Numerical simulation and experimental research on heat transfer and flow resistance characteristics of asymmetric plate heat exchangers

Abstract

The asymmetric plate heat exchanger (APHE) has the possibility of achieving balanced pressure drops on both hot and cold sides for situations with unbalanced flow, which may in turn enhance the heat transfer. In this paper, the single-phase water flow and heat transfer of an APHE consisted of two types of plates are numerically (400≤Re≤12000) and experimentally (400≤Re≤ 3400) investigated. The numerical model is verified by the experimental results. Simulations are conducted to study the effects of N, an asymmetric index proposed to describe the geometry of APHEs. The correlations of the Nusselt number and friction factor in the APHEs are determined by taking N and working fluids into account. It is found that an optimal N exists where the pressure drops are balanced and the heat transfer area reaches the minimum. The comparison between heat transfer and flow characteristics of the APHEs and the conventional plate heat exchanger (CPHE) is made under various flow rate ratios of the hot side and the cold side and different allowable pressure drops. The situations under which APHE may perform better are identified based on a comprehensive index Nu/f^1/3.

Keywords

plate heat exchanger / asymmetric / simulation / correlation / heat transfer enhancement

Introduction

Plate heat exchangers (PHEs) are extensively used in varieties of applications, such as nuclear reactors, air-conditioning systems, steam power plants, petrochemical equipment and so on. They essentially consist of a number of metal plates, with corrugations in herringbone or similar pattern. The cross-corrugated heat exchangers, in which the cross section of the flow channel is complex, are proposed due to their flexible design and high heat transfer [¹]. Many researchers have performed simulations and experiments on this geometry in order to study the thermal and hydraulic characteristics [²–⁴]. Under most situations, the geometry is symmetric, which means hot and cold fluids flow in channels of the same structure [⁵,⁶]. Nilpueng et al. [⁷] presented the heat transfer and pressure drop in PHEs with different surface roughness. They concluded that the chevron angle and surface roughness have a significant effect on the increase in the heat transfer coefficient. Kanaris et al. [⁸] employed a response surface methodology to optimize the PHEs design with respect to blockage ratio, channel aspect ratio, corrugation aspect ratio, and angle of attack. They proposed new correlations for predicting the Nusselt number and friction factor. Zhang and Che [⁹] numerically studied cross-corrugated plates with sinusoidal, isosceles triangular, trapezoidal, rectangular, and elliptic corrugations. They found that optimal structures were among those with smooth corrugation shapes and small inclination angles. Doo et al. [¹⁰] conducted a numerical simulation to analyze modified primary surface geometries for the development of light weight and high thermal efficiency intercoolers. They concluded that the secondary corrugation could reduce pressure drop by approximately 15% with small changes in heat transfer capacity.

The APHE is also one kind of modification of the profile, which can be advantageous in many working conditions. An important distinction between the two kinds of PHEs is that the characteristics of the hot and cold channels in the APHEs are different. For the CPHEs, studies usually focus on one side because of the symmetric characteristic. But the two sides in the APHEs both need to be investigated by using experimental or numerical methods. In traditional CPHEs, both fluids are subject to identical channel geometry since the channels are symmetric. When the flow rates of the hot and the cold fluid are unequal, the used pressure drops are unequal too. Therefore, the allowable pressure drops on both sides cannot be sufficiently optimized in the conventional symmetric PHEs when the allowable pressure drops for both fluids are the same. If the APHE is available, it can be designed that both of the two channels can make full use of the allowable pressure drops. Then both fluids would be individually optimized leading to increasing velocities. Since the increasing velocities of the fluids can enhance the heat transfer coefficient, the heat transfer area can be reduced [¹¹]. The advantages of APHEs can be summarized as follows:

(1) When the operating conditions include unequal flow rates, the hot and cold fluids can make full use of their allowable pressure drops and the heat transfer area can be reduced.

(2) The pressure drop balance can reduce deformation due to the pressure difference, leading to a greater pressure bearing capacity.

(3) If one fluid is easy to scale, such as silt fluids and other impurity fluids, it can flow through the channel with a larger flow area to reduce scaling.

Table 1 summarizes the characteristic of different APHEs in Refs. [¹²–¹⁷]. Focke [¹²] investigated several types of asymmetric profiles which had different geometries. He found that the performance of the APHE is related to the allowable pressure drop and such plates might be profitable in some specific conditions. Kim et al. [¹³] parametrically investigated the APHE that can be used in intercoolers, studied the characteristics of the APHEs using the unitary-cell CFD model, discussed the shape of different asymmetric profiles by a new parameterization method, conducted sizing minimization based on the simulation, and parameterized the models by proposing a structure sinusoidal equation. They found that the effect of the asymmetric profile was highly sensitive to the given operating conditions and the designs. Vestergren [¹⁴] proposed one kind of PHEs whose plates could rotate so that the alternating broad and narrow flow channels could be achieved. Dahlberg [¹⁵] provided an asymmetric brazed PHE with indentations adapted for housing a ridge or groove of a neighboring plate. The APHE had different flow channels and improved pressure handling capabilities. Funke [¹⁶] also invented an unsymmetrical corrugation pattern with a base section arranged between the frustums of pyramids and plate plane. Lee et al. [¹⁷] investigated the resistance and heat transfer correlations of a new type of APHE for ammonia-water heat transfer. The flow areas for water and refrigerant differed a lot. It was found that the asymmetric profiles were significantly superior to the symmetric profiles

Although APHEs have many advantages as listed above, very limited researches on APHEs have been conducted due to manufacturing costs, and the applications of APHEs are thus few. To resolve this problem it is necessary to design cost-effective APHEs. Numerical simulation plays an important role in the design and evaluation of the fundamental mechanisms of PHEs [¹⁸]. The applications of CFD in PHEs were practiced by various researchers [¹⁹–²³]. Zhang et al. [²⁰] presented a simulation methodology of a capsule-type plate channel. They set a cell model which contained three channels of fluids and two plates to investigate the capsule-type profile. The results showed that the capsule-type PHE had a good overheat transfer performance. Computational fluid dynamics (CFD) can be helpful in optimization for the asymmetric PHEs by suggesting modifications.

In the present paper, a new APHE is proposed. Commercial software ANSYS is applied to study the APHE by building a double-pass parallel plate model. The simulation results are compared with the test data of an experimental APHE. Further simulations are performed to investigate the effects of plate geometry on the performance of APHE.

Numerical models and experimental procedure

Physical model of the asymmetric profiles

The two types of profiles consist of sinusoidal corrugation as shown in Fig. 1. If the corrugation depth is the same for adjoining plates, the flow area of both sides will be identical. If one plate has uneven corrugation depths as characterized by (b₁, b₂) in Fig. 1, the areas of adjacent channels will be different. This would also make fluid flow resistance characteristic in two channels distinct.

As Fig. 1 illustrates, for asymmetric profiles, the top plate and the bottom plate are with the sinusoidal corrugation, which are the same as the conventional PHEs. However, the middle plate is different, which has high and low corrugations. By means of two kinds of plates alternately stacking, the channels for two heat-exchanging streams are geometrically distinct. Due to the fact that the ratio of the high corrugation to the low corrugation could be changed, it could achieve the design flexibility. Figure 2 demonstrates the actual model of the asymmetric plates, which are tested in this paper. This structure could make the flow area of the two channels different, achieving channel asymmetry.

Computational models and boundary conditions

Simulations are conducted in a part of the main flow field of the APHE using a commercial CFD code FLUENT 15.0 (ANSYS). The computational domain representing the fluid channel is part of Fig. 2, which is a double-channel model, including three plates. A comparison of the RNG k-e model, the realizable k-e model, and the shear stress transport k-w model are made, as depicted in Fig. 3. For the same mesh base size, the numerical results of the Nusselt number with the shear stress transport k-w model are closest to the experimental results. The shear stress transport k-w model can be advantageous in description of the turbulence shear stress transmission near the wall and flow separation in the negative pressure region [²⁰]. Therefore, the shear stress transport k-w model is adopted for simulations.

The fluid flow is set as the incompressible and steady-state. The properties of working fluids are constant.

The simulation boundary conditions are as follows:

(1) Because the experimental PHE in this paper consists of 40 plates, to be closer to actual conditions, on the top and bottom plate walls, the wall temperature is set as constant (300 K). The middle plate uses the coupling condition. No-slip boundary conditions are employed at the walls.

(2) The velocity-inlet and pressure-outlet boundary conditions are employed in the heat exchangers. The outlet pressures on both sides are set to atmospheric pressure.

(3) The model is simplified into 1/2 of the real model as the symmetry of the plates. Symmetric boundary conditions are used in the symmetric surface.

To reduce computation load and improve precision, the calculated plate length is set as 80 mm. The width of the model, half of the plate width, is equal to 55 mm. The unstructured tetrahedral grid is generated inside the computational domain performed in Gambit. The flow in the PHE channel is characterized by secondary flows, boundary-layer separation, and fluid stream between the braze junctions. Therefore, the features of the fluid channel such as corrugation crests, corrugations faces, and braze junction surfaces are grouped into different regions of the application of varying mesh dimensions in order to represent the actual flow conditions in a better manner. When the grid number increases from 8 × 10⁵ to 12 × 10⁵, the pressure drop in the cold channel increases less than 2%. Considering both the calculation speed and the accuracy, the grid number in this paper is set as around 8 × 10⁵. The turbulence intensity at the inlet is set at 10%. The simulation model of the APHE in the above calculation is constructed, as shown in Fig. 4. The upper and lower plates are conventional symmetric plates, having the same high corrugation height. The middle plate is a specific corrugated plate with unequal corrugation depths. The cross section diagram of this domain can be referred to in Fig. 1(b). In addition, the CPHE model is also set as the above method.

Table 2 lists the operation conditions and the geometry size of the simulated plates. The calculation method of the CPHE is similar to the APHE.

Experimental procedures

To study the performance of the asymmetric channels and validate the numerical model, experimental tests for the APHE are conducted in Jiangsu Baode Heat Exchanger Co. Ltd. The test APHE consists of 40 plates whose geometric parameters are tabulated in Table 3.

The heat transfer coefficient and pressure drop of the APHE are investigated with water on hot and cold sides. To cover the range of the Reynolds number in the PHE as far as possible and reduce the test data assessment, the orthogonal experimental design method is used [²⁴], which is one type of incomplete experimental design. This method cannot only test the index change condition, but also control the number of experiments. Approximately 20 test points are arranged. The flow rates are from 2 m³/h to 8 m³/h for both channels. The water temperature is from 10°C to 60°C.

The system consists of three main components, the hot water cycle, the cold water cycle, and the test section, as exhibited in Fig. 5. A 6 kW electrical heater installed inside a 60-L hot water tank is used to regulate the hot water temperature. The cold water cycle comprises of a 10 kW vapor compression refrigeration system. The temperature of hot and cold fluids is measured by four PT-100 thermometers with an accuracy of±0.1 K. Pressure transmitters (YOKOGAWA, EJA110 A) with an accuracy of±0.1% of its calibrated span are placed near the port of two streams to measure pressure drops. Pumps with needle valves are used to regulate the flow rate. The turbine type digital flow meters are used to measure the flow rates of both sides with an accuracy of±0.20% in the entire span. The energy balance deviation within 5% over a period of 30 min is set as stable premise to make sure that the conditions are in a steady-state.

Data reductions

The Reynolds number is defined as

(1)\mathrm{Re}={\displaystyle \frac{u{D}_{\mathrm{e}}}{\nu }},

where D_e means the hydraulic diameter, calculated by

(2)D_{\mathrm{e}}={\displaystyle \frac{\mathrm{4}{A}_{\mathrm{w}}}{C}},

where A_w is the cross section area, calculated by surface integral in Fluent; and C represents wetted perimeter

(3)C=\mathrm{2}l,

where l is the channel width.

For the CPHE, the hydraulic diameters are the same for both sides. For the APHE, the cross section areas of cold and hot channels are different (hot channels represent wide channels). D_{e, h} and D_{e, c} represent the hydraulic diameters of the hot and cold channels. Therefore, D_{e, h} and D_{e, c} are unequal. At the same fluid velocities and temperatures, Reynolds numbers of the hot channels and cold channels, Re_h and Re_c, are different too.

The friction factor is selected to characterize the flow resistance defined as

(4)f={\displaystyle \frac{\mathrm{2}{D}_{\mathrm{e}}\Delta p}{\rho u^{\mathrm{2}}L}},

where L is the length of the field along the mainstream direction.

The friction factor is evaluated as a piecewise continuous functions of channel. The Reynolds number with correlation constants.

(5)f=bR{e}^{-m}.

The total heat transfer coefficient k and Nusselt number are defined as

(6)k={\displaystyle \frac{q}{\mathrm{LMTD}}},

(7)Nu={\displaystyle \frac{h{D}_{\mathrm{e}}}{\lambda }},

where q is the average wall heat flux, LMTD is the logarithmic mean temperature difference, and h is the heat transfer coefficient of the cold stream or hot stream.

The range of Reynolds numbers of cold channels in the experiments is from 400 to 3400. Because the flow is turbulent when Re>200 in the CPHE [²⁰], it is assumed that the flow in APHEs is also turbulent within the range of the Reynolds number in the experiments and simulations.

The uncertainties of the data are calculated by the standard methods in Ref. [²⁵]. The uncertainties for temperature, pressure drop, flow rate, Reynolds number, Nusselt number, friction factor are presented in Table 4.

Results and discussion

Experimental validations

The accuracy of the double channel APHE model is validated with the experimental data, as shown in Fig. 6. The comparison between the pressure drops on the cold side in different Re_c calculated by the current method and the experiment is performed for validation. Tendency lines of the simulation and the measurement are almost consistent. The simulation pressure drops are in good agreement with the experimental values. The maximum discrepancy is 8.18%. The numerical heat transfer Nusselt number is also compared with the experimental data. The comparison is also made for various Re_c while the velocity in the hot side keeps constant. The simulation underestimates the heat transfer coefficient by 7% on average. In previous simulation validation, Lee and Lee [²⁶] validated the appropriateness of the numerical analysis with a maximum error of 10% for the friction factor between the numerical and experimental results. In Ref. [²⁷], the heat flux had a maximum deviation of 10% between the simulations and the measurements for Re= 821 also could create confidence about the model. The CFD model employed here is proved reliable.

Researches related to the CPHE model are relatively mature. The comparison of frictional factor between Ref. [²⁸] and the simulation results are shown in Fig. 7. The calculated data for the 60°chevron angle plate obtained from the correlation of Martin yield mean absolute deviations of 16.79% with the numerical data.

Comparison of heat transfer and flow characteristics of the APHE and the CPHE

A higher velocity leads to a better heat transfer, but the velocities in the channels are restricted by the allowable pressure drops of hot and cold fluids. With the same pressure drop limitations and fluid inlet parameters, the APHE and the CPHE usually have different heat transfer performances.

To find the advantages of the APHE over the CPHE, simulations are conducted in different conditions on two kinds of PHEs. In the calculation, plate corrugation parameters are the same, except the chevron depth. The allowable pressure drop is set as 60 kPa/m in this calculation, which is in the range of engineering practice.

In the first situation, the flow rate ratio of hot water to clod water (Q_h-c) is 2. The performance comparison of the APHE and the CPHE is given in Table 5.

It is found from Table 5 that the asymmetric plate yields a higher heat transfer coefficient. Although the velocity of hot water in the APHE decreases by 5%, the total heat transfer coefficient is enhanced by 22%. The pressure drops in the CPHE are unbalanced due to the flow rate difference between the two channels. The velocity of the cold water is restricted, but the APHE, in which the cold water flow area is smaller than the hot water flow area can have a higher cold side velocity. When the allowable pressure drops in both sides are equal, the cold side velocity of the APHE will be higher than that of the CPHE by 35% because of the difference of flow area ratio. When the water velocity increases, there will be more recirculation flow of water in the troughs of the corrugated channel. The recirculation flow can produce the turbulence of water flow inside the plate heat exchanger and thus lead to an enhancement of the heat transfer coefficient. Therefore, the overall heat transfer performance of the APHE is greater than that of the CPHE.

In the big difference between the two sides flow rates, the APHE can make full use of the allowable pressure drop while the CPHE cannot. Figure 8 is the contours of the static pressure in the channels of the APHE and CPHE under the condition in Table 5. For flow rate ratio of two, the pressure drops are more balanced in the APHE, which means that both the hot and cold fluids can make full use of their allowable pressure drops. The increase in cold side velocity compensates for the decrease in hot side velocity in the APHE. Therefore, when the flow ratio of hot water to cold water is two, under a similar corrugation configuration, the APHE proves to have a better heat transfer characteristics.

The cold side and hot side have different cross section areas and corrugated structures. To investigate the single-phase flow in the two plate channels of the APHE, the pressure distributions along the cold channel and the hot channel are plotted in Figs. 9 and 10, respectively. When the two channels have the same inlet velocity, typical pressure drop distributions are different. The pressure drops distribute in net shape as can been seen in the results of the CPHE in Ref. [⁸]. The contact points in PHEs have great effects on the flow and heat transfer characteristics. There are more contact points in the cold side, which have strong disturbing effects on the fluid flow. Therefore, the pressure drop is higher in the cold side at the same inlet velocity.

d_k is defined as the numerical total heat transfer coefficient increasing percentage, expressed as

(8){\delta }_k={\displaystyle \frac{k_{\mathrm{a}}-k_{\mathrm{c}}}{k_{\mathrm{c}}}},

where the subscript a and c represent APHE and CPHE, respectively.

Considering the effect of allowable pressure drop in the same flow rate and the chevron structure, it is found in Fig. 11 that the total heat transfer coefficient of the APHE is higher than those of the CPHE by 7.4%, 10.4%, 16.2%, 23%, 30% for allowable pressure drops of 16, 28, 43, 62, 80 kPa/m. The reason for this is that with the increase in the allowable pressure drop, the water velocity in the cold and hot channels of the symmetric plate increases, and the difference between the water velocities in both sides increases. This leads to more space in the available pressure drop that can be fully used by the APHE. Therefore, the APHE is more advantageous as the allowable pressure drop increases.

Besides the effect of pressure drop, the effect of the flow rate ratio of hot water to cold water is also taken into consideration. When the flow rates of hot water and cold water are the same, it can be seen in Table 6 that the asymmetric profile has a lower heat transfer rate compared to conventional symmetric sinusoidal surfaces, not revealing the advantages of the asymmetric profile. As observed from Table 6, because of the unequal flow area, the water velocity on the cold side in the asymmetric profile is higher than the water velocity on the hot side. The narrow channel has a larger pressure drop. To keep the flow rate the same, the velocity of the hot fluid in the APHE must be lower than that of the cold side. The pressure drop of the cold water in the APHE will first reach the limit as the velocity increases, which will result in the fact that the asymmetric profile cannot make full use of the allowable pressure drop in turn. The result shows that in this condition, using the asymmetric profiles will decrease heat transfer coefficient by 20%. Therefore, the conclusion can be reached that when the flow ratio is close to one, it is not recommended to use the APHEs.

The influence of the flow rate ratio of hot water to cold water is displayed in Fig. 12. Based on the present data, asymmetric profiles are more advantageous when the ratio increases. The total heat transfer coefficient of the asymmetric profile is enhanced by 22%, 18%, 15%, 11%, 0.4%, and -19.8% for the ratio of 2, 1.7, 1.5, 1.3, 1.2, and 1. This indicates that the flow rate ratio has an influential effect on the comprehensive performance. It is apparent that the decrease in the advantage of the asymmetric profile is caused by the decrease in the flow rate ratio. For the present asymmetric profile, the suitable application occasion is Q_h-c>1.3, on which the thermal performance is significantly improved. This can be explained by the principle that when allowable pressures on both sides are optimized the heat exchanger can be the most economical. When the flow rate ratio increases, the design is mainly controlled by the cold-side thermal requirement. The heat transfer enhancement of 22% is possible with asymmetrical channels when the flow rate ratio is two, allowing optimization of available pressure drops and maximum thermal efficiency of both fluids.

Influence of asymmetric index on the thermal performance

In this paper, the unequal chevron height plate is used. The important variable in the production of an asymmetric profile is N, the ratio of the low corrugation height to the high corrugation height, which has a value less than or equal to 1. N is defined as the asymmetric index

(9)N={{\displaystyle b}}_{\mathrm{2}}/{{\displaystyle b}}_{\mathrm{1}}.

When N = 1, which means the high corrugation is equal to the low corrugation, the plate is identical to the conventional symmetric plate. As the value of N decreases, the level of asymmetric geometry becomes large. To find the influence of asymmetrical index on the heat transfer performance, five types of plates at different N values of 0.20, 0.38, 0.50, 0.59, and 0.77, are simulated and analyzed. The D_{e, h} and D_{e, c} for different N values are shown in Table 7.

The cross-sectional area of the wide side increases and that of the narrow side decreases as N decreases, as expected. Figure 13 shows the cross-sectional area ratio of wide (A_h) to narrow side (A_c) and D_e for various N values. According to this, a larger range of N values can be analyzed and the flow area ratio is approximately from 1.1 to 1.7 in this paper. These profiles are calculated numerically to predict the hydraulic-thermal performance.

The effect of asymmetric index on the heat transfer performance in the APHE is depicted in Fig. 14. d_A is the reduction percentage of the heat transfer surface area of APHE as compared to CPHE in the same heat capacity when the flow rate ratio of hot water to clod water is 2. d_A is calculated by

(10){\delta }_A={\displaystyle \frac{A_{\mathrm{c}}-A_{\mathrm{a}}}{A_{\mathrm{c}}}}.

The heat transfer surface area (A) is calculated

(11)A={\displaystyle \frac{Q}{k\cdot \mathrm{LMTD}}}.

The heat transfer areas needed for various cases are compared under the same heat capacity and LMTD. It is observed that asymmetric heat exchangers whose N values are more than 0.38, wherein different flow channels exhibit different flow features concerning flow resistance, have less heat transfer area than the CPHE. But when N is 0.20, the heat transfer performance is less than the CPHE in this condition. This means that the heat transfer coefficient can be enhanced in a certain range of N values, which can be explained by Fig. 15. The pressure drops in both sides in the conventional plate heat exchanger are greatly different, the pressure drop in the hot side being almost twice as much as that in the cold side. This will lead to the exchanger designs controlled by the pressure drop in the hot side, and the cold side is inefficient. As the N value decreases, the pressure drops are more balanced. When N = 0.38, there is a difference of 6% between the pressure drop in the hot and cold side. Therefore, the available pressure drops in both sides are nearly utilized and the heat transfer area is substantially decreased, 20.6% less than the CPHE. It is found that the decrease in N values (N≥0.38) causes the decrease in the heat transfer area at the same heat load. The heat transfer area of the asymmetric profiles is decreased by 20.6%, 18.0%, 16.1%, and 14.1% for N values of 0.38, 0.50, 0.59, and 0.77, respectively, compared to that of the CPHE. The reason for this is that with the pressure drop in the cold side being fully utilized, the velocity of the cold water rises, which leads to a high disturbance of water inside the plate heat exchanger. Therefore, the heat transfer coefficient increases when N is decreasing. But it can be seen that when N = 0.20, the exchanger designs will be controlled by the pressure drop in the cold side. The velocity in the hot side will decrease more quickly which leads to a lower heat transfer rate. Therefore, the heat transfer performance of APHEs is not always greater than that of the CPHE, which is decided by the operating condition, heat transfer, and resistance characteristic.

The results in this paper are compared with those in some recent researches. Kim et al. [¹³] studied the application of APHE in the intercooler. A variable (N) was introduced to describe the asymmetry, whose definition differed from that given in this paper. Their results coincided with the conclusions stated earlier, which showed that the asymmetric plate heat exchanger could make the pressure drops in both sides balanced. They found that the weight of the design heat exchanger matrix decreases with the level of asymmetric increasing. They also found that the effect of the asymmetric profile was highly sensitive to the giving operating conditions. It is concerted with the conclusion that the flow rate ratio of hot water to cold water and the allowable pressure drop influence the performance of the APHE in the present paper. Lee et al. [¹⁷] investigated a new type of asymmetric plate exchanger. The similarity between the work of Lee et al. and the present paper is pointing out the distinct characteristics of the two channels in the PHE, although their PHE has been used for the heat pump. The present paper differs from former researches in asymmetric profiles and application. However, these researches indicate the principle that the fluid flow and heat transfer in both sides of the heat exchanger should be kept balanced. Besides, the increase in the value of N can make the hydraulic performance of both sides balanced, leading to the enhancement of the overall heat transfer coefficient. It will be instructive to make wider use of APHEs. Lee et al. [¹⁷] also indicated that the APHEs could be advantageous in phase-change heat transfer. The application prospect is wide in heat pumps and refrigeration. The performance of the asymmetric profiles in phase-change still needs to be explored.

Correlations for the APHE with different size

Previous correlations for the APHEs have limitations because only one type of geometry is considered. Thus, correlations of the Nusselt number and friction factor are obtained for different asymmetrical indexes. The calculation of convection coefficients is crucial in developing the experimental correlations. For the conventional PHEs, the Wilson plot method and the equal Reynolds number method are often used in dealing with the determination of convection coefficients [²⁹]. However, these methods are not suited to the present paper because they all need similar characteristics of the cold and hot channels. As for the APHEs, the geometric structures of both sides are distinct. Therefore, the least square method is used to develop the correlations of Nusselt number for the asymmetric PHEs in the present paper.

The heat transfer coefficient of the hot fluid side can be calculated as

(12)N{u}_{\mathrm{h}}=C_{\mathrm{h}}R{e}_{\mathrm{h}}^{m_{\mathrm{h}}}P{r}_{\mathrm{h}}{}^{\mathrm{0.3}}{\left({\displaystyle \frac{N{d}_{\mathrm{h}}}{d_{\mathrm{0}}}}\right)}^{\mathrm{e}},

where d₀ is a constant which equals 0.008 m.

The heat transfer coefficient of the cold fluid side can be calculated as

(13)N{u}_{\mathrm{c}}=C_{\mathrm{c}}R{e}_{\mathrm{c}}^{m_{\mathrm{c}}}P{r}_{\mathrm{h}}{}^{\mathrm{0.4}}{\left({\displaystyle \frac{N{d}_{\mathrm{c}}}{d_{\mathrm{0}}}}\right)}^f.

The overall heat transfer coefficient K of the APHEs can be expressed as

(14){\displaystyle \frac{\mathrm{1}}{k}}-R={\displaystyle \frac{\mathrm{1}}{h_{\mathrm{h}}}}+{\displaystyle \frac{\mathrm{1}}{h_{\mathrm{c}}}},

where R is the remaining thermal resistance and the subscripts c and h represent the cold and hot water sides.

Substituting Eq. (12) and Eq. (13) into Eq. (14) can give

(15)\begin{array}{l}{\displaystyle \frac{\mathrm{1}}{k}}-R={\displaystyle \frac{\mathrm{1}}{C_{\mathrm{h}}}}{\displaystyle \frac{\mathrm{1}}{({\lambda }_{\mathrm{h}}/d_{\mathrm{h}})P{r}_{\mathrm{h}}^{\mathrm{0.3}}}}{\left({\displaystyle \frac{\mathrm{1}}{R{e}_{\mathrm{h}}}}\right)}^{m_{\mathrm{h}}}{\left({\displaystyle \frac{N{d}_{\mathrm{h}}}{d_{\mathrm{0}}}}\right)}^e+\\ {\displaystyle \frac{\mathrm{1}}{C_{\mathrm{c}}}}{\displaystyle \frac{\mathrm{1}}{({\lambda }_{\mathrm{c}}/d_{\mathrm{c}})P{r}_{\mathrm{c}}^{0\mathrm{.4}}}}{\left({\displaystyle \frac{\mathrm{1}}{R{e}_{\mathrm{c}}}}\right)}^{m_{\mathrm{c}}}{\left({\displaystyle \frac{N{d}_{\mathrm{c}}}{d_{\mathrm{0}}}}\right)}^f.\end{array}

Sixteen calculation points are investigated for each asymmetrical index. The working fluid in the correlations is water. The linear regression of the simulation results is used with the least-squares method to obtain the constant C_h, C_c, m_h, m_c, and e. The derived correlation for Nu_h is

(16)N{u}_{\mathrm{h}}=\mathrm{0.4082}R{e}_{\mathrm{h}}^{\mathrm{0.6546}}P{r}_{\mathrm{h}}{}^{\mathrm{0.3}}{\left({\displaystyle \frac{N{d}_{\mathrm{h}}}{d_{\mathrm{0}}}}\right)}^{\mathrm{0.6118}}.

Similarly, the derived correlation for Nu_c is

(17)N{u}_{\mathrm{c}}\mathrm{=1}\mathrm{.1556}R{e}_{\mathrm{c}}^{\mathrm{0.4833}}P{r}_{\mathrm{c}}{}^{\mathrm{0.4}}{\left({\displaystyle \frac{N{d}_{\mathrm{c}}}{d_{\mathrm{0}}}}\right)}^{-\mathrm{0.0251}}.

The resistance characteristic of both sides is different because of the unequal flow area. The derived correlation for f of the cold side is

(18)f_{\mathrm{c}}=\mathrm{9.157}R{e}_{\mathrm{c}}{}^{-\mathrm{0.090}}{N}^{-\mathrm{0.0777}}.

The derived correlations for f of the hot side is

(19)f_{\mathrm{h}}=\mathrm{9.287}R{e}_{\mathrm{h}}{}^{-\mathrm{0.103}}{N}^{\mathrm{0.2459}}.

The application ranges of the above correlations (Eq. (16)-19) are 400<Re_c<12000, 400<Re_h<12000, 4<Pr_c<10, 4<Pr_h<10, and 0.2<N<0.8.

As shown in Fig. 16, the calculated data of the total heat transfer coefficient are plotted against the fitting data by Eqs. (16) and (17). The predicted results have a good agreement with the simulation data, with a deviation of±15%.

The correlations of Nu_h and Nu_c (N = 0.5) obtained from the simulation results are also compared with the experimental data, as plotted in Fig. 17. At the same Reynolds number, the deviation between the simulation and the experiment is less than 8%. Figures 18 and 19 are the f_c and f_h of different N and their fitting curves. The mean absolute deviations of prediction are 4.32% and 2.89% for the hot side friction factor and the cold side friction factor, respectively.

In Section 3.2, the allowable pressure drops in both sides are set equal. It is found that the balanced hydraulic performance of two watersides could lead to the enhancement of the overall heat transfer coefficient. According to the correlations of f_c and f_h, the actual operating conditions in which the allowable pressure drops in both sides might be unequal can be calculated. The heat transfer will be enhanced to the most extent when both the hot fluid and the cold fluid can make full use of their allowable pressure drops.

As N changes, the channel resistance is one of the factors for performance changing. The influence of N on resistance performance could be found in Fig. 20. By decreasing the asymmetric index, f_c increases but f_h decreases. The different resistance characteristics of both sides make the possibility of the allowable pressure drops to be utilized.

Suitable application range of asymmetric profiles at different N values

The lack of APHEs knowledge might be a reason for their rare application in the market. For the APHE investigated here, two of the most important factors are the flow rate ratio and N value.

In Section 3.2, the flow ratio range in which the APHE (N = 0.5) performs better than the CPHE has been illustrated. A critical flow rate ratio exists when the heat transfer coefficients of the CPHE and APHE are the same. When the actual ratio is higher than the critical ratio, the APHE will have a better performance. For APHE with other asymmetries, the critical ratio may be decided as follows: ① with the allowable pressure drops specified, the maximum velocities in both channels are obtained; ② at a given flow rate ratio, the Nusselt correlations are used to calculate the total heat transfer coefficients for the CPHE and APHE; and ③ different flow rate ratios are tried until the heat transfer coefficients for the CPHE and APHE are equal, the corresponding ratio is deemed as the critical ratio.

For APHEs with various asymmetries, the critical flow rate ratio will be different. The corresponding curve will divide the plane of the flow rate ratio versus N into two parts, as shown in Fig. 21. The allowable pressure drop is set as 60 kPa/m in the simulation. In the blank region (Region 1), APHEs have higher heat transfer coefficients than CPHEs; in the filled region (Region 2), the contrary holds. With the help of Fig. 21, the situations under which APHEs are more efficient could be identified.

Correlations for the APHE with different working fluids

The APHEs have wide application and development prospects in refrigeration equipment and various fluids. Previous investigations on the APHEs have limitations because only water has been used as the working fluid. Different working fluids usually have different thermodynamic characteristics. Thus, five fluids are numerically tested in a larger range of Prandtl number. The N value of the tested APHE is 0.5. Numerical analyses are performed with water (4<Pr<10), ethyl alcohol (10<Pr<20), ethylene glycol (20<Pr<30), gasoil (30<Pr<50), and diesel fuel #2 (50<Pr<70).

The Nusselt number correlation for the working fluids in the wide side is

(20)N{u}_{\mathrm{h}}=\mathrm{0.1428}R{e}_{\mathrm{h}}^{0\mathrm{.701}}P{r}_{\mathrm{h}}{}^{\mathrm{0.38}}.

The Nusselt number correlation in the narrow side is

(21)N{u}_{\mathrm{c}}=\mathrm{0.2842}R{e}_{\mathrm{c}}^{0\mathrm{.656}}P{r}_{\mathrm{c}}{}^{0\mathrm{.48}}.

The application ranges are 400<Re_c<12000, 400<Re_h<12000, 4<Pr_c<70, and 4<Pr_h<70. The coefficient for the determination of the above correlations, R², is greater than 0.9.

The correlations of Nu_h and Nu_c of different fluids obtained from the simulation results are compared with the numerical data, as presented in Fig. 22.

The friction factor is usually independent of the Prandtl number, and therefore, previous correlations for the friction factor can be used [²⁶], which can also be proved by the simulation results as shown in Fig. 23. Equations (22) and (23) can be applied to the APHE using different working fluids (4<Pr<70).

(22)f_{\mathrm{h}}=\mathrm{7.832}R{e}_{\mathrm{h}}{}^{-\mathrm{0.103}},

(23)f_{\mathrm{c}}=\mathrm{9.664}R{e}_{\mathrm{c}}{}^{-\mathrm{0.090}}.

The coefficient for determination is greater than 0.9, and the application ranges are 400<Re_c<12000, and 400<Re_h<12000.

Asymmetrical index evaluations via Nu/f^1/3

To investigate the comprehensive performance of different asymmetric indexes, the index Nu/f^1/3 is used to evaluate the energy efficiency. Nu/f^1/3 has been used as heat exchanger energy efficiency to quantify the comprehensive performance. Because the performances of both sides in APHEs are different, the performance coefficients of hot channels and cold channels are calculated, respectively. Moreover, to comprise with the symmetric plate, the symmetric plate curve is also plotted. A higher Nu/f^1/3 means that the profile will have a higher heat transfer coefficient when the pumping power is identical.

Figure 24 shows that the Nu/f^1/3 at cold sides are larger than that at hot sides, which means that the narrow channels of APHEs usually have a better heat transfer performance than the wide channels. The reason for this is that cold channels have more contact points. As for N, the results show that the increase in N value leads to a higher Nu/f^1/3 at hot sides. This means the increase in low chevron depth (b₂) could enhance the heat transfer performance of hot sides. But the increase in N value only brings a slight decrease in the comprehensive performance in cold sides. This is due to the fact that the change of N leads to similar changes of Nu_c and f_c. Therefore, in the application of different asymmetrical index plates, the flow rate ratio and energy efficiency need to be a comprehensive consideration.

Conclusion

The thermal and hydrodynamic characteristics of one type of APHE are experimentally and numerically investigated. The shear stress transport k-w model is adopted in CFD simulation after validation with the experimental data. The performances of five APHEs at different asymmetric indexes (N = 0.2, 0.38, 0.5, 0.59, and 0.77) are simulated. It is shown that APHE can offer significantly higher heat transfer performance than CPHE if the flow rate ratio of the hot side to the cold side satisfies a certain condition. At N = 0.5 and a flow rate ratio of 2, the enhancement of heat transfer can be up to 22%. For the special case of water-water heat exchange, the critical flow rate ratios have been identified for various N values. When the actual ratio is higher than the critical ratio, APHE performs better than CPHE. Further simulations with more fluid types lead to the development of common correlations for Nusselt number and friction coefficient. Using Nu/f^1/3 as the performance evaluation index, it is found that N has different effects on the performances of hot and cold sides. The present paper illustrates the comparative advantages of APHEs. The results can provide useful guides for the applications of the APHE investigated.

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