Proceedings of the 10th World Congress on Mechanical, Chemical, and Material Engineering (MCM'24)
Barcelona, Spain –August 22-24, 2024
Paper No. HTFF 127
DOI: 10.11159/htff24.127

Performance Characterization of a Small-Scale Pillow Plate Heat
Exchanger Designed with the Effectiveness-NTU Method
Alessandro Dai Pré1, Luca Marchetto1, Maurizio Grigiante2
1

Dept. of Industrial Engineering, University of Trento
via Sommarive 9, 38123, Trento, Italy
alessandro.daipre@unitn.it; luca.marchetto-1@unitn.it
2
Dept. of Civil, Environmental and Mechanical Engineering, University of Trento
via Mesiano 77, 38123, Trento, Italy
maurizio.grigiante@unitn.it
Abstract - In recent years the design of pillow plate heat exchangers (PPHE) is attracting more and more interest both in thescientific community and in relevant industrial sectors. In this manuscript an experimental investigation has been carried out
to study the thermo-hydraulic behaviour of a very compact PPHE designed in collaboration with a manufacturing company.
This has allowed to design a PPHE with the smallest geometric parameters currently achievable with the present
manufacturing technologies. This device mounts two pillow plates 450 mm long and 80 mm wide with an internal inflationof 3mm. In this preliminary analysis water is used as working fluid both for hot and cold channel of the PPHE which has
been installed on a purpose-built laboratory-scale setup. The design of the PPHE has been carried out by implementing theefficiency-NTU number (ε-NTU) method specific to PPHE geometries. The inside h1 and outside h2 heat transfer
coefficients have been determined by correlations available in literature for normal types of PPHEs. The strength of the
adopted approach has been verified by evaluating the errors percentage for the outlet temperatures, the efficiency and theDarcy factor referred to a wide experimental campaign. In terms of errors, the performance for predicted thermal power set
from -15% to 15%, for thermal efficiency from -13% to 10%, and between 15% and 30% underestimation for the Darcy
factor. The proposed procedure looks a promising engineering tool to be implemented for those applications involving small
scale PPHE.
Keywords: Pillow Plate, Effectiveness-NTU method, Heat Exchanger, Heat transfer

1

Introduction

Heat exchangers (HEs) play a pivotal role in power and process industries, as well as in satellite, aviation, and microelectronics cooling applications. The increasing demand for HEs necessitates innovative and flexible design approaches
beyond traditional Shell and Tube (STHE) and Plate Heat Exchanger (PHE) designs. Recent innovations in the HEs industry
can be classified into passive, active, and compound enhancements. Passive methods include special inserts into the channels
or modification of the flow duct with surface extensions, thus creating swirls and vortices that enhance heat transfer
coefficients. This study focuses on a specific class of passive HEs known as "Pillow Plate Heat Exchangers (PPHEs),"
composed of plates, welded and hydroformed in a "pillow" shape, assembled to form two heat transfer fluid (HTF) channels
within and between the inflated plates. PPHEs offer superior design flexibility, compactness, ease of installation, costeffective production, and improved thermal efficiency and hydrodynamic performance compared to conventional HEs [1],[2]. These advantages make PPHEs attractive for applications in food industry, energy and process industries, HVAC andrefrigeration, gas coolers, and distillation process HEs. Despite these benefits, PPHEs have not yet achieved the technologicalreadiness found in conventional equipment. This is primarily because they have not been exhaustively investigated, and thus,
their design procedures are not yet supported by robust and validated design tools [3].
The study of heat transfer problems in PPHEs presents valuable opportunities. The flexibility in manufacturing pillow
plates allows for unlimited variations in their geometrical parameters, making PPHEs ideal for fundamental heat transfer

HTFF 127-1

studies and novel design approaches, for applications involving heat transfer in the context of intensification processes.
Therefore, a new PPHE configuration, designed with the smallest geometric parameters achievable using advanced
technologies, was investigated. The main novelties from this study can be summarised as:-

2

A dedicated procedure, based on the ε-NTU method, was implemented to design and test theperformance of a small-scale PPHE built for this investigation.
Existing correlations for heat transfer coefficients and friction factors for normal PPHEs [4], [5] wereapplied and critically assessed for this PPHE.
Validation tests confirmed the reliability of the proposed design approach for compact PPHEs,
increasing their potential for becoming a reliable high performance heat transfer geometry.

Methodology

2.1

The ε-NTU model for a Small Scale PPHEThe Effectiveness (ε) – Number of Transfer Units (NTU) (ε-NTU method) is recognized as a simple and yet powerfultool for analysing the thermal performance of heat exchangers. In this work this method has been implemented within adedicated procedure and tested as basic design tool to be applied to Small-Scale Pillow Plate Heat Exchanger (SSPPHE)This is achieved by modelling the geometry of the pillow plate through the elaborations proposed in [3] that allows toestimate, by providing as input only the macroscopic geometric parameters, the hydraulic diameters, the exchange area andfurther geometric parameters specific of this type of heat exchanger. All these parameters, suitably elaborated by including
the inlet temperatures, the flow rates and the thermophysical properties values (in this work obtained with the Coolprop
open-source database [6]), allow to determine the dimensionless numbers of Reynolds (Re) and Prandtl (Pr) that, correlated
through empirical equations, provide the Nusselt (Nu) number from which the heat transfer coefficient is finally obtained.
The correlations for Nu, which are commonly used in the model to determine the heat transfer coefficients, were chosen by
the most advised by the extensive review on PPHE by Joybary et al. [7]. The correlation Nu=f(Re,Pr) for the pillow plate
inner channel (IC) consist of Eqs. 1, 2, 3 and 4 from [4]. The parameters a, b, and c are respectively the ratio of thelongitudinal pitch to the transversal pitch of the PP welding spots (a=2sl/st), the ratio of the welding spot diameter to thetransversal pitch (b=dsp/st) and the ratio of the plate internal inflation to the transversal pitch (c=hi/st). Eq.5, named OC2 in[7] is chosen for the outer channel (OC). The validity ranges are stated as 1000<Re<8000, 1<Pr<150 for the IC correlationin Eq1, 9500<Re<30000, 6<Pr<150 for the OC correlations in Eq. 2.

𝑵𝒖 = 𝒏𝟑𝑹𝒆𝒏𝟒 𝑷𝒓𝒏𝟓(1)𝑛3 = −0.163𝑏 + 0.711𝑐 + 0.022(2)

𝑛4 = 0.29𝑏 − 𝑐 + 0.8(3)𝑛5 = 0.4(4)

𝑁𝑢 = 0.06𝑅𝑒 0.745 𝑃𝑟 0.35

(5)

It is important to specify that after the characterisation, all the correlations proposed in [7], suitable for water, were tested
and the ones proposed in this work show the lowest overall error values. The Number of Transfer Units, the effectivenessand the outlet temperatures are therefore as well determined [8],[9]. By requesting only temperatures and flow rates asprocess data, the ε-NTU method appears to be particularly attractive and suitable to be applied to the compact PPHE
investigated in this work. The relation between ε and NTU is expressed in Eq.6, where Cr is the capacity ratio.
𝟏(6)
𝛆 = 𝟏 − 𝐞𝐱𝐩 {( ) 𝐍𝐓𝐔 𝟎.𝟐𝟐 [𝐞𝐱𝐩(−𝐂𝐫 ∙ 𝐍𝐓𝐔 𝟎.𝟕𝟖 ) − 𝟏]}
𝐂𝐫

HTFF 127-2

2.2

SSPPHE Design
To validate the E-NTU design method, a PPHE was designed to replicate a STHE geometry while minimizing production
parameters according to laser welding process constraints, with the objective of providing a characterisation of the smallest
pillow plate geometry possible. The model allowed to forecast the PPHE performance beforehand. To provide more insights
on the performance of Nu correlations for the OC, which are scarce in literature [7],it was decided to investigate lower Re
ranges. The PPHE features two plates, each measuring 450 mm in length (L) and 80 mm in width (W), with a thickness (dp)of 1 mm. A rendering of the plate is shown in Fig. 1. The two-plate configuration is studied to create two ICs and three OCs.
The inflated plate thickness is 5 mm (hi + 2dp), with an internal inflation of 3 mm (hi). The PPHE has two internal and threeexternal channels, with a median surface distance of 8 mm. The longitudinal pitch is 18 mm (sl), and the transversal pitch is
21 mm (st). The welded edges, not inflated, are 3 mm (le) long, and the welding spot diameter is 5 mm (dsp). The compact
design ensures efficient heat transfer in a small footprint, with a heat transfer area over volume ratio β=340m2/m3.

Figure 1: Pillow Plate (left) and Pillow Plate repeating unit (right) with parameters defining of the geometry

2.3

Experimental Setup
The experimental setup was designed to simulate steady-state operating conditions for the PPHE, including water flow
systems, temperature control and measurement, pressure measurement, and flow meters. The system operates by circulating
hot and cold fluids through the heat exchanger. The hot fluid is heated using an electric resistance heater and then pumped
through the heat exchanger. Simultaneously, cold fluid from the reservoir circulates through the exchanger. Heat is rejected
through the water which is discharged and refilled trough the municipal water system, with the reservoirs providing thermal
stability to the set-up. This setup enables a first iteration analysis of the heat transfer and the evaluation of the hydraulicresistance characteristics of the heat exchanger, providing valuable data for optimizing both the PPHE design and its target
working conditions beside providing possible improvements as discussed in the conclusions section.
2.4

Heat Transfer and Pressure Drop Characterization
To characterise the SSPPHE behaviour, two distinct thermal performance characterisation campaigns were carried out,
with fixed inlet temperatures (with small variations attributable to the heat sources). The tests of the first campaign, each one
referred to the identification number (ID) 1&n (n states for the test number), were performed by changing the OC flow rate
within the range of 0.111-0.224 kg/s and keeping the IC flow rate constant around an averaged value of 0.111 kg/s.
Conversely, the second campaign, with ID2&n, was performed by changing the IC flow rate in the range of 0.042-0.330
kg/s. The OC flow rate was set at an average value of 0.180 kg/s. As for the inlet temperature of the fluids (T1i for IC and
T2i for OC), the average value was T1i=324.11K, T2i= 287.77K for the campaign ID1&n, T1i=323.77K and T2i= 285.57K
for ID2&n.
Pressure drop was measured in both the inner and outer channels of the SSPPHE. However, since the pressure drop in
the outer channel was unexpectedly below the sensor's accuracy, only data for the inner channel are here reported. To isolate
the pressure loss within the pillow plates, the pressure drop in the connections was computed using the Blasius correlation,
as described in [10]. The pressure drop relationship with the flow regime is evaluated throughout the Blasius equation, that

HTFF 127-3

correlates the Darcy friction factor (ξ) and Re number, shown in Eq. 7. As seen in Eq. 8-9, both ξ and Re depend from thehydraulic diameter (Dh) and the velocity of the flow (v), which is calculated by dividing the volumetric flow rate by the fluidpassage area (Acs). To determine Dh and Acs from the main geometrical features of the pillow plate, two different approaches
from literature were selected: one from Arsenyeva et al. [11] (also referred in this text as Arsenyeva’s method), a relevant
SSPPHE investigation work, and another from Piper et al. [3] (also referred in this text as Piper’s method), which is used in
the design ε-NTU model to determine the internal volume (V), heat transfer area, Dh, and other relevant geometricalparameters. The mass flux rate (φ) was also computed to display pressure drop against different flow conditions and is
obtained from the volumetric flow rate (Q1), the density at mean temperature (ρ) and Acs. However, since this PPHE is evensmaller, a significant deviation from the available geometrical models was expected and had to be determined before
analysing data from the pressure drop analysis.
𝝃𝑫𝒂𝒓𝒄𝒚 = 𝒏𝟏 ⋅ 𝑹𝒆𝒏𝟐2 ⋅ Δ𝑃 ⋅ 𝐷ℎ
𝜌 ⋅ 𝑣2 ⋅ 𝐿𝑣 ⋅ 𝐷ℎ
𝑅𝑒 =
𝜇
𝑄⋅𝜌
𝜑=
𝐴𝑐𝑠
4 ∙ 𝑉𝑜𝑙𝑢𝑚𝑒
𝐷ℎ =𝐴𝑟𝑒𝑎𝑤𝑒𝑡𝑡𝑒𝑑

𝜉𝐷𝑎𝑟𝑐𝑦 =

(7)
(8)
(9)
(10)
(11)

2.5

Passage Area Measurements
To evaluate theoretical Acs values, the pillow was cut to expose the cross-sectional area at specific points where it is at
the minimum and maximum. Images were then captured with a calibrated microscope at low magnification, and a computer
vision algorithm was applied to measure the area in the pictures, providing a reference range for minimum and maximum
area. To validate these measurements, the inner channel was filled with distilled water, and the pillow was weighed before
and after being filled, to determine the weight gain. Knowing the specific density of water at 20°C, the inner volume wascomputed. By dividing the volume by (L-2le), Acs can be determined, following the approach of calculating the crosssectional area by dividing the volume element per its length (the longitudinal semi-pitch)[3]. The hydraulic diameter wasdetermined using Eq.11, with the wet area (Areawetted) calculated by the SSPPHE design model which implements theapproach from Piper et al. [3]. These values are important as they allow to check the difference between the computed values,
which apply to an infinitely extended pillow plate surface with negligible border effects, and the real values of the studied
small scale pillow plate with the same set of geometrical parameters.

3

Experimental Results and discussion

3.1

Pressure Drop Study
For the characterization of the pressure drop in the Inner Channel (IC) of the Small-Scale Pillow Plate Heat Exchanger(SSPPHE), experiments were conducted over a mass flow ranging from 0.042 to 0.330 kg/s. The temperature in the IC was
intended to be constant, but minor variations due to system instabilities resulted in a mean temperature range of 313.15320.15 K. The Outer Channel (OC) maintained a temperature range of 288.15-290.15 K. The obtained value for averagemeasured area from the volume of contained water and the wet area, is 118.36 mm² (water density at 20°C was considered
998.29 kg/m3). This value is validated against the range defined by the minimum and maximum values of the area measured
from the images, which are 77.60 mm² and 148.60 mm². From the images hi, was found to have a maximum and a minimum

HTFF 127-4

Figure 2: Representation of a) pressure drop data against mass flux; b) Darcy friction against Reynolds number. Bothdiagrams present different curves from the same data, analyzed with different values for hydraulic diameter and passage area of the
PPHE. Sets of points and curves, from top to bottom, blue to green, are respectively calculated with quantities from references[3],[11] and experimental values from this work.

of 2.99 mm and 2.52 mm: the former value indicates a good agreement with the nominal hi value, but also local deviations
from it. The corresponding Dh values were 4.24 mm (Arsenyeva’s method), 4.06 mm (Piper’s method), and 3.32 mm
calculated from the destructive testing (later referred to as “experimental parameters”).
Using the pressure drop data and the determined cross-sectional areas and hydraulic diameters, the Reynolds number,mass flux, and Darcy factor were computed. Figure 2a illustrates the relationship between the pressure drop in the IC andmass flux, while Figure 2b presents the computed Darcy factor for the three different approaches. Based on the measured
points, three different fitting curves were computed using Eq. 7. The resulting curves are shown in Figure 3, with thecoefficients of Eq. 7 found to be: 𝑛1=4.778 (blue line – Arsenyeva’s method), n1=3.922 (orange line – Piper’s method), and
𝑛1=2.135 (green line – experimental parameters), while 𝑛2=−0.116 for all curves. These results are significantly influenced
by the computed areas and hydraulic diameter, underscoring the importance of accurate measurements. Figure 2a shows thepressure drop in the IC plotted against mass flux, and Figure 2b shows the computed Darcy factor against Reynolds number.
These figures demonstrate the influence of the calculation method on the pressure drop and Darcy factor curves. Figure 2a
indicates higher mass flux numbers for the lower computed area at a given pressure drop value, while Figure 1b suggeststhat the actual passage area and hydraulic diameter are significantly lower than those calculated by the geometricalcorrelations. The curves in Figure 2b are, at a given Reynolds number, too high compared to all available pressure drop dataon PPHEs [7]. The experimental results were then compared by plotting the computed Darcy factor using Eq. 8 against theanalytical results derived from the method in [11]. Figure 3a shows four curves derived in [11] for significant PPHE
geometries in the literature, and an orange curve representing the correlation presented in [11] but calculated for the SSPPHE
in this study. This is the curve the measured data is expected to fit. The blue cross marks on the plot represent the experimental
Darcy values, derived from pressure measurements and the average area and hydraulic diameter obtained from experimental
destructive testing explained in section 2.5. Figure 3b provides a direct comparison between the experimental Darcy values
and those computed using the method from [11] with the pillow parameters from this study at the same Reynolds numbers.
This plot shows a deviation of up to 30% from equality, which can be attributed to the border effect not considered by current
geometrical models [3], [11], and the averaging technique used in this study, which does not account for the flow regime
near the narrowly inflated borders of the internal channel of the SSPPHE.

HTFF 127-5

Figure 3: Comparison of Darcy values from experimental data, determined with experimental hydraulic diameter and crosssectional areas from this study. In Fig. 2a) the Experimental data points are plotted against curves drawn with the SSPPHE correlation
for Darcy presented in [11] and the same correlation including the geometrical parameters of this study (ξ_Experimental). Fig. 2b)
Directly compares the experimental values for the Darcy friction factor and the ones determined with the approach proposed in [11]3.2

SSPPHE Characterization and ε-NTU model performance evaluation.
The data set presented by Table 1 reveals critical insights into the behaviour of the heat exchanger across two
experimental campaigns. In the first campaign, the IC conditions were kept constant, and the OC Reynolds numbers (Re2)are varied (524 to 1807), leading to a wide range in the thermal resistance of the OC. In the second campaign, Reynolds
number is varied only in the internal channel. The moderate deviation between experimental and calculated overall heattransfer coefficient values reported in Table 1, suggest the onset of turbulence at low Reynolds number for both the OC and
IC of the SSPPHE. It is generally possible, across the dataset, to appreciate the general fidelity of the predictions of themodel, but also some variability in the stability of the experimental system, since the non-adiabaticity (%NA) does not seemto correlate to the flow regime and thermophysical property variation. In Figure 4, two relevant output values predicted bythe ε-NTU model are plotted against the measured values of two key output parameters of the SSPPHE: the thermal power,
and the outlet temperature for both the inner and outer channel. The comparison of computed vs. measured output powers,
as depicted in Figure 4a, provides critical insights into the performance and accuracy of the ε-NTU model for the PPHE. The
ε-NTU model demonstrates accuracy within specified Reynolds (Re) and Prandtl (Pr) number ranges but exhibits higher
deviations when applied increasingly outside these ranges, particularly in the OC. In Figure 4a it is also possible to see
separate points for IC (hot fluid) and OC (cold fluid), with the gap related to the non-adiabaticity (NA). Positive deviations
indicate underestimation of a computed parameter.
IDs 1&1 and 1&2 show significant positive deviations (13.65%, 15.26%) with Pr2 values that are close to the lower
limit valid range. High deviations are also observed in the second experimental run for IDs 2&9 (6.53%) and 2&7 (5.68%),
indicating underestimation at higher Reynolds numbers in the inner channel.
Negative Deviations indicate overestimation by the model, and IDs 1&3, 1&4, and 1&5 exhibit negative deviations (6.80%, -9.13%, -8.01%) despite being within the valid range for Re1, most likely due to error from OC low Re conditions.
ID 2&1 (Re1 = 914) shows a -7.53% deviation: since the Reynolds value for the IC is below the valid range of the Nusselt
correlation, it is possible to confirm reduced accuracy outside the specified ranges. In Figure 4b, deviation in the predictionof outlet temperatures is shown, completing the information from Figure 4a. The deviations are much smaller for thepredicted outlet Temperatures, which is to be expected since both the computed power and the measured power are derived
from the Temperature differences, the specific heat, and the measured flow rates.

HTFF 127-6

Table 1: Summary of key parameters of influence for performance evaluation: effectiveness (ε), global heat exchange coefficient(U), Non-adiabaticity (NA), Reynolds (Re) and Prandtl (Pr) numbers calculated by the presented ε-NTU model, heat exchange
coefficients (h) calculated by the model, Power (P). The subscript “exp” is for values calculated from experimental data, “comp” is forparameters computed by the model, while 1 and 2 indicate the inner (hot) and outer (cold) channel.
ID
1&1
1&2
1&3
1&4
1&5
2&1
2&2
2&3
2&4
2&5
2&6
2&7
2&8
2&9

εexp

εcomp

(%)
36.78
28.13
23.93
26.86
31.09
50.83
35.34
27.19
27.19
21.91
23.24
23.76
25
25.46

(%)
33.81
25.41
27.12
30.09
34.7
56.04
37.61
28.13
28.02
22.42
23.31
23.93
24.49
24.84

Uexp
(W/m2K)

919
1410
1132
1270
1492
1039
1292
1440
1437
1515
1578
1602
1668
1701

NA
(%)
4.25
3.97
5.31
1.78
2.57
1.92
2.63
3.82
1.59
3.8
1.93
6.04
0.9
3.8

Re1

Re2

Pr1

Pr2

2649
2657
2737
2705
2663
914
1904
2947
2945
4026
5071
6112
7150
8190

524
1000
1121
1351
1807
1269
1327
1354
1346
1368
1368
1368
1370
1380

3.73
3.87
3.82
3.86
3.93
4.26
4.02
3.91
3.92
3.83
3.76
3.74
3.73
3.71

6.51
6.88
7.42
7.62
7.8
8.09
7.91
7.8
7.82
7.73
7.62
7.64
7.6
7.57

h1

h2

Ucomp

(W/m2K)(W/m2K)(W/m2K)

814
1218
1321
1471
1718
4056
6794
9249
9252
11528
13552
15502
17354
19118

936
1537
1703
1970
2460
1909
1962
1984
1977
1995
1989
1990
1990
1999

814
1218
1321
1471
1718
1204
1399
1495
1491
1552
1581
1606
1624
1645

Pcomp
(W)
2727
4160
4736
5316
6180
3759
4982
5605
5564
5991
6219
6395
6472
6614

Given that the highest thermal resistance is in the outer channel, it can be stated that the overall thermal resistance iscloser to its value. However, the model's predictions are reliable within a 15% deviation, allowing the outer channel
correlation to be used down to Re = 1000, thereby extending its applicability. This extension is significant for designing heatexchangers operating under varied flow conditions, ensuring robust thermal performance predictions.

Figure 4: Comparison between a) the computed power and b) the computed temperature, against the respective measured value.
Positive deviation from equality corresponds to underestimation of the quantity by the model.

HTFF 127-7

4

Conclusions

This work investigates the accuracy of a design procedure based on the ε-NTU method to evaluate the hydro-thermal
performances of a compact PPHE. The predictions of the model align well with the experimental data within acceptableerrors ranges. Future works will be focus on extending the experimental activities and enhancing the methodology pertaining
to the calculation of the geometrical parameters. Considering the global performances referred to the actual state of the study,
the obtained model can be used for a preliminary evaluation of PPHH design.

Acknowledgements
The authors would like to express their gratitude to DavCoil srl for supporting this project with the construction of the
SSPPHE and the acquisition of new instrumentation for the experimental runs. Special thanks are also extended to Dr. Mauro
Hueller, Dr. Francesco Negrisolo and Dr. Raffaele De Biasi who provided invaluable assistance in the experimental work.

5

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