The Energy Balance Equation

The energy balance equation states that the rate of increase in specific internal (thermal) energy in a control volume equals the rate of energy addition by conduction plus the rate of energy dissipation. The principle of energy conservation is also described by the first law of thermodynamics see Section 5.2. If a constant density is assumed, the energy equation can be written as  

Eacc is the accumulation term, the convection term, E ond the conduction term, and Ediss the dissipation term. Equation 5.5(c) is an expression of Eourier s law of heat transfer see also Section 5.3.1. In cylindrical coordinates, only the convection, conduction, and dissipation terms change  

In order to obtain the energy conservation equation, Irving and Kirkwood showed that a should be given by 

a represents a sum of the kinetic, external potential ((pent), and intermolecular potential ((/ int) energies at the locator vector r and at a time t. Substituting Eq. (5.60) into Eq. (5.1), and following the same type of manipulations given in Secs. 5.3 and 5.4 leads to (Prob. 5.2) 

substituting Eq. (5.60) into the first term on the right-hand side of Eq. (5.9) and simplifying gives (Prob. 5.3) 

Summarizing our results, the energy conservation equation up to this point is 

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In a similar manner, the energy balance equation can be determined ... 

Equation (22) was obtained, essentially, with examination of the energy balance equation with respect to flows of gas-containing polymer melts. The key moment of this analysis is, in our view, comprehension of the fact that the energy of gas dissolved in the polymer is transformed into the energy of movement of the two-phase medium. 

The kinetic energy attributable to this velocity will be dissipated when the liquid enters the reservoir. The pressure drop may now be calculated from the energy balance equation and equation 3.19. For turbulent flow of an incompressible fluid ... 

If at time t the liquid level is D m above the bottom of the tank, then designating point 1 as the liquid level and point 2 as the pipe outlet, and applying the energy balance equation (2.67) for turbulent flow, then ... 

Then applying the energy balance equation between D and the liquid level in each of the tanks gives ... 

In order to obtain the kinetic energy term for use in the energy balance equation, it is necessary to obtain the average kinetic energy per unit mass in terms of the mean velocity. 

Since in the energy balance equation, the kinetic energy per unit mass is expressed as a2/2a, hence a = 0.5 for the streamline flow of a fluid in a round pipe. 

Methods have been given for the calculation of the pressure drop for the flow of an incompressible fluid and for a compressible fluid which behaves as an ideal gas. If the fluid is compressible and deviations from the ideal gas law are appreciable, one of the approximate equations of state, such as van der Waals equation, may be used in place of the law PV = nRT to give the relation between temperature, pressure, and volume. Alternatively, if the enthalpy of the gas is known over a range of temperature and pressure, the energy balance, equation 2.56, which involves a term representing the change in the enthalpy, may be employed ... 

The energy balance equation can be applied between any two sections in a continuous fluid, If the fluid is not moving, the kinetic energy and the frictional loss are both zero, and therefore ... 

Considering a small filament of liquid which is brought to rest at section 2, and applying the energy balance equation between the two sections, since g Az, Wv, and F are all zero ... 

The value of the integral in the energy balance (equation 11.55) is again given by equation 11.60 [substituting (6S - 8o) for 0 ]. The heat flux q0 at the surface is now constant, and the right-hand side of equation 11.55 may be expressed as (—qa/Cf,p). Thus, for constant surface heat flux, equation 11.55 becomes ... 

A heat exchange term is added to the energy balance. Equation (5.30), to give... 

Compound norms Y arise naturally in connection with the energy balance equation. Their structure seems to be rather complicated. It is desirable to possess a priori estimates for solutions of problems (1) and (83) in the usual energy norms of the spaces Ha and Hr. We proceed to the derivation of such estimates. This amounts to setting any three-layer scheme in the form... 

If specific heat capacities can be assumed constant and the mechanical energy input by the stirrer, Wagit, is significant, the energy balance equation simplifies... 

The application of the energy balance equation to segment n, results similarly in the relationship... 

A similar finite-differenced equivalent for the energy balance equation (including axial dispersion effects) may be derived. The simulation example DISRET involves the axial dispersion of both mass and energy and is based on the work of Ramirez (1976). A related model without reaction is used in the simulation example FILTWASH. 

Neglecting acceleration effects in the metal, the energy balance equation is given by... 

The incremental increase in the water temperature dr is related to the heat transferred dQ by the energy-balance equation ... 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

For isothermal and adiabatic modes of operation the energy balance equations developed above will simplify so that the design calculations are not nearly as tedious as they are for the other modes of operation. In the case of adiabatic operation the heat transfer rate is zero, so equation 10.2.10 becomes... 

Equation 10.3.6, the reaction rate expression, and the design equation are sufficient to determine the temperature and composition of the fluid leaving the reactor if the heat transfer characteristics of the system are known. If it is necessary to know the reactor volume needed to obtain a specified conversion at a fixed input flow rate and specified heat transfer conditions, the energy balance equation can be solved to determine the temperature of the reactor contents. When this temperature is substituted into the rate expression, one can readily solve the design equation for the reactor volume. On the other hand, if a reactor of known volume is to be used, a determination of the exit conversion and temperature will require a simultaneous trial and error solution of the energy balance, the rate expression, and the design equation. 

The heat transfer in the reactor can be determined from the energy balance equation (equation 10.3.6). 

The energy balance equation for adiabatic operation becomes... 

However, the energy balance equation appropriate for use in this illustration differs from that employed in the previous case because thermal losses through the reactor walls must be accounted for. It will be of the same general form as equation 12.7.48, but with the wall heat transfer coefficient replaced by an overall heat... 

In order to provide a periodic check on the energy balance equation, one may use the form of equation 13.1.18, which results from integration between the reactor inlet and a distance L downstream where the fraction conversion is... 

Eleat transfer occurs not only within the solid surface, droplet and vapor phases, but also at the liquid-solid and solid-vapor interface. Thus, the energy-balance equations for all phases and interfaces are solved to determine the heat-transfer rate and evaporation rate. 

The radiative heat transfer across the vapor layer is neglected under the condition that the solid temperature is lower than 700 °C (Harvie and Fletcher, 2001 a,b). On the liquid-vapor interface, the energy-balance equation is... 

The heat load, Q> is obtained from the energy-balance equation (14.3-10) for steady-state operation with constant values for the various parameters ... 

From the energy balance (equation 14.3-10) for adiabatic operation [Q = UAC(TC T) =... 

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