Momentum steady-state flow

Pipe Flow For steady-state flow through a constant diameter duct, the mass flux G is constant and the governing steady-state momentum balance is ... 

Let us consider a simplified flow, that is, a one-dimensional steady-state flow-without viscous stress or a gravitational force. The conservation equahons of continuity, momentum, and energy are represented by rate of mass in - rate of mass out = 0... 

The momentum equation at the nozzle exit is represented by thg du = -A dp,., and dthg = Ofor a steady-state flow at the nozzle. Thus, from Eq. (1.64), one obtains the... 

Fig. B-1 presents a steady-state flow in a combustion wave, showing mass, momentum, and energy transfers, including chemical species, in the one-dimensional space of Ax between Xj and %2- The viscous forces and kinetic energy of the flow are assumed to be neglected in the combustion wave. The rate of heat production in the space is represented by coQ, where ai is the reaction rate and Qis the heat release by chemical reaction per unit mass.

For the solution of this problem, the momentum and continuity equations for the steady-state flow of an incompressible viscoelastic fluid are given by... 

An integrated form of the momentum equation may be derived only under additional restrictive conditions. In steady-state flow with f = 0, if viscous forces are negligible, then equation (20) reduces to pv dv/dx - -dp/dx = 0. If, in addition, v /(p/p) 1, then d In p/dx d n v/dx (that is. 

The simulation program AIOLOS is developed for the numerical calculation of three-dimensional, stationary, turbulent and reacting flows in pulverised coal-fired utility boilers. AIOLOS contains submodels treating fluid flow, turbulence, combustion and heat transfer. In these submodels equations for calculating the conservation of mass, momentum and energy are solved, presupposing high Reynolds-numbers and steady-state flow conditions. It is assumed that the flow field is weakly compressible which means that the density depends only on temperature and fluid composition but not on pressure. 

Two-Phase Region. The two-phase flow and heat transfer are given by the continuity equations for the i and g phases, the momentum equations (Eqs. 9.76 and 9.77), and the energy equation (Eq. 9.80). The two-phase region is assumed to be isothermal by neglecting the effect of the curvature (i.e., saturation) on the thermodynamic equilibrium state. This is justifiable, except for the very small pores (large pc). For the steady-state flow considered here, we have (for the assumed isotropic phase permeabilities)... 

This equation, often referred to as the moment-of-momentum equation is one of the basic tools in the analysis of rotating fluid machines, turbines, pumps, and other devices [5]. In the steady-state flow dLldt) is zero and equals m p.so we have... 

The temperature at any point can be predicted from the equations of mass, momentum, and energy. When a temperature sensor, such as a thermocouple, is inserted into the polymer melt stream to measure the temperature of the melt, the steady-state flow is disturbed, and a new steady state will develop after a short time. Therefore, the measured temperature will be different from the true, undisturbed melt temperature. Thus, in order to determine the true melt temperature, certain corrections have to be made to the measured (disturbed) melt temperature. The factors that have to be taken into account are ... 

In order to circumvent this difficulty, an energy-based approach is suggested instead of the momentum-based approach [1], which usually assumes steady-state flow. 

Figure A3.1.5. Steady state shear flow, illustrating the flow of momentum aeross a plane at a height z.

We will apply the steady state momentum balance to a fluid in plug flow in a tube, as illustrated in Fig. 5-6. (The stream tube may be bounded by either solid or imaginary boundaries the only condition is that no fluid crosses the boundaries other that through the inlet and outlet planes.) The shape of the cross section does not have to be circular it can be any shape. The fluid element in the slice of thickness dx is our system, and the momentum balance equation on this system is... 

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... 

In two-phase flow, most investigations are carried out in one dimension in the steady state with constant flow rates. The system may or may not be isothermal, and heat and mass may be transferred either from liquid to gas, or vice versa. The assumption is commonly made that the pressure is constant at a given cross section of the pipe. Momentum and energy balances can then be written separately for each phase, and with the constraint that the static pressure drop, dP, is identical for both phases over the same increment of flow length dz, these balances can be added to give over-all expressions. However, it will be seen that the resulting over-all balances do not have the simple relationships to each other that exist for single-phase flow. 

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