OEP*

NISTIR 88-3855

Integral Mass~Barances

and Pulse Injection

Tracer Techniques

James Axley and Andrew Persily

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

(Formerly National Bureau of Standards)

National Engineering Laboratory

Center for Building Technology

Building Environment Division

Gaithersburg, MD 20899

October 1988

Sponsored by

U.S. Department of Energy

National Institute of Standards and Technology

"Vo

NISTIR 88-3855

INTEGRAL MASS BALANCES

AND

PULSE INJECTION TRACER TECHNIQUES

James Axley

Andrew Persily

U.S. DEPARTMENT OF COMMERCE

National Institute of Standards and Technology

National Engineering Laboratory

Center for Building Technology

Building Environment Division

October 1988

Sponsored by:

U.S. Department of Energy

U.S. National Institute of Standards and Technology

ABSTRACT

Tracer gas techniques for measuring airflow rates in building systems are considered.

These techniques are classified in terms of tracer gas injection strategy employed and mass

balance relationships used to analyze measured tracer concentration data. The discussion

focuses on one class of tracer techniques - the pulse injection techniques - based upon pulse

injection strategies and integral mass balance relationships. These pulse injection techniques

have not been commonly used in the past yet they provide practically useful means for the

determination of airflow rates in building systems. Pulse injection techniques are presented for

measuring airflows in ducts, and for studying single-zone and multi-zone building airflow

systems. Experimental procedures for these three cases are discussed, and preliminary results

from field applications of these techniques are presented. The possibility of flow variation is

accounted for in all cases, and the sensitivity of the single-zone pulse injection technique to

these flow variations is compared to that of the single-zone constant injection technique. This

comparison leads to integral formulations of the constant injection technique for duct, singlezone, and multi-zone situations that may provide means to improve the accuracy of the

commonly used constant injection tracer technique.

KEY WORDS: airflow, infiltration, integral mass balances, tracer gas techniques, ventilation
ACKNOWLEDGEMENT

This work was supported, in part, by the U. S. Department of Energy through the

Interagency Agreement: I AG: DE AI01-86-CD 21013 Amendment Number 4 . with the

remaining funding by the National Institute of Standards and Technology.

This report was presented at the 9th Air Infiltration and Ventilation Centre Conference,

Effective Ventilation, held at Novotel Gent, Belgium, 12-15 September 1988, and is published in

the conference proceedings.

iii
CONTENTS

Page

ABSTRACT ii ACKNOWLEDGEMENT iii CONTENTS iv

LIST OF TABLES & FIGURES v NOMENCLATURE vi

INTRODUCTION 1 Tracer Injection Strategy 1 Mass Balance Relations 1 Classification of Tracer Techniques 2

Pulse Injection Technique 3

DUCT PULSE TECHNIQUE 4

Theory 4

Experimental Procedures 5

Measured Results 6

BUILDING PULSE TECHNIQUES 6

Single Zone Pulse Technique 7

Theory 7

Experimental Procedures 8

Measured Results 9

Comparison of Single-Zone Pulse and Constant Injection Techniques 9

Theory: Constant Injection Technique 9

Harmonic Flow Variation 11

Pulse Injection Response: 11

Constant Injection Response: 11

Multi-Zone Theory 14

Singularity and Conditioning of the Inverse Formulations 19

Conservation of Air Mass Flow 20

The Tracer and Integral Tracer Matrices 21

Multi-Zone Pulse Tracer Injection Strategy 21

Nonsingularity of the Integral Concentration Matrix 22

Constant Injection Tracer Injection Strategy 23

Nonsingularity of the Concentration Matrix 23

Integral Constant Injection Technique 23

Solution of the Inverse Analysis Equations and Error Evaluation 24

Application of the Multi-Zone Pulse Injection Technique 25

Measurement Method . 25

Measured Results: NBS Office Building Study 26

CONCLUSION 30

REFERENCES 30

iv
LIST OF TABLES & FIGURES

Page

Table 1 Classification of Tracer Techniques 2

Fig. 1 Duct Pulse Injection Technique 4

Fig. 2 Comparison of Duct-Pulse and Hot-Wire Measurements 6

Fig. 3 Single Zone Pulse Injection Technique 7

Fig. 4 Single-Zone Constant Injection Technique 10

Fig. 5 Pulse Injection Response for a= 1.0 12

Fig. 6 Pulse Injection Response for a= - 1.0 12

Fig. 7 Constant Injection Response for a= 1.0 13

Fig. 8 Ratio of Constant Injection Estimated Flow Rate to Mean Flow Rate .... 14

for Long Averaging Time Intervals

Fig. 9 A Three-Building-Zone Idealization 15

Fig. 10 Multi -Zone Pulse Injection Technique 22

Fig. 1 1 Multi-Zone Constant Injection Technique 23

Fig. 12 Multi-Zone Integral Constant Injection Technique 24

Fig. 13 Simplified Section of an Individual Floor of Tower 26

Fig. 14 Office Tower Three-Zone Idealization and Results for 12/1/87 27

Fig. 15 Comparison of Measured and Predicted Response to Pulse Injections .... 28

Fig. 16 Individual Floor Idealization and Results for Repeated Tests of 12/17/87 ... 29

v
NOMENCLATURE

C

Ci

AC

c

G

Gj

I . M

minT

t . w

w

tracer concentration in terms of mass fraction [=] mass-tracer/mass-air

tracer concentration within well-mixed zone i the change of concentration over a given time interval

the average concentration over a given time period

tracer mass generation rate [=] mass-tracer/time

tracer mass generation rate within well-mixed zone i amount of tracer released in an impulse [=] mass-tracer

mass of air within a given well-mixed volume [=] mass-air

mass of air within well-mixed zone "i"

number of building zones

period of harmonic flow variation [=] time per cycle

time

air mass flow rate [=] mass-air/time

air mass flow rate from zone i to zone j the average air mass flow rate over a given time period

a

5(t)

T . 4 •

relative amplitude of flow variation; —1 < a < 1 [=] dimensionless

dirac delta function

mean flow time constant for single-zone system; x = M /w [=] time

an instant in lime within a given time interval [=] time

Vectors and Matrices

{C}

[C]

[1c]

{ert

{G}

[M]

[in

[W]

[t] - [M]’[W]

vector of zone concentrations

instantaneous concentration matrix

integral concentration matrix

the ith unit vector with element i equal to 1 and all other elements zero

vector of zone generation rates

system mass matrix

instantaneous tracer rate matrix

integral tracer matrix

system mass transport rate matrix

steady flow system state matrix

Following subscripts identify well-mixed zones with "0" used to identify, specifically, the

exterior environment. Following superscripts are used to identify data sets. Leading

superscripts are used to identify tracer species.

vi
-

f

'
INTRODUCTION

Indoor air quality and energy use in buildings are both closely related to airflow into, out of,

and within a building system. Consequently, indoor air quality and building energy analysis

both depend critically upon obtaining complete and detailed information about these airflows. In

some special cases these flows will be substantially determined by the design of the HVAC

system, but, more often, due to uncertainties in the actual performance of the HVAC system,

additional uncertainties in envelope infiltration, and the inherently complex nature of inter-zone

airflows, these flows will be unknown. In these cases one may attempt to determine these

flows using network flow analysis methods [Walton 85, Axley 87] or, for existing buildings,

using tracer gas measurement techniques.

Perera [82] and Lagus [85] provide comprehensive reviews of existing tracer gas

techniques for measuring airflows in buildings. This paper reconsiders these tracer techniques

from the point of view of integral mass balance relationships. In this section we classify tracer

techniques by the tracer gas injection strategy employed to excite the building airflow system

and by the form of the mass balance relations used to reduce measured tracer concentration

data to determine flows. It is argued that the tracer techniques based upon integral mass

balance relations - integral tracer techniques - have been largely ignored and need to be

studied more thoroughly. We then focus consideration on one class of techniques based upon

integral mass balance relations, the pulse injection techniques. A second integral tracer

technique, the constant injection integral technique , is briefly presented in a subsequent

section and a third technique, the constant concentration integral technique, is presently under

consideration by the authors.

Tracer Injection Strategy

Tracer gas techniques attempt to determine building airflow rates from the measured tracer

concentration response of building airflow systems to carefully controlled injections of tracer

gases. Inasmuch as the tracer injection strategy employed largely determines the capability

and accuracy of a given tracer gas technique, these techniques can be classified by injection

strategy. Presently, three injection strategies are commonly used;

1. decay: a suitable amount of tracer gas is injected into the building system to establish a uniform concentration within the system (i.e., an initial condition) and the ensuing

concentration decay response is measured,

2. constant injection : tracer gas is injected at a constant rate and the resulting

concentration response is measured (often with the expectation that practically steady

state conditions exist), and

3. constant concentration: tracer gas is injected under instrumental control in an attempt

to realize a constant and uniform tracer concentration throughout the building system.

Mass Balance Relations

The tracer gas concentration response to a given injection strategy may be described by

mass balance equations that relate tracer concentrations to airflows. It is through these mass

balance relations that one is able to estimate airflows from measured concentration data. These

mass balance relations may be formulated in either an instantaneous form, which, for the multizone case, leads to systems of ordinary differential equations, or in an integral form that

accounts for mass conservation over a given interval of time. Researchers have historically

tended to favor the use of the instantaneous mass balance relations in the development of tracer

gas techniques. A few researchers have, nevertheless, considered integral formulations for the

reduction of concentration response data for these common injection strategies [Sinden 78,

Turiel 80, Penman 82, Walker 85, Waters 87]. Jensen [88] has demonstrated the use of

1
integral mass balance relations in the reduction of data collected using an unusual "active"

tracer injection strategy. To date, however, there has been no systematic attempt to reconsider

the common tracer techniques from an integral point of view.

Classification of Tracer Techniques

As a unique tracer technique may be developed, in principal, for each injection strategy

using either instantaneous or integral mass balance formulations, we may classify tracer gas

techniques by both injection strategy and mass balance formulation. It is also useful to

distinguish the application of each technique to either single or multi-zone building idealizations

because of the significant mathematical differences of the corresponding mass balance

relationships. The array of unique tracer techniques that may be considered for the three

common injection strategies discussed above is tabulated below, in Table 1, along with

indications of the capabilities that each technique may offer.

Tracer

Injection Strategy

Mass Balance Formulation

Instantaneous Integral

• Decay

SZ: yields infiltration

MZ: yields all flows

(see Pulse Injection)

• Constant

Injection

SZ: yields infiltration**

MZ: yields all flows**

SZ: yields infiltration*

MZ: yields all flows*

• Constant

Concentration

SZ: yields infiltration

MZ: yields only infiltration

SZ: yields infiltration*

MZ: yields only infiltration*

• Pulse

Injection

(see Decay)

SZ: yields infiltration

MZ: yields all flows

SZ=single-zone; MZ=multi-zone; *=presently under consideration; **=tends to underestimate

Table 1 Classification of Tracer Techniques

The tracer techniques based upon instantaneous formulations of the mass balance relations

have been applied with varying degrees of success. The tracer techniques based upon the

integral formulations have been largely ignored until recently and have yet to be studied

thoroughly.

In particular, the constant injection technique may be applied to single and multi-zone

situations to determine the details of infiltration, exfiltration, and zone-to-zone flows. The

constant injection technique based upon an instantaneous formulation tends, however, to

significantly underestimate infiltration airflows as commonly implemented (i.e., using average

concentrations measured over relatively long time periods, as in the so-called Perfluorocarbon

Tracer (PFT) method [Dietz 87]) [Bohac 87, Sherman 87]. It is believed that the integral

formulation, to be discussed subsequently, will provide a means to mitigate this shortcoming.

The constant concentration technique has proven to be a reliable technique for single and

multi-zone situations providing accurate determinations of outdoor airflow rates into the building

zones [Sandberg 85], but does not provide any information regarding zone-to-zone airflows.

When based upon an instantaneous formulation this technique requires careful control of the

tracer injection rates that is realized using relatively sophisticated instrumental devices. It is believed that the integral formulation of the constant concentration technique, presently under

consideration by the authors, will provide a means to implement this technique without the need

2
for such careful, and therefore expensive, control.

The decay technique may be used to effectively determine infiltration airflows in buildings

that behave as single-zone systems. It has also been applied to determine the details of

infiltration, exfiltration, and zone-to-zone flows in buildings that behave as multi-zone systems.

Several multi-zone decay techniques based upon instantaneous formulations have been

considered including techniques based upon;

a) the measurement of both concentration responses and their first time derivatives

[Sinden 78, Perera 82, Walker 85],

b) the consideration of the tracer gas response at times corresponding to a maximum

concentration in one of the building zones (where the time derivative of concentration

vanishes) [Dick 49], and

c) the intermediate determination of the system eigenmodes, assuming real-valued

eigenvalues (i.e., inverse time constants) and nondegenerate eigenmodes (i.e.,

describing concentration response time histories as the sum of exponential decays)

[Sinden 78], using, in some instances, Prony analysis [I'Anson 82, Irwin 85] to realize

this determination.

Difficulties in measuring the first time derivative of concentration response have limited the success of the first approach [Sinden 78, Walker 85]. The authors simulated the application of

the second approach for three and four zone cases, but it appeared that the approach was not

well -conditioned enough to warrant further consideration. The third approach has yet to be

applied to buildings that behave as more than four-zone systems and falters on the assumption

of a nondegenerate system having real-valued eigenvalues; examples of systems that may

demonstrate both degeneracy and imaginary eigenvalues have been presented [Sinden 78,

Lawrance 87, Waters 87]. Most multi-zone decay techniques rely on data collected very soon

after the tracer gas injection. For this data to be reliable, the tracer gas injection must be wellmixed in each of the target zones. This is often a very difficult initial condition to achieve, and

the accuracy of the results will be degraded by deviations from these assumed initial conditions.

Pulse Injection Technique

In this paper we shall consider the pulse injection technique , that was presented by

Walker as the decay integral method [Walker 85] and further developed by Afonso and his

colleagues [Afonso 86a, 86b, 86c]. This technique is based upon a tracer injection strategy of

separate, short-duration, pulse injections of tracer into each zone of the building system and the

application of integral mass balance equations to the reduction of the measured concentration

response data.

Pulse excitations of flow systems are commonly used in the chemical process industry to determine dynamic characteristics of chemical process systems [Wen 75, Nauman 83,

Westerterp 84] and Sandberg has described similar applications for buildings [Sandberg 83,

84]. The use of pulse excitation here is closer, however, to the tracer injection strategy

suggested by Dick to extend decay methods to multi-zone problems [Dick 49]. In fact, the

pulse injection technique may be considered to be an integral formulation of Dick's multi-zone

decay technique. Although decay techniques employ pulse injections to establish initial

concentrations, they have not used data collected during the time interval of the pulse to solve

for airflows. In the pulse injection techniques we may choose to use data collected during the

time interval of the pulse. It is for this reason that we distinguish pulse injection techniques

from traditional decay techniques.

It will be shown that the pulse injection technique may be applied to single and multi-zone

situations to determine all airflows into, out of, and within the building system. The pulse and

constant injection techniques are the only tracer gas techniques with this capability and,

therefore, these two techniques will be compared. In addition, the pulse injection technique may

be used to determine airflows in HVAC ducts.

3
This paper will first consider the simplest case, the application of pulse injection techniques

to the determination of flows in ducts, then move on to building applications, both single-zone

building idealizations and multi-zone idealizations. Although there is only limited experience

with the pulse injection techniques, we present discussion of experimental procedures and the

results of applications of the techniques to the study of airflows in large buildings.

DUCT PULSE TECHNIQUE

The application of the pulse injection technique to the measurement of airflow in ducts

provides a useful introduction to the pulse techniques in general. The underlying theory is especially straightforward and the utility of the technique appears to be great. Measuring

airflow rates in ducts in building ventilation systems is difficult using traditional airflow rate

measurement techniques (e.g. pitot tubes and hot-wire anemometers), due to insufficient

lengths of straight ductwork for the establishment of fully-developed flow profiles. Constant

injection tracer gas techniques have been used to measure these airflow rates [Lagus 1985],

but they require one to wait for equilibrium and to measure very low tracer gas flow rates. The

duct pulse technique is a simple and quick alternative for measuring these important quantities

in even the most complex duct configurations.

Theory

Consider the duct segment illustrated below in Figure 1. Air flows into the duct from the

left at a time-varying mass flow rate of w(t). We inject a short duration tracer pulse at a rate

G(t) into the duct and measure the time variation of tracer concentration C(t) at the exit.

w(t)

Fi g. 1 Duct Pulse Injection Technique

Assuming that the tracer injection results in only trace concentrations and, therefore, does

not contribute significantly to the air mass flow rate, then the exit air mass flow rate will equal

w(t). Furthermore, if the exit concentration measurement represents a flow-averaged

concentration (e.g., the concentration is well-mixed across the section), then the mass flow rate

of tracer exiting the duct will simply be equal to the product of the flow rate and the exit

concentration, w(t)C(t), where concentration is expressed in terms of the mass fraction of tracer

relative to air. Recognizing that after some time interval, say (t-| , t2), all tracer will be purged

from the duct, we may account for tracer mass conservation through the use of the following

integral mass balance:

J

r

x 2 r

x 2

w(t) C(t) dt = G(t) dt ; w(t) > 0

l

i 'x

i (I)

which simply asserts that the tracer mass leaving the duct segment must equal the amount

4
injected.

We may apply the integral mean value theorem to the expression on the left, as the

concentration variation does not involve a sign change, and simplify to obtain the governing

equation for the duct pulse injection tracer technique:

In words, we may determine the air mass flow rate that occurred at some time, during the

time interval (t-j , t2) by simply computing the ratio of the mass of tracer injected to the integral

of the concentration response downstream from the tracer. Clearly if the air mass flow rate is

constant, the determination will yield this constant value. If the air mass flow rate changes very

little during the interval, then w(£) should be a good estimate of the average flow rate during

that interval.

Experimental Procedures

In applying the duct pulse technique to an actual length of duct there are several practical

considerations. The most important issues are knowing the mass of tracer that is injected and

obtaining an accurate determination of the concentration integral. Since one only requires the

integral of the G(t), the actual injection profile is irrelevant. It is only important to know the

injection mass. This mass can be measured before or during the injection, but it is crucial that

all of the tracer gas is injected into the duct.

The duct pulse measurement technique requires only the determination of the integral of the

concentration at the downstream measurement point, not the concentration time history. The

determination of this integral relies on more than just accurate measurement of tracer gas

concentrations. The concentration in the duct, at the point of measurement, must be varying

only along the length of the duct, not across the duct cross-section. Otherwise, the integral

must be based on a cross-sectional average concentration. A multi-point injection across a duct

cross-section, as opposed to a single point injection, can assist in achieving a uniform

concentration across the cross-section at the concentration measurement point.

Because the concentration response will be relatively short-lived, it will be difficult to determine the concentration integral from numerical integration of the concentration data unless

one's concentration measuring equipment has a high sampling frequency and covers a wide

range of measurable concentrations. Therefore, it is advantageous to determine the

concentration integral through the measurement of the average tracer gas concentration at the

measurement point. This average concentration can be determined by filling an appropriate air

sample container, beginning well before the pulse is injected and continuing until the pulse is completely purged from the duct. The concentration integral simply equals the average

concentration multiplied by the length of the time over which the sample container is filled.

In applying this technique to a particular system there will always be some initial

uncertainty in the amount of tracer gas that should be injected into the ductwork. The primary

requirement is that the average concentration in the air sample container is in the accurately

measurable range of one's tracer gas concentration measurement equipment. Meeting this

requirement depends on choosing an appropriate combination of injection mass and

concentration averaging period. In general, there will be some "trial-and-error" in determining

these quantities. Since each measurement requires only a few minutes, it is not difficult to find

appropriate values for these quantities. In addition, because the time required to make a measurement is so short, an airflow measurement can be repeated several times, thereby

obtaining an estimate of the repeatability of the tests.

-1

C(t) dt

( 2 )

5
Measured Results

Some preliminary applications of the duct pulse technique have been conducted in the

HVAC system of a mechanically ventilated office building. A comparison between the results of

these duct pulse measurements and the airflow rates measured by a hot-wire traverse is shown

in Figure 2. These results lie in three distinct regions, depending on the type of duct that was

studied. In the ducts corresponding to the two lower airflow rates, a premeasured amount of

tracer gas was injected by hand. In the measurements corresponding to the higher airflow

rates, the tracer gas was injected through a calibrated flow meter. In all of these tests, the

concentration integral was based on an average concentration determined by filling an air

sample bag with a battery operated pump over a period beginning at least one minute before the

injection and lasting several minutes after the injection was complete. In these tests, the

injection mass and sampling period were varied to examine the sensitivity of the results to

these variables, and the measurements were repeatable to within about 5%. The agreement

between the duct pulse results and the results of the hot-wire traverses are encouraging given

the uncertainties in the hot-wire readings and additional errors due to the unknown flow profile

at the duct walls and uncertainties in the inside area of the duct. A detailed, laboratory study of

the duct pulse technique is still necessary to provide a rigorous validation of the technique.

Hot-Wire

Measured

Flow

(m3/s)

0.00 2.00 4.00 6.00 8.00 10.00 12.00 14.00

Duct-Pulse Measured Flow (m3/s)

Fi g, 2 Comparison of Duct-Pulse and Hot-Wire Measurements

BUILDING PULSE TECHNIQUES

Building and chemical process flow systems may be thought of as well or partially mixed

zones connected by flow paths (i.e., supply, return, and exhaust ducts in buildings; inlet and

outlet pipes in chemical process systems). One may, then, consider subjecting a flow system to

a) pulse excitations of the flow paths, b) pulse excitations of the zones, or c) a combination of

a) and b). The duct pulse technique, based upon a flow path excitation, is possible because the

inlet and outlet flow paths of a duct are (presumably) known with certainty 1

. The flow paths

1 Note: if the duct pulse technique was applied, as described, to a long leaky duct not all tracer mass transport

would be accounted for and, therefore, errors would result. However, in the case of a leaky duct, one could actually

quantify the amount of duct leakage by employing a series of pulse injections and/or concentration response

measurements along the length of the duct.

6
in chemical process flow systems are also known with certainty and, as a result, flow path

excitation is commonly used for these systems. For buildings, on the other hand, the analyst

will seldom have complete knowledge of all flow paths (e.g., infiltration, exfiltration, and zoneto-zone) and, therefore, to determine both the general topology and magnitude of airflows in the

building airflow system the analyst will have to employ pulse excitation of the zones rather than

the flow paths. As the analyst becomes better acquainted with a given building flow system

then it may be possible to use a combination of flow path and zone pulse excitations.

Here, we formulate the theory relating to pulse excitation of ideally well-mixed zones in

building systems, considering, first, the single-zone idealization then the general multi-zone

idealization of building airflow systems. The theoretical development parallels that presented

for ducts but now we must include the possibility of accumulation of tracer within zones.

Single Zone Pulse Technique

Theory

Consider the single-zone idealization illustrated below in Figure 3. Airflows into the zone

at a mass flow rate of w(t) and is assumed to be instantaneously and uniformly mixed within the zone. A short duration tracer pulse is injected into the zone and the zone concentration

response to the pulse, C(t), is measured.

Pulsg. Imsstiflii Measured Response

J C(t)dt

w(t)

G(t)

1 C(,) *

L , a 7

M W(t)

Fig. 3 Single Zone Pulse Injection Technique

Again we assume that the tracer injection mass is small enough such that the exit air mass

flow rate is practically equal to the inlet rate. We may write an instantaneous mass balance

relation for this single-zone idealization, with M equal to the mass of air within the zone, as:

w(t)C(t) + M = G(t) ; w(t) > 0

(3)

where we have assumed the concentration of tracer outside to be zero. In words, at any instant

in time the mass flow rate of tracer out of the zone, w(t)C(t), plus the accumulation of tracer

within the zone, MdC/dt, is equal to the rate of generation (i.e., injection) of the tracer, G(t).

We may also demand that tracer mass be conserved over any arbitrary time interval, say

(t<j, t2), by directly integrating equation (3) over the time interval to obtain;

J

r

t 2 2

w(t)C(t) dt + M AC = I G(t) dt

*1 K (4)

where AC= C(t2) - C(t-| )

. Again we apply the integral mean value theorem to the first

integral and simplify to obtain the governing equation for the single-zone pulse injection tracer

7
technique:

wft)

C

2

-1 C

2

C(t) dt G(t) dt - M AC

K K ( 5 )

For the single-zone system, then, it is seen that we may determine the air mass flow rate

that occurred at some time, during the time interval (t«j , t£) by simply computing the ratio of

the mass of tracer injected, corrected by the amount of tracer accumulated -MAC, to the integral

of the concentration response downstream from the tracer. Again, if the air mass flow rate is

constant the determination will yield this constant value. If the air mass flow rate changes very

little during the time interval, then w(£) should be a good estimate of the average flow rate

during that interval.

By explicitly accounting for the accumulation, MAC, we are able to consider any time

interval we desire; we do not require complete purging of the tracer as before. This widens the

possible experimental options as discussed below. We may consider a time interval sufficiently

long to allow complete purging, or short time intervals that in the limit approach an instant in

time, which would in theory provide instantaneous determinations of airflow rates.

Experimental Procedures

The use of the single zone pulse technique to measure the air exchange rate of a zone is

based on the assumption that the zone is well-mixed (i.e. the tracer gas concentration in the

zone can be characterized by a single value). To increase the appropriateness of this

assumption, it is important to release the tracer gas as uniformly as possible throughout the

space being tested. As in the duct pulse technique, it is important to know the mass of tracer

gas released, not the injection time history. The gas can be released directly into the space

itself using a multi-point injection scheme or by moving the injection outlet through the space

during the release. Such a "within-the-space" injection can be difficult in a large or complex

zone, in which case the gas can instead be injected into the supply duct serving that zone, if one

exists. Using the supply air distribution system to inject the tracer can provide a uniform

dispersal of the tracer gas, but one must be sure that all of the gas gets to the space (i.e., the

supply ductwork does not leak). In many systems this assumption can not be justified, but as discussed below, one can still use the supply air system for injection by not including the

injection period in the concentration integral.

As with the duct pulse technique, one only needs the integral of the tracer gas concentration

in the zone. This can be determined with real-time sampling or with an average air sample

taken during the integration period. Although the use of this technique employs the assumption

that the concentration is uniform throughout the zone, the tracer gas concentration should be

sampled at several locations in order to verify this assumption. The time interval over which

the integral is determined need not include the tracer gas injection, nor need it last until the

tracer gas concentration goes to zero. If the integral includes the tracer gas injection, then the

injection mass must be known precisely and be well dispersed throughout the test space. If the

injection period is not included in the integral then the injection mass need not be known, though

it needs to be controlled such that the concentration within the zone is in the measurable range

of the tracer gas detector. The integral of G(t) in equation (5) will equal zero, but the value of

MAC will be large due to the significant tracer gas concentration at t=t<j.

The duration of the integration period involves a trade off between one's ability to accurately

measure low concentrations and one's knowledge of the zone mass M. During the tail of the

concentration response, the tracer concentrations will be very low and may be difficult to

measure accurately. One can avoid this source of error in the integral by choosing t2

to be a

8
time when the concentrations are still within a range that can be accurately determined. In this

case, the term MAC may be significant, and if it is, an accurate knowledge of M will be

important. The importance of knowing M accurately depends on the relative magnitudes of the

integral of G(t) and of MAC.

If the concentration response to the pulse is very short lived, corresponding to a high air

exchange rate, then one should employ an average concentration to determine the integral. This

averaging should begin well before the tracer gas injection and continue until the concentration

within the zone has decreased to essentially zero. This approach enables the determination of

the air exchange rate of a single zone based on a single air sample of the average concentration,

but one must be certain that the tracer gas concentration is zero at t=t2 . If it is not, then an

additional air sample must be taken at this time to compute the MAC term. This approach

enables the low-cost determination of single zone air exchange rates with on-site air sampling

and off-site tracer gas concentration analysis

Measured Results

Afonso et. al. [Afonso 86, 87] report the results of single zone pulse tests conducted in a laboratory test facility in which the airflow rate into the zone was measured with nozzles. Four

tests were conducted at four different supply airflow rates into the zone. The measurements of

the space air exchange rate were generally repeatable within 3%, and the agreement with the

supply airflow rates measured with the nozzle ranged from 10 to 17%. The values calculated

from the pulse tests were always less than the measured airflow rates, probably due to air

leakage from the supply ducts. In these experiments, the room mass M was treated as an unknown and was solved for by evaluating equation (5) both before and after the tracer gas

injection.

Comparison of Single-Zone Pulse and Constant Injection Techniques

The pulse and constant injection techniques can both be used to determine the air exchange

rate of a single well-mixed zone. A pulse measurement will generally last only a few building

time constants (inverse of the air exchange rate), while the constant injection technique has

been applied for periods from several days to several weeks. In both cases, the effect of

variations in the air exchange rate during the measurement period will affect the results of the

measurements. The discussion below examines these effects of flow variation in comparing the

pulse and constant injection techniques.

Theory: Constant Injection Technique

Consider, now, for purposes of comparison, a well-mixed zone subjected to a constant

injection of tracer as illustrated below in Figure 4.

We may again apply the instantaneous and integral mass balance relations presented

above, equations (3) and (5), with G(t) = G0 . If the airflow remains steady then eventually a steady state response will be achieved:

C(t—constant ; = 0 ; w(t) = steady

dt (6)

and by the instantaneous mass balance, equation (3), we obtain:

w« = c(fer ; w(,) - stead

* (7)

9
This expression is the basis of the constant injection technique; flow is estimated by the

quotient of the constant injection rate divided by the steady state concentration.

Constant Injection

G(t)

Fig. 4 Single-Zone Constant Injection Technique

In many practical situations, however, the flow will seldom be constant long enough to

achieve a steady state response, therefore, it has become common practice to use average

concentration measurements, in place of the steady state value, in equation (7) to estimate

airflows. The time intervals over which these average measurements are made range, typically,

from a single day to several weeks. Early on it had been hoped that the use of average

concentration measurements in equation (7) would provide acceptable approximations of the

mean airflow occurring over the averaging time interval, but more recently it has been shown

that this approximation may significantly underestimate the mean flow [Bohac 87, Sherman 87].

The question remains, then, if this approximation does not provide an estimate of the mean flow,

what estimate does it provide?

From the integral mass balance relation, equation (5), with G(t) = G0 , we may obtain the

following expression:

ti < ^ < t 2

( 8 )

where C is the mean concentration measured over the interval (ti, t2). Equation (8) defines the

basis of an alternate constant injection tracer technique, discussed below, that we designate as the integral constant injection technique.

For averaging intervals on the order of days to weeks M(AC/At) will generally be negligibly

small and thus we may presume that:

W($) ; At s t2

- t. t, < 4 ^ t 2

(9)

Thus, it is seen that the approximation using average concentration measurements yields an

estimate of flow that occurred at some time, during the averaging interval which, in general,

will not equal the mean value. Ironically the constant injection technique, as practically applied

using long-term averages, is actually an "integral technique" rather than an "instantaneous

technique" as originally presented.

10
Harmonic Flow Variation

By employing the integral mean value theorem we have derived governing equations for

both the pulse injection and constant injection technique that account for flow variation. The

results, equations (5), (8) and (9), leave, however, much to be desired from a practical point of

view. Although they provide a rigorous estimate of flow that occurred at some time, during

the integration time interval, that time is unknown and the results provide no information

regarding how the flow estimate relates to the actual flow variation.

We may gain some insight into the affect of flow variation about a mean value on both the

pulse injection and constant injection techniques by considering the special case of harmonic

flow variation:

w(t) = w (1 + a cos(27tt / T )) (10)

where w is the mean flow, a is the amplitude of flow variation, and T is the period of the

variation. Substituting equation (10) into equation (3) and solving we obtain expressions for

the concentration time histories for both injection strategies.

Pulse Injection Response: For an impulse injection of tracer (i.e., a very short duration,

relative to the system time constant, pulse) of amount I:

G(t) = I 5(t)

; f

G(t)dt = f

Jo Jo

G(t) dt = I 5(t) dt = I ; 5(t=0) = 1 ; S(t*0) = 0

we obtain:

C(,)-M exP |«r

12jtt

sin

(2kz tV

T T II

( 11 )

( 12 )

where z is the system time constant under the mean flow conditions, z = (M / w ). This

equation describes a decay response that includes an oscillatory component as illustrated in

Figures 5 and 6. The difference between the responses in Figures 5 and 6 is that the former

corresponds to the flow rate assuming its maximum value at t = 0 (i.e., a = 1.0), while Figure 6

corresponds to the minimum airflow rate at t = 0 (i.e., a = - 1.0).

Constant Injection Response: For constant injection, G(t) = G0 , we obtain:

(13)

This equation describes a build-up response having an oscillatory component as illustrated in

Figure 7.

11
— Steady — T= 0.5T — T= 2T — T= 81 — T= 32T

1 0.9

0.8

0.7

0.6

£ID 0 5

C /M ) 0 4

0.3

0.2

0.1

0

0 0.5 1 1.5 2 2.5 3 3.5 4

( t/T)

Fi g. 5 Pulse Injection Response for q=1.0

— Steady — T= 0.5T ~~ T= 2T — T= 8T — T= 32 T

0 0.5 1 1.5 2 2.5 3 3.5 4

(t/T)

Fig. 6 Pulse Injection Response for a= - 1.0

12
— T-0.5X — T-2T — T-8X — T« 32 X

7

6

5

_2CL 4

<<V W) 3

2

1

0

0 5 10 15 20 25

(t/T)

Fig. 7 Constant Injection Response for a=1.0

These figures clearly reveal the importance of the period of variation of flow. Specifically,

when the period of variation of flow is small relative to the mean flow time constant, x, then both

the pulse injection response and the constant injection response closely approximate the mean

flow response. We conclude then:

1. For flow variations having periods of variation small relative to the system mean-flow

time constant, both pulse injection and constant injection techniques will provide

accurate estimates of the mean flow rate regardless of the amplitude of flow variation.

At the other extreme, it is also clear that flow variations having periods of variation much

greater than the mean flow time constant may greatly distort the response relative to the mean

flow response. For these long period variations, we may conclude:

2. If integration time intervals are selected to be small relative to the period of flow

variation then both the pulse injection technique and an integral constant injection

technique defined by equation (8) will provide accurate estimates of nearinstantaneous flow rates.

3. If integration time intervals are selected nearly equal to the period of variation (e.g., 0.2

T < At < 5 T ) then the pulse injection and constant injection techniques may under-or

over-estimate the mean flow rate depending upon the actual timing of the integration

interval relative to the flow variation.

4. If integration time intervals are selected to be very large, or a multiple of the period of

flow variation, then for constant injection the average concentration will overestimate

the mean flow steady state concentration and, as a result, the constant injection

technique will underestimate this mean flow. (For the pulse injection technique,

integration time intervals are practically limited to 2 to 4 times the mean flow time

constant and thus this case need not be considered.)

Conclusion 4 may be evaluated numerically by substituting equation (13) into equation (8)

and integrating over one period of variation after steady periodic conditions have been realized.

The results of this exercise are plotted below in Figure 8. From these results it is seen that the

flow determined by constant injection and long-term average concentrations, w(^), may

underestimate the actual mean flow, w, by 10 to 30% for moderate amplitude flow variations

(i.e., for a in the range of 0.5 to 0.7) when the period of flow variation is on the order of ten or

13
more times the mean flow time constant. Bohac [88] and Sherman [87] have both reported

underestimation errors of this magnitude.

0.0 50.0 100.0 150.0 200.0 250.0 300.0

(T/t)

Fig. 8 Ratio of Constant Injection Estimated Flow Rate to Mean Flow Rate

for Long Averaging Time Intervals

Given the importance of long-period flow variations, it is natural to ask if one should expect

to encounter such long-period flow variations in the field. Infiltration time constants for

buildings range, typically from 0.5 to 5 hours (i.e., infiltration rates of 0.20 to 2.0 air changes per

hour may be considered typical), thus, from a practical point of view we should be particularly

concerned with periods of flow variation exceeding from 5 to 50 hours. Flow variations driven

by occupant intervention and HVAC system operation are likely to introduce a diurnal, 24 hour,

component (e.g., night set-back of heating and cooling systems and daily airings of occupied

spaces) that should be expected to be important in buildings having higher infiltration rates. Flow variations driven by wind and ambient temperature variations may be expected to reflect

the spectral content of the wind and thermal environment both of which reveal significant

variations in the semi-diurnal, 12 hour, to several-day time periods (i.e., periods associated with

weather variation) and, therefore, may be expected to important for all buildings. Therefore,

long-period flow variations can be expected to be commonplace in the field and will introduce

errors in the airflow rates determined with the constant injection technique.

Multi-Zone Theory

The development of the theory for multi-zone building idealizations follows that presented

above for single-zone idealizations in that we shall consider, first, instantaneous mass balance

equations and from them derive integral mass balance relations using the integral mean value

theorem. Now, however, we shall have to consider systems of equations and, consequently,

the solution for airflows will involve matrix, rather than scalar, algebraic operations. As a

result, the issues of singularity and conditioning of the resulting equations will become a

central concern and will, largely, determine the success or failure of any proposed multi-zone

tracer technique.

We begin by stating that contaminant dispersal in a multi-zone idealization of a building

airflow system may be described by the following instantaneous mass balance equations (see

14
[Axley 87, 88a, 88b] for complete details):

[WHO + [M]-^ = {G}

(14)

where;

[W] is the system mass transport matrix, {C} is a vector of nodal concentrations (i.e., concentrations at discrete spacial

locations in the building airflow system), which, for our purposes corresponds to a vector of zonal concentrations,

[M] is the system mass matrix, which, for our purposes is a diagonal matrix with

entries equal to the mass of the volume of air contained in each zone,

{G} is the system generation vector, which, for our purposes is a vector of zonal

tracer mass generation rates.

The system transport matrix is, in general, assembled from expressions defining mass

transport due to both flow and nonflow transport processes (e.g., airflow rates and chemical and

physical rate constants). Consequently, one may devise tracer gas techniques to determine the

elements of this matrix from measured concentration data, and thereby determine the

characteristics of both flow and nonflow transport processes that define this matrix. Our

purpose is somewhat more limited; we seek to determine the flow characteristics of a multizone building airflow system using tracers that are affected only by flow transport processes

and leave the determination of the characteristics of nonflow transport processes to future

study.

To this end we shall consider the determination of airflows for a multi-zone building

idealization consisting of n well-mixed building zones and a single well-mixed exterior (i.e.,

outdoor) "zone," with single flow paths linking each of these zone to all others. Furthermore,

we shall assume that airflow from zone-to-zone is practically instantaneous. A three-zone

example of such an idealization is illustrated below in Figure 9.

mo = ~

exterior "zone" 0

Fig. 9 A Three-Building-Zone Idealization

15
It will be convenient to identify the exterior zone as zone "0" and number the building zones

from 1 to n, so that the elements of the concentration vector, {C}, the system mass matrix, [M],

and the system generation vector, {G}, are:

»-

CM o c (15)

1» m2 • mn) (16)

_ T

> 2 , ... Gn } (17)

where Cj , rrij , and Gj are, respectively, the concentration of tracer in zone i (expressed in terms

of mass fraction; mass tracer/mass air), the mass of the volume of air in zone i, and the mass

generation rate of tracer in zone i. (The mass of air in the exterior zone "0" shall be considered

infinite.)

Designating the mass flow rate of air from zone i iq zone j by Wy, we may directly assemble

the system mass transport matrix to obtain:

'Woo Wo, ••• w0n

-

[W] = W, 0 Wln

-Wn0 Wn , Wnn.

( nX woj -w,o - Wn0

\j=0; *0 /

" \ “ w01 I wj " Wn1

\j=o;*i /

1 ^

0 c 1 - w1n £ Wr

Vj=0; *n J(18)

where we admit only positive values for Wy. It should be noted that the diagonal terms are equal to total air mass flow out of each zone. The central objective of multi-zone tracer techniques is to determine the elements of the

mass transport matrix, fW], and thereby determine the airflows, Wjj, in equation (18). With

this objective in mind we may transpose and rewrite equation (14) in the following form:

{C}

T

[W]

T =

{{G} -

which, when expanded, has the form:

(19a)

16
{C0 * Ci , ... Cn }

Woo W, 0

Wo, W„

Wn0

Wnl

Lw0n w1n ... wnn J

= {(Go - m0 ^). (G, - m, ^1). ... (Gn - mn }

(19b)

From the point of view of tracer analysis the Cj's, Gj's , and dCj/dt’s may be considered to

be quantities that are measured at some instant in time with the Gj's defining the tracer

injection at that time and the Cj’s and dCj/dt's the corresponding concentration response.

Together we shall designate these quantities, determined at some instant in time, as an instantaneous data set. To distinguish data set i from all others we shall use the superscript i as follows:

Instantaneous Data Set i

dCo dC', dC'n dt ' dt ' dt }.{Go.G

i1*

From the form of equation (19) it is clear that we shall need (n+1) data sets to define a determined set of equations (i.e., to have sufficient equations to determine the elements of the

[W] matrix):

{C° , C,° , ... Cn}

{c;,c,\...c’} Woo W,o

W0 , W,,

LW0

'

n W,' n L {cS.c",...cS}

W„o

W„,

Wnn J

{(G°-m0 ^),...(G°-m„^)}

dt

{(G^ - mo^O),

.

..(G’-m„^n)}

,, r

n m

dC"

{(G0

- rriQ-^-), (GS-mn ^)} J

(20a)

To simplify notation we shall represent equation (20a) as:

[C] [W]

T = [T]

( 20b)

The ( instantaneous) concentration matrix, [C], would be formed from measured tracer

concentration data; the ( instantaneous) tracer rate matrix, [ T ], would be formed from the

known tracer injection rates corrected by the "mdC/dt" accumulation rate term. This set of

equations defines the instantaneous inverse (contaminant dispersal) problem.

Following the same procedure used above in the single-zone theory we may also formulate

the integral inverse (contaminant dispersal) problem. We begin by integrating equation (19) over an arbitrary time interval, (t-| , t2), and apply the integral mean value theorem to obtain:

{Jc 0 jc„ ...Jc n }[W(K])]

T = {(jGo-m0 AC 0)....(iG n -mn AC n)} (21)

17
where we have introduced the short-hand notation:

|

2

Ci dt ; JG; = |

i ^

/Gj =| Gj dt ; ACj = Cj (t2

) * Cj (ti )

i

Here, we must introduce a separate unknown time, for each element of the mass transport

matrix:.

O o 2* o o /<

—

jTr* o Wn 0 (^n o)

[W( R] )]

T H WoiRoi) W, ,<$,,) ... wn1 (^)

.Won(^n)W, n (^ n) Wn n (^n n).

( 22 )

In the integral formulation the JCj's, /Gj's, ACj's, and, now, the

£jj which depend upon the nature

of the measured data, constitute a data set. As before, to distinguish one data set, i, from all others we shall use a superscript i as follows:

Integral Data Set i

{/Co, Jc'„ ... jcj . {ACp. AC

1,. ... AC^}, {jGo , Jg'„ ... jG„} . [4

1

]

If (n+1) integral data sets are collected satisfying the condition:

[W([50

])] = [W([^])l = [W([52])] = ... [W([5n])] [W] (23)

then, and only then, we may assemble a system of equations to determine the elements of the

mass transport matrix as:

{Ico.jc,

0 ....!^}

{JcJ.Jc,

1 ...Jci}

"WooWio ... Wno'

Woi Wn ... Wn

{(JGq - m0 AC°), ... (Jg® - mn AC°)

}

{(JgJ - m0 acJ), ... (jGn - mn ACn)

}

. {JcS./c" , ...JcS}

W0n W1n ... Wnn . _ {(/Go - m0 aCq), ... (JGn - mn ACfJ) } _ (24a)

To simplify notation we shall represent equation (24a) as:

[/cj fw ] T = f/n (24b)

The integral concentration matrix, [JC], would be formed from measured tracer concentration

data; the integral tracer matrix, [It] , would be formed from the known tracer injection amounts

corrected by the "mAC" accumulation term. This set of equations defines the integral inverse

(contaminant dispersal) problem.

It must be emphasized that the formulation of the integral inverse problem depends critically

on the condition imposed by the integral mean value theorem, equation (23). In practical

18
situations we should expect system airflows to vary with time. If all airflows in the system do

not vary greatly over the time period spanning all integral time intervals used to evaluate the

(n + 1) data sets, then this condition will essentially be met. If the flow variation is a high

frequency variation (i.e., relative to the time period spanning all time intervals used to evaluate

the (n + 1) data sets and the dominant time constant of the system) then we should expect the

condition of equation (23) will be met and, furthermore, the mass transport matrix will

correspond to a mean flow condition in the system.

Practically speaking, then, to be able to apply the integral inverse formulation to tracer

analysis with confidence, we should employ tracer injection and data collection strategies that

may be completed rapidly. Both the pulse injection strategy and a integral constant injection

strategy can meet this objective, especially if multiple tracers are employed.

Singularity and Conditioning of the Inverse Formulations

Although equations (20) and (24) appear to have the form of a determined system of

equations (i.e., (n + 1) equations of (n+1) sets of (n+1) unknowns for (n + 1) sets of right hand

sides), they will yield solutions (i.e., be nonsingular) only if they describe (n + 1) linearly

independent equations. From the theory of linear systems of algebraic equations we know that

linear independence, or nonsingularity, will be realized if the concentration matrix, [C], for

equation (20), or the integral concentration matrix [JC], for equation (24), consists of linearly

independent row vectors. We shall, therefore, consider this a necessary condition that must be

met by any tracer gas technique.

Even if these row vectors can be shown to be linearly independent, in principal, "nearsingularity" may still result, due to the limitations of finite-precision calculations, that will

manifest itself in terms of the conditioning of the inverse problem and result in, possibly,

unacceptable error in the determination of the elements of the mass transport matrix. That is to say, if the row vectors are "nearly" dependent due to measurement error, a poor measurement

and injection strategy, and/or the intrinsic character of the building idealization being studied2

, then the resulting system of inverse equations will be ill-conditioned3 and results obtained

from them may be overwhelmed by error. In principal, a well-conditioned system of inverse equations can be formed if a tracer

injection strategy is chosen that will not only result in linearly independent row vectors but data

sets that satisfy the orthogonality condition:

{C

i }

T

{C j } = 0 ; i.j = 1,2, ...n ; i*j (25)

for the instantaneous form, or

{Jc'}

T

{Jc'} = 0 ; i.j = 1 , 2 , ...n ; i*j (2 6)

for the integral form. One could achieve this objective if one could identify a tracer injection

strategy that would result in measurable concentrations in a single zone and zero

concentrations in all others, that is one that would result in n row vectors of the form:

th

i elem.

i

(C'}

T = c'{ 0 ... 1 ... 0 } = c'{ef

2 For example, idealizing two building volumes having very large air exchange rates with each other as separate,

well-mixed zones, rather than a single well-mixed zone, will lead to a poorly conditioned inverse problem.

3 An ill-conditioned problem is one for which relative errors in the data defining the problem will tend to become

amplified to result in larger relative errors in the solution of the problem.

19
for the instantaneous case, or:

{Jc'}

T = jc'{ 0

ttl

i elem.

I

1 0 } = |c'{e l}

T

for the integral case, where C' and JC' are scalars and {ej is a vector with a unit value for

element i and zero values elsewhere. (For conditions of steady flow the orthogonality

conditions of equations (25) and (26) would be met by utilizing the system eigenvectors, but

steady flow conditions will not in general prevail and the analyst will, in any event, have no a

priori knowledge of these eigenvectors.)

It will not, in general, be possible to achieve this end. One may, however, approach this

goal practically by exciting each zone with an individual tracer injection so that, presumably, the

concentration response in the excited zone will be greater than that in all others. This strategy

must be considered fundamental to any multi-zone tracer gas technique, and has been used as the basis of the multi-zone constant injection tracer technique and the pulse injection technique.

Conservation of Air Mass Flow

Regrettably, one zone, the exterior "zone", which will be involved in all real building

idealizations, can not be excited by tracer injection (due to its practically infinite volume it tends

to act as a constant concentration sink) and, therefore, in applying the individual zone injection

strategy, we will find ourselves short by one data set. By demanding, however, the

conservation of total air mass flow we may obtain this needed (n+1)th equation.

The conservation of total mass flow may be conveniently realized by recognizing that the

mass concentration for air in all zones is unity and unchanging. The instantaneous data set for

air alone is: d “fc,

dt = 0

and, the integral data set for air alone:

t air ajf fair

J Cj = 1At ; A Ci = 0 ; J G| = 0 ; At = (t 2 -t,)

Using these data sets we obtain the first of the required (n+1) equations:

{1 . 1 , ... 1

}

W00 W10

Wqi W„

Wn0

Wn1

LW0n W1n ... Wnn J

= {0, 0, ... 0 }

for the instantaneous form of the inverse analysis equations, and

{1 , 1 , ... 1

}

Woo W10

W01 Wn

LW0n Win

Wn0

Wnl

Wnn J

= { 0 , 0, ... 0 }

(27)

(28)

20
for the integral forms of the inverse analysis equations.

The Tracer Rate and Integral Tracer Matrices

The formation of the terms of the tracer rate and integral tracer matrices, [ T ] and [ Jt ], corresponding to the exterior environment:

dC

(Go-nV^p) and (/Go’ mcACo) i = 1.2, n

presents a problem since the mass of air in the exterior zone, mg, is considered infinite. In

practice, however, it will not be reasonable to inject tracer into the exterior zone directly thus

we may assume Gq to be zero and, therefore, requiring conservation of tracer mass for each

data set we may conclude that:

ITIn m —-— - V (c' = 2* v^i ‘ m

[ dt

H

dt

)

(29)

mcACo = £(jG|-mjACj)

H (30)

That is to say, the infinitesimal change in concentration of the infinite exterior mass is simply

equal to the net (rate of) generation of tracer less the net (rate of) accumulation tracer (i.e., the

first term of each row of the tracer rate and integral tracer matrices is simply equal to the sum of

the other terms in the row which are readily determined from the measured data sets).

This resolves the practical problems of accounting for the infinite mass of the exterior zone

and we are now in a position to consider collecting the additional n equations needed to form a complete inverse problem. Two approaches will be considered; the first based upon a pulse

injection strategy and the second on a constant injection strategy.

Multi-Zone Pulse Tracer Injection Strategy

A multi-zone pulse injection of tracer may be employed to obtain a sufficient number of

equations to formulate a complete inverse analysis problem. The injection and measurement

strategy used for the multi-zone pulse injection technique is illustrated below in Figure 10 for a three-zone case. We first subject one zone to an individual, short-duration tracer pulse and

measure the tracer concentration responses in all zones. A second zone is excited and, again,

we measure the responses in all zones. The process of excitation and response measurement

is continued until all zones have been independently pulsed. These independent zone pulses

may, conceivably, be done in series using a single tracer, done simultaneously using multiple

tracers, or done as a series of multiple-tracer pulses.

If we feel confident that the condition of equation (23) has been realized experimentally

(i.e., airflows in the system have remained more or less constant during the test time period)

we may then directly substitute the measured data into equation (24) to form a complete

inverse problem. As each zone is injected individually, the integral tracer matrix, ifa , will have

generation integral contributions, JG, only along the diagonal; the off diagonal terms will involve

only mAC contributions. As in the single-zone case, for each data set we may consider arbitrary

time intervals of integration - complete purging of the system is not required if we carefully

21
account for the mAC terms.

Zone 1 Pulse Pulse

Zone 3

Pulse

Fi g. 10 Multi-Zone Pulse Injection Technique

Nonsingularity of the Integral Concentration Matrix [JC]: If the pulse injected into zone i

is of a relatively short duration (i.e., the pulse duration is small relative to the smallest system

time constant, approaching, in effect, an impulse excitation) then, given the assumption of wellmixed zones, the pulse will create an initial concentration in zone i equal to the amount of tracer

injected, I', divided by the mass of the air in zone i, mj . That is to say, the pulse will establish an initial concentration condition:

{C(t-o)}

(31)

where the superscript i is used to indicate data set i and, as before, G'(t) is the pulse injection

time history for zone i. From the theory of systems of differential equations we obtain the decay

response of the system to this initial condition, for constant flow conditions, as:

{C'(t)} = exp( — [x ] t

\ mi/ {e;} ; [x] - [M] ' [ W]

(32)

where [X] is sometimes called the state matrix of the system. Integrating this expression

over the time interval (t-j , t2 ) we obtain a general expression for the rows (transposed) of the

integral concentration matrix, [JC], as;

(C'(t)} dt = [X]'1 [[ exp( - [ x ] t, )] - [ exp( - [x ]

t2 )]]

(33)

The state matrix [X ] will be nonsingular for system idealizations that satisfy continuity of

total airflow (i.e., real systems) [see Axley 87, or 88a] and, therefore, it follows that both [X ]

1

and exp(- [T ] t), for all t, will also be nonsingular. As a result, the product matrix on the right

hand side of equation (33) will be nonsingular or, equivalently, will consist of linearly

independent column vectors. The unit vector {ej} acts, then, to select one of these independent

column vectors which when then scaled by the initial concentration ( P/m j) forms the ith row

(transposed) of the integral concentration matrix [JC]. We may conclude then that the integral

22
concentration matrix [JC] will be nonsingular (i.e., have independent rows) when a) flow is steady, b) an equivalent integration time interval (relative to the pulse injection) is used to integrate the concentration responses for all pulses, and c) the independent zone pulse strategy

is employed. In the practical application of the pulse injection technique we select time intervals so that, presumably, condition a) and b) are substantially satisfied.

Constant Injection Tracer Injection Strategy

Another approach to obtain the needed equations to formulate a complete inverse analysis

problem involves the constant injection strategy. In the ideal application of the constant

injection technique each zone is subjected to an independent constant injection of tracer, and

when the concentration in all zones becomes steady, they are measured to form a single

concentration data set. This is repeated for all zones, as illustrated below in Figure 11, the

concentration matrix [C] is formed from this measured data, and equation (20) is solved for the

unknown airflows assuming steady conditions prevail (i.e., setting all mdC/dt terms equal to

zero). To independently inject each zone and yet measure steady state concentrations in all zones for each of the independent injections requires the simultaneous injection of multiple

tracers, one for each zone.

Zone 1 Zone 2 Zone 3

Const. Injection Const. Injection Const. Injection

Fig. 11 Multi-Zone Constant Injection Technique

Nonsingularity of the Concentration Matrix [C] : If flow conditions are steady in the

building system, we may obtain an expression for the concentration response to a constant

injection in zone i, G', directly from equation (14) as:

{C'm = [W]

1 G'fe,} (34)

[W], and hence [W]* 1

, will be nonsingular for system idealizations that satisfy continuity of

total airflow (i.e., real systems) and, therefore, will have independent column vectors. Again

the unit vector, { ej }

,

that mathematically defines the individual zone injection strategy, acts to

select one of the independent columns of [W]'1

, which is then scaled by G' to form one of the

rows (transposed) of the concentration matrix. We may conclude, then, that the concentration

matrix [C] will be nonsingular for the ideal conditions of constant airflow in the building system.

Integral Constant Injection Technique

As discussed earlier, in many practical situations the airflows in the building system will

seldom remain constant long enough to achieve a steady state response. Therefore, it has

become common practice to use average concentration measurements, taken over a relatively

long time period, for the concentration data sets needed in equation (20). Using average values

is completely equivalent to using integral data sets and, therefore, these values should be used

in conjunction with the integral formulation of the problem, equation (24), rather than the

23
instantaneous form equation (20), but to do so we must be careful to satisfy the condition

defined by equation (23) - a condition that, practically speaking, requires consideration of

relatively short integration time intervals (i.e., averaging periods). For situations where airflow

is likely to be nonconstant, then, one should use a integral constant injection tracer technique

based upon equation (24), accounting for the mAC accumulation due to unsteady flow conditions

and employing relatively short integration time intervals to assure the satisfaction of the

condition of equation (23). The injection and measurement strategy used for the multi-zone

integral constant injection technique is illustrated below in Figure 12 for a three-zone case.

Fig. 12 Multi-Zone Integral Constant Injection Technique

Solution of the Inverse Analysis Equations and Error Evaluation:

Errors in the estimation of airflows by tracer techniques may be attributed to; a) an inappropriate idealization of the building system being investigated,

b) uncertainties introduced via flow variations, and/or

c) error introduced via measurement error. The idealization of a given building airflow system may, to a great extent, determine the success or failure of the application of tracer techniques to the determination of airflows in the

building. For example, the idealization of a very well-mixed portion of a building system as a collection of multiple zones will, in itself, result in a poorly conditioned system of inverse

equations that will tend to amplify measurement error. Although we attempt to provide some

guidance in this paper we are forced to admit that the process of system idealization remains an

art that requires considerable experience and skill.

It was shown that in the single-zone case flow variation could result in very large errors in

the estimation of mean airflows. It must be expected that even greater errors will result in

multi-zone cases due to the numerical phenomena of ill-conditioning that is intrinsically

associated with the inverse problems being considered here. It is the primary responsibility of

the analyst, then, to attempt to conduct a given tracer test in such a way that the underlying

assumptions of the tracer technique are satisfied. With this done, numerical techniques exist to

deal with solution errors resulting from measurement error. The inverse problem defined by either equations (20) or (24), or their specific applications

to the pulse injection, constant injection, or integral constant injection techniques must be

expected to be ill-conditioned and, therefore, must be solved with special care to avoid

unnecessary amplification of data errors. Conventional elimination or iterative equation solving

techniques may be expected to fail for very ill-conditioned problems and, thus, the analyst is

well-advised to employ numerically more stable algorithms. Singular value decomposition has

become the method of choice for solving ill-conditioned problems and is recommended here

[Press 86]. (Solution techniques based upon Cramer's Rule are always computational inferior to

the elimination and iterative techniques and should, therefore, not be considered).

Furthermore, as the degree of ill-conditioning that might be associated with any given

problem will not, in general, be evident, the analyst is well-advised to not only compute the

24
solution to the inverse problem; but also compute (and report) a measure of error associated

with the solution. D'Ottavio [87] and Walker [85] have discussed error analysis techniques

relating to the solution of both the constant injection technique and the pulse injection technique

(Walker's decay integral method); their results apply here as well. Three error estimation

techniques are offered; a) error estimation based upon perturbation analysis of systems of linear

equations involving vector and matrix norms, b) error estimation based upon Monte Carlo error

analysis, and c) error estimation based upon first order error analysis using Taylor expansions.

The first approach provides an upper-bound error estimation. As this approach is sensitive to

the scaling of the inverse equations, D'Ottavio employed optimal scaling of the equations,

based upon scaling individual equations by the inverse of their row Euclidean norm, in

conjunction with this approach to provide a (near) minimum upper bound error estimation.

Central to perturbation analysis of systems of linear equations is the so-called condition

number, which, in simple terms, provides an upper bound estimate of the ratio of the maximum

relative solution error to the maximum relative data error (i.e., an error amplification factor).

Thus, reporting the condition number of the concentration matrix, [C], (integral concentration

matrix, [JC] ) provides one means to characterize the error associated with the solution of a given problem. This condition number is conveniently computed as a by-product of singular

value decomposition.

Application of the Multi-Zone Pulse Injection Technique

Measurement Method

The pulse techniques presented above provide flexible tools for the determination of building

airflow rates, however, their successful application in a particular building requires the user to understand the building and its systems. Based on this understanding the user develops an

"idealization" of the building as a series of zones and formulates the pulse experiment in order

to determine specific airflow rates between these zones. This process of applying the pulse

technique to a particular building consists of qualitative and quantitative aspects. The method

begins with a qualitative analysis of the building layout and the ventilation system equipment

and zoning. The major zones and system airflow paths of the building are identified, and the

existence of unexpected or undesired airflows due to envelope leakage, poor system

performance or inadequate separation between zones are investigated. A qualitative airflow

diagnosis, using hand-held instrumentation (e.g. anemometers), "smoke-sticks", or tracer gas

pulses, can serve to further elucidate the building's airflow characteristics. For example,

exhaust airflows may be verified as such or shown to be not flowing in the expected direction.

In addition, certain airflows may be shown to be zero, and need not be included in the

idealization.

Based on the qualitative analysis and the particular airflows that the experimenter is interested in, a system or subsystem idealization of the building is developed. This idealization

consists of a series of well-mixed zones connected by airflow paths. The idealization need not

include every airflow and every zone in the building. In fact, such an all-inclusive model of a building will generally be unmanageably complex from an experimental point of view and involve

the determination of more airflow rates than are necessarily of interest. In certain

circumstances, a selected subsystem of the building can be investigated, providing information

without consideration of the rest of the building.

Once the building idealization has been developed, the quantitative experimental analysis of

this system is conducted. One must define tracer gas injection and air sampling strategies, in

conjunction with one’s data collection and analysis approaches. The injection strategy includes

the manner in which the tracer will be delivered to each zone, the amount of tracer to be injected

and a means for determining this amount, and the timing of the injections into the various zones. The air sampling strategy includes the number and location of air sampling points in each zone,

and the manner in which they will be sampled (real-time monitoring or container averaging).

25
Real-time monitoring must be conducted with consideration given to the sampling frequency of

the tracer gas monitor and the transport of air samples from the sampling locations to the

monitor. Once the data is collected it is converted into the form of equation (24), one row for

each tracer gas injection, which is then solved for the unknown airflows and analyzed to provide

some evaluation of error. In the studies considered below the solution was achieved using the

robust and stable numerical method known as singular value decomposition [Press 86] and the error was characterized by the condition number of the system which is obtained as a byproduct of the singular value decomposition.

Measured Results: NBS Office Building Studies

Two examples of the application of integral pulse techniques to a fifteen-story office building

in Washington, DC are presented. This building has four separate air handlers serving the

fifteen-story tower section, two for the fourteenth and fifteenth floors, and two more serving

floors one through thirteen. These air handlers run on 100% outdoor air and are located in a penthouse mechanical room. The air from the building is exhausted through a relief air system

directly to the outdoors, with no provision for the recirculation of return air. On each floor, air

from the supply air ducts is forced into a ceiling plenum from which it enters the occupied space

through diffusers in the suspended ceiling as shown in Figure 13. Based on an on-site

inspection of the building and its systems, it was noted that there were significant amounts of

supply air leaking from the pressurized ceiling plenum to the relief air shafts, to other service

shafts, and through the exterior envelope. This leakage led to the question of exactly how much

of the supply air was actually reaching the occupied space on the floors. In addition, strong

airflows were noted in the two stairways in the building, flowing up to the penthouse mechanical

room.

nU • *

Supply

Shaft

7A^ 7A^

4, vl' I

s

Relief

4 T> Shaft

Booster Fan

nI/

Fig. 13 Simplified Section of an Individual Floor of Tower

Based on the inspection of the building and its systems, two different idealizations of the

building were investigated with integral pulse tests. As shown in Figures 14 and 16 these

idealizations include one of the whole tower and the penthouse, and the second is of an

individual floor. In the first idealization, the tower is modeled as three building zones, based on

the air handler zoning and the observed importance of the penthouse. Zone 1 consists of the

penthouse mechanical room, the fourteenth and fifteenth floors are modeled as a single zone - zone 2, and zone 3 idealizes floors three through thirteen. It must be emphasized that these

groups of floors behave as single well-mixed zones only for the tracer injection strategy used in

the tower tests (i.e., they respond with practically uniform concentrations for this particular

injection strategy). For other injections or releases of contaminant these floors will, in general,

behave differently. The second model is an example of a building subsystem that enables the

investigation of limited aspects of the building's air exchange characteristics. This model is

26
based upon the characteristics of an individual floor and its response to the tracer injection

strategy used to study the floor. An individual floor is modeled as two zones, a supply zone and

an occupied zone. The supply zone includes the supply air distribution system for a floor and

the occupied zone includes the space below the suspended ceiling. The supply zone is a conceptualization, not a distinct physical volume that is contained between well-defined

boundaries. Therefore, the mass of the supply zone can not be used in analyzing the data, that

is the integrals corresponding to the supply zone injection must begin and end when the supply

air concentration equals zero.

Exterior Zone "0"

Condition Number = 3.9

Fig. 14 Office Tower Three-Zone Idealization and Results for 12/1/87

(all flows m3/s)

The tower model depicted in Figure 14 was investigated using successive pulse injections

of sulfur hexafluoride (SFg) into the penthouse (zone 1), the fourteenth and fifteenth floors

(zone 2), and floors three through thirteen (zone 3). A premeasured amount of SF6 was

injected into the penthouse by hand while walking throughout the space over a period of about

two minutes. The injections into the other two zones were made by injecting SF$ into the air

handlers serving these zones through flowmeters at a known rate for a known length of time

(about one minute). The penthouse concentration response was very short-lived and therefore

the tracer gas injection period was included in the concentration integral. Because of leakage in

the supply air distribution systems serving the building, the injection periods were not included

in the integrals of the concentration response to the injections in zones 2 and 3. The results of

one of the tower pulse tests are shown in Figure 14. The airflow rates from the outdoors (zone

27
0) into zones 2 and 3 include intentional outdoor air intake through the air handlers and the

infiltration of air through leaks in the exterior envelope of the building. The airflow rate through

the air handlers serving zone 2 is about 7 m^/s, as measured with a duct pulse test, but not all of this supply air gets to the zone due to leaks in the supply air system. Therefore the

difference between the measured airflow rate from the outdoors to zone 2, 8.5 m^/s, and the

measured airflow rate through the air handlers is a lower limit on the infiltration airflow into

zone 1, i.e., 1.5 m^/s or about 0.5 air changes per hour. Similarly, the airflow rate through the air

handlers serving zone 3 is about 20 m^/s. Only 18.7 m^/s of outdoor airflow into zone 3 was

measured, and therefore no estimate of the minimum infiltration rate into that zone can be made.

Using the measured flow rates and the known injection time histories, the concentration

response of the tower was determined analytically using the program CONTAM87 [Axley 88a].

Predicted (modeled) concentration time histories are compared below. Figure 15, to measured

values; the close agreement provides an indication of validation of the pulse injection technique.

1000

100

10

[SF6]

(PPb)

0.1 -

— Mexjel: Penttilouse o- Meas: Peiru house

Sv &- A<*\

<v 1

^o*"£

1&

(SF6]

1 (PPb)

[SF6]

, (PPb)

8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00 16.00

Time (hr)

Fig. 15 Comparison of Measured and Predicted Response to Pulse Injections

28
The floor model in Figure 16 was investigated with a pulse test in order to determine the

amount of supply air that was bypassing the occupied space of the floor. In this idealization of a

floor of this building, zone 0 includes the outdoors and the rest of the building, zone 1 is the

supply air distribution system, and zone 2 is the occupied space of the floor. This idealization is appropriate because the floors of this building are well separated from each other in terms of

airflow. During these tests the SF^ concentration was measured on the floors above and below

the floor being tested, and there was essentially no SF6 on the surrounding floors. The injection

into zone 1 was made by hand into the supply air ductwork, and the concentration response in

this ductwork was determined by filling an air sample container to determine the average

concentration. This was essentially a duct pulse test to determine the supply airflow rate to the

floor, but the concentration response was also measured in real-time at four locations in the

occupied space (zone 2). The integral of this concentration response obviously included the

tracer gas injection period. The tracer gas injection into zone 2 was made by hand; a known

amount of SF^ was released throughout the occupied space. Since there was no backflow from

the occupied space into the supply air system there was no need to measure the concentration

response in zone 1. The results for a set of repeated "floor-bypass" tests are presented in

Figure 16. These results are based upon a series of three injection tests; the supply zone was

injected once and the occupied space was subjected to two separate injections. The test A

results were computed using concentration data for the single supply zone injection and the first

of the occupied space injections; the test B results were computed using concentration data for

the supply zone injection and the second of the occupied space injections. A comparison of

these results provides an indication of the uncertainty of the computed flows. During this test,

and during other tests on this and other floors of the building, it was found that a significant

percentage of the supply air intended for this floor does not reach the occupied space. The

supply air "bypasses" the occupied space due to several leaks in the pressurized ceiling supply

air plenum that allow the supply air to flow into the relief air system, service shafts in the

building, and the outdoors instead of through the ceiling diffusers into the occupied space.

1.5 Exterior Zone

3.7 T 22 v

Supply Zone ' *1* Room Zone

*2* / '

N 0.0 0.7—)

2.9

Test A Results

Condition Number = 3.2

Test B Results

Condition Number = 3.4

Fi g. 16 Individual Floor Idealization and Results for Repeated Tests of 12/17/81

(all flows m3/s)

29
CONCLUSION

The pulse injection tracer techniques, based on integral mass balance formulations, provide

useful tools for studying building airflow systems. The duct pulse application provides a rapid

and convenient means of measuring airflows in ducts. The building pulse applications are capable of determining airflow rates in multi-zone building systems in relatively short time

periods (i.e., time periods on the order of the dominant system time constants), and can be

employed with a single tracer gas. Pulse injection determinations of airflow rates may be

expected to be relatively insensitive to variations in airflow rates, and the analysis of data from

field studies to date indicate that the multi-zone pulse injection technique may be expected to

yield relatively well-conditioned equations. In the multi-zone pulse injection technique, as in all multi-zone tracer gas techniques, the manner in which the building airflow system is idealized

as a series of inter-connected zones is pivotal in obtaining a well-conditioned system of

equations and, thereby, reasonable estimates of the system airflow rates.

The development of the pulse injection techniques has implications for other tracer gas

measurement approaches. Integral mass balance relations should be applied to existing tracer

injection strategies to provide alternate formulations and, possibly, to identify strategies of

minimizing errors in the determination of airflow rates. An integral formulation of the constant

injection technique, which, in principal, should avoid some of the problems of the technique as commonly used, was presented and should be considered further. The pulse injection

techniques themselves require additional study in both laboratory and field settings to more

completely examine sources of errors and to better establish experimental procedures for their

practical application.

REFERENCES

Afonso, C.F.A., Maldonado, E.A.B., & Skftret, E., "A Single Tracer-gas Method to Characterize Multi-room Air

Exchanges," Energy and Buildings, Vol. 9, Elsevier Sequoia, The Netherlands, 1986, pp. 273-280

Afonso, C.F.A., & Maldonado, E.A.B., "A Tracer-Gas Procedure for the Simultaneous Evaluation of Effective

Volumes and Multizone Airflows," Proceed. 3rd International Congress on Building Energy Management,

Lausanne, Switzerland, Sept., 1987

Afonso, C.F.A., Maldonado, E.A.B., & SkAret, E., "Measuring Indoor Air Movement: An Essential Task to Characterize Indoor Air Quality," Proceed. IAQ '86, ASHRAE, Atlanta, GA, Apr., 1986, pp. 592-601Axley, James,

Indoor Air Quality Modeling: Phase II Report. NBSIR 87-3661, CBT, National Bureau of Standards, Gaithersburg,

MD, OcL, 87

Axley, J., Indoor Air Quality Modeling: Phase II Report. NBSIR 87-3661, CBT, National Bureau of Standards,

Gaithersburg, MD, October., 87

Axley, J., Progress Toward a General Analytical Method for Predicting Indoor Air Pollution in Buildings: Indoor

Air Quality Modeling: Phase III Report NBSIR 88-3814, CBT, National Bureau of Standards, Gaithersburg, MD,

June., 88

Axley, J., Multi-Zone Dispersal Analysis bv Element Assembly, (submitted for publication to Building and

Environment, May 1988)

Bohac, D., Harrje, D.T., & Homer, G.S., "Field Study Comparisons of Constant Concentration and PFT Infiltration

Measurements," Proceed, of 8th ATVC Confer., Uberlingen, Germany, Sept., 1987

Dick, JJB., "Experimental Studies in Natural Ventilation of Houses," Journal of the Institute of Heating and

Ventilating Engineers, England, Dec., 1949, pp. 421-466

Dietz, R.N., Goodrich, R.W., Cote, E.A., & Wieser, R.F., "Detailed Description and Performance of a Passive

Perfluorocarbon Tracer System for Building Ventilation and Air Exchange Measurements."Measured Air Leakage

of Buildings. ASTM Special Technical Publication 904, Trechsel, H.R., Trechsel, H.R., & Lagus, P.L., Editors,

ASTM, Phil., PA, 1986

D'Ottavio, T.W., Senum, G.I., & Dietz, R.N., "Error Analysis Techniques for Perfluorcarbon Tracer Derived

Multizone Ventialtion Rates," BNL 39867, Brookhaven National Laboratory, June, 1987

I' Anson, S.J., Irwin, C., & Howarth, A.T., "Air Flow Measurement Using Three Tracer Gases," Building and

Environment, Pergamon Press, Great Britain, Vol. 12, No. 4, 1982, pp.245-252

Irwin, C., Edwards, R.E., & Howarth, A.T., "The Measurement of Airflows Using a Rapid Response Tracer Gas

Technique," Budding Services Engineering Research and Technology, Vol. 6, No. 4, 1985, pp. 146-152

30
Jensen, L.t "Determination of Flows and Volumes in Multiple Cell Systems," IEAA AIVC Newletter, Vol. 9, No. 2,

Berkshire, England, Feb., 1988, pp. 4-6

Lagus, P. & Persily, A.K., "A Review of Tracer-Gas Techniques for Measuring Airflows in Buildings," ASHRAE

Transactions, Vol. 91, Pt. 2, 1985, pp. 1075-1087

Lawrance, G.V. & Waters, J.R., "Measurements of Infiltration and Air Movement in Five Large Single-Cell

Buildings," Proc. 8th ATVC Confer., Uberlingen, Germany, Sept, 1987, pp. 2.1-2.18

Nauman, E.B. & Buffham, B.A., Mixing in Continuous Flow Systems. John Wiley & Sons, NY, 1983

Penman, J.M. & Rashid, A.A.M., "Experimental Determination of Air-flow in a Naturally Ventilated Room Using

Metabolic Carbon Dioxide," Building and Environment, Pergamon Press, Great Britain, Vol. 17, No. 4, 1982

Perera, M.D.A.E.S., "Review of Techniques for Measuring Ventilation Rates in Multi-Celled Buildings," Proc. of

EEC Contractors Meeting on Natural Ventilation, Brussels, Sept., 1982

Press, W.H., Flannery, B.P., Teukolsky, S.A., & Vetterling, W.T., Numerical Recipes: The Art of Scientific

Computing. Cambridge University Press, Cambridge, 1986

Sandberg, M., "The Use of Moments for Assessing Air Quality in Ventilated Rooms," Building and Environment,

Vol. 18, No. 4, Pergamon Press, Great Britain, 1983 pp. 181-1978

Sandberg, M., ’The Multi-Chamber Theory Reconsidered from the Viewpoint of Air Quality Studies," Building and

Environment, Vol. 19, No. 4, Pergamon Press, Great Britain, 1984 pp. 221-233

Sandberg, M. & Blomqvist, C., "A Quantitative Estimate of the Accuracy of Tracer Gas Methods for the

Determination of the Ventilation Flow Rate in Buildings,", Building and Environment, Vol. 20, No. 3, Pergamon

Press, Great Britain, 1985, pp. 139-150

Sherman, M.H., "Variation-Induced Bias in Passive Ventilation Measurement," LBL-23088, Lawrence Berkeley Lab,

U.C., Berkeley, CA, 1987

Sinden, F.W., "Multi-Chamber Theory of Air Infiltration," Building and Environment, Vol. 13, Pergamon Press,

Great Britain, 1978 pp. 21 -28

Turiel, I. & Rudy, J., Occupant-Generated COn as an Indicator of Ventilation Rates. Lawrence Berkeley Lab, LBL10496, UC-95d, EEB-Vent 80-15, Berkeley, CA., 1980

Walker, R.R., "Interpretation and Error Analysis of Multiple -Tracer Gas Measurements to Determine Air

Movements in a House," Proc. 6th AIVC Confer., Supplemental Proceedings: Ventilation Strategies and

Measurement Techniques, Het Meerdal, Park, Netherlands, 1985

Walton, G.N., Estimating Interroom Contaminant Movements. NBSIR 85-3229.U.S. DOC, NBS, Gaithersburg, MD,

August, 1985

Waters, J.R. & Simons, M.W., "The Evaluation of Contaminant Concentrations and Air Flows in a Multizone Model

of a Building,"Building and Environment, Vol. 22, No. 4, Pergamon Press, Great Britain, 1987 pp. 305-315

Wen, C.Y., & Fan, L.T., Models for Row Systems and Chemical Reactors. Marcel Dekker, Inc., NY.NY, 1975

Westerterp, K.R., van Swaaji, W.P.M., & Beenackers, A.A.C.M., Chemical Reactor Design and Operation. John Wiley

& Sons, New York, 1984

31
U.S. OKPT. or COMM.

BIBLIOGRAPHIC DATA

1. PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. Publication bate

SHEET (See instructions) NISTIR 88-3855 SEPTEMBER 1988

4. TITLE AND SUBTITLE

Integral Mass Balances and Pulse Injection Tracer Techniques

5. AUTHOR(S)

James Axley and Andrew Persily

6. PERFORMING ORGANIZATION (If joint or other thon NBS. see instructions)

NATIONAL BUREAU OF STANDARDS

U.S. DEPARTMENT OF COMMERCE

GAITHERSBURG, MD 20899

7. Contract/Grant No.

I. Typa of Raport & Pariod Covarad

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. Stole, ZIP)

U.S. Department of Energy

Washington, D.C.

10. SUPPLEMENTARY NOTES

| | Document describes a computer protram; SF-185, FlPS Software Summary, is attached.

11. ABSTRACT (A 200»word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

i- __ • # „ m J M L..4 T oiTfif ome nrn rnncl H

oionogropny or literature survey, mention n nere)

Tracer gas techniques for measuring airflow rates in building systems are considered.

These techniques are classified in terms of tracer gas injection strategy employed and

mass balance relationships used to analyze measured tracer concentration data. The

discussion focuses on one class of tracer techniques - the pulse injection techniques

based upon pulse injection strategies and integral mass balance relationships. These

pulse injection techniques have not been commonly used in the past yet they provide

practically useful means for the determination of airflow rates in building systems.

Pulse injection techniques are presented for measuring airflows in ducts, and for

studying single-zone and multi-zone building airflow systems. Experimental procedures

for these three cases are discussed, and preliminary results from field applications

of these techniques are presented. The possibility of flow variation is accounted for

in all cases, and the sensitivity of the single-zone constant injection technique. Thii

comparison leads to integral formulations of the constant injection technique for duct,

single-zone, and multi-zone situations that may provide means to improve the accuracy

of the commonly used constant injection tracer technique.

12. KEY WORDS (Six to twelve entries ; alphabetical order; capitalize only proper names; and separate key words by semicolons)

building infiltration, building ventilation, tracer gas techniques, integral mass

balances.

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