How Is the Pressure of a Fluid Affected by the Length of the Vessel Through Which It Is

Rhys Carroll

How is the Pressure of a Fluid Affected by the Length of the Vessel through which it is Flowing?

INTRODUCTION:

As blood travels throughout the body in arteries and veins, there is a resulting decrease in pressure due to the flow resistance of theblood vessels. This is of particular interest to me as it allows me to create links between physics and the medical field in which I hope to enter after high school. There are many variables that can affect this loss of blood pressure, but the primary factors impacting this resistance to flow are: vessel diameter, vessel length and viscous resistance. Poiseuille’s law relates the pressure drop of a fluid to the length of a pipe with constant diameter[1], and will be the topic of this investigation. These theories are significant in many aspects of physics beyond just medical sciences; for example, the energy loss in transporting water for hydro-power plants must be taken into account when determining efficiency. Poiseuille’s Law states that: (see page 6 for a further explanation)..

Before carrying out the investigation, it is important to understand the different concepts involved with Poiseuille’s law, including the following definitions. This experiment will use Poiseuille’s law to investigate the drop in pressure of the fluid in a pipe. Fluid pressure is the kinetic energy per unit volume of a fluid[2]. According to Poiseuille’s Law, one variable affecting the pressure change is dynamic viscosity; a measure of internal resistance, or the force per unit area needed to move a fluid.[3]

When studying the flow of a fluid, it is possible to simplify the analysis by making some assumptions about the properties of that fluid. In this case, Poiseuille’s law is most evident under certain conditions; the fluid involved is incompressible and undergoes steady uniform flow. An incompressible fluid is one that, under constant conditions, can be assumed to have constant density. Steady, uniform flow means that the conditions of the fluid (velocity, pressure and cross-section) do not change with the position in the stream or with time.[4] Under these conditions, the analysis of viscous resistance becomes possible as there are no other factors that could impact the flow.

However, in spite of these assumptions, Poiseuille’s law only takes effect when the fluid involved is a Newtonian fluid with laminar flow. A Newtonian fluid is one who’s viscosity is only dependent on temperature; therefore, it is independent of shear stress. This is important in terms of Poiseuille’s law because it means that the viscosity of the fluid is not impacted by the velocity with which it travels through the pipe.[5] Laminar flow is a type of fluid flow with a smooth and regular path with properties similar to that of uniform flow.[6]

Hypothesis/Prediction:

Based on this information, a prediction can be formed that there will be a linear relationship between the length of the tube, and the pressure loss of the fluid. Therefore, if the length of the pipe is doubled, the pressure loss with also double.

INVESTIGATION:

The aim of this investigation is to use Poiseuille’s Law to determine the relationship between the length of a pipe and the pressure loss of a fluid flowing through it. To accomplish this, a water syphon will be used to create a water flow, and the length of the pipe will be altered in order to investigate the water pressure. A bike pump with a reverse valve will be used to create suction and start a flow of water from the reservoir into the secondary bucket. This set up is ideal because it creates a constant water flow that is only dependent on the properties of the pipe; preventing interference of any external factors.The flow rate of the water in the syphon will be calculated by measuring the volume of water transferred from the reservoir in 10 seconds. This raw data will then be manipulated using Poiseuille’s law to calculate the loss in fluid pressure.

Assumptions:

Water is an incompressible fluid under the conditions in the syphon
The water will have steady, uniform flow in the syphon
The pipe is manufactured with a constant diameter throughout
The interior of the pipe is completely smooth so that it has a minimal impact on the flow.
The initial flow caused by the pump does not affect the flow rate in the syphon

Independent Variable:

The independent variable being manipulated in this experiment is the length of the pipe used in the water syphon. Five different lengths will be used decreasing by increments of 0.2m: 1m, 0.8m, 0.6m, 0.4m and 0.2m. By using this range, the results from the experiment should clearly show the relationship between the length of the pipe and the pressure loss.

Dependent Variable:

The dependent variable of this experiment is the flow rate of water through the pipe, and will be measured in m3/s. The pressure loss through the pipe will then be calculated using Poiseuille’s law.

Control Variables:

There are a number of variables that could affect the outcome of the experiment and therefore must be controlled in order to successfully test the independent variables previously mentioned:

Height between the reservoir and transfer bucket: The height from which the fluid is transferred in the syphon can have an effect on the flow rate, and therefore the data collected. Because of this, it must be kept constant during the whole experiment and is set at 0.300m +/- 0.002m (see Figure 2)
Diameter of the pipe: the diameter of the pipe is one of the major factors affecting the resistance to flow. For this experiment, it must remain constant in order to accurately study the impact the length of the pipe has on that resistance. This is controlled during the experiment by using the same typepipes all from one manufacturer with a diameter 0.032m
Type of liquid used in the syphon: different liquids have different properties – such as viscosity – which could affect the flow rate through the syphon. This will be controlled by using the same liquid (water) for all the tests. Water was chosen because it is a Newtonian fluid, and it is easily accessible in large volumes.

Apparatus:

The apparatus required for the experiment are as follows:

32mm diameter Pipes: these must all have constant diameter and be a range of lengths (previously specified). They are used to create the syphon; the pipes on the left must be 0.300m longer than those on the right (see figure 3) to maintain a consistent height between the reservoir and the transfer bucket.
32mm diameter U-joint: this is used to create a 180 degree turn while maintaining a constant diameter.
Silicon: Used to seal the connection between pipes to prevent air escaping which would result in a loss of suction
2 buckets: one to act as the reservoir and one to transfer the water into (see Figure 2). These buckets must be able to hold at least 12 litres of water, and the reservoir bucket must have a scale to measure the volume of water.
2 clamps and stands (1m): this is used to hold the syphon in a vertical position over the buckets during the experiment.
Adjustable platform: this is used to elevate the reservoir bucket 0.300m above the transfer bucket
Bike pump with reverse valve: this is used to create suction to induce the initial flow of water in the syphon. The diameter of the hose should be just under 32mm so that it fits snugly into the end of the pipe.
Measuring tape (1mm increment): this is used to measure the length of the tubes etc.
Water supply: this will need to be easily accessible to keep filling up the buckets for the syphon
Stopwatch: this is used to measure the amount of water transferred over a specific period of time
Pencil and Paper: this will be used to record the results during the experiment which can later be transferred onto a computer for analysis.

Preliminary:

Before carrying out the actual investigation, a small preliminary experiment was completed in order to confirm that the chosen method was effective, and to ensure the proposed experiment was valid. From these tests, a number of potential issues were discovered that needed to be addressed. First of all, it was determined that solid PVC pipes were more suitable for this experiment that soft tubing due to the fact that the diameter of flexible tubing changed when it was bent. Furthermore, it became apparent that the experiment would need to be carried out on a relatively large scale in order to see any significant effect on the water pressure in the syphon. This meant that large amounts of water (12l) would be needed in order to accurately measure the flow rate. Certain alterations to the method itself had to be made. Both ends of the pipe must be fully submerged to prevent any air from entering the syphon and reducing the suction. Finally, these trials revealed that it would be most effective to measure the volume of water leaving the reservoir than the volume being transferred to the transfer bucket.

Method:

Assembling the syphon:

Place the two buckets side by side, one elevated 0.3m above the other using the adjustable platform. (see figure 3)
Arrange the stands so that the clamps are positioned above the buckets.
Attach the first length of pipe (1m) to the U-joint with one end in the reservoir bucket and the corresponding length of pipe (0.3m longer) in the transfer bucket.
Using the clamps, hold the syphon in a vertical position above the two buckets – the end of the pipe should be almost touching the bottom of the reservoir bucket

Carrying out the experiment:

Fill the reservoir bucket (elevated) with 12 litres of water using the scale on the inside of the bucket.
Fill the second bucket (on the ground) with enough water so that the end of the pipe is submerged.
Place the end of the bike pump’s hose into the pipe.
Create suction using the bike pump, and remove the hose once the water is flowing.
Start the timer as soon as the water starts flowing
After 10 seconds, lift the pipes out of the water to stop the flow
Note down the water level in the reservoir bucket (this will later be used to calculate the amount of water transferred in 10 seconds)
Repeat steps 1-7 until you have three repeats with the first length of pipe.
Repeat steps 1-8, changing the length of pipe after three trials until all 5 different lengths have been tested. (you should have 15 water levels recorded)

Risk Assessment:

Despite the fact that there are no hazardous materials or actions required for this experiment, there are still dangers that must be taken into account

During the experiment, buckets containing large volumes of water may need to be manually handled. Due to the heavy weight of these objects this should be done with care to prevent injury or spilling of water. Only move heavy objects when absolutely necessary, and ask for the assistance of another person if required. Serious injuries are unlikely, however seek medical attention if necessary. This is a low risk danger due to the marginal consequences and relatively low likelihood of an incident.

Furthermore, there is the risk of falling objects during the experiment, such as the bucket falling off the platform. This should be avoided by using a flat surface and ensuring all platforms are stable before putting any weight on them. If an incident were to occur, do not attempt to catch the falling object in order to avoid injury. Again, this is a low risk scenario due to the minor consequences and unlikelihood of an occurrence.

Due to the large volumes of water involved with this experiment, there is a risk of electrocution. To avoid such a scenario, the experiment should be carried out outside, away from any electrical ports. If an incident involving electricity were to occur, seek medical assistance immediately. Despite the critical consequences of an incident, the remote chance of an occurrence means that this is a low risk scenario.

Slips, trips and falls are a commonly overlooked danger due to their seemingly trivial consequences, however they can nonetheless result in injury. This should be avoided by clearing the work space of all obstacles, and clearing up any spills whenever possible (understanding that this experiment involves a large volume of water). If injury does occur, seek medical assistance if deemed necessary. Due to the marginal consequences and likelihood of occurrence this danger is medium risk.

DATA COLLECTION:

Raw Data:

The water level in the reservoir was measured before and after the water syphon was allowed to flow for 10 seconds. This data was then used to find the total amount of water transferred in 10 seconds.

Processed Data:

The raw data collected during the experiment (water transferred in 10 seconds) was used to calculate the flow rate in m3/s for each different length of pipe. These results were then manipulated using Poiseuille’s law to calculate the pressure in Pascal’s.

Determining Flow Rate:

To convert the amount of water transferred in 10 seconds into volumetric flow rate, the volume of water must first be converted into meters cubed, and then divided by change in time.

For example: in the first trial with a 0.400m pipe

The total water transfer in 10 seconds was 9.6 litres. Therefore, the water transfer in meters cubed . Finally, the volumetric flow rate = volume of water/time it flowed = 0.00096 m3/s

Calculating Pressure Loss:

Poiseuille’s Law states that [7]

(2.s.f)

[8]

): the dependent variable (see table 2)

kept constant at 0.032m

Rhys Carroll

ANALYSIS AND JUSTIFICATION OF DATA:

Analyzing the data, it is clear that the pressure loss through the syphon decreases as the length of the pipe increases. With a pipe length of0.2 meters, the average pressure loss was 530 Pascal, but when the length was increased to 1 meter, the pressure loss dropped to an average of 410 Pascal. This is a range of 120 Pascal over the 0.8 meter change in length. Unfortunately, this contradictsmy prediction that the pressure loss increases with the length of the pipe. Furthermore, the relationship between the two variables is not linear as stated in the hypothesis; as the length of the pipe continues to increase, there is a smaller change in pressure. The graph shows that there is a more significant change in pressure with the shorter pipes than there is with the longer pipes. For example, when the length of the pipe increases from 0.2m to 0.4m the pressure loss decreases by 60 Pascal. However, when the length increases from 0.8 to 1m, the pressure loss decreases by only 10 Pascal; just 1/6 of the previous change. The original hypothesis states that “there will be a linear relationship between the length of the tube, and the pressure loss of the fluid.” In reality, the data collected in the experiment shows that the actual trend is the complete opposite. The line of best fit reveals that there is an inversely proportional relationship between the pressure loss of the fluid and the length of the pipe: for example with a 0.2 m pipe, the pressure loss was 530 Pa; .

Despite the inaccuracy of my hypothesis, the data collected by the experiment can be justified using Poiseuille’s law. The resistance to flow of a tube is generally explained in terms of the viscosity of the fluid. With laminar flow, the more viscous the fluid; the more resistance there is from the pipe. According to Poiseuille’s law, if the viscosity of the fluid –in this case water – is constant, then the pressure loss caused by the resistance to flow must be a result of the properties of the pipe. Therefore, because it was the only variable that changed during this experiment; the length of the pipe must be what caused the different flow rates.[9] Looking at the results from the experiment, the flow rate decreased significantly as the length of the pipe increased: a total change of 0.00027m3/s. This change is not due to the larger mass of water being moved through the syphon (there is an equal mass on either side so it balances out), but rather due to the increase in friction caused by the viscous resistance. If the water has to flow through a longer pipe than more friction will be created, resulting in a decreasein the flow rate.[10] Furthermore, the decrease in pressure loss can also be explained by closer examination of Poiseuille’s law: . If the flow rate (Q) decreases, this would result in a decrease of pressure loss (ΔP); this result is clearly shown by the graph. The gradual decrease in rate of change of the graph shows that beyond a certain length the pressure change will remain constant and can no longer be explained using Poiseuille’s law as it is not laminar flow.

In conclusion, the data collected from this experiment shows that the pressure loss through a pipe of constant diameter is inversely proportional to the length of that pipe. Increasing the length of the pipe means the pressure change will decrease, however there is a limit to how far the length will have an effect on the pressure.

EVALUATION:

Systematic Errors are “errors which affect all measurements alike, and which can be traced to an imperfectly made instrument or to the personal technique and bias of the observer.” While Random Errors are “errors for which the causes are unknown or indeterminate, but are usually small and follow the laws of chance. Random errors can be reduced by averaging over a large number of observations.” (Department of Physics and Astronomy, n.d.)