Thermal energy equation derivation

By subtracting the mechanical-energy contributions from the total energy equation, a thermal energy equation can be derived. It is this equation that proves to be most useful in the solution of chemically reacting flow problems. By a vector-tensor identity for symmetric tensors, the work-rate term in the previous sections can be expanded as... 

With the convective derivatives eliminated and the properties constant, the thermal-energy equation is completely decoupled from the system. Moreover the energy equation is a simple, linear, parabolic, partial differential equation. 

In deriving the thermal-energy equation it is usually beneficial to introduce enthalpy (not internal energy) as the dependent variable. Doing so, however, also introduces a Dp/Dt term in exchange for a p V-V term. The objective of this exercise is to explore the behavior of certain terms in the alternative formulations of the thermal-energy equation. 

Introducing specific length and velocity scales provides a more intuitive approach to nondi-mensionalization. In this section the thermal-energy equation is also included in the analysis. Assuming constant transport properties and a single-component fluid, a subset of the governing equations is derived from Section 6.2 as... 

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

Even with these simplifications, however, it is rarely possible to obtain analytic solutions for fluid mechanics or heat transfer problems. The Navier Stokes equation for an isothermal fluid is still nonlinear, as can be seen by examination of either (2 89) or (2 91). The Bousi-nesq equations involve a coupling between u and 6, introducing additional nonlinearities. It will be noted, however, that, provided the density can be taken as constant in the body-force term (thus neglecting any natural convection), the fluid mechanics problem is decoupled from the thermal problem in the sense that the equations of motion, (2 89) or (2-91), and continuity, (2-20), do not involve the temperature 0. The thermal energy equation, (2-93), is actually a linear equation in the unknown 6, once the Boussinesq approximation has been introduced. In that case, the only nonlinear term is dissipation, but this involves the product E E and can be treated simply as a source term that will be known once Eqs. (2-89) or (2 91) and (2 20) have been solved to determine the velocity. In spite of being linear, however, the velocity u appears as a coefficient (in the convective derivative term). Even when the form of u is known (either exactly or approximately), it is normally quite a complicated function, and this makes it extremely difficult to obtain analytic solutions for 0 even though the governing equation is linear. 

The governing equation to determine the temperature distribution in the tube is the thermal energy equation, (2-110). To solve this equation, we need to know the form of the velocity distribution in the tube. We have already seen that the steady-state velocity profile for an isothermal fluid, far downstream from the entrance to the tube, is the Poiseuille flow solution given by (3-44). In the present problem, however, the temperature must depend on both r and z, and hence the viscosity (which depends on the temperature) will also depend on position. The dependence on z is due to the fact that heat is added for all z > 0, and thus the temperature must continue to increase with the increase of z. The dependence on r is due to the fact that there must be a nonzero conductive heat flux in the fluid at the tube wall to match the prescribed heat flux through the wall, and thus the temperature must have a nonzero r derivative. It follows that the velocity field will generally differ from Poiseuille flow. 

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... 

From Eq. 15 it is possible to derive the following particular expressions of the thermal energy equation as a function of the fluid considered ... 

Clausius-Clapeyron Equation. This equation was originally derived to describe the vaporization process of a pure liquid, but it can be also applied to other two-phase transitions of a pure substance. The Clausius-Clapeyron equation relates the variation of vapor pressure (P ) with absolute temperature (T) to the molar latent heat of vaporization, i.e., the thermal energy required to vajxirize one mole of the pure liquid ... 

The second step is the molecular dynamics (MD) calculation that is based on the solution of the Newtonian equations of motion. An arbitrary starting conformation is chosen and the atoms in the molecule can move under the restriction of a certain force field using the thermal energy, distributed via Boltzmann functions to the atoms in the molecule in a stochastic manner. The aim is to find the conformation with minimal energy when the experimental distances and sometimes simultaneously the bond angles as derived from vicinal or direct coupling constants are used as constraints. 

In this summary, the local thermal equilibrium model has been used to derive the energy equation. This model is much simpler than the two-phase model however, the local thermal equilibrium model is most likely not adequate to describe the transport of energy when the temperature of the fluid and solid are undergoing extremely rapid changes. Although such extremely rapid temperature changes are not expected, in most RTM, IP, and AP processes the correctness of the local thermal equilibrium assumption can be verified by following the procedure discussed by Whitaker [28]. 

The algebraic equations for the orthogonal collocation model consist of the axial boundary conditions along with the continuity equation solved at the interior collocation points and at the end of the bed. This latter equation is algebraic since the time derivative for the gas temperature can be replaced with the algebraic expression obtained from the energy balance for the gas. Of these, the boundary conditions for the mass balances and for the energy equation for the thermal well can be solved explicitly for the concentrations and thermal well temperatures at the axial boundary points as linear expressions of the conditions at the interior collocation points. The set of four boundary conditions for the gas and catalyst temperatures are coupled and are nonlinear due to the convective term in the inlet boundary condition for the gas phase. After a Taylor series expansion of this term around the steady-state inlet gas temperature, gas velocity, and inlet concentrations, the system of four equations is solved for the gas and catalyst temperatures at the boundary points. 

Hence, we arrive at the conclusion that only in the limit a - 0 the Hookean body is the ideal energy-elastic one (r = 0) and the uniform deformation of a real system is accompanied by thermal effects. Equation (19) shows also that the dependence of the parameter q (as well as to) on strain is a hyperbolic one and a, the phenomenological coefficient of thermal expansion in the unstrained state, is determined solely by the heat to work and the internal energy to work ratios. From Eqs. (17) and (18), we derive the internal energy of Hookean body... 

One important purpose of the energy equation is to describe and predict the fluid temperature fields. The energy equation will be closely coupled to the Navier-Stokes equations, which describe the velocity fields. The coupling comes through the convective terms in the substantial derivative, which, of course, involve the velocities. The Navier-Stokes equations are also coupled to the energy equation, since the density and other properties usually depend on temperature. Chemical reaction and molecular transport of chemical species can also have a major influence on the thermal energy of a flow. 

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. 

Accurate modeling is only possible by the consideration of wavelength-dependent optical and temperature-dependent thermodynamic parameters and the correct application of the thermal accommodation coefficient which is dependent on the ambient particle conditions and is described in detail elsewhere (Schulz et al., 2006 Daun et al., 2007). Moreover, Michelsen (2003) suggested the inclusion of a nonthermal photodesorption mechanism for heat and mass loss, the sublimation of multiple cluster species from the surface, and the influence of annealing on absorption, emission, and sublimation. A more general form of the energy equation including in more detail mass transfer processes has been derived recently by Hiers (2008). For practical use, Equation (1) turns out to be of sufficient physical detail. 

Problem definition requires specification of the initial state of the system and boundary conditions, which are mathematical constraints describing the physical situation at the boundaries. These may be thermal energy, momentum, or other types of restrictions at the geometric boundaries. The system is determined when one boundary condition is known for each first partial derivative, two boundary conditions for each second partial derivative, and so on. In a plate heated from ambient temperature to 1200°F, the temperature distribution in the plate is determined by the heat equation 8T/dt = a V2T. The initial condition is T = 60°F at / = 0, all over the plate. The boundary conditions indicate how heat is applied to the plate at the various edges y = 0, 0[Pg.86]

The derivation of these equations (cf., W) involves the important assumptions that the gas phase inertia is negligible compared with that of the solid, the temperatures of the solid and gas phases have the same local values and the kinetic energy of the system is small compared with the thermal energy. 

In the operator L, the first term represents convection and the second diffusion. Equation (44) therefore describes a balance of convective, diffusive, and reactive effects. Such balances are very common in combustion and often are employed as points of departure in theories that do not begin with derivations of conservation equations. If the steady-flow approximation is relaxed, then an additional term, d(p(x)/dt, appears in L this term represents accumulation of thermal energy or chemical species. For species conservation, equations (48) and (49) may be derived with this generalized definition of L, in the absence of the assumptions of low-speed flow and of a Lewis... 

---