Problem 8. Check to make sure the above solution is a minimum. The gradients of the constraints are linearly independent, so that the Lagrange multipliers exist. The constraints are satisfied the second constraint is 3. The Lagrange multipliers for any inequality constraints are not involved in the problem. They exist for the equality constraints. The steps for part b are the same as Part a, except that the Hessian matrix of L must be positive semi-definite.
You can substitute for V into the objective function, and get P in terms of xi. Then the necessary conditions for an unconstrained function can be tested.
For convexity, you need the further specification that Problem 8. However, this is a problem with three variables and four active constraints. Hence, their intersection, if unique, is the only feasible point, and it is the minimum.
Thus L is not at a stationary point. Because of iv and vi , the sufficient conditions are not met. Thus, the sufficient conditions are not met.
Advantages: i It will remain within the feasible region. Disadvantages: i An initial feasible point must be located. We can examine the difference in terms of the value of r needed. Because of complementary slackness, as term to satisfy g to within a given termination criterion.
The solution strategy is as follows: 1 Assume a feasible x 0ij set. If not, go to step 5. T Problem 8. Thus, it is a feasible point. Now h1 and h2 are not satisfied. The solution is a feasible point, and is used to start the next iteration.
The variables are A, L, and N. Although N is an integer we will assume it to be a continuous variable. Now solve the LP again using these x0ij ' s. The branch and bound analysis can be carried out with Excel. Note that the first noninteger solution achieves a larger value of f than the integer solution, as is expected.
Problem 9. The optimal solution turns generators 1 and 2 on in period 1, and does not use generator 3. This is because generators 1 and 2 have the lowest operating costs, while 3 is much higher. Further, generator 1 is used to capacity, because it has by far the lowest operating cost, and 2 is used to satisfy the remaining demand. Even though generator 3 has the lowest startup cost, its higher operating cost excludes it.
The first year two MW generators are bought, because first year cost is smaller than second year. The second year nothing is bought. In years 3 to 5 only one MW generator is installed. Purchases in years 3 to 5 are deferred as long as possible because costs are declining. A GAMS model for this problem and its solution follows. They are also used to incorporate setup times into the maxtime constraints, and to include setup costs into the objective.
The optimal solution produces product 1 in weeks 1 and 3, and product 2 in weeks 2 and 4. There is unsatisfied demand for product 1 in week 2 and product 2 in week 1, because only one product can be produced in any week. This causes the penalty cost to be incurred, but the backlogged demand is satisfied in the next week The constraint that inventory is zero at the end of week 4 insures that there is no backlogged demand after the fourth week.
All of the 90 available hours are used in week one, but fewer are needed in subsequent weeks. If there are only 80 hours available per week, the problem has no feasible solution. Node 2 gives the MIP optimum. Explicit enumeration is easy in this situation. To each extractor, assign the stream with the least cost for that extractor. Node 5 gives the IP optimum. SteadyState and Dynamic Design, a two-column extractive process is modeled from conceptual design to dynamic simulation.
In two distillation columns, the equimolar feed of Toluene and Heptane is separated using Phenol as a solvent. HYSYS - Conceptual Design is used to calculate the interaction parameters and carry out the preliminary design and optimization of the process. SteadyState, the column is set up and optimized, using the Spreadsheet to model economic factors. Finally, controls are added and various disturbances are introduced to test the effectiveness of the design.
The objective is to maximize the purity of the Heptane and Toluene streams coming off the top of the first and second column, respectively. Using Hyprotech's process simulation software, we can develop a conceptual design, optimize the steady-state process, and develop and test a control scheme.
These are the steps: 1. SteadyState, set up the column configurations in a single flowsheet, using the specifications determined in the previous step.
Dynamics, set up a candidate control scheme and evaluate dynamic operability. Column SubFlowsheet C Extractive distillation is used in the petroleum industry for the separation of aromatics from non-aromatic hydrocarbons.
In general, the presence of the solvent raises the vapour pressures of the key components to different degrees, so that the relative volatility between these key components is increased. The more volatile component is removed in the distillate, and the bottoms mixture solvent and less volatile component is separated in a second distillation column. Toluene-"non-toluene" separation is well-documented. The non-toluene fraction is often a narrow mixture of saturated hydrocarbons, and for the purpose of this study will be represented by n-Heptane.
The objective of this process, therefore, is to maximize the separation of n-Heptane and Toluene. The distillate and bottoms molar purities are 0. Phenol is commonly used as the solvent, due to its effect in significantly increasing the volatility ratio of n-Heptane and Toluene.
Unlike other potential solvents which can also increase the volatility ratio, phenol does not form azeotropes, and is currently inexpensive. It is not particularly dangerous, although there is some concern as to its environmental impact. Since n-Heptane and Toluene do not form an azeotrope, the separation can theoretically be performed without the use of a solvent.
However, the number of stages and reflux ratio is excessive, as shown in the side table. This is due to the fact that these components have similar volatilities. This example is set up in five parts as outlined below.
Some sections can be completed independently, without referring to previous steps. For example, if you wish to do only the Steady-State design, you need only complete steps 3 and 4, using the interaction parameters and column design as predicted in steps 1 and 2. SteadyState - page 26 4. SteadyState - page 33 5. Dynamics - page 55 C Interaction parameters for the three binary pairs are obtained separately and combined in the binary matrix.
Now we will look at the interaction parameters for the three component pairs and if necessary, regress new parameters from experimental data.
New interaction parameters can be regressed from experimental data specifically chosen for the system conditions. The TRC database contains data for over fitted binaries. These sets will be imported into the current Fluid Phase Experiment. Next, check the Herrington Thermodynamic Consistency for each set by selecting the Consistency page tab, and pressing the Calculate Consistency button.
This set has 13 points which is sufficient for our investigation. If we were going to regress the interaction parameters to the experimental data, we would run the Optimizer.
However, we will instead compare the experimental data to the calculated data based on the default interaction parameters. Select the Calculate button — the XY and TXY curves will be constructed based on the default interaction parameters, and the errors will be calculated. The calculated data in this case is the TXY or XY data calculated using the Property Package and current interaction parameters , which is displayed graphically on the Plots page of the Data Set view.
The TXY plot appears as follows: The experimental and calculated points match remarkably well, and thus it is not necessary to regress the interaction parameters for the C7-Toluene pair. We will, however, regress interaction parameters from the available TRC data set Set Drickamer, H. There is only one data set available for the Toluene-Phenol pair. Note that this data, obtained at kPa, has 23 points.
This data set is consistent according to the Herrington test: Now we will calculate new interaction parameters based on this experimental data. Before running the Optimizer, compare the experimental data to the predictions made using the default interaction parameters. By looking at the Plots page, it appears that there is reasonably good agreement between the experimental data and calculated curves.
On the Errors page of the Fluid Phase Experiment view, note that the average and maximum temperature errors are 0. This allows the bij parameters to vary during the optimization process. Before running the optimizer, set up the view so that you can observe the solution progress.
This is best done from the Optimizer page, although you may prefer to remain on the Variables page and watch the progress of the interaction parameters ensure that the Parameters radio button is selected; as well, it is probably more useful to observe the aij parameters. Once you start the optimizer you cannot change pages until the calculations are complete. For this example, we will observe the solution progress from the Optimizer page.
Choose the Optimizer tab, then select the Run Optimizer button. We may be able to get better results using a different Objective Function. The Maximum Likelihood function is the most rigorous from a statistical point of view, but also is the most computer intensive. The convergence time increases when we use this function, but the improved results may be worth it. We obtain the following interaction parameters: The bij parameters are 0.
The temperature errors are now 0. Note, however, that while the toluene composition errors decreased, the phenol composition errors increased.
Nevertheless, we will use these interaction parameters for the Phenol-Toluene pair. The following data taken from Chang, Y. With only the first data set active, optimize using the Activity Coefficients Objective Function. With only the first data set active, optimize using the Maximum Likelihood Objective Function.
With only the second data set active, optimize using the Activity Coefficients Objective Function. With only the second data set active, optimize using the Maximum Likelihood Objective Function. With both data sets active, optimize using the Objective Function which results in the smallest error. Scheme 5 uses the Maximum Likelihood Objective Function. The following table outlines the results of this analysis. In all cases, using the Maximum Likelihood Objective function rather than the Activity Coefficients Objective function resulted in significantly smaller temperature errors, while in most cases the composition errors increased slightly.
In some instances, the average or maximum composition decreased when the Maximum Likelihood Objective function was used see Ave C7 and Max Phenol for Schemes 3 and 4. Therefore, we conclude that the Maximum Likelihood Objective function results in a better fit. For this component set, there is no liquid-liquid region.
A liquid-liquid region is not predicted with our interaction parameters. Although we can be reasonably confident of these results, it is wise to regard the following: 1. Consider defining a weight of zero for outliers data points which deviate significantly from the regressed curve. Check the prediction of liquid-liquid regions. The plots shown below are the TXY diagrams for Phenol-Heptane, comparing the experimental data to the points calculated from the Property Package.
The figure on the left plots the Kolyuchkina experimental data, while the figure on the right plots the Chang experimental data. This requires you to enter a temperature and a pressure. Here, you only enter a pressure. Select the Ternary plot radio button, transfer the three components to the Selected Components group, and enter a temperature and pressure.
Over a range of temperature and pressures, no liquid-liquid region is predicted. As we did in the previous section, we will determine the Interaction Parameters based on TRC and literature experimental data.
The procedure is essentially the same, and is concisely summarized below. The Activity Coefficient Objective Function should not generally be used for Equations of State as results tend to be mediocre for highly polar systems.
Note that you may have to decrease the tolerance or step size in order to obtain adequate convergence in this order of magnitude. The TXY plot using this interaction parameter is shown below, displaying a reasonably good fit. The interaction parameters predicted using the Chang data is very different from the Kolyuchkina data. The TXY plots using an interaction parameter of 0. These plots show that the dew point curve does not match the experimental data very well, and they also indicate a liquid-liquid region.
The VLLE plot at a pressure of 18 psia is shown below. We can avoid the prediction of a liquid-liquid region by setting the n-Heptane-Phenol interaction parameter to 0. However, the calculated curve still does not fit the experimental data very well, and we conclude that the Peng-Robinson Property Package is not acceptable for this example. Note that using the PRSV Property Package results in a better fit, although a two-liquid-phase region is incorrectly predicted under certain conditions.
For the ternary distillation experiment, the column can have two feeds, a sidestream, condenser, reboiler and decanter. The bottoms stream of the first column feeds the second column, and the bottoms stream of the second column is the upper feed to the first column. Open a Ternary Distillation Experiment, set a pressure of 18 psia the average of the top and bottom pressures in the column, 16 and 20 psia , and select the appropriate Fluid Package from the drop down list.
The program will then determine if there are any azeotropes or two-liquid regions: There are no azeotropes or liquid-liquid regions at this pressure, as predicted by the NRTL Property Package using our new interaction parameters. There is no decanter or sidestream. For the remaining streams, we will enter the specifications on the Spec Entry page. Before entering the specifications, set the Reflux Ratio to be 5. Later, we will do a sensitivity analysis in order to estimate an optimum Reflux Ratio.
We know that the upper feed is primarily phenol. As an initial estimate, we will use the following specifications: We have specified the C7 and Phenol mole fractions to be 1E and 0. Next, specify a Distillate C7 mole fraction of 0.
This restricts our range of choice for the remaining specifications. Select the Bottoms radio button. Set the Phenol fraction to be 0. This constrains the C7 mole fraction from 0 to 0. Specify the C7 fraction to be 0. The remaining mole fractions will be calculated based on the overall mole balance. At this point, all that is left is to specify a reflux ratio.
As an initial estimate, set the reflux ratio to be 5. Select the Calculate button. You will see the following message: The optimum value for Omega that which results in the lowest number of total stages is automatically calculated; if you simply press the Calculate button again, the number of stages will be determined using this optimum value.
Alternatively, you could set Omega to any value you wanted on the 2 Feed Omega page. We will always use the optimum value in this example. We could specify a lower heptane fraction in the bottoms — if we define it to be 0. As well, if we respecify the bottoms composition so that the phenol fraction is lower, we will require less stages in this column.
Because we want to take relatively pure toluene off the top of the second column and relatively pure phenol off the bottom, the heptane fraction in the bottoms coming off the first column must be small. Note that as we decrease the phenol composition, we must increase the C7 composition. Also, below a certain point phenol composition » 0. If we were to specify the heptane and phenol compositions to be 0. Note that most of the toluene and heptane in the bottoms stream will exit in the distillate stream of the second column.
The toluene composition would be 1 - 0. This is not adequate; therefore, the heptane composition must be even lower. Specify the heptane and phenol compositions to be 0.
The Heptane to Toluene ratio is now , which should allow the Toluene fraction off the top of the second tower to be about 0. Second Column As before, set the pressure to 18 psia, and select the appropriate Fluid Package. The Reflux Ratio for the second column will initially be set at 5. The Feed specifications, taken from the bottoms stream off the first column, are shown here: At this point, it may take some experimentation to see what stream specifications will result in a converged column.
As well, the phenol composition off the bottoms had to be adjusted to 0. At this point, we must return to the first column, using the new recycle stream specs. In other words, we must use the Bottoms specifications obtained here for the Top Feed of the first column. First Column: Second Pass The recycle stream flow upper feed is now If we keep the recycle compositions as they are, the minimum number of stages required to obtain a heptane composition of 0. Thus we will have to increase the phenol composition to compensate.
As well, there are inherent simplifications, such as the assumption of constant molal overflow. Using these results as a base case, the reflux ratio and product purities are now adjusted in order to determine an optimum configuration.
Two configurations will be proposed, one which has lower purities 0. Higher purities 0. As shown in the table below, increasing the reflux ratio above 5 gives no improvement in the number of stages required for the separation. Decreasing the reflux ratio below five causes the number of stages to increase. We therefore conclude that a reflux ratio of 5 is optimum for the first column. Although we would like a high purity, the number of stages increases substantially as we increase the Heptane Fraction above 0.
At a Heptane fraction of 0. We will go with a Heptane fraction of 0. Heptane Fraction 0. A reflux ratio of 5 is selected as the optimum. Decreasing the ratio to 4 is done at a cost of two extra stages, while increasing the ratio to 10 reduces the number of stages by one.
We will keep the toluene fraction of 0. Toluene Fraction 0. Any increase in the ratio does not decrease the number of stages significantly. Note that there are feed streams at stages 13 and As with the high purity case, we will start with a Reflux Ratio of 5 for both columns. Reflux Ratio Reboil Ratio Column 1 As before, we adjust the reflux ratio, and observe the effect on the number of stages.
When we increase the reflux ratio above 5, there is no improvement in the number of stages required for the separation. A reflux ratio of 4 is selected as the optimum. SteadyState, and obtain a steady-state solution for both column configurations. SteadyState, the c term is the alpha term. Change the Interaction Parameters to match the regressed parameters obtained in Part 1 or copy the.
SteadyState, the number of trays does not include the reboiler. Therefore the number of trays in each column are 24 and 9, not 25 and 10, as predicted in part 2. The Feed locations remain the same. Run the column. Once it solves, replace the Toluene Flow spec with the Heptane Frac spec. Re-run the column do not reset. Once it solves, replace the Toluene Recovery spec with the Toluene Frac spec.
Re-run the column. Whether a certain set of specifications will solve depends in part on the solution history, even if you have Reset the solution. Run but do not Reset the column after each change. We obtain the following Condenser and Reboiler duties: Column 1 Condenser — 1.
At this point, you may want to save the first configuration in a separate file. The pressures and temperature estimates will be defined as before: Pressures Condenser-2 — 16 psia Reboiler-2 — 20 psia Condenser-1 — 16 psia Reboiler-1 — 20 psia Temperature Estimates Temperature Estimate Condenser-1 — F Temperature Estimate Condenser-2 — F The types of specifications are the same as before; therefore it is not necessary to add new specs. Simply change the Heptane and Toluene Fracs to 0.
This will be confirmed in the next section. We obtain the following Condenser and Reboiler duties: Column 1 Condenser — 3. SteadyState to calculate the economics of the process. The methods used here to calculate capital costs, expenses and revenue are relatively simple, but are sufficient to provide a preliminary estimate. The benefit of these methods is that they are easy to implement, and as they are formula-based, can be used in the optimization calculations.
This section is divided into the following parts: Raw Data — Data which is used in the calculation of capital costs, expenses, revenue and net present worth. Capital Cost — Initial equipment and related costs associated with the construction of the process, incurred at time zero.
Annual Expenses — Expenses associated with the operation of the plant, incurred at the end of each year. Revenue — Income obtained from the sale of the process products, namely Toluene and Heptane; incurred at the end of each year. Nomenclature and Constants — A list of the nomenclature and constants used in the various expressions in this section. That is, the equipment column tray sections, reboilers and condensers are sized and priced; all additional costs, such as piping, construction and so on are calculated as a percentage of the equipment cost.
The sum of the costs of these six items is the Equipment cost based on prices. Direct and Indirect Costs The costs of each item below is estimated as the Equipment cost multiplied by the respective Factor for that item. Item Factor Reference Installation 0. Item Factor Reference Contracting 0. Annual Expenses The following table lists the expenses which are considered in the economic analysis of this plant. Calculation of Net Present Worth The following points outline the simplified calculation for net present worth: The total capital investment is the Fixed Capital Investment plus the Working Capital.
This expenditure is the total cash flow for year zero. It is assumed that the life of the process is five years. The revenue and expenses are applied at the end of each year, from years one to five. The Income after tax is the Annual Operating Income multiplied by one minus the tax rate. The Annual Cash Income is the Income after tax plus the Depreciation Expense, which was earlier discounted as an annual expense. It is assumed that there is no salvage value; the Annual Cash Income is exactly the same for years one to five.
Note that the Spreadsheet we are constructing contains some information which is not used in this example, but which may be of generic use. If you want to use this Spreadsheet as a template for other processes, it is a good idea to set it up as a template file, then insert it as subflowsheet in the appropriate case. Create a new template and enter Fluid Package data. Add a Spreadsheet and enter the following information: Note that the Spreadsheet we are constructing contains some information which is not used in this example, but which may be of generic use.
Save the template. Retrieve the process case, and add a subflowsheet, using the previously created file as a template. Import data links into Spreadsheet. As with creating a Case, it is necessary to define a Fluid Package. Enter the Main Environment. Adding the Spreadsheet Simulation Data Column A lists the headings, while column B will contain the data imported from the case file, or the appropriate formula. When we load this template into the case, we will then import the appropriate variables into this Spreadsheet.
Enter the headings as shown below. There are no formulae on this page. There is also some additional Simulation Data in this area H16 - H Note that cells A18 - B26 have already been completed.
All of the cells in column D shown here are formulae - do not enter the values 3. Save it e. Importing Variables into Spreadsheet First, add the subflowsheet; select the Read an Existing Template button when prompted to select the source for the sub-flowsheet. There are a large number of variables to import into the Spreadsheet. In the Cell column, type or select from the drop down list the Spreadsheet cell to be connected to that variable.
When you move to the Spreadsheet page, that variable will appear in the cell you specified. Importing Variables from the Spreadsheet Page Browsing You may also import a variable by positioning the cursor in an empty field of the Spreadsheet and clicking the right mouse button.
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