Report- Monica Salib

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D r . C h r i s S w a r t z Y a n a n C a o

Air Separation Unit Monica Salib There is no doubt that optimization is the key for the new engineering future. It has a significant impact in chemical engineering research. It is the tool that allows us to improve the existing processes in terms of cost, energy and overall efficiency. As chemical engineers, we are expected to develop new computational and mathematical algorithms that can cope up with the world’s needs. This paper studies different ASU (Air Separation Unit) optimization strategies with gPROMS and addresses some of the ASU industrial issues.

Monica Salib Chemical Engineering

Spring-‐2015

08 Fall

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Table of Contents

1. CSTR Steady State Optimization ................................................................................. 1

2. Batch reactor Dynamic Optimization ........................................................................ 2 2.1 Single Scenario Dynamic Optimization .......................................................................... 2 2.2 Multi-‐scenario Dynamic Optimization ........................................................................... 4

3. Air Separation Unit .......................................................................................................... 5 3.1 Overview ................................................................................................................................... 5 3.2 Air Separation Process ......................................................................................................... 6 3.3 Dynamic Simulation .............................................................................................................. 7

4. Steady State Optimization .......................................................................................... 10 4.1 Objective function formulation ....................................................................................... 10 4.2 Constraints ............................................................................................................................. 12 4.3 Results ..................................................................................................................................... 13

5. Dynamic Optimization ................................................................................................ 14

6. Dynamic Optimization Under Uncertainty ........................................................... 15 6.1 Overview ................................................................................................................................. 15 6.2 Objective function formulation ....................................................................................... 15 6.3 Observations ......................................................................................................................... 16

7. Economics Dynamic Optimization .......................................................................... 16 8. Conclusion ....................................................................................................................... 17

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1. CSTR Steady State Optimization

Optimization is highly required to ensure that any given process is operating at its

best feasible conditions. In this example, four isothermal CSTRs are connected in

series. Steady state is assumed throughout the process because of having a fixed

flow rate. The task was to find the most economical operation while staying in the

plant operation limits (constraints), which was ensuring that the summation of all

the volumes is 20 m!. To do so, our controls (decision variables) were the volumes

of each CSTR. The optimization task is summarized in Table 1 below.

Objective Maximize yield of product Constraint ΣV =20 Controls V of each CSTR

Two solution approaches were implemented in order to solve this problem in

gPROMS. The starting point is the same, which requires defining the parameters and

the variables.

First, mole balance equations (one for each CSTR) were written in the MODEL

section in gPROMS as shown below in Eqn1-‐Eqn4.

Eqn 1. Fc! − Fc! − kc!!.!V! = 0

Eqn 2. Fc! − Fc! − kc!!.!V! = 0

Eqn 3. Fc! − Fc! − kc!!.!V! = 0

Eqn 4. Fc! − Fc! − kc!!.!V! = 0

Second, four MODELS were created (one for each CSTR) and then they were linked

together to the upper layer model by defining the inlet concentration of each CSTR

as the outlet of the previous one in series.

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The optimization file was created using the information in Table 1 and the solver

successfully converged giving us same results with both solution strategies.

2. Batch reactor Dynamic Optimization

2.1 Single Scenario Dynamic Optimization

A lot of research has been going on dynamic optimization since it is the more

realistic and applicable type in the existing plants. The Ramirez control problem was

examined and the task was to maximize the yield of species B at the final time by

calculating the optimal temperature profile, T (t) for a batch reactor with the

consecutive reactions shown below are carried out.

A!! B

!! C

The material balances for species A (concentration,x!) and B (concentration, x!) are

shown below in Eqn5-‐Eqn8.

Eqn 5. !!!(!)!"

= −k!(t) ∗ x!(t)

Eqn 6. !!!(!)!"

= k! t ∗ x! t − k!(t) ∗ x!(t)

Eqn 7. k! t = k!" ∗ e!!"!"(!)

Eqn 8. k! t = k!" ∗ e!!"!"(!)

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The desired goal was to maximize the yield of species B at the final time, i.e.,

x! t! − x!(t!).

After setting up this optimization problem, it was solved in two different ways:

(1) Piecewise constant

(2) Piecewise linear

The main difference between both methods is in the controlled variables. In the

piecewise constant method, it is the temperature that is being changed till it reaches

the optimum value. In the piecewise linear case, it is the slope of the temperature

that changes till it shapes the optimum temperature profile for the process. The

results of both strategies are shown below in Figure 1 and Figure 2.

Figure 1. Piecewise constant optimum Temperature profile

Figure 2. Piecewise linear optimum Temperature profile

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It was concluded by looking at the results that the piecewise linear is a better

approach as it converges at a higher yield value even if it takes a little while longer.

2.2 Multi-‐scenario Dynamic Optimization

There is no doubt that a design is more optimum if it can tolerate different reactions

with different parameters and rates. Therefore, taking uncertainty into

consideration in the design stage helps optimize the dynamic performance of the

plant. Using the previous example, nine scenarios were implemented each with a

different combination of k values.

If the changing parameters are the reaction rate constants k! and k! where:

k! ∈ [k!!"#, k!

!"#]

k! ∈ [k!!"#, k!

!"#]

Then, a theta (θ) variable is introduced as θ = k!k! and each scenario is assigned to a

different θ combination as illustrated below.

Scenario 1: θ = k!!"#

k!!"# Scenario 2: θ =

k!!"#


k!!"#

k!!"#

Scenario 4: θ = k!!"#


k!!"#


k!!"#

k!!"#

Scenario 7: θ =k!

!"#


k!!"#


k!!"#

k!!"#

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Generally, the optimization problem follows the same formulation as the single

scenario case. The objective function is the only thing that changes, as it has to

account for all the scenarios’ constraints at the same time. The most common

technique for the objective function reformulation is assigning it to the average of

the summation of the scenarios as shown in Eqn 9.

Eqn9. New_objective_function = !! ∗ ∑!!!!!! objective (i)

Piecewise linear and piecewise constant were again used to solve the optimization.

Piecewise linear proved to be a better approach in solving multi-‐scenario dynamic

optimization problems as it converged at a higher yield. The only disadvantage it

was the time it needed to converge.

Overall, both methods worked successfully in gPROMS and settled at very close

values.

3. Air Separation Unit

3.1 Overview

The air separation industry is essential as it plays an important role in a lot of

markets such as food processing, petrochemicals and healthcare. For a long time in

the air separation industry, the dynamic performance of the plant was assessed by

how capable it is in rejecting disturbances. The idea of switching the operation

points was not relevant due to the usual stability of electricity prices. Recently, this

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has not been the case because of the fluctuations in electricity prices in many

regions. Those deregulations in the price cause unexpected rapid changes in the

plant cost since electricity is the main operating cost for the air separation plant.

Therefore, further techniques that take dynamic rapid changes into account should

be adopted as steady state simulation has their limitations.

In this paper, all the studies and results are based on the cryogenic approach of air

separation. It produces large gas phase quantities of air components (Nitrogen,

Oxygen and Argon) and operates at a low temperature distillation.

3.2 Air Separation Process

A simple schematic of the main process equipment is shown in Fig 3. Further

description of the process can be found in Roffel et al. [2000] and Miller et al.

[2008a].

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Fig 3. A cryogenic air separation plant that produces argon, oxygen and nitrogen. LAr = Liquid Argon;

L02 = Liquid Oxygen; G02 = Gas Oxygen; LN2 = Liquid Nitrogen; GN2 = Gas Nitrogen; PHX = Primary

Heat Exchanger; LC = Lower Column; UC = Upper Column.

3.3 Dynamic Simulation

Step tests were conducted to investigate the effect of certain input changes on the

dynamic performance of the process. Fig 4. Shows the output results of the product

impurity, air feed, reflux and GN2 production upon -‐5% change in inlet volumetric

air feed. Fig 5. Shows the dynamic response of the product impurity, gas draw

fraction rate, reflux and GN2 production after being subjected to a positive step

change in the gas draw rate.

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Figure 4. Dynamic response of selected scaled variables to a negative step change in the air feed

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Figure 5. Dynamic response of selected scaled variables to a negative step change in the air feed.

While keeping all other inputs fixed in the system, as the gas draw fraction

increases, GN2 production increases and reflux rate decreases. The system tries to

reach a new equilibrium steady state but since the reflux rate is much lower than

before relative to the air feed at the new steady state, the product impurity

increases significantly.

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4. Steady State Optimization

4.1 Objective function formulation

This paper focuses on optimizing the air separation unit so that it can handle the

frequent changes in demand and electricity. Normally, the plant runs steadily under

certain operating points.

When a change in demand/electricity price occurs, the plant is adjusted to operate

at a new set of operating conditions. Therefore, the plant is expected to operate at

steady state all the time except that time during the transition between the

operating points.

Since the plant is assumed to be steady, we have to ensure that all operation points

are feasible and optimum. Hence, a steady state economical optimization is

performed, not only to make sure that all operation point are economically

optimum, but also to serve as a guideline in the dynamic transition optimization as

well. The steady state optimization takes the form shown in Eqn 10.

Eqn. 10 maxΦ!! = C!"# F!"# !"#$ + F!"#$ − C!"!#W!"#$ − C!"#$F!"#$

Subject to:

f (x = 0, x, z, u , p) =0

g(x, z, u , p) =0

h (x , z, u , p) < 0

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Where:

x = differential state vector

z = algebraic state vector

u = control input vector

• u!= Inlet volumetric air flow rate under standard conditions

• u!= Liquid molar air flow rate to the column

• u!= Liquid Nitrogen production rate (Distillate)

• u!= Gas draw fraction

• u!= Evaporation rate of liquid nitrogen for unsatisfied demand

p = parameter vector

C!"#= Sales price of gas nitrogen

C!"!#,C!"#$= Costs associated with compression and evaporation

F!"# !"#$= Flow rate of GN 2 produced

F!"#$= Rate of evaporation of pre-‐stored liquid N2

W!"#$= Power consumption of the compressor

A detailed plant configuration with labeled variables, parameters and inputs is

shown in Figure 5 below.

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Figure 5. Plant configuration with labeled decision variables

4.2 Constraints

There are different types of constraints that we need to put in consideration to

make sure that the plant runs without violating the physics/chemical laws or

the operational conditions along with satisfying the customer’s needs. All the

constraints are categorized and summarized in Table 2 below

Table 2. Constraints for steady state optimization

Operational Constraints Product Specification Modeling Constraints

Compressor Surge Demand Satisfaction Pressure in PHX

Flooding No Overproduction Temp diff. in IRC

Product Purity

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Each constraint mentioned above is modelled by an equation that has a certain

tolerance. The goal is always to minimize the tolerance to approach zero

4.3 Results

It was observed that the initial guess plays an important role in the optimization

process in terms of the converging time and value. That is due to the non-‐

convexity from the nonlinear model. Therefore, different initial guesses were

provided and the best point was reported.

Also, the system responds differently in terms of active constraints (when the

final value is very close to one of the bounds) depending on the change it was

subjected to. For instance, as demand increases, both flooding and impurity

constraints are active because the system has to settle at its maximum level of

impurity. However, as demand decreases, the compressor surge constraint is

the active one because of the decrease in the flow rate.

The steady state optimization results upon demand fluctuations (-‐30% to +30%)

are reported in Table 3. Those output results were used again as a target to

dynamic optimization.

Table 3. Steady state optimization results for demand fluctuations

** Data in the Table are scaled values for company confidentiality

-‐30% -‐20% -‐10% 0% 10% 20% 30%

LN2 production rate 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002

Gas draw rate fraction 0.058 0.068 0.0754 0.0758 0.0758 0.076 0.076

Air feed volumetric flow rate 29.998 30 30.4 33.8 37.075 38.2 38.212

Liquid air to the column 6.84 6.5434 6.21 5.356 4.6202 2.5288 2.5288

Evaporation rate 0 0 0 0 0 2.176 5.64

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5. Dynamic Optimization

Steady state optimization provided us with feasible optimal operating points.

However, it did not account for the transitions between those points. Hence,

dynamic optimization is conducted to switch from the base optimal case to the

new operation point upon demand fluctuations.

The objective function is formulated differently as it is no longer a cost based

one but rather a trajectory demand tracking function as shown in Eqn 11.

Eqn 11. minΦ = t! 1−F!"# !"#$ tF!"# !"#$∗

!!!

!! dt+ w! 1−

u! t!u!∗

!!!

!!!

Where 𝑤! represents the weights assigned to the manipulated variables as

shown below:

w!"# !""# = w!"# !"# = w!"# !"#$ = 1

w!"# = 0.1

The problem was solved using 5 control intervals with a tolerance of 1E-‐5. The

number of control intervals is critical in the optimization problem formulation.

It has to be big enough for capturing the control behaviour but not too big for

unwanted oscillations in the results.

The time was divided into three periods with an input slope of 0 to the first and

last period. This ensures that both the start and end point operate at optimal

feasible steady state. Also, a dummy variable was introduced to represent the

slowest variable in the process. By minimizing that variable, we guarantee that

our system converge at a steady state.

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The solution time in the air separation unit is too long compared to the one in

the batch reactor. That concludes that the complexity of the problem is a key

variable that affects the solution time.

6. Dynamic Optimization Under Uncertainty

6.1 Overview

The bigger picture comes into place after the individual demand dynamic

optimization cases have successfully converged. The task gets more complicated

because our objective is not only to optimize the operating conditions and the

trajectory for certain demand percentage change; instead it is finding the

conditions that are optimum overall regardless of the fluctuations.

The approach used to tackle the uncertainty is very similar to the “Multi-‐

scenario dynamic optimization” one in section 2.2 regarding the batch reactor.

6.2 Objective function formulation

Since the solution time increases drastically in uncertainty problems, only two

cases (0% demand change + 10% demand change) were solved simultaneously

to better understand the capabilities and limitations of gPROMS in handling

those complex problems.

The problem formulation was very similar to the single case dynamic

optimization. The difference was mainly including both cases into one single

process and having the new objective function (Φ∗) capturing both of them as

explained by Eqn 12.

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Eqn 12. Φ∗= Φ! +Φ!

Where:

Φ! = Objective function of the 0% demand change case

Φ! = Objective function of the 10% demand change case

All the optimization elements were handled in the same manner as the single

case with only the difference of having two of each manipulated

variable/constraint, one representing each case.

6.3 Observations

Multiple factors were observed to affect the optimization under uncertainty

more than just the number of the control intervals.

Based on the results reported in Table 4, relaxing the bounds seems to be an

important key that affects the ability of the optimization to converge. Also, using

the dynamic optimization results as initial guesses plays an important role in

terms of the solution time.

Table 4. Observations in uncertainty dynamic optimization trials with fixed control intervals

Provided initial guess from previous optimization

Relaxed bounds

Optimization converged

Solution time

Trial 1 ✖ ✖ ✖ -‐

Trial 2 ✓ ✖ ✖ -‐

Trial 3 ✖ ✓ ✓ 6.6 h

Trial 4 ✓ ✓ ✓ 2.6 h

7. Economics Dynamic Optimization

Another economical base approach was tried out to optimize the dynamic

process. The objective function was not formulated to track the demand

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trajectory as before but rather to maximize the accumulation of the profit.

The constraints and controls were handled in the same manner as the

demand trajectory case. The optimization successfully converged in 30 min

for the 10% demand change case and that is promising as the solution time is

very short compared to the demand based one.

8. Conclusion

The research is still ongoing to better optimize the air separation process, as it

is the primary supplier for our everyday vital chemicals. The design has got a

lot of interesting possibilities that should be examined in future work. For

instance, the introduction of external liquid nitrogen and the addition of a vent

stream after the compressor can be some promising alternatives. Also, more

cases should be solved simultaneously to achieve a better operating point.

Report- Monica Salib

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