A perfectly stirred, constant-volume tank has two input streams, both consisting of the same liquid. The temperature and flow rate of each of the streams can vary with time. Tl Stream 1 ------,1 wl '+' T2 Stream 2 w2 T3 Stream 3 w3 Figure E2.1 (a) Derive a dynamic model that will describe transient operation. Make a degrees of freedom analysis assuming that both Streams 1 and 2 come from upstream units (i.e., their flow rates and temperatures are known functions of time). (b) Simplify your model, if possible, to one or more differential equations by eliminating any algebraic equations. Also, simplify any derivatives of products of variables. Notes: w; denotes mass flow rate for stream i. Liquid properties are constant (not functions of temperature).
PHYS 1010 Notes Week 8 February 29 Jan 4 Magnetic fields Magnetic particles are dipoles, and the two poles of a magnet are the North and South poles All magnetic field lines originate from the North Pole and move toward the South Pole We draw magnetic field lines by placing a compass near the magnet and tracing the direction in which the arrow of the compass points (towards the south magnetic pole) Right now the south magnetic pole is at the North Pole of the Earth Earth's poles have switched historically Earth's magnetic poles are not exactly located at the north and South Pole Field lines pass through the magnet and form closed loops Opposite magnetic poles attract each other, and like magnetic poles repel each other Northern lights: magnetic fields crashing down in one place Spin: some particles, like electrons, produce a magnetic field characteristically Permanent magnet: magnetic fields of many particles arranged in a specific manner Some materials let particles float and when they go close to a magnet, the particles will arrange in a way that creates a magnet out of that material Magnetic force only affects moving particles SI unit for magnetic field is tesla (T) The equation for a magnetic field Tells us which direction the force will accelerate the charged particle (perpendicular to both the direction of the velocity and the magnetic field) Shows that magnetic fields tend to make particles move in a circle Magnetic fields can never increase the kinetic energy of a moving particle, only change the direction Magnetic fields cannot do work Magnetic fields only exert force on moving charged particles (moving charge is the same as an electric current) Magnetic force on a currentcarrying wire Running a current through a wire and putting it in a magnetic field will make the wire move back and forth Electrical current produces a magnetic field A loop of current behaves the same way of a bar magnet Loop of current inside a magnetic field is an electric motor: converts electrical current into mechanical energy Works by alternating the magnetic field/current direction Solenoid: wire wrapped in a cylindrical shape to amplify the magnetic field Solenoids start car engines (magnetic field exerts a force that pushes a button that starts the car) Toroid: a solenoid with ends connected to form a donut Create an amplified magnetic field that force charged particles to move in a circular path Used as particle accelerators Magnetism = moving electricity/charge Magnetism creating electricity "Induction": inducing an electric field Magnet being moved into a loop of wire creates electric current because from the magnet's perspective, those particles are now moving Electricity causing electricity Putting a loop of current next to another loop of current creates induction > Wireless communication Lenz's law Current of a wire inserted into a magnetic loop will induce magnetism that is the opposite direction of the original magnet, so it will "push back" at the magnet being inserted Verifies that there are forces being created, and forces can do work Dropping a magnet through a copper pipe Copper conducts electricity Changing magnetic fields create electricity Current/magnetic field in pipe pushes against magnet (Lenz's law) > Slows down magnet as it falls through pipe Electromagnetism Electricity and magnetism are basically the same thing, they just work in opposite directions Induction and energy transfers Rail guns Conducting wire frame creates magnetic field that pushes one side of moving wire out and gets bigger and bigger; used to fire projectiles Eddy Currents "Whirlpool" currents produced when a solid object moves through a magnetic field Result in lost energy Useful for train brakes Eddy currents created by superconductors create gigantic electric currents with no resistance > huge magnetic field > maglev trains Superconductors only conduct at low temperatures, which is an issue because it's lower than the temperature of liquid nitrogen (used to cool it off); superconductors are difficult to keep cold all the time, which is why we don't use them for everything Maxwell's equations Noticed that all of these equations together explain everything about electricity and magnetism 1. Definition of electric field 2. Definition of magnetic field 3. Faraday's law: a changing magnetic field creates an electric field 4. AmpereMaxwell Law: a changing electric field creates a magnetic field Explanation that electromagnetic radiation means that light is made up of waves But waves must travel through a substance so how does light travel through space According to Maxwell, light is electricity and magnetism (a wave that doesn't need to travel through anything) Electromagnetic waves Maxwell's equations 3 and 4 explain that light can move through an empty space because it is creating/driving itself with electricity and magnetism Pushing electric charges up and down creates electric waves/electric field which creates a magnetic field, and on and on We can send signals (radio, etc.) that keep moving forever The speed of light is a constant (never changing) Throwing a baseball 20mph means the speed it 20mph Throwing a baseball 20mph out a car while driving 20mph means the speed is 20mph Turning a flashlight on means the speed is constant, but turning a flashlight on while running also results in a constant speed Concept of mass/time/length units is called into question, changes everything we know about physics End of Midterm Material Transmitting energy using waves Transfer of energy without physically moving stuff from one point to another Opposite of particles All waves have diffraction and interference (particles have neither) Diffraction: moving around an object in its path Interference: overlapping waves create new waves Waves: repeating motion Changes position in time and in space Mechanical waves: some type of physical medium transmits the waves Medium is not necessarily moving left and right, just up and down Electromagnetic waves: don't require a medium Magnetic and electric fields that drive each other Matter waves: waves behaving like particles Properties of waves Amplitude: height of wave Wavelength: distance between peaks of waves Period: time between each wave passing a certain point Frequency: how many waves go by per second (basically same as period) Velocity: wavelength x frequency Creating waves Transverse waves: amplitude is perpendicular to motion Longitudinal waves: amplitude is parallel to motion