Section 1.1 - Basic Sciences (page 2)
Work in the physics sense is a force applied through a distance, or in equation terms W = Fd. It is a scalar (numerical) value found by multiplying two vectors (directional values), the direction of the applied force, and the direction of motion. Since those directions do not have to be the same, the vector product can vary as the cosine of the angle between them, and therefore be zero or negative. For example, if you apply a lifting force to a table, but not enough to raise it off the floor, you do no work in the physics sense, even though your muscles will tell you they are working in the biological sense. If you manage to lift the table, the direction of motion (up) is opposite the direction of gravity (down), and therefore the work done on the table is negative. As odd as that sounds in conversation, the mathematics works out when solving physics problems.
Energy is then defined as the ability to do work. It comes in many forms which can be converted either by natural actions or human devices. As far as we have reliably observed, total energy always remains the same, a principle known as Conservation of Energy. An exception to this might be the hypothetical "Dark Energy" invoked to explain the apparent acceleration in expansion of the Universe. We have not yet reliably observed what Dark Energy is, though. It is at present just a label applied as a placeholder for whatever is causing anomalous redshifts, and we definitely have no way to apply such energy. For practical engineering purposes we will ignore it and treat conservation of energy as a firm principle.
Energy is measured in SI units by Joules, named after a 19th century physicist who helped discover the relationships of energy, work, and heat. Since energy comes in different forms, the Joule has several equivalent definitions. Leaving out the numerical values and only looking at units it can be expressed as:
The first equivalence is to base SI units of kilograms, meters, and seconds. Where N is Newtons, Pa is Pascals, the unit of pressure, and W is Watts, the unit of power, the following expressions are in terms of force times distance, pressure times volume, and power times time. Note that W as work and W as Watts mean different things, and the latter is distinguished by having a quantity attached (ie 100 W meaning 100 Watts). Unfortunately there are more physics concepts than letters of the alphabet, which can be confusing at times. When a formula could lead to such confusion, write out the unit in full rather than abbreviate, or define the symbol in words as we usually do around a formula. Which of the above expressions for energy are appropriate to use depends on the types, and which conversions of it, are involved in a particular situation. One of the forms which energy takes is as matter. Where E is energy, m is mass, and c is the speed of light they are related by the famous equation
Since the speed of light is a large number, by definition exactly 299,792,458 meters per second, and that number is squared in the formula, the energy contained in a given amount of mass is enormous. The conversion of less than 1% of mass in nuclear reactions produces sufficient energy to power stars, atomic bombs, and electric power reactors. Objects which are moving in gravity fields are very common in space projects. We find it useful to define the following two energy quantities based on their motion and position:
Kinetic Energy is the energy an object possesses through its motion. It can also be described as the amount of work required to get a body of mass to move. In mathematical form Work = change in Kinetic Energy, or
This is in addition to the energy by virtue of rest mass. Referring to Newton's first law of motion, an object will retain its kinetic energy unless acted upon by another force such as friction or gravity. For example, objects in outer-space (free of a gravitational field and in a vacuum) will retain their kinetic energy, direction, and velocity. Kinetic energy KE is a function of mass m and velocity v according to the formula
A Reference Frame is a non-accelerating environment. Velocity is measured relative to such a reference frame. Therefore a space station in orbit, and an astronaut inside it, may both have a large velocity relative to the center of the Earth. They then both have a large kinetic energy in an Earth-centered reference frame, in fact sufficient to raise their temperature hotter than the Sun, to 7000 K. Relative to each other, however, their velocity is near zero, so in a reference frame moving with them they have near zero kinetic energy. Since the formula above takes the square of velocity, kinetic energy is always positive, even if the velocity is negative in a given reference frame.
Potential Energy is the difference between the energy of an object in a given position and its energy in a reference position. When work is done against a conservative field, such as gravity, then the energy of that work is converted to potential energy. Setting the reference position as infinity, the gravitational potential is always negative since you must do positive work to lift an object to infinity. Since gravity varies as the inverse square of distance, the integral of the work going to infinity varies as the inverse of distance, giving the potential U as
If no forces besides gravity are acting on an object then the sum of kinetic and potential energy is constant. Thus objects in elliptical orbits are constantly exchanging potential and kinetic energy as their distance r from the body they are orbiting changes. They have more kinetic and less potential energy at the lowest point and so move faster. The velocity v at any point in the orbit can be found from:
- is the distance between the orbiting object and the body it orbits.
- is half the long axis of the elliptical orbit shape, or Semi-major Axis.
Mechanics is the description of the motion of an object under the influence of forces. For space projects this is usually the thrust forces generated by a propulsion system, and the influence of gravity. The motion in a vacuum among massive objects like planets and the Sun is called Orbital Mechanics, which will be covered later. When operating within an atmosphere, an additional force is generated when moving. This force is decomposed into a perpendicular component called Lift, and a parallel component called Drag. When two objects are in contact they additionally generate a force which is decomposed into parallel, or Friction, and perpendicular, or Normal Forces. The combination of all forces, including less common ones not listed in this paragraph, produce a vector sum total force on the object and thus an acceleration in some direction.
Friction forces matter for space systems for things like rovers moving on the surface of a body and for internal parts of motors and pumps. Since they always oppose motion, they require energy to overcome, and thus a source of energy in the system. Friction at a microscopic level is caused by interactions between electrons of the two objects, thus is an electromagnetic effect. At a slightly larger level they cause temporary bonding and physical obstruction due to surface roughness. At human scale these microscopic interactions can be summed up as average values which are proportional to the normal forces and depend on the types of surfaces in contact. The multiplier to the normal forces is called the Coefficient of Friction, with symbol \mu. The friction force, f, is then related to the normal force, Fn by:
Coefficients of friction are determined experimentally, and depend on whether the objects are moving (kinetic friction) or not (static friction). Static friction is typically higher because the objects have time to form atomic bonds and settle into the bumps of surface roughness. The coefficients also depend on type of material and whether any gas or liquid is trapped between them. The ability of skaters to move easily on ice comes from a microscopic layer of water which forms due to pressure of the blades. Vertical lift from contact only requires breaking atomic bonds, and not interlocking of surface roughness. Thus wheels and ball bearings, which vertically separate the contact surfaces, have lower Rolling Friction than sliding contact.
Normal forces act perpendicular to a surface, and have several sources. This includes gravity, magnetic or electrostatic attraction, and gas or liquid pressure. In reality, friction and normal forces are components of the total contact force. Since motion is prevented in the perpendicular direction by the existence of a solid surface, it is easier to calculate the effects by looking at the components separately. When perpendicular motion is not prevented, which happens with liquids and gases, it becomes more complex. The field of Fluid Dynamics is the study of these more complex motions, both against solid surfaces and internally within fluids.
Thrust is the force generated by a vehicle by expelling reaction mass or by interacting with the environment. When something external acts on the vehicle it is referred to as an accelerating force, and often specifically named. The magnitude of the thrust due to expelled mass is given by
where T is the thrust generated (force); is the rate of change of mass with respect to time (mass flow rate of exhaust); and v is the speed of the exhaust measured relative to the vehicle.
Drag is a force component generated by interaction with a fluid medium, such as the Earth's atmosphere. It is parallel to the incoming flow direction, and given by the formula
where FD is the drag force, ρ is the mass density of the fluid, v is the velocity of the object relative to the fluid, A is the reference area, which is the projected area occupied by the vehicle in a plane perpendicular to the motion, and CD is the Drag Coefficient — a dimensionless number. While most of the terms in the above formula are simple to determine, drag coefficient varies in a complex way based on object shape, velocity, and other parameters. This is caused by complex flow conditions such as turbulence, shock waves, heating, and even decomposition at higher velocities.
When a surface moves relative to a fluid, the layer closest to the surface is affected most by the molecules colliding with the surface. They are deflected by the angle of the surface, roughness in the surface, or atomic forces between their respective atoms. This causes that fluid layer to tend to move along with the surface. The surface layer in turn affects farther layers by collisions of the molecules. At lower velocities, this sets up a smoothly varying Boundary Layer near the surface. At higher velocities the deflection is violent enough to create Flow Vortexes, where the fluid develops circular motions perpendicular to the direction of motion, known as Turbulent Flow. Since it takes more energy to create the vortexes, the forces on the surface are higher, increasing friction or drag. These effects happen both externally to a vehicle moving in an atmosphere, and internally to a gas or liquid flowing within an engine.
The ratio of inertial forces, such as the sideways deflection, to the viscous forces caused by shearing or varying speed in the boundary layer is called they Reynolds Number. The transition from smooth, or Laminar Flow, to turbulent flow, and the size of the vortexes, and thus the drag, is found experimentally to depend on Reynolds Number. It is a Dimensionless Number, meaning all the units in the formula cancel out when using consistent units, leaving a pure number. The Reynolds Number, Re, is found by:
where: is the mean velocity of the object relative to the fluid (m/s), is a characteristic linear dimension of the surface (m), is the Dynamic Viscosity of the fluid (Pa·s or N·s/m² or kg/(m·s)), is the Kinematic Viscosity () (m²/s), and is the density of the fluid (kg/m³). The characteristic dimension is defined by convention for various types of shapes, such as the diameter for a sphere. Since the motions within a turbulent fluid are too complex to reduce to a single formula, for early design purposes drag coefficients are usually found from tables and graphs based on Reynolds number, which in turn were developed from experiment or historical data.
In more detailed or important design projects they are measured for the proposed design in a wind tunnel or other experiment, or calculated by detailed numerical simulations, a topic known as Computational Fluid Dynamics, or CFD. In CFD simulations the flow is broken up into sufficiently small volumes that the flow in each volume obeys relatively simple formulas, and thus the total flow in the simulation can be determined reasonably accurately. Historically this needed the largest available computers, and therefore physical testing often proved easier. With the vast increase in computer speed in recent decades this method has become much more practical.
Lift is the other force component generated by interaction with a surrounding medium. It is perpendicular to the incoming flow direction, and given by the formula
where L is lift force, ρ is density of the medium, v is the velocity relative to the medium, A is planform area of the shape, and is the lift coefficient at the desired Angle of Attack (angle between the reference plane of the shape and the velocity direction), Mach Number, and Reynolds Number. Like drag, lift coefficient is determined by complex fluid flow, and depending on the design need is found by looking at a table or graph, physical experiment, or CFD simulation.
The thermodynamic cycle for a liquid rocket booster is a modified Brayton (jet) cycle. A one-dimensional analysis may be performed by assuming the following ideal steps.
- Fuel is injected into a combustion chamber isentropically either through use of pressurized fuel tanks or by a high-pressure pump, increasing the pressure to and increasing the enthalpy.
- Heat is added to the fuel by means of combustion. In an ideal situation, it is assumed that the pressure remains constant during this step, but that the temperature rises. Both enthalpy and entropy increase during this step.
- The combusted fuel expand isentropically to the exit pressure, , as it goes through the nozzle into the surroundings, which is at pressure . Ideally should equal . During this process, the enthalpy decreases from to .
The thrust produced by a rocket is given by
where and are the mass flow rate and exit velocity of the propellant, is the exit area of the nozzle and and are the pressure at the exit point of the nozzle and the atmospheric pressure. The enthalpy represents the internal energy available for work or the potential energy. Thus, the energy change per unit time as the propellant moves from the combustion chamber to the nozzle exit is
Solving for the propellant velocity yields
Let us assume that the combustion mixture of the propellants is an ideal gas. The internal energy per unit mass of an ideal gas is given by
producing an equation for the propellant velocity of
When an ideal gas expands isentropically, a change of temperature and pressure such that the following two relations hold
where M represents the Mach number at the location having static pressure p and temperature T. Using these two equations, we can relate the temperature and pressure ratios as
Thus, we can rewrite the equation for the propellant velocity as
The final analysis step in the one-dimensional analysis is the effects of the nozzle. The previous equation demonstrates that making the ratio as small as possible maximizes the propellant speed, which in turn maximizes the thrust. The nozzle is designed to match the exit pressure as close as possible to the pressure of the atmosphere or the vacuum of space.