Hydrodynamic Equations of the Sea
4.4 Thermodynamics of Seawater
Thermodynamics and Energy Relations 131
of the medium and the need t o derive very accurate values of densities for certain work in oceanography. However, this subject is not alone in that re- gard, for real systems invariably require a retreat to experimental data for most values of material properties.
ume of sea water, measured in cubic meters per kilogram. Or, if several chem- ical substances are added to a sample of water, each of which has a chemical potential, p i , and each in an amount of dn, moles, the internal energy would be changed by C p, dn,; this could be due t o the addition of salts, say, upon which energy is absorbed during the dissolution of the salts. An- other source of change comes about when evaporation occurs, in which event the heat of vaporization would lead to a loss of heat, L , dn, where L , is the heat of vaporization per mole (see Eq. 2.9), and dn is the number of moles evaporated. Another source that can change the internal energy of the water is the induction of an electric dipole moment per unit volume, dP, by an applied electric field, E, which might come from an electromagnetic wave, for example. Both the polar water molecule and the dissociated salt ions can contribute to the electric polarization, as will be discussed in Chap- ter 8 . Other processes could also occur that change the internal energy and which must be entered into the energy budget, so that a more general con- servation equation can be written:
de = dq
-
p d a+
pi dni+
L , dn+
aE.dP+ ... .
(4.4)i
It is worth noting that each of the terms in this expression (with the excep- tion of dq) is the product of an intensive variable giving the intensity of a quantity-pressure, chemical potential, etc.-and an extensive variable, such as (Y or n,, giving the amount of substance present. We will presently see that the application of the Second Law will remove the lack of symmetry for dq.
In this formulation we have left out other types of energy, such as kinetic and potential energy of the fluid flow, since these are mechanical forms. While in principle they can be converted to thermodynamical forms through fric- tion and other dissipative forces, in practice in the sea these changes are small compared to other energy inputs. We shall return t o the mechanical energies at the end of the chapter, when rate equations for the various types of ener- gy will be derived and the energy conversions will be discussed.
The chemical potential may be written as if there were only one chemical species affecting the internal energy; but we have seen that there are at least a dozen. However, by considering the mass fraction of these constituents as a fraction of the total salinity, s, and further considering p as a combined chemical potential of all of the salts relative to the potential of pure water, the chemical potential of the combined system can be expressed as
p ds = p, dn,
.
I
(4.5) This is possible because the proportions of the various salts in the sea vary
Thermodynamics and Energy Relations 133
only slightly and can therefore be represented by a single quantity, the mass fraction or salinity, s.
The internal energy is one of several thermodynamic potentials of use in arriving at measurable thermomechanical coefficients. For fluids, a more con- venient one is the enthalpy per unit mass, h , defined via
h = e + p a , (4.6a)
whose differential is
dh = de
+
p d a+
a d p . (4.6b)If Eq. 4.4 is substituted into this enthalpy expression, the term in p d a im- mediately cancels, leaving the differential a dp representing pressure-volume changes. For fluid systems (as contrasted with gases), the desired coefficients can be determined much more readily at constant pressure than at constant density; hence the preference for enthalpy as a state variable.
Second Law of Thermodynamics
In Eq. 4.4, the heat differential, dq, was not expressed as the product of an extensive and an intensive variable, as were the other quantities; nor is dq a perfect differential. These deficiencies can be rectified by the introduc- tion of the Second Law, which states that the change in entropy per unit mass, dr, when an increment of heat is added, is greater than or equal to dqlT, where Tis the absolute temperature in kelvins. Thus, the Second Law of Thermodynamics is expressed as an inequality, viz:
dq L d q / T . (4.7)
Now entropy is a measure of the disorder possessed by the system and is a state variable/perfect differential of the extensive type that is paired with the absolute temperature as an intensive variable. In fluid dynamics, as we shall see, entropy is proportional to the amplitude of the motion, among other quantities. Strictly speaking, the equality in Eq. 4.7 holds only for re- versible processes, which, in going through a thermodynamic cycle that returns the state variables to their initial condition, also return entropy to its initial value. In the presence of rapid variations, however, the Second Law asserts that the entropy will increase during such a cycle. Calculations with Eq. 4.7 as an equality rather than an inequality will place a lower bound on the entropy, and quantities derived from it will be bounded estimates.
In practice, the circumstances are somewhat unusual wherein the changes in the ocean are so rapid as to render the equation unusable.
By substituting the entropy expression (Eq. 4.7) and the equivalent chem- ical potential (Eq. 4.5) into the First Law, the combined form of the two laws is obtained:
de = T d q - p da
+
p ds+
L, dn+
aE.@+ ... .
(4.8a)The same procedure for the enthalpy expression yields
dh = T d q
+
a dp+
p ds+
L , dn+
a E . d P+ ... .
(4.8b)This latter equation will form the basis for the discussion of the thermodynam- ics of the ocean to follow. In order to apply it to the time-dependent situa- tions encountered in hydrodynamics, time derivatives are required, and these must be considered to be convective derivatives because the thermodynamic fields are both time- and space-dependent. Thus, we must write
Dh DV DP Ds Dn DP
Dt Dt Dt Dt Dt Dt ' (4.9)
- -
- T - + + - + p - + L , - + + E * -and agree that we shall restrict ourselves to time variations that are slow enough to maintain the validity of the entropy equality.
A word about the propriety of using time derivatives in Eq. 4.9 is in or- der, since thermodynamics ordinarily deals with equilibrium states and, in- deed, might more appropriately be. called thermostatics. For a fluid in a nonequilibrium state, the time changes associated with its motion are to be thought of as occurring during a succession of closely related states, in each of which the departure from equilibrium is small. State variables such as den- sity and internal energy clearly have definite values at any time during this process, and are independent of the existence of equilibrium. Additionally, other quantities that depend functionally on p and e, such as q and T, may be defined in nonequilibrium, time-dependent situations through equations relating them to p and e. Thus time-dependent thermodynamics can be de- rived from classical thermodynamics by considering the fluid states to be a succession of near-equilibrium states. In situations such as very high frequency sound waves, or shock conditions, the fundamental premises must be re- examined for correctness, but for most problems in oceanography, the near- equilibrium case may be assumed.
Thermodynamics and Energy Relations 135 4.5 Additional Thermodynamic Equations
Equation 4.9 is an important statement that enlarges our repertoire of dy- namical relationships, but which introduces more new dependent variables (h, 9, p, and T ) than it contributes equations. We must therefore derive ad- ditional thermodynamic relationships from our previous work, in order to obtain as many final equations as unknowns. This task will take us into the complexities and abstractness of partial derivatives of thermodynamic state variables, a thorough understanding of which requires a certain amount of experience with thermodynamical functions. Since our objectives here are to study ocean physics, we shall only outline the derivation of the more im- portant relationships between partial derivatives and refer the reader to the bibliography at the end of the chapter for more detail.
Consider the enthalpy per unit mass, h, to be a function of the variables
9 , p , and s, to coincide with Eq. 4.8:
We will regard the other functions ( L , and E) appearing in the equation for the enthalpy as being fully specified and contributing in a deterministic way to the variables in Eq. 4.10. Then small changes in h can be written as
dh =
(?)
a9 PS dq+ ( g)qs
d p+ ( g)qp
d s , (4.11)where the subscripts indicate which variables are held constant in the partial differentiations. By comparison with Eq. 4.8b, one immediately obtains ex- pressions for the temperature, specific volume, and chemical potential as par- tial derivatives:
and
= ’
=
(El, .
(4.12a) (4.12b)
(4.12c) In this equation, T is measured in kelvins, a in cubic meters per kilogram,
and p in joules per kilogram. These quantities can themselves be further regarded as variables depending on 7, p , and s, with small changes written as
and
It is clear that the nine partial derivatives in Eqs. 4.13 to 4.15 are second order in h, and many are recognizable as being related to the various ther- momechanical coefficients mentioned previously. We want to write these equations in terms of measurable coefficients that may be used as necessary in the equation of state. Furthermore, our interests in changes in p are slight, since variations in the enthalpy of the ocean due to changes in dissolved salt content are very small compared to those, for example, from radiation and pressure. Hence we will neglect Eq. 4.15 and consider p as given when it ap- pears elsewhere; this approximation can easily be removed, if necessary.
The common thermomechanical coefficients for seawater are mdifica- tions of the usual ones valid for s = 0, plus four new ones arising from the salinity derivatives. They are:
Specific Heats
Specific heat at constant volume and salinity,
Gas:
Specific heat at constant pressure and salinity, Cps:
(4.16)
(4.17)
Thermodynamics and Energy Relations
Specific heat at constant volume and pressure, Cap
137
(4.18)
This last specific heat, while definable mathematically, may be operational- l y very difficult, if not impossible, to measure. We will have no further use for it here.
Volumetric Coefficients
IsobaricAsohaline thermal expansion coefficient, a T :
(4.19)
Isothermal/isohaline compressibility coefficient, ap (reciprocal bulk modulus):
1
-
ap = -
(Y
Isobaric/isothermal saline contraction coefficient, a, : a, =
a TP
Halinometric Coefficients Pycnohalinity coefficient, ?ra :
Thermohalinity coefficient, aT:
(4.20)
(4.21)
(4.22)
(4.23)
Clearly ST, and aa, are related much as are the speed of sound and the bulk modulus, i.e., as isentropic and isothermal changes, respectively.
138
Miscellaneous Speed of sound, c:
(4.24)
Ratio of specific heats, Y h :
Y h = cps/ccxs * (4.25)
To arrive at the partial derivatives in Eqs. 4.13 and 4.14 in terms of mea- surable quantities, we first compare the coefficient of dp in Eq. 4.13 with Eq. 4.24, and obtain
(4.26)
(Note that pc is the acoustic impedance of seawater, as will be shown in Chap- ter 7.) Next consider the coefficient of dy in Eq. 4.14, and compare it with Eq. 4.17. From this we obtain
(4.27)
Similarly, a comparison of the coefficients of ds with the halinometric coefficients gives
and
(4.28)
(4.29)
The remaining coefficients are more difficult to derive. Consider first the specific heat, Gas; from Eqs. 4.6 and 4.8 we obtain the well-known ther- modynamic relationship,
(4.30)
G s = =
(
- a *Thermodynamics and Energy Relations
Now when da = 0 = ds, Eq. 4.13 yields
cY2
da = 0 =
( g)ps
dq - - c2 dp139
(4.3 1)
while for the same case, Eq. 4.14 becomes
=
Z d q+
Y d p , (4.32)where for convenience in writing, the coefficients have been abbreviated X , Y, and Z . The parameter Y is, from the equality of the cross-derivatives of h,
y =
(-)
a7 a 2 h aP =(F)
11 PS =(z)7s
=(-)
aP a2h all.
(4.33)Substituting Eq. 4.31 in Eq. 4.32 and solving for dqldT, we obtain
- -
- cas = 1 / ( 1
+
Y 2 / X Z ) Z.
(4.34) TRestoring the values for X and Z and solving for Y 2 gives
(4.35)
Another expression for Y may be obtained using Eqs. 4.33, 4.17, and 4.19:
(4.36)
Equating Eq. 4.35 to the square of Eq. 4.36 results in a useful thermodynamic identity:
y h ( Y h - 1) C,, = c’a: T
.
(4.37a) An alternative form of this relationship that involves only experimentally observed or calculable quantities on the right-hand side may be obtained from the definitions of the coefficients c2, o r , and Cus:= Cps - Ta$/pap
.
(4.37b)The specific heat at constant pressure and the required derivatives may be obtained from an equation of state, the most recent internationally accepted version of which is given ahead. Numerical values for that quantity are near 3994 J (kg “C)
~’
at T = 20°C and s = 35 psu; the specific heat at constant Volume is at most 2% smaller, with the value of Y h at those environmental conditions being approximately 1.014.We are now in a position to write Eqs. 4.13 and 4.14 in terms of the mea- surable coefficients above; at this point of recasting, we shall also generalize them for time-varying problems and immediately obtain two additional equa- tions that contribute to our goal. In addition, we shall write one, the tem- perature equation, in terms of convective heating rates, DqlDt, rather than the entropy derivative, Dv/Dt, but will retain the latter in the equation for specific volume. These useful equatibns give time derivatives of cr and T i n terms of time variations of other state variables that can be calculated-readily:
-
+ - -
(4.38)Da (YaTT Dv Dp 1 Ds
Dt Cps Dt C’ Dt SH, Dt
DT - 1 Dq CYaTT Dp 1 Ds
Dt C p s Dt Cps Dt sxT Dt
and
+ - - + - - .
(4.39)Equation 4.38 will be used in Chapter 7 in the derivation of the acoustic wave equation, while Eq. 4.39 is used immediately below.