**3.2 Numerical optimization of fin shape**

To analyze the exchanger heat transfer problem, a model of heat exchanger is built under some assumptions:


The heat exchanger characteristic dimensions are written in Table 1:


Table 1. Heat exchanger characteristic dimensions.

Calculations are done for circular fin-tube heat exchanger. Three-dimensional models are performed to find heat transfer characteristics between a finned tube and the air for different fin shapes in order to optimize the heat transfer between the air and fin material during the air flow in the cross flow heat exchanger. The model allows considering the heat transfer in three directions. The model is so defined that its output is compared with the results received from correlation formula. Correlation is described in Heading 3.3. Then, the main objective of this research is examined. The performance of a given heat exchanger for different fin profiles, with emphasis on the flow rates, is determined numerically by means of Ansys Workbench program (Ansys 12 Product Documentation).

A fin shape, used for the simulations, is shown in Figure 4 and the dimensions are presented in Table 2 where *Rf* – radial coordinate of fin tip, *R* – radial coordinate of fin base, *Rch* – radial coordinate of chamfer. All profiles have the same radius *Rf* and thickness *<sup>f</sup>* at the fin base (the thickness depends on angle 1 and 2 and changes along the fin height).

Fin-Tube Heat Exchanger Optimization 355

The model sketch, including also an air volume attached to the fin and tube segment, is

1 2

7

6

5

1. inlet area with constant air temperature *TIN* = 300 0C and unanimous air velocity

The cross–flow heat exchanger is often used in process plants. Fins, applied in heat exchangers, assure greater surface area (contact area) per a unit volume and can reduce the size or cost of the unit. The negative feature of fins is the bigger pressure drop in the flowing

Finned tubes may be divided into two categories: low fin and high fin. The ratio of the fin

Similar to plain tubes, heat transfer correlations are based on maximum fluid velocity and additional terms for fin geometry. Average heat transfer coefficient *h* is of more specific interest but it should be underlined that the surface temperature of the fins is not uniform. This is done by including the fin efficiency in deriving the effective heat transfer coefficient

demonstrated in Figure 5 (Wais, 2010):

Fig. 5. Model used for CFD simulation.

3

distribution, normal to the section *vIN* = 4.0 m/s

4

5. inner tube surface with constant temperature *TT* = 70 0C

**3.3 Correlation for external heat transfer in cross-flow heat exchanger** 

height to external diameter of the tube is the determinant (Hewitt et al., 1994):

where:

2. outlet area 3. fin made of steel 4. tube made of steel

6. air volume

fluid.

7. model width equal to pt /2

low-fin tubes: 0.05 0.33

high-fin tubes: 0.2 0.7

(Hewitt et al., 1994).

*f l <sup>D</sup>*

*f l <sup>D</sup>*

Fig. 4. Circular fin in optimization process.


Table 2. Fin and tube dimensions.

The model sketch, including also an air volume attached to the fin and tube segment, is demonstrated in Figure 5 (Wais, 2010):

Fig. 5. Model used for CFD simulation.

where:

354 Heat Exchangers – Basics Design Applications

Symmetry axis

*R f*

*R ch*

*R* 

*1*

*t*

> *2*

(a) (b) (c) (d) (e) (f)

Fin version

R f mm 20,5 20,5 20,5 20,5 20,5 20,5

R ch mm - - - 14,5 14,5 14,5

R mm 12,5 12,5 12,5 12,5 12,5 12,5

t mm 2,0 2,0 2,0 2,0 2,0 2,0

p f /2 mm 1,5 1,5 1,5 1,5 1,5 1,5

1 deg 90 90 90 92,9 95,7 98,5

2 deg 180 180 180 177,1 174,3 171,5

variable at R f d =1,0 at R d =1,2

variable at R f d =0,8 at R d =1,2

variable at R f d =0,6 at R d =1,2

Symmetry plain

constant 0,8

Fig. 4. Circular fin in optimization process.

mm

constant 1,2

Table 2. Fin and tube dimensions.

constant 1


#### **3.3 Correlation for external heat transfer in cross-flow heat exchanger**

The cross–flow heat exchanger is often used in process plants. Fins, applied in heat exchangers, assure greater surface area (contact area) per a unit volume and can reduce the size or cost of the unit. The negative feature of fins is the bigger pressure drop in the flowing fluid.

Finned tubes may be divided into two categories: low fin and high fin. The ratio of the fin height to external diameter of the tube is the determinant (Hewitt et al., 1994):

$$\text{low-fin tubes:} \quad 0.05 > \bigvee\_{f} \text{D}\_{f} > 0.33$$

$$\text{high-fin tubes:} \quad 0.2 > \bigvee\_{f} \text{D}\_{f} > 0.7$$

Similar to plain tubes, heat transfer correlations are based on maximum fluid velocity and additional terms for fin geometry. Average heat transfer coefficient *h* is of more specific interest but it should be underlined that the surface temperature of the fins is not uniform. This is done by including the fin efficiency in deriving the effective heat transfer coefficient (Hewitt et al., 1994).

Fin-Tube Heat Exchanger Optimization 357

*l*

pt

*s*

One sector

*pf*

where

*<sup>f</sup>* – fin efficiency

Fig. 6. Fin–tube geometry.

*t* 

*h* - average heat transfer coefficient *h*' – effective heat transfer coefficient

*T* – effective mean temperature difference, *A* – total external area of the tubes and fins

> *D Df*

Therefore:

$$\text{Re}\_D = \frac{\rho \, v\_{\text{max}} \, D}{\eta} \tag{40}$$

$$
\sigma\_{\text{max}} = \frac{\dot{m}\_f}{S\_{\text{min}} \, \rho} \tag{41}
$$

$$\overline{Nu} = \frac{\overline{h}D}{k\_f} \tag{42}$$

where

*D* – external tube diameter, max *v* – maximum fluid velocity (in minimum flow area), – fluid density, – fluid dynamic viscosity, *S*min – minimum flow area, *<sup>f</sup> k* – fin thermal conductivity, *h* – average heat transfer coefficient

Typical fin-tube geometry, with surface area equation and minimum cross-sectional area, are presented in Figure 6.

Surface area of one sector (consists of fin and tube) are defined as:

Surface area of fins: <sup>1</sup> 2 2 ( ) <sup>2</sup> *A DD D ff f* Surface area of tube between fins: *A Ds <sup>t</sup>* Total surface area: <sup>1</sup> 2 2 ( ) <sup>2</sup> *A D D D Ds f f* Total tube surface (with fin removed): ( ) *A Ds <sup>T</sup>* 

Characteristic fin–tube heat exchanger configurations are shown in Figure 7.

Average heat transfer coefficient *h* is of more specific interest for the whole process, which is correlated with the maximum velocity between tubes max *v* .

Total heat transfer can be calculated taking into consideration fin efficiency:

$$\dot{Q} = \overline{h}\,\Delta T \left(\eta\_f \, A\_f + A\_t\right) = \overline{h}\,' \Delta T \,' A \tag{43}$$

#### where

356 Heat Exchangers – Basics Design Applications

max Re*<sup>D</sup> v D* 

*mf <sup>v</sup> S* 

max

.

min

*f hD Nu*

Typical fin-tube geometry, with surface area equation and minimum cross-sectional area,

 

 

Average heat transfer coefficient *h* is of more specific interest for the whole process, which

( )' *Q h T A A h TA*

(43)

(40)

(41)

*<sup>k</sup>* (42)

Therefore:

where

*D* – external tube diameter,

 – fluid dynamic viscosity, *S*min – minimum flow area, *<sup>f</sup> k* – fin thermal conductivity, *h* – average heat transfer coefficient

are presented in Figure 6.

Surface area of fins: <sup>1</sup> 2 2 ( ) <sup>2</sup>

Total surface area: <sup>1</sup> 2 2 ( ) <sup>2</sup>

Surface area of tube between fins: *A Ds <sup>t</sup>*

– fluid density,

max *v* – maximum fluid velocity (in minimum flow area),

Surface area of one sector (consists of fin and tube) are defined as:

is correlated with the maximum velocity between tubes max *v* .

Total tube surface (with fin removed): ( ) *A Ds <sup>T</sup>*

*A DD D ff f*

 

*A D D D Ds*

Characteristic fin–tube heat exchanger configurations are shown in Figure 7.

Total heat transfer can be calculated taking into consideration fin efficiency:

*ff t*

*f f*


Fig. 6. Fin–tube geometry.

Fin-Tube Heat Exchanger Optimization 359

. ( ) *ff t*

( ) ( )

The value of heat transfer depends on local fluid velocity, fluid properties and details of the tube bank geometry. Correlations that allow calculating average heat transfer coefficient, *h* ,

Having calculated average heat transfer coefficient, *h* , effective mean temperature

( ) *Q hT A A f f t*

The average Nusselt number for low fin tube can be calculated from below correlation

*s l <sup>X</sup> Nu FF F lD D*

*F*1 – factor for fluid property variation (significant only at high temperatures),

0.06 0.11 0.36 0.7 0.36 12 3 0.183 Re Pr *<sup>t</sup> f f*

(51)

*h AA T T m c*

*f f*

. ( ) *ff t*

*h AA T T T*

*m c*

*f f*

.

.

*h AA*

*m c*

*ff t*

*f f*

*ff t*

*f f*

( ) 1 exp

*h AA*

*m c*

*<sup>f</sup>* , the rate of heat transfer equals:

(47)

(48)

(50)

(49)

*Fluid*

*OUT IN*

0

are derived from experimental data and take into account geometrical features.

Pr*Fluid Ave* \_ – Prandtl number of fluid for bulk temperature *TFluid Ave* \_

Pr*S* – Prandtl number of fluid for mean tube and fin surface temperature, *TS*

*IN*

*TT T*

After transformation

difference, *T* , and fin efficiency,

**3.3.1 Correlation for low fin tube** 

0.26

,

\_

Pr *Fluid Ave S <sup>F</sup>*

Pr

(Hewitt et al., 1994):

where

1

finally

It is necessary to find the effective mean temperature difference to evaluate the heat transfer. Since the fluid temperatures change in fluid flow through the tube bank, the fluid temperature difference *TFluid* can be calculated from energy exchanged as:

$$\dot{Q} = \overline{h}\,\Delta T \,\left(\eta\_f \, A\_f + A\_t\right) = \dot{m}\_f \, c\_f \,\Delta T\_{\text{Fluid}}\tag{44}$$

where

$$
\Delta T = \frac{(T\_0 - T\_{OUT}) - (T\_0 - T\_{IN})}{1 \text{n} \frac{T\_0 - T\_{OUT}}{T\_0 - T\_{IN}}} \tag{45}
$$

and

*T*0 – temperature at the external tube surface (for diameter *D* ) *TOUT* – average fluid temperature in the outlet section *TIN* – average fluid temperature in the inlet section

and for *T T IN OUT*

$$
\Delta T\_{Fluid} = T\_{IN} - T\_{OUT} \tag{46}
$$

#### After transformation

$$
\Delta T\_{\text{Fluid}} = \frac{\overline{h} \text{ (}\eta\_f \text{ A}\_f + \text{A}\_t\text{)}}{\dot{m}\_f \text{ c}\_f} \Delta T \tag{47}
$$

$$T\_{OUT} = T\_{IN} - \frac{\overline{h} \left(\eta\_f A\_f + A\_t\right)}{\dot{\iota}\_{mf} c\_f} \Delta T \tag{48}$$

finally

358 Heat Exchangers – Basics Design Applications

Flow direction Flow direction

a) in-line array b) staggered array

temperature difference *TFluid* can be calculated from energy exchanged as:

*Xd*

*T*0 – temperature at the external tube surface (for diameter *D* )

*TOUT* – average fluid temperature in the outlet section *TIN* – average fluid temperature in the inlet section

It is necessary to find the effective mean temperature difference to evaluate the heat transfer. Since the fluid temperatures change in fluid flow through the tube bank, the fluid

( ) *<sup>f</sup> Q hT A A mc T*

0 0 0 0

*TT TT <sup>T</sup> T T*

ln

( )( )

*T T*

*OUT IN OUT IN*

.

(44)

*Xl*

*Xd Xt*

*T TT Fluid IN OUT* (46)

(45)

*f f t f Fluid*

Fig. 7. Fin–tube patterns.

*Xl*

*Xt*

where

and

and for *T T IN OUT*

$$\Delta T = (T\_{IN} - T\_0) \frac{1 - \exp\left(-\frac{\overline{h} \cdot (\eta\_f A\_f + A\_t)}{\cdot \dot{m}\_f c\_f}\right)}{\frac{\overline{h} \cdot (\eta\_f A\_f + A\_t)}{\dot{m}\_f c\_f}}\tag{49}$$

The value of heat transfer depends on local fluid velocity, fluid properties and details of the tube bank geometry. Correlations that allow calculating average heat transfer coefficient, *h* , are derived from experimental data and take into account geometrical features.

Having calculated average heat transfer coefficient, *h* , effective mean temperature difference, *T* , and fin efficiency, *<sup>f</sup>* , the rate of heat transfer equals:

$$
\dot{Q} = \overline{h}\Delta T \left(\eta\_f A\_f + A\_t\right) \tag{50}
$$

#### **3.3.1 Correlation for low fin tube**

The average Nusselt number for low fin tube can be calculated from below correlation (Hewitt et al., 1994):

$$\overline{Nu} = 0.183 \text{ Re}^{0.7} \left( \frac{s}{l} \right)^{0.36} \left( \frac{X\_t}{D\_f} \right)^{0.06} \left( \frac{l}{D\_f} \right)^{0.11} \text{Pr}^{0.36} \cdot F\_1 \cdot F\_2 \cdot F\_3 \tag{51}$$

where

*F*1 – factor for fluid property variation (significant only at high temperatures), 0.26 \_ 1 Pr Pr *Fluid Ave S <sup>F</sup>* , Pr*Fluid Ave* \_ – Prandtl number of fluid for bulk temperature *TFluid Ave* \_

Pr*S* – Prandtl number of fluid for mean tube and fin surface temperature, *TS*

Fin-Tube Heat Exchanger Optimization 361

tanh( 2 /( ) ) 2 /( )

*h k*

*D D D f f D D*

The heat exchange optimization function is defined as the amount of dissipated heat to the heat exchanger weight for a one raw heat exchanger (optimization parameter is the profile shape). The shape of the fin is modified to calculate heat transfer, reduce the total mass that refers to the cost of the whole heat exchanger. The performance of the heat transfer process in a given heat exchanger is determined for different fin profiles, considering the fluid flow. Fin geometry affects the heat transfer phenomenon between the plate itself and the air. Changing the fin profile, the fluid streamline can be modified in a way that it affects the

Numerical analyses are carried out to examine a modified finned tube heat exchanger. The tube material is kept fixed as well as the heat exchanger fin and tube pitches (spacing). No changes are done to the inlet and outlet temperature and pressure values. The shape of the fin and tube is modified to calculate heat transfer for different conditions, reduce the total mass that refers to the cost of the whole heat exchanger. The temperature difference is found

To confirm the correctness of the numerical model, the results of the heat transfer are reviewed and compared with those received from the correlation recommended by Engineering Sciences Data Unit, Equation (52), modified for one row crossflow tube-fin heat exchanger of rectangular profile and fin constant thickness - fin profile (a), (b) and (c).

Comparison, shown in Table 3, should be used only as a reference. Correlations for the heat transfer of air flow are expressed for at least 4 tube raws. Then factors are introduced to recalculate *Nu* number for one raw heat exchanger. The standard deviation of correlation for external flow is about 25% for laminar flow and 15% for turbulent flow (Hewitt et al., 1994). Presented correlation is used to check the model accuracy in relation to fin shape

After model verification, the fin of variable thickness is considered and the optimized

to model (a) thickness (for manufacturing and operating reason) and the mass flow in outlet

near the tube is set up to be constant and equal

*h k*

*f*

 

*f*

1 1 0.35 ln

  *<sup>f</sup>* can be achieved from

(54)

(55)

Calculating the average Nusselt number, the fin efficiency value

*f*

2

temperature changes on the fin surface and heat convection conditions.

numerically and the solid volume is calculated for different fin profile shapes.

(McQuiston & Tree, 1972):

Results are presented in Table 3.

is calculated. The fin thickness

*mf* does not change in different models.

modifications.

function

section .

where

**4. Results** 

*F*2 – factor for number of fin – tube raws for number of raws > 10: *F*<sup>2</sup> = 1.000 for number of raws = 8: *F*<sup>2</sup> = 0.985 for number of raws = 6: *F*<sup>2</sup> = 0.955 for number of raws = 4: *F*<sup>2</sup> = 0.900

*F*3 – factor for staggered arrangement ( = 30 , 45 ,60 *o oo* , see Fig. 5. for definition) *F*3 =1 *F*3 – factor for in-line arrangement 3 <sup>o</sup> for plain tube (in-line) for plain tube staggered array ( 30 ) *Nu <sup>F</sup> Nu* 

$$\begin{aligned} \text{Abbve recommendded correlation is applicable for Reynolds number } 10^3 \le \text{Re} \le 8 \cdot 10^5, \\ 0.19 < \frac{s}{l} < 0.66, \ 1.1 < \frac{X\_t}{D\_f} < 4.92, \ 0.058 < \frac{l}{D\_f} < 0.201 \text{ (Hewitt et al., 1994)} \end{aligned}$$

#### **3.3.2 Correlation for high fin tube**

Recommended correlation to calculate the average Nusselt number for staggered tube banks by Engineering Science Data (Hewitt et al., 1994) and Reynolds number range 3 4 2 10 Re 4 10 , 0.13 0.57 *<sup>s</sup> l* , 1.15 1.72 *<sup>t</sup> l X X* :

$$\sqrt{N\mu} = 0.242 \text{ Re}^{0.658} \left(\frac{s}{l}\right)^{0.297} \left(\frac{X\_t}{X\_l}\right)^{-0.091} \text{Pr}^{V\_3} \cdot \text{F}\_1 \cdot \text{F}\_2 \tag{52}$$

where

*F*1 – factor for fluid property variation (significant only at high temperatures)

*F*2 – factor for number of fin – tube raws

 1.0 for four or more raws, 0.92 for three raws 0.84 for two raws 0.76 for one raw

For high fin-tube and in–line array the correlation that can be applied for Reynolds number 3 5 5 10 Re 10 , and 5 12 *T A A* :

$$
\sqrt{Nu} = 0.30 \text{ Re}^{0.625} \left(\frac{A}{A\_T}\right)^{-0.375} \text{Pr}^{0.333} \tag{53}
$$

where

*AT* – total tube surface area of one sector, ( ) *A Ds <sup>T</sup>* 

$$A \quad \text{- total surface area of one sector}; \quad A \quad = \left[\frac{1}{2}\pi \left(D\_f^{-2} - D^2\right)\right] + \pi D\_f \,\delta + \pi \,D \,\text{s}$$

Calculating the average Nusselt number, the fin efficiency value *<sup>f</sup>* can be achieved from (McQuiston & Tree, 1972):

$$\eta\_f = \frac{\tanh(\sqrt{2\bar{h}}/(\delta \, k\_f) \cdot \psi)}{\sqrt{2\bar{h}}/(\delta k\_f)} \,\,\,\,\,\tag{54}$$

where

360 Heat Exchangers – Basics Design Applications

= 30 , 45 ,60 *o oo* , see Fig. 5. for

(Hewitt et al., 1994)

 for plain tube (in-line) for plain tube staggered array ( 30 )

definition) *F*3 =1

(52)

(53)

*F*3 – factor for in-line arrangement 3 <sup>o</sup>

, 0.058 0.201

*Nu <sup>F</sup> Nu*

> *f l D*

, 1.15 1.72 *<sup>t</sup>*

*F*1 – factor for fluid property variation (significant only at high temperatures)

*<sup>A</sup> Nu*

Above recommended correlation is applicable for Reynolds number 3 5 10 Re 8 10 ,

Recommended correlation to calculate the average Nusselt number for staggered tube banks by Engineering Science Data (Hewitt et al., 1994) and Reynolds number range

> *l X X* :

*<sup>s</sup> <sup>X</sup> Nu F F l X*

 

For high fin-tube and in–line array the correlation that can be applied for Reynolds number

0.625 0.333 0.30 Re Pr *T*

*A*

0.297 0.091 <sup>1</sup> <sup>3</sup> 0.658 1 2 0.242 Re Pr *<sup>t</sup> l*

0.375

 

*A D D D Ds*

*f f*

*F*2 – factor for number of fin – tube raws for number of raws > 10: *F*<sup>2</sup> = 1.000 for number of raws = 8: *F*<sup>2</sup> = 0.985 for number of raws = 6: *F*<sup>2</sup> = 0.955 for number of raws = 4: *F*<sup>2</sup> = 0.900 *F*3 – factor for staggered arrangement (

, 1.1 4.92 *<sup>t</sup>*

**3.3.2 Correlation for high fin tube** 

3 4 2 10 Re 4 10 , 0.13 0.57 *<sup>s</sup>*

*F*2 – factor for number of fin – tube raws 1.0 for four or more raws, 0.92 for three raws 0.84 for two raws 0.76 for one raw

3 5 5 10 Re 10 , and 5 12

*f X D*

*l*

*T A A* :

*AT* – total tube surface area of one sector, ( ) *A Ds <sup>T</sup>*

*<sup>A</sup>* – total surface area of one sector, <sup>1</sup> 2 2 ( ) <sup>2</sup>

0.19 0.66 *<sup>s</sup> l*

where

where

$$\varphi = \frac{D}{2} \left( \frac{D\_f}{D} - 1 \right) \left( 1 + 0.35 \ln \frac{D\_f}{D} \right) \tag{55}$$

#### **4. Results**

The heat exchange optimization function is defined as the amount of dissipated heat to the heat exchanger weight for a one raw heat exchanger (optimization parameter is the profile shape). The shape of the fin is modified to calculate heat transfer, reduce the total mass that refers to the cost of the whole heat exchanger. The performance of the heat transfer process in a given heat exchanger is determined for different fin profiles, considering the fluid flow. Fin geometry affects the heat transfer phenomenon between the plate itself and the air. Changing the fin profile, the fluid streamline can be modified in a way that it affects the temperature changes on the fin surface and heat convection conditions.

Numerical analyses are carried out to examine a modified finned tube heat exchanger. The tube material is kept fixed as well as the heat exchanger fin and tube pitches (spacing). No changes are done to the inlet and outlet temperature and pressure values. The shape of the fin and tube is modified to calculate heat transfer for different conditions, reduce the total mass that refers to the cost of the whole heat exchanger. The temperature difference is found numerically and the solid volume is calculated for different fin profile shapes.

To confirm the correctness of the numerical model, the results of the heat transfer are reviewed and compared with those received from the correlation recommended by Engineering Sciences Data Unit, Equation (52), modified for one row crossflow tube-fin heat exchanger of rectangular profile and fin constant thickness - fin profile (a), (b) and (c). Results are presented in Table 3.

Comparison, shown in Table 3, should be used only as a reference. Correlations for the heat transfer of air flow are expressed for at least 4 tube raws. Then factors are introduced to recalculate *Nu* number for one raw heat exchanger. The standard deviation of correlation for external flow is about 25% for laminar flow and 15% for turbulent flow (Hewitt et al., 1994). Presented correlation is used to check the model accuracy in relation to fin shape modifications.

After model verification, the fin of variable thickness is considered and the optimized function is calculated. The fin thickness near the tube is set up to be constant and equal to model (a) thickness (for manufacturing and operating reason) and the mass flow in outlet section . *mf* does not change in different models.

Fin-Tube Heat Exchanger Optimization 363

Fig. 8. Air streamlines for fin profile (a).

Fig. 9. Air streamlines for fin profile (f).

Fig. 10. Temperature on fin surface and flowing air temperature for fin profile (a).


Table 3. Results from numerical calculation and correlation.

The values of the optimization function are found and presented in Table 4 where:

( ) *IN OUT s T T V* from Equation (39) for constant . *mf* , model ( ) m odel ( ) m odel ( ) *i a i a T T T T* , model ( ) model ( ) model ( ) *i a i a* :


Table 4. Results from numerical calculation for profile modification.

The results illustrate how the fin dimensions and configuration influence the heat transfer. The function *<sup>i</sup>* is higher for models with profile modification.

The same conclusion may also be drawn with flow analysis. Evaluating the streamlines for all models, the influence of fin shape on mass flow distribution is seen. To confirm the observation, the outlet area is divided into sections for which the mass flow distribution is calculated (Wais, 2010). The fin and tube surface orientation also affects the flow route and causes the variation of the air streamlines. In Figure 8 and 9, it is seen that the flow streams vary and change the flow direction depending on fin profile modification that has an impact on the fin surface temperature. The fin surface temperature is shown in Figure 10 and 11.

C] 45.5 44.7 43.9

C] 45.7 43.3 41.4

T\_model [0

T\_correl [0

( ) *IN OUT s*

:

model ( ) model ( ) model ( ) *i a*

 

*a*

*T T V*

The function *<sup>i</sup>*

*i*

ΔT\_correl ΔT\_model ΔT\_correl

Table 3. Results from numerical calculation and correlation.

from Equation (39) for constant .

T\_model [0

Ti

<sup>i</sup>

T\_model (a) [<sup>0</sup>

Table 4. Results from numerical calculation for profile modification.

The values of the optimization function are found and presented in Table 4 where:

(a) (b) (c)

Fin version


(d) (e) (f)

Fin version


5,2% 13,6% 20,9%

C] 42,8 40,9 37,8

The results illustrate how the fin dimensions and configuration influence the heat transfer.

The same conclusion may also be drawn with flow analysis. Evaluating the streamlines for all models, the influence of fin shape on mass flow distribution is seen. To confirm the observation, the outlet area is divided into sections for which the mass flow distribution is calculated (Wais, 2010). The fin and tube surface orientation also affects the flow route and causes the variation of the air streamlines. In Figure 8 and 9, it is seen that the flow streams vary and change the flow direction depending on fin profile modification that has an impact on the fin surface temperature. The fin surface temperature is shown in Figure 10 and 11.

is higher for models with profile modification.

C] 45,5 45,5 45,5

*mf* , model ( ) m odel ( )

*T T*

*T* 

*i*

*T*

m odel ( ) *i a*

*a*

,

Fig. 8. Air streamlines for fin profile (a).

Fig. 9. Air streamlines for fin profile (f).

Fig. 10. Temperature on fin surface and flowing air temperature for fin profile (a).

Fin-Tube Heat Exchanger Optimization 365

*AT* – total tube surface (with fin removed) *<sup>f</sup> c* – fluid (air) specific heat capacity

*h* – average heat transfer coefficient *h*' – effective heat transfer coefficient

*<sup>f</sup> k* – fin thermal conductivity,

 – fluid mass flow rate *ms* – tube and fin mass (solid). *Nu* – average Nusselt number *P* - perimeter (function of x)

*Rch* – radial coordinate of chamfer

*s* – spacing between adjacent fins

*TS* – surrounding temperature *TT* – inner tube surface temperature *Vs* – volume of tube and fin material *vIN* – air velocity in the inlet section

*w* – fin width

*TIN* – fluid temperature in the inlet section

*X <sup>d</sup>* – diagonal tube pitch ( 2 2 *X X t t* )

*TOUT* – average fluid temperature in the outlet section

max *v* – maximum fluid velocity (in minimum flow area)

*X <sup>l</sup>* – longitudinal (parallel to the flow) tube pitch

Re – Reynolds number *S*min – minimum flow area

*Rf* – radial coordinate of fin tip ( / 2 *R D f f* )

*l* – fin height

*<sup>f</sup> p* – fin pitch *<sup>t</sup> p* – tube pitch

*mf* 

.

*D <sup>f</sup>* – diameter of fin tip

*D* – external tube diameter (also diameter of fin base),

<sup>0</sup> *I* – modified, zero-order Bessel function of the first kind <sup>1</sup> *I* – modified, first-order Bessel function of the first kind *K* 0 – modified, zero-order Bessel function of the second kind *K* 1 – modified, first-order Bessel function of the second kind

*Q* – heat flow removed from the fluid to the fin and tube

*R* – radial coordinate of fin base (external tube radius *R D* / 2 )

*T*0 – temperature at the external tube surface (for diameter *D* )

Fig. 11. Temperature on fin surface and flowing air temperature for fin profile (f).
