Turbomakinalarda meridyenel düzlemde hız ve basınç dağılımının hesaplanması

Abstract

Bu çalışmada Thedore Katsanist ve Micheal R.Vanco tarafından geliştirilen eksenel, radyal veya karışık akışlı turbo ma kinada meridyenel akım yüzeyi üzerinde hız ve basınç dağılımını ve de akım çizgilerini hesaplayan fortran programının tanıtılması ve çalıştırılmışı ele alınmıştır. Çalışma dürt bölümde toplanmıştır.Birinci bölümde gerekli denklemler ve sayısal metod tanıtılmıştır. İkinci bölümde hız gradyeni denklemi elde edilmiştir ve bu çalışmada kullanılan eğri uydurma metodu açıklanmıştır. Üçüncü bölümde programa ait bilgiler verilmiştir. Dördüncü bölümde programın örnek girişlerle çalıştırılmasından elde edilen çıkışlar ve bu çıkışlara ait grafikler sunulmuştur.

Turbomachines employing centrifugal effects for increasing fluid pressure have been in use for more than a century. The earliest machines using this principle were, undoubtably hydraulic pumps followed later by ventilating fans and blowers. It is on record that one of the first successful aircraft jet propulsion engines used a centrifugal compressor» much having been learnt from their earlier application in supercharging reciprocating engines. An account of the centrifugal compressor used in the early gas turbine is given by Cheshire and some interesting comprarisons of the relative advantages of axial and centrifugal compressors are available. It is now appreciated that the best efficiencies from centrifugal compressors, even under the most favourable circumtances, are lessthan those of axial compressors designed for the same duty by as much as 3 or 4 %. Howewer, at very low mass flows axial compressor efficiency drops, blading is small and difficult to make acurately, and the advantage appears to lie with the centrifugal compressor in its relative simplicity and cost. Recently, increased interest has been shown in high-pressue-ratio backward-swept centrifugal impeller blades. Centrifugal compressors with backswept impeller blades have the potential of achieving higher effeciencies than those with radial impeller blades. Several methods are available for designingradial-bladed compressors, but has been done on backward-swept impeller blades/ Quasi-three-dimensional methods have been developed for analyzing flow through mixed-flow turbomachines. These methods use streamlines and their normals to establish a grid for the solution. In cases where the distance between hub to shroud is great and there is a large change in flow direction within the rotor, howewer, the normals vary considerably in length and direction during the course of the calculations. Therefore, it becomes difficult to obtain a direct solution on the computer without resporting to indermediate graphical steps. The use of normals, howewer, is not essential to the method, and it appeared possible to overcome this difficulty by the use of a set of arbitrary curves from hub to shroud instead of streamline normals. These arbitrary curves will be hereinafter termed quasi-orthogonals. The quasi-orthogonals are not actually orthogonal to each streamline, but merely intersect every streamline across the width of the passage. The quasi-orthogonals remain fixed regardless of any change of streamlines. By using this technique, it appeared possible to develop a computer that would calculate the velocity and pressure distributions without any intermediate graphical procedures even for turbomachines with wide passages and a change of direction from radial to axial within the rotor blade. In view of these considerations, a method of analysis utilizing quasi-orthogonals in lieu of streamline normals was developed. This report presents the analysis method and contains a discussion of the numerical techniques required for obtaining solutions with a digital computer. In this study, numerical flow analysis has been on centrifugal compressors.The computer program is used that gives the hub to shroud solution of centrifugal compressor. This program will determine the velocities in the meridional plane of a backward swept impeller, a radial impeller or a vaned diffuser. The flow is two dimensional, subsonic, incompressible or nonviscous. The blade may be fixed or rotating. The flow may be axial, radial or mixed. There may be a change in stream- channel thickness in the through flow direction. The velocity gradient equation with the assumption of a hub-to-shroud mean stream surface is solved along arbitrary quasi-orthogonals in the meridional plane. These quasi-orthogonals are fixed straight lines. This report includes the computer program which was written by Micheal R.Vanco.with an expiation of the equations involved also the method of solution and the calculation of velocities. This report has four sections. In the first section of study.developments in the analysis of flow in turbomachines presented. Also the method and given the numerical techniques used to find the flow distribution in the meridional plane of centrifugal compressor arepresented. The general velocity gradient equation is derived along an arbitrary quasi-orthogonals in the meridional plane with the assumption of a hub-to-shroud mean /stream surface. The equations are derived in the second section. VI _dW_ ds dr ds -ü> dX ds K. cosacos ft sxn ft ^., A A A= ; - - - + sinasin/îcos/î- dr _ sinacos ft... A _ de B=- i- + sinotsin/?cos/? r dz dW C= sinacosft- m dm -2a>sin/? + rcos. " fdW^ /3 [dm + 2a>sinc* 1d& dr D= cosacos/? + r cos/3 I + 2cosina I dm I dm dz In this study the total entalphy at inlet h* and prerotation at the inlet ^,ri^ei areassumed constant.Continuity must also be satisfied from hub to tip. The calculated mass flow across fixed line from hub to tip must be qualthe specif iedmass flow. The density is calculated from the isentropic flow equation with a correction in total relative pressure. The first step in the analysis is the numerical evaluation of the paremeters a, ft, r, âö/âr, â9Söz, dr/ds, dz/ds, dW /dm, and dW^/dm for use in the mean m & equations. In order to evaluate the paremeters a, ft, and r a streamline geometry must be established. First fixed straight lines Cquasi-orthogonals} are drawn from hub to shroud along which the velocity gradient foran assumed stream surface will be determined. For an initial approximation to the streamlines, each quasi -orthogonal can be divided into a number of equal spaces. The success of the method is based on the fact that, for a reasonable assumed streamline pattern, the geometrical streamline paremeters Involved are not to different from those the final solution. By means of a spline fit approximation,dr/dz, d r/dz can be determined at each of the points established, oc and r can be calculated when they are c known. For the remaining parameters, the mean stream surface Ö=ÖCr,z) between blades is needed; it must be given in such a manner that dö/dr and dd/dz can be VI i determined at any given point. The spline fit curve can assist in this. When they are known, ft can be calculated. For the initial calculation, W may be assumed constant throughout the rotor. Since the distance along the meridional streamline m is known, dW /dm and dWö/dm can m & then be determined by the spline fit curve. Since dr/ds and dz/ds are determined by the angles of the quasi-or thogonals, all the quantities necessary for the calculation dW/ds from the mean equation, except W itself, are now determined. The next step is the numerical integration of the mean equation, which is in the form dW ds =f CW.s) where f is known only for a finite number of values of s. For a given initial velocity on »say the hub, the velocity distribution along the quasi-or thogonal can be approximated by m W. = W. + \^-\ As İ where the subscripts denote the number of the streamline, and As is the distance along the quasi-orthogonal between streamlines. For an improved estimate, a Runge-Kutta method can be used. Completing this computtation for a quasi-orthogonal from hub to shroud results in the complete velocity on the hub. The numerical Integration can be performed by use of a spline fit approximation. The computed total weight flow is then compared with the actual weight flow. If the computed weight flow is too small, the velocity on the hub is increased, and vice versa. A few iterations will determine the hub velocity that will give the correct weight flow. In the second section of study, derivation of the velocity gradient equation and use of spline fit curves are presented. The velocity gradient equation is derivated from Euler's force equation for a nonviscous fluid and under the assumption that the flow is isentropic.The equation is dV 1 _ =- ?P dt If a set of function values corresponding to a set of arguments is given, there are several ways a curve can be fitted through these values so as to approximate the original function with these values. The classical way is vm by an n degree polynomial for n+1 points. This may not be satisfactory, howewer, for a large number of points» especially for computing derivatives or curvature at end points. Another technique is to use fewer points to determine some of piecewise polynomial» but this does not lead to a smooth curve. A method that has received much attention recently is the piecewise cubic, with matching first and second derivatives, commonly referred to a spline fit curve. Since for small slopes, the second derivative approximates the curvature of a function, the strain energy of a spline can be approximately minimized by minimizing /Cf txl) dx, where fCx] denotes the curve described by the spline. The spline fit curve has this proverty. In the third section of study, the program procedure is expressed. The mean program QUAC andthe subroutines C RUUT, SMOOTH, INTGRL, CONTIN, SPLDER, SPLINE, LININT and SPLINT) and their relation between them are shown. The input quantities for this program consist essentially of mass flow, rotational speed, number of blades, inlet total conditions, loss in relative total pressure, hub-to-shroud profile, mean blade shape, and a normal thickness table. Since the program does not use any constants which depend on the system of units being used, any consistent set of units may be used. The first output of the program is input. Output 2 gives the stagnation speed of sound at the inlet; the radius at which the mean stream surface deviates from the mean blade shape; and a list of number of iterations required to obtain a solution with the corresponding maximum streamline change. Output 3 gives some of important quantities used in the calculation procedure which are also useful for debugging purposes. Output 4 gives the velocities and pressure for every streamline printed out. Output 5 gives the stream-channel coordinates and blade shape coordinates for the hub, mean, shroud. And output 6 gives the inlet flow angle for the hub, mean, and tip. In the fourth section of study, the results of given examples are explained.