9.5. RATE-OF-DRYING CURVES

9.5A. Introduction and Experimental Methods

1. Introduction

In the drying of various types of process materials from one moisture content to another, it is usually desired to estimate the size of dryer needed, the various operating conditions of humidity and temperature for the air used, and the time needed to perform the amount of drying required. As discussed in Section 9.4, equilibrium moisture contents of various materials cannot be predicted and must be determined experimentally. Similarly, since our knowledge of the basic mechanisms of rates of drying is quite incomplete, it is necessary in most cases to obtain some experimental measurements of drying rates.

2. Experimental determination of rate of drying

To experimentally determine the rate of drying for a given material, a sample is usually placed on a tray. If it is a solid material it should fill the tray so that only the top surface is exposed to the drying air stream. By suspending the tray from a balance in a cabinet or duct through which the air is flowing, the loss in weight of moisture during drying can be determined at different intervals without interrupting the operation.

In doing batch-drying experiments, certain precautions should be observed to obtain usable data under conditions that closely resemble those to be used in the large-scale operations. The sample should not be too small in weight and should be supported in a tray or frame similar to the large-scale one. The ratio of drying to nondrying surface (insulated surface) and the bed depth should be similar. The velocity, humidity, temperature, and direction of the air should be the same and constant to simulate drying under constant drying conditions.

9.5B. Rate of Drying Curves for Constant-Drying Conditions

1. Conversion of data to rate-of-drying curve

Data obtained from a batch-drying experiment are usually obtained as W total weight of the wet solid (dry solid plus moisture) at different times t hours in the drying period. These data can be converted to rate-of-drying data in the following ways. First, the data are recalculated. If W is the weight of the wet solid in kg total water plus dry solid and W_S is the weight of the dry solid in kg,

Equation 9.5-1

For the given constant drying conditions, the equilibrium moisture content X* kg equilibrium moisture/kg dry solid is determined. Then the free moisture content X in kg free water/kg dry solid is calculated for each value of X_t:

Equation 9.5-2

Using the data calculated from Eq. (9.5-2), a plot of free moisture content X versus time t in h is made, as in Fig. 9.5-1a. To obtain the rate-of-drying curve from this plot, the slopes of the tangents drawn to the curve in Fig. 9.5-1a can be measured, which give values of dX/dt at given values of t. The rate R is calculated for each point by

Equation 9.5-3

where R is drying rate in kg H₂O/h · m², L_S kg of dry solid used, and A exposed surface area for drying in m². In English units, R is lb_m H₂O/h · ft², L_S is lb_m dry solid, and A is ft². For obtaining R from Fig. 9.5-1a, a value of L_S/A of 21.5 kg/m² was used. The drying-rate curve is then obtained by plotting R versus the moisture content, as in Fig. 9.5-1b.

Another method for obtaining the rate-of-drying curve is to first calculate the weight loss ΔX for a Δt time. For example, if X₁ = 0.350 at a time t₁ = 1.68 h and x₂ = 0.325 at a time t₂ = 2.04 h, ΔX/Δt = (0.350 − 0.325)/(2.04 − 1.68). Then, using Eq. (9.5-4) and L_S/A = 21.5,

This rate R is the average over the period 1.68 to 2.04 h and should be plotted at the average concentration X = (0.350 + 0.325)/2 = 0.338.

2. Plot of rate-of-drying curve

In Fig. 9.5-1b the rate-of-drying curve for constant-drying conditions is shown. At zero time the initial free moisture content is shown at point A. In the beginning the solid is usually at a colder temperature than its ultimate temperature, and the evaporation rate will increase. Eventually, at point B, the surface temperature rises to its equilibrium value. Alternatively, if the solid is quite hot to start with, the rate may start at point A'.This initial unsteady-state adjustment period is usually quite short and it is often ignored in the analysis of times of drying.

From point B to point C in Fig. 9.5-1a the line is straight, and hence the slope and rate are constant during this period. This constant-rate-of-drying period is shown as line BC in Fig. 9.5-1b.

At point C on both plots, the drying rate starts to decrease in the falling-rate period until it reaches point D. In this first falling-rate period, the rate shown as line CD in Fig. 9.5-1b is often linear.

At point D the rate of drying falls even more rapidly, until it reaches point E, where the equilibrium moisture content is X* and X = X* − X* = 0. In some materials being dried, the region CD may be missing completely, or it may constitute all of the falling-rate period.

9.5C. Drying in the Constant-Rate Period

Drying of different solids under different constant conditions of drying will often give curves of different shapes in the falling-rate period, but in general the two major portions of the drying-rate curve—constant-rate period and falling-rate period—are present.

In the constant-rate drying period, the surface of the solid is initially very wet and a continuous film of water exists on the drying surface. This water is entirely unbound water and it acts as if the solid were not present. The rate of evaporation under the given air conditions is independent of the solid and is essentially the same as the rate from a free liquid surface. Increased roughness of the solid surface, however, may lead to higher rates than from a flat surface.

If the solid is porous, most of the water evaporated in the constant-rate period is supplied from the interior of the solid. This period continues only as long as the water is supplied to the surface as fast as it is evaporated. Evaporation during this period is similar to that in determining the wet bulb temperature, and in the absence of heat transfer by radiation or conduction, the surface temperature is approximately the same as the wet bulb temperature.

9.5D. Drying in the Falling-Rate Period

Point C in Fig. 9.5-1b is at the critical free moisture content X_C. At this point there is insufficient water on the surface to maintain a continuous film of water. The entire surface is no longer wetted, and the wetted area continually decreases in this first falling-rate period until the surface is completely dry, at point D in Fig. 9.5-1b.

The second falling-rate period begins at point D when the surface is completely dry. The plane of evaporation slowly recedes from the surface. Heat for the evaporation is transferred through the solid to the zone of vaporization. Vaporized water moves through the solid into the air stream.

In some cases no sharp discontinuity occurs at point D, and the change from partially wetted to completely dry conditions at the surface is so gradual that no distinct change is detectable.

The amount of moisture removed in the falling-rate period may be relatively small, but the time required may be long. This can be seen in Fig. 9.5-1. The period BC for constant-rate drying lasts for about 3.0 h and reduces X from 0.40 to about 0.19, a reduction of 0.21 kg H₂O/kg dry solid. The falling-rate period CE lasts about 9.0 h and reduces X only from 0.19 to 0.

9.5E. Moisture Movements in Solids During Drying in the Falling-Rate Period

When drying occurs by evaporation of moisture from the exposed surface of a solid, moisture must move from the depths of the solid to the surface. The mechanisms of this movement affect the drying during the constant-rate and falling-rate periods. Some of the theories advanced to explain the various types of falling-rate curves will be briefly reviewed.

1. Liquid diffusion theory

According to this theory, diffusion of liquid moisture occurs when there is a concentration difference between the depths of the solid and the surface. This method of transport of moisture is usually found in nonporous solids where single-phase solutions are formed with the moisture, such as in paste, soap, gelatin, and glue. It is also found in drying the last portions of moisture from clay, flour, wood, leather, paper, starches, and textiles. In drying many food materials, the movement of water in the falling-rate period also occurs by diffusion.

The shapes of the moisture-distribution curves in the solid at given times are qualitatively consistent with the unsteady-state diffusion equations given in Chapter 7. The moisture diffusivity D_AB usually decreases with decreased moisture content, so that the diffusivities are usually average values over the range of concentrations used. Materials drying in this way are usually said to be drying by diffusion, although the actual mechanisms may be quite complicated. Since the rate of evaporation from the surface is quite fast, that is, the resistance is quite low, compared to the diffusion rate through the solid in the falling-rate period, the moisture content at the surface is at the equilibrium value.

The shape of a diffusion-controlled curve in the falling-rate period is similar to Fig. 9.5-2a. If the initial constant-rate drying is quite high, the first falling-rate period of unsaturated surface evaporation may not appear. If the constant-rate drying is quite low, the period of unsaturated surface evaporation is usually present in region CD in Fig. 9.5-1b and the diffusion-controlled curve is in region DE. Equations for calculating drying in this period where diffusion controls are given in Section 9.9. Also, Problem 7.1-4 for the drying of clay and Problem 7.1-6 for the drying of wood using diffusion theory are given in the Chapter 7 Problems.

2. Capillary movement in porous solids

When granular and porous solids such as clays, sand, soil, paint pigments, and minerals are being dried, unbound or free moisture moves through the capillaries and voids of the solids by capillary action, not by diffusion. This mechanism, involving surface tension, is similar to the movement of oil in a lamp wick.

A porous solid contains interconnecting pores and channels of varying pore sizes. As water is evaporated, a meniscus of liquid water is formed across each pore in the depths of the solid. This sets up capillary forces by the interfacial tension between the water and the solid. These capillary forces provide the driving force for moving water through the pores to the surface. Small pores develop greater forces than do large pores.

At the beginning of the falling-rate period at point C in Fig. 9.5-1b, the water is being brought to the surface by capillary action, but the surface layer of water starts to recede below the surface. Air rushes in to fill the voids. As the water is continuously removed, a point is reached where there is insufficient water left to maintain continuous films across the pores, and the rate of drying suddenly decreases at the start of the second falling-rate period at point D. Then the rate of diffusion of water vapor in the pores and rate of conduction of heat in the solid may become the main factors in drying.

In fine pores in solids, the rate-of-drying curve in the second falling-rate period may conform to the diffusion law; the curve is concave upward, as shown in Fig. 9.5-2b. For very porous solids, such as a bed of sand, where the pores are large, the rate-of-drying curve in the second falling-rate period is often straight, and hence the diffusion equations do not apply.

3. Effect of shrinkage

A factor often greatly affecting the drying rate is the shrinkage of the solid as moisture is removed. Rigid solids do not shrink appreciably, but colloidal and fibrous materials such as vegetables and other foodstuffs do undergo shrinkage. The most serious effect is that there may be developed a hard layer on the surface which is impervious to the flow of liquid or vapor moisture and slows the drying rate; examples are clay and soap. In many foodstuffs, if drying occurs at too high a temperature, a layer of closely packed, shrunken cells, which are sealed together, forms at the surface. This presents a barrier to moisture migration and is known as case hardening. Another effect of shrinkage is to cause the material to warp and change its structure. This can happen in drying wood.

Sometimes, to decrease these effects of shrinkage, it is desirable to dry with moist air. This decreases the rate of drying so that the effects of shrinkage on warping or hardening at the surface are greatly reduced.