Steady energy balance

Stea.dy-Sta.teFeedforwa.rd, The simplest form of feedforward (FF) control utilizes a steady-state energy or mass balance to determine the appropriate manipulated variable adjustment. This form of feedforward control does not account for the process dynamics of the disturbance or manipulated variables on the controlled variable. Consider the steam heater shown ia Figure 15. If a steady-state feedforward control is designed to compensate for feed rate disturbances, then a steady-state energy balance around the heater yields ... 

When the basic physical laws are expressed in this form, the formulation is greatly facilitated. These expressions are quite often given the names, material balance, energy balance, and so forth. To be a little more specific, one could write the law of conservation of energy in the steady state as... 

The energy balance for a steady-state steady-flow process resulting from the first law of thermodynamics is... 

Material and energy balances are based on the conservation law, Eq. (7-69). In the operation of liquid phase reactions at steady state, the input and output flow rates are constant so the holdup is fixed. The usual control of the discharge is on the liquid level in the tank. When the mixing is adequate, concentration and temperature are uniform, and the effluent has these same properties. The steady state material balance on a reacdant A is... 

Material and energy balances of the steady-state model. 

The steady state energy balance for adiabatic operation is determined as follows ... 

At a location MRT and T are often measured with a sphere or ellipsoid representing the person, as shown in Fig. 5.10. In the diagram the energy balance on the globe at steady state is q q, or... 

Quality-assurance procedures have to be established for the checking of both input and results checks of energy balances, plausibility tests, and comparison with steady-state calculations and with results from similar cases. These checks are demanding and time consuming and thus prone to be omitted but are mandatory for reliable simulations. 

Woou.ATT, E. Trans. Inst. Chem. Eng. 24 (1946) 17. Some aspects of chemical engineering thermodynamics with particular reference to the development and use of the steady flow energy balance equations. 

Steady-state temperatures along the length of a piston flow reactor are governed by an ordinary differential equation. Consider the differential reactor element shown in Figure 5.3. The energy balance is the same as Equation (5.14) except... 

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. 

A dynamic model should be consistent with the steady-state model. Thus, Eqs (1) and (4) should be extended to dynamic form. For the better convergence and computational efficiency, some assumption can be introduced the total amounts of mass and enthalpy at each plate are maintained constant. Then, the internal flow can be determined by total mass balance and total energy balance and the number of differential equations is reduced. Therefore, the dynamic model can be established by replacing component material balance in Eq. (1) with the following equation. 

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... 

Alternatively neglecting the jacket dynamics and assuming that the coolant in the jacket is at some mean temperature, Tjavg as shown in Fig. 3.4, a steady-state energy balance can be formulated as... 

Substituting for Tjout into the steady-state jacket energy balance, solving for Tjavg and substituting Tjavg into the steady-state balance, gives the result that... 

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. 

The model is based on the steady-state energy balance combined with Fourier s law which gives... 

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. 

The situation is illustrated in Figure 4. Assuming steady state, the energy balance over the insulating material is given by [using Eq. (13)]... 

An energy balance will be maintained over the sphere, and it will be assumed that there is no angular dependence on heat transport. The energy balance will be executed over a thin (Ar thickness) spherical shell and solved in essentially the same way as in Section III.A, except that we will work in spherical coordinates. The steady-state energy balance is given by... 

As in Section II,A, a set of steady-state mass and energy balances are formulated so that the parameters that must be evaluated can be identified. The annular flow patterns are included in Regime II, and the general equations formulated in Section II,A,2,a, require a detailed knowledge of the hydrodynamics of both continuous phases and droplet interactions. Three simplified cases were formulated, and the discussion in this section is based on Case I. The steady-state mass balances are... 

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. 

The steady-state form of the energy balance for a continuous stirred tank reactor is given by equation 10.1.4. 

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. 

An energy balance on the PFR operating at steady state is given by equation 10.4.6. For adiabatic operation this equation becomes... 

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. 

At steady state, an energy balance on this same volume element gives... 

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. 

An energy balance over the differential length of reactor for steady-state operating conditions... 

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