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Smith, Adam J.
Resting oxygen consumption rates in divers using diver propulsion devices
h [electronic resource] /
by Adam J. Smith.
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b University of South Florida,
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Thesis (M.S.B.E.)--University of South Florida, 2008.
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ABSTRACT: The Marine Corps Systems Command documented mission requirements that cannot be met by current rebreathers. They need to extend dive times without compromising the stealth and compact design of existing devices. This can be accomplished by reducing the fresh gas flow rate. The current flow rate is adequate to support a diver in heavy work. However, the diver will be utilizing a Diver Propulsion Device (DPD) during a large portion of the mission in question. The assumption, then, is that this portion of the mission will not require "hard work". Thus, a new fresh gas flow rate can be established which is sufficient to sustain a Marine diver using a DPD but is conservative enough to extend the duration of the dive. This experiment was designed for manned testing of the rebreathers in such a way to establish the average oxygen consumption rate for divers using a DPD. Marine divers were fitted with a Divex Shadow Excursion (DSE) rebreather modified with a Draeger C8A PO monitor coupled with a Delta P VR3 dive computer. The DSE is a semiclosed-circuit underwater breathing apparatus that provides a constant flow of mixed gas containing oxygen and nitrogen or helium to the diver. The partial pressure of oxygen (PO) and diver depth were monitored and recorded at ten-second intervals. The Navy Experimental Diving Unit has developed and tested a computational algorithm that uses the PO and depth to compute the oxygen consumption rate. Two techniques were employed to estimate the error in this approach: curve fitting and propagation of error. These methods are detailed and the results are presented. They show that the fresh gas flow rate can be safely reduced while the diver is utilizing the DPD, which consequently, will substantially increase the dive time allowed by the device.
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Advisor: William E. Lee, Ph.D.
Propagation of error
x Biomedical Engineering
t USF Electronic Theses and Dissertations.
Resting Oxygen Consumption Rates in Divers Using Di ver Propulsion Devices by Adam J. Smith A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Biomedical Engineering Department of Chemical & Biomedical Engineering College of Engineering University of South Florida Major Professor: William E. Lee III, Ph.D. John R. Clarke, Ph.D. Roland D. Shytle, Ph.D. Date of Approval: October 29, 2008 Keywords: Oxygen Consumption, Diving, Rebreather, P ropagation of Error, Injection Rate, Nitrox, Semiclosed Copyright 2008, Adam J. Smith
Dedication This thesis is dedicated to my family who have lo ved and supported me throughout my studies. I am blessed to have such g reat role models as my parents.
Acknowledgments First, I would like to express the deepest of grati tude to Dr. John Clarke. After beginning an internship with the Navy Experim ental Diving Unit, Dr. Clarke familiarized me with his experiment and invited me to contribute. I will always be grateful for the hard work he put into the experime ntal design and for bringing me up to speed on diving physiology, a subject which I knew very little about going into this project. This work would have not been p ossible without his continual advisement and support. I would also like to thank the other members of my committee, Drs. Bill Lee and Doug Shytle. Dr. Lee has always taken the time to make sure that I choose courses which compliment my research interes ts and strengthen my education. Dr. Shytle hired me as his research ass istant. Not only did this support my financial requirements, but the experien ce and knowledge gained has already proven to be paramount to my graduate e ducation. Those responsible for the Office of Naval Research: Naval Research Enterprise Intern Program (NREIP) should not be ove rlooked. This is the program which funded my trip to the Navy Experiment al Diving Unit in Panama City Beach, FL. If it were not for this program, I would have never had the opportunity to be involved in this project. Ed Lin senmeyer is the coordinator of the NREIP program at the Naval Surface Warfare Cent er. Because of his hard work and dedication, students like me have the oppo rtunity to experience working for the Department of Defense.
i Table of Contents List of Figures iii Abstract iv Chapter 1 Introduction 1 1.1 Motivation for Thesis 2 1.2 Risks 3 1.3 Contributions to the Field 6 1.4 Thesis Structure 7 Chapter 2 Theoretical Foundations 8 2.1 Diving Physiology 8 2.1.1 Gas Laws 8 2.2 Rebreathers 11 2.3 Governing Equations 15 Chapter 3 Materials and Methods 17 3.1 General 17 3.1.1 Divex Shadow Excursion 17 3.2 Experimental Design 18 3.3 Rebreather Modifications 18 3.4 Test Procedures 20 3.5 Data Analysis 21 Chapter 4 Results 23 4.1 Curve Fits 23 4.1.1 Initial Curve Fit 24 4.1.2 Driver Compilation 26 4.1.3 Passenger Compilation 28 4.1.4 Total Compilation 29 4.2 Propagation of Error 31
ii Chapter 5 Discussion 35 5.1 Curve Trends 35 5.1.1 Driver vs. Passenger 35 5.1.2 Negative Slope 38 5.2 Propagation of Error 39 5.3 Limitations 39 Chapter 6 Conclusion 41 6.1 Recommendation 41 6.2 Next Steps 42 References 43 Appendices 45 Appendix A: Driver Curve Fit Â– Numeric Summary 46 Appendix B: Passenger Curve Fit Â– Numeric Summary 47 Appendix C: Total Curve Fit Â– Numeric Summary 48 Appendix D: Comparison of Variances 49
iii List of Figures Figure 1. Schematic of a Semiclosed Rebreather 14 Figure 2. Dive Computer 19 Figure 3. Curve Fit of Total Compilation of Data Fi les 25 Figure 4. Estimated Driver Oxygen Consumption Rate vs. Time 27 Figure 5. Estimated Passenger Oxygen Consumption Ra te vs. Time 28 Figure 6. Estimated Total Oxygen Consumption Rate v s. Time 30 Figure 7. Symbolic Evaluation of Partial Derivative s 32 Figure 8. Numeric Evaluation of Partial Derivatives 33 Figure 9. Propagation of Error 34 Figure 10. Driver vs. Passenger Comparison 37
iv Resting Oxygen Consumption Rates in Divers Using Di ver Propulsion Devices Adam J. Smith ABSTRACT The Marine Corps Systems Command documented missio n requirements that cannot be met by current rebreathers. They ne ed to extend dive times without compromising the stealth and compact design of existing devices. This can be accomplished by reducing the fresh gas flow rate. The current flow rate is adequate to support a diver in heavy work. Howe ver, the diver will be utilizing a Diver Propulsion Device (DPD) during a large port ion of the mission in question. The assumption, then, is that this porti on of the mission will not require Â“hard workÂ”. Thus, a new fresh gas flow rate can b e established which is sufficient to sustain a Marine diver using a DPD bu t is conservative enough to extend the duration of the dive. This experiment was designed for manned testing of the rebreathers in such a way to establish the average oxygen consumpt ion rate for divers using a DPD. Marine divers were fitted with a Divex Shadow Excursion (DSE) rebreather modified with a Draeger C8A PO2 monitor coupled with a Delta P VR3 dive computer. The DSE is a semiclosed-circuit underwat er breathing apparatus that provides a constant flow of mixed gas containing ox ygen and nitrogen or helium to the diver. The partial pressure of oxygen (PO2) and diver depth were
v monitored and recorded at ten-second intervals. Th e Navy Experimental Diving Unit has developed and tested a computational algor ithm that uses the PO2 and depth to compute the oxygen consumption rate. Two techniques were employed to estimate the error in this approach: curve fitting and propagation of error. These meth ods are detailed and the results are presented. They show that the fresh gas flow rate can be safely reduced while the diver is utilizing the DPD, which consequently, will substantially increase the dive time allowed by the device.
1 Chapter 1 Introduction United States Marine Corps Combatant Divers are tra ined to perform mainly reconnaissance and raid type missions. Thes e divers have proven to be paramount in these types of military applications. Many of the missions require the USMC Combatant Divers to remain undetected by t he enemy. They accomplish this by utilizing rebreathers which, dep ending on the type, either greatly reduce or eliminate bubbles from being emit ted into the water and revealing their location. Until recently, they hav e been able to successfully complete their missions by utilizing the MK 25 Oxyg en Rebreather. However, the Marine Corps System Command has since documented a mission requirement that cannot be met by the curre nt rebreather in use. They intend to replace their inventory with a multi-purp ose, O2 closed-circuit or nitrox semiclosed-circuit rebreather named the Enhanced Un derwater Breathing Apparatus (EUBA). Previously, the Navy Experimenta l Diving Unit was asked to review whether the rigs would meet the mission prof ile. Three UBAÂ’s underwent testing to determine if they could meet the mission profile. It was discovered that, under their current configurations, none of t he devices could meet the mission profile. However, if properly reconfigured all of the UBAs could meet
2 the mission requirements (Clarke 2007). One of the se UBAs, the Divex Shadow Excursion (DSE), was selected for use during this t hesis. The mission profile requires that, during a large p ortion of the dive, the divers will be propelled by a Diver Propulsion Devi ce (DPD). A DPD is a vehicle which can transport two divers underwater and, as a result, allow them to travel longer distances, deliver increased payloads, minim ize fatigue, and maximize endurance (McCarter 2005). Therefore, because the divers will be using a DPD, they will actually be performing very light work. This highlights the key assumption that would permit the extension of the t otal dive time allowed by the DSE (in order to meet the USMC mission requirements ). This assumption is that while the divers are being towed by a DPD, their ox ygen consumption rate is similar in magnitude to the previously documented r esting oxygen consumption rate. This assumption had to be tested and verifie d. Consequently, this study was designed in such a way as to provide confirmato ry measurements of oxygen consumption rates during the towed portion of the m ission. 1.1 Motivation for Thesis The United States Military utilizes rebreathers fo r underwater reconnaissance and raid missions. There are advant ages to the use of rebreathers over conventional open-circuit scuba ri gs. They offer better gas efficiency and near-silent operation with few to no bubbles (depending on the type of rebreather). However, the mission capabili ties are limited by the dive
3 time offered by the device. Semiclosed rebreathers like the ones conventionally used by the military, have a constant fresh gas flo w rate. This flow rate is generally set at 6.0 L/min. This has been shown to meet the oxygen demands of a hard working diver with a common nitrox gas mixtu re (60% oxygen, 40% nitrogen). There have been several reports that sh ow that 3.0 L/min is the maximum oxygen consumption rate (Nuckols, Clarke et al. 1998). Unfortunately, a mission requirement is unable to be met due to th e limited dive time that this, all-encompassing, fresh gas flow rate offers. The Navy Experimental Diving Unit was tasked to test solutions to this problem. 1.2 Risks As with all manned experiments, there were health risks which had to be carefully considered. All necessary precautions we re taken to minimize potential health risks. Marine Combatant divers were require d to use an underwater breathing apparatus (UBA) which was new to the Unit ed States Military. Even though the Divex Shadow Excursion had not yet been certified for use by the military, it had been used by the British Royal Mar ines and the following Navies as a Special Operations UBA: Britain (SAS), Norway, Australia, and Germany. Therefore, the DSEÂ’s safety has been well documente d. The DSE was tested in semiclosed mode with constan t nitrox gas flow. As with all rebreathers, there is a limit to how de ep the diver can safely go. This is due to PO2 changes which will be discussed in the Dive Physio logy section
4 below. However, if the driver lost control of the DPD and went deeper than the limits of the rebreather, this would have jeopardiz ed the safety of both divers. For this reason, all manned testing was conducted a bove a 20 to 30 foot deep hard bottom (along the profile of a beach in Panama City, FL). This eliminated the possibility that the divers might exceed the ma ximum depth allowed by the U.S. Navy Diving Manual based on the configurations of the DSE (Navy Diving Manual 2005). Another risk which is inherent to all semiclosed-c ircuit UBAs is the possibility for hypoxia. Hypoxia is the shortage o f oxygen in the body. Unfortunately, there is usually no warning to the d iver that they are becoming hypoxic. This is because carbon dioxide is usually what causes a person to experience the sensation of Â“oxygen hungerÂ”. Howev er, rebreathers filter the carbon dioxide from the breathing circuit which con sequently, eliminates the bodyÂ’s usual warning of oxygen deprivation. The hu man body is optimized to breath oxygen at a partial pressure of .21 atmosphe res absolute (ATA). If the inspired PO2 drops to a value much less than this, hypoxia ensu es and the body begins to shut down (Strauss and Aksenov 2004). To alleviate this risk, the fresh gas flow rate was set to 6.0 L/min, which has alrea dy been shown to support a hard working diver (Nuckols, Clarke et al. 1998). Because the experiment called for the divers to be using a diver propulsion devic e, they would actually be performing very light work. To further increase th e safety of the divers, an O2 monitor was used to monitor the diverÂ’s oxygen part ial pressure. This is not a
5 standard feature on the DSE. This modification wil l be described in detail in the equipment section. The diverÂ’s PO2 will be displayed on a VR3 dive computer. As needed, the diver can manually add fresh nitrox using the demand valve on the rebreather. Hyperoxia was another potential risk that had to be considered. Hyperoxia occurs when the body receives too much oxygen. Oxy gen, when at high partial pressures, is toxic to the human body. This is oft en referred to as oxygen toxicity. One unfortunate incident is discussed in a case report by a Christopher Lawrence, a forensic pathologist. An experienced d iver used 50% nitrox gas during a dive of 47 meters. This resulted in a par tial pressure of oxygen which reached a staggering 2.9 atmospheres absolute (Lawr ence 1996). This diver died, most likely, from seizures associated with ox ygen toxicity. Acute oxygen toxicity mainly affects the central nervous system. If a diver becomes hyperoxic, they can experience visual and audible disturbances nausea, clumsiness, and finally convulsions (Strauss and Aksenov 2004). Hy peroxia was avoided by using nitrox gas (60% oxygen: 40% nitrogen) and by limiting the diver depth in order to control the partial pressure of oxygen. Finally, there is a risk of hypercapnea in closed-c ircuit rebreathers. Hypercapnea is an increased concentration of carbon dioxide in the blood. Rebreathers have carbon dioxide scrubbers that prev ent carbon dioxide from accumulating in the rig. To mitigate the risk of h ypercapnea, the CO2 scrubbing material, Sofnolime 812 absorbent, was replaced bet ween each dive on the
6 UBA. In addition, the experiment required a low wo rk rate and, consequently, a low CO2 production rate. These precautions resulted in a very low risk of hypercapnea to the diver. Even though numerous precautions were taken to avoi d an accident, diving is inherently risky. Equipment failure is u sually unforeseen. However, the divers were at relatively shallow depths. Also, a medical monitor, standby diver, dive supervisor, and principal investigator were on hand at all times in case something was to go wrong. Additionally, divers we re trained on the DSE in a test pool at NEDU before open water testing. 1.3 Contributions to the Field Until now, no one had documented the oxygen consum ption rate of a diver using a diver propulsion device. Although th ese findings may not be directly applicable to the typical recreational div er, they are of great importance to the United States Military. Knowledge of the ox ygen consumption rate of a DPD-propelled diver could be useful for future devi ce reconfigurations and mission planning. The validity of the methods used by the Navy Experimental Diving Unit to measure a diverÂ’s oxygen consumption with time, although previously documented, was reaffirmed by this study The major benefit of this experiment is to the United States Navy and Marine Corps with the extension of combat mission capabilities through increased dive time of the underwater breathing apparatus.
7 1.4 Thesis Structure This thesis is intended to fully outline the divin g concepts, experimental design, and statistical analyses that were utilized in order to best estimate the oxygen consumption of a diver while using a diver p ropulsion device. The following chapter will begin with physiological con cepts which had to be learned in order to safely design this experiment and fully understand the raw data that was collected. Also to be discussed are the govern ing equations employed to find the oxygen consumption and the statistical con cepts which were later used to draw a conclusion. The remainder of this thesis will detail the exper imental design and the equipment that was used. The test results will be presented and their analysis explained. Next, there will be a discussion of so me of the trends which were identified and possible sources of error. The limi tations of the results will also be disclosed. Finally, the conclusion will be present ed along with the next steps of the study.
8 Chapter 2 Theoretical Foundations 2.1 Diving Physiology Scuba diving began in the 1940s and 50s. Since th en, we have made dramatic leaps in understanding the challenges of g etting the human body deeper underwater, keeping it there longer, and bri nging it back more safely. These challenges would be almost non-existent if it were not for the behavior of gases under pressure. Otherwise, breathing underwa ter would not be much different than breathing at the surface. It is of great necessity that anyone who takes interest in diving understands the fundamenta l gas laws that govern the physiological stresses experienced by divers. 2.1.1 Gas Laws One of the most well-known gas laws is also the mo st basic. BoyleÂ’s law is essential to understanding diving physiology. T his law states that at constant temperature, the absolute pressure and the volume o f gas are inversely proportional (Navy Diving Manual 2005). BoyleÂ’s la w can be observed as a diver descends. All of the air-filled regions in the bod y shrink. The opposite is true as the diver ascends. When a diver breathes compresse d air at depth, they must
9 exhale on the way up as the gas in their lungs cont inually expands in volume as the pressure is reduced. This is not the only way in which BoyleÂ’s law can be observed in diving physiology. Another phenomenon which is governed b y this is barotrauma. Any air-filled, rigid walled cavities are susceptible t o this. Two of the most commonly afflicted regions are the middle ears and the sinus es. Here, the same volume changes occur as the pressure is varied. Almost ev eryone has experienced this phenomenon of swimming to the bottom of a pool or f lying in an airplane. We must equalize (pop our ears) in the same manner as a diver must. When a diver does this, air is forced from their lungs int o their Eustachian tubes and sinus cavities to relieve the pressure and establis h equilibrium. This prevents barotrauma to the middle ears and sinuses. Another gas law that is of great importance to di ving is DaltonÂ’s law of partial pressures. The concept of partial pressure s must be understood to fully utilize the findings presented in this thesis. Dal tonÂ’s law states that the Â“total pressure exerted by a mixture of gases is equal to the sum of the pressures that would be exerted by each of the gases if it alone w ere present and occupied the total volumeÂ” (Strauss and Aksenov 2004). The pres sure exerted by each gas is termed the partial pressure. DaltonÂ’s law is parti cularly useful to diving because it allows one to understand the effect that depth h as on the amount of gas delivered to the body. As a diver descends, the to tal pressure increases and, consequently, so does the partial pressure of each gas. This concept comes
10 into play when determining the safe depth that a di ver can go while breathing different gas mixes. Divers utilize DaltonÂ’s law of partial pressures t o determine which gas mixture is most suitable for their dive. One of th e most important considerations is the partial pressure of oxygen. The United Stat es Navy Diving Gas manual suggests that the safe range of partial pressure of oxygen for semiclosed rebreathers is between 0.2 and 1.2 atmospheres abso lute (ATA) (Nuckols, Clarke et al. 1999). Divers must select a gas mixt ure that, at their target depth and dive duration, will keep the partial pressure o f oxygen well within this range. If the diver is breathing mixed gases, they must al so consider the partial pressure of the other gases. Inert gases such as h elium or nitrogen are usually mixed with oxygen to be used for deep water dives o r dives with a long duration. The purpose of these inert gases is to avoid oxyge n toxicity by keeping the partial pressure of oxygen within a physiologically safe range. This, however, throws another potential problem into the equation: inert gas narcosis. Nitrogen is a narcotic at higher partial pressures. The mos t common solution to this is to use helium as the inert gas dilutent to either repl ace or reduce the amount of nitrogen used in the mix (Elliott 1976). Already m entioned in Chapter 1 of this thesis was the importance of maintaining a physiolo gically safe partial pressure of oxygen. These are ideal examples of the importa nce of DaltonÂ’s law of partial pressures as it relates to diving.
11 Another gas law which is fundamental to diving phy siology is HenryÂ’s law. This law says that the amount of a gas which disso lves into a liquid at a given temperature is a function of its partial pressure. This highlights a physiological truth to diving. The deeper that a diver goes, the higher the partial pressures of the gases and, consequently, the higher the amount that is absorbed into the blood and tissues. This phenomenon is well-known a nd documented. HenryÂ’s law is applicable in directions, ascending and desc ending. Gases that diffuse into the blood and tissues at increased pressures m ust fall back out at decreased pressures. This is why decompression is necessary for deep divers. They must allow time for off-gassing or they could develop complications such as decompression sickness. 2.2 Rebreathers Conventional SCUBA dive gear that the majority of recreational divers use is termed Â“open-circuitÂ”. Some of the gas in the tank is used by the diver and the rest is exhaled directly into the water. For m ilitary applications, rebreathers are much more common for many reasons. Primarily, the military uses them because they eliminate most of the noise that opencircuit SCUBAÂ’s make (few to no bubbles released into the water) and they are much more gas efficient. For example, a closed-circuit rebreather is said to be 20 times more efficient in oxygen use as its open-circuit counterpart (Strauss and Aksenov 2004).
12 Rebreathers can have a closed-circuit or a semiclo sed-circuit. Closedcircuit rebreathers emit no gas. The simplest type s of closed-circuit rebreathers are oxygen rebreathers. These UBAs consist only of pure oxygen tanks. Gas is injected into the device to fill up the breathing b ag. The exhaled carbon dioxide from the diver is absorbed by a carbon dioxide scru bber. When the breathing bag collapses, more oxygen is added to refill the d evice. This type of closedcircuit rebreather, although relatively simple, con strains the diver to very shallow depths to avoid oxygen toxicity. The current rebre ather in use by the USMC, the MK 25, is one example of an oxygen closed-circuit r ebreather. To illustrate the limitations of this type of rebreather, the MK 25Â’s normal working limit is in 25 fsw for 240 minutes (Navy Diving Manual 2005). One way to get around the oxygen closed-circuit re breathersÂ’ limitations is by utilizing a constant PO2 closed-circuit rebreather. These rebreathers have two gas tanks: an oxygen tank and a dilutent gas ta nk to keep the PO2 within a physiologically safe range. Gases are injected int o a breathing bag in concentrations which vary with depth and the diverÂ’ s metabolic oxygen consumption rate. The carbon dioxide that is exhal ed by the diver is absorbed by a carbon dioxide scrubber while the rest of the gas is circulated and Â“rebreathedÂ”. The oxygen is injected at the rate a t which it is consumed, thus, achieving nearly 100 percent efficiency. While thi s may seem like the ideal dive rig, there are many downsides to constant PO2 closed-circuit rebreathers. They have a much higher cost due to the technological co mponents that measure
13 oxygen levels and control the release of fresh gas into the breathing loop. This complicates the device significantly, requiring man y hours of training. These components also make it much more difficult and exp ensive to maintain the device. Accordingly, there are also many more oppo rtunities for equipment to fail and compromise the safety of the diver. For these reasons, many believe that the semiclosed rebreather is a much better alt ernative. As the name implies, the semiclosed rebreather ha s features of both a closed and an open-circuit SCUBA rig. The most com monly used semiclosed UBAÂ’s inject fresh gas at a constant rate from a mi xed-gas tank, the contents of which must be determined before the dive based on t he dive profile. During operation, the semiclosed rebreather emits small am ounts of excess gas into the water while the breathing bag is constantly being r eplenished with fresh gas. Similarly to the closed-circuit rebreathers, the ca rbon dioxide is chemically absorbed using a carbon dioxide scrubber. Please r efer to Figure 1 (below) for a schematic of a standard semiclosed rebreather.
14 Figure 1. Schematic of a Semiclosed Rebreather. ( Nuckols, Clarke et al. 1999) The simple design of these devices keeps the maint enance cost low and reliability high. These qualities make it a more d esirable rebreather for many military applications. Unfortunately, its simplici ty does not come without risk. The partial pressure of oxygen in the breathing bag tends to have much more variance than that of a fully closed-circuit rebrea ther. Hypoxia and hyperoxia are serious concerns with semiclosed rebreathers. Suff icient planning and strict adherence to the planned dive profile can minimize this risk.
15 2.3 Governing Equations There is no need to point out the importance of es timating the oxygen consumption rate of divers. The development of an equation to do so has been ongoing since NEDUÂ’s E.T. Flynn derived the steady state solution of the mass balance equation for semiclosed-circuit rebreathers in 1974 (Flynn 1974). His equation, however, required the knowledge of far to o many variables to be easily utilized during operation. J.R. Clarke of NEDU derived a steady state solutio n for oxygen levels in semiclosed UBAÂ’s. This led to NEDUÂ’s development o f a method to measure the oxygen consumption rate of divers. This was po ssible due to the advent of oxygen sensors and dive computers. The equations t hat were used to estimate the diversÂ’ oxygen consumption rates are described on the next page.
16 ) 33 1(2 2fsw PO FIO + = Equation 1. Inspired Fraction of Oxygen Equation 2. Estimated Oxygen Consumption Rate Where 2O V is the estimated oxygen consumption rate, Vinj is the fresh gas injection rate, FO2 is the fraction of oxygen in the injected gas, FIO2 is the inspired oxygen fraction, PO2 is the partial pressure of oxygen, and fsw is ambient pressure in units of feet sea water. The above oxygen consumption formula is a simplifi ed version of the full time-dependent equation that Clarke originally solv ed for. Because these steady state formulas require some variables to be fixed ( even though they might vary slightly), mathematical corrections were applied as necessary. Also, additional measurements were taken and calibrated in order to ensure the accuracy of the data collected. The result was an equation that co uld be used to estimate the oxygen consumption of a diver during an operational dive by simply measuring the partial pressure of oxygen being inspired by th e diver and the diverÂ’s depth. ) 1( ) (2 2 2 2FIO FIO FO inj V O V =
17 Chapter 3 Materials and Methods 3.1 General In order to best simulate a typical mission scenari o, all tests were performed in full US Marine Corps Combatant Divers dress. Additionally, only trained USMC Combatant divers were used for this ev aluation. This ensured that the experiment would yield results which were optimized for application to the USMC mission protocol. Because human subjects were used for this study, the protocol was reviewed extensively and approved by the NEDU Institutional Review Board (IRB). 3.1.1 Divex Shadow Excursion The Divex Shadow Excursion was selected for this e xperiment for many reasons. By design, the DSE is capable of mounting the gas tanks on the front or back of the diver. By utilizing the DSE in its front-mounted configuration, this enabled the Combatant divers to wear a rucksack on their back. Additionally, the Divex Shadow Excursion can operate in both closed a nd semiclosed-circuit modes. The Navy Experimental Diving Unit has alrea dy established a safe method for monitoring the PO2 of a diver who is using a semiclosed rebreather (Clarke and Southerland 1999). Semiclosed rebreath ers also contribute to the
18 overall safety of the diver. When in semiclosed ni trox mode, a constant mass flow orifice supplies nitrox gas to the breathing l oop. During descent, the automatic demand valve adds gas in order to maintai n adequate lung volume. The diver can also use this demand valve to add fre sh gas in the event that the partial pressure of oxygen drops too low. 3.2 Experimental Design The Divex Shadow Excursion was set to semiclosed-c ircuit nitrox mode. Please refer to Figure 1 for a general schematic of a semiclosed rebreather. This mode was chosen for this study because it can most accurately characterize the oxygen consumption rate since the mass flow rate is constant (assuming the automatic demand valve is not activat ed). Oxygen consumption was estimated over 24 manned dives with the DSE. T he divers rotated between the pilot and passenger positions on the diver prop ulsion device. The data collected was used to determine the estimated oxyge n consumption rates throughout the experiment. 3.3 Rebreather Modifications The Divex Shadow Excursion was modified in order t o determine the diversÂ’ oxygen consumption rates at ten-second inte rvals during testing. The Draeger C8a oxygen monitor, using a Teledyne R22D o xygen sensor was used to measure the partial pressure of oxygen. This de vice was coupled with a Delta
19 P VR3 dive computer which was used as a data logger These modifications made it possible to record the diver depth and part ial pressure of oxygen, updated every ten seconds. Additionally, the recor dings were displayed continuously on the VR3 display making it possible for the divers to make corrections to their depth and ensure that their PO2 was within a physiologically safe range. Figure 2 (below) is a simulation of a dive computer analogous to the one that was used in this experiment. Figure 2. Dive Computer As previously mentioned, the DSE was operating in s emiclosed nitrox mode. The rig was equipped with a 300 bar, two liter oxyg en cylinder and an additional 300 bar, 2 liter nitrox cylinder (60% O2 / 40% NO2). The nitrox fresh gas flow rate was fixed at 6.0 L/min throughout the experime nt. The oxygen cylinder was
20 only used as needed. Typically, it only became nec essary toward the end of the run if the diver Â“wastedÂ” too much of the nitrox be fore commencement of the experiment. 3.4 Test Procedures Four U.S. Marine Reconnaissance Divers stayed in P anama City, FL for the duration of the testing. Over a two day period the divers were trained by the USMC and Divex personnel on the maintenance and use of the Divex Shadow Excursion, dry suits, and the Diver Propulsion Devi ce. The initial training took place in the Navy Experimental Diving Unit (NEDU) t est pool. Following the completion of these training sessions, three days o f open water training commenced. This initially took place at Shell Isla nd, but was moved to St. AndrewÂ’s Bay due to complications from rough waters Testing took place over the course of three days. Two test dives were accomplished each day, one in the A.M. and one in t he P.M. Data was obtained from both the driver and passenger for each dive. The divers were instructed to maintain a target de pth of 20 feet sea water (fsw). Their maximum depth was limited to 30 feet by the sea floor. The total dive time was approximately 60 minutes (30 minutes out, 30 minutes back). The Diver Propulsion Device with an attached safety buo y was boarded with one diver as a pilot and the other, a passenger. Each dive consisted of a run parallel to the beach. After the DPD travels for 30 minutes the divers will reverse
21 positions and travel for 30 minutes in the other di rection. A USMC SAFE boat was used to separate the divers from open water. T he SAFE boat also monitored the bottom depth to ensure that the diver s could not exceed the maximum depth of 30 feet sea water (fsw). The DSEÂ’s carbon dioxide scrubbers were repacked, bottles recharged, divers debriefed, and dive logger data downloaded f ollowing the completion of each run. The downloaded data included the depth o f the diver and the partial pressure of oxygen updated at ten-second intervals. 3.5 Data Analysis The raw data, including diver depth and PO2, was used to calculate the fraction of inspired oxygen (FIO2) and estimated oxygen consumption rate (2O V) for each diver at ten second intervals. This was a ccomplished by employing ClarkeÂ’s equations (Equations 1 and 2) in Chapter 2 of this report. Next, the data was organized so that it could be compiled for stat istical analysis. Two methods were used to analyze the data. First, curve fitting was performed to identify a curve that had the best fit to the plotted data for a maximum F value and the lowest number of fit parame ters (Systat 2000). Next, it was predicted that there would be some error in the results. It was necessary to perform a propagation of error analysis in order to most accurately characterize the error that resulted from the use o f the oxygen consumption
22 formula (Equation 2). These statistical analyses w ill be discussed further in the following chapter.
23 Chapter 4 Results 4.1 Curve Fits Curve fitting was used to identify trends in the d ata and to achieve 95% confidence and prediction intervals. The software package, TableCurve 2D v5.01 (Systat Software), was used to perform the va rious curve fits. The 95% confidence and prediction intervals are re presented in each of the graphs in this thesis. The outer, blue lines r epresent the 95% prediction interval. The prediction limits indicate how accur ately the curve is determined in relation to the next experimentÂ’s expected values. This means that if the experiment were repeated, 95% of the 2O V values would fall in between those two limits. The inner, purple lines represent the 95% confidence interval. The confidence interval is a measure of how accurately the average curve for repeated experiments is determined. More specifica lly, it means that there is a 95% probability that the range contains the true me an value of oxygen consumption. TableCurve color codes the data points based on th e number of standard errors represented by the residual. Data points th at are less than one standard error from the curve are blue. Green points are be tween 1 and 2 standard errors and yellow is between 2 and 3 standard errors. Red dots indicate a deviation of
24 more than three standard errors. Any red dots were considered for removal if determined to be outliers. Specific groups of data were compiled for analysis These groups included all of the drivers, all of the passengers, and a total compilation (both drivers and passengers). The purpose of this was t o identify possible trends in the data, establish a mean and 95% prediction inter vals, and determine to what extent the gas flow rate of the rebreathers can be reduced. 4.1.1 Initial Curve Fit Initially, the full data sets were plotted. Howev er, it was quickly discovered that the full data set was not representative of th e oxygen consumption rate of a diver using a DPD. The result of a curve fit perfo rmed on the total compilation of complete data is presented in Figure 3:
25 Figure 3. Curve Fit of Total Compilation of Data Fi les This initial trial revealed an interesting, but un desirable outcome. It took approximately 10 minutes for the oxygen consumption rate to Â“level offÂ”. Initially, this could be caused by a couple of things. The ne gative 2O V values could be due to the breathing bag of the rebreather filling up with gas. The higher 2O V values can be caused by the divers struggling to ge t into position on the DPD. This would cause an elevated oxygen consumption rat e. Additionally, this would explain why the values seem to approach a steady st ate after 10 minutes from the start of the experiment. Another observation is that the data points near t he end seem to go in the negative direction. Additionally, some data was in cluded beyond the end of the experiment (beyond 30 minutes). These inconsistenc ies do not likely represent
26 the true oxygen consumption rates of the divers whi le riding the diver propulsion device. It was determined that all of the data should be t runcated to eliminate these false readings. Only data points collected b etween 10 minutes and 25 minutes were used to characterize the diversÂ’ 2O V. 4.1.2 Driver Compilation To identify any possible trends in the data, the f iles were grouped into driver and passenger compilations. It was decided that only simple equations should be selected for the curve fits. This is bec ause there is no reason to suspect that the estimated oxygen consumption rate should have a complex relationship with time. As indicated in the previo us section, if the data collected had reached a steady state, a linear regression mod el would be appropriate. However, initial trials indicated that this was not the case. Potential explanations for this will be presented in the discussion of thi s thesis. Nonetheless, in order to ensure that the true trend of the data was modeled, nonlinear equations were considered and chosen if they were statistically be tter fits. The resulting curve of the truncated driver files (n=1092) is depicted in Figure 4.
27 Figure 4. Estimated Driver Oxygen Consumption Rate vs. Time Figure 4 indicates that the mean 2O V of the drivers is close to 0.4 L/min. This is approximately the outcome that was expected In a previous study by the Navy Experimental Diving Unit, the mean resting oxy gen consumption rate was measured as 0.37 L/min (Knafelc 2007). Therefore, these results appear to agree with the hypothesis that the mean oxygen cons umption rate of the divers while using a DPD is near the resting oxygen consum ption rate. Also important in this figure is that the 95% prediction interval indicates that 95% of the data points of a repeat experiment are likely to be with in 1.0 liter per minute or less. That being said, it is important to realize that th is graph only includes data from the drivers of the DPD. The numeric summary of the chosen model from Table Curve can be found in Appendix A of this thesis. This output in cludes all of the statistics of the
28 curve fit. One of the most important quantities is the P-value. For the chosen driver curve fit in Figure 4, this value is 0.0022. Generally, a 95% confidence level is used as the criterion in determining the s ignificance of the model. Because 0.0022 is less than =0.05, it can be concluded that the findings presented in Figure 4 are statistically significant 4.1.3 Passenger Compilation The next compilation of data included only the pas senger files. The resulting graph for 1092 data pairs is depicted in Figure 5: Figure 5. Estimated Passenger Oxygen Consumption Ra te vs. Time This graph also has 95% prediction and confidence l imits. Notice that there are four points which fell more than 3 standard errors from the curve. Although it
29 could be argued that these are outliers, they were kept in the analysis because they did not have a significant effect on the curve fit calculations. Figure 5 indicates that the average passenger oxyg en consumption rate appears to be just over 0.5 L/min. This is a signi ficant increase over that of the drivers. Possible explanations for this will be di scussed in the next chapter. The numerical summary for Figure 5 can be found in Appendix B of this thesis. The P-value was for this model was found t o be less than or equal to 0.0001. This is indicative of a probability that i s much lower than required for statistical significance at the 95% confidence leve l ( =0.05). 4.1.4 Total Compilation Finally, the truncated compilation of driver and p assenger data sets were imported into TableCurve 2D. The results for 2184 data points are presented in Figure 6.
30 Figure 6. Estimated Total Oxygen Consumption Rate v s. Time Figure 6 depicts a model that is as expected after seeing the separate compilations. The average oxygen consumption rate appears to be below 0.5 L/min with a 95% prediction interval just above 1.0 liter per minute. The numerical summary in Appendix C shows that the P-value of the chosen model is less than or equal to 0.0001. This highlights the statistical significance of the findings in Figure 6. This fig ure is paramount in determining how far the fresh gas flow rate can be reduced for the USMC Combatant Divers during missions where the DPD will be utilized. Fu rther discussion of this will take place in the conclusion of this thesis.
31 4.2 Propagation of Error Propagation of error was employed to estimate the uncertainty of the calculated estimated oxygen consumption rate. It w as important to use propagation of error and not just find the uncertai nty in our 2O V calculation because the 2O V was not measured. Thus, the uncertainty of each measurement propagated throughout ClarkeÂ’s oxygen c onsumption rate formula. The first step to finding the propagation of error was to compile all of the data and calculate the mean and uncertainty of each measurement. Next, the partial differential of each variable in the 2O V formula that had error had to be solved for symbolically. In this case, all three o f the variables had error. Figure 8 is a screenshot from MathCAD Professional (Mathso ft, 2001) that shows the symbolic evaluation of the partial differentials.
32 Figure 7. Symbolic Evaluation of Partial Derivative s
33 Next, the mean values for each variable had to be plugged into these symbolic evaluations. This is shown in Figure 8: Figure 8. Numerical Evaluation of Partial Derivativ es Each partial differential represents a sensitivity factor in the propagation of error formula. From these values, it can be conclu ded that the injection velocity is the least sensitive variable. This means that t he error in the measured injection velocities will contribute the least to t he overall uncertainty of the 2O V calculation. This is because the absolute value of the solved partial differential yields the lowest number.
34 The general formula for propagation of error is li sted below in Figure 9. This formula assumes that there is no covariance be tween the variables (Taylor 1982). Figure 9. Propagation of Error The first step in Figure 9 indicates how the uncer tainty in the mean of N measurements was found (Taylor 1982). Even though the injection velocity has a relatively high uncertainty, the sensitivity fact or (partial derivative) is so low that it keeps that variableÂ’s contribution to the total uncertainty minimal. The propagation of error is 0.034 for the estimated oxy gen consumption rate. This is a very reasonable level of uncertainty and serves t o validate ClarkeÂ’s method.
35 Chapter 5 Discussion 5.1 Curve Trends 5.1.1 Driver vs. Passenger After studying the curves, it was clear that there were some interesting trends. First, the driver and passenger graphs wer e compared. In order to ensure that the results from each data set could be compared accurately, an F test was performed to compare the variations for ea ch compilation (driver and passenger). The results of this test can be viewed in Appendix D of this thesis. The F ratio of 1.03 indicates that the variances se en in each compilation are not statistically different. This is very important be cause it allows for accurate conclusions to be drawn from the comparison of the two data sets. Please refer to Figure 10 in order view the two gr aphs together. Also, notice that in this figure, the graphs are scaled i dentically to facilitate simple visual comparison. One might assume that the drive r will likely have a higher oxygen consumption rate because they have to manual ly control the Diver Propulsion Device (DPD). However, the results indi cate that this is not the case. It is clear that the passenger has an average oxyg en consumption rate that is more than 0.1 L/min higher than that of the driver. Upon further examination of the raw data, it was found that the mean estimated oxygen consumption rate of
36 the driver data was 0.3908 L/min. The mean 2O V of the passenger data was 0.5410 L/min. Therefore, the mean passenger 2O V was found to be 38.4% higher than the mean driver 2O V. One possible explanation for this is that the passenger experiences more drag resistance from the water. If this is the case, the passenger might have to work harder to hold on to the DPD which could increase their 2O V.
37 Figure 10. Driver vs. Passenger Comparison
38 5.1.2 Negative Slope Another trend that could be identified in all of t he curve fits is that they have slightly negative slopes. In Figures 4 and 6, this downward trend seemed to appear more toward the end of the experiment. T his could be caused by a number of things. It might be an indication that a n insufficient number of data points were truncated in the tail end of the data s et, leaving some of the false reads from the divers preparing to surface. Althou gh this is a possibility, it is not a likely one. It was expected that there would be a need for truncating the first 10 minutes of the data so that the rebreather could equilibrate. However, it is unlikely that the divers did anything that would af fect the estimated oxygen consumption rate further than 5 minutes prior to th e end of the experiment. They were instructed to maintain their positions for the entire duration (30 minutes) of each leg. The decision was made to truncate the la st 5 minutes in order to be certain that we were using data which accurately po rtrayed their 2O V while using the DPD. That being said, there is not enough just ification to truncate the data set any further. Furthermore, the passenger data s et appeared to show a consistent decline in the estimated oxygen consumpt ion rate throughout the entire duration of the experiment. This seems to i ndicate that it might take a longer period of time for the diversÂ’ 2O V to reach a steady state.
39 5.2 Propagation of Error The propagation of error analysis yielded positive results. The mean 2O V was calculated to be 0.482 L/min (Figure 8). The u ncertainty of this estimated oxygen consumption rate is only 0.034 L/min. This validates the use of ClarkeÂ’s equations (Equations 1 and 2) for this application. However, it was concluded that for the 2O V formula, the FO2 and FIO2 are by far the most sensitive variables. A large uncertainty in either of these two measurements would cause the propagation of error to increase drastically. This can be clearly observed in Figure 8. The partial derivatives of these variabl es are quite high. For this reason, great care should be taken in the measureme nt of these two variables. 5.3 Limitations As mentioned in the introduction, this study was d esigned specifically for the application to the Marine Corps Combatant Diver s. These divers maintain a high level of physical fitness and their metabolic rates have been documented. Thus, the mean oxygen consumption rate of this subj ect group might vary considerably from that of the average diver. One aspect that could not be accounted for was the affect that stress might have on the diversÂ’ 2O V. It is probable that during a real-life mission, the divers might experience significantly more stress t han they did in this experiment. One study reported that hormones which are release d when a person is under
40 stress can cause an increase in the oxygen consumpt ion rate (Weissman, Askanazi et al. 1986). Additionally, the negative slope indicates that th e experimental trials were not long enough for the estimated oxygen consumptio n to reach a steady state. Fortunately, the most likely result of this is that the 2O V is overestimated. It is unlikely that the steady state value will be signif icantly lower. Furthermore, a slight overestimation will contribute to the overal l safety of the new fresh gas flow rate that will be discussed in the conclusion of th is thesis.
41 Chapter 6 Conclusion 6.1 Recommendation This study revealed that the mean oxygen consumpti on rate of Marine Combatant Divers using a Diver Propulsion Device is similar to that of the documented resting oxygen consumption rate. Howeve r, we have added new information concerning the variance of that data du ring DPD operations. These preliminary results indicate that the amount of oxy gen made available to the diver should not fall below 1.0 L/min. Even though the mean 2O V was just 0.482 L/min, there were a significant number of data poin ts that were closer to the 1.0 L/min mark. With all of this taken into considerat ion, the preliminary recommendation is to reduce the fresh gas flow rate in the DSE (semiclosed nitrox mode with 60% oxygen mix) to 2.0 L/min of a 60% O2 (40%N2) mixture for a net O2 injection rate of 1.2 L/min This will triple th e dive time offered by the Divex Shadow Excursion while the divers are utilizi ng the diver propulsion device. In order to understand the significance of this ad justment, one could assume a set of default DSE settings: (2) two liter nitrox tanks (60% oxygen / 40% nitrogen) filled to 300 Bar. Assuming these co nditions, the standard gas flow rate of 6.0 L/min would provide a maximum theo retical dive time of 3 hours.
42 If the flow rate were reduced to 2.0 L/min, the ma ximum dive time would theoretically increase to 9 hours. It is important to note that these dive times are only theoretical. Realistically, the adjusted fres h gas flow rate might sustain a diver for 7-8 hours on a diver propulsion device. Nonetheless, this would significantly increase the mission capabilities of the United States Marine Corps. 6.2 Next Steps These preliminary results will now need to be test ed and verified with reduced fresh gas flow rates. Additionally, the du ration of the testing should be increased in order to most accurately characterize the long term oxygen demands of the divers using a DPD. This would offe r more time for the diversÂ’ bodies to reach a steady state metabolic rate. As stated in the limitations, it is possible that the true resting 2O V is lower than this thesis indicated. If this were determined to be the case, it is possible that the fresh gas flow rate could be reduced further, enabling the United States Marine Corps to achieve even longer dives. Only once substantial testing is completed a nd the flow rate of 2.0 L/min of 60% O2 nitrox is proven to be safe should this be attempt ed in a real mission scenario.
43 References Clarke, J. (2007). "Review of MARCORSYSCOM Operatio nal Requirements for Enhanced Underwater Breathing Apparatus (EUBA) Usin g Simulation Software." Navy Experimental Diving Unit Technical Report 07-11. Clarke, J. R. and D. Southerland (1999). "An Oxygen Monitor for Semi-closed Rebreathers -Design and Use for Estimating Metabo lic Oxygen Consumption." Proceedings of SPIE, the Internationa l Society for Optical Engineering 3711: 123-129. Elliott, D. H. (1976). "Some occupational hazards o f diving." Proc R Soc Med 69(8): 589-93. Flynn, E. T. (1974). "Operational Monitoring of Oxy gen Consumption in SemiClosed-circuit Underwater Breathing Apparatus." NE DU TR 22-74. Knafelc, M. E. (2007). Oxygen Consumption Rate for Different Diver Dress, Navy Experimental Diving Unit. TM 07-04. Lawrence, C. H. (1996). "A diving fatality due to o xygen toxicity during a "technical" dive." Med J Aust 165(5): 262-3. McCarter, M. (2005). "Divers Go Deep with Propulsio n Devices." from http://www.special-operations-technology.com/articl e.cfm?DocID=873. Navy Diving Manual (2005). U.S. Navy Diving Manual. [Washington, D.C.], Naval Sea Systems Command Nuckols, M. L., J. Clarke, et al. (1998). "Maintain ing safe oxygen levels in semiclosed underwater breathing apparatus." Life Su pport Biosph Sci 5(1): 87-95. Nuckols, M. L., J. R. Clarke, et al. (1999). "Asses sment of oxygen levels in alternative designs of semiclosed underwater breath ing apparatus." Life Support Biosph Sci 6(3): 239-49. Strauss, M. B. and I. V. Aksenov (2004). Diving sci ence. Champaign, IL, Human Kinetics.
44 Systat (2000). TableCurve 2D Tutorials, TableCurve 2D. Taylor, J. R. (1982). An Introduction to Error Anal ysis, Oxford Universtiy Press. Weissman, C., J. Askanazi, et al. (1986). "The meta bolic and ventilatory response to the infusion of stress hormones." Ann S urg 203(4): 408-12.
46 Appendix A: Driver Curve Fit Â– Numeric Summary Rank 1 Eqn 7 y=a+bx 3 r 2 Coef Det DF Adj r 2 Fit Std Err F-value 0.0085568870 0.0067360549 0.2945470853 9.4075058440 Parm Value Std Error t-value 95% Confidence Limits P>|t| a 0.431589399 0.016001414 26.97195360 0.400192340 0.462986458 0.00000 b -6.4031e-06 2.08762e-06 -3.06716577 -1.0499e-05 -2.3069e-06 0.00221 Area Xmin-Xmax Area Precision 5.8645483747 0.0000000000 Function min X-Value Function max X-Value 0.3315413517 25.000000000 0.4251862873 10.000018938 1st Deriv min X-Value 1st Deriv max X-Value -0.012005766 25.000000000 -0.001920930 10.000018938 2nd Deriv min X-Value 2nd Deriv max X-Value -0.000960461 25.000000000 -0.000384185 10.000018938 Soln Vector Covar Matrix Direct LUDecomp r 2 Coef Det DF Adj r 2 Fit Std Err Max Abs Err 0.0085568870 0.0067360549 0.2945470853 0.7270770390 r 2 Attainable 0.0273181822 Source Sum of Squares DF Mean Square F Statistic P>F Regr 0.81617626 1 0.81617626 9.40751 0.00221 Error 94.566204 1090 0.086757985 Total 95.38238 1091 Lack Fit 1.789497 89 0.020106708 0.216938 1.00000 Pure Err 92.776707 1001 0.092684023 Date Time File Source Oct 26, 2008 1:39:41 PM c:\users\adam smith\documents\final th
47 Appendix B: Passenger Curve Fit Â– Numeric Summary Rank 1 Eqn 13 y=a+blnx r 2 Coef Det DF Adj r 2 Fit Std Err F-value 0.0363884625 0.0346187443 0.2865126668 41.161217521 Parm Value Std Error t-value 95% Confidence Limits P>|t| a 1.141087729 0.093937594 12.14729567 0.956768759 1.325406699 0.00000 b -0.21212774 0.033063846 -6.41570086 -0.27700373 -0.14725176 0.00000 Area Xmin-Xmax Area Precision 8.1123322574 1.552475e-12 Function min X-Value Function max X-Value 0.4582748639 25.000000000 0.6526451475 10.000018938 1st Deriv min X-Value 1st Deriv max X-Value -0.021212734 10.000018938 -0.008485110 25.000000000 2nd Deriv min X-Value 2nd Deriv max X-Value 0.0003394044 25.000000000 0.0021212694 10.000018938 Soln Vector Covar Matrix Direct LUDecomp r 2 Coef Det DF Adj r 2 Fit Std Err Max Abs Err 0.0363884625 0.0346187443 0.2865126668 1.0998009883 r 2 Attainable 0.0584570732 Source Sum of Squares DF Mean Square F Statistic P>F Regr 3.3789041 1 3.3789041 41.1612 0.00000 Error 89.477564 1090 0.082089508 Total 92.856468 1091 Lack Fit 2.0492132 89 0.023024868 0.26362 1.00000 Pure Err 87.428351 1001 0.08734101 Date Time File Source Oct 26, 2008 1:37:15 PM c:\users\adam smith\documents\final th
48 Appendix C: Total Curve Fit Â– Numeric Summary Rank 1 Eqn 5 y=a+bx 2 lnx r 2 Coef Det DF Adj r 2 Fit Std Err F-value 0.0175016006 0.0166006392 0.3005031636 38.868757991 Parm Value Std Error t-value 95% Confidence Limits P>|t| a 0.539429808 0.013432012 40.16001461 0.513088937 0.565770680 0.00000 b -7.6588e-05 1.22846e-05 -6.23448137 -0.00010068 -5.2497e-05 0.00000 Area Xmin-Xmax Area Precision 6.9906876383 2.420723e-09 Function min X-Value Function max X-Value 0.3853499484 25.000000000 0.5217946517 10.000018938 1st Deriv min X-Value 1st Deriv max X-Value -0.014241093 25.000000000 -0.004292908 10.000018938 2nd Deriv min X-Value 2nd Deriv max X-Value -0.000722820 25.000000000 -0.000582466 10.000018938 Soln Vector Covar Matrix Direct LUDecomp r 2 Coef Det DF Adj r 2 Fit Std Err Max Abs Err 0.0175016006 0.0166006392 0.3005031636 1.0415069878 r 2 Attainable 0.0270350712 Source Sum of Squares DF Mean Square F Statistic P>F Regr 3.5099325 1 3.5099325 38.8688 0.00000 Error 197.03929 2182 0.090302151 Total 200.54923 2183 Lack Fit 1.9119302 89 0.021482361 0.230427 1.00000 Pure Err 195.12736 2093 0.093228554 Date Time File Source Oct 26, 2008 1:41:14 PM c:\users\adam smith\documents\final th
49 Appendix D: Comparison of Variances This is a screen shot from a freely-available execu table posted in the public domain. Only the F-ratio was used for this thesis.