Fluid parcel

The motions of the individual fluid parcels may be overlooked in favor of a more global, or Eulerian, description. In the case of single-phase systems, convective transport equations for scalar quantities are widely used for calculating the spatial distributions in species concentrations and/or temperature. Chemical reactions may be taken into account in these scalar transport equations by means of source or sink terms comprising chemical rate expressions. The pertinent transport equations run as... 

The gradient-flux model to describe turbulent diffusion (Eq. 18-70) has the disadvantage that turbulent diffusivity, Ex, is scale dependent. As discussed in more detail in Chapter 22, in natural systems Ex increases with increasing horizontal scale of diffusion. This means that the speed with which two fluid parcels are separated by turbulence increases the further they are from each other. This is because turbulent structures (eddies) of increasing size become effective when the size of a diffusing patch becomes larger. Typical ranges of turbulent diffusivities in the environment are summarized in Table 18.4. 

FLUID PARCEL, in any fluid, an imaginary portion of that fluid which for theoretical studies muy be considered to have all the basic dynamic and thermodynamic propenies of the fluid but which is small enough so lhal... 

Fluid parcel—A volume of fluid composed of several molecules. 

Until 10 to 15 years ago the combined approach of macromixing and micromixing models was very widely used in the field of CRE but gradually CFD-based strategies have replaced the first mentioned strategy. In this respect it should be noted that this change also introduced big conceptual differences because the traditional CRE approach is usually formulated in the age space of fluid parcels whereas in CFD approaches a Eulerian framework is often adopted. Subsequently a brief overview of CFD-based approaches for reacting flows is presented and the current limitations are also indicated. 

The marker fluid parcels can consist of small particles, bubbles, or droplets but can also be generated in situ by activating molecules constituting the fluid with laser beams, causing them to fluoresce (Gharib et al, 1985). 

In a stratified fluid, buoyancy resists any vertical displacement of a water parcel. For a small displacement, a fluid parcel experiences an acceleration in the opposite direction linearly dependent on the displacement 1 ... 

The Coriolis force becomes particularly important when fluid motion occurs at large scales, as in vast lakes such as Lake Michigan (Fig. 2-6 b), or in atmospheric circulation (Fig. 4-12). Figure 4-12 explains the origin of the Coriolis force in terms of the radial and tangential components of the velocity of a fluid parcel in a rotating mass of fluid. 

It is then assumed that, although the total system is not in equilibrium, there exists within small mass elements a state of local equilibrium, for which the local entropy is the same function of E, p and ujc (or H, p and ujc) as in real equilibrium for a fluid parcel. In particular we assume that the above formulas remains valid written as a time rate of change for a mass element as described earlier. 

The mean circulation time represents an average value for all the fluid elements in the tank and provide information on how fast the fluid parcels are transported around in the tank. 

In fluid mechanics the time rate of change of a vector tp is then written as Dtp/Dt = Dip/Dt + il xtp in analogy to the result from classical mechanics. The above relation for the generalized vector tp is applied to a fluid parcel s position r and then to its velocity v, leading to the relation ... 

Assuming that Dw/Dt is equal to the local force acting per unit mass on a fluid parcel, the apparent forces in the rotating frame are derived. The above derivation can be called a Lagrangian approach since it exploits the concept of the fluid parcel . In this framework we define ... 

The question of what controls the asymptotic decay rate and how is it related to characteristic properties of the velocity field has been an area of active research recently, and uncovered the existence of two possible mechanisms leading to different estimates of the decay rate. Each of these can be dominant depending on the particular system. One theoretical approach focuses on the small scale structure of the concentration field, and relates it to the Lagrangian stretching histories encountered along the trajectories of the fluid parcels. This leads to an estimate of the decay rate based on the distribution of finite-time Lyapunov exponents of the chaotic advection. Details of this type of description can be found in Antonsen et al. (1996) Balkovsky and Fouxon (1999) Thiffeault (2008). Here we give a simplified version of this approach in term of the filament model based... 

Neufeld et al. (1999) have shown that the roughness exponent a of the decaying chemical field is a function of the decay rate and of the Lyapunov exponent of the advection. In the large Peclet number limit we can neglect diffusion and set D = 0 so that the concentration field can be described by the Lagrangian representation (6.13) that follows the chemical dynamics within the fluid parcels advected on chaotic trajectories in the flow... 

To obtain the chemical concentration C(x, t) at a point x at time t one first needs to integrate the advection equation backwards in time to find the past trajectory of the fluid parcel, r(t — r) for 0 < r < t satisfying r(t) = x. Then the second equation in (6.25) for the chemical dynamics along this trajectory can be solved as (compare with (2.6))... 

When the stirring rate is below a certain threshold the synchronized oscillation disappears and the mean concentration is almost constant apart from small irregular fluctuations that are independent of the stirring rate. In this regime the mixing is too slow to strongly couple the oscillatory dynamics of the fluid parcels. Snapshots of the... 

If surface waters are in good gas exchange contact with the atmosphere, then very little He will accumulate due to tritium decay. Once isolated from the surface, this He can accumulate. From the measurement of both isotopes in a fluid parcel, a tri-tium- He age can be computed according to ... 

All terms in the momentum balance have units of momentum per unit time, which is synonymous with the units of force. In this respect, it is necessary to account for external forces (i.e., sources) that act on the fluid within the control volume. In general, these forces are not surface related. They are called body forces because they act volumetrically like the accumulation rate process, which means that each fluid parcel within the system is affected by a body force. The... 

Two-Dimensional Planar Flow. A path within a fluid across which no flow occurs is called a streamline. In other words, fluid parcels move along streamlines. If n is an outward-directed unit normal vector from an arbitrary path within a fluid and d/ is a differential length along this path, then... 

ABSOLUTE VELOCITY - The vector sum of the velocity of a fluid parcel relative to the earth and the velocity of the parcel due to the earth s rotation the east-west component is the only one affected. 

Let V be the volume of the reactor and / the volumetric flow rate. We assume that the CSTR is well mixed, i.e., we make the ideal mixing assumption. Then the average time a fluid parcel spends in the CSTR is... 

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