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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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 Spring '10
 McRaven
 Heat

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