Steady Flow Through Models of Cylindrical Arterial Stenoses
BE 310
Final Project Paper
Steady Flow Through Models of Cylindrical Arterial Stenoses
Group T4:
Libby Bucholz
Mina Wu
Christopher Hack
Monday, May 21, 2001
Abstract:
Six cylindrical stenoses were created to test the relationship between four unitless parameters; Euler’s number, Reynold’s number, and two length parameters defined as L1 equal to the ratio of the diameter of the stenosis to the diameter of the upstream/downstream tubing and L2 equal to the ratio of the diameter of the stenosis to the length of the stenosis. It was found that the important variables in determining Euler’s equation were the L2 parameter and Reynold’s number. L1 did not have a significant effect because there were too few trials. Euler’s number was found to be inversely proportional to Reynold’s number and inversely proportional to the L2 parameter. By experimental means Euler’s number was predicted by the following equation:
Euler’s Number = (0.046+/-0.007)/L2 + (2140+/-120)/Re – (0.7123+/-0.39)
Background:
A stenosis is defined as the narrowing of an artery as a result of some type of tissue accumulation. The average blood velocity across a stenosis is higher than that of the average blood velocity in the upstream and downstream arteries. The pressure drop per unit length across a stenosis is higher than that of the pressure drop per unit length for the upstream and downstream tubing. Assuming the heart can pump blood at the same flow rate as before, it will have to work harder to account for the increased change in pressure, and as a result the blood pressure of arterial stenoses patients is usually larger than normal subjects. Cylindrical tubing is often used to simulate arterial vessels and stenoses alike, as it provides a similar geometric model.
Boundary Layers
In many real flow systems in which a fluid flows through a circular conduit, frictional factors need to be considered. As a result of such forces, a boundary layer forms on the surface of the tube and its thickness increases with its increasing respective distance from the entrance of the conduit. A boundary layer forms on the inside surface and occupies a greater portion of the flow area as its distance downstream from the entrance increases. At some given downstream distance, the boundary layer fills the flow area completely. At this point, the velocity profile ceases to change downstream thus earning the name of fully developed flow. The distance form the pipe entrance to downstream where the flow is considered fully developed, denoted Le, is the called the entrance length. In order to satisfy continuity, the fluid velocity outside the boundary layer must increase as the downstream distance form the entrance increases. The velocity increases in such a way that at the center of the pipe it reaches a value of 2v∞ for fully developed laminar flow.
The entrance length required for a fully developed velocity profile to form in laminar flow:
(Le/D) = 0.0575Re
Where D is the inside diameter of the tube.
This relationship, however, holds only for laminar flow. There is no established relation to relate entrance length to turbulent flow due to the nature of the entrance itself. Diessler, however, came up with a general relation that the turbulent velocity profile becomes fully developed after a minimum distance of 50 diameters downstream form the entrance.
The same entrance distance expressed by method of Bernoulli's equation is found to be the laminar development length, XL, for the flow to become fully developed:
XL = 0.03NrD
Where Nr is the Reynold's number.
Methods and Materials:
Assorted tubing, pipettes and connectors for creating stenoses
Glue
Water tank
Flow valve to normalize flow
Evans Blue Dye
Dye injection machine
Syringe
Graduated cylinders
Stop Watch
Rulers
A water tank was situated at a certain height above the rest of the experimental equipment. A flow stabilizer was placed to reduce the error from variable flow that occurred from the water tank. The stenoses were situated between a monometer that measured the pressure drop across it. The dye injector machine was used to stabilize the flow of the dye such that the point of turbulence could be determined (in theory). Four stenoses were created with defined length parameters. A point that was past the stenosis was marked so that if the flow became turbulent at the point, the flow was determined to be turbulent but if it was not, even if flow further down the tube becomes turbulent the flow was defined as not turbulent. This standardization technique attempted to eliminate the human error in judgment of turbulence, which was a major source of error in previous labs. A graduated cylinder collected the flowing fluid and was timed so that the flow rate could be determined. A picture of the experiment is located below:
Figure 1: Experimental Design
A monometer was used to determine the pressure drop across the stenoses and accompanying pipe. Rulers were used to attain the difference in height. Before the experiment could be created, however, the geometry of the stenoses had to be precisely determined. The unitless length parameters needed to be found, and they are pictured below:
Figure 2: Definition of length parameters
Two independent unitless length parameters were found using the Buckingham Pi theorem. Since the units represented a space of size m=3, and the total number of varying parameters were n=7, the number of unitless parameters found was 4, Reynold’s number, Euler’s number, L1 equal to the ratio of the diameter of the stenosis to the diameter of the upstream/downstream tubing and L2 the ratio of the diameter of the stenosis to the length of the stenosis.
In all, six stenoses were created having the following actual lengths and unitless length parameters:
Table 1: Stenoses dimensions and dimensionless parameters used
As seen from the above table, stenosis 1 and 2 have the same length parameters as do stenosis 5 and 6. Stenosis 3 and 4 were completely independent, as they did not have partners.
In the beginning it was planned that two solutions would be used, 5% by mass sucrose and water, which varied the density of the solution and the kinematic viscosity. While showing that the experiment takes viscosity and density into account in the unitless parameters is important, our lab thought it was more important to tests more stenoses to gain more insight into the relationship between the two uniltess length parameters and how they relate to Reynold’s number and Euler’s number. The densities and viscosities, along with their calculated kinematic viscosities are located below for both solutions, but only water was used:
Table 2: fluid properties
The important equations are as follows:
Poiseuille’s Law
P = 128 QL/(D4) (eq. 1)
Euler’s Equation
Eu = P/(U2) (eq.2)
Reynold’s Equation
Re = UD/ (eq.3)
Boundary Layer Equation
XL = 0.03Re x Ds (eq.4)
Bernoulli’s Equation
P/(g) +{1/(2g)}(V22 – V12) + (z2-z1)+hL =0 (eq.5)
Pressure drop in boundary layer as related to Poiseuille's flow:
(eq.6)
There were two parts to the experiment. One was to determine if given the fact that the two length parameters, L1 and L2 remain the same, to see if Euler’s number and Reynolds number could be plotted together. The other two stenoses that did not contain a pair that kept L1 and L2 constant should not follow this same curve as they had geometrical differences that were not accounted for in Reynold’s number alone. Finding the function that related these parameters was the main goal of the experiment. For this portion, multiple flow rates were taken and air bubbles were minimized. It was important that we attained a large number of small Reynold’s numbers as well as several large Reynold’s numbers to produce a nice spectrum for the Euler’s plot. Given the flow rate, and the change in pressure, Euler’s number verses Reynold’s number could be plotted as well as the change in pressure verses the Reynold’s number, which displayed the actual expected flow rates with the experimental flow rates.
The second part of the experiment was to determine the critical velocity that causes turbulence at the line located in figure 1, object G. This part of the experiment was dropped as it produced incredibly unreliable results even if the same person was used to determine turbulence. The monometer would get in the way of allowing the dye to pass and would consequently weaken the appearance of the dye and make it impossible to determine turbulence from laminar flow.
Results:
Euler’s number was plotted verses Reynold’s number for all the stenoses and the following graph resulted:
Figure 3: All stenoses Euler’s verses Reynold’s number
The blue line is meant to indicate the trend line for stenosis 5 and 6 and the red line is meant to indicate the trend line for stenosis 1 and 2. What is noticeable from the graph is that stenosis 3 and 4 show little variance from stenosis 1 and 2, but stenosis 5 and 6 are clearly and significantly different. It was not possible to draw a realistic trendline for stenosis 3 and stenosis 4 as they are simply too close to the trend line for stenosis 1 and 2.
A multiple linear regression analysis was performed using 1/Reynold’s number, 1/L1 and 1/L2 to estimate Euler’s number. A linear regression was also performed using the same parameters stated above except that L1 was used to fit Euler’s number and not 1/L1. The results are as follows:
Graph 2: Euler fitted by 1/Re, 1/L1 and 1/L2Graph 3: Euler fitted by 1/L1 only
The R2 value for the predicted verses actual for Euler’s using 1/L1 was .9234, whereas without it the R2 value was .9233. Therefore the inclusion of 1/L1 was shown to be mathematically irrelevant in the prediction of Euler’s number. When L1 itself was used the same results were found, that the L1 parameter had little benefit in predicting the Euler’s number of the system.Using multiple linear regression, the experimental equation predicting Euler’s number is presented below:
Euler’s Number = (0.046+/-0.007)/L2 + (2140+/-120)/Re – (0.7123+/-0.39)
Keeping Reynold’s number and L1 constant, the following graph was created for Euler’s number verses L2 parameter:
Graph 4: Euler’s number verses L2 keeping other variables constant
As can be seen by the rough linear approximation, L2 is inversely proportional to Euler’s number. The linear regression should be discounted but the line shows the negative slope that is significantly different from zero. Two more graphs of this nature are presented in the appendix (Graph A1 and A2) to demonstrate the inverse relationship between Euler’s number and the three dimensionless parameters.
Graph 5: Pressure drops of Poseuille flow, Boundary Layer, and Experimental Results
The pressure drop for a developing boundary layer flow was determined by equation 6. As the graph shows, the Boundary Layer pressure drop was greater than that for Poiseuille flow. The experimental pressure drop was the largest. One of the possible reasons for the experimental pressure drops being greater than the predicted boundary layer pressure drops is due to the contribution of Poiseuille flow upstream of the stenosis. Because of the placement of the manometer, we could not eliminate the pressure drop within the upstream tubing of the stenosis. Therefore, the experimental pressure drop is greater than predicted due to this additional factor.
Discussion:
The experimental data in Figure 3 demonstrates the relationship between Euler’s Number and all the three other length parameters (Reynold’s Number, L1, and L2) to be inversely proportional in all three cases. The inverse relationship between Euler’s Number and Reynold’s Number can easily be seen by noting that the curves for all 6 stenoses fell rapidly, resulting in decreasing values for Euler’s Number as the Reynold’s Number increased. The effect of the other two dimensionless parameters, L1 and L2, on Euler’s Number can be determined by observing the shift in curves as these two length parameters are varied. Due to experimental uncertainty, only two significantly different curves could be constructed using the experimental data. Stenosis 1 and 2, which had the same L1 and L2 values, but different actual length dimensions (as described by Table 1), fell on the same red curve. The fact that both data sets for these two stenosis fell on the same curve indicates that Euler’s Number is dependent on the length parameter values, and not on the actual length characteristics. Stenosis 5 and 6, which also had the same L1 and L2 values also collapsed on one, single curve (blue curve). However, this curve was significantly different from the red curve of Stenosis 1 and 2 because the two curves had different L2 values. L2 = 0.0625 for the red curve and L2 = 0.125 for the blue curve, while L1 was the same for both curves (L1=0.5). The red curve, which has a lower L2 value than the blue, was positioned above the blue curve. This signifies that for a given Reynold’s Number and a L1 value, decreasing the L2 parameter increases Euler’s Number, resulting in an inversely proportional relationship between these two parameters. The results for Stenosis 3 and 4 could be not used to determine or support the relationship between the parameters, due to the fact that both data sets fell too close to the red curve for Stenosis 1 and 2. This is logically unreasonable since Stenosis 3 had a different L2 from the red curve and Stenosis 4 had a different L1 from the red curve. Due to the fact that Stenosis 4 fell on the red curve despite its different L1 value, the relationship between Euler’s Number and L1 parameter could not be established using the data in Figure 3. This trend in our data can be attributed to experimental uncertainty, limitations in apparatus (the collection of available tubing was not enough to construct more stenoses of varying L1 values), and the narrow range of Reynold’s Number of the experimental data. Figure 3 shows that all the data for Stenosis 3 fell in the region where Re<1000 while for Stenosis 4, all the data fell in the region where Re>1000. If a wider range of Reynold’s Number for both Stenosis 3 and 4 data were available, more variation would be found between the curves.
Graph 4 and Graph A1 & A2 verify the same relationship using different analysis techniques. In these graphs, two of the four parameters were fixed constant so that the relationship between the remaining two varying parameters could be determined. In Graph 4, L1 and Reynold’s number were kept constant and the relationship between Euler’s Number and L2 yielded a down-sloping curve. The negative slope of the graph indicates an inverse relationship, but the linearity of the slope should not be trusted due to the paucity of data. Similar analysis for graph A1 & A2 yielded the inverse relationships between Euler’s number and L1 and between Reynold’s number and L2, respectively.
The general relationship between the four dimensionless parameters using the experimental data, can be derived theoretically by applying Bernoulli’s Equation (Eqn 5)
to the region, shown in the following diagram, where the inlet is located right before the onset of stenosis and the outlet is positioned near the end of the stenosis.
Assuming that there is no change in height, the flow is in steady-state, there is no accumulation within the control volume, no viscous effects, no shaft work, and no heat being added or generated, Bernoulli’s equation simplifies down to the following form:
P/(g) = HL = (L/D){U2/(2g)}
This is Equation 5.23 of the paper found at the end of Experiment 2 in the lab manual. The entrance and exit velocities were determined using conservation of mass, with the known cross-sectional areas of the blood vessel and the stenosis. Rearranging this term yields:
Eu = P/ (U2) = 1/2(L/D)
Where symbolizes the Darcy friction factor. The relationship between the Darcy friction factor and Reynold’s number can be seen to be inversely proportional to the Reynold’s number, as shown by Figure 5.17 in the same paper. This inverse relationship between and Reynold’s number holds for both Poiseuille and turbulent flow as can be seen by the down-sloping trend of all the curves for the entire Reynold’s number region. Since Euler’s number is shown to be directly proportional to the friction factor by the equation above, then Euler’s number should be inversely proportional to Reynold’s number, due to the inverse relationship between and Reynold’s number. This is conclusion is supported by our experimental data. The equation above also shows that Euler’s number is directly proportional to (L/D) which is the inverse of the L2 parameter. Therefore, Eu Number is inversely proportional to L2 according to the equation and as supported by our experimental data.
Graph 2 is a multi-linear regression graph, showing the correlation between the experimental data and the results as predicted by the theoretical equation shown above. The R2 = 0.93 signifies a high degree of correlation between the experimental and theoretical results, and indicates that the derived equation fits the experimental data very well.
The relationship between Euler’s Number and the L1 parameter is less concrete due to a lack of experimental data. Graph A1 shows an inverse relationship between the two parameters based on two data points only. Graph 3 is a multi-linear regression of Euler’s versus the L1 parameter, yielded low correlation and a weak relationship. Therefore, no concrete conclusion can be drawn concerning the effect of L1 on Euler’s Number. In order to investigate the relationship between Euler’s Number and L1, more data needs to be collected of more stenoses with varying L1 values (but with the same fixed L2 value). However, intuitively, the relationship between Euler’s number and L1 should also be inversely proportional. If the diameter of the stenosis become infinitesimally small, relative to the diameter of the blood vessel (as measured by L1), then the pressure change should become infinitely great, resulting in a large Euler’s number. Euler’s number should increase as L1 decreases, indicating an inverse relationship.