APPLICATION OF THE THERMODYNAMICS OF OPEN 
SYSTEMS TO METEOROLOGY* 
By JACQUES M. VAN MIEGHEM 
University of Brussels 
HOMOGENEOUS SYSTEMS 
Introduction. The systems which are generally con- 
sidered in thermodynamics exchange energy (e.g., heat) 
with their environment; however, they neither give 
up nor receive mass. It is for this reason that they are 
called closed systems. An open system, on the other hand, 
is a system which exchanges not only energy but also 
matter with the surrounding medium [4]. A cloud from 
which precipitation is fallmg is thus an open system. 
Tt will be assumed in this first section that the sys- 
tem is homogeneous, that is to say that its physical 
variables (pressure p and absolute temperature 7’) are 
uniform throughout the volume V under consideration. 
This will always be the case in the atmosphere, pro- 
vided one takes a volume of air which is not too ex- 
tensive. 
Uniformity of the quantities p and 7 involves an 
important consequence: The system cannot be the site 
of irreversible transformations which bring about the 
equalization of temperature or pressure between dif- 
ferent points of the system. Therefore it is natural to 
suppose that physical transformations of the system, 
involving no internal modifications of mass, are re- 
versible; only transformations involving changes of the 
masses m; of certain of its constituents 7 may be irre- 
versible. 
At the foundation of any thermodynamic investi- 
gation lie two functions of state: the mternal energy E 
and the entropy S of the system. With these are gen- 
erally associated the enthalpy H = H + pV and the 
Gibbs thermodynamic potential G = H — TS. Any 
one X of these functions depends on p and 7, and on 
the masses m1, m2, --- of the constituents. According 
to Gibbs, the functions’ of state of a system are homo- 
geneous functions of the first degree in variables m, me, 
---; therefore one has X = Dmiz;, where the specific 
4) 
ox 
functions of state x; (where a —) of the constit- 
j 
uents 7 are homogeneous functions of zero degree in 
the variables m; (7 = 1, 2,3, ---). 
The First Law. The specific internal energy (per 
unit mass) of the system will be designated by e, its 
Specific enthalpy by h, its specific volume by v, the 
heat added to the system from time ¢ to time ¢ + dt 
(dt > 0) by dQ, and the heat added to the unit mass 
during the same time interval by dq. If we let m repre- 
sent the total mass of the system, the following rela- 
tions then hold: H = mh, V = mv, dQ = mdq. 
The intensive (or local) form 
dq = de + pdv = dh — vdp (1) 
* Translated from the original French. 
of the first law still applies when the system under 
consideration is open, provided that we assume the 
heat dq added to a unit mass includes, in addition to 
the heat received by radiation and conduction (the 
case of closed systems), the heat received by convec- 
tion (an exchange of matter with the surrounding 
medium). Upon multiplying the two sides of (1) by 
m, we obtain the extensive form 
dQ + hdm = dH + pdV = dH — Vdp (2) 
of the first law for the case of an open system [7, 9, 14]. 
The sum of the quantity dQ of heat received by the open 
system plus the enthalpy increase hdm of the system, 
resulting from exchange of matter with the surroundings, 
as equal to the increase dE in internal energy E of the 
system plus the term pdV. 
If the masses m; of the constituents of the system 
are introduced, it is clear that m =) mj. It should be 
I 
noted that the total increase dm; of mass m,; of con- 
stituent 7 arises from an increase dm; due to internal 
mass modifications and from an external contribution 
dm; (20) during the same time interval dt. Therefore 
dm;=dim,; + dmj, with the condition dyn =) ) dim; = 
@ 
dam =)idm; ~ 0. Further- 
of 
more the differential of any function of state whatever, 
for example H, assumes the form dH = d;H + d.H, 
with 
0, and as a result dm = 
Ail = 
aH 
= ap Ol oe dp + pe hj di my, 
and 
d.H = 2) hjdem;, 
4) 
where h; is the specific enthalpy of constituent j. It 
then turns out, as a consequence of (2), that dQ can be 
written dQ = dQ + d.Q, with 
dQ = dH — Vdp, dQ = di(h; — h)dam;, (3) 
J 
where the quantity d,Q of heat received by the system 
is associated with internal, physico-chemical changes 
of state which it undergoes during the time interval 
dt, while the quantity d.@Q of heat received is associated 
with exchanges of mass with the surroundings during 
the same lapse of time. However, the separation of 
dQ into d,Q and d.Q was not accomplished as a result of 
the different mechanisms of heat exchange (radiation, 
conduction, or convection) between the system and the 
surroundings, but rather as a result of the effects pro- 
duced by the added heat on the physico-chemical vari- 
ables 7', p, mi, m2, «+ of the system [14]. We observe 
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