INOM
EXAMENSARBETE TEKNIK,
GRUNDNIVÅ, 15 HP
STOCKHOLM SVERIGE 2016,
The shape transformation to a
circular form of a fluid jet exiting a non-circular orifice of a nozzle
IDA BRILAND AND REBECKA DANIELSSON
KTH
SKOLAN FÖR INDUSTRIELL TEKNIK OCH MANAGEMENT
Abstract
Nozzles are used in a wide range of applications. Nevertheless, the geometric of non-circular orifices have not been widely studied. This project has examined fluid jets exiting through a non-circular orifice, in the gravitational direction.
Furthermore, its transformation to a circular cross-section due to a surface tension forces. How the length to a circular cross-section changes with the nozzles geometry and bath depth of the tundish was the main focus of this
studied. A water model and high-speed camera was used to capture the profile of the fluid jet. Four different nozzles were attached one by one to five different tundishes with different bath depths. The result showed that with deeper bath depths the circular cross-section occurred further down from the nozzles orifice.
The length to the circular cross-section also depended on the orifice area, a larger area gave a longer distance than a smaller area. It was shown that the length to circular cross-section followed a quadratic function, when the
measured values were analyzed based on the Weber number. The profile of the fluid jet was dependent on the material of the nozzle, the geometries of the orifice, the bath depth and the surface tension.
Table of content
Introduction ... 1
1 1.1 Nozzles ... 2
1.2 Jet Profile ... 2
1.3 Velocity Parameters ... 3
1.4 Model Medium ... 4
Experiment ... 5
2 2.1 Nozzles ... 5
2.2 Water Model ... 7
2.3 Picture Analyze ... 8
Results... 10
3 Discussion ... 15
4 4.1 Result ... 15
4.2 Experiment Issues ... 18
Conclusion ... 19
5 Future Work... 19
6 Reference ... 21
7 Appendices ... 22 8
1 Introduction
1
A fluid jet exiting a non-circular orifice of a nozzle, which is only affected by gravity, will transform itself to a circular shape due to surface tension forces [1].
This kind of transformation is illustrated in figure 1. The transformation can behave in different ways. The fluid jet can either take a circular form and keep it until breakup into droplets, or as it occurs in many elliptic orifice conditions where the jet takes on an axial-switching phenomenon. When axial-switching the jet alternates between non-circular and circular form due to inertial forces [2].
Figure 1 Change in the cross-sectional shape of a jet exiting from an elliptic nozzle [2].
The profile of which a jet exiting a non-circular orifice takes is dependent on the properties of the fluid such as the surface tension, dynamic viscosity and density as well as the shape of the nozzles orifice. Other parameters that have an effect on a fluid jet includes the changes of fluid flow affected by cavitation in the nozzle, turbulence at the nozzle orifice and the jet velocity profile. However, these are difficult to analyze. Behavior of fluid jets exiting a non-circular orifice is not well documented and literature on the subject is limited [2].
This thesis will focus on the first part of the profile as when a non-circular fluid jet takes its first circular form. More precisely, the length of which it takes the jet to obtain a circular form as a function of two dependent parameters. These are the depth of the tundishes (which is the main property that affects the velocity in this case) as well as the geometry of the nozzle orifice. These parameters have been analyzed as variables by performing water model experiments with nozzles of four different dimensions. More details about the dimensions will be further specified in part 2.1. This work has focused on experiments that compare rectangular orifices with rounded edges and a rectangular orifice. The depth of the tundishes that has been investigated in this project was between 11,4-24,8 cm. The profile of a liquid jet can be calculated by using a one-dimensional Cosserat continuum model, which was showed by Amini–Baziani [2] to give
Chapter 1 I ntroduction
19
Figure 1.5: Change in the cross-sectional shape of a jet exiting from an elliptic nozzle (From Taylor 1960).
that surface tension causes a periodical transformation, and in the absence of surface tension this transformation occurs only once, after which the cross-section continues to flatten. Oscillations occur due to the competition between the surface tension that tries to restore the cross-section to a circular shape and the inertial force that tries to change the shape of the cross-section to an ellipse.
Figure 1.6: Axis-switching of an elliptic jet; a) major axis view, b) minor axis view (From Amini &
Dolatabadi 2011c).
a)
b)
Circular cross-section
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sufficient result for jets exiting elliptic orifices. However, this is outside the scope of this thesis.
1.1 Nozzles
In many different processes nozzles are used to regulate the flow of different substances, liquids or gases. They are attached e.g. to pipes, hoses and tundishes where the liquid or gas substances are released [3]. In steel processes a nozzle is attached to a thundish which the steel goes through when it is transferring to the mold, for example in continuous casting [4]. Nozzles can control the pressures on the substances and the direction or characteristics of the flow. Moreover, they are often used to control the speed, mass and shape of the flow that runs through it [5]. There exists different nozzles for different applications [3] and these can have different orifices. A rectangular orifice is preferred in hypersonic ground test facilities and in a flight vehicle. This is because if a circular nozzle is used it will result in a highly three-dimensional flow, which is not preferable for the quality of the flow [6]. The use of a non-circular orifice can be valuable as a jet exiting an asymmetric orifice may have a shorter distance to breakup than one exiting a symmetric orifice. An elliptic orifice could for instant have beneficial aspects for usage in spraying applications such as pesticide spraying, spray- painting and liquid propellant rocket injectors [2].
What kind of orifice geometry that is best to use is hard to tell and this has been discussed in different reports. The Argonne National Laboratory has performed simulations on nozzles to find out the effect of the orifice geometry on the flow.
Through cavitation simulations they observed that cylindrical nozzles were cavitating with patterns throughout the whole nozzle. However, with conical nozzles the cavitation was almost constrained. They also observed that exit density in the cylindrical nozzle was 4 percent lower and that it had a higher turbulence than that of the conical nozzle [7].
1.2 Jet Profile
Due to surface tension forces, the jet will go from an elliptic to a circular cross- section as to minimize the surface energy. At the surface of a liquid the molecules have fewer close bonding neighbors than the bulk molecules have. This results in that the surface molecules will obtain an extra energy, which in the case of
liquids that interface towards air will be a positive energy. The positivity of the surface energy density is what makes the liquid take a shape that gives the least surface area as to minimize the positive energy, which naturally is a circle [1].
This was confirmed by Kasyap [8] who saw that fluid jets exiting elliptic orifices either form circular cross-sections or display an axis-switching phenomenon depending on the different Weber number (We) ranges. Here, the Weber number
3
is a dimensionless number that is relevant to use in analyzing fluid jets as it takes surface tension into account. It can be expressed as follows:
𝑊𝑒 =𝜌𝑢2𝜎𝐷𝐻= 𝑖𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒
𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑡𝑒𝑛𝑠𝑖𝑜𝑛 𝑓𝑜𝑟𝑐𝑒 (1)
Where ρ is density, u is velocity of the liquid at the orifice exit, DH is the flow geometrical scale, and σ is the surface tension [9].
For Kasyap’s study jets with Weber number in the range of 0,86-1,49 showed no axis-switching phenomena and quickly degenerated to a circular jet, the minor axis expanded at the same time as the major axes contracted. Jets with higher Weber number values than 1,49 showed an axis-switching phenomenon, as shown in figure 2. The elliptic jet first degenerated to a circular form in the same way, as explained previously. However, due to a lateral inertia the jet shape transition had difficulty to stop. Therefore, the jet became an inverted elliptic shape. This phenomenon was repeated over and over [8]. As shown in figure 2, axis-switching will cause the jet to develop more than one circular cross-section.
Figure 2 Axis-switching of an elliptic jet; a) major axis view, b) minor axis view [2].
1.3 Velocity Parameters
In the experiment part of this project the geometry of the orifice and the tundish depth were continuously changed and the results were compered. Both the geometry and the tundish depth are parameters that are relevant for the fluids velocity at the orifice [10]. As discussed earlier, the velocity is a parameter that changes the profile of the fluid jet. As for example the higher the velocity the shorter distance to breakup [11]. That will say that the higher the velocity the more unstable the jet will be.
In the case of a fluid running through a nozzle with no pressure differences between the inlet and outlet, the pillar of fluid above will provide a pressure on the liquid at the orifice. The greater the tundish depth the higher pressure at the orifice and higher the velocity will be, as described by Bernoulli’s equation [10]:
Chapter 1 I ntroduction
19
Figure 1.5: Change in the cross-sectional shape of a jet exiting from an elliptic nozzle (From Taylor 1960).
that surface tension causes a periodical transformation, and in the absence of surface tension this transformation occurs only once, after which the cross-section continues to flatten. Oscillations occur due to the competition between the surface tension that tries to restore the cross-section to a circular shape and the inertial force that tries to change the shape of the cross-section to an ellipse.
Figure 1.6: Axis-switching of an elliptic jet; a) major axis view, b) minor axis view (From Amini &
Dolatabadi 2011c).
a)
b)
Circular cross-section
4
𝜌𝑢02
2 + 𝜌𝑔𝑧0+ 𝑝0 = 𝜌𝑢212+ 𝜌𝑔𝑧1+ 𝑝1 (2)
Therefore the bath depth will probably have an effect on the transformation of the jet. Higher bath depth should result in longer distances for the flow to achieve a circular form.
1.4 Model Medium
Studying a jet flow profile through a model can be done with different mediums.
The choice of medium to model a phenomenon with will for one thing have an impact of the smoothness of the flow. A liquid with a higher viscosity than tap water will have less or even no ruffling on the free surface of the liquid and therefore it can be easier to analyze. To increase the viscosity of water, for example, a mixture of water-glycerol could be used [8].
To use a model for a prototype it is important to have dynamic similarity
between them. As an example, water can in some cases be a good liquid to model steel. Both liquids have the same properties at some temperatures and therefore they have a dynamic similarity. Instead of having steel with a temperature between 1560-1600°C you can use water. This is due to the fact that water at 20°C and steel at 1560-1600°C have approximately the same kinematic viscosity [12]:
𝜐𝑠𝑡𝑒𝑒𝑙 = 0,93 ∙ 10−6 [m2/s] (1560-1600°C) 𝜐𝑤𝑎𝑡𝑒𝑟 = 1,0 ∙ 10−6 [m2/s] (20°C) However, steel has higher surface energy than water. Therefore, if a water model is going to be used to simulate steel the surface energy for the water has to be changed. One simple way of doing this is to add salt in the water, which will increase the surface energy of the water. Also, a somewhat constant temperature is preferred due to the changing of the surface energy with the temperature. If the temperature is lower, the surface energy will increase and vice versa [13]. In this report, cold tap water was used due to limitations of the experiment setup.
Also, the results were analyzed with Weber number to give the research a wider use, as the result can be used when a dynamic similarity condition is met and for instant surface tension differences can be overlooked.
5 Experiment
2
The experiment for this project was divided up in three different parts. The initial part was to establish a set-up for the water model, gather all needed equipment and creating nozzles. There after the modeling took place as the second part and in the third part the result were analyzed.
2.1 Nozzles
Prior to the water model experiments two of the four nozzles, which were used in the water model, were 3D printed using the machine Ultimaker2 extended with the filler PLA, polylactic acid. The two other nozzles that were used for the
experiments were ceramic nozzles made of zirconium oxide that was supplied from a collaborating company. The printed nozzles had orifices that were rectangular (PL-RE) and rectangular with rounded edges (PL-EL), as seen in figure 3. The ceramic nozzles had orifices that were rectangular with rounded edges with different areas, CE-SM (the one with the smaller area) and CE-LA (the larger area), see figure 4.
Figure 3 3D printed nozzles. Left has rectangular orifice with rounded edges and right has rectangular orifice.
6
When using Ultimaker2 extended a virtual model of the nozzle is required in the format STL. In this experiment the program Solidworks was used to model the nozzles. An example of the model sketch is given in appendix A.
When the virtual model was ready, a USB were used to transfer the STL file to the Ultimaker2 extended where the printing took place. By choosing a high quality profile, the printing took 6 hours per nozzle. To save time, one of the nozzles was printed using normal quality profile (took about 3,5h). At the normal quality the meshing inside the nozzles walls was a bit coarser then the high quality because of the difference in layer height. Normal quality layer height was 0,1mm and high quality was 0,06mm. An aspect in the printing is that it scaled down the dimensions, so the finished nozzles were smaller than the dimensions in the model sketch. The dimensions of the orifices are shown in table 1.
Table 1 Dimensions of the nozzles.
Nozzle Length, l (mm)
Width, w (mm)
Area, A (mm2)
DH= (l+w)/2 (mm)
PL-EL 24,0 4,0 96 14
PL-RE 25,0 4,0 100 14,5
CE-LA 26,0 4,5 112,65 15,25
CE-SM 25,5 3,5 86,62 14,5
Figure 5 Zirconium oxide nozzles. Left has larger orifice area and right has smaller orifice area.
Figure 4 Zirconium oxide nozzles. Left has larger orifice area and right has smaller orifice area.
7 2.2 Water Model
The experiment was preformed using a water model that was setup as shown in figure 5. Five different buckets with different bath heights (d=11,4cm, 13,7cm, 18,1cm, 20,5cm, 24,8cm) worked as tundishes. To keep a constant bath height, water was continuously added thorough a hose and a water faucet. The excess water could poor out through a cut in the side of the bucket at the given bath height. The hose was attached so the outlet was close to the bottom of the
bucket. This was to prevent swirls in the water and to prevent the pressure from the hose to affect the outlet at the nozzle. When a steady flow through the nozzle was achieved and the bath height was constant, a high-speed camera was used to capture the flow from two angles. These were the angles that showed the orifice width and length.
Figure 5 Experiment setup.
A MotionBLITZ Cube4 – High-Speed Recording Camera was used in the
experiments. The cameras 1280(H) x 1024(V) resolution allows the user to get 1000 fps, but this can be increased to 93 000 fps by using a reduction of the area of interest [14].
On each of every bucket four different setups were done with the four different nozzles. This resulted in 20 different setups. With the help of the high-speed camera, the axial switch area where the stream had a circular cross-section became possible to capture, as shown in figure 6. As well as capturing the jet profile, the volume flow of the jet was measured by timing how long it took to fill up a 500ml-measuring jar.
8 2.3 Picture Analyze
Following Matlab codes that are mentioned can be seen in appendix B.
With five different buckets and four different nozzles 40 different photos were obtained from the water model experiments. These photos were printed out on A4 papers. To get the actual measurements the diameter of the nozzle in the picture as well as the real diameter were measured to get the scale difference.
The circular cross-sections of the flow were then found by comparing the width and the related length pictures were the jet had identical diameter (D). Based on the picture scale the real length to circular cross-section (L) could be calculated.
See figure 7 for clarification. This resulted in 20 different values.
Figure 6 Example showing position of length and diameter for picture analyze.
Figure 6 Picture example of the flow from CE-LA nozzle, width and length, at a bath depth of 13,7 cm.
7 7
9
The 20 values were plotted with the program Matlab R2015a [15] with one curve per nozzle, which had five different values. This is due to the five different bath depths. The same was done for creating a graph for the length related to the velocity at the orifice. The velocity was calculated from the measured volume flow Q as follows:
𝑄 = 𝑢 ∙ 𝐴 , 𝑣 =1𝑄
⇒ 𝑢 =𝑣𝐴1 (m/s) (3)
Additional steps to process the data out of the experiments as to create a function were taken. The Weber number of the fluid jet for the different setups was calculated according to equation 1. Here the literature value that was used for surface tension of water was 0,073 (N/m) and for the density it was
103 (kg/m3) [8]. The velocity was the same as previously mentioned. The
characteristic length DH that was used in this study was calculated by taking the average of the orifices length and its width, see table 1 for values:
𝐷𝐻= 𝑙+𝑤2 (m) (4)
After determining the Weber number for each setup they were plotted relating to the length (L) in Matlab R2015a. There after a linear, exponential and a quadratic curve was extrapolated to the values related to rectangular orifices with rounded edges (EL, total of 15), using the least square method in Matlab R2015a. The same was done to the values related to rectangular orifice (PL-RE, total of 5). The residual values as well as the error sum of squares were
calculated for all different extrapolations. Also, by the extrapolation a function was determined for the length (L) related to Weber number. Weber number is used so that the function will be able to be implemented with different
geometries of orifices as well as for different surface tensions. The limitation of the function is that the dynamic similarity condition has to be fulfilled.
10 Results
3
As expected, the fluid jet exiting the nozzles in the different experiment set-ups all showed a significant transformation. The axial switching phenomenon was easy too see with the naked eye and it was clear that the bath height had a significant effect on the profile. The profile was captured with the high-speed camera, see figure 8, and the length to circular cross section could be determent for all the different set-ups. The result from the experiment indicated for all four nozzles that with deeper bath depth the longer the length to circular cross section becomes.
Figure 8 PL-EL with bath depth 11,4 cm, 13,7 cm, 18,1 cm and 20,5 cm
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The values from the experiment showed that a fluid jet exiting the nozzle PL-RE as well as the nozzle CE-LA transformed to a circular cross section later than PL-EL and CE-SM. This is visualized in the graph shown in figure 9. With an increased bath depth, the difference between the different orifices gets clearer.
For example if extrapolated values are compared, at a bath depth of 11,4 cm the length for PL-RE to take a circular cross-section is 16% longer than for PL-EL and at 24,8 cm it is 23% longer. Also, the length for CE-LA is 20% longer than that for CE-SM at 11,4 cm and at 24,8 cm it is 28%.
The length (L) was also related to the velocity, see figure 10. Instead of the bath depth, the orifices with rounded edges show a more uniformed trend. This is due to that they increase in a similar way. At the same time, a difference between rectangular orifice and rectangular orifices with rounded edges can be clearly noticed. Together the values related to the nozzle with rectangular orifice have a form resembling a convex shape and the others a more concave shape. This is demonstrated in figure 10 by the dotted lines. Aside from the third value for CE- SM and CE-LA. The rectangular orifices with rounded edges show over all an increase in length to a circular cross-section. The rectangular orifice show an increase for the values related to the lowest four bath depths and for the fifth a sudden decrease. In figure 10 it is also clear that with a larger orifice area the
Figure 9 Matlab plot showing the spread of the lengths related to the bath depths for the different nozzles. See part 2.1 for definitions.
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transformation to circular form occurs further down from the orifice compared to a case with a smaller orifice area. This is true with the exception for PL-RE with rectangular orifice.
Figure 10 Matlab plot showing the spread of the lengths related to the velocity.
The Weber number related to the length (L) showed a similar distribution as the velocity related to the length (L). To create a function for the length when the jet takes a circular form, the Weber number values needed to be extrapolated. Here a linear, exponential and quadratic extrapolation was created, as seen in figures 11, 12 and 13.
13
In figure 11 the extrapolation is linear. One for the values related to the rectangular orifice, PL-RE, and one for the rectangular orifices with rounded edges, EL. In both cases there are values that do not match very well to the curve, as shown in the plot for the residual curve. This shows how much the
extrapolated curve differs from the measured values. For the rectangular orifices with rounded edges it differs considerably for some of the values with a
deviation up to a distance of 1,8 cm. For the rectangular orifice the maximum deviation is measured to 1,1 cm. The error sum of square for the quadratic extrapolation was 15,1 for EL and 0,72 for PL-RE. As the residual values are relative high for the linear extrapolation, other extrapolated function are created, to see if a better fit could be accomplished. In figure 12 the measured values are extrapolated as if they follow an exponential function.
Figure 12 Matlab plot showing an exponential extrapolation of the lengths related to Weber number.
Figure 11 Matlab plot showing a linear extrapolation of the lengths related to Weber number.
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The difference between linear and exponential extrapolation is barley noticeable and the maximum deviation was 1,9 cm for the rectangular orifices with rounded edges and 1,0 cm for the rectangular orifice. The error sum of square for the exponential extrapolation was 15,4 for EL and 3,3 for PL-RE. Another attempt was also made to find a function with a better fit. In figure 13 the measured values are extrapolated as if they follow a quadratic function.
Figure 13 Matlab plot showing a quadratic extrapolation of the lengths related to Weber number.
The quadratic extrapolation showed a better fit for the measured values for the rectangular orifice. It followed the distribution of the values better with a maximum deviation as of 0,6 cm. The curve for the rectangular orifices with rounded edges still has a considerable deviation. The maximum deviation was the same as exponential extrapolation 1,9 cm. See appendix C for the residual curves to each extrapolation. The error sum of square for the quadratic extrapolation was 15,1 for EL and 0,72 for PL-RE.
The biggest deviation for the rectangular orifices with rounded edges is the same in the exponential and the quadratic extrapolations. However, when looking at the error sum of squares it shows that the quadratic extrapolation is a better choice.
From the quadratic extrapolation the following functions were given:
For the rectangular orifices with rounded edges
𝐿(𝑊𝑒) = 7,6 + 18 ∙ 10−4∙ 𝑊𝑒 + 40 ∙ 10−7∙ 𝑊𝑒2± 1,3 (cm) (5) For the rectangular orifice
𝐿(𝑊𝑒) = −0,35 + 35 ∙ 10−3∙ 𝑊𝑒 − 24 ∙ 10−6∙ 𝑊𝑒2± 0,36 (cm) (6)
15 Discussion
4
The experiment gave sufficient results that were close to the original
assumptions such as that an increase in bath depth gives an increase in L. Some issues were encountered during the experiment. However they were not
sufficient to give an impact of the result.
4.1 Result
As seen in the first graph, figure 9, there is an ongoing trend were L
increases with the bath depth. To be more specific, the results vary but still a similar increase can be seen for the four different orifices. When looking at the orifices with a larger area it shows that they have greater distances to a circular cross-section. This is due to an increase in pressure and therefore an increase in velocity. This is also seen in figure 10. It seems that the fluid jet takes on a circular cross-section later as the water molecules fall a greater distance per time at higher velocity.
There is a similarity between the slope of the ceramic nozzles and another similarity between the plastic nozzles. This can come from the different materials, such as that they have different wetting properties and surface smoothness. Ceramics are known to have high-energy surfaces and therefore have properties that make liquids most often accomplish a complete wetting to the surface. Polymers on other hand are known to have low-energy surfaces [16]. The zirconium oxide nozzles and the plastic nozzles had a notable difference in smoothness of the surface interfacial layer. The zirconium oxide nozzles had a very smooth surface and the plastic nozzles hade a slightly ruff surface. These two material characteristic together seemed to have an impact of the fluid flowing through the nozzle. In figure 9 it shows that the plastic nozzle has larger impact on the flow as the PL-EL nozzle has a gentler gradient.
The rectangular orifice differs from the others when looking at the values related to the highest bath depth, it decrease when the other increases. This can be due to turbulence at the edges. With a rectangular orifice it is hard to make the medium fill out the whole nozzle and cavities are easily created in the corners and that can affect the result [7,17]. The turbulence could be seen in some of the pictures as irregularities, the flow became unsteady and ruffles was shown. The rectangular orifice also departs from the rectangular orifices with rounded edges when compering the area of the orifices. It only has a slighter larger area than PL-EL but it behaves as if it would have a significant larger area. It has values that are comparable to the CE-LA even though it is smaller. This could have to do with the geometry. The orifices with rounded edges seam to be better suited for making a jet stream achieving a circular form.
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Moving on to figure 10, CM-LA and PL-EL gets more similar and a trend is shown between the rectangular orifice and the others. When the rectangular curve takes on a more convex shape the other three curves have a concave shape, this difference is still a result from the last value for the rectangular nozzle due to possible turbulence as mentioned earlier. A difference can also be seen in the ceramic nozzles which both have the same geometry, rectangular with rounded edges, however different areas. CE-SM reaches a circular cross-section earlier then CE-LA. This is due to that CE-SM has a smaller circumference then CE-LA.
According to Lautrup B [1] the instability of a fluid jet is related to its
circumference and therefore most certainty even the length to circular cross- section.
An attempt was made to create a function of all the values for the rectangular orifices with rounded edges based on the Weber number. Thereafter, the data was extrapolated as seen in figures 11, 12 and 13. This results in a curve with more values, which will give a more reliable function. In this way the function will also be usable with nozzles of different dimensions.
The least square fitting method extrapolations were a well adjustment to the values, in respect of all functions. When using error sum of squares the solution that is obtained is the one that minimizes the residual vectors norm. This function is adapted to the measurement values in the least square sense. The residual value measures the difference between the model and the measured data [18]. By looking at the error sum of squares for the extrapolation, the most suitable fit was determined as being the quadratic extrapolation. The
extrapolation did not take into account that the last value was misrepresented to the phenomena as the flow seems to not behave laminar for the rectangular orifice. The rectangular orifice extrapolation is only based on 5 measured values.
Therefore, it does not have the same reliability as the rectangular orifices with rounded edges.
In the end, the experiments resulted in two functions. These were both quadratic curves, one positive and the other negative. The curve corresponding to the rectangular orifice (the negative curve), has a more sharped curve than the one for the rectangular orifices with rounded edges. The curve related to the
rectangular orifice also meets the maximum point during the interval. Once again, this is probably due to the turbulence that occurred during the measurement of the last experiment value.
How the length to circular cross-section varies with different surface tension is interesting to think about, as it is a variable in determining the Weber number. In the top graph, in figure 14 an estimate on how steel 316L would vary with bath depths between 11,0 cm and 25,0 cm is shown. The bottom graph, in figure 14
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shows the same for water as to compare fluids with different surface tensions.
Here, using Bernoulli’s equation, see part 1.3 equation 2, the velocity has been approximated for bath depth between 11,0 cm and 25,0 cm. Velocity together with the surface tension (1,784 N/m) and density (7160 kg/m3) [19] for steel 316L as well as the characteristic diameter of CE-SM was calculated into Weber number. The length to circular cross-section has thereafter been calculated with the function from the result, part 3 equation 5. The same procedure went for the graph for water. By studying figure 14 it is clear that with higher surface
tensions the length to circular cross-section will occur closer to the nozzle orifice, as steel has a higher surface tension than water. At a bath depth of 11,0 cm the steel 316 L had a length of 7,8 cm while water would have 9,1 cm. At a bath depth of 25,0 cm the length would be 8,4 cm for steel 316L and 13 cm for water.
Figure 14 Estimate on how steel 316L would transform to circular cross-section if running through a nozzle like CE-SM on top (blue curve) and the same for water on bottom (green curve).
18 4.2 Experiment Issues
In the execution of the water lab some minor problems was encountered. The ceramic nozzles were very difficult to attach to the buckets. To be specific, the glue easily loosened from the ceramic surface. This, in some experiment setups, lead to small leaks in the form of drops around the interface of the nozzle and the container. In some cases it also leads to the nozzle being oblique. Therefore, the fluid jet did not at these times exit the orifice in exactly the gravitational
direction. Other problems with leakage were encountered, such as water seeping out trough the material walls on some of the plastic nozzle set-ups. This was found out to be due to the fact that the print of PL-EL had been made in to low of a print quality and after being used a couple of times were worn out. However, the PL-RE nozzle that was printed in high quality worked fine throughout the whole experiment.
While capturing the profile with the high speed camera the intention was always for it to be straight ahead of the orifice length respectively to the width of the orifice. However this was difficult to determent the accuracy of. Nevertheless, it should not lead to any problem for the result, as the circular cross section will still be able to be measured in the same way. Also, the accuracy of the measured volume flow is hard to determine as it was measured in a simple way. However, the result conform to what is expected related to the orifice area and the bath height.
Other than these minor issues the water lab experiment went well. All of the issues could have been prevented with better equipment and the problems are unlikely to have caused any major disturbance to the result.
19 Conclusion
5
The aim was to compare jets exiting rectangular orifices with rounded edges with jets exiting a rectangular orifice. The purpose was to find a function that describes the length of when a jet takes a circular cross-section, depending on the bath depth and geometry of the nozzle orifice as variables. The findings from the experiment can be summarized as:
A fluid jet exiting from a rectangular orifice transforms to a circular form further from the orifice compared to the ones exiting orifices with
rounded edges. However, jets exiting a rectangular orifice show a decreased laminar flow with deeper bath depth.
Fluid jet exiting orifices with larger areas transforms to a circular form further from the orifice than orifices with smaller areas.
The material that the nozzle is made of has a small influence on the jet profile.
With a higher surface tension it is seen that the transformation to a circular cross-section occurs closer to the orifices. Unlike a flow with a medium that has a lower surface tension value.
Future Work 6
To expand this study of fluid jet exiting a non-circular orifice, topics such as the following could be researched further in the future:
More geometries, studying the difference and impact of other orifices geometries such as elliptical and quadratic orifices. It would be of interest to study if the length to circular cross-section will be smaller if an
elliptical orifice is used due to it being similar to an already circular shape?
To see the impact of other mediums with different viscosity than water on the length to circular cross-section. Would it become longer due to slower velocity with mediums with higher viscosity?
Wither range of bath depths would be interesting as to see if the same trend would still be applied. As well as to get more data to either validate or dismiss the conclusion in this report.
20
Acknowledgment
__________________________________________________________
We want to thank everybody that have been a help in this project Prof. Pär Jönsson
Unit of Processes
School of Industrial Engineering and Management, KTH Arashk Memarpour, PhD.
Manager Meltingand Atomising Höganäs AB
Mikael Ersson, Lecture Unit of Processes
School of Industrial Engineering and Management, KTH Yonggui Xu, PhD. student
Unit of Processes
School of Industrial Engineering and Management, KTH Haitong Bai, PhD. student
Unit of Processes
School of Industrial Engineering and Management, KTH And a special thanks to our supportive families
21 Reference
7
[1] Lautrup B, Physics of continuous matter, second ed, chp 5, Boca Raton, CRC Press, 2011
[2] Amini-Baziani G, Instability of Elliptic Liquid Jets, Lap Lambert Academic Publishing GmbH KG, 2013
[3] http://www.thegreenbook.com/applications-of-nozzles.htm 2016-03-30 [4] Jernkontoret, Utbildningspaket del 4,
http://www.jernkontoret.se/globalassets/publicerat/handbocker/utbild ningspaket/jarn-och-stalframstallning_del4.pdf
[5] Munipally P, Satyanarayana G, Simhachalam N, CFD Analysis on a
Different Advanced Rocket Nozzles, International Journal of Engineering and Advanced Technology, Volume-4 Issue-6, (2015) 14.
[6] Chue R.S.M, Cresci D, Montgomery P, Design and analysis of a rectangular cross-section hypersonic nozzle, Shock waves: 26th international
symposium on shock waves Volym-2 (2007) 949.
[7] http://www.anl.gov/energy-systems/project/effect-nozzle-orifice- geometry 2016-03-01
[8] Kasyap T.V, Sivakumar D, Raghunandan B.N, Flow and breakup
characteristics of elliptical liquid jets, International Journal of Multiphase Flow 35 (2009) 8–19
[9] Massey B, Mechanics of fluids, eighth ed, London, Taylor & Francis, 2006 [10] Bird R.B, Stewart W.E, Lightfoot E.N, Transport Phenomena, second ed,
New York, John Wiley & Sons, Inc, 2002
[11] Wang F, Liquid jet breakup for non-circular orifices under low pressures, International Journal of Multiphase Flow 72 (2015) 248-262
[12] Christoph Beckermann C, Water Modeling of Steel Flow, Air Entrainment and Filtration, Proceeding of the 46th SFSA Technical and operating conference, Paper No. 3,6, Steel Founder’s Society of America, Chicago IL, 1992
[13] Cherkewski L, The Surface Tension of Salt Water http://lucas.ecustom.ca/science-fair/ 2016-03-30
[14] Mikrotron GmbH. 07/2007. MotionBLITZ Cube4 – High-Speed Recording Camera. Landshuter: Retrieved from www.mikrotron.de
[15] MathWorks, http://se.mathworks.com/products/matlab/ 2016-05-09 [16] Schrader M.E, Mordern Approces to wettability theory and applications,
New York, Springer Science+Business Media, 1992
[17] Yu H, Study of axis-switching and stability of laminar rectangular jets using lattice Boltzmann method, Computers and Mathematics with Applications 55 (2008) 1611–1619
[18] Eriksson G, Numeriska Algoritmer med Matlab, Stockholm, KTH, 2008 [19] Memarpour A, personal communication, January 15, 2016.
22 Appendices
8
Appendix A: An example of the shape of the used nozzle Appendix B: Matlab codes used to analyze the pictures Appendix C: Residual curves from the extrapolation
Appendix D: The resulting pictures of the flow from the high-speed camera Appendix E: Measured and calculated values
Appendix A
Appendix B
Appendix C
1 Residual curve to the linear extrapolation
2 Residual curve to the exponential extrapolation
3 Residual curve to the quadratic extrapolation
Appendix D
Bath depth 100 CE-SM
CE-LA
PL-EL
PL-RE
Bath depth 135 CE-SM
CE-LA
PL-EL
PL-RE
Bath depth 170 CE-SM
CE-LA
PL-EL
PL-RE
Bath depth 200 CE-SM
CE-LA
PL-EL
PL-RE
Bath depth 235 CE-SM
CE-LA
PL-EL
PL-RE
Appendix E
Velocity, u (m/s)
Bath depths (cm):
114 137 181 205 248
Pl-El 1,21124031 1,25502008 1,47544854 1,7303433 1,76553672 Pl-Re 1,38504155 1,43678161 1,76678445 1,85185185 2,12765957 Ce-LA 1,25028913 1,70712554 1,77541056 1,90494696 2,03602129 Ce-SM 1,33310368 1,65871809 1,72308625 1,86204482 2,13790332
Density, water (kg/m3): Surface tension, water (N/m):
1000 0,073
Characteristic length DH (m):
Pl-El 0,014
Pl-Re 0,0145
Ce-LA 0,01525
Ce-SM 0,0145
Weber number ( )
Bath depths (cm):
114 137 181 205 248
Pl-El 281,362236 302,069255 417,49695 574,208645 597,80382 Pl-Re 381,040157 410,040414 620,029395 681,173309 899,185772 Ce-LA 326,563689 608,804571 658,483023 758,076018 865,987479 Ce-SM 352,998613 546,500173 589,738089 688,692581 907,864979
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