Field of study: Mining and Power Engineering
Course: Fuel Processing
Jan Drzymala
CALCULATIONS
SI unit
International system of units, in French System Internationale d'Unit, usually abbreviated to SI, consists of seven base quantities.
SI base units
Base quantity / unit / symbolLength / meter / m
Mass / kilogram / kg
Time / second / s
Temperature / kelvin / K
Amount of substance / mole / mol
Electric current / ampere / A
Luminous intensity / candela / cd
Remark: temperature in kelvin (K) from that in Celsius (°C) can be calculated using the equation: K = °C + 273.15.
The base SI units provide numerous derived quantities shown in the next table.
Examples of SI derived units
Quantity / Unit / Symbol / DefinitionForce / newton / N / 1 kg·m·s–2
Pressure, stress / pascal / Pa / 1 kg·m–1·s–2
Energy, work, quantity of heat / joule / J / 1 kg·m2·s–2
Electric charge / coulomb / C / 1 A·s
Electric potential / volt / V / 1 kg·m2·s–3·A–1
Power / watt / W / 1 kg·m2·s–3
Frequency / herz / Hz / 2·rad·s–1
Electric capacity / farad / F / 1 kg–1·m–2 ·s4·A2
Area / square meter / m2
Volume / cubic meter / m3
Speed, velocity / meter per second / m/s
Acceleration / meter per second squared / m/s2
Derived units can be expressed as combination of other units.
Examples of relations between units of SI
Quantity / Relation / Quantity / Relationfarad / F = C/V / amper / A = W/V
volt / V = J/C / pascal / Pa = N/m2
joule / J = N·m / wat / W = J/s
Many relations and laws contain physical constants. Selected useful physical constants are given in the next table.
Selected physical constants (CRC, 1997/98)
Constant or number / Symbol / ValueAvogadro number / NA / 6.02552·1023 molecules per mole
Boltzmann constant / k / 1.38054·10–23 J/K
Elemental electric charge / e / 1.60210·10–19C
Dielectric permeability of vacuum / 0 / 8.854·10–12 F/m (*), (0 = 0–1c–2)
Magnetic permeability of vacuum / 0 / 12.5666370·10–7 N/A2, 0= 4·10–7
Speed of light / c / 299 792 458 m/s
Planck constant / h / 6.626 0755 10–34 J·s
Faraday constant / F / 96 485 309 C/mol
Gas constant / Rg / 8.3145 J·mol–1·K–1
Number pi / / 3.14
*In literature equivalent units can be encountered: F·m–1 = C·V–1·m–1 = C2·J–1·m–1 = C2·N–1·m–2.
Sometimes the values of physical quantities are either small or large. Then, they can be expressed as multiples and submultiples of the base units.
SI prefixes used to form SI units
Name / Factor / Symbol / Name / Factor / Symbolyotta / 1024 / Y / deci / 10–1 / d
zetta / 1021 / Z / centi / 10–2 / c
exa / 1018 / E / mili / 10–3 / m
peta / 1015 / P / micro / 10–6 / µ
tera / 1012 / T / nano / 10–9 / n
giga / 109 / G / pico / 10–12 / p
mega / 106 / M / femto / 10–15 / f
kilo / 103 / K / atto / 10–18 / a
hecto / 102 / H / lepto / 10–21 / z
deka / 101 / Da / yocto / 10–24 / y
Remark: SI base units of mass is kg while initial unit for multiplication is gram
Dealing with numbers
Parameter and constants used in physics and chemistry are ether dimensionless or carry a dimension. The value of a given parameter having a dimension is a product of multiplication of number and unit. For instance density of quartz is 2.65 g/cm3.
Determination of parameter value is usually accomplished by measurement. The value is almost never exact but with certain precision and accuracy. To show accuracy of measurement it is essential to show the average value and the range within the parameter value can occur. The best tool for this is statistics. Typical statistical analysis of measured parameter values provide informationshown in the table on Statistics of measured parameters.
Statistics of measured parameters, ,
Statistics / / / number of data points, n / 93 / 93 / 93
minimum value / 1.52 / 21.49 / 0.15
maximum value / 1.97 / 27.08 / 0.30
average / 1.73 / 24.85 / 0.22
standard deviation, s / 0.092 / 0.826 / 0.029
skewness* / 0.053 / -0.664 / 0.237
outliers / - / exp. 28 / -
*Skewness less than 1.5 indicates nearly normal distribution of variables
To show the value of a parameter, the average value should be given along with the range. Typically the range is within one standard deviation , meaning that 67% of measured data points were in it.
=1.73±0.09
or av ± F.
In practice and for simplicity the value of a parameter and it range is indicated by the number itself, that is the last digit of the number. It is assumed that the accuracy of the measurements is such that only the last digit of the number fluctuates
=1.7 (meaning that 1.7±0.1)
that is
=from 1.6 to 1.8.
Thus, we accept the convention that the last digit is not certain within ±1. However, when such simplification is not acceptable, then the parameter value should be shown as average value plus the error.
In the case when the value of a parameter is not measured but calculated from other parameter, simple and practical rules should be known and applied.
Rounding up numbers
The rules of rounding up numbers
1. When the number subjected to rounding up has the final digit or digits from 0 to 4 or 0 to 49, or 0 to 499, etc., this part of the number is rejected while when the rejected part is 6, 7, 8, 9 (or 51 to 99 or 501 to 999, etc.) the last digit is increased by 1. For instance rounding up number 3.73 down to 2 meaning digits provide 3.7 while number 5.577 is reduced to 5.6.
2. When the number subjected to rounding up has final digit 5 or 50 or 500 etc., the number is rounded up to the nearest even number. For instance 6.650 = 6.6 but 6.75 = 6.8.
Adding and subtracting
Using these functions when dealing with not too many numbers (two or three) the accuracy of the resulting number is the same as that number of the sum which was the least accurate. Least accurate is the number which has at the last digit closer to the left in relation to the one. For instance number 11.71 is during addition less accurate than 0.328. Also 3.31105 is less accurate than 81.
Multiplying and dividing
Using these functions we should apply an approximation that the result should carry such a number of meaning digits as the number having the smallest number of meaning digits.
Meaning digits are all digits starting from the first one not being zero until the last after the decimal point. For instance number 0.0240010 has 6 meaning digits while 24.50000 has 7 digits. When the number has no digits after the decimal point, and when the zeros are not meaningful then the number must be shown in scientific notation. For instance number 158000 should presented as 1.58105 (3 meaning digits), 1.580 105 (4 meaning digits).
Thus, when multiplying 1.6by2.32, the results 3.712. Taking into account that number 1.6 has smaller number of meaning digits (2) than number 2.32 (3 digits), the resulting number should have 2 digits, therefore the final result is 3.7:
A = 1.6 2.32 3.712 = 3.7
Logarithm and roots
When calculating logarithm and roots the number of meaning digits in the calculated value should be the same as in the original number.
Examples
a = 0.745 + 22.352121 + 3.33 + 2.0 ~ 28.427121a = 28.4
b = 172.4 + 0.004 ~ 172.404b = 171.4
c = 9.825 - 9.823 ~ 0.002 c = 0.00
d = 14.2 2.2 233.14 ~ 7283.2936d = 7.3103
e = 14.03 0.00120 1.552 ~ 0.026129472e =0.0261
f = 743 0.11331 ~ 83.16954f = 83.2
log 0.30 = -0.52
log 2.00 1012 =12.301.
Attention: The rules are valid for displaying final number. Do not round up numbers which are used for further calculations.
Project co-financed by European Union within European Social Fund