# Momentum equation

Each of the three time-averaged momentum equations contains three unknown turbulent stresses, pu[u j, commonly termed Reynolds stresses, only six of which are independent. The Reynolds stress pu[ for example, is the rate at which x-momentum, pu[, is being transported in the jy-direction by the velocity fluctuation u 2. A hierarchy of equations for velocity correlation functions, ie, averages of products of ti-, can be obtained, but each equation so derived involves an unknown higher order correlation function and hence the set of equations is not closed. A turbulence model is needed to determine the turbulent transport terms before the set of equations can be solved. Turbulence modeling is concerned with the development and testing of closure assumptions for the Reynolds stresses. A large number of closure models are available. They are usually divided into two groups, eddy viscosity models and Reynolds stress models, according to whether or not the Boussinesq assumption is appHed. [c.102]

Unlike the momentum equation (Eq. [6-11]), the Bernoulli equation is not easily generahzed to multiple inlets or outlets. [c.633]

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 [c.634]

Example 2 Simplified Ejector Figure 6-6 shows a very simplified sketch of an ejector, a device that uses a high velocity primary fluid to pump another (secondary) fluid. The continuity and momentum equations may he [c.634]

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. [c.635]

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 [c.636]

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. [c.639]

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 [c.639]

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. [c.673]

Since the El complex does not yield product P, and I competes with S for E, there is a state of competitive inhibition. By analogy to the Michaelis-Menten equation [c.2149]

Several discrete forms of the conservation of momentum equation, (9.3), can be derived, depending on the type of mesh and underlying assumptions. As an example, assume the equation will be solved on staggered spatial and temporal meshes, in two dimensions, in rectangular geometry, and with the velocities located at the nodes. Assume one quarter of the mass from each adjacent element is associated with the staggered element as shown in Fig. 9.11. [c.333]

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- [c.335]

Gas Turbine Engineering Handbook The Momentum Equation [c.118]

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 ). [c.118]

Combining the energy and momentum equations provides the following relationships [c.121]

[c.24]

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 [c.791]

The y-component and z-component of the momentum equation are given by [c.791]

Equation 11-15 is known as the Michaelis-Menten equation. It represents the kinetics of many simple enzyme-catalyzed reactions, which involve a single substrate. The interpretation of as an equilibrium constant is not universally valid, since the assumption that the reversible reaction as a fast equilibrium process often does not apply. [c.839]

PARAMETERS IN THE MICHAELIS-MENTEN EQUATION [c.839]

Assuming laminar flow for a linear momentum equation in the a direction (an approximation from the Navier-Stokes equations) gives [c.134]

Treating the other terms in a similar manner, the linear momentum equation in a dimensionless form is obtained [c.135]

For simplicity, we define T - and T (A iooTe/At). As explained by Luo and Tanner (1989), the decoupled method requires a suitable variable transfonna-tion in the governing equations (3.20) and (3.21). This is to ensure that the discrete momentum equations always contain the real viscous term required to recover the Newtonian velocity-pressure formulation when Ws approaches zero. This is achieved by decomposing the extra stress T as [c.82]

The momentum and continuity equations give rise to a 22 x 22 elemental stiffness matrix as is shown by Equation (3.31). In Equation (3.31) the subscripts I and / represent the nodes in the bi-quadratic element for velocity and K and L the four corner nodes of the corresponding bi-linear interpolation for the pressure. The weight functions. Nr and Mf, are bi-qiiadratic and bi-linear, respectively. The y th component of velocity at node J is shown as iPj. Summation convention on repeated indices is assumed. The discretization of the continuity and momentum equations is hence based on the U--V- P scheme in conjunction with a Taylor-Hood element to satisfy the BB condition. [c.84]

Algorithm and Solution of the Linearized Equations. The procedure is to solve the finite difference equation sets for each dependent variable (inner iteration) sequentially and repeatedly. Each cycle of calculation (outer iteration) consists of computing the velocity components from the momentum equations using the most recently calculated pressure field (a guessed pressure field is used during the first cycle of iteration). The velocities and the pressure fields are then corrected to satisfy the continuity equation. This is followed by the calculation of the other field variables, eg, turbulence quantities, temperature, and concentration, if these influence the flow field. The procedure is repeated until the solution converges, ie, until the so-called residual source (the sum of the absolute values of the residuals of all the equations) is less than a predetermined value. This procedure is followed in the semi-imphcit method for pressure-linked equation (SIMPLE) algorithm (17). Other variants of the SIMPLE algorithms, SIMPLER, SIMPLEC, PISO, PISOC, etc, differ primarily in the procedure used in correcting for continuity. A large proportion of the computer time is used to solve the individual transport equations. Each of the transport equations consists of a single diagonally dominant equation. Good solvers for handling such equations are available. The alternating direction method, line relaxation method, the Stone algorithm, and the incomplete Cholesky conjugate gradient method are but a few examples. To avoid numerical instabiUties, relaxation parameters are used in that only a fraction of the change of the dependent variables calculated at the current iteration step are appHed to the next iteration step. The relaxation parameters can be different for each of the dependent variables. [c.101]

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 [c.632]

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are [c.634]

An additional turbulence pressure term equal to —V Jcbij, where k = turbulent kinetic energy and 1 i=J and 6, = 0 if i 9 J, is sometimes included in the right-hand side. To solve the equations of motion using the Boussinesq approximation, it is necessary to provide equations for the single scalar unknown [Lt (and /c, if used) rather than the nine unknown tensor components With this approximation, and using the effective viscosity PefF = M- + h, ihe time-averaged momentum equation is similar to the original Navier-Stokes equation, with time-averaged variables and [L f replacing the instantaneous variables and molecular viscosity. However, solutions to the time-averaged equations for turbulent flow are not identical to those for laminar flow because PefF is not a constant. [c.672]

The Michaehs-Menten equation and other similar nonhnear expressions characterize immobihzed enzyme kinetics. Therefore, for a spherical porous carrier particle with enzyme molecules immobilized on its external as well as internal surfaces, material balance of the substrate will result in the following [c.2150]

Equation (26-31) can be integrated directly to yield the mass flux G, provided that D, L, f and ( ) are known, as well as the relationship between pressure and volume. For all-vapor cases, the expansion of the vapor is usually assumed to follow the form Pv = constant (y = Cp/C,) and thus the momentum equation can be analytically integrated. Similaj ly, for all-liquid (nonflashing) flow, the stream specific volume is usually assumed to be constant, thus also providing a direct analytical integration of Eq. (26-31). For two-phase flashing flow, the requisite p-v relationship is usually obtained From flash calculations, and normally requires a numerical integration of Eq. (26-31). In addition to calculating the flow rates through sections of piping in the relief system, there may also exist additional pressure drop constraints in both the inlet and outlet piping if the relief device is a PRV. The designer is referred to the ASME and API references for further information. [c.2293]

Momentum equation for a caloricaly and thermally perfect gas, and one in which the radial and axial velocities do not contribute to the forces generated on the rotor the Adiabatic Energy (ifad) per unit mass is given as follows (Euler Turbine Equation) [c.708]

The Michaelis-Menten Equation 11-15 is not well suited for estimation of the kinetic parameters and Reananging Equation 11-15 gives various options for plotting and estimating the parameters. [c.839]

See pages that mention the term

**Momentum equation**:

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Gas turbine engineering handbook (2002) -- [ c.0 ]