Developing Flows

Using a known solution at the inlet. To provide an example for tins option, let us consider the finite element scheme described in Section 2.1. Assuming a fully developed flow at the inlet to the domain shown in Figure 3.3, v, (dvy/dy) = 0 and by the incompressibility condition (dvx/dx) - 0, x derivatives of all stress components are also zero. Therefore at the inlet the components of the equation of motion (3.25) are reduced to... 

Typically the exit velocity in a flow domain is unknown and hence the prescription of Dirichlet-type boundary conditions at the outlet is not possible. However, at the outlet of sufficiently long domains fully developed flow conditions may be imposed. In the example considered here these can be written as... 

Both wetted-sensor and clamp-on Doppler meters ate available for Hquid service. A straight mn of piping upstream of the meter and a Reynolds number of greater than 10,000 ate generally recommended to ensure a weU-developed flow profile. Doppler meters ate primarily used where stringent accuracy and repeatabiHty ate not requited. Slurry service is an important appHcation area. 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

Since = 0 at y = ff/2, the continuity equation integrates to = 0. This is a direct result of the assumption of fuUy developed flow. 

The pressure drop due to fricdional losses is proportional to pipe length L for fully developed flow and may be denoted as the (positive) quantity AP Pj - P2. 

Entrance and Exit Effects In the entrance region of a pipe, some distance is required for the flow to adjust from upstream conditions to the fuUy developed flow pattern. This distance depends on the Reynolds number and on the flow conditions upstream. For a uniform velocity profile at the pipe entrance, the computed length in laminar flow required for the centerline velocity to reach 99 percent of its fully developed value is (Dombrowski, Foumeny, Ookawara and Riza, Can. J. Chem. Engr, 71, 472 76 [1993])... 

The correclion (Fig- 6-14rZ) accounts for the extra losses due to developing flow in the outlet tangent of the pipe, of length L. The total loss ror the bend plus outlet pipe includes the bend loss K plus the straight pipe frictional loss in the outlet pipe 4fL /D. Note that = 1 for L /D greater than the termination of the curves on Fig. 6-14d, which indicate the distance at which fully developed flow in the outlet pipe is reached. Finally, the roughness correction is... 

Liquids and Gases For cocurreut flow of liquids and gases in vertical (upflow), horizontal, and inclined pipes, a veiy large literature of experimental and theoretical work has been published, with less work on countercurrent and cocurreut vertical downflow. Much of the effort has been devoted to predicting flow patterns, pressure drop, and volume fractious of the phases, with emphasis on hilly developed flow. In practice, many two-phase flows in process plants are not fully developed. 

In some convection equations, such as for turbulent pipe flow, a special correction factor is used. This factor relates to the heat transfer conditions at the flow inlet, where the flow has not reached its final velocity distribution and the boundary layer is not fully developed. In this region the heat transfer rate is better than at the region of fully developed flow. 

Thin boundary layers provide the highest values of heat flow density. Because the boundary layer gradually develops upstream from the inlet point, the heat flow density is highest at the inlet point. Heat flow density-decreases and achieves its final value in the region of fully developed flow. The correction is noted in the equations by means of the quotients d/.L and d/x. 

A5. Alessandrini, A., Bertoletti, S., Gaspari, G. P., Lombardi, C., Soldaini, G., and Zavattarelli, R., Critical heat flux data for fully developed flow of steam and water mixtures in round vertical tubes with an intermediate nonheated section, CISE-R69 (1963). 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

For the common problem of heat transfer between a fluid and a tube wall, the boundary layers are limited in thickness to the radius of the pipe and, furthermore, the effective area for heat flow decreases with distance from the surface. The problem can conveniently be divided into two parts. Firstly, heat transfer in the entry length in which the boundary layers are developing, and, secondly, heat transfer under conditions of fully developed flow. Boundary layer flow is discussed in Chapter 11. 

For the region of fully developed flow in a pipe of length L, diameter d and radius r, the rate of flow of heat Q through a cylindrical surface in the fluid at a distance y from the wall is given by ... 

This expression is applicable only to the region of fully developed flow. The heat transfer coefficient for the inlet length can be calculated approximately, using the expressions given in Chapter 11 for the development of the boundary layers for the flow over a plane surface. It should be borne in mind that it has been assumed throughout that the physical properties of the fluid are not appreciably dependent on temperature and therefore the expressions will not be expected to hold accurately if the temperature differences are large and if the properties vary widely with temperature. 

When a fluid flowing with a uniform velocity enters a pipe, a boundary layer forms at the walls and gradually thickens with distance from the entry point. Since the fluid in the boundary layer is retarded and the total flow remains constant, the fluid in the central stream is accelerated. At a certain distance from the inlet, the boundary layers, which have formed in contact with the walls, join at the axis of the pipe, and, from that point onwards, occupy the whole cross-section and consequently remain of a constant thickness. Fulty developed flow then exists. If the boundary layers are still streamline when fully developed flow commences, the flow in the pipe remains streamline. On the other hand, if the boundary layers are already turbulent, turbulent flow will persist, as shown in Figure 11.8. 

Under streamline conditions, the velocity at the axis will increase from a value u at the inlet to a value 2u where fully-developed flow exists, as shown in Figure 11.9, because the mean velocity of flow u in the pipe is half of the axial velocity, from equation 336. 

The velocity distribution and frictional resistance have been calculated from purely theoretical considerations for the streamline flow of a fluid in a pipe. The boundary layer theory can now be applied in order to calculate, approximately, the conditions when the fluid is turbulent. For this purpose it is assumed that the boundary layer expressions may be applied to flow over a cylindrical surface and that the flow conditions in the region of fully developed flow are the same as those when the boundary layers first join. The thickness of the boundary layer is thus taken to be equal to the radius of the pipe and the velocity at the outer edge of the boundary layer is assumed to be the velocity at the axis. Such assumptions are valid very close to the walls, although significant errors will arise near the centre of the pipe. 

Calculate the thickness of the laminar sub-layer when benzene flows through a pipe 50 mm in diameter at 2 1/s. What is the velocity of the benzene at the edge of the laminar sub-layer Assume that fully developed flow exists within the pipe and that for benzene, p — 870 kg/m3 and p = 0.7 mN s/m2. 

The application of the analogies to the problems of heat and mass transfer to plane surfaces and to pipe walls for fully developed flow is discussed later. 

In fully developed flow, equations 12.102 and 12.117 can be used, but it is preferable to work in terms of the mean velocity of flow and the ordinary pipe Reynolds number Re. Furthermore, the heat transfer coefficient is generally expressed in terms of a driving force equal to the difference between the bulk fluid temperature and the wall temperature. If the fluid is highly turbulent, however, the bulk temperature will be quite close to the temperature 6S at the axis. 

Taking the fluid properties at 310 K and assuming that fully developed flow exists, an approximate solution will be obtained neglecting the variation of properties with temperature. 

Indicate the conditions under which this is consistent with the predicted value A k = 4.1 for fully developed flow... 

Fryer, P.J. 210,229 Fuller, E. N. 584,655 Fully developed flow 61.681 Fundamental units, choice of 12 Further reading, flow and pressure measurement (Chapter 6) 272... 

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