Transient Energy Balance

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. 

This follows by a steady state energy balance of the surface heated by qe, outside the flame-heated region S. It appears that a critical temperature exists for flame spread in both wind-aided and opposed flow modes for thin and thick materials. Tstmn has not been shown to be a unique material property, but it appears to be constant for a given spread mode at least. Transient and chemical effects appear to be the cause of this flame spread limit exhibited by 7 smln. For example, at a slow enough speed, vp, the time for the pyrolysis may be slower than the effective burning time ... 

For steady-state design scenarios, the required vent rate, once determined, provides the capacity information needed to properly size the relief device and associated piping. For situations that are transient (e.g., two-phase venting of a runaway reactor), the required vent rate would require the simultaneous solution of the applicable material and energy balances on the equipment together with the in-vessel hydrodynamic model. Special cases yielding simplified solutions are given below. For clarity, nonreactive systems and reactive systems are presented separately. 

For a single reaction in the CSTR, the transient mass- and energy-balance equations are... 

The energy balance for the geometry represented in Fig. 8.41 results in transient and convective terms, conduction through the thickness and viscous dissipation caused by the through-the-thickness shear components... 

The transient behavior of the particle temperature is defined by the energy balance... 

Local asymptotic stability criteria may be obtained by first solving the steady-state equations to obtain steady states and then linearizing the transient mass and energy balance equations in terms of deviations of variables around each steady state. The determinant (or slope) and trace conditions derived from the matrix A in the set of equations obtained are necessary and sufficient for asymptotic stability. 

Implementing the reactor temperature controller merits some discussion. While Tr is not a true slow variable (it has a two-time-scale behavior, as illustrated in Figure 6.11(a)), as we argued above, the fast transient of the process (and, inherently, of Tr) is stable. We are thus interested in controlling the slow component of the reactor temperature, which in effect governs the behavior of the entire process. To this end, we conveniently chose the coordinate transformation (6.61)—(6.62) so that the energy balance in Equations (6.63) is written in terms of the reactor temperature Tr, rather than in terms of the total enthalpy of the process. 

A model for transient simulation of radial and axial composition and temperature profiles In pressurized dry ash and slagging moving bed gasifiers Is described. The model Is based on mass and energy balances, thermodynamics, and kinetic and transport rate processes. Particle and gas temperatures are taken to be equal. Computation Is done using orthogonal collocation In the radial variable and exponential collocation In time, with numerical Integration In the axial direction. 

The independent functions are transient versions of the Naphtali-Sandholm functions, including the component balances lEq. (4.109)] and the energy balance of each stage ... 

This is the macroscopic population balance, which is a more useful form of population balance for describing transient and steady state particle size distributions in well-mixed vessels. This population balance in conjunction with mass and energy balances gives a complete description of particulate processes in well-mixed vessels. 

Transient Material and Energy Balances The relief rate requirement at any instant during any event is developed on the basis that the total volume of vapor plus liquid is just equal to the vessel vol-... 

One way around this problem is to find a nondestructive test that would still give good indication of firing characteristics. This has been done, using the Rosenthal model (seen earlier) as the basis, with transient-pulse testing or electrothermal response. This procedure can be used on 100% of the units fabricated and find individual bad actors to be weeded out as well as to find systematic or lot to lot shifts at the same time. This procedure is based upon the same heat transfer and energy balance equation we saw earlier... 

Most process systems are conveniently analyzed using one of the two forms of the energy balance equation presented in Sections 7.3 and 7.4. To perform energy balance calculations on other types of processes, such as seraibatch processes or continuous processes that are being started up or shut down, the full transient energy balance equation is required, This equation is discussed in an introductory fashion in Chapter 11. A more thorough treatment of the full equation is deferred to thermodynamics courses and texts. 

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. 

Derive material balance equations and provide initial conditions for well-mixed transient single-unit processes, and derive energy balance equations and provide initial conditions for well-mixed transient single-unit nonreactive processes. 

Suppose sys(f) is the total energy (internal + kinetic + potential) of a system, and ihm and /hout are the mass flow rates of the system input and output streams. (If the system is closed, these quantities each equal zero.) Proceeding as in the development of the transient mass balance equation, we apply the general energy balance equation (11.3-1) to the system in a small time interval from t to t + 1st, during which time the properties of the input and output streams remain approximately constant. The terms of the equation are as follows (see Section 7,4) ... 

The final example illustrates a transient energy balance on a continuous System. 

Throughout this book, we have seen that when more than one species is involved in a process or when energy balances are required, several balance equations must be derived and solved simultaneously. For steady-state systems the equations are algebraic, but when the systems are transient, simultaneous differential equations must be solved. For the simplest systems, analytical solutions may be obtained by hand, but more commonly numerical solutions are required. Software packages that solve general systems of ordinary differential equations— such as Mathematica , Maple , Matlab , TK-Solver , Polymath , and EZ-Solve —are readily obtained for most computers. Other software packages have been designed specifically to simulate transient chemical processes. Some of these dynamic process simulators run in conjunction with the steady-state flowsheet simulators mentioned in Chapter 10 (e.g.. SPEEDUP, which runs with Aspen Plus, and a dynamic component of HYSYS ) and so have access to physical property databases and thermodynamic correlations. 

This relation is valid for both steady and transient conditions, and the surface energy balance does not involve heat generation since a surface does not have a volume. The energy balance for the outer surface of the wall in Fig. 1-18., for example, can be expressed as... 

Starting with an energy balance on a spherical shell volume clement, derive the one-dimensional transient heat conduction equation for a sphere with constant thermal conductivity and no heal generation. 

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