Vacuum Drying

Vacuum drying is a chemical process frequently used in the pharmaceutical and food industries to remove water or an organic solvent from a wet powder. When designing a vacuum drying system, engineers aim to minimize the drying time while maintaining high quality in the product. This model investigates vacuum drying in a Nutsche filter-dryer, which consists of a cylindrical drum filled with wet cake as seen in Figure 1. The top of the cake is exposed to a low pressure head space, and the side and bottom walls are exposed to heating fluid. By operating at a very low pressure and an elevated temperature, the evaporation rate of the liquid increases, thus accelerating the drying process of the powder. By modeling the heat transfer and evaporation of the solvent in the powder, the temperature and liquid phase profiles can be studied.

This example is based on the paper published by Murru and others for a powder dried with n-propanol under static conditions (Ref. 1).

Figure 1: Vacuum drying in an axisymmetric Nutsche filter dryer (Ref. 1).

Model Definition

The cylindrical cake can be modeled using a rectangular geometry in a 2D axisymmetric component. The cake radius is 40 cm and the cake height is 10 cm.

The vacuum drying process involves heat transfer and vapor transport in modeling the evaporation of the solvent. This example simulates the heat transfer through the cake using the Heat Transfer in Solids interface and the volume fraction of solvent using a Coefficient Form PDE interface. A predefined moisture transport in porous media physics interface is available in the Heat Transfer Module.

Heat flux boundary conditions are defined on the side and bottom boundaries to account for the external heating fluid. The evaporation of solvent is captured in two ways: a heat sink term to account for the energy lost from the cake as the liquid solvent changes to vapor phase, and a mass sink term to account for the loss of solvent in the cake.

Liquid phase evaporation

The equation in the Coefficient Form PDE interface solves for the volume fraction of liquid, θL, in the cake:

where DL is the apparent liquid diffusion coefficient,

is the evaporation rate, and ρL is the liquid phase density.

The source term

accounts for the loss of solvent as it evaporates. All boundaries are prescribed as no flux conditions.

The evaporation rate

and head space vapor pressure pG with:

, if

where

is the evaporation rate constant (1/s). In this case, the head space pressure, or the vacuum pressure surrounding the cake, is set to be 15 mbar.

Evaporation should cease once the value of liquid phase reaches zero (fully evaporated) or if the local vapor pressure does not exceed the head space vapor pressure (no driving force for the evaporation). In the model, step functions are used to smoothly ramp the evaporation rate down to zero under these conditions.

The equilibrium vapor pressure of n-propanol, which is related to the temperature (T), can be found using Antoine’s equation:

where A, B and C are constants found in Ref. 2.

The solvent migrates in the cake due to capillary flow. Instead of directly solving for the velocity field of the liquid phase in the porous media, the transport of the solvent through the cake can be approximated as a diffusion process. Therefore, when there is a gradient in the volume fraction of solvent, the liquid phase migrates from the region with a high volume fraction of solvent to the region with a low volume fraction of solvent. The liquid diffusion coefficient is calculated with:

, if

where

is the residual saturation and

is the proportionality constant, both of which are determined experimentally in Ref. 1. When the volume fraction of liquid phase present is below the critical value of

, then diffusion of the solvent in the cake should no longer occur. In the model, a step function is used to smoothly ramp the diffusion coefficient down to zero under these conditions.

As the cake dries, the liquid phase evaporates and is replaced by gas between the solid particulates. The sum of the volume fractions of the three phases should equal unity. Therefore, the volume fraction of gas present (θG) can be calculated with respect to the constant volume fraction of solid powder (θS) and the variable liquid volume fraction (θL) with

Heat transfer

The equations in the Heat Transfer in Solids interface solve for the temperature, T, in the cake:

where ρeff is the density (SI unit: kg/m^3), Cp is the heat capacity (SI unit: J/(kg*K)), λeff is the thermal conductivity (SI unit: W/(m*K)), and Q is the heat source (SI unit: W).

Energy is required for evaporation to occur, which is characterized in a heat source domain condition with

, where

is the latent heat of vaporization (SI unit: J/kg) found in Ref. 3.

The side and bottom boundaries of the filter-dryer are exposed to a heating fluid. In this model, heat flux boundary conditions account for the energy transferred to the cake from

heating fluid given a heat transfer coefficient of 10 W/(m^2*K).

The cake consists of solid powder particulates, a liquid solvent, and a gas that fills the voids between the solid powder particulates. Therefore, the material properties of the cake must take into account the properties of the 3 phases in proportion to their presence in the cake. In this case, the effective density

and effective heat capacity

are calculated with:

where θL, θS, and θG represent the volume fraction of the liquid, solid, and gas phases in the cake, respectively. The material properties of the liquid phase (ρL and

) are taken from Ref. 4 for n-propanol at a temperature of

. Solid powder properties are found in Ref. 1.

The effective thermal conductivity is calculated with:

where

and

are the thermal conductivities of the dry and fully saturated cake, respectively, found in Ref. 1.

Results and Discussion

The following figures show the temperature (Figure 2), volume fraction of liquid phase (Figure 3), and the apparent moisture diffusivity (Figure 4) in the cake after 30 hours. The temperature of the cake approaches the temperature of the heating fluid (

or 333.15 K) at the bottom and side boundaries, as seen in Figure 2. Because the liquid phase evaporates in the heated region, the volume fraction of the liquid phase is lowest near the heated side and bottom boundaries and highest in the center of the cake, as seen in Figure 3. The apparent moisture diffusivity is also highest in the center of the cake where liquid phase has not yet evaporated and approaches zero in areas where the liquid phase has already evaporated, as seen in Figure 4.

The evaporation rate after 10, 20, and 30 hours is shown in Figure 5, Figure 6, and Figure 7. The evaporation front starts near the side and bottom walls where the cake is heated. As the amount of solvent present near these boundaries decreases, the evaporation rate decreases in turn, shifting the evaporation front toward the center of the cake.