# Discharge Coefficients of a Heavy Suspension Nozzle
## 3 Mathematical Equations and Analytical Process Followed to Determine the Physical Variables
\[\frac{dm_{u}}{dt}=\frac{\frac{1}{p_{u}}\frac{dp_{u}}{dt}-\frac{1}{m_{u}c_{v}} \Big{[}\frac{1}{I_{u}}+\frac{1}{Z_{u}}\Big{(}\frac{\partial Z_{u}}{\partial T} \Big{)}_{v}\Big{]}\frac{dQ_{u}}{dt}}{\frac{1}{m_{u}}-\frac{1}{Z_{u}}\Big{(} \frac{\partial Z_{u}}{\partial u}\Big{)}_{T}\frac{\forall v}{(m_{u})^{2}}+ \frac{1}{m_{u}c_{v}}\Big{[}\frac{1}{I_{u}}+\frac{1}{Z_{u}}\Big{(}\frac{ \partial Z_{u}}{\partial T}\Big{)}_{v}\Big{]}\Big{[}RT_{u}^{2}\Big{(}\frac{ \partial Z_{u}}{\partial T}\Big{)}_{v}-(h_{u}+u_{u})\Big{]}}, \tag{1}\]  
\[\frac{dT_{u}}{dt}=\frac{T_{u}}{p_{u}}\frac{dp_{u}}{dt}+\frac{T_{u}}{\forall v _{u}}\frac{d\gamma_{u}}{dt}-\frac{T_{u}}{m_{u}}\frac{dm_{u}}{dt}-\frac{T_{u}} {Z_{u}}\frac{dZ_{u}}{dt}, \tag{2}\]  
\[\frac{dT_{d}}{dt}=\frac{T_{d}}{p_{d}}\frac{dp_{d}}{dt}+\frac{T_{d}}{\forall d} \frac{d\gamma_{d}}{dt}-\frac{T_{d}}{m_{d}}\frac{dm_{d}}{dt}-\frac{T_{d}}{Z_{d} }\frac{dZ_{d}}{dt}, \tag{3}\]  
\[\frac{dm}{dt}=\frac{dm_{d}}{dt}=-\frac{dm_{u}}{dt}. \tag{4}\]  
An in-house computer program was created to solve the preceding Equations (1)-(4) with a Runge-Kutta method based on DVERK from the International Mathematics and statistics library, (IMSL). The air density was determined every time step using the Lee-Kesler equation iteratively as performed by [PERSON] [28]. According to this equation, the compressibility factor as a function of the reduced parameters, can be expressed as:  
\[Z^{(r)}=\bigg{(}\frac{P_{r}\
u_{r}}{T_{r}}\bigg{)}=1+\frac{B}{\
u_{r}}+\frac{ C}{\
u_{r}^{2}}+\frac{D}{\
u_{r}^{5}}+\frac{c_{4}}{T_{r}^{3}\
u_{r}^{2}} \bigg{(}\beta+\frac{\gamma_{1}}{\
u_{r}^{2}}\bigg{)}e^{\frac{-\gamma_{1}}{ \
u_{r}^{2}}}. \tag{5}\]  
The parameters B, C, D, \(c_{4}\), \(\beta\), and \(\gamma_{1}\), for various gases, can be determined from Reference [5]. \(P_{r}\), \(T_{r}\), and \(\
u_{r}\) stand for reduced pressure, reduced temperature, and reduced specific volume, respectively. According to [PERSON] and [PERSON] [14], the compressibility factor can be defined as:  
\[Z=Z^{(0)}+\frac{\omega}{\omega^{(R)}}\Big{(}Z^{(R)}-Z^{(0)}\Big{)}, \tag{6}\]  
where \(Z^{(R)}\) and \(Z^{(0)}\) are the compressibility factors for a reference fluid and simple fluid, respectively, while \(\omega^{(R)}\) and \(\omega\) stand for the acentric factors of the reference and working fluids. Considering known the pressure and temperature in a given location and time, the following steps were used to calculate the fluid compressibility factor. Initially, the reduced upstream pressure and temperature (\(P_{r}\) and \(T_{r}\)) were obtained based on the working fluid critical properties (\(P_{c}\); \(T_{c}\)) and the values of the pressure and temperature. When introducing the values of \(P_{r}\) and \(T_{r}\) in Equation (5), introducing, as well, the values of the parameters given for a simple fluid and obtained from reference [5], the value of \(\
u_{r}\) could be determined. Substituting the value of \(\
u_{r}\) in the same Equation (5), the corresponding compressibility factor for a simple fluid \(Z^{(0)}\) was obtained. Following the same procedure just described, but using the values of the parameters characterizing the reference fluid, which, for the present study, was n-octane, and values obtained from Reference [5], the value of \(Z^{(R)}\) was determined. Substituting the compressibility factors \(Z^{(0)}\) and \(Z^{(R)}\) in Equation (6) and considering the acentric factor values \(\omega^{(R)}\) = 0.3978 and \(\omega\) = 0.039, the compressibility factor for the working fluid could finally be obtained. This procedure allows to determine the compressibility factor and the fluid density at any position and time, and just the values of the pressure and temperature are required at the generic location where the information is needed. To be able to determine the instantaneous mass at each reservoir, the pressure evolution was measured in both reservoirs at any time; the volume of both reservoirs was also known, and the fluid temperature, as well as the compressibility factor, were estimated based on the previous equations.