Laminar flow conservation equations

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 ... 

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. 

Of major interest concerning these problems are influences of turbulence in spray combustion [5]. The turbulent flows that are present in the vast majority of applications cause a number of types of complexities that we are ill-equipped to handle for two-phase systems (as we saw in Section 10.2.1). For nonpremixed combustion in two-phase systems that can reasonably be treated as a single fluid through the introduction of approximations of full dynamic (no-slip), chemical and interphase equilibria, termed a locally homogeneous flow model by Faeth [5], the methods of Section 10.2 can be introduced reasonably successfully [5], but for most sprays these approximations are poor. Because of the absence of suitable theoretical methods that are well founded, we shall not discuss the effects of turbulence in spray combustion here. Instead, attention will be restricted to formulations of conservation equations and to laminar examples. If desired, the conservation equations to be developed can be considered to describe the underlying dynamics on which turbulence theories may be erected—a highly ambitious task. 

For single-phase systems involving laminar flows the conservation equations are firmly established. The mass and momentum conservation equations are respectively given by (Bird et al., (1960) ... 

The governing conservation equations and the boundary conditions in non-isothermal laminar tube flow can be formulated as follows ... 

Next we apply three fundamental laws to this fluid element Conservation of mass, conservation of momentum, and conservation of energy to obtain the continuity, momentum, and energy equations for laminar flow in boundary layers. 

The discussion in the previous section assumed that the velocity field required to calculate the necessary coefficients of the discretized equations was somehow known. However, generally, the velocity field needs to be calculated as part of the overall solution procedure by solving momentum conservation equations. The governing equations are discussed in Chapters 2 to 5. The basic momentum transport equations governing laminar flow are considered here to illustrate the application of the finite volume method to calculation of the flow field. The governing equations can be written ... 

The equations of fluid mechanics originate from the momentum and mass conservation principles. The overall mass conservation or continuity equation for laminar flows is... 

The -component momentum conservation equation for laminar flows is... 

The energy conservation equation for laminar flows can be written in terms of enthalpy as... 

The averaged Eulerian-Eulerian multi-fluid model denotes the averaged mass and momentum conservation equations as formulated in an Eulerian frame of reference for both the dispersed and continuous phases describing the time-dependent motion. For multiphase isothermal systems involving laminar flow, the averaged conservation equations for mass and momentum are given by ... 

Heat transfer to a laminar flow in an annulus is complicated by the fact that both the velocity and thermal profiles are simultaneously developing near the entrance and, often, over the length of the heated channel. Natural convection may also be a factor. It is usually conservative (i.e., predicted heat-transfer coefficients are lower than those experienced) to use equations for the fully developed flow. 

It is worth a short detour into fluid mechanics to explore some details of this approach and how it fits into the reactor conservation equations. For moderate flow velocities the dispersion of a tracer in laminar flow will occur by axial and radial diffusion from the flow front of the tracer and, in the absence of eddy motion, this will be via a molecular diffusion mechanism. However, the net contribution of diffusion in the axial direction can be taken as small in comparison to the contribution of the flow velocity profile. This leaves us with a two-dimensional problem with diffusion in the radial direction and convection in the longitudinal direction the situation is considered in illustrated in Figure 5.7. 

The microfluidic device design and the relative flow rate of sheath and sample play important roles in hydrodynamic focusing. Lee et al. [6] proposed a theoretical model to predict the width of focused center flow inside a microfabricated flow cytometer [6]. Based on potential flow theory, they derived the equation for flow inside a planar microfabricated flow cytometer under the two-dimensional situation shown in Fig. 3a. The flow is considered laminar, and the diffusion and mixing between focused stream and sheath flows is assumed negligible. With these assumptions, conservation of mass yields... 

For Reynolds numbers larger than the critical Reynolds number, the flow becomes turbulent and the fluid velocity at a point can be defined as the mean value of the velocity at this point with respect to time by taking a time interval to larger than the typical time of one turbulent fluctuation and shorter than the typical time of pressure variations. Since mass and momentum must be conserved in turbulent as well as in laminar flow, the conservation equations (Eq. 1) hold equally for turbulent flow, provided the velocities and pressures in these equations are interpreted as instantaneous velocities and pressures of the turbulent field. From a practical standpoint such equations are of little value because there is no interest in knowing the complete and complex history of instantaneous velocities and pressures but only their time-mean values that can be measured and observed (see also > Turbulence in Microchannels). In microchannels with hydraulic diameters less than 500 pm the maximum value that the Reynolds number can assume drops to between 10,000 and 30,000. In fact, in order to reach these values of the Reynolds number, a very large pressure difference... 

The equations governing laminar flames include the two conservation equations of over-all mass and energy (momentum conservation is eliminated by the constant pressure approximation) and three differential equations which describe the processes of thermal conduction, diffusion, and chemical reaction. Taken together with the proper boundary conditions this forms a set of equations whose solution is an eigenvalue. The eigenvalue m itself is the mass flow per unit area which can be directly identified with the burning velocity, v ... 

For nonisothermal systems a general differential equation of conservation of energy will be considered in Chapter 5. Also in Chapter 7 a general differential equation of continuity for a binary mixture will be derived. The differential-momentum-balance equation to be derived is based on Newton s second law and allows us to determine the way velocity varies with position and time and the pressure drop in laminar flow. The equation of momentum balance can be used for turbulent flow with certain modifications. 

Other shapes are also found, where transverse transport cannot be described by a single coordinate. The detailed mass transfer problem then becomes 3D, and simplified descriptions are needed. According to the ID model mentioned earlier, the momentum conservation equation in the axial direction (with velocity component u) for fully developed laminar flow is given by the following equation ... 

Turbulent flow is described by conservation equations of continuity and momentum, known as the Reynolds-averaged Navier-Stokes (RANS) equations. Laminar velocity terms in conservation equations are replaced by the steady-state mean components and time-dependent fluctuating components defined by Equation 6.100. 

Conservation Equations The computational model consists of steady-state continuity equation and momentum equations describing the flow inside the laminar static mixer. 

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