Relationship between discharge coefficient and perimeter of nozzle

Question

Question: In a coke-mentos experiment the coke bottle gains a forward momentum due to the expulsion of gas. If the lid of the bottle, where the gas is exiting, has been cut into different shapes with the same area (therefore different perimeter to area ratios), how would the relationship and acceleration look (e.g. proportional, logarithmic, inversely-proportional)?

Background: I am writing a report on this experiment and I wanted to compare my trendline with the theoretical results. I tried looking at how objects play a role in air resistance but since the expulsion is what I am looking at, I found fluid dynamics to be a better approximation. I found Bernoulli's principle and the following derivation, $Q = CA\sqrt{2\rho\Delta P}$, where $Q$ is the flow rate, $C$ is a discharge coefficient, $A$ is the area, $\rho$ is the density of the air, and $P$ is the pressure across the hole. I also found I could plug this in $F=\frac{\Delta p}{\Delta t}$ to find the force through the equation $F=\frac{2CA\Delta P}{\Delta t}$ (and therefore find the acceleration).

My Problem: Now I can find the relationship between acceleration and the discharge coefficient but I cannot find anything explaining the role of the perimeter to acceleration ratio within the discharge coefficient since it is found experimentally and I am trying to find a theoretical result. I understand that it is probably complex (and way above my level right now) due to the interactions which cause turbulence and many other phenomena but if someone could explain the relationship only between the perimeter and discharge-coefficient/acceleration that would be awesome!

Answer 1

The relationship between the perimeter of the nozzle shapes and the discharge coefficient (and consequently the acceleration of the coke bottle in your experiment) can be understood through principles of fluid dynamics, particularly focusing on how changes in nozzle geometry affect fluid flow characteristics. Understanding Nozzle Geometry and Fluid Flow Nozzle Shape and Flow Characteristics: The shape of the nozzle affects how the fluid (in this case, the gas from the coke) exits the bottle. Different shapes can cause variations in flow patterns, which influence the discharge coefficient 𝐶 C. The discharge coefficient 𝐶 C is a dimensionless number that characterizes the flow efficiency of the nozzle. It accounts for losses mainly due to friction and turbulence as the fluid exits the nozzle. Perimeter-to-Area Ratio: The perimeter-to-area ratio of the nozzle influences the extent of interaction between the fluid and the boundary of the nozzle. A higher perimeter relative to the area increases the boundary layer effects, which can enhance frictional losses and potentially increase turbulence. Turbulence and friction reduce the discharge coefficient because they impede the smooth flow of fluid, converting some kinetic energy of the flow into heat or turbulent energy losses. Theoretical Relationship Between Perimeter and Discharge Coefficient Increased Perimeter-to-Area Ratio: Generally, increasing the perimeter-to-area ratio for a given area increases the potential for turbulent flow, especially if the edges are sharp or irregular. This can decrease the discharge coefficient. A lower discharge coefficient implies that the nozzle is less efficient at directing the flow, which could reduce the thrust (and therefore acceleration) generated by the escaping gas. Effect on Acceleration: The force exerted by the escaping gas can be modeled by 𝐹 2 𝐶 𝐴 Δ 𝑃 F=2CAΔP, where Δ 𝑃 ΔP is the pressure difference driving the flow. If 𝐶 C decreases due to an increase in the perimeter-to-area ratio, the resultant force 𝐹 F, and thus the acceleration of the coke bottle, would decrease. Practical Implications for Your Experiment Experiment Design: When designing your experiment, consider measuring the perimeter and area of each nozzle shape accurately. Record the acceleration of the coke bottle for each nozzle shape to see if there is a trend that correlates with changes in the perimeter-to-area ratio. Data Analysis: Plot acceleration against the perimeter-to-area ratio of each nozzle. A trend might emerge that could suggest an inversely proportional relationship, where higher perimeter-to-area ratios lead to lower accelerations. Analyze if the shapes with smoother or more streamlined perimeters perform better in terms of higher acceleration, indicating a higher discharge coefficient. Conclusion Theoretically, an increase in the perimeter-to-area ratio of a nozzle, for a fixed area, is likely to decrease the discharge coefficient due to increased frictional and turbulent losses. This would result in a decrease in the acceleration of the coke bottle in your experiment. The exact nature of this relationship (whether it is strictly inversely proportional, logarithmic, etc.) would depend on the specific fluid dynamics involved, including factors like the exact shape of the nozzle, the speed of the gas, and the physical properties of the gas. Experimentation and further detailed fluid dynamics analysis would be required to precisely quantify this relationship.