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Lecture.Packet.9.risk

Course: CEE 440, Spring 2011
School: University of Illinois,...
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5. CHAPTER QUANTITATIVE RISK ASSESSMENT (from LaGrega et al., p837-885) My teaching goals for this chapter are for you: 1) to learn what information is required to perform a risk assessment 2) to learn how to calculate the risk associated with a given chemical concentration 3) to appreciate the uncertainty associated with risk assessment RISK: probability of suffering harm or loss If the level of risk can be...

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5. CHAPTER QUANTITATIVE RISK ASSESSMENT (from LaGrega et al., p837-885) My teaching goals for this chapter are for you: 1) to learn what information is required to perform a risk assessment 2) to learn how to calculate the risk associated with a given chemical concentration 3) to appreciate the uncertainty associated with risk assessment RISK: probability of suffering harm or loss If the level of risk can be quantified we can calculate it as follows: risk = (probability) * (severity of consequence) CEE 440 (5.1) 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 1 Example: If Bob smashes his finger while loading boxes he loses 5 days of work Severity of Consequence = 5 days The likelihood of Bob smashing his finger is 10% Probability = 0.1 Hence, Risk = 0.1 * 5 days = .5 person days lost Risk must be divided into background and incremental risk: Background risk: occurs in the absence of a particular source of risk Incremental risk: caused by the source of risk Total risk: Background + Incremental CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 2 Example: U.S. Background Cancer risk = 0.25 Excess lifetime cancer risk deemed acceptable by the US EPA at Superfund sites = 1x10-6 Total risk = 0.250001 Death/Cancer Risks Motor Vehicle Accident Home Accidents Annual risk 0.00024 0.00011 Uncertainty 10% 10% Cigarette Smoking (pack/day) Peanut butter (4 tsp./day) 0.0036 8x10-6 Lifetime risk 1x10-6 Factor of 3 Factor of 3 Uncertainty Factor of 10 to 100 Drinking Water with EPA limit of TCE CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 3 Risk Assessments can affect: 1) waste treatment/disposal options 2) remediation of contaminated site How clean is clean? What remediation technology do we use? How do we protect the people doing the cleaning? 3) minimization of waste generation 4) placement of new treatment facilities 5) developing new products CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 4 Four Steps to Risk Assessment 1) Hazard Identification: which chemicals important 2) Exposure Assessment: where do chemicals go, who might be exposed and how 3) Toxicology Assessment: determine numerical indices of toxicology for computing risk 4) Risk Characterization: estimate the magnitude of risk, and the uncertainty of the estimate CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 5 5.1 HAZARD IDENTIFICATION The potential hazard a chemical represents can be evaluated based on concentration and "danger" One approach is outlined below: 1) Sort the contaminants by medium (i.e. soil, groundwater, etc.) 2) Tabulate mean and range of concentration values of each chemical at site 3) Identify reference doses for non-carcinogens and slope factors for carcinogens for each potential exposure route 4) Determine the toxicity scores for each chemical in each media: For non-carcinogens: TS = Cmax/RfD (5.2) where TS = toxicity score Cmax = maximum concentration RfD = chronic reference dose CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 6 For carcinogens: TS = SF * Cmax where SF = slope factor (5.3) 5) For each exposure route (i.e. air, water, soil) rank the compounds by toxicity score 6) For each exposure route, select those chemicals comprising 99% of the total score Other factors which must be taken into consideration to evaluate toxicity are: - mean concentration - frequency of detection - mobility - persistence in the environment - treatability CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 7 It is not rare to detect as many as 100 different chemicals at a contaminated site Hence, a subset of chemicals (surrogates) is often chosen to be representative of all the chemicals detected. This limits the number of chemicals analyzed and focuses efforts on most significant hazards Surrogates are chosen based on which chemicals best represent the following risks: 1) Most toxic, persistent, and mobile contaminants 2) Most prevalent (w.r.t. spatial distribution and concentration) 3) Most exposure CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 8 5.2 EXPOSURE ASSESSMENT Assessing exposure requires identifying and/or quantifying a) contaminant transport and fate source characterization advection, dispersion, sorption, transformation characterization exposure points (i.e. where does contamination reach?) b) general and sensitive populations who will be contaminant receptors who is coming into contact with contaminants and how? c) doses to population from short and long term exposure routes based on exposure, how much is actually taken into receptor? CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 9 Up to now, CEE440 has been focused on contaminant transport and fate. Example of two simple but common transport and fate scenarios are shown below. One involves groundwater, the other atmospheric transport (Figure 14-1 from LaGrega et al., p 845). CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 10 Below are schematics of potential soil and groundwater transport and fate pathways: (Fig 14-2a,2b,2c from LaGrega, p846-848) CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 11 Once we figure out where the contaminants are going we must determine: Potentially Exposed Populations present population in vicinity of site future population in vicinity of site sub-populations of special concern (i.e. children, fishermen) Identifying these populations requires identifying different exposure scenarios: worker scenario trespasser scenario residential use scenario recreational use scenario construction scenario The last step is to identify: Receptor Doses CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 12 Three exposure routes are commonly considered: ingestion inhalation dermal contact A general formula that represents the relevant parameters to consider for dose is: I = (C * CR * EF * ED) / (BW * AT) (5.4) I = intake (mg/kg of body weight) C = concentration at exposure point (mg/L in water, mg/m3 in air, mg/kg in soil) CR = contact rate (e.g., L/day, m3/day) EF = exposure frequency (day/year) ED = exposure duration (yr) BW = body weight (kg) AT = average time (days) CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 13 Modified forms of this equation have been developed to account for specific exposure pathways. For example, the intake dose from the inhalation of fugitive dust may be calculated as follows: I = (C * CR * EF * ED * RR * ABS) / (BW * AT) (5.5) RR = retention rate (decimal fraction) ABS = absorption into bloodstream (decimal fraction) C = Cs * Pc Cs = concentration of chemical in fugitive dust (mg/mg) Pc = concentration of fugitive dust in air (mg/m3) CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 14 We can also represent the intake via dermal exposure as follows: I = (C * A * (DA/Exposure Event) * ABS * SM * EF * ED) / (BW * AT) (5.6) C = mg chemical/kg of solid A = surface area of skin exposed (say 20% of total skin area in cm2) DA = dust adherence (0.51 mg/cm2 per exposure event) ABS = skin absorption rate (~6%) SM = effect of soil matrix (~15% of contaminant on soil available for dermal contact) EF = frequency of contact (day/year) ED = exposure duration (yr) CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 15 Typical Exposure Parameter Table (if living by landfill) Parameter Average body wt(kg) Skin surface area (cm2) Water ingested (L/day) Air breathed (m3/hr) Retention rate (inhaled air) Absorption rate (inhaled air) Soil ingested (mg/day) Bathing duration (min) Exposure frequency (days) Exposure duration (yrs) CEE 440 Adults 70 18,150 2 0.83 100% 100% 100 30 365 30 Child age 6-12 29 10,470 2 0.46 100% 100% 100 30 365 6 Child age 2-6 16 6,980 1 0.25 100% 100% 200 30 365 4 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 16 Example: Calculate the average daily intake of chloroform for on-site workers over one year from dermal contact of a soil with a mean concentration of 1.12 mg/kg of soil, where 2 exposure events per day occur over a period of 156 work days/year. Solution: I = (C * A * (DA/ExposureEvent) * ABS * SM * EF * ED) / (BW * AT) A = 18,150 cm2 * (0.2) = 3630 cm2 DA = 0.51 mg/cm2 per exposure event ABS = 0.06 SM = 0.15 EF = 2 exp. events/day * 156 days/year ED = 1 year BW = 70 kg AT = 1 year = 365 days I= mg Exp. Ev. day - kg 0.51 mg 1.12 kg * 3630 cm 2 * Exp. Ev. 2 * 0.06 * 0.15 * 2 day * 156 yr * 1yr * 10 6 mg cm [] I = 2.27x10-7 mg/kg-day CEE 440 70[kg]* 365days 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 17 5.3 TOXICOLOGY ASSESSMENT When assessing contaminant transport and fate we evaluated contaminant intake via three different exposure pathways (inhalation, ingestion, dermal contact). We now want to understand the relationship between dose (or intake) and response (or adverse health effect). This involves determining the dose-response relationships for each chemical of interest (surrogate chemicals) and determining the confidence in these relationships. Note: as an engineer you do not have an adequate background to conduct toxicological evaluations. However, in order to understand how toxicological decisions are made you should understand the general methods that toxicologists use. CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 18 Toxicology is, to a large extent, supported by information obtained from animals. Extrapolating from animals at high doses to humans at low doses (i.e. doses usually encountered) carries with it a high degree of uncertainty. Hence, estimates of toxicological risk can be orders of magnitude off. Nevertheless, site cleanup is being more and more driven by a risk based approach and we must understand the basics of toxicology to interpret risk based decisions with any intelligence. Chemicals are classified as non-carcinogens and carcinogens. In hazardous waste management we evaluate the toxic nature of non-carcinogens via the reference dose (RfD) and of carcinogens via the slope (SF). RfD factor = the acceptable daily intake as established by the EPA [mg/kg-day] SF = the slope of the dose-response curve at very low exposures, often referred to as the carcinogen potency factor (CPF) [kg-day/mg) CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 19 Calculating RfD 1. Select the most sensitive species for which adequate studies are available (unless other species more appropriate reference for humans). If human data are available these are given priority 2. Select the principal or critical studies using the appropriate route of exposure. RfDs are route specific (i.e. inhalation, ingestion, dermal). 3. Select supporting studies. These studies may relate effects in animal species of interest to humans, etc. 4. Identify the "No Observed Adverse Effect Level" (NOAEL) or, if NOAELs are not available, the "Lowest Observed Adverse Effect Level" (LOAEL). CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 20 5. Adjust the NOAEL downward by orders of magnitude to reflect uncertainty: Reduce the NOAEL found in humans by an uncertainty factor of 10 to account for variations in the general public Reduce the NOAEL by an additional uncertainty factor of 10 when extrapolating from animals to humans Reduce the NOAEL by an additional uncertainty factor of 10 if the data are derived from a sub-chronic instead of a chronic study If no NOAEL were available (i.e. adverse health effects were observed at all health levels), reduce the LOAEL by a an uncertainty factor of 10. The EPA also applies a modifying factor, which ranges from 1 to 10, to reflect a qualitative professional judgment of uncertainties which are not accounted for by the preceding uncertainty factors. CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 21 Example: In a subchronic oral toxicity study in mice, a LOAEL of 5 mg/kg-day was determined for a specific agent. The quality of the data is given a high rating by the expert evaluating the data. What is the RfD? Area of uncertainty Uncertainty factor Variation within population 10 Extrapolation from animals to humans 10 Extrapolation from subchronic to chronic 10 Extrapolation from LOAEL to NOAEL 10 Modifying factor 1 RfD = 5 mg/kd-day / (10*10*10*10*1) = 0.5 g/kg-day CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 22 Calculating SF 1) The carcinogenic response of an animal to a specific exposure route can be obtained at different doses and plotted as shown below (Figure 5-23, LaGrega, 1994) CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 23 2) In order to obtain any carcinogenic response high doses are typically used. Hence, these must be extrapolated down to low doses to obtain our SF. Note: Extrapolation of data from the 10-90% carcinogenesis range of test animals to 0.0001% carcinogenesis may yield results which are off by several orders of magnitude. The range of error in experimental dose-response extrapolation for low carcinogenic frequencies in potential exposed human populations is indicated below (Figure 5-24 LaGrega, 1994): CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 24 EPA classification system for carcinogenicity A Human carcinogen B1 Probable human carcinogen, limited human data available B2 Probable human carcinogen, no human data available (animal data) C Possible human carcinogen D Not classifiable as to human carcinogenicity E Evidence of noncarcinogenicity for humans Note: approximately 50 human carcinogens are known, 13 of these are found in the environment, and the rest are primarily pharmaceutical products CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 25 Human environmental carcinogens Carcinogen Arsenic Asbestos Benzene Benzidine Bis-chloromethyl-ether Chromium, hexavalent Diethylstilbestrol 2-Naphthylamine Nickel Plutonium-239 Radium-226 Radon-222 Vinyl chloride CEE 440 Cancer Lung, skin Lung, mesothelioma Leukemia Bladder Lung Lung Female Bladder Lung Lung Lung, osteosarcoma Lung Liver, angiosarcoma 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 26 5.4 RISK CHARACTERIZATION The final step in risk assessment is to estimate the risks. This consists in part on calculating estimates of both carcinogenic and noncarcinogenic risks to susceptible receptors for all exposure scenarios considered important at both maximum and average exposure concentrations. It is important to note that the calculated values are only as good as the data used to calculate them. Each piece of data has a confidence level associated with it. And this all factors into the confidence in the final esimate. CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 27 Carcinogenic Risks: Risk = I * SF (5.7) I = chronic daily intake calculated in the exposure assessment (mg/kg-day) SF = carcinogen slope factor (kg-day/mg) The EPA has designated a risk < 1x10-6 acceptable. CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 28 Non-Carcinogenic Risks Normally characterized in terms of a hazard index (HI). HI = I / RfD (5.8) HI = hazard index (dimensionless) I = chronic daily intake (mg/kg-day) RfD = reference dose (mg/kg-day) Note: reference doses are dependent on the route of exposure and may only be used with exposure data for the same route A HI < 1 is acceptable. If greater than one chemical is present, then the sum of the HI for all chemicals should be <1 for the risk to be acceptable. If different chemicals affect different organs, then the HI for all chemicals that affect a particular organ can be summed individually to determine risk. CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 29 Example: Chloroform has reached a drinking water well for a small town and is being consumed by the town occupants. The average concentration of chloroform in the well is 4.30x10-4 mg/L. For an adult, calculate the carcinogenic and noncarcinogenic risk of drinking the groundwater contaminated with chloroform assuming water is drunken daily over a year, the parameters given in the Typical Exposure Parameter Table and I = (C * CR * EF * ED) / (BW * AT). SF = 6.1x10-3 kg-day/mg; RfD = 1.0x10-2 mg/kg-day C = 4.30x10-4 mg/L CR = 2 L/day EF = 365 day/yr ED = 1 yr BW = 70 kg AT = 365 days I = 4.30x10-4 [mg/L] * 2 [L/day] * 365 [day/yr] * 1 [yr] / (70 [kg] * 365 [days]) I = 1.22x10-5 mg/kg-day CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 30 Carcinogenic Risk = I * SF = 1.22x10-5 * 6.1x10-3 = 7.44x10-8 This is below the EPA risk threshold of 1x10-6 so the townsfolk are "safe" Non-carcinogenic Risk = HI = I/RfD = 1.22x10-5 / 1.0x10-2 = 0.00122 This is below 1 so the townsfolk are "safe" Up to now we have only been examining human health effects. There is an increasing emphasis to look at ecological health effects. CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 31 5.5 ECOLOGICAL RISK ASSESSMENT Estimate of the adverse effect to an ecosystem from a hazardous waste site Such assessments at most Hazardous Waste sites are minimal at best. The basic steps in an ecological risk assessment are the same as those previously covered when examining the risks to humans. However, ecological risks assessments are more complicated because: - multiple biological endpoints: multiple species - more complex exposure pathways: determined by the biological endpoints - indirect effects: habitat impairment, disruption of intertrophic relationships - evaluating impacts on ecosystems: very complex environment On the other hand, there are advantages to an ecological risk assessment: exposures and hazards can often be estimated directly on the species of concern CEE 440 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 32 Tiered Approach to Corrective action Objectives (TACO) Initial Site Assessment -Conduct site investigation and identify principal chemical(s) of concern, extent of affected environmental media, and potential migration pathways and receptors. Compare measured values to look-up table values Involves changing input parameters on standard risk equations based on site specific data Site Classification and Initial Response Action -Classify site per specified scenarios and implement appropriate initial response action. -Reclassify site as appropriate following initial response action, interim remedial action, or additional data collection. Interim Remedial Action -Conduct partial source removal or other action to reduce risk(s) and site classification Tier 1 Evaluation -Identify reasonable potential sources, transport pathways, and exposure pathways (use flowchart in Figure ??). -Select appropriate Tier 1 risk-based screening levels (RBSLs) from Tier 1 Look-Up Table, or other relevant criteria (taste, odor thresholds, etc.) Compare these values with site conditions no Chemical(s) of concern concentrations exceed RBSLs? yes no Remediation to Tier 1 RBSLs practicable? Interim remedial action appropriate? yes no Tier 2 Evaluation -Collect additional site data as needed. -Conduct Tier 2 assessment per specified procedures. Compare Tier 2 site-specific target levels (SSTLs) with site conditions. no Chemical(s) of concern concentrations exceed SSTLs? yes no Remediation to Tier 2 SSTLs practicable? Interim remedial action appropriate? yes Involves transport and fate modeling based on site specific data yes yes no Tier 3 Evaluation -Collect additional site data as needed. -Conduct Tier 3 assessment per specified procedures. Compare Tier 3 site-specific target levels (SSTLs) with site conditions. no Chemical(s) of concern concentrations exceed SSTLs? yes Interim remedial action appropriate? yes no Remedial Action Program -Identify cost-effective means of achieving final corrective action goals, including combinations of remediation, natural attenuation, and institutional controls. Implement the preferred alternatives. Continued monitoring required? no yes Compliance Monitoring -Conduct monitoring program as needed to confirm that corrective action goals are satisfied. CEE 440 No Further Action 2011 Charles J. Werth, University of Illinois at Urbana-Champaign. All rights reserved. 33
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Exp_Com_Con_Gloss.qxd7/3/078:24 PMPage 47GlossaryAll key terms appearing in this book (in bold italic) are listedalphabetically in this Glossary for easy reference. If you wantto learn more about a feature or concept, use the Index tofind the term
HCCS - BCIS - 1405
Exploring MicrosoftOffice 2007Computing ConceptsRobert Grauer, Lynn Hogan, Keith MulberyChanges made by Anci Shah @ HCCCommitted to Shaping the Next Generation of IT Experts.1Copyright 2008 Pearson Prentice Hall. All rights reserved.ObjectivesUnd
HCCS - BCIS - 1405
Exploring MicrosoftOffice 2007Computing ConceptsRobert Grauer, Lynn Hogan, Keith MulberyCopyright 2008 Pearson Prentice Hall. AllNextCommitted reserved.to Shaping the rightsGeneration of IT Experts.1ObjectivesUnderstand computer concepts andcom
HCCS - BCIS - 1405
HCCS - BCIS - 1405
From: &quot;Saved by Windows Internet Explorer 7&quot;Subject: Course ContentDate: Mon, 30 Nov 2009 15:41:46 -0600MIME-Version: 1.0Content-Type: multipart/related;type=&quot;multipart/alternative&quot;;boundary=&quot;-=_NextPart_000_000A_01CA71D3.9FE71480&quot;XX-MimeOLE: Prod
Purdue - ME - 509
Purdue - ME - 509
Purdue - ME - 509
Purdue - ME - 509
Purdue - ME - 509
Practice Problems on the Linear Momentum EquationsCOLM_01A frequently used hydraulic brake consists of a movable ram that displaces water from a slightly larger cylinder, asshown in the figure. The cross-sectional area of the cylinder is Ac and the cro
Purdue - ME - 509
Notes on Fluid Mechanics and Gas DynamicsCarl Wassgren, Ph.D.School of Mechanical EngineeringPurdue Universitywassgren@purdue.edu16 Aug 2010Chapter 01:Chapter 02:Chapter 03:Chapter 04:Chapter 05:Chapter 06:Chapter 07:Chapter 08:Chapter 09:C
Purdue - ME - 509
Practice Problems on Fluid Staticsmanometry_01Compartments A and B of the tank shown in the figure below are closed and filled with air and a liquid with aspecific gravity equal to 0.6. If atmospheric pressure is 101 kPa (abs) and the pressure gage rea
Purdue - ME - 509
Practice Problems on Conservation of MassCOM_01Construct from first principles an equation for the conservation of mass governing the planar flow (in the xy plane)of a compressible liquid lying on a flat horizontal plane. The depth, h(x,t), is a functi
Purdue - ME - 509
Practice Problems on Pipe Flowspipe_02A homeowner plans to pump water from a stream in their backyard to water their lawn. A schematic of the pipesystem is shown in the figure.sprinklerinlet pipe-to-pump3 m coupling1 m streamhose-to-hose coupling
Purdue - ME - 509
Purdue - ME - 509
Purdue - ME - 509
172Chapter 3 Integral Relations for a Control VolumeEXAMPLE 3.19A hydroelectric power plant (Fig. E3.19) takes in 30 m3/s of water through its turbine and discharges it to the atmosphere at V2 2 m/s. The head loss in the turbine and penstock system is
Purdue - ME - 509
1. In fluid mechanics, it is the ratio of the area of the vena contracta to the area of the smaller pipe.Answer: A. Contraction coefficient2. When the Reynolds number of a fluid flow is 3500, the flow isAnswer: C. Intermediate between turbulent or lami
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LECTURE NOTES ONINTERMEDIATE FLUID MECHANICSJoseph M. PowersDepartment of Aerospace and Mechanical EngineeringUniversity of Notre DameNotre Dame, Indiana 46556-5637USAlast updatedSeptember 7, 20082Contents1 Governing equations1.1 Philosophy of
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CHAPTER 3FLOW PAST A SPHERE II: STOKES LAW, THEBERNOULLI EQUATION, TURBULENCE, BOUNDARYLAYERS, FLOW SEPARATIONINTRODUCTION1 So far we have been able to cover a lot of ground with a minimum ofmaterial on fluid flow. At this point I need to present to
Purdue - ME - 509
Purdue - ME - 509
Chapter 6SOLUTION OF VISCOUS-FLOW PROBLEMS6.1 IntroductionTHE previous chapter contained derivations of the relationships for the conservation of mass and momentumthe equations of motion in rectangular,cylindrical, and spherical coordinates. All the
Purdue - ME - 509
Purdue - ME - 509
Appendix AVECTORS, TENSORS ANDMATRIX NOTATIONThe objective of this section is to review some of the vector operations that you have already coveredin your MATH and ENGR courses. For more details and examples you should refer to your calculustext unde
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Home &gt; Notes &gt; Equations &gt; Substantial Derivative of scalar field - also known as Total derivative @WolframD = + V Dt tIn rectangular coordinates D = +u +v +w Dt t x y z where V is the fluid velocity field, u is the component of V in the x direction, v
Purdue - ME - 509
Fluids Lecture 10 Notes1. Substantial Derivative2. Recast Governing EquationsReading: Anderson 2.9, 2.10Substantial DerivativeSensed rates of changeThe rate of change reported by a ow sensor clearly depends on the motion of the sensor.For example,