Area Formulas

Area Formulas

Square = a ² 
Rectangle = ab 
Parallelogram = bh 
Trapezoid = h/2 (b₁ + b₂) 
Circle = pi r ² 
Ellipse = pi r₁ r₂ 

triangle =

one half times the base length times the height of the triangle

  

equilateral triangle =

triangle given SAS (two sides and the opposite angle)= (1/2) a b sin C

triangle given a,b,c = [s(s-a)(s-b)(s-c)] when s = (a+b+c)/2 (Heron's formula)

regular polygon = (1/2) n sin(360°/n) S²
   when n = # of sides and S = length from center to a corner

   Units

Area is measured in "square" units. The area of a figure is the number of squares required to cover it completely, like tiles on a floor.

Area of a square = side times side. Since each side of a square is the same, it can simply be the length of one side squared.

If a square has one side of 4 inches, the area would be 4 inches times 4 inches, or 16 square inches. (Square inches can also be written in².)

Be sure to use the same units for all measurements. You cannot multiply feet times inches, it doesn't make a square measurement.

Volume Formulas

Volume Formulas

cube = a ³ 

rectangular prism = a b c 

irregular prism = b h 

cylinder = b h = pi r ² h 

pyramid = (1/3) b h 

cone = (1/3) b h = 1/3 pi r ² h 

sphere = (4/3) pi r ³ 

ellipsoid = (4/3) pi r₁ r₂ r₃ 

Units

Volume is measured in "cubic" units. The volume of a figure is the number of cubes required to fill it completely, like blocks in a box.

Volume of a cube = side times side times side. Since each side of a square is the same, it can simply be the length of one side cubed.

If a square has one side of 4 inches, the volume would be 4 inches times 4 inches times 4 inches, or 64 cubic inches. (Cubic inches can also be written in³.)

Be sure to use the same units for all measurements. You cannot multiply feet times inches times yards, it doesn't make a perfectly cubed measurement.

Fluid Mechanics - Law of Continuity

 Law of Continuity

Any fluid moving through a pipe obeys the Law of Continuity, which states that the product of average velocity (v), pipe cross-sectional area (A), and fluid density (ρ) for a given flow stream must remain constant:

 

Fluid continuity is an expression of a more fundamental law of physics: the Conservation of Mass. If we assign appropriate units of measurement to the variables in the continuity equation, we see that the units cancel in such a way that only units of mass per unit time remain:

 

This means we may define the product ρAv as an expression of mass flow rate, or W:

In order for the product  to differ between any two points in a pipe, mass would have to mysteriously appear and disappear. So long as the flow is continuous (not pulsing), and the pipe does not leak, it is impossible to have different rates of mass flow at different points along the flow path without violating the Law of Mass Conservation. The continuity principle for fluid through a pipe is analogous to the principle of current being the same everywhere in a series circuit, and for equivalently the same reason₁.

We refer to a flowing fluid as incompressible if its density does not substantially change₂. For this limiting case, the continuity equation simplifies to the following form:

Examining this equation in light of dimensional analysis, we see that the product Av is also an expression of flow rate:

Cubic meters per second is an expression of volumetric flow rate, often symbolized by the variable Q:

 

The practical implication of this principle is that fluid velocity is inversely proportional to the cross-sectional area of a pipe. That is, fluid slows down when the pipe’s diameter expands, and visa-versa. We see this principle easily in nature: deep rivers run slow, while rapids are relatively shallow (and/or narrow).

For example, consider a pipe with an inside diameter of 8 inches (2/3 of a foot), passing a liquid flow of 5 cubic feet per minute. The average velocity (v) of this fluid may be calculated as follows:

 

Solving for A in units of square feet:

 

Now, solving for average velocity v:

 

 

₁In an electric circuit, the conservation law necessitating equal current at all points in a series circuit is the Law of Charge Conservation.

₂Although 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 ρ does not substantially change, whether by actual impossibility or by circumstance.

Kepler's Laws

Kepler's Laws

Johannes Kepler, working with data painstakingly collected by Tycho Brahe without the aid of a telescope, developed three laws which described the motion of the planets across the sky.

1. The Law of Orbits: All planets move in elliptical orbits, with the sun at one focus.

2. The Law of Areas: A line that connects a planet to the sun sweeps out equal areas in equal times.

3. The Law of Periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

Kepler's laws were derived for orbits around the sun, but they apply to satellite orbits as well.

The Law of Orbits

All planets move in elliptical orbits, with the sun at one focus.

This is one of Kepler's laws. The elliptical shape of the orbit is a result of the inverse square force of gravity. The eccentricity of the ellipse is greatly exaggerated here

Orbit Eccentricity The eccentricity of an ellipse can be defined as the ratio of the distance between the foci to the major axis of the ellipse. The eccentricity is zero for a circle. Of the planetary orbits, only Pluto has a large eccentricity. Examples of Ellipse Eccentricity Planetary orbit eccentricities Mercury .206 Venus .0068 Earth .0167 Mars .0934 Jupiter .0485 Saturn .0556 Uranus .0472 Neptune .0086 Pluto .25 The Law of Areas A line that connects a planet to the sun sweeps out equal areas in equal times. This is one of Kepler's laws.This empirical law discovered by Kepler arises fromconservation of angular momentum. When the planet is closer to the sun, it moves faster, sweeping through a longer path in a given time.

The Law of Periods

The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit.

This is one of Kepler's laws.This law arises from the law of gravitation. Newton first formulated the law of gravitation from Kepler's 3rd law.

Kepler's Law of Periods in the above form is an approximation that serves well for the orbits of the planets because the Sun's mass is so dominant. But more precisely the law should be written

In this more rigorous form it is useful for calculation of the orbital period of moons or other binary orbits like those of binary stars.

Data: Law of Periods

Data confirming Kepler's Law of Periods comes from measurements of the motion of the planets.

Planet	Semimajor axis (1010m)	Period T (y)	T2/a3 (10-34y2/m3)
Mercury	5.79	0.241	2.99
Venus	10.8	0.615	3.00
Earth	15.0	1	2.96
Mars	22.8	1.88	2.98
Jupiter	77.8	11.9	3.01
Saturn	143	29.5	2.98
Uranus	287	84	2.98
Neptune	450	165	2.99
Pluto	590	248	2.99

The quantity T²/a³ depends upon the sum of the masses of the Sun and the planet, but since the mass of the Sun is so great, adding the mass of the planet makes very little difference.

Data from Halliday, Resnick, Walker, Fundamentals of Physics 4th Ed Extended.

Intercooler

Intercooler

An intercooler is an intake air cooling device commonly used on turbocharged and supercharged engines.

An intercooler cools the air compressed by the turbo/supercharger, reducing its temperature and thereby increasing the density of the air supplied to the engine.

As the air is compressed by a turbo/supercharger it gets very hot, very quickly. As its temperature climbs, its oxygen content (density) drops, so by cooling the air, an intercooler allows denser, more oxygen rich air to the engine, allowing more fuel to be burned, thus improving combustion and giving more power. It also increases reliability as it provides a more consistent temperature of intake air to the engine which allows the air fuel ratio of the engine to remain at a safe and steady level.

There are two types of intercoolers; Air-to-Air and Air-to-Water.

An Air-to-Air intercooler extracts heat from the compressed air by passing it through its network of tubes with cooling fins. As the compressed air is pushed through the intercooler it transfers the heat to the tubes and, in turn to the cooling fins. The cool air from outside, traveling at speed, absorbs the heat from the cooling fins reducing the temperature of the compressed air. The advantages of this system are simplicity, lower cost and light weight. These factors make it by far the most common form of intercooling. The down sides can be a longer intake length (as the intercooler is usually at the front of the car) and more variation in temperature than the Air-to-Water type.

An Air-to-Water intercooler uses water as a heat transfer agent. In this setup cool water is pumped through the air/water intercooler, extracting heat from the compressed air as it passes through. The heated water is then pumped through another cooling circuit (usually a dedicated radiator) while the cooled compressed air is pushed into the engine.

These intercoolers (also known as heat exchangers) tend to be smaller than their Air-to-Air counterparts making them well suited to difficult installations where space, airflow and intake length are an issue. Water is more efficient at heat transfer than air and has more stability so it can handle a wider range of temperatures.

On the downside the Air-to-Water system is complex, heavy and has the added cost of a radiator, a pump, water and transfer lines. Common applications for these are industrial machinery, marine and custom installs that don’t allow the easy fitment of an air to air intercooler, such as a rear-engined vehicle.

Intercooler placementThe best placement for an air to air is in the at the front of the vehicle. The “front-mount” is considered to be the most effective placement.

When the engine layout, or type of the vehicle do not permit the “front-mount” placement, an intercooler can be mounted on top of the engine, or even on its side, but these are not considered as effective as the air flow is not as good and the intercooler can suffer from heat soak from the engine when the external airflow drops. These placements will often require additional air ducts or scoops to route the air directly into the intercooler.

Subaru’s trademark top-mount requires a scoop in the bonnet to route the airflow onto the intercooler.

The Air-to-Water system can be mounted anywhere in the engine bay, as long as the radiator is mounted in a position where it receives good airflow, and/or with a thermo fan attached to it.

The Steady Flow Energy Equation (SFEE)

The Steady Flow Energy Equation (SFEE) is used for open systems to determine the total energy flows.

It is assumed that the mass flow through the system is constant.

It is also assumed that the total energy input to the system is equal to the total energy output.

The energies that are included are;

          internal, flow, kinetic, potential, heat and work.

The equation is shown below where suffix ₁ is the entrance and suffix ₂ the exit from the system.

where:

u          =          internal energy (J)

P          =          pressure (N/m²)

v          =          volume (m³)

C          =          velocity (m/s)

g          =          acceleration due to gravity (m/s²)

Z          =          height above a datum (m)

Q         =          heat flow (J)

W         =          work (J)

The term P .v  represents the displacement or flow energy.

The term  C² / 2   represents the kinetic energy.

The term g. Z  represents the potential energy.

 

In thermodynamics the changes in potential energy are usually small except for example a water reservoir supplying water to a low level turbine.

In the following examples we can omit the potential energy term thus simplifying the equation to;

u₁  +  P₁v₁  +           +  Q   =    u₂  +  P₂v₂  +            +  W

Also the term  ( u  +   P.v ) is also known as specific enthalpy (h), so the equation is now written;

h₁  +            +  Q   =    h₂  +            +  W

Example 1

In a steady flow open system a fluid flows at a rate of 4 kg/s.

It enters the system at a pressure of 6 bar, a velocity of 220 m/s, internal energy

2200 kJ/kg and specific volume 0.42 m³/kg.

It leaves the system at a pressure of 1.5 bar, a velocity of 145 m/s, internal energy

1650 kJ/kg and specific volume 1.5 m³/kg.

During its passage through the system, the fluid has a loss by heat transfer of 40kJ/kg to the surroundings.

Determine the power of the system, stating whether it is from or to the system.

Neglect any change in potential energy.

u₁  +  P₁v₁  +                +    Q   =       u₂  +  P₂v₂  +               +   W

Power from the system (kW)   = Work output (kJ/kg) x mass flow rate of fluid (kg/s)

Work output (kJ/kg) can be found by rearranging the SFEE.

We can work in kilojoules (kJ). This means that the  kinetic energy section of the SFEE will be divided by 10³ to put it in kJ.

W  =   ( u₁  -  u₂)   +  ( P₁v₁  -    P₂v₂ ) +   (                 )  +  Q

2 x 103

W   =   ( 2200  -  1650 )  +  ( 600  x  0.42    -   150  x  1.5 )  +  (                     )   – 40

W  =    550     +      27         +         13.69          -    40

W   =    550.69  kJ/kg

This is positive so it is energy output from the system.

Power from the system (kW)   =           W    x  m

                                   

                                                =          550.69   x   4 kg/s

                                                =          2202.76 kW