Example 2
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1. 1 1 Psat Psatx T T_sample i coeff i very concise d version which deals with all of the polynomial terms of divided differences through backwards nested loop end o Psat Psat 1 01325 conversion from bar to atm AspenTech Flash Calculator EPCflash m User Manual EPCflash m is a powerful flash drum calculator in MATLAB This user manual will walk you through an example flash calculation to get you up and running as soon as possible for your flash calculation needs Before we begin please make sure the following files are in your current directory e Psat1 m This function calculates the vapor pressure for methane CH e Psat2 m Same as above for ethane C H e Psat3 m Same as above for ethylene C H e Psat4 m Same as above for propane C H e flash1 m Performs a flash calculation given z F T P e flash2 m Performs a flash calculation given z F T V e flash3 m Performs a flash calculation given z F T y1 e EPCflash m Integrates all of the above the primary function to use Input z F T P V y1 To get started enter the following in the command window SA A pO 024 0e 0L z is a vector of mole fractions corresponding to the input feed stream It is important that z is always a vector of length four must always contain values between 0 and 1 and must always add up to 1 In this example the mole fraction of our input feed is 0 2 methane 0 4 ethane 0 3 ethy2. available points The estimated error for this approach is computed by generating the 5th order polynomial at the same temperature and taking the difference between the 4th order and 5th order approximations Since the error varies between data points I have listed the error bounds for each function below Species MATLAB Function Valid Temp K Maximum Error atm methane Psat1l m 110 190 0 1 ethane Psat2 m 110 270 0 01 ethylene Psat3 m 110 270 0 009 propane Psat4 m 110 270 0 002 The maximum error is computed by comparing the estimated error to the error inherent in the experimental data whichever contributes the most significant source of error for each function is the maximum possible error The error beyond the temperature ranges cannot be bound in such a manner The data obtained from my vapor pressure functions therefore will be accurate up to the respective maximum error listed within the range of valid temperatures Attached Psatl m Listing 1 Psat1 m Vapor Pressure as a function of Temperature for Methane function Psat Psat1 T SPSAT1 Obtain the partial pressure of methan CH4 as a function of Stemperature in Kelvin Psat Psatl1l T From a given set of divided difference coefficients the appropriate Newton forward divided difference form of interpolatory polynomial is evaluated at the specified T and the appropriate partial pressure is returned The pressure is given in at
3. parameter as follows gt gt yl 0 4 gt gt x y L V P EPCflash z F T 0 0 yl eo 0 0079 0 5406 0 2580 0 1936 ee 0 4000 0 2536 0 3438 0 0026 Les 102 0200 AE 97 9800 P 0 2034 This method uses flash3 m to calculate the remaining values which is done by using Newton s method for solving initial value problems At every iteration the pressure is recomputed until everything converges As you can see in order to obtain 40 methane in the vapor exit stream at 150 K it will be necessary to lower the pressure to 0 2034 atm Of the possible final inputs vapor molar flow rate is easily bounded by the input feed since the output rate cannot exceed the input rate and y1 must lie between 0 and 1 Pressure is difficult to bound and arbitrary selections of temperature and pressure will easily lead to poor flash1 m output values The following table presents general guidelines for valid temperature and pressure ranges Temperature K Pressures atm 120 0 01 0 03 140 0 04 0 06 160 0 13 0 64 180 0 4 3 200 1 4 9 220 3 6 22 240 8 8 42 260 17 86 280 28 140 300 45 210 This data was compiled through trial and error using an input stream composition of 25 methane 25 ethane 25 ethylene and 25 propane The pressure ranges will also vary depending on the input stream composition this table only serves as a guideline for what will actually be computable by the flash programs Even s
4. AspenTech VAPOR PRESSURES Technical Memo No XX 1 Subject Vapor pressure interpolation error analysis To Ms Sam K Safobeen Submitted by Student 2 Date May 11 2010 Summary I have compiled vapor pressure functions for the following four species methane CH ethane C H ethylene C H and propane C3Hg The functions accept temperature inputs in Kelvin and return the vapor pressure of the species in atm The error in each approximation was calculated and presented in tabular form Findings I am writing because I have successfully fit the provided vapor pressure data to separate MATLAB functions Each function accepts a temperature input in Kelvin and will output the corresponding vapor pressure in atm I have enclosed a sample MATLAB function file Psat1 m for methane The approach used for all four functions was a variation of Newton s forward divided difference form of the interpolatory polynomial The algorithm generates a 4th order polynomial using the data points given from the experimental data closest to the temperature input This method is preferred over a more comprehensive polynomial because of how the polynomial generated A large order polynomial is overly concerned with fitting the given data points exactly than observing a general trend in the data Fitting only a handful of points to a polynomial stresses the importance of the points around the input temperature rather than placing equal weight on all of the
5. ar flow This is caused by F being negative or equal to 0 P related errors From flash1 m Error Operating pressure must be positive Please check to s is greater than 0 if the pressur P This is caused by an input pressure P less than or equal to 0 V related errors From flash2 m Error Vapor flow rate cannot exceed that of the input feed molar flow rate Please make sure V is less than F Error Vapor flow rate cannot be negative Please make sure V is a positive number less than the molar flow rate F This is caused by an input vapor molar flow rate V greater than F The vapor flow rate must also be greater than or equal to 0 yl related errors From flash3 m Error Desired mole fraction of methane must be between 0 and 1 Please check to see if this value yl is within this range This is caused by the desired vapor mole fraction of methane y1 not within 0 1 var related errors From EPCflash m Error Not enough inputs in var Please input a vector P V yl of which only one value is defined whereas the others are 0 input a vector P V yl for in which Error No valid var parameter defined Pleas only one out of the thr are defined Either var was not input as a vector of correct length or no value of P V or y1 was detected to be usable The format is strict in order for the program to work Other messa
6. ges From EPCflash m These values are most likely unusable Please Warning x values do not add up to 1 revise input conditions for better results Warning y values do not add up to 1 These values are most likely unusable Please revise input conditions for better results most likely unusable Please Warning Negative x values detected These values ar revise input conditions for better results Warning Negative x values detected These values ar revise input conditions for better results most likely unusable Please These are warnings and not errors because the flash calculations may have taken place and values may be usable to some degree These warnings will almost always accompany every error message to let the user know that the mole fractions most likely do not make sense and should not be used or interpreted to have any meaning In certain cases will the x and y values be usable although a leniency of their sums adding up to 0 995 to 1 005 is allotted anything beyond those bounds trigger the first type of warning message AspenTech EPCQcalc Executive Summary Heat and Flash Drum Calculator Problem When operating a flash drum there are many parameters which must be taken into account Given a handful of operating conditions it is possible to derive and approximate the behavior of such a system This process is tedious and time consuming and the derivation is not easily ob
7. lene and 0 1 propane gt gt F 200 gt gt T 200 SS PvS 37 Here we have set the input feed stream F to be 200 this stream is in units of mol per unit time The operating temperature T is set to 200 K and the pressure P is set to 3 atm We are now ready to perform our first flash calculation This is done by entering the following in the command window gt gt x y L V P EPCflash z F T P 0 0 The last parameter P V yl indicates which method will be called to perform the flash calculation if P is provided then flash1 m will be invoked if V is provided then flash2 m will be used and if yl is provided then flash3 m will be called Only one of the three possible variables should be used when using this program the other two should be marked as 0 The flash calculation should be run correctly and no error messages should be displayed After this you should have these following variables with the corresponding values 0 0141 0 4943 0 2249 0 2666 y 0 2922 0 3532 03 3373 020173 66 3212 aa 133 6788 P 3 The way flash1 m works is by substituting all of the known data into each vapor mole fraction of y and using a bisection method to solve for L the liquid molar flow rate that satisfies the given conditions The extensive mathematical details will not be discussed here x is the liquid mole fraction of the feed and V is the vapor molar flow rate of the exit strea
8. m 1 atm 1 01325 bar Student 2 04 21 2010 JP A P AP P AP P AP P X T given 110 10 190 Temperature K Psat_given 0 884 1 919 3 681 6 422 10 41 15 94 23 81 32 286 sas 45 20 Pressure bar T pivot index_pivot min abs T given T find the closest T within T_given if index_pivot 1 index_pivot 2 elseif index_pivot gt 8 index_pivot 7 end T_sample T_given index_pivot 1 T_given index_pivot T_given index_pivot 1 T_given index_pivot 2 Psat_sample Psat_given index_pivot 1 Psat_given index_pivot Psat_given index_pivot 1l Psat_given index_pivot 2 dim_T size T_sample dim_P size Psat_sample N dim T 2 could easily be dim_P 2 both should be the same valu P_temp Psat_sample zeros N N 1 initialize P_temp first column values are from given initial values It holds all the relevant divided differences oO 2 calculations coeff zeros 1 N initialize the relevant coeffs used in the polynomial coeff 1 Psat_sample 1 First coeff is not even touched by the algorithm and must be set manually for i 2 N for j 222 P_temp i j P_temp i j 1 P_temp i 1 j 1 T_sample i T_sample i j 1 i a 3 coeff i P_temp i j the needed coeffs only occur at the diagonals of the triangular matrix o which is when i j end end end Psat 0 initialize Psat for i length coeff
9. m As we can see here of the 200 mol per unit time initial feed 66 3212 exits as a liquid under 200 K and 3 atm which has a composition of 1 41 methane 49 43 ethane 22 49 ethylene and 26 66 propane Let us perform another flash calculation using a different value for the last parameter Enter clear in the command window and input the same values for z F and T from before Let us find the conditions for an equimolar flow of both liquid and vapor output streams at 200 K by entering the following gt gt V 100 gt gt x y L V P EPCflash z F T 0 V 0 x2 0 0228 0 5117 0 2749 0 1903 y 0 3772 0 2883 01 3251 0 0097 Te 100 er 100 P 3 8048 This method uses flash2 m to run the flash calculation which does so by obtaining the pressure P in a similar manner as flashi m obtains L and then uses Runge Kutta order 4 for systems of initial value problems This is made possible through the method of false transients in which the vapor fraction y is assumed to vary with time and the solution to this problem would be when y reaches steady state From this example we can see that at 200 K and 3 8048 atm we will obtain an equimolar liquid and vapor flow Suppose we want to set a fixed vapor mole fraction for methane We will use the same z and F but let us set T 150 this time just so we are not performing flash calculations at the same temperature every time Input a value for y1 into the last input
10. o at most of the extreme conditions although values will be properly generated most species will be below their boiling points or above their critical temperature and thus the output will not be accurate under these circumstances Essentially if the operating pressure lies well within the guidelines provided the more likely that the flash calculations will approximate realistic scenarios Error Messages When an error occurs almost all return values will be 1 to denote that no flash calculation took place z related errors From flash1 m flash2 m and flash3 m Error Incorrect number of mole fractions please provide the mole fractions in the following format SS oz 0 20 4 0 3 O11 17 where 0 2 corresponds to mole fraction of inlet methane 0 4 corresponds to inlet ethane 0 3 corresponds to inlet ethylene and 0 1 corresponds to inlet propane Error Mole fractions must be within the range of 0 and 1 Please check the mol fractions to s if they are valid and try again BErors Mole fractions provided do not add up to 1 to see if they are valid and try again Please check the mole fractions These are caused by z inputs either not having 4 elements not consisting of real fractions and not totaling up to 1 F related errors From flash1 m flash2 m and flash3 m Error Feed molar flow rate must be positive rate F is greater than 0 Please check to see if the mol
11. sed is the inline function which creates an inline object out of a string Future Changes The program is fixed to run for a set number of species currently 4 and the species them selves are also fixed methane ethane ethylene and propane The next step to improving this program would be to encorporate a database of vapor pressure values and their corresponding re lationship with temperature This would allow for user specification in selecting which species to be used in the flash operation as well as allow the number of active species be specified by the user Temperatures below the boiling point and above the critical temperature have significant impact on individual species These substances would be at either pure liquid or pure vapor compositions such effects are not taken into account in this program This is because the documented boiling point and critical temperatures only apply at 1 atm the dew and bubble points for a species changes with pressure as well Implementing some kind of algorithm which calculates each active species boiling point and critical temperature at the operating pressures and factoring them into the stream composition would also be the next step towards improving the validity of the program s results 11
12. tained analytically Furthermore based on what the given conditions are the approach to solving the rest of the flash drum conditions will vary as well The task is not an easy one and not all operating conditions are guaranteed to be solvable Solution EPCQcalc is a numerical approach to solving flash drum simulations The function sim ulates feeding a mixture into a flash drum at a known temperature and pressure and flashing it to a specified temperature and either pressure volume or vapor mole fraction This process requires heat to either be added or released from the system depending on whether the feed temperature is increased or decreased respectively A pressure drop will also require heat to be added to the system and heat to be removed for a pressure increase EPCQcalc calls upon a sister function EPCflash which tackles on the following flash tank simulations e Feed rate composition molar flow rate temperature and pressure given With this information either the liquid or vapor exit flow rates can be expressed as a single non linear equation solvable by any root solving algorithm The flash program uses bisection method in this case If the algorithm converges a liquid molar flow rate is found and thus the vapor flow rate can be found as well From here both vapor and liquid stream compositions can be found e Feed rate composition molar flow rate temperature and vapor molar flow rate given The pressure of this s
13. ystem can be solved for using the same method as the previous case through a root solving algorithm Bisection method is once again used for this parameter With this a system of non linear equations can be solved for by treating them as initial value problems This contrived system of initial value problems can be solved for by treating one of the mole fractions as changing with time and finding when such a change equals 0 also known as the method of false transients When this system converges the vapor mole fraction composition is complete and is used to calculate the missing liquid mole fraction e Feed rate composition molar flow rate temperature and desired vapor mole fraction of light key given Using the desired vapor mole fraction of the species with the lowest boiling point a system of non linear equations are established This system is composed of the other unknown vapor mole fractions and either vapor molar flow rate or liquid molar flow rate one can be expressed in terms of the other The flash program uses Newton s method for solving non linear systems which involves taking the Jacobian of the set of functions and solving for each unknown until convergence is reached The liquid composition and the missing molar flow rate are then subsequently calculated 10 No built in MATLAB solver functions are required to use this program all algorithms are im plemented using MATLAB commands The only potentially special command u
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