The General Energy Equation

Equation 2.41 is the general energy equation for a steady flow process. 

In deriving the head equation, the general energy Equation 2.41 will be used, The equation can be modified by regrouping and eliminating the z terms, as elevation differences are not significant with gas. 

In Chapter 2, the general energy equation for the flow of a fluid through a pipe has been expressed in the form ... 

Ideally, one would like to use the tensorial approach used In the Section 7.6 to calculate the energy dissipation. The generalized energy equation has a number of terms [58] ... 

Panned (P4) and Jurges (J8) laid the basis for such predictions and more recently analogue computers have been applied to this problem (J6, S6). In principle, such predictions involved the solution of the general energy equation which may be written in the following form for two-... 

The general energy equation Eq. (10.5) and the continuity equation are two important keys to the solution of many problems in fluid mechanics. For compressible fluids it is necessary to have a third equation, which is the equation of state, i.e., the relation between pressure, temperature, and specific volume (or specific weight or density). 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

Refer to the hydraulic jump schematic of Figure 6.9. The general energy equation may be applied between points 1 and 2 producing... 

The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. Eor K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When and... 

The full solution to the general energy equation will be seen to lead to an implicit set of equations for pipe-flow, requiring an iterative solution. However, it is possible to approximate the solution accurately with explicit formulations, and these will be presented at the end of the chapter. 

We begin by recalling the general energy equation for frictionally resisted flow developed in Section 4.2 and repeated below ... 

The general energy equation applied to fluid may be written as in equation (4.7), repeated below ... 

Equation 7.204 has the same form as the Fourier law for conduction heat transfer. However, the equivalent conductivity is highly temperature dependent. For a medium such as a high-temperature solid where convective effects and sources are absent, the general energy equation (Eq. 7.203) reduces to... 

Bernoulli equation = a form of the general energy equation = law of conservation of energy, applied to thermal and fluid flow situations. Particularly, illustrating the interconversion of kinetic (velocity) energy and pressure energy. Also see Venturi, betw = between, billet = See bar, billet, bloom. 

The generalized energy equation for a polymerization system in convective or non-convective domain is (Bird et al., 2007)... 

With these assumptions, the general energy equation (Eq. 5.5) can be simplified to ... 

The "general" energy equation can be simplified in the same way as the continuity equation, since the approximations introduced there are assumed to be equally applicable here. But, whereas mass is generally not diffusing through the wall, heat frequently is. In deriving the one-dimensional model by averaging over the cross section, a boundary condition for heat transfer at the reactor wall has to be introduced for this reason. It is commonly written as... 

Note that (9.1-6) is nothing but (7.Z.4-4) of Chapter 7, obtained by simplifying the general energy equation (7.2.3-1), provided that dr, the reactor diameter, is replaced by dt, the tube diameter. Using the general equations of Chapter 7 as a starting point clearly shows the assumptions used in the derivation of a specific form of the balance. 

The general energy equation for analyzing this system... 

Q - heat - since none will be added or subtracted The general energy equation now becomes ... 

Bernoulli s equation is a special case of the general energy equation (mechanical energy balance)... 

Bernoulli s equation is a special case of the general energy equation (mechanical energy balance) that is probably the most widely used tool for solving fluid flow problems. It provides an easy way to relate the elevation head, velocity head, and pressure head of a fluid. 

Bernoulli s equation results from the application of the general energy equation and the first law of thermodynamics to a steady-flow systan. 

Consider a steady-flow system in which no work is done on or by the fluid, no heat is transferred to or from the fluid, and no change occurs in the internal energy (i.e., no temperature change) of the fluid. Under these conditions, the general energy equation is simplified to... 

The viscous dissipation of mechanical energy results in conversion of energy to heat and therefore a rise in temperature. For a general situation with heat transfer to/from the fluid medium, the generalized energy equation can be derived from the N-S equation. 

The generalized energy equation for fluid flow is expressed as... 

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