Analysis of the twodimensional polydisperse liquid sprays in a laminar boundary layer flow using the similarity transformation method
 Ophir Nave^{1, 2}Email author
https://doi.org/10.1186/s4032301500428
© Nave. 2015
Received: 18 May 2015
Accepted: 30 July 2015
Published: 12 August 2015
Abstract
In this paper we analyze the model that describes the evaporation process of multisize (polydisperse) of fuel droplets in a laminar boundary layer flow. The spray is described using a probability density function and is based on the wellknown sectional approach. The spray is described in two dimensional boundary layer directions that consists of five equations: the continuity equation, the momentum equation, the energy equation, the equation of state and the spray equation. The governing equations are a system of nonlinear partial differential equations (PDE). In order to convert the PDE to ordinary differential equations, we applied the LeesDorodnitsyn similarity transformation and the corresponding similar solution based on the compressible stream function. We then solve the model numerically and compare our results to the sectional approach with an experimental data.
Keywords
Partial differential equations Similarity transformation Polydisperse fuel spray Laminar boundary layerIntroduction
The problem of combustion of polydisperse fuel spray in a laminar boundary layer flow is very important from a practical point of view and has a large number of possible applications. The most common description of this problem is discussed by [1–5] when the spray is modeled by the “sectional approach”. This method is based on dividing the droplet size domain into sections and dealing only with one integral quantity in each section i.e., number, surface area of droplets, or volume. The advantage of this method is that the integral quantity is conserved within the computational domain and the number of conservation equations is reduced to be simply equal to the number of sections. The natural generalization of this method is to describe the size of the droplets with a probability density function as occurred in this paper [6–10]
Most problems in engineering applications are described by a system of partial differential equations (PDE) which is usually difficult to solve. Using the similarity transformation method [11, 12] the system can be converted to an ordinary differential equations (ODE) set by combining two independent variables into a single independent variable [13]. The new set can be solved by a variety of numerical methods. However, with further assumptions that relate to the transport properties, these set of ODEs can be uncoupled mathematically or can have simpler forms, almost similar to the form that is obtained from the incompressible boundary layer analysis. Hence, the simplified ODE set makes it possible to obtain the solution from the already existing solutions of the incompressible analysis and also reduces the computing time of the numerical simulation [14].
In this paper, we apply the LeesDorodnitsyn similarity transformation and the corresponding similar solution based on the compressible stream function to the new continuous model. We solve the model numerically and compare our results to the sectional approach model.
Physical assumptions and governing equations
The physical model is a system of nonlinear partial differential equations in two dimensional x and y, steady state, compressible, laminar boundary layers and consists of a multi size (polydipserse) spray of evaporating droplets. The analysis of the spray is restricted to the no slip condition between the droplets and the boundary layer flow field [1].
Under the above assumptions, the governing equations are:
Nondimensional model and similarity transformation
The model in terms of stream function
Discussion
The purpose of the atomization is to disperse the droplets into the oxidizer and to increase the surface area of the droplets. This process of atomization enhanced the heat and mass transfer during the combustion processes.
Initial surface area distribution
Section number  Droplet diameter (μm)  Uniform  Symmetrical  Nonsymmetrical 

I  <1  0.2  0.5  0.48 
II  1–3  0.2  0.8  0.66 
III  3–5  0.2  0.6  0.32 
IV  5–6  0.2  0.8  0.56 
V  6–9  0.2  0.5  0.63 
Comparison with experimental results
Conclusion
Section number and sectional coefficients normalized by E(T), the surface recession rate of a 65 \(\upmu {\rm m}\) droplets
Section number  Droplet diameter \((\upmu {\rm m})\)  C\(_{j}\)/E(T)  B\(_{j, j+1}\)/E(T) 

I  <1  0.454  0.0692 
II  1–3  0.987  0.0018 
III  3–5  0.657  0.0875 
IV  5–6  0.342  0.0787 
V  6–9  0.566  – 
We compared the theoretical results to the experimental data (in contrast to [1]), as can be seen in Table 2. Additionally, we have rewritten the model that was presented in [1] in a continuous form. This reformulation not only reduced the complexity of the model but also reduced the computation time, which is very important from a practical point of view. This change is particularly evident in the spray equation, which in our model is merely one equation!. In conclusion, we demonstrate in this paper how the surface area of the droplets in polydisperse fuel spray has been changed as a function of the temperature and the distance from the wall.
Nomenclature
 B :

evaporation coefficient
 C :

specific heat capacity
 D :

diffusion coefficient of droplets
 \(E_{i}\) :

frequency of the molecule evaporation from an imer droplet
 h :

enthalpy
 L :

characteristic longitudinal direction
 M :

Mach number
 n :

number of droplets per unit volume
 p :

pressure
 P :

probability density function
 Pr :

Prandtl number
 Q :

droplet moment
 r :

radius of drop
 Re :

Reynolds number
 u :

velocity in the x direction
 U :

longitudinal freestream velocity
 v :

velocity in the y direction
 x :

direction along the surface creating the boundary layer
 y :

direction normal to the surface
Greek symbols
 \(\alpha\) :

0, 1, 2, 3
 \(\gamma\) :

4 0, 1 or 2/3
 \(\kappa\) :

specific heat ratio
 \(\rho\) :

density
 \(\mu\) :

dynamic viscosity
 \(\nu\) :

kinematic viscosity
Subscripts
 e :

the edge of the boundary
 p :

at constant pressure
 v :

at constant volume
 w :

properties at the wall surface
Declarations
Compliance with ethical guidelines
Competing interests The authors declare that they have no competing interest.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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