• Drying method;
  • drying temperature;
  • food drying;
  • solid density;
  • volume;
  • water density


  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

Abstract:  This review presents the concepts involved in determining the density of foodstuffs, and summarizes the volumetric determination techniques used to calculate true density and apparent density in foodstuffs exposed to the drying process. The behavior of density with respect to moisture content (X) and drying temperature (T) is presented and explained with a basis in changes in structure, conformation, chemical composition, and second-order phase changes that occur in the processes of mass and heat transport, as reported to date in the literature. A review of the empirical and theoretical equations that represent density is presented, and their application in foodstuffs is discussed. This review also addresses cases with nonideal density behavior, including variations in ρs and ρw as a function of the inside temperature of the material, depending on drying conditions (X, T). A compilation of studies regarding the density of dehydrated foodstuffs is also presented.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

Microchanges in physical and chemical structure occur during drying, including shrinking (β), porosity (ɛ), changes in true density (ρp), changes in apparent density (ρb), and changes in chemical composition. These changes have been explained in accordance with the period during the drying process in which they occur (constant rate, first and second falling rate periods).

  • 1
    In the constant and first falling rate periods, cellular structure is elastic, allowing shrinking in the empty space created as a result of water evaporation.
  • 2
    In the second falling rate period, cellular structure becomes rigid, favoring or limiting the formation of pores, as well as shrinking, depending on moisture content (X), as well as drying method and conditions (Krokida and Maroulis 2000; Madiouli and others 2007).

The interdependence of physicochemical structure on structural and transport properties has consequences in the quality of the dried material (Zogzas and others 1994; Madiouli and others 2007). During drying, moisture transport is strongly affected by changes in structural properties. Thickness, the formation of channels, and pore distribution are all representing a preferential path for water transport (Van der Zanden 1995; Aguilera and Stanley 1999; Aversa and others 2011).

The quality of a dried material is generally related to its structural (density, porosity, pore size, and specific volume), optical (color and appearance), textural (compression test, stress relation test, and tensile test), thermal (product state: glassy, crystalline, rubbery), sensory (aroma, flavor, and taste), nutritional (vitamins and proteins), and rehydration properties (rehydration rate and rehydration capacity) (Krokida and Maroulis 2000). Many of these properties are interrelated. For example, density properties are related to shrinking and porosity, through sample size. Furthermore, variations in porosity and mean pore size, as well as pore size distribution and pore area, have a significant effect on mechanical, textural, and quality characteristics in the dried material (Huang and Clayton 1990; Farkas 1991; Karathanos and others 1993).

Density is an important structural property in materials. As a physical characteristic, density is necessary in engineering calculations and is a quality parameter in both mid state (at intermediate stages of drying and food processing) and after completion of drying and food processing. As a quality parameter, it is important in the characterization and prediction of the quality of dried and processed products. It is also very important in the development of new industrial products with specific desired properties, and in improvements of the quality of existing products.

The density of dried food systems is essential in the design of food processes and processing equipment (Krokida and Maroulis 2000).

In process modeling, density is used to study transport phenomena during the drying or processing of foodstuffs involving changes to solid-phase volume and the concentration of mobile phases. Phenomenological models describe mass and heat transference better when density is considered to be dependent on temperature and inside moisture content.

As such, it is important to be familiar with the precise determination methods and models available for various foodstuffs, as well as the concepts behind the phenomena that affect these parameters (Ré 1998; Krokida and Maroulis 2000; Barbosa-Cánovas and others 2005).


  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

For solid materials, density is defined at the relationship between mass and volume. Depending on the method used to measure volume, it can be classified as

True density (ρp), defined as the quotient of mass over the volume of a sample, without considering pores in the material (true volume). In the case of granular materials, the terms particle density and particle volume are used.

  • image(1)

Apparent densityb), defined as the relationship between the mass and volume of the material, including pores and water (apparent volume). The terms bulk density and bulk volume are used for granular materials.

  • image(2)

where ms and mw are dry solid mass and water mass, respectively. Vs, Vw, and Va are the solid volume, water volume, and air volume, respectively.

Methods of Measurement

  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

Experimental determination of density is a function of measuring mass, apparent volume, and true volume. Mass is the weight of the sample and volume can be determined by various experimental methods.

True volume is usually measured with a gas stereopycnometer (helium or nitrogen), excluding interparticular volume (Mohsenin 1980; Zogzas and others 1994). It is preferable to use helium, as its small molecular diameter makes it possible to access pores of up to 3.5 Å (Karathanos and Saravacos 1993).

The advantages of this method are the accurate measurement that it yields; however, depending on the type of sample, other aspects should be considered, including the effects of the pressure of the gas on the structure of very porous materials, such as foams.

One disadvantage is that it requires very precise calibration, as very small variations in gas pressure inside the sample chamber may give rise to significant errors in the measurement of volume. Equilibrating time depends on the type of sample, and is quite long for porous materials. Not allowing sufficient time for the sample to reach equilibrium will result in errors in the calculation of true volume.

Apparent volume can be measured using the following methods:

The volumetric displacement method is employed for solid materials that do not absorb liquid easily. There are 2 options for this method:

  • 1
    Using a graduated cylinder or burette. An immersion liquid is used to measure the volume displacement caused by the sample inside the container (Zogzas and other 1994).
  • 2
    Using the buoyant force method. Based on the Archimedes principle, the sample is weighed inside and outside of an immersion liquid of known density. The sample should remain suspended in the liquid, and not touch the bottom or the sides of the container.

The advantages of this method are as follows:

  • • 
    Ease of measurement, given that only the volume displacement of the sample is measured.
  • • 
    Less than 5 min measuring time
  • • 
    Minimal cost of equipment and maintenance.
  • • 
    Applicable for samples of any geometric shape.

The disadvantages of this method are:

  • • 
    The immersion liquid, usually toluene, heptane, mercury, water, alcohol, or tetracloroethylene, among others; the majority of these are toxic (Mohsenin 1970; Zogzas and other 1994).
  • • 
    Theoretically, there is no absorption of the immersion liquid into the sample; however, the possibility of extracting substrates of the sample, according to the polarity of the phases, remains; this can cause a nonquantified error in measurement.
  • • 
    The presence of air bubbles in the sample or the immersion liquid during measurement can cause errors.
  • • 
    This method does not allow for automated measurement, for example, in lyophilized or convection dehydrated materials, where samples must be periodically extracted.

Dimension method

Apparent volume is an average of the dimensions of the sample, assuming a spherical or plate shape (Lozano and others 1983). The advantages of this method are as follows:

  • • 
    It can be automated, as it is not necessary to extract samples from the dryer if high-resolution cameras are used to capture images of the sample, allowing for continuous measurement of area and thickness.

The disadvantages of this method are:

  • • 
    It does not consider warping or deformation of the material.
  • • 
    The software may ignore bright or shadowed areas in the images.
  • • 
    It is inexact, due to irregularities in the material, and subjective, due to the inconsistency of human judgment (Kelkar and others 2011).
  • • 
    If the material is granular, interparticular space is included in the measurement.

Stereopycnometric method

The sample is covered with silicone grease to waterproof the material. Apparent volume is measured with a steropycnometer (Loch-Bonazzi and others 1992).

Recent, highly accurate techniques for measuring apparent volume have been proposed, including laser scanning (Uyar and Erdogdu 2009; Kelkar and others 2011), computerized tomography (Mendoza and others 2007; Kelkar and others 2011), and magnetic resonance imaging (Kelkar and others 2011). These techniques utilize optic devices and specialized software to process and create 3D images of objects, calculating volume with finite differences or finite element. However, the disadvantages of these techniques outweigh their advantages, as a certain background is required, as well as specialized equipment that is far more costly than traditional techniques.

Compilation of Studies

  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

A number of works have addressed the behavior of true and apparent density, both in the drying process and in the rehydration of foodstuffs (Table 1 and 2). These studies evaluate several foods exposed to various drying conditions and in various geometric shapes. The variables that have been studied are as follows: pressure (P), relative humidity (RH), drying air velocity (ν), and drying temperature (T), among others. Pretreatments such as blanching, coatings, and osmotic dehydration, among others, have been reported (Boukouvalas and others 2006).

Table 1–.  True and apparent density as reported in the bibliography.
Material Apparent density (kg/cm 3 ) True density (kg/cm 3 ) Geometry Conditions Source
Amioca1235 to 7501275 to 1500∵X>0.2 1500 to 1450∵X<0.2GranularHydrated with distilled water Marousis and Saravacos 1990
Amioca gel 1275 to 1500∵X>0.2 1500 to 1450∵X<0.2Gelatinized spheres ø= 2 cmGelatinization at 100 °C/30 min Convection (C)ν= 2 m/s, RH = 10%, T= 60 °C Marousis and Saravacos 1990
Apple900 to 5201090 to 1490Cubes λ 18 mmν= 2.5 m/s, T= 70 °C RH = 20%, 30%, 45%, and 60% Zogzas and others 1994
 900 to 450 Disks ø= 2.6 mm λ= 3, 3.5, 4.2, 5.2, 6.5, and 8.6 mm Disks ø= 5.9 mm, λ= 15 mm Convection (C)ν= 3.5 m/s, RH = 12%, T= 70 °C Freeze-drying (F) P1= 20 mbar, P2= 0.05 mbar, T=−25 to 40 °C Moreira and others 2000
 900 to 750(O/C) 900 to 600 (M/C, C) 900 to 300 (V) 900 to 100 (F)1025 to 1620Cylinders ø= 30 mm, λ= 8 mm Convection (C)ν= 2 m/s, T= 70 °C, RH = 7%P= 1000 mbar Vacuum drying (V) P= 33 mbar, T= 70 °C Freeze-drying (F) T =−35 °C, P= 0.04 mbar Osmotic/convection (O/C) Sucrose solution (50%, 40 °C, 10 h) Microwave/convection (M/C) 810 W Krokida and Maroulis 2001
Banana1030 to 1390 Slices ø= 25 mm, λ= 6 mm T= 40, 50, and 60 °C Talla and others 2004
Calamari1060 to 1260∵X/X0>0.2 1260 to 1215∵X/X0<0.21075 to 1340Slabs 10×5 cm T= 70 °C, RH = 15% Rahman and others 1996
Carrot1021 to 12081203 to 1462∵X/X0>0.1 1462 to 1244∵X/X0<0.1Cylinders ø= 1 cm, λ= 4 cmν= 1 m/s, RH = 35%, T= 60 °C Food starch materials 40 °C Lozano and others 1983
 1050 to 13801080 to 1495Cubes λ= 18 mmν= 2.5 m/s, RH = 20, 30, 45, and 60%, T= 70 °C Zogzas and others 1994
Garlic983 to 1018∵X/X0>0.5 1018 to 937∵X/X0<0.51143 to 1378Whole clovesν= 1 m/s, RH = 35%, T= 60 °C Food starch materials T= 40 °C Lozano and others 1983
 1110 to 1300 Whole cloves T= 70 °C Madamba and others 1994
 1007 to 1317 Slices, cutting one clove in half along the longest axisν= 1 m/s, RH = 35%, T= 60 °C Food starch materials 40 °C Lozano and others 1983
 40 °C 1030 to 1345∵X>0.13 1345 to 1308∵X<0.13 50 °C 1030 to 1280∵X>0.1 1280 to 1224∵X<0.1 60 °C 1030 to 1216∵X>0.1 1216 to 1156∵X<0.140 °C 1210 to 1682∵X>0.13 1682 to 1314∵X<0.13 50 °C 1210 to 1527∵X>0.2 1527 to 1334∵X>0.2 60 °C 1210 to 1450∵X>0.1 1450 to 1126∵X<0.1Slices λ= 2 mmν= 1.5 m/s, T= 40, 50, and 60 °C López-Ortiz and others 2012
Hylon gel 1275 to 1500∵X>0.2 1500 to 1450∵X<0.2Gelatinized spheres ø= 2 cmGelatinization at 120 °C/30 min Convection (C)ν= 2 m/s, RH = 10%, T= 60 °C Marousis and Saravacos 1990
Hylon1135 to 7001275 to 1500∵X>0.2 1500 to 1450∵X <0.2GranularHydrated with distilled water Marousis and Saravacos 1990
Onion991 to 11691066 to 1416Slices T= 70 °C Rapusas and Driscoll 1995
Pear991 to 11441091 to 1464Cylinders ø =1 cm, λ= 4 cmν= 1 m/s, RH = 35%, T= 60 °C Food starch materials T= 40 °C Lozano and others 1983
 1010 to 1220 Slices ø= 30 mm, λ= 3 mmSolar drying, and Air chamber at ν= 300 m3/h, T= 30 °C Guiné 2006
Pear991 to 11441091 to 1464Cylinders ø= 1 cm, λ = 4 cmν= 1 m/s, RH = 35%, T= 60 °C Food starch materials T= 40 °C Lozano and others 1983
 1010 to 1220 Slices ø= 30 mm, λ= 3 mmSolar drying, air chamber at ν= 300 m3/h, T= 30 °C Guiné 2006
Potato1054 to 1160∵X/X0>0.2 1160 to 943∵X/X0<0.21117 to 1319Cylinders ø= 1 cm, λ= 4 cmν= 1 m/s, RH = 35%, T= 60 °C Food starch materials 40 °C Lozano and others 1983
 1050 to 12501060 to 1290Cubes λ = 18 mmν= 2.5 m/s, RH = 20%, 30%, 45%, and 60%, T= 70 °C Zogzas and others 1994
 40 °C 1070 to 1380∵X>0.5 1380.1300∵X<0.5 50 °C 1070 to 1320 ∵X>0.2 1320 to 1270∵X>0.2 60, 70 °C 1070 to 1300∵X>0.2 1300 to 1250∵X>0.2 Slices 10×20×45 mmν= 4 m/s T= 40, 50, 60, and 70 °C Wang and Brennan 1995
Sweet Potato1046 to 1157∵X/X0>0.3 1157 to 947∵X/X0<0.31250 to 1530Cylinders ø= 1 cm, λ = 4 cmν= 1 m/s, RH = 35%, T= 60 °C Food starch materials T= 40 °C Lozano and others 1983
Quince1000 to 1020 (FB) 1000 to 1050 (TD) 1000 to 1110 (I) 1000 to 1325 (O/T) Cubes 11×11×11 mm Fluid bed (FB)ν= 1.4 m/s, T= 70 °C Tray Drying (TD)ν= 1.4 m/s, T= 70 °C Infrared/Convective air drying (I/C) T= 70 °C Osmotic/tray drying (O/T) 50% sucrose at 40 °C/6 h ν= 1.4 m/s, T= 70 °C Koç and others 2008
Quince1000 to 400 (F) Cubes 11×11×11 mm Freeze-drying (F)ν= 1.4 m/s, T=−25 °C for 20 min, P= 0.014, Chamber T= 30 °C Koç and others 2008
Table 2–.  True and apparent density of rehydrated products as reported in the bibliography.
Material Apparent density (kg/cm 3 ) True Density (kg/cm 3 ) Geometry Conditions Source
Apple820 to 750(O/C) 650 to 600 (M/C, C) 600 to 300 (V) 300 to 100 (F)1025 to 1620Cylinders ø= 30 mm, λ= 10 mm Convection(C)ν= 2 m/s, T= 70 °C, RH= 7%P= 1000 mbar Vacuum drying (V) P= 33 mbar, T= 70 °C Freeze-drying (F) T=−35 °C, P= 0.04 mbar Osmotic/convection (O/C) Sucrose solution (50%, 40°C, 10 h) Microwave/convective (M/C) 810 W Krokida and Maroulis 2001
Banana1020 to 1750 (C, M/C) 1020 to 1350 (O/C) 850 to 620 (V) 810 to 270 (F)1100 to 1650Cylinders ø= 20 mm, λ= 10 mm  
Potato1050 to 1500 (C) 1050 to 1300 (V) 680 to 400 (M/C) 580 to 200 (F)1080 to 1630Cylinders ø= 20 mm, λ= 10 mm  
Carrot1190 to 1580 (C) 900 to 890 (V) 730 to 500 (MC) 500 to 100 (F)1100 to 1780Cylinders ø= 20 mm, λ− 10 mm  

The effects of drying method on structural properties

Traditional convective drying, microwave drying, osmotic drying, spray-drying, vacuum-drying, and lyophilization are the most commonly used methods; they have been evaluated under both constant and variable operating conditions (Pezzutti and Capriste 1997; Chua and others 2002; Doymaz and Pala 2002). Materials dried by the convective method are characterized by their low porosity and high apparent density (Zogzas and others 1994). Similar characteristics have been reported for materials dried with the microwave method, due to the fact that a combination of these 2 methods is generally used (Krokida and others 2000). Osmotic dehydration leads to an increase in apparent density for some materials, and a decrease for others. This phenomenon has been attributed to the increase in solids during the osmotic process (Krokida and Maroulis 1997). Lyophilization yields products with low density, due to their porosity; this method produces the highest quality end result, as there are no deformations in the material, color and aroma are preserved. Nevertheless, it does have the disadvantage of being costly and requiring long drying periods (Krokida and Maroulis 2001; Doymaz and Pala 2002).

Several structural parameters of foodstuffs have been evaluated during the lyophilization process. Sablani and others (2007) found that apparent density, true density, and porosity are a function of moisture content (X) and plate temperature; however, there is a lack of clarity in these tendencies (Karathanos and others 1996; Sablani and others 2007; Oikonomopoulou and others 2011).

The effects of drying conditions on density

Some authors have found that the effect of relative humidity on true and apparent density is negligible (Zogzas and others 1994). Although the effects of drying air velocity on density have not been studied, it is possible to compile data from various bibliographic sources referring to a single material; such data suggest that true and apparent density are lower at lower drying air velocities (Table 1). However, further research on the topic is needed. Few studies have reported on the effects of the geometry of the material on density, although it has been found to influence apparent density values. For example, the density of whole garlic cloves has a concave-down shape with respect to X; in contrast, garlic cloves sliced in half display a linear, ascendant tendency with respect to the decrease in X (Lozano and others 1983).

The effects of temperature on density have been more widely studied. It has been shown that T strongly influences the characteristics of the dried product. As drying temperature increases, the final product becomes less dense (Wang and Brennan 1995, López-Ortiz and others 2012).

Due to technological advances in temperature controllers and processing control techniques, it has become possible to integrate drying air temperature control strategies (nonisothermal drying), making it possible to follow sinusoidal wave profiles (Figure 1a), square wave or box function profiles (Figure 1b), increasing and decreasing scaled ramp profiles, saw-tooth wave profiles, and trapezoidal wave profiles (Figure 1c), among others. Nonisothermal drying has made it possible to obtain products of higher quality and even shorter drying times than those obtained with constant convective drying (Chua and others 2002). Although various quality properties have been studied in materials submitted to nonisothermal drying, structural properties have not been reported, nor have equations been proposed to describe their behavior (Chua and others 2000; Chua and others 2002). To date, no equations have been proposed to relate changes in the structural properties of materials exposed to variable external conditions.


Figure 1–. Air heating profile for (a) sinusoidal wave, (b) square wave, and (c) trapezoidal wave.

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It has been observed that there is no significant difference between the behavior of true density with respect to the X of the material during convective drying and after being rehydrated to different moisture contents; however, differences have been observed between apparent density with respect to the X of the material during lyophilization and subsequent rehydration to different moisture contents (Krokida and Maroulis 2001).

Three possible tendencies of ρp as a function of X have been found (Figure 2), both linear and nonlinear (concave-down and concave-up). Tendency in Figure 2a shows a linear relationship between ρp and X, which considers reduction in volume to be equal to the volume of the water eliminated from the material (Madamba and others 1994). In tendency of Figure 2b, it can be observed how ρp increases slowly up to a critical point, followed by an exponential decay. This change has been explained by water loss in the material during drying (Lozano and others 1983; Zogzas and others 1994). In tendency in Figure 2c, it is assumed that the increase in ρp to a critical point is due to the fact that the reduction in volume is greater than the reduction in mass; after this critical point, the pores in the material are considered to be closed, and measured volume is greater than true volume (Lozano and others 1983). In the above-described tendencies, it is merely assumed that volume and solid dry mass are constant. However, when materials are heated, they may expand or contract; in such cases, volume is not constant.


Figure 2–. True density as a function of moisture content during the drying process.

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Density Models

  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

Several efforts have been made to predict different tendencies of ρ as a function of X. Generally in these models, the foodstuff is considered to be a binary compound (water–solid). Table 3 shows models for true and apparent density proposed for foodstuffs. The models for predicting ρb and ρp are based on the development of pores during the lyophilization process, as the formation of pores is considered to be a function of ideal conditions, since there is no reduction in the volume of the solid as a result of water sublimation (Karathanos and others 1996).

Table 3–.  Empirical, theoretical, and semitheoretical equations for calculating true density (ρp) and apparent density (ρb).
Equation Parameter Empirical and theoretical equations Material Source
5True density inline image Garlic, carrot, potato, pear Lozano and others 1983
6True density inline image Garlic López-Ortiz and others 2012
7True density inline image Onion Rapusas and Driscoll 1995
8True density inline imageρs and ρw constantsApple, carrot, potato Zogzas and others 1994
9  10  11True density inline image where: inline imageinline image in liquid suspensions (Choi and Okos 1986)Dry solid and water Boukouvalas and others 2006
12True density inline image Potato, quince Madamba and others 1994; Koc and others 2008
13  14  15True density inline image inline image inline image Dry solid and water López-Ortiz and Rodriguez-Ramírez 2011
16Apparent density inline image Potato Madamba and others 1994
17Apparent density inline image Onion Rapusas and Driscoll 1995
18Apparent density inline imageρs,a apparent density evaluated at X= 0Apple, carrot, potato Zogzas and others 1994
19Apparent density inline image inline image Various materials Boukouvalas and others 2006
20Apparent density inline image apparent density a the beginning of processingApple Moreira and others 2000
21  22  23  24  25Apparent density inline image inline image where: inline imageinline imageinline imageVarious materials Khalloufi and others 2010
26  27Apparent density inline image inline image The coefficient r19 is an indicator of the slope of δ(X) Xc is the critical concentration of the collapse function (Levi and Karel 1995)Various materials Khalloufi and others 2010

True density (ρp) models

Various authors have proposed correlations obtained through nonlinear regression for predicting true density and apparent density as a function of moisture content. However, the results not become widespread and are only valid for the material, geometry, and drying conditions used in those works (Table 3).

Effect of composition during drying It has been proposed that the composition of liquid significantly affects the behavior of density. Choi and Okos (1986) propose that in liquid suspensions, density is a function of the concentration and temperature of the material. This relationship was expressed in an equation (Eq. 3) where the functionality of ρi is with respect to air temperature:

  • image(3)

They observed linear behavior of density for various suspensions of pure compounds, such as proteins, fats, carbohydrates, fibers, and ash. These authors compared their models with experimental density values for milk, orange juice, and bratwurst sausage, obtaining a maximum error of 1.45%, and finding a quadratic functionality for water.

The linear functionality of ρs with respect to T from Choi and Okos (1986) in Eq. 11 (Table 3) is valid for suspensions that are not contained by a solid matrix, where different transference phenomena occur than in solids.

Some authors currently use this equation in the theoretical prediction of the real density of solids. However, measurement of the density of liquids is performed at a constant pressure, whereas measurement of solid density must take into account the fact that intracellular pressure varies as a function of the concentration of components and the permeability of cell membranes (turgor pressure).

Although it may be simpler to determine density by using the composition of the solid being studied, to date, no studies have been done comparing the density found in variable pressure systems (such as those present in solids) with that found in constant pressure systems.

The effects of phase changes on the behavior of components in density were studied by Heldman (1982), who showed the influence of freezing on the density of strawberries with high moisture contents (Eq. 4):

  • image(4)

According to Irudayaraj (2001), the effect of moisture content on phase changes is limited by the moisture content in the product, and dry solid density is directly dependent on structure.

Models for solids Zogzas and others (1994) proposed Eq. 8 (Table 3), which includes water density and true dry solid density as constants for predicting true density during drying. The effects of temperature (T) on the sample are considered to be negligible.

Boukovalas and others (2006) retake the theory that the density of a material is dependent on the temperature at which it was processed, and propose using Eq. 9 (Table 4) to calculate ρp, considering ρw and ρs to be functions of T.

The functionality of ρw and ρs with respect to T was previously studied by Choi and Okos (1986) and the nonlinear tendency of ρw with respect to drying temperature has been widely studied; its parabolic behavior is attributed to the change between the molecular spaces that occur in the liquid–solid phase transition. The variation of ρw as a function of T is of an order of magnitude of 1 (Kotz and others 2005).

López-Ortiz and Rodriguez-Ramírez (2011) found nonlinear behavior of ρs with respect to X and T in garlic; this was attributed to contraction or expansion of the volume of the solid material throughout the drying process, probably due to second-order phase changes.

The model proposed by Boukovalas and others (2006) provides a good approximation of the experimental data available in the literature for materials such as apples, bananas, whole garlic, onions, and potatoes; however, the difference found between the real data and that generated by the model was 20% for carrots. A concave-down behavior with respect to X was found in carrots by Lozano and others (1983). All of the models found in the literature fit data with linear or concave-up behavior. However, there are no models available to represent concave-down behavior, which has been reported in various vegetables such as carrots and garlic (Lozano and others 1983, López-Ortiz and others 2012).

At the molecular level, second-order phase changes have been identified that modify the structural and/or molecular arrangements of materials, producing collapse and shrinkage. Levi and Karel (1995) demonstrated that collapse and shrinkage in carbohydrates is a dynamic process that occurs as a consequence of exceeding the glass transition temperature (Tg), on which the velocities of collapse or shrinking strongly depend (T−Tg). Glass transition temperature is the temperature at which a second-order phase change occurs, moving from a glassy state to an amorphous state or vice versa. In convective drying, foodstuffs in a rubbery-amorphous state change (as an effect of the decrease in moisture and the increase of temperature) to a glassy state when the material reaches the glass transition temperature. Structural and/or molecular arrangements are modified after the second-order phase change. As such, chemical composition can all vary as a function of drying conditions.

Franklin (1948) found that chemical composition (hydrogen concentration) in carbon is related to structural changes, and thus, true density. Furthermore, he mentions that carbonized temperature, at the same hydrogen concentration, influenced true density. In foodstuffs (complex materials), variation in true density may be caused by a change in structural and/or molecular arrangement resulting from a second-order phase change, which depends on the thermal history of the material (López-Ortiz and others 2012). These phase changes can also influence ρs through their effects on composition and structural changes in the material.

The effects of salt concentration and pH on density have been confirmed for some liquid food products such as egg yolks (Sousa and others 2007). Hicsasmaz and others (2003) also demonstrated that concentrations of polydextrose influence the true and apparent density of cakes.

Apparent density (ρb) models

According to Krokida and Philippopoulos (2005), apparent density is a function of moisture content, type of solid, and air volume proportion. The majority of the models for predicting apparent density found in the literature are empirical; fundamental or theoretical models have only been proposed by a limited number of authors (Rahman and others 1996). The apparent density models have been formulated based on moisture content (X), without considering the effects of drying temperature; very few consider shrinking, although it is significant in the majority of cases (Moreira and others 2000; Mayor and Sereno 2004).

Zogzas and others (1994) consider that apparent density depends on the volume shrinking coefficient and moisture content (Table 3, Eq. 18), where the volumetric shrinking coefficient (β) is a linear function of moisture content. Boukouvalas and others (2006) mention that the parameters (inline image, β) included in Eq. 18 of Table 3 depend on the drying method and processing conditions; as such, they propose Eq. 19, which considers β to be a function of inline image, ρw, and X.

Moreira and others (2000) hold that apparent density can be obtained by using the apparent density of the solid (inline image) and the mass of water (inline image) at the beginning of drying as a reference for initial volume (Eq. 20, Table 3).

In Eq. 18 and 19, X= 0 is taken as a reference; in Eq. 20, the reference is X0= moisture content at the onset of drying.

The most recent theoretical model (Eq. 21) was put forth by Khalloufi and others (2010). They consider ρb to be a function of the initial porosity of the material (ɛ), X, ρw, ρs, β(X), and collapse [δ(X)].

Given that β(X) and δ(X) are the functions of moisture content, they are dependent on processing conditions, the nature of the product being dried, the drying method, and the drying stage (Khalloufi and others 2010).

Madiouli and others (2007) propose that shrinking may be expressed in terms of specific volume, collapse δ(X), and the volumetric shrinking coefficient β(X). Levi and Karel (1995) suggest that collapse (loss of structure, reduction in pore size, and shrinking) may occur in foodstuffs during convection drying. To identify collapse, they proposed the following equation:

  • image(5)

The collapse equation (Eq. 27) proposed by Khalloufi and others (2009) guarantees that the collapse function always begins in one and ends in zero.

When the material is in a rubbery state, shrinking compensates almost completely for moisture loss, and the volume of the material decreases linearly with moisture content (Mayor and Sereno 2004). However, low-temperature dehydration of foodstuffs prevents the moisture content in the center of the material from ever being much greater than that of the surface, minimizing internal stress, and consequently, cracking (Bai and others 2002).

An erroneous calculation of β(X) and δ(X) will give rise to an error in the calculation of apparent density. This occurs quite frequently, as the majority of authors assume that shrinking is linear (Zogzas and others 1994; Moreira and others 2000; Mayor and Sereno 2004), or they fit it to a second-degree polynomial (Khalloufi and others 2010).

Future models should strive to represent the behavior of apparent density with greater accuracy, taking into account X and the thermal history of the material being dried.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

The tendencies of density with respect to moisture content and drying temperature have been discussed within the framework of currently available theories of second-order phase changes, structural changes, and changes in chemical composition occurring in the mass and heat transference processes. The majority of the empirical and theoretical equations representing true density do not fit the concave-down tendencies of certain materials. Equations for calculating apparent density based on shrinking and collapse are not reliable, as errors arise from failing to consider minimal variations in volume at the end of the drying process. It is recommended that variations in ρs and ρw as a function of the internal temperature of the material, according to the drying conditions (X, T), be included to yield a better fit to the tendencies of density with respect to X.


  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References

The authors would like to thank the CONACyT, SIP, and the COFAA of the Inst. Politécnico Nacional for the facilities provided to carry out this investigation, and their generous financial support of the project SIP-20110358 and CONACyT 123158 and 181980.


Adjustment function


Adjustment function


Adjustment function


Relative humidity (%)


Mass (kg)


Pressure (atm)


Temperature (°C)


Time (min)


Inside temperature of the garlic slice (°C)


Volume (m3)


Moisture content (kg of water/ kg dry solid)


Regression coefficient


Diameter (mm)

Greek Letters 

Shrinking (nondimensional)




Collapse (nondimensional)


Porosity (nondimensional)


Thickness (mm)


Density (kg/m3)


Apparent density evaluated at X = 0 (kg/m3)


Wave period (min)


Air velocity (m/s)












Dry solid










Convection air drying


Freeze drying


Vacuum drying


Osmotic drying


Microwave drying


Fluid bed


Tray drying




  1. Top of page
  2. Abstract
  3. Introduction
  4. Definitions
  5. Methods of Measurement
  6. Compilation of Studies
  7. Density Models
  8. Conclusions
  9. Acknowledgments
  10. References
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