- ... conversions
^{1.1} - An interesting point to make here is the United States did get something right when they designed their monetary system of dollars and cents. This is essentially a metric system of measurement, with 100 cents per dollar. The founders of the USA wisely decided to avoid the utterly confusing denominations of the British, with their pounds, pence, farthings, shillings, etc. The denominations of penny, dime, dollar, and eagle ($10 gold coin) comprised a simple power-of-ten system for money. Credit goes to France for first adopting a metric system of general weights and measures as their national standard.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... alter
^{1.2} - A basic mathematical identity is that multiplication of any quantity by 1 does not change the value of that original quantity. If we multiply some quantity by a fraction having a physical value of 1, no matter how strange-looking that fraction may appear, the value of the original quantity will be left intact. The goal here is to judiciously choose a fraction with a physical value of 1 but with its units of measurement so arranged that we cancel out the original quantity's unit(s) and replace them with the units we desire.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... pressure
^{1.3} - Density figures taken or derived from tables in the CRC Handbook of Chemistry and Physics, 64th Edition. Most liquid densities taken from table on page F-3 and solid densities taken from table on page F-1. Some liquid densities taken from tables on pages E-27 through E-31. All temperatures at or near 20 C.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... derived
^{1.4} - The only exception to this rule being units of measurement for angles, over which there has not yet been full agreement whether the unit of the radian (and its solid counterpart, the steradian) is a base unit or a derived unit.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... system
^{1.5} - The older name for the SI system was “MKS,” representing meters, kilograms, and seconds.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... wonderful
^{1.6} - I'm noting my sarcasm here, just in case you are immune to my odd sense of humor.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... instrumentation
^{1.7} - Relativistic physics deals with phenomena arising as objects travel near the speed of light. Quantum physics deals with phenomena at the atomic level. Neither is germane to the vast majority of industrial instrument applications.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... definition
^{1.8} - A common definition of energy is the “ability to do work” which is not always true. There are some forms of energy which may not be harnessed to do work, such as the thermal motion of molecules in an environment where all objects are at the same temperature. Energy that has the ability to do work is more specifically referred to as exergy. While energy is always conserved (i.e. never lost, never gained), exergy is a quantity that can never be gained but can be lost. The inevitable loss of exergy is closely related to the concept of entropy, where energy naturally diffuses into less useful (more homogeneous) forms over time. This important concept explains why no machine can never be perfectly (
%) efficient, among other things.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... vectors
^{1.9} - A vector is a mathematical quantity possessing both a magnitude and a direction. Force (), displacement (), and velocity () are all vector quantities. Some physical quantities such as temperature (), work (), and energy () possess magnitude but no direction. We call these directionless quantities “scalar.” It would make no sense at all to speak of a temperature being “79 degrees Celsius due North” whereas it would make sense to speak of a force being “79 Newtons due North”. Physicists commonly use a little arrow symbol over the variable letter to represent that variable as a vector, when both magnitude and direction matter. Thus represents a force vector with both magnitude and direction specified, while plain merely represents the magnitude of that force without a specified direction. A “dot-product” is one way in which vectors may be multiplied, and the result of a dot-product is always a scalar quantity.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... wheel
^{1.10} - Note that this calculation will assume all the work of towing this load is being performed by a single wheel on the truck. Most likely this will not be the case, as most towing vehicles have multiple driving wheels (at least two). However, we will perform calculations for a single wheel in order to simplify the problem.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... not
^{1.11} - Consider the example of applying torque to a stubbornly seized bolt using a wrench: the force applied to the wrench multiplied by the radius length from the bolt's center to the perpendicular line of force yields torque, but absolutely no work is done on the bolt until the bolt begins to move (turn).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... gravity
^{1.12} - In practice, we usually see heavy objects fall faster than light objects due to the resistance of air. Energy losses due to air friction nullify our assumption of constant total energy during free-fall. Energy lost due to air friction never translates to velocity, and so the heavier object ends up hitting the ground faster (and sooner) because it had much more energy than the light object did to start.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Law
^{1.13} - Hooke's Law may be written as without the negative sign, in which case the force () is the force applied on the spring from an external source. Here, the negative sign represents the spring's reaction force to being displaced (the restoring force). A spring's reaction force always opposes the direction of displacement: compress a spring, and it pushes back on you; stretch a spring, and it pulls back. A negative sign is the mathematically symbolic way of expressing the opposing direction of a vector.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... vector
^{1.14} - Technically, it is a pseudovector, because it does not exhibit all the same properties of a true vector, but this is a mathematical abstraction far beyond the scope of this book!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...flywheels
^{1.15} - A “flywheel” is a disk on a shaft, designed to maintain rotary motion in the absence of a motivating torque for the function of machines such as piston engines. The rotational kinetic energy stored by an engine's flywheel is necessary to give the pistons energy to compress the gas prior to the power stroke, during the times the other pistons are not producing power.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... advantage
^{1.16} - Technically, mechanical advantage should be defined by the ratio of input motion to output motion, rather than being defined in terms of force. The reason for this is if friction happens to exist in the machine, it will cause a degradation of force but not of motion. Since “mechanical advantage” is supposed to represent the ideal ratio of the machine, it is always safest to define it in terms of motion where friction will not affect the calculation. For a frictionless machine, however, defining mechanical advantage in terms of force is perfectly legitimate, and in fact makes more intuitive sense, since a larger mechanical advantage always corresponds with force multiplication from input to output.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... torque
^{1.17} - “Torque” is to rotational motion as “force” is to linear motion. Mathematically, torque () is defined as the cross-product of force acting on a radius (
).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... illustration
^{1.18} - I am indebted to NASA for this and the rest of the black-and-white gear illustrations found in this section. All these illustrations were taken from NASA technical reports on gearing.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... gear
^{1.19} - Here, each gear is shown simply as a toothless wheel for the sake of simplicity. Truth be told, your humble author has difficulty drawing realistic gear teeth!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... centered
^{1.20} - An interesting feature of many flat-belt sheaves is a slight “crown” shape to the sheave, such that the diameter is slightly larger at the sheave's center than it is at either side edge. The purpose of this crown is to help the belt center itself while in operation. As it turns out, a flat belt naturally tends to find the point at which it operates under maximum tension. If the belt happens to wander off-center, it will naturally find its way back to the center of the sheave as it rotates because that is where the tension reaches a maximum.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... theoretical
^{1.21} - In practice, not all of these 24 “speeds” are recommended, because some of the front/rear sprocket selections would place the chain at an extreme angle as it engaged with both sprockets. In the interest of extending chain life, it should run as “straight” on each sprocket as possible.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... monatomic
^{1.22} - Helium at room temperature is a close approximation of an ideal, monatomic gas, and is often used as an example for illustrating the relationship between temperature and molecular velocity.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Kelvin
^{1.23} - Kelvin is typically expressed without the customary “degree” label, unlike the three other temperature units: (degrees) Celsius, (degrees) Fahrenheit, and (degrees) Rankine.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... combustion
^{1.24} - Animals process food by performing a very slow version of combustion, whereby the carbon and hydrogen atoms in the food join with oxygen atoms inhaled to produce water and carbon dioxide gas (plus energy). Although it may seem strange to rate the energy content of food by measuring how much heat it gives off when burnt, burning is just a faster method of energy extraction than the relatively slow processes of biological metabolism.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... spontaneously
^{1.25} - Heat may be forced to flow from cold to hot by the use of a machine called a heat pump, but this direction of heat flow does not happen naturally, which is what the word “spontaneous” implies. In truth, the rule of heat flowing from high-temperature to cold-temperature applies to heat pumps as well, just in a way that is not obvious from first inspection. Mechanical heat pumps cause heat to be drawn from a cool object by placing an even cooler object (the evaporator) in direct contact with it. That heat is then transferred to a hot object by placing an even hotter object (the condenser) in direct contact with it. Heat is moved against the natural (spontaneous) direction of flow from the evaporator to the condenser by means of mechanical compression and expansion of a refrigerant fluid.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Radiation
^{1.26} - In this context, we are using the word “radiation” in a very general sense, to mean thermal energy radiated away from the hot source via photons. This is quite different from nuclear radiation, which is what some may assume this term means upon first glance.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Kelvin
^{1.27} - Or in degrees Rankine, provided a suitably units-corrected value for the Stefan-Boltzmann constant were used.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... energy-intensive
^{1.28} - Jim Cahill of Emerson wrote in April 2010 (“Reducing Distillation Column Energy Usage” Emerson Process Expert weblog) about a report estimating distillation column energy usage to account for approximately 6% of the total energy used in the United States. This same report tallied the number of columns in US industry to be approximately 40000 total, accounting for about 19% of all energy used in manufacturing processes!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Celsius
^{1.29} - An important detail to note is that specific heat does not remain constant over wide temperature changes. This complicates calculations of heat required to change the temperature of a sample: instead of simply multiplying the temperature change by mass and specific heat (
or
), we must integrate specific heat over the range of temperature (
), summing up infinitesimal products of specific heat and temperature change () over the range starting from temperature through temperature then multiplying by the mass to calculate total heat required. So, the specific heat values given for substances at 25 C only hold true for relatively small temperature changes deviating from 25 C. To accurately calculate heat transfer over a large temperature change, one must incorporate values of for that substance at different temperatures along the expected range.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... assume
^{1.30} - In reality, the amount of heat actually absorbed by the pot will be less than this, because there will be heat losses from the warm pot to the surrounding (cooler) air. However, for the sake of simplicity, we will assume all the burner's heat output goes into the pot and the water it holds.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... container
^{1.31} - We will assume for the sake of this example that the container holding the water is of negligible mass, such as a Styrofoam cup. This way, we do not have to include the container's mass or its specific heat into the calculation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... other
^{1.32} - An alternative way to set up the problem would be to calculate for each term as
, making the iron's heat loss a negative quantity and the water's heat gain a positive quantity, in which case we would have to set up the equation as a zero-sum balance, with
. I find this approach less intuitive than simply saying the iron's heat loss will be equal to the water's heat gain, and setting up the equation as two positive values equal to each other.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... fluid
^{1.33} - This is not far from the hypotheses of eighteenth-century science, where heat was thought to be an invisible fluid called caloric.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... measure
^{1.34} - A useful analogy for enthalpy is the maximum available balance of a bank account. Suppose you have a bank account with a minimum balance requirement of $32 to maintain that account. Your maximum available balance at any time would be the total amount of money in that account minus $32, or to phrase this differently your maximum available balance is the most money you may spend from this account while still keeping that account open. Enthalpy is much the same: the amount of thermal energy a sample may “spend” (i.e. lose) before its temperature reaches 32 degrees Fahrenheit.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... values
^{1.35} - Appealing to the maximum available balance analogy, if we compared the maximum available balance in your bank account before and after a transaction, we could determine how much money was deposited or withdrawn from your account simply by subtracting those two values.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...)
^{1.36} - Following the formula
, we may calculate the heat as (1)(1)() = 45 BTU. This is obviously the same result we obtained by subtracting enthalpy values for water at 170 F and 125 F.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... occurs
^{1.37} - The word “latent” refers to something with potential that is not yet realized. Here, heat exchange takes place without there being any realized change in temperature. By contrast, heat resulting in a temperature change (
) is called sensible heat.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... one
^{1.38} - Latent heat of vaporization also varies with pressure, as different amounts of heat are required to vaporize a liquid depending on the pressure that liquid is subject to. Generally, increased pressure (increased boiling temperature) results in less latent heat of vaporization.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Fahrenheit
^{1.39} - The reason specific heat values are identical between metric and British units, while latent heat values are not, is because latent heat does not involve temperature change, and therefore there is one less unit conversion taking place between metric and British when translating latent heats. Specific heat in both metric and British units is defined in such a way that the three different units for heat, mass, and temperature all cancel each other out. With latent heat, we are only dealing with mass and heat, and so we have a proportional conversion of or left over, just the same as if we were converting between degrees Celsius and Fahrenheit alone.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... paper
^{1.40} - Styrofoam and plastic cups work as well, but paper exhibits the furthest separation between the boiling point of water and the burning point of the cup material, and it is usually thin enough to ensure good heat transfer from the outside (impinging flame) to the inside (water).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... fire
^{1.41} - This is a lot of fun to do while camping!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... pressure
^{1.42} - This may be done in a vacuum jar, or by traveling to a region of high altitude where the ambient air pressure is less than at sea level.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... influences
^{1.43} - The mechanism of this influence may be understood by considering what it means to boil a liquid into a vapor. Molecules in a liquid reside close enough to each other that they cohere, whereas molecules in a vapor or gas are relatively far apart and act as independent objects. The process of boiling requires that cohesion between liquid molecules to be broken, so the molecules may drift apart. Increased pressure encourages cohesion in liquid form by helping to hold the molecules together, while decreased pressure encourages the separation of molecules into a vapor/gas.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Fahrenheit)
^{1.44} - As mentioned previously, a useful analogy for enthalpy is the maximum available balance for a bank account with a $32 minimum balance requirement: that is, how much money may be spent from that account without closing it out.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... useful
^{1.45} - At first it may seem as though the enthalpy of steam is so easy to calculate it almost renders steam tables useless. If the specific heats of water and steam were constant, and the latent heat of vaporization for water likewise constant, this would be the case. However, both these values ( and ) are not constant, but rather change with pressure and with temperature. Thus, steam tables end up being quite valuable to engineers, allowing them to quickly reference heat content of steam across a broad range of pressures and temperatures without having to account for changing and values (performing integral calculus in the form of
for specific heat) in their heat calculations.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... load
^{1.46} - This is not unlike calculating the voltage dropped across an electrical load by measuring the voltage at each of the load's two terminals with respect to ground, then subtracting those two measured voltage values. In this analogy, electrical “ground” is the equivalent of water at freezing temperature: a common reference point for energy level.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... determine
^{1.47} - Applying the maximum available balance analogy to this scenario, it would be as if your bank account began with a maximum available balance of $1287 and then finished with a maximum available balance of $138 after an expenditure: the amount of money you spent is the different between the initial and final maximum available balances ($1287 $138 = $1149).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... point
^{1.48} - When HO is at its triple point, vapor (steam), liquid (water), and solid (ice) of water will co-exist in the same space. One way to visualize the triple point is to consider it the pressure at which the boiling and freezing temperatures of a substance become the same.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... temperatures
^{1.49} - Anywhere between the triple-point temperature and the critical temperature, to be exact.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... point
^{1.50} - The triple point for any substance is the pressure at which the boiling and freezing temperatures become one and the same.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... fixed
^{1.51} - The non-freedom of both pressure and temperature for a pure substance at its triple point means we may exploit different substances' triple points as calibration standards for both pressure and temperature. Using suitable laboratory equipment and samples of sufficient purity, anyone in the world may force a substance to its triple point and calibrate pressure and/or temperature instruments against that sample.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...thermometer
^{1.52} - To be more precise, a propane tank acts like a Class II filled-bulb thermometer, with liquid and vapor coexisting in equilibrium.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... volume
^{1.53} - Steam boilers exhibit this same explosive tendency. The expansion ratio of water to steam is on the order of a thousand to one (1000:1), making steam boiler ruptures very violent even at relatively low operating pressures.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... same
^{1.54} - Class IIA systems do suffer from elevation error where the indicator may read a higher or lower temperature than it should due to hydrostatic pressure exerted by the column of liquid inside the tube connecting the indicator to the sensing bulb. Class IIB systems do not suffer from this problem, as the gas inside the tube exerts no pressure over an elevation.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... simplified
^{1.55} - Circulation pumps and a multitude of accessory devices are omitted from this diagram for the sake of simplicity.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... greater
^{1.56} - This is another example of an important thermodynamic concept: the distinction between heat and temperature. While the temperature of the pressurizer heating elements exceeds that of the reactor core, the total heat output of course does not. Typical comparative values for pressurizer power versus reactor core power are 1800 kW versus 3800 MW, respectively: a ratio exceeding three orders of magnitude. The pressurizer heating elements don't have to dissipate much power (compared to the reactor core) because the pressurizer is not being cooled by a forced convection of water like the reactor core is.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... array
^{1.57} - In this application, the heaters are the final control element for the reactor pressure control system.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... table
^{1.58} - Since the relationship between saturated steam pressure and temperature does not follow a simple mathematical formula, it is more practical to consult published tables of pressure/temperature data for steam. A great many engineering manuals contain steam tables, and in fact entire books exist devoted to nothing but steam tables.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... damage
^{1.59} - An experiment illustrative of this point is to maintain an ice-water mixture in an open container, then to insert a sealed balloon containing liquid water into this mixture. The water inside the balloon will eventually equalize in temperature with the surrounding ice-water mix, but it will not itself freeze. Once the balloon's water reaches 0 degrees Celsius, it stops losing heat to the surrounding ice-water mix, and therefore cannot make the phase change to solid form.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...pressure
^{1.60} - The concept of pressure is also applicable to solid materials: applying either a compressive or tensile force to a solid object of given cross-sectional area generates a pressure within that object, also referred to as stress.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... small
^{1.61} - To give some perspective on this, 1 pascal of pressure is equal to (only) 0.000145 pounds per square inch!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... immediately
^{1.62} - There is actually a speed of propagation to this increase in pressure, and it is the speed of sound within that particular fluid. This makes sense, since sound waves are nothing more than rapidly-changing regions of pressure within a material.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... inch
^{1.63} - Interestingly, the amount of pressure generated by the weight of a fluid depends only on the height of that fluid column, not its cross-sectional area. Suppose we had a column of water the same height (144 feet) but in a tube having an area twice as large: 2 square inches instead of 1 square inch. Twice the area means twice the volume of water held in the tube, and therefore twice the weight (124.8 lbs). However, since this greater weight is distributed over a proportionately greater area at the bottom of the tube, the pressure there remains the same as before: 124.8 pounds 2 square inches = 62.4 pounds per square inch.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... amount
^{1.64} - Suppose a 1 square inch piston were set on the top of this tall fluid column, and a downward force of 20 lbs were applied to it. This would apply an additional 20 PSI pressure to the fluid molecules at all points within the column. The pressure at the bottom would be 82.4 PSI, and the pressure at the middle would be 51.2 PSI.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... temperature
^{1.65} - Usually, this standard temperature is 4 degrees Celsius, the point of maximum density for water. However, sometimes the specific gravity of a liquid will be expressed in relation to the density of water at some other temperature. In some cases specific gravity is expressed for a liquid at one temperature compared to water at another temperature, usually in the form of a superscript such as 20/4 (liquid at 20 degrees Celsius compared to water at 4 degrees Celsius).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... gravity
^{1.66} - For each of these calculations, specific gravity is defined as the ratio of the liquid's density at 60 degrees Fahrenheit to the density of pure water, also at 60 degrees Fahrenheit.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... cumbersome
^{1.67} - A colleague of mine told me once of working in an industrial facility with a very old steam boiler, where boiler steam pressure was actually indicated by tall mercury manometers reaching from floor to ceiling. Operations personnel had to climb a ladder to accurately read pressure indicated by these manometers!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... purposes
^{1.68} - To give some perspective on just how little the liquid level changes in the well, consider a well-type manometer with a 1/4 inch (inside) diameter viewing tube and a 4-inch diameter circular well. The ratio of diameters for these two liquid columns is 16:1, which means their ratio of areas is 256:1. Thus, for every inch of liquid motion in the viewing tube, the liquid inside the well moves only
of an inch. Unless the viewing tube is quite tall, the amount of error incurred by interpreting the tube's liquid height directly as pressure will be minimal – quite likely less than what the human eye is able to discern on a ruler scale anyway. If the utmost accuracy is desired in a well manometer, however, we may compensate for the trifling motion of liquid in the well by building a custom ruler for the vertical tube – one with a
reduced scale (so that
of an inch of liquid motion in the tube reads as exactly 1 inch of liquid column) in the case of the 1/4 inch tube and 4 inch well dimensions.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... usually
^{1.69} - With few exceptions!
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... equal
^{1.70} - The origin of this unit for pressure is the atmospheric pressure at sea level: 1 atmosphere, or 14.7 PSIA. The word “bar” is short for barometric, in reference to Earth's ambient atmospheric pressure.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... pressure
^{1.71} - At sea level, where the absolute pressure is 14.7 PSIA. Atmospheric pressure will be different at different elevations above (or below) sea level.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ... Universal
^{1.72} - It should be noted that many different values exist for , depending on the units of measurement. For liters of volume, atmospheres of pressure, moles of substance, and Kelvin for temperature,
. If one prefers to work with different units of measurement for volume, pressure, molecular quantity, and/or temperature, different values of are available.
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- ... reason
^{1.73} - The conservation law necessitating equal current at all points in a series electric circuit is the Law of Charge Conservation, which states that electric charges cannot be created or destroyed.
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- ... pressure
^{1.74} - Although not grammatically correct, this is a common use of the word in discussions of fluid dynamics. By definition, something that is “incompressible” cannot be compressed, but that is not how we are using the term here. We commonly use the term “incompressible” to refer to either a moving liquid (in which case the actual compressibility of the liquid is inconsequential) or a gas/vapor that does not happen to undergo substantial compression or expansion as it flows through a pipe. In other words, an “incompressible” flow is a moving fluid whose does not substantially change, whether by actual impossibility or by circumstance.
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- ... Bernoulli
^{1.75} - According to Ven Te Chow in Open Channel Hydraulics, who quotes from Hunter Rouse and Simon Ince's work History of Hydraulics, Bernoulli's equation was first formulated by the great mathematician Leonhard Euler and made popular by Julius Weisbach, not by Daniel Bernoulli himself.
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- ... appropriately
^{1.76} - Surely you've heard the expression, “Apples and Oranges don't add up.” Well, pounds per square inch and pounds per square foot don't add up either! A general mathematical rule in physics is that any quantities added to or subtracted from each other must bear the exact same units. This rule does not hold for multiplication or division, which is why we see units canceling in those operations. With addition and subtraction, no unit cancellation occurs.
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- ... results!
^{1.77} - It is entirely possible to perform all our calculations using inches and/or minutes as the primary units instead of feet and seconds. The only caveat is that all units throughout all terms of Bernoulli's equation must be consistent. This means we would also have to express mass density in units of slugs per cubic inch, the acceleration of gravity in inches per second squared (or inches per minute squared), and velocity in units of inches per second (or inches per minute). The only real benefit of doing this is that pressure would remain in the more customary units of pounds per square inch. My personal preference is to do all calculations using units of feet and seconds, then convert pressures in units of PSF to units of PSI at the very end.
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- ... pressure
^{1.78} - A simple approximation for pressure loss due to elevation gain is approximately 1 PSI for every 2 vertical feet of water (1 PSI for every 27.68 inches to be more exact).
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- ...ejectors
^{1.79} - Technically, an eductor uses a liquid such as water to generate the vacuum, while an ejector uses a gas or a vapor such as steam.
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- ...piezometers
^{1.80} - A piezometer tube is nothing more than a manometer (minus the well or the other half of the U-tube).
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- ... energy
^{1.81} - For a moving fluid, potential energy is the sum of fluid height and static pressure.
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- ... represented
^{1.82} - The form of Bernoulli's equation with each term expressed in units of distance (e.g. = [feet] ;
= [feet] ;
= [feet]) was chosen so that the piezometers' liquid heights would directly correspond.
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