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Atmospheric Science 4310 / 7310. Atmospheric Thermodynamics - II By Anthony R. Lupo. Day 1. The work done by an expanding gas Let’s draw a piston:. Day 1.  Consider a mass of gas at Pressure P in a cylinder of Cross section A Now, Recall from Calc III or Physics: - PowerPoint PPT Presentation


  • Atmospheric Science 4310 / 7310Atmospheric Thermodynamics - IIByAnthony R. Lupo

  • Day 1The work done by an expanding gas

    Lets draw a piston:

  • Day 1 Consider a mass of gas at Pressure P in a cylinder of Cross section A

    Now, Recall from Calc III or Physics:

    Work = force x distance or Work = Force dot distance

    So only forces parallel to the distance travelled do work!

  • Day 1Then,

  • Day 1But, we know that:

    Pressure = Force / Unit Area

    So then,

    Force = P x Area

  • Day 1Total work increment now:


    Area x length = Volume

    soo A * ds = dVol

  • Day 1Then we get the result:

    Work : Lets work with Work per unit mass:

    Thus, we can start out with volume of only one 1 kg of gas!!!

  • Day 1 And we know that:

    a = Vol/m

    then.. Note the heavy D

  • Day 1So.

    Very Important!!! This is not path independent, its not an exact differential!

    Thus is is easy to see that if a parcel undergoes expansion, or a2 > a1

  • Day 1Parcel does work expanding against environment!

    But if the parcel undergoes contraction (a2 < a1)

    Environment is doing work on the parcel!

  • Day 1/2A derivation more relevant to an air parcel

    Given a spherical parcel of air with radius R, Surface Area = 4pr2 and the volume is 4/3pr3.

  • Day 2 We calculate dW in expanding the parcel from r to r + dr in an environment where the pressure is p!


    Work = F x dist = Pressure x Area x distance

  • Day 2Well,

    Dist = dr


    Area = A + dA

  • Day 2As with the expanding piston

    DW = p*A*dr = p * dV

    Recall from Calc II (or is it physics?) dV = 4p r2 dr

  • Day 2Then well deevide by the unit mass again to get:

    dw = p da

    Now the short comings in this problem are obvious:

    1.(Surf)Area = A + dA = 4 p r2 + 8p r dr

  • Day 22. The parcel expands and p decreases;

    so: p = p + dp

    So.. the REAL problem is:

    dW = (p+dp)(A+dA) dr

  • Day 2 Which foils out to,

    DW = pAdr + dpAdr + pdAdr + dpDAdr

    then assume;

    Dw = p da + dp da + p dA dr + dp dA dr

  • Day 2Which then reduces to (by scale analysis):

    Dw = p da

  • Day 2The First Law of Thermodynamics: A derivation.

    The First Law of Thermodynamics, is really just a statement of the principle of the conservation of energy.

    This law can be used as a predictive tool: ie, we can calculate changes in T, a (r), or pressure for a parcel.

  • Day 2 The first law of thermodynamics is associated with changes in these quantities whereas the equation of state relates the quantities themselves, at a given place at a given time (or at a given place for a steady state).

    Forms of energy relevant to our treatment of the 1st Law

    Energy is (simply) the capacity to do work!

    for our purposes, a system is an air parcel.

  • Day 2KE (kinetic energy) associated with motion, energy of motion (Definition: Kinetic means the study of motion w/o regard to the forces that cause it)

  • Day 2PE (potential energy) is the energy of position relevant to some reference level, or within some potential (e.g., gravitational) field.

  • Day 2**The location of Z = 0 is arbitrary, but it makes good sense to use sea level.

    Internal energy (IE) Is the energy (KE and PE) associated with the individual molecules in the parcel.

    **So IE is temperature dependent (use Absolute T)!!!

  • Day 2Deriving the First Law: (Hess pp 25 26)

    Statement of Conservation of Energy: Energy of all sorts can neither be created nor destroyed (Of course we ignore relativistic theory E = mc2)

    Written in incremental form (here it is!):

  • Day 2/3The book definition:

    Any increment of energy added to a system is equal to the algebraic sum of the increments in organized KE, PE, IE (thermal), work done by the system on its surroundings, and whatever forms of electrical and magnetic energy may appear

    Dheat = DKE + DPE + DIE + DWork + DE&M

  • Day 2/3 We can neglect electrical and magnetic for atmospheric motions we consider (scale analysis).

    ** Recall: our system is the air parcel!

    Consider the LHS; increment of energy added to a system can be broken down into heat added to the system + work done on the system by the surroundings.

    We want to simplify, if possible to quantify our physical principle (the conservation of energy).

  • Day 3 Kinetic Energy: We first note that increments of Kinetic energy are small compared to changes in internal energy and increments of work done. For example, consider a parcel of air changing speeds from 10 m/s to 50 m/s

    DKE = (50 m/s)2 (10 m/s)2 or 1.2 x 103 m2 sec -2

  • Day 3Next, let us consider a change of 20 C for a parcel of air at a constant volume, a change of internal energy:

    DIE = Cv DT = 717 J K-1 kg-1 x 20 K

    or 1.4 x 104 m2 sec-2

  • Day 3Then, lets consider the change in PE at, oh, say 500 hPa,


    DPE = g*Dz = 10 m s-2 * 60 m = 600 m2 s-2

  • Day 3Pressure work;

    We have dw = p da and pa = RT so dW = (RT/a) da = RT d[ln(a)]

    Since T = Const, we can write:

    = ln 0.1 * 287.04 J/kg K * 280 K

    W = 8.47 x 103 m2 s-2

  • Day 3So, weve shown that:

    Dh/dt = IE + PE + KE - Work done on the system by the surroundings + work done by the system on the surroundings+ EE + ME

    We neglected EE and ME by scale analysis, and we can neglect KE and PE by scale analysis (or do we)?

  • Day 3Well,

    DKE = DKE(horiz) + DKE (vert)

    ***KE vertical is very tiny, by several orders of magnitude (at least 4)!

    Work done on the system by surroundings (W):

    -(Work horiz + Wvert)

  • Day 3 The Horizontal pressure gradient is responsible for changes in wind speed.

    The pressure gradient is in DKE and Wh (which is horizontal wind times distance covered)

    Then, DKE must cancel with Wh!

    How about Wv?

  • Day 3 Well, DPE = gdz


    Wv = vert. PGF x distance = - (1/r) (dp/dz) (dz)

    If hydrostatic balance is assumed to hold then g = -(1/r) (dp/dz).

  • Day 3 Thus, DPE + Wv = gdz g dz = 0 (and these cancel)!

    So in the incremental form of the first law we are left with:

    Dh/Dt = IE + work done by the system on the surroundings (1)

    **(note, the heavy D)

  • Day 3/4/5Then if we

    define heat added as dh/dt (dq/dt) or just dh or dq

    Internal energy as du,

    and work done by the system as dw, then:

    We can quantify first law (this is it)! ?

    Dh = du + Dw or du = dh - dw

  • Day 5The First Law of Thermodynamics and Entropy.

    What happens when heat is added to parcel? 1st Law: dh = du + dw

    Heat can be added (or subtracted, of course)! How?

    Conduction, convection, radiation, or phase change (All Diabatic heating)

  • Day 5Diabatic processes:

    Sensible heating warm to cold by contact

    Radiative heating Short wave in or long wave out

    Latent Heating phase changes of water mass

  • Day 5 So we can say formally if a small quantity of heat* is added to the system (parcel) some goes into increasing the int. energy of the parcel, and some gets expended as the parcel does work (dw) on the surroundings.

    *(Note: heat is NOT temperature!)

    We know the 1st law, however, how do we express in terms of state thermodynamic variables (T, a, or P?) so we can obtain useful expressions for atmospheric calculations?

  • Day 5 Unfortunately, there are no formal, or analytical, mathematical expressions for dh (especially for atmospheric processes these typically parameterized (which is an empirical fudge factor)).

    Thus, we are reduced to solving as a residual, and in the 1st Law leave as dh/dt or dq/dt (Q-dot).Well, we showed that the work done by an expanding gas could be derived by considering a piston (closed controlled system), or an air parcel.,

  • Day 5Thus, dw = p da.

    So the 1st Law is now:

  • Day 5Internal Energy

    With the pressure work term expressed in terms of state variables p and a, we must now consider that internal energy may be dependent on (T and a) or (T and p) (since p and a can be related)

    So: Consider that u = u(T,a) or u = u(T,p)

  • Day 5The expression

  • Day 5Joules Law; and his experiment conceptualize this term

    Consider an insulated container with two chambers, one filled with Gas at pressure P = Po, and the other a vaccuum at P = 0. (connected by valve)

    Insulated chamber, thus Q dot = 0. (No surroundings also, so dw = 0).

  • Day 5Draw:

  • Day 5Then the valve is opened and gas expands into vacuum chamber (let er rip!) P = 0. (No temp change occurs), then from first law:

    Hmm, Dq = du + dw


    1. Dq = 0 (no heat added)2. dw = 0 (since chamber cannot expand)

  • Day 5Thus, du must be 0!!


  • Day 5Thus:

    Buuut, Joules experiment also showed that dT = 0 soooo,

  • Day 5 This (celebrated) result is known as Joules law, which is strictly true for an Ideal gas.


  • Day 5/6 Thus, u = u(T) internal energy is a function of T only!!! Q: So what did Joules experiment show?

    A: Joules experiment, separate from Joules law showed that heat and mechanical energy are two forms of the same thing.

  • Day 6Next, l