Heat Transfer Performance and Flow Characteristics of Helical Baffle–Corrugated Tube Heat Exchanger

Abstract

Heat exchangers are widely used in petrochemical and other industries. Improving the efficiency of heat exchangers to increase energy utilization is crucial. Passive enhanced heat transfer technology is widely studied in heat exchanger research due to its low energy consumption and simple operation. Helical baffle–corrugated tube heat exchangers have not been extensively studied as a promising new class of these devices. This paper investigates the key structural parameters of a helical baffle-corrugated heat exchanger through numerical simulation. This study focuses on the factors affecting heat transfer and flow resistance performance. The results show that reducing the helical angle from 28.42° to 10.81° increases the total heat transfer coefficient by approximately 20%. The overall performance of the heat exchanger is evaluated using the efficiency evaluation coefficient (EEC). Optimal levels of each structural factor are determined for different working conditions based on this evaluation.

1. Introduction

Heat exchangers are extensively used in petroleum, metallurgy, refrigeration, chemical, engineering machinery, and light industries [1]. These systems are crucial in ensuring normal industrial production, waste heat recovery, and energy conservation [2]. Therefore, it is imperative to enhance the efficiency of heat exchanger apparatuses for the secure and reliable functioning of industrial production, whilst also augmenting the rate of energy utilization [3].

Enhanced heat transfer technology can be divided into two types: passive or active enhanced heat transfer technology [4,5,6,7,8]. Among them, active enhanced heat transfer technology refers to the external application of power to a system to achieve the enhancement of the system heat transfer effect, such as mechanical stirring of a fluid applied, an electromagnetic field [9], a nanofluid [10,11,12], etc. Generally, active enhanced heat transfer technology has a high cost, and therefore it is rarely seen in large-scale industrial applications [13]. Passive heat transfer enhancement does not require additional power, only changing the structure of the heat transfer equipment to achieve a change in the state of fluid flow, causing a boundary layer fluid disturbance in order to improve the efficiency of heat transfer. The utilization of passive enhanced heat transfer technology obviates the necessity for additional energy consumption, thereby conferring a cost advantage over active enhanced heat transfer technology [14]. Furthermore, passive enhanced heat transfer is relatively straightforward to operate, which has contributed to its widespread adoption in industrial contexts.

Continuous helical baffles and helical corrugated tubes are innovative variations of baffles and heat exchanger tubes that facilitate efficient heat exchange. However, most of the present experimental studies utilize substitutes of the ideal continuous helical baffles, such as trisection helical baffles [15,16,17], helical trapezoidal baffles [18,19], and helical baffles with central tube [20,21]. Most studies on improving heat transfer in special-shaped tubes have been directed towards heat exchangers with conventional tube bundle support. However, there has been limited research on heat transfer enhancement in special-shaped tubes with continuous helical baffles.

In view of the lack of research on helical baffle–corrugated tube heat exchangers in the previous studies, this study is carried out to design orthogonal experiments by selecting different factors for helical baffle–corrugated tube heat exchangers. The effects of the helix angle of the helical baffle on the integrated heat transfer performance and fluid flow characteristics of the helical baffle–corrugated tube heat exchanger under different working conditions are discussed. Meanwhile, the optimum level combinations of various structural factors under different working conditions are discussed. The results are expected to provide data reference for the future use of helical baffle–corrugated tube heat exchanger.

2. Numerical Simulation and Verification

2.1. 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:

$${\displaystyle \frac{𝜕\rho }{𝜕t}}=-\left({\displaystyle \frac{𝜕\left(\rho u_x\right)}{𝜕x}}+{\displaystyle \frac{𝜕\left(\rho u_y\right)}{𝜕y}}+{\displaystyle \frac{𝜕\left(\rho u_z\right)}{𝜕z}}\right)$$

(1)

(b)

The momentum conservation equation is as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{𝜕\left(\rho u_x\right)}{𝜕t}}+\nabla \cdot \left(\rho u_x\overrightarrow{u}\right)=-{\displaystyle \frac{𝜕p}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{xx}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{yx}}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{zx}}{𝜕z}}+\rho f_x\\ {\displaystyle \frac{𝜕\left(\rho u_y\right)}{𝜕t}}+\nabla \cdot \left(\rho u_y\overrightarrow{u}\right)=-{\displaystyle \frac{𝜕p}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{xy}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{yy}}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{zy}}{𝜕z}}+\rho f_y\\ {\displaystyle \frac{𝜕\left(\rho u_z\right)}{𝜕t}}+\nabla \cdot \left(\rho u_z\overrightarrow{u}\right)=-{\displaystyle \frac{𝜕p}{𝜕z}}+{\displaystyle \frac{𝜕{\tau }_{xz}}{𝜕x}}+{\displaystyle \frac{𝜕{\tau }_{yz}}{𝜕y}}+{\displaystyle \frac{𝜕{\tau }_{zz}}{𝜕z}}+\rho f_z\end{array}\right.$$

(2)

(c)

The energy conservation equation is as follows:

$$\begin{array}{l}{\displaystyle \frac{𝜕\left(\rho T\right)}{𝜕t}}+{\displaystyle \frac{𝜕\left(\rho u_{x}T\right)}{𝜕x}}+{\displaystyle \frac{𝜕\left(\rho u_{y}T\right)}{𝜕y}}+{\displaystyle \frac{𝜕\left(\rho u_{z}T\right)}{𝜕z}}\\ ={\displaystyle \frac{𝜕}{𝜕x}}\left({\displaystyle \frac{k}{c_p}}\cdot {\displaystyle \frac{𝜕T}{𝜕x}}\right)+{\displaystyle \frac{𝜕}{𝜕y}}\left({\displaystyle \frac{k}{c_p}}\cdot {\displaystyle \frac{𝜕T}{𝜕y}}\right)+{\displaystyle \frac{𝜕}{𝜕z}}\left({\displaystyle \frac{k}{c_p}}\cdot {\displaystyle \frac{𝜕T}{𝜕z}}\right)\end{array}$$

(3)

where ρ is the fluid density, t is the time, u_x, u_y, and u_z is the velocity vector component, p is the pressure, τ_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 = $\overrightarrow{g}$.

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 Realizable k-ε 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:

$${\displaystyle \frac{𝜕}{𝜕t}}\left(\rho k\right)+{\displaystyle \frac{𝜕}{𝜕x_j}}\left(\rho k{u}_j\right)={\displaystyle \frac{𝜕}{𝜕x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_k}}\right){\displaystyle \frac{𝜕k}{𝜕x_j}}\right]+P_k+P_b-\rho \epsilon -Y_M+S_k$$

(4)

The turbulent dissipation rate ε transport equation is as follows:

$$\begin{array}{c}{\displaystyle \frac{𝜕}{𝜕t}}\left(\rho \epsilon \right)+{\displaystyle \frac{𝜕}{𝜕x_j}}\left(\rho \epsilon u_j\right)\\ ={\displaystyle \frac{𝜕}{𝜕x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_t}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{𝜕\epsilon }{𝜕x_j}}\right]+\rho C_{1}S\epsilon -\rho C_2{\displaystyle \frac{{\epsilon }^2}{k+\sqrt{v\epsilon }}}+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}{C}_{3\epsilon }{P}_b+S_{\epsilon }\end{array}$$

(5)

where

$$C_1=\max \left[0.43,{\displaystyle \frac{\eta }{\eta +5}}\right],\eta =S{\displaystyle \frac{k}{\epsilon }},S=\sqrt{2S_{ij}{S}_{ij}}$$

In these equations, P_k represents the generation of turbulence kinetic energy due to the mean velocity gradients, and P_b is the generation of turbulence kinetic energy due to buoyancy.

The structural parameters of the continuous helical baffle heat exchanger for the study are as follows: the inner diameter of the shell is 100 mm, the length of the shell is 1400 mm, and the diameters of the tube and shell course inlet and outlet are 40 mm. The tube form is a light tube, the number of heat exchanger tubes is 51, the center distance of the tube is 12.8 mm, the outer diameter of the tube is 10 mm, and the wall thickness is 1 mm. The tube layout is in the form of positive triangular tube distribution, and the helical baffle is used with a helical pitch of 100 mm and a thickness of 2 mm. The thickness is 2 mm. In order to ensure the accuracy and computational efficiency of the numerical simulation, the following simplifications are carried out in the establishment of the geometric overall model: (1) neglect the shell thickness; (2) neglect the bolts, nuts, and other connectors at the continuous helical baffle. ANSYS Meshing is used for meshing, and for the complex structure of the heat exchanger, the mesh is a composite mesh of hexahedral mesh combined with tetrahedral mesh. The 3D model and element model are shown in Figure 1.

The mesh quality affects the accuracy of the numerical simulation results greatly, but too many mesh nodes will increase the computational volume, making the computational cycle much longer with limited improvement in accuracy. Therefore, it is necessary to verify the mesh independence of the finite volume model. Element sizes of 1, 3, and 5 mm (corresponding to the number of nodes 1,721,994, 2,828,456, and 5,958,327, respectively) are selected to verify the mesh independence by comparing the numerical simulation with the experimental shell pressure drop and heat transfer coefficient, and the results are shown in

Table 1. Mesh independence verification (experimental values in parentheses).

Size (mm)	Node No.	Pressure Drop (Pa)	Error (to Exp.) (%)	Heat Transfer Coefficient (W/(K·m2))	Error (to Exp.) (%)
1	1,721,994	11,853.25 (11,107.40)	6.71%	2014.13 (2173.22)	7.32%
3	2,828,456	11,246.15 (11,107.40)	1.25%	2109.58 (2173.22)	2.93%
5	5,958,327	11,107.44 (11,107.40)	0.00%	2173.18 (2173.22)	0.00%

. According to the results in Table 1, the element size in this study is 5 mm.

The discrete partial differential equation method in the numerical calculation is the finite volume method, the solver adopts a pressure-based steady-state solver based on the pressure correction algorithm, and the turbulence model adopts the Realizable k-ε model, which is one of the most commonly used turbulence models in Fluent to simulate turbulence effects in turbulence by solving the turbulence kinetic energy equations and turbulence dissipation rate equations. The Realizable k-ε model has higher accuracy and reliability than the traditional k-ε model and performs better in dealing with high Reynolds number flow, reverse flow, and shear laminar turning [27]. The standard wall function is used to deal with the flow near the wall, the Coupled algorithm is used to carry out the coupling calculation of pressure and velocity, and the second-order windward format is used to calculate the pressure. The momentum, turbulent kinetic energy, turbulent dissipation rate, and energy equations are discretized using the second-order windward format. The convergence accuracy of the energy equation is 1 × 10⁻⁶, and that of other equations, such as the continuity equation and the turbulent kinetic energy equation, is 1 × 10⁻³. The flow model takes the effect of gravity into account, the exit temperatures of the shell and tube courses are set as the reference variables for convergence, and the calculations are stopped when the exit temperatures are stable and the residuals are less than 1 × 10⁻³. The heat transfer medium of the heat exchanger tube and shell processes is saturated water, and the change in physical properties of water with temperature is ignored. The shell process and tube process are used in the mass flow into the mouth conditions, with a shell process inlet temperature of 40 °C and tube process inlet temperature of 65 °C. The outlet is under the pressure outlet condition, and the outlet pressure is 0. The shell process wall and folding plate wall are set as adiabatic walls, and the tube wall is set as a heat conduction coupling wall, which are all non-slip wall conditions. The flow of the numerical simulation is shown in Figure 2.

2.2. Experimental Verification

In this paper, the device shown in Figure 3 was built to verify the validity of the numerical algorithm, and the parameters of the test device are consistent with the parameters in the numerical calculation model. The device consists of a hot- and cold-water circulation system, a continuous helical baffle heat exchanger, a temperature control system, and a data acquisition system. During the experiment, the inlet and outlet temperatures of the heat exchanger, the differential pressure in the tube course, the differential pressure in the shell course, and the volume flow rate of the cold- and hot-water pipelines were recorded.

The experiments were repeated three times for each condition to reduce the effect of errors generated during the experimental process, and the temperature data and pressure data were recorded every 2 s for each set of conditions. Convective heat transfer experiments were carried out with different shell flow rates. The main influence of the continuous helical baffle on the heat exchanger studied in the experiment lies in the shell course, and the flow rate of the tube course remains unchanged during the experiment and simulation. Only the influence of the change of the flow rate of the shell course is investigated, so the heat transfer coefficient of the shell course and the pressure drop are used as the comparative parameters. The variation in the shell heat transfer coefficient with the shell flow rate for the continuous helical baffle heat exchanger in experimental and numerical simulations is shown in Figure 4a, and the variation in the pressure drop with the shell flow rate is shown in Figure 4b.

As demonstrated in Figure 4, the shell heat transfer coefficient and pressure loss exhibit a linear increase with rising shell flow. The experiments conducted using a continuous helical baffle heat exchanger were compared to the experimental and simulation results for the shell heat transfer coefficient and pressure loss under similar conditions. The experimental and simulation results have a maximum deviation of 6.15% for the shell heat transfer coefficient, and the maximum difference in the shell-stage pressure drop between the experiment and simulation is 13.19%, which is within a reasonable range. The mutual verification attests to the reliability of the numerical and experimental outcomes of the simulation model. It is also observed that when the shell flow rate is 3.5 m³/h, the experimentally obtained heat transfer coefficient has a high uncertainty. After verification of the data, we initially judge that this is due to the instability of heat transfer in the experimental equipment at lower flow rates. However, combining the performance at all flow rates, the experiments and simulations are still in good agreement.

3. Results and Discussion

3.1. Helix Angle and Heat Transfer Performance

The structure of a continuous helical baffle plays a crucial role in strengthening the heat transfer performance of a heat exchanger. The continuous helical baffle has a helical surface, and the shell diameter determines the main structural parameters for the helix angle. Changing the same shell diameter under the pitch in essence is to change the helix angle. The definition of a helical angle is shown in Equation (6).

$$\beta =\arctan ({\displaystyle \frac{H_s}{\pi d_s}})$$

(6)

where β is the helix angle, H_s is the pitch of the helical baffle, and d_s is the helical diameter.

In this work, the pitches of the continuous helical baffle are 60 mm, 80 mm, 100 mm, 130 mm, and 170 mm, and the corresponding helix angles are 10.81°, 14.29°, 17.66°, 22.48°, and 28.42°, respectively. The change in helix angle corresponds to the change in baffle structure, which mainly affects the distribution of the heat exchanger shell flow field. The shell-stream volume flow rates of 3.5 m³/h, 4.5 m³/h, 5.5 m³/h, 6.5 m³/h, and 7.5 m³/h were selected, keeping the tube-stream volume flow rate of 4 m³/h unchanged.

Simulations were carried out for different shell volume flow rates, and the results for a shell volume flow rate of 7.5 m³/h are shown in Figure 5. From Figure 5a, it can be seen that in the helical baffle heat exchanger with a larger helical angle, there is a phenomenon of sudden increase in the diagonal distribution velocity at the outlet position due to the sudden reduction in the fluid circulation area and change in the flow direction. While the helical angle is smaller, the axial velocity of the shell fluid in the direction of the extended center axis is smaller and closer to the transverse flow, so the smaller the helical angle of the helical baffle heat exchanger, the smaller the change of flow velocity at the exit. The pressure results of helical baffles with different helical angles are illustrated in Figure 5b, and it can be seen that the smaller the helical angle, the greater the axial pressure drop of the helical baffle heat exchanger.

A comparison of the velocity streamline of helical baffle heat exchangers with different helical angles is shown in Figure 6, and the results show that fluid stagnation zones exist in the inlet and outlet sections of helical baffle heat exchangers with different helical angles. In addition, clear vortices exist near the shell inlet when β equals 17.66°, 22.48°. This is due to the fact that in the inlet section, the fluid flows vertically and axially from the shell process inlet and is affected by the disturbing effect of the tube bundles and the influence of the larger circulation area, which makes the flow velocity decrease rapidly, impacts the tube bundles and the shell process wall, and produces a reflux zone near the inlet area. In the exit section, the shell process fluid, due to the exit of the flow cross-sectional area and the reduction in the exit to produce acceleration, the vast majority of the fluid shows normal outflow. Due to the disturbing effect of the tube bundle, a small portion of the fluid will flow back in the exit section at the shell angle, forming a flow dead zone. Combined with Figure 5a, it can be seen that with the reduction in the helical angle, the different helical angle of the helical baffle heat exchanger shell process fluid retention area is more obvious, but the smaller the helical angle, the smaller the shell process inlet and outlet section of the area, so that the inlet and outlet section of the fluid retention area is smaller.

The total heat transfer coefficient of the heat exchanger is an important evaluation indicator, and the total heat transfer coefficient K_C of the helical baffle heat exchanger can be calculated by Equation (7):

$$\left\{\begin{array}{l}{\displaystyle \frac{1}{K_c}}={\displaystyle \frac{D_h}{d_h{K}_h}}+{\displaystyle \frac{1}{K_s}}+R_w\\ R_w={\displaystyle \frac{b}{\lambda }}\end{array}\right.$$

(7)

where d_h, D_h are the inner and outer diameters of the heat exchanger tube, K_s, K_h are the heat transfer coefficients of the shell and tube, R_w is the thermal resistance of the wall of the tube, b is the thickness of the heat exchanger tube, and λ is the thermal conductivity of the tube.

As can be seen in Figure 7a, the total heat transfer coefficients of helical baffle heat exchangers with different helix angles all increase with the increase in the shell flow rate, and the improvement ranges from 30.80% to 38.91%. When the shell flow rate is the same, reducing the helix angle can effectively improve the total heat transfer coefficient of the heat exchanger. In this study, when the helix angle is reduced from 28.42° to 10.81°, the overall improvement is around 20%. The shell heat transfer coefficients per unit pressure drop corresponding to different helix angles are shown in Figure 7b. The shell heat transfer coefficient per unit of pressure drop decreases with the decreasing helix angle. When the shell flow rate is small, i.e., there is a weaker degree of turbulence, the shell heat transfer coefficient per unit of pressure drop decreases significantly with the decreasing helix angle. In this study, when the shell volume flow rate is increased from 3.5 m³/h to 7.5 m³/h, comparing the helix angle of 28.42° with 10.81°, it is found that the percentage of reduction in the shell heat transfer coefficient per unit pressure drop decreases from 86% to 75%.

It is also found in Figure 7 that when the total heat transfer coefficient is similar, the larger the helix angle is, the larger the shell heat transfer coefficient per unit of pressure drop is. This is because in the helical baffle heat exchanger, the fluid flows in the form of a helical plunger, and the smaller the helix angle is, i.e., the smaller the cross-sectional area of the fluid flow is, the smaller the axial component of its velocity, the larger the radial component, the larger the tangential component, and the stronger the effect of transverse flow. The helical flow of the bundle of the tube scouring is closer to the transverse scouring, which improves the overall turbulence level and achieves the purpose of destroying the boundary layer near the wall of the tube, thus increasing the convection heat transfer intensity of the shell process. However, at the same time, the disturbance of the flow field generated by the tube bundle and the resistance brought by the baffle will also be stronger, resulting in a larger pressure loss.

3.2. Optimization of Helical Baffle–Corrugated Tube Heat Exchanger

There is no heat exchanger constructed from a helical corrugated tube and a continuous helical baffle, and its flow characteristics and heat transfer performance have not been studied yet. In order to expand the application of the continuous helical baffle and develop a new type of high-efficiency heat exchanger, this study constructs a helical baffle–corrugated tube heat exchanger, and numerical simulation methods are used to study the flow and heat transfer performance of the heat exchanger.

The main structural factors of a helical baffle–corrugated tube heat exchanger are the helical angle of the helical baffle, the cross-section shape of the helical groove, and the helical angle of the corrugated tube. Liu et al. [28] investigated a corrugated tube with an inner diameter of 20 mm and corrugated tube pitches of 15 mm, 20 mm, and 25 mm, as well as corrugated tubes with triangular, rectangular, and semicircular cross-sections. In this study, the simulated single-tube heat exchanger shell has an inner diameter of 40 mm, corrugated tube pitches of 20 mm, 30 mm, and 40 mm, and helical baffle pitches of 20 mm, 40 mm, and 60 mm. The shell length and tube length are 200 mm, the diameter of the inlet and outlet of the tubular process is 15 mm, the outer diameter is 19 mm, and the thickness of the baffle is 0.5 mm. The cross-sectional shape of the corrugated tube shown in Figure 8. The orthogonal tables L₉ (3³) are used to arrange the experiments. As shown in

Table 2. Factors and levels.

Level	Factors
	A: Section Shape of Helical Groove	B: Helix Angle of Corrugated Tube (°)	C: Helix Angle of Helical Baffle (°)
1	Triangle	37.07	18.10
2	Rectangle	53.09	35.33
3	Semicircle	67.69	51.07

, three factors are evaluated each time and each factor takes three levels, and the detailed experimental programs are presented in

Table 3. Orthogonal experiment scheme L₉ (3³).

No.	A	B	C
1	Triangle	37.07	18.10
2	Triangle	53.09	35.33
3	Triangle	67.69	51.07
4	Rectangle	37.07	35.33
5	Rectangle	53.09	51.07
6	Rectangle	67.69	18.10
7	Semicircle	37.07	51.07
8	Semicircle	53.09	18.10
9	Semicircle	67.69	35.33

. Simulations were carried out under different conditions for different heat exchanger structures, in which the tube inlet temperature was 60 °C and the shell inlet temperature was 20 °C. The specific condition parameters are shown in

Table 4. Condition parameters.

No.	Velocity of Tube Inlet (m/s)	Reynolds Number of Tube	Velocity of Shell Inlet (m/s)	Reynolds Number of Shell
I	0.270	4000	0.193	4000
II	0.405	6000	0.193	4000
III	0.504	8000	0.193	4000
IV	0.270	4000	0.289	6000
V	0.270	4000	0.385	8000

.

The total heat transfer coefficient and pressure drop for the nine programs under the five operating conditions designed are shown in

Table 5. Numerical simulation results of the 9 programs under 5 conditions.

No.	Total Heat Transfer Coefficient (W/(K·m2))					Pressure Drop (Pa)
	I	II	III	IV	V	I	II	III	IV	V
1	1924.21	2004.43	2065.96	2029.77	2133.15	2412.93	2587.97	2825.65	4792.44	8035.30
2	1925.21	1804.46	1840.36	1847.35	1904.45	650.89	774.80	930.55	1185.08	1887.77
3	1657.01	1720.70	1750.51	1761.90	1829.52	319.22	407.42	520.10	552.27	856.89
4	2267.64	2451.92	2498.81	2424.29	2532.09	668.93	855.98	1101.73	1171.77	1855.14
5	2095.01	2283.82	2393.30	2257.31	2357.66	345.79	472.91	638.03	579.89	883.67
6	1855.67	1975.78	2053.03	1952.99	2038.76	2349.20	2436.36	2541.85	4743.77	7990.80
7	2138.51	2285.93	2378.58	2325.26	2431.43	363.79	513.18	708.61	598.04	901.49
8	2412.39	2633.87	2759.83	2538.50	2639.94	2354.99	2462.44	2600.67	4698.94	8009.70
9	1855.86	2422.98	2532.53	2383.01	2476.16	674.00	664.67	770.72	1084.96	1774.84

.

Due to the orthogonal features, the importance order of each factor could be found through the analysis. These nine test sets have tested all of the pairwise combinations of the independent variables. This demonstrates significant savings in testing efforts over the all-combinations approach. The variance analysis method (i.e., range analysis method) was used to clarify the significance levels of different influencing factors, and the most significant factors could be disclosed based on the result of the range analysis. The range analysis method mainly includes the following factors:

$$\left\{\begin{array}{l}{\overline{K}}_i={\displaystyle \frac{{\displaystyle \sum _{j=1}^{n}S}(i,j)}{n}}\\ R_i=\max {\overline{K}}_i-\min {\overline{K}}_i\\ C{R}_i={\displaystyle \frac{R_i}{{\displaystyle {\sum }_{i=1}^m{R}_i}}}\end{array}\right.$$

(8)

where the average values of each level for each factor were named as ${\overline{K}}_i$ ; the variances between each factor were defined as R_i to analyze the difference between the maximal and minimal value of the four levels for each factor; and CR_i indicates the influence of each factor.

The analysis results for the total heat transfer coefficient and pressure drop are shown in

Table 6. Range analysis of total heat transfer coefficient and pressure drop.

		Total Heat Transfer Coefficient (W/(K·m2))					Pressure Drop (Pa)
		k1	k2	k3	R	CR	k1	k2	k3	R	CR
I	A	1772.66	2072.77	2135.59	362.93	0.457	1127.68	1121.31	1130.93	9.62	0.005
	B	2110.12	2081.39	1789.51	320.61	0.404	1148.55	1117.22	1114.14	34.41	0.017
	C	2064.09	1953.42	1963.51	110.67	0.139	2372.37	664.61	342.93	2029.44	0.979
II	A	1843.20	2237.17	2447.59	604.40	0.642	1256.73	1255.08	1213.43	43.30	0.019
	B	2247.43	2240.72	2039.82	207.61	0.220	1319.04	1236.72	1169.48	149.56	0.067
	C	2204.69	2226.45	2096.82	129.64	0.138	2495.59	765.15	464.50	2031.09	0.913
III	A	1885.61	2315.05	2556.98	671.37	0.665	1425.43	1427.20	1360.00	67.20	0.028
	B	2314.45	2331.16	2112.02	219.14	0.217	1545.33	1389.75	1277.56	267.77	0.113
	C	2292.94	2290.57	2174.13	118.81	0.118	2656.06	934.33	622.25	2033.81	0.859
IV	A	1879.67	2211.53	2415.59	535.92	0.619	2176.60	2165.14	2127.31	49.28	0.012
	B	2259.77	2214.39	2032.63	227.14	0.262	2187.42	2154.64	2127.00	60.42	0.014
	C	2173.75	2218.22	2114.82	103.39	0.119	4745.05	1147.27	576.73	4168.32	0.974
V	A	1955.70	2309.50	2515.84	560.14	0.616	3593.32	3576.54	3562.01	31.31	0.004
	B	2365.55	2300.68	2114.81	250.74	0.276	3597.31	3593.71	3540.84	56.47	0.008
	C	2270.61	2304.23	2206.20	98.03	0.108	8011.93	1839.25	880.68	7131.25	0.988

and Figure 9. As seen from Table 6, the factor influence of the total heat transfer coefficient decreases in the order A > B > C according to the R values; the pressure drop decreases in the order C > B > A. Figure 9 indicates that the semicircle cross-section helical groove has a significant advantage over other shapes for enhancing the heat transfer coefficient. Moreover, the best program of the optimized total heat transfer coefficient is A3, B1, and C1 for the low tube flow rate, low shell flow rate condition (Condition I). With the increase in the tube flow rate and shell flow rate (Conditions II, IV, and V), the best optimization scheme of the total heat transfer coefficient is A3, B1, and C2. It should be noted that when the tube flow rate is much higher than the shell flow rate (Case III), the best optimized solutions for the total heat transfer coefficient are A3, B2, and C1.

Figure 9 also shows that it is always impossible to achieve the optimal flow resistance characteristic optimization at the same time as the optimal heat transfer performance optimization. Therefore, in this paper, the effectiveness evaluation parameter EEC proposed by Liu et al. [28] for the complex flow of heat exchangers is used for the evaluation, and the calculation method is shown in Equation (9):

$$EEC={\displaystyle \frac{Q/Q_0}{(q_v\cdot \Delta P)/(q_{v0}\cdot \Delta P_0)}}$$

(9)

where Q is the heat transfer, $\left(q_v\cdot \Delta P\right)$ is the pump power, and q_v is the volume flow rate. The numerator represents the heat transfer benefit; the denominator represents the pump power loss. Equation (9) directly represents the ratio of heat transfer gain to flow cost.

The analysis results for the EEC are shown in

Table 7. Range analysis of EEC.

		EEC
		k1	k2	k3	R	CR
I	A	4.096	4.260	5.559	1.463	15.10%
	B	5.019	4.820	4.076	0.943	9.73%
	C	1.427	3.778	8.710	7.284	75.17%
II	A	3.774	4.600	5.043	1.269	16.72%
	B	4.441	4.376	4.599	0.224	2.95%
	C	1.471	4.378	7.567	6.096	80.33%
III	A	3.192	3.843	4.137	0.945	16.02%
	B	3.510	3.595	4.068	0.558	9.47%
	C	1.469	3.840	5.863	4.393	74.51%
I	A	1.631	2.055	2.230	0.599	15.53%
	B	2.078	1.948	1.889	0.188	4.88%
	C	0.523	1.800	3.593	3.070	79.59%
V	A	0.810	1.026	1.113	0.303	15.22%
	B	1.046	0.974	0.929	0.117	5.89%
	C	0.248	0.882	1.820	1.572	78.89%

and Figure 10. As seen from Table 7, the factor influence of the EEC decreases in the order C > A > B according to the R values. Furthermore, the contribution of C, B, and A based on the CR values to the EEC influence range are from 74.51% to 80.33%, around 15%, and between 2.95% and 9.73%, respectively.

Figure 10 presents the level of influence of each factor for the EEC. In the case of a low pipe flow rate (Conditions I, IV, and V), the optimum level of the helical baffle–corrugated tube heat exchanger is A3, B1, and C3 in terms of heat exchanger efficiency evaluation coefficients; in the case of large pipe flow rates (Condition II and III), the optimum level of the helical baffle–corrugated tube heat exchanger is A3, B3, and C3.

Heat exchanger research aims to improve equipment performance, but achieving an optimal balance between the heat transfer coefficient and pressure drop is challenging in production. The EEC index can help address this issue by evaluating comprehensive performance. In this study, the EEC index is used to optimize structural parameters, offering practical ideas for production.

4. Conclusions

In this paper, the enhanced heat transfer and flow characteristics of continuous helical baffle–corrugated tube heat exchangers with different structures are investigated by numerical simulation and experimental research, and the main conclusions are as follows:

1.

The results of the study show that when the total heat transfer coefficients of the heat exchangers are similar, the helical baffle heat exchanger with a larger helix angle has a more obvious advantage in the overall heat transfer performance. Therefore, in industrial applications, a helical baffle heat exchanger with a large helix angle with a large shell volume flow rate should be selected as often as possible, so that the heat exchanger has sufficient heat transfer performance while having a small pressure loss.

2.

The efficiency evaluation coefficient (EEC) results show that the helix angle of the helical baffle is the most influential factor. And the optimum levels are A3, B1, and C3 in the case of the low pipe flow rate. The section shape of the helical groove is semicircle, the helix angle of the corrugated tube equals 37.07°, and the helix angle of the helical baffle equals 51.07°. The optimum levels are A3, B3, and C3 in the large pipe flow rate condition. The section shape of the helical groove is semicircle, the helix angle of the corrugated tube equals 67.69°, and the helix angle of helical baffle equals 51.07°.