Accumulation term energy balance

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

The accumulation term in the energy balance equation can be rewritten as... 

All the examples of energy balances considered previously have been for steady-state processes where the rate of energy generation or consumption did not vary with time and the accumulation term in the general energy balance equation was taken as zero. 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

The input and output terms of equation 1.5-1 may each have more than one contribution. The input of a species may be by convective (bulk) flow, by diffusion of some kind across the entry point(s), and by formation by chemical reaction(s) within the control volume. The output of a species may include consumption by reaction(s) within the control volume. There are also corresponding terms in the energy balance (e.g., generation or consumption of enthalpy by reaction), and in addition there is heat transfer (2), which does not involve material flow. The accumulation term on the right side of equation 1.5-1 is the net result of the inputs and outputs for steady-state operation, it is zero, and for unsteady-state operation, it is nonzero. 

Inputs + Sources = Outputs + Sinks + Accumulations where each of these terms may be a quantity or a rate. Inputs and Outputs are accomplished by crossing the boundary of the reference volume. In case of mass transfer this occurs by bulk flow and diffusion. Sources and Sinks are accretions and depletions of a species without crossing the boundaries. In a mass and energy balance, sinks are the rate of reaction, rdVr, or a rate of enthalpy change, AHrpdC. Accumulation is the time derivative of the content of the species within the reference volume, for example, (<9C/3t)dVr or... 

For reactions involving heat effects, the total and component material balance equations must be coupled with a reactor energy balance equation. Neglecting work done by the system on the surroundings, the energy balance is expressed by where each term has units of kj/s. For steady-state operation the accumulation... 

Another potential model simplification involves assuming negligible energy accumulation in the gas phase as compared to that in the solid, which is equivalent to the earlier approximation [Eq. (66)] based on the relative magnitude of the energy accumulation in the gas and solid. For our system, the accumulation of energy in the solid is approximately 250 to 300 times that in the gas phase due to the relative thermal capacitance of the gas [Eq. (65)] and the similarity of the temporal behavior of the gas and catalyst temperatures (e.g., Fig. 19). Thus the accumulation term in the energy balance... 

The energy balance (3.301) is applicable for catalysis, adsorption, and ion exchange. More specifically, in catalysis, where the steady-state condition exists, frequently the accumulation term is zero. In contrast, adsorption and ion exchange operate under unsteady-state condition. The analysis of the energy balance equation for catalytic fixed beds is presented in detail in Section 5.3.4. 

The gas energy and mass balance equations, unlike the corresponding solid balances, do not have a term for accumulation. This is because the high convective flow of gas through the channels of the monolith makes accumulation of heat or reactants in the gas phase negligible. In practice, the accumulation term in the solid mass balance could also be removed as, in general, it also tends to be small. However, it is included in our models as it enables the equations to be solved numerically more easily. 

Ranzi et al. [195] found that the full energy balances, including the accumulation term, have to be considered in order to predict correct dynamic process behavior. Therefore, the differential dynamic energy balance for the liquid phase is applied as follows ... 

Let the system be the whole car. The accumulation term in the energy balance is not zero because the kinetic energy of the vehicle is initially not zero but after stopping is zero. Also, energy (heat) is transferred from the vehicle to the surroundings so that the energy transfer term in the energy balance is not zero. The rest of the terms presumably are zero. Consequently we get (in symbols)... 

In the energy balance the accumulation term is zero because the internal energy of an ideal gas depends only on the temperature, and the temperature is constant. The energy transport terms involve heat and work... 

In deriving the integral mass balance for a closed system in Section 4.2c we eliminated the input and output terms, since by definition no mass crosses the boundaries of a closed system. It is possible, however, for energy to be transferred across the boundaries as heat or work, so that the right side of Equation 7.3-1 may not be eliminated automatically. As with mass balances, however, the accumulation term equals the final value of the balanced quantity (in this case, the system energy) minus the initial value of this quantity. Equation 7,3-1 may therefore be written... 

The procedures for deriving balances on transient systems are essentially those developed in Chapters 4 (material balances) and 7 (energy balances). The main difference is that transient balances have nonzero accumulation terms that are derivatives, so that instead of algebraic equations the balances are differential equations. 

Write an expression for the amount of the balanced quantity in the system (mass, moles of a particular species, energy) and set the accumulation term in the balance equation equal to the derivative of that amount with respect to time. 

As a first approximation, we assume a quasi-steady state for the coolant flow and neglect the accumulation term (i.e., dTJdt = 0). An energy balance on the coolant fluid entering and leaving the exchanger is... 

The first term accounts for accumulation, the second for convective transport with the velocity Uz in the direction z, and the third for chemical reaction. The energy balance gives the following equation for the temperature profile ... 

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