The buoyant plume above a heat source

The boundary-layer type conservation equations of mass, momentum and energy for the steady free turbulent flow in gravitational convection over heat sources are set up for both two-dimensional and axisymmetric cases. These are reduced to ordinary differential equations in a similarity parameter by suitable transformations. The three classical hypotheses of turbulent diffusion-the Constant Exchange Coefficient hypothesis, Prandtl's Momentum Transfer theory and Taylor's Vorticity Transfer theory-are then incorporated into these equations in succession. The resulting equations are solved numerically and the results compared with some experimental results on gravitational convection over heat sources reported by Rouse et al.

Effect of impulsive change of wall velocity and wall temperature in viscous heat-conducting gas

A study of compression waves produced in a viscous heat-conducting gas by the impulsive start of a one-dimensional piston and by the inpulsive change of piston wall temperature is made using Laplace Transform Technique for Prandt1 number unity. Expressions for velocity, temperature and density have also been obtained using small-time expansion procedure in this case. For arbitrary Prandt1 number solutions have been developed using large-time expansion procedure. A number of graphs exhibiting the distribution of the fluid velocity, temperature and density have been drawn.

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Free convection heat transfer in vertical annuli

Free convection heat transfer in vertical concentric, cylindrical annuli is investigated analytically and experimentally. The approximate double boundary layer model used by Emery and Chu for the case of vertical parallel plates is extended to the present case in obtaining heat transfer correlations in laminar free convection. Different correlations for the inner cylinder depending on the radius to the length ratio of the inner cylinder and the Rayleigh number, were used in the derivation of correlations for the annuli. The results for the case of short cylinders inside tubes are in agreement (within about 10 per cent) with the existing correlations. For other cases, namely long cylinders in annuli and wires in annuli, experiments conducted show the agreement of the analysis with experiments.

Steady laminar flow with suction or injection and heat transfer through porous pipe

Following the method due to Bhatnagar (P. L.) [Jour. Ind. Inst. Sic., 1968, 1, 50, 1], we have discussed in this paper the problem of suction and injection and that of heat transfer for a viscous, incompressible fluid through a porous pipe of uniform circular cross-section, the wall of the pipe being maintained at constant temperature. The method utilises some important properties of differential equations and some transformations that enable the solution of the two-point boundary value and eigenvalue problems without using trial and error method. In fact, each integration provides us with a solution for a suction parameter and a Reynolds number without imposing the conditions of smallness on them. Investigations on non-Newtonian fluids and on other bounding geometries will be published elsewhere.

Free convection heat transfer between vertical parallel plates

A new mathematical model for the solution of the problem of free convection heat transfer between vertical parallel flat isothermal plates under isothermal boundary conditions, has been presented. The set of boundary layer equations used in the model are transformed to nonlinear coupled differential equations by similarity type variables as obtained by Ostrach for vertical flat plates in an infinite fluid medium. By utilising a parameter ηw* to represent the outer boundary, the governing differential equations are solved numerically for parametric values of Pr = 0.733. 2 and 3, and ηw* = 0.1, 0.5, 1, 2, 3, 4, ... and 8.0. The velocity and temperature profiles are presented. Results indicate that ηw* can effectively classify the system into (1) thin layers where conduction predominates, (2) intermediate layers and (3) thick layers whose results can be predicted by the solutions for vertical flat plates in infinite fluid medium. Heat transfer correlations are presented for the 3 categories. Several experimental and analytical results available in the literature agree with the present correlations.

Heat transfer in rarefied couette flow on the basis of discrete ordinate method

The method of discrete ordinates, in conjunction with the modified "half-range" quadrature, is applied to the study of heat transfer in rarefied gas flows. Analytic expressions for the reduced distribution function, the macroscopic temperature profile and the heat flux are obtained in the general n-th approximation. The results for temperature profile and heat flux are in sufficiently good accord both with the results of the previous investigators and with the experimental data.

Correlation between terrestrial heat flow and thermal conductivity

The significant correlation coefficient between the terrestial heat flow and thermal conductivity computed from the continental heat flow data by Horai and Nur [1]2) may be explained as a natural consequence of terrestrial heat flow through a random medium. The theory predicts a value of 0.40 for the correlation coefficient. A simple statistical test shows that the majority of the computed coefficients belong to the statistical population whose mean is equal to the theoretical correlation coefficient. There are, however, a few observations of unsually high correlation coefficient which cannot be explained by the above hypothesis.

Free convection heat transfer from vertical cylinders part I: Power law surface temperature variation

This paper reports on the investigations of laminar free convection heat transfer from vertical cylinders and wires whose surface temperature varies along the height according to the relation TW - T∞ = Nxn. The set of boundary layer partial differential equations and the boundary conditions are transformed to a more amenable form and solved by the process of successive substitution. Numerical solutions of the first approximated equations (two-point nonlinear boundary value type of ordinary differential equations) bring about the major contribution to the problem (about 95%), as seen from the solutions of higher approximations. The results reduce to those for the isothermal case when n=0. Criteria for classifying the cylinders into three broad categories, viz., short cylinders, long cylinders and wires, have been developed. For all values of n the same criteria hold. Heat transfer correlations obtained for short cylinders (which coincide with those of flat plates) are checked with those available in the literature. Heat transfer and fluid flow correlations are developed for all the regimes.

Free convection heat transfer from vertical cylinders part II: Exponential surface temperature variation

Investigation on laminar free convection heat transfer from vertical cylinders and wires having a surface temperature variation of the form TW - T∞ = M emx are presented. As in Part I for power law surface temperature variation, the axisymmetric boundary layer equations of mass, momentum and energy are transformed to more convenient forms and solved numerically. The second approximation refines the results of the first upto a maximum of only 2%. Analysis of the results indicates that cylinders can be classified into the same three categories as in Part I, namely, short cylinders, long cylinders, and wires, heat transfer and fluid flow correlations being developed for each case.

Effect of vibration on heat transfer from a horizontal cylinder to a normal air stream

The effect of vibration on heat transfer from a horizontal copper cylinder, 0.344 in. in diameter and 6 in. long, was investigated. The cylinder was placed normal to an air stream and was sinusoidally vibrated in a direction perpendicular to the direction of the air stream. The flow velocity varied from 19 ft/s to 92 ft/s; the double amplitude of vibration from 0.75 cm to 3.2 cm, and the frequency of vibration from 200 to 2800 cycles/min. A transient technique was used to determine the heat transfer coefficients. The experimental data in the absence of vibration is expressed by NNu = 0.226 NRe0.6 in the range 2500 < NRe < 15 000. By imposing vibrational velocities as high as 20 per cent of the flow velocity, no appreciable change in the heat transfer coefficient was observed. An analysis using the resultant of the vibration and the flow velocity explains the observed phenomenon.

Non-linear couette flow with heat transfer using the B-G-K model

In recent years a large number of investigators have devoted their efforts to the study of flow and heat transfer in rarefied gases, using the BGK [1] model or the Boltzmann kinetic equation. The velocity moment method which is based on an expansion of the distribution function as a series of orthogonal polynomials in velocity space, has been applied to the linearized problem of shear flow and heat transfer by Mott-Smith [2] and Wang Chang and Uhlenbeck [3]. Gross, Jackson and Ziering [4] have improved greatly upon this technique by expressing the distribution function in terms of half-range functions and it is this feature which leads to the rapid convergence of the method. The full-range moments method [4] has been modified by Bhatnagar [5] and then applied to plane Couette flow using the B-G-K model. Bhatnagar and Srivastava [6] have also studied the heat transfer in plane Couette flow using the linearized B-G-K equation. On the other hand, the half-range moments method has been applied by Gross and Ziering [7] to heat transfer between parallel plates using Boltzmann equation for hard sphere molecules and by Ziering [83 to shear and heat flow using Maxwell molecular model. Along different lines, a moment method has been applied by Lees and Liu [9] to heat transfer in Couette flow using Maxwell's transfer equation rather than the Boltzmann equation for distribution function. An iteration method has been developed by Willis [10] to apply it to non-linear heat transfer problems using the B-G-K model, with the zeroth iteration being taken as the solution of the collisionless kinetic equation. Krook [11] has also used the moment method to formulate the equivalent continuum equations and has pointed out that if the effects of molecular collisions are described by the B-G-K model, exact numerical solutions of many rarefied gas-dynamic problems can be obtained. Recently, these numerical solutions have been obtained by Anderson [12] for the non-linear heat transfer in Couette flow,

Estimation of the critical viscous sublayer in heat transfer problems

Imagining a disturbance made on a compressible boundary layer with the help of a heat source, the critical viscous sublayer, through which the skin friction at any point on a surface is connected with the heat transferred from a heated element embedded in it, has been estimated. Under similar conditions of external flow (Ray1)) the ratio of the critical viscous sublayer to the undisturbed boundary layer thickness is about one-tenth in the laminar case and one hundredth in the turbulent case. These results are similar to those (cf.1)) found in shock wave boundary layer interaction problems.

Lattice vibrations and specific heat of zinc blende

The dispersion relations, frequency distribution function and specific heat of zinc blende have been calculated using Houston's method on (1) A short range force (S. R.) model of the type employed in diamond by Smith and (2) A long range model assuming an effective charge Ze on the ions. Since the elastic constant data on ZnS are not in agreement with one another the following values were used in these calculations: {Mathematical expression}. As compared to the results on the S. R. model, the Coulomb force causes 1. A splitting of the optical branches at (000) and a larger dispersion of these branches; 2. A rise in the acoustic frequency branches the effect being predominant in a transverse acoustic branch along [110]; 3. A bridging of the gap of forbidden frequencies in the S. R. model; 4. A reduction of the moments of the frequency distribution function and 5. A flattening of the Θ- T curve. By plotting (Θ/Θ0) vs. T., the experimental data of Martin and Clusius and Harteck are found to be in perfect coincidence with the curve for the short range model. The values of the elastic constants deduced from the ratio Θ0 (Theor)/Θ0 (Expt) agree with those of Prince and Wooster. This is surprising as several lines of evidence indicate that the bond in zinc blende is partly covalent and partly ionic. The conclusion is inescapable that the effective charge in ZnS is a function of the wave vector {Mathematical expression}.