3.2: Combining Uncertainty in Function

4. Total combined uncertainty
   1. Many results are a function of multiple measurands and variables with variety of units: \[ r = f(x_{1}, x_{2}, x_{3}, \ldots, x_{n}) \]
   2. Example scenarios:
      1. Strain: \( \varepsilon = \frac{P}{AE_m} \)
      2. Density: \( \rho = \frac{m}{\pi R^{2} L} \)
      3. Natural frequency: \( f = \sqrt{\frac{L}{g}} \)
   3. Cannot combine dissimilar units directly via root sum square: \[ u_{\varepsilon} \neq \pm \sqrt{u_{P}^{2} + u_{A}^{2} + u_{E_{m}}^{2}} \] since the measurands each have unique dimensions
   4. Variance of result is defined: \[ \sigma_{r}^{2} = \lim_{N\rightarrow \infty} \left(\frac{1}{N}\sum_{i=1}^{N} [r_{i} - r']^{2}\right) \]
   5. Use Taylor Series to expand arguments: \[ r_{i} - r' = \sum (x_{k,i} - x_{k}') \frac{\partial r}{\partial x_{k}} + \sum (x_{k,i} - x_{k}')^{2} \frac{\partial^{2} r}{\partial x_{k}^{2}} \ldots \]
   6. Assuming first first order of expansion and substitute variance definitions: \[ \sigma_{x_{k}}^{2} = \lim_{N\to\infty} \frac{1}{N} \sum (x_{k,i} - x_{k}')^{2} \] \[ \text{Covariance: } \sigma_{x_1x_2} = \lim_{N\to\infty} \frac{1}{N} \sum (x_{1,i} - x_{1}')(x_{2,i} - x_{2}') \]
   7. Combined result variance: \[ \sigma_{r}^{2} = \sum \left(\frac{\partial r}{\partial x_{i}}\right)^{2} \sigma_{x_{i}}^{2} + 2\sum \frac{\partial r}{\partial x_{i}} \frac{\partial r}{\partial x_{j}} \sigma_{x_{i}x_{j}} \]
   8. Replacing variance with uncertainty: \[ u_{r}^{2} = \sum \left(\frac{\partial r}{\partial x_{i}}\right)^{2} u_{x_{i}}^{2} + 2\sum \frac{\partial r}{\partial x_{i}} \frac{\partial r}{\partial x_{j}} u_{x_{i}} u_{x_{j}} \]
   9. Total Combined Uncertainty: \[ u_{r}^{2} = \sum_{i=1}^{J} \theta_{i}^{2} u_{i}^{2} + 2 \sum_{i=1}^{J-1}\sum_{j=i+1}^{J} \theta_{i}\theta_{j} u_{i} u_{j} \]

      Where senstivity coefficient is \( \theta_{i} = \frac{\partial r}{\partial x_{i}} \) in order to scale the measurand to the dimension of the result \(r\)

   10. Magnitude of contribution is product of both \( \theta_{i} u_{i} \)
   11. When independent sensors are used, no covariance exists making product of uncertainties: \( u_{i} u_{j} = 0 \)

5. Exercise \(\PageIndex{1}\)

   A project team was estimating major head loss along a pipe length using the fundamental equation: \[  h_{L} = \left(\mathcal{F}\frac{L}{D}\right)\frac{v^{2}}{2g} \] where the velocity was measured from a volumetric flow meter \(Q = v\frac{\pi D^{2}}{4}\). The following measurements and resolutions were collected:

   • Diameter \(D=0.75\) in with \(\frac{1}{16}\)inch resolution
   • Volume flow rate of \(Q = 65\) L/min with resolution of 5 L/min (necessary conversion 1 L = 61in\(^{3}\))
   • Gravity is estimated with resolution at 32.2 ft/sec\(^{2}\)
   • Pipe length was \(L=40.25\) in with same \(\frac{1}{16}\) inch resolution
   • Reynolds limit was reached so friction coefficient \(\mathcal{F} = 0.012\) was assumed a constant with no uncertainty

   Determine the head loss in inches and an estimate of the uncertainty. Which measurand has the least impact on the uncertainty?       

   Answer

   One could substitute for velocity right away or calculate uncertainty of \(Q\) separately; either way both will result in the same conclusion: \[ h_{L} = \left(\mathcal{F}\frac{L}{D}\right)\frac{8Q^{2}}{\pi^{2}D^{4}g} \]

   The uncertainty from each of the measurands is only the resolution:

   • \( u_{D}=u_{L} = \pm 1/32\) in
   • \(u_{Q} = \pm 2.5\) L/min
   • \(u_{g} = \pm 0.05 \) ft/sec\(^{2} \)

   The partial derivatives are then:

   • \(\frac{\partial h_{L}}{\partial L} = \left(\mathcal{F}\frac{1}{D^{5}}\right)\frac{8Q^{2}}{\pi^{2}g} = \frac{h_{L}}{L} \hspace{0.5in}\) 
   • \(\frac{\partial h_{L}}{\partial D} = -5\left(\mathcal{F}\frac{L}{D^{6}}\right)\frac{8Q^{2}}{\pi^{2}g} = 5\frac{h_{L}}{D} \) 
   • \(\frac{\partial h_{L}}{\partial Q} = 2\left(\mathcal{F}\frac{L}{D^{5}}\right)\frac{8Q}{\pi^{2}g} = 2\frac{h_{L}}{Q}\)
   • \(\frac{\partial h_{L}}{\partial g} = -\left(\mathcal{F}\frac{L}{D^{5}}\right)\frac{8Q^{2}}{\pi^{2}g^{2}} = -\frac{h_{L}}{g} \)

   After some conversion from liters to inches, etc, the total combined uncertainty then would be \[ u_{h_{L}} = \sqrt{\left(\frac{h_{L}}{L} u_{L}\right)^{2}+\left(5\frac{h_{L}}{D} u_{D}\right)^{2}+ \left(2\frac{h_{L}}{Q} u_{Q}\right)^{2} + \left(-\frac{h_{L}}{g} u_{g}\right)^{2}} \] with a numerical calculation of \[ u_{h_{L}} =  \sqrt{(0.014476)^{2}+(3.8845)^{2} + (1.4343)^{2}+(-0.028953)^{2}} \] with a final result of \(h_{L} = 18.6\pm 4.14\) in.