9.6. CALCULATION METHODS FOR CONSTANT-RATE DRYING PERIOD

9.6A. Method Using Experimental Drying Curve

1. Introduction

Probably the most important factor in drying calculations is the length of time required to dry a material from a given initial free moisture content X₁ to a final moisture content X₂. For drying in the constant-rate period, we can estimate the time needed by using experimental batch-drying curves or by using predicted mass- and heat-transfer coefficients.

2. Method using drying curve

To estimate the time of drying for a given batch of material, the best method is based on actual experimental data obtained under conditions where the feed material, relative exposed surface area, gas velocity, temperature, and humidity are essentially the same as in the final drier. Then the time required for the constant-rate period can be determined directly from the drying curve of free moisture content versus time.

EXAMPLE 9.6-1. Time of Drying from Drying Curve A solid whose drying curve is represented by Fig. 9.5-1a is to be dried from a free moisture content X1 = 0.38 kg H2O/kg dry solid to X2 = 0.25 kg H2O/kg dry solid. Estimate the time required. Solution: From Fig. 9.5-1a for X1 = 0.38, t1 is read off as 1.28 h. For X2 = 0.25, t2 = 3.08 h. Hence, the time required is

3. Method using rate-of-drying curve for constant-rate period

Instead of using the drying curve, the rate-of-drying curve can be used. The drying rate R is defined by Eq. (9.5-3) as

Equation 9.5-3

This can be rearranged and integrated over the time interval for drying from X₁ at t₁ = 0 to X₂ at t₂ = t:

Equation 9.6-1

If the drying takes place within the constant-rate period, so that both X₁ and X₂ are greater than the critical moisture content X_C, then R = constant = R_C. Integrating Eq. (9.6-1) for the constant-rate period,

Equation 9.6-2

EXAMPLE 9.6-2. Drying Time from Rate-of-Drying Curve Repeat Example 9.6-1 but use Eq. (9.6-2) and Fig. 9.5-1b. Solution: As stated previously, a value of 21.5 for LS/A was used to prepare Fig. 9.5-1b from 9.5-1a. From Fig. 9.5-1b, RC = 1.51 kg H2O/h · m2. Substituting into Eq. (9.6-2), This is close to the value of 1.80 h in Example 9.6-1.

9.6B. Method Using Predicted Transfer Coefficients for Constant-Rate Period

1. Introduction

In the constant-rate period of drying, the surfaces of the grains of solid in contact with the drying air flow remain completely wetted. As stated previously, the rate of evaporation of moisture under a given set of air conditions is independent of the type of solid and is essentially the same as the rate of evaporation from a free liquid surface under the same conditions. However, surface roughness may increase the rate of evaporation.

During this constant-rate period, the solid is so wet that the water acts as if the solid were not there. The water evaporated from the surface is supplied from the interior of the solid. The rate of evaporation from a porous material occurs by the same mechanism as that occurring at a wet bulb thermometer, which is essentially constant-rate drying.

2. Equations for predicting constant-rate drying

Drying of a material occurs by mass transfer of water vapor from the saturated surface of the material through an air film to the bulk gas phase or environment. The rate of moisture movement within the solid is sufficient to keep the surface saturated. The rate of removal of the water vapor (drying) is controlled by the rate of heat transfer to the evaporating surface, which furnishes the latent heat of evaporation for the liquid. At steady state, the rate of mass transfer balances the rate of heat transfer.

To derive the equation for drying, we neglect heat transfer by radiation to the solid surface and also assume no heat transfer by conduction from metal pans or surfaces. Conduction and radiation will be considered in Section 9.8. Assuming only heat transfer to the solid surface by convection from the hot gas to the surface of the solid and mass transfer from the surface to the hot gas (Fig. 9.6-1), we can write equations which are the same as those for deriving the wet bulb temperature T_W in Eq. (9.3-18).

The rate of convective heat transfer q in W (J/s, btu/h) from the gas at T°C (°F) to the surface of the solid at T_W°C, where (T − T_W)°C = (T − T_W) K is

Equation 9.6-3

where h is the heat-transfer coefficient in W/m² · K (btu/h · ft² · °F) and A is the exposed drying area in m² (ft²). The equation of the flux of water vapor from the surface is the same as Eq. (9.3-13) and is

Equation 9.6-4

Using the approximation from Eq. (9.3-15) and substituting into Eq. (9.6-4),

Equation 9.6-5

The amount of heat needed to vaporize N_A kg mol/s · m² (lb mol/h · ft²) water, neglecting the small sensible heat changes, is the same as Eq. (9.3-12):

Equation 9.6-6

where λ_W is the latent heat at T_W in J/kg (btu/lb_m).

Equating Eqs. (9.6-3) and (9.6-6) and substituting Eq. (9.6-5) for N_A,

Equation 9.6-7

Equation (9.6-7) is identical to Eq. (9.3-18) for the wet bulb temperature. Hence, in the absence of heat transfer by conduction and radiation, the temperature of the solid is at the wet bulb temperature of the air during the constant-rate drying period. Thus, the rate of drying R_C can be calculated using the heat-transfer equation h(T − T_W)/λ_W or the mass-transfer equation k_yM_B(H_W − H). However, it has been found more reliable to use the heat-transfer equation (9.6-8), since an error in determining the interface temperature T_W at the surface affects the driving force (T − T_W) much less than it affects (H_W − H).

Equation 9.6-8

To predict R_C in Eq. (9.6-8), the heat-transfer coefficient must be known. For the case where the air is flowing parallel to the drying surface, Eq. (4.6-3) can be used for air. However, because the shape of the leading edge of the drying surface causes more turbulence, the following can be used for an air temperature of 45–150°C and a mass velocity G of 2450–29 300 kg/h · m² (500–6000 lb_m/h · ft²) or a velocity of 0.61–7.6 m/s (2–25 ft/s):

Equation 9.6-9

where in SI units G is νρ kg/h · m² and h is W/m² · K. In English units, G is in lb_m/h · ft² and h in btu/h · ft² · °F. When air flows perpendicular to the surface for a G of 3900–19 500 kg/h · m² or a velocity of 0.9–4.6 m/s (3–15 ft/s),

Equation 9.6-10

Equations (9.6-8)–(9.6-10) can be used to estimate the rate of drying during the constant-rate period. However, when possible, experimental measurements of the drying rate are preferred.

To estimate the time of drying during the constant-rate period, substituting Eq. (9.6-7) into (9.6-2),

Equation 9.6-11

EXAMPLE 9.6-3. Prediction of Constant-Rate Drying An insoluble wet granular material is dried in a pan 0.457 × 0.457 m (1.5 × 1.5 ft) and 25.4 mm deep. The material is 25.4 mm deep in the pan, and the sides and bottom can be considered to be insulated. Heat transfer is by convection from an air stream flowing parallel to the surface at a velocity of 6.1 m/s (20 ft/s). The air is at 65.6°C (150°F) and has a humidity of 0.010 kg H2O/kg dry air. Estimate the rate of drying for the constant-rate period using SI and English units. Solution: For a humidity H = 0.010 and dry bulb temperature of 65.6°C, using the humidity chart, Fig. 9.3-2, the wet bulb temperature TW is found to be 28.9°C (84°F) and HW = 0.026 by following the adiabatic saturation line (the same as the wet bulb line) to the saturated humidity. Using Eq. (9.3-7) to calculate the humid volume, The density for 1.0 kg dry air + 0.010 kg H2O is The mass velocity G is Using Eq. (9.6-9), At TW = 28.9°C (84°F), λW = 2433 kJ/kg (1046 btu/lbm) from steam tables. Substituting into Eq. (9.6-8) and noting that (65.6 − 28.9)°C = (65.6 − 28.9) K, The total evaporation rate for a surface area of 0.457 × 0.457 m2 is

9.6C. Effect of Process Variables on Constant-Rate Period

As stated previously, experimental measurements of the drying rate are usually preferred over using the equations for prediction. However, these equations are quite helpful in predicting the effect of changing the drying-process variables when limited experimental data are available.

1. Effect of air velocity

When conduction and radiation heat transfer are not present, the rate R_C of drying in the constant-rate region is proportional to h and hence to G^0.8 as given by Eq. (9.6-9) for air flow parallel to the surface. The effect of gas velocity is less important when radiation and conduction are present.

2. Effect of gas humidity

If the gas humidity H is decreased for a given T of the gas, then from the humidity chart the wet bulb temperature T_W will decrease. Then, using Eq. (9.6-7), R_C will increase. For example, if the original conditions are R_C1, T₁, T_W1, H₁, and H_W1, then if H₁ is changed to H₂ and H_W1 is changed to H_W2, R_C2 becomes

Equation 9.6-12

However, since λ_W1 ≅ λ_W2,

Equation 9.6-13

3. Effect of gas temperature

If the gas temperature T is increased, T_W is also increased somewhat, but not as much as the increase in T. Hence, R_C increases as follows:

Equation 9.6-14

4. Effect of thickness of solid being dried

For heat transfer by convection only, the rate R_C is independent of the thickness x₁ of the solid. However, the time t for drying between fixed moisture contents X₁ and X₂ will be directly proportional to the thickness x₁. This is shown by Eq. (9.6-2), where increasing the thickness with a constant A will directly increase the amount of L_S kg dry solid.

5. Experimental effect of process variables

Experimental data tend to bear out the conclusions reached on the effects of material thickness, humidity, air velocity, and T − T_W.