## 2 Materials and Methods
### Numerical Analysis  
The Runge-Kutta and Adams-Moulton predictor-corrector methods were used for the numerical solution of the differential equations. The developed simulation algorithm is coded in C++ and uses the REFPROP 10: Thermodynamic and Transport Properties Database developed by the National Institute of Standards and Technology (NIST). This database describes the properties of refrigerants such as density, enthalpy, liquid and vapor-specific heat, entropy, saturation temperature and pressure, thermal conductivity, dynamic viscosity, and surface tension. The numerical simulation is able to determine the thermodynamic parameters and the state parameters of the refrigerant at any point of the evaporator and, of course, at the endpoint of the evaporator with sufficient accuracy. The parameters such as temperature, pressure, vapor quality, and tube wall temperature of the refrigerant are studied as a function of the chilled water temperature. The heat transfer process is also analyzed. Both sections of the thermodynamic process in the horizontal tubes of the evaporator, evaporation, and superheating, are considered. The developed mathematical model is tested by a series of numerical simulations to obtain the values for the refrigerant and the water. The results of the numerical simulation were validated by experimental tests.  
The input data of the mathematical model describe the steady-state operation of the evaporator:  
* Refrigerant R134a
* Mass flow rate of water: \(m_{w}=0.28\) kg/s.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Region** & **Tube Side Pressure Drop Correlation** & **Equation** \\ \hline Single and & \(\Phi=1+\left[\frac{dp}{dz}\right]_{Fr}\cdot\left[\frac{\rho/p_{0}}{(\rho/\rho )^{2.5}}-1\right]\), & (53) \\ two-phase flow: & \(\left[\frac{dp}{dz}\right]_{Fr}=f_{Fr}\cdot\left[x+4\cdot\left(x^{1.8}-x^{10. }f_{Fr}0^{.5}\right)\right]\), & (54) \\ Grönnerud [27] & \(f_{Fr}=Fr0^{3}+0.0055\cdot\left[ln\frac{1}{Fr_{1}}\right]\). & (55) \\  & \(\left(\frac{dp}{dz}\right)_{l}=\frac{2.3\omega_{out}\cdot G}{4\rho_{l}}\), & (56) \\  & \(\lambda_{wall}=\frac{0.316}{Rd^{2.5}}\) & (57) \\ \hline \multicolumn{3}{c}{Shell side pressure drop correlation} \\ \hline  & \(\Delta p_{s}=\left[(N_{B}-1)\cdot\Delta p_{c}\cdot R_{B}+N_{B}\cdot\Delta p_{w} \right]\cdot R_{L}+\) & (58) \\  & \(2\cdot\Delta p_{c}\cdot R_{B}\) & \(\cdot\left(1+\frac{N_{W}}{N_{C}}\right)\). & \\  & The ideal cross-flow pressure drop through one baffle & \\  & space is obtained with the use of Equation (60) & \\  & \(\Delta p_{c}=\left(K_{a}+N_{c}\cdot K_{f}\right)\cdot\left(\frac{\cdot\dot{V}^ {\prime}}{2}\right)\). & (59) \\ Single phase flow: & The window zone pressure drop for \(Re>\)100 & \\ Bell-Delaware [25] & \(\Delta p_{w}=\frac{(2+0.6\cdot N_{w})+m_{w}^{2}}{2\cdot S_{w}\cdot S_{w}\cdot p _{w}}\), & (60) \\  & The number of effective cross-flow rows in the window zone: & (61) \\  & \(N_{cw}=\frac{0.8\cdot L_{s}}{p_{s}}\), & \(N_{c}=\frac{d_{s}\cdot(1-2\cdot L_{s}/D_{s})}{p_{s}}\), \\  & Equation (62) define the net cross-flow area through one & \\  & baffle. & (62) \\  & \(\frac{D_{s}^{2}}{4\cdot}\left[cos^{-1}D_{b}-D_{b}\cdot\left(1-D_{b}\right)^{1 /2}\right]-\frac{\pi}{8}\cdot\left(1-F_{1}\right)\cdot\pi\cdot D_{s}^{2}\) & (63) \\  & \(D_{b}=\frac{D_{s}-2\cdot L_{s}}{D_{s}}\) & & \\ \hline \hline \end{tabular}
\end{table}
Table 2: Pressure drops correlations.  
* Water inlet temperature: 13-19 \({}^{\circ}\)C
* Inner diameter of the shell: \(D_{s}\) = 28 mm.
* Number and diameter of tubes in the bundle: \(5\times\mathcal{O}8\times 1\) mm.
* Tubes material: cooper
* Hydraulically smooth tubes
* Number of baffles: \(N_{b}\) = 15.
* The window of the baffles: 25%.  
The heat exchanger shell is insulated from the outside, so that the heat exchange with the environment is neglected.