**2. Heat transfer from fins**

The analysis of heat transfer from finned surfaces involves solving second-order differential equations and is often a subject of researches including also the variable heat transfer coefficient as a function of temperature or the fin geometrical dimensions. To analyze the heat transfer problem, a set of assumptions is introduced so that the resulting theoretical models are simple enough for the analysis. Analytical investigations and search activities, which allow finding the optimal profile of the fin, are available under assumptions that simplify the problem of heat transfer. These basic assumptions are proposed by Murray (1938) and Gardner (1945) and are called Murray-Gardner assumptions (Kraus et al., 2001):


In general, the study of the extended surface heat transfer compromises the movement of the heat within the fin by conduction and the process of the heat exchange between the fin and the surroundings by convection.

#### **2.1 Straight fin analysis**

Straight fin is any extended surface that is attached to a plane wall (Incropera et al., 2006). It may be of uniform cross–sectional area, or its cross–sectional area may vary with the distance *x* from the wall. The fin of variable thickness is shown in Figure 1.

fin mass by means of changing the shape of the fin. The fin shape modification influences not only the mass of the heat exchanger, but also affects the flow direction that causes the temperature changes on the fin contact surfaces. The air flow is considered in all 3D models. The numerical outcome of heat transfer coefficient is compared to the results received from the empirical equation for the fin-tube heat exchanger of uniform fin thickness. The correlation function is cited and the procedure how to verify the models is described. For modified fin shapes, mass flow weighted average temperatures of air volume flow rate are calculated in the outlet section and compared for different fin/tube shapes in order to optimize the heat transfer between the fin material and the air during the air flow in the cross flow heat exchanger.

The analysis of heat transfer from finned surfaces involves solving second-order differential equations and is often a subject of researches including also the variable heat transfer coefficient as a function of temperature or the fin geometrical dimensions. To analyze the heat transfer problem, a set of assumptions is introduced so that the resulting theoretical models are simple enough for the analysis. Analytical investigations and search activities, which allow finding the optimal profile of the fin, are available under assumptions that simplify the problem of heat transfer. These basic assumptions are proposed by Murray (1938) and Gardner (1945) and are called Murray-Gardner assumptions (Kraus et al., 2001):






In general, the study of the extended surface heat transfer compromises the movement of the heat within the fin by conduction and the process of the heat exchange between the fin

Straight fin is any extended surface that is attached to a plane wall (Incropera et al., 2006). It may be of uniform cross–sectional area, or its cross–sectional area may vary with the

distance *x* from the wall. The fin of variable thickness is shown in Figure 1.







**2. Heat transfer from fins** 

and it remains constant

entire surface of the fin

leaving its lateral surface

and the surrounding medium

and the surroundings by convection.

**2.1 Straight fin analysis** 

neglected

Fig. 1. Straight fin of variable cross section.

Both the conduction through the fin cross section and the convection over the fin surface area take place in and around the fin. When the fin temperature is lower than the base (primary surface) temperature *T*<sup>0</sup> , the heat is transferred from the fin to the surroundings (Shah & Sekulic, 2003).

The fin height is *l* , width is *w* , variable thickness ( ) *x* . Its perimeter for surface convection depends on coordinate *x* and is *Px w x* ( ) 2[ ( )] . Its cross-sectional area for heat conduction at any cross section is *Ax x w* () () , where – fin thickness as a function of *x* , *w* - fin width.

The temperature distribution can be calculated taking into consideration an energy balance on a typical element between *x* and *x dx* , shown in Figure 2.

Fig. 2. Energy balance on a typical element.

Fin-Tube Heat Exchanger Optimization 347

() () *<sup>S</sup>*

– temperature difference between a point on a fin surface and the surroundings, 0C

*d dT dx dx* 

<sup>2</sup> , <sup>2</sup>

*d x dx dx*

and subsequently, the heat transfer rate through the fin (Shah & Sekulic, 2003)

 

(ln ) <sup>0</sup> *k x d d d A*

This second order, linear, homogeneous ordinary differential equation with nonconstant coefficients is valid for any thin fins of variable cross section. Once the boundary conditions and the fin geometry are specified, its solution will provide the temperature distribution

The fin of uniform thickness of circular fin that can be applied on the outside of a tube is shown in Figure 3. Such fins have extensive application in liquid-gas heat exchangers (Mills,

The energy balance on a typical element of circular fin between *r* and *r dr* can be

( 2 ) ( 2 ) 2 (2 ) ( ) 0 *<sup>r</sup> r dr <sup>S</sup> q r q r h r dr T T*

*f*

( ) ( )0 <sup>2</sup> *<sup>S</sup> <sup>d</sup> r q hr T T dr* 

<sup>2</sup> ( ) ( )0 *<sup>S</sup> f*

*<sup>d</sup> dT hr r TT dr dr k*

 

(12)

(14)

(15)

*dT q k dr* (13)

 

*m*

*x Tx T* (9)

(10)

(11)

Because the ambient (surrounding) temperature is assumed to be constant, then:

2

where

and

1995).

where

then:

written as:

**2.2 Circular fin analysis** 

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

*TS* – surrounding temperature

– fin thickness

The energy balance:

$$
\dot{Q}\_x - \dot{Q}\_{x+dx} - d\,\dot{Q}\_{CONV} = 0 \tag{1}
$$

where

$$
\dot{Q}\_x = -k\_f \, A\_{k,x} \, \frac{dT}{dx} \tag{2}
$$

$$\dot{Q}\_{\text{x+dx}} = -k\_f \left( A\_{k,\text{x}} \frac{dT}{d\text{x}} + \frac{d}{d\text{x}} (A\_{k,\text{x}} \frac{dT}{d\text{x}}) \, d\text{x} \right) \tag{3}$$

$$\mathrm{h}\dot{\mathcal{Q}}\_{\mathrm{CONV}} = \mathrm{h}\,\mathrm{A}\_{f}\left(\mathrm{T} - \mathrm{T}\_{\mathrm{S}}\right) = \mathrm{h}\,\mathrm{P}\,\mathrm{d}\,\mathrm{x}\,\left(\mathrm{T} - \mathrm{T}\_{\mathrm{S}}\right) \tag{4}$$

Where

*<sup>f</sup> k* – fin thermal conductivity *h* – heat transfer coefficient

*TS* – surrounding temperature

*Af* – fin surface area

*Ak x*, – cross-sectional area as a function of *k* and *x*

*P* - perimeter (function of *x* )

Then

$$(k\_f \frac{d}{d\mathbf{x}}(A\_{k,\mathbf{x}} \frac{dT}{d\mathbf{x}})) \, d\mathbf{x} = h \, P \, d\mathbf{x} \tag{5} \tag{5}$$

and

$$\frac{d^2T}{dx^2} + \frac{1}{A\_{k,x}}\frac{dA\_{k,x}}{dx}\frac{dT}{dx} - \frac{h}{k\_f}\frac{P}{A\_{k,x}}(T - T\_S) = 0\tag{6}$$

or

$$\frac{d^2T}{dx^2} + \frac{d\left(\ln A\_{k,x}\right)}{dx}\frac{dT}{dx} - m^2\left(T - T\_S\right) = 0\tag{7}$$

where

$$m^2 = \frac{h \ P}{k\_f \ A\_{k,x}} \tag{8}$$

Both *P* and *Ak x*, are the function of *x* or a variable cross section.

To simplify the equation, the new dependent variable is introduced:

$$
\partial \mathbf{\dot{x}}(\mathbf{x}) = T(\mathbf{x}) - T\_S \tag{9}
$$

where

346 Heat Exchangers – Basics Design Applications

0 *Q Q dQ x x d x CONV* (1)

*dT Q kA d x* (2)

*dx dx dx* (3)

() () *CONV <sup>f</sup> S S dQ h A T T h Pdx T T* (4)

*dx dx* (5)

*S*

(7)

*k A* (8)

(6)

*S*

.. .

*x f kx*,

, , ( ( )) *x dx f kx k x dT d dT Q kA A dx*

, ( ) () *<sup>f</sup> k x <sup>S</sup> d dT k A dx h Pdx T T*

<sup>1</sup> ( )0 *k x*

(ln ) ( )0 *k x*

*f k x*, *h P*

.

.

.

*Ak x*, – cross-sectional area as a function of *k* and *x*

<sup>2</sup> ,

2

Both *P* and *Ak x*, are the function of *x* or a variable cross section.

To simplify the equation, the new dependent variable is introduced:

<sup>2</sup> , ,

<sup>2</sup> , <sup>2</sup>

2

*m*

*d x dx dx*

*d T d A dT mTT*

*d x A dx dx k A*

*k x f kx d T d A dT h P T T*

The energy balance:

where

Where

Then

and

or

where

*<sup>f</sup> k* – fin thermal conductivity *h* – heat transfer coefficient *TS* – surrounding temperature

*P* - perimeter (function of *x* )

*Af* – fin surface area

 – temperature difference between a point on a fin surface and the surroundings, 0C Because the ambient (surrounding) temperature is assumed to be constant, then:

$$\frac{d\theta}{d\mathbf{x}} = \frac{dT}{d\mathbf{x}}\tag{10}$$

and

$$\frac{d^2\,\theta}{dx^2} + \frac{d\left(\ln A\_{k,x}\right)}{dx}\frac{d\,\theta}{dx} - m^2\,\theta = 0\tag{11}$$

This second order, linear, homogeneous ordinary differential equation with nonconstant coefficients is valid for any thin fins of variable cross section. Once the boundary conditions and the fin geometry are specified, its solution will provide the temperature distribution and subsequently, the heat transfer rate through the fin (Shah & Sekulic, 2003)

#### **2.2 Circular fin analysis**

The fin of uniform thickness of circular fin that can be applied on the outside of a tube is shown in Figure 3. Such fins have extensive application in liquid-gas heat exchangers (Mills, 1995).

The energy balance on a typical element of circular fin between *r* and *r dr* can be written as:

$$\left. \left( q \begin{array}{c} 2\,\pi \, r \, \delta \right) \right|\_{r} - \left( q \begin{array}{c} 2\,\pi \, r \, \delta \end{array} \right) \Big|\_{r+dr} - 2 \,\ln(2\pi \, r) \, dr \, \left( T - T\_{\rm S} \right) = 0 \tag{12}$$

where

$$q\_{\parallel} = -k\_f \frac{dT}{dr} \tag{13}$$

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

– fin thickness

*TS* – surrounding temperature

then:

$$-\frac{d}{dr}(r\frac{\delta}{2}q) - h\,r\,\left(T - T\_S\right) = 0\tag{14}$$

$$\frac{d}{dr}(r\frac{dT}{dr}) - \frac{2\operatorname{hr}}{\mathcal{S}k\_f}(T - T\_S) = 0\tag{15}$$

Fin-Tube Heat Exchanger Optimization 349

where 0*I* and *K*0 are modified, zero-order Bessel functions of the first and second kind

Assuming the constant and known base temperature and zero heat flow through the tip of

0 00 *<sup>S</sup> rR TT T T*

*f f f r R r R*

*dr dr*

*dT <sup>d</sup> r R*

0 10 2 0

10 1 1 [ ( )] ( ) *<sup>d</sup> C I mr C mI mr*

2 0 2 1 [ ( )] ( ) *<sup>d</sup> C K mr C mK mr*

 11 2 1 11 2 1 ( ) ( ) ( ) ( )0 *f f r R r R <sup>f</sup> <sup>f</sup> <sup>d</sup> C mI mr C mK mr C mI mR C mK mR*

> ( ) ( ) ( )( ) ( ) ( ) ( )( )

> (2 ) *<sup>f</sup> k rR <sup>f</sup> r R*

 (28)

*dT <sup>d</sup> Q kA k R dr dr*

*I mr K mR K mr I mR I mR K mR K mR I mR*

0 1 01 00 1 0 1

(26)

*f f f f*

(27)

*d mr* are modified, first order Bessel functions of

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

From the second condition and according to differentiation rules

*C*1 and *C*2 can be evaluated to find a temperature distribution:

[ ( )] ( ) ( ) *d K mr K mr*

Heat dissipated by the fin and its efficiency can be expressed as:

*d mr* and <sup>0</sup> 1

.

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

0 0

(22)

*C I mR C K mR* () () (23)

*dr* (24)

*dr* (25)

(21)

respectively.

the fin:

where

we obtain:

where

1

*dr* 

0

the first and second kind.

[ ( )] ( ) ( ) *d I mr I mr*

From the first condition:

To simplify the equation, the new dependent variable is introduced:

$$
\theta(\mathbf{x}) = T(\mathbf{x}) - T\_S \tag{16}
$$

and constant value

$$\text{cm}^2 = \frac{h \text{ } P}{k\_f \text{ } A\_k} = \frac{h \text{ } \text{2} \text{ } \text{2} \text{ } \text{2} \text{ } r \text{ } \text{)}}{k\_f \text{ } \text{(2} \text{ } \text{2} \text{ } r \text{)} \delta} = \frac{2 \, h}{k\_f \, \delta} \tag{17}$$

Because the ambient (surrounding) temperature is assumed to be constant, then:

$$\frac{d\,\theta}{dr} = \frac{dT}{dr}, \qquad \frac{d^2\,\theta}{dr^2} = \frac{d^2T}{dr^2} \tag{18}$$

and

$$\frac{d^2\theta}{dr^2} + \frac{1}{r}\frac{d\theta}{dr} - m^2\theta = 0\tag{19}$$

The general solution of the equation is modified Bessel function of order zero:

$$\theta = \mathbb{C}\_1 I\_0(mr) + \mathbb{C}\_2 K\_0(mr) \tag{20}$$

where 0*I* and *K*0 are modified, zero-order Bessel functions of the first and second kind respectively.

Assuming the constant and known base temperature and zero heat flow through the tip of the fin:

$$
\sigma = R \quad \rightarrow \quad T = T\_0 \quad \rightarrow \quad \theta\_0 = T\_0 - T\_S \tag{21}
$$

$$\left.r = R\_f \quad \rightarrow \quad \frac{dT}{dr}\right|\_{r=R\_f} = 0 \quad \rightarrow \quad \frac{d\theta}{dr}\Big|\_{r=R\_f} = 0 \tag{22}$$

where

348 Heat Exchangers – Basics Design Applications

<sup>r</sup> ff 2/DR

Fig. 3. Circular fin of uniform thickness.

rd

and constant value

and

To simplify the equation, the new dependent variable is introduced:

*m*

Because the ambient (surrounding) temperature is assumed to be constant, then:

2

2

The general solution of the equation is modified Bessel function of order zero:

*dr r dr* 

() () *<sup>S</sup>*

<sup>2</sup> 2(2 ) 2 (2 ) *fk f <sup>f</sup> hP h r h*

*kA k r k* 

2 2 , *d dT d d T dr dr dr dr*

<sup>1</sup> <sup>0</sup> *d d m*

10 2 0

   

2 2

 

2

*x Tx T* (16)

2/DR

(18)

*C I mr C K mr* () () (20)

(19)

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

From the first condition:

$$\mathcal{O}\_0 = \mathbb{C}\_1 \, I\_0(mR \text{ \textquotedblleft}) + \mathbb{C}\_2 \, K\_0(mR \text{ \textquotedblright}) \tag{23}$$

From the second condition and according to differentiation rules

$$\frac{d}{dr}[\mathbb{C}\_1 \, I\_0(mr)] = \mathbb{C}\_1 \, m \, I\_1(mr) \tag{24}$$

$$\frac{d}{dr}[\mathcal{C}\_2\mathcal{K}\_0(mr)] = -\mathcal{C}\_2\,m\mathcal{K}\_1(mr) \tag{25}$$

we obtain:

$$\frac{d\left.\theta\right|\_{r=R\_f}}{dr}\Big|\_{r=R\_f} = \left(\mathbf{C}\_1 \, m \, I\_1(\boldsymbol{m} \, r) - \mathbf{C}\_2 \, m \, K\_1(\boldsymbol{m} \, r)\right)\Big|\_{r=R\_f} = \mathbf{C}\_1 \, m \, I\_1(\boldsymbol{m} \, R\_f) - \mathbf{C}\_2 \, m \, K\_1(\boldsymbol{m} \, R\_f) = 0 \tag{26}$$

*C*1 and *C*2 can be evaluated to find a temperature distribution:

$$\frac{\partial}{\partial \theta\_0} = \frac{I\_0(mr)K\_1(mR\_f) + K\_0(mr)I\_1(mR\_f)}{I\_0(mR\_f)K\_1(mR\_f) + K\_0(mR\_f)I\_1(mR\_f)}\tag{27}$$

where

0 1 [ ( )] ( ) ( ) *d I mr I mr d mr* and <sup>0</sup> 1 [ ( )] ( ) ( ) *d K mr K mr d mr* are modified, first order Bessel functions of

the first and second kind.

Heat dissipated by the fin and its efficiency can be expressed as:

$$\dot{Q}\_{\parallel} = -k\_f A\_k \frac{dT}{dr}\Big|\_{r=R} = -k\_f (2\pi R \text{ } \delta) \frac{d\theta}{dr}\Big|\_{r=R} \tag{28}$$

Fin-Tube Heat Exchanger Optimization 351

*dr R R*

2 <sup>2</sup> <sup>0</sup> *<sup>d</sup> dr* 

2

*k fr r r R h R R Rr*

*f f f*

( ) 1 11 3 26 *f f*

The heat flux in a parabolic fin is less sensitive to the variation of the tip temperature than in the case of rectangular and trapezoidal fin profiles. This can be seen after resolving the differential equations analytically. Due to the manufacturing problem, the profile described

The analysis and design of heat exchangers consider problems in which the temperature of the fluid changes as it flows through a passage as a result of heat transfer between the wall and the fluid. For heat transfer and pressure drop analyses, at least the following heat transfer surface geometrical properties are needed on each side of a two-fluid exchanger: minimum free-flow area, core frontal area, heat transfer surface area which includes both primary and fin area, hydraulic diameter, and flow length. These quantities are computed from the basic dimensions of the core and heat transfer surface. Due to the complexity of calculations (heat transfer and flow characteristics) it is necessary to find the best possible design solution taking into consideration certain assumptions. In practice, flow maldistribution is common and influences the heat exchanger performance. It can be induced by heat exchanger geometry or heat exchanger operating conditions (e.g., viscosity, density). The objective function is defined within constraints and resolved afterwards.

The optimization of fin–tube heat exchanger is presented focusing on different fluid velocities and the consideration of aerodynamic configuration of the fin. It is reasonable to expect the influence of fin profile on the fluid streamline direction. In the cross-flow heat exchanger, the air streams are not heated and cooled evenly. The fin and tube geometry

To analyze the heat transfer problem, a set of assumptions is introduced so that the resulting theoretical models are simple enough for the analysis. One of the common assumptions in basic heat exchanger design theory is the uniform fluid distribution at the inlet of the exchanger on each fluid side. Firstly, calculations for circular fin–tube heat exchanger are done. To confirm the correctness of the numerical model, the results of heat transfer (outlet temperature) are reviewed and compared with the proper correlation (Hewitt et al., 1994) modified for one row crossflow tube-fin heat exchanger of rectangular profile and fin

affects the flow direction and has the effect on the temperature changes.

*d*

 

the profile function is derived for the radial fin of least material (Kraus et al., 2001):

0 *f*

(34)

(35)

(36)

and resolving above equation with two differential conditions

2

by Equation (36) is not used.

constant thickness.

**3. Heat exchanger optimization** 

where is the fin thickness.

Then

$$\dot{Q}\_{\parallel} = 2\,\pi R\_{\odot} \,\delta \, k\_f \, \theta\_0 \, m \frac{K\_1(mR\_{\odot})I\_1(mR\_f) + I\_1(mR)K\_1(mR\_f)}{K\_0(mR\_{\odot})I\_1(mR\_f) + I\_0(mR\_{\odot})K\_1(mR\_f)}\tag{29}$$

and fin efficiency

$$\eta\_f = \frac{\dot{Q}}{h2\pi \left(\text{R}\_f^{-2} - \text{R}^{-2}\right)\theta\_0} = \frac{2\,\text{R}}{m(\text{R}\_f^{-2} - \text{R}^{-2})} \frac{\text{K}\_1(m\,\text{R}\,\text{ })\,\text{I}\_1(m\,\text{R}\_f) + \text{I}\_1(m\,\text{R}\,\text{ })\text{K}\_1(m\,\text{R}\_f)}{\text{K}\_0(m\,\text{R}\,\text{ })\,\text{I}\_1(m\,\text{R}\_f) + \text{I}\_0(m\,\text{R}\,\text{ })\text{K}\_1(m\,\text{R}\_f)}\tag{30}$$

This result may be applied for an "active" tip (no zero heat flow through the tip of the fin) if the tip radius *Rf* is replaced by the corrected radius of the form \_ <sup>2</sup> *R R f COR f* (Incropera et al., 2006). The fin tip area can be also neglected, taking into consideration the fact that the heat transfer at the fin tip is small. Some authors propose using simpler

#### **2.3 Circular fin thickness optimization**

expressions for hand calculations (Shah & Sekulic, 2003).

The simple radial fin with a rectangular profile is sketched in Figure 3. The fin profile and its optimization issue is often the subject of research. Different authors eliminate some of Murray-Gardner assumptions in their investigations that make the problem more complex. The literature includes a large number of publications dealing with convective heat transfer for different surface geometry, fluid flow type, fluid composition, and thermal boundary conditions but without considering the fluid flow motion.

For the ideal case, if the convection is considered in a fin heat exchanger and the surrounding temperature is equal to *TS* , the temperature difference between any point on the fin surface and the surrounding temperature can be written as:

$$
\theta = T(r) - T\_S \tag{31}
$$

where:

*T r*( ) is the fin surface temperature that varies from the fin base to the fin tip

The optimized profile of the symmetrical radial fin of least material can be found from the generalized differential equation (Kraus et al., 2001):

$$f(r)\frac{d^2\theta}{dr^2} + \frac{f(r)}{r}\frac{d}{dr}\frac{\theta}{r} + \frac{d}{dr}\frac{f(r)}{r}\frac{d}{dr}\frac{\theta}{r} - \frac{h}{k\_f}\theta = 0\tag{32}$$

assuming that the temperature excess changes linearly:

$$\theta = \theta\_0 \left( 1 - \frac{r - R\_f}{R\_f - R} \right) \tag{33}$$

( )( ) ( ) ( ) <sup>2</sup>

01 0 1

0 01 0 1

2( ) ( ) ( )( ) ( ) ( )

*h R R mR R K mR I mR I mR K mR*

This result may be applied for an "active" tip (no zero heat flow through the tip of the fin) if

the tip radius *Rf* is replaced by the corrected radius of the form \_ <sup>2</sup>

(Incropera et al., 2006). The fin tip area can be also neglected, taking into consideration the fact that the heat transfer at the fin tip is small. Some authors propose using simpler

The simple radial fin with a rectangular profile is sketched in Figure 3. The fin profile and its optimization issue is often the subject of research. Different authors eliminate some of Murray-Gardner assumptions in their investigations that make the problem more complex. The literature includes a large number of publications dealing with convective heat transfer for different surface geometry, fluid flow type, fluid composition, and thermal boundary

For the ideal case, if the convection is considered in a fin heat exchanger and the surrounding temperature is equal to *TS* , the temperature difference between any point on

( )

The optimized profile of the symmetrical radial fin of least material can be found from the

() () ( ) <sup>0</sup>

<sup>0</sup> <sup>1</sup> *<sup>f</sup> f r R R R*

*d d dh f r dfr f r dr dr dr dr r k*

*T r*( ) is the fin surface temperature that varies from the fin base to the fin tip

 

*f f f f Q R K mR I mR I mR K mR*

*K mR I mR I mR K mR*

( )( ) ( ) ( )

(29)

*f f*

*f f*

11 1 1

*Tr T <sup>S</sup>* (31)

*f*

  *f f*

(30)

*R R f COR f*

(32)

(33)

2 ( )( ) ( ) ( )

. 11 11

*Q Rk m K mR I mR I mR K mR*

0

*f*

22 22

 

expressions for hand calculations (Shah & Sekulic, 2003).

conditions but without considering the fluid flow motion.

generalized differential equation (Kraus et al., 2001):

assuming that the temperature excess changes linearly:

2 2

the fin surface and the surrounding temperature can be written as:

where

Then

and fin efficiency

*f*

where:

is the fin thickness.

.

**2.3 Circular fin thickness optimization** 

and resolving above equation with two differential conditions

$$\frac{d\theta}{dr} = \frac{-\theta\_0}{R\_f - R} \tag{34}$$

$$\frac{d^2\theta}{dr^2} = 0\tag{35}$$

the profile function is derived for the radial fin of least material (Kraus et al., 2001):

$$\frac{k\_f \cdot f(r)}{\ln R\_f} = \frac{1}{3} \left(\frac{r}{R\_f}\right)^2 - \frac{1}{2} \left(\frac{r}{R\_f}\right) + \frac{1}{6} \left(\frac{R\_f}{r}\right) \tag{36}$$

The heat flux in a parabolic fin is less sensitive to the variation of the tip temperature than in the case of rectangular and trapezoidal fin profiles. This can be seen after resolving the differential equations analytically. Due to the manufacturing problem, the profile described by Equation (36) is not used.

#### **3. Heat exchanger optimization**

The analysis and design of heat exchangers consider problems in which the temperature of the fluid changes as it flows through a passage as a result of heat transfer between the wall and the fluid. For heat transfer and pressure drop analyses, at least the following heat transfer surface geometrical properties are needed on each side of a two-fluid exchanger: minimum free-flow area, core frontal area, heat transfer surface area which includes both primary and fin area, hydraulic diameter, and flow length. These quantities are computed from the basic dimensions of the core and heat transfer surface. Due to the complexity of calculations (heat transfer and flow characteristics) it is necessary to find the best possible design solution taking into consideration certain assumptions. In practice, flow maldistribution is common and influences the heat exchanger performance. It can be induced by heat exchanger geometry or heat exchanger operating conditions (e.g., viscosity, density). The objective function is defined within constraints and resolved afterwards.

The optimization of fin–tube heat exchanger is presented focusing on different fluid velocities and the consideration of aerodynamic configuration of the fin. It is reasonable to expect the influence of fin profile on the fluid streamline direction. In the cross-flow heat exchanger, the air streams are not heated and cooled evenly. The fin and tube geometry affects the flow direction and has the effect on the temperature changes.

To analyze the heat transfer problem, a set of assumptions is introduced so that the resulting theoretical models are simple enough for the analysis. One of the common assumptions in basic heat exchanger design theory is the uniform fluid distribution at the inlet of the exchanger on each fluid side. Firstly, calculations for circular fin–tube heat exchanger are done. To confirm the correctness of the numerical model, the results of heat transfer (outlet temperature) are reviewed and compared with the proper correlation (Hewitt et al., 1994) modified for one row crossflow tube-fin heat exchanger of rectangular profile and fin constant thickness.

Fin-Tube Heat Exchanger Optimization 353

The temperature difference is found numerically and the solid volume is calculated for different fin profile shapes. The air temperature value is also computed numerically in the

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



Fin and tube pitches

2 and changes along the fin height).

at the


outlet section and the average air temperature is evaluated.



The heat exchanger characteristic dimensions are written in Table 1:

p f (fin pitch), mm 3.0

p t (tube pitch), mm 46

D (tube ext diam), mm 25

D t (fin ext diam), mm 41

of Ansys Workbench program (Ansys 12 Product Documentation).

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

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>*

1 and


**3.2 Numerical optimization of fin shape** 

heat, density, heat conduction …)

uniform over the flow cross section.


Table 1. Heat exchanger characteristic dimensions.

fin base (the thickness depends on angle

some assumptions:

#### **3.1 Optimization and objective function**

The optimization process should lead to project the heat exchanger that meets the stated criteria (for instance heat transfer required, minimum weight, heat exchanger efficiency or performance, allowable pressure drop etc).

There are different types of the optimization for radial fin heat exchangers. The optimization can consider (Kraus et al., 2001):


One of the important issues that should be defined during the design work is the optimization of the heat efficiency, taking into consideration the cost of material and the whole heat exchanger.

As an example, the objective function is to maximize the heat transfer ratio for elementary heat exchanger mass (or volume for known material density) in fin-tube heat exchanger. It means that the fin profile is optimized to find the maximum value of function , defined as the ratio between the heat removed from the tube/fin component to the tube/fin weight:

$$\mathcal{L} = \frac{\dot{\mathcal{Q}}}{m\_s} \tag{37}$$

where

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

*ms* – tube and fin mass (solid).

Introducing *<sup>f</sup> c* – fluid (air) specific heat capacity, *mf* – fluid mass flow rate, *TIN* – fluid temperature in the inlet section, *TT* – internal tube surface temperature, *<sup>s</sup>* – material density of solid (tube and fin), *Vs* – volume of tube and fin material, the ratio is equal:

$$\xi = \frac{\stackrel{\cdot}{m}\_f \, c\_f \, \text{(} T\_{IN} - T\_{OUT}\text{)}}{\rho\_s \, V\_s} \tag{38}$$

If the values of *<sup>f</sup> c* , *<sup>s</sup>* do not change during the air flow, then the optimization problem can be resolved by finding the maximum value of the optimization function, :

$$
\omega = \frac{\dot{m}\_f \left( T\_{IN} - T\_{OUT} \right)}{V\_s} = \frac{\dot{m}\_f \Delta T\_{Fluid}}{V\_s} \to \max \tag{39}
$$

where

*T TT Fluid IN OUT* - difference in fluid temperature between outlet and inlet section.

The optimization process should lead to project the heat exchanger that meets the stated criteria (for instance heat transfer required, minimum weight, heat exchanger efficiency or

There are different types of the optimization for radial fin heat exchangers. The optimization


One of the important issues that should be defined during the design work is the optimization of the heat efficiency, taking into consideration the cost of material and the

As an example, the objective function is to maximize the heat transfer ratio for elementary heat exchanger mass (or volume for known material density) in fin-tube heat exchanger. It

.

*s Q m* 

*<sup>f</sup>* ( ) *<sup>f</sup> IN OUT s s*

*V*

( ) max *f f IN OUT Fluid s s*

*mc T T*

*<sup>s</sup>* do not change during the air flow, then the optimization problem can

(39)

the ratio between the heat removed from the tube/fin component to the tube/fin weight:

(37)

(38)

– fluid mass flow rate, *TIN* – fluid

: *<sup>s</sup>* – material

is equal:

, defined as

means that the fin profile is optimized to find the maximum value of function

temperature in the inlet section, *TT* – internal tube surface temperature,

density of solid (tube and fin), *Vs* – volume of tube and fin material, the ratio

. .

*mT T m T V V*

*T TT Fluid IN OUT* - difference in fluid temperature between outlet and inlet section.

be resolved by finding the maximum value of the optimization function,

**3.1 Optimization and objective function** 

performance, allowable pressure drop etc).



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

Introducing *<sup>f</sup> c* – fluid (air) specific heat capacity, *mf*

can consider (Kraus et al., 2001):

whole heat exchanger.

*ms* – tube and fin mass (solid).

If the values of *<sup>f</sup> c* ,

where

where

.

The temperature difference is found numerically and the solid volume is calculated for different fin profile shapes. The air temperature value is also computed numerically in the outlet section and the average air temperature is evaluated.
