The Momentum Equations

The momentum equation for a Newtonian fluid is yielded from Navier-Stokes equation (3.59) 

As free flow occurs because of density gradients, we will assume a variable density whilst all other material properties shall be taken as constant. Gravity will act as the body force k.j = g,j. In areas where the density is constant, for example far away from a heated wall, hydrostatic equilibrium is reached. Here it holds that 

We assume that the deviations Ap and Ag are small in comparison to the values at hydrostatic equilibrium px and g.  

As an example we will consider the vertical heated wall in Fig. 3.44, at a temperature 0, in front of which there is a fluid that before heating was in hydrostatic equilibrium and had a temperature throughout. After a short time steady temperature and velocity profiles develop. These are sketched in Fig. 3.44. With (3.293), the momentum equation (3.291) is transformed into 

Considering that on the left hand side Ag C gx and on the right hand side according to (3.292) dp /dx = g g is valid, we obtain 

The rate of transfer of momentum through the elementary strip 

In passing from plane 1 -2 to plane 3-4, the mass flow changes by  

A mass flow of fluid equal to the difference between the flows at planes 3 -4 and l -2 (equation 11.3) must therefore occur through plane 2-4, as it is assumed that there is uniformity over the width of the element. 

Since plane 2-4 lies outside the boundary layer, the fluid crossing this plane must have a velocity us in the X-direction. Because the fluid in the boundary layer is being retarded, there will be a smaller flow at plane 3-4 than at 1-2, and hence the flow through plane 2-4 is outwards, and fluid leaves the element of volume. 

It will be assumed that a fluid of density / and viscosity p. flows over a plane surface and the velocity of flow outside the boundary layer is A boundary layer of thickness 8 forms nejff the surface, and at a distance y from the surface the velocity of the fluid is reduced to a value Ux- 

At plane 1-2, mass rate of flow through a strip of thickness dy at distance y from the surface 

This balance is obtained by application of Newton s second law on a moving fluid element. In chemical reactors only pressure drops and friction forces have to be considered in most cases. A number of pressure drop equations are discussed in the chapters on tubular plug flow and on fixed bed catalytic reactors. 

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Here g is the gravity vector and tu is the force per unit area exerted by the surroundings on the fluid in the control volume. The integrand of the area integr on the left-hand side of Eq. (6-10) is nonzero only on the entrance and exit portions of the control volume boundary. For the special case of steady flow at a mass flow rate m through a control volume fixed in space with one inlet and one outlet, (Fig. 6-4) with the inlet and outlet velocity vectors perpendicular to planar inlet and outlet surfaces, giving average velocity vectors Vi and V9, the momentum equation becomes... 

Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generahzed to multiple inlets or outlets. 

Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so that g, = —g where g is the gravitational acceleration (9.806 mVs), the pressure field is given by... 

Application of the momentum equation to ejectors of other types is discussed in Lapple (Fluid and Paiticle Dynamics, University of Delaware, Newark, 1951) and in Sec. 10 of the Handbook. 

For smooth pipe, the friction factor is a function only of the Reynolds number. In rough pipe, the relative roughness /D also affects the friction factor. Figure 6-9 plots/as a function of Re and /D. Values of for various materials are given in Table 6-1. The Fanning friction factor should not be confused with the Darcy friction fac tor used by Moody Trans. ASME, 66, 671 [1944]), which is four times greater. Using the momentum equation, the stress at the wall of the pipe may be expressed in terms of the friction factor ... 

For gradual changes in channel cross section and hquid depth, and for slopes less than 10°, the momentum equation for a rectangular channel of width b and liquid depth h may be written as a differential equation in the flow direction x. 

Solution of the algebraic equations. For creeping flows, the algebraic equations are hnear and a linear matrix equation is to be solved. Both direct and iterative solvers have been used. For most flows, the nonlinear inertial terms in the momentum equation are important and the algebraic discretized equations are therefore nonlinear. Solution yields the nodal values of the unknowns. 

This derivation indicates a strong coupling between the momentum equation and the energy equation, which implies that the momentum and energy balance equations should be solved as a coupled system. In particular, the dis-... 

The momentum equation is a mathematieal formulation of the law of eonservation of momentum. It states that the rate of ehange in linear momentum of a volume moving with a fluid is equal to the surfaee forees and body forees aeting on a fluid. Figure 3-2 shows the veloeity eomponents in a generalized turbomaehine. The veloeity veetors as shown are resolved into three mutually perpendieular eomponents the axial eomponent (FJ, the tangential eomponent (Fg), and the radial eomponent (F ). 

It is possible to determine the x-component of the momentum equation by setting the rate of change of x-momentum of the fluid particle equal to the total force in the x-direction on the element due to surface stresses plus the rate of increase of x-momentum due to sources, which gives ... 

Recall that equations 9.86 and 9.100 have been both derived using only the first-order terms in the Taylor series expansion of our basic kinetic equation (equation 9.77). It is easy to show that if instead all terms through second-order in 6x and 6t are retained, the continuity equation ( 9.86) remains invariant but the momentum equation ( 9.100) requires correction terms [wolf86c]. The LHS of equation 9.100, to second order in (ia (5 << 1, is given by... 

This expression, known as the momentum equation, may be integrated provided that the relation between ux and y is known. 

Obtain the momentum equation for an element of boundary layer. If the velocity profile in the laminar region may be represented approximately by a sine function, calculate the boundary-layer thickness in terms of distance from the leading edge of the surface. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. 

Derive the momentum equation for the flow of a viscous fluid over a small plane surface. Show that the velocity profile in the neighbourhood of the surface may be expressed as a sine function which satisfies the boundary conditions at the surface and at the outer edge of the boundary layer. 

A roughness-viscosity model was proposed to interpret the experimental data. An effective viscosity /tef was introduced for this purpose as the sum of physical and imaginary = Mm(f) viscosities. The momentum equation is... 

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... 

The momentum equations and the continuity equation governing the motion are ... 

Numerical Solution. The momentum Equation 5 is solved simultaneously along with the energy Equation 6 to obtain axial velocity, v, and temperature fields. The continuity equation with the known axial velocity is used to... 

Thermally-Driven Buoyancy Flow. Thermal gradients can Induce appreciable flow velocities in fluids, as cool material is pulled downward by gravity while warmer fluid rises. This effect is Important in the solidification of crystals being grown for semiconductor applications, and might arise in some polymeric applications as well. To illustrate how easily such an effect can be added to the flow code, a body force term of pa(T-T ) has been added to the y-coraponent of the momentum equation, where here a is a coefficient of volumetric thermal expansion. 

VOF or level-set models are used for stratified flows where the phases are separated and one objective is to calculate the location of the interface. In these models, the momentum equations are solved for the separated phases and only at the interface are additional models used. Additional variables, such as the volume fraction of each phase, are used to identify the phases. The simplest model uses a weight average of the viscosity and density in the computational cells that are shared between the phases. Very fine resolution is, however, required for systems when surface tension is important, since an accurate estimation of the curvature of the interface is required to calculate the normal force arising from the surface tension. Usually, VOF models simulate the surface position accurately, but the space resolution is not sufficient to simulate mass transfer in liquids. 

For the liquid phase, Cheremisinoff and Davis (1979) solved the momentum equation using von Karman s and Deissler s eddy viscosity expressions. 

In the absence of friction, the momentum equation for an isentropic annular flow can be written as... 

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