Product mass transfer resistance directly determined during freeze-drying cycle runs using tunable diode laser absorption spectroscopy (TDLAS) and pore diffusion model

Abstract

The pore diffusion model is used to express the dry layer mass transfer resistance, , as a function of the ratio r_e/τ, where r_e is the effective pore radius and τ is the tortuosity factor of the dry layer. Using this model, the effective pore radius of the dry layer can be estimated from the sublimation rate and product temperature profiles measured during primary drying. Freeze-drying cycle runs were performed using the LyoStar II dryer (FTS Systems), with real-time sublimation rate profiles during freeze drying continuously measured by tunable diode laser absorption spectroscopy (TDLAS). The formulations chosen for demonstration of the proposed approach include 5% mannitol, 5% sucrose, 5% lactose, 3% mannitol plus 2% sucrose, and a parenteral nutrition formulation denoted VitaM12. The three different methods used for determination of the product resistance are: (1) Using both the sublimation rate and product temperature profiles, (2) using the sublimation rate profile alone, and (3) using the product temperate profile alone. Unlike the second and third methods, the computation procedure of first method does not need solution of the complex heat and mass transfer equations.

Keywords::

• Tunable diode laser absorption spectroscopy
• freeze drying
• pore diffusion
• product resistance
• mathematical modeling
• mass transfer

Appendix A

Derivation for the product resistance in terms of the pore radius parameters

It is assumed that the heat and mass transfer processes during the entire course of primary drying follow pseudo-steady state. That is, at any given time, the heat transfer rates from the shelf fluid, through the shelf surface, through the frozen product, then to the surface of the ice interface are the same. In addition, the mass transfer rates of water vapor from the ice surface, through the dry layer, through the stopper, then to the chamber are also the same. This assumption suggests that equilibrium in heat and mass transfers is quickly established during the incremental change in drying time Δt. Since freeze-drying is a very slow process, this assumption is probably valid except during fast ramping of the shelf temperature. This assumption is the basis for solving the five coupled heat and mass transfer equations simultaneously, as discussed by Kuu et al.^[3] Since primary drying mathematically is a moving boundary problem, the heat and mass transfer rates will change over time due to increase in the product resistance along with increasing dry layer thickness. However, the effect of the change in the heat and mass transfer rates on the solution of the equations can be managed by the mass balance at each integration time interval. In this report, the integration interval Δt was set at 1 min. The other assumption is the planar moving boundary of the ice interface that was previously used in the literature.^[1,2]

Thus, with the pseudo-steady state assumption, the instantaneous sublimation rate through the dry layer can be expressed by Equation (A1):

A1

where D_e is the effective Knudsen diffusion coefficient and ℓ is the dried-layer thickness at time t; A_p is the area of the product in the vial; C_i and C_v are the concentrations of water vapor at the ice surface and in the headspace of the vial, respectively. The concentrations C_i and C_v in Equation (A1) can be converted to partial pressures of water using the ideal gas law, as expressed by Equations (A2) and (A3).

A2

and

A3

where P_i and P_v are the partial pressures of water at the ice interface and in the headspace of the vial, respectively. M is the molecular weight of water, 18 g mole⁻¹, and R is the gas constant, 82.05 cm³ atm ^oK⁻¹, mole⁻¹. T_i and T_v are the temperatures at the ice interface and in the vial, respectively. The unknown variable T_v may be approximated as equal to T_i, since it only slightly affect the value of P_v. This assumption doesn’t affect the other more important variables, such as m, T_s, T_b and T_i. In addition, since unit of T_v is ^oK in Eq. (A3), the error in C_v with a temperature difference of 10–20°C is not significant. Since the unit of T_v is in ^oK, the value of C_v only changes by 5.7% when T_v is changed from −25°C (298°K) to −10°C (263°K). Thus, substituting C_i and C_v in Equation (A1) with Equations (A2) and (A3), the sublimation rate becomes:

A4

It is clear that the denominator of the right side of Equation (A4) is equivalent to the normalized product mass transfer resistance through the dry layer as used in the literature.^[3] As expressed by Equation (A5).

A5

where the effective Knudsen diffusion coefficient D_e is dependent on thickness, ℓ, due to structural heterogenity in the dry layer.

• : in Torr. hr. g⁻¹.

• R: gas constant, 82.05 cm³ atm mole⁻¹ K⁻¹ = (82.05)(760) cm³ Torr mole⁻¹ K⁻¹ = 6.236 × 10⁴ cm³ Torr mole⁻¹ K⁻¹

• T_i: temperature at the ice surface, in ^oK.

• ℓ: thickness of the dry layer in cm.

Now, it is necessary to derive the effective diffusion coefficient D_e in Equation (A5) in order to expressed in terms of the pore size parameters b₀ and b₁. It is assumed the vapor flow through the pores of the dry layer is dominated by free molecular flow as discussed in the Theoretical section, which is also assumed that the surface diffusion of water vapor does not occur during primary drying. Thus, the diffusion coefficient of water vapor in Knudsen region of free molecular flow can be simplified from the literature, i.e. Equation (29) by Ho and Roseman^[11] or Equation (2.43) on page 41 by Sherwood et al.,^[12] as expressed by Equation (A6):

A6

The effective diffusion coefficient through the dry layer is then obtained by correcting D_K with the porosity, ϵ, and tortuosity of the dry layer, τ, as expressed by Equation (A7)

A7

where T_o is the average temperature in the pores, and r_e is the effective pore radius, in cm; ϵ and τ are dimensionless. After plugging in the pore radius equation, Equation (3) into Equation (A7), followed by plugging the resulting De into Equation (A5), the resistance equation is obtained, as shown in Equation (A8)

A8

During freeze-drying, the product temperature at the bottom-center of the vial, T_b, is usually measured. It was often observed that the intercept of the resistance is not zero at the onset of primary drying when the cake thickness is zero.^[1,2] Thus, an intercept, R₀, is added to Equation (A8) to complete the final resistance equation in Equation (A8)

A9

The Knudsen number is defined in Equation (A10):^[11]

A10

where λ is the mean free path in cm and r_e is the effective pore radius in cm. The mean free path is defined as the distance a molecule travels before colliding with another one, which can be calculated as:

A11

where d is the molecular diameter of water vapor (3.46 Å, or 3.46 × 10⁻⁸ cm) and N is the number of molecules per cubic centimeter, which can be calculated as:

A12

where P is the average pressure in the pores, in atm; Nav is Avogadro’s number, equal to 6.023 × 10⁻²³; R is the gas constant, 0.08205 liter atm ^oK⁻¹ mole⁻¹.

The value of the pressure P in Equation (A12) can be estimated as:

A13

According to the literature,^[11] the criterion for free molecular flow is that the Knudsen number Kn > 3. Since the values of Knudsen number range from 6–60 for all cycle runs performed in this report, it is reasonable to assume free molecular flow.

Appendix B

Determination of product mass transfer resistance using both the sublimation rate and product temperature profiles, without solving the complex heat and mass transfer equations

Since the method for determination of the product resistance using the sublimation rate measured by TDLAS has not be discussed in the literature, it may be necessary to present a detailed derivation. The sublimation rate is related to the mass transfer resistances from the vial to the condenser, as:

B1

where P₀ is the equilibrium vapor pressure of the subliming ice, in mmHg, and P_c is the chamber pressure, in mmHg. R_s and R_p are the stopper resistance and product resistance, in (torr. hr. g⁻¹), respectively. Thus, R_p can be obtained by rearranging Equation (B1), as

B2

Equation (B2) indicates that the accuracy of R_p depends on the difference (P₀ − P_c), and the accuracy of the measured . Hence, Equation (B2) suggests that the accuracy of the determined resistance is higher at a higher shelf temperature where P₀ is significantly higher than P_c. The accuracy of the measured by TDLAS increases with increasing water vapor velocity, and a stabilized velocity profile. Thus, resistance determination using the proposed approach may not be suitable during the shelf temperature ramping period, unless the ramp rate is very slow, since the pseudo-steady state is not established.

In Equation (B2), P₀ can be expressed in terms of the exponential function of the ice surface temperature T_i^:[2]

B3

Thus, the area-normalized resistance , in cm². torr. hr. g⁻¹, becomes:

B4

Since T_i in Equation (B3) is not measurable due to the moving boundary of the ice surface, in order to use Equations (B2) through (B4) to calculate the resistance, the easiest way is to let T_i = T_b. Nevertheless, the conductive heat transfer resistance from the bottom to the top of the frozen product could be significant, especially when the cake is thick. The following procedure can be used to avoid this error. The heat conduction through the frozen product can be expressed by Equation (B5):

B5

where is in cal/s; K_I is effective thermal conductivity of the frozen layer, in cal. s⁻¹ cm⁻¹ °C⁻¹; A_v is vial area (calculated based on the outside diameter), in cm²; X is the thickness of the frozen layer, in cm. The rate of heat transfer can be converted to the instantaneous sublimation rate using Equation (B6):

B6

Thus, the sublimation rate can be obtained by solving Equations (B5) and (B6), as given be Equation (B7):

B7

by which, the temperature T_i is obtained as:

B8

Equation (B8) indicates that T_i can be calculated from T_b, but it is a function of the dry layer thickness ℓ. This can be done using a simple computation source codes such as FORTRAN, or alternatively using a Microsoft Excel^® spreadsheet.

The accumulated mass of sublimation is also computed as follows. The change in the sublimed mass ΔM_t in each computational step is first computed by:

B9

where (Rate)_avg is the average rate of sublimation in g.h⁻¹, between two computational steps. The change in the frozen layer thickness Δℓ is then obtained as:

B10

where ρ is the density of the frozen layer, which is approximately equal to 0.917. (1-y) where y is the fraction of total solid;^[2] d is the inside diameter of the vial. Thus the accumulated dry-layer thickness ℓ and sublimation mass M_t become:

B11

and

B12

The stopper resistance, R_s, is expressed by Equation (B13):^[2]

B13

where S₀ and S₁ are constants, and P_avg is the average pressure in the vial and chamber. The density of the ice, ρ, in the frozen 5% lactose can be expressed as:

B14

The thickness of the ice removed, ℓ, is expressed by the following equation:

B15

where A_p is the area of the frozen product in the vial.

For convenience, the step-by-step calculation procedure for determining b₀ and b₁ is summarized below. Note that Steps (1) through (8) can be performed using a simple computation source code such as FORTRAN, or alternately using the Microsoft Excel spreadsheet.

(1)	Input the following constants or parameters: ℓm, KI, Av, S0, S1, and Pc.
(2)	Input a block of data consisting of the primary drying time, product temperature, and sublimation rate, as expressed by [Time, Tb, ]. The product temperature profile Tb is from the recorded data file of the LyoStar II dryer. The sublimation profile is downloaded from the TDLAS. The following steps (3–9) are performed for each data point in step (2).
(3)	Calculate Ti using Equation (B8).
(4)	Calculate P0 using Equation (B3).
(5)	Calculate Rs using Equation (B13).
(6)	Calculate ΔMt using Equation (B9).
(7)	Calculate Δℓ using Equation (B10).
(8)	Calculate ℓ using Equation (B11).
(9)	Calculate the accumulated mass using Equation (B12).
(10)	Calculate the profile using Equations (B2) and (B4), excluding the shelf temperature ramping period since thermal equilibrium may have not reached, by which a block of data for ∼ ℓ are obtained.
(11)	Select representative values of ∼ ℓ from the results in Step (10) for non-linear parameter estimation, since the data points obtained in this step are too numerous, more than 2000 for each of the cycle runs in Table 1. This was performed using the VLOOKUP function (or macro) in Microsoft Excel®, which can be found in the pull-down help menu of an Excel® spreadsheet. The values of b0 and b1 are then obtained by fitting Equation (4) to the experimental values of ∼ ℓ. This can be done using a nonlinear least-squares algorithm, such as the Marquart-Leverburg compromise or Powell’s SSQMIN algorithm.[3,15]