Nondimensional energy equation

Assuming constant properties, the nondimensional energy equation is... 

E19 - E47 In cell El 9, enter the difference formula for the nondimensional energy equation... 

For this boundary condition, the nondimensional energy equation and the boundary conditions for the flow inside a microtube, including axial conduction and viscous dissipation are... 

A loss is usually expressed as a loss of heat or enthalpy. A eonvenient way to express them is in a nondimensional manner with referenee to the exit blade speed. The theoretieal total head available (i/ioi) is equal to the head available from the energy equation... 

Equations (11.1)-(11.3) respectively express conservation of mass, momentum, and energy. To nondimensionalize these equations, the initial diameter do, the initial centreline velocity Uq, and a characteristic temperature difference To are used as scales. If the heat is deposited uniformly over a unit volume at a constant rate J, for a total time th into the flow. To may be defined as the resultant net temperature change that would result if this heat was deposited To = Jth/ P p ... 

Nondimensionalizing the energy equation based on the above characteristic quantities (using for nondimensional variables), one obtains ... 

Since G and Ch appear only as a product, G = GCh is used as the relevant nondimensional parameter governing heat release in the present studies. The other governing parameters for this flow are Re and Pr, along with the precise strength and distribution of the source term in the energy equation. 

For the purpose of understanding pressure filtering, attention may be restricted to the single-component, constant-property, nonreacting equations for a perfect gas. Introducing the nondimensional variables into the vector forms of the mass-continuity, constant-viscosity Navier-Stokes, and perfect-gas thermal-energy equations yields the following nondimensional system ... 

Write a thermal energy equation that could be used to describe the temperature distribution in the channel. Nondimensionalize the energy equation. Discuss how it could be used to determine the wall heat transfer in sections where the velocity distribution is fully developed, but the wall temperature is varying. 

Fig. 6.3 Nondimensional axial and radial velocity profiles for the axisymmetric stagnation flow in the semi-infinite half plane above a solid surface. The flow is approaching the surface axially (i.e., u < 0) and flowing radially outward (i.e., V > 0). The temperature profile, which is the result of solving the thermal-energy equation, is discussed in Section 6.3.6.

For a constant-property, incompressible fluid, the energy equation can be solved after the velocity profiles have been determined. A nondimensional temperature can be defined as... 

Derive the nondimensional thermal-energy equation for an axisymmetric, semi-infinite stagnation flow of a constant-property incompressible fluid. 

The columns of cells below row 16 contain the values of the dependent variables at the node points. They will all be iterated until a final solution is achieved. The formula in each cell represents an appropriate form of the difference equations. Each column represents an equation. Column B represents the continuity equation, column C represents the radial momentum equation, column D represents the circumferential momentum equation, and column E represents the thermal energy equation. Column F represents the perfect-gas equation of state, from which the nondimensional density is evaluated. The difference equations involve interactions within a column and between columns. Within a column the finite-difference formulas involve the relationships with nearest-neighbor cells. For example, the temperature in some cell j depends on the temperatures in cells j — 1 and j + 1, that is, the cells one row above and one row below the target cell. Also, because the system is coupled, there is interaction with other columns. For example, the density, column F, appears in all other equations. The axial velocity, column B, also appears in all other equations. 

Having established that similarity solutions for the velocity profile can be found for certain flows involving a varying ffeestream velocity, attention must now be turned to the solutions of the energy equation corresponding to these velocity solutions. The temperature is expressed in terms of the same nondimensional variable that was used in obtaining the flat plate solution, i.e., in terms of 8 = (Tw - T)f(Tw -Tt) and it is assumed that 0 is also a function of ij alone. Attention is restricted to flow over isothermal surfaces, i.e., with Tw a constant, and T, of course, is also constant. 

Reconsider the nondimensionalized momentum and energy equations for steady, incompressible, laminar flow of a fluid with constant properties and negligible viscous dissipation (liqs. 6-65 and 6-66). When Pr 1 (wliicli is approximately the case for gases) and dP ldx = 0 (which is the case when, II V = constant in the free stream, as in flow over a flat plate), these equations simplify to... 

The final step in nondimensionalizing is to deal with the thermal energy equation, (6-201). Introducing the standard thin-film scaling for u, w and spatial derivatives with respect to x and z, we find that this equation becomes... 

Let us suppose, based upon the estimates of (9-28), that the dimensionless form (9-24) of the thermal energy equation is valid within the region 1 < r < ()(Pe ). In other words, we suppose that within this so-called inner region, the sphere radius is an appropriate characteristic length scale as we have assumed in the nondimensionalization leading up to (9 24). Hence, within this region, the dimensionless temperature field can be represented in the form (9 25) with (>o = 1 /r, that is,... 

The overbar, such as Ep, means nondimensional energy. The total energy of the system is calculated by summing the above equations. 

Nondimensionalize the energy equation and show the dimensionless form of the energy equation as... 

The system of governing equations contains several parameters. 3 is a nondimensional activation energy and y is a nondimensional... 

Derive a nondimensional system of equations that describes the fluid-flow, thermal-energy, and mass-transfer problem for the ideal rotating-disk problem in the semiinfinite half plane. 

Nondimensionalization of the species- and energy-conservation equations follows a procedure that is analogous to that for the Navier-Stokes equations. For two-dimensional steady axisymmetric flow of a perfect gas, the full equations are given as... 

In Section 5.3.6, activation-energy asymptotics have been applied to the adiabatic version of equation (9) for a particular rate function w burning-rate formulas are given in Section 5.3.6 for this rate function and in Section 5.3.7 for others. Here it is convenient to presume that for L = 0, the burning rate is known on the basis of these results and to employ the known adiabatic mass burning rate for the purpose of nondimensionalization. Thus, in analogy with equation (5-18), we introduce the nondimensional stream wise coordinate = m CpX/X and obtain the equation... 

The combustion mechanism addressed involves inert heat conduction in the solid, surface gasification by an Arrhenius process and a gas-phase deflagration having a high nondimensional activation energy. With the density, specific heat, and thermal conductivity of the solid assumed constant, the equation for energy conservation in the solid becomes... 

FIGURE 9.4. Representative burning-rate response curves obtained from equation (66), showing a nondimensional measure of the amplification rate as a function of the nondimensional frequency for various values of the nondimensional activation energy for gasification, A, with A = IB. 

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