# 07 Uncertainty Analysis

## Elemental Uncertainties
Uncertainty due to elemental contributors, $u_i$:
\begin{align}
u_{e} = \sqrt{\sum_i u_i^2}
\end{align}

begin{align}
u_{overall} & = \sqrt{\sum_{comp} u_{comp}^2}
\end{align}

### Pressure Regulator Calibration 

It is desired to use an Equilibar pressure regulator QPV in gas flow experiment.  The specifications for this regulator are:
(https://www.equilibar.com/electronic-pressure-regulators/)

> accuracy = .25%FS

> repeatability = 0.02%FS

However, the flow meter accuracy is too low for our application and one decides to calibrate it with a flow calibrator that has an accuracy of 0.04 % of FS.

> What will be the accuracy of the calibrated instrument?


### Pressure Measurement

A pressure transducer is connected to a digital data-acquisition board on a computer.  The characteristics of the pressure transducer and the data-acquisition system (DAS) are given below.  All uncertainties are given at 95 % confidence level.  
The environment temperature is uncertain by $\pm 10^\circ$C.


> Pressure Transducer Characteristics: 

Range: 0-1,000 kPa 

 Output: 0-5 V 

 Linearity: $\pm 0.25$ %FS 

 Repeatability: $\pm 0.06$ %FS 

 Thermal-stability error: $\pm 0.01$ %FS/$^\circ$C  

> DAS Characteristics:

 Number of bits: 10 bits

 Input range: $0-10,\, 0-1,\, 0-0.1,\, 0-0.01$ V

 Gain error: $\pm 2$ LSD

 Linearity: $\pm 2$ LSD

Overall Uncertainty for measurement: Combination of Pressure Transducer and DAS

\begin{align}
u_{overall} & = \sqrt{\sum_{comp} u_{comp}^2} \\
& = \sqrt{u_{PT}^2 + u_{DAS}^2}
\end{align}

Uncertainty of presssure transducer
\begin{align}
u_{PT} = \sqrt{\sum_i u_i^2}
\end{align}
with $u_i$ elemental error:
> linearity: $\pm 0.25$ %FS 

> repeatability: $\pm 0.06$ %FS 

> Thermal stability: $\pm 0.01$ %FS/$^\circ$C

LSD: Least Significant Digit

01101: binary word, LSD: bit associated with $2^0$ and MSD (most significant digit): bit associated with $2^{N-1}$ for $N$ bit word.

ie value of the first bin in your digitized word/measurement.

number of bins: $2^N$, $N$: number of bits

size of a bin: resolution, $\Delta V = \frac{V_{max} - V_{min}}{2^N}$


## Uncertainty of a result

\begin{align}
R = f(x_1, x_2, \dots) \quad u_R = \sqrt{\sum_i \left( u_{x_i} \frac{\partial f}{\partial x_i} \right)^2}
\end{align}

### Design of experiment

One wishes to determine the heat transfer coefficient, $h$, of a heated rod of length, $L$, and diameter, $D$, in a flow. By applying the first law of thermodynamics one can relate the input power of the experiment, $W$, to
$h$ by:

\begin{align*}
h = \frac{W}{\pi D L \Delta T}, \textrm{where $\Delta$T is the temperature difference between the rod and the surrounding fluid}
\end{align*}

The relative uncertainties are: $\frac{u_L}{L}=0.1\%$, $\frac{u_D}{D}=0.2\%$, $\frac{u_{\Delta T}}{\Delta T} = 1.0\%$.  One wishes to have a maximum relative inaccuracy of 2% of $h$.

What should be the relative accuracy of the power measurements, $u_W/W$?



```python

```

b2a7d1f40a1301669f4ea033de824b1d44790613
ipynb
Jupyter Notebook
Lectures/07_UncertaintyAnalysis.ipynb
bardet/GWU-MAE3120_2020
0b783b0550edd153231d19460c7575a2aa0a0675
BSD-3-Clause
2019-12-12T02:24:13.000Z
2020-01-17T14:18:28.000Z
Lectures/07_UncertaintyAnalysis.ipynb
bardet/GWU-MAE3120_2020
0b783b0550edd153231d19460c7575a2aa0a0675
BSD-3-Clause
Lectures/07_UncertaintyAnalysis.ipynb
bardet/GWU-MAE3120_2020
0b783b0550edd153231d19460c7575a2aa0a0675
BSD-3-Clause
2019-12-18T15:56:16.000Z
2021-06-02T01:13:22.000Z
Qwen/Qwen-72B
1. YES
2. YES