Application of Equation-Oriented Modeling in solving Diffusion Equation in Different Types of Networks

Nowadays, network structures are found in many natural and engineered systems, e.g., river networks, microchannel networks, plant roots, human blood vessels, etc. Therefore, providing efficient methods for modeling phenomena such as diffusion, advection, etc. is very practical. One of the most common tools for modeling this phenomena is numerical modeling, as mathematics software is well- developed and powerful nowadays. In this research, a new approach called Equation-Oriented Modeling has been presented. In this approach each branch of the network has its own differential equation, and these branches are connected or coupled by boundary conditions. In other words, unlike classical modeling, EOM does not solve through the discretization of the partial differential equation in the whole domain of the network, while in this approach, each branch of the network has its own differential equation with its own specific diffusion coefficient and cross section area, then the problem is solved as a system of PDE. The main point of EOM is to formulate a physical problem in the network into a system of differential equations, which is finally solved by the Method of Lines. MOL is an efficient computational method used to solve partial differential equations or PDE systems. MOL is generally implemented in two steps, in the first step spatial derivatives are replaced by algebraic approximation. In the second step, the ordinary differential equation system is integrated with respect to time using any method, for example, in this research, we use the Runge-Kutta 4th order method. EOM was implemented to solve the diffusion equation in three types of networks, including tree-shaped and loop network. Then modeling results for 3 networks were presented as spatial concentration profiles in different paths in the networks. The model had reasonable results in the boundaries and branches according to the boundary conditions, loading and concentration functions, as well as the continuity of concentrations and loading by diffusion in the output results was reasonable. The boundary conditions that apply at the intersections of the branches include the continuity of concentration and the continuity of loading due to the diffusion phenomenon. The results of test case 3 were compared with another numerical model for validation, and three types of Error Parameters were calculated at different times between these two models. R-Squared (R²) was calculated in path (1-2-3-5-9), and its value was 0.99-1, which was the optimal value. This coefficient shows that the results of the EOM and the other numerical model has the same trend. Then, RMSE and MAE were also calculated and their values were approximately zero for all times. The modeling results for 3 networks were presented as spatial concentration profiles in different paths in the networks. The first advantage of the EOM approach is that the choice of terms in the differential equation is left to the user rather than the software developer, so that a wider range of phenomena can be modeled and the effects of different terms can be seen in the modeling. The second advantage of this approach over classical modeling is that the equations are available to the user as tools and model elements, and modeling complex networks such as tree-shaped, and Loop networks is not as complicated as classical models. The third advantage of EOM is the tools available in mathematical programs for optimization or linking with other programs. Since the heat equation is similar to the diffusion equation, the results of this research can be used for other important topics, such as solving the heat equation in microchannel networks for cooling systems, modeling pollutant transport in river networks, or diffusion modeling of solutes in plant roots.