View source for Example Problem for Experimental Design

See code at [[ToyProblem_cmr.m]]

=Problem Description=

<s>
Problem description: http://files.charlesmartinreid.com/ExperimentalDesign/VUQ_Toy_Problem.pdf
</s>

=Inputs and Outputs=

There are several (polynomial) response surfaces being fit for the Monte Carlo simulations.  This is because there is one response surface for each output or observable.  (Technically, these are all part of one large multivariate response surface, but it is easier to think about them as independent response surfaces).

The variables that are included in the response surface analysis are:

* <math>z_i</math> (for <math>i=1,2,3</math>) - the location of measurement of axial concentrations
* <math>\dot{m} = \dot{m}_1 = \dot{m}_2</math> - the mass flowrate of the inlet streams of A and B
* <math>L_{mix}</math> - mixing length (parameter for the mixing model)
* <math>k(T)</math> - reaction rate for the reaction <math>A + B \rightarrow^{k} P</math>

==Variables: I/U Map==

{|class="wikitable"
!Variable Name
!Input value (I)
!Uncertainty (U)
!Logarithmic Scale?
|-
|<math>z_1</math>
|<math>0.5 m</math>
|<math>\pm 0.02 m</math>
|no

|-
|<math>z_2</math>
|<math>1.5 m</math>
|<math>\pm 0.02 m</math>
|no

|-
|<math>z_3</math>
|<math>2.5 m</math>
|<math>\pm 0.02 m</math>
|no

|-
|<math>\dot{m}_1</math>
|<math>1.0</math>
|<math>\pm 0.05</math>
|no

|-
|
|<math>2.0</math>
|<math>\pm 0.10</math>
|

|-
|<math>\dot{m}_2</math>
|<math>1.0</math>
|<math>\pm 0.05</math>
|no

|-
|
|<math>2.0</math>
|<math>\pm 0.10</math>
|

|-
|<math>k(T)</math>
|<math>1</math>
|<math>10^{0 \pm 2}</math>
|yes

|-
|<math>L_{mix}</math>
|
|<math>0.3-3.0</math>
|yes

|}


==Response Surfaces==

===Product at Exit Response Surface===

This response surface maps the response of the mass fraction of product at the exit, <math>y_{P,exit}</math>.

This response surface is a function of several variables:

<math>
y_{P,exit} = y_{P,exit} \left( \dot{m}, k, L_{mix} \right)
</math>

However, the axial location of measurement of product <math>z_i</math> is not considered, because it does not affect the measurement of P at the exit.

===Product at Axial Location Response Surfaces===

This response surface maps the response of the mass fraction of product at several axial locations, <math>y_{P,z1}, y_{P,z2}, y_{P,z3}</math>.

This response surface is a function of all variables:

<math>
y_{P,zi} = y_{P,exit} \left( \dot{m}, k, L_{mix}, z_i \right)
</math>

(Of note is that only the corresponding <math>z_i</math> will be a response surface independent variable, since other <math>z_i</math> values have no affect).

'''NOTE''': It is easy to get confused about why a parameter dealing with the model output, like the location at which the observable is actually observed, can be part of the input.  However, given some thought, it is easy to see how this is an input variable.

Imagine constructing a response surface for the observable <math>Y_{P,z1}</math> as a function of only one variable, <math>z_1</math>.  This response surface <math>Y_{P,z1}\left( z_1 \right)</math> is simply the concentration <math>Y_P</math> as a function of location.  As <math>z_1</math> is varied, different concentrations <math>Y_P</math> are observed - just as different concentrations are observed when the mixing length is changed, or when the reaction rate constant is changed.  Just because the input parameter <math>z_i</math> is intuitively easier to connect to the observable <math>Y_P</math> doesn't mean that it can't be treated as an input variable!

=See Also=

* [[Experimental Design Lecture]]
* [[Response Surface Methodology]]

{{ExperimentalDesign}}