Sensitivity Analysis

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…from CheLabWiki, an online resource for chemical-engineering laboratories located at www.chelabwiki.org; Site Revision #859; 6 January 2009.

Community Portal → Data Analysis → Sensitivity Analysis

1 Differential Sensitivity
2 Fractional Sensitivity
3 Response to Incremental Changes
4 Caution

In the common analysis situation, we tacitly assume a model experiment in which we measure values for a dependent variable $y$ at precisely controlled values of an independent variable $x$ . But in practice, the variable $x$ can be controlled only to within some tolerance $δ x$ . For example, $x$ might be temperature $T$ and $y$ might be the vapor pressure $P$ of some fluid; then the experiment is to measure $P$ at selected values of $T$ . In such experiments, $T$ is usually controlled by placing the fluid sample in a heat bath. The sample is in thermal equilibrium with the bath, and we control the temperature of the bath using heating elements, a sensing mechanism, and a feedback circuit. When the bath starts to cool, the heating elements are turned on to restore the desired $T$ .

The problem is that, in such an apparatus, the temperature can be controlled to only within $δ T$ of a set-point $T$ : the tolerance might be $δ T$ = 1 K, or 0.1 K, or even 0.01 K. But in any case, the issue is to determine how variations in $T$ affect the dependent variable $P$ : How sensitive is $P$ to small changes in $T$ ?

The answers to these kinds of questions are important to us both in the design of the experiment and in the subsequent analysis of data. During design we need to know how tightly we must control the temperature to achieve a desired level of accuracy in the measured pressures. During analysis we assign uncertainties to the measurements, and for measured pressures, part of the uncertainty is caused by variations in temperature.

Differential Sensitivity

Let $y$ represent a property of the experimental system. The value for $y$ depends on the values of $N$ other properties, $x i$ ,

(1)

$y \ = \ f(x_1, x_2, ... , x_N)$

This is a measurement equation for the property $y$ . Now define the differential sensitivity of $y$ with respect to $x i$ to be the partial derivative,

(2)

$\left ( \frac{\partial y}{\partial x_i} \right )_{x_j \ne x_i} \ = \ \lim_{x_i \to 0} \left ( \frac{\Delta f}{\Delta x_i} \right )_{x_j \ne x_i}$

The subscript $x j$ ≠ $x i$ means that the values of all $x$ s are held fixed except that for $x i$ . So, the differential sensitivity tells how $y$ responds to differential changes in one $x$ when all others are constant. A large value for the derivative in (2) indicates high sensitivity: a small change in $x i$ causes a large response in $y$ , see Figure 1.

Fractional Sensitivity

In some cases it is helpful to have a fractional or relative sensitivity, which is defined by

(3)

$\left ( \frac{\partial \ln y}{\partial \ln x_i} \right )_{x_j \ne x_i} \ = \ \lim_{x_i \to 0} \left ( \frac{\Delta \ln f}{\Delta \ln x_i} \right )_{x_j \ne x_i}$

This allows us to determine the percentage response of $y$ to a given percentage change in $x i$ .

Response to Incremental Changes

Let $Δ x i$ represent a small finite change in $x i$ ; we call this an incremental change. Then $Δ y$ represents the incremental response of $y$ , given by

(4)

$\Delta y \ = \ \left ( \frac{\partial f}{\partial x_i} \right )_{x_j \ne x_i} \Delta x_i$

If all the $x$ s change incrementally, then the total response of $y$ is

(5)

$\Delta y \ = \ \sum_i^N \left ( \frac{\partial f}{\partial x_i} \right )_{x_j \ne x_i} \Delta x_i$

Let $y o$ be the value for $y$ when all the $x$ s have values { $x i o$ }, then the value of $y$ after an incremental change is, from (5),

(6)

$y \ = \ y_o + \sum_i^N \left ( \frac{\partial f}{\partial x_i} \right )_{x_j \ne x_i} \Delta x_i$

This is the Taylor expansion for $y$ , truncated at first-order.

Caution

We caution that in applying these definitions, you must be aware of constraints imposed either by the experimental design or by nature. For example, if a fluid completely fills a rigid vessel, then the experimental design imposes a constant-volume constraint. So you can only evaluate constant-volume measures of sensitivity. Moreover, nature imposes constraints via conservation laws, such as conservation of mass and energy. For example, for a binary mixture, you cannot ask about sensitivity to changes in one mole fraction with the other held fixed.

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