Mass and Energy Balances - Chapter II 1

Chapter 2. Introduction to Engineering Calculations

6. Handling Numbersi. How do handle extremely large and small numbers?

All numbers can be expressed in scientific notation, as N10n,where N is a number between 1 and 10, and n is a positive/negative integer, named the «exponent». See Ex. 1

ii. How do handle arithmetic operations in scientific notation?A) Addition/Subtraction: Write all the terms with the same exponent.

See Ex. 2

B) Addition/Subtraction: Multiply/divide N1 by N2.

Add/substract n1 by n2. See Ex. 3

Mass and Energy Balances - Chapter II 2

iii. Is it possible to obtain exact values of quantities? Except integer quantities, generally NO! For this reason, it is important to indicate the margins of error in a

measurement by clearly indicating the number of «significant figures». Significant figures are the meaningful digits in a measured or calculated

quantity.Ex. A measurement of 1.6 kg indicates that the measured quantitity is between

(and including) 1.5500000000… kg 1.650000000… kg, in which there are infinite numbers of zeros following both numbers.

iv. Guidelines for Significant FiguresA) Any nonzero digit is significant. See Ex. 4B) Zeros between nonzero digits are significant. See Ex. 5C) Zeros to the left of the first nonzero digit are NOT significant. See Ex. 6

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v. Rule of addition/subtraction Answer can be at most PRECISE as the least precise term. Answer cannot have more digits to the right of the decimal point than

either of the original numbers. See Ex. 10vi. Rule of multiplication/division

The number of significant figures in the final answer is equal to the smallest number of significant figures in the original numbers. See Ex. 11

Note that exact numbers are considered to have an infinite number of significant figures. See Ex. 12,13

D) If a number is greater than 1, all zeros to the right of the decimal point are significant. See Ex. 7

E) For numbers that do not contain decimal point, the zeros after the last non zero digit MAY OR MAY NOT BE significant. See Ex. 8,9

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Note on rounding off:Rounding off is a kind of estimating. The simple rule of rounding off is as follows: Find the place of the rounding digit and look at the digit just to the right of it. If that digit is < 5, do not change the rounding digit but drop all digits to the

right of it. If that digit is < 5, add one to the rounding digit and drop all digits to the right

of it. If that digit is = 5, round the number so that rounding digit will be even.Ex. Round off the following numbers to the following number of sig. fig.’s:a. 71.1346 to four s.f. 3 is the rounding digit, and 4 < 5 71.13b. 1.68788 to three s.f. 8 is the rounding digit, and 7 > 5 1.69c. 94.8511 to three s.f. 8 is the rounding digit, and 5 = 5 94.8d. 94.7511 to three s.f. 7 is the rounding digit, and 5 = 5 94.8

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v. Significant Figures in Chain Calculations Theoretically, do NOT round off at intermediate steps. Practically, do NOT round off too much at intermediate steps. See Ex. 14 If the intermediate results are to be reported, i.e. for engineering

purposes, then round off these results.

Every valid equation must be DIMENSIONALLY HOMOGENEOUS, i.e. all additive terms on both sides must have the same dimensions.

Note that dimensional homogeneity is a necessary but not sufficient condition for an equation to be valid.

Ex. Is the following equation dimensionally homogenous?

6. Dimensional Homogeneity

Mass and Energy Balances - Chapter II 6

√ 𝐸𝑚=𝑣+𝑔𝑡 ,

√ 𝐸𝑚=√ 𝐹𝐿𝑚 =√𝑚( 𝐿𝑡 2 )𝐿𝑚

=√ 𝐿2𝑡 2 =𝐿𝑡

𝑣=𝐿𝑡 𝑔𝑡=

𝐿𝑡 2𝑡=

𝐿𝑡

• All terms on the L.H.S. and R.H.S. have the same dimensions, L/t. The above eqn is dimensionally homogeneous.

Dimensionless quantity (group)A pure number (such as ½ of KE = ½mv2), or a multiplicative combination of variables with no net dimensions.Ex. The most «famous» dimensionless quantity in ChE is Reynold’s number,

𝑚𝐿3

𝐿(𝑝𝑖𝑝𝑒𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 )𝐿𝑡

𝑚𝐿𝑡

( 𝑓𝑙𝑢𝑖𝑑𝑣𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦)