# Characteristic Drying Curve Models

## Theoretical Framework

The characteristic drying curve (CDC) model is another approach that can be used to arrive at a unified behavior on how the falling rate of the droplet changes throughout the drying process (Langrish and Kockel, 2001). In the model, the highest possible drying rate, the drying rate corresponding to the wet bulb period of evaporation, is taken as the reference point from which the drying rate is progressively reduced. This highest possible drying rate is given as follows:

Taking this as the upper limit of drying, one would then expect the drying rate to progressively reduce in the secondary drying period. The moisture content of the droplet can then be arbitrarily adopted to delineate the extent to which the drying has progressed. Secondary drying only begins once the droplet reaches critical moisture content, and the lowest possible moisture content is the equilibrium moisture content. Therefore, these two moisture contents can be adopted to normalize the progressively reducing moisture content of the droplet to arrive at a dimensionless reduction factor as follows:

It is noteworthy that at the initial instance of the falling rate period, the reduction factor will be unity. As the drying rate drops when the droplet progresses into the falling rate period, this ratio then drops and eventually reaches zero when the droplet approaches equilibrium moisture content. Combining both equations, the CDC approach to describe the falling rate drying behavior then becomes:

This arbitrary form for the reduction factor is mathematically convenient, as the value N then denotes the shape or, if physically interpreted, the behavior in which the solidification affects the progressive drying retardation. A value of 1 denotes a linear falling rate behavior. A value of greater than 1 for N implies a sudden retardation in the drying rate, most likely due to a significant skin-forming phenomenon. While this mathematical profile for the reduction factor hitherto fits commonly observed drying behavior so far, it is mainly arbitrary; it can be expressed in other forms if it is suitably normalized.

2.3.2 **How **to Obtain the Parameter N from Experiments

Different methods can be used to firstly obtain the data on how the mass of the sample of interest changes over the drying time. This can be from thin film or single droplet or another form of experimental technique. By computing the gradient at each mass data throughout the drying process, a plot of drying rate versus droplet moisture content can then be obtained (Figure 2.3.1). Normalize the drying rate as follows, which is a rearrangement of Equation (2.3.3):

FIGURE 2.3.1 Obtaining the CDC drying kinetics.

The highest possible experimentally measured drying rate should also correspond to the wet bulb drying rate for low initial droplet solid concentration. The critical moisture content can be obtained by examining the moisture content when the drying rate starts to reduce. The equilibrium moisture content can be obtained from available isotherm data.

This procedure should then be repeated by drying the material under different temperatures, humidities, and velocities (different drying conditions) if the experimental rig permits. The normalized data described above from each experimental run can then be collated and the parameter N can then be obtained by fitting using only the set of normalized data at normalized moisture content lower than unity, delineating when the particles have entered the secondary drying period.

TABLE 2.3.1

Characteristic Drying Curve Parameters for Some Common Spray Dried Materials

Materials |
References |
CDC index |
Critical moisture content |

Skim milk |
Langrish and Kockel (2001) |
1 (linear) |
Initial moisture content (80% wt analyzed) |

Sucrose |
Woo |
2.58 |
Initial moisture content (50% wt analyzed) |

Maltodextrin |
Woo |
3.22 |
Initial moisture content (50% wt analyzed) |

Sucrose-maltodextrin (1:1) |
Woo |
1.98 |
Initial moisture content (60% wt analyzed) |

Maltodextrin |
Zbicinski and Li (2006) |
1 (linear) |
From pilot scale conditions: Evaporation rate >8 g/cm Evaporation rate >5 g/cm Evaporation rate >3 g/cm The critical moisture content was close to the initial moisture |

*Source:* Woo, M.W. *Computational Fluid Dynamic Simulation of Spray Dryers - An Engineer's Guide.* CRC Press, Boca Raton, FL, 2016, With permission.

## Compilation of Falling Rate Curves

The CDC. index for some commonly spray dried materials is given in Table 2.3.1. For most applications of the CDC model, the initial moisture content is typically taken as the critical moisture content. The simplifying assumption here is that the initial heating-up time and the constant drying period are very short and rapid relative to the overall spray drying process. It is important to note that the critical moisture content, fundamentally, may be dependent on the initial concentration of the feed solution and the drying rate of the droplets (Zbicinski and Li, 2006).