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 (m 3 /s); C = flow coefficient (m 3 /s/Pa n ; 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.

Pre-press
Post-press
Change 2
Pre-depress
Post-depress
Change
Pre-press
Post-press
Change
Pre-depress
Post-depress
Change
Table 1 – Flow exponent values for individual suites 1
Suite 101
0.59
0.56
-0.03
0.92
0.79
-0.13
Suite 1101
0.52
0.71
+0.19
0.47
0.61
+0.14
Suite 102
0.56
0.67
+0.11
0.80
0.69
-0.11
Suite 1102
0.49
0.56
+0.07
0.46
0.43
-0.03
Suite 301
0.60
0.75
+0.15
0.50
0.60
+0.10
Suite 1103
0.53
0.64
+0.11
0.78
0.59
-0.19
Suite 302
0.68
0.85
+0.17
0.62
0.57
-0.05
Suite 1301
0.74
0.77
+0.03
0.57
0.57
0.00
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.
Suite 303
0.54
0.85
+0.31
0.61
0.37
-0.24
Suite 1302
0.52
0.72
+0.20
0.63
0.47
-0.16
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
P
10
1570 30
2150 50
2450 60
Series 1: n=0.5-1.0 logQ logP
2.97 1
3.2 1.48
3.33 1.7
3.38 1.78 n r-sq Q
0.52 0.99 800
1300
1600
1700
P
10
30
50
60
Series 2: n<0.5 logQ
2.9
3.11 1.48
3.2
3.23 logP
1
1.7
1.78 n
0.43 r-sq
0.99
4
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
2
1
R²  =  0.99676
R²  =  0.99952
1.1   1.2   1.3   1.4   1.5   1.6   1.7   1.8   1.9   log  P
Series  1:  n=0.5-­‐1.0
Linear  (Series  1:  n=0.5-­‐1.0)
Series  2:  n<0.5
Linear  (Series  2:  n<0.5)

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
940 10
1570 30
2150 50
2450 60 logQ
2. 97
3.2 logP
1 n
0.52
1.48
3.33 1.7
3.38 1.78 r-sq
0.99
Series 2: n>1.0
Q
200
1100
2200
3500
P
10
30
50
60 logQ logP n
2.3
3.54
1
3.04 1.48
3.34 1.7
1.78
1.56 r-sq
0.99
3.8
3.6
3.4
3.2
3
2.8
2.6
2.4
2.2
R²  =  0.99671
R²  =  0.99676
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)
210.00
320.00
460.00
Series 2
Q (L/s)
210.00
320.00
460.00
P (Pa)
10.00
30.00
50.00
P (Pa)
10.00
30.00
50.00 logQ
2.32
2.51
2.66 logQ
2.32
2.51
2.66 logP
1.00
1.48
1.70 logP
1.00
1.48
1.70
C (avg.)
69.52
C (avg.)
28.56 n
0.47 n
0.71
Flow @4Pa
(L/s)
133.50
Flow @4Pa
(L/s)
76.47

Pa
4.00
10.00
20.00
30.00
40.00
50.00
60.00
70.00
75.00
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=95.30 n=0.40
163.65
236.09
311.53
366.38
411.06
449.44
483.44
514.19
528.58
C=66.45 n=0.50
132.60
209.66
296.50
363.13
419.31
468.80
513.55
554.70
574.16
C=59.28 n=0.65
97.45
176.78
277.39
361.04
435.27
503.21
566.53
626.23
654.95
C=24.51 n=0.80
72.30
150.48
262.00
362.39
456.17
545.32
630.95
713.76
754.27
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=0.65,  C=39.28   n=0.50,  C=66.45   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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