Download More Carnot Cycle March 4, 2010 Efficiency = W/Qin = Qin

More Carnot Cycle
March 4, 2010
Efficiency = W/Qin = Qin - Qout/Qin = 1-Qout/Qin
W = Qin/Qout
Efficiency = 1-Tc/TH
Quiz Monday March 8th - Kinetic Theory
Test Tuesday March 16th - Thermodynamics + Kinetic Theory
Thermodynamic Cycle
March 2, 2010
A Thermodynamic cycle is a series of processes where the system is eventually
brought back to its original state. See fig 1.1q
In the diagram the gas starts at V1, P1 then it was expanded isothermally to V2, P2
then compressed at constant pressure to V3, P3. In this case, V3 = V1. Heat was added
to bring it back to its original state at constant volume. The total work by the system
is equal to the sum of the works done in each process.
W = ∫P dV
The work done in a complete cycle is the contained area. See fig 1.2
Carnot Cycle
Heat engines are based on the ability of a system to do work, and the carnot engine
is based on the following cycle. See fig 2.1
In this process, the system is expanded along an isotherm, with heat being supplied
by the surroundings (Q1). The system expands further along an adiabat (Q = 0)
Compressed along another isotherm giving heat off to the surroundings (Q2). The
system is compressed further, along another adiabat to bring it back to its original
state.
Thermodynamics
Ideal Gas Law
PV = nRT
n = # of moles of gas
V = volume (m3)
Feb 10, 2010
P = Pressure (Pa)
T = Temperature (K)
1 mole of gas contains 6.022 x 1023 molecules.
n = mtotal/M
mtotal = total mass of gas (grams)
For an Ideal Gas
P1V1/T1 = P2V2/T2 = constant
R = 8.3145 J/mol K
M = molecular mass (g/mol)
∆U = Q - W
∆U = change in internal energy (J)
Q = heat (J)
W = Work done by the system
∆U = Q + W
Note: work is the work done ON the system
Q is positive for heat added to the system and is negative for removed from the
system. The work done by the system is defined as:
W = ∫P dV
(area under pressure (Y) vs volume (X))
There can only be work done by the system if there is a change in volume. If the
volume of gas expands, the work done by the system is positive.
The internal energy U is an indication of the thermal energy state of the system. U is
directly related to the temperature of the system.
Ex Why does an expanding gas like using a container of compressed air feel cold?
∆U = P - UV
expanding gas ∆V > 0
W = ∫P dV > 0
∆U < 0, ∆T < 0
Specific Heats for Ideal Gases
There are two specific heats defined for an ideal gas. One for the gas kept at constant
pressure (Cp), and the other for constant volume (Cv).
Q = nCv∆T
n = number of moles
Cv = molar heat capacity at constant volume
Q = nCp∆T
Cp = molar heat capacity at constant pressure
CP always has to be greater than Cv
Thermodynamic Processes
In an isothermal process a gas is expanded or compressed at a constant
temperature. Q = W
W = ∫P dv, since P = nRT/V ... W = nRT ln(Vf/Vi)
In an isochoric process, heat is added or removed from the system while keeping it
at constant.
V = constant, W = 0, ∆U = Q
In an Isobaric process, a gas is expanded or compressed at constant pressure.
P = constant
∆U = Q - W
Q = nCp∆T
∆U = nCv∆T
W = P∆V
Adiabatic Process
In this process, the system expands or contracts while being thermally insulated. No
heat can be added or removed from the system. Q = 0
∆U = -W
All is equal to the work done on the system.
Kinetic Theory of Gases
The molecular or particle interpretation of temperature and gas laws
Feb 4, 2010
Assumption of Kinetic Theory
1.
here are a large number of molecules N, each of mass m, moving in random
directions with a variety of speeds/
2.
he molecules are on average, far apart from one another.
3.
he molecules are assumed to obey the laws of classical mechanics, and one
assumed to only interact in a collision
4.
ollisions with another particle or wall of the vessel are assumed to be perfectly
elastic
Definitions and Relationships
Distinction between average velocity (v), and the root-mean-square velocity (vrms)
Assume the gas has N molecules
vaverage = v1 + v2 + v3... vn / N
vrms = √( vaverage2), where v2 = v12 + v22... + vn2/N
Ex A container has 5 molecules travelling at the following speeds: 60m/s, 55m/s,
73m/s, 62m/s.
What is the average speed?
What is Vrms?
PV = 2/3N(1/2 mv2) = 2/3N(KE)
PV = nRT, R = 8.315 J/mol l K
Distribution of Molecular Speeds (Maxwell)
Effective Thermal Conductivity
L/K = L1/k1 + L2/k2 .. Ln/kn
Thermal Expansion
Jan 27, 2010
T
T
T
C
It has been observed that solids expand or contract with change in temperature.
Structures are designed to allow for expansion and contraction.
Suppose a rod of material has length LO at some initial temp. When the temperature
changes by ∆T, the length changes by ∆L. It has been observed that heat ∆L is
directly proportional to ∆T.
∆L = LO∆T
∆L = change in length (m)
LO = initial length (m)
∆T = change in temperature (OC, k)
 = linear thermal expression coefficient (1/OC)
L - LO = LO∆T
V = VO(1 + ∆T)
 = coefficient of volume expansion, roughly 3x 
Ex: A pendulum (LO = 2.50m) has a certain time period. Its time period increased by
0.10%. If the pendulum was made of brass, what temperature change did the
pendulum go through?
Thermal Conductivity
Jan 25, 2010
Heat Transfer occurs only between regions that are at different temperatures, and
the direction of heat flow is always from higher to the lower temperature.
The diagram shows a rod of a conducting material with cross sectional area A and
length L. The left end of the rod is kept at constant temperature TH and the right end
at lower temperature TC. The sides of the rod are perfectly translated so that there is
no heat transfer to the environment.
When a quantity of heat (dQ) is transferred through the rod, the ratio of heat flow is
dQ/dt. This is called the heat current, denoted by H (how much heat/second is
moving down the rod)
Experiments show that the heat current is proportional to the cross-sectional area
A, and the temperature difference (TH - TC), and is inversely proportional to the
length (L) of the rod. Introducing a constant of proportionality K, called the thermal
conductivity of the material.
H = dQ/dt = KA(TH - TC)/L
W = k(m)(oc)
k = W/moc
Ex: A cabin has an interior temperature of 21oc. The outside air temperature is 10oc. How much energy is conducted through a window (3m x 1m) in a period of
24h? Assume the thickness is 1.00cm. kglass = 0.84
Answer: 450 MJ
Temperature and Heat
Jan 13, 2010
Temperature is the measure of the average kinetic energy of individual molecules
that make up the system
There are different temperature scales: Kelvin (Absolute), Celsius, and Fahrenheit
scales.
Absolute Zero - it is the coolest temperature possible that cannot be reached (it is a
limit). 0 K.
K = C = 273.15°
C°
F°
0°
32°
100°
212°
F = (9/5)C+ 32
Heat is the transfer of energy, in Joules. There are three different methods of heat
transfer: conduction, convection, and radiation.
Conduction: for heat to be transferred by conduction, there must be contact between
water. Matter or a medium is required.
Convection: Convection relies on mass transport of matter. Eg: air thermals. Hot air
rises cold air sinks.
Radiation: Heat transferred through radiation (electrons - magnetic waves) No
medium is required.
Heat Energy
Heat can be related to the change in temperature of a system
Q = mc∆T
Q = heat (J)
m = mass (kg)
c = specific heat (J/kg/°c)
∆T = Change in temperature (°C)
Specific Heat
Different materials have different values of specific heats. It relates an amount of
heat added or removed from a substance and that substance's change in
temperature.
Applications of Bernoulli's Equation
Jan 7, 2010
P1 + (1/2)pv12 + pgy1 = P2 + (1/2)pv22 + pgy2
A1v1 = A2v2
Ex: An open cylinder (radius 0.4m) is filled with water. 0.8m below the water's
surface there is another opening (radius 0.05m). How fast does the water emerge
from the bottom at this instant:
y1 = 0
y2 = 0.8
P1 = P2 atmospheric pressure
V1 >> V2 *assumption
From the equation above, you can cancel out things to get (1/2)v12 = gy2
v1 = 3.96 m/s
Ex: A flat roof has an area of 400m2. In a wind storm the wind blows at 100km/h
(27.8 m/s) over the top of the roof. What upward force is created by the wind?
P1 + (1/2)pv12 + pgy1 = P2 + (1/2)pv22 + pgy2
y1 = y2 (assume roof thickness is negligable)
(1/2)pv12 = P2 - P1 - ∆P
(1/2)(1.29)(27.8)2 = 498 Pa
P = F/A, find Force.
More Fluids
Jan 5, 2010
When fluids flow around an object there are two kinds of flow: laminar flow and
turbulent flow
Laminar Flow
The fluid flows smoothly around the object
Turbulent Flow
When it doesn't
LAB Buoyancy
Dec 14, 2009
Aim: To determine the density of water and in the process verify assumptions
concerning buoyancy.
Fluids
Dec 4, 2009
Gases and liquids are commonly referred to as fluids
Gases
can flow
Fluids
can flow
conforms to a container
compressible
conforms to a container
incompressible
Density
v =  = m/v
 = volume density (kg/m3)
Specific Gravity
The ratio of the density of a gram of a given substance to the density of water
S.G = specific gravity = x/water
Density of water = 1 g/cm3 = 1000 kg/m3
The density is measured in g/cm3 - the specific gravity
the max density of H2O(L) is at 4.00o C
Pressure
Pressure is defined as the force per unit area. Where the force acts perpendicularly
to the surface.
P = F/A
P = pressure (Pa, Pascal)
In a fluid which is static, the pressure will act perpendicular to any surface.
For Change in Pressure of a Static Fluid (see notes),
P = gh
one is assuming that there is zero pressure acting down on the liquid's surface. If
there were pressure (Po) acting down on the liquid surface, the absolute pressure
(P) would be given by the following
P = gh + Po
P = gh
Always assuming density of liquid remains constant.
Buoyancy & Archimedes Principle
(see notes for derivation)
FB = fVg
Thin Film Interference
Nov 30, 2009
Thin film interference can be observed when looking at soap film (bubble) and oil
films (oil in water). One normally sees the colors of the rainbow. In this case, the
colors observed are a result of constructive interference between light reflected
from two different surfaces.
There is a phase change of a light ray on reflection if the second medium has a
higher index of refraction than the first medium.
If there is a phase change for one of the light rays in the film interference, the
condition for constructive changes to the condition of destructive interference.
If a phase change or reflection occurs for both light rays, there is no relative phase
change. Therefore, the condition for constructive interference would be the normal
one.
Young's Experiment
Nov 16, 2009
Experiment showed that light behaves like a wave.
This demonstrated two source interference of light
For constructive interference
P.D = dsinθ = mλ (m = 0, 1, 2, 3 ...)
For destructive Interference
P.D = dsinθ = (m + 1/2)λ (m = 0, 1, 2, 3 ...)
Diffraction Gratings
There are two basic kinds of diffraction gratings:
Transmission
Reflection
The diffraction grating is commonly used to analyze light spectra.
Transmission Grating
It is created by having a material with multiple parallel grooves or opening for light
to pass through.
Constructive Interference: P.D = dsinθ = mλ (m = 0, 1, 2, 3 ...)
d is distance between grooves.
The "Sonic Boom"
Nov 12, 2009
Whenever the source of the waves travel faster than the waves of the given medium,
this phenomenon is observed.
Beats
Beats can be explained by the superposition Principle - oscillation of loudness rather
than pitch.
Beat frequency - ∆ƒ = |ƒ2 - ƒ1|
Dispersion
Light is separated into its component colors through refraction. The index of
refraction is slightly different for each color and thus the angle of refraction is
different for each color
Polarization of Light
When light is polarized the light waves is oriented in a particular manner.
eg: a choppy sea - un-polarized. breaking waves - polarized
Double Lens System
Nov 3, 2009
To determine where the final image is produced by two lenses. The image produced
by the 1st lens acts as an object for the second lens. The 1st lens is the lens nearest
the initial real object.
In some cases the image produced by the 1st lens may act as a virtual object for the
second lens (do2 < 0). This occurs when the image produced by the first lens
appears on the far side of the second lens.
1/f1 = 1/do1 + 1/di1
1/f2 = 1/do2 + 1/di2
do1 is always positive, do2 could be "+" or "-"
focal length NEGATIVE for diverging lens**
Magnification
MT = M1 x M2
= (-di1/do1)(-di2/do2)