Flow methods, heat

Example Buckingham Pi Method—Heat-Transfer Film Coefficient It is desired to determine a complete set of dimensionless groups with which to correlate experimental data on the film coefficient of heat transfer between the walls of a straight conduit with circular cross section and a fluid flowing in that conduit. The variables and the dimensional constant believed to be involved and their dimensions in the engineering system are given below ... 

Palen, J. W. and Taborek, J., Solution of Shell-Side Flow Pressure Drop and Heat Transfer by Stream Analysis Method (Heat Transfer Research, Inc., Alhambra, CA), AICHE Chemical Engineering Progress Symposium Series No. 92, Vol. 65 (1969), pp. 53-63. 

Computational fluid dynamics (CFD) is the numerical analysis of systems involving transport processes and solution by computer simulation. An early application of CFD (FLUENT) to predict flow within cooling crystallizers was made by Brown and Boysan (1987). Elementary equations that describe the conservation of mass, momentum and energy for fluid flow or heat transfer are solved for a number of sub regions of the flow field (Versteeg and Malalase-kera, 1995). Various commercial concerns provide ready-to-use CFD codes to perform this task and usually offer a choice of solution methods, model equations (for example turbulence models of turbulent flow) and visualization tools, as reviewed by Zauner (1999) below. 

This partial differential equation is most conveniently solved by the use of the Laplace transform of temperature with respect to time. As an illustration of the method of solution, the problem of the unidirectional flow of heat in a continuous medium will be considered. The basic differential equation for the X-direction is ... 

Numerical methods have been developed by replacing the differential equation by a finite difference equation. Thus in a problem of unidirectional flow of heat ... 

Prodanovic V, Fraser D, Salcudean M (2002) On transition from partial to fuUy developed subcooled flow boiling. Int J Heat Mass Transfer 45 4727-4738 Qu W, Mudawar I (2003a) Measurement and prediction of pressure drop in two-phase micro-channel heat sinks. Int J Heat Mass Transfer 46 2737-2753 Qu W, Mudawar I (2003b) Flow boiling heat transfer in two-phase micro-channel heat sink. 1 Experimental investigation and assessment of correlation methods. Int J Heat Mass Transfer 46 2755-2771... 

A number of authors have considered channel cross-sections other than rectangular [102-104]. Figure 2.17 shows some examples of cross-sections for which friction factors and Nusselt numbers were computed. In general, an analytical solution of the Navier-Stokes and the enthalpy equations in such channel geometries would be involved owing to the implementation of the wall boundary condition. For this reason, usually numerical methods are employed to study laminar flow and heat transfer in channels with arbitrary cross-sectional geometry. 

VAN Doormal, ). P., Raithby, G. D., Enhancement of the SIMPLE method for predicting incompressible fluid flows. Numerical Heat Transfer 7 (1984) 147-163. 

The, J. L, Raithby, G. D., Stubley, G. D., Surface-adaptive finite-volume method for solvingfree-sutface flows, Numer. Heat Transfer B26 (1994) 367-380. 

The complex flow pattern on the shell-side, and the great number of variables involved, make it difficult to predict the shell-side coefficient and pressure drop with complete assurance. In methods used for the design of exchangers prior to about 1960 no attempt was made to account for the leakage and bypass streams. Correlations were based on the total stream flow, and empirical methods were used to account for the performance of real exchangers compared with that for cross flow over ideal tube banks. Typical of these bulk-flow methods are those of Kern (1950) and Donohue (1955). Reliable predictions can only be achieved by comprehensive analysis of the contribution to heat transfer and pressure drop made by the individual streams shown in Figure 12.26. Tinker (1951, 1958) published the first detailed stream-analysis method for predicting shell-side heat-transfer coefficients and pressure drop, and the methods subsequently developed... 

Nitrogen adsorption was performed at -196 °C in a Micromeritics ASAP 2010 volumetric instrument. The samples were outgassed at 80 °C prior to the adsorption measurement until a 3.10 3 Torr static vacuum was reached. The surface area was calculated by the Brunauer-Emmett-Teller (BET) method. Micropore volume and external surface area were evaluated by the alpha-S method using a standard isotherm measured on Aerosil 200 fumed silica [8]. Powder X-ray diffraction (XRD) patterns of samples dried at 80 °C were collected at room temperature on a Broker AXS D-8 diffractometer with Cu Ka radiation. Thermogravimetric analysis was carried out in air flow with heating rate 10 °C min"1 up to 900 °C in a Netzsch TG 209 C thermal balance. SEM micrographs were recorded on a Hitachi S4500 microscope. 

The samples were shaped as strips 1 m long and 2 mm wide. The longitudinal steady heat flow method was used to measure the integrated thermal conductivity in the 30-150 mK range. 

An alternative method of solution to these analytical procedures, which is particularly useful in computer-assisted calculations, is the finite-difference technique. The Fourier equation describes the accumulation of heat in a thin slice of the heated solid, between the values xr and x + dx, resulting from the flow of heat through the solid. The accumulation of heat in the layer is the difference between the flux of energy into the layer at x = x Jx> and the flux out of the layer at x = x + dx, iX +dv. Therefore the accumulation of heat in the layer may be written as... 

When using the continuous flow method, however, some additional versatility is available in chemisorption measurements. For example, when data is required at an adsorbate pressure of 0.1 atm, a 10 % mixture of adsorbate, mixed with an inert carrier gas, is passed through the apparatus with the sample cooled to a temperature at which no chemisorption can occur. Upon warming the sample to the required temperature, adsorption occurs producing an adsorbate-deficient peak that is calibrated by injecting carrier gas into the flow stream. Equation (15.9) is then used to calculate the quantity adsorbed. This process is repeated for each concentration required. Caution must be exercised to avoid physical adsorption when the sample is cooled to prevent chemisorption. Should this occur, the adsorption peak due to chemisorption can be obscured by the desorption peak of physically bound adsorbate when the sample is heated. 

The test method for determining the fusibility of coal ash (ASTM D-1857) covers the observation of the temperatures at which triangular pyramids (cones) prepared from coal and coke ash attain and pass through certain defined stages of fusing and flow when heated at a specified rate in controlled, mildly reducing, and where desired, oxidizing atmospheres. 

The higher concentrations of the sample needed for the NMR method compared with other physical methods is a drawback, as also is the lower precision in the determination of rate constants. The latter is usually because the temperature of the sample in the NMR probe is controlled by a flow of heated or cooled nitrogen which does not normally provide highly accurate temperature control and measurement. Sometimes, the need for isotopically labelled substrates and solvents can be an additional drawback. 

O.A. Estrada, I.D. Lopez-Gomez, and T.A. Osswald. Modeling the non-newtonian calendering process using a coupled flow and heat transfer radial basis functions collocation method. Journal of Polymer Technology, 2005. 

E. Mitsoulis and J. Vlachopoulos, The Finite Element Method for Flow and Heat Transfer Analysis, Adv. Polym. Technol., 4, 107-121 (1984). 

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