# Industrial Drying Rate & Scale-Up Sizer Documentation

# 1. Objective

The Industrial Drying Rate & Cycle Time Sizer calculates batch drying turnaround times across the Constant-Rate Period (surface evaporation) and Falling-Rate Period (internal Fickian capillary diffusion). It provides a first-principles thermodynamic engine for deriving constant rate flux ($R_c$) and a Lab-to-Commercial Equipment Scale-Up Engine for scaling Buchner/lab Nutsche filter cake data to plant-scale ANFD, RCVD, and VTD dryers.

# 2. Design Basis & Core Governing Equations

# 2.1 Moisture Balance & Batch Time Equations

The total batch drying time $t_{\text{total}}$ is the sum of the constant-rate phase ($t_c$) and falling-rate phase ($t_f$):

Where:

• $L_s$: Dry solid mass ($\text{kg}$)
• $A$: Effective heated / contact drying area ($\text{m}^2$)
• $R_c$: Constant rate period drying flux ($\text{kg/h}\cdot\text{m}^2$)
• $X_1$: Initial moisture ratio ($\text{kg solvent / kg dry solid}$)
• $X_c$: Critical moisture ratio ($\text{kg solvent / kg dry solid}$)
• $X_2$: Final target moisture ratio ($\text{kg solvent / kg dry solid}$)
• $X_e$: Bound equilibrium moisture ratio ($\text{kg solvent / kg dry solid}$)

# 3. First-Principles Determination of Constant Rate Flux $R_c$

The calculator supports three distinct modes for determining $R_c$:

# 3.1 Direct Input Mode

Manual specification of measured laboratory or pilot plant flux $R_c$ ($\text{kg/h}\cdot\text{m}^2$).

# 3.2 Convective Gas Drying Mode (Fluid Bed / Hot Air Tray Dryers)

Calculates $R_c$ based on gas-to-surface heat transfer:

• $h_c$: Convective heat transfer coefficient ($\text{W/m}^2\cdot\text{K}$)
• $T_{\text{gas}}$: Bulk gas dry-bulb temperature (°C)
• $T_{\text{wetbulb}}$: Wet-bulb surface temperature (°C)
• $\lambda$: Latent heat of vaporization ($\text{kJ/kg}$)

# 3.3 Vacuum Contact Drying Mode (ANFD, RCVD, VTD)

Calculates $R_c$ based on wall/agitator contact heat transfer:

• $U_{\text{contact}}$: Wall-to-cake overall contact heat transfer coefficient ($\text{W/m}^2\cdot\text{K}$)
• $T_{\text{jacket}}$: Heating medium jacket temperature (°C)
• $T_{\text{vac,boiling}}$: Solvent boiling temperature under operating vacuum (°C)

# 4. Lab-to-Commercial Equipment Scale-Up Engine

Commercial filter cake bed depths (50–150 mm) are substantially thicker than lab Buchner / Nutsche cakes (15–30 mm). While surface evaporation scales linearly with mass/area ratio, internal capillary diffusion time scales exponentially with bed depth $h$.

# 4.1 Fickian Bed-Thickness Diffusion Scaling Model

• $h_{\text{lab}}$: Laboratory cake thickness ($\text{mm}$)
• $h_{\text{plant}}$: Commercial plant cake thickness ($\text{mm}$)
• $n_{\text{bed}}$: Bed depth diffusion exponent ($1.5 \le n_{\text{bed}} \le 2.0$, Fickian limit $n=2.0$)
• $f_{\text{agitation}}$: Mechanical agitation / turnover enhancement factor:
  • 1.0: Static tray / unagitated cake
  • 1.3 – 1.8: ANFD smooth wiper agitator blade action
  • 2.0: RCVD double cone tumbling motion

# 5. Equipment Technology Selection Matrix

# 6. Worked Example: 4.5 m² ANFD Commercial Scale-Up

Lab Data:

• $h_{\text{lab}} = 20\text{ mm}$, $h_{\text{plant}} = 90\text{ mm}$
• $X_1 = 0.50$, $X_c = 0.20$, $X_2 = 0.05$, $X_e = 0.015$
• $L_s = 500\text{ kg}$, $A = 4.5\text{ m}^2$
• Contact $U = 65\text{ W/m}^2\cdot\text{K}$, $T_{\text{jacket}} = 70^\circ\text{C}$, $T_{\text{vac,boiling}} = 35^\circ\text{C}$, $\lambda = 2260\text{ kJ/kg}$

Calculated Values:

1. $R_c = \frac{65 \times 35}{2260 \times 1000} \times 3600 = 3.62\text{ kg/h}\cdot\text{m}^2$
2. $t_c = \frac{500}{4.5 \times 3.62} \times (0.50 - 0.20) = 9.21\text{ hours}$
3. $t_f = \frac{500 \times 0.185}{4.5 \times 3.62} \ln(5.286) = 9.53\text{ hours}$
4. Scaled Commercial Falling Time ($n=1.8, f=1.4$):

# 7. Reference Standards

• Mujumdar, A.S.: Handbook of Industrial Drying, CRC Press.
• Perry, R.H. & Green, D.W.: Perry's Chemical Engineers' Handbook, Section 12 (Psychrometry & Drying).
• VDI Heat Atlas: Section L3 (Drying Kinetics of Solid Granules).