Thermal energy equation

This equation is the expression of the conservation of thermal energy (first law of themiodynamics) and is written as 

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Note that the mass flux of component i in the liquid phase changes due to chemical conversion, whereas this flux remains constant in the vapour/gas phase since it has been assumed that no reaction occurs in the vapour/gas phase. For both phases the conservation for thermal energy equation is given by... 

Almost all of the models assume local thermal equilibrium between the various phases. The exceptions are the models of Beming et al., ° who use a heat-transfer coefficient to relate the gas temperature to the solid temperature. While this approach may be slightly more accurate, assuming a valid heat-transfer coefficient is known, it is not necessarily needed. Because of the intimate contact between the gas, liquid, and solid phases within the small pores of the various fuel-cell sandwich layers, assuming that all of the phases have the same temperature as each other at each point in the fuel cell is valid. Doing this eliminates the phase dependences in the above equations and allows for a single thermal energy equation to be written. 

Transient heating effects occur for a small particle over a very short time after heating commences, and then quasi-steady state is reached. The quasisteady temperature distribution within the sphere is governed by the thermal energy equation... 

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... 

In the general vector form, the thermal-energy equation is De ... 

The objective of the following series of manipulations is to replace the internal energy on the left-hand side with the enthalpy, which provides a form of the thermal-energy equation that is usually more convenient. 

Even in cases where V-V is small, pV-V may not be small if p 1. With these substitutions the thermal-energy equation becomes... 

The thermal energy equation now has a single term that involves the viscosity it is called the dissipation function... 

The general thermal-energy equation is commonly written in the form Dh Dp ... 

The thermal-energy equation has no explicit source term to describe the heat release associated with chemical reaction. Nevertheless, as stated, the thermal-energy equation does fully accommodate chemical reaction. As is described subsequently, the thermal effects of chemical heat release are captured in the enthalpy term on the left-hand side. 

In the following sections the thermal-energy equation is further specialized for perfect gases, which is our primary focus here. 

The perfect-gas thermal-energy equation is finally simplified to... 

Finally, the perfect-gas thermal-energy equation can be written explicitly in cylindrical coordinates. 

Beginning with the thermal-energy equation in the form of Eq. 3.203, substitute the general expression for dh as given by Eq. 3.221. The left-hand side of Eq. 3.203 becomes... 

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 ... 

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. 

For the numerical CVD problem (Fig. 3.16), compare the magnitudes of the following terms that appear in the thermal-energy equation ... 

Discuss the behavior of the pressure in the DpjDt term that appears in the thermal-energy equation. 

For high-viscosity fluids (e.g., oils), high rod velocities, or small gaps, the thermal energy generation by viscous dissipation may be important. In this case the steady-state, incompressible, thermal energy equation reduces to... 

The thermal energy equation for the parallel-flow in a duct reduces to... 

Beginning with the full Navier-Stokes and thermal-energy equations equations in differential-equation form, eliminate all appropriate terms. Write out the steady-state differential equations that describe this situation. 

Neglect the effects of viscous work (i.e., viscous dissipation), and compare with the steady-state form of the thermal-energy equation for an incompressible fluid. Are they the same equation If so, why If not, why ... 

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.

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... 

With some significant simplifying assumptions, the species-continuity equation can be put into a form that is analogous to the thermal-energy equation. Specifically, consider that there is no gas-phase chemistry and that a single species, A, is dilute in an inert carrier gas, B. In this case, considering Eq. 3.128, Eq. 6.24 reduces to... 

The equation is nearly analogous to the thermal energy equation with the Schmidt number replacing the Prandtl number, although the density dependence is different. The Schmidt number and Reynolds numbers are defined as... 

The Prandtl number must still be retained as a parameter in the thermal-energy equation. The boundary conditions required to solve the system of equations are... 

Formulate the system of equations to couple the thermal-energy equation (Eq. 6.69) into the shooting algorithm. 

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

The pressure term has been retained in the thermal-energy equation, although in low-speed flow there is very little energy content in this term. Indeed, the term is very often neglected in boundary-layer analysis. For high-speed flow (i.e., Ma > 0.3), however, the thermal energy can be substantially affected by pressure variations. 

The recombination reactions consume free radicals to create stable species, resulting in a net reduction of radicals. Since these recombination reactions are very exothermic, they cause the temperature to increase. The lower panel of Fig. 16.11 shows the contribution of various reactions to the temperature rise. Specifically, it shows the contribution of each reaction i to the heat-of-reaction term in the thermal-energy equation (Eq. 16.98) ... 

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. 

The Macroscopic Energy Balance and the Bernoulli and Thermal Energy Equations, 54... 

We should note that the Navier-Stokes equation holds only for Newtonian fluids and incompressible flows. Yet this equation, together with the equation of continuity and with proper initial and boundary conditions, provides all the equations needed to solve (analytically or numerically) any laminar, isothermal flow problem. Solution of these equations yields the pressure and velocity fields that, in turn, give the stress and rate of strain fields and the flow rate. If the flow is nonisothermal, then simultaneously with the foregoing equations, we must solve the thermal energy equation, which is discussed later in this chapter. In this case, if the temperature differences are significant, we must also account for the temperature dependence of the viscosity, density, and thermal conductivity. 

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