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for which seven coeﬃcients must be determined. Table 3.2 summarizes the

minimum number of measurements needed.

Location of the measurement points

An important issue is the appropriate location of measur ement points. The

set of measurement points is called an experimental design. There are many

possible experimental designs, but they do not give the expected results

with the same accuracy. For example, it is obvious that to ﬁt a linear

model in one dimension only (the straight line modelled by y ¼ ax þ b), the

location of the two measurement points (the minimum number) that gives

the best accuracy for the coeﬃcients a and b is at the extremities of the

experimental domain, i.e. at the minimum and maximum possible values of

the variable, x.

For more sophisticated models or in a larger number of dimensions, the

locations of the best sampling points ar e not so obvious. However, several

tools exist for planning such experiments, which are fo und in the literature

(Fedorov, 1972; Box et al., 1978; Bandemer and Bellmann, 1979; Feneuille

et al ., 1983; Aeschlimann et al., 1986) and are applied below.

In experiments to determine the age of the air, points close to walls do not

represent the inner volume, and the sampling points should not be located too

close to walls or in the corners of the room. In the following, the ‘room’ or the

‘experimental domain’ is a volume that is smaller than the actual measured

space by about 20 per cent in each direction.

Let us take a coordinate system in such a rectangular volume using as the

unit, for each direction, the half-length of that domain in that direction.

Three numbe rs, included in the interval [1; þ1], locate any point in the

‘room’.

The experimental design can be represented by a rectangular matrix with

three columns (one for each coordinate) and as many lines as measurement

points. A general condition is that in order to obtain the coeﬃcients of a

polynomial of degree k, each of the variables x, y and z shall take at least

k þ 1 values in the experimental design, which should have at least k þ 1

levels on each axis.

Table 3.2 Minimum number of measurements needed to obtain the coeﬃcients of

a kth degree polynomial empirical model representing a variable in a two- and

three-dimensional space

Model Linear Interaction Quadratic Cubic 4th degree

dimensions

2 3 4 6 10 15

347102035

Age of Air and Ventilation Eﬃciency 51

1 2 ... 67 68 69 70 71 72 73 74 75 76 77 ... 210 211

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Sharp JX-9400 Technical Information Seite 72