All of the responses of interest in this work, e.g., the experimentally measured and/or computed responses discussed in the previous Sections, can be generally represented in the functional form
, where
is a known functional of the model’s state functions and parameters. As generally shown in [
2], the sensitivity of such as this response to arbitrary variations in the model’s parameters
and state functions
is provided by the response’s Gateaux (G-) differential
, which is defined as follows:
where the so-called “direct effect” term,
, and the so-called “indirect effect” term,
, are defined, respectively, as follows:
where the components of the vectors
are defined as follows:
The system represented by Equation (26) is called the
forward sensitivity system, which can be solved, in principle, to compute the variations in the state functions for every variation in the model parameters. In turn, the solution of Equation (26) can be used in Equation (24) to compute the “indirect effect” term,
. However, since there are many parameter variations to consider, solving Equation (26) repeatedly to compute
becomes computationally impracticable. The need for solving Equation (26) repeatedly to compute
can be circumvented by applying the Adjoint Sensitivity Analysis Procedure (ASAM) formulated in [
2,
3,
4]. The ASAM proceeds by forming the inner-product of Equation (26) with a yet unspecified vector of the form
, having the same structure as the vector
, transposing the resulting scalar equation and using Equation (24). Furthermore, by requiring that the vector
satisfy the following adjoint sensitivity system:
it ultimately results that the “indirect effect” term can be expressed in the form
The system represented by Equation (28) is called the
adjoint sensitivity system, which –notably– is independent of parameter variations. Therefore, the adjoint sensitivity system needs to be solved only once, to compute the adjoint functions
. In turn, the adjoint functions are used to compute
, efficiently and exactly, using Equation (29). As an illustrative example of computing response sensitivities using the adjoint sensitivity system, consider that the model response of interest is the air relative humidity,
, in a generic control volume i, as given by Equation (21). For this model response, the “direct effect” term, denoted as
, is readily obtained in the form:
where:
2.2.1. Verification of the Adjoint Functions for the Outlet Air Temperature Response
When
, the quantities
defined in Equation (25) all vanish except for a single component, namely:
Thus, the adjoint functions corresponding to the outlet air temperature response
are computed by solving the adjoint sensitivity system given in Equation (28) using
as the only non-zero source term; for this case, the solution of Equation (28) has been depicted in
Figure 8.
(a) Verification of the adjoint function
Note that the value of the adjoint function
obtained by solving the adjoint sensitivity system given in Equation (28) is
, as indicated in
Figure 8. Now select a variation
in the inlet air temperature
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Re-writing Equation (42) in the form:
indicates that the value of the adjoint function
could be computed independently if the sensitivity
were available, since the quantity
is known. To first-order in the parameter perturbation, the finite-difference formula given in Equation (41) can be used to compute the approximate sensitivity
; subsequently, this value can be used in conjunction with Equation (43) to compute a “finite-difference sensitivity” value, denoted as
, for the respective adjoint, which would be accurate up to second-order in the respective parameter perturbation:
Numerically, the inlet air temperature
has the nominal (“base-case”) value of
. The corresponding nominal value
of the response
is
= 297.4637
. Consider next a perturbation
, for which the perturbed value of the inlet air temperature becomes
. Re-computing the perturbed response by solving Equations (2)–(13) with the value of
yields the “perturbed response” value
297.6073
. Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity”
. Using this value together with the nominal values of the other quantities appearing in the expression on the right side of Equation (44) yields
. This result compares well with the value
obtained by solving the adjoint sensitivity system given in Equation (28), cf.,
Figure 8. When solving this adjoint sensitivity system, the computation of
depends on the previously computed adjoint functions
; hence, the forgoing verification of the computational accuracy of
also provides an indirect verification that the functions
, were also computed accurately.
(b) Verification of the adjoint function
Note that the value of the adjoint function
obtained by solving the adjoint sensitivity system given in Equation (28) is
, as indicated in
Figure 8. Now select a variation
in the inlet air humidity ratio
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Re-writing Equation (45) in the form:
indicates that the value of the adjoint function
could be computed independently if the sensitivity
were available, since the
has been verified in (the previous)
Section 2.2.1 (a) and the quantity
is known. To first-order in the parameter perturbation, the finite-difference formula given in Equation (41) can be used to compute the approximate sensitivity
; subsequently, this value can be used in conjunction with Equation (46) to compute a “finite-difference sensitivity” value, denoted as
, for the respective adjoint, which would be accurate up to second-order in the respective parameter perturbation:
Numerically, the inlet air humidity ratio
has the nominal (“base-case”) value of
. The corresponding nominal value
of the response
is
. Consider next a perturbation
, for which the perturbed value of the inlet air humidity ratio becomes
. Re-computing the perturbed response by solving Equations (2)–(13) with the value of
yields the “perturbed response” value
. Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity”
. Using this value together with the nominal values of the other quantities appearing in the expression on the right side of Equation (47) yields
. This result compares well with the value
obtained by solving the adjoint sensitivity system given in Equation (28), cf.
Figure 8. When solving this adjoint sensitivity system, the computation of
depends on the previously computed adjoint functions
; hence, the forgoing verification of the computational accuracy of
also provides an indirect verification that the functions
were also computed accurately.
(c) Verification of the adjoint functions and
Note that the values of the adjoint functions
and
obtained by solving the adjoint sensitivity system given in Equation (28) are as follows:
and
, respectively, as indicated in
Figure 8. Now select a variation
in the inlet water temperature
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Since the adjoint functions
and
have been already verified as described in
Section 2.2.1 (a) and (b), it follows that the computed values of adjoint functions
can also be considered as being accurate, since they constitute the starting point for solving the adjoint sensitivity system in Equation (28). Hence, the unknowns in Equation (48) are the adjoint functions
and
. A second equation involving solely these adjoint functions can be derived by selecting a perturbation,
, in the inlet water mass flow rate,
, for which Equation (40) yields the following expression for the sensitivity of the response
to
:
Numerically, the inlet water temperature, , has the nominal (“base-case”) value of , while the nominal (“base-case”) value of the inlet water mass flow rate is . As before, the corresponding nominal value of the response is . Consider now a perturbation , for which the perturbed value of the inlet air temperature becomes . Re-computing the perturbed response by solving Equations (2)–(13) with the value of yields the “perturbed response” value . Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity” .
Next, consider a perturbation
, for which the perturbed value of the inlet air temperature becomes
. Re-computing the perturbed response by solving Equations (2)–(13) with the value of
yields the “perturbed response” value
. Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity”
. Inserting now all of the numerical values of the known quantities in Equations (48) and (49) yields the following system of coupled equations for obtaining
and
:
Solving Equations (50) and (51) yields
and
. These values compare well with the values
and
, respectively, which are obtained by solving the adjoint sensitivity system given in Equation (28), cf.
Figure 8.
2.2.2. Verification of the Adjoint Functions for the Outlet Water Temperature Response
When
, the quantities
defined in Equation (25) all vanish except for a single component, namely:
Thus, the adjoint functions corresponding to the outlet water temperature response
are computed by solving the adjoint sensitivity system given in Equation (28) using
as the only non-zero source term; for this case, the solution of Equation (28) has been depicted in
Figure 9.
(a) Verification of the adjoint function
Note that the value of the adjoint function
obtained by solving the adjoint sensitivity system given in Equation (28) is
, as indicated in
Figure 9. Now select a variation
in the inlet air temperature
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Re-writing Equation (52) in the form:
indicates that the value of the adjoint function
could be computed independently if the sensitivity
were available, since the quantity
is known. To first-order in the parameter perturbation, the finite-difference formula given in Equation (41) can be used to compute the approximate sensitivity
; subsequently, this value can be used in conjunction with Equation (53) to compute a “finite-difference sensitivity” value, denoted as
, for the respective adjoint, which would be accurate up to second-order in the respective parameter perturbation:
Numerically, the inlet air temperature
has the nominal (“base-case”) value of
. The corresponding nominal value
of the response
is
. Consider next a perturbation
, for which the perturbed value of the inlet air temperature becomes
. Re-computing the perturbed response by solving Equations (2)–(13) with the value of
yields the “perturbed response” value
. Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity”
. Using this value together with the nominal values of the other quantities appearing in the expression on the right side of Equation (54) yields
. This result compares well with the value
obtained by solving the adjoint sensitivity system given in Equation (28), cf.,
Figure 9. When solving this adjoint sensitivity system, the computation of
depends on the previously computed adjoint functions
; hence, the forgoing verification of the computational accuracy of
also provides an indirect verification that the functions
were also computed accurately.
(b) Verification of the adjoint function
Note that the value of the adjoint function
obtained by solving the adjoint sensitivity system given in Equation (28) is
, as indicated in
Figure 9. Now select a variation
in the inlet air humidity ratio
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Re-writing Equation (55) in the form
indicates that the value of the adjoint function
could be computed independently if the sensitivity
were available, since the
has been verified in (the previous)
Section 2.2.2 (a) and the quantity
is known. To first-order in the parameter perturbation, the finite-difference formula given in Equation (41) can be used to compute the approximate sensitivity
; subsequently, this value can be used in conjunction with Equation (56) to compute a “finite-difference sensitivity” value, denoted as
, for the respective adjoint, which would be accurate up to second-order in the respective parameter perturbation:
Numerically, the inlet air humidity ratio
has the nominal (“base-case”) value of
. The corresponding nominal value
of the response
is
. Consider next a perturbation
, for which the perturbed value of the inlet air humidity ratio becomes
. Re-computing the perturbed response by solving Equations (2)–(13) with the value of
yields the “perturbed response” value
. Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity”
. Using this value together with the nominal values of the other quantities appearing in the expression on the right side of Equation (57) yields
. This result compares well with the value
obtained by solving the adjoint sensitivity system given in Equation (28), cf.
Figure 9. When solving this adjoint sensitivity system, the computation of
depends on the previously computed adjoint functions
; hence, the forgoing verification of the computational accuracy of
also provides an indirect verification that the functions
were also computed accurately.
(c) Verification of the adjoint functions and
Note that the values of the adjoint functions
and
obtained by solving the adjoint sensitivity system given in Equation (28) are as follows:
and
, respectively, as indicated in
Figure 9. Now select a variation
in the inlet water temperature
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Since the adjoint functions
and
have been already verified as described in
Section 2.2.2 (a) and (b), it follows that the computed values of adjoint functions
can also be considered as being accurate, since they constitute the starting point for solving the adjoint sensitivity system in Equation (28). Hence, the unknowns in Equation (58) are the adjoint functions
and
. A second equation involving solely these adjoint functions can be derived by selecting a perturbation,
, in the inlet water mass flow rate,
, for which Equation (40) yields the following expression for the sensitivity of the response
to
:
Numerically, the inlet water temperature, , has the nominal (“base-case”) value of , while the nominal (“base-case”) value of the inlet water mass flow rate is . As before, the corresponding nominal value of the response is . Consider now a perturbation , for which the perturbed value of the inlet air temperature becomes . Re-computing the perturbed response by solving Equations (2)–(13) with the value of yields the “perturbed response” value . Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity” .
Next, consider a perturbation
, for which the perturbed value of the inlet air temperature becomes
. Re-computing the perturbed response by solving Equations (2)–(13) with the value of
yields the “perturbed response” value
. Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity”
. Inserting now all of the numerical values of the known quantities in Equations (58) and (59) yields the following system of coupled equations for obtaining
and
:
Solving Equations (60) and (61) yields
and
. These values compare well with the values
and
, respectively, which are obtained by solving the adjoint sensitivity system given in Equation (28), cf.
Figure 9.
2.2.3. Verification of the Adjoint Functions for the Outlet Air Relative Humidity Response
When
, the quantities
defined in Equation (25) all vanish except for two components, namely:
Thus, the adjoint functions corresponding to the outlet air relative humidity response
are computed by solving the adjoint sensitivity system given in Equation (28) using
and
as the only two non-zero source terms; for this case, the solution of Equation (28) has been depicted in
Figure 10.
(a) Verification of the adjoint function
Note that the value of the adjoint function
obtained by solving the adjoint sensitivity system given in Equation (28) is
, as indicated in
Figure 10. Now select a variation
in the inlet air temperature
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Re-writing Equation (64) in the form:
indicates that the value of the adjoint function
could be computed independently if the sensitivity
were available, since the quantity
is known. To first-order in the parameter perturbation, the finite-difference formula given in Equation (41) can be used to compute the approximate sensitivity
; subsequently, this value can be used in conjunction with Equation (65) to compute a “finite-difference sensitivity” value, denoted as
, for the respective adjoint, which would be accurate up to second-order in the parameter perturbation:
Numerically, the inlet air temperature
has the nominal (“base-case”) value of
. The corresponding nominal value
of the response
is
. Consider next a perturbation
, for which the perturbed value of the inlet air temperature becomes
. Re-computing the perturbed response by solving Equations (2)–(13) with the value of
yields the “perturbed response” value
. Using now the nominal and perturbed response values together with the parameter perturbation in the finite-difference expression given in Equation (41) yields the corresponding “finite-difference-computed sensitivity”
. Using this value together with the nominal values of the other quantities appearing in the expression on the right side of Equation (66) yields
. This result compares well with the value
obtained by solving the adjoint sensitivity system given in Equation (28), cf.,
Figure 10. When solving this adjoint sensitivity system, the computation of
depends on the previously computed adjoint functions
; hence, the forgoing verification of the computational accuracy of
also provides an indirect verification that the functions
were also computed accurately.
(b) Verification of the adjoint function
Note that the value of the adjoint function
obtained by solving the adjoint sensitivity system given in Equation (28) is
, as indicated in
Figure 10. Now select a variation
in the inlet air humidity ratio
, and note that Equation (40) yields the following expression for the sensitivity of the response
to
:
Re-writing Equation (67) in the form
indicates that the value of the adjoint function
could be computed independently if the sensitivity
were available, since the
has been verified in (the previous)
Section 2.2.3 (a) and the quantity
is known. To first-order in the parameter perturbation, the finite-difference formula given in Equation (41) can be used to compute the approximate sensitivity
; subsequently, this value can be used in conjunction with Equation (68) to compute a “finite-difference sensitivity” value, denoted as
, for the respective adjoint, which would be accurate up to second-order in the parameter perturbation:
Numerically, the inlet air humidity ratio has the nominal (“base-case”) value of . The corresponding nominal value of the response is . Consider next a perturbation , for which the perturbed value of the inlet air humidity ratio becomes . Re-computing the perturbed response by solving Equations (2)–(13) with the value of yields the “perturbed response” value