Thermal energy conservation equation

An abuse model requires (i) materials (mass) balance for the exothermic side reactions, (ii) estimation of the reaction parameters (e.g., heat of reaction) from experimental measurements such as differential scanning calorimetry (DSC) and accelerated rate calorimetry (ARC), (iii) devising the kinetic expressions of the reactions, and (iv) incorporation of the thermal behavior due to these reactions in the energy balance equation (e.g., in terms of volumetric source terms). Specifically, the thermal energy conservation equation is duly modified to include the additional heat generation effects to reflect the specific abuse behavior in terms of heat generation due to side reaction kinetics and/or joule heating. The thermal boundary condition may also include radiative heat transfer to the ambient air. 

Thermal plumes above point (Fig. 7.60) and line (Fig. 7.61) sources have been studied for many years. Among the earliest publications are those from Zeldovich and Schmidt. Analytical equations to calculate velocities, temperatures, and airflow rates in thermal plumes over point and line heat sources with given heat loads were derived based on the momentum and energy conservation equations, assuming Gaussian velocity and excessive temperature distribution in... 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

As mentioned, to include nonisothermal effects, an overall thermal energy balance needs to be added to the set of governing equations. The energy conservation equation can be written for phase k in the... 

For example, for a constant thermal conductivity, k, the energy conservation equation can be written in this form for the temperature, i.e.,... 

For a ternary mixture, equations above can describe thermodynamically and mathematically coupled mass and energy conservation equations without chemical reaction, and electrical, magnetic and viscous effects. To solve these equations, we need the data on heats of transport, thermal diffusion coefficient, diffusion coefficients and thermal conductivity, and the accuracy of solutions depend on the accuracy of the data. 

The thermal model is that of Semenov. Space-averaged variables were used in order to minimize the mathematical difficulties and hence the model does not give an account of the spatial propagation of cool flames. The energy conservation equation used is... 

The mathematical formulation of such a problem begins with the statement of the appropriate equations of change. In Rayleigh s problem, these were the equations of motion, the equation of continuity, and the equation of thermal-energy conservation, together with an appropriate equation of state. In their most general form, these equations are... 

In equation 3, p stands for water density, pi for liquid dynamic viscosity and for relative conductivity. The liquid conductivity is associated to darcean liquid flow, in water mass conservation equation. This term is non-linear because water relative conductivity depends on capillary pressure, which is the main variable associated with water mass conservation equation. Furthermore it is coupled to thermal effects because liquid dynamic viscosity depends on temperature, which is the main variable associated with energy conservation equation. 

Numerical studies have been carried out in three-dimensional (x, y, z) geometry. The system of equations of the mathematical model includes the continuity equation, a generalization of Darcy law for the case of variable density flow, equation of thermal energy conservation, and closing relationships for the calculation of the pore solution density and viscosity. 

The production, conversion, and utilization of energy may lead to ecological cost that includes environmental problems, sueh as air and water pollution, impact on the use of land and rivers, thermal pollution due to mismanagement of waste heat, and global elimate change. As an energy conservation equation, the first... 

The multi-physics of SOFCs are governed by the mass, momentum, and energy conservation equations, and the chemistry and electrochemistry. The governing equations of SOFCs are tightly coupled and changes to one aspect of the fuel cell can drastically affect another. For example, the rate and composition of the fuel flow in the anode will affect the temperature distributions in the cell, which can induce stresses due to mismatches between the coeSicients of thermal expansion of the various layers in the SOFC. The fuel flow wiU also affect the overall performance of the fuel cell based on the distribution of species in the anode and the electrochemical reactions. 

For steady-state cases, the energy conservation equation can be simplified by removing the time-dependent terms, which decreases the computational cost of the thermal solution. However, the general form of Eq. (26.7) is necessary for the simulation of transient operating conditions, such as start-up and load change, when the thermal stresses in the system will be greatest. 

With regard to energy conservation equation, we must consider the two types of fluxes, namely the convective and the diffusive (thermal conductivity of the fluid) ... 

Kroeker et al. [52] investigated the pressure drop and thermal characteristics of heat sinks with circular microchannels using the continuum model consisting of the conventional Navier-Stokes equations and the energy conservation equation. 

The energy conservation equation can be written directly from Eq. 9.4 by simply replacing Q by pCi,r, D by the effective thermal conductivity pQ,K, and Ra by l(—A/fi)Ra, l y neglecting the kinetic and potential energy terms. 

In general we have to deal with a set of simultaneous differential equations written for the conservation of the gaseous reactants and the conservation of the solid reactants, together with the appropriate thermal energy balance equations. 

The thermal calculations are carried out from the core inlet to the core outlet. The inlet coolant temperature and mass flow rate are used as boundary conditions. The temperatures of the coolant in fuel channels and those of the moderator water in the water rods are calculated from the mass and energy conservation equations. The axial power is assumed to follow a cosine distribution. The radial power distribution in the fuel assembly is not considered. The steady-state temperature distributions are assumed in the fuel pellet, fuel cladding, and the gap. The thermal power generated in the reactor is to be consumed among the turbines, the condenser, and the feedwater heaters. The calculations are carried out iteratively until the solutions are convergent to steady-state values. 

In this chapter, we briefly describe fundamental concepts of heat transfer. We begin in Section 20.1 with a description of heat conduction. We base this description on three key points Fourier s law for conduction, energy transport through a thin film, and energy transport in a semi-infinite slab. In Section 20.2, we discuss energy conservation equations that are general forms of the first law of thermodynamics. In Section 20.3, we analyze interfacial heat transfer in terms of heat transfer coefficients, and in Section 20.4, we discuss numerical values of thermal conductivities, thermal diffusivities, and heat transfer coefficients. 

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

The objectives are not realized when physical modeling are applied to complex processes. However, consideration of the appropriate differential equations at steady state for the conservation of mass, momentum, and thermal energy has resulted in various dimensionless groups. These groups must be equal for both the model and the prototype for complete similarity to exist on scale-up. 

This equation may be used as an appropriate form of the law of energy conservation in various pseudo homogeneous models of fixed bed reactors. Radial transport by effective thermal conduction is an essential element of two-dimensional reactor models but, for one-dimensional models, the last term must be replaced by one involving heat losses to the walls. 

However, by the thermally thin approximation, T(x,t) T(t) only. A control volume surrounding the thin material with the conservation of energy applied, Equation (3.45), gives (for a solid at equal pressure with its surroundings)... 

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... 

Discretization of eqs (10) and (11) for, respectively, the liquid phase and the gas/vapour phase together with the discretization of the conservation equation for thermal energy (12) and the energy flux equation (8) leads to a total of (/C1 + Kj ) (2n 2) non-linear equa-... 

---