## **My Very First Experimental Report of a Pressure Sensor**

<div class="pull-center"><center><img src="https://steemitimages.com/DQmSYa23vsqSmLF9UzncwFbPipMwTrrYofWvGit9bsLRZ8j/image.png" /><br/><em><a href="">My System</a></em></center></div>

I am an undergrad studying [Chemical engineering](https://en.wikipedia.org/wiki/Chemical_engineering) and I thought I would share an interesting experiment that I conducted a few weeks back to elaborate the typical job of a student engineer. My experiment was to [calibrate](https://www.merriam-webster.com/dictionary/calibrate) and assess the [accuracy](https://en.wikipedia.org/wiki/Accuracy_and_precision) and [precision](https://en.wikipedia.org/wiki/Accuracy_and_precision) between 2 instruments found in the [laboratory](https://en.wikipedia.org/wiki/Laboratory).

## **Abstract**

<div class="pull-right"><center><img src="https://steemitimages.com/DQmRGuvYSbZsUQeitdi9KoFaFdeCKisYFbkdpVkhLoNswht/image.png" /><br/><em><a href="">A Reference Pressure Gauge Situated above the base of a Perspex cylinder</a></em></center></div>

The aim of this practical is to calibrate the water filled [U-Tube manometer](https://www.engineeringtoolbox.com/u-tube-manometer-d_611.html) and to assess the practicality of the instrument as well as a comparison of its [precision and accuracy](https://en.wikipedia.org/wiki/Precision) by making a few calculations and comparing those answers to a [reference pressure gauge](https://www.sensorsone.com/gauge-reference-pressure/). 

The U-tube manometer was filled using water due to water being both a practical and highly accurate fluid. Application of the manometer is used to evaluate the pressure experienced at a range of marked heights in the cylinder. A [hydrometer](https://en.wikipedia.org/wiki/Hydrometer) was used to evaluate the [density](https://en.wikipedia.org/wiki/Density) of the liquid which was then used for calculations (in Appendix B). A small error of 0.199% was recorded when comparing the literature value of the density to that of the recorded valued. 

All graphs plotted displayed strong [correlation coefficients](https://en.wikipedia.org/wiki/Correlation_coefficient) indicating highly accurate results recorded and calculated. It can be noted that the comparison between that of the calculated pressure data and the pressure gauge value had inaccuracies. This was due to error in reading of data, miscalculations or even flawed [experimental data](https://en.wikipedia.org/wiki/Experimental_data).

## **Introduction**

<div class="pull-centre"><center><img src="https://chemengineering.wikispaces.com/file/view/PressureChart.png/243433139/PressureChart.png" /><br/><em><a href="https://chemengineering.wikispaces.com/Pressure">Table Illustrating The Different Measurements of Pressure</a></em></center></div>

The formal definition of [pressure](https://en.wikipedia.org/wiki/Experimental_data) is the quantity of perpendicularly applied force to a surface. It can be solved algebraically using the equation,
P = F/A.cross sectional 

The [SI units](https://en.wikipedia.org/wiki/International_System_of_Units) of pressure is the [Pascal (Pa)](https://en.wikipedia.org/wiki/Pascal_(unit)) but it is often measured is many other units, for calculations purposes, which include [kilo-Pascal (kPa)](https://www.aqua-calc.com/what-is/pressure/kilopascal), [atmospheres (atm)](https://en.wikipedia.org/wiki/Atmosphere_(unit)) and even [millimeter of mercury (mmHg)](https://en.wikipedia.org/wiki/Millimeter_of_mercury)/torr. Measurement of pressure, in the chemical engineering world is very important and extremely significant as it is an important variable to consider in the designing of equipment. The study of pressure can be used to prevent many [catastrophes](https://www.vocabulary.com/dictionary/catastrophe) from occurring, especially on an engineering site, as it can be used to evaluate failing [equipment](https://en.wikipedia.org/wiki/Equipment) and even detect [explosions](https://en.wikipedia.org/wiki/Explosion) due to the study of pressure buildup.

 Two important pieces of equipment utilized in the study of pressure is a manometer and a pressure gauge. The manometer used in this practical is a water filled U-tube manometer and a pressure gauge, with a range of between (0-60) kPa were required. The main purpose of the manometer is to determine the pressure acting upon the stagnant fluid in the column, by making use of the resulting manometer arm height differences. The innovation behind the manometer was first progressed by[ E. Torricelli](https://en.wikipedia.org/wiki/Evangelista_Torricelli), an Italian scientist, in 1642. In the years to follow, the manometer was improvised, and adapted to become the versatile piece of equipment we use today, by scientists by the likes of [B. Pascal](https://en.wikipedia.org/wiki/Blaise_Pascal) as well as [C. Hygens](https://en.wikipedia.org/wiki/Christiaan_Huygens). A pressure gauge is an instrument used to measure the pressure in a specific area of a system and is compulsory in most faculties of engineering. 

The aim of this practical is to calibrate the U-tube manometer and contrast its accuracy and precision to that of a reference pressure gauge, situated above the cylinders base. The corresponding pressure drops at each height in the cylinder is equal to the product of the fluids density (ρ), the rounded constant for gravitational acceleration (g) and the difference in the manometer arm heights (Δh) into the following equation:                                
ΔP = ρgΔh

The practical consists of the calibration of the manometer (Part A) as well as the measurements at 3 randomly unmarked heights (Part B). 

## **Theoretical Background**

<div class="pull-right"><center><img src="http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/statics/pdefin.gif" /><br/><em><a href="http://www-mdp.eng.cam.ac.uk/web/library/enginfo/aerothermal_dvd_only/aero/fprops/statics/node13.html">Absolute Presure is the sum of Atmospheric and Gauge Pressure</a></em></center></div>

Pressure can be understood to be the force acting upon a surface, per unit of its area. Pressure can be split up into many aspects but we mainly refer to 2, [gauge](https://en.wikipedia.org/wiki/Pressure_measurement) and [absolute](https://physics.stackexchange.com/questions/20460/gauge-pressure-vs-absolute-pressur) pressure. Gauge pressure is referred to as the quantity or amount by which the measured pressure exceeds that of [atmospheric pressure](https://en.wikipedia.org/wiki/Atmospheric_pressure), which is evaluated at 101.325 kPa. Absolute pressure can be referred to as the summation of both pressure mentioned, ie, atmospheric and gauge:
P.absolute = P.atmospheric +  P.gauge 

Other types of pressure include [differential pressure](https://www.wika.us/solutions_differential_pressure_gauges_measure_a_difference_you_can_see_en_us.WIKA), [hydrostatic pressure](https://en.wikipedia.org/wiki/Hydrostatics) and so on.

**U-tube manometer:**
A U-tube manometer is a piece of instrumentation that consists of 2 arms with each of these arms exposed to a different pressure. In this practical, the left arm of the manometer experienced atmospheric pressure while the right arm of the manometer experienced a higher pressure than that of atmospheric pressure as this arm is connected to the [Perspex cylinder](https://www.dems.co.za/Engineering-Plastics). It can be noted that as the water level in the cylinder increased, the manometer height difference also increases. This difference is referred to as the [pressure head](https://en.wikipedia.org/wiki/Pressure_head) and can be used in the [hydrostatic equation](https://en.wikipedia.org/wiki/Hydrostatic_equilibrium) below, 

<div class="pull-center"><center><img src="https://steemitimages.com/DQmVL4HuyooNn3wfHYmhZyr4ySu3hgtVYELqu37xxM6XAeR/image.png" /><br/><em><a href=""></a></em></center>

Where:        
P – pressure [Pa]
 ρ – density of the fluid [kg/m3]
 g – acceleration of gravity [9.81m/s2]
 h - pressure head [m]

**The derivation of the hydrostatic equation is as follows:**
Consider a fluid with a [cross sectional area](https://www.quora.com/How-do-you-calculate-the-cross-section-area-of-cylinder) (A) in a vertical column, with a fluid height (h):
The [volume](https://en.wikipedia.org/wiki/Volume) (V) of the fluid can be calculated by:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmVv4xMW9jQQzFMuWMschHFkvtkJ28jC1SGGErSixygXKf/image.png" /><br/><em><a href=""></a></em></center>


The mass (m) of the fluid, by manipulation of the density equation is:


<div class="pull-center"><center><img src="https://steemitimages.com/DQmbKgZ732jmZDHBp7chccuYfqC2tvm3NzEU3Ks1CQSPPNc/image.png" /><br/><em><a href=""></a></em></center>

Substituting for V:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmWJADmutYuc7Kuu1R5z9jDSXpbEnrTMCmDdnRzWzwfAKs/image.png" /><br/><em><a href=""></a></em></center>
                       
The force (F), by [Newtons Second Law](https://www.physicsclassroom.com/class/newtlaws/Lesson-3/Newton-s-Second-Law), can be expressed as:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmbsyAVXdCnAXigDSmxYAEVHsFcnUvNfSnJqw7Znay7TPv/image.png" /><br/><em><a href=""></a></em></center>


Considering a strictly vertical system, a can be taken as the [gravitational acceleration constant](https://www.physicsclassroom.com/class/1DKin/Lesson-5/Acceleration-of-Gravity) (g):

<div class="pull-center"><center><img src="https://steemitimages.com/DQmdpiNUjuP184GvwBf2aft53FWxKGcuhJJ4Jp8uUuLtigX/image.png" /><br/><em><a href=""></a></em></center>

                              
By replacing m in the above equation yields:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmcMgx9Z2g5ehBq836BNKxTHXLy3WZ3DCGLbzTGhMoyChB/image.png" /><br/><em><a href=""></a></em></center>
                             
Note:                                                                     P = F/A
                                          
A substitution of F from the equation on the previous page yields:


<div class="pull-center"><center><img src="https://steemitimages.com/DQmfHcTcmdStjUJosWzLPNE1wUEduJhuGkUmPdgsudKiYDi/image.png" /><br/><em><a href=""></a></em></center>
                                    
And the removal of the cross sectional area through cancellation yields:


<div class="pull-center"><center><img src="https://steemitimages.com/DQmZ1cj4rBjmgLYfQbrgSjc1rvZ6hQf18Ce29JCuFmFuxXF/image.png" /><br/><em><a href=""></a></em></center>
    
This is the derivation ofthe Hydrostatic Equation.


**Hydrometer:**
The density a liquid can be obtained if the [specific gravity](https://en.wikipedia.org/wiki/Specific_gravity) of that liquid is known. Such an instrument that allows us to obtain a liquids specific gravity is referred to as a hydrometer. A hydrometer has an appearance of a thin, cylindrical glass tube with a bulb connected to one end of the cylinder. A dense material is generally used within the bulb. In the case of this practical, lead was used within the bulb. 

The use of density obtained from this instrument can be used to derive the [Bernoulli Equation](https://www.khanacademy.org/science/physics/fluids/.../what-is-bernoullis-equation) as it is assumed that the fluid has a constant density (which also indicates that we assume the fluid is [incompressible](https://en.wikipedia.org/wiki/Incompressible_flow)). The derivation is as follows:  

A mass element, m, [displaces](https://en.wikipedia.org/wiki/Displacement_(fluid)) itself (X) from position 1 to position 2:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmSkbKgH8PnwNQHgxyJzTvtCAqDKRDUx8ALaCXVYEqPqN2/image.png" /><br/><em><a href=""></a></em></center>
              
By use of density, the mass transfer can be written as,

<div class="pull-center"><center><img src="https://steemitimages.com/DQmTQkDZyLbrNzAB6G48dAiqQgr2ggByUAyoYqwhJmWv1L6/image.png" /><br/><em><a href=""></a></em></center>
as 
<div class="pull-center"><center><img src="https://steemitimages.com/DQmYfXDDvieUm6rnPNcJXbZSLWzUGK4dzYswZQGRJDfoG8R/image.png" /><br/><em><a href=""></a></em></center>

As the volume decreases the [acceleration](https://en.wikipedia.org/wiki/Acceleration) increases. The force difference will be given by:


<div class="pull-center"><center><img src="https://steemitimages.com/DQmaet4oTjXDNCLdzDoxEdeWF2VTuq9rYzVD8aKomhbpHr2/image.png" /><br/><em><a href=""></a></em></center>

The [Net Work](https://www.brightstorm.com/science/physics/energy-and.../work-energy-theorem/), through manipulation of the [Conservation Law of Energy](https://en.wikipedia.org/wiki/Conservation_of_energy) can be expressed as:                        

<div class="pull-center"><center><img src="https://steemitimages.com/DQmWYrhiLDawGp4XC8Bi8pzuXCtGABTvUyf9gomxiqL7dsg/image.png" /><br/><em><a href=""></a></em></center>
	     
By replacing m by ρ∆V, in ∆K and ∆U, we obtain:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmd5TK2LK6P8sFRcsbmpAdJ9Hf2JrtqZymKvzjjyKpq41Y/image.png" /><br/><em><a href=""></a></em></center>


Net Work therefore simplifies to:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmPqofJvNK8Er1881mYF3Tqjeh9qUkwUa7BGDLAF3LCBpH/image.png" /><br/><em><a href=""></a></em></center>
    
Since ΔV is common, it can be removed from the equation. The resultant equation is that of the Bernoulli Equation:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmPMxSLvjcX2T5cHQgUhESu3KgouzrBcGuoYDDr1HFnF1f/image.png" /><br/><em><a href=""></a></em></center>



**Pressure Gauge:**
Pressure gauges are used to measure the pressure at certain points in our system and can be assessed by comparison to that of calculated pressure if the manometer arm height difference is known. Pressure gauges are crucial components of most processing systems. In these environments, a pressure gauge needs to be reliable, accurate and easy to read to help prevent failure in everyday operations.




## **Apparatus**

<div class="pull-left"><center><img src="https://steemitimages.com/DQmbmKEoW3RK7u25BruETkSqGvAW1yEpkcNAbDhgqeV6HFt/image.png" /><br/><em><a href="">Emptying of a Dicharge Valve(connected to the  Perspex cylinder) into a jug</a></em></center></div>

The following set of apparatus were required to complete the practical:

* U-tube manometer (with liquid water inside). One end of the manometer will be opened to the atmosphere to experience atmospheric pressure while the other end will be connected to the latter end of the Perspex cylinder.

* A Perspex cylinder with a height of 1 m and an internal diameter of 0.1 m. The top of the cylinder is open to the atmosphere while at the other end (in line with the pressure gauge) will be connected to the U-tube manometer, mentioned above, via a tube made of flexible, plastic material. 

* A reference pressure gauge with a range of 0-60 kPa. It must be at the same level as the plastic tube connected to the cylinder from the manometer. 

* Hydrometer- an instrument used to measure the liquids specific gravity.


* [Thermometer](https://en.wikipedia.org/wiki/Thermometer)- an instrument to measure the temperature of the liquid. 

* A specialized meter ruler to add to accuracy of results when measuring heights.


* Equipment for filling and discharge of water:
  A jug, a bucket and a [discharge valve](https://www.valveandequipment.com/catalog/valves_Discharge_97.html) for the water.


## **Experimental Procedure**

<div class="pull-right"><center><img src="https://steemitimages.com/DQmVJDaiax9sqWkg6tncP5J1oSfA1ogfaBz5RNeCWGN972D/image.png" /><br/><em><a href="">An emptied Perspex cylinder to prevent contamintion of results</a></em></center></div>

As with any practical, it is imperative that all factors that could negatively affect the practical or contaminate our results, be dealt with before commencing the practical. We must therefore ensure that all/any water found inside the cylinder be removed by the use of the water discharge valve. Once the water is completely emptied from the cylinder, we close the discharge valve and check if there are any [leaks](https://en.wikipedia.org/wiki/Leak) in the system. This is done by observation of the manometer upon the addition of water to the cylinder. We are required to fill water to a level just above the pressure sensor and then note the manometers change in height. If a change in height occurs, as in, the levels in both arms of the manometer are not the same height, it can be concluded that there is no leaks in the system and the practical can continue.

At the outset, we are required to measure the temperature and the density by the use of the thermometer and the hydrometer respectively. Since the pressure sensor is not at the base of the cylinder, we need to measure the height of the sensor from the base to assist us with calculated pressures in our results for comparisons. Once these formalities are completed, we fill the cylinders to the respectively marked heights on the cylinder and record both heights on either arm of the manometer as well as the pressure reading on the pressure gauge for all marked levels. Once the last marked level is reached and all data is required, we use the discharge valve to decrease the water levels in the cylinder to the respective marked levels again, while recording the same data as before. Up till this point, is referred to as the [calibration](https://www.aicompanies.com/education/calibration/what-is-calibration/) part of the practical.

Once the last/minimum level has been reached, with all data recorded, we fill the cylinder up to 3 randomly unmarked heights and once again record the heights as well as the pressure reading. This part of the practical is referred to as the measurement aspect of the practical.

## **Results**

<div class="pull-center"><center><img src="https://steemitimages.com/DQmYB9VYMJWwdY2Vco2c1amXLi6ybyA3oCZzNgTM9tp99xs/image.png" /><br/><em><a href="">U-Tube Manometer with 2 distinct height differences, used for calculaions</a></em></center></div>

The initial conditions and preliminary measurements recorded before commencing the practical 
were tabulated in the table below.

| Factor | Recorded Result  | 
| -------- | -------- | 
| Atmospheric Pressure   | 100.75 kPa    | 
| Density   | 1000 kg/m^3   |
| Temperature     | 20.9°C |
| Inlet Height above Cylinder Base | 11.3 m    |
Table 1: Recorded Initial Conditions


The data that was recorded from the calibration part of the practical are tabulated below. These tables exhibit the results (pressure and manometer arm height differences) when adding water to the cylinder and when removing water from the cylinder respectively. Along with the measured data, I have included the calculated pressure, [=] kPa.



| Height of Liquid from Base (cm) | Manometer Height Difference (cm) | Calculated Pressure (kPa) | Gauge Pressure Reading |
| -------- | -------- | -------- | -------- |
| 25   | 6.5    | 0.63765    | 1.4 |
| 40   | 12.7   | 1.24587    | 2.9 |
| 55   | 17.4   | 1.70694    | 4.1 |
| 70   | 22.2   | 2.17782    | 6.0 |
| 85   | 26.7   | 2.61927    | 7.9 |
Table 2: Data obtained from adding water into cylinder/ increasing manometer heights as per conditions in Table 1

| Height of Liquid from Base (cm) | Manometer Height Difference (cm) | Calculated Pressure (kPa) | Gauge Pressure Reading |
| -------- | -------- | -------- | -------- |
| 85 | 26.7 | 2.61927 | 7.9 |
| 70 | 23.1 | 2.26611 | 5.1 |
| 55 | 17.5 | 1.71675 | 4.5 |
| 40 | 13.2 | 1.29492 | 3.0 |
| 25 | 6.9 | 0.67689 | 1.2 |
Table 3: Data obtained from removing water into cylinder/ decreasing manometer heights as per conditions in Table 1

| Height of Liquid from Base (cm) | Manometer Height Difference (cm) | Calculated Pressure (kPa) | Gauge Pressure Reading |
| -------- | -------- | -------- | -------- |
| 37.8 | 11.9 | 1.16739 | 2.6 |
| 57.5 | 18.5 | 1.81485 | 4.8 |
| 77.3 | 24.7 | 2.42307 | 6.9 |
Table 4: Data obtained from addition of water to 3 random heights as per conditions in Table 1


<div class="pull-center"><center><img src="https://steemitimages.com/DQmdBcENrBu2NocFes9pAG2Svwa4K4aDxui4ZvVNZyZ5iuT/image.png" /><br/><em><a href="">Figure 1: Calculated pressure compared to gauge pressure readings on a graph of pressure as a function of manometer height differneces</a></em></center></div>

## **Post Practical Report**

* Using the data from Part A to calibrate the manometer, we plotted a graph of manometer height difference (cm) vs calculated pressure:

The following graph was plotted from the obtained data in Table 2 and Table 3. A graph of manometer height differences, [=] m, was plotted against calculated pressure, [=] kPa. The calculated pressure values were obtained using the equation: 
P= ρg∆h


<div class="pull-center"><center><img src="https://steemitimages.com/DQmWkSU1LVN9QnKayctgqPTATcWwuxvMNXPopD29bHG1Z44/image.png" /><br/><em><a href="">Figure 2: Calculated pressure as a function of increasing as well as decreasing manometer heights </a></em></center>
    
    
It can be noted that due to R^2 (the correlation co-efficient) being almost perfectly [linear](https://www.investopedia.com/terms/l/linearrelationship.asp), we may conclude that the effects of [hysteresis](https://en.wikipedia.org/wiki/Hysteresis) does not apply to the pressure sensor and this therefore boosts the accuracy of our results.


* Using the calibration plot and the measurements from Part B, we were able to calculate the pressure for the three randomly selected heights.

<div class="pull-center"><center><img src="https://steemitimages.com/DQmb1qurk9vShsw7rmRZ4PSSsQbgzRtM4mYvfgUyzHVZ3Qz/image.png" /><br/><em><a href="">Figure 3: Calculated pressure as a function of increasing random manometer height difference</a></em></center>


The pressures at these points can be calculated using the equation mentioned the point above this and the calculations are as follow, 

1. 	Δh= 11.9 cm 
P =9.81 * ( 11.9/100  )= 1.16739 kPa

2.	Δh= 18.5 cm 
P= 9.81 * ( 18.5/100  ) = 1.81485 kPa

3. 	Δh= 24.7 cm
P= 9.81 * ( 24.7/100) = 2.42307 kPa

From our results obtained, and assuming that the tank is [closed](https://en.wikipedia.org/wiki/Closed_system) and [pressurised](https://study.com/academy/lesson/pressure-systems-types-effects.html), given certain additional information such as the [discharge velocity](https://en.wikipedia.org/wiki/Discharge_(hydrology)), we can calculate the required pressure to be exerted at any point in the column of water. The derivation of the equation used for this calculation is as follows:

By firstly using the [mechanical energy balance](https://www.engineeringtoolbox.com/mechanical-energy-equation-d_614.html), 

<div class="pull-center"><center><img src="https://steemitimages.com/DQmP4dUwGpMAJxnjzwVDvU6B2N5dUMfBbRjiBCUnx4AFmwR/image.png" /><br/><em><a href=""></a></em></center>

And then applying our assumptions to our system, which are:
* Incompressible Fluid (thus constant density)
* No [work](https://en.wikipedia.org/wiki/Work_(thermodynamics)) (example, shaftwork) or [friction](https://en.wikipedia.org/wiki/Friction) 
* Fluid flow is [turbulent](https://en.wikipedia.org/wiki/Reynolds_number), hence α=1 ,

the mechanical balance will reduce to that of the Bernoulli Equation:

<div class="pull-center"><center><img src="https://steemitimages.com/DQmc9vwUCg6f4kdb7Qbdz8E5N7JvRK6nRZzX14EJx6tgXzz/image.png" /><br/><em><a href=""></a></em></center>

We may now apply our knowledge of the [continuity equation](https://en.wikipedia.org/wiki/Continuity_equation) that states: 

<div class="pull-center"><center><img src="https://steemitimages.com/DQmXoQBKcXfudyuTGJEP5v19szGofZfjx9X4tPuKGsvH5J2/image.png" /><br/><em><a href=""></a></em></center>

Since *v*(1)>>*v*(2), it can be assumed that *v*(1)= 0. It should also be noted that since the end of the tank was exposed to the atmosphere, it can also be assumed that P(2) = P.atm = 100.75 kPa.

Thus, the final equation can be written as follows, 

<div class="pull-center"><center><img src="https://steemitimages.com/DQmZ3CS9UnVjLHmHbSYLrsCVS75cgtF1oKDv237d6iKSgRQ/image.png" /><br/><em><a href=""></a></em></center>

Where,

*v*(2) = 0.75 m/s
g = 9.81 m/s^2
Z(2) = Height of cylinder recorded [=] m
Z(1) = 0.113 m, height of pressure gauge outlet above base
Patm = 100.75 kPa, taken from Table 1
ρ= 1000 kg/m3

Z(2) will be the only factor that changes and thus becomes the [independent variable](https://www.thoughtco.com/definition-of-independent-variable-605238) with the calculated pressure been the [dependent variable](https://www.thoughtco.com/definition-of-dependent-variable-604998).


Using the above values and given a discharge velocity of 0.75 m/s, we obtained the pressures at each of the randomly selected heights. The final results after substitution are tabulated below while a sample calculation can be obtained in Appendix B. 



| Z(2) [=] m | P(1) [=] kPa  | 
| -------- | -------- | 
| 0.378     | 101.631     | 
| 0.573     | 105.544     | 
| 0.773     | 107.506     | 
Table 5: Corresponding pressures experienced at the surface of the liquid due to the height of liquid in the cylinder above its base.

* The density recorded during the practical, although may seem accurate, contains a certain [perchange of error](https://www.calculator.net/percent-error-calculator.html).Table 9 in Appendix C contained an accurate, tabulated range of densities at their corresponding temperatures from 0°C to 39.9°C, in 0.1°C increments. The table is evaluated by finding the temperature to its first decimal point and then reading off its density at that temperature. 

Since the temperature at the time of the practical was 20.9°C. By reading along the y-axis and finding 20°C and then reading along that row till the last increment of 0.9°C is reached (20+0.9=20.9°C), we obtain the literature density at that temperature. This value corresponds to 998.014 kg/m3. A comparison between the measured data from the hydrometer and the literature value from the table will produce a [deviation](https://www.dictionary.com/browse/deviation) of 1.986 kg/m3. This indicated an almost minimal experimental error of 0.199% (refer to Appendix B). Due to such a small experimental error, it is unlikely that this small deviation will cause an inconvenience to our calculations or will [contaminate](https://en.wikipedia.org/wiki/Contamination) our conclusions at a later stage.

It is, however, important to discuss reasons as to why this deviation has occurred as we must try to prevent these errors from occurring in the future as it could possibly lead to a contamination of results in the future. A few reasons for errors may include:
* Flawed or damaged instruments 	
* Changes in temperature in the working environment 
* Movement of hydrometer by physical interaction 
* Inaccurate reading of the hydrometer

These errors are often referred to as [systematic errors](https://www.statisticshowto.com/systematic-error-random-error/) and they are often unavoidable. Although these errors do not greatly affect our results, we can further reduce these errors by repeating the procedure.

## **Discussion**

<div class="pull-left"><center><img src="https://steemitimages.com/DQmVKrqpuSiSYbvFAn7KWGK5jxzxpR3mmxU56uZLnZYATup/image.png" /><br/><em><a href="">A Hydrometer</a></em></center></div>

The aim of this practical was calibration of a water fill U-tube manometer and to assess the instruments precision and accuracy to that of a reference pressure gauge with the help of certain specified apparatus.

**Hydrometer-**
A hydrometer is a glass tube made up of a thin cylindrical stem and a weighted bulb that can float vertically upright on the surface of a fluid and is able to measure the specific gravity of that fluid. The bulb of the hydrometer can either consist of lead or mercury. For this practical, the specification of the hydrometer is of a lead shot bulb.

Readings of the hydrometer was done when the [ambient temperature](https://www.dictionary.com/browse/ambient-temperature) was 20.9°C and atmospheric temperature of 100.75 kPa, produced a value of 1000 kg/m^3. The literature value at these conditions however indicated that we should have obtained a value of 998.014 kg/m^3. This produced an almost [infinitesimally small](https://www.thefreedictionary.com/Infinitesimally+small) error of 0.199%. It is important to note that although we may never be able to achieve 100% accuracy, the minor error that we incurred is unlikely to affect our results greatly, thus proving us with confidence in our data. 

Many reasons may occur that may have resulted in the minor difference between our actual density recorded and the literature value, such as:
(1)	Error due to misreading of hydrometer and thermometer.
(2)	Interference with hydrometer while attaining the waters density.
(3)	Hydrometer used could possibly have minor faults. 

* Calibration of the Manometer- 
The calibration of the manometer was obtained by the plotting the manometer arm height differences and the calculated pressures for the respective height differences. The calculated values for pressure are obtained by the use of the equation, ΔP = ρgΔh. 

 It can be noted that the graphs for both increasing and decreasing manometer height differences displayed strong linear relationships, as portrayed in Figure 2. It can also be noted that the reading from the pressure gauge is higher than that of our calculated pressures. There are many possible reasons for this deviation such as human errors in calculation, a malfunctioning pressure gauge or even improper calibration of the U-tube manometer.

## **Conclusion**

The aim of this practical was to calibrate the U-tube manometer and to assess its accuracy as well its precision through practicality. Through assumptions made as well as calculations, the following conclusions can be drawn:

* The obtained value of the density contained a small error percentage of 0.199%. This error is extremely small and unlikely to affect our results drastically therefore we were able to move along with the practical and calculations thereafter. 


* Calibration of the manometer was completed accurately as the graphs in Figure 2 almost overlap each other. This indicates that the hysteresis is extremely low and thus that the accuracy of the instrument is high, reconfirming that calibration was done correctly and accurately.

* From Figure 2, it can also be concluded that there is a linearly correlated relationship between the height difference of the manometer and calculated pressure. This is confirmed via the statistical coefficient, R^2, being almost 1 (refer to Figure 2).


* [Repeatability](https://en.wikipedia.org/wiki/Repeatability) of this practical was confirmed when a linear relationship was observed for the Figure 3.

## **Recommendations**

* If working in a goup, a single group member should be tasked with reading water temperature and density values. The reason for this is due to these [factors fluctuating](https://www.dictionary.com/browse/fluctuation) with time, therefore resulting in different values if another group member takes measurements later on.


* The temperature should only be recorded once “[steady state](https://https://physics.stackexchange.com/.../difference-between-steady-state-and-equilibrium)” is achieved at room temperature (ie) T.water = T.atmosphere. This will prevent any minor differences from occurring in temperature that might affect the waters density. 


* The temperature and density of the water must be recorded simultaneously to increase accuracy of results and [consistency](https://en.oxforddictionaries.com/definition/consistency).


* [Human error](https://en.wikipedia.org/wiki/Human_error), when reading data, is likely to occur. Therefore, implementation of electronic devices, such as an electronic pressure gauge and electronic thermometer, will increase accuracy of results. 

## **Appendix**

* Appendix A: Raw data

 INTIALLY MEASURED DATA:

Atmospheric pressure: 100.75 kPa
Water density: 1000 kg/m3
Temperature: 20.9 ̊C 
Manometer tube inlet height into cylinder: 11.3 cm 

 
 
 **CALIBRATION OF MANOMETER (PART A):**



| Height of liquid from base (cm)  | Pressure gauge reading (kPa) | Height of left arm of manometer (cm) | Height of right arm of manometer (cm) |
| -------- | -------- | -------- | -------- |
| 25     | 1.4     | 40.2 | 33.7 |
| 40     | 2.9     | 43.3 | 30.6 |
| 55     | 4.1     | 45.6 | 28.2 |
| 70     | 6.0     | 48.1 | 25.9 |
| 85     | 7.9     | 50.5 | 23.8 |
Table 6: Raw data recorded for increasing manometer height differences/ addition of water to the cylinder at the initial conditions mentioned in Appendix A and Table 1.


| Height of liquid from base (cm)  | Pressure gauge reading (kPa) | Height of left arm of manometer (cm) | Height of right arm of manometer (cm) |
| -------- | -------- | -------- | -------- |
| 85     | 7.9     | 50.5 | 23.8 |
| 70     | 5.1     | 48.2 | 25.1 |
| 55     | 4.5     | 45.6 | 28.1 |
| 40     | 3.0     | 43.7 | 30.5 |
| 25     | 1.2     | 40.4 | 38.5 |
Table 7: Raw data recorded for decreasing manometer height differences/ removal of water from the cylinder at the initial conditions mentioned in Appendix A and Table 1.

 MEASUREMENTS (PART B)

| Height of liquid from base (cm)  | Pressure gauge reading (kPa) | Height of left arm of manometer (cm) | Height of right arm of manometer (cm) |
| -------- | -------- | -------- | -------- |
| 37.8   | 2.6     | 43.0 | 31.1 |
| 57.5   | 4.8     | 46.3 | 27.8 |
| 77.3   | 6.9     | 49.3 | 24.6 |
Table 8: Raw data recorded for water levels at randomly selected unmarked heights in the cylinder at the initial conditions mentioned in Appendix A and Table 1.


* Appendix B: Sample calculations (in results)

**EXPERIMENTAL ERROR:**
Calculation of experimental error during the calculation of water density was conducted as follows, 

<div class="pull-center"><center><img src="https://steemitimages.com/DQmUwsutMVmUK2oYc3QjcH12m3hcxi8LeGn3SsL3FQFow4m/image.png" /><br/><em><a href=""></a></em></center>

 **CALCULATED PRESSURE:**

The sample calculations for the calculated pressures in Tables 2-4 were calculated as follows:

Let ∆h be the manometer height difference, g be the gravitational acceleration and ρ be the density of the water obtained from the hydrometer. 

For the calculation of pressure for an increasing manometer height difference of 25 cm (Table 2, row 1, column 3): 

<div class="pull-center"><center><img src="https://steemitimages.com/DQmQB2V3Nz1jYy65EcpojMGreXDQScBzLJDY1DEjfDY86fh/image.png" /><br/><em><a href=""></a></em></center>

This method is applied for all calculated pressure entries in Table2-4.

 **PRESSURE EXPERIENCED AT THE SURFACE OF THE LIQUID:**

All data is provided to us, we just need to substitute the appropriate values into the equation and we shall obtain the pressure experienced by the liquid at its surface. 

<div class="pull-center"><center><img src="https://steemitimages.com/DQmPAJuXv92iZ3i65xYcXyCyLkEcFewwpPKjrc2PKVp4KKV/image.png" /><br/><em><a href=""></a></em></center>

By use of this method, the corresponding surface pressures can be solved for all corresponding Z(2) values. 

* Appendix C: Calculation Data (density table)

<div class="pull-center"><center><img src="https://steemitimages.com/DQmc8LRJ6yJUMDdWyvBYvKnSvCViSAd7UDAra3iz1UQ1PRL/image.png" /><br/><em><a href=""></a></em></center>
Table 9: Table of density as a function of temperature for a range from 0°C to 39.9°C. The first vertical column are increments of temperature on 1°C. The first horizontal row are increments of 0.1°C. 

When reading a value, the whole number will correspond to the vertical axis while the increment (the decimal) will correspond to the horizontal axis. The value of the density will be obtained by the intersection of these values .

***Images 3/4 are referenced while all others were taken and done by myself***


***The End***

**References:**

[1]http://www.wika.co.in/landingpage_differential_pressure_en_in.WIKA
[Accessed 17 October 2017].

[2]https://encyclopedia2.thefreedictionary.com/manometer

[3]Pressure Sensing in a Liquid filled Column (2016)

[4]Geankoplis, C.J., Transport processes and Separation Principles (includes unit operations),4th edition, Pearson Education Limited, Harlow (2014)

[5]en.wikipedia.org/wiki/Hydrometer

[6]en.wikipedia.org/wiki/Pressure_sensor

[7]www.khanacademy.org/science/physics/fluids/.../what-is-bernoullis-equation

[8]www.engineeringtoolbox.com/water-density-specific-weight-d_595.html

[9]www.sensorsmag.com/components/manometer-basics