Cross-sectional mean

In this model, energy balances are set up for the reactor and the separator tube separately, and two equations are obtained. The gas holdup can then be obtained from combining these two equations. Details can be found in Zhang et al. [7]. The comparison between the measured and calculated cross-sectional mean gas holdups is shown in Fig. 5. It can be seen that there is a satisfactory agreement between the experimental and calculated gas holdup in the different operating conditions. Therefore, it is reasonable to conclude that the energy balance model used in this work can describe the circulation flow behavior in the novle internal-loop airlift reactor proposed in this work. 

The adiabatic ionization potential (1A) of a molecule, as shown in Figure 4.1, equals the energy difference between the lowest vibrational level of the ground electronic state of the positive ion and that of the molecule. In practice, few cases would correspond to adiabatic ionization except those determined spectroscopically or obtained in a threshold process. Near threshold, there is a real difference between the photoabsorption and photoionization cross sections, meaning that much of the photoabsorption does not lead to ionization, but instead results in dissociation into neutral fragments. 

Figure 1.2. Illustration of longitudinal dispersion of a tracer plane at t = 0 to a dispersed cloud at t = T. C is the cross-sectional mean concentration.

Mean values are important in environmental transport and fate because the environment is not well mixed. To address various applications in the most effective manner, we often consider mean values and the variations from the mean values separately. We will be predominantly using two types of mean values temporal means and cross-sectional means. 

A cross-sectional mean is the mean value of a quantity over a cross section. We will use the cross-sectional means to compute concentration in a system with dispersion. An illustrative example is given in Figure 1.7. For the system visualized in this figure, the cross-sectional mean velocity, U x, t), and cross-sectional mean concentration, C(x, f),would be given by... 

Random fluctuations in C and u are normally smoothed out in time as well, because these cross-sectional mean values also act similar to a low-pass filter in time. U and C can change slowly, but not rapidly, because of the general smoothing character of... 

Figure 1.7. Illustration of cross-sectional mean velocity, U, and cross-sectional mean concentration, C, versus u and C.

Dispersion is another reactor mixing topic that will be discussed in Chapter 6. Dispersion normally is used when cross-sectional mean concentrations and velocities are being computed. A cross-sectional mean concentration is useful for a pipe, stripping tower, river, or groundwater transport. 

Describe the difference between the mathematics of temporal and cross-sectional mean values of velocity and concentration. How does the cross-sectional mean also operate similar to a low-pass filter (removes high-frequency fluctuations) in time ... 

Consider the steady, laminar flow of an incompressible fluid in a long and wide closed conduit channel subject to a linear pressure gradient, (a) Derive the equation for velocity profile, (b) Derive the equation for discharge per unit width and cross-sectional mean velocity, and compare this with the maximum velocity in the channel, (c) Derive the equation for wall shear stress on both walls and compare them. Explain the sign convention for shear stress on each wall. 

An experimentally based rule-of-thumb is that laminar flow often occurs when the pipe Reynolds number, Vdjv, is less than 2,000, or when an open channel Reynolds number, Vhjv, is less than 500, where V is the cross-sectional mean velocity, d is the pipe diameter, v is the kinematic viscosity of the fluid, and h is the channel depth. The diameter or depth that would not be exceeded to have laminar flow by these experimental criteria is given in Table 5.1. 

The river and waste effluent is illustrated in Figure E5.3.1. The river has a discharge of <2o, a cross-sectional mean velocity of U, a mean stream width of b, and an ambient pollutant concentration of Q. The effluent discharge is Qi, with a pollutant concentration of Q. We will solve for a vertical mean concentration, C, which varies spatially across the river (y-axis) and in the direction of flow (x-axis), or... 

Consider the wastewater treatment plant on a river, as illustrated in Figure E6.4.1. The river flows with a cross-sectional mean velocity of U, a discharge of Qr, a DO concentration of Co, and a BOD concentration of Ly. The wastewater treatment plant adds an additional BOD of L, discharging at <2w What will be the DO concentration in the river reach downstream ... 

To convert equation (6.21) to cross-sectional mean values, we will assign... 

In expressing the equations for longitudinal dispersion, we will drop the hat above the cross-sectional mean values. It will be assumed for the remainder of Chapter 6, that if is involved, we are discussing cross-sectional mean concentrations. 

Laminar flow in a pipe (Taylor, 1953) 4SD Rf = radius of tube U = cross-sectional mean velocity D = diffusivity h = depth... 

Rivers are close to the perfect environmental flow for describing the flow as plug flow with dispersion. The flow is confined in the transverse and vertical directions, such that a cross-sectional mean velocity and concentration can be easily defined. In addition, there is less variation in rivers than there is, for example, in estuaries or reactors - both of which are also described by the plug flow with dispersion model. For that reason, the numerous tracer tests that have been made in rivers are useful to characterize longitudinal dispersion coefficient for use in untested river reaches. A sampling of the dispersion coefficients at various river reaches that were... 

If the aforementioned river was modeled as a plug flow reactor with dispersion, what would be the dispersion coefficient and the cross-sectional mean velocity How does the predicted curve compare with those plotted for problem 2 (i.e., plot it) ... 

We could operate our computational transport model with only numerical dispersion (i.e., as a tanks-in-series model). It is often inconvenient to do so in environmental transport applications, however, because the cross-sectional mean velocity, U, can... 

V is cross-sectional mean velocity, v is kinematic viscosity, e is wall roughness, and D is pipC di IUCtel 4Rh. Rh is hydraulic radius, A/P, where A is cross-sectional area and P is wetted perimeter (Moody,... 

W, M Average velocity across a cross-section, mean at a point L T 1 m s-1... 

Under great volume consumption rates of melts in the same experiments, an interesting effect was observed the extrudate became narrowed at the channel slit size 1.9 x 9.9 mm. The extrudate was 2 mm thick but only 5.5 + 0.5 mm wide. Reduction of the extrudate cross-section means transfer to a multilayer flow. It can be assumed that this phenomenon is caused by the existence of temperature cross-gradients along the channel section. This leads to a non-linear distribution of speed gradients and tensions over the channel section and along the flow axis, that in some places conditions for the melts-cohesion disintegration exist (see Sect. 4.4). 

Fig. 1.19 also shows the cross-sectional mean values of the fluid temperatures... 

The INS spectra of ethene [23,24] and propene [24] are discussed in 7.3.2.3 and shown in Fig. 7.16. The spectra are dominated by the effects of molecular recoil. This is less of a problem for propene because it has internal vibrations at lower energy (and hence on low-bandpass spectrometers, lower Q) than ethene. With the much heavier tetrabromoethene [25] this does not occur but the small cross section means that a large (8 g) sample was needed. Tetracyanoethene has been studied by coherent INS [26]. The bicyclic alkene norbomene [27] has been studied by INS because it is the parent compound for a class of advanced composites. 

For a purely axial temperature gradient, if the cross section mean decomposition is again performed, a new mean concentration transport equation is derived... 

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