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An innovative engineer has invented a device to replace the hydraulic jacks found at many service stations. A movable piston with a diameter of \(0.15 \mathrm{m}\) is fitted into a cylinder. Cars are raised by opening a small door near the base of the cylinder, inserting a block of dry ice (solid \(\mathrm{CO}_{2}\) ), closing and sealing the door, and vaporizing the dry ice by applying just enough heat to raise the cylinder contents to ambient temperature \(\left(25^{\circ} \mathrm{C}\right)\). The car is subsequently lowered by opening a valve and venting the cylinder gas. The device is tested by raising a car a vertical distance of \(1.5 \mathrm{m}\). The combined mass of the piston and the car is \(5500 \mathrm{kg}\). Before the piston rises, the cylinder contains \(0.030 \mathrm{m}^{3}\) of \(\mathrm{CO}_{2}\) at ambient temperature and pressure ( 1 atm). Neglect the volume of the dry ice. (a) Calculate the pressure in the cylinder when the piston comes to rest at the desired elevation. (b) How much dry ice (kg) must be placed in the cylinder? Use the SRK equation of state for this calculation. (c) Outline how you would calculate the minimum piston diameter required for any elevation of the car to occur if the calculated amount of dry ice is added. (Just give formulas and describe the procedure you would follow- -no numerical calculations are required.)

Step by step solution

01

Calculating the pressure in the cylinder

To calculate the pressure in the cylinder when the piston comes to rest at the desired elevation, we first need to determine the force exerted by the combined mass of the car and the piston. This is achieved using the formula: \(F = m \cdot g\), where \(m = 5500 \ kg\) is the mass and \(g = 9.81 \ m/s^2\) is the acceleration due to gravity. This will give us the force in newtons. After that, the pressure \(P\) can be found by applying the formula \(P = F/A\), where \(A = \pi \cdot (d/2)^2\) is the area of the piston and \(d = 0.15 \ m\) is the diameter.

02

Calculating the amount of dry ice

The amount of dry ice required can be calculated using the Soave-Redlich-Kwong (SRK) equation of state for gases. This equation is a modification of the ideal Gas Law, encompassing factors like molecular interaction and gas compressibility. The SRK equation is: \(P = \frac{RT}{(v-b)} - \frac{\alpha a}{v(v+b)}\), where \(P\) is the pressure, \(R\) is the gas constant, \(T\) is the temperature, \(v\) is the volume per moles of gas, \(\alpha\) is the correction factor for the attraction between molecules, and \(a\) and \(b\) are constants that are dependent on the particular gas. We will rearrange the equation to solve for \(v\), the combustion volume of carbon dioxide. Once we have \(v\), we can multiply it by the molecular weight of carbon dioxide to get the amount of dry ice required.

03

Outlining the procedure for calculating the minimum piston diameter

To calculate the minimum piston diameter for any elevation of the car to occur if the calculated amount of dry ice is added, we would explore the relationship between piston diameter and pressure as well as the effect of the additional weight of the car. The minimum piston diameter must generate enough pressure to counteract the weight of the car. We would begin from the force equilibrium equation, \(F = P \cdot A\), rearranging it to solve for \(d\), the diameter of the piston : \(d = \sqrt{F / (P \cdot \pi)}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Pressure Calculation

The task of calculating pressure within a cylinder involves understanding the relationship between force and the area over which this force is exerted. In this case, we are dealing with a piston and cylinder setup, where the force applied to the piston must counteract the weight of a car it is intended to lift. To calculate pressure, we begin by determining the force, denoted as \( F \), exerted by the car and piston combination. This is achieved using the formula:- \( F = m \cdot g \), where: - \( m \) is the combined mass (5500 kg) - \( g \) is the gravitational acceleration (9.81 \( m/s^2 \)) Once the force is known, the pressure \( P \) can be calculated using the formula:- \( P = \frac{F}{A} \) Here, \( A \) represents the area of the piston. It is a circular area given by \( A = \pi \cdot (d/2)^2 \), where \( d \) is the diameter of the piston (0.15 m). With these calculations, one can determine the cylinder pressure needed to hold the car steady at a given elevation.

Soave-Redlich-Kwong Equation

The Soave-Redlich-Kwong (SRK) equation is an equation of state that provides a more accurate way to understand and predict the behavior of gases under various conditions than the ideal Gas Law. This is particularly useful when considering the vaporization of dry ice in our piston setup.The SRK equation is expressed as:- \( P = \frac{RT}{(v-b)} - \frac{\alpha a}{v(v+b)} \)Where:- \( P \) is the pressure of the gas.- \( R \) is the universal gas constant.- \( T \) is the absolute temperature.- \( v \) is the molar volume of the gas.- \( \alpha \), \( a \), and \( b \) are specific coefficients for the SRK equation, accounting for molecular interactions and volume exclusion effects.To find the necessary amount of dry ice, use the SRK equation to rearrange and solve for the molar volume \( v \). By understanding how \( v \) is affected by the temperature and pressure within the cylinder, you can estimate the amount of solid \( CO_2 \) (as dry ice) needed to produce a sufficient gas quantity that will exert the calculated pressure, allowing the piston to rise effectively.

Piston Dynamics

In analyzing piston dynamics, we explore how well a piston can move and maintain pressure inside the cylinder to lift objects, such as a car, vertically. This requires an understanding of the relationship between the piston's diameter, the pressure it can create, and the force it needs to counteract.The procedure involves using the equation \( F = P \cdot A \) to ensure that the force exerted by the piston is sufficient to lift the car. Here, \( F \) is the force from the weight of the car and piston combined, and \( A \) is the cross-sectional area of the piston (which varies with diameter).To calculate the minimum piston diameter \( d \) necessary to achieve a lift, we rearrange the equation to:- \( d = \sqrt{\frac{F}{P \cdot \pi}} \)This approach assists in defining the smallest possible piston size that can provide the necessary lift when considering the force required to counteract the car's weight. Optimizing these dynamics ensures efficiency in the process of lifting, as achieving the proper pressure and upward force is critical to the device's operation.