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# Basics and Some Theory of AnTherm

## Conductance

An exact description of the thermal behaviour of a building assembly would by nature be non-linear, and thus very complicated to evaluate. Fortunately, for most construction situations of interest, it is possible to drastically reduce the complexity of the physical model without sacrificing any appreciable accuracy.

A linear description of thermal transmission involving an entire building can be introduced by

•  assuming that the temperature of each environment (space) associated with the building is unique, i.e. independent of position, and
•  forfeiting an exact treatment of radiation exchange within the spaces in favour of an approximation of this factor through adjusted surface transfer coefficients.
 thermal conductance, L3D [W/K] In light of this simplified physical model, it can be stated that the amount of heat which flows from one space to another, Q, is proportional to the temperature difference between the two environments under consideration (i and j). The factor of proportionality is a conductance , Lij, and linearly defined by Q = Lij • ( Ti − Tj ) conductance matrix The analogy of an electrical circuit can be extended to model a building structure more generally as a set of environments thermally connected through a resistance, i.e. the heat-conducting material elements. A complete description of the thermal relationships in a particular structure is given by a conductance matrix of values for all thermal connections, i • j, between the spaces of the model. The conductance matrix defines the heat transfer characteristics of a given model based solely on the geometry and materials (thermal properties) of the resistance, that is, independently of temperature conditions in adjacent spaces. This matrix is symmetrical. thermal transmittance, U-value, L1D [W/m2K] The area-related term "thermal transmittance" (U-value in Wm-2K-1) commonly used in standards to date essentially describes the same conductance as the reciprocal of the sum of resistances of a planar component, that is, resistances in "series": 1/U = R = Rsi + ΣRj + 1/αse = 1/αsi + ΣRj + 1/αse whereby αsi and αse are the surface transfer coefficients of the interior and exterior environments, and ΣRj represents the sum of material resistances of j constituent component layers. The resistance of an individual homogeneous (isotropic) layer is directly proportional to layer thickness, d, and indirectly proportional to the material conductivity, λ: Rj = d/λHowever, this simple formula applies only to the planar regions of a building assembly in which strictly one-dimensional heat flow can be reasonably assumed. length-related conductance, L2D [W/mK] A further type of special situation is given where two-dimensional heat flow patterns are to be expected. Such a region is a stretch of the building assembly which can be evaluated with respect to a two-dimensional section - under the assumption that no heat flow occurs normal to the section plane. In this case, a length-related conductance, L2D [Wm-1K-1], must be calculated for the applicable region. The most general case, of course, is that of three-dimensional heat flow. For the regions of a construction in which no directional assumptions can be made about local heat flow patterns, only the evaluation of the (3D) conductance, L3D [WK-1], provides a reliable indicator of the thermal behaviour. total conductance Due to the linearity of conductances, an entire building can be modelled as a sum of parts, with each part evaluated according to the applicable geometric conditions. Of course, the summation of conductances is only applicable if the temperature difference is the same through all model parts (e.g. one interior and one exterior temperature valid for the whole building).The model is sub-divided by introducing theoretical cut-off planes, which must be located such that any potential heat flow through these planes can be considered negligible. The total thermal conductance of a building thus modelled can be written as whereby lj is the length over which the two-dimensional conductance, L2D, is valid for part j, and Ak is the area of validity for Uk. This reliable and flexible approach to analysing the thermal performance of buildings is referred to in the Standards as the direct method. It requires the implementation of a suitable computer program for attaining two- and three-dimensional conductance results with the necessary precision The program AnTherm makes use of the linear nature of the heat conduction model by first determining a generally applicable calculation model: a characteristic set of temperature-independent base solutions (see also "Method of Analysis").

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