R.H. Sabersky

Heat and momentum transfer in smooth and rough tubes at various prandtl numbers

Results are presented from an experimental investigation of the relation between heat transfer and friction in smooth and rough tubes. Three rough tubes and one smooth tube were formed from electroplated nickel. The rough tubes contained a close-packed, granular type of surface with roughness-height-to-diameter ratios ranging from 0.0024 to 0.049. Measurements of the heat-transfer coefficients (CH) and the friction coefficients (CF) were obtained with distilled water flowing through electrically heated tubes. A Prandtl number range of 1.20–5.94 was investigated by adjusting the bulk temperature of the water. Results were obtained for Reynolds numbers from 6 × 104 to 5 × 106 and from 1.4 × 104 to 1.2 × 105 at the lowest and highest Prandtl numbers respectively. A similarity rule for heat transfer was used to correlate, interpret, and extend the experimental results. The results were compared with previously existing results, both theoretical and experimental. Increases in CH due to roughness of as high as 270 per cent were obtained. These increases were, in general, accompanied by even larger increases in CF. An exception to this general behavior occurs at high Prandtl number in the region of transition between the “smooth” and the “fully rough” CF characteristic.

Heat transfer to flowing granular media

The convective heat transfer resulting from a granular flow over a heated surface is investigated. The specific type of flow considered is that in which adjacent material particles are in physical contact. The qualitative features of this type of flow are discussed, and the existing equations of motion are extended. With regard to the equations of motion, an exact solution is shown which has applications concerning the mass flow rate of granular materials through hoppers. The particular heat transfer problem investigated is convection from a flat plate with its long axis parallel to the flow field. An approximate analytical solution, which takes into account the particulate nature of the medium, is developed and experimental measurements obtained. The theory was found to correctly predict the trends of the experimental data. The results indicate that the Nusselt number for this configuration is influenced substantially, under certain conditions, by the noncontinuous nature of the medium. A semi-empirical correlation is presented, based on experimental results obtained with four different granular materials.

Measurements of Velocity, Velocity Fluctuation, Density, and Stresses in Chute Flows of Granular Materials

Experiments on continuous, steady flows of granular materials down an inclined channel or chute have been conducted with the objectives of understanding the characteristics of chute flows and of acquiring information on the rheological behavior of granular material flow. Two neighboring fiber-optic displacement probes provide a means to measure (1) the mead velocity by cross-correlating two signals from the probes, (2) the unsteady or random component of the particle velocity in the longitudinal direction by a procedure of identifying particles, and (3) the mean particle spacing at the boundaries by counting the frequency of passage of the particles. In addition, a strain-gauged plate built into the chute base has been employed to make direct measurement of shear stress at the base. With the help of these instruments, the vertical profiles of mean velocity, velocity fluctuation, and linear concentration were obtained at the sidewalls. Measurements of some basic flow properties such as solid fraction, velocity, shear rate, and velocity fluctuation were analyzed to understand the characteristics of the chute flow. Finally, the rheological behavior of granular materials was studied with the experimental data. In particular, the rheological models of Lun et al. (1984) for general flow and fully developed flow were compared withthe present data.

Heat transfer in an annular gap

Abstract Heat-transfer coefficients were measured for the flow in the annular space between an inner rotating cylinder and an outer stationary one, with superimposed axial flow. The problem was stimulated by the desire for additional information on the design of cooling systems for electric motors of high power density. The speeds of rotation were such as to include Taylor numbers up to about 106, and the range of Reynolds numbers based on the axial velocity components and the gap distance were extended to 7000. Experiments were performed at three different Prandtl numbers 2.5, 4.5 and 6.5.

Gravity Flow of Granular Materials in Conical Hoppers

An approximate solution to the flow of a cohesionless granular material in a conical hopper is presented. The material is modeled as a perfectly plastic continuum which satisfies the Mohr-Coulomb yield condition. Analytical expressions of the mass flow rate and the wall stress are derived and compared to some experimental data and other analyses.

Funnel Flow in Hoppers

Detailed observations of funnel flows of dry granular materials in wedge-shaped hoppers of different geometries are presented. The variations of the flow regime with changes in the height of material in the hopper/vertical bin configuration, the width of the vertical bin, the hopper angle and the hopper opening width were investigated and a number of specific flow regimes identified (mass flow and several forms of funnel flow). In the first part of the paper particular attention is paid to the conditions for transition from one flow regime to another; in particular it is shown that the existence of a funnel depends not only on the hopper angle but is also strongly dependent on the geometry of the hopper/bin system. In the second part of the paper the variations in the shape of the funnel near the exit opening are explored in detail.

Convective heat transfer to rapidly flowing, granular materials

Convective heat transfer to a rapidly flowing, granular material is experimentally investigated over a flat plate in an inclined chute. Two different sizes of glass beads are used as the granular materials. A technique was developed to measure the average density of the flowing material, which allowed the more accurate determination of the average velocity as well as of parameters depending on the velocity. The results are presented in terms of the Nusselt number and a model is proposed relating this Nusselt number to a Peclet number and a Froude number. The predicted results of the model are compared with the experimental data.

Shear Flows of Rapidly Flowing Granular Materials

Shear flows of granular materials are studied in an open channel. The wall shear is calculated from an open channel momentum equation which includes the density variations in the flow. An experimental technique was developed that allowed the measurement of the average density of the flow at different longitudinal locations in the channel. Two sizes of glass beads are examined and results show the variations in the wall shear as a function of various dimensionless parameters.

Flow regimes in inclined open-channel flows of granular materials

Open-channel flows of fluids may be classified as supercritical or subcritical, depending on whether their average velocity ῡ is larger, equal to or smaller than the propagation rate of small disturbances √(gh) cos α (where g is the gravitational acceleration, h is the flow depth and [alpha] is the channel inclination). Typically, the flow type is classified by the magnitude of the Froude number, ῡ√(gh), relative to its value under critical conditions Fr_c = √(cosα). Supercritical and subcritical flow represent conjugate states of open-channel flow; that is, a given supercritical flow will transition via an hydraulic jump that will transition a subcritical flow back to its corresponding supercritical value. flow. (However, energy considerations prohibit a reverse hydraulic jump that will transition a subcritical flow back to its corresponding supercritical value.) Supercritical flows are unaffected by downstream conditions, as they move faster than the downstream information can propagate upstream. Subcritical flows are strongly affected by downstream conditions. If downstream conditions are relaxed, a subcritical flow may transition back to supercritical flow (although not its conjugate state) via an expansion wave progagating upstream. The existence of a subcritical flow requires that the flow must be critical somewhere downstream before any abrupt expansion of the channel (such as the drop-off at the channels end). The critical state prevents an expansion wave from propagating upstream from the expansion and causing a transition to supercritical flow. (A more complete discussion of these phenomena may be found in most introductory fluid mechanics textbooks; see, for example, Ref. 1, pp. 363-377.)

Transient heat transfer in Bénard convection

Abstract Experimental results are presented for a study of the effects of time-dependent heating on Benard convection, where the fluids were 5cs (centistoke), 100cs and 500cs viscosity grades of silicone oil. Fluid layer depths were 0.00635 m, 0.01270 m and 0.01905 m. For each run the heat flux at the lower surface was approximately constant, and in the several runs a range of fluxes from 9.2 × 10 2 to 1.9 × 10 7 (in dimensionless terms) was covered. The experiments were designed to examine the effects of different heating rates on the onset of convection, on the change of the Rayleigh number with time and on the development of motion. Observations were made from shadowgraph images, which were recorded photographically. As the heat flux at the lower surface is increased, the temperature difference required for the initiation of convection increases while the time to the onset of motion decreases. For the conditions of the present tests a “closed cell” pattern is the first to be observed, shortly after the onset of motion. This pattern does not appear in the steady-state system. Because of the “large” (greater than about 100) Prandtl number, specifying the time and the lower surface heat flux is sufficient to characterize the state of the fluid layer.