Dimensionless time, pressurization

Dimensionless time Pressure Temperature Mean temperature Initial temperature Variance of nif vm at Tso Initial value of Vm Delta function Scalar... 

Once the model fluid and its pressure and temperature are chosen, which sets the gas density and viscosity, there is only one unique set of parameters for the model which gives similarity when using the full set of dimensionless parameters. The dependent variables, as nondimensionalized by Eq. (18), will be the same in the respective dimensionless time and spatial coordinates of the model as the commercial bed. The spatial variables are nondimensionalized by the bed diameter so that the dimensional and spatial coordinates of the model is proportional to the two-thirds power of the kinematic viscosity, as given by Eq. (69)... 

Figure 26.3 The mole fraction of H2 just above the surface (a) and the wall heat flux (6) as functions of the dimensionless time, 2at, for 10% H2-air mixture at a surface temperature of 1100 K. Self-sustained oscillations and stationary solutions are represented by solid and dashed curves, respectively. The pressure is 4 atm and the strain is a = 200 s ...

Fig. 51. A log—log diagram of the velocity autocorrelation function for a Brownian particle in nitrogen as at two different pressures. Both axes are scaled to make the velocity autocorrelation function normalised and the time dimensionless. The pressures were O, 0.1 MPa ", 1.135PMa. At short times, the experimental data fit an exponential...

Finally, in calculating the pressure at the running point x (coordinate) within the forming cavity, the dimensionless time tx nesessary for the x point to be reached by the flow front may turn out to be of use (i.e. the time x is needed for pressure to become non-zero at the x point). According to Eqs. (6.7) and (6.7 a), this time is given by ... 

Fig. 14. Dimensionless pressure as a function of dimensionless time at four points of the mould for plastisol I with gelatination taken into account upon moulding into a preheated mould. Solid lines calculated dotted measured...

One can find the dimensionless time required to accomplish the full pressure change from PL to P j by substituting Eqs. (14) and (15) into Eq. (11) and integrating. The result is... 

As before, one can show that the length of dimensionless time required for the pressure decrease during blowdown is that given by Eq. (17) The dimensionless time definition in this case is given by Eq. (19). Hence the mole fraction profile after the pressure decrease is found by integrating Eq. (20) from t = 0 to... 

Mi M M Mmax 71 O2 T ) X A Yi dimensionless time units [t/t] (-) chemical potential of species i fluid viscosity, (kg/m s), Pa s specific growth rate, s 1 maximum specific growth rate, s 1 total pressure (Pa) variance of a tracer curve or distribution functions (s2) space time CAOV/uCAO (s) stoichiometric coefficient ratio overall fractional yield Life expectancy, s difference 2 / "2 dimensionless variance ct /1 (-) fugacity coefficient (= f,/p)... 

The above ODE is directly integrated to yield the analytical filter in terms of the dimensionless time-variable pressure gradient ... 

Fourth dimensionless parameter, dimensionless separation impedance, E (also known as efficiency ), which embraces retention time, pressure drop, plate number, eluent viscosity and retention factor, is taken as a measure of quality ... 

The rate In a batch reactor must be based on the total amount of substrate In the system. There Is no way to measure the concentration (or moles) of substrate inside the polymer beads It must be computed. Substrate Is depleted from the bulk phase for two reasons Initially the polymer phase contained no substrate, and substrate reacts Inside the polymer beads. Total moles of substrate are easily determined In systems which Involve addition of a gas phase reactant at constant pressure. The reaction rate Is developed using the conversion of substrate. Equation 2 uses the total moles of substrate In the system. A material balance Is performed over the batch reactor at any time t to give the moles of substrate in the bulk and polymer phases. The derivative of this balance may be used to calculate reaction rates by making use of Equation 3. The derivative of the total mole balance with respect to dimensionless time is... 

Shown in Fig. 21.1 is the dependence of molar concentration of water in the drop on dimensionless time r for various values of pressure and initial mass content of methanol 70% and 95%. Eventually the methanol completely evaporates... 

According to Darcy law, the mathematical model of gas emission in the coal particle and its boundary conditions is established, by means of which the dimensionless pressure, the dimensionless time and the dimensionless radius are put forward to simphfy the model. 

For a given position of the interfaces 5, the solution of the mathematical model (Eqs. 15, 16, and 17) for the velocity u(xo) at a given point Xo can be obtained by means of the boundary integral formulation. The equations are non-dimensionalized and, for simplicity, the same notations for the dimensionless velocity, pressure, space, and time variables are used as those for the corresponding dimensional variables from the previous equations ... 

Here, fa is dimensionless time, f is time, is porosity, k is permeability, Oo/w is interfacial tension, IFT, fi , is viscosity of water, and L is block dimension (length). They assumed that gravity effects are negligible, and that the shape of the matrix blocks, wettability, initial fluid distributions, relative permeabilities, and capillary pressures are the same. From equation 4 it is seen that the imbibition rate decreases if interfacial tension decreases. 

For the arbitrary characteristics listed, an arbitrary permeate rate is determined in Example 3.1 with the value V = 12.7056 for an assigned permeate-to-feed (WP) ratio of 0.5, and for component / has the dimensions of overall permeability times pressure (P,Py) that is, it is a permeate flux rate. In other words, considering the expression for the dimensionless fC-value, developed in Chapter 3, if... 

The curve in Figure 6.8, the design chart for reduced saturation vs. the product of dimensionless time (6) and dimensionless pressure (p ), is represented precisely by the two correlations shown in equation (B.2) ... 

Fig. 4 The pressure in the expanding products, of acetylene-oxygen explosion (1) and R for a Lagrangian cell as a function of dimensionless time f — iaofr (oo is the acoustic velocity in the nonexpanded products).

The governing equations were advanced in time with a dimensionless time step. At. On each time step, it was necessary to determine the following set of dimensionless unknowns the acceleration, a, the velocity, Ub, the stream-function, ij/, the vorticity, co, the pressure on the surface of the bubble, p, the coordinate mapping, the volume concentration of surfactant,... 

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