Method and system for controlling a freeze drying process
A method for monitoring and/or controlling and 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, comprises during a primary drying phase of the freeze drying process the steps of: isolating the drying chamber closing an isolating valve thereof and sensing and collecting pressure values inside the drying chamber for a defined pressure collecting time and a shelf temperature of the temperature-controlled shelf (Step 1 ); calculating a product temperature of product and a plurality of process/product related parameters (Step 2 ); calculating a new shelf temperature and a sequence of shelf temperatures up to the end of the primary drying phase, that maximizes a sublimation rate of the product maintaining the product temperature below a maximum allowable product temperature.
1. 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 (p c,mes ) inside said drying chamber for a pressure collecting time (t f ) and a shelf temperature (T shelf ) 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 (T i0 ) at a sublimation interface of product;
mass transfer resistance (R p ) 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 (t f );
heat transfer coefficient (K v ) between said temperature-controlled shelf and said container;
thickness (L frozen ) 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 [m 2 ]
c P
specific heat at constant pressure [J kg −1 K −1 ]
c P,frozen
frozen layer specific heat at constant
pressure [J kg −1 K −1 ]
F leak
leakage rate [Pa s −1 ]
k
thermal conductivity [J m s −1 K]
K v
overall heat transfer coefficient [J m −2 s −1 K]
k frozen
frozen layer thermal conductivity [J m s −1 K]
L
total product thickness [m]
L frozen
frozen layer thickness [m]
M
molecular weight [kmol kg −1 ]
M w
water vapour molecular weight [kmol kg −1 ]
N v
number of containers
p
pressure [Pa]
R
ideal gas Constant [J kmol −1 K]
R p
mass transfer resistance in the dried layer [m −1 s]
T
Temperature [K]
t
time [s]
T B
frozen layer temperature at z = L [K]
V
Volume [m 3 ]
V c
drying chamber volume [m 3 ]
z
axial coordinate [m]
ρ
mass density [kg m −3 ]
ΔH s
enthalpy of sublimation [J kg −1 ]
ρ frozen
mass density of frozen layer [kg m −3 ]
ρ dried
mass density of dried layer [kg m −3 ]
L frozen,n
frozen layer thickness to be calculated in the
actual pressure rise test [m]
L frozen,n−1
frozen layer thickness calculated in a previous
pressure rise test [m]
p c
drying chamber pressure [Pa]
p c0
drying chamber pressure at t = 0 [Pa]
p w
water vapour pressure [Pa]
p w0
water vapour pressure at t = 0 [Pa]
p pin
inert gas pressure [Pa]
p in0
inert gas pressure at t = 0 [Pa]
T B0
frozen layer temperature at z = L and t = 0 [K]
Δt n−1
total time passed between the actual and the
preceding pressure rise test
T shelf
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
[t 0 , t f ] is the interval of Step 1 ;
I.C. are initial conditions, B.C are boundary conditions.
2. Method according to claim 1 , wherein calculating said product temperature and said plurality of process/product related parameters comprises the following step:
assigning values to T i0 , R p parameters (Step 11 );
calculating values of T BO , K v , L frozen parameters respectively by means of equations (eq. 6), (eq. 5), (eq. 10) (Step 12 );
calculating an initial temperature T| t=0 of frozen product by means of equation (eq. 2) (Step 13 );
integrating the equation (eq. 1) in said interval [t 0 , t f ] of Step 1 (Step 14 );
repeating step 12 to 14 up to solve a non-linear least square problem:
min
T
i
0
,
R
P
1
2
p
c
(
T
i
0
,
R
P
)
-
p
c
,
mes
2
2
=
1
2
∑
j
(
p
c
(
T
i
0
,
R
P
)
j
-
(
p
c
,
mes
)
j
)
2
(
eq
.
11
)
so as to determine values of T i0 , R p that fit a simulated drying chamber pressure (p c (T i0 ,R p )) to said pressure values (p c,mes )
where
j index of summation of capital-sigma notation;
calculating said product temperature (T=T(z,t)).
3. Method according to claim 1 , wherein said estimator algorithm further comprises a correction coefficient (f) that takes into account heterogeneity of a batch of said containers, said correction coefficient (f) being defined by the equation:
f
=
∑
j
=
1
N
V
[
A
j
k
1
,
j
L
-
L
frozen
,
j
(
p
i
(
T
i
,
j
)
-
p
w
)
]
N
V
A
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
7
C
)
Where
k 1 effective diffusivity coefficient [m 2 s −1 ]
j index of summation of capital-sigma notation.
4. Method according to claim 3 , wherein calculating said product temperature and said plurality of process/product related parameters comprises the following step:
assigning values to T i0 , R p parameters (Step 11 );
calculating values of T BO , K v , L frozen parameters respectively by means of equations (eq. 6), (eq. 5), (eq. 10) (Step 12 );
calculating an initial temperature T|t=0 of frozen product by means of equation (eq. 2) (Step 13 );
integrating the equation (eq. 1) in said interval [t 0 , t f ] of Step 1 (Step 14 );
repeating step 12 to 14 up to solve a non-linear least square problem:
min
T
i
0
,
R
P
1
2
p
c
(
T
i
0
,
R
P
)
-
p
c
,
mes
2
2
=
1
2
∑
j
(
p
c
(
T
i
0
,
R
P
)
j
-
(
p
c
,
mes
)
j
)
2
(
eq
.
11
)
so as to determine values of T i0 , R p that fit a simulated drying chamber pressure (p c (T i0 ,R p )) to said pressure values (p c,mes );
calculating said product temperature (T=T(z,t));
and said correction coefficient (f) is inserted in equations (eq. 7, eq. 11) of estimator algorithm which are modified in:
ⅆ
p
w
ⅆ
t
=
f
N
v
A
V
c
RT
i
M
w
1
R
P
(
p
i
(
T
i
)
-
p
w
)
for
t
>
0
(
eq
.
7
B
)
min
T
i
0
,
R
P
,
f
1
2
p
c
(
T
i
0
,
R
P
,
f
)
-
p
c
,
mes
2
2
=
1
2
∑
j
(
p
c
(
T
i
0
,
R
P
,
f
)
j
-
(
p
c
,
mes
)
j
)
2
(
eq
.
11
B
)
where
j index of summation of capital-sigma notation.
5. Method according to claim 1 , comprising repeating said Step 1 and Step 2 (at intervals of 30 minutes).
6. Method according to claim 1 , wherein said calculating said new shelf temperature comprises calculating a new shelf temperature and a sequence of shelf temperatures up to the end of the primary drying phase, that maximises a sublimation rate of said product maintaining the product temperature below a maximum allowable product temperature (Step 3 ).
7. Method according to claim 6 , wherein said new shelf temperature and said sequence of shelf temperatures is such as to drive the product to a desired target temperature.
8. Method according to claim 7 , wherein said desired target temperature is lower than said maximum allowable product temperature by an amount ranging from 1 to 3° C.
9. Method according to claim 6 , comprising repeating said Steps 1 to 3 (at intervals of 30 minutes).
10. Method according to claim 6 , wherein said calculating said new shelf temperature and/or said sequence of shelf temperatures is made by means of a control algorithm, based on a numerical code, which implements a non-stationary mathematical model of containers and of freeze dryer apparatus and an optimization algorithm which uses as inputs said product temperature and said plurality of process/product related parameters calculated in a previous step (step 2 ).
11. Method according to claim 10 , wherein said control algorithm comprises a PID type controller for controlling a product temperature and for minimizing an energy consumption during said primary drying phase.
12. Method according to claim 10 , where said control algorithm comprises the following equations:
ⅆ
L
frozen
ⅆ
t
=
-
1
n
~
II
-
n
~
Ie
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
12
)
k
1
=
RT
i
M
L
-
L
frozen
R
P
(
eq
.
13
)
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
=
Δ
H
s
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
14
)
T
B
=
T
shelf
-
1
K
v
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
(
eq
.
15
)
T
SP
(
t
)
:
{
T
SP
,
1
=
T
shelf
(
t
0
)
+
K
OPT
(
T
B
(
t
0
)
-
T
B
,
SP
)
t
0
≤
t
<
t
1
T
SP
,
2
=
T
shelf
(
t
1
)
+
K
OPT
(
T
B
(
t
1
)
-
T
B
,
SP
)
t
1
≤
t
<
t
2
⋮
T
SP
,
N
=
T
shelf
(
t
N
-
1
)
+
K
OPT
(
T
B
(
t
N
-
1
)
-
T
B
,
SP
)
t
N
-
1
≤
t
<
t
N
(
eq
.
16
A
)
min
K
OPT
(
ISE
)
=
min
K
OPT
∫
t
0
t
N
(
T
B
(
t
)
-
T
B
,
SP
)
2
ⅆ
t
(
eq
.
17
A
)
F
=
∫
t
0
t
h
1
t
·
e
2
(
t
)
·
ⅆ
t
or
(
eq
.
17
B
)
T
MAX
>
max
(
T
B
,
SP
)
T
MAX
>
max
(
T
B
(
t
)
)
+
Δ
T
DPE
t
=
t
0
…
t
N
(
eq
.
20
A
)
where the parameters in the equations are:
e
error
k 1
effective diffusivity coefficient [m 2 s −1 ]
K OPT
optimum gain of the controller
K v
overall heat transfer coefficient [J m −2 s −1 K]
L
total product thickness [m]
L frozen
frozen layer thickness [m]
M
molecular weight [kmol kg −1 ]
p
pressure [Pa]
R
ideal gas Constant [J kmol −1 K]
R p
mass transfer resistance in the dried layer [m −1 s]
T
Temperature [K
t
time [s]
T B
frozen layer temperature at z = L [K]
T MAX
maximum allowable temperature for the product
T SP
Set point shelf temperature [K]
ΔT DPE
maximum temperature increase during DPE run
ρ
mass density [kg m −3 ]
ν shelf
cooling or heating rate of the shelf
ΔH s
enthalpy of sublimation [J kg −1 ]
T B,SP
set point value of frozen layer temperature at z = L [K]
ρ II
frozen layer mass density [kg m −3 ]
ρ Ie
effective dried layer mass density [kg m −3 ]
T shelf
temperature of temperature-controlled shelf
subscripts and superscripts are:
I
referred to dried layer
II
referred to frozen layer
e
effective
SP
set point value
i
interface
ISE
integral square error
where the parameters in equation (eq. 17B) are:
e
difference between the bottom product temperature
and its limit [K];
F
cost function;
t
time [s];
t 0
initial time [s];
t h
horizon time [s].
13. Method according to claim 12 , wherein calculating at least said new shelf temperature comprises the following step:
entering said plurality of product/process related parameters and other process/user parameters comprising liquid volume filling each container, number of loaded containers, volume of drying chamber, thermo-physical characteristics of solvent present in product (if different from water), maximum allowable product temperature, control logic selected, horizon and control time;
calculating a relation between L frozen and Ti and a frozen layer temperature by means of equations (eq. 12), (eq. 13), (eq. 14), (eq. 15);
calculating an optimal sequence of set-point temperature values by means of equation (eq. 16A) and equation (eq. 17A) or (eq. 17B) in case of feedback logic, or by means of equation (eq. 16B) in case of feedback logic, and equations (eq. 18), (eq. 19);
calculating an updated product temperature (T B,SP ) and a new shelf temperature (T′ shelf ) by means of equation (eq. 20A).
14. Method according to claim 13 , further comprises the following steps for calculating cooling/heating rates during a cooling/heating step of said primary drying phase:
defining a defined number of temperature intervals where said cooling/heating rates will be calculated;
during said cooling/heating step collecting the shelf temperature throughout all temperature intervals;
calculating the cooling/heating rate for each interval by means of equation:
r
i
=
1
n
∑
j
=
2
n
(
(
T
f
(
j
)
-
T
f
(
j
-
1
)
)
(
t
(
j
)
-
t
(
j
-
1
)
)
)
(
eq
.
21
)
where:
r i :
cooling/heating rate for the temperature interval i, K/min;
n:
number of data acquired in the interval i;
T f :
heating fluid temperature, K;
t:
time, s;
j
index of summation of capital-sigma notationupdating said
cooling/heating rate at least for said defined intervals.
15. Method according to claim 10 , where said control algorithm comprises the following equations:
ⅆ
L
frozen
ⅆ
t
=
-
1
n
~
II
-
n
~
Ie
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
12
)
k
1
=
RT
i
M
L
-
L
frozen
R
P
(
eq
.
13
)
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
=
Δ
H
s
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
14
)
T
B
=
T
shelf
-
1
K
v
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
(
eq
.
15
)
T
SP
(
t
)
:
{
T
SP
,
1
=
T
B
,
SP
-
[
1
-
K
v
(
1
K
v
+
L
frozen
(
t
0
)
k
frozen
)
(
T
B
,
SP
-
T
i
(
t
0
)
)
]
-
1
t
0
≤
t
<
t
1
T
SP
,
2
=
T
B
,
SP
-
[
1
-
K
v
(
1
K
v
+
L
frozen
(
t
1
)
k
frozen
)
(
T
B
,
SP
-
T
i
(
t
1
)
)
]
-
1
t
1
≤
t
<
t
2
⋮
T
SP
,
N
=
T
B
,
SP
-
[
1
-
K
v
(
1
K
v
+
L
frozen
(
t
N
-
1
)
k
frozen
)
(
T
B
,
SP
-
T
i
(
t
N
-
1
)
)
]
-
1
t
N
-
1
≤
t
<
t
N
(
eq
.
16
B
)
T
MAX
>
max
(
T
B
,
SP
)
T
MAX
>
max
(
T
B
(
t
)
)
+
Δ
T
DPE
t
=
t
0
…
t
N
(
eq
.
20
B
)
where the parameters in the equations are:
e
error
k 1
effective diffusivity coefficient [m 2 s −1 ]
K v
overall heat transfer coefficient [J m −2 s −1 K]
L
total product thickness [m]
L frozen
frozen layer thickness [m]
M
molecular weight [kmol kg −1 ]
p
pressure [Pa]
R
ideal gas Constant [J kmol −1 K]
R p
mass transfer resistance in the dried layer [m −1 s]
T
Temperature [K
t
time [s]
T B
frozen layer temperature at z = L [K]
T MAX
maximum allowable temperature for the product
T SP
Set point shelf temperature [K]
ΔT DPE
maximum temperature increase during DPE run
ρ
mass density [kg m −3 ]
ν shelf
cooling or heating rate of the shelf
ΔH s
enthalpy of sublimation [J kg −1 ]
T B,SP
set point value of frozen layer temperature at z = L [K]
ρ II
frozen layer mass density [kg m −3 ]
ρ Ie
effective dried layer mass density [kg m −3 ]
T shelf
temperature of temperature-controlled shelf
subscripts and superscripts are:
I
referred to dried layer
II
referred to frozen layer
e
effective
SP
set point value
i
interface
ISE
integral square error.
16. Method according to claim 15 , wherein calculating at least said new shelf temperature comprises the following step:
entering said plurality of product/process related parameters and other process/user parameters comprising liquid volume filling each container, number of loaded containers, volume of drying chamber, thermo-physical characteristics of solvent present in product (if different from water), maximum allowable product temperature, control logic selected, horizon and control time;
calculating a relation between L frozen and Ti and a frozen layer temperature by means of equations (eq. 12), (eq. 13), (eq. 14), (eq. 15);
calculating an optimal sequence of set-point temperature values by means of equation (eq. 16A) and equation (eq. 17A) or (eq. 17B) in case of feedback logic, or by means of equation (eq. 16B) in case of feedback logic, and equations (eq. 18), (eq. 19);
calculating an updated product temperature (T B,SP ) and a new shelf temperature (T′ shelf ) by means of equation (eq. 20A).
17. Method according to claim 6 , comprising determining the end of primary drying phase by calculating when a frozen layer of said product is reduced to zero.
18. Method according to claim 17 , wherein said determining comprises:
performing a pressure rise test and calculating a current solvent mass flow as the tangent of the pressure rise curve at the beginning of the test;
integrating the solvent mass flow versus time in order to get an actual cumulative sublimated mass curve; the primary drying can be considered finished when the sublimated mass curve reaches a plateau;
calculating a stop coefficient ( r s (i)) that is directly related to the average sublimating mass rate and it is used as reference for establishing whether or not the main drying is finished, taking into account the similarity between curves in different cycles:
r
_
s
(
i
)
=
m
(
i
)
-
m
(
i
-
1
)
m
tot
·
(
t
(
i
)
-
t
(
i
-
1
)
)
·
100
(
eq
.
22
)
where:
m
sublimated solvent mass [kg];
t
time [h];
r s
sublimating mass rate [kg s −1 ];
comparing the current r s with a limit value set by the user, which consists in the percentage variation of the sublimated solvent mass with respect to the total one, to verify if r s is lower than this limit and the primary drying can be considered finished.
19. Method according to claim 1 , wherein said calculating said new shelf temperature comprises calculating a new shelf temperature according to said product temperature so as to maximize a heat flux provided by said temperature-controlled shelf and so as to drive the product to a desired target temperature (Step 3 ).
20. Method according to claim 1 , comprising before said calculating providing parameters and data related to characteristics of freeze drying process, freeze dryer apparatus, product, containers, number of loaded containers, volume of drying chamber, thermo-physical characteristics of solvent present in product, maximum allowable product temperature during primary drying phase.
21. Method according to claim 1 , wherein said collecting pressure values is made at a sampling rate ranging from 5 to 50 Hz.
22. Method comprising performing a primary drying phase of a freeze drying process for freeze-drying a product to be dried 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, said method comprising during said primary drying phase the steps of:
entering a plurality of process/product related parameters comprising liquid volume filling each container, number of loaded containers, volume of drying chamber, thermo-physical characteristics of solvent present in product (if different from water), maximum allowable product temperature, control logic selected, horizon and control time;
calculating at least a product temperature and a new shelf temperature and/or a sequence of shelf temperatures up to the end of the primary drying phase that maximises a sublimation rate of said product maintaining the product temperature below said maximum allowable product temperature; and
adjusting a temperature of said temperature-controlled shelf on the basis of said new shelf temperature for minimising the duration of drying phase and preserving the product quality;
wherein said calculating is made by means of a control algorithm, based on a numerical code, which implements a non stationary mathematical model of containers and of freeze dryer apparatus and an optimization algorithm which uses as inputs said product/process related parameters, and further wherein said control algorithm comprises a PID type controller for controlling a product temperature and for minimizing an energy consumption during said primary drying phase, said control algorithm comprising the following equations:
ⅆ
L
frozen
ⅆ
t
=
-
1
n
~
II
-
n
~
Ie
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
12
)
k
1
=
RT
i
M
L
-
L
frozen
R
P
(
eq
.
13
)
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
=
Δ
H
s
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
14
)
T
B
=
T
shelf
-
1
K
v
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
(
eq
.
15
)
T
SP
(
t
)
:
{
T
SP
,
1
=
T
shelf
(
t
0
)
+
K
OPT
(
T
B
(
t
0
)
-
T
B
,
SP
)
t
0
≤
t
<
t
1
T
SP
,
2
=
T
shelf
(
t
1
)
+
K
OPT
(
T
B
(
t
1
)
-
T
B
,
SP
)
t
1
≤
t
<
t
2
⋮
T
SP
,
N
=
T
shelf
(
t
N
-
1
)
+
K
OPT
(
T
B
(
t
N
-
1
)
-
T
B
,
SP
)
t
N
-
1
≤
t
<
t
N
(
eq
.
16
A
)
min
K
OPT
(
ISE
)
=
min
K
OPT
∫
t
0
t
N
(
T
B
(
t
)
-
T
B
,
SP
)
2
ⅆ
t
(
eq
.
17
A
)
F
=
∫
t
0
t
h
1
t
·
e
2
(
t
)
·
ⅆ
t
or
(
eq
.
17
B
)
T
MAX
>
max
(
T
B
,
SP
)
T
MAX
>
max
(
T
B
(
t
)
)
+
Δ
T
DPE
t
=
t
0
…
t
N
(
eq
.
20
B
)
where the parameters in the equations are:
e
error
k 1
effective diffusivity coefficient [m 2 s −1 ]
K OPT
optimum gain of the controller
K v
overall heat transfer coefficient [J m −2 s −1 K]
k frozen
frozen layer thermal conductivity [J m s −1 K]
L
total product thickness [m]
L frozen
frozen layer thickness [m]
M
molecular weight [kmol kg −1 ]
M w
water vapour molecular weight [kmol kg−1]
p
pressure [Pa]
p w
water vapour pressure [Pa]
R
ideal gas Constant [J kmol −1 K]
R p
mass transfer resistance in the dried layer [m −1 s]
T
Temperature [K
t
time [s]
T B
frozen layer temperature at z = L [K]
T MAX
maximum allowable temperature for the product
ΔT DPE
maximum temperature increase during DPE run
ρ
mass density [kg m −3 ]
ν shelf
cooling or heating rate of the shelf
ΔH s
enthalpy of sublimation [J kg −1 ]
ρ II
frozen layer mass density [kg m −3 ]
ρ Ie
effective dried layer mass density [kg m −3 ]
T SP
Set point shelf temperature [K]
T B,SP
set point value of frozen layer temperature at z = L [K]
T shelf
temperature of temperature-controlled shelf
subscripts and superscripts are:
I
referred to dried layer
II
referred to frozen layer
e
effective
SP
set point value
i
interface
ISE
integral square error
where the parameters in equation (eq. 17B) are:
e
difference between the bottom product temperature
and its limit [K];
F
cost function;
t
time [s];
t 0
initial time [s];
t h
horizon time [s]
or comprising the following equations:
ⅆ
L
frozen
ⅆ
t
=
-
1
n
~
II
-
n
~
Ie
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
12
)
k
1
=
RT
i
M
L
-
L
frozen
R
P
(
eq
.
13
)
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
=
Δ
H
s
M
w
RT
i
k
1
L
-
L
frozen
(
p
i
(
T
i
)
-
p
w
)
(
eq
.
14
)
T
B
=
T
shelf
-
1
K
v
(
1
K
v
+
L
frozen
k
frozen
)
-
1
(
T
shelf
-
T
i
)
(
eq
.
15
)
T
SP
(
t
)
:
{
T
SP
,
1
=
T
B
,
SP
-
[
1
-
K
v
(
1
K
v
+
L
frozen
(
t
0
)
k
frozen
)
(
T
B
,
SP
-
T
i
(
t
0
)
)
]
-
1
t
0
≤
t
<
t
1
T
SP
,
2
=
T
B
,
SP
-
[
1
-
K
v
(
1
K
v
+
L
frozen
(
t
1
)
k
frozen
)
(
T
B
,
SP
-
T
i
(
t
1
)
)
]
-
1
t
1
≤
t
<
t
2
⋮
T
SP
,
N
=
T
B
,
SP
-
[
1
-
K
v
(
1
K
v
+
L
frozen
(
t
N
-
1
)
k
frozen
)
(
T
B
,
SP
-
T
i
(
t
N
-
1
)
)
]
-
1
t
N
-
1
≤
t
<
t
N
(
eq
.
16
B
)
T
MAX
>
max
(
T
B
,
SP
)
T
MAX
>
max
(
T
B
(
t
)
)
+
Δ
T
DPE
t
=
t
0
…
t
N
(
eq
.
20
B
)
where the parameters in the equations are:
e
error
k 1
effective diffusivity coefficient [m2 s −1 ]
K v
overall heat transfer coefficient [J m −2 s −1 K]
k frozen
frozen layer thermal conductivity [J m s −1 K]
L
total product thickness [m]
L frozen
frozen layer thickness [m]
M
molecular weight [kmol kg −1 ]
M w
water vapour molecular weight [kmol kg −1 ]
p
pressure [Pa]
p w
water vapour pressure [Pa]
R
ideal gas Constant [J kmol −1 K]
R p
mass transfer resistance in the dried layer [m −1 s]
T
Temperature [K]
t
time [s]
T B
frozen layer temperature at z = L [K]
T MAX
maximum allowable temperature for the product
ΔT DPE
maximum temperature increase during DPE run
ρ
mass density [kg m −3 ]
ν shelf
cooling or heating rate of the shelf
ΔH s
enthalpy of sublimation [J kg −1 ]
ρ II
frozen layer mass density [kg m −3 ]
ρ Ie
effective dried layer mass density [kg m −3 ]
T SP
Set point shelf temperature [K]
T B,SP
set point value of frozen layer temperature at z = L [K]
T shelf
temperature of temperature-controlled shelf
subscripts and superscripts are:
I
referred to dried layer
II
referred to frozen layer
e
effective
SP
set point value
i
interface
ISE
integral square error.
23. Method according to claim 22 , wherein said calculating comprises calculating a new shelf temperature according to said product temperature so as to maximize a heat flux provided by said temperature-controlled shelf and so as to drive the product to a desired target temperature.
24. Method according to claim 22 , wherein calculating at least said new shelf temperature comprises the following step:
entering said plurality of product/process related parameters and other process/user parameters;
calculating a relation between L frozen and Ti and a frozen layer temperature by means of equations (eq. 12), (eq. 13), (eq. 14), (eq. 15);
calculating an optimal sequence of set-point temperature values by means of equation (eq. 16A) in case of feedback logic, or by means of equation (eq. 16B) in case of feedback logic, equation (eq. 17A) or (eq. 17b) and equations (eq. 18), (eq. 19);
calculating an updated product temperature and a new shelf temperature by means of equation (eq. 20B).
25. Method according to claim 24 , further comprising the following steps for calculating cooling/heating rates during a cooling/heating step of said primary drying phase:
defining a defined number of temperature intervals where said cooling/heating rates will be calculated;
during said cooling/heating step collecting the shelf temperature throughout all temperature intervals;
calculating the cooling/heating rate for each interval by means of equation:
r
i
=
1
n
∑
j
=
2
n
(
(
T
f
(
j
)
-
T
f
(
j
-
1
)
)
(
t
(
j
)
-
t
(
j
-
1
)
)
)
(
eq
.
21
)
where:
r i :
cooling/heating rate for the temperature interval i, K/min;
n:
number of data acquired in the interval i;
T f :
heating fluid temperature, K;
t:
time, s;
updating said cooling/heating rate at least for said defined intervals.
26. Method according to claim 22 , wherein said plurality of product/process related parameters can be received from an estimator tool and/or from a sensor.