Fluid packets

Using a flow-through model, for example, we can follow the evolution of a packet of fluid as it traverses an aquifer (Fig. 2.5). Fresh minerals in the aquifer react to equilibrium with the fluid at each step in reaction progress. The minerals formed by this reaction are kept isolated from the fluid packet, as though the packet has moved farther along the aquifer and is no longer able to react with the minerals produced previously. 

Fig. 26.3. Silica concentration (bold lines) in a fluid packet that cools from 300 °C as it flows along a quartz-lined fracture of 10 cm aperture, calculated assuming differing traversal times At. Fine lines show solubilities of the silica polymorphs quartz, cristobalite, and amorphous silica.

Fig. 26.5. Calculated silica concentration in a fluid packet flowing through a quartz sand aquifer. The fluid descends from the surface (T = 20 °C) to a depth of about 2 km (80 °C) and then returns to the surface (20 °C). Results are shown for time spans At (representing half of the time the fluid takes to migrate through the aquifer) of 0.1, 1, and ten years. In the latter calculation, the fluid remains near equilibrium with quartz.

Within a fluid packet, the net directed fluid velocity V is a mass-weighted average of the individual molecular velocities ... 

Riding along with a fluid packet is a Lagrangian notion. However, in the limit of dt - 0, the distance traveled dx vanishes. In this limit, (i.e., at a point in time and space) the Eulerian viewpoint is achieved. The relationship between the Lagrangian and Eulerian representations is established in terms of Eq. 2.52, recognizing the equivalence of the displacement rate in the flow direction and the flow velocity. In the Eulerian framework the... 

The extra terms appear because in noncartesian coordinate systems the unit-vector derivatives do not all vanish. Only in cartesian coordinates are the components of the substantial derivative of a vector equal to the substantial derivative of the scalar components of the vector. The acceleration in the r direction is seen to involve w2, the circumferential velocity. This term represents the centrifugal acceleration associated with a fluid packet as it moves in an arc defined by the 9 coordinate. There is also a G acceleration caused by a radial velocity. In qualitative terms, one can visualize this term as being related to the circumferential acceleration (spinning rate) that a dancer or skater experiences as she brings her arms closer to her body. 

A fluid packet, like a solid, can experience motion in the form of translation and rotation, and strain in the form of dilatation and shear. Unlike a solid, which achieves a certain finite strain for a given stress, a fluid continues to deform. Therefore we will work in terms of a strain rate rather than a strain. We will soon derive the relationships between how forces act to move and strain a fluid. First, however, we must establish some definitions and kinematic relationships. 

Fig. 2.5 A fluid packet, shown initially as a cylindrical element, deforms continuously as it moves in a velocity field.

Consider first the normal strain rates in the z and r coordinates, ezz and err. By definition, the strain rate is given as the rate at which the relative dimension of a fluid packet changes per unit time. Stated differently, the product of the strain rate and time represents a relative elongation. Consider first the relative elongation in the radial direction owing to the r-direction normal strain,... 

The symmetry of the stress tensor can be established using a relatively straightforward argument. The essence of the argument is that if the stress tensor were not symmetric, then finite shearing stresses would accelerate the angular velocity w of a differential fluid packet without bound—something that obviously cannot happen. 

Since the stress field varies spatially, there are differential forces across the differential element. These net forces serve to accelerate a fluid packet. Determining the net forces on an element of fluid requires understanding how the stresses vary from one face of a differential element to another. Assuming that the stress field is smooth and differentiable, local variations can be expressed in terms of Taylor-series expansions. 

We have discussed stresses and strain rates. A critical objective is to relate the two, leading to equations of motion governing how fluid packets are accelerated by the forces acting on them. Generally, we are working toward a differential-equation description of a momentum balance, F = ma. The approach is to represent both the forces and the accelerations as functions of the velocity field. The result will be a system of differential equations in which velocities are the dependent variables and the spatial coordinates and time are the independent variables (i.e., the Navier-Stokes equations). 

In this form one sees an analogy in the vorticity equation to the other transport equations— a substantial-derivative description of advective transport, a Laplacian describing the diffusive transport, and possibly a source term. It is interesting to observe that the vorticity equation does not involve the pressure. Since pressure always exerts a normal force that acts through the center of mass of a fluid packet (control volume), it cannot alter the rotation rate of the fluid. That is, pressure variations cannot cause a change in the vorticity of a flow field. 

Dynamic segmented fluid packets in 200 Compute Hone l sim ule tion o velocity profile contiguous segmented fluid... 

Flow conditions may cause a more or less broad distribution of residence times for individual fluid packets or molecules. Effects of variable residence times have to be taken into account in the design and operation of large industrial reactors with adequate precautions the chemical engineer can prevent the undesirable effects of a residence time distribution, or utilize them. 

Some researchers use plug-flow reactors (PFRs), also known as packed bed reactors or column reactors (if run vertically) to model natural systems. In an ideal plug-flow or column reactor, fluid is pumped or drained through a packed bed of mineral grains and every fluid packet is assumed to have the same residence or contact time (Hill, 1977). The residence time equals the ratio of the pore volume of the reactor (Vo) divided by flow rate Q. With no volume change in the reaction, radial flow, or pooling of fluid in the reactor (Laidler, 1987), the outlet concentration varies from the inlet concentration according to ... 

The mixing process is modelled as a walk in velocity states. The basic process of the model is the transition of fluid "packets ... 

Whereas Higbie assumed a constant renewal time, Dankwert s (1951) extension of the penetration theory employs a wide spectrum of eddy contact times. These eddy-like fluid packets are assumed to remain in contact with the interface for variable times from zero to infinity. He assumed a surface-age distribution function, which skews the contact times to small values it is... 

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