Heat multi-dimensional

Equation (9.62) contains seven dimensionless parameters accounting for the hydrodynamic and thermal effects. The regimes in which stable flows in a heated capillary are possible correspond to the following interval of the length Zp 0 < Zp < 1. The latter allows us to use Eq. (9.62) to define the domains of the existence of stable and unstable flow regimes. In the multi-dimensional parametric space (Z, i9p Pep Ja Ts ki. c < pL.g) the limiting values of these parameters corre-... 

Sparrow, E. M., and M. A. Ansari A Refutation of King s Rule for Multi-Dimensional External Natural Convection, Int. J. Heat Mass Transfer, vol. 26, p. 1357, 1983. 

In this paper, we will treat the nonlinear couplings, which are linear in the system coordinate but quadratic in the external heat bath coordinates, and show that the GHT is not equivalent to the multi-dimensional TST for the whole solution system if the potential function contains the nonlinear couplings. In Sec. II, we introduce the microscopic Hamiltonian (IIA), evaluate the GH rate expression (IIB), and, in Sec. IIC, we compare it with the multidimensional TST rate for the whole solution system. Finally, the main points are summarized in Sec. III. 

Cooling and heating in multi-dimensional heat flow... 

In geometric multi-dimensional heat flow the temperature fields that have to be calculated depend on two or three spatial coordinates and must satisfy the heat conduction equation... 

The heat released in multi-dimensional heat conduction can also be determined from of the product solutions. According to section 2.3.4... 

Operation. After activation, the sample application and developing procedures are carried out almost exactly as in PC. Normally, sample sizes range from 10-100 pg per spot for analytical TLC, but in preparative TLC when samples are applied as bands, up to 100 mg can be used with a 20 x 20-cm plate. Spots should be 2-5 mm in diameter. Ascending or descending development and one- or multi-dimensional techniques can be used. Because of differences in capillary action and possibly in solvent heat of adsorption, development times are usually faster in TLC than in PC. Depending on the mobile phase and the particle size of the adsorbent, a typical time... 

A wide variety of solvents can be used to extract pitch fractions that will form mesophase upon melting. The properties of the mesophase depend on the pitch, the solvent, the Insoluble fraction, and the time and temperature of the heat-treatment. A multi-dimensional... 

In 1999, Blackmond and Pfalts from the Max-Planck Institute introduced a rapid screening protocol based upon reaction calorimetry. In the absence of significant side-reactions, heat-flow monitoring can provide a rapid measurement of activity and a multi-dimensional kinetic profile of the catalyst candidate for the reaction of interest. 

The generation of current induces fluxes of gases, liquid water, heat and charged particles in a cell. The distribution of the respective parameters (concentrations, fields etc.) is usually very non-uniform. Furthermore, the characteristic scale of parameters variation ranges from several micrometres (the thickness of the catalyst layer) to several metres (the length of the channel). In general, the problem of fuel cell modeling is multi-scale and multi-dimensional. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

Kunz RF, Siebert BW, Cope WK, Foster NF, Antal SP, Ettorre SM (1998) A coupled phasic exchange algorithm for three-dimensional multi-field analysis of heated flows with mass transfer. Computers Fluids 27(7) 741-768... 

The numerical model we used was originally developed by Gill and Clyne [6] and has been modified to handle multi-layered deposits. It is a 1-dimensional model and consists of two parts thermal profile calculation and stress calculation. By regarding the torch motion as a fluctuation in the heat and mass flux onto a reference point on the substrate and assuming biaxial stress state, the program calculates both the through-thickness thermal profile and stress distribution during thermal spray as functions of time. 

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