Energy Balance Procedures

Now consider the properties at 10 bar and 20 bar, both at a temperature of 400°C. Even though the pressure has doubled, the values of U and U change by much less than 1%. Similar results would be obtained for liquid water. The conclusion is that when you need a value of V or H for water (or for any other species) at a given T and P. you must look it up at the correct temperature—interpolating if necessary—but you don t have to find it at the exact pressure. 

Tile next example illustrates the use of the steam tables to solve energy balance problems 

Steam at 10 bar absolute with 190 C of superheat is fed to a turbine at a rate m = 2000 ke/h. The turbine operation is adiabatic, and the effluent is saturated steam at 1 bar. Calculate the work output of the turbine in kilowatts, neglecting kinetic and potential energy changes. 

SOLUTION The energy balance for this steady-state open system is 

From either Table B.6 or B.7, you can find that the enthalpy of saturated steam at I bar is W om(l bar, saturated) = 2675kJ/kg 

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At this point, you can perform energy balance calculations only for systems in which A / (closed system) or AH (open system) can be neglected and for nonreactive systems involving species for which tables of f/ or H are available. Energy balance procedures for other types of systems are presented in Chapters 8 and 9. 

The psychrometric chart for most gas-liquid systems would show a family of adiabatic saturation curves in addition to the families of curves shown on Figures 8.4-1 and 8.4-2. However, for the air-tvater system at 1 atm, the adiabatic saturation curve through a given state coincides with the constant wet-bulb temperature line through that state, so that Tas = Twb- The simple material and energy balance procedure for adiabatic cooling outlined in this section is possible only because of this coincidence. 

The essential differences between sequential-modular and equation-oriented simulators are ia the stmcture of the computer programs (5) and ia the computer time that is required ia getting the solution to a problem. In sequential-modular simulators, at the top level, the executive program accepts iaput data, determines the dow-sheet topology, and derives and controls the calculation sequence for the unit operations ia the dow sheet. The executive then passes control to the unit operations level for the execution of each module. Here, specialized procedures for the unit operations Hbrary calculate mass and energy balances for a particular unit. FiaaHy, the executive and the unit operations level make frequent calls to the physical properties Hbrary level for the routine tasks, enthalpy calculations, and calculations of phase equiHbria and other stream properties. The bottom layer is usually transparent to the user, although it may take 60 to 80% of the calculation efforts. 

Mass and Energy Balances. The formulation of mass and energy balances follows procedures outlined ia many basic texts (2). The use of solubihties to calculate crystal production rates from a cooling crystallizer was demonstrated by the discussion of equations 1 and 2. Subsequent to determining the yield, the rate at which heat must be removed from such a crystallizer can be calculated from an energy balance ... 

Another procedure, which is more accurate for the external-heat-exchanger cases, is to nse an equivalent value for MC (for a vessel being heated) derived from the following energy balance ... 

Compute a new set of values of the T) tear variables by solving simultaneously the set of N energy-balance equations (13-72), which are nonlinear in the temperatures that determine the enthalpy values. When linearized by a Newton iterative procedure, a tridiagonal-matrix equation that is solved by the Thomas gorithm is obtained. If we set gj equal to Eq. (13-72), i.e., its residual, the hnearized equations to be solved simultaneously are... 

Much of the basic theory of reaction kinetics presented in Sec. 7 of this Handbook deals with homogeneous reaclions in batch and continuous equipment, and that material will not be repeated here. Material and energy balances and sizing procedures are developed for batch operations in ideal stirred tanks—during startup, continuation, and shutdown—and for continuous operation in ideal stirred tank batteries and plug flow tubulars and towers. 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

Often in plant operations condensate at high pressures are let down to lower pressures. In such situations some low-pressure flash steam is produced, and the low-pressure condensate is either sent to a power plant or is cascaded to a lower pressure level. The following analysis solves the mass and heat balances that describe such a system, and can be used as an approximate calculation procedure. Refer to Figure 2 for a simplified view of the system and the basis for developing the mass and energy balances. We consider the condensate to be at pressure Pj and temperature tj, from whence it is let down to pressure 2. The saturation temperature at pressure Pj is tj. The vapor flow is defined as V Ibs/hr, and the condensate quality is defined as L Ibs/hr. The mass balance derived from Figure 2 is ... 

This section discusses the principal causes of overpressure in refinery equipment and describes design procedures for minimizing the effects of these causes. Overpressure is the result of an unbalance or disruption of the normal flows of material and energy that cause material or energy, or both, to build up in some part of the system. Analysis of the causes and magnitudes of overpressure is, therefore a special and complex study of material and energy balances in a process system. 

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. 

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. 

If the same procedure is applied to a tube only containing distilled water, the energy balance is shown in the next two equations ... 

A new steam balance can now be calculated, the energy balance around the deaerator revised and the procedure repeated until convergence is achieved. A converged steam balance is shown in Figure 23.24. 

By far, the most suitable method to quantify individual ruminant animal CH4 measurement is by using respiration chamber, or calorimetry. The respiration chamber models include whole animal chambers, head boxes, or ventilated hoods and face masks. These methods have been effectively used to collect information pertaining to CH4 emissions in livestock. The predominant use of calorimeters has been in energy balance experiments where CH4 has been estimated as a part of the procedures followed. Although there are various designs available, open-circuit calorimeter has been the one widely used. There are various designs of calorimeters, but the most common one is the open-circuit calorimeter, in which outside air is circulated around the animal s head, mouth, and nose and expired air is collected for further analysis. 

If the batch reactor operation is both nonadiabatic and nonisothermal, the complete energy balance of equation 12.3-16 must be used together with the iiaterial balance of equation 2.2-4. These constitute a set of two simultaneous, nonlincmr, first-flijer ordinary differential equations with T and fA as dependent variables and I as Iidependent variable. The two boundary conditions are T = T0 and fA = fAo (usually 0) at I = 0. These two equations usually must be solved by a numerical procedure. (See problem 12-9, which may be solved using the E-Z Solve software.)... 

The material and energy balances of a tubular vessel are based on the conservation law, Eq 2.42, applied to a differential ring between r and r+dr and z and z+dz. A material balance is derived, for example, in problem P5.08.01, and is quoted in Table 2.6 along with the heat balance. The result is a pair of second order partial differential equations, usually nonlinear, that must be solved numerically. Table 2.6 indicates one possible procedure, but computer software is plentiful. 

A similar approach was undertaken by Mah et al. (1976) in their attempt to organize the analysis of process data and to systematize the estimation and measurement correction problem. In this work, a simple graph-theoretic procedure for single component flow networks was developed. They then extended their treatment to multicomponent flow networks (Kretsovalis and Mah, 1987), and to generalized process networks, including bilinear energy balances and chemical reactions (Kretsovalis and Mah, 1988a,b). 

The procedure developed by Joris and Kalitventzeff (1987) aims to classify the variables and measurements involved in any type of plant model. The system of equations that represents plant operation involves state variables (temperature, pressure, partial molar flowrates of components, extents of reactions), measurements, and link variables (those that relate certain measurements to state variables). This system is made up of material and energy balances, liquid-vapor equilibrium relationships, pressure equality equations, link equations, etc. 

Q-R factorization is successful in decomposing linear systems of equations. It is also satisfactory when bilinear systems contain component balances and normalization equations. If energy balances are included in the set of process constraints, the procedure has the drawback that only simple thermodynamic relations for the specific enthalpy of the stream can be considered. 

In these cases it gives a suspicious subset of gross errors that includes the simulated ones but is larger than the real set. The user is advised that a unique solution is not possible. To sort out these difficulties, additional information on the process, for example, component or energy balances using fixed values of composition or temperature, may be included in Stage 2 of the procedure. 

Another potential solution technique appropriate for the packed bed reactor model is the method of characteristics. This procedure is suitable for hyperbolic partial differential equations of the form obtained from the energy balance for the gas and catalyst and the mass balances if axial dispersion is neglected and if the radial dimension is first discretized by a technique such as orthogonal collocation. The thermal well energy balance would still require a numerical technique that is not limited to hyperbolic systems since axial conduction in the well is expected to be significant. 

Another procedure often used with less stringent assumptions than the equality of the gas and solid temperatures is the pseudohomogeneous analysis proposed by Vortmeyer and Schaefer (1974). This procedure has proved to be quite effective for simple adiabatic packed bed analyses and involves reducing the energy balances for the gas and catalyst to a single equation using the... 

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