Momentum laminar flow

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

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... 

Assuming laminar flow for a linear momentum equation in the a direction (an approximation from the Navier-Stokes equations) gives... 

Explain the concepts of momentum thickness" and displacement thickness for the boundary layer formed during flow over a plane surface. Develop a similar concept to displacement thickness in relation to heat flux across the surface for laminar flow and heat transfer by thermal conduction, for the case where the surface has a constant temperature and the thermal boundary layer is always thinner than the velocity boundary layer. Obtain an expression for this thermal thickness in terms of the thicknesses of the velocity and temperature boundary layers. 

Let us explain this assertion by an example of the developed laminar flow in a rectangular micro-channel. As is well known (Loitsianskii 1966) this problem reduces to integrating the momentum equation... 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

The principles of conservation of mass and momentum must be applied to each phase to determine the pressure drop and holdup in two phase systems. The differential equations used to model these principles have been solved only for laminar flows of incompressible, Newtonian fluids, with constant holdups. For this case, the momentum equations become... 

This expression applies to the transport of any conserved quantity Q, e.g., mass, energy, momentum, or charge. The rate of transport of Q per unit area normal to the direction of transport is called the flux of Q. This transport equation can be applied on a microscopic or molecular scale to a stationary medium or a fluid in laminar flow, in which the mechanism for the transport of Q is the intermolecular forces of attraction between molecules or groups of molecules. It also applies to fluids in turbulent flow, on a turbulent convective scale, in which the mechanism for transport is the result of the motion of turbulent eddies in the fluid that move in three directions and carry Q with them. 

The foregoing procedure can be used to solve a variety of steady, fully developed laminar flow problems, such as flow in a tube or in a slit between parallel walls, for Newtonian or non-Newtonian fluids. However, if the flow is turbulent, the turbulent eddies transport momentum in three dimensions within the flow field, which contributes additional momentum flux components to the shear stress terms in the momentum equation. The resulting equations cannot be solved exactly for such flows, and methods for treating turbulent flows will be considered in Chapter 6. 

Show how the Hagen-Poiseuille equation for the steady laminar flow of a Newtonian fluid in a uniform cylindrical tube can be derived starting from the general microscopic equations of motion (e.g., the continuity and momentum equations). 

The cases considered so far are ones in which the flow is turbulent and the velocity is nearly uniform over the cross section of the pipe. In laminar flow the curvature of the velocity profile is very pronounced and this must be taken into account in determining the momentum of the fluid. 

If the velocity had the uniform value u, the momentum flow rate would be mfpu2. Thus for laminar flow of a Newtonian fluid in a pipe the momentum flow rate is greater by a factor of 4/3 than it would be if the same fluid with the same mass flow rate had a uniform velocity. This difference is analogous to the different values of a in Bernoulli s equation (equation 1.14). 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

When the inlet length is expressed in terms of number of gap widths , the difference between the flow in a tube and the flow in an annulus of narrow gap differs only by 25% [(0.05 - 0.04)/0.05]. This situation is an indication that the growth of the laminar boundary layers from the wall to the center of the channel is similar in both cases. Because duct friction coefficients, a measure of momentum transfer, do not vary by more than a factor of 2 for ducts of regular cross sections when expressed in terms of hydraulic diameters, the use of the inlet length for tubes or parallel plates can be expected to be a reasonable approximation for the inlet lengths of other cross sections under laminar flow conditions. In the annular denuder, the dimensionless inlet length for laminar flow development, L, can be expressed as... 

Transport phenomena modeling. This type of modeling is applicable when the process is well understood and quantification is possible using physical laws such as the heat, momentum, or diffusion transport equations or others. These cases can be analyzed with principles of transport phenomena and the laws governing the physicochemical changes of matter. Transport phenomena models apply to many cases of heat conduction or mass diffusion or to the flow of fluids under laminar flow conditions. Equivalent principles can be used for other problems, such as the mathematical theory of elasticity for the analysis of mechanical, thermal, or pressure stress and strain in beams, plates, or solids. 

Laminar Flow, Re < 2000. A momentum balance on a differential fluid element yields the laminar velocity profile (see Table 2) and... 

Comparing these equations with the x- and y- Navier-Stokes equations for two-dimensional laminar flow shows that in turbulent flow extra terms arise due to the presence of the fluctuating velocity components. These extra terms, which arise because the Navier-Stokes equations contain nonlinear terms, are the result of the momentum transfer caused by the velocity fluctuating components and are often termed the turbulent or Reynolds stress terms because of their similarity to the viscous stress terms which arise due to momentum transfer on a molecular scale. This similarity can be clearly seen by noting that the x-wise momentum equation, for example, for laminar flow can be written as ... 

Consider next the y-momentum equation. Expressing this in dimensionless form in the same way as was done with the laminar flow equation and neglecting the appropriate mean flow terms, this equation then gives ... 

But v,2/u2 must have the same order of magnitude as the turbulence terms in the x-momentum equation so its order of magnitude is (8/L). Hence, it follows that in turbulent flow, as in laminar flow ... 

This is termed the boundary layer momentum integral equation. As previously mentioned, it is equally applicable to laminar and turbulent flow. In laminar flow, u is the actual steady velocity while in turbulent flow it is the time averaged value. 

The way in which the momentum integral equation is applied will be discussed in detail in the next chapter. Basically, it involves assuming the form of the velocity profile, i.e., of the variation of u with y in the boundary layer. For example, in laminar flow a polynomial variation is often assumed. The unknown coefficients in this assumed form are obtained by applying the known condition on velocity at the inner and outer edges of the boundary layer. For example, the velocity must be zero at the wall while at the outer edge of the boundary layer it must become equal to the freestream velocity, u. Thus, two conditions that the assumed velocity profile must satisfy are ... 

Other boundary conditions for laminar flow are discussed in the next chapter. In this way, the velocity profile is expressed in terms of u and 5. If the wall shearing stress rw is then also related to these quantities, the momentum integral equation (2.173) will allow the variation of S with x to be found for any specified variation of the freestream velocity, u. ... 

Using these and other boundary conditions, some of which for laminar flow will be discussed in the next chapter, the temperature distribution is expressed as a function of St. If qw is then related to the wall thermal conditions and St, the energy integral can be solved, using the solution to the momentum integral equation, to give... 

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