Report of Investigations 9690 

A New Leak Test Method for Enclosed Cab Filtration Systems 

John A. Organiscak and Michael Schmitz 

DEPARTMENT OF HEALTH AND HUMAN SERVICES
 Centers for Disease Control and Prevention
 

National Institute for Occupational Safety and Health
 

Office of Mine Safety and Health Research
 

Pittsburgh, PA • Spokane, WA
	

May 2012 
This document is in the public domain and may be freely copied or 

reprinted. 

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DHHS (NIOSH) Publication No. 2012–145 

May 2012 

SAFER • HEALTHIER • PEOPLETM 
Contents
 

Abstract .............................................................................................................1
 

Introduction .......................................................................................................2
 

Examination of CO2 Instruments .......................................................................3
 

Experimental Test Methods ..........................................................................4
 

Instrument Test Results ................................................................................6
 

Development of New Cab Leakage Sampling Methodology ...........................10
 

Experimental Test Methods ........................................................................11
 

Test Methodology Results ..........................................................................14
 

Discussion of Cab Leakage Measurement Error.........................................17
 

Conclusions.....................................................................................................20
 

References......................................................................................................22
 

Appendix A. Leakage Model for Cab Filtration System ...................................25
 

Appendix B. Initial Experimental Leakage Data From Cab Test Stand ...........28
 

Appendix C. Refined Experimental Leakage Data From Cab Test Stand .......29
 

Appendix D. Propagation of Error Analysis for the Cab Leakage Testing
 

Methodology ...........................................................................................31
 
 

Figures
 

Figure 1. CAF cab test stand setup for comparative instrument testing................5
 

Figure 2. John Deere tractor cab setup for comparative instrument testing..........6
 

Figure 3. Instrument measurements inside CAF cab test stand during test 

series 1............................................................................................................7
 

Figure 4. Instrument measurements inside John Deere tractor cab during test 

series 1............................................................................................................9
 

Figure 5. CAF cab test stand setup for leakage experiments .............................11
 

Figure 6. Leak testing the John Deere tractor cab ..............................................13
 

Figure 7. CO2 concentrations for the first series of initial leakage tests ..............14
 

Figure 8. Initial leakage test measurements made on the CAF test stand ..........15
 

Figure 9. Typical CO2 concentrations during refined leakage tests.....................15
 

Figure 10. Refined leakage test measurements made on CAF test stand ..........16
 

Figure 11. Cab leakage RSDl estimates with a CO2 instrument RSD of 0.05 .....19
 

Figure 12. Cab leakage RSDl estimates with a CO2 instrument RSD of 0.10 .....19
 

Figure A-1. Basic enclosed cab filtration system.................................................25
 

Tables
 

Table 1. Carbon dioxide instrument specifications................................................4
 

Table 2. Initial instrument comparisons with reference gases...............................7
 

Table 3. Linear regression parameters of Telaire and Vaisala instruments as 

compared to the Sable instrument inside the CAF test stand. .........................8
 

Table 4. Instrument comparisons with reference gases after user recalibration. ..8
 

Table 5. Linear regression parameters of Telaire and Vaisala instruments as 

compared to the Sable instrument inside the John Deere tractor cab. ..........10
 

Table 6. Average leakage and confidence intervals for CAF test stand
 

experiments...................................................................................................16
 

Table 7. Average leakage and confidence intervals for John Deere 7820
 

tractor cab......................................................................................................17
 
ACRONYMS AND ABBREVIATIONS
 

USED IN THIS REPORT
 

ASAE American Society of Agricultural Engineers 

ASTM American Society for Testing and Materials 

CAF Clean Air Filter 

CRADA Cooperative Research and Development Agreement 

CO2 carbon dioxide 

HVAC heating, ventilation, and air conditioning 

NIOSH National Institute for Occupational Safety and Health 

RSD relative standard deviation 

UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT
 

ft foot 

ft3 cubic foot 

ft3/min cubic feet per minute 

in inch 

L liter 

L/min liter per min 

m meter 

μm micrometer 

mL milliliter 

mL/min milliliter per minute 

min minute 

psig pounds per square inch gauge 

pressure 

ppm parts per million 

% percent 

sec second 

 

A NEW LEAK TEST METHOD FOR ENCLOSED CAB
 

FILTRATION SYSTEMS
 

1 By John A. Organiscak

1 Mining Engineer, Office of Mine Safety and Health Research, NIOSH 

2 and Michael Schmitz

2 President and Director of Research and Development, Clean Air Filter, Defiance, Iowa 

Abstract 

A new test method has been developed by the National Institute for Occupational Safety and 

Health (NIOSH) and Clean Air Filter (CAF) for quantifying the outside air leakage into 

environmental cab filtration systems. This method uses specially configured filter cartridges to 

remove carbon dioxide (CO2) from the environmental cab’s air filtration system. Real-time gas 

monitors are used to measure the outside and inside cab CO2 concentrations after the cab reaches 

steady-state equilibrium conditions inside an unoccupied cab. Cab filtration system leakage can 

be mathematically determined using the measured cab penetration (inside to outside cab 

concentration ratio) and the special CO2 filter cartridge efficiency. 

Examination of several CO2 sampling instruments for this type of testing showed them to be 

relatively precise, but exhibited noticeable variations in accuracy. These results indicate that 

frequent gas calibration checks of comparative sampling instruments would be needed to ensure 

their accuracy, which is not well-suited for cab field testing. Thus, a single-instrument, multiplesample-location cab testing methodology was devised to eliminate multiple-instrument sampling 

biases and frequent calibrations during testing. This methodology was examined and can provide 

accurate measurements of filtration system leakage into enclosed cabs with a precise instrument 

at or near steady-state test conditions. The new leak test method provides cab manufacturers, cab 

service personnel, and industrial hygienists with a measurement tool to ensure environmental cab 

integrity and minimize worker exposure to outside airborne substances. 
Introduction
 

Enclosed cabs are an engineering control that can provide a safe, comfortable, and healthy 

work environment for equipment operators. Most modern day enclosed cabs have heating, 

ventilation, and air-conditioning (HVAC) systems for maintaining a comfortable temperature 

and a breathable quantity of air for its occupant(s). Various levels of filtration can be 

incorporated into the HVAC system to improve the ventilation quality of the air inside the cab by 

removing outside airborne pollutants such as dusts, chemical aerosols, and vapors. Outside air 

leakage around the intake filter into the HVAC system can notably diminish the cab’s filtration 

system effectiveness [Heitbrink et al. 2003; NIOSH 2008]. A poorly sealed cab HVAC/filtration 

system can be difficult to recognize because of its concealed system components and the 

invisible nature of some airborne pollutants that can penetrate the cab. 

Enclosed cab filtration system effectiveness has been previously studied and can be difficult 

to measure in practice. The American Society of Agricultural Engineers (ASAE) previously 

devised a consensus standard for testing a cab’s particulate reduction factor and specifying a 

cab’s performance criteria for pesticide applications. These procedures used optical particle 

counters inside and outside the cab to examine 2 to 4 μm ambient air particulate penetration into 

the cab as it drives along on an end-use tractor [ASAE 1997; Heitbrink et al. 1998]. An 

alternative test procedure that has been examined is particle counting inside and outside of a 

stationary vehicle cab parked inside a temporary enclosure filled with incense smoke 

contaminants [Moyer et al. 2005]. Other researchers have measured respirable dust mass 

concentrations inside and outside of mining equipment cabs during multiple production shifts 

[Organiscak et al. 2003; Cecala et al. 2003; Cecala et al. 2005]. Several of these studies have 

indicated that inconsistent and low particulate or dust concentrations can yield unreliable cab 

performance results between replicated cab tests [Heitbrink et al. 1998; Organiscak et al. 2003]. 

Also, internal cab particulate generation such as dirty floors, interior surfaces, and abraded 

blower motor brushes can also interfere with measuring external particle or dust penetration into 

the cab [Cecala et al. 2005; Heitbrink and Collingwood 2005]. 

In a response to develop alternative cab performance test methods, NIOSH entered into a 

Cooperative Research and Development Agreement (CRADA) with Clean Air Filter (CAF) of 

Defiance, Iowa to develop field test methods for evaluating environmental integrity of enclosed 

cabs. This research was conducted to develop an expedient, simple, quantitative, and reliable 

field test method for measuring air leakage into enclosed cab filtration systems. A field test was 

sought that used a measurable airborne agent around the test vehicle which would pose minimal 

health and safety risks to the user. Atmospheric gases have these desirable attributes and were 

considered for the leak test medium or tracer. Cab leakage testing research was ultimately 

conducted with carbon dioxide given that instrumentation and gas-absorbent media were readily 

available for filtration and methodology development. This research focused on conducting a 

timely stationary cab leak test for an unoccupied cab. 

The initial development of this new leak testing concept is described in a previously 

published paper [Organiscak and Schmitz 2006] and several patents [Organiscak and Schmitz 

2009; Organiscak and Schmitz 2010]. Enclosed cab carbon dioxide filtration and leakage 

experiments were conducted on a laboratory enclosure test stand to formulate and validate a 

testing methodology. During this research a mass balance mathematical model was developed 

and is shown in Appendix A (Equation A-12) to account for air leakage into filtered 

2
 
environmental enclosures. This model, shown below, describes cab penetration (Pen) in terms of 

fractional intake filter efficiency (ηf), intake air leakage (l), inside concentration (x) and outside 

concentration (c) at steady-state cab conditions. Intake air leakage (l) is defined as the proportion 

of cab intake airflow that bypassed the filter media and is unfiltered. 

f f

l

c

x

Pen  1  

(A-12) 

Results from the initial laboratory experiments indicated that this steady-state atmospheric 

gas testing methodology can detect the amount of air leakage into filtered environmental 

enclosures, but additional research was needed to refine the level of accuracy that can be 

achieved with the instrumentation and test filters [Organiscak and Schmitz 2006]. 

Additional research was conducted to examine several grades of carbon dioxide 

instrumentation and to refine the leakage testing methodology for improving measurement 

accuracy. Tests were conducted to examine the operational differences between three 

instruments, having diverse performance specifications. A multiple-location cab sampling 

methodology was also developed for a single-instrument evaluation of cabs. The singleinstrument sampling methodology accentuates instrument precision characteristics for leak 

testing cabs while diminishing calibration inaccuracies or biases associated with using multiple 

instruments. This paper describes the tests conducted with the carbon dioxide instruments and 

the single-instrument, multiple-location cab sampling methodology development. 

Examination of CO2 Instruments 

Three different types of infrared-sensing carbon dioxide instruments were examined for their 

suitability for the proposed leak test. The instruments chosen were conveniently available to 

NIOSH and CAF for testing. These instruments included Telaire 7001 carbon dioxide monitors 

(Goleta, Calif.), Vaisala GM70 carbon dioxide meters (Helsinki, Finland), and a Sable CA-10a 

carbon dioxide analyzer (Sable Systems International, Las Vegas, Nev.). Table 1 shows the 

manufacturer specifications and approximate cost for each of these products. These only 

represent a few of the many instruments available from other manufacturers, but indicate the 

performance range that can exist between different grades of instruments. The Telaire and 

Vaisala were lower-cost, hand-held passive sampling instruments, and the Sable was a highercost, portable benchtop active sampling instrument. NIOSH had two Vaisala and two Telaire 

instruments available for testing, and these instruments are referred to specifically as Vaisala 1, 

Vaisala 2, Telaire NIOSH1, and Telaire NIOSH2 throughout this paper. CAF had one Sable and 

two Telaire instruments available for testing, and these instruments are referred to specifically as 

Sable, Telaire CAF1, and Telaire CAF2 throughout this paper. 

According to Table 1, these instruments had different specified levels of accuracy, response 

times, and user calibration capabilities. The Telaire was the least accurate instrument (+ 50 ppm 

or + 5% of reading) with a response time specified as less than 60 seconds. It could only be 

zeroed with nitrogen or one-point calibrated to a known gas concentration by the user. The 

Vaisala was the second most accurate instrument (< + 20 ppm + 2% of reading) with a 30 second 

3
 
 

 

 

 

 

response time, and the Sable was the most accurate instrument (better than 1% of calibrated 

span) with a 1⁄2 second response time. Both these instruments could be two-point calibrated 

(zeroed and spanned) by the user. Because the Sable had 11 selectable measurement ranges, it 

was the most flexible in that it could be spanned over a narrow or wide CO2 concentration range. 

Table 1. Carbon dioxide instrument specifications 

Instrument 

Specifications 

Telaire 

Model 7001 

Vaisala 

Model GM70 

Sable 

Model CA-10a 

Sample method Passive diffusion Passive diffusion or pump 

aspiration 

Active sample pump 

400 ml/min 

Measurement range 

0–10,000 ppm display 

0–4,000 ppm analog 

output 

GMP 222 probe 

0–2,000 ppm 

11 Selectable ranges 

Used ranges at or under 

0–0.20% or 2,000 ppm 

Data logging 

capability 0–2 V analog output Internal memory 2,700 points 0–5 V analog output 

Accuracy The larger of + 50 ppm or 

+ 5% of reading 

< + [20 ppm + 2% of 

reading] 

Better than 1% of calibrated 

span 

Resolution 1 ppm 10 ppm 1 ppm up to 2,000 ppm 

User calibration Zero or one point Two-point span Two-point span 

Pressure 

compensation User input of elevation User input of barometric pressure 

Instrument adjusted 

barometric pressure 

Temperature 

compensation None User input of temperature Software adjusted 

Response time < 60 seconds 30 seconds 0.5 second 

Warm up time < 60 seconds 15 minutes Several hours 

Cost Approx. $500 Approx. $2,000 Approx. $6,500 

Experimental Test Methods 

All of these instruments were tested against three known calibration gases and comparatively 

3 tested side-by-side in an enclosure at various CO2 concentration levels.

3 The instrument comparisons made in this study are intended to be illustrative of such categories of instruments 

available for cab/leakage testing, but should not be considered definitive with respect to their performance or 

reliability in other environmental or ventilation monitoring applications. 

4
 

They were not evaluated 

under the NIOSH Guidelines for Air Sampling Analytical Method Development and Evaluation 

[NIOSH 1995]. These tests were performed at CAF’s test facility in Defiance, Iowa. The three 

calibration or reference gases used for instrument calibration, validation, and testing were 0 ppm 

(pure nitrogen), 20.1 ppm (CO2 and nitrogen mix), and 400 ppm (CO2 and nitrogen mix). Before 

these instruments were tested, NIOSH sent their Telaire and Vaisala instruments back to the 

manufacturers for calibration; these instruments were initially tested as calibrated by the 

manufacturer. CAF purchased the Sable and one of their Telaire instruments before testing. 

CAF’s older Telaire was rezeroed, but the newly purchased Telaire was initially tested as 

calibrated by the manufacturer. The Sable was zeroed with nitrogen and spanned with the 400 

ppm reference gas. 
These three types of instruments were used to measure the known gas concentrations flowing 

from the reference gas tanks. The Sable actively drew gas at 400 ml/min through sample tubing 

from the larger opening side of a syringe reservoir packed off with permeable facial tissue that 

was positively pressurized by 500 ml/min of regulated reference gas flow from the tank. This 

procedure avoided over pressurizing the Sable’s sampling pump from the highly pressurized gas 

cylinders. Reference gases were regulated at a positive pressure of 7 psig from the tank into the 

calibration ports of the Telaire or Vaisala instruments. Carbon dioxide (CO2) concentration 

averages of 15 seconds were recorded over 5-minute sampling periods for each instrument and 

reference gas. The Sable and Telaire analog outputs were recorded by a Telog 3307 multichannel 

data acquisition system (Telog Instruments, Inc., Victor, N.Y.). The Vaisala measurements were 

stored in the internal memory of the instrument. 

After the reference gas measurements were completed, instrument tests were conducted 

under more realistic atmospheric sampling conditions inside CAF’s experimental cab test stand. 

Figure 1 shows the cab test stand setup for instrument comparisons. 

 

 

 

 

 

 

CO2 Intake 

Sable 

CAB TEST STAND 

Panel Filter 

Blower Fan 

Cab Airflow 

Discharge Orifice 

Mixing Fan 

Vaisalas 

Telaires 

Figure 1. CAF cab test stand setup for comparative instrument testing. 

5 

The cab test stand is a 

3 simulated plywood cab enclosure with a known volume of 52.5 ft . It does not represent any 

particular cab, but a physical model that can have its filter, airflow, and pressure characteristics 

controlled during laboratory testing. The four Telaire and two Vaisala passive diffusion sampling 

instruments were placed side-by-side inside the center of the cab test stand. The Sable was 

placed outside the enclosure with the inlet of its sampling hose place at the center of the 

enclosure near the other instruments. The interior of the cab test stand was initially sampled at 

atmospheric CO2 concentrations just after its entry door was closed. The CO2 concentrations 

were sequentially stepped to lower concentrations by turning the air filtration system on and off 

during instrument testing. A carbon dioxide panel filter (18-in x 11-in x 2.19-in thick) was fitted 

on the intake air filtration system to lower CO2 concentrations inside the enclosure. CO2 

concentration averages of 15 seconds were again data logged during testing, and instrument 

comparisons were only made during stable 5-min concentration periods with the filtration system 

turned off. A small 6-in-diameter table fan was placed on a small bench inside the back corner of 

the enclosure to ensure adequate mixing and movement of inside air for the passive diffusion 

instruments when the filtration system was off. 
 

Two series of instrument tests were made inside CAF’s experimental cab test stand. Test 

Series 1 was conducted with the instruments after the initial reference gas testing, and Test Series 

2 was conducted after the instruments were recalibrated (Sable and Vaisala instruments) or 

rezeroed (Telaire instruments) by the user. These instruments were further re-evaluated as 

previously described with the reference gases after the second series of cab tests. 

A final examination of these instruments was also conducted inside a John Deere 7820 

tractor cab. Figure 2 shows the tractor cab setup for instrument comparisons. 

Sable 

Vaisalas 

Telaires 

TRACTOR CAB 

CO2 Intake 

Panel 

Filter 

Recirculation Fan 

Cab Airflow Discharge 

Intake Fan 

Leak 

Figure 2. John Deere tractor cab setup for comparative instrument testing. 

The tractor cab 

intake air duct was fitted with a 3-in-diameter tee and straight section of PVC pipe leading to the 

intake filter box fitted with a carbon dioxide panel filter (18-in x 11-in x 2.19-in thick). The 

perpendicular opening of the tee was used for regulating outside air leakage into the filtered 

intake air flowing into the cab. The tee was completely opened to the atmosphere at the start of 

the instrument testing with the tee opening sequentially reduced with PVC fittings to decrease 

outside air leakage around the filter, thus lowering the CO2 concentrations inside the cab. The 

Sable, one Vaisala, and four Telaire instruments were placed side-by-side on the operator’s seat 

in the tractor cab. Concentration averages of 15 seconds were again data logged during testing, 

and instrument measurements were only examined during the stable 5-min concentration periods 

under various leakage conditions at the tee. 

Two series of instrument tests were made inside the John Deere tractor cab. Test Series 1 was 

conducted after the Sable and one Vaisala were recalibrated, and four Telaires were rezeroed by 

the user. Several of the Telaires had to be rezeroed multiple times to improve their agreement 

with the reference gases. Because one of the Vaisalas would not recalibrate, it was excluded 

from the tractor cab testing. Test Series 2 was conducted inside the cab after the Vaisala and two 

Telaires were single-point calibrated to 20 ppm. 

Instrument Test Results 

The instrument concentration averages and standard deviations initially measured during the 

5-minute periods with the different reference gas concentrations are shown in Table 2. As can be 

seen from this table, and consistent with manufacturer representations, the Sable instrument 

6
 
appeared to be the most accurate and precise instrument. The Sable averaged within 2 ppm of the 

reference gas concentrations and had a standard deviation below 1 ppm. The accuracy and 

precision of the other instruments were noticeably less and inconsistent as compared to the Sable 

instrument. The factory-calibrated Telaire and Vaisala instruments showed both higher and lower 

measurement biases as compared to the reference gases. 

Table 2. Initial instrument comparisons with reference gases. 

Instrument Avg. of calibration gas 

concentrations 

Std. deviation of calibration gas 

concentrations 

Pooled Std. 

deviation 

0 ppm 20.1 ppm 400 ppm 0 ppm 20.1 ppm 400 ppm All gases 

Sable CA-10a 2 21 400 0.9 0.4 0.5 0.7 

Telaire CAF1 2 29 395 1.8 3.0 3.2 2.7 

†

Telaire CAF2 0 0 372 0.0 1.2 5.2 3.1 

†

Telaire NIOSH1 11 29 418 3.8 3.6 7.1 5.1 

†

Telaire NIOSH2 4 32 428 0.7 1.0 28.2 16.3 

†

Vaisala 1 11 21 402 2.2 2.2 5.2 3.5 

†

Vaisala 2 -23 -18 377 4.7 16.2 4.9 10.1 

†

Instrument measurements with manufacturer calibration. 

Instrument tests made inside CAF’s experimental cab test stand following the reference gas 

measurements are shown in Figure 3 and Table 3. Figure 3 shows the Telaire and Vaisala 

concentration measurements as compared to the Sable for Test Series 1. 

 

Telaire CAF1 Telaire CAF2 Telaire NIOSH1 Telaire NIOSH2 Vaisala 1 Vaisala 2 

1000 

800 

600 

400 

200 

0

Telaire and Vaisala Concentration, 

ppm 

Unity Line, y = x 

0 200 400 600 800 1000 

Sable Concentration, ppm
 

Figure 3. Instrument measurements inside CAF cab test stand during test series 1.
 

7
 

As can be seen in Figure 

3, the Telaire and Vaisala instruments exhibited linear relationships with the Sable. A unity line 

is also shown on Figure 3 and illustrates that the Telaire and Vaisala instruments exhibited either 

a higher or lower measurement bias as compared to the Sable instrument. Similar linear 

relationships with changed measurement biases were observed for the Test Series 2 after the 

Telaire instruments were rezeroed, and the Vaisala and Sable instruments were recalibrated. 
 

Table 3 shows the linear regression analyses of the Telaire and Vaisala measurements with 

respect to the Sable inside the CAF’s experimental cab test stand. These regression results 

showed a good linear fit of the data with regression slopes (b) and coefficient of determinations 

2 (r ) near 1. However, all of these instruments, with the exception of the Telaire NIOSH1 

instrument during Test Series 1, were not considered to be equivalent to the Sable because their 

regression intercepts (a) and slopes (b) were significantly different from 0 and 1, respectively, at 

the 95% confidence level. Regression intercepts ranged from -21.9 ppm to 22.9 ppm for Test 

Series 1 and ranged from -27.2 ppm to 22.0 ppm for Test Series 2. Noticeable changes to the 

regression intercepts were observed for most of the instruments between the two test series after 

recalibration. Finally as shown in table 3, the Telaire instruments exhibited consistently higher 

regression standard deviations when compared to the Vaisala instruments. 

Table 3. Linear regression parameters of Telaire and Vaisala instruments as compared to the 

Sable instrument inside the CAF test stand.
 

Instrument 

Test Series 1 inside CAF test stand Test Series 2 inside CAF test stand 

Intercept, a Slope, b r 2 s Intercept, a Slope, b r 2 s 

Telaire CAF1 22.9 0.96 0.997 14.2 21.9 0.97 0.996 16.4 

Telaire CAF2 -21.9 0.97 0.997 14.9 22.0 0.99 0.998 11.6 

Telaire NIOSH1 *1.3 *1.00 0.998 12.1 -19.4 0.97 0.997 15.3 

Telaire NIOSH2 3.0 1.03 1.000 6.6 -27.2 0.99 0.997 14.7 

Vaisala 1 22.3 0.97 1.000 4.8 2.1 1.03 1.000 5.0 

Vaisala 2 -11.1 1.00 1.000 4.3 -7.3 1.01 1.000 4.7 

Note: Linear Regression Model is y = a + bx, where y is the Telaire or Vaisala instrument concentration and x is the
 

Sable concentration. 

*The regression intercept (a) and slope (b) parameters were not significantly different from 0 and 1, respectively, at 

the 95% confidence level.
 

Further reference gas evaluations of these instruments were conducted after CAF test stand 

comparisons; these results are shown in Table 4. 

Table 4. Instrument comparisons with reference gases after user recalibration. 

Instrument 

Avg. of calibration gas 

concentrations 

Std. deviation of calibration gas 

concentrations 

Pooled Std. 

deviation 

0 ppm 20.1 ppm 400 ppm 0 ppm 20.1 ppm 400 ppm All gases 

Sable CA-10a 1 20 398 0.4 0.4 0.6 0.5 

Telaire CAF1 19 61 443 3.9 5.0 7.7 5.8 

Telaire CAF2 32 57 435 3.5 2.3 1.9 2.6 

Telaire NIOSH1 2 52 402 1.5 5.1 1.6 3.2 

Telaire NIOSH2 4 0 377 4.6 0.0 4.2 3.6 

Vaisala 1 3 21 390 4.4 3.1 0.0 3.1 

Vaisala 2 0 1 359 0.0 2.2 3.1 2.2 

8
 

As can be seen from this table, the Sable 

instrument appeared to maintain its accuracy and precision. The Sable again averaged within 2 

ppm of the reference gas concentrations and had a standard deviation below 1 ppm. The accuracy 

and precision of the Telaire and Vaisala instruments were noticeably less and inconsistent. Their 

accuracy evidently changed from the first reference gas evaluations in Table 1 after 

rezeroing/recalibration and usage. 
The additional instrument tests conducted inside a John Deere 7820 tractor cab showed 

similar linear measurement characteristics with instrumental biases as observed in the CAF cab 

test stand. Before Test Series 1 inside the tractor cab, the instruments were warmed-up, rezeroed, 

and recalibrated. The Vaisala 1 instrument was removed from the tractor cab testing because it 

was malfunctioning and would not recalibrate. Figure 4 shows the Telaire and Vaisala instrument 

measurements as compared with the Sable instrument made during Test Series 1 inside the 

tractor cab. 

 

Telaire CAF1 Telaire CAF2 Telaire NIOSH1 Telaire NIOSH2 Vaisala 2 

600 

500
 

Unity Line, y = x
 

400 

300 

200 

100 

0 

0 100 200 300 400 500 600 

Sable Concentration, ppm 

Telaire and Vaisala Concentration, 

ppm 

Figure 4. Instrument measurements inside John Deere tractor cab during test series 1. 

As previously observed in the CAF test stand comparisons, the Telaire and Vaisala 

concentration measurements exhibited linear relationships as compared to the Sable. Before Test 

Series 2, only the Telaire CAF2, Telaire NIOSH2, and Vaisala 2 were single-point calibrated to 

the 20.1 ppm reference gas concentration. 

Table 5 shows the linear regression parameters fitted for the hand-held Telaire and Vaisala 

instruments as compared to the Sable instrument for Test Series 1 and 2 inside the tractor cab. 

These regression results once more showed a good linear fit of the data with regression slopes (b) 

2 and coefficient of determinations (r ) near 1. These instruments were again not considered to be 

equivalent to the Sable because all of their regression intercepts (a) and slopes (b) were 

significantly different from 0 and 1, respectively, at the 95% confidence level with the exception 

of the regression slope (b) for the Telaire NIOSH1 instrument during Test Series 1. The most 

noteworthy differences between these instruments were, again, their regression intercepts (a) 

with several instrumental changes observed between Test Series 1 and 2 inside the tractor cab, 

regardless of recalibration status. 

9
 
 

 

 

 

 

 

Table 5. Linear regression parameters of Telaire and Vaisala instruments as compared to the 

Sable instrument inside the John Deere tractor cab.
 

Instrument 

Test Series 1 inside John Deere tractor cab Test Series 2 inside John Deere tractor cab 

Intercept, a Slope, b r 2 s Intercept, a Slope, b r 2 s 

Telaire CAF1 3.4 0.98 0.998 8.6 16.1 1.03 0.998 6.1 

†

Telaire CAF2 -2.3 1.02 0.999 4.8 -4.6 1.03 0.999 4.8 

Telaire NIOSH1 -3.6 *1.00 0.999 5.3 -8.4 1.04 0.999 6.8 

†

Telaire NIOSH2 11.4 1.04 0.998 8.0 -3.2 1.03 0.998 7.1 

†

Vaisala 2 21.0 0.98 0.999 4.7 19.6 1.00 0.999 4.1 

Note: Linear Regression Model is y = a + bx; where y is the instrument conc. and x is the Sable conc. 

*The regression slope (b) parameter was not significantly different from 1 at the 95% confidence level. 

†

These instruments were single-point calibrated at 20.1 ppm of reference gas for Test Series 2. 

These test results have indicated that the Sable, Telaires, and Vaisalas are linear instruments 

with respect to CO2 concentrations. However, and consistent with manufacturer representations, 

the Telaire and Vaisala instruments were found to be less accurate and precise. During our 

testing, the less expensive instruments were also more difficult to accurately recalibrate. It was 

also apparent from this testing that all of these instruments can drift and would need daily checks 

or calibration to ensure their accuracy. Inaccuracy or biases between multiple sampling 

instruments increases their measurement error of cab penetration and/or cab leakage. Because 

maintaining instrument accuracy for field testing cabs can be challenging, a single-instrument, 

cab testing methodology was developed to potentially do away with instrument bias errors and 

field calibrations. 

Development of New Cab Leakage Sampling Methodology 

The linear and relatively precise characteristics of the CO2 instruments tested may be used to 

provide accurate cab leakage field measurements without frequent instrument calibration. One 

way this may be achieved is by measuring relative CO2 concentration differences inside the cab 

filtration system by using the same instrument at or near steady-state conditions. In order to 

accomplish this type of leakage assessment, Equation A-12, previously presented above, is 

reformulated into leakage Equation A-17 in Appendix A and is expressed in CO2 concentrations 

as measured inside the cab, after the intake filter, and outside the cab under steady-state cab test 

conditions. This reformulated leakage Equation A-17 is shown below. 

c i

x i

l







(A-17) 

Where: x = Inside cab concentration (ppm) 

c = Outside cab concentration (ppm) 

i = Immediate concentration after the intake filter (ppm) 

l = Proportion of intake air leakage into the cab 

And: x > i, c > i, and c > x 

10
 
Given the linear and relatively precise nature of the CO2 instruments previously tested, it is 

assumed that instrument inaccuracies can be negated when using the single-instrument sampling 

methodology of Equation A-17. An active CO2 instrument like the Sable would be more suitable 

to take samples from tubing at these multiple locations on the tractor cab. To study this sampling 

methodology, cab filtration system leakage experiments were conducted on CAF’s laboratory 

test stand. 

Experimental Test Methods 

Air leakage testing with the new single-instrument, multiple-location sampling methodology 

was conducted on CAF’s experimental test stand. Figure 5 shows the schematic of the cab air 

leakage test setup. 

 

 

 

 

 

 

Air Sampling Pumps 

CO2 Intake 

Panel Filters 

Leak 

Mass 

Flow 

Discharge Orifice Meter 

Sable Vaisalas 

Telaires 

CAB TEST STAND 

Fan 

Hotwire 

Anemometer 

Two Position Porting Valves 

Cab Sample Lines 

Outside Air 

Samples 

Filter Sample Lines 

Cab Airflow 

Figure 5. CAF cab test stand setup for leakage experiments. 

11
 

Two CO2 intake panel filters (18-in x 11-in x 2.19-in thick) were stacked to 

ensure negligible filter penetration for the leak testing experiments. Because the filtration system 

was well-sealed with gaskets, silicon, and duct seal compound (putty), the only air leakage 

source into the system was presumed to be the controlled and quantifiable leak downstream of 

the filter panels. A TSI Model 4040 Thermal Mass Flowmeter (TSI, Inc., Shoreview, Minn.) 

measured the controlled air leakage around the intake filter, and its data was continuously 

recorded on a Telog data acquisition system (Telog Instruments, Inc., Victor, N.Y.). A TSI 

Model 8345 VELOCICALC Hot Wire Anemometer (TSI, Inc., Shoreview, Minn.) was used to 

measure the cab intake airflow (3-in-diameter pipe centerline measurement) to the cab test stand 

at the beginning and end of each leak test condition. These centerline airflow measurements were 

multiplied by a 0.85 centerline velocity factor to determine the average airflow throughout the 

cross section of the PVC pipe for the high Reynolds Number turbulent flow condition [Knudsen 

and Katz 1958]. The portion of air leakage around the intake filter (l) could be determined by 

dividing the average leakage airflow quantity by the average cab intake airflow quantity for the 

test. 
Three leakage levels were evaluated in the laboratory. The three leakage levels studied were 

no leak, a 1⁄4-in-diameter orifice leak, and a 3⁄4-in-diameter orifice leak. The fan speed was 

electronically adjusted by a variable voltage source to try and maintain about 25 cfm of intake air 

into the cab for the three leakage conditions. This yielded approximate leakage percentages of 

0%, 5.5%, and 11.5% during laboratory testing. 

One Sable, two Vaisala, and two Telaire instruments were used to sequentially sample CO2 

concentrations: (1) inside the cab, (2) immediately after the intake filter, and (3) outside the cab 

at or near-steady operating conditions. To conduct this type of sampling procedure, the 

instruments were located outside the unoccupied cab test stand, and the air samples were actively 

drawn from the multiple locations through 1/8-in inside diameter Tygon tubing. The Sable is an 

active sampling instrument with its own air pump and was found to be ideal for this type of 

sampling procedure. Because the Vaisalas and Telaires are passive sampling instruments without 

pumps, SKC Model 224-PCXR4 universal air sampling pumps (SKC, Eighty Four, Pa.), 

operating at 1.0 L/min, were used to actively pump air from the sampling locations to the Vaisala 

and Telaire instruments. The sampled air, drawn with SKC sampling pumps, was pumped into 

the Vaisala instruments’ calibration hood or into the Telaire instruments’ calibration port. Table 

1 specifies that the Vaisala instruments can be aspirated with a sampling pump, and that the 

Telaire instruments are primarily passive samplers. However, the Telaire instruments were also 

pump aspirated in these experiments to examine their performance for this type of sampling 

procedure. Instrument comparisons with calibration gases before testing showed that the Sable 

operated within a few ppm of the known gas concentrations, although the Vaisala and Telaire 

instruments varied up to 40 ppm from the known gas concentrations. 

Two initial series of air leakage tests were conducted with the CO2 instruments. For Test 

Series 1 the Sable and one of the Vaisalas simultaneously sampled the same cab locations with 

the other Vasiala sampling a different cab location. For Test Series 2, the Sable, one Telaire, and 

one Vaisala simultaneously sampled the same cab locations with another Telaire sampling a 

different cab location. The Sable and one of the other instruments initially sampled the inside cab 

concentrations to detect when they reached their lowest stable level after the cab test stand door 

was closed (at or near steady-state conditions). The CO2 concentration measurements for this 

leak test methodology started only after the cab interior was stable. All the instruments were 

sequentially rotated through the three sampling locations twice. Instrument air samples drawn 

from inside the cab were successively switched to immediately after the intake filter by using 

two-position porting valves (Dwyer Instruments, Inc. Michigan City, Ind.). The outside cab 

location was consecutively sampled by disconnecting the instruments’ sampling hoses from the 

two-position porting valves and later reconnecting these hoses to return to the inside cab 

location. The CO2 concentrations at each sampling location were measured for 5 minutes after 

stabilizing for a 5-minute period between location changes. Thus, 10 minutes of sampling time 

was required at each sampling location with 60 minutes of total time needed for completing two 

rotations through the three sampling locations. This procedure was conducted under the three 

leakage conditions. 

Given that these initial tests were conducted during the winter season, CO2 concentrations 

inside the laboratory were found to fluctuate appreciably with the heating cycles of the building. 

Additional laboratory tests were conducted in late spring to re-examine cab leakage 

measurements under more stable CO2 test conditions without the heating cycle interferences. 

During these tests a refined sampling methodology was devised to help average out some of the 

12
 
outside CO2 concentration variation effects on cab leakage measurements. This methodology 

shortened the instrument stabilization and sampling times at each location and allowed an 

increase in the sampling frequency that can be made at each location for a given test time period. 

The new refined methodology reduced the instrument stabilization time to 2 minutes and the CO2 

concentration measurement interval to 3 minutes. This requires 5 minutes of sampling at each 

location and 15 minutes for instrument rotation through the three sampling locations. Thus, the 

frequency of sampling at each location could be increased for a given test time period. Two 

minutes of stability time was considered adequate to cover the few seconds it would take for air 

samples to reach the instruments through nearly 7 ft of 1/8-in-diameter sample tubing and up to 1 

minute of response time for the Telaire instruments. 

This new refined test methodology was evaluated for the three leakage conditions previously 

tested. However, during these tests the Sable, two Vaisalas, and one Telaire were simultaneously 

sampled inside the cab, immediately after the intake filter, and outside the cab for three complete 

rotations. The leak test measurements again started after the inside cab concentrations stabilized 

and the cab was operating at or near steady-state conditions. Three sampling rotations were 

conducted at all the sampling locations for a total sampling time of 45 minutes during each 

leakage condition. All the leakage conditions were replicated with this methodology during the 

two test series. Because these tests were conducted in late spring, the laboratory concentrations 

were reasonably steady due to less heating cycles and better laboratory ventilation. 

Finally, the use of the refined leak testing methodology was demonstrated on a John Deere 

7820 tractor cab as shown in Figure 6. 

Figure 6. Leak testing the John Deere tractor cab. (Photo courtesy of Clean Air Filter). 

13
 

Because the filtration system leakage area(s) into this 

tractor cab were unknown and air leakage could not be quantified by airflow measurements as 

previously conducted in the laboratory test setup, only the refined CO2 leak testing methodology 

was used. For this test a Clean Air Filter JD60R cylindrical CO2 test filter cartridge was 

constructed with two sealed sampling tubes placed through the filter housing to the immediate 

downstream side of the filter. The intake test filter was inserted into the tractor, and the Sable 

and Vaisala 1 instruments were used to measure concentrations inside the cab, immediately after 

the filter, and outside the cab. The instruments were simultaneously rotated three times through 

these sampling locations as just previously described for the refined test methodology. 
Test Methodology Results 

Air leakage was calculated from the air quantity measurements made during the laboratory 

tests and compared to the CO2 determined leakage as measured by each of the instruments used 

during the tests. The average mass flow meter quantity during the leakage condition was divided 

by the average hot wire intake air quantity measured before and after each leakage test. The air 

leakage measured by each CO2 instrument was determined from using the sequential 

concentrations measured inside the cab, immediately after the intake filter, and outside the cab in 

Equation A-17. The proportional leakages measured are expressed as percentages throughout this 

paper. 

The initial leakage methodology test results are shown in Figures 7 and 8. The data for both 

initial test series are shown in Appendix B. As illustrated by Test Series 1 in Figure 7, the outside 

cab test stand laboratory CO2 concentrations noticeably varied during these initial leakage tests. 

Sable Vaisala 1 Vaisala 2 Leakage Airflow 

0.0 

1.0 

2.0 

3.0 

4.0 

0 

200 

400 

600 

800 

1000 

1200 

Airflow Leakage, ft3 /min 

CO2 Concentration, ppm 

13:25 14:30 15:35 16:40 17:45 

Time, hr:min 

Figure 7. CO2 concentrations for the first series of initial leakage tests. 

14
 

The outside concentration ranged on average by 76 ppm between sampling rotations for each 

leakage condition tested. Because the leakage Equation A-17 assumes steady-state conditions, 

these concentration variations were reflected in the CO2 leakage variations calculated and shown 

in Figure 8 for both Test Series 1 and 2. Figure 8 illustrates the notable variation in CO2 leakage 

determinations around a unity line as compared to the proportional air leakage quantities 

measured. The notable CO2 variations in some of the no-leak tests yielded negative leakages. 
 

 

Sable Vaisala 1 Vaisala 2 Telaire CAF1 Telaire CAF2 

12% 

10% 

CO2 Measured Leakage, % 

Unity Line, y = x 

0% 2% 4% 6% 8% 10% 12% 

8% 

6% 

4% 

2% 

0% 

-2% 

Air Flow Leakage, % 

Figure 8. Initial leakage test measurements made on the CAF test stand. 

The refined leakage test methodology results are shown in Figures 9 and 10. The data for 

both refined test series are shown in Appendix C. As indicated in Figure 9, the outside cab test 

stand laboratory concentrations were more uniform during these leakage tests. 

Sable Vaisala 1 Vaisala 2 Telaire CAF1 Airflow Leakage 

600 3.0 

500 

0.0 

9:00 9:30 10:00 10:30 11:00 11:30 12:00 

2.5 

2.0 

1.5 

1.0 

0.5 

CO2 Concentration, ppm 

Airflow Leakage, ft3

400 

/min

300 

200 

100 

0 

-100 

Time, hr:min 

Figure 9. Typical CO2 concentrations during refined leakage tests. 

15 

The outside 

concentration ranged on average by 17 ppm between sampling rotations for each leakage 

condition tested. The steadier concentrations were reflected in the leakage comparisons shown in 

Figure 10 for both series of CO2 instrument tests. Figure 10 illustrates less variation in Sable and 

Vaisala CO2 leakage determinations around a unity line as compared with Figure 8 for the initial 

tests. The Telaire leakage measurements continued to show wide variations which could be a 

reflection of the Telaire’s slower response to concentration changes and/or larger variations in 

concentration measurements as previously observed during instrument testing. 
 

 

 

Sable Vaisala 1 Vaisala 2 Telaire CAF1
 

14%
 

12% 

10% 

8% 

6% 

4% 

2% 

0% 

-2% 

Air Flow Leakage, % 

CO2 Measured Leakage, % 

Unity Line, y =x 

0% 2% 4% 6% 8% 10% 12% 14% 

Figure 10. Refined leakage test measurements made on CAF test stand. 

Table 6 shows the averages and 95% confidence intervals for both test series of the refined 

CO2 leakage sampling methodology. 

Table 6. Average leakage and confidence intervals for CAF test stand experiments 

Test 

series 

Airflow 

leakage, % 

CO2 Instrument leakage averages and 95% confidence intervals 

Sable CA-10a Vaisala 1 Vaisala 2 Telaire CAF1 

1 

0.0% 

5.6% 

11.5% 

0.2 + 0.7% 

5.1 + 0.2% 

11.1 + 0.7% 

1.0 + 1.0% 

6.1 + 2.9% 

12.4 + 1.1% 

0.0 + 0.8% 

6.0 + 0.4% 

11.9 + 1.4% 

0.0 + 0.0% 

1.7 + 3.0% 

9.1 + 6.1% 

2 

0.0% 

5.4% 

11.4% 

0.4 + 0.7% 

5.0 + 0.7% 

10.7 + 0.6% 

-0.1 + 0.7% 

5.7 + 2.7% 

11.2 + 2.9% 

0.8 + 1.6% 

5.2 + 3.3% 

11.0 + 0.4% 

0.0 + 0.1% 

4.1 + 3.0% 

12.1 + 1.5% 

1&2 

0.0% 

5.5% 

11.5% 

0.3 + 0.3% 

5.0 + 0.2% 

10.9 + 0.3% 

0.4 + 0.7% 

5.9 + 1.1% 

11.8 + 1.1% 

0.4 + 0.7% 

5.6 + 1.0% 

11.5 + 0.6% 

0.0 + 0.0% 

2.9 + 1.8% 

10.6 + 2.4% 

16
 

Each test series has three complete rotations between the 

sampling locations for each leakage condition. As can be seen from this table, the Sable 

instrument yielded the most accurate and precise CO2 leakage measurements. The Sable leakages 

were within 0.7% of the air quantity leakage measurements with confidence intervals up to 0.7%. 

When both test series data (replicates) were averaged together, the Sable leakage confidence 

intervals decreased to < 0.3%. Leakages determined from the Vaisalas showed reasonable 

agreement with the air quantity measurements of leakage. The Vaisala leakages were usually 

within 1.0% of the air quantity leakage measurements with confidence intervals up to 3.3%. 

When both test series data (replicates) were averaged together, the Vaisala confidence intervals 

decreased to < 1.1%. The Telaire instrument had the most notable disagreement with the air 

leakage measurements. The Telaire leakages were greater than 1.0% of the air quantity leakage 

measurements for about half of the tests with confidence intervals up to 6.1%. Averaging both 

tests series data (replicates) reduced the Telaire’s confidence intervals to < 2.4%. 
Table 7 shows the leakage testing methodology demonstrated on the John Deere 7820 tractor 

cab. The intake filter cartridge housing has a limited area for inserting a couple of air sampling 

tubes to the downstream side of the intake filter. Only the Sable and Vaisala 1 instruments were 

used simultaneously during this test. The pressure differential measured across the intake filter 

was 0.74 in of water gauge during cab testing. This pressure differential was equivalent to 37 

3 ft /min of airflow through the filter as measured on the CAF cab test stand. As can be seen in 

Table 7, the intake air filter did not have to remove all of the CO2 to detect and measure outside 

air leakage into the cab. 

Table 7. Average leakage and confidence intervals for John Deere 7820 tractor cab 

CO2 

instrument 

Cab 

concentration. 

(ppm) 

Filter 

concentration 

(ppm) 

Outside 

concentration 

(ppm) 

Cab leakage 

(%) 

Average cab 

leakage 

(%) 

Sable CA-10a 

Vaisala 1 

19 

18 

17 

47 

50 

52 

14 

13 

13 

43 

46 

42 

420 

405 

429 

441 

432 

438 

1.2% 

1.2% 

1.0% 

0.8% 

1.1% 

2.5% 

1.1 + 0.3% 

1.5 + 2.3% 

The Sable instrument appeared to provide the most reliable leakage 

measurements. The Sable instrument detected a 1.1% cab filtration system leak with a 95 

confidence interval of 0.3%. The Vaisala instrument detected a 1.5% leak with a confidence 

interval of 2.3%. These test results demonstrate that the multiple-sampling-location methodology 

is a viable cab leakage test method when using a specially constructed intake test filter to allow 

for CO2 concentration sampling immediately downstream of the filter. Because cab filtration 

system leakage area(s) are usually unknown, cab leakage measurements determined from airflow 

quantities would be impractical. 

Discussion of Cab Leakage Measurement Error 

The above laboratory experiments showed that both instrument precision and outside cab 

concentration variations can have an effect on the cab leakage measurement errors. Although 

using the same instrument for cab leakage testing is expected to eliminate instrument bias or 

accuracy errors, random and/or experimental errors are present in the testing. Random errors are 

primarily a result of instrument precision, whereas experimental errors are a result of systematic 

deviations from steady-state test conditions. In order to examine the size of these errors, a 

propagation of error analysis was conducted for the cab leakage measurement methodology 

[Bevington 1969]. The uncertainty or relative standard deviation (RSD) of the leakage 

methodology was derived in Appendix D and is shown below in Equation D-6. The relative 

standard deviation of this leakage methodology (RSDl) is the standard deviation (sl) of 

proportional leakage measurements divided by its mean (l). On the right side of this equation, the 

first three terms tend to reflect the random errors during testing, whereas the last three terms tend 

to reflect the experimental systematic errors of testing with unstable outside concentrations. At 

steady-state cab test conditions, the last three systematic error terms in the equation become zero, 

because the measured concentrations are uncorrelated and have zero covariances. 

17
 
 

 

     

 

  

  

 

   ixic

xc

s

ixic

s

ixic

xc

s

ic

s

ixic

xc

s

ix

s

l

s

RSD

xc ci

x i c xi

l

l































2

2

2

2

2

2

2

2

2 2

2

2

2

2

2

1

2

2

1 1

(D-6) 

where l = mean of proportional leakage, 

sl = standard deviation of proportional leakage, 

RSDl = relative standard deviation of leakage estimate, sl /l, 

x = mean cab concentration, 

sx = standard deviation of cab concentration, 

i = mean downstream filter concentration, 

si = standard deviation of downstream filter concentration, 

c = mean outside cab concentration, 

sc = standard deviation of outside cab concentration, 

 = covariance between x and i, 

= covariance between x and c, 

 = covariance between c and i. 

This equation can be useful for estimating the relative standard deviation of the CO2 cab 

leakage testing methodology given the means, standard deviations, and covariances of the 

concentrations. It was used to examine the primary effects of instrument precision (i.e., random 

error) and intake filter efficiency on cab leakage measurement error. In this analysis the 

systematic error terms were neglected, assuming steady-state test conditions. The RSDl estimates 

were determined for a series of cab leakages from 0.25% to 12%, using instrument RSDs of 0.05 

and 0.10 and filter efficiencies of 90%, 95%, and 99%. A mean outside cab concentration of 400 

ppm was used for this analysis because it represents an approximate outside atmospheric 

concentration within which a cab would be tested. Cab penetration leakage Equation A-12 was 

applied to determine the expected mean cab concentrations for the series of cab leakages at the 

three filter efficiencies. The expected mean downstream filter concentrations were determined by 

using the three filter efficiencies and the outside concentration. Mean concentrations inside the 

cab, outside the cab, and downstream of the filter were assumed to randomly vary with 

instrument precisions (RSDs) of 0.05 and 0.10. Their standard deviations were estimated by 

multiplying their expected means by these RSDs. 

Figures 11 and 12 show the results of the cab leakage and RSDl analysis for instrument RSDs 

of 0.05 and 0.10, respectively, at filter efficiencies of 90%, 95%, and 99%. The individual points 

shown in Figure 11 are the average leakages and RSDls of the instrument test replicates measured 

on the laboratory test stand (Appendix C) and on the John Deere 7820 tractor cab (Table 7) for 

the refined leakage testing methodology. These experimental data points are included in Figure 

11 because they were measured with outside CO2 levels near steady-state conditions with outside 

concentration RSDs near or below 0.05. 

Both Figures 11 and 12 clearly show that using a more efficient intake filter and a more 

precise instrument reduces the cab leakage measurement error (RSDl). The RSDl estimates for a 

2% leak were 0.43, 0.23, and 0.09 for the 90%, 95%, and 99% filter efficiencies, respectively, 

assuming an instrument RSD of 0.05. When assuming an instrument RSD of 0.10, the RSDl 

18
 
estimates for a 2% leak basically doubles to 0.86, 0.46, and 0.19 for the 90%, 95%, and 99% 

filter efficiencies, respectively. Both figures also show that the RSDl estimates for the 90% and 

95% filters appear to rapidly increase and diverge from the 99% efficient filter RSDl estimates 

when leakage is less than 2%. Conversely, the RSDl estimates for the 90% and 95% filters 

converge closer to the 99% filter efficiency estimates when leakage is above 2%. This analysis 

indicates that cab leakage measurement errors can be reduced by using a higher efficiency filter 

(> 95% efficiency) and the most precise instrument at or near steady-state conditions. Lower 

efficiency intake filters and more variable outside concentrations would likely require additional 

cab testing replicates to increase the confidence in the measurements. 

 

RSDl 1.8 

1.6 

1.4 

1.2 

1.0 

0.8 

0.6 

0.4 

0.2 

0.0 

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 

90% Filter 95% Filter 99% Fillter 

Sable Vaisala 1 Vaisala 2 

Telaire CAF1 Sable/Tractor Vaisala 1/Tractor 

Air Leakage 

Figure 11. Cab leakage RSDl estimates with a CO2 instrument RSD of 0.05. 

 

RSDl 3.5 

3.0 

2.5 

2.0 

1.5 

1.0 

0.5 

0.0 

90% Filter 95% Filter 99% Fillter 

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12%
 

Air Leakage 

Figure 12. Cab leakage RSDl estimates with a CO2 instrument RSD of 0.10. 

19
 
Figure 11 further illustrates the similarities between the estimated leakage errors, assuming 

steady-state test conditions and the actual Sable and Vaisala leakage measurement errors 

conducted near steady-state conditions. The efficiency of the stacked panel filters used on the 

laboratory test stand was greater than 98% as measured by the Sable (Appendix C). The filter 

efficiency of the cylindrical filter used during the tractor test was greater than 96% as measured 

by the Sable instrument (Table 7). As can be seen in Figure 11 the Sable performed closest to the 

99% filter efficiency line. The Vaisala overall performed better than the 90% filter efficiency 

line. The Telaire instrument, in general, did not perform as well as the other instruments. Again 

this is reflective of the precision of the instruments, with the Sable being the most precise 

instrument of the three tested and the Vaisala being the second most precise instrument tested. 

Finally, the instruments response time can also influence cab leakage measurement error 

under non-steady-state test conditions. The quicker response time of the Sable (1/2 sec) allows 

the instrument to more quickly measure concentration changes, while the Vaisala (30 sec) and 

the Telaire (< 60 sec) are slower to respond to the concentration changes. The slower responding 

instruments will tend to over or under measure concentration changes that occur more frequently 

than their response time. Therefore, non-steady-state test conditions can cause additional cab 

leakage measurement errors for the slower responding instruments because they may never fully 

complete their measurement response to concentration variations. 

Conclusions 

Several CO2 instruments were examined for their suitability in leak testing cab filtration 

systems. Variations in measurement accuracy and precision were observed during instrument 

testing. Consistent with manufacturer representations, the Sable was found to be the most 

accurate and precise instrument tested, within 2 ppm of the reference gases and with a standard 

deviation less than 1 ppm. It was clear from the testing that all the CO2 instruments could drift 

and would need daily checks or calibration to ensure accuracy between the multiple instruments 

used for cab testing. 

A single-instrument, multiple-location cab sampling methodology was devised and examined 

to eliminate multiple-instrument sampling biases and frequent calibrations during testing. Initial 

laboratory testing of this sampling methodology showed notable cab test stand leakage 

discrepancy with changing CO2 concentrations outside the cab during testing. A more refined 

test methodology, that reduced the sampling time and increased the sampling frequency at each 

cab location reduced leakage variations measured under more stable test concentrations outside 

the cab. 

The Sable, being an active sampling instrument, was ideal for sampling the multiple cab 

filtration system locations and was shown to be the most accurate and precise instrument for the 

refined cab testing methodology. The Sable provided average leakage measurements within 

0.7% of the air quantity measured leakages with 95% confidence intervals < 0.7% for the three 

complete sampling location repetitions per leakage condition. When the leakage conditions for 

both test series (replicates) were averaged, the confidence interval was reduced to < 0.3%. The 

Vaisala instruments had to be pump-aspirated from the sampling locations and provided average 

leakage measurements usually within 1.0% of the air leakage measurements with confidence 

intervals < 3.3%. When both test series under the same leakage conditions (replicates) were 

20
 
averaged together, the Vaisala instruments’ confidence intervals decreased to < 1.1%. The 

Telaire instruments also had to be pump-aspirated and provided average leakage measurements 

greater than 1.0% of the air quantity leakage measurements for about half of the tests conducted 

with confidence intervals up to 6.1%. Averaging both tests series data (replicates) reduced the 

Telaire’s confidence levels to < 2.4%. When compared to the other instruments tested, data 

indicated that the Telaire was the least suitable for the refined test methodology proposed in this 

report due to lower accuracy, slower response times, and passive mode of operation. 

Further leakage testing with the refined sampling methodology was demonstrated on a John 

Deere 7820 tractor cab with a special filter cartridge containing sampling ports to the 

downstream side of the filter. The Sable and Vaisala measured the tractor cab leakage at 1.1 + 

0.3% and 1.5 + 2.3%, respectively, during three sampling rotations through the cab filtration 

system locations. The tractor cab leakage measurement errors for these particular instruments 

had relatively similar 95% confidence intervals as compared to the cab test stand. 

Both the CAF cab test stand experiments and tractor cab results demonstrate that the refined 

multiple-location cab sampling methodology can be a viable cab leakage test method that does 

not require frequent CO2 instrument calibrations during testing. Measuring cab filtration system 

leakage in the field from airflow quantities is impractical because the filtration system leakage 

area(s) are usually unknown. Testing cabs for leakage with the CO2 sampling methodology is 

best conducted in well-ventilated locations or outside where the CO2 concentrations are 

reasonably steady for approximating the steady-state condition assumptions of the mathematical 

model. A more precise and responsive instrument provides a greater level of measurement 

certainty when quantifying cab leakages less than or equal to 2%. When less precise instruments 

are used, or outside cab test conditions fluctuate, additional sampling repetitions may be needed 

to reduce measured cab leakage confidence intervals. In order to field test cab filtration systems 

with this sampling methodology, a sampling port needs to be installed within the filter or HVAC 

system to allow for CO2 sampling immediately downstream of the filter. Because it was shown 

that a very high efficiency filter (> 99% efficiency) is not necessary to measure cab leakage 

using the multiple-location sampling methodology, test filters can be reused multiple times for 

leakage testing at a somewhat reduced CO2 absorbent media efficiency performance (> 95% 

efficiency). The new leak test method provides cab manufacturers, cab service personnel, and 

industrial hygienists with a measurement tool to ensure environmental cab integrity and 

minimize worker exposure to outside airborne substances. 

21
 
References
 

ASAE (American Society of Agricultural Engineers) [1997]. Agricultural cabs– 

environmental air quality, Part 1: Definitions, test methods, and safety practices, ASAE Standard 

S525-1.1, St. Joseph, MI. 

Bevington PR [1969]. Data reduction and error analysis for the physical sciences. New York: 

McGraw-Hill Book Company, Inc., pp. 56–64. 

Cecala AB, Organiscak JA, Heitbrink WA, Zimmer JA, Fisher T, Gresh RE, Ashley JD 

[2003]. Reducing enclosed cab drill operator’s respirable dust exposure at a surface coal 

operation using a retrofitted filtration and pressurization system. In: Yernberg WR, ed. 

Transactions of the Society for Mining, Metallurgy, and Exploration, Inc. Vol. 314. Littleton, 

CO: Society for Mining, Metallurgy, and Exploration, Inc., pp. 31–36. 

Cecala AB, Organiscak JA, Zimmer JA, Heitbrink WA, Moyer ES, Schmitz M, Ahrenholtz 

E, Coppock CC, Andrews EH [2005]. Reducing enclosed cab drill operator’s respirable dust 

exposure with effective filtration and pressurization techniques. JOEH 2:54–63. 

Hartman HL [1961]. Mine ventilation and air conditioning. New York: John Wiley & Sons, 

pp. 37–38. 

Heitbrink WA, Hall RM, Reed LD, Gibbons D [1998]. Review of ambient aerosol test 

procedures in ASAE standard S525. J. Agric. Saf. Health 4(4):255–266. 

Heitbrink WA, Moyer ES, Jensen PA, Watkins DS, Martin Jr. SB [2003]. Environmental 

agricultural tractor cab filter efficiency and field evaluation. AIHA Journal 64:394–400. 

Heitbrink WA, Collingwood S [2005]. Aerosol generation by blower motors as a bias in 

assessing aerosol penetration into cabin filtration systems. JOEH 2(1):45–53. 

Knudsen JG, Katz DL [1958]. Fluid dynamics and heat transfer. New York: McGraw-Hill 

Book Company, Inc., pp. 148–149. 

Moyer ES, Heitbrink WA, Jensen PA [2005]. Test for the integrity of environmental tractor 

cab filtration systems. JOEH 2:516–523. 

NIOSH [1995]. Guidelines for air sampling and analytical method development and 

evaluation. By Kennedy ER, Fischbach TJ, Song R, Eller PM, Shulman SA. U.S. Department of 

Health and Human Services, Centers for Disease Control and Prevention, National Institute for 

Occupational Safety and Health, DHHS (NIOSH) Publication No. 95-117. 

NIOSH [2008]. Key design factors of enclosed cab dust filtration systems. By Organiscak 

JA, Cecala AB. Pittsburgh, PA: U.S. Department of Health and Human Services, Centers for 

Disease Control and Prevention, National Institute for Occupational Safety and Health, Report of 

Investigations No. 9677, DHHS (NIOSH) Publication No. 2009–103. 

Organiscak JA, Cecala AB, Thimons ED, Heitbrink WA, Schmitz M, Ahrenholtz E [2003]. 

NIOSH/Industry collaborative efforts show improved mining equipment cab dust protection. In: 

Yernberg WR, ed. Transactions of Society for Mining, Metallurgy, and Exploration, Inc. Vol. 

314. Littleton, CO: Society for Mining, Metallurgy, and Exploration, Inc., pp. 145–152. 

22
 
Organiscak JA, Schmitz M [2006]. A new concept for leak testing environmental enclosure 

filtration systems. J. ASTM Int. 3(10):11 pp. 

Organiscak JA, Schmitz M [2009]. Method for leak testing an environmental enclosure. 

Australian Patent No. 2003286587. 

Organiscak JA, Schmitz M [2010]. Method for leak testing an environmental enclosure. 

United States Patent No. 7,727,765. Washington, DC: U.S. Patent and Trademark Office. 

Shanks ME, Gambill R [1973]. Calculus: analytical geometry/elementary functions. New 

York: Holt, Rinehart, and Winston, Inc., pp. 352–360. 

23
 
24
 
x Qc dt Qc  dt Qx dt Vc d l 1  ff  i

Appendix A. Leakage Model for Cab Filtration System
 

Development of the cab filtration system leakage model is based on a time-dependent mass 

balance model of airborne substances in a control volume. Equation A-1 below is a differential 

equation describing the mass balance of an airborne substance for a cab filtration system control 

volume as shown in Figure A-1. 

F 

i 

l 

t 

e 

r 

Ql 

Qi 

Qf 

i 

c 

Vc 

x 

CAB ENCLOSURE 

ηf 

l 

Fan 

Figure A-1. Basic enclosed cab filtration system. 

This is a reformulation of the basic equation for general dilution 

ventilation [Hartman 1961]. The left-hand part of the equation describes the mass of a particular 

airborne substance in the control volume. The positive terms in the right-hand part of the 

equation describe the addition of the mass substance into the control volume from filter 

penetration and intake air leakage. The negative term describes the removal of the mass 

substance from the control volume by intake air dilution. 

Mathematical model: (A-1) 

Model assumptions: 

1.	 Outside airborne substance concentration is constant. 

2.	 Intake filter removes airborne substance. 

3.	 Airborne substance leakage into the filtration system is proportional to the air quantity 

leakage around the intake filter. 

4.	 Inside cab volume static pressure is greater than the outside static and wind velocity
 

pressure, so as to keep the outside airborne substance from infiltrating into the cab 

structure (control volume).
 

where	 Vc = cab volume, 

x = inside cab substance concentration, 

c = outside cab substance concentration, 

i = immediate substance concentration after the filter, 

25
 
ηf = intake filter efficiency, fractional, (1i/c) 

Qf = filtered air quantity, volume per time, 

Ql = air quantity leakage around the intake filter, volume per time, 

Qi = intake air quantity discharged into the cab, volume per time, 

l = proportion of cab intake air leakage, or Ql /Qi , 

t = time, 

and Pen= cab penetration, ratio, x/c. 

 

 

V dx Qlc dt  lQc  dt Qx dt c i i 11  f  i

 

V dx Qc  l dt Qx dt c 1 fi  f  i

 

 

  Qcu 1  ffi  Qxl

i

 du Q dx 

i

 

 

 

 

 

Qc 1  ffi  Qxl

i  0 

Because: Qi = Ql + Qf and Ql = l Qi ; Qf = Qi (1  l) 

Substitute in model: (A-2) 

Simplify: (A-3) 

Rearrange: 

 

 





2

1

1

1

t

t

x

x ffi ci

dt

Qc VQxl

dx

o



(A-4) 

Let: and 

Substitute: 







2

1

11

t

c t

u

i u

dt

Vu

du

Q o

(A-5) 

Integrate: 

   

c

o

i V

tt

uu

Q

12

lnln

1 



(A-6) 

Rearrange: 

c

i

o V

Q t

u

u  

ln 

(A-7) 

Substitute for u: 

 

  c

i

ioffi

ffi i

V

tQ

Qc Qxl

Qc Qxl 











1

1

ln

(A-8) 

Antilog of equation: 

 

 















 







c

i

V

tQ

ioffi

ffi i

e

Qc Qxl

Qc Qxl





1

1

(A-9) 

- The steady-state mathematical solution as t; e 0
 

Reduces to: (A-10)
 

26
 
 

 

 

 

 

 

 

Rearrange: 

 

i

i f f

Q

cQ l

x

  



1

(A-11) 

Simplify: 

f f

l

c

x

Pen   1   

(A-12) 

Solve for l: 

 



f

f

c

x

l



 1

(A-13) 

This leakage equation can be expressed in CO2 concentrations measured inside the cab, after the 

intake filter, and outside the cab as shown below. 

Substitute for f : 



  



c

i

c

i

c

x

l

1

1 1

(A-14) 

Simplify: 







c

i

c

i

c

x

l

1

(A-15) 

Rearrange: 







c

c i

c

x i

l

(A-16) 

Simplify: 

c i

x i

l







(A-17) 

Where: x > i, c > i, and c > x 

Note: This equation is of indeterminate form (0/0) for the particular case where no intake filter 

is used and all the concentrations are equal (x = i = c). In this particular case l’ Hospital’s rule 

can be applied as shown below for this indeterminate form [Shanks and Gambill 1973], yielding 

a limit of 1 or 100% leakage without a filter. 

 

 

 

 

1

1

1

lim lim lim lim 



















ic ic ic ic g i

f i

c i

x i

g i

f i

27
 
Appendix B. Initial Experimental Leakage Data from Cab Test Stand
 

(Test Conditions @ 972 to 973 hPa, 69.5 to 75.3F, 22.8 to 24.3% RH) 

Test 

series 

Airflow 

leakage 

(%) 

CO2 

instrument 

Cab 

concentration 

(ppm) 

Filter 

concentration 

(ppm) 

Outside 

concentration 

(ppm) 

CO2 calculated 

Air leakage (%) 

0.0% Sable 7 8 923 -0.2% 

1 6 766 -0.6% 

1 0.0% Vaisala 1 46 46 983 0.0% 

39 44 824 -0.6% 

0.0% Vaisala 2 -6 -4 1023 -0.2% 

-3 2 824 -0.5% 

5.5% Sable 59 14 912 5.0% 

53 9 1005 4.4% 

1 5.5% Vaisala 1 92 40 952 5.7% 

93 44 1052 4.9% 

5.5% Vaisala 2 50 1 900 5.5% 

50 -5 985 5.6% 

10.8% Sable 93 10 1081 7.8% 

113 16 901 11.0% 

1 10.8% Vaisala 1 139 52 1108 8.2% 

157 66 947 10.3% 

10.8% Vaisala 2 112 17 1018 9.5% 

82 13 927 7.5% 

0.1% Sable 2 1 477 0.2% 

0 3 511 -0.6% 

0.1% Telaire CAF1 2 1 490 0.2% 

4 3 528 0.2% 

2 0.1% Telaire CAF2 9 18 525 -1.9% 

14 23 549 -1.5% 

0.1% Vaisala 2 -17 -24 463 1.5% 

-24 -20 496 -0.9% 

5.5% Sable 33 16 461 3.9% 

30 24 487 1.3% 

5.5% Telaire CAF1 38 26 480 2.8% 

46 36 519 2.0% 

2 5.5% Telaire CAF2 51 35 539 3.3% 

66 58 550 1.7% 

5.5% Vaisala 2 20 -5 447 5.5% 

15 9 476 1.3% 

10.9% Sable 52 6 530 8.8% 

53 8 486 9.3% 

10.9% Telaire CAF1 45 7 544 7.1% 

61 12 506 9.1% 

2 10.9% Telaire CAF2 75 24 561 9.5% 

66 38 541 5.7% 

10.9% Vaisala 2 38 -15 516 5.5% 

37 -10 475 1.3% 

28
 
Appendix C. Refined Experimental Leakage Data from Cab Test Stand
 

(Test Conditions @ 961 to 964 hPa, 64.4 to 72.9F, 45.2 to 52.5% RH) 

Test 

series 

Airflow 

leakage 

(%) 

CO2 

instrument 

Cab 

concentration 

(ppm) 

Filter 

concentration 

(ppm) 

Outside 

concentration 

ppm 

CO2 calculated 

Air leakage (%) 

0.0% Sable 4 5 460 -0.2% 

3 2 419 0.3% 

3 2 415 0.3% 

0.0% Vaisala 1 47 43 498 0.9% 

48 45 474 0.6% 

1 47 41 468 1.4% 

0.0% Vaisala 2 -30 -29 428 -0.2% 

-31 -30 393 -0.2% 

-27 -28 393 0.4% 

0.0% Telaire CAF1 1 1 445 0.0% 

1 1 412 0.0% 

1 1 408 0.0% 

5.3% Sable 23 2 412 5.2% 

23 1 425 5.0% 

23 3 411 5.0% 

5.3% Vaisala 1 61 40 461 5.0% 

73 41 477 7.3% 

1 66 40 472 6.0% 

5.3% Vaisala 2 -8 -33 384 6.0% 

-5 -30 398 5.8% 

-4 -30 388 6.2% 

5.3% Telaire CAF1 4 1 414 0.7% 

14 1 418 3.0% 

9 3 411 1.5% 

10.9% Sable 45 1 412 10.8% 

46 1 396 11.4% 

45 1 399 11.1% 

10.9% Vaisala 1 91 40 459 12.1% 

91 38 444 12.9% 

90 40 449 12.2% 

1 10.9% Vaisala 2 19 -30 388 11.8% 

17 -30 378 11.4% 

21 -30 375 12.6% 

10.9% Telaire CAF1 29 2 413 6.7% 

37 2 400 8.8% 

48 2 398 11.6% 

0.0% Sable 7 4 440 0.7% 

5 4 427 0.2% 

5 4 420 0.3% 

0.0% Vaisala 1 46 48 485 -0.4% 

44 45 481 -0.2% 

43 42 470 0.2% 

2 0.0% Vaisala 2 -26 -33 414 1.5% 

-27 -30 396 0.8% 

-29 -30 392 0.2% 

0.0% Telaire CAF1 1 1 434 0.0% 

2 2 423 0.0% 

2 2 414 0.1% 

29
 
Test 

series 

Airflow 

leakage 

(%) 

CO2 

instrument 

Cab 

concentration 

(ppm) 

Filter 

concentration 

(ppm) 

Outside 

concentration 

(ppm) 

CO2 calculated 

Air leakage (%) 

5.1% Sable 23 4 410 4.7% 

24 4 401 5.2% 

24 4 394 5.2% 

5.1% Vaisala 1 66 37 461 6.9% 

61 42 448 4.7% 

2 62 39 446 5.5% 

5.1% Vaisala 2 -11 -28 383 4.1% 

-11 -31 374 4.9% 

-8 -34 367 6.7% 

5.1% Telaire CAF1 13 2 405 2.7% 

20 2 397 4.6% 

21 2 385 5.1% 

10.8% Sable 44 3 398 10.4% 

46 3 397 10.9% 

46 3 398 10.8% 

10.8% Vaisala 1 83 43 444 10.0% 

90 40 447 12.3% 

2 90 45 443 11.3% 

10.8% Vaisala 2 10 -34 373 10.9% 

15 -30 373 11.2% 

10 -35 371 11.1% 

10.8% Telaire CAF1 45 2 386 11.4% 

49 2 386 12.2% 

50 2 389 12.6% 

30
 
 

Appendix D. Propagation of Error Analysis for the Cab Leakage 

Testing Methodology 

Instrument imprecision and unsteady test conditions increases the propagation of 

measurement error for the single-instrument, multiple-location cab sampling leakage method. In 

order to examine these measurement errors, the following uncertainty analysis for this testing 

methodology was conducted below to show the relative standard deviation of cab leakage 

measurements [Bevington 1969]. 

Cab Leakage Equation: 

 

ic

ix

cixfl





,,

(A-17) 

The variance for this three variable cab leakage equation is: 

















































































 



















 



















 





















c

l

x

l

s

i

l

x

l

s

c

l

s

i

l

s

x

l

ss xl i c xi xc

2 2

2

2

2

2

2

22

2 2











































i

l

c

l

sci

2

2

(D-1) 

where 	 l = mean of proportional leakage, 

sl = standard deviation of proportional leakage, 

x = mean cab concentration, 

sx = standard deviation of cab concentration, 

i = mean downstream filter concentration, 

si = standard deviation of downstream filter concentration, 

c = mean outside cab concentration, 

sc = standard deviation of outside cab concentration, 

 = covariance between x and i,
 

= covariance between x and c,
 

 = covariance between c and i.
 

RSDl = relative standard deviation of leakage estimate, sl /l, 

and 

     

   ic ic

ic

ic

ixic

x

l

































 001 1

2 2

     

   

 

  2 2 2

1 1

ic

xc

ic

ixci

ic

ixic

i

l





































     

 

 

  2 2

10

ic

ix

ic

ixic

c

l































31
 
 

 

 

 

 

 

 

 

 3

2

3

2

4

2

2

4

2

2

2

22

2 2

1

ic

ix

s

ic

xc

s

ic

ix

s

ic

xc

s

ic

ss xl i c xi xc





























   

2

3

2

2

3

2

2 2









































 









































ix

ic

ic

ix

s

ix

ic

ic

xc

s xi xc  

 

     

 

  2

2

2

2

2 2

2

2

2

2

2

2

2

2

1 1

ixic

xc

s

ic

s

ixic

xc

s

ix

s

l

s

RSD x i c xi

l

l





















 

Substitute: 

 

 

 

 

   

 

  

















































































 



















2

2

2

2

2

2

2

2

2

22 1

2

1

ic

xc

ic

s

ic

ix

s

ic

xc

s

ic

ss xl i c xi  

 

 

 

 

 

  















































































 2 2

2

2

2

2

1

2

ic

xc

ic

ix

s

ic

ix

ic

s xc ci

(D-2) 

Simplify: 

  

 4

2

2

ic

xcix

sci







(D-3)
 

 

 

 

 

 

2

4

2

2

2

4

2

2

2

2

2

2

2

2 1

























 























 























ix

ic

ic

ix

s

ix

ic

ic

xc

s

ix

ic

ic

s

l

s

RSD x i c

l

l

  

 

2

4

2

2











































ix

ic

ic

xcix

sci

Re-expressed symmetrically: 

(D-4)
 

Simplify: 

  

 

   ixic

xc

s

ixic

sxc ci









 2

2 2

2

1

2

32
 

(D-5)
 
 

 

     

 

  

  

 

c i x i

c x

s

c i x i

s

c i x i

c x

s

c i

s

c i x i

c x

s

x i

s

l

s

RSD

xc ci

x i c xi

l

l

 





 



 









 







 

2

2

2

2

2

2

2

2

2 2

2

2

2

2

2

1

2

2

1 1

Take the square root to solve for the relative standard deviation of the leakage: 

(D-6) 

33