Example Problem for Experimental Design

Problem Description

Problem description: http://files.charlesmartinreid.com/VUQ_Toy_Problem.pdf

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:

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

Variables: I/U Map

Variable Name	Input value (I)	Uncertainty (U)	Logarithmic Scale?
$ z_1 $	$ 0.5 m $	$ \pm 0.02 m $	no
$ z_2 $	$ 1.5 m $	$ \pm 0.02 m $	no
$ z_3 $	$ 2.5 m $	$ \pm 0.02 m $	no
$ \dot{m}_1 $	$ 1.0 $	$ \pm 0.05 $	no
	$ 2.0 $	$ \pm 0.10 $
$ \dot{m}_2 $	$ 1.0 $	$ \pm 0.05 $	no
	$ 2.0 $	$ \pm 0.10 $
$ k(T) $	$ 1 $	$ 10^{0 \pm 2} $	yes
$ L_{mix} $		$ 0.3-3.0 $	yes

Response Surfaces

Product at Exit Response Surface

This response surface maps the response of the mass fraction of product at the exit, $ y_{P,exit} $.

This response surface is a function of several variables:

$ y_{P,exit} = y_{P,exit} \left( \dot{m}, k, L_{mix} \right) $

However, the axial location of measurement of product $ z_i $ 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, $ y_{P,z1}, y_{P,z2}, y_{P,z3} $.

This response surface is a function of all variables:

$ y_{P,zi} = y_{P,exit} \left( \dot{m}, k, L_{mix}, z_i \right) $

(Note, however, that only the corresponding $ z_i $ will be a response surface independent variable, since other $ z_i $ values have no affect).

Dealing with Multimodal Variables

The variable $ \dot{m} $ is an excellent example of modal variables - that is, variables that have certain modes, or distinct sets of values.

In the case of $ \dot{m} $, the two modes are $ \dot{m} = 1.0 \pm 5% $ and $ \dot{m} = 2.0 \pm 5% $.