Method for monitoring and/or controlling a freeze drying process in a freeze dryer apparatus provided with a drying chamber having a temperature-controlled shelf supporting containers of a product to be dried, said drying chamber being connected to a condenser chamber, comprising during a primary drying phase of said freeze drying process the steps of:
- isolating, for a time period, said drying chamber from said condenser chamber by closing an isolating valve thereof and sensing and collecting pressure values (pc,mes) inside said drying chamber for a pressure collecting time (tf) and a shelf temperature (Tshelf) of said temperature-controlled shelf (Step 1) (Pressure rise test);
- calculating a product temperature (T) of product and a plurality of process/product related parameters (Step 2), said calculating comprising calculating:
- product temperature (Ti0) at a sublimation interface of product;
- mass transfer resistance (Rp) in a dried portion of product;
- product temperature (T=T(z,t)) at an axial coordinate (z) and at a time (t) during said pressure collecting time (tf);
- heat transfer coefficient (Kv) between said temperature-controlled shelf and said container;
- thickness (Lfrozen) of a frozen portion of product;
- mass flow in the drying chamber;
- remaining primary drying time;
- calculating a new shelf temperature using said calculated product temperature and said calculated process/product related parameters (Step 3) and adjusting a temperature of said temperature-controlled shelf on the basis of said new shelf temperature;
- wherein said calculating said product temperature and said plurality of process/product related parameters is made by means of an estimator algorithm, which implements an unsteady state model for mass transfer in said drying chamber and for heat transfer in the product, and comprises the following equations:
- ∂
- T
- ∂
- t
- =
- k
- frozen
- ϱ
- frozen
-
- c
- P
- ,
- frozen
-
- ∂
- 2
-
- T
- ∂
- z
- 2
-
-
- for
-
-
- t
- >
- t
- 0
- ,
- 0
- <
- z
- <
- L
- frozen
- (
- eq
- .
-
- 1
- )
- T
-
- t
- =
- 0
- =
- T
- i
-
-
- 0
- +
- z
- k
- frozen
-
- Δ
-
-
- H
- s
- R
- P
-
- (
- p
-
- (
- T
- i
-
-
- 0
- )
- -
- p
- w
-
-
- 0
- )
-
-
- I
- .
- C
- .
- :
-
-
- t
- =
- 0
- ,
- 0
- <
- z
- <
- L
- frozen
- (
- eq
- .
-
- 2
- )
- k
- frozen
-
- ∂
- T
- ∂
- z
-
- z
- =
- 0
- =
- Δ
-
-
- H
- s
- R
- P
-
- (
- p
-
- (
- T
- i
-
- )
- -
- p
- w
- )
- B
- .
- C
-
- .1
-
- :
-
-
- t
- ≥
- 0
- ,
- z
- =
- 0
- (
- eq
- .
-
- 3
- )
- k
- frozen
-
- ∂
- T
- ∂
- z
-
- z
- =
- L
- =
- K
- v
-
- (
- T
- shelf
- -
- T
- B
- )
- B
- .
- C
-
- .2
-
- :
-
-
- t
- ≥
- 0
- ,
- z
- =
- L
- frozen
- (
- eq
- .
-
- 4
- )
- K
- v
- =
- [
- T
- shelf
- -
- T
- i
-
-
- 0
- Δ
-
-
- H
- s
- R
- P
-
- (
- p
-
- (
- T
- i
-
-
- 0
- )
- -
- p
- w
-
-
- 0
- )
- +
- L
- frozen
- k
- frozen
- ]
- -
- 1
- (
- eq
- .
-
- 5
- )
- T
- B
-
-
- 0
- =
- T
- i
-
-
- 0
- +
- L
- frozen
- k
- frozen
-
- Δ
-
-
- H
- s
- R
- P
-
- (
- p
-
- (
- T
- i
-
-
- 0
- )
- -
- p
- w
-
-
- 0
- )
- (
- eq
- .
-
- 6
- )
- ⅆ
- p
- w
- ⅆ
- t
- =
- N
- v
-
- A
- V
- c
-
- RT
- i
- M
- w
-
- 1
- R
- P
-
- (
- p
- i
-
- (
- T
- i
- )
- -
- p
- w
- )
- for
-
-
- t
- >
- 0
- (
- eq
- .
-
- 7
- )
- p
- c
- =
- p
- w
- +
- p
- in
- =
- p
- w
- +
- F
- leak
- ·
- t
- +
- p
- in
-
-
- 0
- for
-
-
- t
- ≥
- 0
- (
- eq
- .
-
- 8
- )
- p
- w
-
- ❘
- t
- =
- 0
- =
- p
- c
-
-
- 0
- -
- p
- in
-
-
- 0
- I
- .
- C
- .
- :
-
-
- t
- =
- 0
- (
- eq
- .
-
- 9
- )
- ϱ
- frozen
-
- AL
- frozen
- ,
- n
- +
- ϱ
- dried
-
- A
-
- (
- L
- -
- L
- frozen
- ,
- n
- )
- =
- ϱ
- frozen
-
- AL
- frozen
- ,
- n
- -
- 1
- -
- K
- v
-
- A
- Δ
-
-
- H
- s
-
- (
- T
- shelf
- -
- T
- B
-
-
- 0
- )
- ×
- Δ
-
-
- t
- n
- -
- 1
-
-
- where
-
-
-
- T
- =
- T
-
- (
- z
- ,
- t
- )
- ,
- T
- i
- =
- T
-
- (
- t
- )
-
- z
- =
- 0
- ,
- T
- B
- =
- T
-
- (
- t
- )
-
- z
- =
- L
- ,
- T
- i
-
-
- 0
- =
- T
-
- z
- =
- 0
- ,
- t
- =
- 0
- ;
- (
- eq
- .
-
- 10
- )
- and the parameters in the equations are:
- A
- internal cross surface of the container [m2]
- cP
- specific heat at constant pressure [J kg−1 K−1]
- cP,frozen
- frozen layer specific heat at constant
- pressure [J kg−1 K−1]
- Fleak
- leakage rate [Pa s−1]
- k
- thermal conductivity [J m s−1 K]
- Kv
- overall heat transfer coefficient [J m−2 s−1 K]
- kfrozen
- frozen layer thermal conductivity [J m s−1 K]
- L
- total product thickness [m]
- Lfrozen
- frozen layer thickness [m]
- M
- molecular weight [kmol kg−1]
- Mw
- water vapour molecular weight [kmol kg−1]
- Nv
- number of containers
- p
- pressure [Pa]
- R
- ideal gas Constant [J kmol−1 K]
- Rp
- mass transfer resistance in the dried layer [m−1 s]
- T
- Temperature [K]
- t
- time [s]
- TB
- frozen layer temperature at z = L [K]
- V
- Volume [m3]
- Vc
- drying chamber volume [m3]
- z
- axial coordinate [m]
- ρ
- mass density [kg m−3]
- ΔHs
- enthalpy of sublimation [J kg−1]
- ρfrozen
- mass density of frozen layer [kg m−3]
- ρdried
- mass density of dried layer [kg m−3]
- Lfrozen,n
- frozen layer thickness to be calculated in the
- actual pressure rise test [m]
- Lfrozen,n−1
- frozen layer thickness calculated in a previous
- pressure rise test [m]
- pc
- drying chamber pressure [Pa]
- pc0
- drying chamber pressure at t = 0 [Pa]
- pw
- water vapour pressure [Pa]
- pw0
- water vapour pressure at t = 0 [Pa]
- ppin
- inert gas pressure [Pa]
- pin0
- inert gas pressure at t = 0 [Pa]
- TB0
- frozen layer temperature at z = L and t = 0 [K]
- Δtn−1
- total time passed between the actual and the
- preceding pressure rise test
- Tshelf
- temperature of temperature-controlled shelf
- the subscripts and superscripts in the equations are:
- 0
- value at z = 0
- frozen
- frozen layer
- dried
- dried layer
- c
- chamber
- i
- interface
- in
- inert gas
- mes
- measured
- shelf
- heating shelf
- w
- water vapour
- [t0, tf] is the interval of Step 1;
- I.C. are initial conditions, B.C are boundary conditions.