Compressible flow cross section

Flow behaviour of polymer melts is still difficult to predict in detail. Here, we only mention two aspects. The viscosity of a polymer melt decreases with increasing shear rate. This phenomenon is called shear thinning [48]. Another particularity of the flow of non-Newtonian liquids is the appearance of stress nonnal to the shear direction [48]. This type of stress is responsible for the expansion of a polymer melt at the exit of a tube that it was forced tlirough. Shear thinning and nonnal stress are both due to the change of the chain confonnation under large shear. On the one hand, the compressed coil cross section leads to a smaller viscosity. On the other hand, when the stress is released, as for example at the exit of a tube, the coils fold back to their isotropic confonnation and, thus, give rise to the lateral expansion of the melt. 

As shown in Fignre 17.12, the flow of water is in a serpentine configuration the direction of the arrows indicates the direction of flow. This configuration allows the satisfaction of the contact time. Since the channel flow cross section is compressed into a smaller cross-section area, this scheme approximates a ping flow. Note that the figure shown is only one of the contact tank compartments in this plant. One of the other compartments is partly shown at the top of the figure. 

In many cases solids progressively soften during extraction. This is particularly true when solids tend to decompose because of solubilization reactions. Flow pressure drop can strongly compact such solids. In such cases, compression permeability mensurements should be used to determine the flow resistance characteristics of the solid bed. If flow-induced compaction is excessive it may be necessary to reduce bed depth or use a larger than normal flow cross-sectional area. 

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

Adiabatic Frictionless Nozzle Flow In process plant pipelines, compressible flows are usually more nearly adiabatic than isothermal. Solutions for adiabatic flows through frictionless nozzles and in channels with constant cross section and constant friction factor are readily available. 

Figure 2.3. A rigid piston drives a shock wave into compressible fluid in an imaginary flow tube with unit cross-sectional area. The shock wave moves at velocity U into fluid with initial state 0, which changes discontinuously to state 1 behind the shock wave. Particle velocity u is identical to the piston velocity.

For the discharge of compressible fluids from the end of a short aiping length into a larger cross section, such as a larger pipe, vessel, or atmosphere, the flow is considered adiabatic. Corrections are applied to the Darcy equation to compensate for fluid property changes due to the expansion of the fluid, and these are known as Y net expansion factors [3]. The corrected Darcy equation is ... 

Reciprocating compressors compress gases by a piston moving backwards and forwards in a cylinder. Valves control the flow of low-pressure gas into the cylinder and high-pressure gas out of the cylinder. The mechanical work to compress a gas is the product of the external force acting on the gas and the distance through which the force moves. Consider a cylinder with cross-sectional area A containing a gas to be compressed by a piston. The force exerted on the gas is the product of the pressure (force per unit area) and the area A of the piston. The distance the piston travels is the volume V of the cylinder divided by the area A. Thus ... 

Consider a volume of gas flowing into the compressor. Compression work W, is required to force the gas into the system. The constant force exerted on the gas is P At, where A is the cross-sectional area of the inlet duct. The distance through which the gas is forced as it enters the system is - V / A. The negative value of - V /A results from the force acting from the surroundings on the system. Thus ... 

Flere APf(, is the pressure drop due to gas only flow (i.e., the gas flowing alone in the full pipe cross section). Note that if the pressure drop is less than about 30% of Pi, the incompressible flow equations can be used to determine APf(j by using the average gas density. Otherwise, the compressibility must be considered and the methods in Chapter 9 used to determine A P(q. The pressure drop is related to the pressure ratio P /Pi by... 

When a compressible fluid, ie a gas, flows from a region of high pressure to one of low pressure it expands and its density decreases. It is necessary to take this variation of density into account in compressible flow calculations. In a pipe of constant cross-sectional area, the falling density requires that the fluid accelerate to maintain the same mass flow rate. Consequently, the fluid s kinetic energy increases. 

Due to the change in the average velocity w, it is more convenient in calculations for compressible flow in pipes of constant cross-sectional area to work in terms of the mass flux G. This is the mass flow rate per unit flow area and is sometimes called the mass velocity. If the mass flow rate is constant, as will usually be the case, then G is constant when the area is constant. The relationship between G and u is given by... 

In principle, this is the same as for single-phase flow. For example in steady, fully developed, isothermal flow of an incompressible fluid in a straight pipe of constant cross section, friction has to be overcome as does the static head unless the pipe is horizontal, however there is no change of momentum and consequently the accelerative term is zero. In the case of compressible flow, the gas expands as it flows from high pressure to low pressure and, by continuity, it must accelerate. In Chapter 6 this was noted as an increase in the kinetic energy. 

In the normal mechanism the reaction runs simultaneously over the entire cross-section of the tube the curves presented in 11.5 illustrate the change in pressure, temperature and composition. We axe fully justified in using an approach in which we consider all quantities characterizing the state to be dependent only on the distance of the point from the shock wave front. In the case of the SM, in the mechanism which we have proposed here for rough tubes, in each intermediate cross-section part of the substance has not reacted at all (the core of the flow) and part of the substance has completely reacted (the peripheral layers) the states of the two parts— composition, temperature, specific volume—are sharply different. The only common element is the pressure, which is practically identical in a given cross-section in the two parts of the flow (in the compressed, but unreacted mixture and in the combustion products), but which changes as combustion progresses from one cross-section to another. 

We now analyze mechanical-displacement flow in a straight channel of constant cross-sectional area, as shown in Fig. 4.11 (with the upper plate at rest). A column of compacted solids of length L is compressed between two rams. The one on the left exerts a force Fo on the solids and it is opposed by a smaller force FL on the right. Thus, friction on the channel walls also opposes the applied resultant force. 

In the extruder, not only shear flow is present, but also extensional flow occurs as well. This is illustrated in Fig. 3.20 for the deformation of a fluid element. Wherever cross-sections narrow, such as at the tips or between kneading blocks and the wall, the fluid elements are compressed and extended. This effect is particularly relevant for non-homogenous polymer melts, e.g., immiscible blends, in which the disperse phase can be split by extensional deformation. For more details, see Chapter 9. 

Equations (7.14), (7.15), and (7.20), combined with the relations between the thermodynamic properties at constant entropy, determine how the velocity varies with cross-sectional area of the nozzle. The variety of results for compressible fluids (e.g., gases), depends in part on whether the velocity is below or above the speed of sound in the fluid. For subsonic flow in a converging nozzle, the velocity increases and pressure decreases as the cross-sectional area diminishes. In a diverging nozzle with supersonic flow, the area increases, but still the velocity increases and the pressure decreases. The various cases are summarized elsewhere.t We limit the rest of this treatment of nozzles to application of the equations to a few specific cases. 

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