Element of fluid

The mathematical theory is rather complex because it involves subjecting the basic equations of motion to the special boundary conditions of a surface that may possess viscoelasticity. An element of fluid can generally be held to satisfy two kinds of conservation equations. First, by conservation of mass. 

We defined the equation of motion as a general expression of Newton s second law applied to a volume element of fluid subject to forces arising from pressure, viscosity, and external mechanical sources. Although we shall not attempt to use this result in its most general sense, it is informative to consider the equation of motion as it applies to a specific problem the flow of liquid through a capillary. This consideration provides not only a better appreciation of the equation of... 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

Danckwerts [Jnd. Eng. Chem., 42, 1460(1951)] proposed an extension of the penetration theoiy, called the surface renewal theoiy, which allows for the eddy motion in the liquid to bring masses of fresh liquid continually from the interior to the surface, where they are exposed to the gas for finite lengths of time before being replaced. In his development, Danckwerts assumed that every element of fluid has an equal chance of being replaced regardless of its age. The Danck-werts model gives... 

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

The fluid is regarded as a continuum, and its behavior is described in terms of macroscopic properties such as velocity, pressure, density and temperature, and their space and time derivatives. A fluid particle or point in a fluid is die smallest possible element of fluid whose macroscopic properties are not influenced by individual molecules. Figure 10-1 shows die center of a small element located at position (x, y, z) with die six faces labelled N, S, E, W, T, and B. Consider a small element of fluid with sides 6x, 6y, and 6z. A systematic account... 

For the small element of fluid ABCD the volume flow rate dQ % given by... 

This may be compared to the situation in the extruder where the fluid is being dragged along by the relative movement of the screw and barrel. Fig. 4.8 shows the position of the element of fluid and (4.2) may be modified to include terms relevant to the extruder dimensions. 

Also, for the element of fluid of depth, dy, at distance, y, from the centre line (and whose velocity is V) the elemental flow rate, dQ, is given by... 

Referring to the element of fluid between the screw flights as shown in Fig. 4.8, this equation may be rearranged using the following substitutions. Assuming e is small, T = nD tan ... 

Consider the forces acting on an element of fluid as shown in Fig. 5.4. 

Consider an element of fluid between parallel plates, T wide and spaced a distance H apart. For unit width of element the forces acting on it are ... 

Fig. 5.10 shows an annular element of fluid of radius r and thickness dr subjected to a shear stress in the capillary. When the element of fluid emerges from the die it will recover to the form shown by ABCD. 

Consider the annular element of fluid shown in Fig. 5.12. The true tensile strain Sr in this element is given by... 

There are two possible kinds of force acting on a fluid cell internal stresses, by which an element of fluid is acted on by forces across its surface by the rest of the fluid, and external forces, such as gravity, that exert a force per unit volume on the entire volume of fluid. We define an ideal fluid to be a fluid such that for any motion of the fluid there exists a pressure p(x, t) such that if 5 is a surface in the fluid with unit normal vector n, the stress force that is exerted across S per unit area at x at time t is equal to —p x,t)h. An ideal fluid is therefore one for which the only forces are internal ones, and are orthogonal to 5 i.e. there are no tangential forces. ... 

Ruckenstein shows (R9) that the duration t0 of the contact between the element of fluid and the bubble may be estimated for a group of bubbles as 2a/ U [compare Eq. (83)]. 

Figure 2.9. Forces acting on element of fluid in a vortex...

Thus, the energy per unit mass increases with radius r and is independent of depth In the absence of an agitator or mechanical means of rotation energy transfer will take place to equalise j/ between all elements of fluid. Thus the forced vortex tends to decay into a free vortex (where energy per unit mass is independent of radius). 

Figure 3.36. Shear stress and normal stresses on element of fluid...

When the pressure at some point in a fluid is changed, the new condition takes a finite time to be transmitted to some other point in the fluid because the state of each intervening element of fluid has to be changed. The velocity of propagation is a function of the bulk modulus of elasticity e, where e is defined by the relation ... 

The pitot tube, in which a small element of fluid is brought to rest at an orifice situated at right angles to the direction of flow. The flowrate is then obtained from the difference... 

Frequently, stirred tanks are used with a continuous flow of material in on one side of the tank and with a continuous outflow from the other. A particular application is the use of the tank as a continuous stirred-tank reactor (CSTR). Inevitably, there will be a vety wide range of residence times for elements of fluid in the tank. Even if the mixing is so rapid that the contents of the tank are always virtually uniform in composition, some elements of fluid will almost immediately flow to the outlet point and others will continue circulating in the tank for a very long period before leaving. The mean residence time of fluid in the tank is given by ... 

Figures 8.5 and 8.6. The liquid is carried round in the spaces between consecutive gear teeth and the outer casing of the pump, and the seal between the high and low pressure sides of the pump is formed as the gears come into mesh and the elements of fluid are squeezed out. Gear pumps are extensively used for both high-viscosity Newtonian liquids and non-Newtonian fluids. The lobe-pump (Figures 8.7 and 8.8) is similar, but the gear...

Convection. Heat transfer by convection arises from the mixing of elements of fluid. If this mixing occurs as a result of density differences as, for example, when a pool of liquid is heated from below, the process is known as natural convection. If the mixing results from eddy movement in the fluid, for example when a fluid flows through a pipe heated on the outside, it is called forced convection. It is important to note that convection requires mixing of fluid elements, and is not governed by temperature difference alone as is the case in conduction and radiation. 

The work of Higbie laid the basis of the penetration theory in which it is assumed that the eddies in the fluid bring an element of fluid to the interface where it is exposed to the second phase for a definite interval of time, after which the surface element is mixed with the bulk again. Thus, fluid whose initial composition corresponds with that of the bulk fluid remote from the interface is suddenly exposed to the second phase. It is assumed that equilibrium is immediately attained by the surface layers, that a process... 

The procedure adopted here consists of taking a momentum balance on an element of fluid. The resulting Momentum Equation involves no assumptions concerning the nature of the flow. However, it includes an integral, the evaluation of which requires a knowledge of the velocity profile ux = f(y). At this stage assumptions must be made concerning the nature of the flow in order to obtain realistic expressions for the velocity profile. 

The equilibrium is considered of an element of fluid bounded by the planes 1 -2 and 3 4 at distances x and x + dx respectively from the leading edge the element is of length l in the direction of flow and is of depth w in the direction perpendicular to the plane 1 -2-3-4. The distance l is greater than the boundary layer thickness (Figure 11.5), and conditions are constant over the width w. The velocities and forces in the X-direction are now considered. 

It is assumed here that the fluid in contact with the surface is at rest and therefore h(j must be zero. Furthermore, all the fluid close to the surface is moving at very low velocity and therefore any changes in its momentum as it flows parallel to the surface must be extremely small. Consequently, the net shear force acting on any element of fluid near the surface is negligible, the retarding force at its lower boundary being balanced by the accelerating force at its upper boundary. Thus the shear stress Ro in the fluid near the surface must approach a constant value. 

Consider the equilibrium set up when an element of fluid moves from a region at high temperature, lying outside the boundary layer, to a solid surface at a lower temperature if no mixing with the intermediate fluid takes place. Turbulence is therefore assumed to persist right up to the surface. The relationship between the rates of transfer of momentum and heat can then be deduced as follows (Figure 12.5). 

Consider the movement of an element of fluid consisting of n molar units of a mixture of two constituents A and B from a region outside the boundary layer, where the molecular concentrations are CAs and CBs, to the surface where the corresponding concentrations are CAw and CBw. The total molar concentration is everywhere Cr- The transfer is effected in a time t and takes place at an area A of surface. 

With respect to reaction rates, an element of fluid will behave in the ideal tubular reactor, in the same way, as it does in a well-mixed batch reactor. The similarity between the ideal tubular and batch reactors can be understood by comparing the model equations. 

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