Energy balances simultaneous solution

For an isothermal system the simultaneous solution of equations 30 and 31, subject to the boundary conditions imposed on the column, provides the expressions for the concentration profiles in both phases. If the system is nonisotherm a1, an energy balance is also required and since, in... 

The couphng equation is a vapor mass balance written at the vent system entrance and provides a relationship between the vent rate W and the vent system inlet quahty Xq. The relief system flow models described in the following section provide a second relationship between W and Xo to be solved simultaneously with the coupling equation. Once W andXo are known, the simultaneous solution of the material and energy balances can be accomplished. For all the preceding vessel flow models and the coupling equations, the reader is referred to the DIERS Project Manual for a more complete and detailed review. 

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. 

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. 

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. 

The performance of propints is a unique function of the temp of the hot reaction products, their compn and their pressure. The pro-pint bums at constant pressure and forms a set of products which are in thermal and chemical equilibrium with each other. The multiplicity of the reaction products requires that the combustion chamber conditions be calcd from the solution of simultaneous equations of pressure and energy balances. This calcn is best performed by computer, although the manual scheme has been described well by Sutton (Ref 14) and Barr re et al (Ref 10). The chamber conditions determine the condition in the nozzle which in turn characterizes the rocket engine performance in terms of specific impulse and characteristic exhaust velocity... 

Remark 4 The presented optimization model is an MINLP problem. The binary variables select the process stream matches, while the continuous variables represent the utility loads, the heat loads of the heat exchangers, the heat residuals, the flow rates and temperatures of the interconnecting streams in the hyperstructure, and the area of each exchanger. Note that by substituting the areas from the constraints (B) into the objective function we eliminate them from the variable set. The nonlinearities in the in the proposed model arise because of the objective function and the energy balances in the mixers and heat exchangers. As a result we have nonconvexities present in both the objective function and constraints. The solution of the MINLP model will provide simultaneously the... 

Industrial design problems often occur in tubular reactors that involve the simultaneous solution of AP, energy, and mass balances. 

The solution to a multi-component, multi-phase, multi-stage separation problem is found in the simultaneous or iterative solution of the material balances, the energy balance and the phase equilibrium equations (see Chapter 1). This implies that a sufficient number of design variables are specified so that the number of remaining unknown variables exactly equals the number of independent equations. When this is done, a separation process is said to be specified. 

The energy balances are not solved in the same manner as the component or total material balances. With some solution methods, they are simultaneously solved with other MESH equations to get the independent cc umn variables in others they are used in a more limited manner to get a new set of total flow rates or stage temperatures. 

Eqs. (11.92) and (11.96), along with the boundary conditions, constitute a pair of simultaneous ordinary differential equations in F and 0. However, the value of Vs must be found in order to derive the solution. To do this, it is noted that if the flow up to any value of x from the top of the plate is considered, the overall energy balance requires ... 

Write the material- and energy-balance expressions for the reactor. This problem must be solved by simultaneous solution of the material- and energy-balance relationships that describe the reacting system. Since the reactor is well insulated and an exothermic reaction is taking place, the fluid in the reactor will heat up, causing the reaction to take place at some temperature other than where the reaction rate constant and heat of reaction are known. 

After counting all the equations and variables in Tables 3.4.1 and 3.4.2, we find that we now have zero degrees of fieedom. Thus, we have defined the problem, and we can now outline the solution procediu e. The twenty-two equations are decoupled, i.e., it is not necessary to solve all them simultaneously. By inspection we find that we can solve the mole balance equations independently of the energy balance. This frequently occurs, usually when the temperatures in some of the lines are known. Furthermore, in this case, we do require an iterative calculation procedure. We again obtained a solution procedure by inspection, which is given in Table 3.4.3. 

Some of the full simulation flow-sheeting packages can also be used to calculate the material balance without simultaneous solution of the energy balance, or use of the equipment design routines. They should be used in this mode for the initial, scouting, flow-sheet calculations, to economise on computing costs. 

Since the reaction rate expression now contains the independent variable T, the material balance cannot be solved alone. The solution of the material balance equation is only possible by the simultaneous solution of the energy balance. Thus, for nonisothermal reactor descriptions, an energy balance must accompany the material balance. 

To solve the mass balance, it must be accompanied by the simultaneous solution of the energy balance (i.e., the solution of Equation (9.2.9)). To do this, Equation (9.2.9) can be written in more convenient forms. Consider that the enthalpy contains both sensible heat and heat of reaction effects. That is to say that Equation (9.2.10) can be written as ... 

The material and energy balance equations must be solved simultaneously. A convenient form for solution by numerical techniques is ... 

The solution of this differential equation is straightforward and is shown in Figure 9.4.2. For the nonisothermal case, the material and energy balances must be solved simultaneously by... 

Solution of Equation (10.2.1) provides the pressure, temperature, and concentration profiles along the axial dimension of the reactor. The solution of Equation (10.2.1) requires the use of numerical techniques. If the linear velocity is not a function of z [as illustrated in Equation (10.2.1)], then the momentum balance can be solved independently of the mass and energy balances. If such is not the case (e.g., large mole change with reaction), then all three balances must be solved simultaneously. 

Example 10.2.1 illustrates the simultaneous solution of the mass and energy balances for an adiabatic, fixed-bed reactor with no fluid density changes and no transport limitations of the rate, that is, rj = 1. Next, situations where these simplifications do not arise are described. 

Another class of problems involves processes for which the heat input and outlet temperature are specified but the extent of reaction and product composition are not. Solving such problems requires the simultaneous solution of material and energy balance equations, as the next example illustrates. 

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. 

One feasible network would correspond to the cold streams Cl, C8, and C9 diverted to suitable jacketed reactor compartments, as the simple network in Fig. 14 shows. The hot streams not shown in this network are matched directly with cooling water (CW), and the amount of steam used here is very small. Note that this network would require the same minimum utility consumption predicted by the solution of (PIO). It can be inferred that the network in Fig. 14 is equally suitable for both the simultaneous and sequential solutions. In fact, Balakrishna and Biegler (1993) showed that, for exothermic systems in which the reactor temperature is the highest process temperature, the pinch point is known a priori as the highest reactor temperature (in this case, the feed temperature) and the inequality constraints in (PIO), Qh 2h () ). F G P. can be replaced by a simple energy balance constraint. This greatly reduces the computational effort to solve (PIO). 

Chapter 5 on the simultaneous solution of material and energy balance problems has been completely revised. It now focuses on the use of flowsheeting codes to solve problems. The problem set for Chapter 5 emphasizes the use of FLOWTRAN and PROCESS for complex problems, codes that have clear manuals and are available at most departments of chemical engineering. 

The behavior of a battery is normally hard to predict due to the complex chemical and physical processes inside the battery. A very detailed model implies the solution of the material and energy balances in the battery, and therefore the simultaneous solutions of partial... 

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