Method for monitoring the secondary drying in a freeze-drying process
A method to monitor a secondary drying phase of a freeze-drying process comprises initial steps in which is provided to perform pressure rise tests at different time and to calculate a respective value of experimental desorption rate of product (steps 1 to 3). Subsequently, the method provides to estimate initial conditions and kinetic constants of a kinetic model of the process (step 4) and to calculate at time t=t 2 a respective residual moisture content and a respective desorption rate (step 5). The method can be performed in a freeze-dryer apparatus which includes a drying chamber that contains a product to be dried and can be isolated to perform pressure rise tests.
1. Method for calculating a residual moisture concentration and/or desorption rate of a product during a secondary drying phase of a freeze-drying process in a freeze-dryer apparatus including a drying chamber and a condenser chamber where vapour generated by drying process flows, said apparatus being provided with a pressure sensor that can be isolated for performing pressure rise tests and measuring a total pressure inside said drying chamber, said method comprising the steps of:
a) setting a desired final residue moisture concentration and/or a desired final desorption rate of said product;
b) measuring initial residual moisture concentration and/or desorption rate and estimating kinetic constants of a kinetic model of the drying process, said kinetic model being suitable for calculating the residual moisture concentration and/or desorption rate of said product;
c) closing a valve placed on a duct connecting said drying chamber to said condenser chamber for a preselected period of time;
d) measuring a pressure change in said drying chamber;
e) calculating a desorption rate from the closing time period of step c) and the pressure change of step d);
f) repeating steps c)-e) to calculate residual moisture concentrations and/or desorption rates at pre-specified time intervals;
g) integrating the calculated residual moisture concentrations and/or desorption rate from said initial conditions to the calculated residual moisture concentrations and/or desorption rates of the pre-specified time intervals of step e) to determine a current residual moisture concentration of the drying product;
wherein if said current residual moisture concentration and/or said current final desorption rate is lower than or equal to said desired residual moisture concentration and/or to said desired desorption rate then said secondary drying phase ends, and wherein
a first pressure rise test at time t=t 0 at the beginning of the secondary drying phase is performed and a first value of experimental desorption rate (DR exp, 0 ) of said product is calculated using the equation:
D
R
exp
=
VM
w
RT
(
d
P
d
t
)
t
=
t
0
100
m
dried
(
eq
.
4
)
where:
DR exp : experimental desorption rate, [% water and/or solvent s −1 ]
P: measured pressure, [Pa]
t: time, [s]
t 0 : time instant at the beginning of the pressure rise test, [s]
R: gas constant [8,314 J mol −1 K −1 ]
T: temperature of the vapour, [K]
V: (free) volume of drying chamber, [m 3 ]
M w : molecular weight of water and/or solvent, [kg mol −1 ]
m dried : mass of the dried product, [kg]
a second pressure rise test at a successive time t=t 1 is performed and a second value of experimental desorption rate (DR exp, 1 ) of said product is calculated using the equation:
DR
exp
=
VM
W
RT
(
dP
dt
)
t
=
t
0
100
m
dried
;
(
eq
.
4
)
a third pressure rise test at time t=t 2 is performed and a third value of experimental desorption rate (DR exp, 2 ) of said product is calculated using the equation eq. 4
DR
exp
=
VM
W
RT
(
dP
dt
)
t
=
t
0
100
m
dried
;
(
eq
.
4
)
and wherein
said initial conditions (C S,0 ) and kinetic constants (k 0 , k 1 , k 2 ) of a kinetic model of the drying process, said kinetic model being suitable for calculating a residual moisture content (C S ) and/or desorption rate (DR theor ) of said product are calculated using the equations:
DR exp,0 =DR theor,0 =−k 0 C S,0 (eq. 13)
DR exp,1 =DR theor,1 =−k 1 C S,0 e −k 1 (t 1 −t 0 ) (eq. 14)
DR exp,2 =DR theor,2 =−k 2 C s,0 e −k 1 (t 1 −t 0 ) e −k 2 (t 2 −t 1 ) (eq. 15)
where:
DR exp,j : experimental desorption rate at time tj, [% water and/or solvent s −1 ]
DR theor,j : desorption rate (theoretical value) at time tj [% water and/or solvent s −1 ]
C S,0 : value of the residual moisture [% water and/or solvent over dried product] at the beginning of the secondary drying phase (t=t 0 );
k j : kinetic constant of the process at time t=t j (with j=0, 1, 2), [s −1 ];
and a minimisation algorithm to solve the minimum least square problem described by equation:
min
C
S
,
0
,
k
i
∑
i
=
0
2
(
DR
exp
,
i
-
DR
theor
,
i
)
2
(
eq
.
16
)
calculating at time t=t 2 a respective residual moisture content (C S,2 ) and a respective desorption rate (DR theor, 2 ) (step 5) respectively using the equations:
C
S
=
C
S
,
0
∏
i
=
1
j
-
1
e
-
k
i
(
t
i
-
t
i
-
1
)
e
-
k
j
(
t
-
t
j
-
1
)
(
eq
.
11
)
DR
theor
=
-
k
j
C
S
,
0
∏
i
=
1
j
-
1
e
-
k
i
(
t
i
-
t
i
-
1
)
e
-
k
j
(
t
-
t
j
-
1
)
.
(
eq
.
12
)
2. Method according to claim 1 , further comprising after step 5 the steps of:
comparing said residual moisture content (C S,2 ) and/or said desorption rate (DR theor,2 ) calculated at time t=t 2 respectively with a desired final residual moisture concentration (C S,f ) and/or a desired final desorption rate (DR f ) (step 6); if said residual moisture content (C S,2 ) is lower than, or equal to, said final residual moisture concentration (C S,f ) or said desorption rate (DR theor,2 ) is lower than, or equal to, said final desorption rate (DR f ), then the secondary drying phase is considered ended; if not the method further comprising the steps of:
estimating a final time (t f ) at which said final residual moisture concentration (C S,f ) or said final desorption rate (DR f ) is obtained (step 7);
performing a further pressure rise test at time t=t j and calculating at said time t=t j a respective residual moisture content (C S,j ) and a respective desorption rate (DR theor,j ) (step 8);
estimating initial conditions (C S,0 ) and kinetic constants (k 0 , k 1 , k 2 , . . . , k j ) of said kinetic model (step 9);
calculating at said time t=t j a respective residual moisture content (C S,j ) and/or a respective desorption rate (DR theor,j ) (step 10);
comparing said residual moisture content (C S,j ) and/or said desorption rate (DR theor,j ) calculated at said time t=t j respectively with said final residual moisture concentration (C S,f ) and/or said final desorption rate (DR f ) (step 11); if said residual moisture content (C S,j ) is lower than, or equal to, said final residual moisture concentration (C S,f ) or said desorption rate (DR theor,j ) is lower than, or equal to, said final desorption rate (DR f ) then the secondary drying phase is considered ended; if not steps 7 to 11 are repeated.
3. Method according to claim 2 , wherein said experimental desorption rates (DR exp,O , DR exp,1 , DR exp,2 ) are calculated using the equation:
DR
exp
=
VM
W
RT
(
dP
dt
)
t
=
t
0
100
m
dried
(
eq
.
4
)
where:
DR exp : experimental desorption rate, [% water and/or solvent s −1 ]
P: measured pressure, [Pa]
t: time, [s]
t 0 : time instant at the beginning of the pressure rise test, [s]
R: gas constant [8,314 J mol −1 K −1 ]
T: temperature of the vapour, [K]
V: (free) volume of drying chamber, [m 3 ]
M w : molecular weight of water and/or solvent, [kg mol −1 ]
m dried : mass of the dried product, [kg].
4. Method according to claim 3 , wherein said kinetic model comprises mathematical equations suitable for modeling the dependence of the desorption rate (DR) on the residual moisture content (C S ) in the product.
5. Method according to claim 3 , wherein said desorption rate is assumed to depend on said residual moisture content in said product according to the equation:
DR=− kC S (eq. 5)
where:
DR: desorption rate, [% water and/or solvent s −1 ]
k: kinetic constant of the process, [s −1 ]
C S : residual moisture content, [% water/solvent over dried product].
6. Method according to claim 5 , wherein a time evolution of said residual moisture concentration (C S ) at time t=t j is given by the integration of the following differential equation:
d
C
S
d
t
=
D
R
j
=
-
k
j
C
S
(
eq
.
7
)
where:
DR j : desorption rate at time t=t j , [% water and/or solvent s −1 ]
t: time, [s]
k j : kinetic constant of the process at time t=t j , [s −1 ].
7. Method according to claim 6 , wherein said calculating a residual moisture content (C s ) is made by means of the equation:
C
S
=
C
S
,
0
∏
i
=
1
j
-
1
e
-
k
i
(
t
i
-
t
i
-
1
)
e
-
k
j
(
t
-
t
j
-
1
)
(
eq
.
11
)
where:
C S,0 : value of the residual moisture [% water and/or solvent over dried product] at the beginning of the secondary drying phase (t=t 0 );
k r : kinetic constant of the process at time t=t r (with r=1, 2, . . . , j), [s −1 ].
8. Method according to claim 7 , wherein said calculating a desorption rate (DR theor ) is made by means of the equation:
D
R
theor
=
-
k
j
C
S
,
0
∏
i
=
1
j
-
1
e
-
k
i
(
t
i
-
t
i
-
1
)
e
-
k
j
(
t
-
t
j
-
1
)
.
(
eq
.
12
)
9. Method according to claim 8 , wherein said estimating initial conditions (C S,0 ) and kinetic constants (k 0 , k 1 , k 2 , . . . , k j ), at time t=t j , is made by means of the following equations:
D
R
exp
,
0
=
D
R
theor
,
0
=
-
k
0
C
S
,
0
(
eq
.
13
)
D
R
exp
,
1
=
D
R
theor
,
1
=
-
k
1
C
S
,
0
e
-
k
1
(
t
1
-
t
0
)
(
eq
.
14
)
D
R
exp
,
2
=
D
R
theor
,
2
=
-
k
2
C
s
,
0
e
-
k
1
(
t
1
-
t
0
)
e
-
k
2
(
t
2
-
t
1
)
(
eq
.
15
)
D
R
exp
,
j
=
D
R
theor
,
j
=
=
-
k
j
C
S
,
0
∏
i
=
1
j
-
1
e
-
k
i
(
t
i
-
t
i
-
1
)
e
-
k
j
(
t
j
-
t
j
-
1
)
(
eq
.
15
bis
)
and solving the following non-linear least square problem:
min
C
S
,
0
,
k
i
∑
i
=
0
j
(
D
R
exp
,
i
-
D
R
theor
,
i
)
2
.
(
eq
.
16
bis
)
10. Method according to claim 7 , wherein said final time (t f ) is calculated, assuming that temperature of said product does not change, by means of the following equation, resulted from (eq. 11):
t
f
=
t
j
-
1
k
j
ln
(
C
S
,
f
C
S
,
j
)
(
eq
.
18
bis
)
where:
C S,f : final residual moisture concentration [% water and/or solvent over dried product];
C S,j : residual moisture concentration at time t=t j [% water and/or solvent over dried product].
11. Method for calculating a residual moisture concentration and/or desorption rate of a product during a secondary drying phase of a freeze-drying process in a freeze-dryer apparatus including a drying chamber and a condenser chamber where vapour generated by drying process flows, said apparatus being provided with a pressure sensor that can be isolated for performing pressure rise tests and measuring a total pressure inside said drying chamber, said method comprising the steps of:
a) setting a desired final residue moisture concentration and/or a desired final desorption rate of said product;
b) measuring initial residual moisture concentration and/or desorption rate and estimating kinetic constants of a kinetic model of the drying process, said kinetic model being suitable for calculating the residual moisture concentration and/or desorption rate of said product;
c) closing a valve placed on a duct connecting said drying chamber to said condenser chamber for a preselected period of time;
d) measuring a pressure change in said drying chamber;
e) calculating a desorption rate from the closing time period of step c) and the pressure change of step d);
f) repeating steps c)-e) to calculate residual moisture concentration and/or desorption rate at pre-specified time intervals;
g) integrating the calculated residual moisture concentration and/or desorption rate from said initial conditions to the calculated residual moisture concentration and/or desorption rates of the pre-specified time intervals of step e) to determine a current residual moisture of the drying product;
wherein if said current residual moisture concentration and/or said current final desorption rate is lower than or equal to said desired residual moisture concentration and/or said desired desorption rate then said secondary drying phase ends, and wherein
a first pressure rise test at time t=t 0 at the beginning of the secondary drying phase is performed and a first value of experimental desorption rate (DR exp, 0 ) of said product is calculated using the equation:
DR
exp
=
VM
W
RT
(
dP
dt
)
t
=
t
0
100
m
dried
(
eq
.
4
)
where:
DR exp : experimental desorption rate, [% water and/or solvent s −1 ]
P: measured pressure, [Pa]
t: time, [s]
t 0 : time instant at the beginning of the pressure rise test, [s]
R: gas constant [8,314 J mol −1 K −1 ]
T: temperature of the vapour, [K]
V: (free) volume of drying chamber, [m 3 ]
M w : molecular weight of water and/or solvent, [kg mol −1 ]
m dried : mass of the dried product, [kg]
a second pressure rise test at a successive time t=t 1 is performed and a second value of experimental desorption rate (DR exp, 1 ) of said product is calculated using the equation:
DR
exp
=
VM
W
RT
(
dP
dt
)
t
=
t
0
100
m
dried
;
(
eq
.
4
)
a third pressure rise test at time t=t 2 is performed and a third value of experimental desorption rate (DR exp, 2 ) of said product is calculated using the equation eq. 4
DR
exp
=
VM
W
RT
(
dP
dt
)
t
=
t
0
100
m
dried
;
(
eq
.
4
)
and wherein
said initial conditions (C S,0 ) and kinetic constants (k 0 , k 1 , k 2 ) of a kinetic model of the drying process, said kinetic model being suitable for calculating a residual moisture content (C S ) and/or desorption rate (DR theor ) of said product are calculated using the equations:
DR exp,0 =DR theor,0 =−k 0 ( C S,0 −C S,eq,0 ) (eq. 27)
DR exp,1 =DR theor,1 =−k 1 {C S,0 e −k 1 (t 1 −t 0 ) ++k 1 C S,eq,1 [ t 1 −t 0 e −k 1 (t 1 −t 0 ) ]− C S,eq,1 } (eq. 28)
DR exp,2 =DR theor,2 =−k 2 {C S,1 e −k 2 (t 2 −t 1 ) ++k 2 C S,eq,2 [ t 2 −t 1 e −k 2 (t 2 −t 1 ) ]− C S,eq,2 } (eq. 29)
where
DR exp,j : experimental desorption rate at time tj, [% water and/or solvent s −1 ]
C s,eq,j equilibrium moisture concentration at time tj [% water and/or solvent s −1 ]:
C S,0 : value of the residual moisture [% water and/or solvent over dried product] at the beginning of the secondary drying phase (t=t 0 );
k j : kinetic constant of the process at time t=t j (with j=0, 1, 2), [s −1 ]. and a minimisation algorithm to solve the minimum least square problem described by equation:
min
C
S
,
0
,
k
i
∑
i
=
0
2
(
DR
exp
,
i
-
DR
theor
,
i
)
2
(
eq
.
30
)
calculating at time t=t 2 a respective residual moisture content (C S,2 ) and a respective desorption rate (DR theor, 2 ) (step 5) using for calculating the residual moisture content (C S,2 ) the equations:
C S =C S,j−1 e −k j (t−t j−1 ) ++k j C S,eq,j [ t−t j−1 e −k j (t−t j−1 ) ] (eq. 21)
C S,j−1 =C S,j−2 e −k j−1 (t j−1 −t j−2 ) ++k j−1 C S,eq,j−1 [ t j−1 −t j−2 e −k j−1 (t j−1 −t j−2 ) ] (eq. 22)
C S,j−2 =C S,j−3 e −k j−2 (t j−2 −t j−3 ) ++k j−2 C S,eq,j−2 [ t j−2 −t j−3 e −k j−2 (t j−2 −t j−3 ) ] (eq. 24)
C S,1 =C S,0 e −k 1 (t 1 −t 0 ) +k 1 C S,eq,1 [ t 1 −t 0 e −k 1 (t 1 −t 0 ) ] (eq. 25)
where:
C S,0 : value of the residual moisture [% water and/or solvent over dried product] at the beginning of the secondary drying phase (t=t 0 );
k r : kinetic constant of the process at time t=t r (with r=1, 2, . . . , j), [s −1 ];
C s,eq,r : equilibrium moisture concentration at time t=t r with r=1, 2, . . . , j), [% water and/or solvent over dried product];
and using for calculating the respective desorption rate (DR theor, 2 ) the equation:
DR theor =−k j {C S j−1 e −k j (t−t j−1 ) ++k j C S,eq,j [ t−t j−1 e −k j (t−t j−1 ) ]− C S,eq,j } (eq. 26).
12. Method according to claim 11 , wherein said desorption rate (DR theor ) is assumed to depend on said residual moisture content (C S ) in said product according to the equation:
DR=− k ( C S −C S,eq ) (eq. 19)
where:
DR: desorption rate, [% water and/or solvent s −1 ]
k: kinetic constant of the process, [s −1 ]
C S : residual moisture concentration, [% water and/or solvent over dried product]
C s,eq : equilibrium moisture concentration, [% water and/or solvent over dried product].
13. Method according to claim 12 , wherein a time evolution of said residual moisture concentration (C S ) at time t=t j is given by the integration of the following differential equation:
d
C
S
d
t
=
D
R
j
=
-
k
j
(
C
S
-
C
S
,
eq
,
j
)
(
eq
.
20
)
where:
DR j : desorption rate at time t=t j , [% water and/or solvent s −1 ]
t: time, [s]
k 1 : kinetic constant of the process, [s −1 ],
C s,eq,j : equilibrium moisture concentration at time t=t j , [% water and/or solvent over dried product].
14. Method according to claim 13 , wherein said calculating a residual moisture content (C S ) at time t=t j is made by means of the following equations:
C S =C S,j−1 e −k j (t−t j−1 ) +—k j C S,eq,j [ t−t j−1 e −k j (t−t j−1 ) ] (eq. 21)
and
C S,j−1 =C S,j−2 e −k j−1 (t j−1 −t j−2 ) ++k j−1 C S,eq,j−1 [ t j−1 −t j−2 e −k j−1 (t j−1 −t j−2 ) ] (eq. 22)
C S,j−2 =C S,j−3 e −k j−2 (t j−2 −t j−3 ) ++k j−2 C S,eq,j−2 [ t j−2 −t j−3 e −k j−2 (t j−2 −t j−3 ) ] (eq. 24)
C S,1 =C S,0 e −k 1 (t 1 −t 0 ) +k 1 C S,eq,1 [ t 1 −t 0 e −k 1 (t 1 −t 0 ) ] (eq. 25)
where:
C S,0 : value of the residual moisture [% water and/or solvent over dried product] at the beginning of the secondary drying phase (t=t 0 );
k r : kinetic constant of the process at time t=t r (with r=1, 2, . . . , j), [s −1 ];
C s,eq,r : equilibrium moisture concentration at time t=t r with r=1, 2, . . . , j), [% water and/or solvent over dried product].
15. Method according to claim 14 , wherein said calculating a desorption rate (DR theor ) is made by means of the equation:
DR theor =−k j {C S,j−1 e −k j (t−t j−1 ) ++k j C S,eq,j [ t−t j−1 e −k j (t−t j−1 ) ]− C S,eq,j } (eq. 26)
16. Method according to claim 15 , wherein said experimental desorption rates (DR exp,0 , DR exp,1 , DR exp,2 ) are calculated using the equation:
D
R
exp
=
VM
w
RT
(
d
P
d
t
)
t
=
t
0
100
m
dried
(
eq
.
4
)
where:
DR exp : experimental desorption rate, [% water and/or solvent s −1 ]
P: measured pressure, [Pa]
t: time, [s]
t 0 : time instant at the beginning of the pressure rise test, [s]
R: gas constant [8,314 J mol −1 K −1 ]
T: temperature of the vapour, [K]
V: (free) volume of drying chamber, [m 3 ]
M w : molecular weight of water and/or solvent, [kg mol −1 ]
m dried : mass of the dried product, [kg]
and wherein said estimating initial conditions (C S,0 ) and kinetic constants (k 0 , k 1 , k 2 , . . . , k j ), at time t=t j , is made by means of the following equations:
DR exp,0 =DR theor,0 =−k 0 ( C S,0 −C S,eq,0 ) (eq. 27)
DR exp,1 =DR theor,1 =−k 1 {C S,0 e −k 1 (t 1 −t 0 ) ++k 1 C S,eq,1 [ t 1 −t 0 e −k 1 (t 1 −t 0 ) ]− C S,eq,1 } (eq. 28)
DR exp,2 =DR theor,2 =−k 2 {C S,1 e −k 2 (t 2 −t 1 ) ++k 2 C S,eq,2 [ t 2 −t 1 e −k 2 (t 2 −t 1 ) ]− C S,eq,2 } (eq. 29)
DR exp,j =DR theor,j =−k j {C S j−1 e −k j (t j −t j−1 ) ++k j C S,eq,j [ t j −t j−1 e −k j (t j −t j−1 ) ]− C S,eq,j }(eq. 29ter)
and solving the following non-linear least square problem:
min
C
S
,
0
,
k
i
∑
i
=
0
j
(
D
R
exp
,
i
-
D
R
theor
,
i
)
2
.
(
eq
.
30
bis
)
17. Method according to claim 16 , wherein said final time (t f ) is calculated, assuming that a temperature of said product does not change, by means of the following equation, resulted from (eq. 21):
C S,f =C S,j e −k j (t f −t j ) ++k j C S,eq,j [ t f −t j e −k j (t f −t j ) ] (eq. 31bis)
where:
C S,f : final residual moisture concentration [% water and/or solvent over dried product];
C S,j : residual moisture concentration at time t=t j [% water and/or solvent over dried product].