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From post #24 and the equation from the Wikipedia article

$$\dot{m}(t)=\left\{\begin{matrix}

0.007902C_d & \textrm{if } 0\leq t\leq 1.8963\\

C_dA \sqrt{2\, \rho_1\, p_{atm}\left ( \frac{\gamma}{\gamma-1} \right )\left [ \left ( \frac{p(t)}{p_{atm}} \right )^\frac{2}{\gamma}-\left ( \frac{p(t)}{p_{atm}} \right )^\frac{\gamma+1}{\gamma} \right ]} & \textrm{if }1.8963< t

\end{matrix}\right. \tag{3}$$$$C_d=0.6740$$

I then plug the pressure from the experimental data into the piecewise function above in excel to get the mass flow rate.

I then get the mass at each data point by doing integration by the trapezoidal rule

$$\sum_{n=1}^{\infty}m_n(t_n)=\frac{\dot m(t_n)-\dot m(t_{n-1})}{2(t_n-t_{n-1})}+m(t_{n-1})\tag{4}$$From post #7

$$m_0=n_iM=\frac{P_iV_TM}{RT_i}=0.02350\, \textrm{kg}\tag{5}$$

I then calculate the pressure at each data point using equation 2 from post 17:

$$p=\frac{RT_i}{V_T}(\gamma n-(\gamma-1)n_i)\tag{6}$$

$$\sum_{n=1}^{\infty}p_n(t_n)=\frac{RT_i}{V_T}(\gamma \frac{m_n(t_n)}{M}-(\gamma-1)\frac{m_0}{M})\tag{7}$$Finally, I look at the percent error at each data point to see how well this model fits to the experimental data:

$$\textrm{% Error}=\left |\frac{p_n(t_n)-p_{exp}(t_n)}{p_{exp}(t_n)} \right |\cdot 100$$

Looking at the data, I got the max error to be 2.62% with an average of 1.064%. Thus, I feel that these equations accurately model the experimental data. I did try the other equation that you recommended with the expansibility factor but it lead to more error.

Thus, the governing equations for ##p<53530.58\mathrm{Pa}##:

$$\frac{dm}{dt}=C_dA\sqrt{\gamma \rho_{atm}P_{atm}\left (\frac{2}{\gamma+1}\right )^{\frac{\gamma+1}{\gamma-1}}}\tag{8}$$Integrating and plugging ##m_0## from eq. 5 into yields:

$$m(t)=C_dA\sqrt{\gamma \rho_{atm}P_{atm}\left (\frac{2}{\gamma+1}\right )^{\frac{\gamma+1}{\gamma-1}}}t+\frac{P_iV_TM}{RT_i}\tag{9}$$ Which is then put into equation 6 giving the final governing equation for ##p<53530.58\mathrm{Pa}##:

$$p(t)=\frac{RT_i}{V_T}\left (\frac{\gamma C_dA}{M}\sqrt{\gamma \rho_{atm}P_{atm}\left (\frac{2}{\gamma+1}\right )^{\frac{\gamma+1}{\gamma-1}}}t+\frac{\gamma P_iV_T}{RT_i}-(\gamma-1)n_i \right )\tag{10}$$ To get the other half, we have to plug in the other equation but that is a function of pressure and I'm not quite sure how to integrate that.

Starting this experiment, I had no idea the complexity involved with finding the governing equations and thus far it has been an amazing learning experience.