Question

During the inflation and deflation of a safety airbag in an automobile, the gas enters the airbag with a specific volume of \(15 \mathrm{ft}^{3} / \mathrm{lbm}\) and at a mass flow rate that varies with time as illustrated in Fig. P5-165E. The gas leaves this airbag with a specific volume of \(13 \mathrm{ft}^{3} / \mathrm{lbm},\) with a mass flow rate that varies with time, as shown in Fig. P5-165E. Plot the volume of this bag (i.e., airbag size) as a function of time, in \(\mathrm{ft}^{3}\).

Short answer

Question: Determine the volume of the airbag (in ft³) as a function of time using the given specific volumes and mass flow rate data in Fig. P5-165E. Answer: The volume of the airbag (in ft³) as a function of time is given by the equation \(V(t) = m(t)v(t)\), where \(m(t)\) is the mass of the gas in the airbag at time \(t\), and \(v(t)\) is the weighted average specific volume of the gas at time \(t\). To find the mass and volume, use the mass flow rate data provided in Fig. P5-165E.

Step-by-step solution

Find the mass in the airbag at any given time

We need to find the mass of gas in the airbag at any given time \(t\). To do this, we need to subtract the mass of gas that left the airbag from the mass of gas that entered it. To find the mass of gas that entered and left the airbag, we need to multiply the mass flow rates (from Fig. P5-165E) by the elapsed time and the specific volumes. $$ m_{in}(t) = \int_0^t m_{in,rate}(\tau) d\tau $$ $$ m_{out}(t) = \int_0^t m_{out,rate}(\tau) d\tau $$ Then, the mass of gas in the airbag at any given time is: $$ m(t) = m_{in}(t) - m_{out}(t) $$

Calculate the volume of the airbag at any given time

The volume of the airbag depends on the mass of gas it contains and the specific volume of the gas it contains. Since the gas entering the airbag has a different specific volume than the gas leaving the airbag, we will use a weighted average specific volume of the gas in the airbag. We can define the weighted average specific volume \(v(t)\) as: $$ v(t) = \frac{m_{in}(t)v_{in} - m_{out}(t)v_{out}}{m(t)}$$ Now, the volume \(V(t)\) of the airbag at any given time is the product of the weighted average specific volume and mass of gas in the airbag: $$ V(t) = m(t)v(t) $$

Plot the volume of the airbag as a function of time

Now, using the given mass flow rate data from Fig. P5-165E, calculate the volume of the airbag \(V(t)\) at different time points, and plot the results on a graph. Please note that the actual values of mass flow rate as a function of time are not provided and can only be taken from Fig. P5-165E by approximation. The volume of the airbag (in \(\mathrm{ft}^3\)) as a function of time is the plot of the equation \(V(t)\) derived above.