# Ideal Fluids

The ideal fluid is a frictionless liquid or gas. Properties are mass, volume, density, pressure and temperature. Consequences of Newton's 2nd Law are called "hydrostatics." The hydrostatic equation is a first order equation. The Ideal Gas Equation is introduced.

## 2.00 Ideal Fluids

Of the behaviors of phases of matter, fluids (as the Greeks classified liquids and gases) have been the most difficult to describe. An early approximation of fluid behavior (in disregard of fact) was to ignore that flowing fluid matter experiences friction.

## 2.01 Mass, Density and Volume

The subject of Newtons First and Second Laws is
physical reality modeled as a BODY.

In 1687 Newton published two axiomatic statements, three Laws of Motion and much else. His first axiom stated the obvious existence of measurable mass.

## LNG Tanker

The photo of a Norwegian tanker is shown. The tanker capacity (all five of its spherical tanks completely full) is 128,000 cubic meters of liquid natural gas (LNG). A scale profile of the tanker is available.

Estimate the length of the tanker.

## Sphere, Tank and Water

The diameter of the sphere suspended above the tank containing water is one foot. When the sphere is lowered into the tank the water contained rises to a final depth of two feet. No water spills over.

Calculate the mass of water in the tank.

## Concrete Mixture

At the simplest level a mixture is classified as homogeneous or non-homogeneous. Homogeneous means uniform through out. The text example discussed here (Sonntag et al. Example 2.1) treats its mixture as being homogeneous but the figure provided by the authors shows clearly that the mixture is not uniform throughout. Indeed, since air is included as a system component, the mixture cannot possibly be uniform throughout.

## Hurricane Wilma

This very large hurricane set some records in October, 2005. At times its winds reached 185 miles per hour. Measurements showed the barometric pressure in her eye to be 882 mbars of mercury. These pressure measurements were "static."

## Torricelli's Barometer

Torricelli, a hydraulics engineer, established a number for the pressure of the atmosphere. The idea of pressure was known to many but Torricelli made that property quantitative.

He and fellow engineers (1645) were puzzled that their best water pumps, used in mining, failed completely to draw water columns any higher that 34 feet. At that height, "vacuum was created" and the flow abruptly failed. Thinking about water and liquids, Torricelli rationalized...

## 2.02 Standard Atmosphere

A properly installed barometer can be used with the hydrostatic equation to quantify the pressure of the atmosphere. With the barometer, according to Torricelli's discovery, the pressure in the void above mercury is zero (the first quantified pressure).

We let the hydrostatic principle start "in the void" and be applied along a path downward through liquid mercury to the atmosphere (at zero elevation and sea level). Thus the magnitude of term (1) (previous page) is zero and the result of the calculation will be the pressure of the atmosphere.

## 2.03 Hydrostatic Equation

Consider the simple physical scenario of an expanse of air (a fluid) that exists above the Earth surface. Locations within the fluid have the properties position, temperature, density and pressure. An equation that relates there properties has been derived using Newton's method of system analysis. Below that derivation is presented and explained. Without loss of generality, let the fluid be atmospheric air.

## 2.04 Hydrostatic Equation Solutions

The mathematics of the hydrostatic equation is that the Momentum Equation (Newton's Second Law of Motion) applied to a small "element" of static fluid. Static does not mean "not moving." It means "not disturbed." It means the sum of forces equals zero. The sum of forces are zero with no motion and with uniform motion.

## Pascal's Experiment

In 1648, the brother-in-law of Blaise Pascal carried a mercury barometer from the base to the top of the French volcanic formation, Puy de Dome. The peak of Puy de Dome is 4803 feet above its base.

Approximately what difference in heights of mercury in the barometer were observed?

## Density of Fluid "A"

The glass U-shaped tube contains Fluid "A" and water. Since the fluids are static, the hydrostatic equation applies.

Calculate the density of fluid A.

## Oil, Cable Support a Rod

A cylindrical steel bar of length "L" and a cross-sectional area, "A" is suspended by a cable. The upper third of the bar is surrounded by atmospheric air; the lower two-thirds of the bar is submerged in oil. Determine an equation for the tension of the cable?

## Water Barometer

A sketch is shown being the configuration of a barometer but the fluid is water not mercury. The "pressure of the void" for water is the vapor pressure of water at 80°F. This pressure is about 0.5 psi. Use the schematic.

Calculate the height of the column of water, H.

## Piezometer

The "piezometer" is a pressure measuring device of antiquity (and with a few applications now). How it works is our interest.

The image depicts water flowing through a long horizontal pipe. The pipe elevation is above sea level; the water flows "to the sea." The hydrostatic principle can be used to determine the pressure of the "flowing" water in the pipe.

## Manometer, Bourdon Gage

The ele­vation of the horizontal pipe is slightly above sea level. Water flows through a pipe that has been fitted with a manometer and a Bourdon gage. Use the readings of the manometer and Bourdon gage:

Calculate the pressure of water flowing through the pipe.

## 2.05 Piston Cylinder Apparatus

The class­ical "piston and cylinder" is an artifice often used to assist in understanding physical reality. The geometry is a circular piston of finite mass that fits... (incomplete)

## 2.06 Pressure in a Fluid Jet

In most cases the path used for pressure determination is obliged to cross (at least one) fluid interface. On one side of an interface there is say "Fluid A" while on the other side exists "Fluid B." How does pressure change as a consequence of passing through an interface, as from "Fluid A" to "Fluids B?"

It will not be proved here, but if the interface is flat, there is no pressure change. However, for spherical droplets in air and for the very small diameter jets of water (as with of cutting machines) in air, the pressure of water inside its substantially greater than atmospheric. Consider water that issues downward from a faucet.

## 2.07 Pressure Terminology

Terms of pressure technology can confuse the beginner. Clarification of the many terms is made certain by concentration on the fact: pressure belongs to a fluid at a point. The numerical values of pressure of any fluid are not read on the faces of gages. The readings of gages are not pressures; they are indicators of the pressure at a location within a fluid.

## Pressure Calculations

The sketch shows a gage connected to a tank. The gage has "vacuum" written on its face and its reading (GR) is 5.8 psi. Also attached to the tank is a manometer containing oil with a specific gravity (S.G.) of 0.9. The ambient pressure is known to be 14.5 psi. The schematic also shows a mercury barometer and a water barometer beside the tank.

## Pressure of the Void

To high precision, standard atmospheric pressure equals 101.32 kPa and will support a barometric column of mercury 760 millimeters in height. These numbers are defined to be (assumed as) the surface-of-Earth average.

Torricelli assumed the void at the top of the column of a mercury barometer to be "vacuum." He also assumed vacuum meant numerically zero pressure. The actual, non-zero pressure, "in the void" of a mercury barometer is called the vapor pressure of mercury at the prevailing temperature. Use the information of the sketch. Calculate the "Pressure of the Void."

## Cave Diver

On average, 8 persons a year drown while exploring spring-fed underwater caves in Florida. In one cave a local dive club posted warning signs and installed a mercury barometer. The schematic shows a diver's view of the barometer and its location relative to the spring surface.

Calculate the depth of the diver beneath the surface of the spring.

## Wilma's Surge

Many persons who have experienced the eye of a hurricane have witnessed its calmness, its winds with speeds and also its very high "tides." These are not tides. The seemingly "tide" phenomenon is more precisely called "storm surge." High water levels that occur in the eye of a hurricane are not directly related to location of the moon. The water levels are forced by atmospheric air that surrounds the storm at a distance. Estimate the height of water surge within the eye of Hurricane Wilma.

## Off-Shore Natural Gas

An image of an off-shore natural gas well is shown. In normal operation natural gas flows from a gas reservoir beneath the ocean floor upward through pipes to the production rig then through pipes leading to shore facilities. Occasionally valves are closed such the gas flow is stopped. The "stopped no flow" flow" condition is our interest. Use the information provided to estimate the well-head pressure of the natural gas.

## U-Tube

A glass U-tube is shown in its "Initial Condition," 1). The tube is filled with mercury to the level "0 - - 0", its legs are open at their tops to the atmosphere. The cross-sectional areas, left and right, are AL and AR. Beside the U-tube is a cup containing mw grams of water.

## Bubble-Tube Depth Gage

The sketch show a bubble-tube apparatus. An open tube extends to very near the bottom of a tank of oil. In operation, compressed nitrogen is supplied to the tubing at low pressure. The nitrogen bubbles from the bottom of the tube into the oil so slowly that the pressure in the gas at depth equals the oil pressure adjacent to it (at B). Use the information given. Calculate the depth from seal level to the surface of the oil.

## Monster Tires

The load carried by the six large tires of a fully-loaded strip-mine hauler is 360 tons. Carrying such a great load, we expect that the air pressure in the tires to be great. It is not possible to calculate the pressure of the air in the tires of a given truck. However knowing the hub dimensions (given below), a calculation is possible.

Estimate the least possible air pressure required within each tire to support the truck and its load.

Most of

## Volume of a Sculpture

Many replicas of a sculpture are to be made. The completed model is given to a technician to determine the volume of precious metal each replica will require. The material of the model is compact and its density is known. Although the model cannot be immersed in water, two precision scales are available.

## Arctic Shelf-Ice

The image shows an edge of the Ross Ice Shelf northeast of Canada. From its top to sea level the ice slab measures about twenty meters. Presently about 5 million square miles of similar shelf-ice float on the seas of Earth. In many places that ice is melting. As floating ice melts into the sea, calculate the height of rise of sea level.

## 2.8 Ideal Gas and its Temperature

A brief explanation of how the ideal gas equation was developed (about 250 years ago) is helpful in learning to use it. Studies of typical solids and liquids (the condensed phases) showed their densities to be inconsequentially influenced by changes of pressure and/or temperature. But for gases, every condition not standard exhibits a different density. Concurrent with studies of gases were development of temperature scales and thermometers.

## Ideal Gas Calculations

The specific energy of a single phase state of a pure substance is least as solid, greater as liquid and greatest as a gas. In general, the greater the temperature or lower the pressure (or both trends combined) the more likely a gas is to be Ideal. Although we discuss the first, extensive form of the ideal gas equation, the results apply to either form.

## Casting Rail Car Wheels

A somewhat elaborate apparatus is used to cast the steel wheels for railroad cars. The process begins when a large "ladle" of molten steel is placed in a chamber which is then covered with a heavy air-tight lid. Ceramic wheel molds are arranged on top of the chamber. A "refractory tube" or pipe extends downward from each mold to the bottom of the pool of molten steel. When the chamber is pressurized with gaseous nitrogen, molten steel flows from the ladle upward into the molds.

## Professor Foghorn's Gage

Professor Foghorn constructed a super-sensitive gage for his Arctic Laboratory. He claims the gage can measure in the range of 1 megaPascal to an accuracy of
±1 Pascal.

Foghorn set up the apparatus as shown and asked you what his gage should read. Neglect the density of the ambient air and the air in the tank.

## Abandoned Car

While cleaning trash from an estuary, students discovered an abandoned car exposed only at low tide. They decided to remove the eyesore.

Their plan was to insert inverted steel washtubs inside the car at low tide. With the bottoms of the tubs against the roof, the incoming tide would trap air in the them and float the vehicle. Once floated, the car body would be dragged to much deeper water then sunk from sight.

## First Equestrian Ascent

At the World Fair (Paris ~ 1798) the crowd watched curiously as Louis Testu-Brissy assembled his hot air balloon with its platform-shaped gondola. His aides lighted the "lifting flame," the balloon inflated, became spherical and then, to the astonishment of all, the balloon and horse (with Louis astride) slowly ascended high above the fairground.

## Otto Von Guernike

German scientist, inventor and engineer.

At a fair in 1654 Otto amazed the spectators. He showed them two hemispherical brass shells that fit together (to form a hollow sphere) but fell apart when not held. Next, (using the air pump he invented) Otto removed air from the inside of the sphere. Each hemisphere had built-in ring to which a rope was attached. The other ends of the ropes were attached to harnesses of two teams of horses. The horses, urged to pull, strained and snorted but, to the surprise of everyone, they were unable to pull the hemispherical shells apart.

## Victor Regnault

This careful and meticulous scientist (1810 - 1878) was one of the first to measure the density of atmospheric air. A master glass blower, Victor crafted two near identical flasks; one being a test flask, the other being a "counterpoise." Both flasks were filled with dry (standard) atmospheric air. Next, an air pump was used to extract air from the test flask until the inside air pressure became 560 mm Hg barometric. Both flasks were placed on a beam balance...

## Roof-Top Exit

A building 1200 feet in height will have a doorway for rooftop exit. Air interior to the building will be maintained at 70°F and prevailing local pressure. The worst case of air outside in a light breeze is 40°F and 14.7 psi.

On the roof is a powerful fan which when turned on, will act to force air from the building and into the outside air. At ground level the building has a typical door which we assume to be air-tight such that air cannot leak past the seals along its sides, top and bottom. Furthermore assume the entirety of the rest of the building has no "leaks" or paths for air to enter or leave the building - the single exception is the exhaust duct of the fan on the roof.

## Two Tanks

Each tank is square in cross-section and has a valve for filling with water (or venting of air) at its top. The tanks are connected at their bottoms by a pipe with a valve.

Two cases are to be studied.  The sketch shows the setup for both: all valves are closed, the depth of water in tank A is 5 meters and the water depth in tank B is zero. Events initiate when valves are opened. For each event, calculate the final depth of water in each tank.

## Ocean of Air - its Average Density

Earth is spherical, and above Earth, our atmosphere is a spherical shell of gases. The Earth diameter is about 6,380 kilometers and the depth of the atmosphere above its surface is about 30 kilometers. The standard atmospheric pressure, is assumed to have an average over the Earth surface of 101.3 kPa (enough precision for now). Estimate the average density of the atmosphere of Earth.

## Hand Lotion Event

An air-traveler packed a 1/2 liter, 1/4 full, rigid plastic bottle of hand lotion in a "bottom-up" position in her bag with clothing (shown in the sketch). The little spout of the bottle was in the open position but lotions hardly flow at all. The bottle was loaded into the cargo hold of the aircraft which then climbed to 35,000 feet where pressure and temperature were 30 kPa and -30°C. When the plane landed, approximately what volume of lotion will remain in the bottle?

## Ideal Gas Equation Validity

More substances are not ideal gases than are. Thus the ideal gas equation can be used ONLY "after the gas has been shown to have ideal behavior" or "by assumption that the equation applies." Is water at STP an ideal gas?

## Fuel Cell Proportions

A fuel-cell is a battery-like chemical device into which flows oxygen and hydrogen. In the fuel cell these species react to produce electrical potential and current. The exhaust of the fuel cell is water. As one comes to understand basic thermodynamics, one is obliged to learn some basic chemistry.

## Concreted Nuclear Waste

Fast-Breeder Reactors produce fuel that is used for other reactors or for weapons. Spent fuel (just out of the reactor) is agitated, hot, and lethal. To calm down, it is placed in cooling ponds for several years. From the pond it is solidified and mixed into cubical concrete blocks which are buried...