Thermal energy balance

Referring to the schematic diagram in Figure 2-81, it is clear that the thermal energy balance should include terms AH and AH, which represent the sensible heats necessary to raise the temperatures of reactants and products from to T,. Thus,... 

The preceding calculation of the thermal energy balance of a planet neglected any absorption of radiation by molecules within the atmosphere. Radiation trapping in the infrared by molecules such as CO2 and H20 provides an additional mechanism for raising the surface temperature - the greenhouse effect. The local temperature of a planet can then be enhanced over its black body temperature by the atmosphere. 

For the discrete bubble model described in Section V.C, future work will be focused on implementation of closure equations in the force balance, like empirical relations for bubble-rise velocities and the interaction between bubbles. Clearly, a more refined model for the bubble-bubble interaction, including coalescence and breakup, is required along with a more realistic description of the rheology of fluidized suspensions. Finally, the adapted model should be augmented with a thermal energy balance, and associated closures for the thermophysical properties, to study heat transport in large-scale fluidized beds, such as FCC-regenerators and PE and PP gas-phase polymerization reactors. 

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... 

The fundamental material and thermal energy balance equations have been set out by Scott et al. (1964) and by Nienow and Rowe (1975) amongst others. Referring to Figure 5.5, the mass balance is... 

Before we develop the energy balances for our reactors, it is worthwhile to define some quantities from thermocfynamics because the energy balances we need are thermal energy balances. We begin with the First Law of Thermodynamics,... 

Adiabatic trajectories are typical of the cure of large parts, where the rate of heat diffusion is much lower than the rate of heat generation. A thermal energy balance may be stated as follows ... 

The following thermal energy balance may be written for an arbitrary volume surrounding the measuring point ... 

The adiabatic temperature rise method may also be used if the reaction is not very fast, by taking into account corrections for heat losses. The thermal energy balance may be written as (Rojas et al., 1981) ... 

To obtain the evolution of the concentration of functional groups and temperature, the constitutive equations for the reaction rates must be introduced into mass balances (in fact, balances of functional groups) and in a thermal energy balance. The form of these balances depends on the characteristics of the reactor in which the thermosetting polymerization is taking place (the reactor may be a DSC pan, a heated mold, a flow reactor, etc.). 

If, instead of isothermal conditions, the cure is performed at a constant heating rate, the constitutive equations for the reaction kinetics are not altered, but a thermal energy balance must be included to describe the system. In this case, this balance may be reduced to... 

If the same polymerization is carried out in a heated mold, concentrations and temperature vary with both position and time. For a one-dimensional system (for example, when variations along the thickness are only relevant), mass and thermal energy balances are written as... 

The thermal energy balance for this case may be written as... 

Integration of Eq. 2.9-11 leads to the macroscopic mechanical energy balance equation, the steady-state version of which is the famous Bernoulli equation. Next we subtract Eq. 2.9-11 from Eq. 2.9-10 to obtain the differential thermal energy-balance... 

The physical mechanisms that can bring about melting or heating of any substance are included in the terms of the thermal energy balance, Eq. 2.9-14... 

Alternative heating mechanisms to conduction, such as dielectric or ultrasonic energies, have also been attempted. These mechanisms can be dissipated by polymer solids, creating volumewide homogeneous heat sources. With these mechanisms, the governing form of the thermal-energy balance becomes... 

Adiabatic Compression Heating Melting of polymers by adiabatic compression has been shown to be feasible for processes such as injection molding (2). Discuss this method, in principle, in terms of an order-of-magnitude analysis of the terms of the thermal energy balance for an amorphous (PS) and a semicrystalline polymer (LDPE). Use the data in Appendix A. 

In batch reactors, for thermally simple types of reactions, that is, ones that can be attributed to a single reaction step, generally applicable to the propagation step of polymerization reactions, we can write the following thermal energy balance (6)... 

One of the assumptions made in solving the flow in the fully developed region of the capillary was that of constant fluid temperature throughout the flow region. This is not a valid assumption for the flow of very viscous fluids at high rates of shear in which a nonuniform temperature field is created. As we have already mentioned in connection with the thermal-energy balance (Section 2.2), the rate of viscous heating per unit volume ev is... 

As discussed in Chapters 2 and 3, in the integral method it is assumed that the boundary layer has a definite thickness and the overall or integrated momentum and thermal energy balances across the boundary layer are considered. In the case of flow over a body in a porous medium, if the Darcy assumptions are used, there is, as discussed before, no velocity boundary layer, the velocity parallel to the surface near the surface being essentially equal to the surface velocity given by the potential flow solution. For flow over a body in a porous medium, therefore, only the energy integral equation need be considered. This equation was shown in Chapter 2 to be ... 

Manns M, Maurer E, Steinbach B, Evering HG. Thermal energy balance during in vitro continuous veno-venous hemofiltration. ASAIO J 1998 44 M601-M605. 

This relationship is known as the mechanical energy balance and is a direct consequence of Newton s second law. Substituting for Du2/Dt in (2-45) using (2 16) and recalling that T is symmetric, we obtain the so-called thermal energy balance ... 

An alternative is to express the thermal energy balance, (2-47), in terms of the specific enthalpy h ... 

The change of enthalpy in transforming from one phase to the other is known as the latent heat per unit mass, and may be denoted as either AH or L. The thermal energy balance at the interface thus requires that there be an imbalance in the heat flux to and from the interface by conduction that is just equal to the rate of release (or adsorption) of enthalpy as the material changes phase. 

The thermal energy balance at the interface then takes the (still complicated) form... 

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