**Scientific Foundations for the Design of Energy Efficient Buildings**

By Yu.A. Tabunschikov, M.M. Brodatch

**INTRODUCTION**

The task of designing energy efficient buildings falls under the category of the so-called "system analysis" problems or "operations research" problems, the search of designs for which involves making a choice among the alternatives and requires to make an analysis of a body of complex information of various physical nature [1, 2]. The system analysis or operations research methods aim at producing preliminary quantitative justification of optimum designs. "Optimum designs" are meant here to be those that are preferable to all others by certain criteria.

Operations research includes three main steps:

- construction of a mathematical model, i.e., mathematical description of the process;
- selection of a goal function. This will include producing the definition of the boundary conditions and formulation of the optimization problem;
- solving the specific optimization problem.

It should be noted that making the final decision will go beyond the operations research area falling within the terms of reference of the responsible person (more often, a group of responsible persons) who is (are) authorized to make the final choice and will bear responsibility for it. In making the choice, the responsible person(s) can take into account, along with recommendations following from the mathematical analysis, some other qualitative or quantitative considerations that were not part of the analysis.

The mathematical model and the goal function

for the energy efficient building

In accordance with the system analysis methodology, it is advisable to present the mathematical model of the thermal conditions of a building viewed as a single thermal system in the form of three interrelated models that are more convenient to study [3, 4, 5]:

- a mathematical model of the effect of outdoor thermal conditions on the building;
- a mathematical model of thermal accumulation properties of the building shell;
- a mathematical model of the thermal balance inside the building.

A detailed description of the mathematical models of individual building components and the building as a single energy system is given in [3, 4, 5].

The optimization problem for the energy efficient building has the contents as follows: **defining the indicators of architectural and engineering designs of the building, which provide for the minimization of energy consumption needed to create a proper microclimate inside the building.** In a generalized mathematical form, the goal function for the energy efficient building can be written as:

Q_{min} = ?(?_{i}),

where Q_{min} - minimum energy consumption for the creation of a proper microclimate inside the building;

a_{i} - indicators of architectural and engineering designs used in the building, which provide for the minimization of energy consumption.

In doing a real design, the energy efficient building in most cases will not be realized due to certain restrictions imposed by the specific construction situation or certain qualitative or quantitative considerations that were not taken into account in constructing the mathematical model. If such is the case, it is advisable to introduce a indicator characterizing the extent to which the realized design differs from the optimum one. In other cases the same indicator could serve as a criterion for assessing the designer's craft. Let's call this ?, a *"thermal efficiency indicator of the desig"*; therefore, by definition:

? = Q_{eff} / Q_{acc},

where Q_{eff} - energy consumption for the creation of a proper microclimate inside the energy efficient building; Q_{acc} - energy consumption for the creation of a proper microclimate inside the building accepted for design.

Given the assumed subdivision of the mathematical model of the thermal conditions of the building viewed as a single thermal system into three interrelated submodels, we obtain:

? = ?_{1}·?_{2}·?_{3},

where ?_{1} - indicator of the thermal efficiency resulting from the optimum account of the effect of outdoor climatic conditions on the building; ?_{2} - indicator of the thermal efficiency resulting from the optimum selection of thermal-proof and solar-proof properties of enclosing structures; ?_{3} - indicator of the thermal efficiency resulting from the optimum selection of the systems to provide for a proper microclimate.

Optimization of the thermal effect of the outdoor

climatic conditions on the building thermal balance

Thermal effect of the outdoor climatic conditions on the building thermal balance can be optimized by selecting the building shape (for rectangular buildings, parameters such as dimensions and orientation will be taken into account), arrangement and area of fenestration, regulation of filtration flows. For instance, proper selection of orientation and dimensions of a rectangular-shaped building will allow to reduce the effect of solar radiation on the building shell during the warm months, thus reducing the cooling costs, and to increase the effect of solar radiation on the building shell during the cold months, thus reducing the heating costs. Similar results will be obtained by properly selecting the orientation and dimensions of the building with respect to the wind effect on the building thermal balance.

The design methodology for heating, ventilation and air conditioning systems is based on the analyses of thermal and air balances of the building for the characteristic time intervals of the year. For Russia, such periods include: the coldest five days, the heating season, the warmest month, the cooling season, and the design-basis year. In this case, optimization of the thermal effect of the outdoor climatic conditions on the building thermal balance by selecting its shape and orientation will produce results as follows:

*for the coldest five days*- reduced installed capacity of the heating system;*for the heating season*- reduced heat consumption for heating;*for the warmest month*- reduced installed capacity of the air conditioning system;*for the cooling season*- reduced energy consumption for cooling the building;*for the design-basis year*- reduced energy consumption for building heating and cooling.

Generally, it is possible to optimize the thermal effect of the outdoor climatic conditions on the building thermal balance for any characteristic time interval.

It is important to note that changing the shape or dimensions and orientation of the building in order to optimize the effect of the outdoor climatic conditions on its thermal balance will not require changing the area or volume of the building, which will remain the same.

[7] gives a solution of the problem of selecting the optimum shape of the building.

Table 1.Building Thermal Efficiency Upgrading by Optimizing the Effect of Outdoor Climatic Conditions on Building Thermal Balance |
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Design Basis Interval | Thermal Efficiency Upgrading, %% | |

Moscow | Rostov-on-Don | |

Coldest Five Days | 7 | 8 |

Heating Season | 12 | 15 |

Cooling Season | 22 | 25 |

Warmest Month | 15 | 18 |

[5] gives a solution of heat energy optimization of the shape of the building in a more general way. It also gives the solution of a "thermal efficiency indicator of the design", which from one side displace the possibilities for decreasing the energy consumption of the building, from other side proves the skill of a designer in designing a building according to the climate.

The authors conducted a study of the thermal effect of outdoor climatic conditions on the building thermal balance by optimizing the dimensions and orientation of the building. The analysis was made for the climatic conditions of the cities of Moscow (56° NL) and Rostov-on-Don (48° NL). Initial orientations assumed were latitudinal, meridional and diagonal. Studied was a rectangular-shaped building with a 1,440 m^{2} usable area. Minimization of energy consumption for building heating during the cold season or building cooling during the hot season was assumed as a goal function. The purpose of the study was to determine the quantitative growth of the building thermal efficiency indicator by optimizing the account of the effect of outdoor climatic conditions on the building thermal balance. Table 1 shows the results of the study.

Thermal protection optimization of the enclosing structures

Traditionally, thermal protection optimization of the enclosing structures of buildings is understood as a method of calculation "by the minimum of the present worth costs" of how thick the thermal insulation of a structure should be. The mathematical model of the present worth costs will generally include two items, structure manufacturing costs (onetime costs) and structure utilization costs (operating costs). Thermal insulation design "by the minimum present worth cost" is an objective, internationally accepted method; however, inherent in it there is a latent risk that reflects the reality of the current economic situation in Russia, which may be an insurmountable barrier to practical implementation of the method. This is because the method uses energy and material costs.

Some works [3, 4, 6] show a possibility of solving this problem in its current perspective and using state-of-the-art methods. By saying current perspective we mean that a solution will be achieved that is the most preferable given the assumed restrictions, and state-of-the-art methods are the operations research methodology. Let's elaborate on this.

Generally, there is quite a few requirements specified for the outer enclosing structures. These include high thermal protection during the cold season under heat transfer close to the stationary operating conditions; high thermal stability during warm and cold seasons under heat transfer close to the periodic operating conditions; low energy intensity of the inner layers under variation of the heat flow inside the room; high air-proofness; low moisture absorption, etc.

The main focus in the designing process is certainly on meeting the principal requirements, in the first place. As practical experience shows, there are, as a rule, not more than two such requirements. First of all, these are thermal protection and thermal stability. There are vast opportunities for their optimization. Such an optimization will imply designing, using the operations research method, an enclosing structure to optimally meet the thermal protection and thermal stability requirements (standards).

[6] provides a solution of the problem of optimization of the arrangement of the layers of materials in a multi-layer enclosing structure. Also, a detailed solution of the problem is offered and it is shown that the thermal stability can vary three-fold depending on the order of arrangement of the layers of material in the structure.

[3] provides a solution of the problem of selection of material for a multi-layer enclosing structure of pre-specified thickness, which ensures the maximum attenuation of the outdoor heat effect. The solution obtained is as follows: material with lower heat conductivity and higher volumetric heat capacity will provide the maximum attenuation. Hence, it is advisable to select a structure with lower heat conductivity of the materials for hot climate regions and the one with a higher heat gain coefficient of the materials for cold climate regions.

Table 2.Room Warming Up: Energy Consumption Analysis |
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Alternatives | Warming Up Time (?, hr) and Energy Consumption (Q, W·hr) for Room Warming Up At Various Heat Convection Coefficients | |||||||||||

a_{1}=3,5 W/(m^{2}·°C) |
a_{2}=10,5 W/(m^{2}·°C) |
a_{3}=21 W/(m^{2}·°C) |
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a | Q | Energy Saving, %% |
a | Q | Energy Saving, %% |
a | Q | Energy Saving, %% |
||||

ES_{1} |
ES_{2} |
ES_{1} |
ES_{2} |
ES_{1} |
ES_{2} |
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Brick masonry, solid brick on the sand cement mortar | 9,7 | 58100 | 0 | 0 | 3,5 | 20970 | 64 | 0 | 1,2 | 7160 | 88 | 0 |

Expanded-clay concrete | 2,5 | 35200 | 0 | 40 | 0,9 | 12560 | 64 | 40 | 0,31 | 4330 | 88 | 40 |

Sandwich panel with a warmth-keeping jacker of plate foamed plastic | 0,6 | 15650 | 0 | 56 | 0,2 | 4715 | 70 | 62 | 0,08 | 1940 | 88 | 55 |

[4] offers a solution of the problem of determination of the limit values of thermal protection of the outer enclosing structures of a room not equipped with an air conditioning unit at a given solar-proof value of the windows and a given air change rate. Following are some interesting findings that resulted from the solving:

- thermal protection of the enclosing structures has no effect on the temperature conditions of a room at certain values of solar-proofness of the windows and the air change rate;
- if thermal protection of the windows is inadequate and the air change rate is low, then increasing thermal protection of the outer enclosing structures causes worsening of the heat conditions in the room.

The latter finding requires the designers to pay special attention to specifying the outer enclosing structures with an effective thermal insulation for buildings to be constructed in warm climate regions.

[3] contains some interesting approaches to the optimization of thermal protection of the outer enclosing structures of buildings with air conditioning, for windows with heat-reflecting film, for buildings with periodic heating, etc.

Optimizing the heat load on the air heating of a building

It is obvious for a specialist involved in designing and making analysis of the heating, ventilation and air conditioning systems that the objective of the design and analysis is determination of two interrelated indicators: amount of energy and method of its distribution (dispensing). What is actually meant here is analyzing and designing a system for control of consumption and distribution of energy that will be able to provide minimum energy consumption during operation. Therefore, a problem of optimization of the thermal load on the system for the provision of the specified heat conditions inside the building falls under the category of the so-called optimum control problems, and it will have the meaning as follows: seeking the control of consumption of energy Q(t) needed to provide proper heating, which will satisfy the thermal balance equation of the room and appropriate initial and final thermal conditions, for which energy consumption:

I = ?Q(t)dt

has the lowest possible value.

Control Q(t), that produces the solution of the set problem is called the optimum control while the respective indoor air temperature variation path is called the optimum path.

The solution of the problem was obtained by the authors and is described in [5].

Minimizing the room warming time is the sum and substance of the solution.

Since a real room is a combination of heat gaining enclosing structures and heat gaining indoor equipment (furniture), the warming process implies the temperature rise of the whole collection of the room components, i.e., enclosing structures and equipment. Components with high heat accumulation will require longer time for their warming up. Therefore, minimizing the room warming time can be achieved by minimizing the warming time for components with high heat accumulation. We can easily point out two simple cases: the room warming time will tend to the minimum where the materials of the inner surfaces of the enclosing structures feature low heat gain coefficients, as well as where high intensity of heat convection takes place between the indoor air and inner surfaces of the enclosing structures. Optimally, the two cases coincide.

That the above solution is correct was further corroborated during the discussion of the presentation made by the authors on this subject-matter in the Technical University of Denmark. According to the Danish specialists, during the restoration of a Roman Catholic cathedral, they decided to arrange the warming up in such a way that the process begins with warming up the massive stone arm-chairs for the parishioners using electric heaters with a purpose to save power given the lower indoor air temperature during the night hours. The resulting power saving reached 30-35%.

The authors made a power consumption analysis for a 24 m^{2} area, 72 m^{2} volume room with two outer enclosing structures and a 3 m^{2} double-glazed window. Three alternatives were considered for the outer enclosing structure:

- a 0.56 m thick brick masonry;
- a 0.23 m thick expanded-clay concrete panel;
- a 0.052 m thick sandwich panel, with a warmth-keeping jacket of plate foamed plastic, coated with metal sheets on both sides.

For comparing the analytical results, the above enclosing structures have the same thermal resistance. The assumed air change rate is 3 1/hour.

The outdoor air temperature is minus 5°С.

The initial conditions are as follows: indoor air temperature is 10°С, temperature of the inner surfaces of the enclosing structures is 10°С.

The final conditions are as follows: indoor air temperature is 22°С, temperature of the inner surfaces of the enclosing structures is 14°С.

In order to provide for the minimization of the warming up time, it is assumed that the warming up is done with convection heat streams spreading over the inner surfaces of the enclosing structures (Fig. 3). The heat convection intensity was equal to the following three values of the heat convection coefficient: a_{1}=3,5 W/(m^{2}·°C); a_{2}=10,5 W/(m^{2}·°C); a_{3}=21 W/(m^{2}·°C).

Table 2 shows the results of the analysis.

Table 2 uses the following symbols: Q - energy consumption for the warming up, including heat loss through the windows and that due to air change; ES1 - energy saving through increased heat convection intensity, the enclosing structure being the same; ES2 - energy saving through reduced heat accumulation of the enclosing structure (reduced heat gain coefficient).

The result obtained looks incredible if seen in the "common sense" perspective: the maximum energy saving in warming up the room so as to minimize the warming up time reaches 97%.

This result was achieved by selecting the optimum energy distribution strategy for a room where energy is consumed; that is to say, the heating started with warming up the high heat gain enclosing structures. Consideration of Table 3 allows to make the following conclusions:

- energy saving in the warming up of a room through a 3-fold increase of the heat convection intensity reaches 64-70%; if the same increase is 6-fold, then the energy saving reaches 88%, with the warming up time being shortened 3 times, on the average;
- energy saving in the warming up of a room through a 2.4-fold reduction of the heat gain of the enclosing structure (reduced heat gain coefficient) reaches 40%; if the same reduction is 10.4-fold, then the energy saving reaches 55-62%, with the warming up time being shortened, on the average, 3.8 times and 16 times, correspondingly.

References:

- Моисеев Н.Н. Математические задачи системного анализа. - М.: Наука, 1981./Moiseev. Mathematical problems in system analysis/
- Вентцель Е.С. Исследование операций. Задачи, принципы, методология. - М.:Наука, 1988./Wentzel. Operations research. Problems, principles, methodology/
- Табунщиков Ю.А. Основы математического моделирования теплового режима здания как единой теплоэнергетической системы. Докторская диссертация. - М.: НИИСФ, 1983./Tabunschikov. Foundations of mathematical modeling of thermal conditions of the building as a single thermal system. The D.Sc. Thesis/
- Табунщиков Ю.А., Хромец Д.Ю., Матросов Ю.А. Тепловая защита ограждающих конструкций зданий и сооружений. - М.: Стройиздат, 1986./Tabunschikov, Khromets, Matrosov. Thermal protection of enclosing structures of buildings/
- Tabunschikov Y. Mathematical models of thermal conditions in buildings, CRC Press, USA 1993.
- Jurobic S.A. An investigation of the minimization of building energy load through optimization techniques. Los Angeles scientific center, IMB Corporation, Los Angeles, California.
- Бродач М.М. Изопериметрическая оптимизация солнечной энергоактивности зданий. - Гелиотехника 2, Ташкент, 1990. /Brodatch. Isoperimetric optimization of solar energy activity of buildings/
- Бродач М.М. Энергетический паспорт зданий / АВОК, 1993, № 1/2.
- Klaus Daniels, "The Technology of Ecological Building", Birkhauser-Verlag fur Arhitektur, Basel, 1997. /Brodatch. Detail energy inspection records of buildings/.

*Yu.A. Tabunschikov
M.M. Brodatch
Moscow University of Arhitecture
Rozhdestvenka, 11
Moscow 103754, RUSSIA *