## 4 Oil Film Layer Thickness Estimation  
In this section, we describe the processing of the data obtained by simulation and measured weight data of the anticorrosive liquid ANTICORIT DFV 2001, hereinafter referred to as \"oil\", on the individual one-meter pipe sections, with a total considered length of 6 m. Our goal is to obtain a lower (pessimistic) estimate of the thickness of the oil film layer to ensure compliance  
Figure 5: Average values of experimental measurements for all pipe diameters, i.e., 15, 25, and 35 mm. Trendlines demonstrate different behaviour of oil microparticles in tubes of different diameters. Flow rate was constant at all measurements. with the FLV-F3 standard derived from our own FUCHS laboratory method, i.e., the areal density of the oil film \(\mu_{c}=1.4\) g/m\({}^{2}\). The areal density in the given position will be equal to \(\mu=\varrho\cdot h\), where \(h\) is the layer thickness, and \(\varrho=830\) kg/m\({}^{3}\) is the oil volume density, in compliance with the DIN 51 757 standard. The critical value of \(\mu_{c}\) corresponds to the critical layer thickness of \(h_{c}=1.68675\times 10^{-6}\) m = 1.68675 \(\upmu\)m.  
While we expect a cylindrical symmetric thickness of the oil layer, let us suppose that the oil is angular, uniformly distributed in the pipe, so the oil film thickness depends only on the distance \(x\) from the pipe's end. Let it be denoted \(h(x)\), \(x\in[0,l]\), where \(l\) is the length of a pipe. Then, the total oil mass from the pipe's beginning at \(x=0\) to the point \(x\) is defined by the relation  
\[m(x)=\pi\varrho\int_{0}^{x}\left[r^{2}-(r-h(\xi))^{2}\right]\,d\xi=\pi\varrho \int_{0}^{x}h(\xi)\big{[}2r-h(\xi)\big{]}\,d\xi, \tag{1}\]  
where \(r=d/2\) is the pipe's inner radius. The oil mass at the \(i\)-th pipe section between the points \(x_{i-1}\) and \(x_{i}\), \(i=1,2\),..., \(n\) will then be equal to  
\[m_{i}=m(x_{i})-m(x_{i-1}). \tag{2}\]  
For example, the oil mass in the 2 nd section of a 1-m length, established in an experimental measurement, will be equal to \(m_{2}=m(2)-m(1)\). Given the small number of data, we have decided to determine the appropriate, not very complex, dependence of \(h(x)\) so that the theoretical mass values \(m_{i}\) correspond quite well to the measured values, but do not exceed them, especially in sections with a lower assumed areal density of the oil layer, i.e., further from the injection point. Due to the big relative differences between measured values for different sections, measured values logarithms are shown in Figure 5. Logarithms of measured values can be approximated also by linear functions with similar precision as is the precision of the shown trendlines. Therefore, we decided to use exponential functions \(h(x)\) first. Another reason for a choice of exponential functions is that the function \(m(x)\) in (1) can be calculated analytically for such functions. If the oil film thickness \(h(x)\) has an exponential form  
\[h(x)=h_{0}\,e^{-a\,x},\qquad a>0, \tag{3}\]  
then using few commands of an open-source Computer Algebra System (CAS) **wxMaxima** (version 16.04.2, General Public License) [17]:  
assume(x>0); h:h0*exp(-a*x); h:h*exp(-a*x); M:%pi*rho*integrate((r^2-(r-subst(t,x,h))^2),t,0,x); tex(%);  
one arrives at a formula  
\[m(x)=\frac{\pi\,\varrho\,e^{-2\,a\,x}\,\left(\left(4\,h_{0}\,r-{h_{0}}^{2} \right)e^{2\,a\,x}-4\,h_{0}\,r\,e^{a\,x}+{h_{0}}^{2}\right)}{2\,a}\,. \tag{4}\]  
The oil masses within individual one-meter sections are:  
\[  
\[m_{5}=m(5)-m(4) = \frac{\pi\,\rho\,h_{0}\,\left(1-e^{-a}\right)}{2\,a}\,\left[4\,R\,e^ {-4\,a}-\left(1+e^{a}\right)\,h_{0}\,e^{-9\,a}\right],\] \[m_{6}=m(6)-m(5) = \frac{\pi\,\rho\,h_{0}\,\left(1-e^{-a}\right)}{2\,a}\,\left[4\,R\,e^ {-5\,a}-\left(1+e^{a}\right)\,h_{0}\,e^{-11\,a}\right],\]  
where \(\rho\) is the oil density. It is evident that \"theoretical\" masses (5) decrease almost exponentially, similarly to the oil film thickness \(h(x)\).