Rotational Mach Number Research Articles

Modelled performance of non-chlorinated substitutes for CFC11 and CFC12 in centrifugal chillers

A number of partially fluorinated alkanes and ethers were identified from earlier technical publications and from a joint Electric Power Research Institute/Environmental Protection Agency (EPRI/EPA) project as potential alternatives for chloroflurocarbon (CFC) and hydrogen-containing CFC refrigerants. These larger molecules, by virtue of their more elaborate structure, have larger vapour phase heat capacities ( C ps) and larger molecular weights than currently used refrigerants, which result in decreased volumetric capacities and greater flash gas losses in simple cycle applications. The compounds were fitted to the Lee-Kessler-Plöcker and Carnahan-Starling-DeSantis equations of state, and refrigerant property routines based on these equations were used to simulate their performance in a single-stage centrifugal chiller as pure refrigerants and nearly azeotropic refrigerant mixtures. Consideration was given to the effects of acoustic velocity in the refrigerant, rotational mach numbers, the application of superheat prior to the compressor inlet to avoid ‘wet isentropic compression’, and liquid subcooling before isenthalpic expansion. Results indicate that several chlorine-free compounds give modelled chiller performance comparable to R11 and R123 and better than R12 and R134a. Blends of these refrigerants may be required to mitigate the inflammability of alternatives that show the best performance. Modifications to the basic chiller cycle — such as liquid subcooling and suction gas superheat — may offer unique advantages for more complicated, larger refrigerant molecules. An algorithm based on molecular bond energies and vapour phase C ps was used to estimate the flammability of these alternatives and of blends made from them. Depending on desirable and permissible deviations from the currently used chiller operating conditions and on the acceptability of flammable refrigerant combinations, ideal coefficients of performance comparable to R11 and 5–10% better than R12 are indicated.

Higher approximations in the asymptotic theory of propeller noise

This paper extends earlier work on the asymptotic theory of the acoustic radiation from a single-rotatio n propeller (in the many-blade limit B —> oc) in two ways. First, a correction is found to the dominant Mach radius field in supersonic operating conditions; the correction is associated with the blade tip and its inclusion brings the asymptotic prediction into fine agreement with numerical evaluation of the radiation integrals, in addition to giving physical insight and the possibility of control, at negligible computing expense. Second, an expression is derived that provides a transition across the Mach condition (from subsonic to supersonic radiation characteristics), and from this, composite expressions are given, valid across the entire range of operating conditions. Nomenclature Ai = Airy function B = number of blades JmB = Bessel function of first kind and order mB Mrt = tip relative Mach number Mt = tip rotational Mach number Mx = flight (axial) Mach number m = harmonic index Pm = normalized complex amplitude of far-field pressure in rath harmonic S(z) = normalized radial source function z — normalized radial (spanwise) coordinate z0 = normalized hub radius z* = Mach radius, defined by Eq. (2) 0, = sech-'Cl/z*), seeEq. (16) A = coordinate defined by Eq. (18) 0 = angular location of observer at emission time A, - sec-!(l/z*), see Eq. (14) JJL = tip-source exponent

Discretization errors inherent in finite difference solution of propeller noise problems

Global errors arising from the use of a finite difference approximation to solve propeller noise problems are investigated. It is found that the calculated waveforms of the finite difference solutions are subjected to severe distortion due to dispersive effects. This is so even when the Fourier amplitudes of the different blade-passage harmonics of the acoustic disturbance of the propeller are adequately predicted. In addition, it is found that finite difference solutions can produce spurious acoustic radiation. The spurious radiation is a form of aliasing error. The relative intensity of the spurious acoustic radiation can become significant at high subsonic flight Mach number and at high subsonic blade-tip rotational Mach number. Also, the relative intensity is higher for the higher order blade-passage harmonics.

Sonic eddy - A model for compressible turbulence

A new model is proposed for entrainment in supersonic turbulence. The central assumption is that only those eddies whose rotational Mach number is unity directly engulf fluid. With the additional assumption that a Kolmogorov spectrum of eddy scales exists for all subsonic eddies, the theoretical effect of the sonic eddy on entrainment and structure is compared to observation in shear layers and wakes

Time-averaged intensity spectral density of acoustic radiation due to time-varied lift force of rotating blades.

The time-averaged intensity spectral density function of the acoustic radiation from rotating blades is theoretically derived by replacing the blades with rotating dipoles. Dimensional analysis reveals two non-dimensional parameters which play an important role in generating the blade-passing frequency tone and its multiples. Increasing the number of rotor blades widens the peak under the condition that the non-dimensional parameter of the rotational speed is fixed. Moreover, the acoustic spectral density function is not stationary even if the inflow turbulence is homogeneous and isotropic. Further, the time variation of the propagation path due to the rotation should be considered in the computation of the spectral density function. For instance, in the rotor specifications of the present study, the rotor radius is approximately 0.3m and the rotational speed Mach number is approximately 0.2.

Noise and performance of a counter-rotation propeller

The noise and aerodynamic performance of a 3 X 3 counter-rotating propeller scale model were quantified experimentally in anechoic environments with a velocity of 68 m/s incoming flow on. The model diameter was 400 mm and the aft blade rotational speeds and the rotational Mach numbers were varied up to 9000 rpm and 0.55, respectively, while the front blades were kept at 9000 rpm. The reinforcement of counter-rotating harmonics was observed even at high blade passage frequencies. The axial separation between two rotors had a favorable effect on the noise levels, whereas an adverse effect on the performance was found for the large spacing with axial velocity acceleration attendant. The spike due to the overturning of tangential velocity was found near the tip section. A difference of the rotational speeds yielded the spinning sound waves with a beat frequency of three times the rps difference. The split frequencies could be identified on the measured spectrum when the two rotors had such nonsynchronous rotational speeds. Nomenclature A = area B - number of blades D — diameter of propeller / = frequency H = enthalpy or total head / = rothalpy J - advance ratio k = load harmonic t = axial spacing between two rotors M = Mach number m — sound harmonic N = rotational speed P = pressure r - radius s - static T = thrust t = total U = axial velocity W - tangential velocity fl = angular velocity /3 = blade setting angle p = density oo = downstream infinity

Axisymmetric compressible flow in a rotating cylinder with axial convection

The steady compressible flow of an ideal gas in a rotating annulus with thermally conducting walls is considered for small Rossby number ε and Ekman number E and moderate rotational Mach numbers M. Attention is focused on nonlinear effects which show up when σ and εM2 are not small (σ = ε/HE½, H is the dimensionless height of the container). These effects are not properly predicted by the classical linear perturbation analysis, and are treated here by quasi-linear extensions.The extra work required by these extensions is only the numerical solution of one ordinary differential equation for the pressure.Numerical solutions of the full Navier–Stokes equations in the nonlinear range are presented, and the validity of the present approach is confirmed.

The structure of the Stewartson layers in a gas centrifuge. Part 2. Insulated side wall

The Stewartson E½- and E¼-layers in a rapidly rotating compressible fluid are considered within the framework of linearized equations and the boundary-layer method. The fluid is contained in a cylinder made of a thermally insulated side wall and conducting top and bottom end plates. The end plates and the side wall rotate with slightly different angular velocities. The case of an incompressible fluid was discussed by Stewartson, who found that the flow is restricted to the side-wall boundary layers. In the case of a compressible fluid, however, the solutions are strongly dependent upon the thermal boundary conditions assumed on the side wall. In particular, if the wall is insulated the fluid in the inner core is dragged along too since it is coupled strongly to the flow in the side-wall Stewartson layers. The critical parameter governing the solutions is found to be $(\gamma -1) PrG_0 E^{-\frac{1}{3}}/4\gamma$, where γ is the ratio of specific heats, Pr the Prandtl number, G0 the square of the rotational Mach number and E the Ekman number.

超音速軸流圧縮機の動翼内の流れに関する研究 Ｉ 超音速動翼の設計と垂直衝撃波前後の半径平衡

The radial equilibrium of pseudo-axisymmetric flows in supersonic rotors which have no normal shock was considered and two examples were calculated with the following results.[1] The supersonic rotor blades which have flows with no turning and satisfying radial equilibrium conditions in the entire passages get thicker at the tip radius than at the hub.[2] If the deceleration combined with the appropriate turning through the supersonic rotor passage are taken into consideration, the constant absolute total pressure may be expected at the exit.The flow condition for which the radial equilibrium relations could be satisfied both before and behind the normal shock was described. An expression to establish the above was written by using three basic variables, which are the relative Mach number, the rotational Mach number of rotor (the ratio of rotational speed of rotor to sound velocity) and the relative flow angle, before the normal shock. In order to solve the equation, two variables or two equivalent conditions must be specified. The approach in this report seems to be useful in designing the rotor passage containing the normal shock.

Thermally, mechanically or externally driven flows in a gas centrifuge with insulated horizontal end plates

The axisymmetric motion of a compressible fluid in a rapidly rotating cylinder is considered using a linear approach and boundary-layer techniques. The deviation of the fluid motion from rigid-body rotation is caused either by an applied temperature distribution on the side wall, a differential rotation of the top and bottom end plates or sources and sinks of fluid distributed on the end plates. The horizontal end plates are assumed to be thermally insulated, while the side wall is conducting. The critical parameter governing the problem is found to be E−½(γ − 1) PrG0/4γ, where E is the Ekman number, γ the ratio of specific heats, Pr the Prandtl number and G0 the square of a rotational Mach number. If this parameter is larger than unity, the coupled effect of the compressibility of the fluid and the thermal condition on the end plates suppresses the flow in the cylinder. The flow in the inner inviscid core is strongly coupled with the Ekman-layer flow through the boundary conditions on the end plates, something which does not occur if the fluid is Boussinesq nor if the fluid is compressible and the end plates are conducting.

A similarity approach to the general circulation of planetary atmospheres

Similarity and dimensional considerations and some thermodynamic arguments are applied to planetary atmospheres to estimate such characteristics of their general circulations as the total kinetic energy of the circulation, the rate of generation (or dissipation) of the kinetic energy, the efficiency of an atmosphere in transformation of the supplied energy into energy of atmospheric motions, and the characteristic temperature difference which drives the circulation. As known external parameters there are the energy supply to an atmosphere, the mass of the atmosphere, its heat capacity, the radius of the planet, the angular velocity of rotation, and the gravitational acceleration, g. The energy balance in an atmosphere is maintained by longwave radiation into space which requires one universal constant, Stefan's constant. From these seven dimensional parameters only four have independent dimensions; therefore one may construct three nondimensional complexes which are the criteria of similarity. The first of these is the ratio of the atmospheric scale height to the planetary radius. It is always small; therefore one may propose the first self-similarity hypothesis of the circulation with respect to g; i.e., the exact value of g is not important for the theory. The second criterion of similarity may be interpreted in various ways; it is also small for all the planets, with the possible exception of Mercury (if it has an atmosphere). The self-similarity with respect to this parameter turns out to be the independence on the mass of the atmosphere, if only the mass is large enough. The third criterion of similarity is the ratio of the linear velocity of an atmosphere at the equator to the sound velocity (rotational Mach number). This criterion is small for Venus(and Mercury), of order unity for the Earth and Mars, and large (about 15) for Jupiter and Saturn. If one neglects rotation, then it is possible to derive an expression for the total kinetic energy of an atmosphere where a nondimensional constant enters, which even for the Earth and Mars is of order unity. The lifetime of the circulation τ u is introduced using radius, mean velocity, and a multiplier β which is a measure of the degree of organization of atmospheric motions. The value of β is of the order of unity for slowly rotating planets, but it is very large for the giant rapidly rotating planets. Knowledge of π u allows us to estimate the mean rate of dissipation of the kinetic energy and the efficiency of the atmospheric heat engine. Considering the last quantity to be proportional to δT, the circulation driving temperature difference, one may estimate δT as well. Possible modifications of the theory are proposed for rarified atmospheres, and for rapidly rotating planets. This approach gives estimates which agree well with the observational data for the Earth's atmosphere and with the results of numerical experiments for the Martian atmosphere. For Venus, averaged throughout the atmosphere, the velocities of circulation are estimated to be of the order of 1m/sec and δT ⋍ 2° K . From the kinetic energy of visual motions of Jupiter and Saturn, which can be estimated from observations, it is found that β should be very large. This should lead to a very large lifetime of the overall circulation and of major atmospheric hydrodynamic formations. In this connection a suggestion is proposed that the Great Red Spot might be a long-lived eddy. Attention is paid to the fact that the energy of the visual motions on Saturn is greater by an order of magnitude than on Jupiter, while the similarity criteria are almost equal.

Subsonic Compressibility Corrections for Propellers and Helicopter Rotors

The blade element theory is applied to subsonic flow by introducing the linearized compressibility correction, while the firstorder compressibility effect on the axial inflow is obtained from the simple momentum theory. The subsonic compressibility correction for the lightly loaded propeller is found to be expressed in terms of the forward flight Mach Number and the Mach Number based on the rotational speed of the propeller t ip. These relations are di rec t^ applicable to a helicopter rotor in vertical flight and reduce to a simpler form depending only upon the rotational Mach Number at the rotor tip. I t is shown that for the helicopter rotor a satisfactory approximation, for tip Mach Numbers less than 0.8, may be obtained by using the simple Prandtl-Glauert correction with the Mach Number given by the rotational speed at the three-quarter radius.