The expediency of using heat-insulating construction products from a straw at the erecting of energy-efficient envelope constructions is considered in the paper. The hierarchical model in the form of an inference tree of the factors influencing the target function – reliability of energy efficiency of heat-insulating building products made of straw has resulted. A fuzzy matrix of knowledge is proposed which reflects the influence of thermophysical, physic-mechanical and durability parameters on the target function. The hierarchical connections between classified factors proceeded by apparatus of fuzzy logic and linguistic variables. A system of fuzzy logical equations which describes linguistic expressions of input variables according to the corresponding terms is proposed. In the research the expressions which describe the objective function – reliability of energy efficiency of heat-insulating construction products made of straw were obtained. It was made with the of membership functions following linguistic variables, by taking into account both qualitative as well as quantitative factors of influence. Represented in the paper model can be used as the design and engineering tool for the prediction of thermal performance of any multilayered wall assembly at the design stage of the project to assess complex energy efficiency parameters, which could be applied in practice during the decision-making process.

The concept of “energy efficiency” has many interpretations, but if to express the most typical, it obviously will be the degree or measure of efficient use of any energy resource that is brought to the object that is the consumer of these resources.

One of the quantitative definitions of energy efficiency of a building can be the Energy conversion efficiency, _{ useful output} / E _{input}_{useful output}_{input}

According to the authors of the review [

The maximum energy efficiency will have those buildings in which at the same rate of the energy conversion efficiency, the amount of total investment in the installation of thermal insulation and engineering systems to ensure optimal indoor climate will have a minimum value at their maximum durability. Significant energy consumption to maintain the thermal comfort of the premises does not increase the energy efficiency of buildings.

In modern scientific research solving, the problem of decision-making in uncertainty conditions where the choice of alternatives requires the analysis of complex information of different physical nature in search of the best solution is called a methodology of systems analysis. The essence of system analysis is multicriteria (decision) analysis (MCDA). That is why, when designing energy-efficient buildings, there is a need for a systematic analysis of the factors that affect the multicriteria (as usual) value of the objective function. With the help of system analysis, it is possible to substantiate the optimal design solution to reduce energy costs for the construction and operation of the building as a single energy and environmental system.

The concept of “reliability” is used very often in various branches of technics. In the material of this paper, the authors consider the term “reliability” as an integral indicator that numerically expresses the probability of providing several guaranteed thermophysical, physic-mechanical and durability characteristics in the design of thermal insulation products.

The author [

Ensuring the reliability of the thermal insulation shell guarantees the maintenance of optimal microclimatic conditions of the premises at the actual consumption of thermal energy for heating. When substantiating the need for construction of low-rise buildings using multilayered thermal insulation products made of straw, there is a need to assess the reliability of their energy efficiency, by taking into account both quantitative and qualitative parameters of impact.

The scarcity of fossil energy reserves, especially oil, natural gas and coal, leads to a significant increase in their value. This requires the implementation of advanced up to dated technologies aimed at reducing energy consumption to ensure the affordable thermal regime of buildings [

The authors of the study [

The materials are recommended for the installation of thermal insulation shell by the Ukrainian Code of thermal insulation [

In paper [

In the researches [

The conducted analysis of literature sources [

There is also no methodology in Ukrainian Building Codes for assessment of their energy efficiency, during the erecting of the thermal insulation shell. All of the abovementioned encouraged the authors to dedicate the present research to the developing of a mathematical model for assessing the reliability of energy efficiency of thermal insulation construction products made of straw using a fuzzy knowledge base.

To do this, the authors propose to solve the following problems:

Construction of a hierarchical classification of quantitative and qualitative factors influencing the reliability of energy efficiency of thermal insulation building materials from straw using an expert fuzzy knowledge base.

Development of a mathematical model for complex assessment of the reliability of thermal insulation building materials made of straw, based on linguistic variables by fuzzy rules and fuzzy logical operations.

In assessing the reliability the concept of the structural-probabilistic model was used [

– uniform distribution of reliability;

– weights;

– indefinite Lagrange multipliers;

– matrix method of reliability calculation;

– Delphi method;

– method of statistical modelling, etc.

The reliability assessment method’s selection is based on the following criteria: the object life cycle, the failure criteria, etc.

The reliability of the efficiency ensuring of the thermal insulation construction products can be determined by the probability of their complex parameters failure-free during the life cycle of the building by the formula [

Evaluation of the objective function – the reliability of energy efficiency of thermal insulation construction products from straw using equation (

According to this mathematical apparatus, the fuzzy set by which the term _{1}, _{2},…, _{n}

_{F}_{i}_{i}

The logical conclusion between cause and effect is described by a system of fuzzy logical statements, respectively:

For the MIN operation

For the MAX operation

Fuzzy logical operations

The rule of distribution of belonging degrees according to the normalization (μ_{1} + μ_{2} +… _{n}

where _{i}_{F}_{i}_{i}

_{i}_{F}_{i}

The degree of membership _{F}_{i}_{i}_{1} / r_{j}_{ij}

By the known line elements of the matrix (_{ij}_{i}_{j}_{kj}_{k}_{j}

In this matrix (

In the research [

Taking into account the results of research [

Hierarchical model in form of the inference tree for assessment of the energy efficiency reliability of heat-insulating construction products from straw

The root of the logical inference tree corresponds to the value of the objective function – the assessment of the reliability of energy efficiency of thermal insulation products made of straw, and hanging tops – quantitative and qualitative thermo-physical, physicmechanical parameters as well durability indicators as linguistic variables, respectively.

According to the principle of linguistic variables, the causal relationships between the influencing factors of the model are described using fuzzy terms [

Reliability of efficiency of heat-insulating building products from a straw for envelope designs according to researches [

where _{1}, _{2}, _{3} – linguistic variables, which describe thermo-physical, physic-mechanical and durability indicators as linguistic variables, influencing the reliability of ensuring the energy efficiency of heat-insulating construction products made of the straw.

The meaningful interpretation of the factors influencing the reliability of ensuring the energy efficiency of heat-insulating construction products made of straw and the corresponding set of linguistic evaluations is described by relations (

A linguistic variable describing the thermo-physical factors influencing the reliability of the energy efficiency of heat-insulating construction products made of straw can be represented by the ratio_{11} – specific heat capacity of the insulating building product c is determined on the universal sets _{11}) = (800; 1300; 1800) (J/(kg×K). Linguistic values of this factor are given by the term set

T(_{11}) = <low, medium, high>;

_{12} – thermal conductivity of heat-insulating construction product is determined on the universal set _{12}) = (0.07; 0.10; 0.15; 0.20; 0.25), (W/m×K). The linguistic variable of this factor is given by the term set _{12}) = <low, below average, average, above average, high>;

_{13} – heat absorption coefficient of thermal insulation material of a construction product is determined on the universal set U(_{13}) = (0.5; 1.0; 1.5) (W/m^{2}K). The linguistic variable of this factor is given by the term set T(_{13}) = <low, medium, high>;

_{14} – vapor permeability coefficient of thermal insulation material of the construction product is determined on the universal set _{14}) = (0.02; 0.04; 0.06) (mg/m× h×Pa). The linguistic variable of this factor is given by the term set _{14}) = <low, medium, high>;

_{15} – the coefficient of air permeability of the thermal insulation material of the construction product is determined on the universal set _{15}) = (0.15; 0.35; 0.45) (kg/m^{3}h). The linguistic variable of this factor is given by the term set _{15}) = <low, medium, high>;

_{16} – the dimensionless indicator of thermal inertia ^{2}×Κ);

_{i}

_{i}

_{i}^{3}.

The dimensionless indicator of thermal inertia _{16}) = (2; 6; 10). The linguistic variable of this factor is given by the term set _{16}) = <low, medium, high>.

A linguistic variable describing the physic-mechanical factors influencing the reliability of the efficiency of the heat-insulating building materials from a straw can be represented by the ratio (_{21} is the density of the heat-insulating construction product determined on the universal set _{21}) = (25; 50; 75; 100; 125) (kg/m^{3}). Linguistic values of this factor are given by the term set _{21}) = <low, below average, average, above average, high>;

_{22} – compressive strength of the thermal insulation material of the construction product is determined on the universal set _{22}) = (1, 2, 3, 4, 5) (arbitrary units). The linguistic variable of this factor is given by the term set _{22}) = <low, below average, average, above average, high>;

_{23} – frost resistance of thermal insulation material of a construction product is determined on the universal set _{23}) = (1, 3, 5) (arbitrary units). The linguistic variable of this factor is given by the term set _{23}) = <low, medium, high>;

_{24} – sound absorption of thermal insulation material of a construction product is determined on the universal set _{24}) = (20; 30; 40) (dB). The linguistic variable of this factor is given by the term set _{24}) = <low, medium, high>.

The linguistic variable that describes the durability indicators as factors influencing the reliability of the energy efficiency of heat-insulating materials made of straw can be represented by the equation (_{31} – fire resistance of heat-insulating material of a construction product is determined on the universal set _{31}) = (15; 30; 45; 60; 90) (minutes). The linguistic variable of this factor is given by the term set _{31}) = <low, below average, average, above average, high>;

_{32} – chemical resistance of heat-insulating material of a construction product is determined on the universal set _{32}) = (1, 3, 5) (arbitrary units). The linguistic variable of this factor is given by the term set _{32}) = <low, medium, high>;

_{33} – biological resistance of heat-insulating material of a construction product is determined on the universal set _{33}) = (1, 3, 5) (arbitrary units). The linguistic variable of this factor is given by the term set _{33}) = <low, medium, high>.

The knowledge matrix for the approximation of dependence (

Fuzzy knowledge matrix for equation (

IF | THEN | ||
---|---|---|---|

L | L | L | L |

L | L | BA | |

L | BA | L | |

BA | L | L | |

L | BA | BA | BA |

BA | L | BA | |

BA | BA | L | |

BA | BA | BA | |

A | A | A | A |

BA | A | A | |

A | BA | A | |

A | A | BA | |

AA | AA | AA | AA |

AA | AA | A | |

AA | A | AA | |

A | AA | AA | |

AA | H | H | H |

H | H | H | |

H | AA | H | |

H | H | AA |

Linguistic expressions, given in the knowledge matrix for the approximation of the dependence (

According to the mathematical apparatus of fuzzy logic [

The membership function is a set of values of μ (μ_{i}_{i}

In this paper, the algorithm of the membership function’s calculation is explained in the example of the influence factor _{21} – density of heat-insulating material. As it was revealed above, this factor is characterized by the corresponding universal set _{21}) = (25; 50; 75; 100; 125) (kg/m^{3}). It should be mentioned that the designing method of these membership functions is based on the pairwise comparison, as described in [

The linguistic evaluation used the term set _{21}) = <low, less than average, medium, more than average, high>. The matrix _{low}_{21}), which characterizes the pairwise comparisons of different values of the density of the heat-insulating material in terms of their proximity to the term “Low”, is filled as below (

According to the matrix (_{1}, _{2}, _{3}, _{4} and _{5} to the term “low” is performed as follows:

According to the abovementioned method of finding the membership degree to the term “Low”, matrices of pairwise comparisons for the terms “Below the Average”, “Average”, “Above the average” and “High” and the individual membership degrees are calculated respectively.

To facilitate the comparison of all membership degrees, their normalization by dividing to the maximum value is performed as shown below in formula (

Thus, the normalized values of the terms “Low”, “Below the Average”, “Average”, “Above the Average” and “High” calculated by formulas (_{“Density”}_{“Density”}

Graphic representation of membership functions for linguistic variable _{21} (“Density”) as the factors of influence on the objective function is shown in Fig. _{21} “Density” have the following values. The term “Low” is characterized by an inverse direct proportional dependence of the fuzzy set on the density of the material. The impact factor has the highest value at 25 kg/m^{3}, the minimum for 125 kg/m^{3}. The term “High” is characterized by a directly proportional relationship, with the lowest value for 25 kg/m^{3}, and the highest for 125 kg/m^{3}. For the factors of influence “Below the Average”, “Average” and “Above the Average” the nature of the dependence is curvilinear. The highest value for μ(_{21}) = 1 for a density of 50 kg/m^{3}, for the term “Average” – 75 kg/m^{3}, and “Above the Average” – 100 kg/m^{3}.

Membership functions for the linguistic variable “Density”

The considered technique allows constructing graphic dependences for the other factors of influence resulting in the inference tree (see Fig.

From Fig.

It is obvious, that proposed hierarchical model of the inference tree (see Fig.

In the authors’ opinion, the more real data could be collected, the more objective and comprehensive will be an assessment of the energy efficiency reliability ensuring in the terms of abovementioned influence factors.

With the aid of apparatus of the fuzzy logic and linguistic variables, a hierarchical model in the form of a logical inference tree is proposed, which contains the main quantitative and qualitative factors influencing the objective function – reliability of energy efficiency of heat-insulating building materials made of straw.

The calculation of the parameters of the mathematical model for evaluating the reliability of ensuring the energy efficiency of heat-insulating building materials made of straw on the example of the linguistic variable “Density” is conducted. Numerical values of its fuzzy set with a graphic interpretation of results are calculated. Further researches in this field should be conducted and could be dedicated to a real assessment of the reliability and fine-tuning of this previous model data, with that, obtained from the results of the real field research.

According to the authors’ opinion, the proposed fuzzy-probabilistic approach can be used as an additional tool in Building Energy Modeling (BEM) as well in the decision-making process to estimate the construction and operation cost of buildings.

This will allow evaluating, and, if necessary, increasing the reliability of energy efficiency of multilayered thermal insulation envelope structures of any type of materials, including straw.