Geopotential Height ICAO Standard Atmosphere Lecture Ch. 2a

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International Standard Atmosphere
History of the Standard Atmosphere
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With a little digging, you can discover that the Standard Atmosphere can be
traced back to 1920. The constant lapse rate of 6.5° per km in the troposphere
was suggested by Prof. Toussaint, on the grounds that
– … what is needed is … merely a law that can be conveniently applied and which is
sufficiently in concordance with the means adhered to. By this method, corrections due
to temperature will be as small as possible in calculations of airplane performance, and
will be easy to calculate. …
– The deviation is of some slight importance only at altitudes below 1,000 meters, which
altitudes are of little interest in aerial navigation. The simplicity of the formula largely
compensates this inconvenience.
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•
•
The above quotation is from the paper by Gregg (1920). The early motivations for
this simplified model were evidently the calibration of aneroid altimeters for
aircraft, and the construction of firing tables for long-range artillery, where air
resistance is important.
Unfortunately, it is precisely the inaccurate region below 1000 m that is most
important for refraction near the horizon. However, the Toussaint lapse rate,
which Gregg calls “arbitrary”, is now embodied in so many altimeters that it
cannot be altered: all revisions of the Standard Atmosphere have preserved it.
Therefore, the Standard Atmosphere is really inappropriate for astronomical
refraction calculations. A more realistic model would include the diurnal changes
in the boundary layer; but these are still so poorly understood that no satisfactory
basis seems to exist for realistic refraction tables near the horizon.
http://mintaka.sdsu.edu/GF/explain/thermal/std_atm.html
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The ISA model divides the atmosphere into layers with linear temperature distributions.[2] The other
values are computed from basic physical constants and relationships. Thus the standard consists of a table
of values at various altitudes, plus some formulas by which those values were derived. For example, at sea
level the standard gives a pressure of 1.013 bar and a temperature of 15°C, and an initial lapse rate of 6.5 °C/km. Above 12km the tabulated temperature is essentially constant. The tabulation continues to
18km where the pressure has fallen to 0.075 bar and the temperature to -56.5 °C.[3][4]
U.S.
U.S.
U.S.
U.S.
Extension to the ICAO Standard Atmosphere, U.S. Government Printing Office, Washington, D.C., 1958.
Standard Atmosphere, 1962, U.S. Government Printing Office, Washington, D.C., 1962.
Standard Atmosphere Supplements, 1966, U.S. Government Printing Office, Washington, D.C., 1966.
Standard Atmosphere, 1976, U.S. Government Printing Office, Washington, D.C., 1976.
Geopotential Height
ICAO Standard Atmosphere
Lecture Ch. 2a
Internal Energy vs. Enthalpy
• Energy and its properties
– State functions or exact differentials
– Internal energy vs. enthalpy
• First law of thermodynamics
• Heat/work cycles
– Energy vs. heat/work?
– Adiabatic processes
– Reversible “P-V” work
• Homework problem Ch. 2, Prob. 2
Curry and Webster, Ch. 2 pp. 35-47
Van Ness, Ch. 2
• Difference b/w U and H
– U depends on v
– H depends on p
• Specific heats [a.k.a. heat capacity]
– cv is constant v
– cp is constant p
1
Heat Capacity
For an ideal gas
• Simplify to
• [Types of processes]
– Constant pressure
– Constant volume
Lord Kelvin
(a.k.a William Thomson)
Other Kinds of Energy
• In addition to changes in internal energy, a
system may change
– Potential energy for height change Δz
– Kinetic energy for velocity change Δv
– Nuclear energy for mass change Δm
James P. Joule
• The First Law of Thermodynamics
1
ΔE = ΔU ( p,V,T ) + mgΔz + mΔv 2 − c 2Δm = Q + W
2
• Consequences
Uniqueness of work values
Definition of energy
Conservation of energy
Wrev = − ∫ pdv
Q = 0 ⇒ ΔE = W
Q = 0, W = 0 ⇒ ΔE = 0 ⇒ E2 = E1
Impossibility of perpetual motion machine
(Relativity)
Q = 0, ΔE = 0 ⇒ W = 0
ΔE = mc 2
Reversible
Adiabatic
State function
See also 2nd law!
Proof follows..
if ΔE ≈ ΔU ( p,V,T ), then ΔU ( p,V,T ) = Q + W
Van Ness, p. 13
€
Work
• Expansion work W=-pdV or w=-pdv
– Lifting/rising
– Mixing
– Convergence
• Other kinds of work?
– Electrochemical (e.g. batteries)
2
Cycles
Exact Differentials
• State functions are exact differentials
• Work and heat are path-dependent
transfers
– W work
– Q heat
• State functions are unique “states”
– U internal energy
– H enthalpy
– η (also S) entropy
– A Helmholtz free energy
Heat/Work Cycles
Carnot was an engineer in
Napoleon’s defeated army
with an interest in engines.
• The efficiency with which work is accomplished in a reversible cyclic process
depends only on the temperature of the reservoirs to which heat is
transferred
THE CARNOT CYCLE
T
Q
1
STEP 1: Expand isothermally and reversibly at T 1
W1 = Q1 = RT1 ln
1
PA
PB
STEP 2: Expand adiabatically and reversibly
FLUID
Q
W
W = Cv (T2 − T1 )
STEP 3: Compress isothermally and reversibly at T 2
W2 = Q2 = RT2 ln
PC
PD
2
STEP 4: Compress adiabatically and reversibly
T
2
W = Cv (T1 − T2 )
Efficiency:
η = 1−
TCold
THot
3
Nikolaus
Otto
developed
the Otto
cycle in
1876.
P-V diagrams of work
Rudolf
Diesel
developed
the Diesel
cycle in
1892.
Other Work Cycles
• Work is determined by pathway
The Otto Cycle works
by compressing a
mixture of air and fuel
in a piston and then
igniting the mixture
with a spark.
The Diesel Cycle works
by compressing air and
then adding fuel directly
to the piston. The
compressed air then
combusts the mixture.
The compression ratio of the Diesel Cycle ranges
from 14:1 to 25:1, while the Otto Cycle range is
significantly lower, from 8:1 to 12:1.
Efficiency:
η = 1−
Efficiency:
TB − TC
T A − TD
1  T − TC 

η = 1 −  B
γ  T A − TD 
γ =5
4 Steps of Carnot “Engine”
3 for monatomic ideal gas
Hurricane as Carnot Cycle
3:Lose Heat
(isothermally)
2:Adiabatic
4:Adiabatic
1:Add Heat
(isothermally)
1:Add Heat
(isothermally)
2:Adiabatic
3:Lose Heat
4:Adiabatic
(isothermally)
Ideal Gases
Reversible-Adiabatic-Work
W = − pdv
Reversible
mass is conserved
Frictionless
Low P,
Low T
Ideal Gas
Adiabatic
p1v1
pv
=R= 2 2
T1
T2
Q =0
thick walls
First Law
Internal Energy
 ∂h 
dh  ∂h 
cp ≡   =
= 
 ∂T  p dT  ∂T  v
€
Δu = Q + W
Δu = c vdT
Reversible, Adiabatic
T2  P2 
= 
T1  P1 
R
cp
€
4
Reversible Processes
grain of sand
W rev = − pdv
Reversible
mass is conserved
Frictionless
€
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Always at or infinitesimally close to equilibrium
Infinitesimally small steps
Infinite number of steps
Each step can be reversed with infinitesimal force
Homework Ch. 2 Problem 2
Chapter 1a
5