Energy balance equation, steady-state

As for the mass and energy balance equations, steady-state conditions are obtained when the rate of change of momentum in the system is zero and... 

Energy balance. For steady-state operation and no shaft work. Equation (S-56) can be rewritten in terms of catalyst weight as the spatial variable, that is,... 

From Material and Energy Balances to Steady-State Design Equations (Steady-State Mathematical Models)... 

In its simplest form, a feedforward controller merely proportions the corrective action to the size of the disturbance. In other words, the control equation is merely a gain based on steady state, i.e. mass or energy balance at steady state. This does not take into account any of the process dynamics of the system. If there is a difference, or lag, in the speed of the process response to the control action when compared with that of the disturbance, then it may be necessary to introduce some dynamic compensation into the control equation. The dynamic compensation correctly times the control action and response, thus giving increased accuracy in the feedforward control. 

The coupling of the component and energy balance equations in the modelling of non-isothermal tubular reactors can often lead to numerical difficulties, especially in solutions of steady-state behaviour. In these cases, a dynamic digital simulation approach can often be advantageous as a method of determining the steady-state variations in concentration and temperature, with respect to reactor length. The full form of the dynamic model equations are used in this approach, and these are solved up to the final steady-state condition, at which condition... 

Using the digital simulation approach to steady-state design, the design calculation is shown to proceed naturally from the defining component balance and energy balance equations, giving a considerable simplification to conventional text book approaches. 

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. 

This section treats the material and energy balance equations for a plug flow reactor. For steady-state operation the energy balance analysis leading to equation 10.1.4 is appropriate. 

At steady state the rate of transformation of energy by reaction must be equal to the rate of thermal energy loss. This implies that the intersection ) of the curves given by equations 10.6.6 and 10.6.8 will represent the solution(s) of the combined material and energy balance equations. The positions at which the intersections occur depend on the variables appearing on the right side of equations 10.6.6 and 10.6.8. Figure 10.3 depicts some of the situations that may be encountered. 

For isothermal systems this equation, together with an appropriate expression for rv, is sufficient to predict the concentration profiles through the reactor. For nonisothermal systems, this equation is coupled to an energy balance equation (e.g., the steady-state form of equation 12.7.16) by the dependence of the reaction rate on temperature. 

The heat load, Q> is obtained from the energy-balance equation (14.3-10) for steady-state operation with constant values for the various parameters ... 

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... 

We can also obtain these expressions from the energy-balance equation for the steady-state PFTR by simply transforming dzju dt with A,/ V replacing Pw/At. The solutions of these equations for the batch reactor are mathematically identical to those in the PFTR, although the physical interpretations are quite different. 

Recall from Chapter 6 that the steady-state energy-balance equations in a CSTR can be reduced to a single equation, which we wrote by considering the rates of heat generation and removal. We wrote these as... 

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. 

Determining the conversion of monomer can only be as accurate as the method of quantifying the heat liberated from the reaction. The usual method is to take the difference between the inlet and outlet jacket water temperatures multiplied by the specific heat and flow rate of the water. This steady-state energy balance equation is ... 

Analysis of the effect of permeation, temperature profile and sweep gas will be proposed hereafter considering a steady-state MR modeled by a 1D, first-order model. The model can be extracted from the mass and energy balance, Equations 13.14... 

Integration of Eq. 2.9-11 leads to the macroscopic mechanical energy balance equation, the steady-state version of which is the famous Bernoulli equation. Next we subtract Eq. 2.9-11 from Eq. 2.9-10 to obtain the differential thermal energy-balance... 

Consider an exothermic irreversible reaction with first order kinetics in an adiabatic continuous flow stirred tank reactor. It is possible to determine the stable operating temperatures and conversions by combining both the mass and energy balance equations. For the mass balance equation at constant density and steady state condition,... 

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. 

Starting with the open system balance equation, derive the steady-state mechanical energy balance equation (Equation 7.7-2) for an incompressible fluid and simplify the equation further to derive the Bernoulli equation. List all the assumptions made in the derivation of the latter equation. 

The steady-state energy balance equation may be obtained by setting dT/dt equal to zero in Equation 11.3-12. 

In this section we apply die general energy balance [Equation (8-22)] to the CSTR and to the tubular reactor operated at steady state. We then present example problems showing how the mole and energy balances are combined to size reactors operating adiabadcally. 

In an ideal continuous stirred tank reactor, composition and temperature are uniform throughout just as in the ideal batch reactor. But this reactor also has a continuous feed of reactants and a continuous withdrawal of products and unconverted reactants, and the effluent composition and temperature are the same as those in the tank (Fig. 7-fb). A CSTR can be operated under transient conditions (due to variation in feed composition, temperature, cooling rate, etc., with time), or it can be operated under steady-state conditions. In this section we limit the discussion to isothermal conditions. This eliminates the need to consider energy balance equations, and due to the uniform composition the component material balances are simple ordinary differential equations with time as the independent variable ... 

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