Constant velocity input

The third factors to control and keep constant are the gas pressure and superficial gas velocity. This probably will involve gas recirculation with either a small compressor, or through a hollow shaft or some other pumping device. As seen before, the bubble diameter, the mass transfer area, the gas hold-up, and the terminal bubble-rise velocity, all depend on the superficial velocity of the gas and the power input per unit volume. When these are kept constant, the various mass transfer resistances in the pilot plant and in the large unit will be the same, hence the global rate will be conserved. The last factor is the input power to the agitator. As required for mass transfer, the scale-up must be made on the basis of constant power input per unit volume. If turbulent conditions and geometrical similarity prevail, this rule imposes the following relationship ... 

For homogeneous (completely flowing) open systems a steady-state point becomes unique and stable at a very high constant velocity of the flow [35]. In this case the concentrations of gas-phase components rapidly become almost constant and their ratios are close to those for the input mixture. This fact is independent of a concrete type of the w(c) function. To confirm this postulate, let us consider eqns. (125) for a balance polyhedron Da. Since vm is very high, the inequality (127) is fulfilled automatically and we can write... 

Fig. 5 shows the bubble plume in front view (y-z plane) and profile (z-x plane). In both cases the photo is accompanied by a drawing schematizing the observed flows. For a constant gas input, the total gas/liquid contact surface is greater for small bubbles than for larger bubbles and the gas-liquid interactions are also greater. It was observed that the smaller the bubbles, the greater the liquid velocity, the more agitated and more turbulent the bubble plume, and the more spread out the plume at the air/liquid interface. 

Early SBCR models were reviewed by Ramachandran and Chaudhari (5) and by Deckwer (9). They require hold-up correlations as an input and do not compute flow patterns. The most complete and useful of these models applied to the Fischer-Tropsch (F-T) conversion of synthesis gas in a SBCR is that of Prakash and Bendale (79). They sized commercial SBCR for DOE. They gave syngas conversion and production as a function of temperature, pressure and space velocity. Input parameters with considerable uncertainty that influenced production rates were the gas hold-up, the mass transfer coefficient and the dispersion coefficient. Krishna s group (77) extended such a model to compute product distribution using a product selectivity model. Air Products working with Dudukovic measured dispersion coefficients needed as an input into such model. The problem with this approach is that the dispersion coefficients are not constant. They are a function of the local hydrodynamics. 

Initial conditions that must be specified for a simulation include (1) the distribution of all components, (2) the pK and mobility values of the buffer and sample constituents, (3) the diffusion coefficients and charge tables of the proteins (GENTRANS only), (4) the electroosmotic input data [constant velocity... 

Here is yet another example. A projectile hits a wall ( armor ). Fig. 7.12. The projectile is composed of Lemiard-Jones atoms (with some Sp and p. 347), and we assume the same for the wall (for other values of the parameters, let us make the wall less resistant than the projectile Sw < Sp and Ve,w > fe,p ) All together, we may have hundreds of thousands (or even miUions) of atoms (i.e., there are millions of differential equations to solve). Now, we prepare the input Rq and vq data. The wall atoms are assumed to have stochastic velocities drawn from the MaxweU-Boltzmaim distribution for room temperature. The same for the projectile atoms, but additionally, they have a constant velocity component along the direction pointing to the wall. At first, nothing particularly interesting happens-the projectile flies toward the wall with a constant velocity (while aU the atoms of the system vibrate). Of course, it is most interesting when the projectile hits the wall. Once the front part of the projectile touches the wall, the wall atoms burst into space in a kind of eruption, the projectile s tip loses some atoms, and the spot on the wall hit by the projectile vibrates and sends a shock wave. 

Figure 4 shows the variation of K a with power input at different gas. velocities in XTN 3. At the lowest power input (45 W/m ), K a increases almost two-fold upon doubling gas velocity, whereas at higher constant power inputs it increases by a factor not higher than 1.5. A similar effect was observed in XTN 5 in which at the lowest power input (85 W/m ), K a increased almost three times when gas velocity was doubled. This behavior is not observed in XTN 1 and XTN 2 as shown in curves 1 and 2 of Figure 3. Beyond 120 W/m , K a increases by the combined effect of power input and gas flow rate, although to different extents. At a given gas velocity, a six-fold increase in power input results in an increase of Kj a of the order of 2.5. Within the range from 120 to 700 W/m, K a changes from about 0.001 to 0.005 S" , which represents a more considerable improvement of mass transfer than that in low viscosity XTN and CMC solutions.

The bond graph of the full model is also given in Fig. 2.5 and has six ideal energy elements, two of each type (/, C, and / ), and their parameters are given in Table 2.1. Note that the model includes only the system dynamics in the vertical direction the constant forward speed is used only to convert the spatial road description Zr x) into a temporal vertical velocity input, Vr(t), at the road/tire interface as shown in... 

In this first study, the vehicle model is exercised as it travels over the curb at a constant forward speed, Vf = 5 m/s. This high forward speed generates a severe velocity input that approximates an impulse function the duration of the input is only 0.1 s. The activity is calculated as a function of time by setting the lower bound, Ti, to zero and varying the time window, T, of the integration in (2.2). As shown in Fig. 2.7, the activities remain at zero until the vehicle hits the curb, at which point power starts to flow into the system. The activities increase due to the nonzero power flow until they approach a steady-state value as the system transients die out. Note the discontinuity in the slope of the activities (especially for tire stiffness and damping) at around 1.5 s. The high forward speed causes the wheel to lift off as the vehicle drives over the curb and contact is restored at about 1.5 s. This causes an impact force that results in the rapid increase in the activities. 

The heated-thermocouple anemometer measures gas velocity from the cooling effect of the gas stream flowing across the hot junctions of a thermopile supplied with constant electrical power input. Alternate junctions are maintained at ambient temperature, thus compensatiug for the effect of ambient temperature. For details see Bunker, Proc. Instrum. Soc. Am., 9, pap. 54-43-2 (1954). 

Here Q(t) denotes the heat input per unit volume accumulated up to time t, Cp is the specific heat per unit mass at constant pressure, Cv the specific heat per unit mass at constant volume, c is the sound velocity, oCp the coefficient of isobaric thermal expansion, and pg the equilibrium density. (4) The heat input Q(t) is the laser energy released by the absorbing molecule per unit volume. If the excitation is in the visible spectral range, the evolution of Q(t) follows the rhythm of the different chemically driven relaxation processes through which energy is... 

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