Differential element

The nonconvective energy flux across the boundary is composed of two terms a heat flux and a work term. The work term in turn is composed of two terms useful work deflvered outside the fluid, and work done by the fluid inside the control volume B on fluid outside the control volume B, the so-called flow work. The latter may be evaluated by imagining a differential surface moving with the fluid which at time 2ero coincides with a differential element of the surface, S. During the time dt the differential surface sweeps out a volume V cosdSdt and does work on the fluid outside at a rate of PV cos dS. The total flow work done on the fluid outside B by the fluid inside B is... 

Integral Formulation The zone method has the purpose of dodging the solution of an integral equation. If in Eq. (5-126) the zone on which the radiation balance is foriTUilated is decreased to a differential element, that equation becomes... 

First, a mechanism is assumed whether completely mixed, plug flow, laminar, with dispersion, with bypass or recycle or dead space, steady or unsteady, ana so on. Then, for a differential element of space and/or time the elements of a conservation law. 

When a hot fluid stream and a cold fluid stream, separated by a conducting wall, exchange heat, the heat that is transferred across a differential element can be represented by the following expression (refer to Figure 1) ... 

Figure 1. Heat exchange across a differential element in a heat exchanger.

A enters element 1 by eonveetion and dispersion. The output term is the rate of passage of A through the eross-seetion at 1 -i- 61. The disappearanee by reaetion is the volumetrie rate of disappearanee (-r ) times the volume of the differential element 27ir6r6l. 

Consider the differential element of a laterally and axially loaded beam as in Figure D-1. There, the axial force, shear force, moment, and lateral load are depicted along with the pertinent changes that occur along the length of the differential element. 

The plate buckling equations inherently cannot be derived from the equilibrium of a differential element. Instead, the buckling problem represents the departure from the equilibrium state when that state becomes unstable because the in-plane load is too high. The departure from the equilibrium state is accompanied by waves or buckles in the surface of the plate. That is, the plate cannot remain flat when the... 

Figure 3.10. Configuration for radiative exchange between two differential elements.

In this discussion, we will limit our writing of the Pfaffian differential expression bq, for the differential element of heat flow in thermodynamic systems, to reversible processes. It is not possible, generally, to write an expression for bq for an irreversible process in terms of state variables. The irreversible process may involve passage through conditions that are not true states" of the system. For example, in an irreversible expansion of a gas, the values of p. V, and T may not correspond to those dictated by the equation of state of the gas. 

Thus, we can conclude that, within the neighborhood of every state in this thermodynamic system, there are states that cannot be reached via adiabatic paths. Given the existence of these states, then, the existence of an integrating denominator for the differential element of reversible heat, Sqrev, is guaranteed from Caratheodory s theorem. Our next task is to identify this integrating denominator. 

E2.16 For an ideal gas under reversible conditions, a differential element of heat can be expressed in the form... 

Since the heat generated is proportional to the neutron flux, the heat dQ developed per unit time in a differential element of the fuel rod of length dx may be written as ... 

FIGURE 1.4 Differential element in a piston flow reactor. 

Unlike Qp, Na is not a conserved quantity and varies down the length of the tube. Consider a differential element of length and volume h.zAc. The molar flow entering the element is IV (z) and that leaving the element is lV (z+Az), the difference being due to reaction within the volume element. A balance on component A gives... 

Note that since there are two independent variables of both length and time, the defining equation is written in terms of the partial differentials, 3C/dt and 3C/dZ, whereas at steady state only one independent variable, length, is involved and the ordinary derivative function is used. In reality the above diffusion equation results from a combination of an unsteady-state mass balance, based on a small differential element of solid length dZ, combined with Pick s Law of diffusion. 

Thus considering a small differential element of liquid volume, dV, and depth, dZ, the balance equation becomes... 

Consider a differential element of column volume, AV, height AZ and cross-sectional area, Ac, such that AV=Ac AZ. Component mass balance equations can be written for each of the liquid phases, where... 

Figure 4.23. Differential element of height, AZ, for a liquid-liquid extractor with axial mixing in both phases.

Heat balances on a small differential element of heat transfer surface area, AA, give... 

Fig. 4.26 shows the temperature changes over a small differential element of exchanger length AZ. 

Applying Equation 15.39 across the differential element in Figure 15.5b ... 

Figure 2.17 Force balance on a differential element of a curved surface. 

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. 

If a material balance is written over the differential element of pore length shown in Figure 12.1, one finds that at steady state ... 

For a second-order reaction, a material balance on a differential element of pore length leads to the following differential equation... 

---