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2. Measurement Process Characterization
2.5. Uncertainty analysis
2.5.5. Propagation of error considerations

Propagation of error for many variables

Example from fluid flow with a nonlinear function Computing uncertainty for measurands based on more complicated functions can be done using basic propagation of errors principles. For example, suppose we want to compute the uncertainty of the discharge coefficient for fluid flow (Whetstone et al.). The measurement equation is

$$ C_d = \frac{\dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$ where

$$ \begin{eqnarray*} C_d &=& \mbox{discharge coefficient} \\ \dot{m} &=& \mbox{mass flow rate} \\ d &=& \mbox{orifice diameter} \\ D &=& \mbox{pipe diameter} \\ \rho &=& \mbox{fluid density} \\ \Delta P &=& \mbox{differential pressure} \\ K &=& \mbox{constant} \\ F &=& \mbox{thermal expansion factor (constant)} \\ \end{eqnarray*} $$

Assuming the variables in the equation are uncorrelated, the squared uncertainty of the discharge coefficient is

$$ s^2_{Cd} = \left[ \frac{\partial C_d}{\partial \dot{m}} \right]^2 s^2_{\dot m} + \left[ \frac{\partial C_d}{\partial d} \right]^2 s^2_d + \left[ \frac{\partial C_d}{\partial D} \right]^2 s^2_D + \left[ \frac{\partial C_d}{\partial \rho} \right]^2 s^2_{\rho} + \left[ \frac{\partial C_d}{\partial \Delta P} \right]^2 s^2_{\Delta P} $$

and the partial derivatives are the following.

$$ \frac{\partial C_d}{\partial \dot{m}} = \frac{\sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^2 F \sqrt{\rho} \sqrt{\Delta P}} $$

$$ \frac{\partial C_d}{\partial d} = \frac{-2\dot{m} d}{K F D^4 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} - \frac{2 \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{K d^3 F \sqrt{\rho} \sqrt{\Delta P}} $$

$$ \frac{\partial C_d}{\partial D} = \frac{2 \dot{m} d^2}{K F D^5 \sqrt{\rho} \sqrt{\Delta P} \sqrt{1-\left( \frac{d}{D} \right) ^4}} $$

$$ \frac{\partial C_d}{\partial \rho} = \frac{- \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{2 K d^2 F \rho^{\frac{3}{2}} \sqrt{\Delta P}} $$

$$ \frac{\partial C_d}{\partial \Delta P} = \frac{- \dot{m} \sqrt{1-\left( \frac{d}{D} \right) ^4}}{2 K d^2 F \sqrt{\rho} (\Delta P)^{\frac{3}{2}}} $$

Software can simplify propagation of error Propagation of error for more complicated functions can be done reliably with software capable of symbolic computations or algebraic representations.

Symbolic computation software can also be used to combine the partial derivatives with the appropriate standard deviations, and then the standard deviation for the discharge coefficient can be evaluated and plotted for specific values of the secondary variables, as shown in the comparison of check standard analysis and propagation of error.

Simplification for dealing with multiplicative variables Propagation of error for several variables can be simplified considerably for the special case where:
  • the function, \(Y\), is a simple multiplicative function of secondary variables, and
  • uncertainty is evaluated as a percentage.

For three variables, \(X, Z, W\), the function

$$ Y = X \cdot Z \cdot W $$

has a standard deviation in absolute units of

$$ \begin{eqnarray*} s_Y & = & \sqrt{(ZW)^2 s_x^2 + (XW)^2 s_z^2 + (XZ)^2 s_w^2} \\ & = & Y \sqrt{\frac{s_x^2}{X^2} + \frac{s_z^2}{Z^2} + \frac{s_w^2}{W^2}} \\ \end{eqnarray*} $$

In percent units, the standard deviation can be written as

$$ \frac{s_Y}{Y} = \sqrt{\frac{s_x^2}{X^2} + \frac{s_z^2}{Z^2} + \frac{s_w^2}{W^2}}$$

if all covariances are negligible. These formulas are easily extended to more than three variables.

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