# Heat Transfer Performance and Flow Characteristics of Helical Baffle-Corrugated Tube Heat Exchanger
## 2 Numerical Simulation and Verification  
### Numerical Simulation Theory  
In order to ensure the calculation accuracy and improve the calculation efficiency, the following assumptions are made for the simulation: (1) the fluid in the tube side and the shell side is water, and both are incompressible Newtonian fluid; (2) it is assumed that the density, viscosity, and other physical parameters of the fluid remain unchanged during the heat transfer process; (3) the influence of gravity is considered; (4) only the heat transfer between the shell side and the tube side is considered, and the remaining heat loss is not considered. For calculations, the governing equations are shown in Equations (1)-(3) [22; 23; 24]. (a) The mass conservation equation is as follows:  
\[\frac{\partial\rho}{\partial t}=-\left(\frac{\partial(\rho u_{x})}{\partial x }+\frac{\partial\big{(}\rho u_{y}\big{)}}{\partial y}+\frac{\partial(\rho u_{ z})}{\partial z}\right) \tag{1}\]  
(b) The momentum conservation equation is as follows:  
\[\left\{ \begin{array}{l}\frac{\partial(\rho t_{x})}{\partial t}+\
abla\cdot \left(\rho u_{x}\overset{\rightarrow}{u}\right)=-\frac{\partial p}{\partial x }+\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial\tau_{yx}}{\partial y}+ \frac{\partial\tau_{yx}}{\partial z}+\rho f_{x}\\ \frac{\partial}{\partial t}+\
abla\cdot\left(\rho u_{y}\overset{\rightarrow}{u }\right)=-\frac{\partial p}{\partial y}+\frac{\partial\tau_{xy}}{\partial x}+ \frac{\partial\tau_{yy}}{\partial y}+\frac{\partial\tau_{yy}}{\partial z}+ \rho f_{y}\\ \frac{\partial(\rho u_{z})}{\partial t}+\
abla\cdot\left(\rho u_{z}\overset{ \rightarrow}{u}\right)=-\frac{\partial p}{\partial z}+\frac{\partial\tau_{xy}}{ \partial x}+\frac{\partial\tau_{yx}}{\partial y}+\frac{\partial\tau_{yx}}{ \partial z}+\rho f_{z}\end{array} \right. \tag{2}\]  
(c) The energy conservation equation is as follows:  
\[ \begin{array}{l}\frac{\partial(\rho T)}{\partial t}+\frac{\partial(\rho u_{ x}T)}{\partial x}+\frac{\partial\big{(}\rho u_{y}T\big{)}}{\partial y}+ \frac{\partial(\rho u_{z}T)}{\partial z}\\ =\frac{\partial}{\partial x}\left(\frac{k}{c_{p}}\cdot\frac{\partial T}{ \partial x}\right)+\frac{\partial}{\partial y}\left(\frac{k}{c_{p}}\cdot\frac{ \partial T}{\partial y}\right)+\frac{\partial}{\partial z}\left(\frac{k}{c_{p} }\cdot\frac{\partial T}{\partial z}\right)\end{array}  \tag{3}\]  
where \(\rho\) is the fluid density, \(t\) is the time, \(u_{x}\), \(u_{y}\), and \(u_{z}\) is the velocity vector component, \(p\) is the pressure, \(\tau_{ij}\) is the stress due to viscosity during fluid flow, and \(f_{i}\) is the force per unit mass in each direction. Only gravity is considered here, and the direction of gravity extends the \(y\)-axis in the positive direction; hence, \(f_{x}=f_{z}=0\), \(f_{y}=\overset{\rightarrow}{g}\). As the continuous helical baffle heat exchanger structure is more complex, and the main stream of the shell is always in a complex flow state, this paper chooses the Realizablek-\(\epsilon\) turbulence model to solve the calculations. The transport equations for the model are shown in Equations (4) and (5) [25; 26]. The turbulent kinetic energy \(k\) transport equation is as follows:  
\[\frac{\partial}{\partial t}(\rho k)\ +\ \frac{\partial}{\partial x_{j}}\big{(}\rho ku _{j}\big{)}\ =\ \frac{\partial}{\partial x_{j}}\Bigg{[}\bigg{(}\mu+\frac{\mu_{t}}{\sigma_{k}} \bigg{)}\frac{\partial k}{\partial x_{j}}\Bigg{]}+P_{k}+P_{b}-\rho\varepsilon- Y_{M}+S_{k} \tag{4}\]  
The turbulent dissipation rate \(\epsilon\) transport equation is as follows:  
\[ \begin{array}{c}\frac{\partial}{\partial t}(\rho\varepsilon)\ +\ \frac{ \partial}{\partial x_{j}}\big{(}\rho\varepsilon u_{j}\big{)}\\ =\ \frac{\partial}{\partial x_{j}}\bigg{[}\Big{(}\mu+\frac{\mu_{t}}{\sigma_{k}} \Big{)}\frac{\partial\varepsilon}{\partial x_{j}}\bigg{]}+\rho C_{1}S \varepsilon-\rho C_{2}\frac{\epsilon^{2}}{k+\sqrt{\alpha}}+C_{1}\epsilon \frac{\epsilon}{k}C_{3\varepsilon}P_{b}+S_{\varepsilon}\end{array}