Low Biot Number Research Articles

Dryout stability and inception at low flow rates

A theoretical model for the heat flux required to maintain a stable hot patch on a heated surface cooled by a falling liquid film is developed. The theoretical model is based on an existing analytical solution of the two-dimensional heat equation with boundary conditions supplied by heat transfer coefficient correlations appropriate to the fluid conditions at the hot dry patch and at the wet region from the patch. Using physically reasonable correlation forms for the dry patch and wet region heat transfer coefficients, the heat flux is expressed as a function of the liquid film Reynolds number. Existing experimental data were compiled, collected, and used to obtain the correlating lines. The data cover the one-dimensional (low Biot number) region, and indicate that laminar and turbulent liquid film conditions occur. Corresponding correlating equations are obtained from the analytical results. For laminar flow, the dryout heat flux is q d = 0.27 Re 0.5, and for turbulent flow q d = 0.017 Re 0.9, with an uncertainty of the order of ± 50%.

Modeling of multidimensional solidification of an alloy

The solidification of an alloy and pure metal is distinct because of the morphological differences between the respective interfaces. For certain conditions of imposed heat extraction, this dis-tinction can lead to large differences in the times required for complete solidification of the alloy over the pure metal. These conditions are examined in this paper. To do this, a powerful numerical technique to model alloy solidification is introduced which allows for the precise integration of the Scheil equation. The model takes into account the nonlinearity of the fraction liquid with temperature as well as the interface nonlinearity in the heat flow. As a test for the model, accurate temperature profiles from a previously published experiment are numerically simulated. The numerical results are noted to closely match experimental values, and as a con-sequence, contact heat transfer values are tabulated. One- and two-dimensional solidification of aluminum-copper alloys are now simulated for different cooling conditions (i.e., varying Biot numbers). The results obtained indicate the essential differences for alloy solidification at low and high Biot numbers and highlight the importance of properly accounting for the mushy zone.

Effect of supercooling on freezing time due to dendritic growth of ice crystals

A finite-difference procedure was used to calculate the freezing time of a slab when supercooling occurs. Crystallization is assumed to take place at a definite nucleation temperature, followed by dendritic growth of crystals. Resistance to mass transfer is neglected. The effects of various parameters were examined. It was shown that the effect of supercooling on freezing time may be significant at combinations of low Biot number, low nucleation temperature and high freezing-medium temperature in situations where ice forms as dendrites.

Drying kinetics of Stuart oil shale

The drying kinetics of Stuart oil shale were investigated over the temperature range 100–250 °C in both fluidized bed and convection oven heating. Experimental drying rates of cylindrical samples were determined gravimetrically and from the moisture content of the gases leaving the dryer. The observed kinetics were described to 90% completion by a diffusion-controlled progression of a sharp drying front through the sample for decoupled heat and mass transfer. For both the convection oven (low Biot number, Bi) and fluidized bed drying (high Bi), the drying kinetics over the whole temperaure range were modelled using an estimated apparent diffusivity of 2.1 × 10 − 7 m 2 s − 1 (standard deviation = 28 %) for steam in the shale matrix.

Predicting Temperature vs Time Behavior During the Freezing and Thawing of Rectangular Foods

Based on the effective heat capacity characteristics of biological materials during freezing we have derived a short-cut equation for predicting their temperature vs time behavior during freezing and thawing. The equation can be used in the low Biot number range for rectangular bodies with similar or dissimilar heat-transfer coefficients at their different surfaces. It has been experimentally and numerically tested for rectangular blocks of chopped beef with two or four exposed sides for Biot numbers ranging between 0 and 3. In this range it predicts temperature vs time behavior with errors of less than 5 to 10% in terms of overall processing time. By using numerical methods, the approach used can be extended to handle problems involving time-varying boundary conditions and many types of non-linear, physical-property behavior.

Transient heat conduction at low Biot numbers: A supplement to Heisler's charts

Charts for trasient heat conduction in a cylinder of infinite length, an infinite flat plate, and a sphere are presented for a range of Biot numbers ( 1 100 to 1 1000 ) relevant to problems which involve cooling by natural convection. These charts may be easily combined to solve for temperature histories of different bodies, including cylinders of finite length. Analyses of the minute temperature gradient within finite cylinders for the range of Biot numbers studied and the accuracy of a simplified solution, the lumped capacity analysis, are presented in tables for the infinite plate and infinite cylinder; these solutions can be combined to solve for cylinders of finite length.

Laminar Heat Transfer in an Externally Finned Circular Tube

An analysis has been carried out for laminar heat transfer in a circular tube subjected to an axially periodic variation of the external heat transfer coefficient (or of its dimensionless counterpart, the Biot number). This thermal boundary condition serves to model flow in a tube fitted with a regularly spaced array of annular fins. It was found from numerical solutions that external finning yields substantial heat transfer enhancement compared with an unfinned tube, especially when the Biot number of the unfinned tube is low. The degree of enhancement is quite insensitive to the period of the Biot number variation provided that the overall proportions of the unfinned and finned surface are maintained and the Biot number level is also maintained. An increase in the interfin spacing tends to diminish the enhancement. A constant-Biot-number model intended to approximate the heat transfer results for the periodic-Biot-number case was devised and tested. The predictions of this simplified model are especially accurate at low Biot numbers. The results of the laminar flow analysis have suggested ways of modeling turbulent heat transfer in externally finned tubes.

Power required to form a floating zone and the zone shape

The relation between the power required to produce a molten zone and the zone shape is investigated by using a simplified model. The Laplace equation which governs the temperatures both in the rod and in the zone is computationally solved. The temperature distribution is characterized by five dimensionless numbers: the Biot number, the conductivity ratio between liquid and solid, the ratios of the rod length and the width of the heated region to the crystal radius, and the non-dimensionalized power. The temperature distributions and zone shapes for various Biot numbers and conductivity ratios are obtained as functions of the power supplied to the zone. For low Biot number, the power supplied to the zone is mainly dissipated by conduction through the interfaces, while for high Biot number, it is mainly dissipated by radiation from the zone surface. The comparison between the one-dimensional model and the present indicates that the former is no longer valid for high Biot number.

Heat transfer to a sphere immersed in a shallow fluidized bed

By recording the transient response of spherical, calorimetric probes, of low Biot number, when plunged into shallow, heated, air-fluidized beds, heat transfer coefficients have been measured at bed temperatures of up to 1100°C. Bed heating was by means of premixed gas combustion, but all measurements were carried out in its absence. Particles of silica sand, zircon sand and silicon carbide, ranging in size from 200 μm to 800 μm, were investigated. A correlation, fitting some 400 data points to within ±10%, was obtained as Nu max = 0.365 θ 0.82 Ar 0.22 where θ = absolute bed temperature in o K/273 .