Dimensional analysis for the heat transfer characteristics in the corrugated channels of plate heat exchangers

Dimensional analysis for the heat transfer characteristics in the

corrugated channels of plate heat exchangers

☆

J.H. Lin, C.Y. Huang, C.C. Su

⁎

Department of Mechanical Engineering, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan, ROC Available online 16 January 2007

Abstract

Using the Buckingham Pi theorem, this study derives dimensionless correlations to characterize the heat transfer performance of the corrugated channel in a plate heat exchanger. The experimental data are substituted into these correlations to identify the flow characteristics and channel geometry parameters with the most significant influence on the heat transfer performance. Simplified correlations by omitting the factors with less influence are then obtained. The results show that Nuxis affected primarily by Re, R/

Dh, x/Dh, andβ. Neglecting the minor effect of factors on Nux, it is shown that Numis determined primarily by Re, R/Dhandβ.

© 2007 Published by Elsevier Ltd.

Keywords: Corrugated channel; Buckingham Pi theorem; Dimensionless correlation; Nusselt number; Forced convection

1. Introduction

Plate heat exchangers (PHEs) are widely applied throughout industry and are commonly designed with a corrugated channel surface resulting in the enhanced heat transfer performance by increasing the area over which heat transfer takes place and generating a vigorous mixing effect within the working fluid.

In the literature[1,2], it is shown that the use of a corrugated channel results in a more complex flow structure and improves the heat transfer by as much as two or three times compared to a conventional straight channel. In[3–6], the authors demonstrated that sinusoidal wavy plate arrangements and channel geometries improved the heat transfer performance by increasing the surface area and prompting the formation of vortexes in the flow. The symmetric arrangement yields a superior heat transfer performance to an asymmetric arrangement. Unfortunately, the geometric parameters are not expressed clearly. For this reason, in[7,8], the authors conducted a systematic investigation into the relative effects of W,λ, β, and the ratio of the radius of curvature of the channel to the length of the straight section. The results revealed that Nuxincreased with Re, but decreased with the extension of the axial.

In[9,10], the flow field and heat transfer characteristics within fully developed region of corrugated channels were analyzed numerically and the predicted pressure drop within the corrugated channel was in good agreement with experimental observations. In[11,12], the authors compared the performance of asymmetric and symmetric channels

☆_{Communicated by W.J. Minkowycz.}

⁎ Corresponding author.

E-mail address:chinchiasu@ntu.edu.tw(C.C. Su).

0735-1933/$ - see front matter © 2007 Published by Elsevier Ltd. doi:10.1016/j.icheatmasstransfer.2006.12.002

and showed that the influence of the asymmetric channel geometry on the flow resistance and heat transfer characteristics of the channel flow gradually diminished as the undulation and curvature of the channel surface were reduced. In[13], Wang and Chang identified that the flow field is characterized by laminar-flow and swirl-flow regimes with Reynolds Number and corrugated aspect ratio.

The studies presented above provide important insights into the correlation between the heat transfer and the geometry characteristics of the corrugated channel. However, the definitions of the channel geometry parameters are generally different in each study and the quantities of the investigated channels were not enough slightly. Accordingly, the present study performs experimental trials and applies the Buckingham Pi theorem to develop a set of dimensionless correlations relating Nuxand Numto the flow conditions and the geometry parameters of the corrugated channel. The experimental measurements of Nuxand Numare then compared with the results calculated from the correlations in order to analyze the relative errors of the correlations.

2. Experimentation

Fig. 1presents the principal geometry parameters and illustrates the experimental setup established to investigate the heat transfer characteristics in the corrugated channel for different flow conditions. The basic components of the experimental apparatus include a water loop, an air loop, and a measurement system. The water loop comprises a water

Nomenclature

A Cross-sectional area, m2 As Surface area, m2

cp Specific heat at constant pressure, kJ kg− 1K− 1 Dh Hydraulic diameter, m

f Friction factor H Height of channel, m

hx Local heat transfer coefficient, kJ m− 2K− 1

j Coburn factor

k Thermal conductivity, kJ m− 1K− 1 m˙ Mass flow rate, kg s− 1

Num Average Nusselt number Nux Local Nusselt number P Wetted perimeter, m Pr Prandtl Number Q Heat transfer rate, kW R Radius of curvature, m Re Reynolds number Tb Bulk temperature, K Tm Mean temperature, K Tw Wall temperature, K V Average velocity, m s− 1 W Width of channel, m x, y Coordinates λ Wave length, m β Corrugated angle ρ Density of air, kg m− 3 μ Viscosity of air, N s m− 2

Δθfw Temperature difference between flow and wall, K ϕ Coefficient of determination

tank containing a heater, a pump, a flow meter, and a temperature controller. Importantly, all of the components in the water system are thermally insulated such that the wall temperature of the corrugated channel can be maintained at a nearly constant temperature. The air loop consists of the test section containing the corrugated channel, a blower, two air flow meters, and a number of valves which enable the flow rate to be adjusted. Additionally, a flow straightener is installed at the entrance of the test section to maintain a uniform inlet flow. The test section is constructed from two hollow stainless steel 304 cases, each with a corrugated surface on one side and a flat surface on the other. The two cases are clipped between two horizontal, metal plates to form the corrugated channel through which the working fluid (air) is passed. During the experiments, hot water was flowed through the two hollow cases to maintain the channel surfaces at an approximately constant temperature and T-type thermocouples wrapped in copper tubes and inserted into upper metal plates were used to record the corresponding air temperature. The side effect was avoided by specifying a small aspect ratio for the channel such that variations in the channel height could be neglected. In the experiments, the temperature distribution in the horizontal, middle plane of the channel was monitored using thermocouples positioned at more than 200 different locations along the length of the channel. The precise number of measuring points was varied in accordance with the parameters W, R andβ. For each test condition, two to three measurements were made in this study.

3. Dimensional analysis and data reduction

Using the Buckingham Pi theorem, a set of dimensionless correlations were established to investigate the heat transfer characteristics within the corrugated channel. The following procedure was adopted to determine the dimensionless parameters,π: (1) list the corresponding parameters; (2) apply the MLtθ system; (3) list the dimensions of all the parameters; and (4) letρ, μ, Dhand k denote the repeating variables. The local heat transfer coefficient, hx, is given by:

hx¼ f ðq;l; Dh; k;V; x; R; b; cp; DhÞ ð1Þ

where the hydraulic diameter is defined as:

Dh¼

4A

P ¼

2WH

Wþ H ð2Þ

The dimensionless parameter groupsΠ1–Π7listed inTable 1are obtained through the procedures described above. The dimensionless correlation of the heat transfer performance within the channel can be expressed as:

j1¼ adjb2dj c 3dj d 4dj e 5dj f 6dj g 7 ð3Þ

where Π1,Π4,Π6, and Π7are calculated from the measured temperature data within the fully developed region. Applying the differential control volume concept and the energy conservation law, it can be shown that:

dQ¼



Combining Eqs. (4)–(6), the local heat transfer coefficient can be expressed as: hx¼



mcp½ðTmÞ_xþdx−ðTmÞ_x

ðTw−TbÞdAs

ð7Þ Therefore, it can be shown that:

Nuxu hxDh k ¼



In Eq. (10) presented inTable 2a, the value of constant a and the exponents b–g of the Π parameters are obtained

by substituting the experimental data into Eq. (3) using STATISTICA statistical software. Subsequently, omitting each parameter in turn, Eqs. (11)–(16) can be obtained. Comparing Eq. (11) to Eq. (16), it is observed that the effects ofΠ6andΠ7onΠ1are minor. Accordingly, Eq. (17) is obtained by neglecting the effects ofΠ6andΠ7in Eq. (10). Fig. 2compares the measured Nuxwith those predicted using Eq. (10). From inspection,ϕ is found to be 0.914 and the range ofξ varies from −30% to 30%. Here, ϕ is a relative comparison criterion between the measured and the

Table 1

Dimensionless∏ groups

Groups Definition Effect Range

Π1 hxDh/k Local Nusselt number –

Π2 ρVDh/μ Reynolds number 300–7000

Π3 R/Dh Geometry 1.21–3.25

Π4 x/Dh Location 1–14.5

Π5 β Geometry π/12–π/4

Π6 ρ2Dh2kΔθ/μ3 Temperature difference 1.328E + 11–10.507E+11

Π7 μcp/k Prandtl number 0.703–0.706

predicted data and ξ indicate the maximum deviation of the experimental data from the predicted values. As ϕ approaches a value of 1, the predicted data converge to the measured data. The mathematic definitions ofϕ and ξ are given respectively by:

/ ¼ P ðj1;pred−j P 1;measÞ2 P ðj1;meas−j P 1;measÞ2 ð18Þ n ¼ðj1;meas_j− j1;predÞ 1;pred ð19Þ

whereΠ1,measis the measured data ofΠ1,Π1,predis the predicted data calculated from Eq. (3), and Π¯1,meas is the average value of the measured data ofΠ1.

4. Results and discussion

In the straight duct, the Nuxconverges to a constant value within the fully developed region. However, in the case of the

corrugated channel,Fig. 3 indicates that the value of Nux reduces with increasing x/Dh. Furthermore, it is evident that Nux

attains a local peak at the individual crests of the channel surface. This result is very different from that observed in a straight duct. Therefore, the heat transfer characteristics discussed below are related to the fully developed region, as defined by Lee et al. [7].

Table 2a

Dimensionless correlation for Nux:∏1= a·∏2b·∏3c·∏4d·∏5e·∏f6·∏7g

a b c d e f g Eq. number Π1 10− 2.79 0.912 0.334 −0.282 0.198 0.104 0.010 (10) Π1,a 10− 8.53 – 0.820 0.068 0.241 0.813 0.246 (11) Π1,b 10− 0.72 0.922 – −0.337 0.174 −0.066 0.029 (12) Π1,c 10− 6.13 0.893 0.475 – 0.221 0.376 0.003 (13) Π1,d 10− 2.58 0.914 0.307 −0.292 – 0.083 0.015 (14) Π1,e 10− 1.54 0.919 0.251 −0.333 0.189 – 0.014 (15) Π1,f 10− 2.82 0.913 0.338 −0.282 0.200 0.106 – (16) Π1,g 10− 1.52 0.921 0.255 −0.333 0.190 – – (17)

4.1. Nuxand Num

Observing the exponents in Eq. (10), it is seen that Nuxincreases with increasing Re, R/Dh,β, Δθfwand Prx, but decreases with

increasing x/Dh. As shown inTable 1,Π8corresponds to Num, and is computed with Eq. (9). Therefore, the effect ofΠ4,Π6, andΠ7

can be neglected in developing an expression forΠ8, i.e.

j8¼ AdjB2djC3djD5 ð20Þ

The experimental data obtained for Numin the fully developed region are substituted into Eq. (20) using statistical software to

yield Eq. (21), shown inTable 2b. From the exponents in Eq. (21), it can be seen that Numincreases with increasing Re, R/Dhandβ.

Fig. 4compares the measured Numwith those predicted from Eq. (21). From inspection,ϕ is 0.985 and ξ varies between −20% and

30%. From the variations of the exponents in Eqs. (10) and (21), it can be inferred that Nuxand Numare affected by the factors listed

inTable 1, respectively. 4.2. Effect of Reynolds number

With a value of 0.912, exponent b ofΠ2is the highest of all the exponents in Eq. (10). This not only indicates that Nuxincreases

with increasing Re, but also implies that the effect of Re on Nuxis more significant than that of any of the other factors. As shown in Fig. 3. Variation of Nuxalong channel length.

Fig. 3, at a low value of Re, the variation of Nuxwith increasing x/Dhis relatively minor with increasing R/Dhandβ.For a given

value of Dh, a higher Re prompts the formation of flow vortexes around the crests of the channel surface with increasing inlet

velocity. These vortexes cause the flow to become turbulent, resulting in an enhanced mixing effect. Consequently, the value of Nux

increases. Furthermore, at high Re, Nuxincreases notably with increasing R/Dhandβ. The exponent B in Eq. (21) has a value of

0.914, which is in good agreement with the results ofFig. 5, which shows that Numincreases strongly with increasing Re. When Re

is omitted in Eq. (11),ϕ reduces to 0.133. Hence, the dimensionless analysis confirms that Re has the most significant effect on Nux

and Num.

4.3. Effect of radius of curvature

In Eq. (10), the exponent c ofΠ3has a value of 0.334, which indicates that Nuxincreases with R/Dh. As shown inFigs. 3 and 5,

for the case of Re = 7000 andβ=π/6, the R/Dhincrease from 1.75 to 2.30 implies that the ratio of the length of the straight section of

the channel to the total length of the channel decreases. Therefore, by implication, the ratio of the length of the corrugated section of the channel to the total length of the channel increases. The flow in the straight section of the channel tends to become laminar. However, the fluid experiences a greater disturbance in the corrugated region of the channel with increasing R/Dh. The reduction in

the wave amplitude of the corrugated section of the channel with increasing R/Dh weakens the vortexes generated around the

corrugated section. Nevertheless, the overall result show that Nuxstill increases by approximately 20–40% and Numincreases by

approximately 30%. These results are consistent with the numerical results presented by Yang and Chang[14]. As shown in Eqs. (10) and (21), the effect of R/Dhon the Nusselt number is greater than that ofβ. An increased value of β induces vortexes within the

channel as a result of the flow separation. The enhanced mixing effect enhances the heat transfer rate on the one hand, but reduces the effective heat transfer area resulting in a reduction in the heat transfer rate on the other. The overall result is that the effect of R/Dhon

the heat transfer performance is greater than that ofβ. The exponent C in Eq. (21) has a value of 0.338, which confirms that Num

increases with increasing R/Dhand has a greater effect on Numthanβ. When R/Dhis omitted,ϕ reduces from 0.914 to 0.875 and ξ

varies between−38% and 38%. Fig. 6 compares the predicted Numusing Eq. (21) with those calculated by Lee [15]. From

inspection,ξ varies between −20% and 40%. 4.4. Effect of distance from entrance

Exponent d ofΠ4in Eq. (10) has a value of−0.282, which implies that Nuxdecreases with increasing x/Dh. As shown inFig. 3, as the

thermal boundary layer of the fluid in the entrance region grows, the temperature gradient of the fluid near the wall decreases with increasing x/Dh. Therefore, Nuxreduces with reducing hxas x/Dhincreases. In a straight duct, for a given value of Re, Nuxconverges to a Table 2b

Dimensionless correlation for Num:∏8= A·∏B2·∏3C·∏5D

A B C D Eq. number

Π8 10− 1.747 0.914 0.338 0.258 (21)

constant value in the fully developed region, and hence the effect of x/Dhneed not be considered. However, in the present corrugated

channel, the vortexes generated near the crests of the channel cause a local increase in Nux. Note that these results are consistent with the

numerical findings presented by Rush et al.[5]. In other words, x/Dhhas an important influence on Nuxin the corrugated channel. When

the effect of x/Dhis neglected in Eq. (13),ϕ reduces from 0.914 to 0.889 and the range of ξ varies from −35% to 38%.

4.5. Effect of corrugated angle

Exponent e ofΠ5in Eq. (10) has a value of 0.198, while exponent D ofΠ6in Eq. (21) has a value of 0.258. These results indicate

that both Nuxand Numincrease with increasingβ. The abrupt change induced in the direction of flow of the fluid as β increases

enhances the heat transfer rate. As shown inFig. 3, for given values of Re, R/Dh and x/Dh, Nux increases with increasingβ.

Similarly, in the case of Re = 3500 and R/Dh= 2.30 shown inFig. 5, Numincreases by approximately 25% asβ increases from π/6 to

π/4. The increased value of β causes the laminar flow within the channel to transit to turbulent flow at a lower value of Re. For example, for a constant value of R/Dh= 2.30, Numin the case ofβ=π/4 and Re=1900 is roughly the same as that in the case of β=π/

6 and Re = 2500. Therefore, omitting factorβ in Eq. (14) causes ϕ to reduce from 0.914 to 0.895 and ξ to vary between −32% and 35%. Hence, it is apparent that the corrugated angle of the channel has a significant effect on the value of the Nusselt number. 4.6. Effect of local temperature difference and local Prandtl number

Exponent f ofΠ6in Eq. (10) has a value of 0.104, and hence it can be inferred thatΔθfwof the flow entering the fully developed

region converges gradually to a constant. This implies that the increases in Nuxin the regions around the crests of the channel are

caused primarily by variations in R/Dhandβ rather than by that in Δθfw. WhenΔθfwis omitted in Eq. (15),ϕ decreases from 0.914

to 0.910 andξ varies from −35% to 33%. Exponent g of Π7in Eq. (10) has a value of 0.010 and is the lowest of any of the

exponents. Therefore, it can be surmised that the effect of Prxon Nuxis less than that of any of the other factors. This result is

reasonable since this study uses air as the working fluid. The variation ofμ, cpand k for air in the measured range is less than one

percent, and therefore the effect of Prxon Nuxand Num, respectively, investigated in Eq. (21) can be neglected. When Prxis omitted

in Eq. (16),ϕ decreases slightly from 0.914 to 0.911 and the range of ξ varies between −35% and 30%.

5. Conclusions

This study has developed dimensionless correlations for analyzing the heat transfer characteristics of the corrugated channel used in a PHE. The major contributions and analytical findings of this study can be summarized as follows: 1. Dimensionless correlations for Nux and Num have been developed based on experimental results and the Buckingham Pi theorem. Compared with the experimental results, the accuracies of the dimensionless correlations are found to be acceptable.

2. The dimensionless analysis reveals that Nuxis determined primarily by Re, R/Dh, x/Dhandβ. Conversely, Prxand Δθfwdo not have a significant effect on the heat transfer performance.

3. The value of the mean Nusselt number, Num, is determined principally by Re, R/Dhandβ. Acknowledgement

The current authors gratefully acknowledge the financial support provided to this study by the National Science Council of Taiwan under Contract No.SC90-2212-E-002 -201.

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