Tech Talk: (1) Instrument Measurement Basics

I. The SI System of Measurement Units

The SI system has been adopted by most countries in the developed world, though within English-speaking countries, this has not been universal. The United Kingdom has officially adopted a partial metrication policy, with no intention of replacing imperial units entirely. The United States and Canada do not use metric units outside of science, medicine and the government and extensively use imperial units throughout the engineering disciplines.

The core of the SI system is built on seven base units defined in an absolute way without reference to any other units.

Property	Name	Symbol
Length	Metre	m
Mass	Kilogram	kg
Time	Second	s
Electrical current	Ampere	A
Temperature	Kelvin	K
Luminous intensity	Candela	cd
Quantity	Mole	mol

SI derived units with special names and symbols acceptable in SI

Property	Name	Symbol	SI units
Plane angle	Radian	rad
Electrical capacitance	Farad	F	m−2 kg−1 s4 A2
Electrical charge	Coulomb	C	A s
Electrical conductance	Siemens	S	m−2 kg−1 s3 A2
Electrical inductance	Henry	H	m2 kg s−2 A−2
Electrical potential	Volt	V	m2 kg s−3 A−1
Electrical resistance	Ohm	Ω	m2 kg s−3 A−2
Force	Newton	N	kg ms−2
Frequency	Hertz	Hz	s−1
Power or radiant flux	Watt	W	kg m2 s−3
Pressure	Pascal	Pa	kg/(ms2) = (N/m2)
Radioactivity	Bequerel	Bq	s−1
Work, energy, heat	Joule	J	m2 kg s−2

Commonly used SI derived units described in terms of acceptable SI units

Property	Symbol	SI base units
Acceleration	m/s2	m s−2
Area	m2	m2
Coefficient of heat transfer	W/(m2 K)	kg s−3K−1
Concentration	mol/m3	mol m−3
Density (mass density)	kg/m3	kg m−3
Electrical charge density	C/m3	m−3 s A
Force	N or J/m	m kg s−2
Heat flow rate	W or J/s	m2kg s−3
Magnetic field strength	A/m	A m−1
Molar energy	J/mol	m−2 kg s−2 mol−1
Molar entropy	J/(mol K)	m−2 kg s−2 K−1 mol−1
Moment of force (or torque)	N m	m2 kg s−2
Moment of inertia	kg m2	kg m2
Momentum	kg m/s	kg m s−1
Power	kW	10−3 m2 kg s−3
Pressure	kPa	10−3 m−1 kg s−2
Specific heat capacity	J/(kg K)	m2 s−2 K−1
Specific volume	m3/kg	m3 kg−1
Stress	MPa	10−6 m−1 kg s−2
Surface tension	N/m	kg s−2
Thermal conductivity	W/(m K)	m kg s−2 K−1
Torque	N m	m2 kg s−2
Velocity	m/s	m s−1
Viscosity, absolute or dynamic	Pa s	m−1 kg s−1
Viscosity, kinematic	m2/s	m2 s−1
Volume	m3	m3
Work	J or N m	m2 kg s−2

II. SI Prefixes

Numbers in the form 10⁷ cannot be conveniently represented in computer programs, so the scientific notation is used where the 10⁷ is now E07. Note E is not related to the mathematical constant e or the exponential function. The following conventions avoid errors in allocating the correct number of zeros where fractions such as 1/10 = 10⁻¹, 1/100 = 10⁻² and so on.

Factor		Prefix	Symbol
1018	E18	Exa	E
1015	E15	Peta	P
1012	E12	Tera	T
109	E09	Giga	G
106	E06	Mega	M
103	E03	Kilo	k
102	E02	Hecto	h
101	E01	Deca	da
10−1	E-01	Deci	d
10−2	E-02	Centi	c
10−2	E-03	Milli	m
10−6	E-06	Micro	µ
10−9	E-09	Nano	n
10−12	E-12	Pico	p
10−15	E-15	Femto	f
10−18	E-18	Atto	a

Note 1 Caution: Refinery industry sometimes uses MM to signify 10⁶.

III. Unit Equation

In an equation, the units on each side are the same and should be checked for consistency. Consider an object uniformly increasing its speed (m/s) from u to v in t s, (v − u)/t represents the change in speed over t seconds defined as acceleration a (m/s²).

$$\begin{array}{ccc}\text{Acceleration formula}& \text{a}=(\text{v}-\text{u})/\text{t}& \text{LHS m}/{\text{s}}^2(\text{m}{\text{s}}^{-2})=\text{RHS m}/\text{s}\times 1/\text{s}=\text{m}/{\text{s}}^2\end{array}$$

$$\begin{array}{ccc}\text{Distance travelled s in time t}& \text{s}=\text{ut}+\text{t}(\text{v}-\text{u})/2& \text{LHS m}=\text{RHS}(\text{m}/\text{s}\times \text{s})+(\text{s}\times \text{m}/\text{s})=\text{m}\end{array}$$

$$\begin{array}{ccc}\text{substituting for}(\text{v}-\text{u})& \text{s}=\text{ut}+\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.{\text{at}}^2& \text{LHS m}=\text{RHS}(\text{m}/\text{s}\times \text{s})=(\text{m}/{\text{s}}^2\times {\text{s}}^2)=\text{m}\end{array}$$

IV. Unit Conversion

Unit converters are available from many sources so are not shown here. A typical example for energy is shown. The unit to be converted from is multiplied by the factor shown in column with the desired unit.

From	To
	Btu	Joule	kWh	therm
Btu	1	1.055E03	0.2931E−03	10E−06
Joule	0.948E−03	1	0.2778E−06	9.48E−09
kWh	3.412E03	3.6E06	1	34.12E−03
therm	100E03	105.5E06	29.31	1

1000 Btu = 10³ Btu = 1E03 Btu = 1.055E06 joule = 0.2931 kWh = 10E−03 therm

A. Temperature Conversion

To convert from °F to °C we have: °F = 9/5 °C + 32

A temperature of −10 °C is equivalent to °F = (1.8 × −10) + 32 = 14

B. Pressure Conversion

Absolute pressure p_a is the pressure above a total vacuum and gauge pressure p_g is the pressure above or below atmospheric pressure p_atm giving:

$$\begin{array}{l}{\text{p}}_{\text{a}}={\text{p}}_{\text{g}}+{\text{p}}_{\text{atm}}\text{for}{\text{p}}_{\text{g}}>{\text{p}}_{\text{atm}}\\ {\text{p}}_{\text{a}}={\text{p}}_{\text{g}}-{\text{p}}_{\text{atm}}\text{for}{\text{p}}_{\text{g}}<{\text{p}}_{\text{atm}}\end{array}$$

At atmospheric pressure p_atm (kg/m²), the absolute pressure p_a (kg/m²) at the bottom of a column of liquid with a density ρ (kg/m³) and height H (m) is:

$${\text{p}}_{\text{a}}={\text{H}}_{\text{ρ}}+{\text{p}}_{\text{atm}},\text{check units RHS}=\text{kg}/{\text{m}}^3\times \text{m}=\text{kg}/{\text{m}}^2=\text{LHS}$$

To convert p_a from kg/m² to bar:

We know 1 kg/cm² = 0.98065 bar and 1 cm² = 0.0001 m² giving 10⁴ kg/m² = 0.98065 bar, and 1.0197 × 10⁴ kg/m² = 1 bar leading to p = Hρ/(1.0197 × 10⁴) bar. And p_atm = 1.01325 bar, so a p_g of 5 barg is equivalent to p_a of 6.01325 bar and a vacuum p_g of 0.5 barg is equivalent to p_a of 0.51325 bar.

V. Accuracy

The accuracy of a measurement relates to the nearness of that measurement relative to a National or International Standard. Measuring instrument accuracy is expressed in a variety of ways:

• As a percentage of the full-scale reading at any reading;

• As a percentage of the calibrated span at any reading;

• As a percentage of the actual reading.

Consider a flow transmitter, with a calibrated range of 100–1000 Nm³/h, reading 800 Nm³/h with an accuracy of 1% (basis unspecified):

• As a percentage of the full-scale reading at any reading;

  $$\text{Full scale}=1000{\text{Nm}}^{\text{3}}/\text{h giving}\pm 1\%\text{of}1000=\pm 10{\text{Nm}}^3/\text{h};$$

• As a percentage of the calibrated span at any reading;

  $$\text{Calibrated span is}1000-100=900{\text{Nm}}^3/\text{h}\pm 1\%\text{of}900=\pm 9{\text{Nm}}^3/\text{h}$$

• As a percentage of the actual reading at any reading

  $$1\%\text{of}800=\pm 8{\text{Nm}}^3/\text{h}$$

It can be seen from the above that the best accuracy is obtained as a percentage of the actual measurement reading.

VI. Measurement Loop Accuracy

Where a number of devices are connected in series, the overall loop accuracy can be obtained by the root mean square method ensuring all device accuracies are on the same basis.

$$\text{Primary measuring element accuracy}=\pm 2\%$$

$$\text{Transmitter}/\text{Converter accuracy}=\pm 1\%$$

$$\text{Analogue input card channel accuracy}=\pm 0.5\%$$

$$\mathbf{\text{Loop accuracy}}=\sqrt{({\mathbf{2}}^2+{\mathbf{1}}^2+\mathbf{0}\mathbf{.}{\mathbf{5}}^2}=\pm \mathbf{2}\mathbf{.}\mathbf{29}$$

VII. Repeatability or Reproducibility

This is an important factor to consider for production processes in the application of a measuring instrument over its service life. Calibration errors and drift can be caused by physical changes affecting the sensor and its electronics. Factors to consider are ambient conditions, the effect of process fluid adhesion and wear or stress on wetted or moving parts may all influence the measurement uncertainty over time, often indicating the need for frequent calibration checks.

The repeatability coefficient is a precision measure which represents the value below which the absolute difference between two repeated test results may be expected to lie with a probability of 95%. The standard deviation under repeatability conditions is part of precision and accuracy.

VIII. Instrument Scaling

Electronic transmission in instrumentation almost universally uses an elevated zero signal of 4 mA dc with a span of 16 mA dc giving a full-scale signal of 20 mA dc. Below are some examples of applied scaling:

• A temperature transmitter calibrated 50–250 °C is reading 75 °C

  $$\text{The}\%\text{calibrated range}=100\times 75/(250–50)=37.5\%$$
  $$\text{The transmitted signal will be}(20–4)\times 37.5/100+4=10\text{mA}.$$

• A flow transmitter, with an orifice plate primary element, is calibrated 0–2500 mm wg for 0–25,000 kg/h has a reading of 12,500 kg/h on a receiving device having a square root scale. The orifice plate differential pressure (dp) and the transmitted signal are determined as follows:

  At maximum flow (W_Max) the orifice plate dp is 2500 mm wg (referred to as h_Max). For a transmitter signal of 4 − 20 mA the following relationship applies:

  $$\begin{array}{l}\text{W}/{\text{W}}_{\text{Max}}=\sqrt{(\text{h}/{\text{h}}_{\text{Max}})}\text{which leads to h}={\text{h}}_{\text{Max}}{(\text{W}/{\text{W}}_{\text{Max}})}^2\\ \text{h}=2500{(12,500/25,000)}^2=625\text{mm wg}\end{array}$$

The resulting transmitter signal is therefore = (0.25 × 16) + 4 = 8 mA.