where is the branch resistance because here is the

where is the branch resistance because here ρ is the density of the transported fluid; is the sum of the local resistance coefficients. Also, where
Thus, to obtain for a pipeline of any type and purpose, it renin inhibitor is enough to specify (see Table 1). The hydraulic characteristics of the active branches are, in general, defined by algebraic polynomials of the form when
However, they may differ in exponents b, in the number of and in the composition of terms (for the mandatory assignment of for some q). A piecewise approximation of a single characteristic by different polynomials may also be applied. Previously cases of type (2) were attributed to the HC class with variable parameters for which the procedure of double-loop iterations [1] was applied to calculate the flow distribution. By this procedure, the functions () are recalculated in the outer loop, and the basic nodal pressure or loop flow methods are used in the inner loop for the fixed values. The above-described technique for the derivatives of flow-rate-implicit relationships allows to avoid the outer iteration loop, which drastically reduces the computational costs [4]. The flow rate may be iteratively calculated from the given pressure difference for type (2) implicit functions, for example with the methods of simple iteration, where or Newton\'s method, according to which where k is an index of an iteration for calculating the flow distribution, t is an index of an iteration for searching for , is a residual. For t = it is assumed that . Fig. 1 illustrates the convergence processes of the three suggested options for a pipeline with = 5.3085 m H2O, d = 300 mm, l = 1 000 m; the λ factor was found using the Shevelev formula for new steel pipes (A1= 0.0159; A0= 1; C = 0.684; m = 0.226), at the initial approximation = 800 kg/s and the solution = 71 kg/s. It is apparent that using the combined method allows a considerable gain, while the solution with the acceptable accuracy ( 10–4m H2O) was obtained in only three iterations. Therefore, for implicit functions (), the PM calculation pattern remains unaltered, except for applying special procedures when calculating and .
Pressure-dependent closing relations Sometimes it is impossible to reduce the hydraulic characteristics of pipeline system elements to the traditional form . For example, a certified specification of a gas pumping unit (GPU) connects the pressure ratio (the compression degree) rather than the pressure difference with the flow volume ( are the pressures before and after the GPU). The simplest analytical form of this characteristic [11] is where are the coefficients obtained by the ɛ2(x) approximation of the GPU certified performance data; x is a capacity regarded here as an input condition. Other types of ɛ(x) or ɛ2(x) functions for GPUs are described in literature; these are obtained for various powers of approximating polynomials. For example, Ref. [12] suggests the following function
The functions were also examined throughout the study, where are the coefficients of the approximating polynomials for ɛ(x). To generalize the examined cases of the hydraulic characteristics (including the conventional and the flow-rate-implicit), let us introduce a closing relation of the following functional form: where are the n-dimensional pressure vectors, respectively, at the start and at the end of the HC branches. Indeed, as , then instead of , we have
Fig. 2, а shows some examples of classical closing relations in x, y coordinates. As previously stated, in order to secure a single solution of the flow distribution problem it is necessary to satisfy the requirement of monotonicity of the closing relations [1], which may be extended to the reviewed case (7) by representing the classical relations in the and coordinates (see Fig. 2, b and c). As shown in Fig. 2, a, the requirement that y(x) should monotonously increase is equivalent to simultaneously satisfying the requirements that (x) (Fig. 2, b) monotonously increase and (x) (Fig. 2, c) monotonously decrease, which, in turn, is equivalent to the requirement that the conditions