T

Fig. 2.5 Entropy cycles for an idealized Carnot engine and a non-Carnot engine.

transition from state 4 to state 1, entropy of the system remains constant. Therefore, there is no net change of entropy during the complete cycle. An important point to note is that the amount of entropy increase due to the transition from state 1 to 2 is exactly balanced by the transition from state 3 to 4; thus there is no net entropy production in the system and the environment.

On the other hand, if the engine is not a perfect engine, there will be an extra amount of entropy production associated with each step in the cycle. As indicated by the dashed arrows in Figure 2.5, irreversible processes in the engine make the entropy cycle different from the simple one denoted by the solid arrows. In order to make the system return to the same initial state 1 as that of the perfect Carnot cycle, the entropy produced must be removed and sent to the environment, indicated by the slightly longer arrow denoting the entropy flux for the transition from state 3 to 4, as compared with that during the transition from state 1 to 2. As a result, the total amount of entropy for the isolated system, i.e., the engine and its environment, increases due to these irreversible processes.

2.3.5 Energy versus entropy

Although energy and entropy are two very common quantities in physics, there is something worth examining more closely. These two thermodynamic quantities are closely linked to each other. According to the first law of thermodynamics, the same amount of heat transformation at different temperatures is considered to be equal. However, according to the second thermodynamic law, thermal energy exchange between the system and its environment may have different qualities, so that the same amount of heat flux at different temperatures is considered to have different effects on the system. In fact, heat transport at high temperature is considered to be energy flux of high quality, whereas heat transport at low temperature is considered to be of low quality.

The quality of energy can be illustrated in the following imaginary experiment. Assume that two perfect Carnot engines are driven by the same amount of heat flux from heat reservoirs with different temperatures T2 > T1, and they are cooled by the same cold reservoir with temperature T0. According to the discussion above, the engine associated with heat flux with higher temperature should have a higher efficiency than the engine working with heat flux with lower temperature. The difference between these two engines is due to the difference in the quality of heat fluxes they received.

In addition, a simple example related to our daily life is as follows. Two pots each containing a sunflower are exposed to the same conditions, such as the pot soil, air temperature, the amount of water and fertilizer, except for the light. One pot is in the sunlight, but the other one is under a fluorescent light. The sunlight is radiation with high equivalent temperature on the order of several thousands of degrees; on the other hand, the fluorescent light has a much lower equivalent temperature. Although the amount of light energy for these two pots is maintained at the same level, the sunflower in the sunlight should grow much better than that under the fluorescent light because of the difference in the quality of energy in the light.

In summary, entropy can be used as a quality index for the energy transported between systems. As a thermodynamic system, the oceanic circulation should be subject to both the first and second laws of thermodynamics, i.e., balance of both energy and entropy should be essential for the study of the oceanic general circulation.

2.4 Thermodynamics of seawater

Seawater is a mixture of pure water and many different kinds of chemical component, and the mass fraction of each component is denoted as mi. Other essential thermodynamic variables include temperature, salinity, and pressure; these state variables are denoted as (T, S, P) in this book; the lower-case letter p will be reserved for pressure in the dynamic analysis.

2.4.1 Basic differential relations of thermodynamics

Thermodynamic relations of seawater can be established in different ways. For example, they can be defined in terms of entropy or the Gibbs function. We first introduce thermodynamics from entropy. We will then use the Gibbs function to establish a unified system of defining the thermodynamics of seawater.

Basic differential relations for a multiple-component system Thermodynamics of a multiple-component system can be established from the definition of specific entropy n. Two other crucial variables of a system, including temperature and specific chemical potential, can be defined as follows

T V9 e) vm \d mi'e,v where e is the specific internal energy, v is the specific volume, mi and ^ are the mass fraction and chemical potential for the i-th component of the seawater.

During a reversible adiabatic process (which means that neither heat exchange with the environment nor change in entropy exist there), change in the specific internal energy is balanced by the pressure work, i.e.

de = -Pd v where P is pressure; this relation can be rewritten as

V9v)nm

A convenient way to derive some of these thermodynamic relations is to use the Jacobian expression df, g) df dg dg df d (x, y) dx d y d x d y For example, combining Eqn. (2.43) with Eqn. (2.42) leads to

= d (n, e) = d(n, e) d (n, v) = / de\ (dA = P (244) dv) em ^(v, e) ^(n, v)^(v, e) \dv) nm\de) vm T '

Using Eqns. (2.42), (2.43), and (2.44), the specific entropy satisfies the following Gibbs relation

1 P v—v fi dn = — de +— dv - > —dmi (2.45)

Another very useful thermodynamic function is the Gibbs function g = e + Pv - Tn (2.46)

Differentiating Eqn. (2.46) leads to n d g = - ndT + vdP + ^ fidmi (2.47)

Since T and P are intensive variables, they are homogeneous everywhere in an equilibrium system. g is an extensive variable of state of the system; thus it should be a linear function in the mass fractions n d g g (T, P m}) = £ dmidmi i= 1 1

From Eqn. (2.47), we have the Euler relation n e + Pv - Tn = Y_I fm (2.48)

Taking the derivative of Eqn. (2.48) and comparing it with Eqn. (2.47), we obtain the Gibbs-Duhem equation n ndT - vdP + Mid m = 0 1=1

Seawater as a two-component system In general, the thermodynamics of a multi-component system is quite complicated. However, many aspects of these complications are not essential for the study of dynamical oceanography; thus it is desirable to simplify the thermodynamics of seawater. One of the common practices in dynamical oceanography is to assume that the ratios of different chemical components remain constant in the world's oceans. And seawater can be treated as an equivalent two-component system, comprising salt and water.

For such a two-component system, we will use the following notations for the mass fractions of salt and water in seawater ms = s, mw = 1 - s and dmw = -ds (2.50)

The common expression of salinity is the practical salinity unit (psu), which is defined in parts per thousand (by weight), i.e., S = 1,000 * s; however, in this section we will use s for consistency of notation. Thus, Eqns. (2.48) and (2.45) are reduced to

dn = — de + — dv----ds = — de + — dv - — ds (2.52)

where \x = \xs - \xw is the specific chemical potential of seawater, and \xs and \xw are the partial chemical potential for the salt and water in seawater.

Three types of energy

In addition to the internal energy, there are two commonly used thermodynamic functions: specific enthalpy is defined as h = e + Pv (2.53)

and specific free enthalpy (Gibbs function), defined in Eqn. (2.46) as g = e + Pv - Tn

Using Eqn. (2.50), this relation is reduced to g = s^s + (1 - s)ixw (2.46')

Thus, there are three types of energy frequently used, including the specific internal energy e, specific enthalpy h, and specific free enthalpy g (Gibbs function). The meaning and connections between these forms of energy are discussed at the end of this section.

In the above approach we build the seawater thermodynamics on the basis of specific entropy. Another way to set up the seawater thermodynamics is to start from the Gibbs function. For example, both specific entropy and specific chemical potential can be defined in terms of the Gibbs function n =

d Tjs,P

The specific chemical potential of pure water is negative infinite by definition, so it is cumbersome to deal with. However, as will be shown later, the term really relevant to problems in dynamical oceanography is the partial chemical potential for the water in seawater, nw.

Assuming that the second derivatives are continuous, the order of partial differentiation can be exchanged; cross-differentiation of Eqn. (2.54) leads to the following relation

Using specific entropy, specific volume, and mass fraction of salt as independent variables, Eqns. (2.51), (2.52), and (2.53) lead to the following differential relations where cp is specific heat capacity. Assuming that the second derivatives are continuous, the order of partial differentiation can be exchanged; and we have the following relations de = Td n - Pd v + /ids dh = Td n + vdP + /ds d g = -ndT + vdP + /ds

From the above discussion, we have the following Maxwell relations:

From the above discussion, we have the following Maxwell relations:

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