Turbomakinalarda akışın numerik metodlarla analizi

Abstract

ÖZET Bu çalışmada Katsanis tarafından geliştirilen bilgi sayar programı ele alınmış ve isler hale getirilmiştir. Program kanatlar arası akım yüzeyinde eksenel, radyal ve karışık akımlı turbomakinalarda çift sıra kanat için hız dağılımını ve akım çizgilerini hesaplamaktadır. Çalışma 7 bölümde toplanmıştır. ilk bölümde turbomakinalar içindeki akis teorisinde günümüze kadar olan gelişmeler anlatılmıştır. ikinci bölümde ise turbomakinalar içindeki akis için temel aer oter modi namik bağıntılar çıkarılmış ve kanatlar arası akım yüzeyinde 3 boyutlu analiz için gerekli denk lemler verilmiştir. Üçüncü bölümde ise Wu nun 1952 yaptığı çalışmadan yola çıkılarak, akım fonksiyonu eşitliği kanatlar arası a kım yüzeyinde uygulanır hale getirilmiştir. Akım fonksi yonu eşitliğinin çözümü için sonlu farklar yöntemi kulla nılmış ve gerekli sınır koşulları verilmiştir. Dördüncü bölümde ise kullanılan bilgisayar programı nın yapısı ve alt programların işlevleri gösterilmiştir. Ayrıca programda oluşan hata durumları açıklanmıştır. Besinci bölümde ise programın herhangi bir türboma kina tipi için kullanmak isteyene yol göstermek amaciyle, programda kullanılan giriş bilgileri açıklanmıştır. Ay rıca verilen girişlerin hazırlanmasında uyulacak talimat lar ver ilmi şti r. Altıncı bölümde ise bilgisayar programı» eksenel gaz türbini, karışık akımlı çark, santrifüj pompa ve eksenel kompresör için hazırlanmış uygun giriş değerleriyle çalış tırılarak sonuçlar elde edilmiş ve bu sonuçlar grafikler üzerinde gösterilmiştir. Son bölümde ise türbomakina çarkları içinde kanatlar arası akım yüzeyinde elde edilen sonuçların kısa bir açık laması yapılmıştır. vii

SUMMARY ANALYSIS OF FLOWS IN TURBÛMACHINES WITH NUMERICAL METHODS In this study, numerical.flow analysis was done tandem bides of turbomachine. A computer program was used that gives the blade-to-blade solution of tandem or slotted turbomachine blades. The flow is two dimensional» subsonic, compressible Cor incompressible), or non-viscous The form of cascade is circular or straight infinite. The blades 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 blades may be overlapping or non-overlapping in the meridional flow direction. Shortly, the program input consists of blade and stream channel geometry, total flow conditions, inlet and outlet flow angles, blade to blade stream channel weight flow and the portion of this weight flow that passes between the front and rear tandem blades. Also, the output includes blade surface velocities, velocity magni tude and direction at all interior mesh points and streamline coordinates throughout the passage. The report includes the computer program which was writen Katsanis, T., with an explanation of the equations involved. Also the method of solution and the calculation of velocities. This report consist of seven sections. In the first section of study devolepments in the analysis of flow in turbomachines presented. The most important investigation about three dimensional flow in turbomachinery was done by Wu, C.. [13 in 1QS2. This theory based on stream function. In the second section of study, a general theory of steady three dimensional flow of a non=viscous fluid in subsonic and supersonic turbomachines having a finite number of blade is presented. The solution of three dimensional problem is obtained by investigating an appropiate combination of flows on relative stream surfaces. The equations obtained to describe the fluid flow on blade to blade surface. This also lead to a solution of the three dimensional problem in a two dimensional manner through iteration. The theory is applicable to both irrotational and rotational absolute f low. In the section of basic aero- thermodynamic relations, the motion and energy equations of a non-viscous compressible fluid in a rotating blade row are expressed. The equations expressed in terms of the vanexpressed. The equations expressed in terms of the velocity components and two basic thermodynamic properties of the fluid. This properties are entropy and modified total enthalpy for flow in rotating blade rows. Estimated entropy changes due to shock waves» heat transfer or viscous effects can be easily accomodated in the calculation. Besides» in order to solve the steady three dimensional flow» an approach is taken to obtain the three dimensional solution by combination of two dimensional flows on two different kinds of relative stream surface is one whose intersection with a z-plane either upstream of the blade row or midway in the blade row forms a circular arc. The second kind of relative stream surface is one whose intersection with a z-plane either upstream of the blade row or somewhere inside the blade row forms a radial line. These two kinds of relative stream surfaces will be represented SI and S2. In the third section of study mathematical analysis of tandem blade rows was done by using theory of Wu and simplifying the equations. In the design of tandem or slotted blade rows for compressors or turbines, an analysis is desirable which will give velocity` distributions from blade to blade, and particulary over the blade surfaces. It is desired to determine the flow distribution through a stationary or rotating cascade of tandem blades on a blade to blade surface. Some simplifying assumptions are used in deriving the equations and in obtaining a solution. 1-The flow is steady relative to blade 2- The fluid is a perfect gas or is incompressible 3-The fluid is nonviscous 4,-There is no loss of energy S-The flow is absolutely ir rotational 6- The blade to blade surface is surface of revolation 7-The velocity component normal to the blade to blade surface is zero 8- The stagnation temperature is uniform across the inlet 9- The velocity magnitude and direction is uniform across both the upstream and downstream boundaries lû-The relative velocity is subsonic is everywhere The flow may be axial, radial or mixed and there may be a variation in the stream channel thickness in through flow direction. The proportion of flow between the front and rear blades must be specified as an input in the program. This input may be difficult for the user to estimate. Correlation with experimental work may yield more reliable values. The coordinates of stream surfaces are defined r, Ö, z. Since the variables r and z are not independent on a stream surface, one variable can be eliminated. It is IXbetter to use the meridional streamline distance m in place of r and z as an independent variable. Then, m and ö two basic independent variables. A stream channel is therefore defined by specifying a meridional streamline radius r and a stream channel thickness b at several locations m. For the mathematical formulation of the problem the stream function is used In fact the flow through stationary or rotating cascades cannot be axisymmetric if the flow must exert a moment on the blades. Although it has been shown that arbitrary steady absolute flows cannot have axisymmetric stream surfaces, a special type of flow exists where this is possible. It will be shown that such flows can be two dimensional. Incompressible flows in radial cascades has axisymmetric stream surfaces. Such flows can be analyzed by means of stream functions. The stream function is expressed Wu C1952). In this report stream function u is related to stream function v* defined in Katsania CI 969) by u=-v/`w. The stream function equation under the given assumptions. âibpy âm öu 2bpd> = sina Cl> ifm w Also, the derivates of the stream function satisfy the equations, du bp itm We C2> du bpr = W C3> «*e For the solution of stream function equation a finite region is considered. Since the stream function iselliptic, all boundary conditions over the blade surfaces must be given. That's why boundary conditions are given for solving the stream function equations. The simultaneous, nonlineer, finite difference equations of the stream function are solved by using two major levels of iteration. The equations are nonlineer since the coefficients involve the densty. The inner iteration consists of the solution of simultaneous lineer equations by successive ovei - relaxation, using an estimated optimum over-relaxation factor. The outer iteration then changes the coefficients of the simultaneous equations to correct for compressibility. After computing a numerical solution to stream function equation in a given flow region, the velocity at any point can be computed from derivates of stream function by using numerical differentiation. Then the streamlines are located by the contours of equal stream function values. In the fourth section of study, the program procedure wsa expressed. Also the facilities of subroutines was given. The program is segmented into seven main parts. CINPUT. PRECAL, COEF. SOR. SLAX. TANG, and VELOCY) All the subroutines and their relation between them was shown. The same set of variables was used in subroutines. In the fifth section of study, detailed inputs for program was given. The program requires two types of variables, geometric and nongeometric. These variables are given as an input. The program needs as input a geometrical description in m, ö coordinates of the tandem blade segments, a description in m, r coordinates of the stream channel through the blades, appropiate gas constants, and operating conditions such as inlet temperature and densty, inlet and outlet flow angles, weight flow, and rotational speed. An estimate of the portion of the weight flow which passes between the tandem blades must also be given. Output obtained from the program includes velocity magnitude and direction at all interior mesh points in the blade to blade passage, blade surface velocities, stream function values throughout the blade-to-blade region of solution, and streamline locations. In this section, also instructions for preparing input was given. The internatinal system of units was used throughout this report. However, the program does not use any constants which depend on the system of units being is used. Therefore, any consistent set of units may be used in preparing input for the program. Since any consistent set of units can be employed, the output is not xilabeled with any units. Furthermore, the upper and lower surfaces of the front and rear tandem blades are each defined by specifying three things: leading- and trailing edge radii. angles at which these radii are tangent to the blade surfaces, and m- and Ö- coordinates of several points along each surface. These angles and coordinates are used define a cubic spline curve fit to the surface. A finite-difference mesh is used for the solution of equation ( 1 J. The mesh spacing and the extent of the upstream and downstream regions are determined by the values of MBI. MBO. MBI2. MBÛ2. and MM of the input. The mesh spacing must be chosen so that there are not more than 2000 unknown mesh points. If the user does not change the mesh indexes between runs, even though blade geometry or other input does change, he may use the final estimate of optimum over-relaxation factor in the input. While the program is written for compressible flow, it can be easily used for incompressible flow. To do this. GAM=1.S. AR=1000, TIP=1 000 000 must be as an input. This results in a single outer iteration of the program to obtain the stream function solution. Also the program is easily applied to straight infinite cascades as to circular cascades. Furthermore, for a two-dimensional cascade with constant stream-channel thickness, only two points are required for defining stream channel. In the sixth section of study, to illustrate the use of program and the type of results which can be obtained four numerical examples are given. These examples are axial-flow turbine, mixed-flow impeller, centrifugal pump, and axial compressor. Also the graphics of streamline locations and blade surface velocities was drawn. In the seventh section of study, the results of given examples are explained. xii