature of 30 °C. The system was run for one hour before performing any experiments such that any changes in the inlet air humidity and temperature were minimal. The inlet air humidity (Yin) during experiments
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varied between 1 to 1.5 g of water per kg of dry air, the difference between batches was mainly a
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function of the ambient humidity, and the inlet air humidity remained approximately constant for
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each experiment.
The dehumidified air was passed into a heater to increase its temperature to 40-80 °C, with the
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temperature being varied depending on the experimental conditions. For all experiments, the mass flow rate was measured using a Proline thermal mass flow meter (Endress+Hauser), the air
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flow rate (G) was set to 55 ± 1.4 kg hr-1. This corresponded to a superficial velocity of 1.6 m s-1 (calculated using a bed diameter of 100 mm). It was calculated that approximately 22 kJ was needed to heat the dryer from ambient temperature (25
) to 30
(using the equipment mass and
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heat capacity listed in Tables 1 and 2).
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Figure 1 – Schematic showing set-up of pilot-scale dryer used in the current work.
The bed temperature and air humidity were measured using a humidity and temperature meter (Vaisala), as shown schematically in Figure 1. Bed temperatures were measured to an accuracy of
0.2 °C between 0 – 60 °C.
The accuracy of the bed humidity measurements was a function of both the relative humidity and temperature, with the accuracy decreasing at high temperatures and high values of the relative humidity. For example, at a temperature of 40 °C the accuracy in the relative humidity measurement was 0.49-1.61 g of water per kg of dry air for 0-100% relative humidity, respectively (VAISALA 2013). The bed temperature and air humidity were recorded every 10 s
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by an automated data logging system. A Mettler Toledo Halogen Moisture Analyzer was used to measure the moisture content on a wet-basis of the yeast before and after drying. All the measured moisture contents had an uncertainty of ± 0.1%.
At the start of each experiment, a known mass (400 or 600 g) of yeast was added to the dryer. The average moisture content of particles after compression was recorded to be approximately 67-70%, and the final moisture content of particles was 2.3-6.4%, depending on the final inlet air
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temperature. Each experiment continued until the difference between the exhaust and inlet air
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humidity was less than 1 g of water per kg of dry air or until the bed temperature reached 60 °C. Using the current experimental set-up, it was not possible to measure bed temperatures greater
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than 60 °C. Such conditions are likely to be highly damaging to the yeast and hence are not likely
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to be of significant interest. The maximum possible inlet air temperature was 110 °C. In the first stage of this research, the inlet air temperature was set to 50, 60, 70 and 80 °C and the
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initial mass of the yeast was 400 and 600 g. Another set of experiments was performed with stepchanges in the inlet air temperature from 80 and 70 °C to 40 and 50 °C after 10 and 15 min. This set of experiments aimed to provide better control over the bed temperature and further validate
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the model by ensuring it could accurately predict the results of a step change in the inlet air
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temperature.
To investigate the repeatability of the system, the inlet temperature was fixed at 50 °C and the
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flow rate was 55 kg hr-1, the same conditions were used for experiments performed in three different days. It was found that the measured values of the bed temperature, outlet air humidity and final moisture content were highly reproducible, with a difference of not more than 2.5 °C, 1.5 g of water per kg of dry air, and 1% in the bed temperature, outlet air humidity and the moisture content, (wet-basis), respectively. The results are comparable with the uncertainty of the measurements. 2.3
Particle size measurements
To quantify the extent of shrinkage during the drying process a separate experiment was performed using a fixed inlet temperature of 55 °C and an air flow rate of 55 kg hr-1. Samples 7 Page 7 of 35
were collected from the dryer 0, 10, 15, 20, 25 and 30 minutes after the start of drying. Images of the yeast particles were then taken with a stereomicroscope (Olympus, SZ61). At each time point, a total of 20 particles were measured; the reported results are the average diameter with the error bars denoting one standard deviation about the mean.
3
Numerical modelling
The following assumptions and simplifications were made in the model development: It was assumed that the system was well mixed.
•
It was assumed that gradients of both temperature and moisture content within the
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•
•
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particles were minimal (i.e. a lumped parameter model was used).
The pressure drop across the bed was calculated (using the Ergun equation (Ergun 1952))
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to be of the order 1-2 kPa. Since this is relatively small it was assumed that the process was operated at a constant pressure of 101.3 kPa (one atmosphere). All thermodynamic
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properties were calculated at this pressure.
It was assumed that the equipment and the bed were at thermal equilibrium.
•
Ideal gas behavior was assumed. This was thought to be reasonable due to the operating
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•
•
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pressure (101.3 kPa) and temperatures (30-80 °C) used. Constant values were used for the heat capacities as these do not change appreciably over
•
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the temperature range examined (30-80 °C). The mass of dry yeast particles was assumed to be constant (i.e. there was no mass loss due to particle attrition). •
The density and heat capacity of the yeast was assumed to be equivalent to that of liquid water (Türker et al. 2006).
•
It was assumed that the inlet and outlet mass flow rates of dry air were equal (i.e. no chemical reactions took place).
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•
It was assumed that the particles shrunk uniformly in all directions.
•
It was assumed that the bed temperature (Tb) was equal to the outlet air temperature (Tg,out). This assumption is reasonable due to the relatively high rates of mass transfer between the gas and particle. It is possible to derive the model equations without making this assumption, as detailed in the Supplementary Material this has a relatively small impact on the model predictions. It was found (see the Supplementary Material) that inclusion of terms for the sensible
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•
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heat of water vapor and the thermal mass of the gas had minimal effect on the model predictions due to their small values. Hence, they have been neglected from the gas-phase
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heat balance.
constant for the duration of the drying.
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Experimentally measured values of the inlet air were used in the model, these values were
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A schematic outlining the simultaneous heat and mass transfer occurring in the system is
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presented in Figure 2.
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Figure 2 – Schematic showing the flows of mass and energy in the drying system.
3.1
Mass and energy balances
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The unsteady mass balance for the moisture content (dry-basis) of the solid can be expressed as:
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(1)
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where X is the moisture content of the solids (dry-basis), t is the time and J is the rate of evaporation. The driving force for the external mass transfer may be expressed as the difference between the air humidity at the external surface of the particle and in the bulk of the gas phase: (2)
where k is the gas-particle mass transfer coefficient, ρg is the density of the gas, Yr is the humidity at the particle surface, Yout is the outlet humidity and a is the interfacial area for mass transfer.
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This type of equation is very similar to the Reaction Engineering Approach (REA), which was proposed by Chen and Xie (1997). This approach has been used to model the drying of a wide range of foods including skim milk, kiwi, apple pieces and potato slices (Langrish 2009; Chen 2008; Chen, Pirini, and Ozilgen 2001). The water mass balance for the gas phase is:
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(3)
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where Vb is the volume of the bed, ε is the void fraction and mS is the mass of dry solids. The energy balance around the system can be used to calculate the rate of temperature change for
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the outlet gas (Tg,out), this is equivalent to the bed temperature. This equation accounts for the
(4)
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thermal mass of the equipment as well as heat loss to the environment.
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where cp,w and cp,eq are the specific heat capacities of liquid water and the equipment, respectively. λ is the enthalpy of latent heat for the water, meq is the mass of the equipment,
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hoverall is the overall heat transfer coefficient from the dryer to the environment, Ω is the external surface area of the dryer and Tamb is the ambient temperature. 3.2
Heat and mass transfer equations
Here we have used a published (Clift, Grace, and Weber 1978) correlation which is valid for cylindrical particles at Reynolds numbers up to 1000 to calculate the Sherwood number: (5)
The Sherwood number (Sh) was defined subsequently:
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S
(6)
where dp is the characteristic length (here the diameter of the cylindrical yeast particles) and D is the diffusivity of water vapor in air (2.6 × 10-5 m2 s-1) (Coulson et al.). The Reynolds (Re) and Schmidt (Sc) numbers are defined as follows:
(8)
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(7)
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where µg is the viscosity of the gas, U is the gas superficial velocity, UG is the gas velocity and UP is the velocity of the yeast particle. In practice precisely determining the particle velocity
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relative to the gas (i.e. UG – UP) is challenging. The superficial gas velocity gives a reasonable approximation to the actual gas velocity due to the high bed voidage; this varies between 1.6 and 0.3 m s-1 due to the changes in the dryer cross sectional area. The particle velocity (UP) can be
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reasonably estimated by determining the terminal velocity of the particles, calculated values of the particle terminal velocity are of the order 0.1 m s-1. Here for convenience we have taken the
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arithmetic average of the superficial gas velocity and the inlet and outlet to calculate the gas velocity (UG), while neglecting the particle velocity in the calculation of the Reynolds number,
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this gave a relative velocity between the gas and particle of the order 1 - 1.1 m s-1. Using this value, the Reynolds number varied between 15 < Re < 40. Values calculated for the Schmidt number were 0.5 < Sc < 0.8. This gave values of the gas-particle mass transfer coefficient (k) between 0.2 and 0.4 m s-1.
Here we have used the GAB isotherm to model the relationship between the equilibrium moisture content (Xe) and the relative humidity (RH) as it is generally thought to be the most suitable for describing the sorption of water by food products (Van den Berg and Bruin 1978; Bizot 1983) including yeast (Debaste et al. 2008). The GAB equation is usually presented in the following form: 12 Page 12 of 35
(9)
where Xe is the equilibrium moisture content (dry-basis), parameters. The parameter surface,
and
and
are characteristic
is related to the number of active sites for adsorption on the
corresponds to the number of multilayers, and
is related to the interaction of water
and the surface (Quirijns et al. 2005). In this study kG, cG and xm were estimated from the GAB
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isotherm as 0.88, 4.5, and 0.28, respectively. These parameters were fitted to experimental data using the generalized reduced gradient method (Lasdon et al. 1978) with an inlet air temperature of 50 °C, bed temperatures from 10-60 °C, and a bed mass of 400 g. The parameters were used
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for all other simulations.
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The Antoine equation was used to calculate the saturated vapor pressure for water (p*), this is necessary to convert the relative humidity value from Equation (9) to the moisture content (dry-
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basis) at the surface of the solid (Yr):
(10)
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(11)
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(12)
where A, B and C are constants in the Antione equation, having values of 8.107, 1750.286, and 235, respectively (Felder and Rousseau 2005). Mw and Mg are the molar masses of water vapor and air respectively and Ysat is the saturation humidity. The overall heat transfer coefficient from the dryer to ambient (hoverall) is made up of three terms: (13)
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The rate of heat transfer through the wall is inversely proportional to its thickness (Lw), and directly proportional to its thermal conductivity kw. Here the wall thickness was 4.5 mm and we have used a thermal conductivity of 0.19 W m-1 K-1 for polycarbonate (Ellis and Smith 2008). Conductive losses will also occur through the base of the dryer, however due to their relatively small magnitude and the difficulty in accurately determining their value they have been omitted from the model. Heat transfer from the bed to the wall (ho) was determined using a correlation proposed for
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particles in group B of the Geldart classification (which includes baker’s yeast) (Zabrodskiĭ
(14)
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1966; Rhodes and Rhodes 2008):
The heat transfer from the bed to the environment
was calculated using Equation (16).
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This correlation was recommended by Churchill and Chu (1975) for a vertical flat plate geometry as it is applicable to all range of Rayleigh numbers (Churchill and Chu 1975), and it
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can be used for a vertical cylinder when:
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is the average diameter of the fluidised-bed dryer.
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where
(15)
(16) where kg is the thermal conductivity of the gas. The Rayleigh (Ra) and Grashof numbers (Gr) are defined as: 14 Page 14 of 35
(17)
(18)
where
is acceleration due to gravity, and
is the kinematic viscosity of air. The Grashof
number was calculated at the film temperature (Tf) using the thermal expansion coefficient for an
(19)
(20)
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ideal gas (β):
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To find the heat loss the characteristic length was taken as the height ( ) of the bed (0.43 m), and the outer surface area of the bed (Ω) was 0.629 m2. . The heat-transfer
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The total heat flow rate at the inlet of the bed varied between 500 and 1000
coefficient from particles to the bed, through the wall and from the exterior of the dryer to
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ambient during all experiments were 700-1000, 42, and 3-6 W m-2 K-1, respectively. Hence, the rate of heat loss was between 5-120 W, this was up to 10% of the inlet heat flow and for this
3.3
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reason heat loss was included in the model. Particle shrinkage model and interfacial area
Other authors (Spreutels et al. 2013) have found that the particle diameter is a linear function of the moisture content. To determine whether this was the case experimental measurements were performed as described in Section 2.3. As shown in Figure 3 this was found to be correct, hence we have used the following equation to calculate the particle diameter as a function of the moisture content (dry-basis): (21)
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where L is the particle length, dp0, L0 and X0 are the initial values of the particle diameter, length and moisture content (dry-basis), respectively. Ψ is the particle shrinkage coefficient, here a value of 0.3 was used, this value being calculated from a fit to the experimental data. The particle surface area (a) was calculated as:
(22)
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Pr
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pr
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where the empirical factor of 9/8 accounts for the particle end area (given L = 4dp)
Figure 3 – Plot showing the particle diameter as a function of the moisture content on a dry-basis. The abscissa scale is the reverse of the norm (i.e. drying occurs from left to right).
The void fraction was calculated by measuring the height of the bed to calculate the bed volume and then using the known mass and density of the yeast to determine its volume and hence the bed voidage. Calculated values ranged between 0.93 and 0.96, for the sake of simplicity a fixed value of 0.95 was used in the model. It was found that the varying the value of the volume fraction across the experimentally measured range had minimal impact on the model predictions. A summary of the physical properties and other parameters used in the model is given in Tables 1 and 2. Equations (1), (3) and (4) were solved using the ordinary differential equation solvers 16 Page 16 of 35
available in Matlab (R2015a). Initial values of the particle moisture content on a dry-basis, outlet air temperature and outlet air humidity ranged between 2.13-2.32 g water per kg of dry solid, 1421 °C and 8-12 g of water per kg of dry air, respectively. These values were obtained from the experimental measurements. Differences between the model predictions and the experimental measurements were quantified
(23)
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using the Root-Mean-Square Error (RMSE):
pr
where n was the total number of measurements and χ was the variable being compared. Here we have reported the final moisture content on a wet basis (W), this was calculated from the
(24)
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Pr
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moisture content on a dry basis (X, kg moisture per kg dry solid) in the following manner:
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Value
Reference
Density of dry air,
[kg m-3]
(Smith and Van Ness 1975)
Specific heat of dry air,
[J kg-1 K-1]
(Çengel and Ghajar 2015)
Dynamic viscosity of air,
[kg m-1 s-1]
(Reid and Sherwood 1966)
Kinematic viscosity of air,
[m2 s-1]
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Parameter
f
Units
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Table 1 – Summary of physical properties used in the model
Thermal conductivity of air,
Molar mass of dry air,
Density of water,
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28.97
[kg kmol-1]
(Treybal 1988)
18.02
[kg kmol-1]
(Treybal 1988)
1000
[kg m-3]
(Smith and Van Ness 1975)
4,180
[J kg-1 K-1]
(Smith and Van Ness 1975)
2257000
[J kg-1]
2.56 × 10-5
[m2 s-1]
(Coulson et al.)
1300
[J kg-1 K-1]
(Ellis and Smith 2008)
0.19
[W m-1 K-1]
(Ellis and Smith 2008)
101,325
[Pa]
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Specific heat capacity of water,
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Latent heat of vaporization for water,
Mass diffusivity of water in air, Specific heat capacity of equipment (polycarbonate), cp,eq Thermal conductivity of polycarbonate Atmospheric pressure,
(Reid and Sherwood 1966) (Smith and Van Ness 1975)
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Molar mass of water,
2015)
[J kg-1 K-1]
287.05
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Gas constant for dry air,
[W m-1 K-1]
(Çengel and Ghajar
(Oosthuizen and Naylor 1999)
(Smith and Van Ness 1975)
18 Page 18 of 35
Table 2 – Summary of parameters used in the model
Value
Units
Ambient temperature, Tamb
25
[°C]
Initial particle diameter, dp0
600
[µm]
External surface area of dryer, Ω
0.629
[m2]
Bed volume, Vb
0.0092
[m3]
Mass of equipment, meq
3.4
Height of bed, H
430
Wall thickness, Lw
4.5
Bed voidage
0.95
Particle shrinkage coefficient, Ψ
0.3
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GAB isotherm parameter, kg
[kg]
[mm] [mm] [-] [-]
0.88
[-]
4.5
[-]
Pr
GAB isotherm parameter, cG
0.28
[kg moisture kg dry solids-1]
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GAB isotherm parameter, xm
f
Parameter
19 Page 19 of 35
4
Results and discussion
In order to validate the model experiments were performed with a broad range of inlet air temperatures (50-80 °C) and initial bed masses of 400 and 600 g. Taking a value of 2% for the inlet air relative humidity the calculated values of the wet-bulb temperature were 19, 23, 26 and 30
for inlet temperatures of 50, 60, 70 and 80
respectively. Results are shown in Figure 4
for an initial bed mass of 400 g and in Figure 5 for an initial bed mass of 600 g.
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As expected, three stages of drying were observed. Firstly a brief (approximately three minutes) warming up period was observed where the inlet temperature reached the desired value (from an initial value of 30 °C). Inclusion of this delay in the model by ramping the inlet air temperature
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from 30 °C to the set point over a period of 3.5 minutes noticeably improved predictions of the bed temperature. After the warming up period the unhindered drying period occurred, the
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duration of this period depended on both the inlet temperature and the mass of the bed, with the model correctly predicting these trends (i.e. longer unhindered drying periods for lower inlet
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temperatures and higher initial bed masses). During this period the model appears to over-predict the outlet humidity. However, this difference is comparable with the uncertainty in the
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experimental measurements, meaning it is not possible to state with certainty that this is the case. It was found for all of the cases examined that the air passing through the dryer was largely
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saturated during the unhindered drying phase. This meant that the driving force for mass transfer (see Equation (2)) was relatively low, explaining the relatively long duration of this period. Such
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results have been observed by others for similar processes (Spreutels et al. 2014). At the end of the unhindered phase the surface of the particles is no longer fully wetted, and moisture was transferred from the interior of the particle to the surface until the equilibrium value was achieved. During this portion of the drying the bed temperature increased, as shown in Figures 4(a) and 5(a) this rise was quite steep for higher values of the inlet air temperature and may be damaging to the yeast cells. The bed temperature continued to increase during this portion of the drying, while remaining below the inlet temperature due to residual drying and heat losses. In all instances the model correctly predicted the experimentally observed trends, while also offering good agreement with the experimental data. RMSEs for bed temperature, air humidity, 20 Page 20 of 35
and the final moisture content (wet-basis) were 2.62 °C, 1.75 g of water per kg of dry air, and 1.69%, respectively for an initial bed mass of 400 g and 3.07 °C, 1.71 g of water per kg of dry air, and 1.06%, respectively for a bed mass of 600 g. Such results demonstrate that the model developed is capable of correctly predicting the drying behavior for a broad range of operating
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conditions.
21 Page 21 of 35
f oo pr ePr al rn Jo u Figure 4 – Plot comparing the experimentally measured values of bed temperature (a), outlet humidity (b) and final moisture content on a wet-basis (c) with the model predictions. Results are for an initial bed mass of 400 g and at fixed inlet air temperatures of 50, 60, 70 and 80 °C. All results are for a fixed inlet air flow rate of 55 kg hr-1. The final moisture content (wet-basis) in the experimental data had an uncertainty of ± 0.1%.
22 Page 22 of 35
f oo pr ePr al rn Jo u Figure 5 – Plot comparing the experimentally measured values of bed temperature (a), outlet humidity (b) and final moisture content on a wet-basis (c) with the model predictions. The final moisture content (wet-basis) in the experimental data had an uncertainty of ± 0.1%. Results are for an initial bed mass of 600 g and at fixed inlet air temperatures of 50, 60, 70 and 80 °C. All results are for a fixed inlet air flow rate of 55 kg hr-1.
23 Page 23 of 35
Running the drying process using a fixed inlet temperature is unlikely to be feasible from an industrial perspective as the increase in the bed temperature during the hindered drying phase is likely to be damaging to the yeast. For example, Bayrock and Ingledew (1997) found that the cell viability decreased when the temperature of the yeast cells exceeded 40 °C. To avoid this and to further validate the model additional experiments were performed where the inlet temperature was changed. Initially a relatively high (70 or 80 °C) inlet air temperature was used, this was reduced to a lower value (40 or 50 °C) after 15 and 10 minutes for initial temperatures of 70 or
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unhindered drying at that temperature (see Figure 4).
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80 °C, respectively. These times were chosen as they correspond to the duration of the
Comparison between the model predictions of the bed temperature and outlet humidity are
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presented in Figure 6, while comparison between the experimentally measured and predicted final moisture content (wet-basis) is given in Figure 7. Again, the agreement between the model
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predictions and experimental measurements was good for the range of operating conditions examined, with the exception of the outlet humidity in the unhindered drying phase. However, as
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previously noted the discrepancy between the model predictions and the data is of similar magnitude to the experimental uncertainty. Maximum values of the RMSE were of the order 2.5
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°C for the bed temperature and 1.6 g of water per kg of dry air for the outlet humidity, indicating
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that the model gave good predictions for the conditions examined.
24 Page 24 of 35
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Figure 6 – Plot comparing the experimental measurements and model predictions of the bed temperature (a) and (b) and the outlet humidity (c) and (d). Results in the first column (a) and (c) were for an initial gas inlet temperature of 80 °C which was reduced to 40 and 50 °C after ten minutes; the results in the second column (b) and (d) are for an initial gas inlet temperature of 70 °C which was reduced to either 40 or 50 °C after 15 minutes. All results are for an initial bed mass of 400 g and an air flow rate of 55 kg hr-1. Times where the air inlet temperature was changed are denoted with a dotted vertical line.
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Figure 7 – Comparison between experimentally measured values of the final solids moisture content-wet basis (W) and the model predictions for experiments performed using varying inlet temperatures. The final moisture content (wet-basis) in the experimental data had an uncertainty of ± 0.1%.
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5
Conclusions
The drying of yeast was performed in a pilot-scale fluidized-bed batch dryer over a range of experimental conditions from 40 to 80 °C and mass of bed 400-600 g. A range of data is presented in this work including measurements of the particle size, bed temperature, outlet humidity and final moisture content. A numerical model including a comprehensive heat and mass transfer description was developed based on differential equations to allow the bed
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temperature, air humidity and moisture content of yeast to be predicted during drying. The model
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developed has a strong physical basis, uses a minimum of fitted parameters and offer good predictions across a range of operating conditions. Hence, the model developed in this work has
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the potential to be broadly applied to the fluidized-bed drying of biological materials at both laboratory and commercial scales. In the second part of this work (Soltani et al. 2019) the model
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developed is further extended to model the viability of the yeast to give a comprehensive model
Nomenclature
Description
Symbol
Description
a
Specific area [m2 of external surface kg of dry solid-1]
ms
Mass of dry solid [kg]
A
Antoine constant [-]
n
Number of measurements [-]
Antoine constant [°C]
p*
Saturated vapor pressure [Pa]
C
Antoine constant [°C]
P
Pressure [Pa]
cG
Constant in GAB isotherm [-]
Qin
Rate of heat flow at the inlet of the fluidised-bed [W]
cp,eq
Specific heat capacity of equipment [J kg-1 K-1]
Qout
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B
al
Symbol
Pr
of the drying process
Rate of heat flow at the outlet of the fluidised bed [W]
27 Page 27 of 35
kg-1 K-1]
fluidised-bed [W]
Specific heat capacity of dry air cp,g
[J kg-1 K-1]
Qloss
Rate of heat loss from the bed to the environment [W]
Rg
Universal gas constant [J kg-1 K-1]
Specific heat capacity of water vapor [J kg-1 K-1]
RH
db
Average dryer diameter [m]
t
dp
Particle diameter [m]
dp0
Initial particle diameter [m]
D
Diffusivity of water vapor in air [m2 s1 ]
g
Gravitational acceleration [m s-2]
G
Mass flow rate of air [kg hr-1]
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Pr
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T
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hloss
Heat transfer coefficient between dryer and ambient [W m-2 K-1]
-2
-1
Time [s] Temperature [°C]
Tamb
Ambient temperature [°C]
Tb
Temperature of fluidised-bed [°C]
Tf
Film temperature [K]
Tg,i
Gas inlet temperature [°C]
Tp
Temperature of particle [°C]
U
Superficial velocity [m s-1]
UG
Gas velocity [m s-1]
Overall heat transfer coefficient
hoverall
Relative humidity [-]
pr
cp,wv
oo
[J kg-1 K-1]
f
Specific heat capacity of liquid water cp,w
[W m K ] Bed-wall heat transfer coefficient ho
-2
-1
[W m K ]
28 Page 28 of 35
H
Height of dryer [m]
UP
Particle velocity [m s-1]
Vb
Bed volume [m3]
Drying rate J
[kg water kg dry solid-1 s-1]
Moisture content (wet basis) Average dryer diameter [m]
W
f
Gas-particle mass transfer coefficient xG
[m s-1] Thermal conductivity of gas
kG
X
e-
[W m-1 K-1]
Pr
kg
Constant in GAB isotherm
Constant in GAB isotherm [-]
X0
al
Thermal conductivity of dryer wall [W m-1 K-1]
Moisture content, dry-basis [kg of water kg of dry solid-1]
Initial particle moisture content, dry-basis [kg of water kg of dry solid-1] Equilibrium moisture content, dry-basis
Xe
rn
kw
[kg of water kg of dry solid-1]
pr
k
[kg water kg solid-1]
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db
[kg of water kg of dry solid-1]
xm
Constant in GAB isotherm [kg of water kg of dry solid-1]
Characteristic length [m]
Yin
humidity of air at the inlet of the dryer [g of water kg of dry air-1]
Lw
Wall thickness [m]
Yout
L0
Initial particle length [m]
Yr
Lc
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Particle length [m]
L
Humidity of air in the bulk of gas phase [g of water kg of dry air-1] Humidity of air at the outer surface of solids
29 Page 29 of 35
[g of water kg of dry air-1] Saturation humidity Mass of equipment [kg]
Tg,out
Gas outlet temperature [°C]
Ysat
[g of water kg of dry air-1]
oo
f
meq
Dimensionless groups
pr
Greek symbols Description
Group
β
Coefficient of expansion [K-1]
ε
e-
Symbol
Description Grashof number
Bed voidage [-]
Nu
Nusselt number
λ
Latent heat of vaporization [J kg-1]
Pr
Prandtl number
µG
Gas dynamic viscosity [kg m-1 s-1]
Ra
Rayleigh number
νG
Gas kinematic viscosity [m2 s-1]
Re
Reynolds number
Gas density [kg m-3]
Sc
Schmidt number
ρs
Solids density [kg m-3]
Sh
Sherwood number
Ψ
Shrinkage factor [-]
Ω
External surface area of the fluidizedbed dryer [m2]
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ρG
Pr
Gr
30 Page 30 of 35
Acknowledgements This research was supported by the Australian Research Council (ARC) Industrial Transformation Training Centre (IC140100026). We would also like to acknowledge the valuable input of Dr Hadi Khabbaz (University of Technology Sydney) for his help with the
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MATLAB programming.
31 Page 31 of 35
References
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Bayrock, D., and W. M. Ingledew. 1997. 'Mechanism of viability loss during fluidized bed drying of baker's yeast', Food Research International, 30: 417-25. Beker, Martin J, and Alexander I Rapoport. 1987. 'Conservation of yeasts by dehydration.' in, Biotechnology Methods (Springer). Bizot, H. 1983. 'Using the'GAB'model to construct sorption isotherms', Physical properties of foods: 43-54. Çengel, Yunus A., and Afshin J. Ghajar. 2015. Heat and mass transfer : fundamentals & applications. Chen, XD, W Pirini, and M Ozilgen. 2001. 'The reaction engineering approach to modelling drying of thin layer of pulped Kiwifruit flesh under conditions of small Biot numbers', Chemical Engineering and Processing: Process Intensification, 40: 311-20. Chen, XD, and GZ Xie. 1997. 'Fingerprints of the drying behaviour of particulate or thin layer food materials established using a reaction engineering model', Food and Bioproducts Processing, 75: 213-22. Chen, Xiao Dong. 2008. 'The basics of a reaction engineering approach to modeling airdrying of small droplets or thin-layer materials', Drying Technology, 26: 627-39. Churchill, Stuart W, and Humbert HS Chu. 1975. 'Correlating equations for laminar and turbulent free convection from a horizontal cylinder', International journal of heat and mass transfer, 18: 1049-53. Clift, R. , J. R. Grace, and M. E. Weber. 1978. Bubbles, Drops and Particles (Academic Press: New York). Coulson, J. M., J. F. Richardson, J. R. Backhurst, and J. H. Harker. "Coulson and Richardson's Chemical Engineering Volume 1 - Fluid Flow, Heat Transfer and Mass Transfer (6th Edition)." In.: Elsevier. Debaste, F., V. Halloin, L. Bossart, and B. Haut. 2008. 'A new modeling approach for the prediction of yeast drying rates in fluidized beds', Journal of Food Engineering, 84: 335-47. Ellis, Bryan, and Ray Smith. 2008. Polymers: a property database (CRC Press). Ergun, Sabri. 1952. 'Fluid flow through packed columns', Chem. Eng. Prog., 48: 89-94. Felder, R.M., and R.W. Rousseau. 2005. Elementary Principles of Chemical Processes, 3rd Edition 2005 Edition Integrated Media and Study Tools, with Student Workbook (Wiley). Goderska, Kamila. 2012. 'Different methods of probiotics stabilization.' in, Probiotics (InTech). Köni, M., M. Türker, U. Yüzgeç, H. Dinçer, and H. Kapucu. 2009. 'Adaptive modeling of the drying of baker's yeast in a batch fluidized bed', Control Engineering Practice, 17: 503-17.
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rn
al
Pr
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pr
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Langrish, TAG. 2009. 'Multi-scale mathematical modelling of spray dryers', Journal of Food Engineering, 93: 218-28. Lasdon, Leon S, Allan D Waren, Arvind Jain, and Margery Ratner. 1978. 'Design and testing of a generalized reduced gradient code for nonlinear programming', ACM Transactions on Mathematical Software (TOMS), 4: 34-50. Mujumdar, Arun S. 2014. Handbook of industrial drying (CRC press). Oosthuizen, Patrick H, and David Naylor. 1999. An introduction to convective heat transfer analysis (McGraw-Hill Science, Engineering & Mathematics). Porter, HF, GA Schurr, DF Wells, and KT Semrau. 1984. "Solids drying and gas-solid systems." In, 20-4 to 20-13, and 20-59 to 20-74. McGraw-Hill New York. Quirijns, Elisabeth J., Anton J. B. van Boxtel, Wilko K. P. van Loon, and Gerrit van Straten. 2005. 'Sorption isotherms, GAB parameters and isosteric heat of sorption', Journal of the Science of Food and Agriculture, 85: 1805-14. Reid, R.C., and T.K. Sherwood. 1966. The properties of gases and liquids: their estimation and correlation (McGraw-Hill). Rhodes, Martin J, and Martin Rhodes. 2008. Introduction to particle technology (John Wiley & Sons). Smith, J.M., and H.C. Van Ness. 1975. Introduction to chemical engineering thermodynamics (McGraw-Hill). Soltani, Behdad, Dale D. McClure, Farshad Oveissi, Timothy A.G. Langrish, and John M. Kavanagh. 2019. 'Experimental investigation and numerical modeling of pilotscale fluidized-bed drying of yeast: Part B – viability measurements and modelling', Food and Bioproducts Processing. Spreutels, L., F. Debaste, R. Legros, and B. Haut. 2013. 'Experimental characterization and modeling of Baker's yeast pellet drying', Food Research International, 52: 275-87. Spreutels, Laurent, Benoît Haut, Jamal Chaouki, Francois Bertrand, and Robert Legros. 2014. 'Conical spouted bed drying of Baker's yeast: Experimentation and multimodeling', Food Research International, 62: 137-50. Strumiłło, C., and Tadeusz Kudra. 1986. 'Drying: Principles, Applications, and Design.' in (Gordon and Breach Science Publishers). Treybal, R.E. 1988. Mass-transfer Operations (McGraw-Hill). Türker, Mustafa, Ali Kanarya, Uğur Yüzgeç, Hamdi Kapucu, and Zafer Şenalp. 2006. 'Drying of baker's yeast in batch fluidized bed', Chemical Engineering and Processing: Process Intensification, 45: 1019-28. VAISALA. 2013. "Vaisala HUMICAP® Humidity and Temperature Transmitter Series HMT330." In. Finland Van den Berg, C, and S Bruin. 1978. "Water activity and its estimation in food systems." In Proceedings Int. Symp. Properties of Water in Relation to Food Quality and Stability, Osaka, 1978. Van Engeland, C, L Spreutels, R Legros, and B Haut. 2019. 'Convective drying of baker’s yeast pellets containing a carrier', J Drying Technology, 37: 1405-17. 33 Page 33 of 35
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Yuzgec, U., M. Turker, and Y. Becerikli. 2004. "Modelling of batch fluidized bed drying of baker yeast for cylindrical pellets." In Mechatronics, 2004. ICM '04. Proceedings of the IEEE International Conference on, 7-12. Yüzgeç, Uğur, Mustafa Türker, and Yasar Becerikli. 2008. 'Modelling spatial distributions of moisture and quality during drying of granular baker's yeast', The Canadian Journal of Chemical Engineering, 86: 725-38. Zabrodskiĭ, Sergeĭ Stepanovich. 1966. Hydrodynamics and heat transfer in fluidized beds (Massachusetts Institute of Technology).
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Yeast drying model with a strong physical basis developed
•
Model validated against a wide range of operating conditions
•
Model gives good predictions of bed temperature, moisture content and air humidity
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•
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PII:
S0960-3085(19)30335-9
DOI:
https://doi.org/doi:10.1016/j.fbp.2019.11.008
Reference:
FBP 1177
To appear in:
Food and Bioproducts Processing
Received Date:
16 April 2019
Revised Date:
11 November 2019
Accepted Date:
12 November 2019
Please cite this article as: Soltani, B., McClure, D.D., Oveissi, F., Langrish, T.A.G., Kavanagh, J.M.,Experimental investigation and numerical modeling of pilot-scale fluidized-bed drying of yeast: Part A - drying model development and validation, Food and Bioproducts Processing (2019), doi: https://doi.org/10.1016/j.fbp.2019.11.008
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Experimental investigation and numerical modeling of pilot-scale fluidizedbed drying of yeast: Part A - drying model development and validation Behdad Soltani, Dale D. McClure, Farshad Oveissi, Timothy A.G. Langrish§, John M. Kavanagh§*
*
: Equal senior authors
: Corresponding Author:
John Kavanagh (Email: [email protected])
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§
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The University of Sydney, School of Chemical and Biomolecular Engineering, Sydney, 2006, Australia
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Abstract
In this work we have developed an extensive experimental data-set for the fluidized-bed drying
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of yeast. A range of experimental conditions including inlet air temperatures from 40 to 80 °C, bed temperatures from 10 to 60 °C and bed masses of 400-600 g were used. This data was used to validate a mathematical model of the drying process. The model developed uses the reaction
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engineering approach to estimate the drying kinetics and has a strong physical basis with the
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only fitted parameters being those used for the GAB isotherm. Agreement between model predictions and experimental data was excellent, the maximum root-mean-square errors were 3.1
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°C, 1.8 g of water per kg of dry air, and 2.3% in the bed temperature, air humidity and the final moisture content on a wet-basis, respectively. Such results demonstrate that the model developed can be used to predict the behavior of the drying process across a broad range of operating conditions. Results from this work can be applied to the drying of yeast and other starter cultures used in the food industry.
Keywords: fluidised-bed dryer, yeast, lumped modeling, drying, reaction engineering approach
1 Page 1 of 35
1
Introduction
Drying is one of the oldest techniques used to preserve the quality of food. It is also used to prolong the shelf life of many food starter cultures, including yeast used for baking and those used in the dairy industry (Goderska 2012). The principle of drying is based on reducing the water content in the product to minimize microbial growth and enzymatic activity. Drying also has the advantage of reducing the mass of the product, thereby lowering transport costs (Mujumdar 2014).
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In yeast drying, fluidized-beds using warm dry air are the most common, whilst for dairy starter cultures low temperature air is used (Mujumdar 2014). Fluidized-bed dryers have the advantages of providing good mixing of the solids as well as having large interfacial areas which leads to
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relatively high rates of heat and mass transfer (Strumiłło and Kudra 1986). Drying of microorganisms is particularly challenging as it is necessary to remove sufficient water to
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improve the shelf life of the product (dried yeast typically has a final moisture content of the order 4-6 wt%) while avoiding damage to the cells. It is known that both the temperature and
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duration of drying are key factors in this regard (Beker and Rapoport 1987), with temperatures above 40 °C being reported to be damaging for baker’s yeast (Bayrock and Ingledew 1997). An ideal drying process would remove sufficient moisture to ensure the shelf life of the product
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while minimizing its exposure to damaging temperatures. Such considerations are particularly
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important in the drying of microorganisms as damaging conditions reduces the viability of the cells which is essential to their use in baking or as starter cultures.
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The drying rate (J) is defined as the rate of moisture removal from the solid per unit time. During a typical drying process, there are three periods: a warming up period, an unhindered (constant rate) drying period and an hindered (falling rate) drying period (Porter et al. 1984). In the unhindered drying period, the rate of drying is limited by the rate of mass transfer from the wetted film on the particle to the air. If the external conditions (i.e. the gas temperature, velocity and humidity) are constant drying will occur at a fixed rate and the temperature of the bed will be approximately constant. As the solids dry a point is reached where the rate of moisture transfer from the interior of the particle to the surface is less than the gas-particle mass transfer rate. It is at this point unhindered drying stops and the hindered drying phase commences. This part of the drying curve continues, with the moisture content approaching the equilibrium value. The bed 2 Page 2 of 35
temperature can rapidly increase during the hindered stage, such increases can be difficult to control. As previously noted high temperatures can be damaging to the viability of starter cultures and hence should ideally be avoided to maintain the product quality. Being able to predict such increases in the temperature is obviously desirable from the perspective of both controlling and optimizing drying processes for microorganisms including yeast. It is for this reason a number of studies into the modeling of yeast drying have been carried out in the last decade (Debaste et al. 2008; Spreutels et al. 2013; Spreutels et al. 2014;
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Yuzgec, Turker, and Becerikli 2004; Türker et al. 2006; Yüzgeç, Türker, and Becerikli 2008; Van Engeland et al. 2019). Köni et al. (2009) developed a black box model using an artificial neural network and adaptive neural network-based fuzzy inference system structures. Their
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model gave good predictions of the moisture content (d.b) as a function of time. However, given that these were black box models, the physical process understanding incorporated in the model
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is unclear.
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The majority of authors (Debaste et al. 2008; Türker et al. 2006; Spreutels et al. 2014) have developed models using mass and energy balance equations. A range of approaches have been used to calculate the drying rate, Türker et al., (2006) used a kinetic approach. It was found that
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this modelling approach gave good agreement for small (0.6 mm diameter) yeast particles. However, agreement was less good for larger diameter (1.2 mm) particles. This difference was
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attributed to the fact that diffusive transport of water within the particles became significant as
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the particle diameter increased, hence it was necessary to account for this in the model. Spreutels et al. (2014) modelled the flux of two different ‘types’ of water; corresponding to extracellular and intracellular moisture. A kinetic model was used to model the drying rate for intracellular water, while the drying rate of extracellular water was modelled using a reaction engineering type approach where diffusion of moisture inside the yeast particles was also accounted for. Debaste et al. (2008) used a similar approach, using the GAB isotherm to calculate the relative humidity. It was found this model gave good agreement in the constant rate period, but poor agreement during the falling rate portion of the drying. To address this shortcoming, they replaced the GAB isotherm with a new model which accounts for the diffusion of moisture through the yeast particle during the falling rate period.
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Based on the previous work it may be necessary to model the diffusion of moisture within the yeast particles to accurately predict drying during the falling rate period. Ideally, any model should have a minimum number of adjustable parameters that are specific to a particular experimental configuration and the model should also be validated against a broad range of experimental data. The aim of this study was to develop and validate a quantitative model capable of accurately predicting the behavior (temperature and moisture content) as a function of time during the
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fluidized bed drying of yeast. Such a model can be used to design and optimize industrial drying processes for starter cultures. A key focus in the development of the model was minimizing the number of configuration specific fitted parameters, in order to ensure that the model would have
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a broad range of application. Results from this paper will be used in the second part of the work to develop a model where the drying conditions are related to the viability of the yeast (Soltani et
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al. 2019). Conclusions from this work can be applied to a broad range of drying processes in the
2 2.1
Materials and methods Yeast preparation
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food industry.
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Cream yeast (79-82% moisture content, wet-basis) was supplied by an industrial source. The
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yeast suspension was stored at 4 °C prior to use. An homogenized dispersion of Sorbitan monostearate (SMS) was added to the yeast suspension at the industrially recommended rate of 1
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wt% (calculated on the mass of dry matter). The mixture of yeast and SMS was then filtered under a vacuum of 50 kPa for one hour to concentrate the cream yeast to 70% moisture content, on a wet-basis. The yeast cake was pressed for 30 min and extruded through ∼0.6 mm diameter holes at a pressure of 500-600 kPa using a pneumatic press. The prepared samples were weighed before being placed into the dryer. In this paper a yeast particle is defined as the cylindrical pellets produced by the extrusion of the yeast cake. 2.2
Pilot-scale dryer
All experiments were performed in a pilot-scale fluidized-bed dryer as shown schematically in Figure 1. This unit was made of polycarbonate (4.5 mm wall thickness) and has a height of
4 Page 4 of 35
670 mm with a diameter of 100 mm at the base, increasing to 250 mm at the top of the conical section. Cool air (4-6 °C) was heated using a heat exchanger before being passed through a Munters desiccant dehumidifier to achieve a relative humidity of 2 to 3% and a temperature of 30 °C. The system was run for one hour before performing any experiments such that any changes in the inlet air humidity and temperature were minimal. The inlet air humidity (Yin) during experiments
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varied between 1 to 1.5 g of water per kg of dry air, the difference between batches was mainly a
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function of the ambient humidity, and the inlet air humidity remained approximately constant for
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each experiment.
The dehumidified air was passed into a heater to increase its temperature to 40-80 °C, with the
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temperature being varied depending on the experimental conditions. For all experiments, the mass flow rate was measured using a Proline thermal mass flow meter (Endress+Hauser), the air
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flow rate (G) was set to 55 ± 1.4 kg hr-1. This corresponded to a superficial velocity of 1.6 m s-1 (calculated using a bed diameter of 100 mm). It was calculated that approximately 22 kJ was needed to heat the dryer from ambient temperature (25
) to 30
(using the equipment mass and
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heat capacity listed in Tables 1 and 2).
5 Page 5 of 35
f oo pr ePr al rn
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Figure 1 – Schematic showing set-up of pilot-scale dryer used in the current work.
The bed temperature and air humidity were measured using a humidity and temperature meter (Vaisala), as shown schematically in Figure 1. Bed temperatures were measured to an accuracy of
0.2 °C between 0 – 60 °C.
The accuracy of the bed humidity measurements was a function of both the relative humidity and temperature, with the accuracy decreasing at high temperatures and high values of the relative humidity. For example, at a temperature of 40 °C the accuracy in the relative humidity measurement was 0.49-1.61 g of water per kg of dry air for 0-100% relative humidity, respectively (VAISALA 2013). The bed temperature and air humidity were recorded every 10 s
6 Page 6 of 35
by an automated data logging system. A Mettler Toledo Halogen Moisture Analyzer was used to measure the moisture content on a wet-basis of the yeast before and after drying. All the measured moisture contents had an uncertainty of ± 0.1%.
At the start of each experiment, a known mass (400 or 600 g) of yeast was added to the dryer. The average moisture content of particles after compression was recorded to be approximately 67-70%, and the final moisture content of particles was 2.3-6.4%, depending on the final inlet air
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temperature. Each experiment continued until the difference between the exhaust and inlet air
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humidity was less than 1 g of water per kg of dry air or until the bed temperature reached 60 °C. Using the current experimental set-up, it was not possible to measure bed temperatures greater
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than 60 °C. Such conditions are likely to be highly damaging to the yeast and hence are not likely
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to be of significant interest. The maximum possible inlet air temperature was 110 °C. In the first stage of this research, the inlet air temperature was set to 50, 60, 70 and 80 °C and the
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initial mass of the yeast was 400 and 600 g. Another set of experiments was performed with stepchanges in the inlet air temperature from 80 and 70 °C to 40 and 50 °C after 10 and 15 min. This set of experiments aimed to provide better control over the bed temperature and further validate
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the model by ensuring it could accurately predict the results of a step change in the inlet air
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temperature.
To investigate the repeatability of the system, the inlet temperature was fixed at 50 °C and the
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flow rate was 55 kg hr-1, the same conditions were used for experiments performed in three different days. It was found that the measured values of the bed temperature, outlet air humidity and final moisture content were highly reproducible, with a difference of not more than 2.5 °C, 1.5 g of water per kg of dry air, and 1% in the bed temperature, outlet air humidity and the moisture content, (wet-basis), respectively. The results are comparable with the uncertainty of the measurements. 2.3
Particle size measurements
To quantify the extent of shrinkage during the drying process a separate experiment was performed using a fixed inlet temperature of 55 °C and an air flow rate of 55 kg hr-1. Samples 7 Page 7 of 35
were collected from the dryer 0, 10, 15, 20, 25 and 30 minutes after the start of drying. Images of the yeast particles were then taken with a stereomicroscope (Olympus, SZ61). At each time point, a total of 20 particles were measured; the reported results are the average diameter with the error bars denoting one standard deviation about the mean.
3
Numerical modelling
The following assumptions and simplifications were made in the model development: It was assumed that the system was well mixed.
•
It was assumed that gradients of both temperature and moisture content within the
oo
f
•
•
pr
particles were minimal (i.e. a lumped parameter model was used).
The pressure drop across the bed was calculated (using the Ergun equation (Ergun 1952))
e-
to be of the order 1-2 kPa. Since this is relatively small it was assumed that the process was operated at a constant pressure of 101.3 kPa (one atmosphere). All thermodynamic
Pr
properties were calculated at this pressure.
It was assumed that the equipment and the bed were at thermal equilibrium.
•
Ideal gas behavior was assumed. This was thought to be reasonable due to the operating
al
•
•
rn
pressure (101.3 kPa) and temperatures (30-80 °C) used. Constant values were used for the heat capacities as these do not change appreciably over
•
Jo u
the temperature range examined (30-80 °C). The mass of dry yeast particles was assumed to be constant (i.e. there was no mass loss due to particle attrition). •
The density and heat capacity of the yeast was assumed to be equivalent to that of liquid water (Türker et al. 2006).
•
It was assumed that the inlet and outlet mass flow rates of dry air were equal (i.e. no chemical reactions took place).
8 Page 8 of 35
•
It was assumed that the particles shrunk uniformly in all directions.
•
It was assumed that the bed temperature (Tb) was equal to the outlet air temperature (Tg,out). This assumption is reasonable due to the relatively high rates of mass transfer between the gas and particle. It is possible to derive the model equations without making this assumption, as detailed in the Supplementary Material this has a relatively small impact on the model predictions. It was found (see the Supplementary Material) that inclusion of terms for the sensible
f
•
oo
heat of water vapor and the thermal mass of the gas had minimal effect on the model predictions due to their small values. Hence, they have been neglected from the gas-phase
pr
heat balance.
constant for the duration of the drying.
e-
Experimentally measured values of the inlet air were used in the model, these values were
Pr
A schematic outlining the simultaneous heat and mass transfer occurring in the system is
Jo u
rn
al
presented in Figure 2.
9 Page 9 of 35
f oo pr ePr
Figure 2 – Schematic showing the flows of mass and energy in the drying system.
3.1
Mass and energy balances
al
The unsteady mass balance for the moisture content (dry-basis) of the solid can be expressed as:
rn
(1)
Jo u
where X is the moisture content of the solids (dry-basis), t is the time and J is the rate of evaporation. The driving force for the external mass transfer may be expressed as the difference between the air humidity at the external surface of the particle and in the bulk of the gas phase: (2)
where k is the gas-particle mass transfer coefficient, ρg is the density of the gas, Yr is the humidity at the particle surface, Yout is the outlet humidity and a is the interfacial area for mass transfer.
10 Page 10 of 35
This type of equation is very similar to the Reaction Engineering Approach (REA), which was proposed by Chen and Xie (1997). This approach has been used to model the drying of a wide range of foods including skim milk, kiwi, apple pieces and potato slices (Langrish 2009; Chen 2008; Chen, Pirini, and Ozilgen 2001). The water mass balance for the gas phase is:
f
(3)
oo
where Vb is the volume of the bed, ε is the void fraction and mS is the mass of dry solids. The energy balance around the system can be used to calculate the rate of temperature change for
pr
the outlet gas (Tg,out), this is equivalent to the bed temperature. This equation accounts for the
(4)
al
Pr
e-
thermal mass of the equipment as well as heat loss to the environment.
rn
where cp,w and cp,eq are the specific heat capacities of liquid water and the equipment, respectively. λ is the enthalpy of latent heat for the water, meq is the mass of the equipment,
Jo u
hoverall is the overall heat transfer coefficient from the dryer to the environment, Ω is the external surface area of the dryer and Tamb is the ambient temperature. 3.2
Heat and mass transfer equations
Here we have used a published (Clift, Grace, and Weber 1978) correlation which is valid for cylindrical particles at Reynolds numbers up to 1000 to calculate the Sherwood number: (5)
The Sherwood number (Sh) was defined subsequently:
11 Page 11 of 35
S
(6)
where dp is the characteristic length (here the diameter of the cylindrical yeast particles) and D is the diffusivity of water vapor in air (2.6 × 10-5 m2 s-1) (Coulson et al.). The Reynolds (Re) and Schmidt (Sc) numbers are defined as follows:
(8)
pr
oo
f
(7)
e-
where µg is the viscosity of the gas, U is the gas superficial velocity, UG is the gas velocity and UP is the velocity of the yeast particle. In practice precisely determining the particle velocity
Pr
relative to the gas (i.e. UG – UP) is challenging. The superficial gas velocity gives a reasonable approximation to the actual gas velocity due to the high bed voidage; this varies between 1.6 and 0.3 m s-1 due to the changes in the dryer cross sectional area. The particle velocity (UP) can be
al
reasonably estimated by determining the terminal velocity of the particles, calculated values of the particle terminal velocity are of the order 0.1 m s-1. Here for convenience we have taken the
rn
arithmetic average of the superficial gas velocity and the inlet and outlet to calculate the gas velocity (UG), while neglecting the particle velocity in the calculation of the Reynolds number,
Jo u
this gave a relative velocity between the gas and particle of the order 1 - 1.1 m s-1. Using this value, the Reynolds number varied between 15 < Re < 40. Values calculated for the Schmidt number were 0.5 < Sc < 0.8. This gave values of the gas-particle mass transfer coefficient (k) between 0.2 and 0.4 m s-1.
Here we have used the GAB isotherm to model the relationship between the equilibrium moisture content (Xe) and the relative humidity (RH) as it is generally thought to be the most suitable for describing the sorption of water by food products (Van den Berg and Bruin 1978; Bizot 1983) including yeast (Debaste et al. 2008). The GAB equation is usually presented in the following form: 12 Page 12 of 35
(9)
where Xe is the equilibrium moisture content (dry-basis), parameters. The parameter surface,
and
and
are characteristic
is related to the number of active sites for adsorption on the
corresponds to the number of multilayers, and
is related to the interaction of water
and the surface (Quirijns et al. 2005). In this study kG, cG and xm were estimated from the GAB
oo
f
isotherm as 0.88, 4.5, and 0.28, respectively. These parameters were fitted to experimental data using the generalized reduced gradient method (Lasdon et al. 1978) with an inlet air temperature of 50 °C, bed temperatures from 10-60 °C, and a bed mass of 400 g. The parameters were used
pr
for all other simulations.
e-
The Antoine equation was used to calculate the saturated vapor pressure for water (p*), this is necessary to convert the relative humidity value from Equation (9) to the moisture content (dry-
al
Pr
basis) at the surface of the solid (Yr):
(10)
rn
(11)
Jo u
(12)
where A, B and C are constants in the Antione equation, having values of 8.107, 1750.286, and 235, respectively (Felder and Rousseau 2005). Mw and Mg are the molar masses of water vapor and air respectively and Ysat is the saturation humidity. The overall heat transfer coefficient from the dryer to ambient (hoverall) is made up of three terms: (13)
13 Page 13 of 35
The rate of heat transfer through the wall is inversely proportional to its thickness (Lw), and directly proportional to its thermal conductivity kw. Here the wall thickness was 4.5 mm and we have used a thermal conductivity of 0.19 W m-1 K-1 for polycarbonate (Ellis and Smith 2008). Conductive losses will also occur through the base of the dryer, however due to their relatively small magnitude and the difficulty in accurately determining their value they have been omitted from the model. Heat transfer from the bed to the wall (ho) was determined using a correlation proposed for
oo
f
particles in group B of the Geldart classification (which includes baker’s yeast) (Zabrodskiĭ
(14)
e-
pr
1966; Rhodes and Rhodes 2008):
The heat transfer from the bed to the environment
was calculated using Equation (16).
Pr
This correlation was recommended by Churchill and Chu (1975) for a vertical flat plate geometry as it is applicable to all range of Rayleigh numbers (Churchill and Chu 1975), and it
al
can be used for a vertical cylinder when:
rn
is the average diameter of the fluidised-bed dryer.
Jo u
where
(15)
(16) where kg is the thermal conductivity of the gas. The Rayleigh (Ra) and Grashof numbers (Gr) are defined as: 14 Page 14 of 35
(17)
(18)
where
is acceleration due to gravity, and
is the kinematic viscosity of air. The Grashof
number was calculated at the film temperature (Tf) using the thermal expansion coefficient for an
(19)
(20)
e-
pr
oo
f
ideal gas (β):
Pr
To find the heat loss the characteristic length was taken as the height ( ) of the bed (0.43 m), and the outer surface area of the bed (Ω) was 0.629 m2. . The heat-transfer
al
The total heat flow rate at the inlet of the bed varied between 500 and 1000
coefficient from particles to the bed, through the wall and from the exterior of the dryer to
rn
ambient during all experiments were 700-1000, 42, and 3-6 W m-2 K-1, respectively. Hence, the rate of heat loss was between 5-120 W, this was up to 10% of the inlet heat flow and for this
3.3
Jo u
reason heat loss was included in the model. Particle shrinkage model and interfacial area
Other authors (Spreutels et al. 2013) have found that the particle diameter is a linear function of the moisture content. To determine whether this was the case experimental measurements were performed as described in Section 2.3. As shown in Figure 3 this was found to be correct, hence we have used the following equation to calculate the particle diameter as a function of the moisture content (dry-basis): (21)
15 Page 15 of 35
where L is the particle length, dp0, L0 and X0 are the initial values of the particle diameter, length and moisture content (dry-basis), respectively. Ψ is the particle shrinkage coefficient, here a value of 0.3 was used, this value being calculated from a fit to the experimental data. The particle surface area (a) was calculated as:
(22)
Jo u
rn
al
Pr
e-
pr
oo
f
where the empirical factor of 9/8 accounts for the particle end area (given L = 4dp)
Figure 3 – Plot showing the particle diameter as a function of the moisture content on a dry-basis. The abscissa scale is the reverse of the norm (i.e. drying occurs from left to right).
The void fraction was calculated by measuring the height of the bed to calculate the bed volume and then using the known mass and density of the yeast to determine its volume and hence the bed voidage. Calculated values ranged between 0.93 and 0.96, for the sake of simplicity a fixed value of 0.95 was used in the model. It was found that the varying the value of the volume fraction across the experimentally measured range had minimal impact on the model predictions. A summary of the physical properties and other parameters used in the model is given in Tables 1 and 2. Equations (1), (3) and (4) were solved using the ordinary differential equation solvers 16 Page 16 of 35
available in Matlab (R2015a). Initial values of the particle moisture content on a dry-basis, outlet air temperature and outlet air humidity ranged between 2.13-2.32 g water per kg of dry solid, 1421 °C and 8-12 g of water per kg of dry air, respectively. These values were obtained from the experimental measurements. Differences between the model predictions and the experimental measurements were quantified
(23)
oo
f
using the Root-Mean-Square Error (RMSE):
pr
where n was the total number of measurements and χ was the variable being compared. Here we have reported the final moisture content on a wet basis (W), this was calculated from the
(24)
Jo u
rn
al
Pr
e-
moisture content on a dry basis (X, kg moisture per kg dry solid) in the following manner:
17 Page 17 of 35
Value
Reference
Density of dry air,
[kg m-3]
(Smith and Van Ness 1975)
Specific heat of dry air,
[J kg-1 K-1]
(Çengel and Ghajar 2015)
Dynamic viscosity of air,
[kg m-1 s-1]
(Reid and Sherwood 1966)
Kinematic viscosity of air,
[m2 s-1]
pr
Parameter
f
Units
oo
Table 1 – Summary of physical properties used in the model
Thermal conductivity of air,
Molar mass of dry air,
Density of water,
e-
28.97
[kg kmol-1]
(Treybal 1988)
18.02
[kg kmol-1]
(Treybal 1988)
1000
[kg m-3]
(Smith and Van Ness 1975)
4,180
[J kg-1 K-1]
(Smith and Van Ness 1975)
2257000
[J kg-1]
2.56 × 10-5
[m2 s-1]
(Coulson et al.)
1300
[J kg-1 K-1]
(Ellis and Smith 2008)
0.19
[W m-1 K-1]
(Ellis and Smith 2008)
101,325
[Pa]
rn
Specific heat capacity of water,
Jo u
Latent heat of vaporization for water,
Mass diffusivity of water in air, Specific heat capacity of equipment (polycarbonate), cp,eq Thermal conductivity of polycarbonate Atmospheric pressure,
(Reid and Sherwood 1966) (Smith and Van Ness 1975)
al
Molar mass of water,
2015)
[J kg-1 K-1]
287.05
Pr
Gas constant for dry air,
[W m-1 K-1]
(Çengel and Ghajar
(Oosthuizen and Naylor 1999)
(Smith and Van Ness 1975)
18 Page 18 of 35
Table 2 – Summary of parameters used in the model
Value
Units
Ambient temperature, Tamb
25
[°C]
Initial particle diameter, dp0
600
[µm]
External surface area of dryer, Ω
0.629
[m2]
Bed volume, Vb
0.0092
[m3]
Mass of equipment, meq
3.4
Height of bed, H
430
Wall thickness, Lw
4.5
Bed voidage
0.95
Particle shrinkage coefficient, Ψ
0.3
oo pr
e-
GAB isotherm parameter, kg
[kg]
[mm] [mm] [-] [-]
0.88
[-]
4.5
[-]
Pr
GAB isotherm parameter, cG
0.28
[kg moisture kg dry solids-1]
Jo u
rn
al
GAB isotherm parameter, xm
f
Parameter
19 Page 19 of 35
4
Results and discussion
In order to validate the model experiments were performed with a broad range of inlet air temperatures (50-80 °C) and initial bed masses of 400 and 600 g. Taking a value of 2% for the inlet air relative humidity the calculated values of the wet-bulb temperature were 19, 23, 26 and 30
for inlet temperatures of 50, 60, 70 and 80
respectively. Results are shown in Figure 4
for an initial bed mass of 400 g and in Figure 5 for an initial bed mass of 600 g.
oo
f
As expected, three stages of drying were observed. Firstly a brief (approximately three minutes) warming up period was observed where the inlet temperature reached the desired value (from an initial value of 30 °C). Inclusion of this delay in the model by ramping the inlet air temperature
pr
from 30 °C to the set point over a period of 3.5 minutes noticeably improved predictions of the bed temperature. After the warming up period the unhindered drying period occurred, the
e-
duration of this period depended on both the inlet temperature and the mass of the bed, with the model correctly predicting these trends (i.e. longer unhindered drying periods for lower inlet
Pr
temperatures and higher initial bed masses). During this period the model appears to over-predict the outlet humidity. However, this difference is comparable with the uncertainty in the
al
experimental measurements, meaning it is not possible to state with certainty that this is the case. It was found for all of the cases examined that the air passing through the dryer was largely
rn
saturated during the unhindered drying phase. This meant that the driving force for mass transfer (see Equation (2)) was relatively low, explaining the relatively long duration of this period. Such
Jo u
results have been observed by others for similar processes (Spreutels et al. 2014). At the end of the unhindered phase the surface of the particles is no longer fully wetted, and moisture was transferred from the interior of the particle to the surface until the equilibrium value was achieved. During this portion of the drying the bed temperature increased, as shown in Figures 4(a) and 5(a) this rise was quite steep for higher values of the inlet air temperature and may be damaging to the yeast cells. The bed temperature continued to increase during this portion of the drying, while remaining below the inlet temperature due to residual drying and heat losses. In all instances the model correctly predicted the experimentally observed trends, while also offering good agreement with the experimental data. RMSEs for bed temperature, air humidity, 20 Page 20 of 35
and the final moisture content (wet-basis) were 2.62 °C, 1.75 g of water per kg of dry air, and 1.69%, respectively for an initial bed mass of 400 g and 3.07 °C, 1.71 g of water per kg of dry air, and 1.06%, respectively for a bed mass of 600 g. Such results demonstrate that the model developed is capable of correctly predicting the drying behavior for a broad range of operating
Jo u
rn
al
Pr
e-
pr
oo
f
conditions.
21 Page 21 of 35
f oo pr ePr al rn Jo u Figure 4 – Plot comparing the experimentally measured values of bed temperature (a), outlet humidity (b) and final moisture content on a wet-basis (c) with the model predictions. Results are for an initial bed mass of 400 g and at fixed inlet air temperatures of 50, 60, 70 and 80 °C. All results are for a fixed inlet air flow rate of 55 kg hr-1. The final moisture content (wet-basis) in the experimental data had an uncertainty of ± 0.1%.
22 Page 22 of 35
f oo pr ePr al rn Jo u Figure 5 – Plot comparing the experimentally measured values of bed temperature (a), outlet humidity (b) and final moisture content on a wet-basis (c) with the model predictions. The final moisture content (wet-basis) in the experimental data had an uncertainty of ± 0.1%. Results are for an initial bed mass of 600 g and at fixed inlet air temperatures of 50, 60, 70 and 80 °C. All results are for a fixed inlet air flow rate of 55 kg hr-1.
23 Page 23 of 35
Running the drying process using a fixed inlet temperature is unlikely to be feasible from an industrial perspective as the increase in the bed temperature during the hindered drying phase is likely to be damaging to the yeast. For example, Bayrock and Ingledew (1997) found that the cell viability decreased when the temperature of the yeast cells exceeded 40 °C. To avoid this and to further validate the model additional experiments were performed where the inlet temperature was changed. Initially a relatively high (70 or 80 °C) inlet air temperature was used, this was reduced to a lower value (40 or 50 °C) after 15 and 10 minutes for initial temperatures of 70 or
oo
unhindered drying at that temperature (see Figure 4).
f
80 °C, respectively. These times were chosen as they correspond to the duration of the
Comparison between the model predictions of the bed temperature and outlet humidity are
pr
presented in Figure 6, while comparison between the experimentally measured and predicted final moisture content (wet-basis) is given in Figure 7. Again, the agreement between the model
e-
predictions and experimental measurements was good for the range of operating conditions examined, with the exception of the outlet humidity in the unhindered drying phase. However, as
Pr
previously noted the discrepancy between the model predictions and the data is of similar magnitude to the experimental uncertainty. Maximum values of the RMSE were of the order 2.5
al
°C for the bed temperature and 1.6 g of water per kg of dry air for the outlet humidity, indicating
Jo u
rn
that the model gave good predictions for the conditions examined.
24 Page 24 of 35
f oo pr ePr al rn Jo u
Figure 6 – Plot comparing the experimental measurements and model predictions of the bed temperature (a) and (b) and the outlet humidity (c) and (d). Results in the first column (a) and (c) were for an initial gas inlet temperature of 80 °C which was reduced to 40 and 50 °C after ten minutes; the results in the second column (b) and (d) are for an initial gas inlet temperature of 70 °C which was reduced to either 40 or 50 °C after 15 minutes. All results are for an initial bed mass of 400 g and an air flow rate of 55 kg hr-1. Times where the air inlet temperature was changed are denoted with a dotted vertical line.
25 Page 25 of 35
f oo pr ePr al rn
Jo u
Figure 7 – Comparison between experimentally measured values of the final solids moisture content-wet basis (W) and the model predictions for experiments performed using varying inlet temperatures. The final moisture content (wet-basis) in the experimental data had an uncertainty of ± 0.1%.
26 Page 26 of 35
5
Conclusions
The drying of yeast was performed in a pilot-scale fluidized-bed batch dryer over a range of experimental conditions from 40 to 80 °C and mass of bed 400-600 g. A range of data is presented in this work including measurements of the particle size, bed temperature, outlet humidity and final moisture content. A numerical model including a comprehensive heat and mass transfer description was developed based on differential equations to allow the bed
f
temperature, air humidity and moisture content of yeast to be predicted during drying. The model
oo
developed has a strong physical basis, uses a minimum of fitted parameters and offer good predictions across a range of operating conditions. Hence, the model developed in this work has
pr
the potential to be broadly applied to the fluidized-bed drying of biological materials at both laboratory and commercial scales. In the second part of this work (Soltani et al. 2019) the model
e-
developed is further extended to model the viability of the yeast to give a comprehensive model
Nomenclature
Description
Symbol
Description
a
Specific area [m2 of external surface kg of dry solid-1]
ms
Mass of dry solid [kg]
A
Antoine constant [-]
n
Number of measurements [-]
Antoine constant [°C]
p*
Saturated vapor pressure [Pa]
C
Antoine constant [°C]
P
Pressure [Pa]
cG
Constant in GAB isotherm [-]
Qin
Rate of heat flow at the inlet of the fluidised-bed [W]
cp,eq
Specific heat capacity of equipment [J kg-1 K-1]
Qout
rn
Jo u
B
al
Symbol
Pr
of the drying process
Rate of heat flow at the outlet of the fluidised bed [W]
27 Page 27 of 35
kg-1 K-1]
fluidised-bed [W]
Specific heat capacity of dry air cp,g
[J kg-1 K-1]
Qloss
Rate of heat loss from the bed to the environment [W]
Rg
Universal gas constant [J kg-1 K-1]
Specific heat capacity of water vapor [J kg-1 K-1]
RH
db
Average dryer diameter [m]
t
dp
Particle diameter [m]
dp0
Initial particle diameter [m]
D
Diffusivity of water vapor in air [m2 s1 ]
g
Gravitational acceleration [m s-2]
G
Mass flow rate of air [kg hr-1]
rn
al
Pr
e-
T
Jo u
hloss
Heat transfer coefficient between dryer and ambient [W m-2 K-1]
-2
-1
Time [s] Temperature [°C]
Tamb
Ambient temperature [°C]
Tb
Temperature of fluidised-bed [°C]
Tf
Film temperature [K]
Tg,i
Gas inlet temperature [°C]
Tp
Temperature of particle [°C]
U
Superficial velocity [m s-1]
UG
Gas velocity [m s-1]
Overall heat transfer coefficient
hoverall
Relative humidity [-]
pr
cp,wv
oo
[J kg-1 K-1]
f
Specific heat capacity of liquid water cp,w
[W m K ] Bed-wall heat transfer coefficient ho
-2
-1
[W m K ]
28 Page 28 of 35
H
Height of dryer [m]
UP
Particle velocity [m s-1]
Vb
Bed volume [m3]
Drying rate J
[kg water kg dry solid-1 s-1]
Moisture content (wet basis) Average dryer diameter [m]
W
f
Gas-particle mass transfer coefficient xG
[m s-1] Thermal conductivity of gas
kG
X
e-
[W m-1 K-1]
Pr
kg
Constant in GAB isotherm
Constant in GAB isotherm [-]
X0
al
Thermal conductivity of dryer wall [W m-1 K-1]
Moisture content, dry-basis [kg of water kg of dry solid-1]
Initial particle moisture content, dry-basis [kg of water kg of dry solid-1] Equilibrium moisture content, dry-basis
Xe
rn
kw
[kg of water kg of dry solid-1]
pr
k
[kg water kg solid-1]
oo
db
[kg of water kg of dry solid-1]
xm
Constant in GAB isotherm [kg of water kg of dry solid-1]
Characteristic length [m]
Yin
humidity of air at the inlet of the dryer [g of water kg of dry air-1]
Lw
Wall thickness [m]
Yout
L0
Initial particle length [m]
Yr
Lc
Jo u
Particle length [m]
L
Humidity of air in the bulk of gas phase [g of water kg of dry air-1] Humidity of air at the outer surface of solids
29 Page 29 of 35
[g of water kg of dry air-1] Saturation humidity Mass of equipment [kg]
Tg,out
Gas outlet temperature [°C]
Ysat
[g of water kg of dry air-1]
oo
f
meq
Dimensionless groups
pr
Greek symbols Description
Group
β
Coefficient of expansion [K-1]
ε
e-
Symbol
Description Grashof number
Bed voidage [-]
Nu
Nusselt number
λ
Latent heat of vaporization [J kg-1]
Pr
Prandtl number
µG
Gas dynamic viscosity [kg m-1 s-1]
Ra
Rayleigh number
νG
Gas kinematic viscosity [m2 s-1]
Re
Reynolds number
Gas density [kg m-3]
Sc
Schmidt number
ρs
Solids density [kg m-3]
Sh
Sherwood number
Ψ
Shrinkage factor [-]
Ω
External surface area of the fluidizedbed dryer [m2]
al
rn
Jo u
ρG
Pr
Gr
30 Page 30 of 35
Acknowledgements This research was supported by the Australian Research Council (ARC) Industrial Transformation Training Centre (IC140100026). We would also like to acknowledge the valuable input of Dr Hadi Khabbaz (University of Technology Sydney) for his help with the
Jo u
rn
al
Pr
e-
pr
oo
f
MATLAB programming.
31 Page 31 of 35
References
Jo u
rn
al
Pr
e-
pr
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Yeast drying model with a strong physical basis developed
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Model validated against a wide range of operating conditions
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Model gives good predictions of bed temperature, moisture content and air humidity
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