Laminar flow pressure drop

As in the case of Newtonian fluids, one of the most important practical problems involving non-Newtonian fluids is the calculation of the pressure drop for flow in pipelines. The flow is much more likely to be streamline, or laminar, because non-Newtonian fluids usually have very much higher apparent viscosities than most simple Newtonian fluids. Furthermore, the difference in behaviour is much greater for laminar flow where viscosity plays such an important role than for turbulent flow. Attention will initially be focused on laminar-flow, with particular reference to the flow of power-law and Bingham-plastic fluids. 

Fig. 6. Examples of types of meshes developed to resolve laminar flow around particles (a) Chimera grid. Reprinted, with permission, from the Annual Review of Fluid Mechanics, Volume 31 1999 by Annual Reviews www.annualreviews.org (b) Unstructured grid with layers of prismatic cells on particle surfaces. Reprinted from Chemical Engineering Science, Vol. 56, Calis et al., CFD Modeling and Experimental Validation of Pressure Drop and Flow Profile in a Novel Structured Catalytic Reactor Packing, pp. 1713-1720, Copyright (2001), with permission from Elsevier.

Prandtl-Nikuradse treatment and considered the liquid film to involve a laminar and a turbulent layer. Anderson and Mantzouranis work is an extension of the equations presented by Dukler and Bergelin (D5), and makes use of the von Karman universal velocity profile. In general, if any two of pressure drop, liquid flow rate, or film thickness are known, the third quantity can be calculated. 

In general, problems of turbulent flow have been too difficult to study mathematically, at least in their initial stages. If, therefore, one is to develop a relationship between pressure drop and flow rate which is valid for the turbulent- as well as the laminar-flow region, the approach must be empirical—and that of dimensional analysis would seem to be eminently suited to the problem at hand. 

V/D of interest, it may be used for the calculation of the relationship between pressure drop and flow rate in a pipe line of any size, provided only that the flow is laminar and that the laboratory data are at the correct temperature. 

In laminar flow elements, the pressure drop and flow are in a linear relationship. The laminar flow element can be used in combination with either a differential-pressure- or a thermal-type flow detector. These flowmeters provide better rangeability at about the same cost as sonic nozzles. 

The size of resin hends nad the pressure drop generally used in industrial applications result in laminar flow conditions in fixed beds of resin. The characteristic of such flow is that (he pressure drop varies lineally with flow rale. Actual measurements of pressure drop over beds of reain include the effect of flow distribution manifolds which may show quadratic dependence of pressure drop against flow rate. 

With the average velocity and channel size now expressed in terms of the measurable parameters Vq, Dp, and e, thp channel model can be used to predict the form of the correlation for pressure drop. For flow at very low Reynolds numbers, the pressure drop should vary with the first power of the velocity and inversely with the square of the channel size, in accordance with the Hagen-Poiseuille equation for laminar flow in straight tubes, Eq. (5.16). The equations for V and ), are used in Eq. (5.16), and a correction factor Aj is added to account for the fact that the channels are actually tortuous and not straight and parallel ... 

When the pressure drop along a macropore is large, a pressure-induced laminar flow can arise. This additional contribution to mass transfer is know as Poiseuille flow. 

An understanding of the energy requirements of static mixers is necessary both with respect to the establishment of installed pressure drop and flow rate requirements and to objective performance comparisons of different mixer types. In laminar flow, the pressure drop-flow rate characteristics of static mixers are simple and analogous to pipe flow. 

The utilization of a three-parameter rheological model to describe the rheology of clay suspensions conducted to a development of new dimensionless parameters named here as generalized Reynolds and Hedstrom numbers permits the analytical relationship between pressure drop and flow rates for the laminar flow in annular flow. 

A further factor which limits particle size is the pressure drop over the column. This is, of course, increased by small particles. Since with a low pressure drop the flow tends much more towards the laminar, it is desirable to operate the entire column with a flow which supplies optimum separation efficiency. 

For evaluation of pressure drop for flow through a pipe one needs to know the friction factor. In laminar flow regime the friction factor is a function of Reynolds number only, and in the case of turbulent flow the friction factor is a function of Reynolds number and also the relative roughness factor. Blasius showed analytically... 

The first term (AQ) is the pressure drop due to laminar flow, and the FQ term is the pressure drop due to turbulent flow. The A and F factors can be determined by well testing, or from the fluid and reservoir properties, if known. 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

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 ... 

Friction Coefficient. In the design of a heat exchanger, the pumping requirement is an important consideration. For a fully developed laminar flow, the pressure drop inside a tube is inversely proportional to the fourth power of the inside tube diameter. For a turbulent flow, the pressure drop is inversely proportional to D where n Hes between 4.8 and 5. In general, the internal tube diameter, plays the most important role in the deterrnination of the pumping requirement. It can be calculated using the Darcy friction coefficient,, defined as... 

Steady-state, laminar, isothermal flow is assumed. For a given viscometer with similar fluids and a constant pressure drop, the equation reduces to 77 = Kt or, more commonly, v = r /p = Ct where p is the density, V the kinematic viscosity, and C a constant. Therefore, viscosity can be determined by multiplying the efflux time by a suitable constant. 

Noncircular Channels Calciilation of fric tional pressure drop in noncircular channels depends on whether the flow is laminar or tumu-lent, and on whether the channel is full or open. For turbulent flow in ducts running full, the hydraulic diameter shoiild be substituted for D in the friction factor and Reynolds number definitions, Eqs. (6-32) and (6-33). The hydraiilic diameter is defined as four times the channel cross-sectional area divided by the wetted perimeter. For example, the hydraiilic diameter for a circiilar pipe is = D, for an annulus of inner diameter d and outer diameter D, = D — d, for a rectangiilar duct of sides 7, h, Dij = ah/[2(a + h)].T ie hydraulic radius Rii is defined as one-fourth of the hydraiilic diameter. 

The hydrauhc diameter method does not work well for laminar flow because the shape affects the flow resistance in a way that cannot be expressed as a function only of the ratio of cross-sectional area to wetted perimeter. For some shapes, the Navier-Stokes equations have been integrated to yield relations between flow rate and pressure drop. These relations may be expressed in terms of equivalent diameters Dg defined to make the relations reduce to the second form of the Hagen-Poiseulle equation, Eq. (6-36) that is, Dg (l2SQ[LL/ KAPy. Equivalent diameters are not the same as hydraulie diameters. Equivalent diameters yield the correct relation between flow rate and pressure drop when substituted into Eq. (6-36), but not Eq. (6-35) because V Q/(tiDe/4). Equivalent diameter Dg is not to be used in the friction factor and Reynolds number ... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

Steady state, fuUy developed laminar flows of viscoelastic fluids in straight, constant-diameter pipes show no effects of viscoelasticity. The viscous component of the constitutive equation may be used to develop the flow rate-pressure drop relations, which apply downstream of the entrance region after viscoelastic effects have disappeared. A similar situation exists for time-dependent fluids. 

Economic Pipe Diameter, Laminar Flow Pipehnes for the transport of high-viscosity liquids are seldom designed purely on the basis of economics. More often, the size is dictated oy operability considerations such as available pressure drop, shear rate, or residence time distribution. Peters and Timmerhaus (ibid.. Chap. 10) provide an economic pipe diameter chart for laminar flow. For non-Newtouiau fluids, see SkeUand Non-Newtonian Flow and Heat Transfer, Chap. 7, Wiley, New York, 1967). 

---