Fluids Lab Report_ The Force of a Jet

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Stevens Institute Of Technology *

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342

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Mechanical Engineering

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Apr 3, 2024

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pdf

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Lab 2: The Force of a Jet I pledge my honor that I have abided by the Stevens Honor System.

Introduction: The objective of this experiment was to understand the fluid forces on solid objects that deflect the momentum of an impinging jet. This was accomplished by theoretically and experimentally calculating the force of a circular fluid jet as it acted on several differently shaped deflecting surfaces. The forces were calculated by comparing physical data and theoretical models derived from Linear Momentum Equations, Newton’s 2nd Law, and Reynolds Transport Theorem. The experiment utilized a Controllable Flow Apparatus to shoot a stream of water with varying pressure at a flat plate at 30, 45, 60, and 90 degrees, a cone at 45 and 60 degrees, and a bucket at 160 and 90 degrees (Shown below in Figure 1). The controllable flow apparatus has a flow control valve, pressure gage, and a digital flow meter which aids in conducting and performing the needed calculations. Water flow pressures of 0, 30, 40, and 50 psi were used to find the volumetric flow rate and force both theoretically and experimentally. The experiment took place in a water bucket with a cover to prevent any water from spilling out during testing as well as to provide a controlled environment. A force balance was connected to the test base to calculate the force of the jet. These recorded values, although not 100% accurate to theoretical models, are a close representation and provide physical evidence to theories learned in Fluid Mechanics. The hypothesis that this lab intends to prove is that force increases as larger vertical angles are introduced to an impinging jet of water, hence, it is expected to see the 160° pelton bucket to have the largest momentum transfer out of all the test specimens. Theory: In order to predict the behaviors produced by the fluid within these experiments - Newton’s 2nd Law, the Linear Momentum equations, and Reynolds transport theorem must be used to derive equations that are capable of determining the expected forces experienced by the various plates. Therefore the following is a comprehensive derivation that expresses the theoretical forces for each plate used in the laboratory. Figure 1: Plate, Bucket, and Cone Flow Diagrams

The significant equation behind all of these relationships is the linear momentum equation: This equation can be manipulated to determine other equations that will help represent the correlations between multiple variables of each solid surface at various angles. To simplify this equation, the time derivative on the right side can be eliminated because the control volume does not change with time. Another way to simplify the equation is to remove the F(B) term because no body forces were present in this lab. The weight of the water in the stream was negligible, and the weight of the deflecting surface was ignored because the scale was reset to zero after the plate was placed on the stand. Now that the equation can be simplified, it is much easier to derive equations to demonstrate relationships between variables. As seen above in figure 1, the first schematic shows a flat plate inclined at some value 𝜃 and has an inlet volume flow of 𝑸 . When the inlet flow splits into two streams, each stream has the same velocity but unequal volume flows, 𝛂𝑄 and (1- 𝛂 ) 𝑄 where 𝛂 ∈ [0,1], due to the fact that for frictionless flow, the fluid can exert no tangential force Ft on the plate. A relationship between 𝜃 and 𝛂 can be derived as follows: → There are no body forces. 𝐹 ?? = 0 → There are no surface forces acting in the x-direction because the flow is frictionless 𝐹 𝑆? = 0 → , The volume does not change over time → ? 1 = 𝑉???θ → ? 2 = ? 3 = 𝑉 → Divide both sides by ρ𝑄 − 𝑉???θ − 𝑉(1 − α) + 𝑉α = 0 → Divide both sides by 𝑉 − ???θ − (1 − α) + α = 0 ⇒ − ???θ − 1 + 2α = 0

⇒ α = (1 + ???θ)/2 (2) The linear momentum equation can also be used to derive an equation to formulate the magnitude of the vertical force Fy to hold the flat plate in place as a function of incoming volume flow rate 𝑄 , fluid density ρ, get area A, and inclination angle . θ

After the data was obtained, the applied force was plotted on a graph as a function of the square of the mass flow rate. A line of best fit was found from the data points and the slope of the line was found. Next, in order to determine the uncertainty in the force measurements (σ y ), the principle of least squares was applied, and the following formula was used: In this formula, N represents the number of data points, and B represents the slope of the line of best fit made from data points. Next, the uncertainty in the slope (σ B ) was calculated using the following formula: Materials and Methods:

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