Considering an atomizer, such as an air-blast atomizer, that produces flat or conical liquid sheets at the atomizer exit, the atomization process starts at the atomizer exit. As the liquid proceeds downstream, the thickness of the liquid sheet decreases; and instabilities set in, which break up the liquid sheet into ligaments and finally droplets. Therefore, the atomization ends at some downstream location where droplets are formed. Here the spray is assumed steady and isothermal. The control volume, as shown in
Figure 1, is taken from the atomizer exit to the droplet-formation plane.
2.1 Entropy analysis
According to the second law in thermodynamics, we have,
For the steady state steady flow process, the rate of entropy increase within the control volume vanishes. Here, the mass flow rates may be different between the two states (i.e., the state at the nozzle exit and at the droplet formation plane downstream) considering the possible mass exchange between the system and the surrounding (condensation or evaporation). The difference can be denoted as
, which is commonly referred to as the mass source term, and the corresponding specific entropy is denoted as
ssource.
Then Eq. (1) can be recast as
Or in the per unit mass flow rate form
where
is the dimensionless source term.
At the nozzle exit, the flow at state 1 is in bulk liquid form with free surface produced. The entropy at the nozzle exit is due to the liquid bulk and the surface tension effect, hence,
. Since the control volume is around the liquid,
ssource =
s(l) ≅
s1(l) because the liquid is assumed isothermal. Eq. (3) can be rewritten as
In the breakup region (state 2), a multitude of droplets forms and the total surface area is increased dramatically. Liquid phase exists inside each of the droplets. The total entropy is composed of two parts. One part is associated with the liquid bulk and is similar to that at the state 1; therefore may be denoted as
. The other part is due to the existence of the numerous individual droplets, represented as
, which is directly associated with the particle nature of the droplets. Hence,
The formation of droplets is not deterministic but stochastic. The entropy at state 2 should be associated with the probability distributions of the droplet sizes. Different distribution results in different amount of entropy. To evaluate entropy quantitatively, an analogy between the present case and that of Gibbs ensemble in statistical thermodynamics may be made here. The entropy due to the droplet nature can be derived as, with details given elsewhere [
6]
where
is the probability of finding droplets with diameter
Di in the spray.
ṅi is the number of droplets produced per unit time and with diameter
Di.
Therefore, the rate of entropy generation in this irreversible atomization process can be expressed as
According to the Gibbs equation for a simple compressible substance with free interfaces, the liquid bulk entropy change can be expressed as
where
a is the surface area of the free interface per unit mass,
σ is the surface tension. All other symbols have their usual meanings in thermodynamics, i.e.
T is temperature,
s(l) is entropy,
u internal energy,
p pressure and
v specific volume. Since the atomization process is assumed isothermal, the internal energy, only a function of temperature, is unchanged. Then Eq. (8) becomes
where
is the isothermal compressibility of the liquid, a thermodynamic property whose dependence on pressure is negligible for liquids.
Entropy is a thermodynamic property that depends only on the state of the system. Hence integrating Eq. (10) over the isothermal process results in,
At the atomizer exit, a thin liquid sheet forms from the annular air-blast atomizer, and the liquid pressure is almost the same as the surrounding air pressure since the curvature effect is small and negligible. However, a circular liquid jet forms for solid-cone sprays produced by small orifice atomizers, the liquid pressure is larger than the ambient air pressure due to the surface tension effect. However, the specific value of
p1 will not affect the determination of the droplet size distribution as shown later. Therefore, for simplicity, we assume
For liquid pressure at state 2, let’s consider only one liquid droplet. For a liquid droplet in the air, the pressure difference across the free interface is related to the surface tension effect as follows:
Substituting Eq. (13) into Eq. (11) gives,
At state 2, the total entropy flow rate for the liquid inside all the droplets is therefore
The entropy generation in the atomization process is therefore
where
A and
B are unknown constant for a given spray because
Ṅtotal is often unknown, which could be determined by the relation
. This equation may then be rewritten in the unit mass flow rate form as follows
where
.
2.2 Droplet size distribution model
At state 2, the droplet size distribution is in reality the initial distribution for droplets just formed in a spray. There are infinite sets of the probability distribution Pi that can satisfy the global constraints on the atomization process such as the conservation laws. In accordance with irreversible thermodynamics, the least biased (or the most realistic) distribution is the one that maximizes the amount of entropy generated for the naturally occurring atomization process.
For the present study the constraints imposed on the atomization process are the conservation of liquid mass and the normalization condition. The normalization condition is
stating the fact that the total probability for all the droplets in a spray should be equal to one.
The mass conservation under steady state requires that the sum of all droplets produced per unit time be equal to the mass of the liquid sprayed per unit time, plus the mass source term (
Ṡm), which represents the rate of mass transfer between the liquid and the gas phase, i.e. the condensation or evaporation, during the atomization process. The expression for the conservation of mass can be written as:
Eq. (19) is nondimensionalized as,
where
denotes the dimensionless mass source term.
To maximize the entropy generation
sgen under the constraints of Eq. (18) and Eq. (20), the Lagrange’s method is adopted here. That yields,
where
To obtain the probability of finding the droplets whose volume is between
and
, we have to evaluate
It is generally regarded that the droplet size and velocity in sprays vary continuously rather than discretely. Therefore, the subscript
i can be dropped, and the summation form of Eq. (24) becomes an integral over the droplet size; that is
The mean volume in a spray can be expressed in terms of mass mean diameter,
Thus the non-dimensional droplet volume becomes
Substituting Eq. (27) into Eq. (25) leads to,
where
and
are the droplet diameters corresponding to the droplet volume
and
respectively; and
f is the continuous droplet size probability density function (pdf). Thus,
The unknown Lagrangian multiplier
α0 and
α3 can be determined from the normalization and conservation of mass equations; and
α1 and
α2 from Eqs. (22) and (23) if
D30 is known. On the other hand, the above formulation is equivalent to the distribution from the MEP using an extended set ofconstraints. In addition to the normalization condition and the conservation of mass, two other constraints are written on the basis of the definition of the mean diameters
D10 and
D20. All the constraints that are needed to determine the unknown coefficients
αi’s are listed below.
Here, it is worth noting that the model is only applicable to the immediate vicinity of the breakup region, where droplets are just formed in a spray.