Reynolds number friction factor, differences

This second method does not lend itself to the development of quantitative correlations which are based solely on true physical properties of the fluids and which, therefore, can be measured in the laboratory. The prediction of heat transfer coefficients for a new suspension, for example, might require pilot-plant-scale turbulent-flow viscosity measurements, which could just as easily be extended to include experimental measurement of the desired heat transfer coefficient directly. These remarks may best be summarized by saying that both types of measurements would have been desirable in some of the research work, in order to compare the results. For a significant number of suspensions (four) this has been done by Miller (M13), who found no difference between laboratory viscosities measured with a rotational viscometer and those obtained from turbulent-flow pressure-drop measurements, assuming, for suspensions, the validity of the conventional friction-factor—Reynolds-number plot.11 It is accordingly concluded here that use of either type of measurement is satisfactory use of a viscometer such as that described by Orr (05) is recommended on the basis that fundamental fluid properties are more readily determined under laminar-flow conditions, and a means is provided whereby heat transfer characteristics of a new suspension may be predicted without pilot-plant-scale studies. 

The main value of data describing turbulent energy requirements is in the computation of pressure drop-flow rate characteristics for installed plant but there are also examples of performance evaluation using energy data -. As with laminar flow characteristics, although different, those for turbulent flow are relatively simple and easily described in terms of the friction factor-Reynolds number relationship used to describe empty tube. 

There is a possible source of confusion in the definitions of friction factor used in different works but in all cases the product of friction factor (0) and Reynolds number (Re) is approximately constant. For the definition of friction factor used by Wilkinson and Cliff and more recently by Pahl and Maschelknautz the empty pipe value of the product 0 Re is 64. 

The friction factor, which is plotted against the modified Reynolds number, is Pi/pu, where R is the component of the drag force per unit area of particle surface in the direction of motion. R can be related to the properties of the bed and pressure gradient as follows. Considering the forces acting on the fluid in a bed of unit cross-sectional area and thickness /, the volume of particles in the bed is /(I — e) and therefore the total surface is 5/(1 — e). Thus the resistance force is R SH — e). This force on the fluid must be equal to that produced by a pressure difference of AP across the bed. Then, since the free cross-section of fluid is equal to e ... 

These dimensionless groups also appear in empirical correlations of the turbulent flow region. Although even in the approximate Eq. (9) of Table 6.7, group He appears to affect the friction factor, empirical correlations such as Figure 6.5(b) and the data analysis of Example 6.10 indicate that the friction factor is determined by the Reynolds number alone, in every case by an equation of the form, / = 16/Rc, but with Re defined differently for each model. Table 6.7 collects several relations for laminar flows of fluids. 

It is stated slightly differently, with f being the friction factor (recall that both heat and mass transfer have a Reynolds number dependency). For mass transfer, they developed the following expression ... 

Microscale heat transfer has attracted researchers in the last decade, particularly due to developments and current needs in the small-scale electronics, aerospace, and bioengineering industries. Although some of the fundamental differences between micro and macro heat transfer phenomenon have been identified, there still is a need for further experimental, analjdical and numerical studies to clarify the points that are not yet understood, such as the effect of axial conduction, friction factors, compressibility effects, critical Reynolds number, and accommodation coefScients less then unity. 

Example 6.10 indicate that the friction factor is determined by the Reynolds number alone, in every case by an equation of the form, / = 16/Re, but with Re defined differently for each model. Table 6.6 collects several relations for laminar flows of fluids. 

FIGURE 23.7 Variation of the friction factor with Reynolds number. Two different flat sheet BOs were tested. (From Zhang, Q. and Cussler, E.L., J. Membr. Sci., 23, 333, 1985. With permission.)... 

If the characteristic linear dimension of the flow field is small enough, then the measured hydrodynamic data differ from those predicted by the Navier-Stokes equations [79]. With respect to the value in macrocharmels, in microchannels (around 50 microns of section) (i) the friction factor is about 20-30% lower, (ii) the critical Reynolds number below which the flow remains laminar is lower (e.g., the change to turbulent flow occurs at lower linear velocities) and (iii) the Nusselt number, for example, heat transfer characteristics, is quite different [80]. The Nusselt number for the microchannel is lower than the conventional value when the flow rate is small. As the flow rate through the microchannel is increased, the Nusselt number significantly increases and exceeds the value for the fully developed flow in the conventional channel. These effects have been investigated extensively in relation to the development of more efficient cooling devices for electronic applications, but have clear implications also for chemical applications. 

A detailed review of other explicit equations is given by Gregory and Fogarasi [7]. Different piping materials are often used in the chemical process industries, and at a high Reynolds number, the friction factor is affected by the roughness of the surface. This is measured as the ratio e/D of projections on the surface to the diameter of the pipe. Glass and plastic pipe essentially have e = 0. Values of e are shown in Table 3-3. 

Table 3-17 gives the Reynolds number, friction factor, and pressure drop of catalyst pellets of 0.25 inch and at different particle length. Table 3-18 shows a typical input data and computer output with PL = 0.25 inch. The simulation exercise gives a pressure drop of 68.603 Ib/in. The results show that the pressure drop in a packed bed depends on size and shape of the particles. 

The asymptotic nature of the friction factor is clearly brought out in Fig. 10.24, which shows the measured fully established friction factors taken in three tubes of differing diameters as a function of the Weissenberg number based on the Powell-Eyring relaxation time for fixed values of the Reynolds number for aqueous solutions of polyacrylamide. The critical Weissenberg number for friction (Ws), is seen to be on the order of 5 to 10. When the Weissenberg number exceeds 10, it is clear that the fully developed friction factor is a function only of the Reynolds number. 

In the experiments, the fluid flow rates on both sides of the exchanger are set constant at predetermined values. Once the steady-state conditions are achieved, fluid temperatures upstream and downstream of the test section on both sides are measured, as well as all pertinent measurements for the determination of the fluid flow rates. The upstream pressure and pressure drop across the core on the unknown side are also recorded to determine the hot friction factors.7 The tests are repeated with different flow rates on the unknown side to cover the desired range of the Reynolds number. 

This equation is an inverted form of the core pressure drop in Eq. 17.65. For the isothermal pressure drop data, p, = p = l/(l/p)m. The friction factor thus determined includes the effects of skin friction, form drag, and local flow contraction and expansion losses, if any, within the core. Tests are repeated with different flow rates on the unknown side to cover the desired range of the Reynolds number. The experimental uncertainty in the/factor is usually within 5 percent when Ap is measured accurately within 1 percent. 

However, the curve of the sphere drag coefficient has some marked differences from the friction factor plot. It does not continue smoothly to higher and higher Reynolds numbers, as does the / curve instead, it takes a sharp drop at an of about 300,000. Also it does not show the upward jump that characterizes the laminar-turbulent transition in pipe flow. Both differences are due to the different shapes of the two systems. In a pipe all the fluid is in a confined area, and the change from laminar to turbulent flow affects all the fluid (except for a very thin film at the wall). Around a sphere the fluid extends in all directions to infinity (actually the fluid is not infinite, but if the distance to the nearest obstruction is 100 sphere diameters, we may consider it so), and no matter how fast the sphere is moving relative to the fluid, the entire fluid cannot be set in turbulent flow by the sphere. Thus, there cannot be the sudden laminar-turbulent transition for the entire flow, which causes the jump in Fig. 6.10. The flow very near the sphere, however, can make the sudden switch, and the switch is the cause of the sudden drop in Q at =300,(300. This sudden drop in drag coefficient is discussed in Sec. 11.6. Leaving until Chaps. 10 and 11 the reasons why the curves in Fig. 6.22 have the shapes they do, for now we simply accept the curves as correct representations of experimental facts and show how to use them to solve various problems. 

As in the case of flow in pipes, there are several different friction factors in common usage for flowing porous media, all differing by a constant. The choice between these is completely arbitrary in this text we drop the 5 in Eq. 12.9 and the in Eq. 12.10 to find our working forms of the friction factor and Reynolds number for porous media ... 

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