9.9. DRYING IN FALLING-RATE PERIOD BY DIFFUSION AND CAPILLARY FLOW

9.9A. Introduction

In the falling-rate period, the surface of the solid being dried is no longer completely wetted, and the rate of drying steadily falls with time. In Section 9.7 empirical methods were used to predict the time of drying. In one method the actual rate-of-drying curve was numerically or graphically integrated to determine the time of drying.

In another method an approximately straight line from the critical free moisture content to the origin at zero free moisture was assumed. Here the rate of drying was assumed to be a linear function of the free moisture content. The rate of drying R is defined by Eq. (9.5-3):

Equation 9.5-3

When R is a linear function of X in the falling-rate period,

Equation 9.7-5

where a is a constant. Equating Eq. (9.7-5) to Eq. (9.5-3),

Equation 9.9-1

Rearranging,

Equation 9.9-2

In many instances, however, as mentioned briefly in Section 9.5E, the rate of moisture movement in the falling-rate period is governed by the rate of internal movement of the liquid by liquid diffusion or by capillary movement. These two mechanisms of moisture movement will be considered in more detail and the theories related to experimental data in the falling-rate region.

9.9B. Liquid Diffusion of Moisture in Drying

When liquid diffusion of moisture controls the rate of drying in the falling-rate period, the equations for diffusion described in Chapter 7 can be used. Using the concentrations as X kg free moisture/kg dry solid instead of kg mol moisture/m³, Fick's second law for unsteady-state diffusion, Eq. (7.10-10), can be written as

Equation 9.9-3

where D_L is the liquid diffusion coefficient in m²/h and x is distance in the solid in m.

This type of diffusion is often characteristic of relatively slow drying in nongranular materials such as soap, gelatin, and glue, and in the later stages of drying of bound water in clay, wood, textiles, leather, paper, foods, starches, and other hydrophilic solids.

A major difficulty in analyzing diffusion drying data is that the initial moisture distribution is not uniform throughout the solid at the start if a drying period at constant rate precedes this falling-rate period. During diffusion-type drying, the resistance to mass transfer of water vapor from the surface is usually very small, and the diffusion in the solid controls the rate of drying. Then the moisture content at the surface is at the equilibrium value X*. This means that the free moisture content X at the surface is essentially zero.

Assuming that the initial moisture distribution is uniform at t = 0, Eq. (9.9-3) may be integrated by the methods in Chapter 7 to give the following:

Equation 9.9-4

where X = average free moisture content at time t h, X₁ = initial free moisture content at t = 0, X* = equilibrium free moisture content, x₁ = the thickness of the slab when drying occurs from the top and bottom parallel faces, and x₁ = total thickness of slab if drying occurs only from the top face.

Equation (9.9-4) assumes that D_L is constant, but D_L is rarely constant; it varies with moisture content, temperature, and humidity. For long drying times, only the first term in Eq. (9.9-4) is significant, and the equation becomes

Equation 9.9-5

Solving for the time of drying,

Equation 9.9-6

In this equation, if the diffusion mechanism starts at X = X_C, then X₁ = X_C. Differentiating Eq. (9.9-6) with respect to time and rearranging,

Equation 9.9-7

Multiplying both sides by −L_S/A,

Equation 9.9-8

Hence, Eqs. (9.9-7) and (9.9-8) state that when internal diffusion controls for long periods of time, the rate of drying is directly proportional to the free moisture X and the liquid diffusivity, and that the rate of drying is inversely proportional to the thickness squared. Or, stated as the time of drying between fixed moisture limits, the time varies directly as the square of the thickness. The drying rate should be independent of gas velocity and humidity.

EXAMPLE 9.9-1. Drying Slabs of Wood When Diffusion of Moisture Controls The experimental average diffusion coefficient of moisture in a given wood is 2.97 × 10−6 m2/h (3.20 × 10−5 ft2/h). Large planks of wood 25.4 mm thick are dried from both sides by air having a humidity such that the equilibrium moisture content in the wood is X* = 0.04 kg H2O/kg dry wood. The wood is to be dried from a total average moisture content of Xt1 = 0.29 to Xt = 0.09. Calculate the time needed. Solution: The free moisture content X1 = Xt1 − X* = 0.29 − 0.04 = 0.25, X = Xt − X* = 0.09 − 0.04 = 0.05. The half-slab thickness x1 = 25.4/(2 × 1000) = 0.0127 m. Substituting into Eq. (9.9-6), Alternatively, Fig. 5.3-13 for the average concentration in a slab can be used. The ordinate Ea = X/X1 = 0.05/0.25 = 0.20. Reading off the plot a value of 0.56 = , substituting, and solving for t,

9.9C. Capillary Movement of Moisture in Drying

Water can flow from regions of high concentration to those of low concentration as a result of capillary action rather than by diffusion if the pore sizes of the granular materials are suitable.

The capillary theory (P1) assumes that a packed bed of nonporous spheres contains a void space between the spheres called pores. As water is evaporated, capillary forces are set up by the interfacial tension between the water and the solid. These forces provide the driving force for moving the water through the pores to the drying surface.

A modified form of Poiseuille's equation for laminar flow can be used in conjunction with the capillary-force equation to derive an equation for the rate of drying when flow is by capillary movement. If the moisture movement follows the capillary-flow equations, the rate of drying R will vary linearly with X. Since the mechanism of evaporation during this period is the same as during the constant-rate period, the effects of the variables gas velocity, temperature of the gas, humidity of the gas, and so on, will be the same as for the constant-rate drying period.

The defining equation for the rate of drying is

Equation 9.5-3

For the rate R varying linearly with X, given previously,

Equation 9.7-9

Equation 9.7-8

We define t as the time when X = X₂ and

Equation 9.9-9

where ρ_S = solid density kg dry solid/m³. Substituting Eq. (9.9-9) and X = X₂ into Eq. (9.7-8),

Equation 9.9-10

Substituting Eq. (9.6-7) for R_C,

Equation 9.9-11

Hence, Eqs. (9.9-10) and (9.9-11) state that when capillary flow controls in the falling-rate period, the rate of drying is inversely proportional to the thickness. The time of drying between fixed moisture limits varies directly as the thickness and depends upon the gas velocity, temperature, and humidity.

9.9D. Comparison of Liquid Diffusion and Capillary Flow

To determine the mechanism of drying in the falling-rate period, the experimental data obtained for moisture content at various times using constant drying conditions are often analyzed as follows. The unaccomplished moisture change, defined as the ratio of free moisture present in the solid after drying for t hours to the total free moisture content present at the start of the falling-rate period, X/X_C, is plotted versus time on semilog paper. If a straight line is obtained, such as curve B in Fig. 9.9-1 using the upper scale for the abscissa, then either Eqs. (9.9-4)–(9.9-6) for diffusion or Eqs. (9.9-10) and (9.9-11) for capillary flow are applicable.

If the equations for capillary flow apply, the slope of the falling-rate drying line B in Fig. 9.9-1 is related to Eq. (9.9-10), which contains the constant drying rate R_C. The value of R_C is calculated from the measured slope of the line, which is −R_C/x₁ρ_SX_C, and if it agrees with the experimental value of R_C in the constant-drying period or the predicted value of R_C, the moisture movement is by capillary flow.

If the values of R_C do not agree, the moisture movement is by diffusion and the slope of line B in Fig. 9.9-1 from Eq. (9.9-6) should equal −π²D_L/4. In actual practice, however, the diffusivity D_L is usually less at small moisture contents than at large moisture contents, and an average value of D_L is usually determined experimentally over the moisture range of interest. A plot of Eq. (9.9-4) is shown as line A, where ln(X/X₁) or ln(X/X_C) is plotted versus D_Lt/. This is the same plot as Fig. 5.3-13 for a slab and shows a curvature in the line for values of X/X_C between 1.0 and 0.6 and a straight line for X/X_C < 0.6.

When the experimental data show that the movement of moisture follows the diffusion law, the average experimental diffusivities can be calculated as follows for different concentration ranges. A value of X/X_C is chosen at 0.4, for example. From an experimental plot similar to curve B, Fig. 9.9-1, the experimental value of t is obtained. From curve A at X/X_C = 0.4, the theoretical value of (D_Lt/)_theor is obtained. Then, by substituting the known values of t and x₁ into Eq. (9.9-12), the experimental average value of D_L over the range X/X_C = 1.0–0.4 is obtained:

Equation 9.9-12

This is repeated for various values of X/X_C. Values of D_L obtained for X/X_C > 0.6 are in error because of the curvature of line A.

EXAMPLE 9.9-2. Diffusion Coefficient in the Tapioca Root Tapioca flour is obtained from drying and then milling the tapioca root. Experimental data on drying thin slices of the tapioca root 3 mm thick on both sides in the falling-rate period under constant-drying conditions are tabulated below. The time t = 0 is the start of the falling-rate period. X/XC t (h) X/XC t (h) X/XC t (h) 1.0 0 0.55 0.40 0.23 0.94 0.80 0.15 0.40 0.60 0.18 1.07 0.63 0.27 0.30 0.80     It has been determined that the data do not follow the capillary-flow equations but appear to follow the diffusion equations. Plot the data as X/XC versus t on semilog coordinates and determine the average diffusivity of the moisture up to a value of X/XC = 0.20. Solution: In Fig. 9.9-2 the data are plotted as X/XC on the log scale versus t on a linear scale and a smooth curve is drawn through the data. At X/XC = 0.20, a value of t = 1.02 h is read off the plot. The value of x1 = 3 mm/2 = 1.5 mm for drying from both sides. From Fig. 9.9-1, line A, for X/XC = 0.20, (DLt/)theor = 0.56. Then substituting into Eq. (9.9-12), Figure 9.9-2. Plot of drying data for Example 9.9-2.