Unit: B the end of this lecture ou should be able to do the following: Bapco Refiner has the capacit to refine 6, barrel/da of crude oil value + unit. Understand the meaning of unit and dimension.. Identif the meaning of a quantit. 3. Understand the Difference between measured and calculated dimension. 4. Understand the meaning of conversion factor. 5. Use the conversion factors table. 6. Convert between different units 7. Define Sstems of units, compound units, derived units, and unit prefi. 8. Know Unit prefies. Dimension: is a propert that can be: Measured Calculated Dimension is a propert that can be: Measured dimensions such as Calculated dimensions such as Units can be treated like algebraic variables when quantities are added, subtracted, multiplied, or divided: 4 cm cm = 3 cm 4 = 3 measured calculated Mass velocit 3 cm + mm =?? 3 + =?? Time volume 3 N 4 m = N. m 3 4 =. m m = 4 m = 4 Length densit 5 m / s = 5 m/s 6 m / 3 m/s = s Temperature???? g / g = (dimensionless)
A measured quantit can be epressed in terms of an units having the appropriate dimension: 6 km/h = km/min = 37.3 mile/h = 7 m/s = 55 ft/s Convert cm to its equivalent value epressed in mm? mm cm = cm X cm = cm X cm / cm = mm / Conversion Factor Quantit Equivalent values Mass kg lb m = g =. metric ton =.46 lb m = 35.739 oz = 6 oz = 5 4 ton = 453.593 g =.453593 kg Length m ft = cm = mm = 6 microns = angstroms = 39.37 in = 3.88 ft =.936 d =.64 mile = in. = /3 d =.348 m = 3.48 cm Volume m 3 ft 3 = L = 6 cm 3 = 6 ml = 35.345 ft 3 =.83 imperial gallons = 64.7 gal = 56.68 qt = 78 in 3 = 7.485 gal =.837 m 3 = 8.37 L = 837 cm 3 Force N lb f = kg.m/s = 5 dnes = 5 g.cm/s =.48 lb f = 3.74 lb m.ft/s = 4.448 N = 4.448 5 dnes Pressure atm =.35 5 N/m (Pa) =.35 kpa =.35 bar =.35 6 dnes/cm = 76 mm Hg at o C (torr) =.333 m H O at 4 o C = 4.696 lb f /in (psi) Energ J = N. m = 7 ergs = 7 dne.cm =.778 7 kwh =.39 cal =.7376 ft. lb f = 9.486 4 Btu Power W = J/s =.39 cal/s =.7376 ft.lb f /s = 9.486 4 Btu/s =.34 3 hp Eample.- (TXB) Convert an acceleration of cm/s to its equivalent in km/r cm km 36 s 4 h 365 da s cm m h da r Common Sstems of Units: SI Units CGS Sstem English Units Write the given quantit (36 X 4 X 365 ) km = = 9.95 X ( X ) r 9 km/r Determine the proper conversion factor Cancel the original units Multipl and divide numbers to get quantit with new units International Sstem of Units: m, kg, s. Centimeter gram second sstem: cm, g, s American Engineering Sstem: ft, lb m, s Base units units for the dimensions of mass, length, time, absolute temperature, electric current, luminous intensit, and amount of substance. Multiple units which are multiples or fractions of base units used for convenience (ears instead of seconds, kilometers instead of meters, etc.). Derived units Compound units: obtained b multipling and dividing base or multiple units ( cm, ft/min, kg.m/s ). As defined equivalents of compound units (N= kg.m/s, pascal = N/m ) A sstem of units has the following components: Sstem of units Base Units Multiple Units Derived Units Compound Units Defined Equivalents of Compound units
A sstem of units has the following components:. Base units: for mass, length, time, temperature, electrical charge and light intensit.. Multiple Units: multiple or fractions of base units; Second Dimension SI sstem CGS sstem Mass Kilogram (kg) American Engineering Sstem Gram (g) Pound mass (lb m ) Length Meter (m) Centimeter(cm Foot (ft) ) Time Second (s) Second (s) Second (s) millisecond minutes hour Temperature Kelvin (K) Kelvin (K) Rankin ( R) 3. Derived units: a. Compound Units: b multipling and dividing base or multiple units; e.g. cm, ft/min, m/s b. Defined Equivalents of Compound units: e.g. newton kg.m/s, J N.m = kg.m /s Electric current Intensit of light Ampere (A) Candela (cd) Ampere (A) Candela (cd) Ampere (A) Candela (cd) /8/6 4 Table.3 (TXB) Table.3 (TXB) Multiple unit Prefies Table.3 (TXB) Wh mastering unit conversion skills is important? Human error blamed for Orbiter loss The Mars Climate Orbiter now in pieces on the planet's surface B BBC News Online Science Editor Dr David Whitehouse. September 4, 999. On later article b the same author: The Mars Climate Orbiter burnt up when entering the planet's orbit: a mi-up of English and metric units used in calculating its trajector sent the spacecraft too close to Mars. /8/6 8 3
Conversion Between Sstem of Units Eample.3 (TXB) Convert 3 lb m.ft/min to its equivalent in kg.cm/s 3 lb m.ft kg in.54 cm min min. lb m ft in 6 s (3 X X.54) kg.cm = (. X 6 ) s =.885 kg.cm/s Newton s second law of motion F = m. a Natural force units kg.m/s (SI) g.cm/s (CGS) lb m.ft/s (American Eng Ss) Derived force units have been defined: SI: Newton (N) kg.m/s, N is the force required to accelerate a mass of kg at a rate of m/s N a = m/s Object Mass = kg dne a = Object cm/s Mass = g Derived force units have been defined: SI: Newton (N) kg.m/s, N is the force required to accelerate a mass of kg at a rate of m/s CGS: dne g.cm/s dne is the force required to accelerate a mass of g at a rate of cm/s lbf a = 3.74 ft/s Object Mass = lbm American engineering sstem: lb f (pound force) is the force eerted b earth on an object that has a mass of lb m and falling at acceleration of gravit at sea level and 45 o latitude (which is 3.74 ft/s ). lb f 3.74 lb m.ft/s Derived Force Units Derived force units have been defined: SI: Newton (N) kg.m/s N is the force required to accelerate a mass of kg at a rate of m/s CGS: dne g.cm/s dne is the force required to accelerate a mass of g at a rate of cm/s American engineering sstem: lb f (pound force) is the force eerted b earth on an object that has a mass of lb m and falling at acceleration of gravit at sea level and 45 o latitude (which is 3.74 ft/s ). lb f 3.74 lb m.ft/s 4
Derived Force Units Derived force units have been defined: SI: Newton (N) kg.m/s N is the force required to accelerate a mass of kg at a rate of m/s Derived Force Units American engineering sstem: lb f 3.74 lb m.ft/s CGS: dne g.cm/s dne is the force required to accelerate a mass of g at a rate of cm/s Eample: The force required to accelerate a mass of 4. lb m at a rate of 8.435 ft/s is 3.74 lb m.ft/s. Epress the force in the unit of lb f. lb f American engineering sstem: lb f 3.74 lb m.ft/s lb f (pound force) is the force eerted b earth on an object that has a mass of lb m and falling at acceleration of gravit at sea level and 45 o latitude (which is 3.74 ft/s ). or 3.74 lb m.ft l lb f s 3.74 lb m.ft/s = lb f The weight of an object is the force eerted on the object b gravitational attraction. Mass vs. Weight.. Eample: If ou have a mass of 7 kg then how much ou weigh? Assume g = 9.866 m/s = 98.66 cm/s = 3.74 ft/s W = m. g = 7 kg 9.866 m/s = 686.46 kg.m/s l N = 686.46 N kg.m/s Mass vs. Weight Eample: Water has a densit of 6.4 lb m /ft 3. How much does. ft 3 of water weigh at a location where the gravitational acceleration is 3.39 ft/s? (Epress the answer in lb f ) Ans: 4.7 lb f 5
B the end of this lecture understand the following: ou should be able to. Mathematical Operation and Significant Figures.. and accurac 3. Validating Results. 4. Estimation of Measured Values: Sample Mean. 5. Sample Variance of Scattered Data. 6. Process Data Representation and Analsis 7. Two Point Linear Interpolation.5a Scientific Notation, Significant Figures, and Scientific Notation: Ver large or ver small numbers are not convenient 5,,.5 =.5 8 =.5-5 Scientific Notation Scientific notation is used in which the number is epressed as the product of another number(usuall between. and ) and a power of..5a Scientific Notation, Significant Figures, and Significant Figures: of a number are the digits from the first nonzero digit on the left of the number to either: - the last digit (zero or nonzero) on the right if there is a decimal point (e.g. 86.3 or 8.63 X has 5 sig. fig.) Significant Figures Significant Figures: of a number are the digits from the first nonzero digit on the left of the number to either: or - the last nonzero digit of the number if there is no decimal point (e.g., has sig. fig.) the last digit (zero or nonzero) on the right if there is a decimal point (e.g. 86.3 or 8.63 has 5 sig. fig.) the last nonzero digit of the number if there is no decimal point (e.g., has sig. fig.) /8/6 34 Eample.5a Scientific Notation, Significant Figures, and Wh a number is reported as 3. instead of 3??,. 4 : The No. of significant figures in a given value is an indication of the precision with which the quantit is known... 3 The more significant figures the more precise is the value.. 4,3.3 4 4.. -3.. - 3 /8/6 36 6
.5a Scientific Notation, Significant Figures, and So Ahmed Can sa that he is 68.8 kg But Can Not claim that he is 68.8 kg In other words, his mass Could Be somewhere between 68.75 68.85 kg.5a Scientific Notation, Significant Figures, and Mathematical Operation and Significant Figures 4.3.444 =.5,.5,.59, or.59? 4.3 +.444 = 6.7, 6.74, 6.744, or 6.744? The of the final answer can be no greater than the least precise measurement Multiplication and Division: 4.3.444 =.5 (3 sig. fig.) (4 sig. fig.) (3 sig. fig.) lowest sig. fig. No. of sig. fig. of the result = the lowest sig. fig..5a Scientific Notation, Significant Figures, and Mathematical Operation and Significant Figures Eample.53X -3 X 3.5 = 46.3 =?????? 46 (5 sig. fig.) ( sig. fig.) (3 sig. fig.) lowest sig. fig..5a Scientific Notation, Significant Figures, and Mathematical Operation and Significant Figures Addition and Subtraction: The result has lowest 4.3 +.444 =??? No. of digits after D.P digits after the D.P. 3 digits after the D.P. 4.3 +.444 = 6.74 lowest No. of digits after D.P. D.P. = Decimal Point digits after the D.P..5a Scientific Notation, Significant Figures, and Mathematical Operation and Significant Figures Eample:. +.3 = 3. The precision of the final answer can be no greater than the least precise measurement Eample 8.76-7 =.76 =?????? ( digits after D.P.) (No digits after D.P.).5a Scientific Notation, Significant Figures, and Eample.3 +.3 +.66 =.94 =??????.9 (5 digits after D.P.) (4 digits after D.P.) ( digits after D.P.) Eample. - +.4 - =??????. +.4 =.34 =.3?????? =.3 - ( digits after D.P.) (3 digits after D.P.) or. - +.4 - =??????. - +.4 - =.34 - =??????.3 - ( digits after D.P.) ( digits after D.P.) A good rule to follow is to epress all numbers in the highest power of ten. 7
Accurac vs..5a Scientific Notation, Significant Figures, and Mathematical Operation and Significant Figures Note If ou are rounding off numbers in which the digit to be dropped off is a 5 -make the last digit even:.35.4.5. Accurac vs. : YouTube movie In other cases, rounded to the closest number:.34.3.6.3 Accurac vs. In the fields of engineering, industr and statistics: The accurac of a measurement sstem is the degree of closeness of measurements of a quantit to its actual true value. Accurac How well a measurement agrees with an accepted value. The precision of a measurement sstem, also called reproducibilit or repeatabilit, is the degree to which repeated measurements under unchanged conditions show the same results. How well a series of measurements agree with each other. Precise /8/6 45.5b Validating Results If ou have a problem: before solving the problem ou should ask: How should I solve the problem? after solving the problem ou should ask: Is the answer right? How do I know it s right? Validate our answer b: Back Substitution Accurate Precise Not accurate Precise Not accurate Not precise Order-of-Magnitude Estimation Test of Reasonableness 8
.5b Validating Results.5b Validating Results Back Substitution before solving the problem ou should ask: How should I solve the problem? after solving the problem ou should ask: How do I know it s right? Validate our answer Order-of-Magnitude Estimation If ou have a problem: Is the answer right? Test of Reasonableness /8/6 49 /8/6 5.5b Validating Results.5b Validating Results Back Substitution 3. = I found that = Order-of-magnitude: obtain an approimate answer and compare to our solution 3.5..5 = calculator solution is = 4. 3() = I substitute back 3. = I make crude calculation = So I am right = /3 = 4 So I am right Procedure for checking an arithmetic calculation b Order Of Magnitude Estimation is as follows:. Substitute simple integers for all numerical quantities, using powers of ten(scientific Notation) for ver large and ver small quantities.. If a number is added to a second, much smaller, drop the second number in the approimation. 3. Do the resulting arithmetic calculation b hand, continuing to round off intermediate answers. 4. If the correct answer (obtained using a calculator) is of the same magnitude as the estimate, ou can be reasonabl confident that ou haven t made a gross error in calculation..5b Validating Results Order-of-magnitude: Some rules 3.5 4 4. 5 Eample.5-: The Calculator Solution = 4.54 6.89 5, or 43,535 4 4.555 6-5 You were calculating volume flow rate of a stream and ou ended up with the following equation: 54.879 6.4 3.866 6.4 3.345 6 The correct answer is.3 or.3x 3 9
.5b Validating Results 54.879 6.4 3.866 6.4 3.345 6 5 5 6 4 6 5 5 4 6 5 4 5 5 5.5b Validating Results Reasonabilit of the Number: You calculated: the area of a countr to be m. Is that right? the temperature in the street to be 5 o C. Is that right? Atmospheric pressure on top of a mountain to be.8 atm. Is that right?.. B the end of this lecture understand the following:. Estimation of Measured Values: Sample Mean.. Sample Variance of Scattered Data. 3. Process Data Representation and Analsis 4. Two Point Linear Interpolation 5. Fitting a straight line. 6. Fitting nonlinear data. 7. Logarithmic coordinates. ou should be able to The conversion after 4 min X is a random variable changing in an unpredictable manner from one run to another run at the same eperimental conditions. Run 3 4 5 6 7 8 9 X (%) 67. 73. 69.6 67.4 7 68. 69.4 68. 68.7 7. We estimate the true value of X for the given eperimental conditions as the sample mean X= ( X X... X N ) N X= (67.73.... 7.) 69.3% Conversion (X%) 8 75 7 65 6 55 5 Mean (69.3%) 4 6 8 Run
.5c Estimation of Measured Values: Sample Mean The mean is 69.3%. How scattered are the values around their mean value? Run X (%) 3 4 5 6 7 8 9 67. 73. 69.6 67.4 7 68. 69.4 68. 68.7 7. Range: R = X ma X min R = 73. 67.= 6% Sample Variance: S X [(67. 69.3) (73. 69.3)... (7. 69.3) ] 3.7 Sample Standard Deviation: S X [( X X ) ( X X )...( X N X ) ] N.5d Sample Variance of Scattered Data Three quantities -the range, the sample variance, and the sample standard deviation-used to epress the etent to which values of a random variable scatter about their mean value. Range Sample Variance Sample Standard Deviation S X S X 3.7.8 Measure of Scatter:. Sample Variance :. Range: is a much better measure for the degree of scatter. is the difference between maimum and minimum measured variable 3. Sample Standard Deviation S : However, range : a) is the crudest measure of scatter. b) gives no indication whether or not most of the values cluster close to the mean or scatter widel around it. isanothermeasureforthedegreeof scatter epressed as the square root of the sample variance. /8/6 63.5d Sample Variance of Scattered Data Run 3 4 5 6 7 8 9 X (%) 67. 73. 69.6 67.4 7 68. 69.4 68. 68.7 7. The mean is 69.3. How scattered are the values around their mean value? Range: Sample Variance: S X 73. 67. 6% Sample Standard Deviation: S X [( X X ) ( X X )...( X N X ) ] N [(67. 69.3) (73. 69.3)... (7. 69.3) ] 3.7 S X S X 3.7.8 Conversion (X) Conversion (X) Set 8 75 7 65 6 55 5 5 5 Run Set 8 7 6 5 5 5 Run Range: R = 75-55=% Mean: 65% Sample Standard Deviation: S = 5. % Range: R = 75-55=% Mean: 65% Sample Standard Deviation: S = 8. %
Consistent units Length Length Length Dimensionall Homogeneous Consistent units Inconsistent unit Dimensionall Homogeneous Dimensionall Homogeneous Quantities can be added and subtracted ONLY if their units are the same. Dimensional Homogeneit in Equations: Ever valid equation must be dimensionall homogeneous; that is all additive terms on both sides of the equation must have the same dimensions. Consistent in its unit v(m/s) = v (m/s) + a(m/s ).t(s) Dimensionall Homogeneous Length/time = Length/time + Length/time Inconsistent in its unit v(m/s) = v (cm/s) + a(m/s ).t(s) Dimensionall Homogeneous Length/time = Length/time + Length/time v(m/s)= v (cm/s) m + a(m/s ).t(s) cm v(m/s)= v (cm/s)/ + a(m/s Can be fied b appling the ).t(s) appropriate conversion Factor Dimensionless Quantities: Relative Values: e.g. relative densities: (the ratio of densit of a substance to a densit of a standard) 3 D ( cm ) u ( cm / s ) ( g / cm ) Dimensionless Group: e.g. Renolds Number: Re ( g / cms) Eponents: e.g.in Transcendental Functions: e.g. log, ln, ep e, sin Arguments of Transcendental Functions: e.g.xinlog(x)orsin(x) ft ft Log () log( min), or log () m Sin (3.4) sin (3.4 N)
General Procedure for rewriting an equation in terms of new variables having the same dimensions but different units: Eample : Convert 3 s to min.. Eample : Convert 5 m s to ft. 3..... Eample.6 (REF) Microchip etching roughl follows the following relation: 6. 6.. where d is the depth of the etch in micron () and t is the time of the etch in seconds.. What are the units associated with the numbers 6., 6.,.?. Convert the relation so that d becomes epressed in inches and t can be used in min. Solution 6.. d = 6.38 X 4 [ ep(.6. t)] where d is in (in) and t in (min)..... &..................... The operation of an chemical process is ultimatel based on the measurements of process variables: T, P, F flow rates, C i Generall: Indirect Techniques are used to measure those variables. Eample: FTIR measures light absorbance Then, absorbance is correlated to concentration. Fourier Transform Infrared Spectroscop (FTIR) The relationship between C A and light absorbance is determined b calibration eperiment in which solutions of known concentration are prepared and absorbance is measured for each solution. Concentration Absorption 4 35 mole/l 3 3.6 5 9.95 5 5.5 5.7..4.6.8 8.88 Need C A when Abs. =.? Have to do interpolation Conc. of Acetic Anhdride, mole/l Absorption Intensit Need C A when Abs. =.95? Have to do etrapolation 3
Two Point Linear Interpolation Graphical Interpolation Interpolation and Etrapolation Methods Curve fitting Depends on the nature of the relationship between the two variables (e.g. C A and Abs., or an and ).7a Two-Point Linear Interpolation The equation of the line through (, ) and (, ) on a plot of versus is: Concentration Absorption ( ) () mole/l () Eample: 3.6 9.95 What is the value of C A if the absorbance is.? 5.5 8..6 6 3 (93) 4 (.95.6, ) C A = = 5.98 mole/l Conc. of Acetic Anhdride, mole/l 8 6 4 (, )..4.6 Absorption Intensit.7a Two-Point Linear Interpolation The equation of the line through (, ) and (, ) on a plot of versus is: ( ) 5 (, ) Conc. of Acetic Anhdride, mole/l 5 (, )..4.6 Absorption Intensit.7b Fitting a Straight Line. Slope intercept Intercept b Slope a.7b Fitting a Straight Line The equation of straight line is: a slope b intercept a a Eample: (, ) = (.8, 5) ( Fit the C A vs. Abs. data to a straight line equation., ) = (.95, 9) 5 9 a 3.48 4.8.95 35 b 9 (3.5)(.95). (, ) CA 3.5 ( abs). What is C A at abs. =.? C A Conc. of Acetic Anhdride, mole/l 3.48 (.). 6. mole/l 3 5 5 5 ab (, )..4.6.8 Absorption Intensit.7c Fitting Nonlinear Data If the data looks like: 75 5 5 5 5 Series And if I want to find a value of using a value of that is not in the Data Table: Can I use a fitted linear equation? Can I use linear interpolation or etrapolation? 4
.7c Fitting Nonlinear Data You could still fit some nonlinear data to a straight line..7c Fitting Nonlinear Data You could still fit some nonlinear data to a straight line. : 8.5 3.5. 75 5 5 4 6 8 4 6 5 5 45.8 3.8 9.8 3.8 5.8 5.8 33.8 59.8 Series : 8.5 3.5. 6 5 4 3 4 6 8 ( 8.5) 75 5 5 5 5 Series : 8.5 3.5..7c Fitting Nonlinear Data You could still fit some nonlinear data to a straight line. 6 5 4 3 4 6 8 ( 8.5) 75 5 Series 5 5 5.7c Fitting Nonlinear Data Conclusion: If an Two Quantities are related b an equation of the form:. Slope intercept 4 6 8 4 6 45.8 3.8 9.8 3.8 5.8 5.8 33.8 59.8. 4.3.3 6.3.3.3.3 3.3 56.3 Intercept b Slope a.7c Fitting Nonlinear Data Eample The following equations ield straight lines: a b If ( ) is plotted vs. ( ) a b If ( ) is plotted vs. ( ) If ( ) is plotted vs. ( ) ae a b b If ( ln ) is plotted vs. ( ) If ( ln ) is plotted vs. ( ln ).7c Fitting Nonlinear Data Eample The chemical engineering department bought a device that produces fresh water from tap water. The device manual states that the production rate of fresh water [ ]isaffectedbthe ambient temperature [T ( o C)] according to the following equation:.. Can ou using straight-line plot verif this formula and determine the constants a and b. 5
.7c Fitting Nonlinear Data Eample.. T ( o C) 4 45 4.7.4 7.73 9.3.. What 3.6to plot? 4.47 6.35 6.7................7d Logarithmic Coordinates Eample: a ep(b) 8 6 4 5 5 5 ln ()...3..8. 3..79 What to plot? 4 6.79 5 7.83 6 45 3.8 7 4.79 8 33 5.8 9 9 6.8,5 7.8 6,5 8.78.7d Logarithmic Coordinates Eample: 5 ln a ep(b) ln ln a b 3 5 ln ()...3..8. 3..79 What to plot? 4 6.79 5 7.83 6 45 3.8 7 4.79 8 33 5.8 9 9 6.8,5 7.8 6,5 8.78.4 9,84 9.9.4 9,84 9.9.7d Logarithmic Coordinates Eample: a ep(b) ln ln a b Y. Ln..7..7..3.8.3.8.3 Parallel ais.7d Logarithmic Coordinates Eample: a ep(b) 4. 3.5.4.37.7 7.39 3. 4 54.6 Log Y-ais Rectangular X-ais 6
.7d Logarithmic Coordinates Eample: a ep(b) Rectangular Y-ais 6 5 4 3 6 4 4 6 Rectangular X-ais Log Y-ais Rectangular X-ais.7d Logarithmic Coordinates Eample (Problem.3 (a)) Aplotof versus ields a straight line on a semilog plot (the vertical ais is a logarithmic scale while the horizontal ais is a rectangular scale). The line passes through (8.68,.) and (,). Sketch the plot and calculate the equation (). a ep(b) ln ln a b Solution: =4e -.69.7d Logarithmic Coordinates = 3.97e.689 R² = Eample (Problem.3 (a)) Aplotof versus ields a straight line on a semilog plot (the vertical ais is a logarithmic scale while the horizontal ais is a rectangular scale). The line passes through (8.68,.) and (,).. Sketch the plot and. calculate the equation ().. 3 4 5 6 Solution: =4e -.69 4 6 8... :.,. &,.......... Use Eq. to calculate a......7d Logarithmic Coordinates Eample: Log Y-ais Use Eq. for checking Finall give the equation with calculated values of a and b.. Log X-ais 7
..... 3 4 5 6 7 8 9. 3 4 5 6 7 8 9.7e Fitting a Line to a Scattered Data 4 3..4.6.8 3..3.5.7 The fitting of a straight line to a series of versus data points (linear regression) can be accomplished using different techniques:. method of least squares, R. robust regression 8