# Numerical and Experimental Study of a Large-Scale Natural Circulation Helium Loop
## 3 Mathematical Model
### Solution Algorithm  
The idea of combining the control volume and particle-tracking approaches has its origins in the following thoughts. The main objective was to achieve the most profitable accuracy/computational speed ratio. The computational accuracy is closely related to the Courant number resulting from the time-step value, grid density, and the speed of information propagation. The grid has to be dense enough to capture propagating information. Usually, fast-spreading pressure waves do not require a dense grid. However, in the current application, the accuracy of the results is closely bound with a thermal wave, which moves relatively slowly in contrast with pressure waves. This fact led to the higher mesh density, so the thermal wave was captured properly. To preserve both the accuracy and the low grid density, the conservation of mass (1) and momentum (2) was solved in a coarse fixed mesh, and the conservation of energy (3) was calculated in a floating mesh. The solution procedure with one time step was as follows. Firstly, the pressure and mass flow rate were calculated from Equations (1) and (2) on a fixed grid. Secondly, the velocity and density are interpolated from the fixed to the floating grid, and each of the floating grid nodes was moved for a distance of \(\Delta\zeta\) (Equation (7)) along the flow direction. \[\Delta\zeta_{k}=\Delta t\,v_{k} \tag{7}\]  
Thirdly, the pressurewasis interpolated from the fixed to the floating mesh, and the temperature value was calculated in the floating grid according to Equation (8), which stemmed from the solution of the differential Equation (3). \[T_{k,j}=T_{ext}+\left(T_{kj-1}-T_{ext}\right)\exp\left(-\frac{4\,D_{e}\,k_{A} \,\Delta t}{d^{2}\,\rho_{kj,j}\,c_{p}}\right) \tag{8}\]  
Lastly, the temperature was interpolated from the floating to the fixed grid. The relation between two parallel grids is illustrated in Figure 3.  
The signs used were the following: \(x\)--fixed grid spacial coordinate, \(\zeta\)--floating grid spacial coordinate, \(i\)--spacial fixed grid index, \(k\)--spacial floating grid index, \(j\)--time index,  
Figure 3: The fixed and floating computational grids. \(k_{A}\) coefficient of heat flux through the pipe wall, \(T_{ext}\)--environment temperature, and \(D_{e}\)--external diameter. However, the heat exchanger temperature was treated differently. Thus, its outlet temperature was calculated from Equation (9), which stemmed from Equation (5). \[T_{out,j}=\frac{\Delta t}{m_{s}\,c_{p,s}}\Big{[}-\dot{m}_{fl,j-1}\,c_{p,fl}\big{(} T_{out,j-1}-T_{in,j-1}\big{)}-k_{A}\,A_{ext}\big{(}T_{out,j-1}-T_{ext}\big{)}+ \dot{Q}_{flux,j}\Big{]} \tag{9}\]  
Equations (1) and (2) were discretized on the fixed grid by using a semi-implicit scheme, as described in [19]. The finite difference method was used for the spatial discretization. The semi-implicit time matching was based on integrating the time derivation according to Equation (10), where the right side of the equation \(X_{j-\theta}\) is written as (11). The semi-implicit parameter \(\theta\) represents the ratio of explicit to implicit terms in the equations, where the \(\theta\) value lies inside the interval from 0 to 1 (\(\theta\) = 0 means a fully implicit formulation and \(\theta\) = 1 means a fully explicit formulation). \[\left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)_{j-\theta} =\left(1-\theta\right)X_{j}+\theta\,X_{j-1} \tag{10}\] \[X_{j-\theta} =\left(1-\theta\right)X_{j}+\theta\,X_{j-1} \tag{11}\]  
Nevertheless, this hybrid concept did not properly conserve the mass due to the sequential solving order. Therefore, the algorithm was enhanced with a pressure correction step, which manipulated the pressure to match the reference medium mass. The estimation of minor losses in pipe segments was based on an assumption of hydraulically smooth pipes. Then, the coefficients were evaluated according to the formulas of [PERSON] and [PERSON] (depending on the flow regime).