Equations for momentum

The three most widespread methods for solving the transport equations for momentum, matter and heat are the finite-difference [76], the finite-element [77,... 

Momentum balance equations are of importance in problems involving the flow of fluids. Momentum is defined as the product of mass and velocity and as stated by Newton s second law of motion, force which is defined as mass times acceleration is also equal to the rate of change of momentum. The general balance equation for momentum transfer is expressed by... 

The analogy also extends to Newton s equation for momentum transport, where... 

For small amplitude waves it is assumed that the density p after passage of the wave equals the initial density p0, and that any terms containing the particle velocity u can be neglected. With these assumptions, the conservation equations for momentum and mass become ... 

Another class of derived quantities is represented by the coefficients in transfer equations for momentum, mass and heat. They were also established by the respective physical laws - therefore they are often called defined quantities - and can only be determined via measurement of their constituents. 

The coordinate transformation also enables a nondimensional version of the equation for momentum conservation to be written from equations (1-2) and (1-5). For this purpose, Prandtl numbers based on the first and a modified second coefficient of viscosity, respectively, may be defined as Pri = /iooCp, /A and Pf2 = + Koo)Cp,ooMoo- To allow for... 

The simplification of equation (1) represented by equation (2) is relatively unimpressive, but in the average of the equation for momentum conservation, an appreciably greater compactness is obtained in the inertial terms the... 

It is of interest also to consider the x -component of linear momentum for the free-particle solutions (3.4). According to Eq (2.24), the eigenvalue equation for momentum should read... 

Reynolds-averaged equations for momentum transport, are already discussed in Chapter 3. For modeling reactive flow processes, in addition to the solution of overall mass conservation equation described in Chapter 3, it is necessary to solve conservation equations for individual species. Following the practices of Reynolds averaging, an... 

The nodal unknows a are to be chosen so as to satisfy the governing equations in an integral sense this can be done by using a Galerkin weighted residual formulation of the conservation equations for momentum and energy transport ... 

Shaw and Schumann [576] were the first to apply LES to the canopy environment in attempts to reproduce characteristics of the flow through and above a deciduous forest. The code numerically solves the basic conservation equations for momentum and heat with options for additional scalars. For flow through a horizontally homogeneous canopy on uniform terrain, the momentum equation appears in the following manner ... 

Not all the terms in these equations have the same importance in determining the flow solution in chemical reactors. The only body force considered in most reactor models, gj (per unit mass), is gravitation which is the same for all chemical species, g. The model equations for momentum and energy can then be simplified. In the momentum equation Pc c = f cS = PS-In the energy equation Xlc=i(jc Sc) = Sc=i jc S = 0- Furthermore, in most multicomponent flows, the energy or heat flux contributions from the interdiffusion processes are in general believed to be small and omitted in most applications, ft-cV jc 0 (e.g., [148], p. 816 [89], p. 198 [11], p. 566). 

After all the modeled terms were substituted into the averaged conservation equation for momentum, the balance equation became ... 

The source term is the mass added to the continuous phase, coming from the dispersed phase due to droplet evaporation. The general form of the equation for momentum conservation is... 

The flow of heat and momentum in the electric furnace was assumed to occur under steady state conditions, so that only steady-state solutions were sought. Further, the flow was mainly laminar, but where it was turbulent the conventional k-e model (9) was applied. Hence, the steady-state transport equations for momentum and heat were solved in three dimensions. The generalized steady-state three-dimensional equation for a conserved variable, q>, is ... 

Thus a momentum and a thermal boimdaiy layer will develop simultaneously whenever the fluid stream and the inunersed surface are at different temperatures (Figure 7.3). The momentum and energy equations are coupled, because the physical properties of non-Newtonian fluids are normally temperature-dependent. The resulting governing equations for momentum and heat transfer require numerical solutions. However, if the physical properties of the fluid do not vary significantly over the relevant temperature interval, there is little interaction between the two boimdaiy layers and they may both be assumed to develop independently of one another. As seen in Chapter 6, the physieal properties other than apparent viscosity may be taken as constant for commonly encountered non-Newtonian fluids. 

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