Design of a Vacuum Dryer Based on Experimental Data From Atmospheric Drying

To ensure long-term storage of feed pellets, it is essential to reduce their water content below the required level to prevent the potential biodegradation of the material [1]. Mechanical methods, such as ultrasonic vibration [2], can partially reduce water content. However, this approach may not be suitable for all materials, and a significant amount of moisture may still remain in the material. For this reason, it is usually necessary to integrate drying into the feed pellet production process. During drying, it is important not to exceed the maximum material temperature at which nutrient degradation may occur [3]. One option is using an infrared dryer, which may be suitable for thin materials. However, according to the results of Huang et al. [4], it may still affect the quality of the product. Another option is indirect vacuum drying, which effectively ensures compliance with the condition of not exceeding the maximum material temperature because the pressure set in the dryer determines the saturation temperature of the water vapor, which is close to the temperature of the material being dried. Therefore, the pressure set in the dryer can significantly affect the drying process. Another advantage of indirect drying compared to commonly used air drying is significantly lower energy consumption [3].

A horizontal paddle dryer is a suitable configuration for this application. An agitated dryer combining contact convective heating is described in Refs. [5,6]. The drying efficiency for dryers with agitators increases with the utilization of waste heat for the heating jacket [7]. The values of specific energy consumption of indirect dryers are lower than those of direct (hot air) dryers [8]. Energy-efficient and cost-effective drying can be achieved by utilizing free solar energy [9–11]. In solar dryers, stationary materials are dried, and specific models are used for calculations, as seen in Ref. [12]. The efficiency of these dryers can be enhanced by incorporating a heat pump [13]. However, these methods typically result in long drying times. To increase the production rate, a different heat source and agitation are necessary to improve the drying rate (DR).

The underlying fundamental problem of heat transfer from the wall to the mechanically agitated bed is usually described by continuous models, specifically by the penetration model, which is recommended for practical use [14]. The penetration model is considered the current industrial standard [15]. However, this model requires the use of adequate experimental data; therefore, the design of indirect dryers is often solved experimentally [8,14]. To properly design an indirect vacuum dryer, a significant number of experiments must be performed. The behavior of the material must be tested for a specific dryer under variable drying conditions, including different pressures, which is time-consuming and technically demanding. Therefore, finding and generalizing the relationship between the operating pressure and the parameters describing the drying process would simplify the design and testing of the vacuum indirect dryer due to the reduced number of experiments that need to be performed.

For this purpose, the penetration model was used to determine the overall heat transfer coefficient (HTC) for various temperature differences and operating pressures. The penetration model is very complex and provides a deep insight into the drying process; however, due to its iterative approach, its use for the design of dryers in industrial practice is quite challenging. The model was developed in Ref. [16] and then improved to be valid for indirect drying in a vacuum atmosphere in Ref. [17]. Other authors extended the penetration model to include the particular characteristics of the material and the different drying conditions in Refs. [18–24]. Most of the previous articles were focused on materials with a spherical shape. There are other studies [25–31], in which different materials can be found. Penetration model was used to simulate the heat and mass transfer and drying kinetics in the paddle dryer for sludge in Refs. [5,32]. The sludge flowrate, heated wall temperature, dryer slope, and sludge density have significant influences on the drying process. On the contrary, the stirring speed has a negligible effect [5,32]. The conductive heat transfer to the particle bed is independent of the particle type, mixer geometry, and stirrer speed [5,6]. In an indirect heat-agitating dryer, the clearance between the heating wall and stirrer blades has a negative effect on the heat transfer. The particle shape is also important for the heat transfer coefficient [6]. Recent articles on indirect drying can be found in Refs. [33–35], focusing on steam-heated rotary dryers for wood chip drying. A rotary kiln utilizing superheated steam is used for wood chip drying in Ref. [36] and for food drying in Ref. [37]. None of these articles focus on the simplification of the penetration model for practical use.

In the literature review, no methods were found regarding how the parameters, such as overall HTC and other drying characteristics, evaluated from the experimental data of atmospheric drying, could be used for the design of a vacuum dryer. This article aims to examine the relationship between the pressure and the overall HTC to propose an equation that could predict the value of the overall HTC and the DR for various temperatures and pressures based on experiments performed under atmospheric conditions. The proposed equation will significantly reduce the number of experiments needed for the design of a vacuum dryer and verification of its function, which could be performed under atmospheric conditions, and the effect of lower pressure on the design parameters of the dryer and on the drying characteristics of the material could be calculated using the proposed equation. The reliability of the proposed equation will be verified experimentally. Drying experiments will be carried out on a horizontal paddle dryer that allows both atmospheric and vacuum operations for a direct comparison between drying under these conditions. The materials chosen for the experiments are feed pellets.

The drying process can be divided into three periods, as described in Fig. 1. The initial period is when the temperature of the material rises at the same rate as the DR. This period can be brief and may not be noticed during the experiment. The second period, known as the constant rate period, is characterized by a constant DR. The temperature of the material rises very slowly, and the moisture content curve falls linearly or is slightly arched. In this period, both the overall HTC and DR remain stable. In the third period, known as the falling rate period, the DR typically decreases, and the material's temperature begins to rise rapidly, which can be problematic, particularly for temperature-sensitive materials (see Fig. 1). The article focuses on the cases where the drying process occurs in a constant rate period.

The penetration model was created by Schlünder and Mollekopf [17]. It was designed for the drying of mechanically agitated particulate materials in a pure vapor atmosphere. The full description of the model can be found in Refs. [16,17].

The analytical determination of the dependence of the overall HTC on the operating pressure from the above equations is very complex; therefore, the analysis of this dependence will be solved numerically in the next chapter.

With this procedure, it is possible to design an indirect dryer, while the penetration model allows to determine the overall HTC for various temperature differences ΔT and operating pressures. However, using the penetration model is very demanding for the practical design of dryers, thus experimental determination of overall HTC is often preferred. In the case of vacuum drying, when the pressure can vary within wider limits, the range of necessary experiments is quite considerable and demanding. For this reason, a numerical analysis of the effect of the operating pressure on the overall HTC was carried out using the penetration model and a simple empirical dependence was proposed, according to which the size of the overall HTC determined at atmospheric pressure could be recalculated to a lower pressure. The proposed relationship was then experimentally verified.

Porous pellets made from a cereal mixture were used for the calculation of the overall HTC by penetration model and for drying tests (see Fig. 2). Table 1 shows the properties of the material.

A horizontal paddle dryer (see Fig. 3) was used for the experimental verification of the relationship proposed for calculating the value of the overall HTC according to pressure. It is a cylindrical thermally insulated vessel made from structural steel with a centrally located agitator made of stainless steel. The agitator is made up of two parts. The first part is a driveshaft, and the second is a shaft with multiple paddles. These two parts are connected by a demountable clutch. The driveshaft is supported by two ball bearings, and the removable shaft with the paddles is supported by a plain bearing. The agitator is made of two parts; therefore, the part with paddles is easily removable and there is no issue with the sealing of the vessel. The heat is supplied by electrically heated jackets. The heating temperature is monitored by T-type thermocouples and regulated by a proportional–integral–derivative (PID) controller. Type-T thermocouples offer a measuring accuracy of ±0.5 °C. The dryer can be run atmospherically or under vacuum. The vacuum is obtained using a vacuum pump. This pump has a nominal flowrate of 10 m³/h and a nominal pressure of 0.040 bar. Pressure is measured using the MB801 pressure gauge, which has an accuracy class of ±1% of the measuring span. The dryer is placed on a tensometric scale, which allows a direct measurement of the weight of the evaporated water, and the moisture content of the material being dried can be calculated. The scale measures with high accuracy within 2 g. The scale is equipped with a Utilcell SWIFT weighing indicator with an analog output signal. The signals from the scale and thermocouples are gathered in the DAQ Yokogawa DX100. The electricity consumption to heat the dryer to calculate the amount of heat required for drying is measured on an Eleman DDS-1Y electricity meter. The basic parameters of the dryer are described in Table 2. Figure 4 presents a schematic of the dryer, including the positions of the measuring equipment.

The initial moisture content of the material used for experiments was evaluated by drying in the laboratory dryer according to a standardized procedure [38].

A set of experiments was carried out to dry the feed pellets in a horizontal paddle dryer under vacuum and atmospheric pressure and different heating temperatures. From the record of the weight loss of evaporated water during the measurements, the drying curves were evaluated and then the DRs and the overall HTCs were determined using Eqs. (11), (10), and (9) for different pressures.

The material's weight inside the dryer is measured using a tensometric scale with an accuracy of ±2 g. The amount of evaporated water is calculated as the difference between two weight measurements, resulting in a possible variation of ±4 g. This variation directly affects the determination of moisture content.

Pressure is measured using an analog pressure gauge with an accuracy class of ±1% of the measuring span. Type B uncertainty for pressure measurement was determined to be ±0.016 bar. The pressure measurement uncertainty primarily influences the determination of the saturation temperature. Lower pressures lead to higher uncertainties, with a maximum uncertainty of ±1.7 °C.

Temperature is measured using a Type-T thermocouple with an accuracy of ±0.5 °C. This results in a potential variation in the temperature difference of up to ±2.2 °C. Variations in temperature difference and pressure significantly affect the determination of the overall heat transfer coefficient. The uncertainty in the overall heat transfer coefficient ranges from ±0.48 to ±1.56 W m⁻² K⁻¹, depending on the experimental conditions. Both low pressure and low temperature differences amplify measurement uncertainties.

The uncertainty in the drying rate ranges from ±0.07 to ±0.08 kg m⁻² h⁻¹. This is influenced solely by the uncertainties in determining the initial moisture content and the accuracy of weight measurements.

The determination of moisture content uncertainty is driven by the accuracy of the initial moisture content measurement and the weight measurement. Error bars are included in the drying curve figures to visually represent data variability and measurement uncertainty.

The penetration model was used to calculate the overall HTC of a specific material. The calculation was done for several heating temperatures and several pressure levels. The heating temperature was set in relation to an operating pressure to maintain the same temperature difference $ΔT$ in the evaluated calculation. The temperature difference $(ΔT)$ is taken between the heating temperature of the dryer wall and the saturation temperature of the medium being evaporated from the material for the respective operating pressure, and the temperature difference remains constant during the period of constant DR. The data are listed in Table 3.

The temperature difference $ΔT$ has a weak influence on the overall HTC, and increasing $ΔT$ from 30 °C to 90 °C increases the overall HTC by less than 10%. On the other hand, the influence of the operating pressure is strong. Lowering the operating pressure while maintaining the same temperature difference $ΔT$ negatively affects the overall HTC. It may drop by up to 40%. This could be explained by analyzing heat transfer in the indirect dryer. The overall HTC is the sum of three basic heat transfer methods: convection, conduction, and radiation. Heat transfer by radiation is dependent on temperature difference, which remains the same; therefore, it does not affect the overall HTC. Heat conduction between the wall and the material depends on the contact surface which is not affected by the operating pressure. In the case of convection, heat is transported to the material by the gas (steam) surrounding the material. The heat is carried by the particles of the gas, and if the pressure is lowered, then the amount of the gas particles is lowered too; therefore, the heat transfer by convection must be lowered, and subsequently the overall HTC.

If the equation is used only for calculating the overall HTC from the atmospheric pressure with the condition that atmospheric pressure is 1 bar (a), then the pressure ratio could be replaced by $pvac$ expressed in bar (a).

Table 4 shows the values of the overall HTC calculated by Eq. (15). The deviation of the equation can be assessed by comparing the results in Table 3 with those in Table 4. Within the pressure range of 1–0.1 bar, the values are nearly identical, with only minor differences of less than 0.3 W m⁻² K⁻¹ observed at pressures between 0.1 bar and 0.8 bar, resulting in a percentage error of less than 1.6%. However, at a pressure of 0.01 bar, the predicted values in Table 4 are consistently lower than the calculated values in Table 3 ranging from 0.8 to 1.1 W m⁻² K⁻¹. The correlation between the penetration model and the proposed equation is satisfactory. The prediction is highly accurate up to 0.1 bar. Below this pressure, drying of this material is considered ineffective due to the challenges of maintaining such low pressure. Figure 5 presents a graphical comparison of the overall HTC values obtained from the penetration model and those calculated using the proposed equation (15) for ΔT = 30 °C.

Equation (15) can be used to predict the overall HTC based on data obtained from atmospheric drying of used cereal feed pellets. The exponent b could be determined for other materials based on the methodology described above.

The penetration model and Eq. (15) demonstrate that atmospheric drying is more effective due to the higher overall HTC compared to vacuum drying. Additionally, atmospheric drying eliminates the need for a vacuum pump, simplifying the process. Consequently, vacuum drying should be reserved for temperature-sensitive materials or those with specific requirements.

Due to the complexity of the drying process, several parameters must be considered, including the initial and final moisture content, the internal structure of the material, and the material size. The porous structure of the material plays a significant role in determining the duration of the constant rate period during drying. During this period, the surface of the material remains saturated with moisture, and evaporation occurs at a nearly constant rate, driven by heat supply and moisture migration from the internal parts of the material. Materials with larger, more interconnected pores, such as feed pellets, allow for faster moisture movement to the surface, where evaporation takes place. Conversely, materials with smaller or poorly connected pores experience slower moisture migration [39]. When the surface of the material is dry and the falling rate period begins, the intensity of the drying process significantly decreases and the governing principle for drying intensity is migration of the moisture within the particle of the material. The proposed equation (15) is valid for materials preheated to the drying temperature, provided that drying occurs primarily during the constant drying rate period, as shown in Fig. 1. In this period, the effect of the material's internal composition is negligible. The equation is applicable when the initial moisture content is sufficiently high to prevent a rapid transition into the falling rate period, and the final moisture content is not low enough to conclude drying in the falling rate period. According to the penetration model, particle size significantly influences the overall HTC, with smaller particles exhibiting higher HTC values. However, the dependence of the overall HTC on pressure remains unchanged. Table 5 presents a comparison of data modeled using the penetration model for two material sizes, differing from the initial size, with the results from Eq. (15). In both cases, Eq. (15) demonstrates similar accuracy in describing the overall HTC.

The results obtained from the proposed equation are compared with those from experimental data. Calculated values of the overall HTC are used to determine the DR and predict the moisture content during drying. The set of experiments contains six tests, two of them were run under atmospheric conditions and four of them under vacuum conditions. The conditions for the individual experiments are listed in Table 6.

Table 7 summarizes DRs and overall HTCs. DRs and overall HTCs are determined from the experiments (marked exp in the table). The results are compared with DRs and overall HTC calculated by Eqs. (15) and (9), (10), and (11) (marked calc in the table). The overall HTCs from the atmospheric test are input values for the calculation of overall HTCs in a vacuum.

Experiments at ΔT = 30 °C (experiments 1 and 3) had significantly lower DRs than those with higher $ΔT$ due to lower heat flux. The values of DRs for the set of experiments run under vacuum at ΔT = 50 °C (experiments 4, 5, and 6) were in good agreement with the prediction based on data from the atmospheric experiment (experiment 2). The comparison of the evaluated overall HTC and drying rates shows that calculated and experimental values largely align, with deviations ranging from 0.5% to 5.2% for HTC. In experiment 3, the HTC deviation is approximately 4.5%, while experiments 4 and 6 exhibit minimal deviations of about 0.5% and 0.9%, respectively. Experiment 5 shows a higher HTC deviation at about 5.2%. For drying rates, the deviations range from 1.1% to 4.1%, with experiment 4 at 1.3%, experiment 5 at 4.1%, and experiment 6 at 1.1%. This indicates that the calculations are generally reliable, demonstrating small deviations from experimental results.

The following figures show a visual comparison of drying curves determined with Eqs. (9), (10), (11), and (15) and directly measured during the experiments.

Figure 6 shows the experimentally determined drying curve for operating pressure 0.2 bar and ΔT = 30 °C and the predicted drying curve based on data obtained from the atmospheric experiment. The comparison between the experimentally determined drying curve and the calculated curve shows a very good correlation. The calculated curves closely follow the experimentally determined curve, with the only difference occurring during the initial period, which may be influenced by the heating phase.

Figures 7 and 8 show the experimentally determined drying curves for operating pressures 0.2 or 0.475 bar and ΔT = 50 °C and the predicted drying curves based on the data obtained from the atmospheric experiment. In this case, four experiments were run in total. Three experiments were carried out under vacuum pressure, two of them at a pressure of 0.2 bar and one at an increased pressure of 0.475 bar. The last one is at atmospheric pressure. Figure 7 demonstrates not only the correlation of the curves but also serves as an indicator of the consistency in the experiments, given that they were conducted under similar operating conditions. The only variation between the experiments is the initial moisture content, which does not influence the slope of the curves. Since the slope is consistent across both experiments, the repeatability is confirmed. The experimental curve with a pressure of 0.475 bar also correlates very well with the predictions of the proposed equation.

Increasing the temperature difference $ΔT$ does not significantly affect the precision of the proposed equation. Comparison of drying curves measured directly with the calculated ones is in good agreement. Figures show that it is possible to estimate the behavior in a vacuum dryer based on data obtained from the atmospheric experiment.

Figure 9 in addition to Table 7 compares theoretical and experimental values of the DR. It shows a good match between the predicted and experimentally measured values of DR. In general, the farther the data points are from the center line in the graphs, the lower the accuracy of the model. Given that nearly all points lie close to this straight line, it can be concluded that the proposed equation estimates the overall HTC in the contact dryer sufficiently accurately.

The proposed equations (15) and (16) can be effectively applied to optimize drying when it is necessary to adjust process parameters. For example, if the current operating pressure of a dryer is set to 0.475 bar (a), the temperature of the feed pellets is approximately 80 °C (case 1). At this temperature, the Maillard reaction may occur, leading to a degradation of the nutritional value in the feed pellets. To prevent this, the process must be adjusted to a different pressure that results in a lower feed pellet temperature, all while maintaining the same production rate from the dryer.

The new pressure is set to 0.2 bar (a), which reduces the temperature of the material to around 60 °C. Initially, the heating temperature was 130 °C. To maintain the same temperature difference of 50 °C, a new heating temperature would be 110 °C (case 2). However, due to the lower pressure, the overall HTC decreases, which in turn lowers the heat flux and the drying rate. Therefore, the temperature difference must be increased to counteract the reduction in overall HTC and accommodate differences in latent heat of evaporation at the new pressures (case 3).

Table 8 demonstrates the application of the proposed equation. The first column shows experimental results (a reference case—case 1). When the pressure is lowered while maintaining the same temperature difference, the drying rate decreases, as shown in case 2. In this context, both the drying rate and the overall HTC are known; however, in a real-world situation, these values would be unknown, necessitating the use of the proposed equation. To maintain the same production rate, an increase in the temperature difference is required. The third column applies the proposed equation (Eq. (16)) to calculate the overall HTC, which is used to determine the necessary increase in temperature (case 3). The calculations indicate that the temperature difference must be increased by 7 °C to achieve the same production rate as initially observed.

In other words, the proposed equation allows to set the appropriate drying pressure and heating temperature to achieve the desired DR while maintaining the temperature of the dried material.

A theoretical analysis of the influence of the operating pressure during indirect drying on both the overall HTC and the DR was done.

The overall HTC is strongly dependent on the operating pressure inside the dryer, significantly less on the difference between the heating temperature and the temperature of the material being dried, which proves to be a more suitable parameter for assessing the overall HTC than the heating temperature itself when the drying pressure changes. Lowering the pressure negatively affects the overall HTC, which could be explained by lower convective heat transfer between the material and surroundings due to fewer particle concentration in the surrounding gas that mediate heat transfer.

The relation between operating pressure and the overall HTC was found in the form of Eq. (15). The results of this equation were compared with the experiments. The correlation between the results of the experiments and the proposed equation is satisfactory. The portability of the results from atmospheric drying to various pressure levels was proven possible. The exponent in the proposed equation (15) represents material properties and is valid for cereal feed pellets. The exponent b could be determined for other materials based on the methodology used in this article.

The results are useful for an indirect vacuum dryer design. Experimental determination of the overall HTC can be carried out under atmospheric pressure, and the decrease in its value when the drying pressure is reduced can be estimated according to the proposed equation. The number and complexity of experimental verification tests required for the design of a new dryer can thus be significantly reduced. The proposed equation can also be used for optimization of the drying process of an existing dryer, which can reduce drying time and save energy.

This work was supported by the project from Research Center for Low Carbon Energy Technologies, project number: CZ.02.1.01/0.0/0.0/16_019/0000753. We would like to gratefully acknowledge support from this grant.

There are no conflicts of interest.

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

$m$ =

weight $(kg)$

$m˙$ =

drying rate $(kgm−2s−1)$

$p$ =

operating pressure $(Pa,bar)$

$q˙$ =

heat flux $(Wm−2)$

$t$ =

drying time (s)

$A$ =

covered surface of the heating wall $(m2)$

$M~$ =

molar mass $(kgmol−1)$

$R~$ =

universal gas constant $(Jmol−1K−1)$

$T$ =

temperature $(K;∘C)$

$X$ =

moisture content $(kgwkgdry−1)$

$cp$ =

specific heat capacity at constant pressure $(Jkg−1K−1)$

$dchar$ =

characteristic dimension of the particle $(m)$

$tR$ =

stagnant period (s)

$CW,bed$ =

overall radiation coefficient $(–)$

$DR$ =

drying rate $(kgm−2s−1)$

$HTC$ =

heat transfer coefficient $(Wm−2K−1)$

$OHTC$ =

overall heat transfer coefficient $(Wm−2K−1)$

$Δhv$ =

latent heat of evaporation $(Jkg−1)$

$α$ =

heat transfer coefficient $(Wm−2K−1)$

$γ$ =

accommodation coefficient (–)

$δ$ =

surface roughness of particles $(m)$

$ε$ =

emissivity (–)

$ζ$ =

dimensionless position of phase change front (–)

$l$ =

modified mean free path of gas molecules $(m)$

$λ$ =

thermal conductivity $(Wm−1K−1)$

$ρ$ =

density $(kgm−3)$

$σ$ =

Stefan–Boltzmann constant $(Wm−2K−4)$

$φ$ =

surface coverage factor (–)

$atmos$ =

atmospheric

$bed$ =

bed of the material

$d$ =

dryer

$dry$ =

dry matter

$db$ =

dry basis

$g$ =

gas

$in$ =

in

$out$ =

out

$rad$ =

radiation

$vac$ =

vacuum

$w$ =

water

$wet$ =

wet matter

$W$ =

heated wall

$WP$ =

wall-particle

$WS$ =

contact