Math and Science Background for

Process Instrumentation

1.0 Introduction to Process Control Instrumentation 1.1 A Brief History of Control Systems and the Use of Feedback

1.2 Industrial Control System Types

1.3 Batch Processing vs. Continuous Manufacturing Processes

1.3.1 Batch Processing

1.3.2 Continuous Manufacturing Processes

2.0 Some Math Background

2.1 Review of Algebra and Common Algebraic Manipulations 2.1.1 Introduction to Algebra and Its Uses

2.1.2 Solving for One Unknown

2.1.3 Order of Operations

2.1.4 Manipulating Equations

2.2 Review of Graphs 2.2.1 Line Graphs

2.2.2 Scatter Plots

2.2.2.1 Best Fit Straight Line and Positive (Direct) Relationships

2.2.2.2 Best Fit Straight Line and Negative (Inverse) Relationships

2.2.2.3 Best Fit Curved Line Fit Relationships

2.3 Linear Equations, Linear Interpolation and Regression 2.3.1 Linear Equations Review

2.3.2 Linear Interpolation

2.3.3 Linear Regression

2.3.3.1 Linear Regression Equations – Least Squares Method

2.4 Some Common Equations 2.4.1 Area Equations

2.4.2 Volume Equations

2.5 Unit Conversions 2.5.1 Length Unit Conversions and Unit Analysis

2.5.2 Area Unit Conversions 2.5.3 Volume Unit Conversions 2.5.4 Mass Unit Conversions 2.5.5 Temperature Unit Conversions

3.0 Test Instruments Used with Process Control Instrumentation 3.1 Pressure Calibrators

3.2 Temperature Calibrators

3.3 Current Calibrator

4.0 Some Science Background and Review 4.1 Some Definitions

4.1.1 Matter

4.1.2 Mass

4.1.3 Energy

4.1.4 Power

4.1.5 Force

4.1.6 Weight

4.1.7 Solid

4.1.8 Fluid

4.1.9 Liquid

4.1.10 Gasses

4.1.11 Vessels

4.1.11.1 Closed Vessel

4.1.11.2 Open Vessel

4.2 Force of Gravity

4.3 Density

4.4 Relative Density and Specific Gravity

4.5 Pressure

4.5.1 Hydrostatic Pressure

4.5.2 Absolute Pressure, Gage Pressure and Differential Pressure

4.5.2.1 Absolute Pressure

4.5.2.2 Gage Pressure

4.5.3 Inches of Water Column, inH2O (Pressure)

4.5.4 Hydrostatic Pressure and Closed Vessels

4.6 Archimedes Principal

5.0 Basic Process Instrumentation Concepts and Terminology

5.1 Process Variables and Process Instruments

5.2 Process Instrumentation

5.3 P&IDs

5.4 Standard Signals (The Short Story)

5.6 Percent (%) Value of a Process Variable

5.7 Elements/Sensors

5.8 Transmitters

5.8 Transmitters (cont.)

5.9 Indicators

5.10 Recorders

5.11 Standard 4-20mA Signals and Systems

5.12 Standard 3-15 PSI Signals and Systems

1.0 Introduction to Process Control Instrumentation This class is concerned with the instrumentation used to measure, indicate and control the

process variables affecting a product as it is manufactured. The instrumentation covered

in this course is collectively referred to as process and control instrumentation.

1.1 A Brief History of Control Systems and the Use of Feedback

One of the most notorious early examples of an automatic control system is the float

regulator used for liquid level control; this was developed in the 1740s. The float

regulator is an example of a control system that employs the use of feedback. A float

regulator uses a float as a sensor to provide level (height) feedback to the mechanism that

controls the level of a tank.

Water

Source

Flow

Valve

Float

Drain

Figure 1.1.1 Float Regulator

Between World War I & II, the use of feedback control methods made automated

equipment became more common, often in place of manually operated equipment.

World War II lead to rapid advances feed back control systems due to the high level of

sophistication required by military weapons. After WW II much of this knowledge and

technology was transferred to the field of industrial control systems.

1.2 Industrial Control System Types The types of industrial control systems found in manufacturing can be separated into two

main groups, motion control systems and process control systems. Motion control

systems are concerned with measuring, indicating and controlling (regulating) the

position and velocity of a product as it is being manufactured. Process control systems

are concerned with controlling (regulating) all other variables affecting a product as it is

manufactured. The four process variables that we will focus on most in this class are

temperature, pressure, level and flow.

1.3 Batch Processing vs. Continuous Manufacturing Processes Manufacturing processes can be categorized as either batch processes or continuous

processes.

1.3.1 Batch Processing A batch process is a manufacturing process where a timed sequence of operations is

performed on a product as it is manufactured.

Example 1.3.1.1 Making Apple Cider

1. Grind up the apples.

2. Squeeze the juice out of the ground apples.

3. Pour the apple juice into wooden barrels and add yeast, then allow contents of

barrel to ferment until completed (until it stops bubbling).

4. Transfer cider to kegs ready for shipment.

Apples

Grinder

Ground

Apples

1. Grind up the apples. 2. Squeeze the juice out of the ground apples.

Cider

Press

3. Pour the apple juice into wooden

barrels and add yeast, then allow

contents of barrel to ferment until

completed (until it stops bubbling).

Raw

Cider

Wooden

Barrel

Raw

Cid

er

Wooden

Barrel Keg

4. Transfer cider to kegs ready for shipment.

Figure 1.3.1.1 Batch Process Example, Apple Cider Production

1.3.2 Continuous Manufacturing Processes In a continuous manufacturing process, one or more operations are performed on a

product simultaneously as it moves continuously through a process as it is manufactured.

Example 1.3.2.1 Making Paper

When making paper, each of the following process operates on the source

material simultaneously as paper is produced. Debarked logs continuously enter

the chipper where they arc chipped. The chips pass through a digester where the

wood chips are chemically broken down into pulp. The resulting pulp is washed,

bleached and cleaned. The clean pulp enters the paper machine where it is

converted from a slurry to paper which is wound using a winder.

Chipper

Digester

Washer

Bleaching

Tower

Cleaning

Stage

Head

Box

Winder

Dryer

Paper Machine

Debarked Logs

Figure 1.3.2.1 Continuous Process Example, Paper Production

2.0 Some Math Background

2.1 Review of Algebra and Common Algebraic Manipulations

This section is a Brief Review of Algebra. 2.1.1 Introduction to Algebra and Its Uses Math is used to solve problems in many professions including technical fields. It is

important to express a mathematical concept in a way that can easily be documented and

communicated to other professionals in the same field. One way to document a

mathematical concept takes the form of a word equation.

Example 2.1.1.1: While working for a food service, a word equation to

calculate the total quantity of fruit purchased by a customer might be written as

follows:

Fruit = Apples + Mangos + Bananas (equation 2.1)

Given a customer who purchased 3 apples, 4 mangos and 0 bananas, the quantity

of fruit can be calculated by replacing the words in the word equation with the

quantity of each item:

Fruit = 3 + 4 + 0

Fruit = 7

From this point on each word in a word equation will be referred to as a variable because

it represents a numeric value which is not fixed. A variable is called an unknown

variable unless it is given a value which replaces the word with a number. Since it is

often cumbersome or inconvenient to write large word equations the variables are

typically abbreviated to a single alphabetical symbol in place of the entire word in the

equation.

Example 2.1.1.2

Given: F = Fruit, A = Apples, M = Mangos, B = Bananas and equation 2.1

Therefore:

F = A + M + B (equation 2.2)

2.1.2 Solving for One Unknown When asked to solve an equation with one unknown it is the task of the problem solver is

to somehow calculate the value of the unknown variable. In the some cases such as

example 2.1.1.1 this may be as simple as replacing all the know values of variables into

an equation and calculating the result.

Example 2.1.2.1

A = B + C, B = 2 and C = 3

A = 2 + 3

A = 5

Example 2.1.2.2

z = u - v, u = 2 and v = 6

z = 2 - 6

z = -4

Example 2.1.2.3

w = x - y, x = 1 and y = -4

w = 1 – (-4)

w = 5

Example 2.1.2.4

i = j ∙ k, j = 3 and k = 8

i = 3 * 8 (* and ∙ mean multiply)

i = 24

Example 2.1.2.5

P= Q / r, Q = 12 and r = 4

P = 12 / 4 (/ means divide)

P = 3

2.1.3 Order of Operations

When solving an equation with one unknown variable, but more that two known

variables and/or numbers, it is traditional to perform your calculations from left to right if

all of the operators are the same:

Example 2.1.3.1

a = b + c + 7 + 1, b = 5 and c = 6

a = 5 + 6 + 7 +1

a = 11 + 7 + 1

a = 18 + 1

a = 19

Example 2.1.3.2

a = b ∙ c ∙ 4 ∙ 5, b = 2 and c = 3

a = 2 ∙ 3 ∙ 4 ∙ 5

a = 6 ∙ 4 ∙ 5

a = 24 ∙ 5

a = 120

When more than type of operation is present, the order that they are performed may be

different from simply moving from left to right. Some operations are more “important”

and are to be performed before lower ranking operations. The order of operations is

determined by the following rules in the following order:

1) Perform all operations inside brackets.

2) Perform all multiplies and divides from left to right.

3) Perform all additions and subtractions from left to right.

It does not matter if upper or lower case letters are used to write an equation, just take

care not to change the case of a letter in an equation while solving a problem in order to

avoid confusion.

Example 2.1.3.3

u = v + w x, v = 3, w = 5, x = 4

u = 3 + (5) (4) (A brackets mean multiply by the following number.)

u = 3 + 20

u = 23

Example 2.1.3.4

u = (v + w) x, v = 3, w = 5, x = 4

u = (3 + 5) (4) (Calculate the contents of the brackets first.)

u = (8) (4) (The brackets mean multiply by the following number.)

u = 32

2.1.4 Manipulating Equations

Often the unknown variable does not appear on the left hand side of an equation as we

would like. In this case the equation needs to be manipulated before it can be solved.

For equations that only involve sums and/or differences, the known values are eliminated

from the side of the equation that contains the unknown value. This is done by either

adding or subtracting each known value from the both sides of the equation that in a way

that separates it from the unknown variables.

Example 2.1.4.1

Given: u = v + w, u = 6, v=? and w = 9

Find: v=?

Solution 1:

u = v + w (Substitute the known values.)

6 = v + 9 (Next, subtract 9 from both sides of equation.)

6 – 9 = v + 9 – 9 (The struck through 9s are eliminated from the next line.)

-3 = v + 0 = v

Now rewrite the equation with the unknown on the left:

v = -3

or

Solution 2:

u = v + w

Subtract w from both sides:

u – v = v + w – w (All struck through w are eliminated from the next line.)

u – v = w (Now rewrite the equation with the unknown on the left.)

w = u - v (Substitute the known values.)

w = 6 – 9

If the unknown variable in an equation is multiplied by a coefficient (multiplier) it will be

necessary to divide both sides of the equation by this coefficient.

Example 2.1.4.2

Given: y = m x + b, y = 12, m=3, b = 6 and x = ?

Find: x=?

Solution 1:

y = m x + b (Substitute the known values.)

12 = 3 x + 6

12 - 6 = 3 x + 6 - 6 (Subtract 6 from both sides of equation.)

6 = 3x (Now rewrite the equation with the unknown on the left)

3x = 6

3 x / 3 = 6 / 3 (divide by 3 on both sides)

x = 2

or

Solution 2:

y = m x + b (Subtract b from both sides.)

y – b = m x + b – b (All struck through b are eliminated from the next line.)

y – b = m x

(y – b) / m = m x / m = x (Divide both sides by m, then rearrange equation.)

x = (y – b) / m

Substitute the known values.

x = (12 – 6) / 3 = 6 /3 (Substitute the known values.)

x = 2

2.2 Review of Graphs Graphs are a pictorial representation of the relationship between variables. Graphs

typical are intended to show how a quantity known as the dependent variable occurs as a

function of the independent variable. Graphs can be used to represent data obtained

experimentally or from an equation.

2.2.1 Line Graphs A line graph is used to visually represent the relationship between two variables. The

parts of a line graph include:

X-Axis - A horizontal line which is marked with numbers representing values

between the maximum and minimum values of the independent variable.

Y-Axis - A vertical line which is marked with numbers representing values

between the maximum and minimum values of the independent variable.

Origin - The origin is the point on the graph where the x-value and where the y-

axis both equal zero. General this is where X-Axis and the Y-Axis cross each

other, but there can be exceptions.

Points - Points are used to visually represent the relationship between a particular

value of one variable and value of a second variable.

Lines - The straight lines that connect the points on the graph.

Graph of X vs. Y

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

X-AXIS

Y-A

XIS

Figure 2.2.1 Line Graph

2.2.2 Scatter Plot A scatter plot is used to visually represent the relationship between two variables,

especially when exploring relationship between them and when it is assume that one

variable is dependent on the other. A scatter plot is similar to a line graph except that

there are no lines connected point to point on the graph.

Scatterplot of X vs. Y

0

1

2

3

4

5

6

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X-AXIS

Y-A

XIS

Figure 2.2.2 Scatter Plot

Whenever the points of a scatter plot seem form an approximate straight line it is said that

these variables are correlated, such as the point on the graph above.

2.2.2.1 Best Fit Straight Line and Positive (Direct) Relationships If the points on the scatter plot seem to form a straight with a positive slope, the graph is

said to show a positive or direct relationship between its data points the line. In this case

it is common to add a line called a best fit line which must appear to fit evenly between

the data points. A best fit straight line can be drawn by hand with a ruler or the equation

for the best fit line can be calculated using a special math procedure known as regression.

Scatterplot of X vs. Y

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X-AXIS

Y-A

XIS

Figure 2.2.2.1 Scatterplot with Best Fit Straight Line

2.2.2.2 Best Fit Straight Line and Negative (Inverse) Relationships

When the points of a scatterplot forms a straight line that has a negative slop it is said that

the data points have a negative or inverse relationship. In this case a best fit line is still

possible.

Scatterplot of X vs. Y

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X-AXIS

Y-A

XIS

Figure 2.2.2.2 Scatter Plot with Best Fit Straight Line (Negative Slop or Inverse)

2.2.2.3 Best Fit Curved Line Fit Relationships

Sometimes there is a relationship between two variables that does conform to a strait best

line, this relationship is referred to as non-linear relationship. A curved line that fits a

non-linear scatter plot is referred to as a best fit curved line.

Scatterplot of X vs. Y

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X-AXIS

Y-A

XIS

Figure 2.2.2.3 Scatter Plot with Best Fit Curved Line

Figure 2.2.2.4 Scatter Plot with an Outlier Sometimes when collecting experimental data, one or more pieces of data do not seem to

be a good fit for the best fit line or best fit curve. These misfit points are known as

outliers. Outliers are generally considered to be points so far off from the best fit line or

curve than can not be accounted for by chance. Most of the outlier can be attributed to

human error.

Scatter Plot of X vs. Y

0

1

2

3

4

5

6

7

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

X-AXIS

Y-A

XIS

Figure 2.2.2.4 Scatter Plot with an Outlier

Outlier

2.3 Linear Equations, Linear Interpolation and Regression The relationship between the inputs and outputs of many of the instruments and processes

in the industrial process controls field can be predicted using linear equations. The two

main techniques used in this course that make use of linear equations are interpolation

and regression.

2.3.1 Linear Equations Review In short, a linear equation is an equation for a straight line. Below is one example that

illustrates the general look of a graph of a linear equation:

1 2 3 4

1

2

3

4

0

0

y = 0.5 x + 1

x

y

Figure 2.3.1.1 Sample Linear Equation Graph

One of the most common forms that a linear equation can take is known as the Slope

Intercept Form:

y = m x + b (Slop Intercept Form) (equation 2.3)

x - Known as the independent variable, it is the input of the equation.

y - Known as the dependent variable, it is the output of the equation.

m - Known as the slope, it is the steepness of the line.

b – Known as the y-intercept, the y value where the line crosses y –axis.

Example 2.3.1.1

Find: m = ? and b = ? using the graph.

Solution:

The slope and y-intercept can be determined from the graph as follows:

1 2 3 4

1

2

3

4

0

0

y = 0.5 x + 1

x

y

b =

1

Run = 2

Rise = 1

m = Rise

Run= 1 / 2 = 0.5

Figure 2.3.1.2 Sample Linear Equation Graph

Another common form is called Two Point Form:

y – y1 = [(y2 –y1)/(x2 – x1)] (x –x1) (equation 2.4)

Or more commonly expressed as:

y = [(y2 –y1)/(x2 – x1)] (x –x1) + y1 (Modified two point form) (equation 2.5)

x1, y1 - coordinates that represent a point on a straight line,

x2, y2 - coordinates that represent a second point on a straight line.

x - the independent variable.

y - the dependent variable.

Example 2.3.1.2

Find: x1, y1, x2 and y2 using the graph.

Solution:

The slop and y-intercept can be determined from the graph as follows:

The values of x1, y1, x2 and y2 can be taken directly from a table or from a graph:

1 2 3 4

1

2

3

4

0

0

y = 0.5 x + 1

x

y

(X1, y1) = (2.0, 2.0)

(X2, y2) = (3.0, 2.5)

x y

0.0 1.0

1.0 1.5

2.0 2.0

3.0 2.5

4.0 3.0

Table

x1, = 2.0, y1 = 2.0, x2, = 3.0, y2 = 2.5

Figure 2.3.1.2 Sample, Finding Two Points on Graph or Using Table

The expression [(y2 –y1)/(x2 – x1)] is equivalent to the slope m, therefore:

m = [(y2 –y1)/(x2 – x1)] (equation 2.6)

y = m (x –x1) + y1 (equation 2.7)

2.3.2 Linear Interpolation Linear interpolation is a technique used to approximate the unknown value of a

dependent variable of a non-linear equation or curve given the value of an independent

variable. In order to use interpolation, two points on the non-linear curve must be known

on either side of the interpolated point. This method works because even though a non-

linear curve may have a slope that varies widely from one end of a curve to its other end,

on the average the slope of the non-linear curve will change little between two points on

the curve that are close together. A straight line that connects two points on a non-linear

curve is called a cord.

1 2 3 4

1

2

3

4

0

0

x

y

Known Points

Chord

Non-Linear

CurveApproximate

Value

True Value

(Not Actually Known)

Known

Value

Figure 2.3.2.1 Linear Interpolation

The equation for the cord, thus the interpolation equation is the same as the modified two

point form equation:

y = [(y2 –y1)/(x2 – x1)] (x –x1) + y1 = m (x –x1) + y1 (Modified two point form equation.)

Example 2.3.2.1

Given the table of square roots below, find the square root of 9.5:

x y

8.0000 2.8284

Table

9.0000 3.0000

10.0000 3.1623

11.0000 3.3166

0.0

1.0

2.0

3.0

4.0

5.0

0.0 2.0 4.0 6.0 8.0 10.0 12.0 14.0 16.0

Figure 2.3.2.2 Square Root Graph

The variable x is the independent variable and y is the dependent variable we are

estimating, in this case the y will be approximately the square root of x.

y = [(y2 –y1)/(x2 – x1)] (x –x1) + y1 = m (x –x1) + y1

Let x = 9.5000, x1 = 9.000, y1 = 3.0000, x2 = 10.0000 & y2 = 3.1623

m = [(y2 –y1)/(x2 – x1)] = [(3.1623 –3.0000)/(10.0000 – 9.00001)] = 0.1623

y = m (x –x1) + y1 = (0.1623) (x – 9.0000) + 3.0000

y = (0.1623) (9.500 – 9.0000) + 3.0000

y = 3.0812

The actual square root of 9.5 with 4 digits of accuracy is 3.0822. The difference

between the actual value and the estimate is called error and is calculated as

follows:

error = estimated value – true value = 3.1612 - 3.1622 = -0.0010

The error can also be expressed as a percentage of the true value:

%error = error / (true value) * 100% = -0.0010/ 3.1622 *100%

%error = -0.032%

Figure 2.3.2.3

Table of Square Roots

2.3.3 Linear Regression (Optional)

Linear regression is a mathematical procedure used to determine the equation of the best

fit line given a set of experimental data. Linear regression assumes a linear relationship

exists between an independent variable and its dependent variable. Linear regression is

useful when it is necessary to determine the unknown value of an independent variable

given the known value of an independent variable. The independent variable is known as

the predictor and the dependent variable is known as the target or response.

2.3.3.1 Linear Regression Equations - Least Squares Method Least squares method uses a particular set of equations that calculate the slope and

intercept of a best fit line. These equations work by reducing the least squares distances

between a set of data points on a scatter plot and the best fit line. Below are the least

square equations:

b = n

n Σ(xy) - Σx Σy

n Σ(x2) - (Σx)

2m =

(equation 2.8) (equation 2.9)

Σy – m Σx

m = slope

b = intercept

n = number of samples used

The values m and be are substistuted into the slope intercept form linear equation:

y = m x + b (equation 2.3):

Example 2.3.3.1.1 Least Squares Best Fit Equation Calculation

Use the data of column x and column y to find the parameters m and b for the best

fit line:

x y x y x2 y

2

0.00 1.19 0.00 0.00 1.41

0.50 1.37 0.69 0.25 1.88

1.00 1.26 1.26 1.00 1.59

1.50 1.73 2.60 2.25 2.99

2.00 2.25 4.50 4.00 5.05

2.50 2.17 5.41 6.25 4.69

3.00 2.29 6.86 9.00 5.23

3.50 2.92 10.23 12.25 8.55

4.00 3.14 12.56 16.00 9.85

N Σx Σy Σ(x y) Σ(x2) Σ(y

2)

9 18.00 18.31 44.10 51.00 41.25

M

0.498355

B

1.038165

Result: y = (0.4984) x + 1.0382

Best Fit Line

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 4.50

X

Y

Figure 2.3.3.1.1.1 Scatter Plot of Source Data

2.4 Some Common Equations

2.4.1 Area Equations Area is the measure of the total amount surface of a figure (shape).

Square: A = a b = ℓ w (equation 2.10)

Area

a

b Area

ℓ

w

Circle: A = πr2 =πd

2/4 (equation 2.11)

r

d

Area

2.4.2 Volume Equations Volume is a measure of the total space that a three dimensional object occupies.

Cylinder: Rectangular Prism:

V = A h = πr2 d = πd

2 h / 4 (equation 2.12) V = A h = a b h (equation 2.13)

A

h

db

a

h

A

2.5 Unit Conversions

A system of measurement is a group of units that can be used to measure quantities such

as temperature, length volume, mass and many other quantities. There are several systems

of units that have developed over history. In modern times the Imperial, US, SI and

Metric systems are dominant. Unfortunately this means that from time to time it will be

necessary convert measured or calculated values from one unit to another.

2.5.1 Length Unit Conversions and Unit Analysis There are a large number of length units to choose from. A short list of length unit

equivalencies is listed in the table below:

Length Equivalencies

1 ft (1') = 12 inch (12")

1 inch (1") = 2.54 cm

1 m = 3.281 ft

1 m = 100 cm

1 m = 1000 mm

1 yard = 3 ft

Figure 2.5.1.1 Length Equivalencies

The expression 1 inch = 2.54 cm states that one inch is the same as 2.54 cm. This

expression can be converted to a fraction that is used as a conversion factor between

inches and cm or to a factor to convert cm to inches:

example 2.5.1.1

inches to cm: 1 inch = 2.54 => (2.54cm/1inch) = 2.54 inch/cm

cm to inches: 1 inch = 2.54 => (1 inch/2.54cm) = 0.392701 cm/inch

When it is necessary to convert a length measured in inches to its equivalent length in cm,

the length needs be multiply by the appropriate conversion factor.

example 2.5.1.2

Convert 4.5 inches to x = ?cm

x = (4.5 inches)(2.54 cm/inch) = 11.43 cm.

Note that after the conversion calculation above that inches are eliminated (cancelled out)

from the result, this leaves only cm as the final unit. The process that examines which

units are left and which are eliminated to verify the correctness of a conversion is called

unit analysis.

example 2.5.1.3

Convert 6.0 cm to x = ? inches

x = (6.0 cm)(0.392701 inches/cm) = 2.3562 inches

At times it is necessary to use more than one conversion factor. If a direct conversion

factor between two units is unknown, then it is necessary two use two separate factors

that share a common unit.

example 2.5.1.4

Given:

cm to inches: 1 inch = 2.54 => (1 inch/2.54cm) = 0.392701 cm/inch

inches to ft: 1 ft = 12 inches => (1 ft/12 inches) = 0.083333 ft/inch

Convert 5.0 cm to x = ? ft

x = (6.0 cm)(0.392701 inches/cm)( 0.083333 ft/inch) = 0.196351 ft

All units except for ft are eliminated; this verifies that the correct conversion

factors were chosen.

2.5.2 Area Unit Conversions Conversions between units of area are calculated the same way as conversions between

length units. Below is a sort table of area equivalencies:

Area Conversion

1 sq. ft (1 ft2) = 929 sq. cm (929 cm2)

1 sq. yd = 9 sq. ft (9 ft2)

1 sq. inch = 6.452 sq. cm (6.452 cm2)

1 sq. m (1 m2) = 10.76 sq. ft (10.76 ft2)

1 sq. m (1 m2) = 1550 sq. inch (1550 inch2)

1 sq. m (1 m2) = 10,000 sq. cm (10,000 cm2)

Figure 2.5.2.1 Area Unit Equivalencies

Converting area unit equivalencies to area unit conversion factors is the same procedure

as is used for length unit conversions factors:

Example 2.5.2.1 Convert sq. yd to ft

2: 1 sq. yd = 9 ft

2 => (1 sq. yd / 9 ft

2) = 0.111111 sq. yd / ft

2

Convert ft2 to sq. yd: 1 sq. yd = 9 ft

2 => (9 ft

2 / 1 sq. yd) = 9.0 ft

2 / sq. yd

Note that the prefix sq. placed in front of a unit is the same as saying that unit squared.

The prefix sq. is used when it is not possible to use a supper script 2 following a unit

(unit2).

example 2.5.2.2

Convert 3 ft2 to x = ? sq. yd

x = (3 ft2)(0.111111 sq. yd/ft2) = 0.333333 sq. yd.

example 2.5.2.3

Convert 5 sq. yd to x = ? ft2

x = (5 sq. yd) (9 ft2

/ 1 sq. yd) = 45 ft2

Sometimes more than one conversion factor is needed for area unit conversion

calculations if a single conversion factor between two units is unknown (just like with

length unit conversion).

example 2.5.2.4

Convert 2 sq. yd to x = ? sq. inch

1 ft2 = (12 inch)(12 inch) = 144 sq. inch => (144 sq. inch/1 ft2) = 144 sq. inch/ft2

X = (2 sq. yd)(9 ft2

/ 1 sq. yd)( 144 sq. inch/ft2) = 2592 sq. inch

If a conversion factor between area units is required which is unknown, it can be derived

squaring the related length conversion factor such as in the example below.

example 2.5.2.5

Given: 1ft = 12 inch

Find: 1ft2 = ? inch

2

(1ft) (1ft) = (12 inch) (12 inch)

1ft2 = 144 inch

2

2.5.3 Volume Unit Conversions The procedure to convert between units of volume is the same as for converting length or

area units, however, keep in mind that the conversion factors values are not the same:

Volume Conversion

1 cu. Ft (1 ft3) = 28,320 cu. cm (cm3)

1 cu. Ft (1 ft3) = 1,728 cu. inch (inch3)

1 cu. Yd = 27 cu. ft (27 ft3)

1 cu. M (1 m3) = 1,000,000 cu. cm (cm3)

1 cu. M (1 m3) = 1.308 cu. Yd ( Yd3)

1 cu. M (1 m3) = 1000 L

1 L = 1000 mL

1 L = 1000 cm3

1 mL = 1 cm3

1 gallon = 3.785 L

Figure 2.5.3.1 Volume Unit Equivalencies

Converting area unit equivalencies to area unit conversion factors is the same procedure

as is used for length unit conversions factors:

example 2.5.3.1

Convert mL to L: 1000 mL = 1 L => (1000 mL / 1 L) = 1000 mL / L

Convert L to mL: 1000 mL = 1 L => (1L / 1000 L) = 0.001 L / mL

example 2.5.3.2

Convert 15 cu. Yd to cu. ft.

1 cu. Yd = 27 cu. ft => (27 cu. ft / 1 cu. Yd) = 27 cu. ft / cu. Yd

X = (15 cu. Yd) (27 cu. ft / cu. Yd) = 404 cu. ft / cu. Yd

If a conversion factor between different volume units is required which is unknown, it

can be derived cubing the related length conversion factor such as in the example below.

example 2.5.3.3

Convert 100 cu. inch to ft3, given 1 ft = 12 inch.

(1 ft) (1 ft) (1 ft) = (12 inch) (12 inch) (12 inch) => 1 ft3 = 1728 inch

3

(1 ft3 / 1728 inch

3) = 0.00057870 ft

3 / inch

3

X = (100 cu. inch) (0.00057870 ft3 / inch

3) = 0.057870 ft

3

2.5.4 Mass Unit Conversions

The procedure to convert between units of mass is the same as for converting length, area

and volume units. A few mass unit equivalencies are listed below:

Mass Conversion

1 kg = 1000 g

1 kg = 2.2046 lb

1 long ton = 2240 lb

1 short ton = 2000 lb

Figure 2.5.4.1 Volume Unit Equivalencies

A few examples of mass conversions are given below.

example 2.5.4.1

Convert 25 g to Kg, given 1kg = 1000g

1kg = 1000g => (1kg / 1000g) = 0.001 kg / g

X = (25 g) (= 0.001 kg / g) = 0.025 kg

example 2.5.4.1

Convert 2.0 short ton to long ton,

Given: 1 short ton = 2000 lb and 1 long ton = 2240 lb

1 short ton = 2000 lb => (2000 lb / 1 short ton) = 2000 lb / short ton

1 long ton = 2240 lb => (1 long ton / 2240 lb) = 0.000446429 long ton / lb.

X = (2.0 short ton) (2000 lb / short ton) (0.000446429 long ton / lb)

X = 1.7857 long ton

2.5.5 Temperature Unit Conversions Temperature is commonly experienced as the hotness or coldness of a substance or an

environment. The actual numerical value of a temperature measurement taken using an

instrument such as a thermometer is a measure of the average kinetic energy of the

particles that make up a substance. The four main units of temperature are listed below:

°C - Celsius

K - Kelvin

°F - Fahrenheit

°R or Ra - Rankine scale

Conversion between temperature units required more effort than simply multiplying by a

conversion factor. Fortunately the units above have simple linear relationships between

them. The most common conversion equations used for temperature unit conversions are

listed below.

K = 273.15 + C C = K – 273.15

C = (F-32) * (5/9) F = C * (9/5) + 32

R = F +459.67 F =R - 459.67

3.0 Test Instruments Used with Process Control Instrumentation This section describes several test instruments used in this class. This is a brief

introduction, for a detailed description of these devices and their operation see the

operating manual for these devices and the lab instructions.

3.1 Pressure Calibrators

Figure 3.1.1 Pressure Calibrator

A pressure calibrator is a device that measures the differential pressure between two

pressure sources with a high degree of accuracy. Typically, the pressure value displayed

by a pressure calibrator is compared to the indicated pressure a second pressure

measuring device to verify the second devices accuracy. When using the pressure

calibrator as a reference for pressure measurements comparison it is referred to as a

reference or standard.

Pressure Calibrator

H L

Pressure

Measuring

Device

(Reference, Standard)(Device Under Test)

Pressure

Source(ex, tank)

Pressure

Regulator

Figure 3.1.2

3.1 Pressure Calibrators (continued) A pressure calibrator has two ports, the port labelled H is called the high side port and

port labelled L is called the low side port. Typically the high side port is connected to the

pressure source to be measured while the low side port is connected to a reference

pressure source. If atmospheric pressure is to be used as a reference then the low side

port is left open to the atmosphere, this is known as a gage pressure measurement.

Measured

Pressure

Source

Pressure

Calibrator

Reference

Pressure

Source

H L

Measured

Pressure

Source

Pressure

Calibrator

H L

Figure 3.1.3 Differential Pressure Figure 3.1.4 Gage Pressure

3.2 Temperature Calibrators

Figure 3.2.1 Temperature Calibrators

A temperature calibrator is a device used to produce (to source) an electrical signal that

simulates the output signal of a temperature sensor. A temperature calibrator is often

called a reference or a standard because of the high accuracy of its output signal. A

temperature calibrator is typically used when calibrating a device called a temperature

transmitter. A temperature transmitter is used to convert the non standard electronic

signal to a standard 4-20mA signal or 3-15 psi signal. If you want to simulate a sensor

that is immersed in an environment that has temperature of 35°C, then press 35°C into the

calibrators keyboard and an electrical signal will appear at the calibrators output that

corresponds to 35°C.

TT+

-

+

-24 VDC

TI

+ -

Temperature Transmitter

Temperature Indicator

Temp.

Sensor

Remove

Temperature

CalibratorAdd

Figure 3.2.2 Temperature Calibrator Simulating Sensor Output

3.2 Temperature Calibrators (continued) A temperature calibrator can also be used to receive output signal from a temperature

sensor and convert this signal to a temperature reading on its display.

Indicates temperature of

sensors environment.

Temperature

Calibrator

Temp.

Sensor

Measures temperature of

environment.

Figure 3.2.3 Temperature Calibrator Used to Display Temperature of Sensor

Often the sensor is placed in a an environment which has a known temperature such as

ice water (0°C) or boiling water at atmospheric temperature (100°C) in order to verify the

correct accuracy of a sensor.

Indicates temperature of

sensors environment.

Temperature

Calibrator

Temp.

Sensor

Ice water

(0 °C)

Indicates temperature of

sensors environment.

Temperature

Calibrator

Temp.

Sensor

Boiling Water

(100 °C)

Heater

Figure 3.2.4 Using Boiling Water Figure 3.2.5 Using Ice Water

as a Temperature Reference as a Temperature Reference

3.3 Current Calibrator

Figure 3.3.1 Current Calibrator

A current calibrator is a device used to provide an accurate source of current ranging

between 0mA and 20 mA to simulate output signal from an instrument. A current

calibrator allows its operator to see the effect that a 4-20 mA signal has on an instrument

receiving a 4-20mA signal (such as a temperature indicator) on the test bench or in the

field without actually having it connected to a transmitters output. This gives the

operator the advantage of controlling the current directly at will rather than having to

attempt to control the current indirectly through the sensor.

+

-

`24 VDC

TI

+ -Temperature

Transmitter

Temperature Indicator

Temp.

SensorTT

+

-

Current

Calibrator

+-

+

-

Figure 3.3.2 Current Calibrator Used as a Current Source

3.3 Current Calibrator (continue) A current calibrator can also be used to accurately measure the 4 – 20mA output signal

generated by an instrument, controller or other signal source.

Current

Calibrator

+

-

+

-

Instrument with

4-20 mA Current

Output Signal

+

-

3.3.3 Current Calibrator Used to Measure an Instruments Output Signal

Many modern currents calibrator also are able to measure and source 0-10 V signals as

well as current signals. This is because 0-10V signals are commonly found in industry.

Since the current calibrators used in this class are used for teaching purposes, an inline

fuse has been added to the current calibrator cord to protect it from overload and short

circuit damage. Always replace the fuse with a 1/8 amp slow blow fuse. A picture of

the cord is shown below:

3.3.4 Current Calibrator Used to Measure an Instruments Output Signal

4.0 Some Science Background and Review Most of the process control instrumentation used today seems quite modern, sleek and

sophisticated; however, many of the principals of operation have been well understood

and used in practice since ancient times. This section covers the theory necsisary to

understand the operation of the process control instrumentation studied in this class.

4.1 Some Definitions

4.1.1 Matter

Matter can be described as the stuff or material that solid, liquids and gasses are made of.

All things that exist that we can touch or feel (water, rocks, the wind, oil,…) are made of

matter. Matter is some times called the “stuff of the universe”. A few examples of things

that may be considered to exist but are not made of matter include energy, ideas, the

mind, corporate entities and friendship.

4.1.2 Mass Mass is a measure of the amount of “stuff” or matter that a thing (object) or substance

has. The amount of mass is not change as an object is deformed and it is not changed

when it is pushed or acted on in any way (heated, kicked, dropped, yelled at,…). Some

units of mass have already been covered in section 2.5.4.

4.1.3 Energy Energy is a measure the ability to do work. The more energy available, the more work

can be done. A couple of common units for measuring energy include Jouls (J) and Btu.

1 Btu(60°F) = 1054.68 J

4.1.4 Power Power is the rate at which energy is consumed or equivalently the rate at which work is

done. A few common units of power are watts (W), horse power (hp) and Btu/h.

4.1.5 Force A force can be considered to be an influence that tends to cause an object to accelerate or

decelerate. A force is commonly experienced as a push or a pull. A few units of force

include Newtons (N), pound-force (lb-f) and kilograms-force (kg-f).

4.1.6 Weight Weight is the force acting on an object due to gravity. The term weight is often

mistakenly used synonymously with the term mass. When weighing an object at the

surface of the Earth the numerical value of the mass of an object is the same as the

numerical value of its weight. If an object was placed on another planet or on the moon

its weight would change but its mass stays the same. Note that even though kg-f and lb-f

are used as units of force, it is still more conventional to write units of weight (force due

to gravity) simply as kg and lb.

4.1.7 Solid A solid is a material that holds its own shape, resists changes in shape and resists changes

in volume (compression). A solid will not flow to assume the shape of its container.

4.1.8 Fluid A fluid is a substance that will flow to conform to the shape of its container. Two

examples of fluids are liquids and gasses.

4.1.9 Liquid A liquid is a fluid that resists changes in volume (fluids are not comprisable). Liquids

tend to have densities similar to solids.

4.1.10 Gasses A gas is a fluid that is compressible. A gas will tend to expand in all directions to fill a

vessel that contains it (and conform to its shape). Gasses have densities many times

lower than liquids because they have many times more average distance between their

particles.

4.1.11 Vessel A vessel is an object used as a container. Vessels are used in industrials setting may

contain liquid and or solids used for applications that include but are not limited to

storage blending/mixing, dissolving, separating, cooking, separation, reaction and

polymerization.

4.1.11.1 Closed Vessel A closed vessel is a vessel that prevents its contents from escaping because it is sealed

from its environment. The contents of a closed vessel may be at the same pressure as the

surrounding environment or the contents may be at a higher or lower pressure with

respect to the surroundings.

4.1.11.2 Open Vessel An open vessel is a vessel that is not sealed from its environment. The only thing that

holds the contents inside an open vessel is the force of gravity. The contents of a closed

vessel are at the same pressure as the surroundings at the contents surface. The only

pressure experienced by the contents of an open vessel is the pressure due to atmospheric

pressure and the pressure due to gravity (hydrostatic pressure).

4.2 Force of Gravity The force of gravity is an effect that is commonly experienced as a force pushing an

object toward the ground (the surface of the Earth). The same gravitational effect can be

observed when an object is placed near any massive body such as the moon or the sun.

Fg

Surface of Earth

object

Figure 4.2.1 Force Acting On an Object near Earths Surface.

The arrow and letters Fg signify the direction of force due to the

Earths gravity acting on an object.

The effect which is referred to as gravitational force, force of gravity or gravitational

attraction occurs because every object in the universe is attracted to every other object in

the universe. Gravitational forces are in fact the reason that the moon is attracted to the

Earth and the Earth is attracted to the sun.

Equation 4.1Sun

Fg Fg Earth

Fg = G m1 m2

r2

Figure 4.2.2 Gravitational Forces Between the Sun, Earth and moon.

The arrows and letters Fg indicate the direction of gravitational forces

acting between these bodies.

Equation 4.1 is known as the universal gravitation equation and is used to determine the

magnitude of the gravitational force attracting two bodies toward each other. The

variables m1 and m2 are the masses of each body, r is the distance between the bodies and

G is the universal gravity constant.

4.2 Force of Gravity (continued) The force of gravity acting on an object at the surface of a large body (such as Earth or

the moon) in can be calculated using the abbreviated version of equation 4.1 shown

below:

Fg = m go (equation 4.2)

In the above equation:

Fg - force of gravity

m - mass of the object

go - constant known as standard gravity

Equation 4.2 assumes that the value of go = G m2/ r2 is relatively constant as long as the

object is stays near the surface.

The value of g0 depends on which planet, sun or moon surface the object is near. If an

object is near the surface of the Earth, then g0 = 9.807 N/kg and is commonly expressed

as ge = 9.807 N/kg. If an object is near the surface of the moon, then g0 = 1.622 N/kg and

is commonly expressed as gm = 1.622 N/kg.

When using metric units, equation 4.2 assumes Newtons (N) as a unit of force and kg as

its mass unit.

example 4.2.1

Given a box with a mass of 10 kg, find the force of gravity acting the box near the

surface of the moon.

moon

10 kg

Fg = ?

Figure 4.2.1

Fg = m go = m gm = (10 kg)( 1.635 N/kg) = 16.35 N

example 4.2.2

Given a TV with a mass of 10 kg, find the force of gravity acting on the TV near

the surface of the Earth.

Earth

10 kg

Fg = ?

Figure 4.2.2

Fg = m go = m gm = (10 kg)(9.807 N / kg) = 98.07 N

Since the term weight is synonymous with the term force of gravity, equation 4.1 is also

expressed as:

W = m go (equation 4.3)

In the above equation:

W - weight (force of gravity)

m - mass of the object

go - constant known as standard gravity

Again, it is more colloquial to express weight in kg or lb, but it is still useful to use kg-f

and lb-f when the distinction between mass and weight might be unclear in a calculation

(particularly while learning about the subject).

example 4.2.3

Given a box with a mass of 5 lb, find the force of gravity acting the box near the

surface of the moon (gm = 1.635 N/kg = 1/6 lb-f/lb).

W = m go = m gm = (5 lb)( 1/6 lb-f / lb) = 0.8333 lb (lb-f)

example 4.2.4

Given a box with a mass of 5 lb, find the force of gravity acting the box near the

surface of the Earth (ge = 1.635 N/kg = 1.000 lb-f/lb)

W = m go = m ge = (5 lb)( 1.000 lb-f / lb) = 5 lb (lb-f)

4.3 Density

Most things in the universe that have mass also occupy space (except for black holes,

black holes will not be discussed further). Density is a measure of a substances mass per

unit volume. The more mass that is squeezed into the same volume, the more dense the

substance is. The symbol ρ (greek letter rho) is used to represent density in equations.

Density is calculated using the following equation:

ρ = m / V (equation 4.4)

In the equation above:

ρ - is density

m - is mass

V - is volume

example 4.3.1

Given a volume of 1.3 L of water with a weight 1.3 kg, calculate the density of

water.

1.3 L

Water

1.3 kg

Weight Scale

Figure 4.3.1

ρwater = m / V = 1.3 kg / 1.3 L = 1.0 kg/L = 1.0 g / mL

The units for density are derived units that reflect the mass and volume units used in the

density calculations. A few useful density unit equivalencies are listed below:

1 g/cm3 = 0.036127292 lb / in3

1 g/cm3 = 1 g / mL

1 g/cm3 = 1 kg/L

Figure 4.3.2 Density Unit Equivalencies

4.3 Density (continued)

example 4.3.2

Given that the cylinder below filled completely filled with water and the

density of water ρwater = 1 g/cm3 = 0.0361 lb / in3 = 1 g / mL, calculate the

mass of the water contained inside the cylinder.

10 in2

10

in

Water

Figure 4.3.2

(1) ρwater = m / V

(2) V = A h

(3) m = ρwater V from (1)

(4) m = ρwater A h plug (2) into (3)

(5) m = (0.036127 lb / in3) (10 in2) (10 in)

(6) m = 3.6127 lb

4.4 Relative Density and Specific Gravity Relative density is the ratio between the density of a specific material and the density of a

second material which is used as a reference. Relative density is typically represented

symbolical as RD in the relative density equation:

RD = ρsubstance / ρreference (equation 4.5)

Specific gravity is also the ratio between the density of a specific material and the density

of a second reference material, however, the reference is always pure water at 4°C and 1

atm. Each material has a specific gravity that is particular to that material, i.e. corn oil

(SG = 0.923) has a different specific gravity than 5W-30 motor oil (SG = 0.860). SG is

typically used as the symbol for standard gravity in the standard gravity equation:

SG = ρsubstance / ρwater (equation 4.6)

Since relative density and specific gravity are both ratios of like dimensions (same units),

relative density and specific gravity is dimensionless (without units) because the density

units cancel out in the equation 4.5 and equation 4.6.

4.4 Relative Density and Specific Gravity (continued) Since specific gravity is a dimensionless ratio of densities, specific gravity can be used to

calculate the density of a material in any unit of density through equation equation 4.5.

example 4.4.1

Given equation 4.6, the specific gravity of 5W-30 motor oil (SG5W-30 = 0.860) and

the density of water is known to be ρwater = 1.0 g/cm = 62.4 lb/ft3 = 8.34 lb/gallon,

calculate the density of 5W-30 in units of g/cm3, lb/ft

3 and lb/gallon.

(1) SG = ρsubstance / ρwater

(2) ρwater * SG = ρwater * ρsubstance / ρwater

(3) ρsubstance = SG ρwater

ρ5W-30 = SG 5W-30 ρwater

ρ5W-30 = (0.860) (1.0 g/cm) = (0.860) (62.4 lb/ft3) = (0.860) (8.34 lb/gallon)

ρ5W-30 = 0.860 g/cm = 56.2 lb/ft3 = 7.17 lb/gallon

4.5 Pressure Pressure is the force per unit area exerted on a surface. In this class pressure is typically

the result of force per unit area exerted on a surface by a fluid.

F

FF

F

F

F

F

F

F

Inner TubePSI

Pressure

Gauge

F

Figure 4.5.1 Cross section of an inner tube illustrating that air pressure inside

the tube caused a force a force be applied against its inner surface.

Whenever a fluid is “squeezed” it pushes back against the surface that contains it

resulting in an increased pressure. The equation that relates pressure, forces and area of

the surface that contains it is:

P = F / A (equation 4.7)

Where

P – is pressure,

F – is force

A – is area

4.5 Pressure (continued)

Pressure is a derived unit. Equation 4.6 can be used to calculate the pressure that results

when a force is applied to the bicycle tire pump handle or to the plunger of a syringe:

Example 4.5.1

Given the syringe shown below, calculate the pressure that results from pressing

its plunger with a force of F =4.0 lb-f.

A

Pressure

Gauge

F

A = 2 in2 Gas or Liquid

Figure 4.5.1 Pressure produced by a syringe due to force applied to its plunger.

P = F / A = 4.0 lb-f / 2 in2 = 2.0 lb/in

2 = 2.0 psi (2.0 pounds per square inch)

Example 4.5.2

Given the information available form the diagram below, calculate the force

exerted by the piston of the cylinder in the upward direction.

Weight

Pressure

GaugeA = 10cm

2

P =10 kPa

Figure 4.5.2 Produced by a Hydraulic Cylinder

First convert area and pressure to units that will make our calculations easier.

A = 10 cm2 = 0.0010 m

2, P = 10 kPa = 10,000 Pa = 10,000 N/ m

2

P = F / A => F = P A = (0.0010 m2) (10,000 N/ m

2)

F = 10.0 N

4.5.1 Hydrostatic Pressure Hydrostatic pressure is defined as the pressure at a point in a liquid due to gravity.

Pressure has been defined as force per unit area. It follows that hydrostatic

pressure can be defined as the force due to gravity acting on a column of liquid

divided by that liquid columns cross-sectional area. The derivation for an

equation to calculate hydrostatic pressure is given below.

A

Column of

Liquid

h

Figure 4.5.1.1

Given:

(1) m = ρliquid V

(2) V =A h

(3) P = F/A

(4) F = mgo

(5) m = ρliquid A h Plug (2) into (1).

(6) F = ρliquid A h go Then plug (5) into (4).

(7) P = F/A = (ρliquid A h go)/ (A) Then plug (6) into (3).

Notice that area is eliminated in (7), therefore

(8) Phydrostatic = ρliquid go h (equation 4.8)

Note that the equation 4.8 is not limited to finding hydrostatic pressure for

rectangular liquid columns. Equation 4.8 will work for any column with any

shape as its base. It is possible to prove that equation 4.8 will work even for

vessels that do not have a constant cross-sectional area, but this proof is beyond

the scope of this program. In a hydrostatic system, the datum is a vertical

reference point from which a level and/or pressure is measured.

h

DatumPressure

Gauge

Figure 4.6.1 The Datum of a Pressure Gauge Used to

Measure Hydrostatic Pressure

4.5.1 Hydrostatic Pressure (continued)

Example 4.5.1.1

Given the information presented in the diagram below and ρwater =

1.0g/cm3 = 0.0361 lb /in

3, find the pressure at the datum in psi (lb/in2).

h =

25

in

ch

P=?

Figure 4.5.1.1 Hydrostatic Pressure Due to Gravity Acting On Water

Phydrostatic = ρliquid go h = (0.0361 lb /in3) (1.000 lb-f/lb)(25in)

Phydrostatic = 0.9025 lb/in2

Example 4.5.1.2

Given the information presented in the diagram below and ρwater =

1.0g/cm3 = 0.0361 lb /in

3, find height of the waters surface above the datum

in cm.

h = ? cm

P=2 kPa

Water

Figure 4.5.1.2 Hydrostatic Pressure Due to Gravity Acting On Water

Phydrostatic = ρliquid go h => h = Phydrostatic / (ρliquid go)

h = Phydrostatic / (ρliquid go) = 2kPa / (1.0g/cm3*9.8N/kg)

h = (2000 N/m2 ) / (1000 kg/ m

3*9.8N/kg)

h = 19.6m = 1960 cm

4.5.2 Absolute Pressure, Gage Pressure and Differential Pressure

4.5.2.1 Absolute Pressure Absolute pressure is a pressure measurement with reference to a vacuum. For example, if

an astronaut were to use a pressure gauge to measure the pressure of an O2 tank using the

vacuum of space as a reference; this would be an absolute pressure measurement.

PI

Pressure

Indicator O2

Tank

Astronaut

Star

Figure 4.5.2.1.1 Absolute Pressure

In order to indicate that a pressure measurements is an absolute pressure measurement the

letter „a‟ is often added to the end of the unit such, as psia in place of psi.

4.5.2.1.1 Atmospheric Pressure (Absolute Pressure of the Atmosphere) When measuring the absolute pressure of the atmosphere with reference to a vacuum the

measurement is referred to as the atmospheric pressure. On the average the pressure of

the atmosphere is 14.7 psi = 1101,325 Pa = 1atm. Atmospheric pressure pressures

typically measured using an instrument known as a barometer.

PI

Vacuum

Atmosphere

Mercury

Vacuum

Figure 4.5.2.1.1.1 A Barometer and its Equivalent Pressure Indicator in a Vacuum (using

the Vacuum as a Reference)

4.5.2.2 Gage Pressure Gauge pressure is the difference between an absolute pressure measurement and the

pressure of the atmosphere, where the atmospheric pressure is the reference. The letter

„g‟ is typically added to a pressure unit to indicate that is a gage pressure measurement,

such as psig for psi. An example of an atmospheric pressure measurement is a tire

pressure measurement. If the absolute pressure inside a tire is 44.7 psia, and the

atmospheric pressure is 14.7psia, then the gauge pressure is 44.7psia - 14.7psia = 30 psig.

Inner Tube

P =

30 psig

Pressure

Gauge

P = 44.7psia

P = 14.7psia

(Atmosphere)

Figure 4.5.2.2.1 Gage Pressure

4.5.2.3 Differential Pressure Differential pressure is the simply the pressure difference measured between two pressure

sources as is illustrated below:

ΔP =

2 psid

H L

PH = 7 psig PL = 5 psig

Tank Tank

Figure 4.5.2.3.1 Differential Pressure

Differential pressure is typically measured using a two port instrument with ports labelled

H (or +) for high side and L (or -) for low side. The differential pressure measured and/or

indicated by such a device is expressed as:

ΔP = PH – PL (equation 4.9)

4.5.3 Inches of Water Column, inH2O (Pressure) Inches of water column is a unit of pressure. The inches of water column unit is

normally written as inH2O. The „2‟ in is not written as a subscript, this probably goes

back to the days so it could be easily typed using a manual typewriter. In short, 1 inH2O

is equal to the amount of hydrostatic pressure produced by a 1 in high column of water

(on Earth only):

h = 1in

P =

1 inH2O

Water

Figure 4.5.3 1 in of water column height producers 1inH20

inH2O can be used as a unit for liquid or gas pressure measurements, but it was made

popular due to a differential pressure measurements device called a manometer which is a

liquid column gauge used to measure air pressure using a U shaped tube.

3 in

Air Tank

P = 3 inH2O

Manometer

Water

Figure 4.5.3 2 in of water column height producers 1inH20

Inches water column is a relatively sensitive scale compared to psi as can be seen by the

equivalency below:

1 psi = 27.68 inH2O

4.5.4 Hydrostatic Pressure and Closed Vessels Pressure measured at the bottom of an open vessel containing a liquid is due entirely to

the force of gravity acting on the liquid inside the container (hydrostatic pressure). In a

closed vessel the total pressure at the bottom of the tank is due to the hydrostatic pressure

plus the pressure of the gas or vapour filling space between the surface of the liquid and

the top of the closed vessel.

Ptotal =PHydrostatic + PGas (equation 4.10)

example 4.5.4.1

Verify that the pressure gauge in Figure 4.5.4.1 is correct by calculate the total

hydrostatic pressure expected at the indicate datum of the pressure gauge.

Datum

Water

Gas, Vapor

3 inH2O

4 in

Water

Wet Leg

7 inH2O

Figure 4.5.4.1

Ptotal =PHydrostatic + PGas

Ptotal = 4inH2O + 3inH2O

Ptotal = 7inH2O

4.5.4 Hydrostatic Pressure and Closed Vessels (continued)

If a differential pressure gauge were used to measure the difference between

the datum of the tank (and gauge) and gas/vapour pressure of the gas above

the liquid in the vessel, the result would be a measurement of the hydrostatic

pressure only:

example 4.5.4.2

Verify that the differential pressure gauge in Figure 4.5.4.2 is correct by calculate

the total hydrostatic pressure expected at the indicate datum of the pressure gauge.

Datum

Water

Gas, Vapor

3 inH2O

4 in

Water

Wet Leg

4 inH2O

Ga

s, V

ap

or

H

L

Dry Leg

Figure 4.5.4.2

PH = PHydrostatic + PGas

PH = 4inH2O + 3inH2O

PH = 7inH2O

PL = PGas

PL = 3inH2O

ΔP = PH – PL = (PHydrostatic + PGas) – (PGas) = PHydrostatic

ΔP = (4inH2O + 3inH2O) – (3inH2O)

ΔP = 4inH2O

4.6 Archimedes Principal Archimedes Principal states that when an object is partially or completely immersed in a

fluid it is buoyed (floated, forced) upward by a force equal to the weight of the liquid it

displaces. In more simple words, when a an object is immersed in a liquid it will appear

to loose weight, the amount of weight loss is equal to the weight of the liquid pushed out

of the way by the object.

liquid

V = A h

A

h

Weight

Scale

Sample

Material

example 4.6.1

The volume of the water displaced by the object is equal to the volume of the immersed

object, therefore:

V = Vdisplaced-liquid = Vobject

According to Archimedes the weight loss of the object is equal to the weight of the

displaced liquid, therefore:

Wweight-loss = Wdisplaced-liquid = m go = V ρliquid go (because m = V ρliquid)

It follows that the weight of the immersed object becomes the initial dry mass of the

object minus the weight of the immersed liquid:

Wimmersed = Wdry-object – Wweight-loss = Wdry-object - V ρliquid go (equation 4.11)

4.6 Archimedes Principal (continued)

example 4.6.1

Given a sample of metal with a volume of V =2cm3 that has a weight of Wdry-object

= 15.0 g before immersion in water, calculate the weight of the sample after it is

immersed in water:

liquid

V = A h

A

h

Weight

Scale

Sample

Material

example 4.6.2

Solution:

The volume of the water displaced by the object is equal to the volume of the

immersed object, therefore:

V = Vdisplaced-liquid = Vobject = 2cm3

Wimmersed = Wdry-object – Wweight-loss = Wdry-object - V ρliquid go

Wimmersed = 15.0 g – (2cm3)(1g/cm

3)(1g-weight/g-mass)

Wimmersed = 13.0 g

Archimedes principal can also be used to determine the standard gravity and the density

of an object immersed in a liquid if the liquid is water and the object dry weight and

immersed weight are measured:

SG = Wdry-object / (Wdry-object - Wimmersed) (equation 4.12)

Proof:

Let (Wdry-object - Wimmersed) = Wweight-loss = Wdisplaced-water

SG = Wdry-object / Wdisplaced-water

SG = (V ρobject go)/(V ρwater go) (because V = Vdisplaced-water = Vobject)

SG = (V ρobject go)/(V ρwater go)

SG = ρobject / ρwater

4.6 Archimedes Principal (continued)

example 4.6.2

Given the object in example 4.6.1, find the standard gravity and the density of the

object:

SG = Wdry-object / (Wdry-object - Wimmersed) = 15.0g / (15.0g – 13.0g)

SG= 7.5

SG = ρobject / ρwater => ρobject = SG ρwater = (7.5)(1.0g/mL)

ρobject = 7.5 g/mL

5.0 Basic Process Instrumentation Concepts and Terminology This section will introduce the basic concepts and terminology required to understand and

communicate concepts regarding process instrumentation.

5.1 Process Variables and Process Instruments Process variables are the variables that have an effect on a product as it is manufactured.

There are many process variables that can affect manufacturing processes, however the

process variables that this course will focus on are:

- pressure of liquids and gasses.

- level (height, volume) of liquids and solids

- temperature.

- flow of liquids and gasses.

These four variables will be the focus of the course because they are the most commonly

measured and controlled process variables during the manufacturing process. A few

industries where these variables affect manufacturing processes are pulp and paper, food,

petrochemical and pharmaceutical manufacturing. In the field of process instrumentation

it is common to represent a process variable with the letters PV and sometimes as %PV.

5.2 Process Instrumentation Process instrumentation is the name given to equipment used measure, transmit, indicate,

record and control the values of process variables. Process instrumentation is generally

divided into the following categories:

-Elements / Sensors, devices that measure process variables.

-Transmitters that converts measurements from elements to standard signals.

-Indicators indicators to display the value of process variables.

-Recorders to records process variables over time on paper or electronically.

-Controllers used to keep a controlled process variable at its desired value.

5.3 P&IDs

The relationship between process instrumentation and the manufacturing equipment is

documented using process instrumentation diagrams, also referred to as P&ID diagrams

for short. In P&IDs instruments are generally represented as a circle with an acromym

typed in its center, a few of these acronyms are listed below:

FE – Flow Element, FT – Flow Transmitter, FI – Flow Indicator,

FR-Flow Recorder, FC-Flow Controller

TE – Temperature Element, TT – Temperature Transmitter, TI –

Temperature Indicator, TR- Temperature Recorder, TC- Temperature

Controller

PT – Pressure Transmitter, PI – Pressure Indicator, PR- Pressure Recorder,

PC-Pressure Controller

LT – Level Transmitter, LI – Level Indicator, LR- Level Recorder, LC-

Level Controller

Below is a sample P&ID showing the relationship between a heat exchanger and its

related instruments:

H L

FT

TIC

FE

Steam

Source

FIRTT

TE PI

Product In

Return

Product Out

Figure 5.1, Heat Exchanger and Related Instruments

5.4 Standard Signals (The Short Story) Standard process instrumentation signals are used to transmit (send) the values of process

variables from one instrument to another, just as USB is used to send documents and

images from a computer to a printer. Standard signals are used to represent value of

process variables in a way that is common to most process instruments, even if they are

built by different manufacturers. The two most common standard signals are 4-20mA

and 3-15 psi signals. In P&IDs electronic signals such as 4-20mA signals are represented

a dashed lines and pneumatic signals such as 3-15 psi signal are represented as solid lines

with double hash marks (see heat exchanger drawing above).

5.5 ZERO, Upper Limit and SPAN Every measuring device has an upper limit and lower limit to its measurement range.

The maximum possible measurement (or signal) value is referred to as the Upper Limit

(U) and the lowest possible measurement value is referred to as the ZERO of the

measurement range. The SPAN is defined as the difference between the upper limit and

the lower limit (SPAN = U – ZERO). ex. Given a thermometer with an upper limit of 120C° and a lower limit of -60C°,

find U, ZERO and SPAN, then plot them on a number line:

-60 C° 60 C°30 C° 90 C°0 C°-30 C° 120 C°-90 C° 120 C°

SPAN = U – ZERO = 120 C° - (- 60 C°) = 180 C°

ZERO = -60 C° U = 120 C°

Figure 5.2, SPAN, ZERO and Upper Limit

5.6 Percent (%) Value of a Process Variable Process variable measurements are typically represented using engineering units. For

example, temperature is often measured in C° or F°, pressure can be measured in psi, kPa

and inH20, level can be measured in ft, m and inch, and flow can be measured in gallons

per min, tonne per hour, etc. At times it is not convenient represent process variables in

engineering units.

Another common, convenient and easily understood method to represent and display a

process variable is to represent its value as a percentage. In cases where the lower limit of

the measurement range is actually ZERO = 0.0, then the common sense notion of

percentage applies where %PV = 100% * PV / U.

ex. Given an air tank with a maximum pressure of 120 psi, but only a measured value

of PV = 60 psi, common sense would lead you to calculate the % pressure as follows:

%PV = 100% * PV / U = 100% * 60 psi / 120 psi = 50%

120 psi

Max

60 psi

Measured

Air Tank

Pressure

Gauge

Figure 5.3, Percent Value of a Process Variable

Sometimes process variables do not have a have a ZERO actually equal to 0.0, in those

cases the common sense notion of percentage does not apply. Whenever the ZERO ≠ 0.0

conditions exists, then the equation %PV = 100% * (PV – ZERO) / SPAN is used to take

account of ZERO ≠ 0.0 conditions.

Ex. For a thermometer with an upper limit of 120C°, a lower limit of -60C° and a

measured temperature of PV =60 C°, calculate the measured temperature as a

percentage value (%PV = ?):

SPAN = U – ZERO = 120 C° - (- 60 C°) = 180 C°

%PV = 100% * (PV – ZERO) / SPAN

%PV = 100% * [60 C° - (- 60 C°)] / 180 C°

%PV = 66.666%

-60 C° 60 C°30 C° 90 C°0 C°-30 C° 120 C°

Temperature = 60 C°

Thermometer

Figure 5.4

The graph below shows the temperature vs. %PV for the thermometer shown above:

-60 C° 60 C°30 C° 90 C°0 C°- 30 C° 120 C°-90 C°

SPAN = 180 C°

U = 120 C°ZERO = - 60 C°

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

Figure 5.5, Temperature vs. %PV

5.7 Elements/Sensors Elements and sensors are the instruments that directly measure or detect the value of a

process variable that is being measured. An element/sensor will generally represent the

measured value of a process variable as an electronic sign, a pressure signal or some

other type of signal. Some examples of elements/sensors and their symbols are shown

below:

Name Process Variable

P&ID Symbol Special /

Electronic Symbol

Picture Output Signal

RTD (Resistive Temperature

Detector) Temperature TE

RTD

Electronic

Ohms, change in resistance is affected by temp.

TC (Thermocouple)

Temperature

TE

TC

Electronic

Volts, voltage produced in proportion to temp.

Orifice Plate Flow

(LPM,GPM,...)

Differential Pressure, increases as a function of flow.

Venturi Tube Flow

(LPM,GPM,...)

FE

Differential Pressure, increases as a function of flow.

5.8 Transmitters Transmitters are devices used to convert the weak, non-standard analog signals from

elements and sensors into standard electronic or pneumatic signals which are transmitted

(sent) to another instrument, often at a different location. Some types of transmitters

have their associated element/sensors built in (integrated). Most of the time the output is

a standard 4-20 mA signal or a 3-15 psi signal.

5.8 Transmitters (cont.) Transmitter examples:

Name Process Variable

P&ID Symbol Two Wire Transmitter

Electrical Symbol Picture

Output Signal

Ultrasound Level

Tansmitter

Level (cm, ft,...)

LT

LT

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Pressure Tansmitter

Pressure (psi, kPa, inH2O,...)

PT

PT

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Differential Pressure

Tansmitter

Pressure difference

between two sources

(psi, kPa, inH2O,...)

DP

H

L

DP

H

L

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Temperature Transmitter

Resistance, Voltage or

other signal from

temperature element

TT

TT

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Flow Transmitter for use with Orifice Plate

(DP cell)

Differential Pressure cross Orifice Plate

(psi, kPa, inH2O,...)

FT

H

L

FT

H

L

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Most of the time the output is a standard 4-20 mA signal or a 3-15 psi signal.

5.9 Indicators Indicators are devices that are used to display the current value of a given process

variable. Some indicators such as pressure gauges measure a process variable directly

and display the measurement. Other indicators display process variables measurements

sent from standard signals from transmitters in remote locations, some examples include

remote pressure indicators, temperature indicators, level indicators and flow indicators.

A few examples of indicators and their symbols are shown below:

Name Process Variable

P&ID Symbol Electrical, Pneumatic or

Tubing Symbols Picture

Input Signal

Pressure Indicator

Pressure (psi, kPa,...)

PI

PI

tube

Direct Measurement

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Flow Indicator

Flow (LPM, GPM,...)

FI

FI

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Flow Indicator

Flow (LPM, GPM,...)

FI

FI

tube

Pneumatic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Temperature Indicator

Temperature

TI

TI

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

5.10 Recorders Recorders are devices that are used to record a permanent record the value of process

variables as they change over time. Some recorders record process variables on paper and

some recorders store measurements electronically. Usually recorders record process

variable measurements received in the form of standard signals from transmitters in

remote locations. Some examples include pressure recorders, temperature recorders,

level recorders and flow recorders. A few examples of indicators and their symbols are

shown below:

Name Process Variable

P&ID Symbol Electrical, Pneumatic or

Tubing Symbols Picture

Input Signal

Pressure Recorder

Pressure (psi, kPa,...)

PR

PR

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Flow Recorder

Flow (LPM, GPM,...)

FR

FR

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Flow Recorder

Flow (LPM, GPM,...)

FR

FR

tube

Pneumatic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

Temperature Transmitter

Level (ft, in, m,…)

LR

LR

+

-

Electronic

4-20mA, 3-5 psi,

0-10V or non-

standard signal.

5.11 Standard 4-20mA Signals Standard 4-20mA signals are the most common standard electronic signals used to

represent and transmit the analog value of a process variable. A 4-20mA signal varies

linearly as a function of the value of the variable that the signal represents. The graph

below shows the relationship between percent value of a variable and a 4-20mA signal

that represent it:

0 mA

18 mA

16 mA

14 mA

12 mA

10 mA

8 mA

6 mA

4 mA

2 mA

20 mA

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Lo

op

Cu

rre

nt (m

A)

%PV

Figure 5.6, Loop Current vs. %PV

The equation to convert from %PV to loop current is Iloop = (16mA) (%PV/100%) + 4mA, to

convert from loop current back to %PV is %PV = (100%) (Iloop – 4mA) / 16mA.

The drawing below shows a generic schematic that represent the relationship between the

instruments in a typical process variable measuring, displaying and recording system.

Note that the interconnections between the electronic instruments make up a simple series

circuit known as a 4-20mA current loop

Transmitter

+

_

Element

Indicator Recorder

+

-

+24 VDC

Supply

+ - + -

4-20mA 4-20mA

4-20mA

4-20mA

Figure 5.7, 4-20mA Current Loop with a Two Wire Transmitter

5.11 Standard 4-20mA Signals (continued) If the transmitter is a two terminal device such as the one shown above, then the

transmitter is referred to as a two wire transmitter. Some transmitters require a constant

current greater than 4mA all times just to keep the internal electronics operating, so a

third terminal in introduced to the transmitter, this type of transmitter is referred to as a

three wire transmitter. The additional terminal is connected to the negative terminal of

the power supply through a return line as shown below:

Transmitter

+

out

Element

Indicator Recorder

+

-

+24 VDC

Supply

+ - + -

4-20mA 4-20mA 4-20mA

-

Return

Source

Figure 5.8, 4-20mA Current Loop with a Two Wire Transmitter

5.12 Standard 3-15 PSI Signals and Systems The standard 3-15 PSI signal is the most common pneumatic signal standard used to

represent the values of process variables. The pressure of 3-15 PSI pressure signal varies

linearly as a function of the value of the process variable it is representing as shown in the

graph below:

0 mA

12 psi

9 psi

6 psi

3 psi

15 mA

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Pre

ssu

re (

psi)

%PV

Figure 5.9, %PV vs. Pressure for a 3-15 PSI Signal

5.12 Standard 3-15 PSI Signals and Systems (continued) Usually a 3-15psi signals is used to represent a variable that is transmitted from one

instrument to at least one or more other instruments that receive the signal. The generic

schematic below shows the relationship between the instruments in a typical 3-15psi

signal based pneumatic based measuring, indicating and recording system:

TransmitterElement

Indicator Recorder

Air Supply, 20 psi

3 – 15 psi Signal

Figure 5.10, Generic Pneumatic Measurement, Indicating and Recording System