A coupled modeling framework for investigating flow maldistribution in cross-flow-corrugated plate heat exchangers Open Access

Plate heat exchangers have shown benefits over other heat exchangers in terms of high heat transfer characteristic, compact structure, and ease of cleaning.^1,2 These features make plate heat exchangers promising candidates for many applications in, for example, the chemical industry, electric power systems, medicine, food production, chemical fiber manufacture, shipping, and heating.³ In these applications, the most favorable size and configuration are of the first concern, with the aim being to provide the required heat transfer performance within a specified pressure drop.⁴ The cross-flow-corrugated plate heat exchanger is designed under an assumption of uniformity, ignoring the negative effects of flow maldistribution, but this is not realistic for real-world operating conditions.⁵ High flow maldistribution conditions are inevitably accompanied by an interior temperature maldistribution⁶ and increased pressure drop,^7,8 resulting in deterioration of capacity and efficiency.^9,10 For example, a low or moderate degrees of air maldistribution will cause a 5%–10% deterioration in effectiveness and a 10%–20% increase in pressure drop.¹¹ Thus, precise prediction of the thermo-hydraulic behavior caused by flow maldistribution is essential for effective thermal design and optimization of heat exchangers.

Considering the key role of flow maldistribution in performance deterioration, much effort has been dedicated to the development of methods for predicting the impact of flow maldistribution.^12–15 Beckedorff et al.¹³ analyzed the flow and heat transfer characteristics of a plate and shell heat exchanger. They established a thermo-hydraulic model to predict the effects of flow maldistribution on the heat transfer coefficient. Their results showed that flow maldistribution causes an increase in the heat exchanger pressure drop and a deterioration in heat transfer performance, with a deterioration in effectiveness of up to 4%. Zhang et al.¹⁶ investigated eight plate heat exchangers with different distributor angles and hole diameters using both experimental and numerical approaches. Tokhtarov et al.¹⁷ analyzed the impact of flow maldistribution on multichannel systems. Their results indicated that improper distribution of fluid flow reduced heat transfer and disrupted the biphasic system that they considered. However, previous mathematical models have been based on parallel-flow-corrugated or counter-flow-corrugated heat exchangers, and models of cross-flow-corrugated plate heat exchangers remain scarce. In addition, the effects of flow maldistribution of one side on the performance of the other side and on the heat exchanger as a whole also remain uninvestigated. Thus, a more effective and simple mathematical model for a cross-flow-corrugated plate heat exchanger that can quantitatively estimate the influence of the side stream distribution on thermo-hydraulic performance is urgently needed.

In the present work, an efficient coupled mathematical model, consisting of a heat transfer model and a hydraulic model, is established to quantitatively calculate the effects on thermo-hydraulic performance of flow maldistribution in a cross-flow-corrugated plate heat exchanger. This coupled model provides a complete solution for the heat transfer characteristics, heat exchanger effectiveness, and pressure drop. An experimental system is also designed to validate the mathematical model.

Furthermore, the effects of the nonuniformity of one side stream on the other hand and on the overall performance are investigated for different inlet Reynolds numbers and inlet temperatures. Finally, analyses are also conducted of flow distribution nonuniformity and of thermo-hydraulic performance, including deteriorations in heat exchanger effectiveness and deviations in pressure drop. The results are expected to pave the way for optimizing the design of a cross-flow-corrugated plate heat exchanger to reduce the effects of flow maldistribution.

The structure of the core of a plate heat exchanger is presented in Fig. 1. It consists of two types of plates flanged in different directions, with each plate having 12 cylindrical protuberances. The folding edges ensure that the two fluids can flow into different channels, forming a cross-flow configuration. There are 20 air channels and 20 gas channels, and the each air channel is sandwiched between two gas channels.

As illustrated in Fig. 2, two 3D physical models with nonuniform inlet air and gas flows are constructed for numerical simulation. Both of these models are composed of 40 plates, constituting 20 gas channels and 20 air channels, and each plate has 12 cylindrical protuberances of diameter 20 mm. The plate size is 500 × 415 mm², and the plate spacing is 12 mm. The header length is 320 mm and the diameter of the inlet duct is 168 mm. The arrangement of the fluid flow is such that one stream flows through the layer sandwiched between the other stream.

The following assumptions are made:^18,19 (1) the volume flow rate is constant on both sides; (2) the fluids are incompressible Newtonian fluids; (3) the effects of buoyancy lift and of gravity are negligible on both sides; (4) thermal radiation and viscous dissipation are negligible; (5) the solid surface is adiabatic; (6) there are no thermal energy sources or thermal sinks; (7) there is no flow stratification, flow bypassing, or flow leakage on either side.

The heat exchanger effectiveness ε is adopted to evaluate the thermal performance of the heat exchanger.^20,21 Thus, an ε-NTU-based heat transfer model is established to analyze the effect of flow maldistribution on heat exchanger effectiveness. Two operating conditions are considered in the modeling procedure, namely, maldistribution of the hot fluid and maldistribution of the cooling fluid.

1. Nonuniform hot fluid. In this case, ε can be expressed as follows:

   $ε=(mCp)h(Th,i−Th,o)(mCp)min(Th,i−Tc,i),$

   (2)

   where C_p is the specific heat at constant pressure, m is the mass flow rate, subscripts h and c indicate the hot and cold fluids, respectively, and subscripts i and o indicate the inlet and outlet, respectively. The average outlet temperature can be calculated as follows:

   $Th,ō=∑j=120m(j)hT(j)h,o∑j=120m(j)h,$

   (3)

   where j denotes a pass of the hot fluid. Therefore, ε due to maldistribution of the hot fluid flow can be expressed as follows:

   $εh,non=Th,i−Th,ōTh,i−Tc,i,$

   (4)

   $εnon=εh,non,(mCp)h=(mCp)min,εh,non/C*,(mCp)h=(mCp)max,$

   (5)

   $C*=(mCp)min(mCp)max.$

   (6)
2. Nonuniform cold fluid. In this case, ε can be expressed as follows:

   $ε=(mCp)c(Tc,o−Tc,i)(mCp)min(Th,i−Tc,i),$

   (7)

   with the average outlet temperature being given by

   $Tc,ō=∑j=120m(j)cT(j)c,o∑j=120m(j)c,$

   (8)

   where j denotes a pass of the cold fluid.
   Therefore, ε due to the maldistribution of the cold fluid flow can be expressed as follows:

   $εc,non=Tc,ō−Tc,iTh,i−Tc,i,$

   (9)

   $εnon=εc,(mCp)c=(mCp)min,εc/C*,(mCp)c=(mCp)max.$

   (10)
   According to Mochizuki et al.,²² when the fluids are in cross-flow and distributed uniformly, the effectiveness can be expressed as follows:

   $εuni=1−exp(−NTU)−exp[−(1+C*)NTU]×∑n=1∞C*nPn(NTU),$

   (11)

   $Pn(y)=1(n+1)!∑j=1nn+1−jj!yn+j,$

   (12)

   $NTU=(Th,i−Th,o)/ΔTm,(mCp)h=(mCp)min,(Tc,o−Tc,i)/ΔTm,(mCp)c=(mCp)min.$

   (13)
   Therefore, the rate of deterioration of heat exchanger effectiveness due to flow maldistribution can be expressed as

   $Δε=εuni−εnonεuni.$

   (14)

The following boundary conditions are applied:

1. Nonuniform inlet air flow. Velocity inlet and pressure outlet boundary types are adopted. For the air side, T_in-air = 300 K. For the gas side, u_in-gas = 1 m/s and T_in-gas = 823, 923, 1023 K.

2. Non-uniform inlet gas flow. Velocity inlet and pressure outlet boundary types are adopted. For the gas side, T_in-gas = 1023 K. For the air side, u_in-air = 1 m/s and T_in-air = 290, 300, 310 K.

Non-slip velocity boundary conditions are applied at all the solid surfaces of the computational region. The shell walls are adiabatic. Symmetry boundary conditions are adopted at the center plane of the heat exchanger. The plates are assumed to be made of steel. The properties of the working fluids and plate material are presented in Table I.

The governing equations for both sides are discretized by the finite volume method (FVM). In consideration of the improved predictions of near-wall flows, the renormalization group (RNG) k-ε model is adopted in the simulation.²⁴ The convection terms are discretized using a second-order upwind scheme. There is a huge difference in temperature, and so radiation has to be considered in the analysis and the discrete ordinates (DO) model is adopted on the coupled wall and the shell wall. A coupled pressure-based solver is adopted, and the coupling of velocity and pressure is solved with the semi-implicit method for a pressure-linked equation (SIMPLE).

In consideration of the accuracy and the time cost of the CFD simulations, a grid-independence study is performed. The inlet air and gas temperatures are set as 300 and 573 K, respectively, and the inlet air and gas velocities as 10 and 12 m/s, respectively.

Figure 3(a) shows the boundary grid in the computational domain, where the meshes are unstructured tetrahedral meshes. Figure 3(b) presents a comparison of the results from different grids for the velocity at the outlet of the air side, where the X coordinate ranges from 0 to 0.25 m. It demonstrates that there is mesh independence for grid sizes of 2 295 840, 2 920 800, 3 814 320, and 7 144 640, since the maximum error in the velocity is 6.587%. In addition, Table II presents the mesh dependence for these four different mesh sizes. A mesh is characterized as “good” if the maximum skewness is less than 0.7 for the hexahedrons and tetrahedrons. Therefore, 2 920 800 grids are used in the numerical simulations, considering the balance between accuracy and workload.

The experimental apparatus consists of (1) the plate heat exchanger, (2) the air system, (3) the gas system, and (4) the measurement equipment. A schematic representation of the experimental system is presented in Fig. 4. The heat exchanger has 40 plates, with 20 gas channels and 20 air channels, each plate having cylindrical protuberances with a diameter of 20 mm. The plate size is 500 × 415 mm², with a plate spacing of 12 mm. Furthermore, the entire test rig is contained in a room with constant temperature and constant humidity.

The volume flow rate passing through the plate heat exchanger is measured by a highly sensitive vortex street flowmeter. The pressure drop is measured by pressure transducers. The inlet and outlet temperatures are measured using two pt-100 thermocouples inserted into the flow line. Twelve K-type thermocouples are welded onto different positions to measure the wall temperature distribution. The uncertainties in the temperature, pressure drop, air flow rate, and gas flow rate are 0.866%, 2.043%, 1.289%, and 1.289%, respectively. The uncertainties in the indirectly measured data are calculated using the method proposed by Mochizuki et al.²² 

Moreover, the simulated center flow rate is higher than the experimental value, while the simulated flow rate at the fringe is less than the experimental value, as shown in Fig. 5. These results can be attributed to the fact that in the experiment, the entrance of air is affected by thermal radiation from the gas, and the air is diffused to the surroundings, resulting in an increase in the edge velocity. However, in the simulation process, the shell walls of both sides are set to be adiabatic. Because of the heat dissipation in the experiment, the temperature difference of the gas in the experiment is larger than that in the simulation, while on the air side, the difference is smaller than the simulated value; thus, P is slightly overestimated, while R is underestimated. Therefore, the model used in the present study is reliable and can be used to predict heat exchanger performance.

To analyze the effect of the air maldistribution and the gas maldistribution on the performance of the heat exchanger, a physical model with one inlet duct for either the air side or the gas side is used in the numerical simulation. That is the air/gas on one side is maldistributed at the inlet, while that on the other hand is uniform at the inlet.

Yang and co-workers^25,26 reported that an increase in the inlet Reynolds number resulted in an increase in the maldistribution parameter. Therefore, in this subsection, the nonuniformity is changed by changing the inlet Reynolds number. In Fig. 7, 30 points, indicated by the red dots, are located on the outlet of the core of the air and gas flow to obtain nonuniform air and gas velocity distribution data.

Figure 8 shows the variation of the air flow distribution nonuniformity parameter with the inlet gas Reynolds number for different inlet air temperatures. The inlet gas temperature is 1023 K, with inlet gas Reynolds numbers of 1153, 1537, 1922, 2306, and 2691. The inlet air temperatures are 310, 300, and 290 K. It can be seen that the air flow distribution nonuniformity parameter is high, ranging from 0.566 to 0.570, indicating that gas maldistribution will result in a strongly nonuniform air distribution. However, the air flow distribution nonuniformity parameter shows no obvious response to with an increase in the inlet gas Reynolds number, with the variation being below 2%. The results here also reveal that the air nonuniformity caused by gas maldistribution first decreases and then increases, with a minimum value at Re = 1922. Thus, we conclude that the air velocity distribution is affected by the nonuniformity of the gas, since the gas maldistribution causes plate temperature maldistribution, leading to air temperature nonuniformity in different channels. A channel with a larger gas flow rate corresponds to a channel with a higher air temperature as well as a lower density. Furthermore, because of the uniform inlet air flow rate, the lower the density, the greater is the velocity, resulting in maldistribution on the air side.

Figure 9 shows the variation of the gas flow distribution nonuniformity parameter with the inlet air Reynolds number for different inlet gas temperatures. The inlet air temperature is 300 K, with inlet air Reynolds numbers of 5980, 8971, 11 961, 14 951, and 17 942. The inlet gas temperatures are 1023, 923, and 823 K. It can be seen that the gas flow distribution nonuniformity parameter is small, ranging from 0.069 to 0.074 for all the inlet air Reynolds numbers, and so the gas flow distribution appears to be affected little by the air maldistribution. Meanwhile, the gas flow distribution nonuniformity parameters changes little with an increase in the inlet air Reynolds number. The results here indicate that nonuniformity of the air has little effect on the gas velocity distribution and will not cause maldistribution on the gas side. This could be explained by the fact that as the plate temperature is mainly controlled by the gas fluid, maldistribution on the air side has little effect on the density and velocity of the gas. Therefore, the distribution on the gas side is not affected by air nonuniformity.

Figure 10 shows the variation of the heat exchanger effectiveness deterioration rate Δε with the inlet gas Reynolds number for inlet air temperatures of 310, 300, and 290 K. The value of Δε varies in the range of 10%–30% when the inlet gas Reynolds number increases. Δε changes little when the air temperature changes. As discussed above, when the inlet velocity of the gas increases, the air nonuniformity parameter changes little (see Fig. 8); therefore, there is no significant variation in Δε with an increase in the inlet gas velocity. Δε first increases and then decreases as the inlet gas Reynolds number increases, with a peak at Re = 1922. The reason for this is that the air nonuniformity caused by gas maldistribution first decreases and then increases, with a minimum value at Re = 1922 (see Fig. 8), and with increasing inlet air temperature, the nonuniformity increases and the loss of heat decreases, offsetting the effect of the deterioration in the heat transfer rate. Therefore, Δε differs little at different inlet air temperatures for the same inlet gas Reynolds number.

Figure 11 shows the variation in Δε with the inlet air Reynolds number for inlet gas temperatures of 1023, 923, and 823 K. The value of Δε changes from 16% to 43% when the inlet gas temperature is 1023 K, from 10% to 32% for 923 K, and from 2.5% to 23% for 823 K. It is can be seen that Δε increases when the inlet air Reynolds number increases, implying that the heat transfer performance deteriorates when the air nonuniformity parameter increases. Further, Δε increases with increasing temperature difference between the gas and air flows. As indicated in Fig. 9, inlet nonuniformity on the air side has little effect on the distribution of the gas flow. When the air inlet velocity distribution is nonuniform, the air flow rate is high in the middle. Therefore, most of the energy of the edge region of the gas has not been fully used, resulting in heat loss and leading to a deterioration in effectiveness. As the inlet air velocity increases, the nonuniformity on the air side increases and less heat of the gas is used, resulting in an increase in Δε. Moreover, because of the limited heat absorption capacity of the air, the greater the temperature difference, the more heat is lost, leading to the larger Δε.

Figure 12(a) shows the relationship between the pressure drop on the air side and the inlet gas Reynolds number at inlet air temperatures of 310, 300, and 290 K. The uneven gas flow causes an increase in pressure drop on the air side, while the pressure drop shows little response to changes in the inlet gas Reynolds number. Moreover, the difference in the pressure drop on the air side decreases with decreasing inlet air temperature. The pressure drop difference changes from 12.4% to 3.26% with a change in the inlet air temperature from 310 to 300 K, and from 3.26% to 1.05% when the inlet air temperature changes from 300 to 290 K. When the temperature difference between the gas and the air increases, the pressure drop difference on the air side decreases, which is in accordance with the conclusion drawn in Sec. III C. Furthermore, an increase in the temperature difference leads to a decrease in air nonuniformity, resulting in a reduction in the pressure drop difference. Figure 12(b) shows the pressure drop increase on the gas side caused by the gas flow maldistribution. Compared with an idealized uniform gas flow, the total pressure drop increases by ∼13%–54% for all the considered inlet air temperatures. Furthermore, the curves in Fig. 12(b) exhibit an increasing trend with increasing inlet gas Reynolds number. The inlet air temperature has little effect on the pressure drop on the gas side. Comparison of Figs. 12(a) and 12(b) shows that the pressure drop increase caused by own-side maldistribution is considerably larger than the increase caused by the other side.

Figure 13(a) shows the relationship between the pressure drop on the gas side and the inlet air Reynolds number for inlet gas temperatures of 1023, 923, and 823 K. The pressure drop on the gas side with a nonuniform air flow is higher than that when the air distribution is uniform, but the inlet air Reynolds number has little effect on the pressure drop on the gas side. When the inlet gas temperatures are 1023, 923, and 823 K, the pressure drop increases by ∼5.55%, 5.43%, and 5.33%, respectively. Thus, the temperature difference does not affect the pressure drop. This could be explained by the fact that maldistribution on the air side has little effect on gas flow nonuniformity, and so the pressure drop increase on the gas side is small. Figure 13(b) shows the relationship between the pressure drop on the air side and the inlet air Reynolds number. As the inlet air Reynolds number increases, the change in the air pressure drop exhibits an upward trend. Further, compared with an idealized uniform air flow, the total pressure drop increases by 40.85%–58.67%, 41.02%–58.23%, and 40.75%–58.60% for inlet gas temperatures of 1023, 923, and 823 K, respectively. Thus, the inlet gas temperature has little effect on the pressure drop on the air side. Comparison of Figs. 13(a) and 13(b) shows that the increase in pressure drop caused by own-side maldistribution is considerably larger than the increase caused by the other side.

In this work, a coupled mathematical model of a cross-flow-corrugated plate heat exchanger, consisting of a thermal model and a hydraulic model, has been developed to quantitatively evaluate the impact of flow maldistribution on thermo-hydraulic performance. In contrast to previous work focusing on counter-flow or parallel-flow heat exchangers, an effective and simple mathematical model for a cross-flow-corrugated plate heat exchanger has been developed here. The numerical results are in good agreement with experimental data, with the errors of 9.04%, 11.14%, and 11.39% for the velocity, thermal effectiveness P, and heat capacity rate ratio R, respectively. A flow distribution nonuniformity parameter has been proposed to facilitate evaluation of the flow nonuniformity of the working fluids. Using the coupled model, a flow maldistribution analysis and a thermal performance analysis considering heat exchanger effectiveness deterioration and pressure drop deviation have been conducted. The CFD results confirm that inlet flow maldistributions have considerable effects on heat exchanger performances.

The coupled model incorporates a complete solution for the heat transfer characteristics, heat exchanger effectiveness, and pressure drop. The effects of nonuniformity of one side stream on the other hand and the overall performance of cross-flow-corrugated heat exchangers have been investigated for different inlet Reynolds numbers and inlet temperatures.

In summary, the following conclusions can be drawn from this study:

1. Gas flow maldistribution will result in air flow maldistribution when the inlet gas Reynolds number is in the range 1100–2700, whereas a nonuniform air flow has no effect on the gas side for inlet air Reynolds numbers ranging from 6000 to 18 000. The effects of temperature differences between the two working fluids on the gas and air flow maldistributions are below 7% and 2%, respectively.

2. When the gas flow is nonuniform, the heat exchanger effectiveness deterioration rate Δε is in the range 10%–30% for 1100 < Re < 27 000. Temperature differences between the two working fluids have little effect on Δε. The heat exchanger effectiveness deterioration rate increases with increasing air flow nonuniformity for 6000 < Re < 18 000, and it increases with increasing temperature difference.

3. When the distribution of the two working fluids is nonuniform, the pressure drops on the gas and air sides increase by ∼5% and less than 12.4%, respectively. Furthermore, the increase in pressure drop caused by own-side maldistribution is considerably larger than the increase caused by the other side. Temperature differences between the two working fluids have little effect on hydraulic performance.

The mathematical model proposed in the present paper can provide a valuable theoretical basis for the analysis and optimization of heat exchanger design, contributing in particular to the optimum design of cross-flow-corrugated heat exchangers with thermal and hydraulic performance taken into consideration. In the future, further work based on the coupled thermo-hydraulic model proposed here will be performed to investigate the effect of the entrance header, and a corresponding optimization study will be conducted.

This work was sponsored by the National Natural Science Foundation of China (Grant Nos. 52376037 and U22A20209), NSFC-BRFFR (Grant No. 52211530451), the Key Research and Development Program of Zhejiang Province (Grant Nos. 2022C01067 and 2022C03036), the Scientific Research Fund of Zhejiang Provincial Education Department (Grant No. Y202352498), and BRFFR-NSFC (Grant No. T23KI-033).

The authors have no conflicts to disclose.

Haowen Li: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (lead); Validation (lead); Writing – original draft (lead). Xiaomei Guo: Funding acquisition (equal); Resources (equal); Writing – review & editing (equal). Marat A. Belotserkovsky: Investigation (equal); Resources (equal). Aleksandr N. Grigorchik: Methodology (equal); Writing – review & editing (equal). Vladimir A. Kukareko: Funding acquisition (equal); Resources (equal); Writing – review & editing (equal). Zheng Bo: Supervision (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

A

Area (m²)

C_p

Specific heat at constant pressure [kJ/(kg K)]

D_h

Hydraulic diameter (m)

f

Fanning friction factor

G

Mass velocity

h_f

Protuberance height (m)

L

Length (m)

l_f

Protuberance diameter (m)

m

Mass flow rate (kg/s)

NTU

Number of transfer units

r_h

Hydraulic radius (m)

S

Generalized source

s

Protuberance spacing (m)

t_f

Protuberance thickness (m)

$T̄$

Mean temperature (K)

U

Wetted perimeter (m)

$v̄$

Average velocity (m/s)

Γ

Generalized diffusion coefficient

ΔT_m

Log average temperature difference (K)

α

Expansion angle (°)

ε

Heat exchanger effectiveness

ζ

Local resistance coefficient

λ

Friction resistance coefficient

A

Area (m²)

C_p

Specific heat at constant pressure [kJ/(kg K)]

D_h

Hydraulic diameter (m)

f

Fanning friction factor

G

Mass velocity

h_f

Protuberance height (m)

L

Length (m)

l_f

Protuberance diameter (m)

m

Mass flow rate (kg/s)

NTU

Number of transfer units

r_h

Hydraulic radius (m)

S

Generalized source

s

Protuberance spacing (m)

t_f

Protuberance thickness (m)

$T̄$

Mean temperature (K)

U

Wetted perimeter (m)

$v̄$

Average velocity (m/s)

Γ

Generalized diffusion coefficient

ΔT_m

Log average temperature difference (K)

α

Expansion angle (°)

ε

Heat exchanger effectiveness

ζ

Local resistance coefficient

λ

Friction resistance coefficient