This example shows how to sample and explore a design space using the Sensitivity Analysis tool. You explore the design of a Continuously Stirred Tank Reactor (CSTR) to minimize product concentration variation and production cost. The design must also account for the uncertainty in the temperature and concentration of the input feed to the reactor.

You explore the CSTR design by characterizing model parameters using probability distributions. You use the distributions to generate random samples and perform Monte-Carlo evaluation of the design at these sample points. You then create plots to visualize the design space and select the best design. You then use the best design as an initial guess for optimization of the design.

Continuously Stirred Tank Reactors (CSTRs) are common in the process industry. The Simulink model, `sdoCSTR`

, models a jacketed diabatic (i.e., non-adiabatic) tank reactor described in [1]. The CSTR is assumed to be perfectly mixed, with a single first-order exothermic and irreversible reaction, . , the reactant, is converted to , the product.

In this example, you use the following two-state CSTR model, which uses basic accounting and energy conservation principles:

, and - Concentrations of A in the CSTR and in the feed [kgmol/m^3]

, , and - CSTR, feed, and coolant temperatures [K]

and - Volumetric flow rate [m^3/h] and the density of the material in the CSTR [1/m^3]

and - Height [m] and heated cross-sectional area [m^2] of the CSTR.

- Pre-exponential non-thermal factor for reaction [1/h]

and - Activation energy and heat of reaction for [kcal/kgmol]

- Boltzmann's gas constant [kcal/(kgmol * K)]

and - Heat capacity [kcal/K] and heat transfer coefficients [kcal/(m^2 * K * h)]

Open the Simulink model.

```
open_system('sdoCSTR');
```

Assume that the CSTR is cylindrical, with the coolant applied to the base of the cylinder. Tune the CSTR cross-sectional area, , and CSTR height, , to meet the following design goals:

Minimize the variation in residual concentration, . Variations in the residual concentration negatively affect the quality of the CSTR product. Minimizing the variations also improves CSTR profit.

Minimize the mean coolant temperature . Heating or cooling the jacket coolant is expensive. Minimizing the mean coolant temperature improves CSTR profit.

The quality of the feed to the reactor can differ amongst suppliers. Thus, the design must allow for variations in the supply feed concentration, , and feed temperature, . The quality of the feed differs from supplier to supplier and also varies within each supply batch.

Open the Sensitivity Analysis Tool. In the Simulink model from the **Apps** tab, click **Sensitivity Analyzer** under **Control Systems**. The tool opens with an empty Sensitivity Analysis session.

Create a parameter set that includes the CSTR design variables `A`

and `h`

and the feed variation parameters `FeedConc0`

and `FeedTemp0`

. You randomly generate multiple values for these parameters to evaluate the CSTR design.

In the

**Sensitivity Analysis**tab, in the**Select Parameters**drop-down menu, select**New**.In the dialog box, select

`A`

,`FeedCon0`

,`FeedTemp0`

, and`h`

.Click

**OK**. An empty parameter set,`ParamSet`

is created in the**Parameter Set**area of the tool browser.

Specify the parameter distributions and correlations. `ParamSet`

will be populate with parameter values selected randomly from the specified distributions:

Sample

`A`

from a uniform distribution with lower bound 0.2 m^2 and upper bound 2 m^2.Sample

`h`

from a uniform distribution with lower bound 0.5 m and upper bound 3 m.Sample

`FeedConc0`

from a normal distribution with mean 10 kgmol/m^3 and standard deviation 0.5 kgmol/m^3.Sample

`FeedTemp0`

from a normal distribution with mean 295 K and standard deviation 3 K.Specify

`FeedCon0`

and`FeedTemp0`

as negatively correlated with covariance 0.6.

To generate 100 parameter values using the above distribution and correlation information, click **Generate Values**, and select **Generate Random Values**. For repeatability of the example reset the random number generator.

```
rng('default')
```

In the **Generate Random Parameter Values** dialog box, specify the following:

Set the number of samples to 100

For parameter

`A`

select**Uniform**distribution, set the lower bound to 0.2 and upper bound to 2.For parameter

`FeedCon0`

select**Normal**distribution, set`mu`

to 10 and`sigma`

to 0.5, and check`cross-correlated`

.For parameter

`FeedTemp0`

select**Normal**distribution, set`mu`

to 295 and`sigma`

to 3, and check`cross-correlated`

.For parameter

`h`

select**Uniform**distribution, set the lower bound to 0.5 and upper bound to 3.In the

**Correlation Matrix**tab, set the`FeedCon0`

,`FeedTemp0`

covariance to -0.6.Click

**OK**to generate the parameter values.

The `ParamSet`

table is updated with the generated parameter values. Note that you can manually edit the generated parameter values in the `ParamSet`

table.

To plot the parameter set click `ParamSet`

in the **Parameter Sets** area of the tool browser. In the **Plots** tab, select **Scatter Plot** in the plots gallery. The plot shows the histogram of the generated parameters on the diagonal and pair-wise parameter values on the off-diagonal.

Note that due to the random number generator the specific values in the plots and tables below may differ from what you get when running the example.

Each marker on the plot represents one row of the `ParamSet`

table, with each row being simultaneously displayed on all the plots. The correlation between `FeedCon0`

and `FeedTemp0`

can be seen on the plot.

The CSTR design is required to minimize the variation in residual concentration and minimize the mean coolant temperature. Select **New Requirement** and click **Signal Property** to create a requirement to minimize the residual concentration variation.

In the **Create Requirement** dialog box, specify the following fields:

In the

**Property**drop-down list, select**Signal Variance**.In the

**type**drop-down list, select**Minimize**.In the

**Select Signals**area, select a logged signal to apply the requirement to. To do so click**+**. A`Create Signal Set`

dialog box opens where you specify the logged signal. In the Simulink model, click the signal at the`CA`

output of the`CSTR`

block. The dialog box now displays this signal. Add the signal to the signal set and click**OK**.

Close the Signal Property requirement dialog by clicking the x in the upper right dialog corner.

Close the **Create Requirement** dialog box. A new requirement, `SignalProperty`

is listed in the **Requirements** area of the tool browser

Right-click

`SignalProperty`

, select**Rename**; rename the requirement to`ConcVar`

.

Select **New Requirement** and click **Signal Property** to create a requirement to minimize the coolant mean (output of block `sdoCSTR/Controller`

) temperature.

In the **Create Requirement** dialog box, specify the following fields:

In the

**Property**drop-down list, select**Signal Mean**.In the

**Type**drop-down list, select**Minimize**.In the

**Select Signals**area, add the`sdoCSTR/Controller`

signal to the requirement.

Close the **Create Requirement** dialog box. A new requirement, `SignalProperty`

is created in the **Requirements** area of the tool browser. Rename the requirement `CoolMean`

.

In the **Sensitivity Analysis** tab, click **Select for Evaluation**. By default, all requirements are selected to be evaluated. Click **Evaluate Model** to evaluate the `ConcVar`

and `CoolMean`

requirements for each row of parameter values in `ParamSet`

. Note you can speed up evaluation by using parallel computing if you have the Parallel Computing Toolbox (TM), or by using fast restart. For more information, see "Use Parallel Computing for Sensitivity Analysis" and "use Fast Restart Mode During Sensitivity Analysis" in the Simulink Design Optimization™ documentation.

A results scatter plot showing each parameter vs each requirement is updated during model evaluation. At the end of evaluation a table with the evaluation results is created, each row in the evaluation result table contains values for `A`

, `FeedCon0`

, `FeedTemp0`

, `h`

and the resulting requirement values `ConcVar`

and `CoolMean`

. The evaluation results are stored in the `EvalResult`

variable in the **Results** area of the tool.

The results scatter plot for `EvalResult`

shows that `CoolMean`

is inversely correlated with `h`

(increasing `h`

decreases `CoolMean`

) and that low values of `h`

can result in high values for `ConcVar`

. The plot shows that low values of `A`

can result in high values for `ConcVar`

, but it is not clear from the plot how `A`

is correlated with `ConcVar`

or `CoolMean`

or which parameter influences `ConcVar`

the most. To investigate further, in the **Statistics** tab, select all the analysis methods and types and click **Compute Statistics**. This performs analysis on the evaluation results and creates a tornado plot. The tornado plot shows the influence of each parameter on each requirement:

`h`

is inversely correlated with`CoolMean`

, and is the parameter that influences`CoolMean`

the most.`A`

is inversely correlated with`CoolMean`

.`FeedCon0`

and`FeedTemp0`

are inversely correlated with`CoolMean`

.`A`

is inversely correlated with`ConcVar`

, and is the parameter that influences`ConcVar`

the most.`h`

is inversely correlated with`ConcVar`

.`FeedCon0`

and`FeedTemp0`

have mixed correlation with`ConVar`

, but have minimal correlation with`ConcVar`

.

The analysis shows that choosing a large `h`

, to reduce `CoolMean`

and choosing a large `A`

to reduce `CoolVar`

would appear to be a good design choice. You can confirm this by creating a contour plot of `CoolMean`

and `CoolVar`

versus `h`

and `A`

. Select `EvalResult`

from the **Results** area of the tool browser, and in the **Plots** tab, in the plots gallery click **Contour plot**. On the contour plot select `h`

for the **Y parameter**, note that large `h`

medium values of `A`

give low values for both `ConcVar`

and `CoolMean`

.

Sort the Evaluation Result table by decreasing `h`

, and select a row that has low `ConcVar`

and `CoolMean`

values. Right-click the selected row and click **Extract Parameter Values**. The extracted parameter values are saved in the `ParamValues`

variable in the **Results** area of the tool browser. These parameter values are used as the initial guess for optimization.

Use the data in the Sensitivity Analysis tool to create an optimization problem to optimize `A`

and `h`

. In the **Sensitivity Analysis** tab click **Optimize**, and select **Create Response Optimization Session**. This opens a dialog to import data from Sensitivity Analysis to Response Optimization tool.

Select both the

`ConcVar`

and`CoolMean`

requirements to import.Select

`ParamValues`

to import as design variables for optimization.Select

`EvalResult`

to import as uncertain variables to use during optimization.Click

**OK**to import the data to the Response Optimization tool

Configure the Response Optimization tool to optimize the CSTR design:

Click the pencil icon to edit the

`ParamValues`

design variable set, and remove the`FeedCon0`

and`FeedTemp0`

variables from the design variable set.

Select

`EvalResult`

as the uncertain variable set, click the pencil icon to edit`EvalResult`

, and remove`A`

and`h`

from the uncertain variable set.

Add iteration plots to see how the variables `ParamValues`

(`A`

and `h`

), and optimization requirements `ConcVar`

and `CoolMean`

change during optimization.

Select the variables in the

`Data to Plot`

drop-down list, and select`Iteration Plot`

in the`Add Plot`

drop-down list.Click

**Optimize**.

The optimization minimizes `CoolMean`

and `ConcVar`

in the presence of varying `FeedCon0`

and `FeedTemp0`

.

To learn how to explore the CSTR design space using the `sdo.evaluate`

command, see Design Exploration Using Parameter Sampling (Code).

[1] Bequette, B.W. *Process Dynamics: Modeling, Analysis and Simulation.* 1st ed. Upper Saddle River, NJ: Prentice Hall, 1998.

Close the model

```
bdclose('sdoCSTR')
```