Transcript Corporate Profile - University of Oklahoma

METR 2413
18 February 2002
Thermodynamics
II
Review
State variables: p, ρ, T
2
1N
 mV 
Pressure p  nm  u  
3V
2
2
1
 mV 
Temperature T 
3k
Equation of state: p = NkT/V = ρ Rd T
Virtual temperature Tv = T (1 + 0.61 r)
Hydrostatic balance
Consider air between two horizontal surfaces of area A located
at level z and level z + Δz.
Then the downward force on the upper surface is -p(z + Δz)A
and the upward force on the lower surface is p(z)A.
An additional force acting on the air between the two surfaces
is the downward weight force due to gravity -mg = -ρΔzAg
z
A
p(z + Δz)
p(z)
Hydrostatic balance
Assuming the air is in equilibrium and experiencing no
vertical acceleration, then the net force must be zero, so
– p(z + Δz)A + p(z)A – ρΔzAg = 0
p(z + Δz) – p(z) = – ρgΔz
(p(z + Δz) – p(z))/ Δz = – ρg
Now, taking the limit Δz →0 gives dp    g
dz
Pressure in the atmosphere increases towards the surface due
to the weight of the air above.
Hydrostatic balance
Hydrostatic balance has the vertical pressure gradient of the
air balance exactly by the weight due to gravity.
It does not preclude vertical motion, but it does preclude
vertical acceleration.
Weather systems with strong vertical motion, such as
thunderstorms, mountain waves or tornadoes, usually have
strong non-hydrostatic vertical accelerations and hydrostatic
balance does not hold.
Synoptic-scale weather systems usually have weak vertical
acceleration and hydrostatic balance holds quite well.
Hydrostatic balance
Integrate the hydrostatic equation in the vertical

z
0

z
z  pg
dp
dz    gdz 
dz
0
0
dz
Rd T
p( z)
p0
z
1
p( z )
g
dp  ln(
) 
dz
0
p
p0
Rd T
g
p( z )  p0 exp(
Rd

z
0
1
dz)
T
For an isothermal atmosphere, T constant,
Rd T
gz
p( z )  p0 exp(
)  p0 exp( z / H ), H 
Rd T
g
Hydrostatic balance
Pressure decreases approximately
exponentially with height,
decreasing faster with height near
the ground than higher up.
For an isothermal atmosphere, the
scale height H is the height over
which the pressure decreases to 1/e
of its original value.
(e=2.72, 1/e=0.37)
For T=288K, H~7.3 km
Zeroth Law
Zeroth Law of Thermodynamics
Two Systems individually in thermal equilibrium
with a third system are in thermal equilibrium with
each other.
System - A collection of objects upon which attention
is focused.
Surroundings - Everything else
Zeroth Law
Thermal Equilibrium occurs when there the net heat flow
between 2 systems = 0.
Heat flow ==> 0 when objects are at the same temperature.
Consider the three boxes at temperatures T1, T2, and T3:
1
2
3
If T1 = T2 and T2 = T3, then boxes 1,2, and 3 are in thermal
equilibrium satisfying the Zeroth law.
Conservation of Energy
First Law of Thermodynamics –
Conservation of Energy
Energy can be exchanged between a system and its
surroundings, but the total energy of the system and the
surroundings is constant.
“You can’t get something for nothing” (you can’t get more
energy out of a system than you put into it)
Experiments by James Joule (in the mid- to late-1800s)
showed that heat and work are both forms of energy that can
be transferred between a system and the surroundings
Conservation of Energy
Energy
Energy can neither be created nor destroyed.
Energy can be converted among various forms, such as:
• Potential energy (e.g. gravity, PE = mgh)
• Kinetic energy (KE = ½ m v2)
• Mass (E = mc2)
We can also define:
Thermal energy – total of the kinetic energy of all molecules in a
substance
Internal energy – sum of the kinetic and potential energy of
molecules and atoms from which a substance is made
Conservation of Energy
Heat
Heat is a measure of energy transfer by means of temperature
differences.
Heat, Q, was originally defined quantitatively as:
“1 kilocalorie of heat = the amount of energy required to raise the
temperature of 1 kilogram of water from 14.5 to 15.5°C”
Substances differ considerably from one another in the quantity
of heat needed to produce a given rise of temperature in a given
mass.
Conservation of Energy
Heat Capacity and Specific Heat
Heat capacity, C = ΔQ/ΔT
Heat capacity simply relates the amount of heat added to obtain a
rise in the temperature of some unspecified amount in a
substance.
Specific heat, c = heat capacity = ΔQ
mass
m ΔT
Specific heat of liquid water at 0°C = 4218 J K-1 kg-1
Specific heat of dry soil ~ 800-2000 J K-1 kg-1
Conservation of Energy
Specific heat of air
In order to uniquely define the specific heat of a gas, we must
specify the conditions under which the heat ΔQ is added to the
substance, e.g. constant pressure
 ΔQ 
cp  

 mT  p const
 ΔQ 
or constant volume cv  

 mT v const
For dry air, cp = 1004 J K-1 kg-1 and cv = 717 J K-1 kg-1
Conservation of Energy
Latent heat
is the energy given up or taken up by a system to cause a change
of phase, such as water vapor condensing into liquid water.
It is a key to understanding weather because latent heat is a major
source of energy for thunderstorms and hurricanes.
For evaporation, energy is transferred to liquid water molecules
(from the soil or from solar radiation) so that they can speed up
and change to water vapor. Since energy can’t be created, the
substance that loses energy cools down.
For condensation, energy is lost from the vapor molecules to the
surrounding air as they condense to liquid, heating up the air.
Latent heat of condensation, Lc = 2,500 kJ kg-1