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This system of 27 equations when combined with the system of Equation 2.9

can be solved in various ways to provide the six main airﬂow rates and

potentially ten parasitic ﬂow rates. This global system of equations contains

more equations than unknowns. There are several ways to address this

situation, and we have found that some methods are better than others for

application to air handling units. Therefore we present the tested methods

below.

Least square solution

The system of equations from 2.9 to 2.19 is over-determined: there are 26

equations for calculating 16 airﬂow rates. In zones where

6¼ 0, the system

could be solved by least square ﬁt:

¼½C



1

ð2:20Þ

where C

is C transposed. The resulting ﬂow vector is the one that best satisﬁes

the set of equations. However, the injection rate vector

, back-calculated

using Equation 2.7 with the resulting ﬂow vector

QQ and the measured con-

centration will not be equal to the actual one. This method always provides a

solution, but, depending on the condition of the system of equations, this

solution could be far from the reality.

At nodes where the tracer i is not injected, the system can only provide

linear combinations of airﬂow rates, as far as the determinant jC

j¼0.

Eliminating some equations

Combining some of the equations of system 2.9 to 2.19 allows avoidance of the

measurement of some concentrations. A system having as many equations as

unknown airﬂow rates can be solved using:

QQ ¼ C

1

II ð2:21Þ

Experience showed that this interpretation method often leads to poorly condi-

tioned systems of equations. Results are then very sensitive to slight changes of

input data.

Looking for the best conditioned system

A set of N equations (N being the number of unknown airﬂow rates, in this

case 16) can be selected to give the best accuracy, or the smallest sensitivity

to variations or errors of injection rates and concentrations. This set can be

theoretically selected by calculating the con dition number (see Chapter 3,

‘Condition of the model matrix’) of all possible sets of equations extracted

from the full system, and taking the set with the smallest condition number.

This selection could be tedious: there are 13,037,895 sets of 16 equations that

can be extracted from the system 2.9!

Airﬂow Rates in Air Handling Units 27

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