The level of performance properties of fuels and lubricants (FL) is determined [1,2,3] by comparing the values of the parameters of performance properties of the test FL specimens with the parameters of a comparison (reference) specimen, which are obtained under fixed test conditions or by comparing with standard values prescribed in standardization documents. In most cases, the so-called point assessment of performance properties is used for the comparison. Point assessment of performance properties (indicators) of FL represents experimentally established or calculated “individual” value (or confidence interval) of the parameter (rate, result, etc.) of the pertinent chemmotological process. The value of point assessment complies with a specific combination of set values of the determining factors of the chemmotological process, which are associated in the tests (calculations) in different degrees with the parameters of the operation conditions and conditions of performance of the equipment. As a result of comparison of point assessments of performance properties of the FL test specimen and the reference specimen (standard value) we get the point assessment of the level of performance properties of the FL.

Point assessment of the level of GL performance properties can be expressed through the value of the parameter (indicator) of the pertinent chemmotological process, which is obtained with a given probability at fixed (set constant) values of the determining factors of the chemmotological process. In other words, the point assessment of the FL performance property is a unit value (confidence interval) or else a partial case of dependence of the rate and/or of the result of the chemmotological process on four groups of factors [3, 4] at specific values of each of the factors:

$${\Pi }_{\text{pp}}({\Pi }_{\text{cp}})=f({\Phi }_{\text{m}},{\Phi }_{\text{c}},{\Phi }_{\text{d}},{\chi }_{\Sigma }),$$
(1)

where Ppp(Pcp) - parameter of performance property (parameter of the pertinent chemmotological process); Fm - factors that take account of the mechanism of the phenomena accompanying the studied chemmotological process; Fc - determining factors of the conditions of occurrence of the chemmotological process; Fd - factors that take account of the equipment design features; cS - aggregate of FL properties.

With change of conditions of experimental determination of performance property parameters, the values of point assessments of performance properties, in general, also change. Because of this, often arise situations where the test FL specimen meets the requirements with respect to the level of a specific property in some conditions of use and, conversely, does not meet the requirements in other conditions. This is particularly typical with respect to appearance of FL properties as a function of various factors when additives are used. An example illustrating one of the types of possible dependencies of the parameters of performance properties Ppp on the studied factors F1 and F2, which do not permit unambiguous determination of the best option of combination of values of the factors (conditions) for getting a certain optimum value of Ppp, is presented in Fig. 1. In similar situations, various procedures that allow selection of preferable alternatives, including performance of FL tests [5] under drastic conditions, are proposed. Nevertheless, the question of assessment of the level of performance properties of FL with respect to the whole possible range of values of the determining factors of chemmotological process conditions (condition of FL use in equipment) not infrequently remains open. In experiment plan, in general, this is fraught with difficulties that stem from the complexity, long time span, and costliness of determination of the required large number of point assessments of the performance properties that “cover” to a sufficient extent the interesting range of actual values of the determining factors of the conditions of use of the FL in equipment, including their various combinations.

Fig. 1
figure 1

Example of one of the possible types of dependencies of parameters of performance properties pp on the studied factors F1 and F2, which do not permit unambiguous determination of the best option of combination of values of the factors for getting a particular optimum Ppp value

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Thus, to get initial data for compatible assessment of the dependence of the chosen performance property parameters on the four factors, each of which is specified, for example, with five values, it is necessary to conduct 54 experiments, i.e., to get 625 point assessments, with a whole set of combinations.

Obviously, for efficient use of time and resources, it is essential to correctly solve, among others, the problem of determination of the required and adequate number of values by which each of the factors should be set in the region of their change when FL is used in the equipment. In the methodological plan for getting the desired generalized assessment of the level of FL performance properties it is necessary to synthesize a sufficiently simple, but nontrivial procedure for joint presentation of an array of experimental values of point assessments of FL performance properties, each of which corresponds to a variable set of conditions within fixed limits.

Consequently, for getting generalized (integral in terms of the method of analytical expression) assessments of the level of FL performance properties it is necessary to take account of all possible (within reasonable limits) combinations of values of the determining factors and results of the studied chemmotological processes. In this context, it is necessary to consider the whole range of variation of values of the determining factors of the chemmotological processes and to present in a “nutshell” the obtained multitude of values of the parameters of FL performance properties that corresponds to the referred set of combinations of values of the factors.

The essential feature of the method of integral assessment of the level of FL performance properties consists in presentation of the correlation of performance property parameters (parameters of chemmotological processes) and factors of FL use conditions identified from the results of FL tests or by other means as a continuous (elementary) function, calculations of a specific integral (in the Riemann sense) from this function, multiple number of factors, the range of integration of which is set by limits of variation of the values of the factors, and relating the integration results received for the test FL specimen with the results of integration for the comparison (reference) specimen.

The structure of the method of integral assessment of the level of performance properties of FL includes three successive stages. In the first stage, for the adopted type of function (model) of correlation of parameters of chemmotological processes (parameters of performance properties) and factors of conditions of FL conversion (conditions of operation and conditions of performance of the equipment), the parameters of the model are identified from the results of tests of the FL specimen and the comparison (reference) specimen. In the second stage, the integrals are calculated from this function (model) for the test FL specimen and for the comparison (reference) specimen. In the third stage, the values of the integrals from the referred function for the chosen FL specimens are compared and the criteria of integral assessment of the level of FL performance properties are calculated as a ratio of the values of the integrals for the test FL specimen and the reference specimen.

For getting an integral assessment of the level of FL performance properties, the function that most fully represents the correlation of the performance property parameters (chemmotological process parameters) and the factors of conditions of use or testing of the FL must specify correlations of the general type (1). In specific conditions and with due regard for the known limitations, correlations of type (1) can be represented with a certain degree of adequacy by various mathematical models, including algebraic, differential, integral, and integrodifferential equations, their systems, and various combinations. In general, k of the determining factors of the conditions of FL use (test) can be regarded as components of the vector X = (x1, …, xk) and the functional link of the determining factors of the chemmotological processes in equation (1) may be denoted through F(X):

$$F(X)=F({x}_{1},\dots .{x}_{k})=f({\Phi }_{\text{m}},{\Phi }_{\text{c}},{\Phi }_{\text{d}},{\chi }_{\Sigma }).$$
(2)

The simplest and most appropriate representation of the desired correlations recommendable for wide use of integral assessment of the level of performance properties of FL are predictive mathematical models of the type of algebraic polynomial of an order not higher than the second, which can be obtained from the results of modeling of chemmotological processes (determination of performance properties of FL):

$${\Pi }_{\text{pp}}({\Pi }_{\text{cp}})=F(X)={b}_{0}+\sum_{i=1}^{k}{b}_{i}{x}_{i}+\sum_{i=1}^{k}\sum_{j=1}^{k}{b}_{ij}{x}_{i}{x}_{j}+\sum_{i=1}^{k}{b}_{ii}{x}_{i}^{2}+{\epsilon,}$$
(3)

where X = (x1, …, xk) - aggregate of k of the studied determining factors of chemmotological processes, i.e., the vector of the test conditions; b - parameters of the model; e - error indicating the influence of uncounted random factors.

Modeling of chemmotological processes caried out for getting an integral assessment of the level of FL performance properties includes physical modeling of FL use conditions [4] with the involvement of the methods of mathematical theory of experiment planning [6,7,8], which is essential for constructing mathematical models of the type (3) and for minimizing experiment time and resources. In congruence with the experiment planning theory for getting relationships of the (3) type of FL performance properties, for example, of the four independent factors, it is enough to set each of the factors at three levels and to conduct only 27 experiments. While applying the theory of experiment planning, the variables in equation (3) are integrated as follows: xi - coded values of the factors (i = 1, …, k) determined (normalized) as xi = xi = (Xin – Xi0n)/∆Xin; Xin – values of the factors in natural variables; Xi0n – coordinate of the experiment center (zero level) Xi0n = 0,5(Ximaxn+Ximinn);∆Xin – half interval of change of factors in the experiment ∆Xin = 0,5(Ximaxn - Ximinn); Ximaxn И Ximinn – upper and lower levels of variation of the factors, respectively; b0, bi, bij, bii - regression coefficients, i < j; k - number of factors.

Thus, for getting integral assessment of the level of FL performance properties from the results of physical and mathematical (polynomial) modeling of chemmotological processes (assessment of FL properties), equation (1) is constructed in approximation to (3) and identification is made, i.e., numerical values and parameters of the obtained nonlinear mathematical models are determined.

In this option, which is most common from the point of availability of initial data, the integral assessment Int(Ppp) of the level of FL performance properties can be obtained by the equation [9, 10]:

$${\text{Int}}({\Pi }_{\text{pp}})=\int \dots {\int }_{D}F(X)dX,$$
(4)

where F(X) - k-dimensional function, for example, of type (3), X = (x1, …, xk); D - region of integration of function F(X). For a function of type (3) the integration region D coincides with the experimentation region and is set in the form of a k-dimensional cube or a k-dimensional parallelepiped by the boundaries of variation of factors xi (i = 1, …, xk) in the experiment.

For conditions of second-order extreme experiment planning [8, 11, 12] and full quadratic mathematical model (3), equation (4) can be transformed to get a suitable equation for calculation of integral assessments:

$${\text{Int}}({\Pi }_{\text{pp}})=(2\Delta {x}_{i}){2}^{(k-1)}({b}_{0}+1/3\sum_{i=1}^{k}{b}_{ii}),$$
(5)

where Dxi - half interval of variation of factors, Dxi = 0,5(xiu – xil) = const, xiu, xil - upper and lower levels of variation of factors in coded variables (i = 1, …, k); k - number of factors. For use of equation (5) it is necessary that the range of variation of factors [xil, xiu] contained within it a “zero” point in coded variables.

To compare integral assessments, we introduce the criterion (Kpp) of integral assessment of the level of FL performance properties, which is the ratio of the integral assessment of the test FL specimen Int(Ppp)N to the corresponding integral assessment obtained for the comparison (reference) specimen Int(Ppp)E:

$${\kappa }_{\text{pp}}={\text{Int}}{({\Pi }_{\text{pp}})}_{N}{[{\text{Int}}{({\Pi }_{\text{pp}})}_{E}]}^{-1}.$$
(6)

The numerical values of the integral assessments (4) and the criteria (6) make it possible to get the predictive assessment of relative advantages (to assess the level of performance properties) of one FL specimen of a specific type in comparison with other specimens. Here, it is obvious that, depending on the requirements imposed for performance properties, the integral value of the criterion Kpp could be both higher and lower than 1. For instance, if it is necessary to minimize the tendency toward formation of deposits, the value of the criterion (6) for a test FL specimen could be less than 1.

The integral assessment is calculated by equation (4) in commonly encountered situations where information about the actual distribution of probabilities of operation conditions and conditions of use of the equipment (factors of chemmotological process conditions) over time is inadequate or it is unavailable. In that case, the hypothesis about the independence and equal probability of all operation conditions and conditions of performance of the equipment is accepted by default, and time is not included in an explicit form in the totality of the studied (determining) factors of chemmotological processes, i.e., it is not included in the components of the vector X = (x1, …, xk), Then, based on the accepted assumptions, variations of values of each of the determining factors xi of chemmotological processes in the region D of experimentation are set by functions of continuous uniform distribution F(xi) and/or by corresponding functions of probability density f(xi):

$$F({x}_{i})=\{\begin{array}{c}1\text{\hspace{0.17em}}{\text{a}}{\text{t}}\text{\hspace{0.17em}}{x}_{i}>{x}_{i}^{\text{u}}\\ \frac{{x}_{i}-{x}_{i}^{1}}{{x}_{i}^{1}-{x}_{i}^{\text{u}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{a}}{\text{t}}\text{\hspace{1em}}{x}_{i}\in [{x}_{i}^{1},{x}_{i}^{\text{u}}]\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{a}}{\text{t}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{i}<{x}_{i}^{\text{u}}\end{array}\},$$
(7)
$$f({x}_{i})=\{\begin{array}{c}\frac{1}{{x}_{i}^{1}-{x}_{i}^{\text{u}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{a}}{\text{t}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{i}\in [{x}_{i}^{1},{x}_{i}^{\text{u}}]\\ 0\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\text{a}}{\text{t}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{x}_{i}\notin [{x}_{i}^{1},{x}_{i}^{\text{u}}]\end{array}\},$$
(8)

where [xil, xiu] - range of variation of factors when the experiment plans, for example, of the second-order [8, 13], are implemented for identification of the parameters of model (3).

In fact, distribution of probabilities of operation conditions and conditions of performance of the equipment is not uniform over time. If information presented in a definite fashion were available on empirical (selective) distribution, the integral assessments of the level of FL performance properties and the corresponding criteria, contrary to (4) and (6), could be obtained in a more general form:

$${\text{Int}}_{Q}({\Pi }_{\text{pp}})=\int \dots {\int }_{G}F(X\text{,}t)Q(X\text{,}t)dXdt;\text{\hspace{1em}}0\le Q(X\text{,}t)\le 1;\text{\hspace{1em}}\int \dots {\int }_{G}Q(X\text{,}t)dXdt\equiv 1;\text{\hspace{0.17em}}{K}_{\text{pp}}^{Q}={\text{Int}}_{Q}{({\Pi }_{\text{pp}})}_{N}{[{\text{Int}}_{Q}{({\Pi }_{\text{pp}})}_{E}]}^{-1},$$
(9)

where Q(X, t) - a certain weight function; t - time; G - region of determination (integration) of functions F(X, t) and Q(X, t); subscript Q indicates weighted assessments and criteria.

In this case, it is obvious that the weighted average value of integral assessment of the level of FL performance properties obtained for the entire region of determination of the functions F(X, t) and Q(X, t) will be:

$$\langle {\text{Int}}({\Pi }_{\text{pp}})\rangle =\frac{\int \dots {\int }_{G}F(X,t)Q(X,t)dXdt}{\int \dots {\int }_{G}Q(X,t)dXdt}.$$
(10)

In the most common case, for getting an integral assessment of the level of FL performance properties, the joint distribution of values of the parameters of operation conditions and conditions of performance of the equipment (factors of chemmotological process implementation conditions) can be represented by the functional L[xh(t)] of distribution of probabilities of k-dimensional vectorial random function (vectorial random process) X(t), the components of which are random functions X1(t), …, Xk(t) of these parameters (factors) and time:

$$\Lambda [{x}_{h}(t)]=P({X}_{h}(t)<{x}_{h}(t),\text{\hspace{0.17em}}h=1,\dots ,k).$$
(11)

Then, the laws of distribution of the referred components of the random function X(t) can be obtained by knowing its characteristic functional y[φ(t)], which is an exhaustive probability characteristic for X(t) [14, 15]:

$${\psi}[\upvarphi (t)]={\psi}[{\upvarphi}_{1}(t),\dots ,{\upvarphi}_{k}(t)]=M[\mathrm{exp}\{i\sum_{h=1}^{k}\int \dots {\int }_{T}{\upvarphi}_{h}(t){X}_{h}(t)dt\}],$$
(12)

where M - symbol of mathematical expectation; i - imaginary unit; the integral covers the whole region T of change in the argument t.

As a result, there appears an option that can be used for calculation of integral assessments instead of Q(X, t) in equation (9) of multidimensional function of density of probability of distribution of vector components of determining factors of chemmotological processes over time p(X, t), which, in principle, can be found by k-multiple reverse transformation of Fourier functional y[φ(t)].

The technology of simulative modeling of complex systems is emerging as another methodological foundation and another source of required initial data for integral assessment of the level of performance properties of FL [16,17,18]. In this context, it is recommended to use, for example, the provisions of the method of simulative modeling of chemmotological processes [19] with setting up of a simulative experiment, in which probabilistic-statistical models of distribution of conditions of use of various types of equipment over time are put in a computer in conformity to multidimensional field of results (rate) and then are processed together.

Some of the recent results [20,21,22] of use of integral assessments and criteria (4)-(6) obtained by use of mathematical models of chemmotological processes (3) are presented below as an illustration of separate potentials of the proposed method. Thus, for a generalized characteristic and prediction of the effectiveness of FL use in an equipment, integral assessments were obtained in options (7) and (8) relying on statistical hypothesis about uniform distribution of dependencies of the parameters of the working process and conditions of operation of the equipment on time [20, 21] and in option (9) based on weighted assessments and criteria [22].

In studies of carbon deposition in the cylinder area of land-based diesel equipment [20], analysis of integral assessments and criteria of affinity of fuels and their composites with additives to formation of high-temperature deposits made it possible to confirm, in particular, that depressing additives significantly increase (as much a 60 % and higher) the potential capacity of the fuel for deposit formation in the whole range of conditions, but it was observed that in isolated narrow ranges of maximum values of determining factors these additives even reduce the amount of deposits. To neutralize the adverse effect of depressants, 2-ethylhexyl nitrate and oligoorganosiloxane were added to the fuel, which led, as comparison of integral assessments showed, to a decrease of deposits of fuel composites in comparison with base fuel by up to 35 %.

Determination of the stability of properties of 7-50S-3, ASTIM and AMG-10 hydraulic fluids for aviation equipment submitted to mechanical damage and high-temperature oxidation made it possible to predict, by use of the integral assessment method, the service life of the fluids under various conditions of use and to take judicious decision about their use [21]. Thus, by comparing integral assessments and criteria of stability of properties of hydraulic fluids under the condition that 7-50S-3 fluid, which is least submitted to mechanical damage, was taken as the standard (reference) it was shown that the potential affinity of the fluids to viscosity degradation increases in the sequence 7-50S-3 < ASTIM < AMG-10, and was also determined how much the stability of viscosity of the standard exceeds the level of this property of the other fluids. During these investigations, it was also found that if, based on the conditions of performance of hydraulic fluid systems, one had to focus on a fluid that has the maximum resistance to oxidation at high temperatures, then the fluid AMG-10, the integral evaluation of which in terms of thermal oxidizability (acid number) is minimum, it turned out to be the best as a standard.

In investigations of the problem of prolonged storage of automotive fuels [22] we obtained weighted predictive values of integral assessments Int(Wevap) of the tendency of the fuels and their composites to evaporation losses in the whole range of storage conditions with due regard for the variation of these conditions over time. In this case, the degree of activeness in terms of indicators of evaporability of one fuel (component) during storage vis-à-vis others was determined from the criterion Kevap; for fuels: DT-3-K5 < TS-1 < Regular-92 (sample No. 1) < Regular-92 (sample No. 2) < Regular-92 (sample No. 3); for components: alkylate < reforming gasoline < catcracking gasoline < straight-run gasoline < isomerisate < MTBE (methyl tertiary-butyl ether) < isopentane fraction.

In conclusion, it should be noted that point assessment of FL performance properties can be regarded as an analog of instantaneous density at various “points” of random field of results of chemmotological processes and integral assessments of the level of FL performance properties, as an analog of the mass of irregular multidimensional objects, the dimensions of which are confined by regions of variation of determining factors of chemmotological processes.

In content, integral assessment of the level of FL performance properties is a quantitative characteristic of the limit of potential tendency toward transformations in a fixed range of values of determining factors of the conditions of use. In this sense, the integral assessment serves as a variant of performance property because its limiting value remains unchanged for any combination of factors in the studied range. Integral assessments and criteria of performance properties can be interpreted as results of multidimensional “convolution” of all quantitative information about performance properties obtained during the tests (/use) of FL and represented in mathematical models of chemmotological processes. Calculations of integral assessments of performance properties and corresponding criteria make it possible to get and present in a nutshell the information that takes full account of the aggregate variations of results of FL conversions in the equipment and allows quantitative prediction of the level of performance properties of FL under real conditions of equipment use. In general, it can be stated that integral assessment of the performance properties provides a generalized numerical expression of the maximally possible effect of use of a specific FL specimen in the FL-equipment-performance system.