2. Terminology of the louvered fin

secondary flow patterns introduced by sudden velocity changes. On the other hand, interrupted surfaces achieve heat transfer enhancement by the continuous growth and destruction of laminar boundary layers on the interrupted portion of the geometry. One of the mostly used example of the interrupted surface is the louvered fin. The louvered fins were firstly investigated by Kays and London [2] in 1950s, and the popularities of the louvered fins

Figure 1. Fin geometries for plate fin heat exchangers: (a) plain triangular fin; (b) plain rectangular fin; (c) wavy fin; (d)

Today, the use of louvered fins has become popular in the fields of automotive, heating, cooling, air conditioning, power plants and food industry. Typical structure of the louvered fin is shown in Figure 2. The efforts of maximize the heat transfer and minimize the pressure drop in heat exchanger design are rapidly increasing due

have been maintained.

offset strip fin; (e) multi-louver fin; (f) perforated fin [1].

62 Heat Exchangers– Advanced Features and Applications

Figure 2. Typical structure of a louver fin geometry [4].

The terminology of a louvered fins was firstly created by Kays and London [2] as shown in Figure 3. Each louvered fin configurations is designated by two figures. The first indicates the length of the louvered fin in the flow direction and the second indicates the fin pitch per inch transverse to the flow. Therefore, the meaning of "1/2–6.06" is that each louver has length of 1/2 inch in the flow direction and 6.06 fins in per inch.

Another terminology used for louvered fins with a flat tube is illustrated in Figure 4. It is shown from the cross-sectional view that the gap between two louvered fins is called fin pitch (Fp). The length of the flow from the leading edge up to the end of the fin is called flow depth (Fd). The vertical length of the fin and the louver are called fin height and louver height and designated by Fh and Lh, respectively. The horizontal length of the gap between each louver is called louver pitch (Lp), and each louver has an angle of Lα.

Figure 3. Terminology of the louvered fin [2].

Figure 4. Terminology of the louvered fin [6].

#### 3. Flow phenomenon in louvered fin arrays

The structure of the flow passes through the louvered fins can be identified with flow visualization technique using dye injection and hydrogen bubble. Numerical analysis is also a method to identify the flow phenomenon of louvered fins. Several experimental and numerical studies indicate that the geometric design and the free velocity of the flow effect the direction of the flow through the louvers. As shown in Figure 5, the flow enters from the leading edge of the first louver and then directed by the louvers or the fins. If the greatest proportion of the flow is passing between the louvers, it is called "louver directed flow". Similarly, if the greatest proportion of the flow is passing through the gap between the fins, it is called "fin or duct directed flow" [7].

Figure 5. Possible flow direction of flow through the louvered fins.

Figure 6. Definition of the flow efficiency.

3. Flow phenomenon in louvered fin arrays

Figure 4. Terminology of the louvered fin [6].

Figure 3. Terminology of the louvered fin [2].

64 Heat Exchangers– Advanced Features and Applications

directed flow" [7].

The structure of the flow passes through the louvered fins can be identified with flow visualization technique using dye injection and hydrogen bubble. Numerical analysis is also a method to identify the flow phenomenon of louvered fins. Several experimental and numerical studies indicate that the geometric design and the free velocity of the flow effect the direction of the flow through the louvers. As shown in Figure 5, the flow enters from the leading edge of the first louver and then directed by the louvers or the fins. If the greatest proportion of the flow is passing between the louvers, it is called "louver directed flow". Similarly, if the greatest proportion of the flow is passing through the gap between the fins, it is called "fin or duct This definition has revealed another concept called flow efficiency (η) [3]. Figure 6 provides a visual definition of the flow efficiency, and it is calculated by Eq. (1)

$$\eta = \frac{N}{D} = \frac{\text{Actual transverse distance}}{\text{Ideal transverse distance}}\tag{1}$$

According to the Eq. (1), if the flow efficiency is equal to 1, the flow is parallel to the louvers. If it is equal to zero, the flow is axial through the louvered fin array which is 100% duct flow [3]. Figure 7 shows the structure of the flow between the louvers. The photographs of the flow through the louvered fin for a louver angle of 26° at a Reynolds number of 500 are obtained using dye injection and hydrogen bubbles techniques. It can be seen that the significant proportion of the flow is directed by the louvers under these circumstances. It is observed that the boundary layers exist on both the upper and lower surfaces of the louver. Flow separation is observed on the back side of the inlet louvers and boundary layer exists on both the upper and lower surfaces of the louvers [3].

Additionally, a large adverse pressure gradient exists on the downstream side of the louvered fin, and the flow separates at the leading edge and forms a large recirculation bubble. Figure 8a clearly shows this recirculation zone which is called "first recirculation zone" for a louver angle of 25° at a Reynolds number of 510 using naphthalene sublimation technique by DeJong and Jacobi [8]. This large recirculation zone circulates in a clockwise direction, resulting in a region of very high shear near the trailing edge of the louver where flow that passes downstream between fins interacts with the separation bubble. The high shear results in the formation of a small "secondary recirculation zone" with counter-clockwise rotation in the wake just downstream of the end of the louver. Except at very low Reynolds numbers, a

Figure 7. Visualization of the flow in the louvered fin array [3].

Figure 8. Flow structure around louvered fins: (a) Re = 510, (b) Re = 820 [8].

"third recirculation zone" forms on the upstream side of each louver. This third recirculation zone is caused when the flow passes through the gap between the first recirculation zone and the next louver downstream. This gap is narrow near the leading edge and much larger near the trailing edge. Flow in this gap must decelerate as it passes between the louvers. At all but the lowest Reynolds numbers where the first separation zone is small, the adverse pressure gradient in the inter-louver gap is large enough to cause separation from the upstream face of the next louver. The result is the third recirculation zone with counter-clockwise circulation [8].

Figure 8b shows flow through the same geometry at a higher Reynolds number (820) where the flow has become unsteady. The third recirculation zone has grown to become as large as the first zone, and the small second zone is no longer clearly evident. Two counter-rotating cells are present between the louvers. Fluid is periodically entrained in the recirculation zones and then ejected in the form of vortices which are carried downstream [8].

Today, the visualization techniques are developing parallel with the rapidly increasing technology. Besides the numerical analysis, infrared technology is a method to identify the flow characteristics of any flow. An open-circuit wind tunnel equipped with an infrared thermovision is illustrated in Figure 9. Infrared temperature measurement is achieved using an infrared camera. The electromagnetic energy radiated in the infrared spectral band by an object is transformed into an electronic signal by each of the thermo-vision sensors and is obtained simultaneously across the whole field of view, which depend on the optical focal length and the viewing distance [9].

In Figure 10, comparison of the experimental results obtained by the infrared measurements and the results of numerical analysis for the same louvered fin geometry is illustrated. The

Figure 9. Wind tunnel test equipped with an infrared thermo-vision [9].

"third recirculation zone" forms on the upstream side of each louver. This third recirculation zone is caused when the flow passes through the gap between the first recirculation zone and the next louver downstream. This gap is narrow near the leading edge and much larger near the trailing edge. Flow in this gap must decelerate as it passes between the louvers. At all but the lowest Reynolds numbers where the first separation zone is small, the adverse pressure gradient in the inter-louver gap is large enough to cause separation from the upstream face of the next louver. The result is the third recirculation zone with counter-clockwise circulation [8]. Figure 8b shows flow through the same geometry at a higher Reynolds number (820) where the flow has become unsteady. The third recirculation zone has grown to become as large as the first zone, and the small second zone is no longer clearly evident. Two counter-rotating

Figure 7. Visualization of the flow in the louvered fin array [3].

66 Heat Exchangers– Advanced Features and Applications

Figure 8. Flow structure around louvered fins: (a) Re = 510, (b) Re = 820 [8].

Figure 10. Comparison of the temperature distribution of an infrared thermographs and numerical simulation results for a louvered fin array [9].

louvered fin has a louver angle of 20° and the flow velocity is 1.0 m/s. It is observed that both methods give similar temperature distributions across the entire louvered fin.

#### 4. Data reduction in a louvered fin heat exchanger

In this section, the calculation method of the performance of a louvered fin heat exchanger is summarized. The equations will be written by considering the following assumptions.

I. Cold flow is the external flow of the louvered fin heat exchanger.

II. Hot flow is the internal flow of the louvered fin heat exchanger.

III. Thermophysical properties of both fluid are constant.

IV. The louvered fins are attached to a mini-channel flat tube.

V. Cold flow is uniform at the inlet of the louvered fin for the numerical analysis.

VI. The temperature of the tube wall is constant for the analytical solutions.

The main problem is the determination of external side heat transfer coefficient for the louvered fin surfaces by experimentally. Effectiveness-NTU method is generally used to determine the external side heat transfer coefficient by following the Kim and Bullard [10] method.

The average heat transfer rate can be expressed as

$$
\dot{Q} = (\dot{Q}\_c + \dot{Q}\_h)/2 \tag{2}
$$

where Q\_ <sup>c</sup>and Q\_ <sup>h</sup> are the heat transfer rates of cold and hot fluid, respectively. The heat transfer rates of each fluid can be calculated with Eqs. (3) and (4)

$$
\dot{Q}\_c = \dot{m}\_c c\_{p,c} (T\_{c,o} - T\_{c,i}) \tag{3}
$$

$$
\dot{Q}\_h = \dot{m}\_h \varepsilon\_{p,h} (T\_{h,i} - T\_{h,o}) \tag{4}
$$

The effectiveness of the heat exchanger for one row configuration can be calculated using the following equation for both fluid unmixed

$$\varepsilon = 1 - \exp\left[\frac{NTU^{0.22}}{\mathbb{C}\_r} \{ \exp\left( -\mathbb{C}\_r NTU^{0.78} \right) - 1 \} \right] \tag{5}$$

where

$$
\varepsilon = \dot{\mathbb{Q}} / \dot{\mathbb{Q}}\_{\text{max}} \tag{6}
$$

$$\mathbf{C}\_{\mathbf{r}} = \frac{(\dot{m}c\_p)\_{\text{min}}}{(\dot{m}c\_p)\_{\text{max}}} \tag{7}$$

We can obtain overall heat transfer coefficient (UA) for the heat exchanger as

Comprehensive Study of Heat Exchangers with Louvered Fins http://dx.doi.org/10.5772/66472 69

$$
\Delta LA = (\dot{m}c\_p)\_{\text{min}} NTLI \tag{8}
$$

where

louvered fin has a louver angle of 20° and the flow velocity is 1.0 m/s. It is observed that both

In this section, the calculation method of the performance of a louvered fin heat exchanger is

The main problem is the determination of external side heat transfer coefficient for the louvered fin surfaces by experimentally. Effectiveness-NTU method is generally used to determine the external side heat transfer coefficient by following the Kim and Bullard [10] method.

<sup>c</sup>and Q\_ <sup>h</sup> are the heat transfer rates of cold and hot fluid, respectively. The heat transfer

<sup>f</sup> exp <sup>ð</sup>−CrNTU<sup>0</sup>:<sup>78</sup>Þ−1<sup>g</sup>

" #

The effectiveness of the heat exchanger for one row configuration can be calculated using the

Cr <sup>¼</sup> <sup>ð</sup>mc\_ <sup>p</sup>Þmin ðmc\_ <sup>p</sup>Þmax

<sup>c</sup> <sup>þ</sup> <sup>Q</sup>\_ <sup>h</sup>Þ=<sup>2</sup> (2)

<sup>c</sup> ¼ m\_ ccp, <sup>c</sup>ðTc, <sup>o</sup>−Tc,iÞ (3)

<sup>h</sup> ¼ m\_ hcp,hðTh,i−Th, <sup>o</sup>Þ (4)

<sup>ε</sup> <sup>¼</sup> <sup>Q</sup>\_ <sup>=</sup>Q\_ max (6)

(5)

(7)

summarized. The equations will be written by considering the following assumptions.

V. Cold flow is uniform at the inlet of the louvered fin for the numerical analysis.

<sup>Q</sup>\_ ¼ ðQ\_

Q\_

Q\_

NTU<sup>0</sup>:<sup>22</sup> Cr

We can obtain overall heat transfer coefficient (UA) for the heat exchanger as

VI. The temperature of the tube wall is constant for the analytical solutions.

methods give similar temperature distributions across the entire louvered fin.

4. Data reduction in a louvered fin heat exchanger

68 Heat Exchangers– Advanced Features and Applications

III. Thermophysical properties of both fluid are constant.

The average heat transfer rate can be expressed as

following equation for both fluid unmixed

rates of each fluid can be calculated with Eqs. (3) and (4)

ε ¼ 1−exp

where Q\_

where

IV. The louvered fins are attached to a mini-channel flat tube.

I. Cold flow is the external flow of the louvered fin heat exchanger. II. Hot flow is the internal flow of the louvered fin heat exchanger.

$$
\Delta UA = \dot{\mathbb{Q}} / \Delta T\_m \tag{9}
$$

ΔTm is the logarithmic mean temperature difference and that is

$$
\Delta T\_m = \frac{(T\_{h,i} - T\_{c,o}) - (T\_{h,o} - T\_{c,i})}{\ln \left( (T\_{h,i} - T\_{c,o}) / (T\_{h,o} - T\_{c,i}) \right)} \tag{10}
$$

The external (cold) side heat transfer coefficient (hc) can be obtained from the following equation by experimentally

$$\frac{1}{\eta\_c A\_c h\_t} = \frac{1}{\text{LIA}} - \frac{\delta\_t}{k\_t A\_t} - \frac{1}{A\_i h\_i} \tag{11}$$

hi is the internal side heat transfer coefficient, and it can be obtained using empirical relations for duct flow. The surface effectiveness (ηc) for a dry surface is

$$\eta\_c = 1 - \frac{A\_f}{A\_c} (1 - \eta\_f) \tag{12}$$

where η<sup>f</sup> is the efficiency of the fin as given in Eq. (13)

$$\eta\_f = \frac{\tanh(ml)}{ml} \tag{13}$$

and

$$m = \sqrt{\frac{h\_c P\_f}{k\_f A\_{f,c}}}\tag{14}$$

Equation (14) can be expressed more explicitly using the following equation

$$m = \sqrt{\frac{2h\_c}{k\_f \delta} \left(1 + \frac{\delta}{F\_d}\right)}\tag{15}$$

For the calculation of external side heat transfer coefficient (hc) by analytically, the flow regime is very important to set up the analytical model. Therefore, Reynolds number is the mandatory parameter for the analytical solution. The flow can be assumed to be laminar at ReLp <1300 for the louvered fin arrays [3]. The characteristics length of louvered fin array is the louver pitch, so that the Reynolds number is calculated based on the louver pitch

$$Re\_{L\_p} = \frac{\mu\_{\max} L\_p}{\nu} \tag{16}$$

umax is the maximum velocity of the external fluid due to the narrowing section of the louvered fin arrays relatively to the inlet of the louvered fin. Therefore, the free velocity of the external fluid (u) at the inlet of the louvered fin is transformed to maximum velocity using the following equation

$$
\mu\_{\text{max}} = \mathfrak{u} F\_p / (F\_p - t) \tag{17}
$$

where Fp and t are the fin pitch and the material thickness, respectively.

If the tube wall temperature is assumed constant for the numerical analysis, the heat transfer can be calculated by the heat gain of the external (cold) fluid as given in Eq. (3). Thus, the heat transfer coefficient of the external side (hc) can be obtained from Eq. (18) by numerically

$$h\_{\mathfrak{c}} = \dot{\mathbb{Q}}\_{\mathfrak{c}} / (A \Delta T\_m) \tag{18}$$

In Eq. (18), A is the total external side heat transfer area and ΔTm is the logarithmic mean temperature difference under constant wall temperature condition given by Eq. (19) [11]

$$
\Delta T\_m = \frac{(T\_w - T\_{c,o}) - (T\_w - T\_{c,i})}{\ln \left( (T\_w - T\_{c,o}) / (T\_w - T\_{c,i}) \right)} \tag{19}
$$

#### 5. Performance evaluation criteria of the louvered fin heat exchangers

In the heat exchanger literature, some dimensionless parameters are used as a performance criteria. The commonly used thermal performance criteria are Stanton number and Colburn jfactor given as

$$St = \frac{h\_c}{\rho u c\_p} \tag{20}$$

$$j = StPr^{2/3} \tag{21}$$

respectively. After the calculation of hc by experimentally or analytically, Stanton number and Colburn j-factor can be obtained to indicate thermal performance of the louvered fin heat exchanger in a dimensionless form as given with Eqs. (20) and (21). Another performance criteria is the friction factor which is the dimensionless form of the pressure drop for the external side of a louvered fin heat exchanger expressed as

$$f = \left(\frac{A\_c}{A}\right) \left(\frac{2\Delta P}{\rho u^2}\right) \tag{22}$$

An alternative equation for the friction factor can be used by considering the entrance, exit and acceleration effects

$$f = \left(\frac{A\_c}{A}\right) \left(\frac{\rho\_m}{\rho\_1}\right) \left(\frac{2\rho\_1 \Delta P}{G\_c^2} - (k\_c + 1 - \sigma^2) \text{-} 2\left(\frac{\rho\_1}{\rho\_2} - 1\right) + (1 - \sigma^2 - k\_c) \frac{\rho\_1}{\rho\_2}\right) \tag{23}$$

where Ac is the minimum free flow area for the external side, and kc and ke are the coefficients of pressure loss at the inlet and the outlet of the heat exchanger. kc and ke can be evaluated according to Kays and London [2]. The overall performance of the louvered fin heat exchangers can be evaluated with another perspective. The ratio of the j-factor to the f, the ratio of the j-factor to the f 1/3 and JF are the overall performance criteria of the louvered fin heat exchangers used in the literature. j/f is known as "area goodness factor"[12, 13] and j/f 1/3 is known as "volume goodness factor"[6, 14, 15]. JF number which is related with the volume goodness factor can be obtained by Eq. (24) [16, 17]. These parameters are dimensionless numbers of the larger—the better characteristics. It is expected that these parameters can effectively evaluate the thermal and dynamic performance of a heat exchanger since it includes both the j- and the f-factor

fluid (u) at the inlet of the louvered fin is transformed to maximum velocity using the follow-

If the tube wall temperature is assumed constant for the numerical analysis, the heat transfer can be calculated by the heat gain of the external (cold) fluid as given in Eq. (3). Thus, the heat transfer coefficient of the external side (hc) can be obtained from Eq. (18) by numerically

In Eq. (18), A is the total external side heat transfer area and ΔTm is the logarithmic mean temperature difference under constant wall temperature condition given by Eq. (19) [11]

<sup>Δ</sup>Tm <sup>¼</sup> <sup>ð</sup>Tw−Tc, <sup>o</sup>Þ−ðTw−Tc,i<sup>Þ</sup>

5. Performance evaluation criteria of the louvered fin heat exchangers

In the heat exchanger literature, some dimensionless parameters are used as a performance criteria. The commonly used thermal performance criteria are Stanton number and Colburn j-

> St <sup>¼</sup> hc ρucp

respectively. After the calculation of hc by experimentally or analytically, Stanton number and Colburn j-factor can be obtained to indicate thermal performance of the louvered fin heat exchanger in a dimensionless form as given with Eqs. (20) and (21). Another performance criteria is the friction factor which is the dimensionless form of the pressure drop for the

An alternative equation for the friction factor can be used by considering the entrance, exit and

where Ac is the minimum free flow area for the external side, and kc and ke are the coefficients of pressure loss at the inlet and the outlet of the heat exchanger. kc and ke can be evaluated according to Kays and London [2]. The overall performance of the louvered fin heat

<sup>−</sup>ðkc <sup>þ</sup> <sup>1</sup>−σ<sup>2</sup>

ρu<sup>2</sup> � �

> <sup>Þ</sup>−<sup>2</sup> <sup>ρ</sup><sup>1</sup> ρ2 −1 � �

!

<sup>f</sup> <sup>¼</sup> Ac A � � 2ΔP

external side of a louvered fin heat exchanger expressed as

ρ1

� � 2ρ1ΔP

G2 c ðTw−Tc, <sup>o</sup>Þ=ðTw−Tc,iÞ

hc <sup>¼</sup> <sup>Q</sup>\_

ln �

where Fp and t are the fin pitch and the material thickness, respectively.

umax ¼ uFp=ðFp−tÞ (17)

<sup>c</sup>=ðAΔTmÞ (18)

<sup>j</sup> <sup>¼</sup> StPr<sup>2</sup>=<sup>3</sup> (21)

þ ð1−σ<sup>2</sup>

−keÞ ρ1 ρ2

� (19)

(20)

(22)

(23)

ing equation

70 Heat Exchangers– Advanced Features and Applications

factor given as

acceleration effects

<sup>f</sup> <sup>¼</sup> Ac A � � ρ<sup>m</sup>

$$JF = \frac{\text{j}/\text{j}\_R}{\text{(}f/f\_R)^{1/3}}\tag{24}$$

where jR and fR are the reference values of Colburn j-factor and friction factor, respectively.

In light of these explanations, thermal and hydraulic characteristics of the heat exchangers with louvered fins are presented using numerical and experimental studies in the literature.

In 1990s, 2D numerical models were preferred rather than 3D models due to the run time and limited computing power. Nevertheless, 3D models are necessary because of the high compatibility with the experimental results. The velocity and the temperature field of a louvered fin heat exchanger for two different Reynolds numbers are presented in Figure 11 as a result of 2D numerical model.

It is observed that significant proportion of the air flows through the channels between the fins rather than between the louvers, as indicated by the presence of high velocity streaks in the channels at a Reynolds number of 100 (Figure 11a, b). The temperature of the air reach the fin temperature before it leaves the fin, therefore, the heat transfer performance of the second half of the fin is poor. In fact, second half of the fin only causes a pressure loss without any heat transfer at low Reynolds numbers. At a higher Reynolds number of 1600 (Figure 11c, d), the boundary layer of the louvers are much thinner, and therefore, the air is directed through the louver passages. A temperature difference is maintained between the air and the fin surface and so every part of the louvered fin contributes to the heat transfer. However, the 2D models is enough for the characteristics of the flow over the louvered fins, it is not possible to say same thing for the thermal performance. The comparison of 2D, 3D and the measured thermal and hydraulic performance of a louvered fin heat exchanger are presented in Figure 12.

It can be seen that the 2D model yields reasonably accurate predictions of friction factor, but poor predictions of Stanton number. An obvious way to identify the reasons of the error in the heat transfer is to consider the practical features which are missing from the 2-D model. Two important features which are missing are the tube surfaces and the resistance. The tube surfaces would add to the heat transfer area but would not add significantly to the overall heat transfer rate, because of the thick boundary layer growth on the tubes. The fin resistance would lower the temperature across the fin, and thus the heat transfer from the fin. Another reason of the over prediction of the thermal performance of the louvered fin is that the efficiency of the louvered fin cannot be calculated exactly. Generally, the experimental hc value is obtained using plate fin surface of the fin efficiency even for the louver fin efficiency due to the absence of the base area of the fin in 2D models. In the literature, 2-D models only consider

Figure 11. Computed velocity and temperature field for the 2-D model; Fp = 2.54 mm, Lp = 1.4 mm, L<sup>α</sup> = 25.5°. (a) Velocity, ReLp = 100, (b) Temperature, ReLp = 100, (c) Velocity, ReLp = 1600, (d) Temperature, ReLp = 1600 [18].

Figure 12. Comparison of the computed 2-D model and measured friction factor and Stanton number: Fp = 2.05 mm, Lp = 1.4 mm, L<sup>α</sup> = 25.5°, Tp = 11 mm and δ = 0.05 mm [18].

the cross section of the louvers and the numerical hc value is used directly without any fin efficiency formula. However, the slopes of the 2D numerical results are comparable and agree with the experimental results, heat transfer coefficient (hc) is overpredicted with the assumption of constant fin temperature and neglecting the tube surface effect [6, 18, 19]. It is these factors which led to the development of the 3D models.

In 2000s, 3D models have come to the forefront with the increasing computing power. Researchers have begun to compare 2D and 3D results with their own experimental results to validate the compatibility of the numerical models. In Figures 13 and 14, the variation of Colburn j-factor and the friction factor with respect to the Reynolds number for a louvered fin heat exchanger is presented. The results are also compared with correlations in the literature. It is observed that the CFD results for the 2D models are overpredicted by 80% compared to the experimental results [19]. This is consistent with the study of Atkinson et al. [18]. The slopes of the experimental results are comparable and agree well with the correlated data. Colburn jfactor and the friction factor decrease with the increasing of Reynolds number.

Figures 12–14 demonstrate the compatibility of the numerical results with experimental results and the effect of Reynolds number on the thermal and hydraulic performance of the louvered fin heat exchanger. In addition to the effect of Reynolds number, the researchers spent great efforts to determine the optimum geometric parameters of the louvered fin. The numerical studies have a great importance in this field, because the testing of the every geometric variation is very difficult in terms of both time and cost. The prior geometric parameter is the louver angle which has the significant influence on the flow regime over the louvered fins. The variation of the thermal performance of a louvered fin heat exchanger for different louver pitch

Figure 13. Comparison of computed and measured j-factor [19].

Figure 11. Computed velocity and temperature field for the 2-D model; Fp = 2.54 mm, Lp = 1.4 mm, L<sup>α</sup> = 25.5°. (a) Velocity,

Figure 12. Comparison of the computed 2-D model and measured friction factor and Stanton number: Fp = 2.05 mm, Lp =

1.4 mm, L<sup>α</sup> = 25.5°, Tp = 11 mm and δ = 0.05 mm [18].

72 Heat Exchangers– Advanced Features and Applications

ReLp = 100, (b) Temperature, ReLp = 100, (c) Velocity, ReLp = 1600, (d) Temperature, ReLp = 1600 [18].

Figure 14. Comparison of computed and measured f-factor [19].

with respect to the louver angle is demonstrated in Figure 15. It is observed that the thermal performance is increasing up to the louver angle of 28.5° and then decreasing for all the louver pitches. The louver angle of 28.5° has the maximum thermal performance within the considered cases of the numerical study. Average heat transfer coefficient is about 200 W/m2 K at the minimum louver pitch of 0.81 mm. It decreases about to 185 W/m<sup>2</sup> K at the maximum louver pitch of 1.4 mm [20].

In the study of Atkinson et al. [18], the louvered fin has uniform louver angle. It can be possible to create a louvered fin having non-uniform louver angles as shown in Figure 16. Figure 17 shows the effect of non-uniform louver angle on the thermal and hydraulic performance of a louvered fin heat exchanger.

Figure 15. The effect of louver angle on the thermal performance of a louvered fin heat exchanger for different louver pitches [23].

Comprehensive Study of Heat Exchangers with Louvered Fins http://dx.doi.org/10.5772/66472 75

Figure 16. Five different cases of successively increased or decreased louver angle (+2<sup>o</sup> , +4o , −2o , −4<sup>o</sup> , and uniform angle 20<sup>o</sup> ) [24].

Figure 17. Effect of the non-uniform louver angle on the (a) Colburn j-factor and (b) friction factor [24].

with respect to the louver angle is demonstrated in Figure 15. It is observed that the thermal performance is increasing up to the louver angle of 28.5° and then decreasing for all the louver pitches. The louver angle of 28.5° has the maximum thermal performance within the consid-

In the study of Atkinson et al. [18], the louvered fin has uniform louver angle. It can be possible to create a louvered fin having non-uniform louver angles as shown in Figure 16. Figure 17 shows the effect of non-uniform louver angle on the thermal and hydraulic performance of a

Figure 15. The effect of louver angle on the thermal performance of a louvered fin heat exchanger for different louver

K at the

K at the maximum louver

ered cases of the numerical study. Average heat transfer coefficient is about 200 W/m2

minimum louver pitch of 0.81 mm. It decreases about to 185 W/m<sup>2</sup>

Figure 14. Comparison of computed and measured f-factor [19].

74 Heat Exchangers– Advanced Features and Applications

pitch of 1.4 mm [20].

pitches [23].

louvered fin heat exchanger.

In Figure 17, Colburn j-factor and the friction factor were normalized with Case E as shown in Figure 16. It is seen that the non-uniform louver angle patterns applied in the heat exchangers could effectively enhance the heat transfer performance. Case B has 18% heat transfer enhancement with respect to the Case E about at Re = 440. On the other hand, it has a negative effect of %19 on the friction factor.

In most cases, the geometric effects on the performance of a louvered fin heat exchanger are not monotonic. The combined relationship between the louver angle, louver pitch, fin pitch, tube pitch, etc., corrupts the linearity between the geometric parameters and the performance of the heat exchanger. An example is given with Figure 18. The variation of the heat transfer coefficient with respect to the frontal air velocity for different tube pitches is illustrated. It is observed that the heat transfer coefficient does not vary with the tube pitch linearly (Figure 18a). In Figure 18b, the effect of the fin pitch on the heat transfer coefficient is shown. It is seen that the heat transfer coefficient is increasing with decreasing of the fin pitch at a frontal velocity greater than 5.5 m/s. This statement is not valid for the frontal velocity smaller than 5.5 m/s.

Figure 18. Variation of the heat transfer coefficient with respect to the frontal velocity for different (a) tube pitches (Fp = 1.5 mm, Lp = 1.2 mm, L<sup>α</sup> = 26o ) and (b) fin pitches (Tp = 9.6 mm, Lp = 1.2 mm, L<sup>α</sup> = 26o ) [25].

Figure 19. Schematic of a wind tunnel test [10].

Researchers make great efforts to identify the real performances of the louvered fin heat exchangers by experimentally. Investigation of the performance of the louvered fins is commonly performed with the wind tunnel tests. In the open literature, several wind tunnel test can be found in different designs. A typical wind tunnel is shown in Figure 19.

As shown in Figure 19, internal fluid of the heat exchanger is water and it is regulated by a constant temperature bath. External fluid of the heat exchanger is air, and it is sucked by a fan and wind tunnel is placed in a constant temperature and humidity chamber to regulate the air flow. Dry and wet bulb temperatures of the air are measured with thermocouples at the inlet and the exit of the heat exchanger. One of the most comprehensive performance data of the louvered fin heat exchangers is presented by this wind tunnel test in the early of 2000s. The effects of the geometric dimensions of the louvered fins and the Reynolds number on the Colburn j-factor and friction factor is identified. Similarly to the previous studies, j-factor and f-factor are decreasing with the increasing of Reynolds number due to its definition as shown in Figure 20.

Comprehensive Study of Heat Exchangers with Louvered Fins http://dx.doi.org/10.5772/66472 77

Figure 20. Variations of j-factor and f-factor with respect to the louver angle and Reynolds number for (a) Fd = 16 mm and (b) Fd = 20 mm [10].

The effect of louver angle is more specifically at a flow depth of 20 mm for a constant fin pitch of 1.40 mm. Friction factor increases with the increasing of louver angle, however, the effect of the louver angle is diminishing for the louver angles larger than 21°. The effect of the louver angle on the j-factor varies with the flow depth. The increasing in the j-factor with the louver angle at a flow depth of 20 mm is more obvious than that of the flow depth of 16 mm. j-factor is increasing especially with the louver angle of greater than 23° due to the louver directed flow for such a small fin pitch of 1.40 mm.

The heat transfer and the pressure drop behaviour of the louvered fin heat exchangers for greater flow depths and Reynolds numbers is available in the open literature. The variations of the j-factor and f-factor with respect to the frontal air velocity for different geometric dimensions are given in Figure 21. As shown in Figure 21, the considered flow depth and Reynolds number range are 36.0–65.0 mm and 200–2500, respectively.

Researchers make great efforts to identify the real performances of the louvered fin heat exchangers by experimentally. Investigation of the performance of the louvered fins is commonly performed with the wind tunnel tests. In the open literature, several wind tunnel test

Figure 18. Variation of the heat transfer coefficient with respect to the frontal velocity for different (a) tube pitches (Fp =

) [25].

) and (b) fin pitches (Tp = 9.6 mm, Lp = 1.2 mm, L<sup>α</sup> = 26o

As shown in Figure 19, internal fluid of the heat exchanger is water and it is regulated by a constant temperature bath. External fluid of the heat exchanger is air, and it is sucked by a fan and wind tunnel is placed in a constant temperature and humidity chamber to regulate the air flow. Dry and wet bulb temperatures of the air are measured with thermocouples at the inlet and the exit of the heat exchanger. One of the most comprehensive performance data of the louvered fin heat exchangers is presented by this wind tunnel test in the early of 2000s. The effects of the geometric dimensions of the louvered fins and the Reynolds number on the Colburn j-factor and friction factor is identified. Similarly to the previous studies, j-factor and f-factor are decreasing with the increasing of Reynolds number due to its definition as shown

can be found in different designs. A typical wind tunnel is shown in Figure 19.

in Figure 20.

1.5 mm, Lp = 1.2 mm, L<sup>α</sup> = 26o

76 Heat Exchangers– Advanced Features and Applications

Figure 19. Schematic of a wind tunnel test [10].

In Figure 21a, c, the effect of fin pitch at a constant flow depth and fin height on the thermal and hydraulic performance is shown. It is observed that the fin pitch has a significant effect on the thermal and hydraulic performance and j-factor and f-factor are decreasing with the increasing of fin pitch. For Fp = 2.0 mm and Fd = 65 mm, Colburn j-factor is maximum for all Reynolds numbers and it decreases about 0.0105–0.0072 as shown in Figure 21a. The cause of this situation is that the hydraulic resistance against the flow increases when the fin pitch decreases at the flow passage. Therefore, the flow tends to more to being louver directed. As a result of this phenomenon, the air flow can be mixed well by the louvers, so the heat transfer and the pressure drop increase. As shown in Figure 21b, j-factor and f-factor increase when the fin height increases. The possible reason is that the proportion of the air flow directed by the louvers increases with the fin height. Figure 21d shows the effect of the flow depth on the jfactor and f-factor. It is obvious that the flow depth has more significant effect on the j-factor and f-factor. The study of Kim and Bullard [10] indicates the same results. In addition to the jfactor and f-factor, the variation of volume goodness factor denoted by j/f 1/3 versus Reynolds number is illustrated in Figure 22 as a performance criteria of the louvered fin heat exchangers.

Figure 21. Variations of j-factor and f-factor with respect to the frontal air velocity [15].

It is seen that the geometry which has the smallest flow depth (36.0 mm), smallest fin pitch (2.00 mm) and the biggest fin height (10.0 mm) has the maximum value of j/f 1/3 = 0.032. The

Figure 22. The variation of j/f1/3 [15].

Figure 23. The variation of j/f1/3 at low Reynolds number [6].

It is seen that the geometry which has the smallest flow depth (36.0 mm), smallest fin pitch

1/3 = 0.032. The

(2.00 mm) and the biggest fin height (10.0 mm) has the maximum value of j/f

Figure 21. Variations of j-factor and f-factor with respect to the frontal air velocity [15].

78 Heat Exchangers– Advanced Features and Applications

Figure 22. The variation of j/f1/3 [15].

effect of the Reynolds number to the j/f 1/3 ratio decreases with a flow depth of 65.0 mm. The most important result is the non-monotonic behaviour of the j/f 1/3 ratio with the geometric dimensions. The variation of the j/f 1/3 ratio has a complicated behaviour with respect to the fin pitch and the fin height. In particular, the low Reynolds number region which has the significant changes of j/f 1/3 ratio for the smaller flow depths can be illuminated with another study by Erbay et al. [6] as shown in Figure 23.

The j/f 1/3 ratios of the study of Erbay et al. [6] are obtained for a constant flow depth of 20 mm by numerically. It seen that the j/f 1/3 ratio for all the geometries has a similar trend with respect to the Reynolds number; however, there is not linear relationship for the geometric parameters. As a result, such a performance criteria which considers both the thermal and hydraulic performance is necessary to design a heat exchanger.

Some of the researches working on the performance evaluation of the louvered fin heat exchangers have developed correlations for Stanton number, Colburn j-factor and friction factor. The basic form of these correlations is

$$a = \mathbb{C}\_1 \mathrm{Re}^{\mathbb{C}\_2} \tag{25}$$

where a represents the performance criteria, and C1 and C2 are dependent on the dimensions of the louvered fin heat exchangers. Some of the correlations for the performance criteria of the louvered fin heat exchangers are listed by considering the studies in the open literature.

• Achaichia and Cowell [7], 150 < ReLp < 3000

$$St = 1.54 Re\_{L\_p}^{-0.57} \left(\frac{F\_p}{L\_p}\right)^{-0.19} \left(\frac{T\_p}{L\_p}\right)^{-0.11} \left(\frac{L\_h}{L\_p}\right)^{0.15} \tag{26}$$

$$f = \text{533.42Re}\_{L\_p}^{0.318(\log \text{Re}\_p - 2.25)} F\_p^{0.22} L\_p^{0.25} T\_p^{0.26} L\_h^{0.33} \,. \tag{27}$$

• Kim and Bullard [10], 100 < ReLp < 600

$$j = R e\_{L\_p}^{-0.487} \left(\frac{L\_\alpha}{90}\right)^{0.257} \left(\frac{F\_p}{L\_p}\right)^{-0.13} \left(\frac{F\_h}{L\_p}\right)^{-0.29} \left(\frac{F\_d}{L\_p}\right)^{-0.235} \left(\frac{L\_h}{L\_p}\right)^{0.68} \left(\frac{T\_p}{L\_p}\right)^{-0.279} \left(\frac{\delta}{L\_p}\right)^{-0.05} \tag{28}$$

$$f = Re\_{L\_p}^{-0.781} \left(\frac{L\_a}{90}\right)^{0.444} \left(\frac{F\_p}{L\_p}\right)^{-1.682} \left(\frac{F\_h}{L\_p}\right)^{-1.22} \left(\frac{F\_d}{L\_p}\right)^{0.818} \left(\frac{L\_h}{L\_p}\right)^{1.97}.\tag{29}$$

• Dong et al. [15], 200 < ReLp < 2500

$$j = 0.26712 Re\_{L\_p}^{-0.1944} \left(\frac{L\_a}{90}\right)^{0.257} \left(\frac{F\_p}{L\_p}\right)^{-0.5177} \left(\frac{F\_h}{L\_p}\right)^{-1.9045} \left(\frac{F\_d}{L\_p}\right)^{-0.2147} \left(\frac{L\_h}{L\_p}\right)^{1.7159} \left(\frac{\delta}{L\_p}\right)^{-0.05} \tag{30}$$

$$f = 0.54486 Re\_{L\_{\eta}}^{-0.3068} \left(\frac{L\_{\alpha}}{90}\right)^{0.444} \left(\frac{F\_{p}}{L\_{\eta}}\right)^{-0.9925} \left(\frac{F\_{h}}{L\_{\eta}}\right)^{-0.5458} \left(\frac{F\_{d}}{L\_{\eta}}\right)^{-0.0688} \left(\frac{L\_{h}}{L\_{\eta}}\right)^{-0.2003} \tag{31}$$

• Ryu and Lee [26], 100 < ReLp <3000

$$j = \text{Re}\_{L\_p}^{(-0.484 - 1.887/\ln \text{Re}\_{L\_p})} \left(\frac{F\_d}{L\_p}\right)^{0.157} \left(2.24 - 0.588 \ln \left(\frac{F\_p}{L\_p \sin L\_d}\right)\right) \tag{32}$$

$$f = Re\_{L\_p}^{-0.433} \left(\frac{F\_d}{L\_p}\right)^{0.185} \left(1.10 + 4.31 \left(\frac{L\_a}{90}\right)^2 + 0.836 \frac{\ln(F\_p/L\_p)}{\left(F\_p/L\_p\right)^2}\right). \tag{33}$$

In addition to above correlations, a large data bank as shown in Table 1 was used by Chang and Wang [20], Chang et al. [22], and Park and Jacobi [27] to develop a more sensible correlations for j-factor and f-factor. These correlations which are the more comprehensive for the louvered fin heat exchangers are listed below.

• Chang and Wang [20], 100 < ReLp < 3000

$$j = R \bar{e}\_{L\_p}^{-0.49} \left(\frac{L\_a}{90}\right)^{0.27} \left(\frac{F\_p}{L\_\rho}\right)^{-0.14} \left(\frac{F\_h}{L\_\rho}\right)^{-0.29} \left(\frac{T\_d}{L\_\rho}\right)^{-0.23} \left(\frac{L\_h}{L\_\rho}\right)^{0.68} \left(\frac{T\_p}{L\_\rho}\right)^{-0.28} \left(\frac{\delta}{L\_\rho}\right)^{-0.05}.\tag{34}$$

• Chang et al. [22], 100 < ReLp < 3000


• Achaichia and Cowell [7], 150 < ReLp < 3000

80 Heat Exchangers– Advanced Features and Applications

• Kim and Bullard [10], 100 < ReLp < 600

Lα 90

<sup>f</sup> <sup>¼</sup> Re<sup>−</sup>0:<sup>781</sup> Lp

• Dong et al. [15], 200 < ReLp < 2500

Lp

<sup>f</sup> <sup>¼</sup> <sup>0</sup>:54486Re<sup>−</sup>0:<sup>3068</sup>

• Ryu and Lee [26], 100 < ReLp <3000

<sup>f</sup> <sup>¼</sup> Re<sup>−</sup>0:<sup>433</sup> Lp

louvered fin heat exchangers are listed below.

• Chang and Wang [20], 100 < ReLp < 3000

Lα 90

• Chang et al. [22], 100 < ReLp < 3000

� �<sup>0</sup>:<sup>27</sup> Fp

Lp

� �<sup>−</sup>0:<sup>14</sup> Fh

Lp

� �<sup>−</sup>0:<sup>29</sup> Td

Lp

� �<sup>−</sup>0:<sup>23</sup> Lh

Lp

� �<sup>0</sup>:<sup>68</sup> Tp

Lp

� �<sup>−</sup>0:<sup>28</sup> δ

Lp � �<sup>−</sup>0:<sup>05</sup>

<sup>j</sup> <sup>¼</sup> Re<sup>−</sup>0:<sup>49</sup> Lp

� �<sup>0</sup>:<sup>257</sup> Fp

Lα 90

Lp

� �<sup>0</sup>:<sup>257</sup> Fp

<sup>j</sup> <sup>¼</sup> Re<sup>ð</sup>−0:484−1:887=lnReLp <sup>Þ</sup> Lp

> Fd Lp � �<sup>0</sup>:<sup>185</sup>

Lp

Lα 90

� �<sup>−</sup>0:<sup>13</sup> Fh

� �<sup>0</sup>:<sup>444</sup> Fp

Lp

� �<sup>0</sup>:<sup>444</sup> Fp

Lα 90 Lp

Lp

� �<sup>−</sup>0:<sup>5177</sup> Fh

Lp

Fd Lp � �<sup>0</sup>:<sup>157</sup>

� �<sup>−</sup>0:<sup>29</sup> Fd

� �<sup>−</sup>1:<sup>682</sup> Fh

Lp

� �<sup>−</sup>0:<sup>9925</sup> Fh

<sup>1</sup>:<sup>10</sup> <sup>þ</sup> <sup>4</sup>:<sup>31</sup> <sup>L</sup><sup>α</sup>

In addition to above correlations, a large data bank as shown in Table 1 was used by Chang and Wang [20], Chang et al. [22], and Park and Jacobi [27] to develop a more sensible correlations for j-factor and f-factor. These correlations which are the more comprehensive for the

Lp

Lp

� �<sup>−</sup>1:<sup>9045</sup> Fd

Lp

90 � �<sup>2</sup>

<sup>j</sup> <sup>¼</sup> Re<sup>−</sup>0:<sup>487</sup> Lp

<sup>j</sup> <sup>¼</sup> <sup>0</sup>:26712Re<sup>−</sup>0:<sup>1944</sup>

St <sup>¼</sup> <sup>1</sup>:54Re<sup>−</sup>0:<sup>57</sup> Lp

<sup>f</sup> <sup>¼</sup> <sup>533</sup>:42Re<sup>0</sup>:318ðlogReLp <sup>−</sup>2:25<sup>Þ</sup>

Fp Lp

� �<sup>−</sup>0:<sup>19</sup> Tp

Lp <sup>F</sup><sup>−</sup>0:<sup>22</sup>

Lp

� �<sup>−</sup>0:<sup>11</sup> Lh

<sup>p</sup> L<sup>0</sup>:<sup>25</sup> <sup>p</sup> T<sup>0</sup>:<sup>26</sup> <sup>p</sup> L<sup>0</sup>:<sup>33</sup>

� �<sup>−</sup>0:<sup>235</sup> Lh

� �<sup>−</sup>1:<sup>22</sup> Fd

Lp

Lp

� �<sup>−</sup>0:<sup>2147</sup> Lh

Lp

� � � �

þ 0:836

Lp

� �<sup>−</sup>0:<sup>5458</sup> Fd

2:24−0:588ln

!

� �<sup>0</sup>:<sup>68</sup> Tp

� �<sup>0</sup>:<sup>818</sup> Lh

Lp

Lp � �<sup>1</sup>:<sup>97</sup>

Lp

� �<sup>−</sup>0:<sup>0688</sup> Lh

Fp LpsinL<sup>α</sup>

lnðFp=LpÞ ðFp=LpÞ 2

� �<sup>1</sup>:<sup>7159</sup> δ

Lp � �<sup>−</sup>0:<sup>2003</sup>

� �<sup>−</sup>0:<sup>279</sup> δ

Lp � �<sup>0</sup>:<sup>15</sup>

(26)

(28)

(30)

(31)

(32)

: (33)

: (34)

<sup>h</sup> : (27)

Lp � �<sup>−</sup>0:<sup>05</sup>

Lp � �<sup>−</sup>0:<sup>05</sup>

: (29)



Source Lp(mm) Fp(mm) Fh(mm) Lh(mm) Lα(

82 Heat Exchangers– Advanced Features and Applications

o

CW(5)a, b [20] 1.42 2.00 19.00 17.18 28.0 22.0 22.0 24.00 0.160 2 CW(6)a, b [20] 1.42 2.20 19.00 17.18 28.0 22.0 22.0 24.00 0.160 2 CW(7)a, b [20] 1.48 1.80 16.00 12.78 28.0 26.0 26.0 21.00 0.160 2 CW(8)a, b [20] 1.48 2.00 16.00 12.78 28.0 26.0 26.0 21.00 0.160 2 CW(9)a, b [20] 1.48 2.20 16.00 12.78 28.0 26.0 26.0 21.00 0.160 2 CW(10)a, b [20] 1.53 1.80 19.00 16.07 28.0 26.0 26.0 24.00 0.160 2 CW(11)a, b [20] 1.53 2.00 19.00 16.07 28.0 26.0 26.0 24.00 0.160 2 CW(12)a, b [20] 1.53 2.20 19.00 16.07 28.0 26.0 26.0 24.00 0.160 2 CW(13)a, b [20] 1.69 1.80 16.00 12.15 28.0 32.0 32.0 21.00 0.160 2 CW(14)a, b [20] 1.69 2.00 16.00 12.15 28.0 32.0 32.0 21.00 0.160 2 CW(15)a, b [20] 1.69 2.20 16.00 12.15 28.0 32.0 32.0 21.00 0.160 2 CW(16)a, b [20] 1.55 1.80 19.00 16.17 28.0 32.0 32.0 24.00 0.160 4 CW(17)a, b [20] 1.55 2.00 19.00 16.17 28.0 32.0 32.0 24.00 0.160 4 CW(18)a, b [20] 1.55 2.20 19.00 16.17 28.0 32.0 32.0 24.00 0.160 4 CW(19)a, b [20] 1.86 1.80 19.00 15.25 28.0 38.0 38.0 24.00 0.160 4 CW(20)a, b [20] 1.86 2.00 19.00 15.25 28.0 38.0 38.0 24.00 0.160 4 CW(21)a, b [20] 1.86 2.20 19.00 15.25 28.0 38.0 38.0 24.00 0.160 4 CW(22)a, b [20] 1.59 1.80 16.00 13.18 28.0 44.0 44.0 21.00 0.160 2 CW(23)a, b [20] 1.59 2.00 16.00 13.18 28.0 44.0 44.0 21.00 0.160 2 CW(24)a, b [20] 1.59 2.20 16.00 13.18 28.0 44.0 44.0 21.00 0.160 2 CW(25)a, b [20] 1.53 1.80 19.00 16.84 28.0 44.0 44.0 24.00 0.160 4 CW(26)a, b [20] 1.53 2.00 19.00 16.84 28.0 44.0 44.0 24.00 0.160 4 CW(27)a, b [20] 1.53 2.20 19.00 16.84 28.0 44.0 44.0 24.00 0.160 4 PSU(1)<sup>a</sup> [20]f 1.00 1.124 8.00 6.50 30.0 16.0 16.0 9.60 0.157 2 PSU(2)<sup>a</sup> [20]f 1.016 1.954 9.22 6.858 27.0 20.32 20.32 11.11 0.0508 2 PSU(3)<sup>a</sup> [20]f 1.016 1.588 9.22 6.858 27.0 20.32 20.32 11.11 0.0508 2 PSU(4)<sup>a</sup> [20]f 1.016 1.270 9.22 6.858 27.0 20.32 20.32 11.11 0.0508 2 PSU(5)<sup>a</sup> [20]f 0.94 1.114 9.15 7.62 27.0 16.26 16.26 11.11 0.127 2 AC(1)a, b [7] 1.40 2.02 9.00 8.50 25.5 41.6 32.0 11.00 0.05 2 AC(2)a, b [7] 1.40 3.25 9.00 8.50 25.5 41.6 32.0 11.00 0.05 2 AC(3)a, b [7] 1.40 1.65 9.00 8.50 25.5 41.6 32.0 11.00 0.05 2 AC(4)a, b [7] 1.40 2.09 9.00 8.50 21.5 41.6 32.0 11.00 0.05 2 AC(5)a, b [7] 1.40 2.03 9.00 8.50 28.5 41.6 32.0 11.00 0.05 2 AC(6)a, b [7] 1.40 2.15 9.00 8.50 25.5 20.8 16.0 11.00 0.05 1 AC(7)a, b [7] 1.40 1.70 9.00 8.50 25.5 20.8 16.0 11.00 0.05 1 AC(8)a, b [7] 0.81 2.11 9.00 8.50 29.0 41.6 32.0 11.00 0.05 2

) Fd(mm) Td(mm) Tp(mm) Δ(mm) NLB


<sup>a</sup> Correlated data considered by Chang and Wang [20].<sup>b</sup> Correlated data considered by Park and Jacobi [27].c Park and Jacobi [27] used half of these pitches because of two fin stocks and a splitter plate between the tubes.<sup>d</sup> Park and Jacobi [27] and Chang and Wang [20] assumed that the tube diameter is 1.50 mm due to the lack of information.<sup>e</sup> Chang and Wang [20] assumed that the tube diameter is 5.00 mm due to the lack of information.<sup>f</sup> Correlated data which was unpublished by R. L. Webb considered by Chang and Wang [20].

Table 1. Geometrical dimensions of the louvered fin heat exchangers in the database [20, 27].

$$f = f\_1 f\_2 f\_3 \tag{35}$$

for ReLp <150

$$f\_1 = 14.39 \text{Re}\_{L\_p}^{\left(-0.805 \frac{F\_p}{L\_p}\right)} \left(\log\_\varepsilon \left(1.0 + \left(\frac{F\_p}{L\_p}\right)\right)\right)^{3.04} \tag{36}$$

$$f\_2 = \left(\log\_{\epsilon}\left(\left(\frac{\delta}{F\_p}\right)^{0.48} + 0.9\right)\right)^{-1.435} \left(\frac{D\_h}{L\_p}\right) (\log\_{\epsilon}(0.5 \text{Re}\_{L\_p}))^{-3.01} \tag{37}$$

#### Comprehensive Study of Heat Exchangers with Louvered Fins http://dx.doi.org/10.5772/66472 85

$$f\_3 = \left(\frac{F\_p}{L\_h}\right)^{-0.308} \left(\frac{F\_d}{L\_h}\right)^{-0.308} \left(e^{-0.1167 \frac{\tau\_p}{D\_{\rm tr}}}\right) L\_\alpha^{0.35} \tag{38}$$

for 150 < ReLp <5000

$$f\_1 = 4.97 Re\_{L\_p}^{0.6049 - 1.064/L\_x^{0.2}} \left( \log\_\varepsilon \left( \left( \frac{\delta}{F\_p} \right)^{0.5} + 0.9 \right) \right)^{-0.527} \tag{39}$$

$$f\_2 = \left(\left(\frac{D\_h}{L\_p}\right) \log\_e(0.3 \text{Re}\_{L\_p})\right)^{-2.966} \left(\frac{F\_p}{L\_h}\right)^{-0.7931 \frac{V\_p}{T\_h}}\tag{40}$$

$$f\_3 = \left(\frac{T\_p}{D\_m}\right)^{-0.0446} \log\_e \left(1.2 + \left(\frac{L\_p}{F\_p}\right)^{1.4}\right)^{-3.553} L\_a^{-0.477}.\tag{41}$$

• Park and Jacobi [27], 27 < ReLp < 4132

$$\begin{array}{l} j\_{cor} = 0.872 j\_{Re} j\_{low} j\_{lower} L\_{\alpha}^{0.219} N\_{LB}^{-0.0881} \left( \frac{r\_h}{L\_p} \right)^{0.149} \left( \frac{F\_d}{F\_p} \right)^{-0.259} \left( \frac{L\_h}{F\_h} \right)^{0.54} \\\ \times \left( \frac{F\_h}{F\_p} \right)^{-0.902} \left( 1 - \frac{\delta}{L\_p} \right)^{2.62} \left( \frac{L\_p}{F\_p} \right)^{0.301} \end{array} \tag{42}$$

where

$$j\_{Re} = \text{Re}\_{L\_p}^{[-0.458 - 0.00874 \cosh(F\_p/L\_p - 1)]} \tag{43}$$

$$j\_{low} = 1 - \sin\left(\frac{L\_p}{F\_p} L\_a \right) \left[ \cos h \left( 0.049 Re\_{L\_p} - 0.142 \frac{F\_d}{N\_{LB}} \right) \right]^{-1} \tag{44}$$

$$j\_{lower} = 1 - (-0.0065 \tan L\_{it}) \left(\frac{F\_d}{N\_{LB} F\_p}\right) \cos \left[2\pi \left(\frac{F\_p}{L\_p \tan L\_{\alpha}} - 1.8\right)\right] \tag{45}$$

$$f\_{av} = 3.69 f\_{Re} N\_{LB}^{-0.256} \left(\frac{F\_p}{L\_p}\right)^{0.904} \sin\left(L\_a + 0.2\right) \left(1 - \frac{F\_h}{T\_p}\right)^{0.733} \times \left(\frac{L\_h}{F\_h}\right)^{0.648} \left(\frac{\delta}{L\_p}\right)^{-0.647} \left(\frac{F\_h}{F\_p}\right)^{0.799} \tag{46}$$

where

f ¼ f <sup>1</sup>f <sup>2</sup>f <sup>3</sup> (35)

<sup>ð</sup>logeð0:5ReLp ÞÞ<sup>−</sup>3:<sup>01</sup> (37)

(36)

Fp Lp

� � � � � � <sup>3</sup>:<sup>04</sup>

Dh Lp � �

for ReLp <150

<sup>f</sup> <sup>1</sup> <sup>¼</sup> <sup>14</sup>:39Re <sup>−</sup>0:805Fp

δ Fp � �<sup>0</sup>:<sup>48</sup>

f <sup>2</sup> ¼ loge

by R. L. Webb considered by Chang and Wang [20].

Source Lp(mm) Fp(mm) Fh(mm) Lh(mm) Lα(

84 Heat Exchangers– Advanced Features and Applications

o

KB(10)<sup>b</sup> [10] 1.70 1.40 8.15 6.40 27.0 24.0 – 10.15 0.10 2 KB(11)<sup>b</sup> [10] 1.70 1.40 8.15 6.40 29.0 24.0 – 10.15 0.10 2 KB(12)<sup>b</sup> [10] 1.70 1.10 8.15 6.40 23.0 20.0 – 10.15 0.10 2 KB(13)<sup>b</sup> [10] 1.70 1.10 8.15 6.40 23.0 24.0 – 10.15 0.10 2 KB(14)<sup>b</sup> [10] 1.70 1.20 8.15 6.40 23.0 20.0 – 10.15 0.10 2 KB(15)<sup>b</sup> [10] 1.70 1.20 8.15 6.40 23.0 24.0 – 10.15 0.10 2 KYL(1)<sup>b</sup> [14] 2.90 2.82 16.50 12.50 20.0 44.0 – 21.20 0.15 2 KYL(2)<sup>b</sup> [14] 2.90 2.42 16.50 12.50 20.0 44.0 – 21.20 0.15 2 KYL(3)<sup>b</sup> [14] 2.90 2.03 16.50 12.50 20.0 44.0 – 21.20 0.15 2 KYL(4)<sup>b</sup> [14] 2.90 2.82 16.50 12.50 25.0 44.0 – 21.20 0.15 2 KYL(5)<sup>b</sup> [14] 2.90 2.42 16.50 12.50 25.0 44.0 – 21.20 0.15 2 KYL(6)<sup>b</sup> [14] 2.90 2.03 16.50 12.50 25.0 44.0 – 21.20 0.15 2 KYL(7)<sup>b</sup> [14] 2.90 2.82 16.50 12.50 30.0 44.0 – 21.20 0.15 2 KYL(8)<sup>b</sup> [14] 2.90 2.42 16.50 12.50 30.0 44.0 – 21.20 0.15 2 KYL(9)<sup>b</sup> [14] 2.90 2.03 16.50 12.50 30.0 44.0 – 21.20 0.15 2 KYL(10)<sup>b</sup> [14] 2.90 2.82 16.50 12.50 35.0 44.0 – 21.20 0.15 2 KYL(11)<sup>b</sup> [14] 2.90 2.42 16.50 12.50 35.0 44.0 – 21.20 0.15 2 KYL(12)<sup>b</sup> [14] 2.90 2.03 16.50 12.50 35.0 44.0 – 21.20 0.15 2

) Fd(mm) Td(mm) Tp(mm) Δ(mm) NLB

Fh � �

þ 0:9

! ! <sup>−</sup>1:<sup>435</sup>

Table 1. Geometrical dimensions of the louvered fin heat exchangers in the database [20, 27].

Lp loge 1:0 þ

<sup>a</sup> Correlated data considered by Chang and Wang [20].<sup>b</sup> Correlated data considered by Park and Jacobi [27].c Park and Jacobi [27] used half of these pitches because of two fin stocks and a splitter plate between the tubes.<sup>d</sup> Park and Jacobi [27] and Chang and Wang [20] assumed that the tube diameter is 1.50 mm due to the lack of information.<sup>e</sup> Chang and Wang [20] assumed that the tube diameter is 5.00 mm due to the lack of information.<sup>f</sup> Correlated data which was unpublished

$$f\_{R\varepsilon} = \left(\text{Re}\_r \frac{F\_p}{L\_p}\right)^{-0.845} + 0.0013 \text{Re}\_{L\_p}^{[1.26(\delta/F\_p)]} \tag{47}$$

However, the correlations of Stanton number, Colburn j-factor and friction factor are defined for a large range of Reynolds number and geometric descriptions for heat exchangers with multi-louvered fins, it is necessary that the performance of every new type of heat exchanger is analysed individually due to the complexity of the combined effects of geometrical and operational parameters [33].
