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Astro and Aeronautic Equations

Table of Contents

Ideal Gas Law

Hydrostatic Equation

Euler Equation

Tropospheric and Tropopause Relations

Bernoulli's Equation

Reynolds' Number

Drag and Lift

Escape Velocity

Rocket Thrust

Rocket Velocity Change

Orbital Mechanics

Aircraft Range

The Ideal Gas Law

This is the most basic endeavor to model the thermodynamic behavior of a gas. Its derivation starts with the assumption that the gas is made up of an extremely large number of hard sphere particles that interact only when they colide. Their speed is directly proportional to their temperature, and when they interact with a boundary surface, they bounce back and provide, collectively, the pressure of the gas. Faster motion results in harder collisions, thus higher pressure. From this there are two immediate consequences: the greater the number of particles, the higher the pressure, and the greater the temperature, the higher the pressure. The constant of proportionality has been experimentally determined to be Boltzmann's Constant, k. Therefore

Ex. At what volume is an ideal gas with 3.011*10²³ molecules, at 500K, and 70kPa?

Perfect Gas Law

The perfect gas law comes from a special case of the ideal gas law where density of the gas is assumed to be constant---a very good assumption in the case of a closed, fixed volume. This allows the ideal gas law to be used with density measurements.

This is the most basic endeavor to model the thermodynamic behavior of a gas. Its derivation starts with the assumption that the gas is made up of an extremely large number of hard sphere particles that interact only when they colide. Their speed is directly proportional to their temperature, and when they interact with a boundary surface, they bounce back and provide, collectively, the pressure of the gas. Faster motion results in harder collisions, thus higher pressure. From this there are two immediate consequences: the greater the density of the particles, the higher the pressure, and the greater the temperature, the higher the pressure. The constant of proportionality has been experimentally determined to be Boltzmann's Constant, k. Therefore

P = ρ k T.

Of course, ρ, the particle density, is just the number of particles in the volume, so it translates easily to another common form,

P V = n k T.

This equation is usually sufficient when a gas is sufficiently rarified that the interatomic spacing is large and the particles interact very weakly. For example, it would be quite reasonable to use in describing sub-atmospheric pressure helium, but very likely to fail in describing high pressure steam.

Hydrostatic Equation

The Hydrostatic Equation: is used to desecribe the relationship between the change in height and the change in pressure. It demonstrates that the difference in pressure is caused by the weight of the fluid between the two levels.
If we examine a box of infinitesimally small sides, dx, dy, dz, filled with some type of fluid, the following derivation applies:
The relationship shows that as you increase your altitude the pressure will decrease.

When one stacks weights, increasing layers of material naturally increases the force on the bottom of the pile---and, proportionate to the area of the stack, the pressure caused by the pile. It is just the same with liquids, except that the layers can be treated as infinitely thin slices stacked upon one another.

Euler's Equation

describes the change in pressure of a moving fluid in relation to its velocity. If we examine another cube of dimensions dx, dy, dz, and assume fluid moving through it with a velocity of v, Euler's equation can be derived in the following manner:

The equation describes how pressure will decrease as velocity increases.

Tropospheric and Tropopause Relations

In many respects these equations are an application of the hydrostatic and Euler equations shown above: the pressure at a given altitude is the result of the weight of the atmosphere above it; the additional pressure imposed by a layer at a given altitude is proportional to the pressure imposed upon it. From these considerations it follows that the change in pressure with altitude is itself a function of altitude, and so the closed-form solution will have an exponential dependence with some scale factor. Pressure will behave similarly.

Of course, this neglects the importance of temperature. The equations presented address this by making temperature an explicit function of altitude, according to the experimentally determined atmospheric lapse rate (α) and scaling accordingly. Thus, in the troposphere

Above the troposphere, however, the explicit temperature dependence is unnecessary, and the equations simplify to

.

Bernoulli's Equation

The Bernoulli Equation applies to a scenario of a steady state, inviscid, incompressible fluid flow in which there is one inlet and one outlet. In this flowing fluid are streamlines, which are lines that are tangent to the velocity vector of the fluid at all points in the fluid. The Bernoulli Equation, , is a statement of conservation of mechanical energy and describes how the static pressure and dynamic pressure are related throughout one of these streamlines. P represents the static pressure throughout a streamline. This is the pressure exerted by the fluid when it is not moving or while moving with the fluid so that the measuring device is static relative to the moving fluid. ρ is the density of the fluid, and V is the velocity at which the fluid is moving.  represents the dynamic pressure of a point along a streamline. This term accounts for the momentum of the fluid and the pressure required to bring it to rest. If the total pressure, or stagnation pressure, and the static pressure are known, the velocity of the fluid can be determined. The arrow below shows a point of stagnation pressure. This is an area in the fluid at which the velocity of the fluid is zero.

The total pressure, which is the sum of the dynamic and static pressures, never changes throughout a streamline. The static pressure is inversely related to the dynamic pressure. This relationship is the foundation of the Bernoulli Equation.

Example: As the air flows around a surface, like a ball or an airfoil, the streamlines are compressed and accelerated to a higher velocity. This increases the dynamic pressure of the fluid, which in turn causes a decrease in the static pressure, because total pressure never changes. This is caused by the conservation of energy which says that

,

or that the pressure, kinetic, and potential energies may change, but the total energy does not. The simplified version of Bernoulli's Equation assumes no change in height or potential energy, so it can be written as

.

Reynolds Number

The Reynolds number, represented by Re, is a dimensionless ratio between the inertial and viscous forces of a fluid flow.

In this equation, the numerator contains:

• ρ represents the density of the fluid
• V is the velocity of the fluid
• d is the characteristic length of the surface across which the fluid is flowing.

It represents the inertial force of the fluid. The viscous forces are determined by μ, which is the viscosity of the fluid. Viscosity is a specific property of individual fluids. It is a measure of how easily a fluid flows. In essence, the Reynolds number is a comparison between the forces which propel the fluid forward and the forces which slow the fluid down. Reynolds number is used to determine the type of flow that is occurring in a system. Low Reynolds numbers indicate that the flow is laminar, but high Reynolds numbers indicate a turbulent flow. The point at which the flow transitions from laminar to turbulent is known as the Critical Reynolds number. This is more of a theoretical value, because it is difficult to determine, but it represents the location at which the flow separates from the surface over which it is flowing.

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*"Reynolds number." McGraw-Hill's Essential American Slang Dictionary. McGraw-Hill Companies, Inc., 2007. Answers.com 12 Aug. 2013.

The numerical value for the critical Reynolds number varies greatly with the geometry of the system as a whole and the nature of the fluid boundaries. A rough surface can achieve the onset of turbulence at much lower Reynold’s numbers than a smooth surface. In addition, as Blanford and Thorne remark, the choice of characteristic length and velocity can also contribute to ambiguity about the system. It must also be borne in mind that the Reynold’s number is frequently calculated on the assumption that the flow is generally the same throughout; in the case of well-developed boundary layers, an assumption which may not be accurate. Blanford and Thorne suggest the estimation of this boundary layer thickness with

where x is a position relative to the beginning of the boundary layer; δ(0)=0.

Drag and Lift

In aircraft design, drag and lift are vital components to consider in the design process. Drag accounts for the friction forces that oppose the motion of an object in a fluid, and lift includes the forces exerted by the fluid to raise the object. Because bodies in a fluid come in all shapes and sizes, it is helpful to characterize their effects on drag and lift. Drag and lift coefficients are commonly determined through experimental testing or analysis. The drag and lift coefficient equation

is simply the drag or lift equation rearranged and solved for the coefficient. The coefficients are dimensionless values that are dependent upon the force of drag or lift, the dynamic pressure of the flowing fluid, and the planform surface area. The coefficients of lift and drag allow for the translation between small test samples in a wind tunnel to the actual size component. The size of the actual airfoil is irrelevant to the coefficients of lift and drag. A specific airfoil shape will have the same coefficients no matter what the scale of the airfoil may be

Escape Velocity

The escape velocity is the velocity required for an object to leave the sphere of influence, or gravitational effects, of the Earth or another planet.

G represents the gravitational constant which is unique to each individual planet. It should not be confused with the other gravitational constant---also denoted G---which is found in Newton's Law of Gravitation and is universal. M is the mass of Earth, or the mass of any other planet that may require an escape velocity. Likewise, the r is the radius of the Earth or other planet.

Whether the space craft is being launched or is already in orbit, an addition of kinetic energy is required for the space craft to escape the gravitational pull of Earth---which means to have more kinetic energy than the gravitational potential that would be overcome in moving the craft from its position to a distance infinitely far from the body.

Rocket Thrust

On a conceptual level, there is no difference between the thrust of a Saturn V and the thrust produced by a Cessna propeller: both work by using Newton's third law of motion between a working fluid and the craft. The propeller impacts molecules in air and knocks them backward; a rocket carries fuel and oxidizer, burns them, and uses the collisions of the high-temperature molecules as they flow out the nozzle. Exterior factors are irrelevant, because the third law applies between two bodies and does not have any information about material far from the craft body; as long as a rocket has reaction mass, it will work perfectly well in a vacuum.

Emphasis belongs upon the necessity of reaction mass---the momentum it carries in one direction must be matched by the change in momentum of the rocket in the opposed direction. Thus the rocket equation:

In the rocket thrust equation, the ? represents the mass flow rate of the burned oxidzer/fuel mixture that is expelled from the rocket engine.  is the velocity at which the exhaust is propelled from the nozzle.  is the pressure of the exhaust at the exit of the nozzle, while the  is the pressure of the surrounding air outside the rocket nozzle.  is the area of the nozzle through which the exhaust is expelled. The  is the velocity of the exhaust that would be needed to produce the same amount of thrust in a vacuum environment. Conveniently,  and  both produce units of force . As stated before, the force of the rocket thrust is a product of Newton's Third Law of Motion. As the rocket blows the exhaust out the nozzle, there is an equal and opposite force applied on the rocket that propels it forward. The rocket thrust equation is derived from the concept of the conservation of momentum, or  where p is linear momentum. This is statement is interchangeable with . As the rocket burns fuel, the mass changes over time, hence the need for incorporating the mass flow rate. Along with the pressure differential, this accounts for the propulsion of the rocket.

Orbital Mechanics

= period of orbit

a= semi-major axis

G= gravitational constant

=Mass of earth

r= distance to the satellite

e= eccentricity

v= true anomaly

Orbital mechanics is the easiest way in which we can classify and describe different orbits. In order to grasp the equations, we much first have a basis of each of the variables.

Because most orbits that we deal with are elliptical, we must define some references and descriptions to aid us in our orbit classification. An ellipse has two different axes. The major axis dissects the ellipse through the longest section. The minor axis cuts across the shorter direction. The semi-major axis that we are concerned with is half of the major axis or the distance from the centroid of the ellipse to the furthest edge of the ellipse.

The radial distance, r, is defined as the distance from the center of Earth or the planet to the satellite. In a circular orbit, r and the semi-major axis a are the same distance. In an elliptical orbit, the distance to the satellite changes over the entire period.

The eccentricity describes the shape of the orbit in a numerical term. Orbits take on 4 different shapes known as conic sections; circular, elliptical, parabolic, and hyperbolic. A circular orbit has an eccentricity of 0. An ellipse has an eccentricity between 0 and 1. A parabolic orbit has an eccentricity equal to 1, while a hyperbolic orbit has an eccentricity greater than 1.

True anomaly is an angle which helps us describe a satellite's instantaneous location within an orbit. True anomaly is defined as the angle between perigee and the satellite. In order to determine this angle, we draw imaginary lines from the center of the Earth's mass to the satellite, and we draw a line from the center of Earth's mass to the perigee point of the orbit. Perigee is the point in an orbit at which the satellite comes closest to earth.

The first equation is used to determine the period of the orbit or the time it would take for the satellite to make one revolution through the orbit. The period is important to know in order to rendezvous with other satellites or spacecraft. The only way to change the period of the orbit is to change the size of the orbit. Naturally, a large orbit will have a larger period.

Rocket Velocity Change

ΔV= change in velocity

= effective velocity

=initial mass

= final mass

The velocity change from a rocket is due to the amount of energy produced by the fuel and the amount of fuel used. The effective velocity can also be written as the acceleration due to gravity at earth€™s surface multiplied by the specific impulse of the fuel. The specific impulse has units of seconds, and it is a measure of how long the fuel could hold the spacecraft in a stationary vertical position. Therefore, the velocity change equation can also be written:

As a rocket propels itself forward, the fuel is burned and expelled as exhaust. Thus, the mass of the spacecraft is not constant from the start of the burn to the end of the burn. The natural log term in the equation accounts for this change of mass. The initial mass is the mass of the entire spacecraft and fuel before the burn initiates. The final mass is the initial mass minus the mass of the fuel that was burned. This comes from analyzing the change in momentum of the rocket.

The amount of velocity change is crucial to reaching specific orbits or escaping earth€™s gravitational field. The design of the rocket must provide more than enough velocity change to reach the desired target. Either the specific impulse or the amount of fuel can be changed to increase or decrease the amount of velocity change that can be produced from a rocket.

Aircraft Range

R= range

V= velocity

= force of lift

= force of drag

c= specific fuel consumption

= initial weight

= final weight

The range of a jet is based upon the amount of fuel it can carry, the efficiency with which the engine burns the fuel, the lift to drag ratio, and the speed at which it can fly. All of these provide the theoretical distance that an aircraft can fly. It assumes a steady state flight of constant velocity in order to do so.

The specific fuel consumption describes how efficiently the engine burns its fuel. It relates similar to a car€™s gas mileage. The more efficiently the engine burns fuel, the further the vehicle can travel on the same amount of fuel. A low value for specific fuel consumption demonstrates high efficiency. Specific fuel consumption technically describes the mass of fuel burned over a given time at a specified thrust.

The ratio of initial and final weights in the equation accounts for how much fuel is burned over the range of flight. Because the weight changes over time, the fuel consumption per unit time also changes. The force of drag accounts for all forces opposing the motion of the aircraft, and the force of lift accounts for all forces keeping the aircraft as a constant altitude.

This equation can be applied to any aircraft despite its size or designed function. All of these give the distance in which an aircraft can fly in miles or kilometers.

Works Cited

Benson, Tom. "Ideal Rocket Equation." Ideal Rocket Equation. NASA, 13 Sept. 2012. Web. 27 Aug. 2013.

Moran, Michael J. Introduction to Thermal Systems Engineering: Thermodynamics, Fluid Mechanics, and Heat Transfer. New York: Wiley, 2003. Print.

Philpot, Timothy A. Mechanics of Materials: An Integrated Learning System. Hoboken, NJ: John Wiley, 2011. Print.

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