One of the parameters that is used to evaluate the

An E after a problem number indicates that it is similar to the preceding problem except for the use of English units. Tho resistors, Rl = 100.0 0.2 nand R2 = 50.0 0.1 n, are connected (a) in series and (b) in parallel. Calculate the uncertainty in

Orifice meters are used to measure the flow rate of a fluid. In an experiment, the flow coefficient K of an orifice is found by collecting and weighing water flowing through the orifice during a certain interval while the orifice is under a constant head. K is calculated from the following formula: K= M tAp(2g Ilh )112 The values of the parameters have been determined to be as follows, with 95 % confidence: Mass Time Density Diameter Head M = 393.00 0.03 kg t = 600.0 1 s p = 1000.0 0.1 % kglm3 d = 1.270 0.0025 em (A is area) Ah = 366.0 0.3 em Find the value of K, its uncertainty (with 95% confidence), and the maximum possible error.

Orifice meters are used to measure the flow rate of a fluid. In an experiment, the flow coefficient K of an orifice is found by collecting and weighing water flowing through the orifice during a certain interval while the orifice is under a constant head. K is calculated from the following formula: K = M tAp(2gAh)tl2 The values of the parameters have been determined to be as follows, with 95 % confidence: Mass TlDle Density Diameter Head M = 865.00 0.05 Ibm t = 600.0 1 s p = 62.36 0.1 % Ibm/ttl d = 0.500 0.001 in. (A is area) Ah = 12.02 0.01 ft Find the value of K, its uncertainty (with 95% confidence), and the maximum possible error.

Variables RJ, R2, R3, and 14 are related to three independent variables Xl> X2, and X3 by the formulas Rt = aXt + bX2 + eX3 R2 = d(Xt)(X2)(X3) e(Xt)(x2) R3 =--X3 R4 = f(Xt)g(X2) h (X3)i where a, ... , i are constants. For each case, derive the uncertainty of the result, WR, in terms of the uncertainties of individual variables (the w/s). Generalize your results to take account of the effect of summation and multiplication on the propagation of errors.

A simple spring is used to measure force. The spring is considered to be linear, so that F = kx, where F is the force in newtons, k is the spring constant in newtons per centimeter, and x is the displacement in cm. If x = 12.5 1.25 cm and k = 700 18 Nlcm, calculate the maximum possible error and the uncertainty of the measured force in absolute (dimensional) and relative (%) terms.

A simple spring is used to measure force. The spring is considered to be linear, so that F = kx, where F is force in pounds, k is the spring constant in pounds per inch, and x is the displacement in inches. If x = 5 0.5 in. and k = 20 0.5 lbflin., calculate the maximum possible error and

A simple spring is used to measure force. The spring is considered to be linear, so that F = kx, where F is the force in newtons. k is the spring constant in newtons per centimeter, and x is the displacement in cm. If x = 20.0 1.25 cm and k = 800 16 N/cm, calculate the maximum possible error and the

A mechanical speed control system works on the basis of centrifugal force, which is related to angular velocity through the formula F = mrw2 where F is the force, m is the mass of the rotating weights, r is the radius of rotation, and w is the angular velocity of the system. The following values are measured to determine w: r = 20 0.02 mm, m = 100 0.5 g, and F = 500 0.1 % N. Find the rotation

A mechanical speed control system works on the basis of centrifugal force, which is related to angular velocity through the formula F = mrw2 where F is the force, m is the mass of the rotating weights, r is the radius of rotation, and w is the angular velocity of the system. The following values are measured to determine w: r = 25 0.02 mm. m = 120 0.5 g. and F = 600 0.2% N. Find the rotatio

Young's modulus of elasticity, E, relates the strain, 8LIL, in a solid to the applied stress, FIA, through the relationship FIA = E(8L/L). To determine E, a tensile machine is used, and F, L, 8L, and A are measured. The uncertainties in each of these quantities are 0.5%, 1 %,5%, and 1.5%, respectively, with 95% confidence. Calculate the uncertainty in E in percentage form. Which of these measurements has

The variation of resistance with temperature is expressed by the relationship R = Ro[l + a(T -To)], where Ro is the resistance at the reference temperature To and a for the resistor material has been determined to be 0.0048 0.1 %/C. In the range 0 to 100C, in which we are calibrating this resistor, temperature measurements have shown a standard deviation of 0.1C. The systematic uncertainty of the temperature-measurement device is known to be 0.1 dc. Calculate the percent uncertainty of R at

One of the parameters that is used to evaluate the performance of an engine is its brakespecific fuel consumption (bsfc), defined as mfuel bsfc = -- 27TNT where, for a given engine, the parameters on the right-hand side of the equation are measured and found to be as follows: Mass flow rate Rotational speed Torque mfuel = 5.0 X 10-4 kgls N = 50 revls (3000 rpm) T = 150N-m We can tolerate a total uncertainty of 1 % in the measurement of bsfc. (a) Calculate the rated bsfc. Problems 233 (b) Determine the maximum tolerable uncertainty in each measurement. This will occur when the uncertainties of the other measurements are negligible. (c) For a total uncertainty of 1 % in bsfc, calculate the relative (%) uncertainty of the measured parameters if all of them have the same relative uncertainty. (d) Calculate the absolute value of the

One of the parameters that is used to evaluate the performance of an engine is its brakespecific fuel consumption (bsfc). defined as b mfuel s c = 27TNr where, for a given engine, the parameters on the right-hand side of the equation are measured and found to be as follows: Mass flow rate Rotational speed Torque mfuel = 2.S X 10-4 kgls N = SO revls (3000 rpm) r = 7SN-m We can tolerate a total uncertainty of 1.S% in the measurement of bsfc. (a) Calculate the rated bsfc. (b) Determine the maximum tolerable uncertainty in each measurement. This will occur when the uncertainties of the other measurements are negligible. (c) For a total uncertainty of .S% in bsfc, calculate the relative (%) uncertainty of the measured parameters if all of them have the same relative uncertainty. (d) Calculate the absolute value of the

In using a temperature probe, the following uncertainties were determined: Hysteresis Linearization error Repeatability Resolution error Zero offset O.lC 0.2 % of the reading 0.2C O.OsoC O.1C Determine the type of these errors (random or systematic) and the total uncertainty due to these effects for a temperature reading of 120C.

In using a temperature probe, the following uncertainties were determined: Hysteresis Linearization error Repeatability Resolution error Zero offset O.lC 0.2% of the reading O.lC O.lCDetermine the type of these errors (random or systematic) and the total uncertainty due to these effects for a temperature reading of 100C.

A digital scale is used to measure the mass of a product on a manufacturing line. The scale has a range of 0-2 kg and uses a 12-bit AID converter. In addition, it has an accuracy of 2 % of its reading. List the uncertainties and categorize them as systematic or random. Calculate the uncertainty in a measurement of 1.25 kg by this instrument.

A digital scale is used to measure the mass of a product on a manufacturing line. The scale has a range of 0-5 kg and uses a 12-bit AID converter. In addition, it has an accuracy of 1 % of its reading. List the uncertainties and categorize them as systematic or random. Calculate the uncertainty in a measurement of 3.20 kg by this instrument.

A digital-output pressure-measuring system has the following specifications: Range Accuracy Resolution Temperature stability Oto 1000kPa 0.5% ofrange l kPa 2 kPa (0 to 50C) Specify the types of these errors and calculate the uncertainty of pressure measurement with this transducer. For a nominal pressure of 500 kPa, calculate the absolute and relative uncertainties.

A digital-output pressure-measuring system has the following specifications: Range Accuracy Resolution Temperature stability Oto 500kPa 0.3 % of range 1kPa l kPa (0 to 50C) Specify the types of these errors and calculate the uncertainty of pressure measurement with this transd

In a measurement of temperature (in 0c) in a duct, the following readings were recorded: 248.0,248.5,249.6,248.6,248.2,248.3,248.2,248.0, 247.5, 248.1 Calculate the average temperature, the standard deviation of the sample, and the random

A manufacturer of plastic pipes uses a scale with an accuracy of 1.5% of its range of 5 kg to measure the mass of each pipe the company produces in order to calculate the uncertainty in mass of the pipes. In one batch of 10 parts, the measurements are as follows: 1.93,1.95,1.96,1.93,1.95,1.94,1.96,1.97,1.92,1.93 (kg) Calculate (a) the mean mass of the sample. (b) the standard deviation of the sample and the standard deviation of the mean. (c) the total uncertainty of the mass of a single product at a 95% confidence level. (d) the total uncertainty of the average mass of the product at a 95% confidence level.

A manufacturer of plastic pipes uses a scale with an accuracy of 1.0% of its range of 10 lb to measure the mass of each pipe the company produces in order to calculate the uncertainty in mass of the pipes. In one batch of 10 parts, the measurements are as follows: 2.90,2.95,2.96,2.92,2.95,2.94,2.96,2.97,2.98,2.91 (lb) Calculate (a) the mean mass of the sample. (b) the standard deviation of the sample and the standard deviation of the mean. (c) the total uncertainty of the mass of a single product at a 95% confidence level. (d) the total uncertainty of the average mass of the product at a 95% confidence level.

In Problem 7.21, if the measurement is performed on, say, 50 pieces of pipe, and the same mean and standard deviation are obtained, repeat parts (b), (c), and (d) of that problem

In a cheese factory, 4.5-kg blocks of cheese are cut manually. For a large number of blocks, the standard deviation of the cutting process is measured and found to be 0.10 kg. The measurement was done with a scale with an accuracy of 1.5% of the full scale of 12 kg. Calculate the total uncertainty of the weight of the blocks of cheese at a 95 % confidence level.

The managers in the cheese factory of Problem 7.24 have decided to reduce the uncertainty in the mass of cheese and reduce their labor as well by automating the cheesecutting line. They will use a 12-bit online digital scale with a calibration accuracy of 1 % of its reading and a range of 12 kg. Calculate the acceptable standard deviation of the cheese blocks that will reduce the uncertainty in the mass of cheese blocks to 1.5% of the mass of each block.

In a yogurt-filling line, the containers are filled with 1 kg of yogurt. The mass of the yogurt is measured while the filling nozzle is open. The nozzle flow rate is 0.25 kg per second. Several factors affect the flow rate, including the density and the viscosity of the yogurt, which, combined, can introduce an uncertainty of about 1 % of the flow, based on statistical analysis of the data. The dispensing time of the yogurt is controlled mechanically through a cam system. The uncertainty in the time during which the nozzle is open was established to be 0.1 sec in a calibration process. (a) Calculate the filling time of each container. (b) Categorize each of the foregoing uncertainties as systematic or random. (c) Calculate the uncertainty in the mass of the filled yogurt containers. (d) If the plant managers decide to reduce the uncertainty in the mass of the yogurt, on what do you suggest they concentrate?

In a yogurt-filling line, the containers are filled with 2 pounds of yogurt. The weight of the yogurt is measured while the filling nozzle is open. The nozzle flow rate is 0.5 pound per second. Several factors affect the flow rate, including the density and the viscosity of the yogurt, which, combined, can introduce an uncertainty of about 1 % of the flow, based on statistical analysis of the data. The dispensing time of the yogurt is controlled mechanically through a cam system. The uncertainty in the time during which the nozzle is open was established to be 0.1 sec in a calibration process. (a) Calculate the filling time of each container. (b) Categorize each of the foregoing uncertainties as systematic or random. (c) Calculate the uncertainty in the weight of the filled yogurt containers. Chapter 7 Experimental Uncertainty Analysis (d) If the plant managers decide to reduce the uncertainty in the weight of the yogurt, on l what do you suggest they concentrate? :.

To improve the accuracy of measuring the heating value of natural gas in Examples 7.4 and 7.5, a new calorimeter with the same range, but with an accuracy of 1 % of the full range, is used to make 15 measurements. The average heating value of natural gas is measured to be 49,200 kJ/kg, and the standard deviation is 450 kJ/kg. Calculate the total uncertainties in the estimates of (a) the mean heating value and (b) a single measurement of the heating value.

The following information about a linear submersible depth pressure transmitter is available: Range Output 0-20mH 20 4-20 rnA Accuracy (including linearity, hysteresis, and repeatability) 0.2% Span Zero balance Thermal effects 2% Span 1.5% Span (a) What will be the nominal output of the device (in rnA) at a depth of 15 m of water? (b) Estimate the uncertainty due to each of the error sources that can be determined from the specifications. Express the uncertainties in rnA, in m H20, and as a percentage of the reading. (c) Calculate the total output uncertainty in rnA, in m H20, and as a percentage of the output. 7

The following information about a linear submersible depth pressure transmitter is available: Range Output Accuracy (including linearity, hysteresis, and repeatability) Zero balance Thermal effects 0-25mH 20 4-20 rnA 0.1% Span 1.5% Span 1.0% Span (a) What will be the nominal output of the device (in rnA) at a depth of 15 m of water? \ (b) Estimate the uncertainty due to each of the error sources that can be determined from the specifications. Express the uncertainties in rnA, in m H20, and as a percentage of the reading. (c) Calculate the total output uncertainty in rnA, in m H20, and as a percentage of the output.

The following information is from an instrument manufacturer's catalog for a load cell: Range (full scale, FS) Operating temperature Signal output Excitation voltage Linearity 0-500N -50 to 120C 3 m V N excitation voltage, nominal 10 volts DC 0.1% Span

Hysteresis Repeatability Temperature effect 0.08% Span 0.03% Span 0.002% SpanfOC (20C reference) Problems 237 The device is operated in an environment where the temperature range is 10-40e. (a) For an unamplified output of 20 mY, what is the applied load (assuming that the device is linear)? (b) Estimate the uncertainty due to each of the error sources that can be determined from the specifications. Express the uncertainties in m V, in N, and as percentage of the reading at 20 m V. (c) Calculate the total output uncertainty in m V, in N, and as a percentage of the output

The following information is from an instrument manufacturer's catalog for a positivedisplacement flowmeter for high-viscosity fluids: Maximum capacity 10 lpm (liters per min) Turndown ratio* 10: 1 Accuracy Repeatability 6% of flow rate 1 % of flow rate *The turndown ratio is the ratio of the maximum to the minimum value of the measurand that the instrument can be used to measure. (a) Estimate the uncertainty due to each of the error sources that can be determined from the specifications for readings of 2 and 10 lpm. (b) What are the total uncertainties in the measured flow rates of 2 and 10 lpm?

In a calibration process, it has been determined that a thermocouple is accurate to 0.2C with 95% confidence. To determine the total uncertainty of temperature measurement in a vessel, a sample of 15 measurements was taken. The average temperature is 250.0C, and the standard deviation of the measurements is calculated to be 0.2e. (a) Calculate the random uncertainty and the total uncertainty of the average temperature at a 95% confidence level. (b) Calculate the random uncertainty and the total uncertainty of a single reading of the temperature at a 95% confidence level. (c) If another thermocouple, which is accurate to O.lC is used, will it significantly affect the total uncertainty of temperature measurement?

In Example 7.6, if the temperature reading is 600C and the wall temperature is 545C, calculate the temperature correction and the uncertainty in the correction. All other parameters remain the same.

Strain is to be measured remotely in a structure by means of a strain gage. To estimate the total uncertainty in the measurement of strain, the strain gage and the transmission line are tested separately. Ten measurements of the strain gage output under the same loading produce a standard deviation of 0.5 m V in an average output of 80 m V. Fifteen measurements of the transmitted voltage produced a standard deviation of 1 m V. Determine the random uncertainty of the voltage measurement of strain produced by the strain gage at a 95 % confidence level.

Strain is to be measured remotely in a structure by means of a strain gage. To estimate the total uncertainty in the measurement of strain, the strain gage and the transmission line are tested separately. Ten measurements of the strain gage output under the same loading produce a standard deviation of 0.8 m V in an average output of 100 m V. Ten measurements of the transmitted voltage produced a standard deviation of 1 m V. Determine the random uncertainty of the voltage measurement of strain produced by the strain gage at a 95 % confidence level.

The pressure-measuring system of Problem 7.18 is used to measure the pressure in a compressed air line. To determine the uncertainty in the nominal pressure of 800 kPa, 15 measurements were made, yielding a standard deviation of 5 kPa. Determine the total uncertainty of each pressure measurement in the line at a 95% confidence level

In Problem 7.38, suppose that you make over 30 measurements and still obtain the same standard deviation for the pressure in the line. Will the uncertainty in each measurement of the pressure change? Why? Discuss your results.

In measuring the power of a three-phase electrical motor running a pump, the following data were obtained (based on one-phase current measurement): v (volts) I (amps) PF 460 30.2 0.78 459 31.3 0.80 458 30.4 0.79 460 32.0 0.82 461 31.7 0.81 462 30.7 0.77 460 30.8 0.78 459 31.2 0.80 For a three-phase motor, the power (P) is related to the voltage (V), current (1), and power factor (PF) by the formula P = V X I X PFV3 (a) Calculate the mean, the standard deviation, and the random uncertainty of each parameter (V, I, PF, and P) at a 95% confidence level. (b) Calculate the mean value of the power and the random uncertainty of the mean power at a 95% confidence level. (c) If all the measured parameters have an accuracy of 1 % of the reading of the measuring device, calculate the systematic uncertainty in the power measurement at a 95% confidence level. (d) Calculate the total uncertainty in the power measurement at a 95% confidence level.

In Example 7.8, to improve the accuracy of the temperature measurements, a temperature sensor with a better degree of repeatability is to be used. For the new probe, which has an accuracy of 0.2C, a standard deviation of O.SOC was obtained in 15 measurements. Further tests performed on data transmission resulted in a standard deviation of O.3C in a total of 30 tests. Calculate the systematic and random uncertainties and the total uncertainty in the temperature measurement. Perform the calculations at a 95% confidence level.

In measuring the flow rate of diesel fuel into a test engine, a graduated burette is used. The burette has a resolution of 1 cc. A technician measuring time with a stopwatch produced measurements with an uncertainty of approximately 0.3 s. Discuss the type of these errors, and determine the uncertainty in the volumetric flow rate of fuel (Q = AVIAt, where V = volume and t = time) if the volume measured is 100 cc and the time is 25 s. What other errors may be involved in this process? 7

As detailed in an instrument catalog, a power quality analyzer has the following specifications: Voltage (V): Current (1): Power factor (PF): 600 V (rms) range, accuracy 1 % of reading 1-1000 A, 1 % of reading 2% of reading Using the formula P = V x I x PFv'3, (a) Calculate the uncertainty in power measurement as a percentage of reading due to the measurement device for measured values of 460 V, 36 amp, and a power factor of 0.81. (b) In logging data with a high sampling rate, the calculated standard deviations of the readings are 3.00 V, 2.00 A, and 0.03 for the power factor, while the mean values are 460 V, 35.00 A, and 0.81 for the power factor. Determine the random and systematic uncertainty of the measurements. (c) What is the total uncertainty in the power measurement at a 95% confidence level?

As detailed in an instrument catalog, a power quality analyzer has the following specifications: Voltage (V): 600 V (rms) range, accuracy l % of reading Current (1): 1-500A, l % of reading Power factor (PF): 1 % of reading Using the formulaP = V x I x PFV3, (a) Calculate the uncertainty in power measurement as a percentage of reading due to the measurement device for measured values of 460 V, 50.0 amp, and a power factor of 0.81. (b) In logging data with a high sampling rate, the calculated standard deviations of the readings are 3.00 V, 1.00 A, and 0.03 for the power factor, while the mean values are 460 V. 49.50 A, and 0.80 for the power factor. Determine the random and systematic uncertainty of the measurements. (c) What is the total uncertainty in the power measurement at a 95% confidence level?

A pressure transducer with the following specifications (taken from an instrument catalog) is used for measuring water-pressure discharge from a pump: Range 0-700 kPa Accuracy Linearity 0.5% of span Hysteresis 0.1 % of span Repeatability 0.1 % of span Stability 0.3% of span Temperature error (reference temperature, 20C) On zero 0.04% of spanfOC On span 0.03% of spanlC (a) Calculate the total uncertainty of the transducer in measuring the fluid pressure at 15C. (b) The same transducer is used to measure water pressure. The standard deviation of the measurement is 3.5 kPa, with a mean value of 550 kPa. Calculate the total uncertainty in measuring pressure with this transducer, assuming 95% confidence. Is the uncertainty due more to the accuracy of the transducer or the fluctuation of the fluid pressure? (c) The signal from the transducer is transmitted to a 12-bit DAS with a range of 1400 kPa. Does the quantization error have any significant effect on the uncertainty of the final result? (d) If the noise induced in the transmission signal is estimated to 2% of the value of th

A pressure transducer with the following specifications (taken from an instrument catalog) is used for measuring water-pressure discharge from a pump: Range 0-100 psi Accuracy Linearity 0.5% of span Hysteresis 0.1 % of span Repeatability 0.1 % of span Stability 0.3% of span Temperature error (reference temperature, 70F) On zero 0.022 % of spanlF On span 0.016% of spanlF (a) Calculate the total accuracy of the transducer in measuring the fluid pressure at 60F. (b) The same transducer is used to measure water pressure. The standard deviation of the measurement is 0.50 psi, with a mean value of 80.00 psi. Calculate the total uncertainty in measuring pressure with this transducer, assuming 95% confidence. Is the uncertainty due more to the accuracy of the transducer or the fluctuation of the fluid pressure? (c) The signal from the transducer is transmitted to a 12-bit DAS with a range of 200 psi. Does the quantization error have any significant effect on the uncertainty of the final result? (d) If the noise induced in the transmission signal is estimated to 2 % of the value of the signal, is the signal-transmission error significant in the uncertainty of the final result?

One of the methods for measuring the power (P) of rotating machinery is to measure the rotational speed (with a magnetic pickup, for example) and the rotational shaft torque (with a shaft torque meter, for example) and then calculate the power transmitted through the shaft. The formula for power is P = T X w, where T is the torque, w( = 271'N) is the rotational speed in radians per second, and N is the number of revolutions per second. From the measurement of the power of a small engine, the following information is available: As per the manufacturers' information, the accuracy of the torque meter is 0.7 N-m and the accuracy of the rotational speed-meter is 5 rpm. The values of the measured torque and rotational speed are 165 N-m and 3000 rpm, respectively. In repeating the speed and torque measurements, it is found that the standard deviations of these measurements are 4 N-m and 5 rpm, respectively. (a) Calculate the power of the engine. (b) If the number of samples used for calculating the standard deviations of the torque and speed are 10 and 20, respectively, calculate the standard deviation of the power. (c) Calculate the random and systematic uncertainty of the power. (d) Calculate the total uncertainty of each power measurement at a 95% confidence level.

In Problem 7.47, if the same standard deviations are obtained for torque and rotational speed through a large number of measurements, (a) calculate the power of the engine. (b) calculate the standard deviation of the power. (c) calculate the random and systematic uncertainty of the power. (d) calculate the total uncertainty of each power measurement at a 95% confidence level.

In Problem 7.47, with the same instruments, 10 independent measurements of the engine power are performed. The average and standard deviation of the power are calculated to be 51.5 kW and 0.75 kW, respectively. (a) Calculate the systematic and random uncertainty of the power measurement at a 95 % confidence level. (b) Calculate the total uncertainty of the power measurement at a 95% confidence level.

To measure the efficiency of a pump, the parameters in the following formula are usually measured and applied: 1/ = Qt1PfW. In this formula, 1/ is the pump's efficiency, Q is the volumetric flow rate of the pump, t1P is the pressure differential between the inlet and outlet of the pump, and W is the power input into the pump. The following equipment is used: Differential pressure gage Flowmeter Range 0-1200kPa Accuracy 0.2% of span (includes linearity, hysteresis, and repeatability) Stability Range 0.2% of span 1200 lpm (liters per minute) Accuracy 1.5 % of reading The power is measured through the input of an electrical motor (with specified efficiency), with an expected accuracy of 0.07 kW. In repeating the measurements, the average values of the pressure differential, flow rate, and power are found to be 700 kPa, 340 lpm, and 5 kW, respectively. The standard deviations of 10 kPa, 5.6lpm, and 0.15 kW are calculated for the respective parameters. If the number of measurements made to calculate the mean values and standard deviations is 15, (a) calculate the efficiency of the pump. (b) calculate the standard deviation of the efficiency of the pump. (c) calculate the random and systematic uncertainty of the efficiency of the pump. (d) calculate the total uncertainty of the efficiency of the pump at a 95% confidence level.

To measure the efficiency of a pump, the parameters in the following formula are usually measured and applied: TJ = QI1PIW. In this formula, TJ is the pump's efficiency, Q is the volumetric flow rate of the pump, I1P is the pressure differential between the inlet and outlet of the pump, and W is the power input into the pump. The following equipment is used: Differential pressure gage Flowmeter Range 0-170 psi Accuracy 0.2% of span (includes linearity, hysteresis, and repeatability) Stability Range 0.2 % of span 40 ft3/min Accuracy 1.5% of reading The power is measured through the input of an electrical motor (with specified efficiency), with an expected accuracy of 0.10 horsepower. In repeating the measurements, the average values of the pressure differential, flow rate, and power are found to be 100.0 psi, 12.0 ft3/min, and 6.50 horsepower, respectively. The standard deviations of 1.5 psi, 0.2 ft3/rnin, and 0.20 hp are calculated for the respective parameters. If the number of measurements made to calculate the mean values and standard deviations is 15, (a) calculate the efficiency of the pump. (b) calculate the standard deviation of the efficiency of the pump. (c) calculate the random and systematic uncertainty of the efficiency of the pump. (d) calculate the total uncertainty of the efficiency of the pump at a 95% confidence level.

In Problem 7.51, if the same standard deviations and mean values are obtained through a large number of measurements, (a) calculate the efficiency of the pump. (b) calculate the standard deviation of the measurements of pump efficiency. (c) calculate the random and systematic uncertainty of the measurements of pump efficiency at a 95% confidence level. (d) calculate the total uncertainty of the efficiency of the pump at a 95% confidence level.

For the pump experiment of Problems 7.51 and 7.52, with the same instruments, a large number of independent measurements of the efficiency of the pump is performed. The mean and standard deviation of the efficiency are determined to be 0.82 and 0.01, respectively. (a) Calculate the systematic and random uncertainties of the measurements. (b) Calculate the total uncertainties of the measurements at a 95% confidence level.

In Example 7.10, 10 engines are tested, and the average value of 30.8% with a standard deviation of 0.25 is obtained for the efficiency of the engine. Calculate the uncertainty of the average efficiency of the engines that are produced.

The drag coefficient measured in model testing is often used to estimate the drag of actual systems (e.g., airplanes, cars, and ships). The drag force F is related to the drag coefficient CD, density p, velocity V, and frontal area A via the formula C _ F D -0.5pV2A

In Example 7.12, a new natural-gas supplier is chosen. Fifteen tests of the heating value of the natural gas produced an average value of 47,SOO kJ/kg, with a standard deviation of 300 kJ/kg. Twenty new tests of the engine produced an average power of 48.S kW, with a standard deviation of 0.2 kW. All other parameters of the test remained the same. Calculate the total uncertainty in the measurement of the efficiency of the engine.

Use the DAS of Example 7.13 for a pressure transducer with the following specifications: Range 0 to 2000 kPa Output 0 to S V Linearity and hysteresis (combined) 0.2S% FS Repeatability Thermal span uncertainty O.03% FS O.003%FS/OC The temperature is uncertain to 100C. Select the DAS input range that gives the best accuracy. Estimate the uncertainty of a pressure measurement made with this system.

In Example 7.14, a reevaluation of the data and test equipment showed a loading error of 2SC (instead of 1.0e) and steam-line temperature scatter with a standard deviation of 2.0e. Estimate the zeroth-, first-, and Nth-order uncertainty limits.

What is the difference between a single-measurement test and a multiple-measurement test? Discuss the advantages and disadvantages of each, together with an example.