Stream variables

Many process simulators come with optimizers that vary any arbitrary set of stream variables and operating conditions and optimize an objective function. Such optimizers start with an initial set of values of those variables, carry out the simulation for the entire flow sheet, determine the steady-state values of all the other variables, compute the value of the objective function, and develop a new guess for the variables for the optimization so as to produce an improvement in the objective function. 

Validate routine methods, i.e., define the conditions under which the assay results are meaningful.115 To do that, one must select samples that are truly representative of the product stream. This may be a difficult task when the process is still under development and the product stream variable. The linearity of detector response should be defined over a range much broader than that expected to be encountered. Interference from the sample matrix and bias from analyte loss in preparation or separation often can be inferred from studies of linearity. Explicit detection or quantitation limits should be established. The precision (run-to-run repeatability) and accuracy (comparison with known standards) can be estimated with standards. Sample stability should be explored and storage conditions defined. 

Stream Variable Measurement Stream Variable Measurement... 

Variability may be defined as reflecting fluctuations in the atmosphere, of natural origin, with both temporal and spatial scales examples are diurnal, seasonal, solar activity-related variations impulsive events such as volcano eruptions and solar proton events fluctuations linked to some peculiar meteorological conditions, for example, intense cyclonic activities and jet streams. Variability by itself is a whole program to be conducted ideally on a four-dimensional basis (latitude, longitude, altitude, and time) by space vehicles, for example, satellites or from the space shuttle. This area of research is certainly the most urgent one to be de-... 

The composition of poplar wood was usedasamodel for the feedstock composition however, as used in this simulation, the poplar is modeled as consisting of only cellulose, xylan, and lignin, with compositions of 49.47, 27.26, and 23.27%, respectively. Laboratory results for carbonic acid pretreatment are relatively scarce, so for the purpose of this comparative study, stoichiometry of pretreatment reactions was assumed to be equal to those used in the comparison model (3) cellulose conversion to glucose 6.5% xylan conversion to xylose 75 and lignins solubilized 5%. Thus, economic comparisons made with this model assess different equipment and operating costs but not product yields. For the successful convergence of the carbonic acid model, the simulation required initial specification of several variables. These variables included initial estimates for stream variables and inputs for the unit operation blocks. 

Sequential Modular. By far the most experience with flowsheeting systems has been with the sequential modular architecture (59- 3). It is this architecture that is most easily understood by the process engineer. Each module calculates all output streams from input streams subject to module parameters. Generally, the stream variables consist of component flows, temperature (or enthalpy) and pressure as the independent variables. Other dependent variables such as total flow, fraction vapor and total enthalpy (or temperature) are often carried in the stream. 

Minimum number of stream variables (iteration stream parameters). 

Direct prices do not take into account the effect a decision in one part of a plant may have on the irreversibilities in another. Marginal and shadow prices do this but are more complicated to compute. They depend upon the system of equations (and their first derivatives with respect to the variables of interest) rather than upon only the states of various zones. The mathematical description of a thermodynamic process requires the specification of a set of "equations of constraint", represented here by the set, [4>.=0]. The thermodynamic performance and stream variables are divided into two sets, state and decision variables, represented by [x.] and [y ], and each of the defining functions, [4.], is expressed in terms of these variables. If the objective function, 4, (whether it is an energy objective or a cost objective) is similarly expressed, a Lagrangian may be defined according to ... 

Some constraints, such as < >2 use stream variables to describe the thermodynamic state of the working fluid. Numerical values for steam properties are generated using a computer subroutine called STEAM (14). Thus not all constraint equations are in analytical form. 

Implementation of dynamic simulators has led to interesting research issues. For example, many have been implemented in a sequential modular format. To carry out the integration correctly from the point of view of correctly assessing integration errors, each unit model can receive as input a current estimate for the state variables (variables x), the unit input stream variables, and any independent input variables specified versus time... 

The executive system can then solve the models in an appropriate sequence to converge the stream variables involved in a process recycle. Once these are converged, the executive can then use the RHSs to integrate simultaneously the differential equations for all the units in this recycle loop. 

The total number of stream variables, the number of equations, and the number of equipment parameters can be summed, and the total degrees of freedom (unknowns minus equations) then determined. A unique solution to a problem exists only when the numbers of unknowns and equations are equal. Therefore, a number of variables equal to the number of degrees of freedom must be given values so that there will be a unique solution. 

Twenty stream variables need to be specified in order for a unique solution to exist. In principle, any 20 stream variables could be supplied however, the usual solution strategy requires that the process feed streams be specified. Specifying the flow rate of each of the eight components, plus the temperature and pressure of the two feed streams (ethylbenzene and steam) reduces the number of variables to 150. Hence, a unique solution is available. 

For the methanol synthesis process illustrated in Fig. 4-1, Example 1, assume that there are algorithms for calculating the outputs of each process unit from the inputs. Determine how many stream variables must be specified and decide what these should be so that a unique solution exists for the mass and energy balances. Identity all recycle loops, tear streams for these loops, and a calculation sequence. 

For a process flow sheet obtained in Problem 3, assume that algorithms are available to calculate the outputs from each process unit from known inputs. Determine the number of stream variables that must be specified, decide what they should be, identify all recycle loops, select tear streams for these loops, and establish a calculation sequence. 

All material balance problems are variations on a single theme given values of some input and output stream variables, derive and solve equations for others. Solving the equations is usually a matter of simple algebra, but deriving them from a description of a process and a collection of process data may present considerable difficulties. It may not be obvious from the problem statement just what is known and what is required, for example, and it is not uncommon to... 

The stream variables of primary interest in material balance problems are those that indicate how much of each component is present in the stream (for a batch process) or the flow rate of each component (for a continuous process). This information can be given in two ways as the total amount or flow rate of the stream and the fractions of each component, or directly as the amount or flow rate of each component. 

Assign algebraic symbols to unknown stream variables [such as m (kg solution/min), x (Ibm N2/lbm), and n (kmol C3H8)] and write these variable names and their associated units on the chart. For example. If you did not know the flow rate of the stream described in the first illustration of step 1, you might label the stream... 

The output gas is analyzed and is found to contain 1.5 mole% water. Draw and label a flowchart of the process, and calculate all unknown stream variables. 

YOURSELF of the labeled stream variables. The solution of the first problem is given as an illustration. 

Observe now that the masses (but not the mass fractions) of all streams could be multiplied by a common factor and the process would remain balanced moreover, the stream masses could be changed to mass flow rates, and the mass units of all stream variables (including the mass fractions) could be changed from kg to g or Ibn, or any other mass unit, and the process would still be balanced. 

Indicate the balances you would write and the order in which you would write them to solve for the unknown stream variables in the following process ... 

Sources of equations relating unknown process stream variables include the following ... 

Draw a flowchart and fill In all known variable values, including the basis of caicuia-tion. Then label unknown stream variables on the chart. 

The 8%-92% benzene split between the product streams is not a stream flow rate or composition variable nevertheless, v/e wriic it on the chart to remind ourselves that it is an additional relation among the stream variables and so should be included in the degree-of-freedom analysis. 

The procedure for material balance calculations on multiple-unit processes is basically the same as that outlined in Section 4.3. The difference is that with multiple-unit processes you may have to isolate and write balances on several subsystems of the process to obtain enough equations to determine all unknown stream variables. When analyzing multiple-unit processes, carry out degree-of-freedom analyses on the overall process and on each subsystem, taking into account only the streams that intersect the boundary of the system under consideration. Do not begin to write and solve equations for a subsystem until you have verified that it has zero degrees of freedom. 

A labeled flowchart of a chemical process involving reaction, product separation, and recycle is shown in Figure 4.5-1. Note the distinction between the fresh feed to the process and the feed to the reactor, which is the sum of the fresh feed and recycle stream. If some of the stream variables shown in Figure 4.5-1 were unknown, you could determine them by writing balances on the overall process and about the reactor, separator, and mixing point. 

If molecular species balances are used to determine unknown stream variables for a reactive process, the balances on reactive species must contain generation and/or consumption terms. The degree-of-freedom analysis is as follows ... 

List possible reasons for the differences between the design predictions and the experimental values of the output stream variables and for the failure of the experimental system balance to close. 

A process stream on a flowchart is completely labeled if values or variable names are assigned to one of the following sets of stream variables (a) total mass flow rate or total mass and component mass fractions (b) mass flow rates or masses of each stream component (c) total molar flow rate or total moles and component mole fractions and (d) molar flow rates or moles of each stream component. If a total amount or flow rate or one or more component fractions are known for a stream, use (a) or(c) to incorporate the known values into the labeling. If neither the total nor any fractions are known, using (b) or (d) (component amounts or flow rates) often leads to easier algebra. Volumetric quantities should be labeled only if they are either given or requested in the problem statement. A flowchart is completely labeled if every stream is completely labeled. 

In the remainder of the problem, you will be given values of measured feed gas stream variables / i(psig), i(mm), and / i], the maximum allowed SO2 mole fraction in the... 

Given a description of a multiple-unit process, determine the number of degrees of freedom, identify a set of feasible design variables, and if there are cycles in the flowchart, identify reasonable tear stream variables and outline the solution procedure. Draw a sequential modular block diagram for the process, inserting necessary convergence blocks. 

Thus, nine process stream variable values must be specified for the given system, at which point balances can be used to determine the remaining six variables. 

As we noted at the beginning of this chapter, there are two broad approaches to the automated solution of the balance equations for a process system the sequential modular approach and the equation-based approach. This section outlines the first of these methods. The balance equations (and any other equations that may arise from physical considerations or process specifications) for each unit are written and solved. If there are no recycle streams, the calculation moves from one unit to another, until all units have been covered. If there is a cycle (the conventional term for a recycle loop in a process flowchart), a trial-and-error procedure is required values of one or more stream variables in the cycle are assumed the balance equations for units in the cycle are solved, one unit at a time, until the values of the assumed variables are recalculated new variable values are assumed and the procedure is repeated until the assumed and calculated values agree. 

The simulation program contains a built-in subprogram corresponding to each block type. To simulate a process, you would use the simulation program to build a flowchart and then enter known block and stream variable values in forms provided by the program. When you subsequently run the simulation, a series of calls to the block subroutines would lead to the solution of the process material and energy balance equations. 

Create a spreadsheet that would determine the product stream variables from given values of the feed stream variables. 

Equations 1 to 6 may be solved sequentially for the product stream variables. 

A spreadsheet program that solves the six system equations is easy to construct. With a little more effort, a flowchart can be imbedded into the spreadsheet in such a way that the product stream variables are automatically updated if any of the input stream variables are changed. In the example that follows, the following heat capacities are used ... 

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