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straightforward, takes time and has its cost, the theory of experimental

design (Box et al., 1978) may help in providing a maximum of information

through a minimum of measurements.

Minimum number of measurements

A map of any scalar variable, v, in a three-dimensional room is in principle

obtained by measuring the variable at each node of a network and interpo-

lating between these nodes. Such measurements are, however, very expensive

and may be unfeasible. If only ﬁve values are taken on each axis, at least 125

measurements are re quired, meaning analysis of the air every few minutes at

125 locations over a couple of hours. Therefore, it makes sense to apply a

method that needs a minimum number of measurements points. This

minimum number depends on the objective of the mapping experiment, or

more precisely on the required mapping details. Since the interpolation

between meas urement points needs a model, the mapping network

indeed depends on the empirical model chosen to represent the map of the

variable, v.

Any inﬁnitely derivable function (as v is assumed to be) can be developed

in a Taylor series around a given point. This gives a polynomial, which can

be approximated by its k þ 1 ﬁrst terms, k being the degree of the poly-

nomial. In the following, models of degree 1 and 2 are considered. If a linear

model is ado pted (degree 1), such as:

v ¼ a þ

ð3:29Þ

where x

are the three coordinates of the measured point, only four measure-

ments are neede d to obtain a set of coeﬃcients ( a; b

). If more meas urements

are made, the coeﬃcients may be obtained by a least square ﬁt procedure

provided there is no (or negligible) uncertainty on the coordinates. If their

coordinates diﬀer for the other points, these supplementary measurement

points give information on the validity of the model used.

If the linear model does not appear to be valid, higher degree models may

be used. For example, a quadratic model:

v ¼ a þ

i 6¼j

ð3:30Þ

that contains ten coeﬃcients, can be chosen. Such a model may already ﬁt

many practical situations and present minimal and maximal value(s). To

determine its coeﬃcients, measurements taken at ten locations are the

minimum necessary.

An intermediate model is the interactions model:

v ¼ a þ

i 6¼j

ð3:31Þ

50 Ventilation and Airﬂow in Buildings

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