Methods - Penn Engineering
Aneurysm Experiment Done with Different Viscosity Sucrose Solutions
Group M7:
Stephen Waid
Henock Abebe
Kristin Bateman
Jason Yuen
December 16, 2004
ABSTRACT
The primary purpose of this experiment was to study the effects of viscosity on Reynolds Number for two different aneurysm models. The definition of NRe based on the length parameter b, the radius of the aneurysm, gave the best approximation for the onset of turbulent flow. The average NRe of the small and large aneurysm models using the length parameter b were 6693.96 + 3211.45 and 9906.26 + 651.87 across all solutions, respectively. In addition, the average NRe using parameter a/b (inlet flow tube radius to radius of aneurysm) for the large and small aneurysms were 16437.68 + 1019.53 and 60643.63 + 29093.98, which shows that the geometry of the aneurysm model has a direct affect on NRe. For large and small aneurysm models, Reynolds Numbers with varying viscosity were within 14% and 23.3% error, respectively. Thus, varying viscosity did not affect the Reynolds Number. The primary source of error in this experiment involved small fluctuations in the temperature which caused slight changes in the viscosity.
INTRODUCTION
The Reynolds number is calculated to determine the viscous effect of a solution in steady flow. The onset of turbulent flow can be determined by calculating the critical Reynolds number, which will yield values above 4000. Reynolds number values below 2000 represent laminar flow, while transitional flow is characterized by Reynolds numbers between 2000 and 4000.[1] The Reynolds number is inversely proportional to the solution’s viscosity, but proportional to flow rate. Viscosity is defined as a fluid’s resistance to flow[2]. Thus, as the viscosity of a fluid increases, the flow rate will decrease because there is a greater resistance to the flow.
One example of a practical application utilizing the calculation of Reynolds Number is an aneurysm. An aneurysm is an abnormal local swelling of a blood vessel that commonly occurs in the aorta. Ruptures of aneurysms, which often lead to death, can be a result of turbulent flow. Turbulence in the blood flow can weaken the aneurysm wall, which increases the chance of rupture of the aneurysm. Thus, calculating the onset of turbulent flow in aneurysms is key in studying their ruptures.
HYPOTHESIS
In this experiment, three definitions of NRe are used, based on different length parameters of b, a/c, and a/b, where a is the radius of the inlet and outlet tube, b is the radius of the aneurysm model, and c is the width of the aneurysm model. The variables a and c are constant for both models, whereas b varies for the large and small model. Therefore, the first hypothesis of this experiment is that the definition of NRe, based on the length parameter b, should give the best approximation of the onset of turbulent flow. Secondly, the larger aneurysm model will have a higher Reynolds Number than the small aneurysm model when based on the length parameter b. Thirdly, varying viscosity should not change the critical Reynolds number for a given model when calculated for a specific length parameter.
METHODS AND MATERIALS
Small and large aneurysm models were tested. The parameters for each model are shown in the following table:
|Model |Inner Tube Radius a (cm) |Aneurysm Radius b (cm) |Aneurysm Width c (cm) |
|Small |0.31 |1.85 |3.0 |
|Large |0.31 |4.35 |3.0 |
TABLE 1. Small and Large Aneurysm Model parameters.
A sample aneurysm model is shown in Figure 1. Each aneurysm model has three parameters: a, b, and c.
[pic]
FIGURE 1. Sample Aneurysm Model with parameters a, b, and c.
Using these parameters and measured critical flow rate, the Reynolds Number (NRe) can be calculated from the following formula:
NREcrit = [pic] Equation 1
[pic] = critical mean velocity through upstream and downstream tubes
[pic] = fluid kinematic viscosity of solution
[pic] = length parameter depending on calculation (b, a/b, a/c)
[pic]
FIGURE 2. Original aneurysm apparatus.
The original apparatus is shown in Figure 2. The experimental setup was modified accordingly due to the amount of sucrose solution that would be needed throughout the experiment. Thus, the system was changed. Instead of having the substance pass through the initial tube and collect in a bucket under the end tube, the system was changed to a recirculating system. This allowed for the use of a smaller tank size and a smaller amount of sucrose needed. The apparatus is shown in Figure 3.
[pic]
FIGURE 3. Water from the tank is pumped through the aneurysm and is collected back into the tank allowing for recirculation.
For this experiment, the chosen sucrose solutions were as follows:
|T = 25 degrees C | |
|% sucrose |density |viscosity |
|0 |997.0433 |0.857855 |
|10 |1036.735 |1.16675 |
|20 |1079.371 |1.710987 |
|30 |1125.197 |2.783864 |
TABLE 2. Sucrose viscosity for selected % sucrose solutions.
Depending on the temperature of the solution, accurate viscosity data is needed for calculations and is shown in the appendix. The chosen solutions were based upon previous groups’ experiences. The pump is not effective in pumping sucrose solutions of high viscosities. Based on this information, sucrose solutions up to 30% were chosen.
Solutions were made by percent weight. For example, a 30% solution would require 3 grams of sucrose for every 7 grams of water. Thus for a 1 kg solution, 300 grams of sucrose were mixed with 700 grams of water. Masses of each solution were kept constant at 1 kg. Solutions were mixed thoroughly before experimenting using a magnetic stirrer.
After pouring the solution into the tank, the pump was started and the aneurysm was filled with the solution, making sure that air bubbles were minimized. The syringe pump was set to 0.5 ml/min, which was determined through experimentation. Since the setup was modified to allow recirculation, injected dye was recirculated after each trial for a particular solution.
The flow rate was adjusted by turning the valve and the dye streamline was observed to identify turbulent flow. Using a ruler positioned at the center of the aneurysm model as the point of reference, the flow rate was adjusted so that the turbulent flow of the streamline would land at the reference point. After the turbulent flow matched the reference point, the syringe pump was stopped.
The critical flow rate was then measured using a graduated cylinder and stopwatch. Also, the temperature was recorded often so that the correct viscosity of sucrose could be used in calculations. The data was recorded and the experiment was repeated for 5 trials per sucrose solution. After each trial, the flow rate was reset by turning the valve down. This allowed for independent trials. The same procedure was then repeated for each solution. With every change of the solution except for water, the system was flushed out for accurate data. The experiment was repeated for both the large and small aneurysms.
RESULTS
Calculations of Reynolds Numbers using Equation 1 were analyzed for the type of flow. As stated above, Reynolds Numbers below 2000 and above 4000 were considered laminar and turbulent flow, respectively. For the aneurysm models, laminar flow was present through inflow tube (parameter a). The inflow tube NRe of the small and large aneurysm models are 1121.69 + 538.13 and 705.96 + 46.46 across all solutions, respectively. Turbulent flow was observed in the aneurysm (parameter b) and also in the outflow tube (parameter c). The average NRe of the small and large aneurysm models using b are 6693.96 + 3211.45 and 9906.26 + 651.87 across all solutions, respectively. Finally, the outflow tube NRe of the small and large aneurysm models are 10855.06 + 5207.75 and 6831.90 + 449.57 across all solutions, respectively.
The difference of the aneurysm radius directly affects the Reynolds Number at the onset of turbulence. Comparing both models for parameter b, Table 3 shows that the Reynolds Numbers for the large aneurysm model are greater than the small aneurysm model by almost a factor of 2, except for the 30% sucrose solution. This suggests that as the radius of the aneurysm increases, the flow becomes more turbulent.
|Small Aneurysm Model | |Large Aneurysm Model |
| |Parameter |b | | |Parameter |b |
|% Sucrose |Avg NRe |Std | |% Sucrose |Avg NRe |Std |
|0 |4560.25 |366.75 | |0 |10717.57 |401.20 |
|10 |5125.22 |383.63 | |10 |10095.53 |408.27 |
|20 |5623.49 |236.15 | |20 |9205.76 |610.42 |
|30 |11466.87 |631.40 | |30 |9606.16 |470.15 |
TABLE 3. Calculated Reynolds Numbers for parameter b for both aneurysm models.
Analysis of the Reynolds Numbers on the ratio of the inflow tube and aneurysm radius was performed to determine its effect by the geometry of the aneurysm models. In Table 4, NRe of the small aneurysm are greater than the large aneurysm because of a higher ratio of a/b. Thus, based on the geometry of the aneurysm, as the ratio of a/b increases, NRe increases proportionally. The values of NRe for the large and small model using parameter a/b, as shown in Table 4, are shown to be significantly different (p Ql
Us > Ul
ls < ll
Based on the results of Qcrit between the two models, it is expected that NRe values calculated using parameters a, c, and a/c should be larger for the smaller model, where a and c are the interior radius and aneurysm width, respectively. This is because a and c are equivalent between the two models, and thus at a constant sucrose concentration and similarly constant viscosity, the NRe will be proportional to the critical flow rate. Since the results show that the small model had higher average flow rate at each sucrose concentration except 0%, NRe values should similarly be higher in the small model. This was seen in Table 5, where average NRe is larger at each viscosity in the small model by the same factor as in average critical flow rate (Table 6).
Viscosity of solutions should not affect the Reynolds Numbers significantly. This is consistent with the data based on the range of error, which proves the third hypothesis. When the viscosity of the solution increases, data shows that the result deviated from the previous trial for the small model, but not by a substantial amount. Data for the small aneurysm were within 23.3% error. For the large aneurysm model, deviations were much smaller and within 14% error.
There are a number of errors in this experiment which may have altered our results, and as such, there are improvements that can be made to help future lab groups obtain more accurate data and enhance their learning experience.
A major source of error in this experiment involves the temperature of the solution during each trial. Before each trial, an accurate temperature reading is necessary for calculations. With the exact temperature recording, the appropriate viscosity can be used for that solution. To decrease variability in this experiment, the large and small aneurysms should be set up simultaneously. This would allow for each particular solution trial to be performed at a similar temperature. This would give a more constant temperature for the solution for each model and would improve our ability to accurately compare the data.
In this experiment, the solution was recirculated through the model for each trial. With dye being injected into the solution to study the onset of turbulence, the dye was mixed with the solution and recirculated. The dye may have affected the viscosity of the solution and hence, our calculations of the onset of turbulence. Although its effects may have been minimal, a small change in viscosity might alter calculations of the Reynolds Number and flow rate. One possible improvement would be to remake the dye for each trial so that the viscosity of the dye would equal the viscosity of the particular solution for each trial. Time limitations may prevent this improvement from being implemented, but this suggestion would give more accurate data for the experiment.
A third improvement to this experiment would be to use a stronger pump. Using the current pump prevented viscosities above 30% from being studied, and in addition, data from the 30% viscosity solution seemed inaccurate due to its inconsistency. Implementing this improvement would allow future lab groups to study Reynolds numbers and flow rates at viscosities above 40% and obtain more accurate data for the 30% viscosity solution.
The pump was also responsible for another inconsistency in the experiment. While flushing out the system, tubes sometimes disconnected which resulted in a loss of solution. The pump also introduced air bubbles during the runs, which would interrupt the flow and collect in the aneurysm model. The air bubbles' effect on the flow, combined with the other errors involved in the experiment, resulted in slight errors in the calculated critical Reynolds number. Thus, a possible improvement to the experiment would be to use a pump that would reduce the production of air bubbles.
As mentioned above, these improvements may decrease the variations in the small aneurysm model, which were generally larger than the large aneurysm model. Different sources of error such as air bubbles affect the small aneurysm model more directly than it would affect the large. This is due to the geometry of the model.
The conclusion that varying viscosities of sucrose does not affect the Reynolds Numbers can be applied towards blood flow in body. Blood viscosity varies throughout the body due to factors such as changes in red blood cell properties. Based on our experimental conclusion, it can be assumed that blood flow through fusiform aneurysms with equivalent dimensions will have similar onsets of turbulence regardless of such variations in blood viscosity. Determination of onset of turbulence is of critical importance because turbulent blood flow causes formation of dangerous emboli. Blood clots can block smaller arteries such as those that supply the kidneys or other organs, leading to permanent damage to these organs.[3] When applying the models to actual physiological properties, the Large Aneurysm Model can be applied to fusiform aneurysms in the descending thoracic aorta and the Small Aneurysm Model can be applied to berry aneurysms. In the comparison of NRe between the models (for parameter b), it was found that the small aneurysm model had lower NRe values. The application of this conclusion is seen through the treatment of berry aneurysms. Blood flow in berry aneurysms is less turbulent and this explains why they are not likely to rupture and may not require treatment.[4]
APPENDIX
|T = 20 degrees C | | |T = 23 degrees C | |
|% sucrose |density |viscosity | |% sucrose |density |viscosity |
|0 |998.2031227 |0.966271342 | |0 |997.5372 |0.898553 |
|10 |1038.115089 |1.32407981 | |10 |1037.314 |1.225671 |
|20 |1080.977467 |1.967028255 | |20 |1080.037 |1.806487 |
|30 |1127.032457 |3.262490567 | |30 |1125.95 |2.961306 |
| | | | | | | |
|T = 21 degrees C | | |T = 24 degrees C | |
|% sucrose |density |viscosity | |% sucrose |density |viscosity |
| |997.9913786 |0.942742003 | |0 |997.2951 |0.877791 |
| |1037.856975 |1.289835906 | |10 |1037.029 |1.195591 |
| |1080.67176 |1.911020053 | |20 |1079.708 |1.757674 |
| |1126.678443 |3.157016655 | |30 |1125.577 |2.870443 |
| | | | | | | |
|T = 22 degrees C | | |T = 25 degrees C | |
|% sucrose |density |viscosity | |% sucrose |density |viscosity |
|0 |997.7693191 |0.920187027 | |0 |997.0433 |0.857855 |
|10 |1037.589728 |1.257060478 | |10 |1036.735 |1.16675 |
|20 |1080.358134 |1.85755607 | |20 |1079.371 |1.710987 |
|30 |1126.317749 |3.056730075 | |30 |1125.197 |2.783864 |
-----------------------
[1]
[2]
[3]
[4]
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Flow Valve
Pump
Dye w/ Syringe Pump
Aneurysm Model
Flow Valve
Pump
Dye w/ Syringe Pump
Aneurysm Model
Water Tank
b
c
a
a
a
................
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