Modeling gradients

In a one-dimensional model, gradients of cA and T at the bed level are allowed only in the axial direction of bulk flow. In a two-dimensional model, gradients at the bed level in both the axial and radial directions are taken into account. 

Wisel A, Schmidt-Traub H, Lenz J, Strube J. Modelling gradient elution of bioactive multicomponent systems in non-linear ion-excahnge chromatography. Journal of Chromatography A 2003 1006 101-120. 

Figure 9.2 Freshwater-marine mixing models for hydrogen, oxygen, sulfate, and dissolved inorganic carbon (DIC) and nitrogen (DIN) across salinity gradients. Modeled gradients for sulfate are based on riverine-marine ratios of 60 pm versus 28 pm (data from Kendall and Coplen, 2001). (Modified from Fry, 2002.)...

C Trondheim Bubble Column Model Gradient of a vector... 

When modeling gradient nonperturbed flow around a body, the boundary conditions at infinity (remote from the body) must be taken in the following form the fluid velocity components tend to the corresponding components of the above gradient flows as R - oo. 

Making statements about significance of effects implies being interested in considering the confidence bounds for the Crow-AMSAA model gradient (the... 

GAL Gallow, K.C., Jhon, Y.K., Tang, W., Genzer, J., and Loo, Y.-L., Cloud point suppression in dilute solutions of model gradient copolymers with piespecffied composition profiles, J. Polym. Sci. Part B Polym. Phys., 49, 629, 2011. 

The pressure waveforms shown in Fig. 1 illustrate a peculiar feature of the ideal detonation model. Gradients of thermodynamic properties are infinite at the point just behind the detonation in cylindrical (and also spherical) geometries. This singular behavior makes the flow in the vicinity of the detonation difficult to accurately simulate with standard finite difference numerical solution methods. In addition, the shock... 

Figure 5.12 Experimental and modeled gradient profiles for 0.03% AIBN in MMA for various times (1.75, 4.50, and 6.00 h) for the cure temperature of 47 - 52°C (The experimental and modeled gradient maxima were...

The gradient model has been combined with two equations of state to successfully model the temperature dependence of the surface tension of polar and nonpolar fluids [54]. Widom and Tavan have modeled the surface tension of liquid He near the X transition with a modified van der Waals theory [55]. 

The gradient model for interfacial tension described in Eqs. III-42 and III-43 is limited to interaction potentials that decay more rapidly than r. Thus it can be applied to the Lennard-Jones potential but not to a longer range interaction such as dipole-dipole interaction. Where does this limitation come from, and what does it imply for interfacial tensions of various liquids ... 

IHP) (the Helmholtz condenser formula is used in connection with it), located at the surface of the layer of Stem adsorbed ions, and an outer Helmholtz plane (OHP), located on the plane of centers of the next layer of ions marking the beginning of the diffuse layer. These planes, marked IHP and OHP in Fig. V-3 are merely planes of average electrical property the actual local potentials, if they could be measured, must vary wildly between locations where there is an adsorbed ion and places where only water resides on the surface. For liquid surfaces, discussed in Section V-7C, the interface will not be smooth due to thermal waves (Section IV-3). Sweeney and co-workers applied gradient theory (see Chapter III) to model the electric double layer and interfacial tension of a hydrocarbon-aqueous electrolyte interface [27]. 

The relationship between mean squared phase shift and mean squared displacement can be modelled in a simple way as follows This motion is mediated by small, random jumps in position occurring with a mean interval ij. If the jump size in the gradient direction is e, then after n jumps at time the displacement of a spin is... 

Figure Bl.14.13. Derivation of the droplet size distribution in a cream layer of a decane/water emulsion from PGSE data. The inset shows the signal attenuation as a fiinction of the gradient strength for diflfiision weighting recorded at each position (top trace = bottom of cream). A Stokes-based velocity model (solid lines) was fitted to the experimental data (solid circles). The curious horizontal trace in the centre of the plot is due to partial volume filling at the water/cream interface. The droplet size distribution of the emulsion was calculated as a fiinction of height from these NMR data. The most intense narrowest distribution occurs at the base of the cream and the curves proceed logically up tlirough the cream in steps of 0.041 cm. It is concluded from these data that the biggest droplets are found at the top and the smallest at the bottom of tlie cream.

Most gradient optimization methods rely on a quadratic model of the potential surface. The minimum condition for the... 

Analytic teclmiques often use a time-dependent generalization of Landau-Ginzburg ffee-energy fiinctionals. The different universal dynamic behaviours have been classified by Hohenberg and Halperin [94]. In the simple example of a binary fluid (model B) the concentration difference can be used as an order parameter m.. A gradient in the local chemical potential p(r) = 8F/ m(r) gives rise to a current j... 

Dalibard J and Cohen-Tannoudji C 1989 Laser cooling belowthe Doppler limit by polarization gradients simple theoretical models J.Opt.Soc.Am. B 6 2023-45... 

The forces in a protein molecule are modeled by the gradient of the potential energy V(s, x) in dependence on a vector s encoding the amino acid sequence of the molecule and a vector x containing the Cartesian coordinates of all essential atoms of a molecule. In an equilibrium state x, the forces (s, x) vanish, so x is stationary and for stability reasons we must have a local minimizer. The most stable equilibrium state of a molecule is usually the... 

The starting point for developing the model is the set of diffusion equations for a gas mixture in the presence of temperature, pressure and composition gradients, and under the influence of external forces." These take the following form... 

Now the force per unit volume exerted on the porous medium by the pressure gradient in the gas is -grad p, where p, as distinct from is the physical pressure of the gaseous mixture. This is the force which must be balanced in our model by the external forces acting on the dust particles, so... 

The complete problem with composition gradients as well as a pressure gradient, may be regarded as a "generalized Poiseuille problem", and its Solution would be valuable for comparison with the limiting form of the dusty gas model for small dust concentrations. Indeed, it is the "large diameter" counterpart of the Knudsen solution in tubes of small diameter. 

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