In this paper we developed a new mathematical model for the flow inside cascade impactors and via this simplified model, we determined the particle size distribution by a fast and low cost computational method. Using cascade impactors for determining the particle size distribution, one can use comprehensive CFD methods to fully simulate the particle traces. Although the results from those CFD analyses can be very accurate, usually that is not a time and cost efficient routine. In contrast, we showed that by using our proposed calculation we can estimate the particle size distribution very fast and yet with the slight error — comparing to the results from CFD method. Cascade impactors are being used to measure the range of substances moving through an opening and determine the particle size of distributed substances. Air flow containing aerosol entering in each stage, after colliding vertically with a plate will deviate 90 degrees from its original direction. Larger (massive) particles cannot follow the flow because of their larger linear momentum. Hence, they will deviate from the flow and deposit on the plate instead. The mass difference before and after the experiment represents the deposited mass in each stage. By integrating multiple uniquely designed stages into one impactor, we can determine size of particles in the flow. Typical cascade impactors consist of up to ten stages in which different size of aerosols are being separated. This paper presents a simple model for the flow in one single stage of a cascade impactor. Flow inside cascade impactor is approximated by stagnation point potential flow with the stream function of Psi = Axy, and particles are tracked by velocity verlet algorithm. Absorbed particles are associated with unit value; otherwise they are associated with zero. It is assumed that particles in entrance have random size distribution and location. Drag, Saffman and Brownian forces are taken into account in this model for different particle sizes. The results are discussed in detail and compared with data driven from different approaches in the literature.

Many theories and mathematical simulations have been proposed concerning urine concentrating mechanism (UCM). The WKM and region approach are the two most valuable methods for compensating the effect of tubule’s architecture in renal medulla. They both have tried to simulate tubule’s confinement within a particular region mathematically in one spatial dimension. In this study, continuity, momentum and species transport equations along with standard expressions for transtubular solutes and water transports on tubule’s membrane were solved numerically in three spatial dimensions which practically is the main significance of our novel approach. Model structure has been chosen as simple as possible to minimize the effect of other factors in tubule’s solute and water exchange. It has been tried to simulate the preferential interaction between tubules by introducing different diffusion coefficients for solutes in the intermediate media in order that changing this physical parameter directly could influence tubule’s confinement with respect to each other. The results have been discussed in detail and then the effect of solute’s diffusivity on UCM has been investigated subsequently. In overall, it has been found out that this simulation can validate the integrity of our proposed approach for further investigation in this field.

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