Energy balance derivation

Starting with a basic energy balance, derive an expression for the effectiveness of a heat exchanger in which a condensing vapor is used to heat a cooler fluid. Assume that the hot fluid (condensing vapor) remains at a constant temperature throughout the process. 

The development of the semi-batch reactor energy balance follows directly from the CSTR energy balance derivation by setting Q = 0. The main results are summarized in Table 6.9 at the end of this chapter. Note in particular that Equations 6.81-6.83 in the semi-batch reactor Table 6.9 are identical to the corresponding Equations 6.72-6.74 in the CSTR Table 6.8. 

For an ideal CSTR or PFR operating at steady state, df/gys/dt = 0. The energy balances derived above (Eqns. (8-5)-(8-8)) can be applied directly by setting the right-hand side to zero. For a PFR, the shaft work will be zero, and for a CSTR the shaft work usually can be neglected. 

Using mass and energy balances, derive suitable equations for the liquid yield and work per unit mass compressed for a Collins helium liquefier utilizing two expanders. Assume all of the components of the system except the expansion valve are ideal and that the work from the expanders is utilized in the compression process. Derive similar equations for a Collins helium liquefier that uses a nitrogen bath to precool the compressed helium gas before entering the first expander. 

The essential differences between sequential-modular and equation-oriented simulators are ia the stmcture of the computer programs (5) and ia the computer time that is required ia getting the solution to a problem. In sequential-modular simulators, at the top level, the executive program accepts iaput data, determines the dow-sheet topology, and derives and controls the calculation sequence for the unit operations ia the dow sheet. The executive then passes control to the unit operations level for the execution of each module. Here, specialized procedures for the unit operations Hbrary calculate mass and energy balances for a particular unit. FiaaHy, the executive and the unit operations level make frequent calls to the physical properties Hbrary level for the routine tasks, enthalpy calculations, and calculations of phase equiHbria and other stream properties. The bottom layer is usually transparent to the user, although it may take 60 to 80% of the calculation efforts. 

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. 

Another procedure, which is more accurate for the external-heat-exchanger cases, is to nse an equivalent value for MC (for a vessel being heated) derived from the following energy balance ... 

The energy conservation equation is not normally solved as given in (9.4). Instead, an evolution equation for internal energy is used [9]. First an evolution equation for the kinetic energy is derived by taking the dot product of the momentum balance equation with the velocity and integrating the resulting differential equation. The differential equation is... 

This derivation indicates a strong coupling between the momentum equation and the energy equation, which implies that the momentum and energy balance equations should be solved as a coupled system. In particular, the dis-... 

Often in plant operations condensate at high pressures are let down to lower pressures. In such situations some low-pressure flash steam is produced, and the low-pressure condensate is either sent to a power plant or is cascaded to a lower pressure level. The following analysis solves the mass and heat balances that describe such a system, and can be used as an approximate calculation procedure. Refer to Figure 2 for a simplified view of the system and the basis for developing the mass and energy balances. We consider the condensate to be at pressure Pj and temperature tj, from whence it is let down to pressure 2. The saturation temperature at pressure Pj is tj. The vapor flow is defined as V Ibs/hr, and the condensate quality is defined as L Ibs/hr. The mass balance derived from Figure 2 is ... 

The following derives an energy balance for a CFSTR based on the following assumptions (Figure 6-2) ... 

Griffith derived a similar equation using an energy balance approach, equating stored energy with the energy required for crack propagation ... 

Neurotransmitter and biogenic amine derived from the amino acid histidine synthesized in hypothalamic tuber-omamillary neurons (TMN) to maintain wakefulness, feeding rhythms, energy balance, neuroendocrine autonomic control, and memory functions prominent immu-nomodulator and proinflammatory signal released from mast cells in response to allergic reactions or tissue damage. 

General equations of momentum and energy balance for dispersed two-phase flow were derived by Van Deemter and Van Der Laan (V2) by integration over a volume containing a large number of elements of the dispersed phase. A complete system of solutions of linearized Navier-Stokes equations... 

The time derivative is zero at steady state, but it is included so that the method of false transients can be used. The computational procedure in Section 4.3.2 applies directly when the energy balance is given by Equation (5.28). The same basic procedure can be used for Equation (5.25). The enthalpy rather than the temperature is marched ahead as the dependent variable, and then Tout is calculated from Hout after each time step. 

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... 

A detailed derivation of the energy balance is given in various textbooks (e.g., Aris, 1989 and Fogler, 1992). 

Whenever changes in temperature are to be calculated, an energy balance is needed. With the assumption of constant Cp and constant p, as derived in Sec. 1.2.5, the balance becomes... 

The energy balance, for element AV of the reactor again follows the generalised form, derived in Sec. 1.2.5. Thus... 

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. 

Fig. 7. Total kinetic energy release derived from velocity map images of 0(3P2) and D(2S) fragment atoms following photodissociation of OD at 226 and 243 nm, respectively. The initial vibrational state of OD is determined from energy balance with TKER = hv + E(vib)oD — Do(OD). The bar graphs show the calculated photodissociation yields for OD X2Il(v) at a vibrational temperature of 1700 K. (From Radenovic et al.97)...

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. 

An energy balance on the reactor can be derived from equation 10.1.1 by omission of appropriate terms. 

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