Poiseuille approximation

In order to calculate the effective adsorption time teff which is necessary to compare the determined surface tension dependence with data from other methods an exact determination of the so-called deadtime is required. From some easy assumption the Poiseuille approximation results which yields [178]... 

Glass and silicon tubes with diameters of 79.9-166.3 iim, and 100.25-205.3 am, respectively, were employed by Li et al. (2003) to study the characteristics of friction factors for de-ionized water flow in micro-tubes in the Re range of 350 to 2,300. Figure 3.1 shows that for fully developed water flow in smooth glass and silicon micro-tubes, the Poiseuille number remained approximately 64, which is consistent with the results in macro-tubes. The Reynolds number corresponding to the transition from laminar to turbulent flow was Re = 1,700—2,000. 

Currently used equations for water flow in unsaturated soil are based on the assumption that soils are similar to a bundle of capillary tubes and that water flow can be approximated by the Hagen-Poiseuille equation.58 While it is obvious that the pore space in soil is not the same as a bundle of capillary tubes, the concept has proven highly useful and is currently used in mathematical descriptions of water flow in soil. 

Figure 5.16 shows the product Rej/ / as a function of Rev. For Rev less than approximately 2, the wall-injection has very little effect. In this case the wall friction approaches that of the Hagen-Poiseuille flow (i.e., Reyf = 24). For Rev greater than approximately 2, the V velocity serves to skew the axial velocity profile and thus affect the wall stress. For Rev greater than approximately 20, Re / approaches a linear relationship as... 

Laminar fluid flow in tubes has been described by Levich [ 3 ]. An entry length, le, is necessary to establish Poiseuille flow, given approximately by... 

There are some fundamental investigations devoted to analysis of the flow in tubular polymerization reactors where the viscosity of the final product has a limit (viscosity < >) i.e., the reactive mass is fluid up to the end of the process. As a zero approximation, flow can be considered to be one-dimensional, for which it is assumed that the velocity is constant across the tube cross-section. This is a model of an ideal plug reactor, and it is very far from reality. A model with a Poiseuille velocity profile (parabolic for a Newtonian liquid) at each cross-section is a first approximation, but again this is a very rough model, which does not reflect the inherent interactions between the kinetics of the chemical reaction, the changes in viscosity of the reactive liquid, and the changes in temperature and velocity profiles along the reactor. 

For a first approximation, when we assume a Poiseuille parabolic profile, the necessary length Lp is two times larger ... 

The influence of pores and leaks in a package on the total permeation depends primarily on whether or not the package is vacuum packed or at atmospheric pressure (Becker, 1965). For the case of vacuum packaging, the Hagen-Poiseuille equation for laminar gas flow can be used as a first approximation ... 

The first test is to compare a turbulent channel flow studied in the previous section and a laminar flow. A three dimensional Poiseuille flow in a pipe geometry was used as test case. The flow is laminar and the Reynolds number based on the bulk velocity and diameter is approximately 500. The bound-... 

Water is conducted to and across the leaves in the xylem. It then moves to the individual leaf cells by flowing partly apoplastically in the cell walls and partly symplastically (only short distances are involved, because the xylem ramifies extensively in a leaf). The water potential is usually about the same in the vacuole, the cytosol, and the cell wall of a particular mesophyll cell (see values in Table 9-3). If this were not the case, water would redistribute by flowing energetically downhill toward lower water potentials. The water in the cell wall pores is in contact with air, where evaporation can take place, leading to a flow along the cell wall interstices to replace the lost water. This flow can be approximately described by Poiseuille s law (Eq. 9.11), which indicates that a (very small) hydrostatic pressure decrease exists across such cell walls. 

Assume Poiseuille flow in the burner tube. The gas velocity is zero at the stream boundary (wall) and increases to a maximum in the center of the stream. The linear dimensions of the wall region of interest are usually very small in slow-burning mixtures such as methane and air, they are of the order of 1 mm. Since the burner tube diameter is usually large in comparison, as shown in Pig. 31, the gas velocity near the wall can be represented by an approximately linear vector profile. Figure 31 represents the conditions in the area where the flame is anchored by the burner rim. Further assume that the flow lines of the fuel jet are parallel to... 

Schofield showed in his model Equation 19.30 (or Equation 19.23) that Toc PAP for viscous flux and /oc AP for Knudsen flux, and obtained a correlation Jm = aP AP that approximates the Knudsen-Poiseuille transition of DGM. However, the Schofield model has the advantage that the exponent b indicates the extent of Knudsen diffusion and Poiseuifle flow contributes to the permeability, while such approximations are not possible from K(, and Bq used in the DGM. Schofield tested his model on membranes having different pore sizes ranging from 0.10 to 0.45 p,m and estimated the values of in a range of 0.1-0.6, which suggest that both the mechanisms (Knudsen and viscous) play an important role in MD flux. On the other hand, Schofield s model has two main disadvantages compared to the DGM model. One is, the components a and b are dependent upon the gas used. The other is, reference pressure P f is chosen in such a way that the dimensionless pressure becomes close to unity (Pr 1), and a is evaluated at P f, hence the parameters a and b also depend on the reference pressure chosen. 

Most attempts at describing the Knudsen-viscous Poiseuille transition involve a combination of Eqs. (9.2) and (9.6). For single gases this is a good approximation and after some reorganisation and integration of Eqs. (9.2) and (9.6) and assuming a linear pressure drop across the membrane this yields the expression for the total flux /j ... 

When air is absent in the pores and the pore diameter is substantially greater than the mean free path, X, of the diffusing water molecules (distance traveled between collisions with other molecules), water molecules collide more frequently with each other than with the pore walls and the Poiseuille flow relationship applies. The mean free path can be calculated using Eq. (7), where kb is the Boltzmann constant and a is the collision diameter of the water molecule. However, for pore diameters in the range of membranes that are suitable for OD applications and for a gas-phase pressure attributable to water vapor alone at ambient temperature (approximately 20 mm Hg, 2.7 kPa), the mean free path is significantly greater than the pore diameter. This results in more frequent collisions of the water molecules with the pore walls than with each other, and Knudsen diffusion predominates. 

Obviously, as t - cxd, this solution reverts to the steady-state Poiseuille profile. To obtain other details of this velocity profile, it is necessary to evaluate the infinite series numerically for each value of t and r. A typical numerical example of the results is shown in Fig. 3-10, where uz has been plotted versus r for several values of 7. It can be seen that the initial profile for 7 = 0.05 is flat, with uz approximately independent of r except for r very close to the tube walls. Right at the tube wall, uz = 0, and it can be seen that this manifestation of the no-slip condition gradually propagates across the tube by means of the diffusion process discussed earlier. The region in which the wall is felt increases in width at a rate proportional to Jvt as is typical of diffusion or conduction processes with diffusivity v. 

Hence, for the analysis in this section, we seek a steady-state solution of (2-110) with velocity approximated as the isothermal Poiseuille flow solution. Because the upstream temperature profile and boundary conditions on the temperature (or temperature gradients) are all independent of the azimuthal angle, we assume that the temperature 6 will be a function of only the cylindrical coordinates r and z. Thus Eq. (2 110) can be written in the form... 

Thus, in the limit R, -> 0, the problem reduces to a quasi-steady Poiseuille flow with an instantaneous pressure gradient sin 7. In view of the analysis in Chap. 3, this result is not surprising, but we do note that the solution (4-4) was easier to obtain in this case in which we directly approximated the differential equation rather than first solving the exact problem and then approximating the solution. 

As a starting point, we recall that the limit a/R = 0 corresponds to a straight circular tube, with the flow described by the Poiseuille flow solution w = (1 — r2), u = v = 0. In the present context, we consider small, but nonzero, values of a/R, and recognize the Poiseuille flow solution as a first approximation in an asymptotic approximation scheme. In particular, if we assume that a solution exists for u in the form of a regular asymptotic expansion,... 

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