One-dimensional flows

For the one-dimensional analysis, the equation of motion is described by Eqs. 7.194 and 7.195. The constitutive relationship, the power law equation, is written as  

Equation 7.246 combined with Eq. 7.195 describes the basic problem. It is convenient to write the resulting equation in dimensionless form. For this purpose, the following dimensionless quantities are defined the dimensionless depth = y/H, the dimensionless down-channel velocity v = v /Vi,, and a reduced pressure gradient Fr. The reduced pressure gradient is defined as  

Variable A, in Eq. 7.248 represents the location where the shear rate is zero, which is also the location of the extremum in the velocity profile. This value needs to be known to eliminate the absolute value in Eq. 7.248. For the time being, only positive pressure gradients g will be considered. If the extremum occurs in the screw channel, then 0 A, 1. When A, Eq. 7.248 can be written as  

By integration and by using the appropriate boundary condition v°(l) = 1, the following expression is obtained  

At = A, the two velocities from Eqs. 7.250 and 7.251 should be the same. From this equality, the following equation for X results  

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The most widely used approach to channel flow calculations assumes a steady qua si-one-dimensional flow in the channel core, modified to account for boundary layers on the channel walls. Electrode wall and sidewall boundary layers may be treated differently, and the core flow may contain nonuniformities. 

In the flow models these effects are confined to the boundary layers, maintaining the vaHdity of the qua si-one-dimensional flow model. The flow is... 

One-dimensional Flow Many flows of great practical importance, such as those in pipes and channels, are treated as onedimensional flows. There is a single direction called the flow direction velocity components perpendicmar to this direction are either zero or considered unimportant. Variations of quantities such as velocity, pressure, density, and temperature are considered only in the flow direction. The fundamental consei vation equations of fluid mechanics are greatly simphfied for one-dimensional flows. A broader categoiy of one-dimensional flow is one where there is only one nonzero velocity component, which depends on only one coordinate direction, and this coordinate direction may or may not be the same as the flow direction. 

Convergent/Divergent Nozzles (De Laval Nozzles) During frictionless adiabatic one-dimensional flow with changing cross-sectional area A the following relations are obeyed ... 

Figure 2.12. A flow tube used to derive one-dimensional flow equations in Lagrangian coordinates. Internal surfaces are massless, impermeable partitions to aid in visualizing elements of fluid in Lagrangian coordinates.

The Euler equation, assuming simple one-dimensional flow theory, is the theoretieal amount of work imparted to eaeh pound of fluid as it passes through the impeller, and it is given by... 

The general features of two-dimensional flow with evaporating liquid-vapor meniscus in a capillary slot were studied by Khrustalev and Faghri (1996). Following this work we present the main results mentioned in their research. The model of flow in a narrow slot is presented in Fig. 10.16. Within a capillary slot two characteristic regions can be selected, where two-dimensional or quasi-one-dimensional flow occurs. Two-dimensional flow is realized in the major part of the liquid domain, whereas the quasi-one-dimensional flow is observed in the micro-film region, located near the wall. 

A steady homogeneous model is often used for bubbly flow. As mentioned previously, the two phases are assumed to have the same velocity and a homogeneous mixture to possess average properties. The basic equations for a steady one-dimensional flow are as follows. 

Flow through abrupt expansion Using the one-dimensional flow assumption for a single-phase incompressible fluid, the energy equation becomes... 

Initial air shock pressures for one-dimensional flow depend primarily on the enthalpy of the driver fluid. 

The velocity field in turbulent flow can be described by a local mean (or time-average) velocity, upon which is superimposed a time-dependent fluctuating component or eddy. Even in one-dimensional flow, in which the overall average velocity has only one directional component (as illustrated in Fig. 6-3), the turbulent eddies have a three-dimensional structure. Thus, for the flow illustrated in Fig. 6-3, the local velocity components are... 

In reacting systems, transfer of matter and heat occurs by bulk flow and diffusion or conduction. Usually transfer in an axial direction is appreciable by bulk flow only. In a rectangular region the various elements of a material balance in one dimensional flow are,... 

As described above, spatial transport in an Eulerian PDF code is simulated by random jumps of notional particles between grid cells. Even in the simplest case of one-dimensional purely convective flow with equal-sized grids, so-called numerical diffusion will be present. In order to show that this is the case, we can use the analysis presented in Mobus et al. (2001), simplified to one-dimensional flow in the domain [0, L (Mobus et al. 1999). Let X(rnAt) denote the random location of a notional particle at time step m. Since the location of the particle is discrete, we can denote it by a random integer i X(mAt) = iAx, where the grid spacing is related to the number of grid cells (M) by Ax = L/M. For purely convective flow, the time step is related to the mean velocity (U) by16... 

FIGURE 6.17 Burning of a flat fuel surface in a one-dimensional flow field. 

Two common types of one-dimensional flow regimes examined in interfacial studies Poiseuille and Couette flow [37]. Poiseuille flow is a pressure-driven process commonly used to model flow through pipes. It involves the flow of an incompressible fluid between two infinite stationary plates, where the pressure gradient, Sp/Sx, is constant. At steady state, ignoring gravitational effects, we have... 

The dispersion model is one of the frequently used models. It describes the dispersion of the residence time of the phases according to Fig. 9.16, for example, in one-dimensional flows by superimposing the plug profile of the basic flow with a stochastic dispersion process in axial direction, which is constructed by analogy to Pick s first law of molecular diffusion ... 

Conservation Equations at a Steady State in a One-Dimensional Flow Field... 

B. 1 Conservation Equations at a Steady State in a One-Dimensional Flow Field 473 The momentum conservation equation is then represented by... 

The conservation equations described in Section B.l show the mass, momentum, energy, and chemical species equations at a steady state in a one-dimensional flow field. Similarly, the conservation equations at a steady-state in two- or three-dimensional flow fields can be obtained. The results can be summarized in a vector form... 

Plug Flow with Dispersion - Plug flow with dispersion is a concept that is often used to describe one-dimensional flow systems. It is somewhat more flexible in computational models because the mixing within the system is not dependent on reactor size, as with complete mix tanks in series. Plug flow with dispersion will be described in the second half of this chapter because special techniques are needed for the analysis. 

Dunkle (Ref 7) reports that the CJ Theory deals with adiabatic transformations in steady non-viscous, one-dimensional flows in stress tubes or ducts of constanc cross-section. This theory interprets deton waves as shock waves in which a continuing degradation of mechanical (shock) energy into heat is balanced by... 

Since the equations of continuity, momentum, energy, and state do not suffice to determine the five unknowns, it. is necessary to inquire into the conditions under which solutions exist and whether solns are unique. The information which has thus far been omitted is a specification of the flow field of die reaction products, that is to say, since this section is restricted to one--dimensional flow, of the rear boundary condition. Before discussing the question of determinancy it is necessary to deduce from the equations of Section II of Ref 66, the general properties of flow ahead and behind reaction waves. To do this the Hugoniot curve for the products... 

Figure 6.11 depicts the numerical solutions for the time dependence of the resin pressure profile in the vertical direction for one-dimensional flow in the vertical direction (corresponding to an edge-dammed laminate). The laminate is 1.4-in thick (z direction) and is a unidirectional lay-up. Values for the specific permeability in the z direction, (kz in.2), the... 

Figure 6.11 Resin pressure profiles in the laminate thickness direction (vertical) in a 1.4-in. thick unidirectional graphite-epoxy laminate for one-dimensional flow (edge-dammed) under conditions indicated in the figure...

Matrix flow relative to the reinforcing fibers is caused by thermal expansion of the fiber-matrix mass within the confines of the die and by the geometrical constriction of the die taper. Once the matrix flow distribution is known, the matrix pressure distribution may be determined using a flow rate-pressure drop relationship. One-dimensional flow models of thermoset pultrusion have been reasonably well verified qualitatively [15-17]. A onedimensional flow model of thermoplastic pultrusion [14,18] has similarly been compared with experimental data and the correlation found to be encouraging [19]. 

For steady, one-dimensional flow without body forces, with local mean velocity u(x) in a channel of constant cross-sectional area A, the energy conservation equation becomes, approximately (13) ... 

To quantify this treatment of migration as influenced by kinetics, a model has been developed in which instantaneous or local equilibrium is not assumed. The model is called the Argonne Dispersion Code (ARDISC) ( ). In the model, adsorption and desorption are treated independently and the rates for adsorption and desorption are taken into account. The model treats one dimensional flow and assumes a constant velocity of solution through a uniform homogeneous media. 

This expression describes the fastest and most important mode of transport in groundwater. In fact, an important task of the hydrologist is to develop models to predict the effective velocity u (or the specific flow rate q). Like the Darcy-Weis-bach equation for rivers (Eq. 24-4), for this purpose there is an important equation for groundwater flow, Darcy s Law. In its original version, formulated by Darcy in 1856, the equation describes the one-dimensional flow through a vertical filter column. The characteristic properties of the column (i.e., of the aquifer) are described by the so-called hydraulic conductivity, Kq (units m s"1). Based on Darcy s Law, Dupuit derived an approximate equation for quasi-horizontal flow ... 

Obviously, everything that has been said in Chapter 24 on dispersion in rivers also applies to the one-dimensional flow in aquifers. 

For the purpose of an order-of-magnitude analysis to compare the relative magnitudes of the two terms, consider a one-dimensional flow in which the velocity in given as u. In this case, rearranging Eq. 3.69, we seek situations for which... 

For the boundary-layer equations, where two-dimensional flow is retained, the continuity equation must retain both terms as order-one terms. Otherwise, a purely onedimensional flow would result. Certainly there are situations where one-dimensional flow... 

The study of combustion in a flow field presents many complex problems. At the present time, a complete description of the various mechanisms involved seems too complicated to obtain even in the case of the simplest first-order reactions in one-dimensional flow. Simplifying assumptions arc necessary in order to obtain cither an exact solution to the approximate problem or an approximate solution to the exact problem. This paper discusses some of these assumptions and categorizes the various combustion phenomena which exist in subsonic and supersonic flow. 

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