Consider the problem of inflating the helium balloon, as...

Consider the problem of inflating the helium balloon, as described in Problem $3.79 .$ For a control volume that consists of the helium inside the balloon determine the work done during the filling process when the diameter changes from $1 \mathrm{m}$ to $4 \mathrm{m}$

Key Concepts

Pressure-Volume Work

Pressure-volume work is the work done by or on a system when it undergoes a change in volume at a given pressure. In many thermodynamic processes, especially those involving gases such as the helium in a balloon, the work is calculated by integrating the pressure with respect to the change in volume, typically expressed as W = ? P dV. This relationship is central to understanding how energy is transferred during expansion or compression processes.

Quasi-static Processes

A quasi-static process is one that occurs slowly enough for the system to remain in approximate equilibrium at each stage of the process. This assumption simplifies the calculation of work because it allows the pressure to be well defined at every instant, enabling one to use differential relationships like dW = P dV accurately. This concept is often used when analyzing controlled expansions such as inflating a balloon.

Control Volume Analysis

A control volume is a defined region in space through which mass and energy may flow. In thermodynamic problems, analyzing a control volume allows one to apply the conservation laws (mass, momentum, energy) to determine how the properties of the fluid or gas within the volume change over time, including the work interactions with the surroundings.

Geometric Calculation of Volume Change

In problems involving balloon inflation, the change in volume is determined by the geometry of the balloon, which is typically modeled as a sphere. The relationship between the diameter (or radius) and the volume of a sphere is used to compute the change in volume during the inflation process. This geometrical understanding is crucial when applying the pressure-volume work relationship to the problem.

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