Fluid flow in a pipe

This is the basis for establishing the condition or type of fluid flow in a pipe. Reynolds numbers below 2000 to 2100 are usually considered to define laminar or thscous flow numbers from 2000 to 3000-4000 to define a transition region of peculiar flow, and numbers above 4000 to define a state of turbulent flow. Reference to Figure 2-3 and Figure 2-11 will identify these regions, and the friction factors associated with them [2]. 

The theory of pressure losses can be established by developing Bernoulli s theorem for the case of a pipe in which the work done in overcoming frictional losses is derived from the pressure available. For a fluid flowing in a pipe, the pressure loss will depend on various parameters. If... 

The velocity over the cross-section of a fluid flowing in a pipe is not uniform. Whilst this distribution in velocity over a diameter can be calculated for streamline flow this is not possible in the same basic manner for turbulent flow. 

Jones, C. and Hermges, G. Brit. J. Appl. Phys. 3 (1952) 283. The measurement of velocities for solid-fluid flow in a pipe. 

The conclusion that dimensionless numerical values are universal is valid only if a consistent system of units is used for all quantities in a given equation. If such is not the case, then the numerical quantities may include conversion factors relating the different units. For example, the velocity (F) of a fluid flowing in a pipe can be related to the volumetric flow rate (Q) and the internal pipe diameter (D) by any of the following equations ... 

Let us see how to represent changes in properties for a system volume to property changes for a control volume. Select a control volume (CV) to be identical to volume V t) at time t, but to have a different velocity on its surface. Call this velocity, w. Hence, the volume will move to a different location from the system volume at a later time. For example, for fluid flow in a pipe, the control volume can be selected as stationary (w = 0) between locations 1 and 2 (shown in Figure 3.4, but the system moves to a new location later in time. Let us apply the Reynolds transport theorem, Equation (3.9), twice once to a system volume, V(t), and second to a control volume, CV, where CV and V are identical at time t. Since Equation (3.9) holds for any well-defined volume and surface velocity distribution, we can write for the system... 

Shear stress at the pipe wall against flow characteristic for a non-Newtonian fluid flowing in a pipe... 

For steady flow in a pipe or tube the kinetic energy term can be written as m2/(2 a) where u is the volumetric average velocity in the pipe or tube and a is a dimensionless correction factor which accounts for the velocity distribution across the pipe or tube. Fluids that are treated as compressible are almost always in turbulent flow and a is approximately 1 for turbulent flow. Thus for a compressible fluid flowing in a pipe or tube, equation 6.4 can be written as... 

If one considers fluid flowing in a pipe, the situation is highly illustrative of the distinction between shear rate and flow rate. The flow rate is the volume of liquid discharged from the pipe over a period of time. The velocity of a Newtonian fluid in a pipe is a parabolic function of position. At the centerline the velocity is a maximum, while at the wall it is a minimum. The shear rate is effectively the slope of the parabolic function line, so it is a minimum at the centerline and a maximum at the wall. Because the shear rate in a pipe or capillary is a function of position, viscometers based around capillary flow are less useful for non-Newtonian materials. For this reason, rotational devices are often used in preference to capillary or tube viscometers. 

In the earlier days of the petroleum age, many pipe experiments were conducted. In the quest for the magic formula, one was found to be the closest to utopia even to this day, called the Darcy formula. The Darcy formula is derived manually from the Bernoulli principle, which simply describes the energy balance between two points of a fluid flowing in a pipe. This energy equation is also applicable to a static condition of no flow between the two points. The classic Bernoulli energy equation [1] is ... 

Figure 6.1 Bernoulli analysis of simple fluid flow in a pipe.

Please note the Cv value used should be at a midrange of the valve s opening. The vendor s maximum Cv value should never be used for controlling a fluid flow in a piping system. Rather, a midrange C valve opening should be used to size the valve. Good... 

For a fluid flowing in a pipe of radius a, length l with a central core of radius r, a force balance gives ... 

Velocity meters measure the velocity v of fluid flow in a pipe of known cross section, thus yielding a signal linearly proportional to the volume flow rate Q. Mass meters provide signals directly proportional to the mass flow rate m = pQ, where p is the mass density. Coriolis meters, which are true mass meters, can be used only for liquids. Thermal-type flow meters use a heating element and determine the rate of heat transfer, which is proportional to the mass flow rate. This type of device is used mostly for gas measurements, but liquid flow designs are also available. 

To illustrate the steps described above, we will consider the problem already introduced at the begirming of this chapter, which was concerned with the pressure drop and heat transfer of an incompressible Newtonian fluid flowing in a pipe. 

The Reynolds number is a dimensionless group defined for a fluid flowing in a pipe as... 

The amount of mass flowing through a cross section of a flow device per unit time is called the mass flow rate, and is denoted by rii. A fluid may flow in and out of a control volume through pipes or ducts. The mass flow rate of a fluid flowing in a pipe or duct is proportional to the cross-sectional area of... 

The flow of a fluid through a pipe or duct can often be approximated to be one dimensional. That is, the properties can be assumed to vary in one direction only (llie direction of flow). As a result, all properties arc assumed to be uniform at any cross section normal to the flow direction, and the properties are assumed to hAvQbidk average values over the. entire cross section. Under the one-dimensional flow approximation, the mass flow rate of a fluid flowing in a pipe or duct can be expressed as (big. 1-16)... 

Will the kinetic energy per unit mass of an incompressible fluid flowing in a pipe increase, decrease, or remain the same if the pipe diameter is increased at some place in the line ... 

To understand the mechanism of the turbulent mixing process occurring in pipe reactors, we have to consider first some of the properties of fluid flow in pipes. Resistance to fluid flow in a pipe has two components, the viscous friction of the fluid itself within the pipe, which increases as the fluid viscosity increases, and the pressure differential caused by a liquid level difference or a pressure difference between the two vessels. 

Making a force balance on the fluid flowing in a pipe of diameter D over a length L yields... 

The molecules of these other species get in the way of the molecules of species 1 (say) and, in effect, exert a drag on them in much the same way that a pipe exerts a frictional drag on the fluid flowing through it. The analogy with pipe-flow does not end here an analysis of diffusion may be carried out in essentially the same way that we may derive, for example, Poiseuille s equation for the rate of fluid flow in a pipe—through the application of Newton s second law. 

Input Data and Computer Output for Two-Phase Fluid Flow in a Pipe Line... 

FLOW PATTERNS OF COMPRESSIBLE FLUID FLOW IN A PIPE (Subsonic, Sonic Supersonic). 

Turbulent Flow. The dimensionless Reynolds number Re of a Newtonian fluid flowing in a pipe of radius R can be defined by... 

Bourdon gauges are commonly used to measure the static pressure of a fluid flowing in a pipe. To reduce fhe pofen-tial for error, it is important that static pressure taps are... 

To have a fluid in convective flow usually requires the fluid to be flowing by another immiscible fluid or by a solid surface. An example is a fluid flowing in a pipe, where part... 

Reynolds analogy. Reynolds was the first to note similarities in transport processes and relate turbulent momentum and heat transfer. Since then, mass transfer has also been related to momentum and heat transfer. We derive this analogy from Eqs. (6.1-4)-(5.1-6) for turbulent transport. For fluid flow in a pipe for heat transfer from the fluid to the wall, Eq. (6.1-5) becomes as follows, where z is distance from the wall ... 

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