Abstract

Compact heat exchangers (HXs) have gained attention in recent years in various fields such as solar and nuclear power generation, oil and gas, and refrigeration due to their low cost, high power density, and robustness in high-pressure and/or high-temperature environments. However, the large difference between a compact HX's overall dimensions (∼m) and the much smaller scale of its channels (∼mm) makes it challenging to model the entire HX at once, due to computational limitations. In this work, we treat the channeled region of a compact HX as a porous medium (PM) to circumvent the need to model/mesh each individual channel. This allows us to simulate the entire HX, including both the header and channeled regions while maintaining the computational cost at a practical level. Although the porous medium approach has been used to model heat exchangers, its validity is still questionable because (1) the resistance coefficients are heavily data-based and thus difficult to be applied to new heat exchangers and (2) the validation has been focused on matching the overall pressure drop in the channel region, which does not address whether such model can predict detailed pressure and velocity field. For the first time, this work addresses under what circumstances and with what uncertainty does the PM approach work for hydrodynamics modeling in compact HXs. By answering these questions, we introduce the PM approach as a powerful tool for HX hydrodynamics modeling that can predict not only the overall pressure drop but also the detailed pressure and velocity distributions.

Introduction

Compact heat exchangers (HXs) have gained increasing attention recently in various industries, such as concentrated solar power [1,2], nuclear power plants [3,4], refrigeration [5], automobiles [6], and chemical plants [7,8]. Typical compact HXs, such as printed circuit heat exchangers (PCHEs) [911], are comprised of a stack of thin plates with millimeter or submillimeter-sized channels in each individual plate. The closely packed small channels, typically of the order of 1 mm diameter, result in a high surface area density, and thus less volume of material for the same heat transfer rate [1214]. This not only makes it easier to manufacture, but also reduces the cost significantly considering that a HX's total cost can be dominated by the material cost [1,15].

Although a large number of closely packed channels is attractive from the standpoint of heat transfer and cost, it presents a challenge from the modeling perspective. Consider a typical Heatric PCHE as an example [16]. There is a three order of magnitude difference between the overall scale of the HX (∼m) and the small scale of its channels (∼mm). This makes it impractical to simulate the entire HX with computational fluid dynamics (CFD), because of the incredibly dense mesh that would be required to resolve the flow in each channel. A promising approach has been developed by conducting highly resolved simulations at the pore-level using representative elemental volumes (REVs) [17]. A REV is a small volume that correctly mimics the porous structure in which the flow and/or heat transfer occurs. Many works have been conducted on geometric idealization and pore-level calculations on REVs [1822]. Most of these works are performed for porous soil, sand, ceramics, foams, packed beds of particles, etc., where the void space of a REV is interconnected. These works are important because they provide invaluable detailed information at the pore-scale and such information can be used to calculate integral quantities that can be used in the porous-continuum (volume-averaged) calculations. For HXs, although the void space, i.e., the fluid channels, is not interconnected, an approach similar to the REV method has been extensively used because of the periodic pattern of the HX core. In this approach, a unit cell is selected as the analysis domain, which includes one hot fluid channel, one cold fluid channel, and the solid structure surrounding these two channels [10,2327]. Periodic boundary conditions are used to effectively account for the repeated pattern of the HX core. However, the problem with this approach and the REV approach is that it does not allow one to examine the flow in the headers, and specifically how the distribution of flow will vary across the array of channels. It is intuitive for most practical header geometries that assuming a uniform distribution would not be correct, i.e., flow maldistribution in the HXs exists, as confirmed by both experiments [28] and simulations [2932]. It has also been found that the flow maldistribution reduces the HXs' effectiveness [3335]. In order to optimize the HXs' performance, much work has been devoted to studying flow maldistribution. Experimental [28] and numerical [36] studies were performed to study the effects of header geometry on the flow maldistribution in a plate-fin heat exchanger. Pasquier et al. [37] analyzed four different header configurations of a PCHE using CFD simulation, and found that an integrated header-core design reduced the flow nonuniformity by 91%. Shao et al. [38] inserted two tubes with several holes into the inlet and outlet headers to uniform the flow distribution in a double tube-passes shell-and-tube heat exchanger with rectangular header. Chu et al. [39] used CFD to analyze the flow nonuniformity of a straight-channel PCHE condenser with four different inlet headers and proposed to use a hyperbolic inlet header to reduce the flow nonuniformity. Zou et al. [40] experimentally studied R134a and R410A distribution in a microchannel heat exchanger with vertical header, and found that the cooling capacity is reduced up to 40% for R410A and 50% for R134a compared to the uniform case. Kumaran et al. [41] conducted experiments to investigate the effect of header shape (rectangular and triangular) on flow maldistribution using numerical simulations and found that a triangular inlet header provides better flow distribution; whereas, for the case of an outlet header, the trapezoidal header provides uniform flow distribution. However, only reduced-scale HXs have been studied in these works to make the computational or experimental cost affordable. Although these studies are valuable in understanding the flow maldistribution in the header region, it remains questionable whether the results obtained from reduced-scale HXs can be used for HXs at an industrial scale. The most direct way to answer this question is to experimentally measure the flow maldistribution of HXs at such a scale. However, such measurements would be cost prohibitive [42].

Therefore, it would be an important breakthrough to have a modeling approach capable of simulating the entire HX, including both the header region and the channel region, while maintaining the computational cost at a practical level. The porous medium (PM) approach [43,44], also called the distributed resistance approach [45], provides a promising potential solution. This approach was originally developed to model shell-and-tube HXs by assuming that the shell is uniformly filled with fluid through which a solid medium is distributed that provides a resistance to fluid motion [44]. It allows the use of a much coarser mesh than that is required for a direct full geometry simulation. Since its birth, the PM approach has been used to analyze a variety of geometries, such as pebble beds [46], filters [47], rock and metal foam [48], tube banks [49,50], and HXs [51,52].

A conceptual schematic illustration of the use of the PM approach for a HX simulation is shown in Fig. 1. Since this work is focused on the hydrodynamics, only the fluid region from one of the two streams is shown in this figure. The fluid enters the inlet header from the left, distributes into the channel region, and leaves the HX on the right header. Only five channels are shown in Fig. 1(a) as a simplified illustration of the HX geometry. Regions A, B, C, D, and E represent the inlet tube, inlet header, channel/PM, outlet header, and outlet tube region, respectively. The diameter (Dt) and length (Lt) of the inlet and outlet tubes, the length of the inlet and outlet headers (Lh), the HX height (H), and the channel diameter (Dc) are marked on the figure. Conceptually, it seems intuitive that as the channel number is increased, eventually the channel region would be well approximated as a PM. By replacing the channel region with a PM as shown in Fig. 1(b), the entire HX can be simulated with a coarser mesh, which then allows for explicit study of the headers.

Fig. 1
Schematic illustration of the PM approach: (a) multichannel simulation and (b) porous medium simulation
Fig. 1
Schematic illustration of the PM approach: (a) multichannel simulation and (b) porous medium simulation
Close modal
For the numerical simulations described herein, the effect of the PM region is accounted for by an additional resistance term in the volume-averaged Reynolds-averaged Navier–Stokes (RANS) equations. The additional resistance terms are heuristic models to capture the additional viscous and form drag due to the presence of the solid structures impeding the flow. The additional source term for the ith momentum equation is expressed as follows [53]:
Si=(jDijμvp,j+jCij12ρ|vp|vp,j)
(1)

The viscous loss is modeled by the first term on the right-hand side of Eq. (1), where D is the viscous resistance factor, μ is the fluid viscosity, and vp is the mean fluid velocity in the PM, also called superficial velocity. The inverse of D is also known as the permeability. The second term on the right-hand side of Eq. (1) represents the inertial loss due to form drag, where C is the inertial loss coefficient and ρ is the fluid density.

Application of the PM approach requires proper determination of the resistance factors D and C so that the behavior of the PM is equivalent to that of the channel region. These parameters can be deduced from experimental data or from models of the pressure drop in the channeled region [44,52,54,55]. Although pressure drop characteristics for channels of different geometries are readily available, the characteristics of the entire channel region can be different due to flow maldistribution [2832] and are usually not available for new HX designs. Therefore, determining the resistance factors is still challenging for compact HXs.

In addition, validation of the PM approach in the past has focused on predicting the overall pressure drop in the HX core region [49,51,52,5456]. While we recognize the importance of matching the pressure drop, it is also important to recognize that the pressure drop itself does not provide a complete description. This is because the pressure drop (or pressure gradient) is evaluated based on only two points in the complicated flow field of the HX. It could only be used to prove that by properly choosing the equivalent resistance factors, the PM approach could predict the overall pressure drop reasonably well. Yet it remains questionable whether the more detailed pressure distribution between the two points, and even beyond (e.g., in the header region), could also be accurately predicted by the PM approach. Answering this question is of great importance because the pressure distribution essentially determines the flow distribution, which affects the HX's performance. Equally important is whether the PM approach can capture the correct velocity distribution, which provides direct information about the flow maldistribution.

We acknowledge the importance of heat transfer in HXs; however, in this work the flow problem is from heat transfer because (1) the flow performance itself is important to a heat exchanger. For example, pressure drop determines the cost of pumping power, which can be a major portion of the HX cost. Flow maldistribution not only affects HX heat transfer performance, but also can cause uneven wear in different regions of the HX, thereby reducing its life period. (2) The complexity of the simulating the entire HX justifies the decoupling of the flow problem from the heat transfer. Separately studying the fluid problem could help obtain a deeper understanding of it without the complications from the coupling of heat transfer. Therefore, the key goals of this study were to determine (1) whether the PM approach could accurately predict the velocity and pressure distribution in a HX; (2) the conditions under which the PM approach work (e.g., how many channels are required to converge to a PM approximation); (3) the equivalent resistance factors for the PM; and (4) and the uncertainty associated with the PM approach.

Validation Strategy for the Porous Medium Approach

The resistance factors D and C should be determined so that the PM exhibits the same effective resistance to flow as the channeled region. The pressure drop through the equivalent PM should be equal to the pressure drop through the channel region. Although the resistance factors may be more accurately determined based on the pressure drop of the corresponding channeled region, this calculation is not realistic when the channel number is high (e.g., PCHEs), which is exactly when the PM approach is needed. For the case when the channel number is sufficiently large, the flows tend to be more uniformly distributed into each channel, and the pressure drop of the channeled region should be close to that of each individual channel. Therefore, the pressure drop in a single channel may be used to determine the resistance factors of the equivalent PM.

A key concern addressed by this work is whether the PM resistance factors based on single channel pressure drop characteristics are appropriate for HXs with different channel numbers. If not, then the threshold channel number below which the PM approach ceases to accurately describe the flow behavior within the channel region needs to be determined. A flowchart for the validation and quantification of the PM approach is shown in Fig. 2. ansysfluent simulation of molten chloride salt flow in a single channel has first been used to determine the equivalent resistance factors of the PM. Molten salt was used as the fluid for the simulations because this work is part of a project to develop high temperature molten salt heat exchangers for concentrated solar power. These resistance factors were then used to perform a full-geometry HX simulation with different number of channels using the PM approach, as shown in Fig. 1(b). Multichannel (MC) simulations have also been conducted using the corresponding HX geometry, as shown in Fig. 1(a). Results from the PM simulations were then compared with the MC simulations to validate, and to quantify the uncertainty associated with the PM approach.

Fig. 2
Flowchart for validation and quantification of PM approach
Fig. 2
Flowchart for validation and quantification of PM approach
Close modal
The pressure drop in a single channel can be written as follows:
Δp=fLcDh12ρvc2
(2)
where f is the Darcy friction coefficient, Lc is the channel length,vc is the channel velocity, which connects with the superficial fluid velocity in Eq. (1) as follows:
vc=vc/α
(3)
where the porosity α is equal to the flow area divided by the cross-sectional area of the entire HX. Dh in Eq. (2) is the channel hydraulic diameter defined as below:
Dh=4AP
(4)
where A is the cross-sectional area of the flow, and P is the wetted perimeter of the cross section. For a two-dimensional channel, the hydraulic diameter Dh is twice of the channel diameter. Once the friction coefficient in Eq. (2) is obtained, e.g., from a single channel calculation, then the resistance factors C and D can be determined so that Eq. (1) reproduces the same relationship between pressure drop and velocity as Eq. (2). Based on a CFD simulation of flow in a single channel, the dependence of the friction coefficient on the Reynolds number can be determined as shown in Fig. 3. In laminar flow regime, the simulation results perfectly match with the theoretical model f = 96/Re. In turbulent flow regime, the simulation results in bounded by an analytical model [57] and an empirical correlation [58] within 30% error. The friction coefficient function can be described as follows:
f=mRe+n
(5)
Fig. 3
Change of the friction coefficient with Reynolds number based on a single-channel simulation
Fig. 3
Change of the friction coefficient with Reynolds number based on a single-channel simulation
Close modal
The Reynolds number associated with the channel Re is defined as
Re=ρvcDhμ
(6)
The variables m and n are given by
m={96Re<200080.78Re>2000}
(7)
n={0Re<20000.026Re>2000}
(8)
The resistance factors C and D in the PM approach can then be easily obtained from Eqs. (1) and (2):
D=m2αDh2
(9)
C=nDhα2
(10)

Equations (8) and (9) indicate that the resistance factors depend on flow conditions (i.e., the Reynolds number) and the HX geometry (i.e., the porosity and channel hydraulic diameter). These resistance factors can then be used for simulations using the PM approach for HXs with different channel numbers. A unique feature of the PM for HXs is that it can be highly anisotropic. The fluid flows from left (i.e., inlet) to right (i.e., outlet), but cannot flow from one channel to another. Although there can indeed be flow within each channel that is not aligned parallel to the channel axis due to turbulence effects, the detailed microscopic characteristics of flow in each channel are less important the overall performance of the HX compared with the overall flow maldistribution, and therefore are not broached in this work. To address the anisotropy of the PM, the flow resistance perpendicular to the main flow direction along the channel axis is set to be 1000 times larger than that of the main flow direction so that all perpendicular velocity components were suppressed inside the PM. Further discussion on the anisotropy of the PM is presented later in Section Anistropy of Porous Mediium in HX. After the resistance factors were determined, PM simulations were conducted for HXs with different number of channels and different inlet conditions. The results were then compared with MC simulations using corresponding HX geometries and flow conditions.

Numerical Modeling

Figure 4 shows an example of the HX simulation domain for the MC study (Fig. 4(a)) and PM study (Fig. 4(b)). The diameter of the inlet tube was 0.1 m, the HX height H was 0.5 m and the lengths of the inlet/outlet tubes, inlet/outlet headers, and the channel region were 0.2 m, 0.2 m, and 1.5 m, respectively. A wide range of channel numbers and flow conditions were investigated, as shown in Table 1. The channel Re (Rec) and inlet Re (Rein) are defined based on the velocity and hydraulic diameter of the channel and inlet pipe, respectively. In the simulations, the Rein was varied from 5000 to 10,000 (corresponding to the Rec from 100 to 2000) for laminar flow conditions, and from 100,000 to 600,000 (corresponding to the Rec from 2040 to 120,000) for turbulent flow conditions. For each inlet condition, the channel number was varied from 5 to 49 to investigate the effect of channel number on the validity of the PM approximation.

Fig. 4
Simulation domain and mesh of (a) MC study and (b) PM study
Fig. 4
Simulation domain and mesh of (a) MC study and (b) PM study
Close modal
Table 1

Channel number and flow conditions for PM validation

Flow regime in channel regionChannel numberPorosityRe in HX inletRe in HX channels
Laminar flow5–49∼0.55000–10,000100–2000
Turbulent flow5–49∼0.5100,000–600,0002040–120,000
Flow regime in channel regionChannel numberPorosityRe in HX inletRe in HX channels
Laminar flow5–49∼0.55000–10,000100–2000
Turbulent flow5–49∼0.5100,000–600,0002040–120,000
The simulations of both MC and PM study were performed using ansysfluent. Kw-SST turbulent model was selected because of its capability in predicting flow separations under adverse pressure gradients, which is a major complication in the compact HX simulation. It uses a kw formulation in the inner part of the boundary layer, which enables its usage as a low-Re turbulence model without any extra damping functions. In the free stream, it switches to a kɛ model and thereby avoids the common kw model weakness that the model is too sensitive to the inlet free-stream turbulence properties. When the channel region was laminar, Gamma transport equation was activated to simulate the transitional behavior of laminar-to-turbulent flow. It is worth to mention that in the PM study, turbulence in the flow medium was treated as though the solid medium has no effect on the turbulence generation or dissipation rates. This is a reasonable assumption because the scale of the turbulent eddies in PM region under the turbulent flow conditions analyzed in this work is less than 20% of the geometric scale of the medium, i.e., the channel diameter. The scale of the turbulent eddies le can be calculated by le = 0.4δ99 [53] where δ99 is the 99% boundary layer thickness and can be approximated by the following equation for turbulent flow along a flat plate channel [59]:
δ99=0.37min(x,H/2)Rep0.2
(11)

where Rep is the Reynolds number in the PM, x is the distance from the starting point of the PM region (point M in Fig. 1) along the flow direction, and H/2 is the half height of the HX. The scale of the turbulent eddies is calculated to be <0.6 mm, while the smallest channel diameter studied in this work is 1.67 mm when the channel number is 150. If the channel size becomes smaller and comparable with the scale of the turbulent eddies, then the turbulence effect in the flow medium should be actively suppressed in the simulation.

Second-order upwind scheme with double precision was used in the solver. The governing equations were solved in the Cartesian coordinate system with a finite volume method. The convergence residual was set to 10−6 in the simulations. Uniform velocity was applied as the inlet condition, and pressure outlet was used as the outlet condition. No-slip conditions were used as the boundary conditions for all the walls.

Figure 4(a) shows a mesh used in the MC simulation when the channel number is 29 and inlet Re is 100,000. A zoomed-in view of the meshes at the inlet header corner, the interface between the header and channel regions, and the fully developed channel region, are also overlaid on this plot. Ten inflation layers were used to capture the boundary layer region. Finer mesh with a bias factor of 40 was used in the channel entrance region to resolve the abrupt change of flow behavior in this region while uniform mesh was used in the center region. A grid independence study was performed with nine different meshes and the result is shown in Fig. 5. As the number of mesh elements increase, the pressure gradient in the channel region converges to within 1% error. The computational efficiency of the PM approach significantly surpasses that of the MC approach due to the reduced number of mesh elements. For example, for the HX with 29 channels at Rein of 100,000, the CPU time of the PM simulation is 12.89 s, which is ∼30X lower than that of the MC simulation (380 s). The comparison results between PM and MC simulations are discussed in detail in the Results & Discussion section.

Fig. 5
Result of mesh sensitivity study
Fig. 5
Result of mesh sensitivity study
Close modal

Results and Discussion

Once the PM and MC simulations were completed, the pressure gradient in the channel region/PM region was used to assess the validity of the PM approximation. The pressure gradient was evaluated at the center points along the inlet and outlet of the channel/PM region (points M and N in Fig. 1) as follows:
dp/dl=pMpNLc
(12)

where Lc is the length of the channel region.

The error of the PM approach in predicting the pressure gradient was calculated as the relative error of the pressure gradient between the PM and MC simulations:
δdp=(dp/dl)PM(dp/dl)MC(dp/dl)MC
(13)

Figure 6(a) shows the comparison of the pressure gradient between PM and MC simulations when the channel flow was turbulent. When the channel number was small (e.g., for five or seven channels), the PM approach overpredicts the pressure gradient with an uncertainty larger than 100%. However, as channel number increased, the pressure gradients from the PM simulation converged to the MC simulation results and at 49 channels the PM simulation results agreed with the MC simulation within 10%. These results not only confirm the initial hypothesis that a PM is an accurate representation of the channel region when the channel number is sufficiently large; they also quantify how many channels are required and the corresponding uncertainty. Figure 6(b) shows the relative error with respect to the number of channels, which dropped quickly as the channel number increased. The error was also a function of Re, whereby more deviation was observed for higher Reynolds numbers. Overall, when the flow in the channel region was turbulent, the PM approach could predict the pressure drop to within 10% uncertainty when the channel number was larger ∼ 30. The PM and MC simulation results when the channel flow was laminar were compared in Figs. 6(c) and 6(d). Similar trends were observed for laminar flow, although the PM simulations tended to converge faster when the channel flow was laminar. In this case, the PM approach predicted the pressure gradient to within 10% when the channel number was larger than 12.

Fig. 6
Comparison between PM and MC simulations: (a) pressure gradient when the channel flow was turbulent, (b) relative error when the channel flow was turbulent, (c) pressure gradient when the channel flow was laminar, and (d) relative error when the channel flow was laminar
Fig. 6
Comparison between PM and MC simulations: (a) pressure gradient when the channel flow was turbulent, (b) relative error when the channel flow was turbulent, (c) pressure gradient when the channel flow was laminar, and (d) relative error when the channel flow was laminar
Close modal

Although important, an accurate prediction of the pressure gradient is not sufficient to prove the validity of the PM approach since the pressure gradient was evaluated based at only two points in the complicated flow field. Such a simple analysis does not reveal whether the pressure distribution elsewhere could also be accurately calculated by the PM approximation. A more detailed evaluation is important because the pressure distribution essentially determines the flow distribution, and maldistribution in flow is an important factor affecting the HX performance. It is also important to establish whether the PM approach can predict the velocity field, especially in the header region, because a central objective is to use the PM approximation as a proxy for the channeled region to enable explicit study of maldistribution. Thus, for the PM approximation to be used in this desired context, the PM approximation would need to reproduce the same velocity field in the header as the MC simulation.

To address this, the velocity fields in the HX obtained from PM and MC simulations were first compared. Results for HXs with three channel numbers (5, 9, and 19) are shown in Fig. 7. The inlet Rewere kept the same (at 100,000) for all these three cases. In each subplot, the top figure was the result obtained from the MC simulation, and the bottom figure was the result obtained from the corresponding PM simulation. As shown in Fig. 7(a), with five channels, the MC simulation predicted that most of the fluid entered the center channel. Although there was much less flow in the near-wall channels, the fluid could still enter these channels and flow from left to right. However, this feature was not be captured by the PM simulation. In the PM simulation results, no flow could enter the near-wall region from the inlet header. After the fluid entered the inlet header, most of the fluid flowed through the near-center region, with a small portion of the fluid recirculating in the inlet header. Although the recirculation vortex from the PM simulation qualitatively reflected the reality, this simulation left no fluid entering the near-wall region. Consequently, backward fluid flow (moving from right to left) was observed in the near-wall region from the PM simulation results, which was shown by the dashed arrows (not to scale) in the white box. Overall, with five channels, the PM simulation did not capture the velocity field as predicted by the MC simulation.

Fig. 7
Comparison of the velocity field from MC and PM simulations: (a) 5 channels, (b) 9 channels, and (c) 19 channels. Top subplot: MC simulation results and Bottom subplot: PM simulation results.
Fig. 7
Comparison of the velocity field from MC and PM simulations: (a) 5 channels, (b) 9 channels, and (c) 19 channels. Top subplot: MC simulation results and Bottom subplot: PM simulation results.
Close modal

As the channel number increased to nine (as shown Fig. 7(b)), the recirculation vortex predicted by the PM simulation became stronger so that more fluid was squeezed into the near-wall region. Although not observable from the velocity vectors due to their small magnitudes in this region, a closer view of the near-wall flow field confirmed that the fluid flow indeed proceeded from left to right as shown by the dashed arrows in the white box. When the channel number was increased to 19, the agreement between PM and MC simulation results becomes even better. As shown in Fig. 7(c), the PM simulation captured nearly all of the details of the flow field predicted by the MC simulation, including the magnitude of the flow as well as the flow pattern. Both simulations showed the same behavior: after the fluid entered the inlet header, the flow redistributed within the header. Most of the fluid entered the center region of the HX, with a small portion migrating to the near-wall region. Some fluid also just recirculated within the header without contributing to the net flow rate.

Similar agreement was also obtained for the pressure field. A comparison of the pressure field between the MC and PM simulations is shown in Fig. 8 when the channel number was 19. The PM approach captured very fine details in the header region, such as the high pressure near the center at the inlet of the channel region, and the isopressure contour lines corresponding to the recirculation zone. When comparing the channel region with the PM region, it was difficult to visually compare the two plots directly because the isopressure contour lines from the MC simulation were scattered due to the constraints of the channels. To allow for easier comparison, three isopressure contour lines were artificially connected with continuous curves. The comparison revealed surprisingly good predictability of the PM approach. The pressure distribution in the channel region was captured by the PM approach even though the channel region had been replaced with a uniformly distributed PM. Therefore, the results in Figs. 7 and 8 confirm that the PM approach can not only capture the overall pressure difference across the channeled region, but once the number of channels is sufficiently large, it also recovers the full velocity and pressure fields.

Fig. 8
Comparison of the pressure field between MC and PM simulations
Fig. 8
Comparison of the pressure field between MC and PM simulations
Close modal
To quantify how well the PM approximation predicts the detailed velocity field, the x-component velocity (u) at the header-channel interface was used for error assessment since the velocity at this location directly describes the flow maldistribution, which is a key flow parameter determining the HX performance. The velocity distribution error was defined as follows:
δv=1N|uPM,iuMC,i|uMC,i
(14)

where the subscript i represents the ith channel and N is the total number of channels.

Figure 9 shows the comparison of the channel x-component velocity between the PM and MC simulations and the calculated error based on Eq. (12). The PM approach tended to overestimate the velocity in the center region and underestimate this velocity in the near wall region. Yet the degree of deviation decreases significantly as the channel number increased. As shown in Fig. 10, the velocity error reduced below 10% when the channel number was larger than 30.

Fig. 9
Comparison of the velocity distribution along the header-channel interface for a HX: (a) five channels, (b) nine channels, (c) 19 channels, (d) 29 channels, and (e) 49 channels
Fig. 9
Comparison of the velocity distribution along the header-channel interface for a HX: (a) five channels, (b) nine channels, (c) 19 channels, (d) 29 channels, and (e) 49 channels
Close modal
Fig. 10
Error of pressure distribution along the HX's centerline in the flow direction
Fig. 10
Error of pressure distribution along the HX's centerline in the flow direction
Close modal

Similarly, the error of the pressure distribution was evaluated based on the pressure along the centerline of the HX. This pressure was important because it provided an overall description of the pressure distribution in the HX and provided valuable information for pump/compressor design. Figure 11 shows the pressure distribution for different channel numbers. The pressure increased in the inlet header due to sudden expansion and decreased in the outlet header due to sudden contraction. The pressure decreased nearly linearly in the inlet/outlet tubes and the channel region, as would be expected. It was interesting to note that even with very few channels, the PM approach captured the trend of the pressure change in the inlet/outlet header/tube regions (regions A, B, D, and E) very well because in these regions both the PM and MC simulations solved the traditional RANS momentum equation. It was in the channel/PM region (region C) where most deviations were observed. When the channel number was small (e.g., N = 5), the pressure drop in this region was also small due to the relatively large channel size. A sudden drop in pressure occurred at the interface between the inlet header and the channel region (B–C interface) due to sudden contraction. However, when using the PM approach, this feature cannot be reproduced. The pressure reaches a maximum at the B–C interface and then dropped linearly along the channel region. This is because the sudden contraction at the B–C interface was not reflected in the PM modeling. Therefore, the pressure in the channel region was overestimated by the PM approach. As the channel number increased, the pressure deviation in the channel decreased because the PM became a better representation of the channel region. The fluid entered each channel almost indifferently as if it flows through a uniform PM. When the channel number was 49 (as shown in Fig. 11(c)), the pressure distribution predicted by the PM approach agreed well with that obtained from the MC simulation.

Fig. 11
Distribution of static pressure along the HX's centerline in the flow direction: (a) five channels, (b) 19 channels, and (c) 49 channels
Fig. 11
Distribution of static pressure along the HX's centerline in the flow direction: (a) five channels, (b) 19 channels, and (c) 49 channels
Close modal
The error of pressure distribution was defined as
δp=1N|pPM,jpMC,j|pMC,j
(15)

where subscript j represents the jth evaluation node along the centerline and N is the total node number. The change of the pressure distribution error with respect to the channel number is shown in Fig. 12. When the channel number was larger than 30, the PM approach predicted the pressure distribution to within 10% error.

Fig. 12
Error of the velocity distribution along the header-channel interface
Fig. 12
Error of the velocity distribution along the header-channel interface
Close modal

Anisotropy of Porous Medium.

As mentioned earlier, it is important to recognize that the equivalent PM for a HX is highly anisotropic. Therefore, a large resistance ratio (R=Dy/Dx=Cy/Cx)between the cross-flow and main-flow directions should be used. In order to visualize this effect, a simulation was performed with nonisotropic isotropic resistance factors separately. The resulting distribution of velocity field shown in Fig. 13 indicated that the PM simulation with isotropic resistance factors could not capture the most basic flow characteristics. The flow recirculation pattern was not predicted. In addition, the fluid flowed in the vertical direction in the channel region. This is an unphysical result, since it would mean the fluid could move from one channel to another.

Fig. 13
Velocity field from the PM simulation using (a) nonisotropic resistance factors, R = 1000 and (b) isotropic resistance factors, R = 1
Fig. 13
Velocity field from the PM simulation using (a) nonisotropic resistance factors, R = 1000 and (b) isotropic resistance factors, R = 1
Close modal

When using the PM approach, it is important to know at what resistance ratio the anisotropic feature of the PM can be properly captured. Since the analysis above confirmed that the uncertainty of the PM approach was below 10% when the channel number was larger than 30, PM simulations were performed with different resistance ratios for channel numbers of 30, 50, and 150. The resistance ratio was varied from 1, which represented an isotropic PM, up to 10,000, which represented a highly anisotropic PM. As shown in Fig. 14, when the resistance ratio was above 100, the anisotropic feature of the PM in the HX could be well captured and the pressure gradient could be predicted within a 10% error.

Fig. 14
Error of pressure gradient using different resistance ratio in the PM approach
Fig. 14
Error of pressure gradient using different resistance ratio in the PM approach
Close modal

Summary

Systematic comparisons between the porous medium (PM) simulations and multichannel (MC) simulations helped elucidate the conditions under which the PM approach can substitute for MC simulations of HXs. In this work, we focused on the hydrodynamics modeling of HXs, namely, the pressure field and the velocity field. Two remaining problems for the PM approach have been addressed in this work: (1) The use of the PM approach in prior work relies on previously obtained experimental data or models of the pressure drop in the channeled region, but such data or models are usually not available for new HX designs. Thus, determining the resistance factors of the equivalent porous medium is still a major challenge. (2) The validation of the PM approach in prior work was focused on matching the overall pressure drop between the porous medium and the channel region, but the pressure drop evaluated based on only two points (the inlet and the outlet) does not provide a complete description of the HX performance. By systematically comparing the performance of the PM and MC simulations of HXs with channel numbers ranging from 5 to 50, this work showed that the resistance factors obtained from pressure drop characterizations of a single channel can be used with confidence for PM simulations with different channel numbers. In general, and as expected, the accuracy of the PM approach increased as the number of channels in the HX increased. For HXs with more than 30 channels, the PM approach not only predicted the overall pressure drop but also the detailed pressure and velocity distributions in the HXs, and generally within ∼10%. A large resistance ratio between the cross-sectional and the main-flow directions of the HX is also needed to properly account for the fluid's inability to cross the channel boundaries in the perpendicular directions. A sensitivity study of the resistance ratio indicated that the PM approach predicted the pressure gradient to within 10% error when the resistance ratio is above 100. Therefore, we introduce the PM approach as a powerful tool for HX hydrodynamics modeling because it allows for simulation of the entire HX, including both the header region and the channeled region, while maintaining the computational cost at a practical level. It is worth to mention that the current validation was performed with two-dimensional (2D) HX geometries because of the massive mesh in three-dimensional (3D) HXs. The general trends found in this work are expected to also apply for 3D HX geometries. 3D validation is suggested as future work to further validate the capability of the PM approach. In addition, further validation of heat transfer calculations using the PM approach is needed for a complete HX analysis.

Funding Data

  • U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) under Solar Energy Technologies Office (SETO) (Agreement No. EE0008369; Funder ID: 10.13039/100006115).

Nomenclature

A =

cross-sectional area of the flow

C =

inertial resistance factor of the porous medium

D =

viscous resistance factor of the porous medium

dp/dl =

pressure gradient in the HX core region

Dc =

channel diameter

Dh =

hydraulic diameter

Dt =

diameter of inlet and outlet tubes

F =

Darcy friction coefficient

H =

heat exchanger height

Lc =

channel length

Lh =

length of inlet and outlet headers

Lt =

length of inlet and outlet tubes

P =

wetted perimeter of the cross section

pM =

static pressure at center point M along the inlet header-core interface

pN =

static pressure at center point N along the core-outlet header interface

R =

resistance ration between the cross-flow and main-flow directions

Re =

Reynolds number

S =

addition pressure gradient due to flow resistance of the porous medium

vc =

mean channel velocity

vp =

mean fluid velocity in the porous medium

α =

porosity

μ =

fluid viscosity

ρ =

fluid density

Abbreviations
HX =

heat exchanger

MC =

multichannel

PCHE =

printed circuit heat exchanger

PM =

porous medium

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