METROLOGÍA E INSTRUMENTACIÓN • Instrumentación Industrial por Antonio Creus – 8va edición • Fundamentals of Industrial Instrumentation and Process Control by William C. Dunn UNIT 1 Fundamentals of Metrology and Metrological Instruments INTRODUCTION At the beginning of the industrial age, measurements did not need higher precision, the manufactured products were controlled manually. This involved a person in charge of controlling the process. Currently, variables must be measured more accurately and quickly, in addition to, the process control is done automatically through instrumentation and controls in such a way that people are needed only to supervise the process. Some types of instruments for measuring variables in the process are: length, pressure, temperature, level, flow, humidity, pH, etc. INTRODUCTION VOCABULARY RELATED TO MEASUREMENT AND CONTROL RANGE (campo de medida) (The measurement field) is the spectrum or set of values of the measured variable that fall within the upper and lower limits of the instrument's measurement, reception or transmission capacity. It is expressed by establishing the two extreme values. RANGEABILITY Quotient between the upper and lower measurement value of an instrument. SPAN (alcance) Algebraic difference between the upper and lower values of the instrument's measuring range Span: 10 Span: 450 Span: 16mA ERROR Measure of the deviation presented by the practical measures of a process variable in relation to the theoretical or ideal measures, as a result of the imperfections of the equipment and the parasitic variables that affect the process. That is to say: Absolute error = Read value - True value Relative error = Absolute error/True error ERROR When a measurement is carried out with the participation of several instruments, placed one after the other, the final value of the measurement will be determined by the errors inherent in each one of the instruments. Ex: 4 instruments: A, B, C, D. 4 errors: ±a, ±b, ±c, ±d. Maximum possible error a+b+c+d Maximum probable error ± 𝑎2 + 𝑏 2 + 𝑐 2 + 𝑑2 ERROR Calcular: • Error máximo posible • El error máximo probable 2% 0.5% 0.5% 0.5% HYSTERESIS The hysteresis (hysteresis) is the maximum difference that is observed in the values indicated by the index or the pen of the instrument or the output signal for the same value of the measurement field, when the variable travels the entire scale in both directions , ascending and descending. It is expressed as a percentage of the span. This is due to stresses induced into the material of the instrument by changing its shape in going from zero to full-scale deflection. HYSTERESIS Compute the hysteresis of this manometer True pressure (psi) 80G100auge reading pressure (psi) Difference Hysteresis (%) 0 20 40 60 80 100 80 60 40 20 0 1.2 19.5 37.0 57.3 81.0 104.2 83.0 63.2 43.1 22.5 1.5 0.3 3.0 6.1 5.9 2.0 0 0.3 3.0 6.1 5.9 2.0 0 ACCURACY (exactitud) Quality of a measuring instrument by which it tends to give readings close to the true value of the measured quantity. It is the difference between the indicated value and the actual value. Accuracy depends on linearity, hysteresis, offset, drift, and sensitivity. The resulting discrepancy is stated as a ± deviation from the true value, and is normally specified as a percentage of full-scale reading or deflection (%FSD). Accuracy can also be expressed as the percentage of span, percentage of reading, or an absolute value. ACCURACY (examples) a) Percentage of the span, (range). Example: range goes from 100°C to 300°C, for a reading of 150°C and an accuracy of ± 0.5%, the actual temperature value will be between 150 ± 0.5 × 200/100 = 150 ± 1, that is, between 149°C and 151°C. b) Directly, in units of the measured variable. Example: accuracy ± 1°C. c) Percentage of the reading made. Example: accuracy of ± 1% of 150°C, that is, ± 1.5°C. d) Percentage of the maximum value of the measurement field. Example: accuracy ± 0.5% of 300°C = ± 1.5°C. e) Percent of the scale length. Example: If the scale length of the instrument in Figure 1.3 is 150 mm, the accuracy of ± 0.5% will represent ± 0.75 mm on the scale. PRECISION Quality of an instrument by which it tends to give readings very close to each other, that is, it is the degree of dispersion of the same. DEAD ZONE or DEAD BAND Is the field of values of the variable that does not vary the indication or the output signal of the instrument, that is, it does not produce its response. It is given as a percentage of the span. For example: dead zone is ± 0.1% and span is 200°C, then, dead zone in values is 0.1 × 200/100 = ± 0.2°C. OTHER TERMS Measuring Range with Zero Elevation.- It is that measurement field in which the zero value of the measured variable or signal is greater than the lower value of the field. For example, -10°C to 30°C. Campo de medida con supresión de cero.- It is that measurement field in which the zero value of the measured variable or signal is less than the lower value of the field. For example, 20°C to 60°C. Zero elevation.- It is the amount by which the lower value of the field exceeds the zero value of the variable. It can be expressed in units of the measured variable or in % of the span. For example, 20°C in the 20°C to 60°C range of the instrument, that is (20/40) × 100 = 50%. OTHER TERMS Drift (deriva).- It is a variation in the output signal that occurs in a given period of time while keeping the measured variable and all environmental conditions constant. The drift from zero (variation in the output signal for the zero value of the measurement attributable to any internal cause) and the thermal drift from zero (variation in the output signal at zero measurement, due to the unique effects of temperature). Drift is usually expressed as a percentage of the full scale output signal at room temperature, per unit, or per range of temperature variation. For example, the zero thermal drift of an instrument under room temperature conditions for 1 month was 0.2% of span. OTHER TERMS Reliability .- The probability that an instrument will continue to behave within specified limits of error over a specified time and under specified conditions. Resolution.- Smallest amount of a variable that an instrument can resolve. Noise.- Any electrical disturbance or accidental unwanted signal that modifies the transmission, indication or recording of the desired data. Reproducibility.- Ability of an instrument to repeatedly read the same signal over time, and give the same output under the same conditions. An instrument may not be accurate but can have good reproducibility OTHER TERMS Repeatability.-Is the reproducibility of the positions of the pen or of the index or of the output signal of the instrument, when repeatedly measuring identical values of the variable under the same operating conditions and in the same direction of variation, going all over the field. Repeatability is synonymous with precision. σ 𝑥𝑖 − 𝑥 𝑁 2 OTHER TERMS Sensitivity.- Ratio between the increase in the output signal or the reading and the increase in the variable that causes it, after reaching the rest state. For example, if in a 0-10 bar electronic transmitter, the pressure goes from 5 to 5.5 bar and the output signal goes from 11.9 to 12.3 mA DC, the sensitivity is the quotient: 12.3 − 11.9 20 − 4 = ±0.5𝑚𝐴𝑐𝑐/𝑏𝑎𝑟 5.5 − 5 10 Traceability.- Property of the result of measurements carried out with an instrument or with a standard, such that it can be related to national or international standards, through an uninterrupted chain of comparisons and with all determined uncertainties. UNCERTAINTY It is the dispersion of the values that can be reasonably attributed to the true value of the measured quantity. Sources of uncertainty: • Influence of environmental conditions. • Different readings of analog instruments made by operators. • Variations in repeated observations of the measure under apparently identical conditions. • Inaccurate values of the standard instruments. • Product sample not representative. For example, in temperature measurement with a glass standard thermometer, the mass of the bulb changes the temperature of the process sample whose temperature is to be measured. UNCERTAINTY For the comparison to be correct, the general procedure is that the measurement standard is sufficiently more precise than that of the device being calibrated (ratio 4:1). The measuring (mesurando) can be measured directly (for example, the temperature of a body with a thermometer) or indirectly from other mathematically or functionally related magnitudes. There are two uncertainties A and B present in the measurement. The A's are related to random sources of error and can be evaluated from statistical distributions (readings on the instrument), while the B's are associated to systemic errors and correspond to the uncertainty of the calibrator, the resolution of the instrument and the influence of other magnitudes (temperature, external fields, humidity, position, etc.) UNCERTAINTY Combined uncertainty Uc 𝑢𝑐 = 𝑢𝐴2 + 𝑢𝐵2 Expanded uncertainty Uexpanded Uexpanded=kuc K = Coverage or safety factor that is determined according to the confidence level of the uncertainty, STUDENT'S T-VALUES FOR DIFFERENT LEVELS OF CONFIDENCE AND DEGREES OF FREEDOM STUDENT'S T-VALUES FOR DIFFERENT LEVELS OF CONFIDENCE AND DEGREES OF FREEDOM UNCERTAINTY type “A” Thus, in a series of repeated measures of the variable, the value is given x is given by the arithmetic mean or average of the observed values: 𝑛 1 𝑥 = 𝑥𝑖 𝑛 𝑖=1 The estimated value of the experimental variance: 1 1 2 2 2 𝜎 𝑥 = 𝑥𝑖 − 𝑥ҧ 𝑜𝑟 𝑏𝑒𝑡𝑡𝑒𝑟, 𝜎 𝑥 = 𝑥𝑖 − 𝑥ҧ 𝑛−1 𝑛 𝑛−1 Its positive square root is the experimental standard deviation of the 𝑠(𝑥) arithmetic mean that is equal to the standard uncertainty. 𝑈 𝑥ҧ = 𝑛 2 MULTIPLYING FACTOR OF THE NUMBER OF MEASUREMENTS When the number of repetitive measurements is less than 10, the standard deviation must be multiplied by a multiplying factor. UNCERTAINTY type “B” It is determined based on the information available from various sources, such as: • Data from previous measurements. • Experience and knowledge of the instruments. • Manufacturer's specifications. • Uncertainty values from technical manuals. UNCERTAINTY type “B” coverage factor=1.65 (= 0.95 × 3) and confidence level=95%: 𝑢 𝑥𝑖 = 𝑠(𝑥) 𝑢 𝑥𝑖 = 𝑏−𝑎 12 2 o 𝑏−𝑎 3 𝑢 𝑥𝑖 = 𝑏−𝑎 24 2 𝑜 𝑏−𝑎 6 UNCERTAINTY AND PROPAGATION OF ERRORS Si z=f(x,y) 𝑢𝑧2 𝜕𝑧 = 𝑢𝑥 𝜕𝑥 2 𝜕𝑧 + 𝑢𝑥 𝜕𝑥 𝐴 2 𝜕𝑧 + 𝑢𝑦 𝜕𝑦 𝐵 2 𝜕𝑧 + 𝑢𝑦 𝜕𝑦 𝐴 2 𝐵 EXCERCISE 3 Determinar la incertidumbre en la medición de la densidad de una sustancia. Instrumentos usados: DATOS PESO (g) 1 23.28 2 23.279 3 23.279 4 23.28 5 23.281 6 23.278 7 23.28 8 23.279 9 23.279 10 23.28 PROMEDIO 23.280 VARIANZA 7.22222E-07 DESVIAC. 0.000849837 DATOS VOLUMEN (ml) DENSIDAD 24.75 0.94060606 25.00 0.93116 24.75 0.94056566 25.25 0.9219802 25.00 0.93124 25.00 0.93112 25.00 0.9312 24.75 0.94056566 25.00 0.93116 25.00 0.9312 24.95 0.025 0.158113883
