The flow of fluids

Two properties of fluids influence the way fluids behave. They are density and viscosity. Most gases have a relatively low density and low viscosities. On the other hand liquids can display a range of densities and viscosities, for instance the density and viscosity of light organic liquids are relatively low, but other liquids such as mercury have a high density and liquids with a high viscosity include fuel oils and treacle. 

Viscosity is not apparent till the fluid is in motion. For a fluid in motion a force is required to maintain flow. In order to spread a viscous paint on a solid surface the necessary force is applied by the paint brush. In simple terms the bottom of the paint adheres to the surface while the layers of paint remote from the surface adhere to the brush. The force applied through the brush - a shear force -maintains the layers of paint between the brush and the surface in motion. The brush may be moved at constant speed, but the layers of paint move at different speeds. It may be visualised that the molecules of paint adjacent to the surface are stationary while there is a gradual increase in velocity through the individual layers till the layer in contact with the brush moves at the speed (or velocity) of the brush. In other words a velocity gradient is established throughout the layers. 

The velocity gradient is proportional to the shear force per unit 

In order to make relationship 5.1 into an equation a constant of proportionality is required, i.e. 

The viscosity is defined as the shear force per unit area necessary to achieve a velocity gradient of unity. Equation 5.2 applies to the majority of fluids, and they are generally known as Newtonian fluids, or fluids that display Newtonian behaviour. There are exceptions, and some fluids (usually liquids) do not conform to Equation 5.2, and these are generally classified as non-Newtonian fluids although within this grouping there is a sub classification with distinctly different viscosity behaviour for the fluids within the different groups. 

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The previous sections have considered the flow of fluid to the wellbore. The productivity index (PI) indicates that as the flowing wellbore pressure (Pwf) reduces, so the drawdown increases and the rate of fluid flow to the well increases. Recall... 

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

Reynolds Number. The Reynolds number, Ke, is named after Osborne Reynolds, who studied the flow of fluids, and in particular the transition from laminar to turbulent flow conditions. This transition was found to depend on flow velocity, viscosity, density, tube diameter, and tube length. Using a nondimensional group, defined as p NDJp, the transition from laminar to turbulent flow for any internal flow takes place at a value of approximately 2100. Hence, the dimensionless Reynolds number is commonly used to describe whether a flow is laminar or turbulent. Thus... 

Gate valves are used to minimize pressure drop in the open position and to stop the flow of fluid rather than to regulate it. The problem, when the valve is closed, of pressure buildup in the bonnet from cold liquids expanding or chemical action between fluid and bonnet should be solved oy a relief valve or by notching the upstream seat ring. 

Motorized sluice valve - To throttle the flow of fluid Figure 6.38 Conventional throttle control... 

Variable-speed drives are essential for many industrial applications requiring variable operating parameters during the course of operation. Such variations can be in the flow of fluid and pressure of air or gas etc. The con-... 

The D Arcy equation is the accepted equation for the flow of fluid through a packed bed and its derivation will be found in most physics textbooks. If a capillary column... 

Crystals suspended in liquors emerging from crystallizers are normally passed to solid-liquid separation devices such as gravity settlers or thickeners that may subsequently feed filters to remove yet more liquid prior to drying. Here the transport processes of particle motion and the flow of fluids through porous media are important in determining equipment size, the operation of which may be intensified by application of a centrifugal force. 

The flow of fluid leaving a choke is in the form of a high-velocity jet. For this reason it is desirable to have a straight run of pipe of at least ten pipe diameters downstream of any choke prior to a change in direction, so that the jet does not impinge on the side of the pipe. 

Bernoulli and Euler dominated the mechanics of flexible and elastic bodies for many years. They also investigated the flow of fluids. In particular, they wanted to know about the relationship between the speed at which blood flows and its pressure. Bernoulli experimented by puncturing the wall of a pipe with a small, open-ended straw, and noted that as the fluid passed through the tube the height to which the fluid rose up the straw was related to fluid s pressure. Soon physicians all over Europe were measuring patients blood pressure by sticking pointed-ended glass tubes directly into their arteries. (It was not until 1896 that an Italian doctor discovered a less painful method that is still in widespread... 

The valve loss 1 is the loss due to the flow of fluid back through the valve during closing. This is of the order of 0.02 to 0.10 depending on the valve design. 

The purpose of a hydraulic pump is to supply the flow of fluid required by a hydraulic system. The pump does not create system pressure. System pressure is created by a combination of the flow generated by the pump and the resistance to flow created by friction and restrictions within the system. 

Although most fluid power motors are capable of providing rotary motion in either direction, some applications require rotation in only one direction. In these applications, one port of the motor is connected to the system pressure line and the other port to the return line. The flow of fluid to the motor is controlled by a flow control valve, a two-way directional control valve or by starting and stopping the power supply. Varying the rate of fluid flow to the motor may control the speed of the motor. 

A valve is defined as any device by which the flow of fluid may be started, stopped, regulated or directed by a movable part that opens or obstmcts passage of the fluid. Valves must be able to accurately control fluid flow, system pressure and to sequence the operation of all actuators within a hydraulic system. 

Flow control valves are used to regulate the flow of fluids. Control of flow in hydraulic systems is critical because the rate of movement of fluid-powered machines or actuators depends on the rate of flow of the pressurized fluid. 

Directional control valves are designed to direct the flow of fluid, at the desired time, to the point in a fluid power system where it will do work. The driving of a ram back and forth in its cylinder is an example of when a directional control valve is used. Various other terms are used to identify directional valves, such as selector valve, transfer valve, and control valve. This manual will use the term directional control valve to identify these valves. 

Convection requires a fluid, either liquid or gaseous, which is free to move between the hot and cold bodies. This mode of heat transfer is very complex and depends firstly on whether the flow of fluid is natural , i.e. caused by thermal currents set up in the fluid as it expands, or forced by fans or pumps. Other parameters are the density, specific heat capacity and viscosity of the fluid and the shape of the interacting surface. 

General laws for the flow of fluids were determined by Reynolds, who recognized two flow patterns, laminar and turbulent. In laminar flow the fluid can be considered as a series of parallel strata, each moving at its own speed, and not mixing. Strata adjacent to walls of the duct will be slowed by friction and will move slowest, while those remote from the walls will move fastest. In turbulent flow there is a general forward movement together with irregular transfer between strata. 

Considering the flow of fluid between two plates of unit width, a distance da apart, as shown in Figure 3.12, then for the equilibrium of an element ABCD, a force balance may be set up in a similar manner to that used for flow through pipes to give ... 

If Ihe particles tend to form a bed, they will be affected by the lateral dispersive forces described by Bagnold<69-70). A fluid in passing through a loose bed of particles exerts a dilating action on the system. This gives rise to a dispersion of the particles in a direction at right angles to the flow of fluid. 

The energy required for conveying can conveniently be considered in two parts that required for the flow of the air alone, and the additional energy necessitated by the presence of the particles. It should be noted, however, that the fluid friction will itself be somewhat modified for the following reasons die total cross-sectional area will not be available for the flow of fluid the pattern of turbulence will be affected by the solids and the pressure distribution through the pipeline will be different, and hence the gas density at a given point will be affected by the solids. 

The flow of fluid over a plane surface, heated at distances greater than. to from the leading edge, is now considered. As shown in Figure 11.11 the velocity boundary layer starts at the leading edge and the thermal boundary layer at a distance o from it. If the temperature of the heated portion of the plate remains constant, this may be taken as the datum temperature. It is assumed that the temperature at a distance y from the surface may be represented by a polynomial of the form ... 

A solids-stabilized water-in-oil emulsion may be used either as a drive fluid for displacing hydrocarbons from the formation or to produce a barrier for diverting the flow of fluids in the formation. The solid particles may be formation solid particles or nonformation solid particles, obtained from outside the formation (e.g., clays, quartz, feldspar, gypsum, coal dust, asphaltenes, polymers) [228,229]. 

After the completion of the fracturing treatment, the fluid viscosity should decrease to allow the placement of the proppant and a rapid fluid return through the fracture. It is important to control the time at which the viscosity break occurs. In addition, the degraded polymer should produce little residue to restrict the flow of fluids through the fracture. 

Momentum balance equations are of importance in problems involving the flow of fluids. Momentum is defined as the product of mass and velocity and as stated by Newton s second law of motion, force which is defined as mass times acceleration is also equal to the rate of change of momentum. The general balance equation for momentum transfer is expressed by... 

In the loop interface the fluid from the extraction cell passes continuously through an injection loop and into a collection vessel. Injection of a predetermined fraction of the extract onto a packed column is made by switching the loop so that it is in-line with the flow of fluid to the analytical... 

Permeability (k) is the transport coefficient for the flow of fluids through a porous medium and has the units of length squared. NMR measures the porosity and the... 

In order to design such an efficient and effective device, one must understand the mechanisms by which drug is transported in the ocular interior. One issue debated in the literature for some time has been the relative importance of transport by passive diffusion versus that facilitated by the flow of fluid in the vitreous (see, e.g., Ref. 226). To predict the geometric distribution even at steady state of drug released from an implant or an intravitreal injection, one must appreciate which of these mechanisms is at work or, as appropriate, their relative balance. 

Figure 26.9 illustrates Darcy s law, the basic equation used to describe the flow of fluids through porous materials. In Darcy s law, the coefficient k, hydraulic conductivity, is often called the coefficient of permeability by civil engineers. 

The term tubular flow reactor is used generically to refer to a reactor in which the flow of fluid is essentially in one direction (e.g., axial flow in a cylindrical vessel) without any... 

A reactor model based on solid particles in BMF may be used for situations in which there is deliberate mixing of the reacting system. An example is that of a fluid-solid system in a well-stirred tank (i.e., a CSTR)-usually referred to as a slurry reactor, since the fluid is normally a liquid (but may also include a gas phase) the system may be semibatch with respect to the solid phase, or may be continuous with respect to all phases (as considered here). Another example involves mixing of solid particles by virtue of the flow of fluid through them an important case is that of a fluidized bed, in which upward flow of fluid through the particles brings about a particular type of behavior. The treatment here is a crude approximation to this case the actual flow pattern and resulting performance in a fluidized bed are more complicated, and are dealt with further in Chapter 23. 

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