To memory of Professor Jan Szargut

Accepted Manuscript To Memory of Professor Jan Szargut Applications of The Orthogonal Least Squares Adjustment in Energy Engineering

Z. Kolenda, T. Styrylska PII:

S0360-5442(18)31411-7

DOI:

10.1016/j.energy.2018.07.114

Reference:

EGY 13379

To appear in:

Energy

Received Date:

23 May 2018

Accepted Date:

17 July 2018

Please cite this article as: Z. Kolenda, T. Styrylska, To Memory of Professor Jan Szargut Applications of The Orthogonal Least Squares Adjustment in Energy Engineering, Energy (2018), doi: 10.1016/j.energy.2018.07.114

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ACCEPTED MANUSCRIPT

TO MEMORY OF PROFESSOR JAN SZARGUT APPLICATIONS OF THE ORTHOGONAL LEAST SQUARES ADJUSTMENT IN ENERGY ENGINEERING Z. Kolenda1*, T. Styrylska2 1)Department

of Fundamental Research in Energy Engineering AGH - St. Staszic University of Science and Technology, Al. Mickiewicza 30, 30-059 Kraków, Poland 2)Department of Environmental Engineering, Cracow University of Technology, ul. Warszawska 24, 31-655 Kraków Poland

Keywords: least squares adjustment, mass and energy balances, heat and mass transfer, mathematical model validation ABSTRACT This paper presents general formulation, mathematical methods and numerical examples of least squares adjustment (reconciliation) of the results of the measurement of thermal energy processes. This paper contains results of research of two groups supervised by Professors T. Styrylska and Z. Kolenda. Almost all problems discussed in the paper were suggested by Professor Jan Szargut a mentor and tutor of the authors. The methods presented have been successfully applied in mathematical modeling in both a global and local sense. A list of selected publications is collected in the references. Motto: “Observations are useless until they have been interpreted…. The analysis of experimental data forms a critical stage in every scientific inquire – a stage which can be responsible for most of the failures and fallacies of the past (from E.B Wilson; An introduction to Scientific Research)” NOMENCLATURE A,B Jacobi matrices of condition equations or matrices of coefficients in linear system of equations c molar concentration cp specific heat capacity under constant pressure D molar diffusion coefficient H enthalpy [i] molar fraction of component i k Lipschitz constant k reaction rate constant L vector of experimental results (observations) M diagonal matrix of the a priori errors n number of moles molar flow rate n q vector of free terms in condition equations q intensity of internal heat source v qi

Q

t T

intensity of internal mass source of component i heat transfer rate time temperature

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V V W xi X Y Greek letters α β Γ Ω ν μ ∇ Subscripts A F d s

vector of corrections to experimental results lower heating value vector of residuals in condition equations cartesian coordinates (i=1,2,3) vector of unknowns (general notation) vector of corrections to the approximations of unknowns vector of estimates of experimental results vector of estimates of unknowns boundary surface domain stoichiometric coefficient measurement error nabla operator Air Fuel Combustion gases Solid

INTRODUCTION The method of least squares is the standard method used to obtain unique values for physical parameters from redundant measurement of those parameters or variables related to them by a known mathematical functional description. Development of the method of least squares has profoundly influenced both theory and practice mainly in mathematical modeling of real processes and in time simulations [1]. Physical quantities can never be measured perfectly. There will always be a limit in the precision of measurements beyond which either the mathematical model describing the physical quantities or the resolution of the measuring instrument or both will fail. Beyond this limit redundant measurements will not agree one another. That is they will not be consistent. In no case do we know the true value of the physical quantity. All we can do is make an “estimate” of the true value. We want this estimate to be unique and to have some idea of how good this estimates are. The method of least squares is based on minimizing the sum of squares of the inconsistences and applying to problems either linear or nonlinear mathematical models. To explain mathematical basis of the least square method, consider this system of linear equations [1]. AX=W (1) Unique nontrivial solutions exist only if W≠0 -no homogeneous system rank of A = dimension of X, rank of A/L = rank of A (consistent system). In the case where there are no-redundant equations the solution is given by X=A-1W (2) When there are redundant equations A will not be square, the solutions are X=(ATA)-1ATW

(3)

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Where ATA is square and nonsingular. The solution (3) is obtained when consistency criterion minimizing the sum of squares of the residuals is applied. As shown above, the least squares adjustment method should be used when [1]: - mathematical models are incomplete, - physical measurements are inconsistent, - redundant measurements exist. In this paper three group of problems will be discussed: - adjustment of measurement results in the material and energy balances of chemical processes, - validation of the mathematical models of heat and mass transfer processes with supplementary information (data), - least squares evaluation of enthalpy of formation. Theoretical considerations are illustrated with examples discussed in theoretical chapters 2,3 and 4. MATHEMATICAL AND ENERGY BALANCES OF CHEMICAL PROCESSES The problem concerning the necessity of the adjustment of measurement results to satisfy material and energy balances was formulated by Szargut [2]. Mathematical, rigorous method of adjustment procedure was given in the paper by Szargut and Kolenda [3,16]. The material and energy balances are based on the laws of conservation of mass and energy. These laws – when applied to the principal chemical elements to the process and to the energy – lead to the system of nonlinear algebraic equations. They contain directly measurable variables which characterize the mass flow rates, the chemical compositions and specific enthalpy of substrates and products of the process. In the description of real processes, the number of balance equations is greater than number of unknowns (not directly measurable quantities) and because of inevitable errors of measurement, the system of equations is not thoroughly satisfied. The measured variables must be corrected to satisfy the required solutions. Such a correction is called adjustment of the material and energy balances. Neglecting it can lead to the following negative consequences [3]; - A result obtained by evaluating a non-measured quantity depends on the way in which the calculation was carried out. Depending on the theoretically correct way, different values can be obtained in each case. - The values of unknowns calculated from some equations do not satisfy the other equations not used in the calculation. The value of a non-measured quantity of the energy balance (e.g. environment heat loss) depends on the method used in evaluating the chemical enthalpy. It is just as important to monitor whether the errors of measurements do not exceed the assumed limits. This system of balance equations (called conditions equations) includes the balances equations written for the principal chemical elements of the process, the equations for the sum of fractions of all substances, the relations resulting from taking into account the mineral composition of some substances and energy balances for all parts of the system [4]. The general formulation of the mass balance of the i-th principal element leads to the following equation: nti  nfi  ns ,i (4) where: nti , nfi - molar flow rates of i-th principal element entering the system with substrates and leaving with products ns ,i - increase of i-th element in the system

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nsi  nsci  nsbi ncsi nsbi

(5)

- number of moles of i-th principal element in the system at the end and beginning of the

experiment. For steady – state processes nsi  0

Equations of energy balances take the general form E  E in

out

(6)  Es

(7)

where: Ein - energy flow rate supplied to the system, E out

- energy flow rate released from the system,

E s

- changes of internal energy of the system. In the case of steady state processes Es  0

(8)

Ein  Eout

(9)

and Eq. (7) is simplified to the form

VALIDATION OF THE MATHEMATICAL MODELS OF HEAT AND MASS TRANSFER PROCESSES WITH SUPPLEMENTARY DATA. Conventional methods of mathematical modeling of heat and mass transfer processes are commonly based on schemes leading to unique solutions. For example, solving initial- boundary value problems, the mathematical model consists of the governing equations and initial and boundary conditions. In the classical approach each variable or physical property (thermal conductivity, thermal emissivity and diffusivity coefficient in mass transfer) and model parameters (heat transfer and mass transfer coefficients) describing the thermodynamic state of system are treated as exactly known and simplifications in the model are usually neglected. Simplifications can be of different kinds. For example, a linear system of differential equations is solved instead of a nonlinear system, the convective heat transfer mechanism is neglected, or one or two dimensional problems are solved instead of three. In such a case the real state of physical system will be different from the mathematical model solution mainly because of model simplifications, measurement errors of the directly measured variables (observations) and errors of evaluation of values of physical properties and model parameters. Although the solution obtained in this way uniquely describes the behavior of the system, the degree of accuracy is usually difficult to establish. If, however, the mathematical model contains more information (called here “supplementary data”) than necessary a unique solution, it becomes possible to check its accuracy and, what is equally significant, to determine the influence of the measurement errors, model simplification, and errors of model parameters on the accuracy of the general model. Statements such as “The analytical results agree satisfactorily with the corresponding experimental data”, “the experimental results agree well with the theoretical results” or “comparison with data shows reasonable agreement between the present theory and experimental results” can often be encountered as the final conclusion of many research papers, without any explanation of the meaning of “agree satisfactorily”, “agree well” and “reasonable agreement” and such statements should be avoided [5]. Validation of the mathematical models of heat and mass transfer is possible by introducing additional supplementary data not necessary for a unique solution. Using the least squares method, the most probable unique

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solution is possible for model validation purposes. Schematically, the proposed procedure is presented in Fig 1.

Fig. 1. Classical approach and least squares adjustment with supplementary data. There are several advantages to using the least-squares method. Especially with the calculation results of unknown variables (non-measured quantities) involving model parameters and constants that are independent of the method of calculation, the most accurate values of the measurement results are obtained, and the accuracy of the measurement results is greater (a posteriori errors are less than a priori errors). Also, systematic errors of measurements can be detected and the mathematical model can be validated and model simplifications can be analyzed [4,5]. An example of least squares adjustment procedure to the solution of an initial boundary - value problem of heat conduction in solids with different types of supplementary data is presented in Fig 2 [7,15]. Boundary Surface Γ

Known Solutions at separated boundary points Known local solutions Heat Conduction Equations +

Boundary conditions of first second and third kind known simultaneously

Domain Ω

Initial Conditions

Fig. 2. Initial – boundary value problem of heat conduction in solids with various types of supplementary data. LEAST SQUARES EVALUATION OF ENTHALPY OF FORMATION

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Tables of standard enthalpy of formation are based on the measurements of standard enthalpy of chemical reactions ( Im ) and are computed from the first law of thermodynamics [6] Im 

i  i H ni  j  j H ni

where: i,j – number of substrates and products, and products,  i

j

H ni

,

H ni

(10)

- standard enthalpy of formation for substrates

– stoichiometric coefficients.

Because the number of possible reactions is much greater than the number of reactants, it is possible to calculate values of enthalpy of formation using different sets of mathematically independent equations. Because of the fact that the values of enthalpy of chemical reactions are obtained from direct measurements they are known with measurement errors and the system of equations (10) used in calculations are internally contradicted theoretically. The problem is identical as in the case of mathematical modeling with supplementary data presented in Chapter 3 for heat and mass transfer processes. To adjust the measurement results, different mathematical methods can be used – singlestage and multi-stage. Examples of calculations are presented in Chapter 4 (Example 4.4) The problem was formulated, solved and analyzed in the paper [6]. MATHEMATICAL METHODS OF LEAST SQUARES ADJUSTMENT Let no denote a minimum of independent variables necessary for a unique solution of the mathematical model and let n be a number of functionally independent observations. When n is greater than no, the redundancy or statistical degrees of freedom defined as r=n-no is said to exist and an adjustment becomes necessary in order to obtain a unique solution of the mathematical model. Because of the statistical properties of the experimental results, redundant observations are not compatible with the model and any arbitrarily chosen subset of experimental results can be used to satisfy the model equations. In a case like this a unique solution when redundant measurements are considered can only be obtained when an additional criterion is imposed [4,5,6]. Let I denotes a vector of all experimental results and let α be a vector of estimates that satisfies the model equations. In general, the values of α are different from I and the difference vector: V=α-I

(11)

which has been termed as either corrections or residuals, plays an important role in calculations. Due to the redundancy the number the number of estimates for α and V is infinite. To calculate the most probable solution consistent with the model, the least squares principle is commonly used as an additional criterion. The least squares principle requires the condition:  i    minimum i  1  i  n

f  V  =V TM-1 V= 

(12)

to be satisfied simultaneously with the model equations where M is the weight matrix of the observations (experimental results). The weight matrix M is square and diagonal and of an order equal to the number of observations. Let us assume that the mathematical model can be performed by the following system of algebraic linear equations: Aα  Bβ  q

(13)

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where α =[α1,…,αs]T and β=[β1,…,βn]T represent directly measured variables and unknowns, respectively, A and B are r x s and r x n matrices of coefficients, q is r x 1 vector of free terms and r is the number of model equations. The set of equations (13) can also be written in the form: AV  BY  W

(14)

V  α  I, Y  β  x, W  q  AI  Bx

(15)

where

V is s x 1 vector of corrections to the measured quantities, Y is n x 1 vector of corrections to the preliminary estimates of unknowns, I is s x 1 vector of measurement results, x is n x 1 vector of preliminary estimates of unknowns. The least squares procedure can now be formulated as follows: minimize   V   V TM-1 V (16) subject to the model equations AV  BY  W

(17)

The Lagrange multipliers method leads to the final solution in the form of [4,5,7] Y  G-1B TF -1 W

(18)

V  MA TF 1  W  BY 

(19)

where F=AMAT and G=BTF-1B. If the accuracy of the solution of a lineared problem is not sufficient, the iterative procedure must be applied. In such cases to get the solution of an original nonlinear problem the values of elements of matrices A and B are continuously corrected at each iteration step. The solution (18) is now used to calculate a posteriori error of directly measured variables, unknowns and any function containing model variables. Using the law of error propagation, the expressions for covariance matrices can be derived in the forms [4,5,6]. M  M  CAM

(20)

M  B TF 1B 

(21)

C  MAT F 1 E  BG1BTF 1 

(22)

and

where

and E is the unit diagonal matrix. A condition for the adequacy of the model can be formulated in different ways but the Lipschitz condition αl  k

or

(23)

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vi  k 

(24)

(where k is the Lipschitz constant) seems to be the most effective from numerical point of view. The mathematical model is then accepted if the Lipschitz condition is satisfied for all experimental results. Usually k=2 is chosen (confidence level - 95%). Unified approach In many practical problems, the necessary (from mathematical point of view) condition of the rank of matrix A to be equal to the number of model equations is not satisfied. Such a situation is observed when the number of directly measured variables is less than the number of equations. The solution (19) cannot be used as an inverse matrix F-1=(AMAT)-1 does not exist [det(AMAT)=0] and another method of solution must be used. One of the possibilities is a unified least squares method proposed by Mikhail and Ackermann [17] for surveying problems and adopted by Szargut and Skorek [9] and Kolenda [15]. Several problems using unified approach have been analyzed and solved by Szega [21,22]. The basic principle and the most important assumption in the unified approach is that all variables in the mathematical model are observations (results of direct measurement) which means that unknown variables (solution of the model) are treated numerically in the same manner as directly measured variables but with sufficiently large error values in comparison to a priori errors of measurement results. The model equation (17) can now be written in a simplified form: AV  W

(25)

A=[A,B]

(26)

and

and V   V T , Y T 

T

(27)

The corresponding a priori weight matrix is M1 0  M 1    1 0 MY 

(28)

where no correlation is assumed between the two vectors I and x. New weight matrix MY 1 is the inverse of a priori covariance (diagonal) matrix for unknowns (zero matrix in the classical approach). Using, as previously, the Lagrange multipliers method the final solution becomes [13] V  M A T F 1 W

(29)

where F  A M A T with the covariance matrix for all variables. A posteriori error matrix is M  M  M A TF

1

A M

Separation of the calculated variables into measured and unknowns is not necessary.

(30)

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The methods described above can be effectively used when transient problems are considered and a unified approach is especially recommended for numerical calculations. Multigroup Method In the multigroup technique [11,12] all matrices are divided into block matrices. Then the vector V of corrections to the measured quantities, the matrix M of covariances a priori, and the matrix A are divided as follows (for brevity the ashes have been omitted) V   V1T , V2T ...., VmT 

T

(31)

M1    M M2   MM 

(32)

A   A1 , A 2 ...., A m 

(33)

and the system (25) can be rewritten in the form m

m

AV  j j

1

 q   A jI j  W

j

j=1, 2, …, m

j 1

(34)

where Aj, Vj are r x sj, sj x 1 matrices, respectively. Using the method of Lagrange multipliers, the solution becomes VJ  MJ A Tj D1 W

j=1, 2, …, m

(35)

where D 

m

Dj  j

T , D j  A j Mj A j

1

(36)

The matrix of covariances of the corrected measurements is G j  H j MHT j

where  j

H j 

I





 lj  Vj I

(37)

  H

1



vj





, Hvj1 ,...., I j  Hvjj ,...., Hvjm  

(38)

and according to (35) and (36) HvJi  M j ATj D1 A j  H j A j

j=1, 2, …..,m

(39)

HvJi  M j ATj D1 A i  H j A i

i  j;i = 1, 2,.....m.

(40)

Finally



 



i j Hvji Mi H

G j  I j  Hvjj M j I j  Hvjj



 

G j  I j  Hvjj M j I j  Hvjj



T

m

i 1, 

iT vj





T



m

i j Hvji Mi HvjiT

(41)

i 1, 





 I  Hj A j Mj I  Hj A j



T



m

i j H j Mi H

i 1, 

T j

(42)

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Multi-stage method When transient problems are considered, the number of equations can be very large and the methods described above are not effective from computational point of view. In problems like this a multi-stage method is recommended [12,14]. A solution for each time level is obtained using a posteriori covariance matrices M instead of a priori M . The values of residuals W are modified at each following time step using solutions from the previous time level, l=1,2,…..,N, separately, to give A1V  W1 for l = 1 (43) A2 V  W2

for l = 2

(44)

A N V  WN

for l = N

(45)

The procedure is as follows: the first step is that the least squares principle is used to adjust a subset of the model equation A1V  W1 (l=1) with a diagonal a priori covariance matrix M . The solution is given by Eq. (29) with a posteriori matrix M given by Eq. (30). At the second step the next subset of model equations A2 V  W2 is adjusted with new covariance matrix M which is now treated as the a priori covariance matrix for the time level l = 2. To obtain the global solution, the procedure as above is repeated l=N times. The partial solution for an l-time level is. Vl  Ml 1 A l

T

Fl ,l11 Wl

(46)

T

(47)

where Fl ,l 1  A l Ml 1 A l

and Ml  Ml 1  Ml 1 A l

T

Fl ,l 11 A l Ml 1

(48)

The final solution is given by the sum V

N

 Vl

(49)

l 1

If new independent variables appear in the consecutive subset of model equations, the a priori covariance matrix must be enlarged by the non-zero elements at the main diagonal only. The procedure presented above is very effective from a computational point of view. It can easily be simplified further to avoid calculations of inverse matrices Fl,1l 1 at each step. It is achieved only if the single equation, instead of the subset of equations, is added at the next time step. NUMERICAL EXAMPLES Example 4.1 Material and energy balances of the combustion process. Consider combustion of natural gas (CH4), Fig.3.

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n F

, Fuel, TF, CH4

n D

, Combustion Gas, Td,

(CO2, CO, O2, N2, H2O) n A ,

Air, TA, (O2,N2,H2O)

Fig.3. Scheme of combustion of natural gas

QS

Mass and energy balances are [4]:  Carbon, C

nF CH 4   nd  CO 2   CO  

 Oxygen, O2

(50)



0.21nA  0.5nA [H 2O ]A  nd CO 2   0.5 CO   O 2   0.5 H 2O d



(51)

 Hydrogen, H2 2nF  n

A

H 2O  A  nd H 2O d

(52)

 Nitrogen, N2 0.79nA  nd N 2 

(53

CO 2   CO   O 2   N 2   1.0

(54)

HF  H A  HCG  QS

(55)

Molar fractions of combustion gases  Energy balance equation where

HF  nF VCH 4  cpCH 4TTFN  T F  T N 

(56)

H A  nAcp ATTNA T A  T N 

(57)



n





i 1



HCG  nd VCO CO    i  cp i Td TN T d  T N  

(58)

The system of equations (50-54), and (55-58) is the basis for an adjustment procedure. These equations contain directly measured variables – molar flow rate nF , chemical compositions of combustion gases CO2, CO,H2O, O2, N, VCH4 and VCO lower heating values of CH4 and CO, temperature of natural gas, air and

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combustion gases TF, TA, Td, respectively. Variables

nA

,

nd

and

QS

are unknown variables as their direct

measurement is not possible. Linearization equations (50-54) and (55) lead to the matrix form

AV  BY  W

(59) where vectors V and Y represent corrections to the measurement results and approximation to the unknowns, respectively. Final solutions are given by Eqs. (18) and (19) with a posteriori error (Eqs .(20) and (21)). Example 4.2 Consider steady state heat conduction process in solids, known as the boundary-value problem. The process is described by the governing equation in the form of nonlinear partial differential equations [13].

  k T  T  x i    qv T , x i   0

(60)

This equation must be satisfied at every internal point of domain Ω. To apply the least squares procedure, boundary conditions and supplementary data must be included (see chapter 3, Fig. 2). Schematically the problem is explained below.

+

+

Least square principle

Fig.4. Scheme of boundary value problem. Because governing equations and boundary conditions are usually given in nonlinear mathematical forms, a solution can only be obtained by numerical methods. Such an approach leads directly to the system of condition equations in the matrix form

AV  BY  W

(61) Further calculations follow either classical method or unified method (chapter 5). The method is presented and described in detail in [4].

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Example 4.3 Consider the mathematical model of mass transfer processes during autocatalytic dissolution of metals in oxygen saturated solutions. All available experimental data are included in the model equations, mainly through supplementary conditions which allow for the evaluation of chemical reactions orders and reaction rate constants. Chemical reactions can be written in the form [5]:

As + B  E

(62)

As + E  D

(63)

D +B  E

(64) and are studied under the assumption that the substrate E in the heterogeneous reaction (64) is continuously reproduced in a very fast homogenous reaction (63) making the overall process autocatalytic. The mathematical model consists of a set of partial differential equations [5] v    ci   Di   ci   q i (65) written for all species, i, responsible for the rate of mass transfer processes where Ci=Ci(xi,t) qi(ci,t),Di=DI(ci) and v=(u,v,w) (velocity vector). Boundary conditions describe the mechanism of physical and chemical processes taking place at the boundary surfaces. Supplementary data are based on the assumption that the main changes in concentration of all species i occur inside a very limited space of the boundary layers. On the basis of experimental study (the rotating disc technique was employed) the following sequences of chemical reactions are considered: - on the copper disc surface Cu  4NH 3  0.5O 2  H 2O  Cu (NH 3 )24   2OH  (66) and Cu  Cu NH 3 4  2Cu (NH 3 )2 2

-

(67)

in the bulk of solution 2Cu NH 3 2  4NH 3  0.5O 2  H 2O  2Cu NH 3 4  2OH  2

2

(68)

According to the general scheme, Eqs. (66), (67) and (68), Cu refers to A(s), oxygen to B, Cu(II) complexes to E and Cu(I) complexes to D. To verify the model an additional equation resulting from the mass balance for Cu(II) complexes in the bulk of solution is added to the classical boundary - value formulation. Mathematically this means that the first, second and third kinds of boundary conditions are known simultaneously. As the problem is nonlinear, the solution can be obtained with numerical method, similar to the solution of heat transfer model discussed in chapter 3. The final and most probable solution is obtained by the method discussed in chapter 5. The results of the calculations show that the method is very effective and creates new possibilities for more rigorous analysis. Supplementary data allows us to obtain estimates (most probable values) of the calculation results and experimental data. This method also makes it possible to conduct a sensitive analysis of the model equations. The method is presented in details in paper [5]. More mathematically advanced models describing mass and heat transfer processes with chemical reactions (coal gasification process) and solidification of binary alloys have been presented in [10, 18-20].

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Example 4.4 Determining the most probable values of enthalpy of formation of manganese, oxygen and carbon compounds (Mn2O3, MnO2, MnCO3) on the basis of directly measured standard enthalpies of the following reactions (with a priori errors) [6]: C s , graphite   O 2 g   CO 2 g 

+393.785  0.13

Mn (S )  0.5O 2 g   MnO s 

+389.8  1.3 +1408.9  1.3 -104.88  1.68

Mn (S )  2O 2 g   Mn 3O 4 s  3Mn 2O 3S   2Mn 3O 4 s   0.5O 2 g  3MnO 2S   2Mn 3O 4 s   O 2 g  3MnCO 3S   0.5O 2  Mn 3O 4 s   3CO 2 g 

MnO S   CO 2 g   MnCO 3s  4MnO 2S   Mn 2O 3s   O 2 g 

(69)

-164.1  12.5 -110.9  4.6 +118.5  3.3 -139.8  5.0

(values are given in MJ/kmol). Condition equations (69) written in the matrix form are as follows: AV  BX  W (70) where V represents corrections to the experimental results (standard enthalpies of reactions (69)), X is the vector matrix of the most probable values of unknowns (enthalpies of formation of Mn2O3, Mn2O and MnCO3). As the first three equations of (69) are reactions of formation, matrices in Eq. (70) become 0

0 2 1 0 0 0 0

0 0 1 0 1 0 0 0 A  3 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 0

0

0

0 0 0 0 1

3 0 B 0 0

0 0 3 0 0 3 0 1

2 4

0

 HMnO3 X   HMnO2  HMnCO3

W4 W5 W  W6 W7 W8

(71)

where wi are residuals of equations for reactions 4 to 8 in (69). Values of V, X and a posteriori errors are calculated from Eqs.(18), (19), (20) and (21), respectively (see chapter 5). The final results after adjustment are (with errors a posteriori): H nCO  393.785  0.126 2

H n MnO  389.62  0.6

H n Mn O  1408.88  0.8 3 4

(72)

H n Mn O  974.23  0.8 2 3

H n MnO  522.26  1.3 2

H n MnCO  900.64  1.4 3

In the set (69) there are three unknowns and five measurement results – therefore two measurements are in excess. As a consequence of adjustment procedure standard deviations of measurement results decrease. The decrease of standard deviations is proportional to the excess of the number of measurements above those necessary to determine unknowns.

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Example 4.5 We examined the efficiency, stability and accuracy considering 2D unsteady temperature field with the unknown initial condition. The surfaces of the plate x  0 , y  0 were thermally insulated and on x  1 , y  1 we imposed Bi  1.0 and Tf  1.0 . The temperature of the surface was measured in just

one point (0,1) every  Fo  0.1 with uncertainty of ± 0.025. We assumed the unknown initial condition to be as high as T0  0.8  0.8 (very rough assumption). We solved this problem employing unified approach and multistage-multigroup procedure.

Fig.5. The difference mesh. The solution given by the multistage-multigroup method was correct, stable and more accurate both from physical and statistical view-points. The multistage-multigroup technique exploits both the real data from all time steps and the structure of constraint equations. Using this approach there is no need to transform the model equations to the standard form, either. Moreover, the way of solving adjustment problem makes approach consistent numerically and reduces calculation time. The multistage-multigroup technique enables us to solve both well- and ill-posed multidimensional heat transfer problems [18]. CONCLUSIONS This paper presents both theory and applications of the advanced least squares adjustment (reconciliation) of experimental results in energy engineering. The method, widely used in physics, chemistry, astronomy and surveying engineering has been adopted to energy problems by Szargut (first publication in 1952). Further applications (in the next 60 years), are concerned with problems in thermodynamics (for example the adjustment of standard enthalpy of formation), mathematical modeling of heat and mass transfer processes with supplementary data and solutions of the inverse initial - boundary value problems. A more thorough look into the problem of application of the least squares method, both from theoretical and practical points of view can be found in monograph [4] written under Professor Szargut supervision.

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Acknowledgments Authors are very grateful to Michał Dudek (M.Eng) for his assistance in paper preparation. REFERENCES [1] Wells D.E., Krakiwsky E.J., The method of least squares, University of New Brunswick Press, Lecture Notes, 1971,18,1-179. [2] Szargut J., Ryszka F., Necessity of coordination of material balances, Research Papers IM. 1952;5;385399 (in Polish). [3] Szargut J., Kolenda Z., Theory of coordination of material and energy balances in metallurgical chemical processes, Archiwum Hutnictwa 1968; XIII; 2;153-169. [4] Szargut J.,(Ed)Least Squares method in thermal engineeering, Ossolineum Press, 1984, 1-215 (in Polish). [5] Kolenda Z., Zembura M., Donizak J., Zembura Z., An application of unified least squares method to the mathematical modeling of autocatalytic reactions, J.Electroanalitical Chem. 1995; 382; 1-15 [6] Styrylska T., Application of the multi – stage least squares method for the evaluation of the enthalpies of formation, Archiwum Termodynamiki , 7, 1986, 3 ,129-139. [7] Kolenda Z., Styrylska T., On the mathematical modeling of physical and chemical processes applying orthogonal least squares principle, Seminar on Contemporary Problems in Thermal Engineering, Silesian University of Technology, Gliwice, 2003, 43-59. [8] Styrylska T., Pietraszek J., Numerical modeling of non-steady state temperature fields with supplementary data ,ZAMM,1992 ,72, 6, 537-539. [9] Szargut J., Skorek J., Influence of the preliminary estimation of unknowns on the results of coordination of material and energy balances, Bull. Polish Academy of Science 1990; 3912,335-393. [10] Kolenda Z., Donizak J., Escobedo Bocardo J., Least squares adjustment of mathematical model of heat and mass transfer processes during solidification of binary alloys, Metal. Mat, Trans. B 1999,30,505-513. [11] Styrylska T., Nowarski J., An application of the multigroup least squares method to the estimation of VLE parameters, ZAMM 1991; 71,768-770. [12] Styrylska T., The multistage and Multigroup Adjustment of the Measurement Results, Cracow University of Technology, Monograph 121 Cracow 1991; 1-164. [13] Kolenda Z., Allman J.S., Coordination of energy balances in heat transfer, Bull Polish Academy of Science 1974; 22, 33-37. [14] Guzik A. Styrylska T., Applying the multistage-multigroup least squares procedure for numerical modeling of unsteady temperature fields, Bull. Pol. Acad. Sci. ,Tech. Sci.1998; 46,4, 399-408. [15] Kolenda Z., Styrylska T., Donizak. J, Guzik A., Numerical and experimental mathematical modeling of heat and mass transfer processes using unified least squares method. Energy Convers. Mgmt. 1998; 399 (16-18) 1763-1772. [16] Szargut J., Kolenda Z., Styrylska T., Justification of measurement results in thermal technology Proc., ECOS’96, 1996, Stockholm, 413-418. [17] Mikhail E., Ackermann F., Observations and least squares, IEP-A, Dun Donnaley Publisher, NY, 1976 [18] Guzik A., Styrylska T., Solving initial-boundary heat transfer problems using the least squares filtering, ZAMM 2000; 80, 682-683

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[19] A. Ściążko Y. Komatsu, G.Brus, S. Kimijima, J.S. Szmyd, A novel approach to improve the mathematical modeling of the internal reforming process for solid oxide fuel cells using orthogonal least squares method. International Journal of Hydrogen Energy 2014; 39, 16372-16389. [20] A. Ściążko Y. Komatsu, G.Brus, S. Kimijima, J.S. Szmyd, A novel approach to the experimental study on methane steam reforming kinetics using the Orthogonal Least Squares method. Journal of Power Sources 2014; 262, 245-254. [21] Szega M., An improvement of measurements reliability in thermal processes by application of the advanced data reconciliation method with the use of fuzzy uncertainties of measurements, Energy 2017; 141, 2490-2498. [22] Szega M., Advantages of an application of the generalized method of data reconciliation in thermal technology Archives of Thermodynamics 2009; 30 (4) 219-232

ACCEPTED MANUSCRIPT Highlights

1. 2. 3. 4. 5.

Theory and applications of the advanced least squares adjustment. Mathematical modeling of heat conduction in solids. Solutions of the inverse initial - boundary value problems. Adjustment of standard enthalpy of formation. Adjustment of mass and energy balances.