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Properties of the ﬂow matrix

The total outdoor airﬂow rate to each zone, i, is easily obtained by summing the

columns of the ﬂow matrix:

j ¼1

ð1:31Þ

And the total exﬁltration airﬂow rate from each zone, i, is the sum of the lines of

the ﬂow matrix:

i ¼1

ð1:32Þ

If there is no totally isolated chamber in the measured system, and if there is

some air exchange with outside (as is the case with any usual building), the

ﬂow matrix determinant, jQj, is positive and Q has an inverse, Q

1

The elements of this inverse Q

1

are given by:

jQj

ð1:33Þ

where A

are the cofactors of the element Q

in Q.

Transfer of contaminants between zones

The basic equations applied to the case where a constant ﬂow rate, I

,ofa

contaminant, k, is applied in each zone, i, leads to an equilibrium concentration

(for constant airﬂow rates) that is:

Cð1Þ ¼ Q

1

I ð1:34Þ

It follows that the equilibrium concentration in room, j, resulting from a

contaminant, k, released only in room, i, is:

ð1Þ ¼ W

ð1:35Þ

and the non-diagonal elements of Q

1

are hence the transfer indexes deﬁned in

Sandberg (1984).

Using a simple inversion of the ﬂow matrix, much info rmation on the

possible spreading of contaminants can be obtaine d.

Age matrix and mean age of air

The  matrix is deﬁned as:

 ¼ Q

1

M ð1:36Þ

or, under the assumptions of constant, uniform temperature:

 ¼ q

1

V ð1:37Þ

Where q is the volume ﬂow matrix and V a diagonal matrix with the volumes

of room i on the diagonal. It is shown (Sandberg, 1984) that the row

10 Ventilation and Airﬂow in Buildings

1 2 ... 26 27 28 29 30 31 32 33 34 35 36 ... 210 211

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