THE RELATIONSHIP BETWEEN FLOW EXPONENTS AND FLOW
VALUES AND ASSOCIATED IMPLICATIONS FOR AIR LEAKAGE
TESTING USING FAN (DE)PRESSURIZATION METHODOLOGY.
AUTHORS
Robin Urquhart. MBSc., MNRES.
Russell Richman. PhD., P. Eng.
ABSTRACT
Fan pressurization and depressurization is the established method for single and multi-unit air leakage
testing in buildings. Tests are conducted at high pressures relative to normal building operating pressures.
The values are the extrapolated by using the power law equation. This paper examines the theoretical
relationship between flow exponents and changing leakage path geometry and the effect of different flow
exponent values on extrapolated data in order to develop a better understanding of the nature of the linear
relationship between flow rates and pressure difference
INTRODUCTION
Fan (de)pressurization has been established as the method of choice for air leakage testing in both the
residential and commercial building sectors. Fan depressurization testing relies on creating an artificial
and constant pressure boundary across which air leakage is measured. Pressure differences of 50Pa and
75Pa are preferred for residential and commercial tests respectively. The advantage to a higher pressure
difference (i.e. 50 - 75Pa) is that it functionally eliminates background pressure “noise” caused by stack
effect, wind and mechanical ventilation systems. The result is a reasonably simple test procedure yielding
consistent and repeatable results.
It has been postulated that testing at a variety of pressure differences will yield more accurate results than
testing at a single pressure difference. The rationale behind this postulation rests in the fact that leakage
path geometry changes at various pressure differences. By establishing a linear relationship between
pressure differences, a more accurate understanding of air leakage through the enclosure is achieved.
Data can then be extrapolated to give air flows at various pressure differences, most importantly at
building operating pressures (~4 – 10 Pa).
However, the nature of the linear relationship established by testing at multiple pressure differentials is
understudied. It is difficult to ascertain whether data extrapolated to 4Pa or 10Pa is an accurate reflection
of air flow through the enclosure. The linear relationship is established using the power law equation and
the calculation of the flow exponent. The effect of flow exponents on extrapolated data is a subject that
as yet has not been discussed at length.
This paper will examine the theoretical relationship between flow exponents and changing leakage path
geometry and the effect of different flow exponent values on extrapolated data in order to develop a better
understanding of the nature of the linear relationship between flow rates and pressure difference. The
results point to careful consideration of flow exponents during testing and results calculation to ensure
accurate data is achieved.
MATERIALS AND METHOD
The analysis and theoretical investigation of the relationship between flow exponents, leakage pathway
geometry and flow rates are based on results from a case study of a multi-unit residential building
(MURB) in Vancouver, Canada. The building, constructed in 1987, received a complete enclosure
retrofit in 2012. The retrofit, in addition to energy saving measures focused on enclosure air tightness.
Air leakage testing was conducted by the author in conjunction with RDH Building Engineering Ltd.
using a neutralized (guarded-zone) fan (de) pressurization method both pre-retrofit and post-retrofit.
Pressure differences of 20Pa, 30Pa, 50Pa and 60Pa were used during testing. More information on the
testing methodology used and research results are available through Ryerson University and the
University of Waterloo. Additional information on the case study building is available from the Canadian
Mortgage and Housing Corporation and RDH Building Engineering Ltd.
The data analysis procedure assumes that uncontrolled air leakage is through leakage pathways formed by
cracks and joints in the building envelope. It is reasonable to assume for the purposes of this study that
there are no large openings and that leakage through porous materials is minimal. It is possible to apply
known pressures and measured flow rates for the aggregate of openings by using the power law equation
(Hutcheon & Handegord, 1995; ASHRAE, 2005).
𝑄 = 𝐶 ∆𝑃
!
[1]
Where, Q = airflow through opening (m3/s); C = flow coefficient (m3/s/Pan; P = pressure difference (Pa)
between interior (reference pressure) and exterior (baseline pressure); n = flow exponent (dimensionless).
C and n can be calculated by recording multiple measurements over a range of pressure differences (10,
30, 50 and 60Pa). An unweighted log-linearized linear regression technique of the flow rates at different
pressures to calculate the flow exponent (n).. The value for n determines the relationship between flow
and pressure. The R-squared is calculated at the same time, by assessing the fit of the linear least squares
to the data points. R-squared values less than 0.95 indicate potential data corruption and are omitted
(Hutcheon and Handegord, 1995). The value of C is algebraically determined by substituting n and Q
into the equation.
In addition, theoretical values have been used to illustrate hypothetical situations that would affect flow
exponent values.
THEORY/CALCULATION
The value for the flow exponent (n) of the power law equation (refer to Equation 1) determines the
relationship between air flow and pressure difference and must be in the range of 0.5 – 1 (Hutcheon and
Handegord, 1995). If the value for n is 0.5, the flow is turbulent. If the value is 1, the flow is laminar.
Air flow through a building enclosure is typically a mixture of laminar flow and turbulent flow (Finch,
2009). Extant literature suggests, when not calculating n, to use a value of 0.62-0.66 (most commonly
0.65) to model the mixture of laminar and turbulent flow (Hutcheon and Handegord, 1995; Price et al.,
2006; Straube and Burnett, 2006).
Flow exponent values (n) calculated from results of the case study air testing are presented below.
Table 1 – Flow exponent values for individual suites 1
Pre-press
Post-press
Change2
Suite 101
0.59
0.56
-0.03
Suite 102
0.56
0.67
+0.11
Suite 301
0.60
0.75
+0.15
Suite 302
0.68
0.85
+0.17
Suite 303
0.54
0.85
+0.31
0.92
0.79
-0.13
0.80
0.69
-0.11
0.50
0.60
+0.10
0.62
0.57
-0.05
0.61
0.37
-0.24
Suite 1101
0.52
0.71
+0.19
Suite 1102
0.49
0.56
+0.07
Suite 1103
0.53
0.64
+0.11
Suite 1301
0.74
0.77
+0.03
Suite 1302
0.52
0.72
+0.20
0.47
0.61
+0.14
0.46
0.43
-0.03
0.78
0.59
-0.19
0.57
0.57
0.00
0.63
0.47
-0.16
Pre-depress
Post-depress
Change
Pre-press
Post-press
Change
Pre-depress
Post-depress
Change
Flow exponent trends
There are two interesting and seemingly contradictory trends in the flow exponent values. First, the postretrofit pressurization flow exponent values increased over the pre-retrofit pressurization values. The
increase suggests that the flow is becoming more laminar and that the leakage pathway geometry has
changed such that the flow has moved towards more fully developed/long pipe laminar flow (Sherman et
al., 1986; Hutcheon and Handegord, 1995). The average difference for the pressurization flow exponent
between pre-retrofit and post-retrofit is +0.13. This is the result one would expect from an improvement
in enclosure air tightness.
However, the depressurization phase showed a less pronounced, contrary trend. The average difference
between pre-retrofit and post-retrofit flow exponent values for the depressurization phase is -0.07 and
suggests a move to more turbulent flow and larger leakage pathways. The results data show that the flow
rates decrease between pressurization and depressurization (refer to section 6.3), but the flow exponent
suggests that leakage pathways are getting larger. This trend makes sense when one considers that the
passive exhaust vents were untouched during pre-retrofit and post-retrofit testing. The area of the vent
leakage would remain constant while, the smaller leakage pathways (i.e., cracked window sealant) were
remediated. So, relative to the pre-retrofit testing, the leakage pathways of the post-retrofit enclosure
indicative of less well developed, orifice flow.
1
pre-press = pre-retrofit pressurization, post-press = post-retrofit depressurization, pre-depress = pre-retrofit depressurization,
post-depress = post-retrofit depressurization
2
positive change indicates a move towards laminar flow (smaller cracks), negative change indicates a move towards turbulent
flow (larger cracks)
ABNORMAL FLOW COEFFICIENT VALUES
Abnormal flow coefficient values in the study
On six occasions, the flow exponent value drops below the theoretical limit of 0.5, while the R-squared
value remains at an acceptable level. According to the power law, flow increases as pressure increases,
and the function of their relationship, n, determines the degree of increase. If n is below the theoretically
possible value (Bernoulli’s limit), the power law relationship is no longer valid (Walker et al. 1998;
ASTM E779-10).
Theoretical flow exponent values
There could be many scenarios that cause invalid flow exponent values, including a window opening, a
seal failing, or a stuck passive exhaust vent suddenly releasing, etc. However, in these cases one would
expect the R-squared to show a sudden change in the test data. In the cases presented in Table 1, the Rsquared values are well within threshold levels. The most likely explanation is that the higher pressures
are engaging one or more components and/or leakage paths in the enclosure assembly at a constant and
proportional rate. Two examples are included below to better explain the phenomenon.
For the purposes of illustrating the observed phenomenon, various values for Q have been selected to
represent flow through a theoretical enclosure at different pressure levels.
Table 2 - Theoretical flow exponent values n<0.5
Q
940
1570
2150
2450
P
10
30
50
60
Series 1: n=0.5-1.0
logQ
logP
n
2.97
1
0.52
3.2
1.48
3.33
1.7
3.38
1.78
r-sq
0.99
Q
800
1300
1600
1700
P
10
30
50
60
4 3.8 3.6 log Q 3.4 R² = 0.99676 R² = 0.99952 3.2 3 2.8 2.6 2.4 2.2 2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 log P Series 1: n=0.5-­‐1.0 Series 2: n<0.5 Linear (Series 1: n=0.5-­‐1.0) Linear (Series 2: n<0.5) 1.9 Series 2: n<0.5
logQ
logP
2.9
1
3.11
1.48
3.2
1.7
3.23
1.78
n
0.43
r-sq
0.99
Figure 1 – Graphical representation of log Q as a function of log P for Table 2 - Theoretical flow exponent values n<
Table 3 - Theoretical flow exponent values n>1.0
Series 1: n=0.5-1.0
Q
P
logQ
940
10
2. 97
1570
30
3.2
2150
50
3.33
2450
60
3.38
logP
1
1.48
1.7
1.78
n
0.52
Series 2: n>1.0
Q
P
200
10
1100
30
2200
50
3500
60
r-sq
0.99
logQ
2.3
3.04
3.34
3.54
logP
1
1.48
1.7
1.78
n
1.56
r-sq
0.99
3.8 3.6 R² = 0.99671 3.4 log Q 3.2 R² = 0.99676 3 2.8 2.6 2.4 2.2 2 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 log P Series 1: n=0.5-­‐1.0 Series 2: n>1.0 Linear (Series 1: n=0.5-­‐1.0) Linear (Series 2: n>1.0) Figure 2 - Graphical representation of log Q as a function of log P for Table 3 - Theoretical flow exponent values n>1.0
What the data shows is that there is a ratio tolerance between flow and pressure. As pressure increases
flow must also increase and by a certain degree. The ratio between the two is described by the power law.
However, in the examples listed above for abnormal flow values, the ratio between flow and pressure is
either too large or too small to conform to the power law even though the data points are almost perfectly
in line (R-Sq >0.99).
The importance of an abnormal flow exponent is in showing the researcher that the leakage path geometry
of the test space enclosure is changing to a large degree. As flow velocity increases, there is a natural
progression towards turbulence. However, in the case of a flow exponent below 0.5, it suggests that
elements of the test boundary are engaging at higher pressures, reducing the small crack leakage pathway
geometry and moving beyond the turbulent flow threshold. In the case of a flow exponent over 1.0, the
opposite phenomenon is occurring. Establishing the presence of these phenomena can be very important
when considering the integrity of the data gathered through fan (de)pressurization testing.
It may also be valuable to know simply that the test enclosure is changing at different pressure
differences. In forensic investigations this may help to explain certain problems or issues in a building’s
performance.
However, extrapolating data based on the power law equation with derived flow exponent values should
be cautioned. As will be discussed in the next section, even acceptable flow exponent values can have
large effects on extrapolated data.
EFFECT OF ABNORMAL FLOW EXPONENT VALUES FOR DATA EXTRAPOLATION
Data that is derived from a theoretically impossible n value is not accurate data as it does not conform to
the power law between the flow rate and all pressure differences. It does not give an indication of the
geometry or area of the leakage pathways at all pressures, as those leakage pathways are changing relative
to pressure. If the linear relationship is invalid, it is impossible to extrapolate air leakage data to pressures
other than those that were tested directly.
There are two severe implications of this error. First, building energy simulations, which use normal
operating pressures in their models, may have inaccurate estimates of the air leakage rate through the
enclosure. Second, some air leakage standards call for buildings to meet specific targets at 4Pa or 10Pa
(CMHC, 2013). As noted previously, it is inherently difficult to test air leakage at these pressures, and
fan pressurization is used instead. If the flow exponent value is incorrect, so will be the air leakage data
at unmeasured pressures. The fallout is a standard based on inaccurate data. Example 3 illustrates the
misleading results using inaccurate flow exponent values.
The data used for Example 3 was fabricated for the purposes of this discussion. In Series 1 the flow
exponent value has been calculated based on all three data points. In Series 2 the flow exponent value has
been calculated based on only the last two data points. The R-squared test is within thresholds for both
series. The different flow exponent values, in turn, affect the flow coefficient values (C) as noted below.
By substituting the pressure difference of 4Pa into the equation (see Equation 1), it is possible to derive
flow rates at this pressure difference.
Table 4 - Example 3: Flow exponent effect on extrapolated data
Series 1
Q (L/s)
P (Pa)
logQ
logP
C (avg.)
n
210.00
10.00
2.32
1.00
69.52
0.47
320.00
30.00
2.51
1.48
460.00
50.00
2.66
1.70
Q (L/s)
P (Pa)
logQ
logP
C (avg.)
n
210.00
10.00
2.32
1.00
28.56
0.71
320.00
30.00
2.51
1.48
460.00
50.00
2.66
1.70
Flow @4Pa
(L/s)
133.50
Series 2
Flow @4Pa
(L/s)
76.47
It is evident from Example 3 that the effect of different flow exponent values on flow rates at 4Pa is large.
This discrepancy would affect any models that rely on air leakage values as a part of their programming.
A related issue is the substitution of flow exponent values when the calculated flow exponent is
theoretically impossible or has not been calculated. This is a precedent set in a variety of literature
(Hutcheon and Handegord, 1995; ASHRAE, 2005; Straube and Burnett, 2006). The value substituted is
usually in the range of 0.62 to 0.66, in an attempt to capture a mix of turbulent and laminar flow through
the enclosure (Price et al., 2006).
However, as identified above, different flow exponent values can dramatically affect extrapolated data.
For illustrative purposes, Example 4 shows how the substitution of different flow exponent values affects
extrapolated leakage rates. In Example 4, the corresponding flow coefficient (C) has also been changed
to reflect the change in the flow exponent value. The data for flow rates (Q) and pressure differences (P)
were taken directly from measurements for one of the suites of the case study building during a
depressurization test. The original flow exponent value for this particular data set was 0.52.
Table 5 - Example 4: The effect of flow exponent and flow coefficient substitution
C=66.45
C=59.28
C=24.51
Pa
n=0.40
n=0.50
n=0.65
n=0.80
4.00
163.65
132.60
97.45
72.30
10.00
236.09
209.66
176.78
150.48
20.00
311.53
296.50
277.39
262.00
30.00
366.38
363.13
361.04
362.39
40.00
411.06
419.31
435.27
456.17
50.00
449.44
468.80
503.21
545.32
60.00
483.44
513.55
566.53
630.95
70.00
514.19
554.70
626.23
713.76
75.00
528.58
574.16
654.95
754.27
Flow rate (L/s) C=95.30
800.00 600.00 400.00 200.00 0.00 0.00 20.00 40.00 60.00 80.00 Pressure difference (Pa) n=0.40, C=95.30 n=0.50, C=66.45 n=0.65, C=39.28 n=0.80, C=24.51 Figure 3 – Substituting flow exponent and flow coefficient values
It is evident from the graph that the disparity in extrapolated data for different flow exponent values is
more pronounced at higher and lower pressure differences. Additionally, as can be noted in the table
above, the percentage influence is greatest at the lower pressure levels. For example, at 4Pa the flow rate
at n=0.65 is 40% different from the flow rate at n=0.40, whereas at 75Pa the flow rate at n=0.65 is 19%
different from the flow rate at n=0.40. Similarly, the difference between theoretically possible flow
exponent values is also quite large at 4Pa. Simple substitution of a flow exponent value of 0.62-0.65
when the flow exponent is unknown may have a significant impact on data extrapolation and result
accuracy.
IMPLICATIONS FOR RESEARCH AND INDUSTRY OF FLOW EXPONENT VALUES
There are four major observations based on the above discussion.
First, it has been shown that R-squared values cannot be relied on as indicators of flow exponent
accuracy. Flow exponent values can vary greatly and even become theoretically impossible, while Rsquared values remain nearly perfect. Without a reliable test it is difficult to ascertain the accuracy of
flow rates at extrapolated pressure differences.
Second, the higher or lower a flow exponent value is the more effect it will have on extrapolated flow
rates. This makes it difficult to justify maintaining an abnormally low (n<0.5) or abnormally high (n>1.0)
flow exponent.
Third, it is difficult to justify the substitution of a flow exponent value for one that is abnormal or where a
flow exponent value is not present. The effect on data extrapolation of flow exponent substitution is
dramatic, especially at building operating pressures.
Fourth, the elimination of data points may resolve theoretically impossible data points; however, there is
no agreed upon method for determining data outliers or their elimination – save trial and error. In the
absence of a rigorous method for determining data outliers, it is inadvisable to simply start eliminating
data.
There are significant implications for the industry of calculated flow exponent values:
•
•
•
•
Energy models using extrapolated flow rates at 4Pa and 10Pa may not be accurate in terms of air
leakage through the enclosure and the effect this will have on building operation.
Testing a building enclosure at various pressure differences and establishing a linear relationship
between the pressure differences may not actually provide a more accurate understanding of air
leakage through the enclosure. It may be important to test the building at various pressure
differences to understand where enclosure elements are engaging; however, the linear relationship
may not be so critical.
A comparison of air leakage rates between buildings using substituted flow exponent values may
be inaccurate.
The linear relationship between flow rates at various pressure differences may be important when
comparing enclosure leakage values of a building pre-retrofit and post-retrofit. The linear
•
relationship will give a better understanding of changing leakage path geometry of the postretrofit enclosure relative to the pre-retrofit enclosure.
Abnormal flow values may be important for forensic investigations in understanding when an
element of the building enclosure is engaging, or how the building enclosure leakage path
geometry is changing. Abnormal flow exponent values are not simply useless data points, but
may indicate significant areas of interest about a test space.
CORRECTIONS
Pre-test corrections
Abnormal flow exponent values are caused by a change in the geometry of the leakage pathways through
the building enclosure at various pressure differentials. One possible mitigation is to create a high
pressure difference prior to the commencement of each testing phase in an attempt to pre-engage elements
of the enclosure assembly that are noticeably engaging at higher pressure differences. Another option is
to reverse the process of both pressurization and depressurization phases. In that case, the test phase
would commence at the highest pressure differential and work its way down to the lowest. It is noted that
elements may only engage at the higher pressure differentials and the mitigations suggested above may
have little effect on the data.
In-test corrections
Flow exponent values should be checked at the termination of each test phase along with the R-squared
values. In the event of an abnormally low or high flow exponent value with an adequate r-squared value,
the enclosure should be checked for any missed leakage pathways, or for any enclosure elements that may
be engaging only at higher pressure differences, such as a sticky vent flap. The elements should be fixed
and the test phase run again.
Post-test corrections
While the least recommended correction, in the case of corrupted data and no provision for retesting, the
following correction is outlined:
1) Data points at the highest pressure difference should be eliminated and effect on the flow
exponent noted. If the flow exponent returns to a theoretically acceptable level, the data
points may be considered incongruous and omitted from the data set.
2) In the event that data point elimination could not reconcile the flow exponent value, then
a value for the flow exponent of 0.65 (ASHRAE 2005) may be substituted and the value
for the corresponding flow coefficient changed accordingly. The data should be
extrapolated to 50Pa using the substituted flow exponent value rather than the original
abnormal flow exponent value.
3) Data should be presented at the 50Pa level to avoid the magnified disparity evident in
higher and lower pressure differentials (ie. 75Pa and 4Pa) and also to conform to industry
reporting practices.
CONCLUSIONS
The paper examined the calculation of flow exponent values and what they indicate about the nature of
the test boundary at different pressure levels. Examples for scenarios where flow exponents surpass the
theoretical thresholds of the power law equation were provided. These examples show how the test
boundary is reacting to higher pressures and may be useful in forensic investigations and in determining
data integrity for fan (de)pressurization tests.
Examples were also provided showing the effect on data extrapolation of different flow exponent values.
The large disparity in flow rates at building operating pressures of different flow exponents from testing
at high pressures (i.e., 50Pa) may give caution to energy modellers or others using this information to
determine normal building air leakage.
The paper suggests a method for checking and correcting abnormal flow exponent values, which may
help the industry attain more accurate results for fan (de)pressurization tests.
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