Suppose we pressurize a new tennis ball with helium (species A) at a gauge pressure of...

  

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Suppose we pressurize a new tennis ball with helium (species A) at a gauge pressure of pa-Po 1.0 atm, where pm and pou are the total pressures inside and outside the ball. Over time, the helium will leak out of the ball via diffusion through its rubber wall (species R), and the ball will be considered deflated when its gauge pressure is po-pose<0.8 atm. The material properties of the ball and the rubber wall are given below. The internal radius of the ball is a, and the wall thickness is w. Typical values of a and w are a = 3.3 cm. and w=0.3 cm. Symbol DAIL K₁ Quantity Diffusivity of helium in rubber Partition coefficient of helium in rubber, (mol/em' in rubber divided by mol/cm' in gas phase) We will use a quasi-steady state approximation to determine the time it takes for a tennis ball to deflate when it is sitting in air. The ball is at room temperature T = 298K, and exposed to air at atmospheric pressure p 1 atm. We will assume the permeability of oxygen and nitrogen into the rubber is much smaller than that of helium, and that there is a negligible amount of helium in the air outside the ball. (a) Draw a schematic and state the assumptions you will use to calculate the deflation time of the ball. DO NOT ASSUME THE WALL IS PLANAR. (b) (i) Use conservation of mass to derive an expression for the total rate of helium leaving the ball (in units of mol/s), denoted by symbol WA. Your answers should be a function of the variables listed in the table, the ball radius a, the wall thickness w, the total pressures Pin and poor, and the temperature T. To help you in your derivation, you can use the equation WANA area constant, where NA is the molar flux. Integrating this equation across the thickness of the rubber should help you determine the rate of helium loss. (ii) Oftentimes, manufacturers introduce another gas (e.g., SF6) into the ball in order to extend the ball's shelf life. This gas typically does not dissolve as easily in the rubber phase as helium (i.e.. has a lower partition coefficient and lower diffusivity). If the partition coefficient decreases by a factor of three, and the diffusivity in rubber decreases by a factor of two, how would WA decrease from the calculation you made? (c) Set up the differential equation to determine the pressure inside the ball as a function of time, using the variables DAR KA, poat, T. a. and w. Solve the differential equation to get the pressure inside the ball as a function of time. Suppose we pressurize a new tennis ball with helium (species A) at a gauge pressure of pa-Po 1.0 atm, where pm and pou are the total pressures inside and outside the ball. Over time, the helium will leak out of the ball via diffusion through its rubber wall (species R), and the ball will be considered deflated when its gauge pressure is po-pose<0.8 atm. The material properties of the ball and the rubber wall are given below. The internal radius of the ball is a, and the wall thickness is w. Typical values of a and w are a = 3.3 cm. and w=0.3 cm. Symbol DAIL K₁ Quantity Diffusivity of helium in rubber Partition coefficient of helium in rubber, (mol/em' in rubber divided by mol/cm' in gas phase) We will use a quasi-steady state approximation to determine the time it takes for a tennis ball to deflate when it is sitting in air. The ball is at room temperature T = 298K, and exposed to air at atmospheric pressure p 1 atm. We will assume the permeability of oxygen and nitrogen into the rubber is much smaller than that of helium, and that there is a negligible amount of helium in the air outside the ball. (a) Draw a schematic and state the assumptions you will use to calculate the deflation time of the ball. DO NOT ASSUME THE WALL IS PLANAR. (b) (i) Use conservation of mass to derive an expression for the total rate of helium leaving the ball (in units of mol/s), denoted by symbol WA. Your answers should be a function of the variables listed in the table, the ball radius a, the wall thickness w, the total pressures Pin and poor, and the temperature T. To help you in your derivation, you can use the equation WANA area constant, where NA is the molar flux. Integrating this equation across the thickness of the rubber should help you determine the rate of helium loss. (ii) Oftentimes, manufacturers introduce another gas (e.g., SF6) into the ball in order to extend the ball's shelf life. This gas typically does not dissolve as easily in the rubber phase as helium (i.e.. has a lower partition coefficient and lower diffusivity). If the partition coefficient decreases by a factor of three, and the diffusivity in rubber decreases by a factor of two, how would WA decrease from the calculation you made? (c) Set up the differential equation to determine the pressure inside the ball as a function of time, using the variables DAR KA, poat, T. a. and w. Solve the differential equation to get the pressure inside the ball as a function of time.

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A Schematic and Assumptions The schematic of the tennis ball being pressurized with helium is shown in Figure 1 We will assume that the rubber wall is impermeable to other gases except helium and that ... View the full answer