Steady-state momentum balance

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 ... 

Steady-state momentum balance over the element 1 - 2 - 3 - 4... 

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... 

Based on a differential cylindrical control volume, derive steady-state momentum balances for the axial and circumferential directions, i.e., the Navier-Stokes equations. 

It is often desirable to know the pressure drop along the length of the channel. In this case the momentum equation may be solved, from which the pressure variation can be determined. By assumption, there is only one velocity component, the axial velocity. The steady-state momentum balance for the system is given as... 

We consider first a cylindrical reactor of uniform diameter D. To derive an expression for the pressure drop, we write the steady-state momentum balance equation for a reactor element of length dL and cross-sectional area A ... 

Steady-State Momentum (Force) Balance Equation... 

The line 9 is given by the steady-state, back-pressure drive flame propagation theory [29], which assumes the momentum flux balance between the upstream and downstream positions on the center streamline and the angular momentum conservation on each streamline. 

Bernoulli s equation on a center streamline ahead of and behind the flame and the momentum flux conservation across the flame front however, the steady-state, backpressure drive theory [29] used only the momentum flux balance across the flame front. These resulted in the -v/2 difference between Equation 4.2.10 and the first term of Equation 4.2.7. 

As for the mass and energy balance equations, steady-state conditions are obtained when the rate of change of momentum in the system is zero and... 

Note that because momentum is a vector, this equation represents three component equations, one for each direction in three-dimensional space. If there is only one entering and one leaving stream, then = m0 = m. If the system is also at steady state, the momentum balance becomes... 

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. 

Although we analyze most polymer processes as isothermal problems, many are non-isothermal even at steady state conditions. The non-isothermal effects during flow are often difficult to analyze, and make analytical solutions cumbersome or, in many cases impossible. The non-isothermal behavior is complicated further when the energy equation and the momentum balance are fully coupled. This occurs when viscous dissipation is sufficiently high to raise the temperature enough to affect the viscosity of the melt. 

The previous section used the constant strain three-noded element to solve Poisson s equation with steady-state as well as transient terms. The same problems, as well as any field problems such as stress-strain and the flow momentum balance, can be formulated using isoparametric elements. With this type of element, the same (as the name suggests) shape functions used to represent the field variables are used to interpolate between the nodal coordinates and to transform from the xy coordinate system to a local element coordinate system. The first step is to discretize the domain presented in Fig. 9.12 using the isoparametric quadrilateral elements as shown in Fig. 9.15. 

The jet is governed by four steady-state equations representing the conservation of mass and electric charges, the linear momentum balance, and Coulomb s law for the E field [9]. Mass conservation requires that... 

Even though the composition of the atmosphere can be considered constant, it is not at equilibrium. In fact, the atmosphere exchanges matter and momentum with the Earth s surface through the cycles mentioned above. A better description would be that some components in the atmosphere are in a steady-state condition. The term steady state describes the balance between the input and output of an atmospheric constituent. If Fm and Fou( are the fluxes inward and outward of the atmosphere, respectively, a steady-state condition requires that... 

Cyclic Steady State is the condition whereby die state at the end of each cycle is identical to that at its beginning. For a non-isotiiermal adsorption systan this may be represented by a mathematical model, comprising material, energy and momentum balances as well as adsorption equilibrium and kinetic models, the CSS can be repressed by ... 

The general form of the mechanical energy balance can be derived starting with the open-system balance and a second equation expressing the law of conservation of momentum, a derivation beyond the scope of this book. This section presents a simplified form for a single incompressible liquid flowing into and out of a process system at steady state. 

The momentum balance over a small element of fluid for incompressible, steady-state fully developed flow in the z-direction results in the following partial differential equation (PDF) ... 

The calculated values required to evaluate the performance index are determined from the solution of differential equations describing flow. Pressure and velocity values are obtained by solving the locally volume-averaged equation of continuity and differential momentum balance (Eqs (1) and (2)). For steady-state conditions, and incompressible flow, the equation of continuity becomes... 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

The Franck-Condon principle implies locality in phase space x, p. It states that the coordinate and momentum of a massive dot stay unchanged during the electron transitions, i.e. electron tunneling in our case. For detailed balance, the tunnel coupling Ta has to be invoked as the main prerequisite for the Coulomb blockade. The additional relaxation due to molecular collisions and electromagnetic radiation may have to be favored for the steady state. This is similar to the dissipation necessitated in the orthodox theory of the Coulomb blockade [Kulik 1975 Averin 1991],... 

In all these cases, the correct design must grow from the equations of mass, energy, and momentum balance to which we now turn in the next few sections. From these we proceed to the design problem (Sec. 9.5) and hence to elementary considerations of optimal design (Sec. 9.6). The stability and sensitivity of a tubular reactor is a vast and fascinating subject. Since the steady state equations are ordinary differential equations, the equations describing the transient behavior are partial differential equations. This... 

The state of the art procedure for design of cyclic PSA or TSA processes using activated carbon adsorbents is to simultaneously solve the partial differential equations describing the mass, the heat, and the momentum balance equations for each step of the process using the appropriate initial and boundary conditions. These numerical calculations are carried out over many cycles for the process until a cyclic steady-state performance solution is achieved. Many different numerical integration algorithms are available for this purpose. The core input variables for the solution are multicomponent gas adsorption equilibria, heats, and kinetics for the system of interest [37]. 

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