Lecture 3 - 7B Lecture 3 John McRaven What I am...

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Unformatted text preview: 7B Lecture 3 John McRaven What I am going to teach you   Linear Transport   Heat Flow   Diffusion   Mathematically the same as fluids and circuits   Capacitors   Exponential Change   The last two lectures have been more focused on equations and using models. The concepts are the focus of this lecture Heat Flow   Heat: The transfer of thermal energy from one body to another   Thermal energy always flows from the higher temperature body to the lower temperature body   Energy can be transferred primarily through three methods   Radiation   Conduction   Convection Radia6on   All bodies radiate energy. The         amount and color depends on their temperature. The sun emits a lot of energy in the “visible light” spectrum because of its temperature (~5777 K) Humans can be seen at night with infrared goggles because we emit mostly infrared light Metal becomes “red hot” when it gets so hot that we can actually see the light it emits This is covered in more detail in 7C Radia6on Convec6on and Conduc6on   Conduction: Transfer of thermal energy through direct contact   When molecules come in to contact, they can exchange thermal energy   This is the primary method of heat flow through a material   Convection: Transfer of thermal energy through fluid motion   Running water over something will cool it quicker because the water absorbs the heat then takes it away   Fans cool you down because of convection Radia6on, Conduc6on, Convec6on Radia6on, Conduc6on, Convec6on Radia6on, Conduc6on, Convec6on Heat Flow: Mathema6cs   ΔT: Difference in temperature between two points (units: K°)   L: distance between the two points (units: m)   P: Power. Amount of thermal energy per second flowing (units: W=J/s)   k: Thermal conductivity. How well a material conducts thermal energy (units:W/(m K))   A: surface area of material that heat is being conducted through (units: m2) ΔT = − IR I=P 1L R= kA A P = −k ΔT L Heat Flow Example: Ice Mel6ng   Ice at 0° C is in a cubic Styrofoam container with sides 2 m long and walls 1 cm thick   A = 22*6=24 m2   L = .01 m   k = .03 (see pg 28 of notes)   Temperature outside is 30° C   ΔT = 30   So:   P = .03 * 24 * 30 / .01 = 2160 W Diffusion   Diffusion: The net transport of particles from a higher concentration to a lower concentration due to random motion   This will happen due to thermal motion, but can be sped up by, for example, stirring   This happens entirely due to random motion. There is no force driving the particles away from each other Random Mo6on: Simple 1D Example   If you have a lot of particles moving in one dimension, then on average, half the particles will move one way, half the particles move the other way   In 3D, particles can move 6 different ways   This example is 1D Diffusion: Mathema6cs   Δc: Difference in concentration between two points (units: # of particles / m3)   L: distance between the two points (units: m)   j: particle flux. (units:# of particles / m2 s)   D: Diffusion constant. How well the particles travel through the material (units:m2 / s)   A: surface area of material that particles are diffusing through (units: m2) Δc = − IR I j= A 1L R= DA Δc j = −D L Linear Transport: PuHng it all together   Fluids, Circuits, Heat Flow, and Diffusion are all mathematically described by linear transport   This basically means that something flows in proportion to the difference in something else   The general equation is: I Δφ j = = −k A Δx What Flows Difference in this quantity drives the flow Proportionality Equation constant Fluids Fluid Pressure Viscosity Bernoulli’s Circuits Charged particles Voltage σ I= σ A ΔV / L ΔV= ­IR Heat Thermal Energy Temperature k P=k A ΔT / L Diffusion Particles D j=D Δc / L € Concentration Linear Transport: What you need to know   You should understand fluid flow and electric circuits in detail   You should understand the concepts behind heat flow and diffusion, but the details aren’t extremely important   You do need to understand the equations though   You should still be capable of doing a complicated heat flow or diffusion problem by understanding the analogy to circuits and fluids Exponen6al Change   When the change in a quantity is proportional to the quantity itself, the process is described by an exponential   y d dt € = λy   Linear:   y(t) = λ t+y0   Quadratic:   y(t) = λ t2+y0   Exponential:   y(t) = y0 eλ t • y is some quantity that changes with time • y0 is the value of y at t=0 • λ is some constant • e = 2.718 Exponen6al Change Example: Water Hea6ng Up   Water at temperature T (T < 30° C) is in a cubic Styrofoam container with sides 2 m long and walls 1cm thick   A = 22*6=24 m2, L = .01 m   m (mass) = 8000 kg, Cp=4181 J/kg K   k = .03 (see pg 28 of notes)   Temperature outside is 30° C   ΔT = 30 ­T   So:   P = k A ΔT / L = .03 * 24 * ΔT/ .01 = 2160 – 72 T   This is the amount of energy entering the water per second ( J/s) at some given temperature. You can see that heat will enter at a slower rate as the temperature of the water approaches the outside temp Exponen6al Change Example: Water Hea6ng Up   In some small amount of time (dt), a small amount of heat will be gained (dQ), and the water will increase a small amount in temperature (dT)   P = dQ / dt = m C dT / dt = 33448000 dT / dt   Combining this with our previous equation, we get:   2160 ­72 T=33448000 dT/dt   The rate at which temperature changes is dependent on the temperature itself − λt T ( t ) = ( 30 − 30e ) t c = 1 / λ = 464556 s Exponen6al Change Example: Water Hea6ng Up   The graph of the temperature over time is exponential   It changes quickly at first, and slowly at the end   tc is the time constant. It is the amount of time it takes to get to 63.2% of it’s final value 30 25 20 T 15 10 5 200 400 t (hours) 600 800 Half Life   If a quantity is decreasing towards 0:   The half life is the time it takes for something to lose half of its value (50% of its initial value)   If you started out with 64 of something, and the half life was t1/2=5s, you would have   32 after 5s   16 after 10s   8 after 15s   4 after 20s Clicker Ques6on   If you started out with 80 of something, how many half ­lives would pass before you only had 10 remaining?   A)1   B)2   C)3   D)4   E)5 Half Life   If the quantity is increasing or decreasing:   The half ­life is the time it takes to get half way between its initial and final value   If you had 0 of something, and were increasing exponentially towards 100 of it, and t1/2=5s, you would have   50 after 5s   75 after 10s   87.5 after 15s   93.75 after 20s Time Constant   The “ Time Constant” or “1/e time” is the amount of time it takes to reach   tc=1/λ   It is the same idea as the half life, except instead of a 50% change, it is a 36.8% change   e ­1=.368   Half ­lives: 50%,25%,12.5%   Time Constants:36.8%,13.5%,4.97% Capacitors   Capacitors are another circuit element   It consists of two parallel metal plates   As current flows through a circuit, electrons are taken from one metal plate to the other   With a negative charge on one plate, and a positive charge on the other, a voltage difference occurs between the two plates   Electrons can’t go between the metal plates, since air does not conduct, but they can go around the wire.   As the negative plate gets more negative, it becomes harder to place electrons on it, which will result in exponential change Capacitance   “Capacitance” is a measure of how easily charge can be added to the capacitor. The units of capacitance is Farads (F)   A: Area of the plates. The capacitance increases as the area of the plates get bigger   d: Distance between the two plates. The capacitance increases as the distance between the plates decreases   When placing an electron on the negative plate, it feels the repulsion of the negative charges, but also the attraction of the positive charges on the other plate   ε: A property of the material in between the capacitor. A C =ε d Capacitors in Circuits   The voltage drop across a capacitor depends upon its charge Q ΔV =   C   For a circuit with a battery, capacitor and resistor, a loop around the circuit gives us:   ΔV = 0 = E − IR − Q C €   If the capacitor is uncharged, Q=0, so I=E/R   If the capacitor is fully charged, its voltage is equal to € that of the battery, so I = 0/R=0   The current decreases as the charge gets closer to the charge of the battery, which slows down the charging   This is exponential change Discharging Capacitor   If we fully charge the capacitor, then remove the battery, the capacitor provides energy to the circuit, and we have:   V = 0 = Q − IR Δ C   Current is the flow of charge, and can be € described as I   = dQ dt   We can solve for Q in terms of an exponential: € dQ 1 =− Q dt RC 1 λ= RC Q( t ) = Q0e − t / RC I ( t ) = I0e − t / RC V ( t ) = V0e − t / RC Finding the Time Constant I ( t ) = I0e − t / RC V ( t ) = V0e − t / RC V01.0 0.8 € 0.6 V0/2 0.4 V0/e 0.2 0.5 1.0 1.5 t/tc 2.0 2.5 3.0 Clicker Ques6on   If the voltage of the battery is 6V, and the resistance is 2 Ω, what is the maximum current in the circuit?   A) 6A   B) 3A   C) 2A   D) There cannot be a current in the circuit Clicker Ques6on   If the voltage of the battery is 6V, and the resistance is 2 Ω, then at the instant when the current is 2A, what is the voltage across the capacitor?   A)  ­6V   B)  ­4V   C)  ­2V   D) 0V Effec6ve Capacitance   Capacitors add in the opposite way that resistors do   This is because of the opposite relations between area and length   Ignore resistors when calculating effective capacitance   Capacitors in series: 1 1 1 =+ + ... CS C1 C2   Capacitors in parallel: € C p = C1 + C2 + ... Clicker Ques6on   If each capacitor in the circuit is 4F, what is the equivalent capacitance of the circuit?   A) 0F   B) 2F   C) 4F   D) 8F Quiz Topics   Next Week: Circuits   No capacitors or linear transport   Two Weeks: Linear Transport/Capacitors   Three Weeks: Entirely new topic (vectors and momentum) Conclusions   Linear Transport can be used to describe many phenomenon   Exponential Change happens when the change in a quantity depends on the quantity itself   Capacitors are a new circuit element that exhibit exponential change ...
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