# A Novel Approach of Unit Conversion in the Lattice Boltzmann Method
## 1 Introduction
This effort appears in increasing the trials that needed to achieve a converge numerical solution, in other words, finding the appropriate values for assumed parameters. Therefore, the second method, i.e., the principle of corresponding states, comes to the picture to deal with those issues. The principle of corresponding states is also called the scaling method. In most numerical simulations the governing equations are known. Therefore, the non-dimensionalization process is to set a list of references for the variables (i.e., length, velocity, temperature, pressure, etc.). Those references must be known and  
Figure 1: A hierarchy of scales and simulation levels. constants. For instance in the multiphase problems, a critical state of density, volume, temperature, and pressure is taken as a reference to non-dimensionalize the property and reduce the number of parameters. For instance, the R-K EoS is as follows:  
\[P=\frac{\rho RT}{1-b\rho}-\frac{a\rho^{2}}{\sqrt{T}(1+b\rho)}. \tag{2}\]  
The equation includes six unknown parameters (\(P\) is pressure, \(\rho\) is density, \(T\) is temperature, \(R\) is gas constant, \(a\) is a parameter characterizing the attraction of gas particles, and \(b\) is effectively a minimum molar volume [8; 9]). The variables \(a\) and \(b\) can be found by taking the first and second derivatives of the R-K EoS with respect to density at the critical state and equating them to zero. Solving the obtained system of equations yields:  
\[a=\frac{0.42748R^{2}T_{c}^{2.5}}{P_{c}}=1.28157\frac{RT_{c}\sqrt{T_{c}}}{\rho_ {c}}, \tag{3}\]  
and  
\[b=0.08664\frac{RT_{c}}{P_{c}}=\frac{20}{77\rho_{c}}, \tag{4}\]  
where the subscript \(c\) refers to the critical state. Using a critical state of each property yields the following dimensionless variables:  
\[P_{r}=\frac{P}{P_{c}},\ T_{r}=\frac{T}{T_{c}},\ and\ \rho_{r}=\frac{\rho}{ \rho_{c}}, \tag{5}\]  
where the subscript \(r\) refers to the reduced state. By substituting Equations (3)-(5) into Equation (2), the R-K EoS becomes:  
\[P_{r}=\frac{T_{r}\rho_{r}}{0.33356-0.08664\rho_{r}}-\frac{{\rho_{r}}^{2}}{ \sqrt{T_{r}}(0.26028+0.067604\rho_{r})}. \tag{6}\]  
The number of unknown parameters is reduced to half. It is worth mentioning that the R-K EoS is one of the most accurate EoS, which is adequate for the calculation of gas-phase properties when \(\frac{P}{P_{c}}<\frac{T}{2T_{c}}\)[10]. A flow in a lid-driven cavity is another example, where the lid velocity (\(u_{lid}\)) is used to scale the velocity field; the height of the cavity (H) is used to scale the length, etc. The non-dimensional governing equation can be written as:  
\[\frac{\partial U}{\partial t^{*}}+U.\
abla U=-\Delta P+\frac{1}{Re}\
abla^{2}U, \tag{7}\]  
where \(t^{*}\) is non-dimensional time scaled by \(H/u_{lid}\), and \(Re=u_{lid}H/\
u\). However, another example, utilizing natural convection in a differentially heated cavity, the velocity scale can be defined as either the kinematic viscosity (\(\
u\)) divided by the length scale or the thermal diffusion coefficient (\(\alpha\)) divided by the length scale. The open literature is full of examples of non-dimensionalized equations using the above-mentioned method. Note, the proper scaling is different than non-dimensionalizing. However, for numerical simulation, it does matter. However, there are no reference states of some properties (e.g., specific heat capacity, viscosity, thermal diffusivity, coefficient of expansion, enthalpy, etc.). Usually, the researchers [5; 11; 12; 13; 14; 15; 16] couple the Buckingham \(\pi\) theorem with the principle of corresponding states (scaling) method to overcome the mentioned disadvantages. [PERSON] et al. [17] mentioned that not every equation or variable could be easily converted by the Buckingham \(\pi\) theorem or scaling method. Consequently, they [17] tried to solve the whole problem by utilizing the Planck unit system. They tested their approach on a 2D convective heat transfer problem in tube banks.