We suggest two interconnected approaches to solve these prob

We suggest two interconnected approaches to solve these problems:
The following types of results are discussed:
The traditional model and the flow distribution problem The traditional model of steady isothermal flow distribution in a HC includes Kirchhoff\'s laws and closing relations: where A is a incidence matrix of nodes and branches of the pattern of calculation with the elements , if j is the start (or end) protein kinase inhibitor for branch i and , if branch i is not incident to node j; x, y are the n-dimensional vectors of flow rates and pressure drops on the branches of the pattern of calculation; f(x) is the n-dimensional vector function with elements accounting for the dependences of pressure drop on flow rate (flow laws) on HC branches; Q is the (m – 1)-dimensional nodal-flow-rate vector with the elements > for inflow to the node < for bleed-off in the node j, and , if the node j is a simple point connecting the branches; P is the ()-dimensional nodal pressure vector. The problem is finding the vectors x, y, P for the given matrix A, vector Q, the given form of () for and the given pressure in one of the nodes; we set equal to zero for simplicity. Numerous methods and algorithms have been proposed for solving this problem, with the most basic ones, as demonstrated in monograph [1], being the contour flow-rate and the nodal pressure methods. Both are based on the Newton method but with a preliminary depression of the linearized systems of Eq. (1). Then, the classical nodal pressure method (PM) [1] involves searching for a solution of (1) in the space of nodal pressures and is reduced to organizing the process as where for each kth iteration the correction is obtained from the solutions of the system where ; ; ; is the diagonal matrix of partial derivatives ∂/∂, , in a point is the vector function inverse to f with the elements (). It is evident from this that the PM calculation pattern does not depend on the form of functions () that must only be monotonically increasing to ensure that the problem of flow distribution has a single solution [1]. The nuances of PM implementation, however, are closely related to the specific form that () and, respectively, ∂/∂ and () take. For the conventional case, where is a hydraulic resistance of the branch, > is a pressure increment on the active branches (e.g., with pump stations) and is the pressure increment on the passive branches (e.g., for pipeline sections). Here we have the explicit expressions
Below we discuss the two main cases of the unconventional closing relations, starting with the case of implicit functions ().
Flow-rate-implicit closing relations To illustrate the reasons for the variety of closing relations let us consider the well-known Darcy–Weisbach formula [6] for pressure head loss in a pipeline: where d, l are a diameter and a length of the pipeline; g is the free fall acceleration, is fluid flow rate, x is a mass rate of this flow. Here the hydraulic resistance factor λ depends on the Reynolds number , where ν is the kinematic viscosity that is assumed to be constant for isothermal fluid flow. There is also a plethora of formulas to calculate λ that account for the the pipeline purpose, type, inner coating material, service life, fluid flow conditions (laminar, transitional, turbulent), etc. Some examples of the formulae for the λ factor and its rate derivatives commonly used in Russia and abroad are listed in Table 1. Pressure head loss in local resistances is determined by the Weisbach formula [6]: where approximating functions may be required to determine the local resistance coefficient ζ when regulating elements are present. For example, the following function is suitable for the case with a sluice valve: where r is the degree of valve closure; a, b, c are the parameters obtained by approximating (including piecewise) the relationship set in the table. Summarizing the cases when local resistances (including the regulated ones) exist in a pipeline of any type, punctuated equilibrium follows for the ith passive HC branch that