Global Momentum Balance Research Articles

Experimental investigations of hydrodynamic characteristics of a hybrid fluidized bed airlift reactor with external liquid circulation

The paper presents a simple analytical model of the hydrodynamics of a hybrid airlift apparatus with an external liquid circulation loop. The apparatus consists of two sections: a two-phase fluidization column and a barbotage section. The advantage of such a configuration is that there is no contact between the gas phase and the solid phase, which is relevant in case of processes involving a biofilm immobilization on fine carrier particles. Then, the shear stresses produced by passing bubbles do not damage the surface of a biofilm. The proposed model was derived based on the global momentum balance. It allows to determine basic hydrodynamic parameters of the apparatus including liquid and gas velocity, gas hold-up, and porosity and height of the fluidized bed. The model was verified experimentally and the hydrodynamics of the selected zones of the apparatus was simulated using Computational Fluid Dynamics (CFD).

A simple hydrodynamic model of a laminar free-surface jet in horizontal or vertical flight

A useable model for laminar free-surface jet evolution during flight, for both horizontal and vertical jets, is developed through joint analytical, experimental, and simulation methods. The jet’s impingement centerline velocity, recently shown to dictate stagnation zone heat transfer, encompasses the entire flow history: from pipe-flow velocity profile development to profile relaxation and jet contraction during flight. While pipe-flow is well-known, an alternative analytic solution is presented for the centerline velocity’s viscous-driven decay. Jet-contraction is subject to influences of surface tension (We), pipe-flow profile development, in-flight viscous dissipation (Re), and gravity (Nj = Re/Fr). The effects of surface tension and emergence momentum flux (jet thrust) are incorporated analytically through a global momentum balance. Though emergence momentum is related to pipe flow development, and empirically linked to nominal pipe flow-length, it can be modified to incorporate low-Re downstream dissipation as well. Jet contraction’s gravity dependence is extended beyond existing uniform-velocity theory to cases of partially and fully developed profiles. The final jet-evolution model relies on three empirical parameters and compares well to present and previous experiments and simulations. Hence, micro-jet flight experiments were conducted to fill-in gaps in the literature: jet contraction under mild gravity-effects, and intermediate Reynolds and Weber numbers (Nj = 5–8, Re = 350–520, We = 2.8–6.2). Furthermore, two-phase direct numerical simulations provided insight beyond the experimental range: Re = 200–1800, short pipes (Z = L/d · Re ≥ 0.01), variable nozzle wettability, and cases of no surface tension and/or gravity.

An advanced switching moving boundary heat exchanger model with pressure drop

This paper presents an advanced heat exchanger model based on the moving boundary approach. Significant improvements have been made to overcome the deficiencies of the extant models. Air flow propagation is taken into account to provide a more accurate prediction of air side heat transfer for multi-row coils. Refrigerant pressure drop is calculated in a reasonably simple manner by solving the global momentum balance equation. The choice of state variables shows the benefits of mass conservation and good computational efficiency. Generalized switching schemes capable of supporting dynamic transition between all possible flow configurations are developed. Model integrity and stability are verified through simulations. The model is applied to explore the start-up transients of an R410a flash tank vapor injection system. Favorable agreement between simulation results and experimental data demonstrates that the proposed model can adequately capture the main transient heat transfer and fluid flow phenomena of the system.

Planar channel flow in Braginskii magnetohydrodynamics

Braginskii magnetohydrodynamics (MHD) is a single-fluid description of large-scale motions in strongly magnetised plasmas. The ion Larmor radius in these plasmas is much shorter than the mean free path between collisions, so momentum transport across magnetic field lines is strongly suppressed. The relation between the strain rate and the viscous stress becomes highly anisotropic, with the viscous stress being predominantly aligned parallel to the magnetic field. We present an analytical study of the steady planar flow across an imposed uniform magnetic field driven by a uniform pressure gradient along a straight channel, the configuration known as Hartmann flow, in Braginskii MHD. The global momentum balance cannot be satisfied by just the parallel viscous stress, so we include the viscous stress perpendicular to magnetic field lines as well. The ratio of perpendicular to parallel viscosities is the key small parameter in our analysis. When another parameter, the Hartmann number, is large the flow is uniform across most of the channel, with boundary layers on either wall that are modifications of the Hartmann layers in standard isotropic MHD. However, the Hartmann layer solution predicts an infinite current and infinite shear at the wall, consistent with a local series solution of the underlying differential equation that is valid for all Hartmann numbers. These singularities are resolved by inner boundary layers whose width scales as the three-quarters power of the viscosity ratio, while the maximum velocity scales as the inverse one-quarter power of the viscosity ratio. The inner wall layers fit between the Hartmann layers, if present, and the walls. The solution thus does not approach a limit as the viscosity ratio tends to zero. Essential features of the solution, such as the maximum current and maximum velocity, are determined by the size of the viscosity ratio, which is the regularising small parameter.

Thermoacoustic streaming in a tube with isothermal outer surface

Thermoacoustic oscillations at a cycle-steady state in a tube with an isothermal outer wall, and with one end closed and the other end connected to a wave generator, is analyzed based on a linearized theory. From the global mass conservation, an analytical solution has been obtained for the cross-sectional and cycle averaged axial velocity. It is shown that this averaged velocity is non-vanishing due to the mass streaming effect. By analyzing the global momentum balance, it is found that the cycle-averaged pressure depends on the momentum streaming and friction force, and a conservation relationship exists between the momentum streaming and the cycle-averaged pressure for the flow oscillating at a high frequency in a wide tube. An investigation of the global energy balance leads to an expression for thermoacoustic energy streaming. Furthermore, it is shown that the refrigeration effect is mainly caused by the non-vanishing mean velocity, and therefore the mass streaming and the energy streaming are intimately connected.

The retraction of the edge of a planar liquid sheet

Our objective is to investigate the motion of a free two-dimensional edge of a liquid sheet, which recedes and accumulates fluid as it is pulled back toward the bulk of the sheet due to surface tension. In the long time limit, the velocity of this edge reaches a constant value given by Taylor [Proc. R. Soc. London, Ser. A 253, 313 (1959)] and Culick [J. Appl. Phys. 31, 1128 (1960)], independent of the fluid viscosity. The way this value is reached, however, depends on the viscosity. In order to follow quantitatively the acceleration process, time-dependent numerical simulations of the two-dimensional Navier–Stokes equations have been performed. For inertia-dominated flow situations, the results show that the motion of the rim is relatively well characterized by a global momentum balance. However, if viscous effects are dominant, the rim is shown to extend far out into the film. In this case, the decrease in film thickness toward the undisturbed film can be described quite precisely by another one-dimensional model. By comparison of the individual contributions to the energy balances, following from numerical computation and from the global momentum balance, a closer look has also been taken on the inner mechanism of the rim formation. Finally, on the basis of the performed simulations, a stability criterion for a moving liquid curtain of finite length is derived, which determines whether a newly formed free rim leads to a break-up of the curtain.

A PV View of the Zonal Mean Distribution of Temperature and Wind in the Extratropical Troposphere

The dependence of the temperature and wind distribution of the zonal mean flow in the extratropical troposphere on the gradient of pontential vorticity along isentropes is examined. The extratropics here refer to the region outside the Hadley circulation. Of particular interest is whether the distribution of temperature and wind corresponding to a constant potential vorticity (PV) along isentropes resembles the observed, and the implications of PV homogenization along isentropes for the role of the tropics. With the assumption that PV is homogenized along isentropes, it is found that the temperature distribution in the extratropical troposphere may be determined by a linear, first-order partial differential equation. When the observed surface temperature distribution and tropical lapse rate are used as the boundary conditions, the solution of the equation is close to the observed temperature distribution except in the upper troposphere adjacent to the Hadley circulation, where the troposphere with no PV gradient is considerably colder. Consequently, the jet is also stronger. It is also found that the meridional distribution of the balanced zonal wind is very sensitive to the meridional distribution of the tropopause temperature. The result may suggest that the requirement of the global momentum balance has no practical role in determining the extratropical temperature distribution. The authors further investigated the sensitivity of the extratropical troposphere with constant PV along isentropes to changes in conditions at the tropical boundary (the edge of the Hadley circulation). It is found that the temperature and wind distributions in the extratropical troposphere are sensitive to the vertical distribution of PV at the tropical boundary. With a surface distribution of temperature that decreases linearly with latitude, the jet maximum occurs at the tropical boundary and moves with it. The overall pattern of wind distribution is not sensitive to the change of the position of the tropical boundary. Finally, the temperature and wind distributions of an extratropical troposphere with a finite PV gradient are calculated. It is found that the larger the isentropic PV gradient, the warmer the troposphere and the weaker the jet.

Pressure Drop Due to Inertia Forces in Step Bearings

A study of the pressure distribution in a step bearing is presented, including inertia effects, by assuming that (1) a simplified pressure differential equation, (2) a global momentum balance or (3) an energy equation can be written in the vicinity of the step. All three methods give comparable results for small and moderate film thickness ratios, while satisfactory agreement with existing numerical data and some experimental data may be emphasized. It is pointed out that under certain circumstances, such as strong reverse flow occurring for larger film thickness ratios, the flow is decelerated instead of being accelerated across the step. As a consequence a pressure rise instead of a pressure drop may occur due to inertia, reaching a limiting value equal to half of the dynamic head corresponding to the sliding velocity (when the minimum film thickness in the land area becomes very small).