Paralel olmayan az eğimli eksenel simetrik plaklar arasında hızın düşey bileşeninin radyal akıma etkisi
Başlık çevirisi mevcut değil.
- Tez No: 55932
- Danışmanlar: PROF.DR. GÜLGÜN YALÇINKAYA
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1996
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 52
Özet
ÖZET Eksenel simetrik plaklar arasında radyal akım bir çok araştırmacı tarafından incelen miştir. Basit teori, hareket denklemlerinde atalet terimlerini ve hızın eksenel bileşe nini ihmal etmektedir. Peube, incelemesinde atalet terimlerini ve eksenel hızı gözönünde tutarak paralel plaklar arasındaki akımı ele almıştır. Katsayıları, sınır şartlarım ve süreklilik denklemine göre belirlenen bir geometrik diziye dayalı kapsamlı bir metod vermiştir. Gans, deneylerde basınçların basit analizin öngördüğü değerlere göre, yaklaşık iki kat daha büyük olduğunu gözlemlemiş ve bu olayı yatağın küçük ortalama yüzey eğimine bağlamıştır. Peube'ün çalışmasında, plak içbü- keyliğinin gözönüne alınmadığını ve Gans'ın eksenel hızı hesaba katmadığını burada belirtmek ilginç olacaktır. Eldeki tez, bu iki özelliği analize dahil etmeyi ve ilgili sonuçlan çıkarmayı amaçlamaktadır. Tez çalışmasında, içbükeyliğe ilaveten eksenel hız da hareket denklemlerine dahil edildikten sonra fark alkmak suretiyle iteratif yöntem kullanılarak, bu eksenel hızın daimi radyal akım üzerindeki etkisi, radyal hızlar ve basınçlar hesaplanarak ortaya konmuştur, sonuçta, eksenel hızın basınçları ortalama %43 arttırdığı anlaşılmış ve deneylerde, basit analize göre gözlenen iki kat artışın sadece içbükeylikten değil ve fakat aynı zamanda eksenel hızdan da ileri geldiği belirlenmiştir. vıı
Özet (Çeviri)
SUMMARY RADIAL FLOW BETWEEN AXISYMMETRIC NONPARALLEL PLATES OF SMALL SLOPE TAKING INTO ACCOUNT THE AXIAL VELOCITY The radial flow between axisymmetric plates was investigated by several authors. The simple theory neglects the axial component of the velocity and the inertia terms in the motion equation. Peube, in his study took care of the axial velocity and the inertia terms and considered the flow between the parallel plates. He gave a comprehensive method based on geometric series whose coefficients are determined according to the boundary conditions and the continuity equation. Gans observed in the experiments that the pressures are approximately two times greater than the values predicted by the conventional analysis and ascribed this fact to the slight mean face slope of the bearing. Thus, it is interesting to note that the work of Peube lacks the concavity of the plates and Gans omits the axial velocity. This thesis attempts gathering these two particularities in the analysis and deducing the related consequences. The calculations are performed conforming to the Gans pattern, using an iterative process. The pressures and the uplift forces are expressed in the form of a series. Its first term that is achieved at the end of the first step of the iteration is denoted as linear term {po). The second obtained at the following step is called the nonlinear term {p\). Added to the first one it leads to a better approximated result and would be theoretically followed by other nonlinear correction terms constituting the normal approach to the solution of a convergent series. Unfortunately, the obtention even of the second term is confronted with mathematical difficulties and only a numerical but not analytical solution was attainable. The effect of the vertical component of the velocity is enormous. Disregarding it caused to Gans an incomplete judgement, that is, the reason of the two times greater pressures observed in the experiments compared to the results of the simplified analysis are due only to the very slight mean face slope of the bearing, whereas in fact the axial velocity also contributed to it in a great extent. A comparison with the Gans formula at the end of Vlllthe iterations first step, showed an excess of forty percent regarding the pressures. In the analysis it is dealt with a nondimensional representation. The inclusion of the axial velocity, resulted in the differential equation of the motion in two supplementary terms, which cause the resolution, being not mathematically practical to attain. That's why only the governing term was kept, the other being left temporarily with the intention to evaluate its effect later. After getting in this way the solution, it is recognized that it has the opposite sign to the first term and reduces it approximately in the ratio of forty-three percent. As a result, the values given by the analysis were assigned in the following the factor 0,574. Peube, in his analysis expressed forcefully, the necessity to satisfy the three conditions below in order to make his formulas valid: - the radius of the disks must be large enough. - the flow must be laminar - the boundary conditions must be appropriately realized. The second condition implies that the fraction - be smaller than 22. This r constraint translated into our notation means that x> 0,754. When investigating the compliance of the conclusions of our analysis with the technical literature and especially with Peube's work, which involves the vertical component of the velocity, it is of prime importance to consider this lower limit for x. Thus, it becomes possible to certify the parallelism between the curves for the pressures and the velocity, drawn accordingly to the present analysis and those of Peube and Gans. The distribution of the radial velocity along the vertical axis is exactly parabolic when the axial component of the velocity is disregarded. This is the case in the Gans analysis. Peube states that, due to the consideration of the axial velocity, the distribution is not anymore parabolic. To have a distribution curve near the parabolic shape it is necessary to the flow to be laminar. The turbulent flow makes this curve to move away from the parabolic shape. The concavity of the upper plate also deforms somewhat the symmetry of the parabola. IXUnder these considerations, the results of the present analysis, illustrated with tables, diagrams and curves are in conformity with those of Gans and Peube, except understandable small differences due to the effects of the concavity and the axial velocity. It is thus proved, that the omission of the axial velocity is not admissible and it should be included in the calculations, despite it makes them lengthy and difficult to cope with. On the subject of integration of the Navier-Stokes equations, it would be pertinent to discuss some of their general properties. In our study we have restricted ourselves to incompressible viscous fluids. Until the present time no general methods have become available for the integration of these equations. Furthermore, solutions which are valid for all values of viscosity are known only for some particular cases, as Poiseuille or Couette flows. In order to simplify considerably the mathematical problam, it was attacked by first tackling limiting cases concerning the viscosities. However, the case of moderate viscosities cannot be interpolated between extreme values. That's why, due to these great mathematical difficulties, the research into viscous fluid motion proceeded to a large extent by experiment. In this connexion the Navier-Stokes equations furnish very useful hints which point to a considerable reduction in the quantity of experimental work required. We remind that the Gans experiments were carried out in the wake of the observations concerning the pressures in the radial flow between to discs. The Navier-Stokes equations become further simplified in the case of incompressible fluids (p^constant) even if the temperature is not constant. Since temperature variations are, generally speaking, small, the viscosity may be taken constant (u=constant). This condition is more nearly satisfied in gases than in fluids. The equation of state as well as the energy equation become superfluous as far as the calculation of the field of flow is concerned. The field of flow can be considered independently from the equations of thermodynamics. The solutions of the mentioned equations become fully determined physically when the boundary and initial conditions are specified. In the derivation of the motion equations it is assumed that the normal and shearing stresses are linear xfunctions of the rate of deformation, which is evidently completely arbitrary and not certain a priori, but confirmed later by experiments. The enormous mathematical difficulties encountered when solving the Navier-Stokes equations have so far prevented from obtaining a single analytical solution in which the convective terms interact in a general way with the friction terms. In my thesis, I departed from the simplified form of the motion equation, but added the axial component of the velocity which led to a more realistic solution and made clear the explanation given by Gans about his observation on the pressures, precising the respective parts of the concavity and the axial velocity in the results obtained. It is necessary to remark that also the dynamic properties, oscillations and stability of the system considered have to be investigated, despite the casual observations made by Gans, suggest great stability, which tentatively was ascribed by him to lift off at minimum pressure,rather than at an excess over running one. Finally Gans noted that the behaviour of this system is not particularly sensitive to the exact form of the concavity. This is in marked contrast to the care that must be taken to insure parallelism when that is the aim. Later, concerning this conclusion, Hayashi specified that for the purpose of insuring parallelism of two plates, it is useful to measure the central pressure and /or the lift off pressure by the system shown and compare them with the predicted values. It should be recognized that the insensitivity of the central pressure or the lift off pressure, i.e., the load carrying capacity, to the exact form of concavity, only comes from the average effect. The pressure distribution will be remarkably influenced by the exact form of the surface. From the view point of stability, it is recommended that the bearing has no recess. In this case, in order to realize the small lift off pressure, i.e., the large load carrying capacity, which is one of the main aims of this bearing, it becomes necessary to increase the concavity of surface, i.e., to increase the additional spacing at the center of the bearing. On the other hand, the coefficient of damping may decrease and become negative at all as the additional spacing increases. Therefore the form of concavity should be carefully chosen. Moreover, the bearing performances may be sensitive to the imperfection of the outer rim of plate, so its recommended giving some extent parallel region in the outer part of bearing. XI
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