Analysis of Corrugation Pitch Influence on Pressure Distribution and Flow Maldistribution in Chevron-Type Plate Heat Exchangers

Plate heat exchangers (PHEs) play a pivotal role in energy conservation and efficient energy utilization. They are recognized for their notably high area density. The high area density can be described as the ratio of surface area’s of heat transfer to the heat exchanger’s volume. This characteristic results in a smaller hydraulic diameter for fluid flow that leads to higher efficiency than conventional tubular heat exchangers, all within a much smaller volume. In recent years, PHEs have gained widespread adoption across various industries including heating, cooling, and heat-regeneration. These heat exchangers are chosen for their beneficial features, such as high overall heat transfer rates, enhanced heat transfer coefficients and areas, excellent efficiency, suitability for sanitary applications in industries like brewing, and food production, as well as ease of maintenance.

Mueller and Chiou [1] conducted a thorough review focusing on the challenges associated with maldistribution. Meanwhile, Bassiouny and Martin [2, 3] conducted an in-depth analysis of PHEs, encompassing the derivation of parameters such as axial velocity, total pressure drops, and pressure distributions in the entry and exit ports. They also explored flow distribution within the channels. Their investigation resulted in the establishment of a generalized parameter, designated as m², which represents flow maldistribution. This parameter was formulated by considering mass and momentum equations for both entry and exit port flows across different PHEs. In his research, Bajura [4] conducted a thorough analysis of flow distribution systems in inlet and exit ports. He created a mathematical framework to elucidate the flow dynamics at specific junctions within the manifold, emphasizing the importance of momentum balance. Acrivos et al. [5] conducted research on the partitioning of fluid streams within manifolds, focusing on pressure alterations caused by wall friction and fluctuations in fluid momentum. Meanwhile, Martin [6] studied the effects of the largest flow path and the interaction between crossings and longitudinal flow. Through his analysis, he derived a simple equation for the friction factor that considers parameters such as the chevron angle and Reynolds number. This equation provides a plausible depiction of the system's behavior. Kumar and Singh [7] studied the hydraulic performance analysis of PHE with a chevron angle of 60°. Yong et al. [8] investigated the PHE's flow characteristics and developed a PHE resistance calculation model.

Mulley and Manglik [9] undertook experimental research focusing on turbulent flow heat transfer and pressure drop within the PHE. Their study investigated various corrugation angles to understand their impact on heat transfer and pressure drop features. They considered equal flow rates in channels for their studies. Tereda et al. [10] studied conduit-to-channel uneven division of fluid flow in PHEs with a fixed number of plates and corrugated angles, while varying port diameters. Gulenoglu et al. [11] conducted experimental research focusing on the thermo-hydraulic performance of three distinct plate. Further, with experimentation they introduced a corelation for novel correlation to describe the relationship between the Nusselt number and friction factor. This correlation provides a mathematical representation that connects these essential parameters, aiding in the understanding and prediction of heat transfer and fluid flow characteristics within the studied system. Faizal and Ahmed [12] focused on the pressure drop and heat transfer in PHEs with varying spacings between corrugated plates. Han et al. [13] performed numerical simulations and experimental analyses to investigate the temperature, pressure, and velocity fields in corrugated PHEs. Their observations included temperature distributions and pressure reductions along the flow direction. Focke et al. [14] carried out experimental investigations to assess how corrugation inclination angles affect the thermohydraulic performance of PHEs. They assumed uniform flow within each channel. Their study provides insights into the correlation between corrugation angles, heat transfer efficiency, and fluid flow characteristics within these exchangers. Khan et al. [15] studied the effects of heat transfer characteristics in single-phase flow across a range of Reynolds numbers, chevron angles, and corrugation depths. Nilpueng and Wongwises [16] examined heat transfer coefficients and water flow’s pressure drops within PHEs featuring rough surfaces, comparing them with smooth surfaces. Bobbili et al. [17] and Rao and Das [18] conducted experimental investigations on flow maldistribution in PHEs, considering various plate’s package sizes and corrugation angles. They observed that flow maldistribution increased with overall pressure drop. Fernandes et al. [19] examined flow characteristics in corrugated-type PHEs, different chevron angles, aspect ratios, and fluid viscosities at lower Reynolds numbers. They observed correlations between friction factors, aspect ratios, and chevron angles. Agarwal [20] examined the temperature distribution, effectiveness, and total heat transfer coefficient for various Reynolds numbers in PHEs.

The literature review identifies a significant research gap concerning the impact of corrugation pitch as a crucial parameter in a PHE. Specifically, there is a shortage of comprehensive studies exploring how corrugation pitch influences the channel frictional factor (CFF), flow maldistribution, channel pressure drop, and overall nondimensional pressure drop. The present study aims to fill these gaps.

The plate configuration utilized for analytical computations is presented in Figure 1. These plates, densely packed and featuring sine-wave corrugations, constrain open flow passages intersecting at an angle of β, resulting in a complex flow pattern and temperature distribution. This setup holds promises for achieving high heat transfer rates while maintaining relatively low pressure drops. Analytical studies of sine waves suggest that the length of the curve between two fixed points varies depending on both the amplitude and the period of the wave.

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Figure 2. Sinusoidal corrugated plate

For demonstrating the changes in curve length between two fixed points having variations in amplitude and period, the sine waves geometry is shown in Figure 2. The plate's geometry resembles a sinusoidal curve, enabling the expression of the corresponding sine wave equation as shown in Figure 3.

$y=\frac{P}{2} \sin \frac{2 \pi}{P_C} x$         (1)

The length of the curve described by the equation {y=f(x), z=0} between x=a and x=b is given by:

$L_\lambda=\int_a^b \sqrt{1+y^{, 2}} d x$    (2)

The chevron pattern's wavelength aligns with the corrugated pitch, as depicted in Figure 1. In this representation, the sinusoidal corrugation's amplitude is labeled as p, while the plate thickness is denoted as t. Considering the sinusoidal curve of the wavy surfaces, the length of a sine curve can be expressed as:

$L_\lambda=\int_0^{p_c} \sqrt{1+\left(\frac{\pi p}{p_c}\right) \cos ^2\left(\frac{2 \pi x}{p_c}\right)} d x$          (3)

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Figure 3. Sine curve for plate geometry

Eq. (3) represents an elliptical integral, and therefore, its solution can be obtained as follows:

$L_\lambda=\int_0^{p_c} \sqrt{1+\left(\frac{\pi p}{p_c}\right) \cos ^2\left(\frac{2 \pi x}{p_c}\right)} d x=\frac{\sqrt{\frac{\pi p}{p_c}+1}}{\frac{2 \pi}{p_c}} E\left[\frac{\frac{\pi p}{p_c}}{\frac{\left(\frac{\pi p}{p_c}\right)}{\left(\frac{\pi p}{p_c}+1\right)}}\right]$       (4)

The incomplete elliptical integral solution of the second kind is provided by Brown [21].

$\begin{aligned} E\left(\frac{\phi}{m}\right)=\int_0^\phi(1- & \left.m \sin ^2 \theta\right)^{\frac{1}{2}} d \theta =\frac{2 \phi}{\pi} E+\sin \phi \cos \phi\left[\frac{1}{2} A_2 m\right. +\frac{1}{2 * 4} A_4 m^2+\frac{1 * 3}{2 * 4 * 6} A_6 m^3  +\cdots\end{aligned}$       (5)

where, $A_2=\frac{1}{2}$, $A_4=\frac{3}{2 * 4}+\frac{1}{4} \sin ^2 \phi$, $A_6=\frac{3 * 5}{2 * 4 * 6}+\frac{5}{4 * 6} \sin ^2 \phi+\frac{1}{6} \sin ^2 \phi$ and $\begin{aligned} & E=1+\frac{1}{2}\left\{\ln \left[4(1-m)^{\frac{-1}{2}}\right]-\frac{1}{1.2}\right\}(1-m)+\frac{1^2 * 3}{2^2 * 4}\left\{\ln \left[4(1-m)^{\frac{-1}{2}}\right]-\frac{2}{1.2}-\frac{1}{3 * 4}\right\}(1-m)^2+\\& \frac{1^2 * 3^2 * 5}{2^2 * 4^2 * 6}\left\{\ln \left[4(1-m)^{\frac{-1}{2}}\right]-\frac{2}{1.2}-\frac{2}{3 * 4}-\frac{1}{5 * 6}\right\}(1-m)^3+\cdots\end{aligned}$.

The enlargement factor (φ) can be expressed as:

$\varphi=\frac{L_\lambda}{p_c} \int_0^{p_c} \sqrt{1+\left(\frac{2 \pi p}{p_c}\right) \cos ^2\left(\frac{2 \pi x}{p_c}\right)} d x$       (6)

The surface enlargement factor for sinusoidal corrugation can be approximately determined using the following method:

$\varphi \approx \frac{1}{6}\left\{1+\sqrt{1+x^2}+4 \sqrt{1+\frac{x^2}{2}}\right\}$        (7)

where, $x=\frac{\pi p}{p_c}$.

It is crucial to emphasize that the Reynolds number for a PHE is expressed in terms of D_h (hydraulic diameter) which can be expressed as follows:

$\begin{aligned} & R e=\frac{u_{c h}}{v}\left(D_h\right) \\ & D_h=2 b L w /(b+L w \varphi) . \text { Since } b<L w, D_h=2 b / \varphi\end{aligned}$                (8)

The fluid mean velocity in the channel can be expressed as:

$u_{c h}=\frac{V}{w b n}$         (9)

Martin [6] provided one of the correlations for the friction factors of PHEs with chevron patterns. The Fanning friction factor can be expressed as:

$f=\left[\frac{\cos \beta}{\left(0.045 \tan \beta+0.09 \sin \beta+\frac{f_o}{\cos \beta}\right)^{\frac{1}{2}}}+\frac{1-\cos \beta}{\sqrt{3.8 f_1}}\right]^{-0.5}$     (10)

where, $f_o=\frac{16}{R e}$ for $R e<2000$ and $f_o=(1.56 \ln R e-3.0)^{-2}$ for $R e>2000$  $f_1=\frac{149.25}{R e}+0.9625$ for $R e<2000$ and $f_1=\frac{9.75}{R e^{0.289}}$ for $R e>2000$.

To validate the analytical results, the friction factor correlation for the existing PHE was compared with experimental data [18].

$f=21.41 R e^{-0.301}$        (11)

The total pressure drop, comprising the channel’s pressure drop, port’s pressure drop, (with an inner diameter of 25.4 mm), and the pressure drop in the passages, can be calculated using the following equation:

$\Delta P_{t p}=\Delta P_{c m}+\Delta P_{\text {port }}+\Delta P_{e c}$         （12）

The calculation of the port’s pressure drop relies on the total flow rate and is determined by empirically established equation [17].

$\Delta P_{\text {port }}=1.5 \rho \frac{V^2 p}{2}$       （13）

The pressure drops attributed to bends, sudden contractions, and sudden expansions at the entry and exits are determined using the following formula:

$\Delta P_{e c}=K_{e c} \rho \frac{V^2{ }_p}{2}$       (14)

where, $K_{e c}$ represents the total pressure loss coefficient associated with bends, sudden contractions, and sudden expansions at the entry and exit of the conduits.

The pressure drops resulting from friction in the corrugated passage are evaluated from flow rate, employing an empirical formula.

$\Delta P_{c h m}=f_{c h} \frac{L_{c h}}{d_h} \rho \frac{V_{c h}^2}{2}$     (15)

The value of m² is determined using the equation provided by Bassiouny and Martin [2] for similar entry and exit port dimensions (Details are given in the Appendix).

$m^2=\left(\frac{n A_C}{A_P}\right)^2 \frac{1}{\xi_C}$        (16）

Here, $\xi_C$ represents the total frictional resistance of the channel and is equal to $\xi_C=f L_{c h} / d_h$ plus other minor losses. The total nondimensional pressure drop of the PHE is calculated as:

$p_{\text {in }}-p_0=\left(\frac{m^2}{\tanh ^2 m}\right)\left(\frac{A_P}{n A_c}\right)^2 \frac{\xi_c}{2}$        (17)

The variation in pressure drop from the first to the last channel is determined using an equation from Bassiouny and Martin [2], which is based on the flow maldistribution parameter (m²).

$\frac{\Delta P}{o W_o^2}=\left(\frac{A_p}{n A_c}\right)^2 \frac{\xi_c}{2} m^2 \frac{(\cosh m(1-z))^2}{(\sinh m)^2}$     (18)

The analytical presentation of flow maldistribution, and pressure drop considers a broad range of corrugation pitch, Reynolds number, and aspect ratio. The study investigates the effects of corrugation pitch and Reynolds number on CFF, flow maldistribution and channel pressure drop in PHEs. It is observed that corrugation pitch has an increasing effect on the hydraulic performance of a PHE, with higher aspect ratios leading to larger pressure drops. However, a larger corrugation pitch tends to decrease the channel pressure drop. The analysis reveals that pressure drop increases with the rise in Reynolds number for all corrugation pitch values. Additionally, the flow maldistribution parameter increases with a decrease in channel aspect ratio for the same corrugation pitch and the flow rate in a PHE. Furthermore, it is found that flow maldistribution parameter increases with increasing corrugation pitch and Reynolds number, while maintaining a fixed channel aspect ratio and water flow rate in the PHE. The nondimensional pressure drop decreases with increasing corrugation pitch for three different channel aspect ratios.

It is noted that for a given channel aspect ratio, increasing water flow rate leads to decreased pressure drops in the channels but increases the uneven fluid flow in the channel. Overall, the study highlights the need to balance Reynolds number, corrugation pitch, and aspect ratio to optimize heat exchanger performance.