Abstract
The article deals with simultaneous identification of heat transfer coefficient and ambient temperature. The general functional properties of thermal models are seeking for the unique estimation of two interrelated parameters. To substantiate the preservation of the one-to-one correspondence the simple approach is used. It is shown that the simultaneous identification of the heat transfer coefficient and ambient temperature is not possible unless the class of the functions to which they belong is suitably narrowed. Typical thermal models are considered and the narrowed classes of the sought properties are revealed. The idea can be applied for a wide class of inverse engineering problems.
Introduction
One of the practically important problems in heat engineering is the estimation of a heat transfer coefficient. Usually the problem is complicated by the troubles of the ambient temperature determination caused by the difficulties of the access to the device space. This can be exemplified by the heat processes in gas-turbine engines, steam plants, cryogenic reservoirs etc. At the same time, a specification of the ambient temperature determines the heat-exchange intensity. This raises the problem of the identification of not only the heat transfer coefficient but also the ambient temperature, which is difficult to be directly measured.
In the theory of the inverse thermal problems the simultaneous identification of specific heat capacity and thermal conductivity has been extensively studied Citation[1–Citation6]. The further development of the inverse problem methodology makes it actual to discuss the simultaneous estimation of other practically important thermal process characteristics [Citation7]. Reconstruction of the heat transfer coefficient and ambient temperature facilitates new approaches for many thermal experiments.
The interest to this problem is also enhanced by the fact that the heat transfer coefficient and ambient temperature are functionally interrelated. In this respect, the investigation of the conditions for the preservation of one-to-one correspondence has apparent theoretical importance. Also, the mathematical aspect study is important because, in many cases, the possibility to increase the number of simultaneously identified quantities is not obvious. The invariance properties [Citation8], inherent to mathematical models, should be taken into account.
The idea behind this study is grounded on the fact that unidentifiable states represent a separate class of direct problem solutions [Citation9]. Their exception in the further inverse problem formulation can extend the number of sought quantities. This idea makes the determination of the unidentifiable class of direct problem solutions by the main subject of the approach offered. This distinguishes it from the standard viewpoint Citation[10–Citation12], where the nonuniqueness is established and hereupon the study comes to the end. Additionally, the idea [Citation9] reveals a class of nonunique inverse problem solutions using only a conventional direct problem formulation. We specially underline that any additional boundary conditions or samples are not required and the complete inverse problem formulation is not considered to determine the conditions of the identifiability violation. Statements of direct problems have sufficient initial data to elicit the number of basic identifiability peculiarities. The results obtained guarantee the experiment against unidentifiable states. In total, the complex invariant properties of mathematical models are investigated with the help of the simple expedients. The present investigation develops the idea [Citation9] from a viewpoint of the study of heat process regularities and seeks the conditions for the simultaneous estimation of two interrelated thermal parameters. The investigation notions are introduced and discussed in [Citation7].
The article is organized as follows. In Section 1 the simplest thermal model is considered. The basic peculiarities of the simultaneous identification are revealed. Section 2 is devoted to the study of typical one-dimensional thermal models. Conditions of the violation of the one-to-one correspondence are sought. In Section 3 we will describe the expedient to preserve the required identifiability, if the functional dependencies of sought quantities do not generally provide for the one-to-one correspondence. Section 5 summarizes the results obtained.
1. Model with lumped parameters
Let us consider the identifiability of the heat transfer coefficient and ambient temperature using a model with lumped parameters. If we neglect the spatial distribution of a specimen temperature, then its state will be described by the following ordinary differential equation
(---619--1)
This simplest kind of mathematical model allows us to represent all details of the approach developed. Its investigation indicates how the substantiation of the simultaneous estimation can be executed with the help of the simple expedient.
First, we study the identifiability of the heat-exchange parameters α = α (t) and uam = const, frequently encountered in practice. The idea is grounded on the determination of the class of the model state u* generating the nonunique solution of the inverse problem in the sense that certain sets of sought quantities give the same solution of the direct problem. Let us assume that there exists a temperature field u* for the different heat-exchange parameters {α prime;, u′am} } and {α ″, uPrime;am }. This supposition results in the existence of the family
(---619--2)
From here it follows that the function υ(t) is represented within an arbitrary constant V in the form υ = α ′′ − V(α ′ − ς ). Consequently, the solution of Eq. Equation(1)
(---619--1) for arbitrary α (t) and uam = const is described by the function
(---619--3)
Substituting Equation(3)
(---619--3) into Equation(1)
(---619--1) , we obtain
(---619---1)
Solving this equation, we are able to reveal the function ς (t) that determines the family Equation(2)
(---619--2) . The ultimate form of the above family looks like this:
(---619--4)
where u0 denotes the initial temperature. Using the expression Equation(3)
(---619--3) we obtain the following general solution of Eq. Equation(1)
(---619--1)
(---619---2)
Hence, every temperature field satisfying model Equation(1)
(---619--1) will be invariant under the transformation Equation(4)
(---619--4) of the heat-exchange parameters. This means that the arbitrary variation of the constant ambient temperature and choice of the heat-transfer coefficient in accordance with Equation(4)
(---619--4) cannot change the slab temperature field in each point of the domain of the state function. Therefore, the sought parameters {α (t), uam = const} are unidentifiable on a whole with model Equation(1)
(---619--1) , if the problem formulation is based on their general functional dependencies.
Also, the identification of the functions α (t) and uam (t) is impossible. The necessary condition for the preservation of the one-to-one correspondence will be satisfied if the constant parameters α = const, uam = const are sought and u ≠ const.
2. One-dimension model
We now consider the one-dimensional equation with the heat-exchange on an object surface
(---619--5)
The identifiability of the heat-exchange parameters {α, uam } will be justified as above by the one-to-one correspondence analysis between the temperature field u(x,t) and its parameters {α, uam }. Let us suppose that the same temperature field u* corresponds to the various parameters {α ′, u′am} and {α ″, u″am }. Subtracting Eq. Equation(5)
(---619--5) with these values from one another we obtain α ′(u* − u′am ) = α ″(u*− u″am ). This expression shows that the nonunique selection of the sought parameters results from the invariance of the heat flux with respect to arbitrary variations in the heat-exchange parameters. Therefore, the unidentifiable field satisfies the following condition
(---619--6)
If the heat-exchange parameters are time-dependent, i.e., α = α (t) and uam = uam (t), then the right side of condition Equation(6)
(---619--6) will also be a time-dependent function. Hence, the one-to-one correspondence between the field u(x, t) and parameters {α (t), uam(t) } remains valid. Similarly, if the heat-exchange parameters depend on the length of a specimen, α = α (x) and uam = uam (x), then the one-to-one correspondence remains valid also.
In general, if the parameters {α (x, t), uam (x, t)} are sought without narrowing of their functional type, then condition Equation(6)(---619--6) shows that for the arbitrary α ′ ≠ α ″ the constant values included, and for the variable u′am (x, t) different ambient temperature can be specified
(---619---3)
or else for the arbitrary u′am ≠ u″am including constant values, and for the variable α ′(x, t) the heat transfer coefficient
(---619---4)
can be assigned in the way that different {α ′, u′am } and {α ″, u″am} will correspond to the same solution u* of Eq. Equation(5)
(---619--5) .
Thus, the heat-exchange parameters of the one-dimensional heat equation cannot be identified, if one of the sought quantities is the time- and space-dependent function. One-variable functions {α, uam } are identifiable on a whole. The latter notion is discussed in [Citation7].
Notice that despite the unidentifiability of the heat-exchange parameters in the general case α = α (x, t) and uam = uam (x, t), the preservation of the one-to-one correspondence can be guaranteed, if supplementary physical information is involved. This case will be studied separately in Section 3.
Let us now study the identifiability of heat-exchange parameters from the boundary conditions. Consider the following model
(---619--7)
where the heat-exchange parameters {α (t), uam (t)} at the slab boundaries are to be equal.
The arguments above bring us to the following expressions
(---619--8)
(---619--9)
in which {α ′, u′am } and {α ″, u″am } denote the parameters generating the same temperature field. The right members of conditions Equation(8)
(---619--8) and Equation(9)
(---619--9) provide the necessary condition for the one-to-one correspondence being violated
(---619--10)
To prove its sufficiency we employ Equation(8)(---619--8) to determine the coefficient
(---619----1) . Substituting this value into Equation(9)
(---619--9) , we obtain
(---619----2) , t > 0. It therefore follows that the satisfaction of condition Equation(10)
(---619--10) is also sufficient for the violation of the one-to-one correspondence between the temperature field satisfying model Equation(7)
(---619--7) and its heat-exchange parameters {α (t), uam (t)}.
For example, if the direct problem Equation(7)(---619--7) has the arbitrary parameters c, ρ, λ = const ≠ 0, volumetric heat source f = cρ − 2λ and initial state u0 = x(x − ℓ ), then its solution will not be changed for different parameters {α, uam } satisfying the condition α (uam − t) = λ ℓ.
Thus, the time-dependent functions {α (t), uam(t) } are identifiable on a whole, if the process simulation is based on the one-dimensional transient model that generate the field with difference between the boundary temperatures. The existence of the nonunique class of the sought quantities is regarded with the special features of temperature fields.
It is necessary to note that the conditions for the preservation of the one-to-one correspondence are insufficient to ensure the precision of the numerical solution. The simultaneous reconstruction of the several unknown functions requires the additional restrictions. The numerical regularization should be engaged and considered as the individual investigation. From this viewpoint the stable estimation of the heat-exchange parameters and numerical simulation of heat process identification with noisy data is studied in [Citation13].
3. Two-variable unknowns
We now analyze the one-to-one correspondence in a two-dimensional mathematical model with two variable unknown functions. In the previous section it was shown that generally the identification of all the heat-exchange parameters is impossible if their functional types are not narrowed. The most effective way is the diminishing of the number of independent variables. Obviously, this method simplifies significantly the inverse problem formulation.
Thereby the next question is raised. How can the violation of the one-to-one correspondence be avoided without loss of functional generality of the desired quantities?
Let us consider a thermal object that is practically important, namely the reservoir with a cryogenic liquid. Its outer side surface is supposed to be heat-insulated. Also, one of the boundary conditions is assumed to be homogeneous, say, because of object symmetry. The mathematical model, which completely reflects the cryogenic processes in the reservoir, is expressed by a complex system of differential equations. It can be simplified if the heat-exchange conditions between the reservoir wall and the liquid are supposed to be known. The following model can be used
(---619--11)
So, the simultaneous identification of the wall heat transfer coefficient α (y, t) and cryogenic liquid temperature uam (y, t) is required.
The general form of desired functions makes their unique identification impossible. Actually, a new heat transfer coefficient can be determined (---619----3) , for which the solution of the problem Equation(11)
(---619--11) is kept with arbitrary variations of α ′, u′am and u″am. The reduction of the number of independent variables, for example {α (t), uam (t)}, is inadmissible procedure from a viewpoint of physical features of the process studied. Therefore, another way of the function narrowing should be considered.
To justify the identifiability on a whole, the specific character of the reservoir filling with cryogenic liquids will be checked. The heat-exchange process goes on within the gas phase until the liquid boundary comes up to a given point and the interaction with the liquid phase starts. A similar variation is approximated satisfactorily by the functions
(---619--12)
(---619--13)
where Ai(t) and Bi(t) are arbitrary functions to be found. This parameterization ensures the function narrowing.
The functional dependencies narrowing are widely used to assure the unique solutions of inverse problems Citation[10–Citation12]. The restrictions being examined in these and analogous papers are based on diminishing the number of independent variables in the desired quantities. We aim to restrict a domain of admissible solutions by special parameterization of sought quantities wherein a priori information should reflect the regularities of the object under study. Consequently, the build up of the identifiability of a given model reduces to the selection of those functions that would, on one hand, satisfactorily approximate sought quantities, and on the other, would keep the one-to-one correspondence between the object state and unknown parameters.
One-to-one correspondence between the temperature field satisfying Equation(11)(---619--11) and functions Equation(12)
(---619--12) , Equation(13)
(---619--13) will be studied as above by the method of contradiction. Let us assume that for the sought properties Equation(12)
(---619--12) and Equation(13)
(---619--13) there exists a coefficient transformation
(---619--14)
which does not change the field u*(x, y, t) during the variations of ς i and υ i. The conservation of the heat flux on the boundary
(---619----4) , gives the condition
(---619---5)
Expanding this expression, the general functional form of the assumed unidentifiable temperature field is defined as
(---619--15)
where V, W and F are the functions different from zero.
Further, we find the derivative of function Equation(14)(---619--14) relative to the variable y. In accordance with the given model Equation(11)
(---619--11) the latter should be equal to zero at the coordinate y = y0. From this it follows that the function F is identical to zero. This does not agree with the dimensionality of the temperature field defined by model Equation(11)
(---619--11) . Therefore, the existence of invariant transformation Equation(14)
(---619--14) contradicts the functional properties of the solution of the direct problem Equation(11)
(---619--11) , whose coefficients are approximated by the functions Equation(12)
(---619--12) and Equation(13)
(---619--13) . This confirms the preservation of the one-to-one correspondence between the heat-exchange parameters Equation(12)
(---619--12) , Equation(13)
(---619--13) and temperature field described by model Equation(11)
(---619--11) .
Note that the substantiation of the one-to-one correspondence between the temperature field and its parameters Equation(12)(---619--12) , Equation(13)
(---619--13) expresses the necessary condition of the identifiability. The parameterization Equation(12)
(---619--12) and Equation(13)
(---619--13) sufficiency can be ultimately resolved by further analysis of the one-to-one correspondence between the sought functions and set of discrete samples. Here we show that it is possible to preserve the one-to-one correspondence using additional information about physical features of the process studied.
We are now going to examine the question of parameterization Equation(12)(---619--12) and Equation(13)
(---619--13) to be sufficient for the identifiability of the thermal model Equation(11)
(---619--11) in the context of the more general boundary conditions assumptions. We abandon the symmetry condition used above at the coordinate y = y0 and consider the temperature v0 (x, t) given at a certain coordinate y = y0. No restrictions on its form are imposed.
Let us substitute the function Equation(15)(---619--15) into the origin heat equation Equation(11)
(---619--11) . The rearrangement can yield the equations satisfied by the functions V, W, F and G from expression Equation(15)
(---619--15) . Among them the functions F and G should satisfy the equation
(---619---6)
Expanding the expression in powers of y, we find that F ≡ 0. Hence, temperature measurements on an arbitrary reservoir segment are identifiable on a whole relative to the sought heat-exchange parameters.
Summarizing the results obtained, the following conclusion can be made. If the general assumptions result in the identification ambiguity, then the involvement of a priori information ensures the preservation of the one-to-one correspondence between the object state and desired functions. In this case it is very important that the prior information is not based on an arbitrary parameterization, but is to express the test object physical properties. The representations Equation(12)(---619--12) and Equation(13)
(---619--13) of heat-exchange parameters for the reservoir filling with a cryogenic liquid illustrate how the idea proposed provides the correct identification of the process.
Conclusions
Simultaneous estimation should allow for the duality of the mathematical model identifiability. The violation of the one-to-one correspondence between the direct problem solution and desired quantities must be kept in mind. At the same time, there are conditions to identify the general mathematical model parameters simultaneously. The study of the violation of the one-to-one correspondence and further discrimination of the violation conditions from the identification problem formulation gives the constructive approach for the inverse engineering problems.
The estimation of both the heat transfer coefficient and ambient temperature grounding on the observation of the specimen temperature field requires the agreement of functional dependence of sought parameters with the form of mathematical model under study. The reduction of the number of independent variables in the heat-exchange parameters is one of the means to preserve the one-to-one correspondence. Identifiability can also be provided by seeking the desired parameters within a preliminarily narrowed character of functional dependencies. The investigation demonstrates that the study of the one-to-one correspondence reveals a number of practically important conditions to reconstruct generally unidentifiable properties.
Acknowledgment
The author thanks the referees for useful comments that help to state the article with greater clarity and completeness.
Additional information
Notes on contributors
M. Romanovski *
E-mail: [email protected]; http://mywebpage.netscape.com/mromanovski/IP.htm
Notes
E-mail: [email protected]; http://mywebpage.netscape.com/mromanovski/IP.htm
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