1. Development of An ERROR ESTIMATE P M V Subbarao Professor Mechanical Engineering Department A Tolerance to Error Generates New Information….

2. Development of Fundamental Knowledge The Basic Knowledge: Evaluation of properties of engineering materials. Equation of State: A first scientific and engineering revolution. EOS for gases and vapours -- Initiated by: Boyle's Law was perhaps the first expression of an equation of state. In 1662, the noted Irish physicist and chemist Robert Boyle performed a series of experiments. In 1787 the French physicist Jacques Charlesfound that oxygen, nitrogen, hydrogen, carbon dioxide, and air expand to the same extent over the same 80 Kelvin interval. Later, in 1802, Joseph Louis Gay-Lussac published results of similar experiments.

3. Information with Error V, ml T, K

4. Every Knowledge is A Geometry!!!

5. The Ultimate Result A mathematical Model of EOS of Gas, known as an Ideal Gas. Lead to development of Pfaffian Differential Equation. Johann Friedrich Pfaffian was a German mathematician. He was described as one of Germany's most eminent mathematicians during the 19th century. A Pfaffian Total Differential Equation:

6. Equation of State for Vapors -- Gases

7. Reduced Pressure p R = p/p c Reduced Temperature T R = T/T c

8. Measurement equation The case of interest is where the quantity Y being measured, called the measurand. It is never measured directly, but is determined from N other quantities X 1, X 2,..., X N through a functional relation f, often called the measurement equation Y = f(X 1, X 2,..., X N ) Included among the quantities X i are corrections (or correction factors), as well as quantities that take into account other sources of variability, such as different observers, instruments, samples, laboratories, and times at which observations are made (e.g., different days).

9. Thus, the function f of equation should express not simply a physical law but a measurement process. In particular, it should contain all quantities that can contribute a significant variation to the measurement result. An estimate of the measurand or output quantity Y, denoted by y, is obtained from previous equation using input estimates x 1, x 2,..., x N for the values of the N input quantities X 1, X 2,..., X N. Thus, the output estimate y, which is the result of the measurement, is given by y = f(x 1, x 2,..., x N ). The measure of a measurand is not only different from true value, but also random !!!!

10. Uncertainty "A parameter associated with the result of a measurement, that characterizes the dispersion of the values that could reasonably be attributed to the measurand“ The word uncertainty relates to the general concept of doubt. The word uncertainty also refers to the limited knowledge about a particular value. Uncertainty of measurement does not imply doubt about the validity of a measurement; On the contrary, knowledge of the uncertainty implies increased confidence in the validity of a measurement result.

11. Error and Uncertainty It is important to distinguish between error and uncertainty. Error is defined as the difference between an individual result and the true value of the measurand. Error is a single value. In principle, the value of a known error can be applied as a correction to the result. Error is an idealized concept and a single number, which cannot be known exactly. Uncertainty takes the form of a range, and, if estimated from an analytical procedure and a defined sample type, may apply to all determinations so described. In general, the value of the uncertainty cannot be used to correct a measurement result.

12. The difference between error and uncertainty should always be borne in mind. The result of a measurement after correction can unknowably be very close to the unknown value of the measurand, and thus have negligible error, Even though it may have a large uncertainty

13. The Uncertainty The uncertainty of the measurement result y arises from the uncertainties u (x i ) (or u i for brevity) of the input estimates x i that enter equation. Components of uncertainty may be categorized according to the method used to evaluate them.

14. Components of Uncertainty “Component of uncertainty arising from a random effect” : Type A These are evaluated by statistical methods. “Component of uncertainty arising from a systematic effect,”: Type B These are evaluated by other means.

15. Definitions Let x be the value of a measured quantity. Let u x be the uncertainty associated with x. When we write x = x measured ± u x (20:1) we mean that x is the best estimate of the measure value, x measured is the value of x obtained by correction of the measured value of x, u x is the uncertainty at stated odds. Usually 20:1 odds

16. Representation of uncertainty components Standard Uncertainty Each component of uncertainty, is represented by an estimated standard deviation, termed standard uncertainty u i, and equal to the positive square root of the estimated variance Standard uncertainty: Type A An uncertainty component obtained by a Type A evaluation is represented by a statistically estimated standard deviation  i, Equal to the positive square root of the statistically estimated variance  i 2. For such a component the standard uncertainty is u i =  i.

17. Repeated Measurand of Peak Pressure in Diesel Engine No. of Measurements p peak

18. PDF of Measurand of Peak Pressure 5.15 – 5.16 5.29 – 5.30

19. Mean and standard deviation Consider an input quantity X i whose value is estimated from n independent observations X i,k of X i obtained under the same conditions of measurement. In this case the input estimate x i is usually the simple mean

20. and the standard uncertainty u(x i ) to be associated with x i is the estimated standard deviation of the mean

21. Standard uncertainty: Type B This uncertainty is represented by a quantity u j, May be considered as an approximation to the corresponding standard deviation; which is it is equal to the positive square root of u j 2. u j 2 may be considered an approximation to the corresponding variance and which is obtained from an assumed probability distribution based on all the available information. Since the quantity u j 2 is treated like a variance and u j like a standard deviation, for such a component the standard uncertainty is simply u j.

22. Evaluating uncertainty components: Type B A Type B evaluation of standard uncertainty is usually based on scientific judgment using all of the relevant information available, which may include: previous measurement data, experience with, or general knowledge of, the behavior and property of relevant materials and instruments, manufacturer's specifications, data provided in calibration and other reports, and uncertainties assigned to reference data taken from handbooks. Broadly speaking, the uncertainty is either obtained from an outside source, or obtained from an assumed distribution.

23. Uncertainty obtained from an outside source Procedure: Convert an uncertainty quoted in a handbook, manufacturer's specification, calibration certificate, etc., Multiple of a standard deviation A stated multiple of an estimated standard deviation to a standard uncertainty. Confidence interval This defines a "confidence interval" having a stated level of confidence, such as 95 % or 99 %, to a standard uncertainty.

24. Uncertainty obtained from an assumed distribution Normal distribution: "1 out of 2" Procedure: Model the input quantity in question by a normal probability distribution. Estimate lower and upper limits a- and a+ such that the best estimated value of the input quantity is (a+ + a-)/2 There is 1 chance out of 2 (i.e., a 50 % probability) that the value of the quantity lies in the interval a- to a+. Then u j is approximately 1.48 a, where a = (a+ - a-)/2 is the half-width of the interval.

25. Uncertainty obtained from an assumed distribution Normal distribution: "2 out of 3" Procedure: Model the input quantity in question by a normal probability distribution. Estimate lower and upper limits a- and a+ such that the best estimated value of the input quantity is (a+ + a-)/2 and there are 2 chances out of 3 (i.e., a 67 % probability) that the value of the quantity lies in the interval a- to a+. Then u j is approximately a, where a = (a+ - a-)/2 is the half-width of the interval

26. Normal distribution: "99.73 %" Procedure: If the quantity in question is modeled by a normal probability distribution, there are no finite limits that will contain 100 % of its possible values. Plus and minus 3 standard deviations about the mean of a normal distribution corresponds to 99.73 % limits. Thus, if the limits a- and a+ of a normally distributed quantity with mean (a+ + a-)/2 are considered to contain "almost all" of the possible values of the quantity, that is, approximately 99.73 % of them, then u j is approximately a/3, where a = (a+ - a-)/2 is the half-width of the interval.

27. For a normal distribution, ± u encompases about 68 % of the distribution; for a uniform distribution, ± u encompasses about 58 % of the distribution; and for a triangular distribution, ± u encompasses about 65 % of the distribution.

28. Propagation of Uncertainty in Independent Measurements