CENG0006: Coursework
Process Flowsheeting with Recycle and Nonlinear Equations
Exercise - 1: Nonlinear Equation - van der Waals Equation of State [20]
Using GAMS (as a programming language) solve for V the following van der Waals equation of state for ammonia at 10 atm and 250°C by using Secant Method where two initial guesses for V are 2.50 and 2.51.
(P + a/V2) (V- b) = RT
a = (27/64) (R2 Tc2/Pc)
b = RTc/(8Pc)
Given: R = 0.08206 atm l K-1 mol-1, P = 10 atm, T = 523 K, Tc = 407.5 K, Pc = 111.3 atm, a = 4.238448, b = 0.037556.
Provide the formulation of the problem, the solution obtained, the GAMS (.gms) code and progress of the iterations in the report.
Resolve the problem from different initial guesses. Analyse and discuss the effect of initial guesses on the final solution and on the convergence to the final solution. These values of initial guesses have to be chosen by you. GAMS codes for this part are not required to be submitted.
Exercise - 2: Simulation of an Equilibrium Reactor [15]
Consider the following reactor where stream 2 is pure ethane.
Ethane is dehydrogenated to ethylene and acetylene in the following pair of catalytic reactions:
C2H6 ↔ C2H4 + H2
C2H6 ↔ C2H2 + 2H2
Let A represent ethane, B ethylene, C acetylene and D hydrogen. The reactions proceed such that the product gas composition satisfies the following equilibrium conditions:
where y denotes mole fraction. Let Y1and Y2 denote extent of first and second dehydrogenation reactions respectively. Assume 100 kmol/hr of ethane is entering the reactor.
(i) Provide mass balance equations for the reactor in terms of component flowrates, and Y1and Y2 .
(ii) Write equations for y, the mole fractions, in terms of the component flowrates, and Y1and Y2 , and substitute the equations obtained into the equilibrium conditions given above to obtain equations in terms of Y1and Y2 . Formulate the resulting equations in GAMS and solve the equations: (a) using the CNS solver to compute values of Y1and Y2 , and (b) using Newton’s method by using GAMS as a programming language to compute values of Y1and Y2 . Provide the GAMS (.gms) codes in the report.
Exercise - 3: Process Flowsheeting with Recycle [40]
In this exercise you have to solve the mass balance for the following flowsheet:
The reactor considered in this flowsheet is the same as the reactor in the previous Exercise except that the assumption of 100 kmol/hr of ethane entering the reactor does not hold due to the recycle (stream-5). A 100 kmol/hr feed (stream-1) of pure ethane is mixed with the recycle (stream-5) containing pure ethane, and the mixed stream (stream-2) goes to the reactor where the dehydrogenation reactions as explained in the previous Exercise take place. The outlet (stream-3) from the reactor goes to the separator which separates 95% of unreacted ethane from ethylene, acetylene, and hydrogen and recycles the separated ethane to the mixer.
PART A: Write the mass balance for this problem and submit it as apart of the report. Carry out degrees of freedom analysis and if it is not zero then make it zero by specifying some variables or removing some specification, as appropriate. Submit degrees of freedom analysis in the report.
PART B: Sequential Modular Process Flowsheeting - using GAMS as a programming tool
Solve the problem by using GAMS by using stream 5 as the torn stream. Plot ε, the error, as a function of iteration counter k, where
where F5(k)(i) denotes the flowrate of component i in stream 5 computed in iteration number k.
In this case you could define the unknown variables as PARAMETERS in GAMS. To start the calculations, guess initial values for stream 5 component flowrates and then sequentially calculate the remaining stream component flowrates. Prepare a short report containing the plot mentioned above and the values of the converged flowrates of all the components in all the streams. Decide a sensible value of ε as the convergence criteria - if you cannot decide, then try different values and discuss your observations in the report.
Provide the GAMS (.gms) code in the report.
PART C: Equation Oriented Process Flowsheeting - using GAMS as a modelling tool
Resolve this problem using the Equation Oriented approach in GAMS using the CNS solver. Provide the GAMS (.gms) code in the report.
Exercise - 4: Solving Simultaneous Nonlinear Equations [25]
Consider the following three simultaneous nonlinear algebraic equations in three variables:
Carry out one iteration of the Secant method by hand, using the starting point:
Do not invert the Jacobian matrix and instead calculate the step vector; you may calculate the step vector by hand or use GAMS to calculate the step vector - if you use GAMS then provide the GAMS (.gms) code in the report.