Fluid, flow

Fluid flow, also known as fluid mechanics, momentum transfer and momentum transport, is a wide-ranging subject, fundamental to many aspects of chemical engineering. It is impossible to cover the whole field here, so we will concentrate upon a few aspects of the subject. 

Some other types of flow situation are also mentioned. Sample calculations are given in Section 3.10, illustrating the use of some of the principles presented here. 

There are many standard texts of fluid flow, e.g. Coulson Richardson,1 Kay and Neddermann2 and Massey.3 Perry4,5 is also a useful reference source of methods and data. Schaschke6 presents a large number of useful worked examples in fluid mechanics. In many recent texts, fluid mechanics or momentum transfer has been treated in parallel with the two other transport or transfer processes, heat and mass transfer. The classic text here is Bird, Stewart and Lightfoot.7 

Chemical engineering in general, and fluid flow in particular, utilises many dimensionless groups, which are discussed in more detail in Chapter 6 11 Scale-up in Chemical Engineering . Since we will use a piping system as an example in this chapter, we will now consider the pertinent dimensionless groups for pipe flow. 

Monophasic fluid flow in capillary-scale ducts is characterized by a low Reynolds number, the flow in capillary-scale microreactors is generally laminar and transport 

The diffusion time t is defined as the time taken by a molecule to travel distance x by diffusive processes  

This means that for reactions limited by diffusion, reaction time is proportional to the square of the rate-limiting distance. Therefore, a reaction in a 10 cm diameter flask could take 1000 000 times less if undertaken in a 100 pm diameter microreactor. Dramatically reduced reaction times have, arguably, been the most potent driving force behind research in microreactor technology. 

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

UV Alumina ion. One phase containing a the dynamic interface beteen packets is illustrated 

E = Head loss due to friction in feet of flowing fluid 

In Equation 1 Ah is called the velocity head. This expression has a wide range of utility not appreciated by many. It is used as is for 

Orifices will be discussed under Metering in this chapter. 

For compressible fluids one must be careful that when sonic or choking velocity is reached, further decreases in downstream pressure do not produce additional flow. This occurs at an upstream to downstream absolute pressure ratio of about 2 1. Critical flow due to sonic velocity has practically no application to liquids. The speed of sound in liquids is very liigh. See Sonic Velocity later in this chapter. 

Still more mileage can be gotten out of Ah = uV2g when using it with Equation 2, which is the famous Bernoulli equation. The terms are 

K = Cp/Cv, the ratio of specific heats at constant pressure to constant volume g = 32.2 ft/sec  

T = Absolute temperature, °R Pi, P2 = Inlet, outlet pressures, psia 

In this chapter, we will primarily focus on fluid flow, heat, and mass transport through gas flow channels and in solid porous electrodes, and its effect on the mass transfer loss. Solid-phase diffusion, charge transport in electrolyte membrane, and ohmic loss will be discussed in Chapter 7. Water transport will also be discussed in Chapter 7. 

Fluid flow and pressure variation in a fuel cell play a critical role in the distribution of reactant gas concentration at electrochemical reaction sites and, hence, in the distribution of local current densities and cause mass transfer loss. The governing equations for reactant gas flows in gas flow channels and in porous electrode-gas diffusion layers are given by conservation of mass and momentum equations. Solutions to these equations result in the distribution of pressure, P, and velocity field, which is also referred to as the bulk motion in the gas flow channels and porous electrode-gas diffusion layers. 

and mass species transport in a tri-layer fuel cell. 

Before considering the effect of fluid flow on the fuel cell, we will briefly review some of the basic principles and relations of fluid flow, heat, and mass transport. 

The steady flow created by an infinite disk rotating at a constemt eungular velocity in a fluid with constant physical properties Weis first studied by von Kirmfin. The solution was sought by using a separation of variables using a dimensionless distance 

Upon introduction of equations (11.74), (11.75), and (11.76), the equation of continuity and the Navier-Stokes equations can be solved numerically. As shown by Cochran, the variables F, G, and H can be written as two sets of series expansions for small and large values of respectively. The series solutions for small values of are especially relevant to the mass-transfer problem. In particular, the derivatives at = 0 are essential in order to determine the first coefficient of the series expansions. The other coefficients are deduced from the first one by using the equation of continuity eind the Navier-Stokes equations. 

The mathematical models for the convective-diffusion impedance associated with convective diffusion to a disk electrode are developed here in the context of a generalized framework in which a normalized expression accoxmts for the influence of mass transfer. 

Substitution of the definition for concentration (equation (11.44)) into the expression for conservation of species i (equation (11.3)) in one dimension yields 

Upon cancellation of the steady-state terms and division by the term e , equation (11.80) can be expressed as 

It is essential that the engineer involved in industrial ventilation have a good foundation in the subject of fluid mechanics, which involves the study of fluids at rest or in motion. 

The fields of application are wide involving computational fluid dynamics (CFD), flow in ducts and pipes, pumps, fans, collection devices, pollution dispersal, and many other applications. 

The flow of compressible and non-compressible liquids, gases, vapors, suspensions, slurries and many other fluid systems has received sufficient study to allow definite evaluation of conditions for a variety of process situations for Newtonian fluids. For the non-Newtonian fluids, considerable data is available. However, its correlation is not as broad in application, due to the significant influence of physical and rheological properties. This presentation is limited to Newtonian systems, except where noted. 

Primary emphasis is given to flow through circular pipes or tubes since this is the usual means of movement of gases and liquids in process plants. Flow through duct systems is treated with the fan section of Compression in Volume 3. 

The basis for single-phase and some two-phase friction loss (pressure drop) for fluid flow follows the Darcy and Fanning concepts. The exact transition from laminar or viscous flow to the turbulent condition is variously identified as between a Reynolds number of 2000 and 4000. 

Any Pressure Level Above Atmospheric (gauge or absolute = (gauge) 1- (barometer)) 

Atmospheric Pressure (pBr), variesl with Geographical Attitude Location, called 1 Local Barometric Pressure, Pb. 

Polymer transport processes that are relevant to the FRRPP process are fluid flow, heat transport, and mass transfer. Fluid flow is evident during fluid conveying and mixing of reactor fluids. Heat transfer is always relevant not only for dissipation of the heat of polymerization but also in an unusual heat trapping effect found in FRRPP systems. Finally, mass transfer is relevant to the diffusion of reactants into polymerization reactive sites, in measurement of spinodal curves, and in study of the evolution of composition profiles during phase separation. 

In FRRPP systems, pertinent fluid flow topics include pipe flow and fluid mixing in reactors. Since the main objective is usually a uniform distribution of components and fast conveying of materials, turbulent flow behavior is the aim. This flow pattern is traditionally characterized by the so-called Reynolds Number, Re, and defined as 

Fluid mixing inside reactor vessels is usually characterized through another type of Reynolds number  

Values of the scale-up exponent n are used, based on the following considerations  

Equal mass transfer rate or equal power/volume n = 2/3 

Modern industrial process plants are connected by a complex network of pipes, valves, pumps, and tanks. Centrifugal and positive displacement pumps are used to transfer fluids from place to 

Fluids assume the shape of the container they occupy. A fluid can be a liquid or a gas. When a liquid is in motion, it remains in motion until it reaches its own level or is stopped. Fluid flow is a critical concept used in the day-to-day operation of all plants. 

Re nolds Number - (Inside Diameter of Pipe) (Density of Fluid) 

Industry uses four different ways to express a fluid s heaviness density, specific gravity, baume gravity, and API gravity. 

The density of a fluid is defined as the mass of a substance per unit volume. Density measurements are used to determine heaviness. For example, one gallon of water weighs 8.33 lb, one gallon of crude oil weighs 7.20 lb, and one gallon of gasoline weighs 6.15 lb. 

It is neGessary to be able to calculate the energy and momentum of a fluid at various positions in a flow system. It will be seen that energy occurs in a number of forms and that some of these are influenced by the motion of the fluid. In the first part of this chapter the thermodynamic properties of fluids will be discussed. It will then be seen how the thermodynamic relations are modified if the fluid is in motion. In later chapters, the effects of frictional forces will be considered, and the principal methods of measuring flow will be described. 

The characteristics and complexity of flow pattern are such that most flows are described by a set of empirical or semi-empirical equations. These relate the pressure drop in the flow system as a function of flow rate, pipe geometry, and physical properties of the fluids. The aim in the design of fluid flow is to choose a line size and piping arrangement that achieve minimum capital and pumping costs. In addition, constraints on pressure drop and maximum allowable velocity in the process pipe should be maintained. These objectives require many trial and error computations, which can be performed well by a computer. 

V = velocity of fluid, ft/s g = dimensional constant, 32.174(lb /lbf)(ft/s ) Z = elevation of fluid, ft subscript 1 = condition at initial point subscript 2 = condition at final point 

The first, second, and third terms in Equation 3-2 represent pressure head, velocity head, and static differences respectively. Equation 3-2 is used for investigating energy distributions or determining pressure differentials between any two points in a pipeline. Incorporating the head loss due to friction, hL, with constant pipe diameter, i.e., V, = V2, Equation 3-2 becomes 

Equation 3-3 shows that the head loss, hL, is generated at the expense of pressure head or static head difference. The static head difference can be either negative or positive. However, for a negative static head difference. 

In general, pressure loss due to flow is the same whether the pipe is horizontal, vertical, or inclined. The change in pressure due to the difference in head must be considered in the pressure drop calculation. 

When solving the problem analytically, it is also possible to use a boundary condition at r = 0 that says the velocity is finite. Both conditions give the same result, and in fact one can be derived from the other. Note that Ap is a positive number. 

Note the r dr in the integrand, which is necessary because cylindrical polar coordinates are used. The relationship between the average velocity and the pressure gradient can be obtained by integrating Eq. (9.7) using Eq. (9.8) this is the Hagen-Poiseuille law. The peak velocity is twice the average velocity in pipe flow  

When the flow is between two wide, flat plates instead of inside a cylinder, Eq. (9.6) becomes 

The peak velocity is 1.5 times the average velocity for flow between parallel plates. These formulas are provided here because they provide a good benchmark against which to check any numerical solution by integrating over boundaries. 

When the fluid is non-Newtonian, it may not be possible to solve the problem analytically. For example, for the Brrd-Carreau fluid (Bird et al., 1987, p. 171) the viscosity is 

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The entering fluid flows downward in a spiral adjacent to the wall. When the fluid reaches the bottom of the cone, it spirals upward in a smaller spiral at the center of the cone and cylinder. The downward... 

The pores between the rock components, e.g. the sand grains in a sandstone reservoir, will initially be filled with the pore water. The migrating hydrocarbons will displace the water and thus gradually fill the reservoir. For a reservoir to be effective, the pores need to be in communication to allow migration, and also need to allow flow towards the borehole once a well is drilled into the structure. The pore space is referred to as porosity in oil field terms. Permeability measures the ability of a rock to allow fluid flow through its pore system. A reservoir rock which has some porosity but too low a permeability to allow fluid flow is termed tight . 

An important safety feature on every modern rig is the blowout preventer (BOP). As discussed earlier on, one of the purposes of the drilling mud is to provide a hydrostatic head of fluid to counterbalance the pore pressure of fluids in permeable formations. However, for a variety of reasons (see section 3.6 Drilling Problems ) the well may kick , i.e. formation fluids may enter the wellbore, upsetting the balance of the system, pushing mud out of the hole, and exposing the upper part of the hole and equipment to the higher pressures of the deep subsurface. If left uncontrolled, this can lead to a blowout, a situation where formation fluids flow to the surface in an uncontrolled manner. 

To a large extent the reservoir geology controls the producibility of a formation, i.e. to what degree transmissibility to fluid flow and pressure communication exists. Knowledge of the reservoir geological processes has to be based on extrapolation of the very limited data available to the geologist, yet the geological model s the base on which the field development plan will be built. 

Laminae of clay and clay drapes act as vertical or horizontal baffles or barriers to fluid flow and pressure communication. Dispersed days occupy pore space-which in a clean sand would be available for hydrocarbons. They may also obstruct pore throats, thus impeding fluid flow. Reservoir evaluation, is often complicated by the presence of clays. This is particularly true for the estimation of hydrocarbon saturation. 

Shallow marine/ coastal (clastic) Sand bars, tidal channels. Generally coarsening upwards. High subsidence rate results in stacked reservoirs. Reservoir distribution dependent on wave and tide action. Prolific producers as a result of clean and continuous sand bodies. Shale layers may cause vertical barriers to fluid flow. 

In many cases faults will only restrict fluid flow, or they may be open i.e. non-sealing. Despite considerable efforts to predict the probability of fault sealing potential, a reliable method to do so has not yet emerged. Fault seal modelling is further complicated by the fact that some faults may leak fluids or pressures at a very small rate, thus effectively acting as seal on a production time scale of only a couple of years. As a result, the simulation of reservoir behaviour in densely faulted fields is difficult and predictions should be regarded as crude approximations only. 

When fluid flow in the reservoir is considered, it is necessary to estimate the viscosity of the fluid, since viscosity represents an internal resistance force to flow given a pressure drop across the fluid. Unlike liquids, when the temperature and pressure of a gas is increased the viscosity increases as the molecules move closer together and collide more frequently. 

Oil viscosity is an important parameter required in predicting the fluid flow, both in the reservoir and in surface facilities, since the viscosity is a determinant of the velocity with which the fluid will flow under a given pressure drop. Oil viscosity is significantly greater than that of gas (typically 0.2 to 50 cP compared to 0.01 to 0.05 cP under reservoir conditions). 

This section will consider the behaviour of the reservoir fluids in the bulk of the reservoir, away from the wells, to describe what controls the displacement of fluids towards the wells. Understanding this behaviour is important when estimating the recovery factor for hydrocarbons, and the production forecast for both hydrocarbons and water. In Section 9.0, the behaviour of fluid flow at the wellbore will be considered this will influence the number of wells required for development, and the positioning of the wells. 

One of the major differences in fluid flow behaviour for gas fields compared to oil fields is the mobility difference between gas and oil or water. Recall the that mobility is an indicator of how fast fluid will flow through the reservoir, and is defined as... 

On a microscopic scale, the most important equation governing fluid flow in the reservoir is Darcy s law, which was derived from the following situation. 

Figure 8.14 Single fluid flowing through a section of reservoir rock...

For a single fluid flowing through a section of reservoir rock, Darcy showed that the superficial velocity of the fluid (u) is proportional to the pressure drop applied (the hydrodynamic pressure gradient), and inversely proportional to the viscosity of the fluid. The constant of proportionality is called the absolute permeability which is a rock property, and is dependent upon the pore size distribution. The superficial velocity is the average flowrate... 

Introduction and Commercial Application Section 8.0 considered the dynamic behaviour in the reservoir, away from the influence of the wells. However, when the fluid flow comes under the influence of the pressure drop near the wellbore, the displacement may be altered by the local pressure distribution, giving rise to coning or cusping. These effects may encourage the production of unwanted fluids (e.g. water or gas instead of oil), and must be understood so that their negative input can be minimised. 

There will be some uncertainty as to the well initials, since the exploration and appraisal wells may not have been completed optimally, and their locations may not be representative of the whole of the field. A range of well initials should therefore be used to generate a range of number of wells required. The individual well performance will depend upon the fluid flow near the wellbore, the type of well (vertical, deviated or horizontal), the completion type and any artificial lift techniques used. These factors will be considered in this section. 

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... 

The purpose of the well completion is to provide a safe conduit for fluid flow from the reservoir to the flowline. The perforations in the casing are typically achieved by running a perforating gun into the well on electrical wireline. The gun is loaded with a charge which, when detonated, fires a high velocity jet through the casing and on into the formation for a distance of around 15-30 cm. In this way communication between the wellbore and the reservoir is established. Wells are commonly perforated after the completion has been installed and pressure tested. 

An interesting question that arises is what happens when a thick adsorbed film (such as reported at for various liquids on glass [144] and for water on pyrolytic carbon [135]) is layered over with bulk liquid. That is, if the solid is immersed in the liquid adsorbate, is the same distinct and relatively thick interfacial film still present, forming some kind of discontinuity or interface with bulk liquid, or is there now a smooth gradation in properties from the surface to the bulk region This type of question seems not to have been studied, although the answer should be of importance in fluid flow problems and in formulating better models for adsorption phenomena from solution (see Section XI-1). 

Chen S and Doolen G D 1998 Lattice Boltzmann method for fluid flows Ann. Rev. Fluid Meoh. 30 329-64... 

Note that in equation system (2.64) the coefficients matrix is symmetric, sparse (i.e. a significant number of its members are zero) and banded. The symmetry of the coefficients matrix in the global finite element equations is not guaranteed for all applications (in particular, in most fluid flow problems this matrix will not be symmetric). However, the finite element method always yields sparse and banded sets of equations. This property should be utilized to minimize computing costs in complex problems. 

Petera, J., Nassehi, V. and Pittman, J.F.T., 1989. Petrov-Galerkiii methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Ini.. J. Numer. Meth. Heat Fluid Flow 3, 205-222,... 

Townsend, P. and Webster, M. I- ., 1987. An algorithm for the three dimensional transient simulation of non-Newtonian fluid flow. In Pande, G. N. and Middleton, J. (eds). Transient Dynamic Analysis and Constitutive Laws for Engineering Materials Vul. 2, T12, Nijhoff-Holland, Swansea, pp. 1-11. 

Bell, B.C. and Surana, K. S, 1994. p-version least squares finite element formulations for two-dimensional, incompressible, non-Newtonian isothermal and non-isothcmial fluid flow. hit. J. Numer. Methods Fluids 18, 127-162. 

Keunings, R., 1989. Simulation of viscoelastic fluid flow. Tn Tucker, C. L. HI (ed.), Computer Modeling for Polymer Proces.sing, Chapter 9, Hanser Publishers, Munich, pp. 403-469. 

Nichols, B. D., Hirt, C. W. and Hitchkiss, R. S., 1980. SOLA-VOF a solution algorithm for transient fluid flow with multiple free surface boundaries. Los Alamos Scientific Laboratories Report No. La-8355, Los Alamos, NM. 

Fluid flowing towards the ap between blade and roll surfaces... 

Diffusional interception or Brownian motion, ie, the movement of particles resulting from molecular collisions, increases the probability of particles impacting the filter surface. Diffusional interception also plays a minor role in Hquid filtration. The nature of Hquid flow is to reduce lateral movement of particles away from the fluid flow lines. 

Capacity Limitations. The fluid flow capacity of a bubble tray may be limited by any of three principal factors. 

In the moving-bed system of Figure 7, soHd is moving continuously ia a closed circuit past fixed poiats of iatroduction and withdrawal of Hquid. The same results can be obtained by holding the bed stationary and periodically moving the positions at which the various streams enter and leave. A shift ia the positions of the iatroduction of the Hquid feed and the withdrawal ia the direction of fluid flow through the bed simulates the movement of soHd ia the opposite direction. 

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