The post Integration of formulas with examples appeared first on ChemE blog.

]]>Integration is the process of determining the area under the curve of an equation. This is the opposite of differentiation.

The antiderivative (F) is a function whose derivative turns out to be the original function (f).

The standard notation of an integral is formed by a sign that looks like the letter S. The total area (A) under the curve of a graph is then calculated between the lower (a) and upper (b) border of x. Integrals with borders over a specific interval are called definite integrals.

The antiderivative is calculated by integration of the original function. Also note that no x interval is considered when you determine the antiderivative equation. General solutions of integrals without borders are called indefinite integrals.

The original equation is returned if you take the derivative of the antiderivative.

The total area (A) can be approximated by dividing the graph into small rectangles and adding these up.

The total sum of all the small rectangles is approximately equal to the definite integral.

The integration of basic formulas take the following form. Also note that the constant (C) of the antiderivative (F) would drop out of the equation after differentiation to the original formula (f).

X to the power of

Fractions

Power equations

Exponentials

Natural logarithmic

Logarithmic

Trigonometric

The post Integration of formulas with examples appeared first on ChemE blog.

]]>The post What is the derivative of a function? appeared first on ChemE blog.

]]>The derivative of a function is defined as the change of the function (y) relative to a small change of the variable (x). The derivative is written in the form of y’ or dy/dx. The process of calculating the derivative is called differentiation.

An example from the introduction to balances was a population balance of the Netherlands. So can we therefore now apply the concept of derivatives to the concept of balances? Indeed, yes we can.

The general framework of balances consisted of accumulation, in, out, and production.

Accumulation is just another word for change of the variable over time. Then this takes the form of the time derivative. We will denote the number of people with N and the time with t.

Filling in the numbers (see picture) gives the value for the time derivative as a result.

This example served to convey the idea that the derivative is ratio of the change of the function (number of people) over the change the variable (time). However, the accuracy of the derivative is dependent on the step size of the variable. Therefore more accurate values of the derivative would we obtained by taking the numbers on a monthly, daily, or even hourly basis.

Linear equations consist of a slope (a) and an intersect (b).

.

The derivative is equal to the change of y relative to the change of x. Or to the slope in the case of linear equations.

Let’s test this principle with an example. What is the derivative of the following function?

We can also use the definition of the derivative to check this result. This can be done by calculating the rate of change of y over a small change of x. We can for example take the difference between x2 = 2.1 and x1 = 2.

The definition of the derivative gives also a value of 6. This equation and it’s derivative can also be plotted in a graph. It can be seen that the function is changing with a constant speed of 6 and that the derivative of the function is always equal to 6.

Polynomial equations are characterized by containing a variable x that is raised to a power of 2 or higher. For example a second order polynomial:

Then the derivative takes the general form:

This indicates that the derivative of polynomial equations is dependent on the value of x. Let’s study this concept further with an example of a second order polynomial.

The derivative of the function appears to be linear with a slope of 12 and an intersect of 4.

The function at point x = 2 is equal to 35.

The derivative at point x = 2 is equal to 28.

This means that the original function is changing with a slope of 28 at point x = 2. Although other values would be obtained for different values of x. Polynomial equations therefore differ from linear equations by having a variable slope. This result can be approximated by calculating the difference of the function over a very small step around x = 2.

We could approximate the actual value of 28 more accurately by taking a smaller step size around x = 2.

The derivative of the first derivative is called the second derivative.

The derivation of the second derivative proceeds in similar way as the first derivative. The blue line in the following graph indicates that the function proceeds with an increasing slope. This can also be seen by noting that the red line or first derivative is changing linearly over x. This linearity means that the second derivative is a constant.

Higher order polynomials contain the variable x that is raised to a power bigger than 2.

The general form of the derivative can be computed by the following form:

We could for example have the following polynomial:

Then the first derivative would be:

And proceeding the differentiation even further:

The third derivative is a constant.

The post What is the derivative of a function? appeared first on ChemE blog.

]]>The post Introduction to mass, energy and momentum balances appeared first on ChemE blog.

]]>The general form of a balance is made up out of accumulation, inflow, outflow, production and consumption. A balance is always made on a control volume. This control volume specifies the borders of the balance. The in- and outflowing terms pass this border and production/accumulation terms take place inside these bordes.

These terms can be combines into the following equation:

Accumulation = in – out + production – consumption

The consumption term is sometimes treated as a negative production term. The equation therefore simplifies into:

Accumulation = in – out + production

An example is the best way to illustrate the concept of balances. The Netherlands, for example, had a population of 16 900 726 people in 2016 (source: cbs). If we want to determine the population growth of the Netherlands with the help of balances, then we have to choose the entire country as our control volume.

The immigration has been a loaded subject in recent years. Today, in our balance, it will serve as the inflowing term. A total of 230 739 people (legally) entered the Netherlands in the year 2016. The inflowing term therefore amounts 230 739 people/year. A total of 151 545 people left the country in 2016 by emigration. This number represents the outflowing term. A total of 172 520 babies were born in 2016. This represents the production term.The production was counteracted by a death rate of 148 977 people. We can treat this term as consumption. There are of course other minor influences as people illegally entering the country or people moving on holiday. However, effective engineers make assumptions to make problems more manageable. We therefore neglect these terms.

Now we can setup our balance to determine population growth:

Accumulation = in – out + production – consumption

Population growth = immigration – emigration + birthrate – death rate

Population growth = 230730 – 151545 + 172520 – 148 997 = 102 717 people/year

This means that the population of people grew with 102 717 people in the year 2016. This represents a growth rate of 102 717/16 900 726 = 0.6%.

Balances can also be used for solving mass, energy and momentum problems. This, however, will be covered in future posts.

The post Introduction to mass, energy and momentum balances appeared first on ChemE blog.

]]>The post Units of temperature appeared first on ChemE blog.

]]>These temperature scales share two features:

- Each scale uses a reference point as zero degrees
- And each scale changes with a certain size

The following graph displays both degrees Celsius and degrees Fahrenheit in terms of the Kelvin temperature scale. Note both units differ in their zero point and slope.

The Celsius temperature scale uses the freezing point of water, namely zero degrees Celsius, as a reference point. Furthermore, hundred degrees Celsius corresponds with the boiling temperature of water. These temperature scales are always defined at standard atmospheric pressure, because the boiling point of a compound changes with pressure.

The following formula displays units conversion from degrees Fahrenheit and Kelvin to Celsius temperature unit.

The Fahrenheit units of temperature defines the freezing point of water at 32 degrees Fahrenheit. In addition, the boiling point of water is defined at a temperature of 212 degrees Fahrenheit (standard atmospheric pressure).

The following formula displays temperature unit conversion to degrees Fahrenheit.

We finally discuss the Kelvin units of temperature, mainly used in scientific calculations.The zero degrees of Kelvin point describes the lowest possible temperature. All molecules will show minimal movement at this temperature. And the second reference point corresponds with the triple point of water, namely 273,15 K. This triple point describes the conditions where the gas, liquid and solid form of a component are in equilibrium.

Above conversions can be simplified for calculations with energy differences. The Kelvin and degrees Celsius temperature differences are actually equal to each other. Furthermore, energy difference in Fahrenheit is converted to difference in Kelvin by dividing the Fahrenheit difference with 1,8.

I’m confident above conversions prepares you for the usages of the units of temperature of degrees Celsius, degrees Fahrenheit and Kelvin.

The post Units of temperature appeared first on ChemE blog.

]]>The post Units of weight appeared first on ChemE blog.

]]>This article will review the mass units used in the UK imperial, US customary and metric system. The imperial and US customary system consists of three subsystems. Namely the avoirdupois, apothecary and troy weight systems. In addition, the metric weight units scale up and down with a factor of 10. Finally, each table contains the conversion factor to the SI unit of weight kilogram (kg).

The people in the United states and United Kingdom uses the avoirdupois weight units for day-to-day usage. Avoirdupois derives from the French word for goods of weight. Furthermore the abbreviation of av indicates the use of the avoirdupois system.

The weight and measurement act (WMA) of 1963 agreed upon the definition of 1 pound av as 0,453 592 37 kg. Unit conversion becomes a possibility as a result of this agreement.

People in the United Kingdom makes use of the ounce, pound, stone, hundred weight and UK ton units of weight. Although units from the metric system are also used. See table 1 for internal and SI units of weight conversions.

Firstly a pound (lb av) is built out of 16 ounces (oz av).

Furthermore a stone (st av) consists of 14 pound.

In addition a hundred weight (cwt) is the equivalent of 8 stone.

Finally a UK ton consists of 20 hundred weights.

By continuation of the proceeding factors other conversion can be derived.

The people in the United States uses more avoirdupois units of weight than people in the United Kingdom. Since the grain and dram expands their vocabulary. They also use two definitions of the hundredweight (cwt) and ton. Namely the short variant (US) and long edition (UK).

A grain (gr av) is the smallest weight unit in the US customary weight measurement and equivalents 64,8 milligram (mg) of the metric system.

A dram (dram av) furthermore consists of 875/32 = 27,34 grains.

In addition a US short hundred weight equivalents 100 pounds (instead of 14*8 = 112 for the UK long edition).

Therefore a short US ton consists of 20 US short hundredweights or 20*(100/112) = 17,86 UK long hundredweights.

Furthermore a metric tonne equivalents 1000 kg. Therefore you should always use unit abbreviations. For example: “I’d like to order 1 US ton of pancake batter”. Not specifying the type of ton could saddle you up with an 12% surplus of UK ton.

Table 2 contains the internal and SI units of weight conversion factors.

Apothecaries in the United Kingdom and United states used special weight units in the past for their small measurements. They have however become obsolete after prohibition by the governments.

The Troy units of weight accommodates the units used for measurement of precious metals and valuable gemstones. The traders in the United Kingdom converted to the metric system. However, the traders in the United States (and therefore international traders) still use this units of weight system.

The name Troy derives from the town of Troyes in Northern France. This town was well-known for its commercial fairs during the medieval times.

Table 3 contains the Troy units of weight, namely: grain, pennyweight, ounce and pound.

One troy pound (lb troy) equivalents 0,3732 SI kilogram. It’s therefore significantly different from the avoirdupois pound (lb av), which equivalents 0,4536 kilogram.

A troy grain (gr troy) equals the avoirdupois grain (gr av).

Furthermore a pennyweight (dwt) consists of 24 grains.

Also one troy ounce (oz troy) equivalents 20 pennyweight. However, the troy ounce specifies a different amount of mass than the avoirdupois ounce (oz av).

And finally the troy pound (lb troy) consists of 12 troy ounce.

The carat unit is used in the branch of jewelery and indicates how pure gold is. Namely 24 carats equivalents 100 wt% pure gold. Furthermore 12 carats equals 50 wt% and so on.

The metric carat equivalents 3 grains in the imperial and 200 mg in the metric unit system.

The metric units of weight scale up and down in a similar method as in previous articles about lenth, area and volume

The tonne, kilogram, gram and milligram are the most used metric units of weight. However other units are also used in specific field. Table 4 displays the metric weight units with conversion to SI kilogram and imperial counterparts.

The post Units of weight appeared first on ChemE blog.

]]>The post Volume units of measurement appeared first on ChemE blog.

]]>This article will describe the volume units of measurement from the imperial, US customary and metric system. We will learn that the imperial and US customary volume units can be divided into linear, liquid and dry measurement. We will also discuss conversion to the SI unit of cubic meter.

Table 1 contains the three dimensional volume unit conversion factors for the imperial and US customary system. We can make a distinction between cubic inch, cubic feet, cubic yard and cubic statute mile. Also, we have already seen one dimensional length and two dimensional area unit conversion factors from previous articles. So by raising the one dimensional conversion factors to the power of three the unit conversion becomes possible.

One feet, for example, equals 12 inches. Therefore one cubic feet equals 12^3 = 1 728 cubic inches

Furthermore, a yard is the equivalent of 0,9144 meter, according to the weight and measurement act of 1963 (WMA 1963). So one cubic yard equals 0,9144^3 = 0,765 cubic meter (m3). The other SI equivalent units can also be calculated with upcoming conversion factors.

The United States and United Kingdom use liquid volume units for commercial measurement of liquids. We can hereby distinguish between fluid ounce, pint, quart and gallons.

Both countries use the same names for the the liquid units. However, these units don’t represent the same amount of liquid. Therefore it is important to indicate the unit system when using liquid volume units.

Table 2 displays the imperial liquid volume units of measurement. One pint in the UK imperial system consists of twenty fluid ounces. Therefore one quart (UK) equals forty fluid ounces or two UK pint. Also, one gallon (UK) is the equivalent of 160 fluid ounce (UK), 8 UK pint or 4 UK quart.

The UK gallon is the equivalent of 4,546 082 L or 0,004 546 082 cubic meter. Therefore imperial – SI conversion factors can be defined by combining the previous internal conversion factors.

Table 3 displays the US customary liquid volume units of measurement. These units are different from their imperial counterparts. One US liquid gallon for example equals 3,785 L from the metric system. This is significantly different from the UK gallon, which equals metric 4,546 liter.

The oil industry also uses the barrel as a volume unit. One barrel equals 42 US liquid gallon, 168 US quart, 336 US pint or 5 376 US fluid ounces. From the preceding conversion factors the definition of the barrel follows as 159 liters.

Companies from the United Kingdom don’t use dry volume units anymore. However, these units are still used for powdered materials in the United States. We can distinguish between the dry pint, dry quart, US dry gallon, peck and bushel (see table 4).

Table 5 contains the commonly used volume units of measurement from the metric system. The metric volume units includes cubic millimeter, cubic centimeter, cubic decimeter, cubic meter, cubic decametre, cubic hectometer and cubic kilometer.

The metric system also makes use of the liter volume unit. One liter equals 1 cubic decimeter or 0,001 cubic meter.

The post Volume units of measurement appeared first on ChemE blog.

]]>The post Units of area appeared first on ChemE blog.

]]>The area of an object specifies the two-dimensional space it covers. For instance, rectangular areas are calculated by multiplying the length and width of the object. For spherical, circular and other objects different formula’s apply.

This two dimensional relationship between area and length affects the conversion factors for two dimensional area. The one dimensional length was discussed in the units of length article. Squaring (multiplying with itself) these conversion factors gives the conversion for units of area.

The Imperial and US customary system uses the same units of measure for area. Table 1 displays the area conversion factors for these systems.

From length conversions we know that 1 foot consists of 12 inch. Therefore one square foot must consist of 12^2 = 144 square inch.

We also can define the square yard as the equivalent of 3^2 = 9 square feet.

Furthermore one square rod is built from 5.5^2 or about 30 square yard.

Finally one square mile consists of 320^2 = 102 400 square rod.

For surveyors units of area only the acre and square feet ares still used in the United States and United Kingdom. The square feet was already discussed in the previous section and is the equivalent of 0.0929 square meter. The acre is calculated by multiplying a square feet with a factor of 43 560. The conversion factors for surveyors area are supplied in table 2.

Finally, the area unit conversion for the metric system proceeds in similar fashion. From one dimensional length we already knew that these convert in steps of 10. Therefore area will convert in steps of 10^2 = 100. Some of the metric units of area are displayed in table 3. Although only square meter and hectare (square hectometer) are the commonly used.

The post Units of area appeared first on ChemE blog.

]]>The post Units of length appeared first on ChemE blog.

]]>The imperial and US customary system divides the length measurement into three classes, namely:

- Linear
- Nautical
- Surveyor length

The conversion in these unit systems can become quit complicated, as we will find out later. The metric system in contrast uses only scales of 10 the convert between the units. This makes unit conversion less comprehensive. However, the imperial system approaches the real length of the used goods better.

The linear length units used in the United Kingdom include in increasing order of length: line, inch, feet (foot), yard, statute mile (land mile) and statute league (land league). The people in the United States, however, only uses inch, feet (foot), yard and statute league (land league).

An inch consists of 12 lines (line).

Furthermore, a feet (ft) consists of 12 inch.

And yard (yd) is the equivalent of 3 feet.

The land mile (st. mi) consists of 1760 yard.

And finally the land league (st. lg) is the equivalent of 3 land miles.

So how can we convert to the metric system and back? The Weight and Measurement act of 1963 (WMA 1963) defined the feet as an equal length of 0.3048 meter of the metric system. Therefore unit conversion becomes a possibility. See table 1 for internal en SI conversion factors.

A yard consist of 3 feet, as stated before. However, of how many feet does a yard consists? Also, a feet consists of 12 inches. So a yard consists of 12 x 3 = 36 inches.

A line consists of 12 points (pt). This means that the point is the equivalent of 0.000176 m (0.176 mm). Quite pointless….

In the navigation on sea the fathom and nautical mile units of length are used by both the UK and US. Other nautical length measurements are not used anymore and will not be reviewed in this article.

A nautical mile (naut. mi) equals 1853.184 meters of the SI system.

Furthermore, a nautical mile consists of about 3040/3 (or about 1013) fathom (fath) units.

The surveyors units of length used by people in the United States include: link, chain, rod and furlong. People in the United Kingdom, however, only used the furlong (horse races).

We can define 1 Ramsden’s chain as 30.48 meter in the SI system.

Furthermore we know that a rod (rd) consists of 16.5 link (lk).

Also, the chain (ch) consists of 100 links.

And finally a furlong (fur) is the equivalent of 6.6 chain.

The metric system is based on the meter (m) and scales up and down with a factor of 10. The most used units are millimeter (mm), centimeter (cm), decimeter (dm), meter (m) and kilometer (km). Although some specific units are used in specialized fields.

I hope you learned something out of this article. Let me know in the comment section below!

The post Units of length appeared first on ChemE blog.

]]>The post Introduction to units and dimensions appeared first on ChemE blog.

]]>Did you know that NASA once crashed a Mars Orbiter due to a mix up in the units of measurement?

How could this happen?

The NASA engineers were working with both the imperial and metric system of units. This caused mistakes leading to the crash of the Mars Orbiter. Hopefully, something like that will never happen to you after reading this article.

Also, knowledge of units and dimensions gives you much more options in life.

Large parts of the engineering data out there is (or was) measured in different units than you are used to. Try searching on engineering toolbox and sooner or later you will come across an article you can’t use because you don’t know the units.

Your world will be so much bigger if you know more units to measure that size in. Whether you measure it in meters, inches or furlongs.

A recent study showed this:

This was a joke (just in case)

A unit of measurement is basically a definition of magnitude and quantity. But the way how to define these is not universally agreed upon.

First of all there is the imperial system of measurement that originated from the British empire. The countries on the British Isles mainly converted to the metric system but some units of the imperial system are still in use.

Secondly, there is also the US customary system. This system originated from the imperial system due to the colonial history of the United States. The imperial and US customary system are different from each other because of some changes in later history.

There are for example different ways to express the weight in ton, namely the US ton and UK ton. Also, the metric system uses tonne for weight measurement. That’s why it is important to always be specific in your unit quantities.

This system finds its origin in the French Republic of Napoleon and is the most used system of measurement. The metric system uses decimal multiples to express units. This means the units scale up and down in multiples of 10. Many of the standard units in the metric system are the same as in the SI system.

The international system of units (SI) is the modern form of the metric system and is used in scientific documentation. It is built on 7 base units:

- kilogram (kg) for mass
- meter (m) for length
- second (s) for time
- kelvin (K) for temperature
- mole (mol) for amount
- ampere (A) for electric current
- candela (cd) for luminous intensity

The SI units system will be used for the engineering calculations in this blog.

In this articles we have shortly introduced the imperial, US customary, metric and international system of units. In upcoming articles we will review the differences between these systems in measuring specific quantities like length, mass and volume.

I hope this article got you motivated to learn more about units and dimensions. Click the link in the box below for articles about units and dimensions.

If you like to read more about the history of units and measurement I can recommend the “Encyclopedia of scientific units, weights and measures” written by Francois Cardarelli. It’s a very good book that learned me a lot about units and dimensions.

Coulson and Richardson volume 1 “Fluid flow, heat transfer and mass transfer” chapter 1 and Perrys “chemical engineerings handbook” section 1 also covers this material.

How do you feel about the several units of measurement? Which system do you prefer? Please share your thoughts in the comment section below

The post Introduction to units and dimensions appeared first on ChemE blog.

]]>