**Tadeusz Piecuch, Jacek Piekarski, Grażyna Malatyńska**

Politechnika Koszalińska

**Abstract**

This paper provides a theoretical analysis of a filtration process with an inflow of a mixture onto a filtration barrier (in this case, a mesh). In practice, such a process takes place in vacuum or pressure filters, filtration presses or belt filters, which is presented, though in a simplified way, on Figure 1. The general equation for the flow of the liquid through the porous layer as per notation (1) was adapted to a filtration process description as a result of certain assumptions and a conducted theoretical analysis, leading to formulation of a universal equation of mixture filtration with sedimentation on the filtrating mesh, as described in many publications interested readers are referred to [37,39,40]. This publication is a continuation of paper [43], but for sediments with lower compressibility s_{O}, i.e. 1/4, 1/5 and 1/6. In notation (3), both output equations, i.e. (2) and (3) differ in presentation of the filtration sediment resistance (1-s_{O}). Equation (2) is a derivative of the theoretical analysis of filtration process conducted by T. Piecuch, inter alia [37,39,40], where (2) the general filtration sediment resistance was inserted without any separate calculation of the filtration resistance towards the general parameter of the compressibility coefficient sO. According to the assumptions made by T. Piecuch, in equation (2) pressure ∆P, which exists both in the numerator and in the denominator, is the filtration process dispositive pressure, in the literature referred to as the motor pressure, too. This makes for the basic difference with the former theoretical approaches, where the pressure placed in the denominator represented the pressure drop in the filtration sediment layer, and thus, only a part of the so- called dispositive pressure for the course of the given filtration process. In turn, in notation of equation (3), the second component of the denominator, which refers to the sediment resistance, includes a formula, which results from a separate solution of a differential equation of the filtration sediment resistance, by integrating that equation towards the compressibility coefficient s_{O}.

That approach is typical of the French school classical theory of filtration formulated a 100 years ago, which was transferred to the Polish literature, for instance [7] quite uncritically, while a solution for that resistance differential equation, or leading out a final notation of the resistance equation in the second part of the denominator in equation (3) can be found by the reader on monograph [50].

In equation (3), pressure ∆P, which appears in both the numerator and the denominator, is the dispositive pressure for the given filtration process. Moreover, the simplifying assumption is adopted that the entire suspension (the solid phase), which forms the mixture provided for the filtration process, will take the form of the filtration sediment on the filtrating mesh (V_{N}∙β_{N}), and hence the filtrate will be a mechanically clean liquid.

The calculations discussed here are based on the filtration equation as per notation (3), where the assumption is that this is going to be filtration at variable values of the motor pressure ∆P, which, in the general mathematical notation of the filtration, presented as variable dx, being a differential equation. Coming over to the simplified form of the filtration equation as per notation (3), as constant values A, B and C were adopted respectively. Therefore, as a result of transformation of the equation as per notation (3) up to the general mathematical formula, it is going to take the form (7) where A, C – certain constant values. A solution of the integral as per notation (7) for individual cases of parameter B, which in a physical notation is related to the compressibility coefficient s_{O}, is presented below. Thus, the derived in this paper final equations, with the physical notations (14), (20) and (28), are the final filtration equations for filtration of a mixture flowing onto the filtrating mesh and forming on it a compressible sediment, having the compressibility coefficient s_{O}, as appropriate to values 1/4, 1/5 or 1/6. The reader will find equations derived from the same basic formula of filtration with notation (3), in assumption of a steady flow and variable motor pressure of the process for compressibility coefficient s_{O} equal the values of 1/3, 1/2 and 2/3, respectively, in a publication by the authors, in Annual Set The Environment Protection (vol. 15. 2013) [43]. Filtration equations for compressibility coefficient s_{O} equal 1/4, 1/5 or 1/6 for the basic equation of filtration as in notation (2) , in assumption of a steady flow and variable motor pressure of the process, can be found in the authors’ publication in Mineral Resources Management (No 3. 2014) [48]. Filtration equations for compressibility coefficient s_{O} of higher values, equal to 1/3, 1/2 or 2/3, respectively, based on the basic filtration equation as per notation (2), can be found in the authors’ article published in Archives of Environmental Protection (Vol. 39. no 1. 2013). This cycle of these four publications of the authors [34] will be continued by a series based on the general filtration equations as per notations (2) and (3) for the same respective values of the compressibility coefficient sO, and thus: for sediments of conventionally high compressibility, where s_{O} amounts to 1/3, 1/2 or 2/3, or conventionally for sediments of low compressibility , where the compressibility coefficient s_{O} is 1/4, 1/5 or 1/6, but on the assumption that the process motor pressure parameter ∆P, which appears in those equations, is to be constant, while the flow of the liquid in the filtration process is the variable factor.

The theoretical analysis of the filtration process, as conducted in this paper, confirmed the general conclusions presented in previous papers on this problem related to determination of final equations for filtration processes with various compressibility coefficients [34,43,48]. Having the intention to introduce the possibility to use these final equations to the designing practice, it is necessary to create algorithms appropriate to them, and, on their basis, numeric applications, which is going to be the subject of the authors’ further works in the years to come.

**Słowa kluczowe**

filtracja, siatka filtracyjna, metody numeryczne

**Filtration at Steady Flows of Mixtures Forming Sediments of Low Compressibility**

**Keywords**

filtration, filter mesh, numerical methods

**Pełny text / Full text**

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