Space Transport and Engineering Methods/Physics2



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6.0 - Energy
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&emsp;Work in the physics sense, W, is a force F applied through a distance d, or W = F * d. It is a scalar (undirected) amount found by multiplying two vectors - the direction of the applied force, and the direction of motion. Those directions do not have to be the same. Their product varies as the cosine of the angle between them, which can be zero or negative.

&emsp;For example, if you apply a lifting force to a chair, 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 lift the chair, the direction of motion (up) is opposite the direction of gravity (down). The work done by gravity opposing your lift is then negative, while your own lifting work is positive. As odd as this sounds in conversation, the mathematics works out when solving physics problems.

&emsp;Energy is defined as the ability to do work. It comes in many forms, and can be converted among them by natural or intentional actions. As far as we know, total energy always remains the same, a principle known as Conservation of Energy. An exception to this might be Dark Energy. That is the name for the unknown cause of the Universe's accelerating expansion. There is no known way to use this energy. It can be ignored for engineering projects, and conservation of energy treated as a firm principle.

&emsp;The SI unit of energy is the Joule, named for a 19th century physicist who helped discover the relationships of energy, work, and heat. Since energy has different forms, the Joule has different ways to be calculated. These include:


 * $$\rm 1 J = {}\rm 1 \frac{kg \cdot m^2}{s^2} = 1 N \cdot m = \rm 1 Pa \cdot m^3={}\rm 1 W \cdot s$$

&emsp;The first way uses the base SI units of kilograms, meters, and seconds. Where N is Newtons, the unit of force, Pa is Pascals, the unit of pressure, and W is Watts, the unit of power, the other ways are in terms of force × distance, pressure × volume, and power × time. Note that W as work and W as Watts mean different things. Watts are indicated by having a quantity attached (ie 100 W means 100 Watts).

&emsp;There are more physics concepts than letters of the alphabet, which can be confusing at times. To avoid confusion, write out the unit name in full rather than the symbol, or define the symbol in words as we usually do around a formula. Which of the above ways to use to calculate energy depends on energy types, and which conversions are involved in a particular situation.

&emsp;One of the forms which energy takes is matter. Where E is energy, m is mass, and c is the speed of light they are related by the famous equation


 * $$ E = m c^2 $$

&emsp;The speed of light in a vacuum is defined as exactly 299,792,458 meters per second. That number is squared in the above formula, so the energy contained in a given amount of mass is enormous. The conversion of less than 1% of mass to other forms of energy in nuclear reactions powers stars, atomic bombs, and nuclear power plants.

 6.1 - Kinetic Energy
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&emsp;Space systems very often involve objects moving in gravity fields. It is useful to define two energy types, kinetic and potential, based on the object's motion and position. Kinetic Energy (KE) 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 m to move at velocity v. In mathematical form work W is the change in kinetic energy from KE1 to KE2, or


 * $$W = \Delta KE = KE_2 - KE_1$$

&emsp;If the bound energy of matter (mc2) is not changing, it can be left out of a given calculation. Referring to Newton's First Law of Motion, an object will retain its kinetic energy unless acted upon by another force. Objects free of any gravity field and in a vacuum with no friction would keep their kinetic energy, direction, and velocity forever. This never actually happens. Gravity has no distance limit, and no vacuum is perfect, though some parts of space come very close. But in some cases gravity and friction are so weak they can be ignored. In general, kinetic energy is one half of mass times velocity squared, or


 * $$KE =\tfrac{1}{2} mv^2 $$.

&emsp;An Inertial Reference Frame is one that is not accelerating. Velocity can be measured relative to such a reference frame. A space station in orbit, and an astronaut inside it, both have a large velocity relative to the center of the Earth. So they both have a large kinetic energy in an Earth-centered reference frame. In fact it is enough to raise their temperature hotter than the Sun, to 7000 K if it were instantly converted to heat. This conversion happens by friction when objects enter the atmosphere, but not all at once so peak temperatures are somewhat lower.

&emsp;Relative to each other, the station and astronaut's velocity is near zero. So in a reference frame moving with them they have near zero kinetic energy, and bumping into the station walls releases very little energy. The formula above includes the square of velocity. So kinetic energy is always positive, even if the velocity is negative in a given reference frame.

 6.2 - Potential Energy
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&emsp;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 Force such as gravity, then the energy of that work is converted to potential energy. By convention the reference position for gravity is infinitely far away. The gravitational potential is always negative since you must do positive work to lift an object to infinity.

&emsp;Gravity varies as the inverse square of distance, so the sum of the work going to infinity varies as the inverse of current distance. Where G is the Gravitational Constant that relates gravity to mass, m1 is the mass being moved, and M2 is the large mass producing gravity, then the potential energy U is:


 * $$U = -G \frac{m_1 M_2}{r} $$

&emsp;If no forces besides gravity are acting on an object then the sum of kinetic and potential energy is constant. Colliding with a planet or its atmosphere involves other forces. But an object in an elliptical orbit which does not collide is free to repeat its motion, constantly exchanging potential and kinetic energy as the distance r from the center of the body changes. Planets in the Solar System have been doing this for billions of years.

&emsp;An object has more kinetic and less potential energy at the lowest point so it moves faster. If the sum of positive kinetic and negative potential energy is above zero, the object has enough total energy to reach infinity with some velocity left, so it escapes from the large body rather than orbit it.

&emsp;The velocity v at any point in a repeating orbit can be found from:




 * $$v=\sqrt{GM\left({2\over{r}}-{1\over{a}}\right)}$$

where r is the distance between the orbiting object and the large body of mass M it orbits, and a is half the long axis of the orbit shape, or Semi-major Axis. An orbit may be circular, but most repeating orbits are elliptical to some degree (Figure 1-1). The large body is located at one focus F of the ellipse. When the orbiting object is large enough relative to the larger body, such as the Moon is to the Earth, they both orbit their common Center of Mass. For the Earth and Moon that center is about 3/4 of the way from the Earth's center to the surface in the direction of the Moon.

7.0 - Mechanics
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&emsp;Mechanics is the branch of physics that relates the motions of objects to their mass and the forces applied to them. For space systems these are often the Thrust forces generated by a propulsion system, and the force of Gravity. The motion in a vacuum by objects only affected by gravity is called Orbital Mechanics, which is the main subject of Chapter 2.

&emsp;When an object moves within an atmosphere, or wind moves past it, an additional force is encountered. This force can be divided into a portion perpendicular to a surface called Lift, and a portion parallel to the surface called Drag.

&emsp;When solid objects in contact are moving they produce a force parallel to the contact area called Friction. Whether moving or not there is a perpendicular force called the Normal Force. This may be zero, such as the case of two upright books on a shelf that are touching but not leaning on each other. When standing on flat ground, the normal force is what is keeping you from falling into the Earth while gravity is trying to pull you down. If they are equal and opposite, you don't move up or down.

&emsp;The combination of all forces, including ones not mentioned here, produce a vector sum net force on an object. If the sum is not zero, the object will accelerate in some direction. Buildings are typically not accelerating, so we know the sum of all forces on them are zero.

 7.1 - Friction
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&emsp;Friction forces affect space hardware like rovers moving on the surface of a body, and their internal parts like motors and bearings. Since friction always opposes motion, they need energy and an energy source to overcome it. Friction at a microscopic level is an electromagnetic effect of interacting electrons from the two objects. At a slightly larger scale they cause temporary bonding, and surface roughness makes motion bumpy rather than smooth.

&emsp;At the component or system scale, these microscopic interactions can be summed up as average values. They are proportional to the normal forces and depend on the types of surfaces in contact and other factors. The multiplier to the normal force is called the Coefficient of Friction, $$\mu$$. A component friction force f is then related to a particular normal force Fn by


 * $$f_{n} = \mu F_{n}\,$$

and total friction is the sum of all internal and external friction forces.

&emsp;Coefficients of friction are determined experimentally, and depend on whether the objects are moving (sliding or kinetic friction) or not (static friction). These are given subscripts k and s to distinguish them. 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 types of materials and whether any gas or liquid is trapped between them. For example skaters can slide easily on ice because there is a microscopic layer of water on the surface.

&emsp;Vertical lift from contact only requires breaking atomic bonds, and not overcoming the interlocking of surface roughness. So wheels and ball bearings, which vertically separate the contact surfaces, have lower Rolling Friction than sliding contact.

 7.2 - Normal Forces
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&emsp;A Normal Force acts perpendicular to a surface, and has several sources. Normal here refers to the perpendicular direction in geometry, not common or average. They include gravity, magnetic or electrostatic attraction, and gas or liquid pressure. 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. Fluid Dynamics is the study of these more complex motions, both against solid surfaces and internally within fluids.

 7.3 - Thrust
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&emsp;Thrust is the force generated by a vehicle 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


 * $$\mathbf{T}=\frac{dm}{dt}\mathbf{v}$$

where T is the thrust generated; $$ \frac {dm} {dt} $$ 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.

 7.4 - Drag
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&emsp;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:


 * $$F_D\, =\, \tfrac12\, \rho\, v^2\, C_D\, A$$

where FD is the drag force, $$\rho\,$$ (Greek letter rho) 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 find, drag coefficient varies in a complex way based on object shape, speed, and other parameters. This is caused by complex flow conditions such as turbulence, shock waves, heating, and even chemical changes at high speeds. So drag coefficient is often found by measurement rather than calculation.

&emsp;When a surface moves relative to a fluid, the fluid layer closest to the surface is affected most by the molecules colliding with the it. They are deflected by the angle of the surface, its roughness, or by atomic forces between their respective atoms. This tends to make that nearest layer move along with the surface. That first layer in turn affects farther fluid layers by collisions of the molecules.

&emsp;At lower speeds this sets up a smoothly varying Boundary Layer near the surface. At higher speeds the deflection is violent enough to create flow Vortexes, where fluid regions revolve around axis lines rather than move in smooth layers. This kind of fluid motion is called Turbulence. It takes more energy to create vortexes, so 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 system components.

&emsp;The ratio of inertial forces, such as the sideways deflection, to the viscous forces caused by shearing (varying speed) in the boundary layer is called the 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 Quantity, meaning all the units in the formula cancel out in a coherent system like SI, leaving a pure number.

&emsp;The Reynolds Number, Re, is found by:


 * $$Re = {{\rho {\mathbf \mathrm v} L} \over {\mu}} = {{{\mathbf \mathrm v} L} \over {\nu}}$$

where: $${\mathbf \mathrm v}$$ is the mean velocity of the object relative to the fluid (m/s), $${L}$$ is a characteristic linear dimension of the surface (meters), $${\mu}$$ is the Dynamic Viscosity of the fluid (Pa·s, N·s/m², or kg/(m·s)), $${\mathbf \nu}$$ is the Kinematic Viscosity ($${\mathbf \nu} = \mu /{\rho}$$) (m²/s), and $${\rho}\,$$ 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.

&emsp;The motions within a turbulent fluid are too complex to reduce to a simple 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 fluid forces like lift and drag can be 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.

&emsp;In CFD simulations the flow is broken up into sufficiently small volumes that the flow in each volume obeys relatively simple formulas. The total flow in the simulation can then 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 simulations have become more practical with smaller computers.

 7.5 - Lift
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&emsp;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


 * $$L = \tfrac12\rho v^2 A C_L$$

where L is lift force, $${\rho}\,$$ is density of the medium, v is the velocity relative to the medium, A is planform (projected from above) area of the shape, and $$C_L$$ is the lift coefficient at a particular Angle of Attack between a reference line of the shape and the oncoming flow. Lift Coefficient also depends on Mach Number, and Reynolds Number. Like drag, lift coefficient is a result of complex fluid flow. Depending on project needs it can be found from a table or graph, physical experiment, or CFD simulation.

8.0 - Thermodynamics
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&emsp;Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. For space projects it becomes important for things like the operation of rocket engines, aerodynamic heating during reentry, and thermal radiators to dissipate excess heat.

&emsp;Thermodynamic variables and formulas are simplifications applied to bulk amounts of material. Their underlying cause is the microscopic behavior of very large numbers of atoms and molecules. Thermodynamics is also related to Chemistry, because chemical reactions can give off or or absorb heat. A prime example is a chemical rocket engine, where the propellants (fuel and oxidizer) react to produce very high temperature gas.

&emsp;Thermodynamics is a complex subject, so we cannot fully explore it here. We refer you to more detailed sources like the open textbook Thermodynamics and Chemistry 2nd Edition by Howard DeVoe. To give you a flavor for the topic, and because chemical rockets are very important in current space projects, we will describe their combustion cycle here, but it is not a complete description and many specialized terms are used:

&emsp;The Thermodynamic Cycle for a liquid rocket engine is a modified Brayton Cycle. A linear (one-dimensional) analysis of the engine may be performed by assuming the following ideal steps:


 * 1) Propellants (fuel and oxidizer) are injected into a Combustion Chamber either through use of pressurized fuel tanks or by a high-pressure pump, increasing the pressure to $$p_c$$ and increasing the Enthalpy.
 * 2) 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.
 * 3) The combusted fuel expands to the exit pressure, $$p_e$$, as it goes through the nozzle into the surroundings, which is at pressure $$p_0$$. Ideally $$p_e$$ should equal $$p_0$$.  During this process, the enthalpy decreases from $$h_c$$ to $$h_e$$.

&emsp;The thrust T produced this kind of engine is given by


 * $$T = \dot {m_p}v_e + A_{exit}*(p_e - p_0)$$

where $$\dot {m_p}$$ and $$v_e$$ are the mass flow rate and exit velocity of the propellant, $$A_{exit}$$ is the exit area of the nozzle and $$p_e$$ and $$p_0$$ are the pressure at the exit point of the nozzle and the atmospheric pressure.

&emsp;Enthalpy is the sum of internal energy available for work and pressure times volume. The energy change per unit time as the propellant moves from the combustion chamber to the nozzle exit is


 * $$\dot {m_p}(h_c - h_e) = {1 \over 2} \dot {m_p} v_e^2$$

&emsp;Solving for the propellant velocity yields


 * $$v_e = \sqrt{2(h_c - h_e)}$$

&emsp;Let us assume that the combustion mixture of the propellants is an Ideal Gas. The internal energy h per unit mass of an ideal gas is given by


 * $$h = \hat{c}_V T$$

where $$\hat{c}_V$$ is the Heat Capacity at Constant Volume producing an equation for the propellant velocity of


 * $$v_e = \sqrt{2\hat{c}_V(T_c - T_e)} = \sqrt{2\hat{c}_VT_c \left (1 - {T_e \over T_c} \right )}$$

&emsp;When an ideal gas expands by an Isentropic Process, the initial temperature T1 and pressure p1 change according to:


 * $${p_1 \over p} = \left (1 + {\gamma - 1 \over 2} M^2\right )^{\gamma/(\gamma - 1)}$$; and


 * $${T_1 \over T} = 1 + {\gamma - 1 \over 2} M^2$$;

where M is the Mach Number at the position having static pressure p and temperature T and $$\gamma$$ is the Heat Capacity Ratio. Using these two equations, we can relate the temperature and pressure ratios as


 * $${T_1 \over T} = \left ({p_1 \over p} \right )^{(\gamma - 1)/\gamma}$$

&emsp;We can rewrite the equation for the propellant velocity as


 * $$v_e = \sqrt{2\hat{c}_V(T_c - T_e)} = \sqrt{2\hat{c}_VT_c \left [1 - \left ({p_e \over p_c} \right)^{(\gamma - 1)/\gamma} \right ]}$$

&emsp;The final analysis step in the one-dimensional analysis is the effects of the nozzle. The previous equation demonstrates that making the ratio $${p_e / p_c}$$ 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. A one-dimensional analysis such as this is only a first approximation for rocket engine design.