# Heat conduction in extended surface

Figure 1: Fin configurations: (a) straight fin of uniform cross-section on plane wall, (b) straight fin of uniform cross-section on circular tube, (c) annular fin, and (d) straight pin fin

The total thermal resistance includes two convective thermal resistances and one conduction thermal resistance. For the cases that one of the convection thermal resistances is dominant (i.e., significantly greater than the conduction thermal resistance and the other convective thermal resistance), one can increase the heat transfer coefficient (hi or ho) or the heat transfer area (A1 or A2) to enhance the overall heat transfer. Since increasing the heat transfer coefficient is constrained by the type of working fluid and the power required driving the flow, increasing the heat transfer area becomes a natural choice. The increase of surface area can be done by using fins that extend into the fluid. Figure 1 shows some examples of fin configurations. It can be seen that the fins can have either uniform or variable cross-sectional areas. For cases where the fluid is cooling the fin, the fin temperature is the highest at the base (x = 0) and decreases with increasing x as convection takes place on the fin surface. The degree of heat transfer enhancement can be maximized by minimizing the temperature variation, which can be achieved by using fin materials with large thermal conductivity. The solution of heat transfer from extended surfaces (fins) will provide (a) temperature distribution in the fin, and (b) total heat transfer from the finned surface.

Figure 2: Heat conduction in a fin with variable cross-section

Figure 2 shows the generalized physical model for heat transfer form an extended surface. It is a steady-state conduction problem in variable cross-sectional area. It is different from the preceding subsection in that convection occurs on the extended surface. The following assumptions are made to simplify the problem:

2. The thermal conductivity of the fin is independent from the temperature,

3. There is no internal heat generation in the fin,

4. Both convective heat transfer coefficient and fluid temperature are constants,

5. The heat transfer is one-dimensional, i.e., the temperature is uniform in the same cross-sectional area.

Heat transfer in a fin can be modeled as a steady-state one-dimensional conduction with internal heat generation, described by

$\frac{1}{{A(s)}}\frac{d}{{ds}}\left( {A(s)\frac{{dT}}{{ds}}} \right) + \frac{{q'''}}{k} = 0,{\rm{ }}{s_1} < s < {s_2}$

if the convection from the extended surface is treated as an equivalent internal heat source. For the differential control volume (dV = Adx) shown in Fig. 2, the convective heat transfer from the side is

$d{q_{conv}} = hPdx\left[ {{T_\infty } - T(x)} \right]$

where P is the perimeter of the fin. The equivalent intensity of the internal heat source in the control volume due to the convection on the extended surface is

$q''' = \frac{{d{q_{conv}}}}{{dV}} = \frac{{hP}}{A}\left[ {{T_\infty } - T(x)} \right]$

Therefore, the energy equation becomes

$\frac{1}{A}\frac{d}{{dx}}\left( {A\frac{{dT}}{{dx}}} \right) - \frac{{hP}}{{Ak}}(T - {T_\infty }) = 0 \qquad \qquad(1)$

or

$\frac{{{d^2}T}}{{d{x^2}}} + \left( {\frac{1}{A}\frac{{dA}}{{dx}}} \right)\frac{{dT}}{{dx}} - \frac{{hP}}{{Ak}}(T - {T_\infty }) = 0 \qquad \qquad(2)$

which requires two boundary conditions. At the base of the fin, the temperature is

$T = {T_0},{\rm{ }}x = 0 \qquad \qquad(3)$

There are commonly three kinds of different boundary conditions at the tip of the fin:

1. When the fin is sufficiently long, the temperature at the tip can be assumed to be equal to the fluid temperature,

$T = {T_\infty },{\rm{ }}x = L \qquad \qquad(4)$

2. It is assumed that the heat transfer at the tip of the fin is negligible,

$\frac{{dT}}{{dx}} = 0,{\rm{ }}x = L \qquad \qquad(5)$

3. When the fin is not long enough, the tip temperature is higher than the fluid temperature and convection occurs at the fin tip,

$- k\frac{{dT}}{{dx}} = h(T - {T_\infty }),{\rm{ }}x = L \qquad \qquad(6)$

Heat transfer from an extended surface is a non-homogeneous problem because eq. (1) or (2) is not homogonous. If convection at the fin tip is considered, eq. (6) is also non-homogeneous. Defining excess temperature $\vartheta (x) = T(x) - {T_\infty }$

Equations (2) – (6) become

$\frac{{{d^2}\vartheta }}{{d{x^2}}} + \left( {\frac{1}{A}\frac{{dA}}{{dx}}} \right)\frac{{d\vartheta }}{{dx}} - \frac{{hP}}{{Ak}}\vartheta = 0 \qquad \qquad(7)$
$\vartheta = {\vartheta _0},{\rm{ }}x = 0 \qquad \qquad(8)$
$\vartheta = 0,{\rm{ }}x = L \qquad \qquad(9)$
$\frac{{d\vartheta }}{{dx}} = 0,{\rm{ }}x = L \qquad \qquad(10)$
$- k\frac{{d\vartheta }}{{dx}} = h\vartheta ,{\rm{ }}x = L \qquad \qquad(11)$

which are all homogeneous. The total amount of heat transfer from a fin can be obtained by using Fourier’s law at the base of the fin

${q_f} = - k{\left( {A\frac{{d\vartheta }}{{dx}}} \right)_{x = 0}} \qquad \qquad(12)$

or accounting for convective heat transfer throughout the fin surface by

${q_f} = h\int_0^L {P\vartheta dx} \qquad \qquad(13)$

Under steady state condition, the heat conduction from the base of the fin obtained from eq. (12) is identical to the total convective heat transfer obtained from eq. (13), however, eq. (12) is much easier to apply than eq. (13). The performance of a fin is usually measured by the fin efficiency, ηf, which is defined as the ratio of the actual heat transfer rate of a fin to the heat which would be transferred if the entire fin area is at base temperature, i.e.,

${\eta _f} = \frac{{{q_f}}}{{{q_{f,\max }}}} = \frac{{{q_f}}}{{h{\vartheta _0}\int_0^L {Pdx} }} \qquad \qquad(14)$

While the general solution of the heat conduction in a fin with arbitrarily variable cross-sectional area cannot be obtained, analytical solutions for some special cases are possible. For fins with uniform cross-sections, the second term in eq. (7) is dropped and the energy equation becomes

$\frac{{{d^2}\vartheta }}{{d{x^2}}} - {m^2}\vartheta = 0 \qquad \qquad(15)$

where

${m^2} = \frac{{hP}}{{Ak}} \qquad \qquad(16)$

The general solution of eq. (15) is

$\vartheta = {C_1}{e^{mx}} + {C_2}{e^{ - mx}} \qquad \qquad(17)$

where C1 and C2 are integral constants that need to be determined by boundary conditions at the base and the tip of the fin. The temperature distributions, heat transfer rate, and fin efficiency for the three boundary conditions represented by eqs. (4) – (6) are:

1. For the case that the fin’s top temperature equals the fluid temperature:

$\frac{\vartheta }{{{\vartheta _0}}} = {e^{ - mx}} \qquad \qquad(18)$
${q_f} = \sqrt {hPkA} {\vartheta _0} \qquad \qquad(19)$
${\eta _f} = \frac{1}{{mL}} \qquad \qquad(20)$

2. For the case that the fin tip is adiabatic

$\frac{\vartheta }{{{\vartheta _0}}} = \frac{{\cosh [m(L - x)]}}{{\cosh (mL)}} \qquad \qquad(21)$
${q_f} = \sqrt {hPkA} {\vartheta _0}\tanh (mL) \qquad \qquad(22)$
${\eta _f} = \frac{{\tanh (mL)}}{{mL}} \qquad \qquad(23)$

3. For the case of a convective fin tip

$\frac{\vartheta }{{{\vartheta _0}}} = \frac{{\cosh [m(L - x)] + (h/mk)\sinh [m(L - x)]}}{{\cosh (mL) + (h/mk)\sinh (mL)}} \qquad \qquad(24)$
${q_f} = \sqrt {hPkA} {\vartheta _0}\frac{{\sinh (mL) + (h/mk)\cosh (mL)}}{{\cosh (mL) + (h/mk)\sinh (mL)}} \qquad \qquad(25)$
${\eta _f} = \frac{1}{{mL}}\frac{{\sinh (mL) + (h/mk)\cosh (mL)}}{{\cosh (mL) + (h/mk)\sinh (mL)}} \qquad \qquad(26)$

Among the three possible boundary conditions, the first case is valid only if the fin is sufficiently long. For this reason it cannot be applied in most cases. While the convective fin tip is most accurate for many applications, its results are very complicated. Practically, one can use the adiabatic fin tip for the case with convection on the fin tip by using a corrected fin length, Lc, to replace L in eqs. (21) – (23). The corrected length is Lc = L + t / 2 for a rectangular fin [see Fig. 1 (a) and (b), t is fin thickness], and Lc = L + D / 4 for a pin fin [see Fig. 1(d), D is pin fin diameter]. This treatment is based on the assumption that the amount of heat transfer from the fin tip can be considered by an equivalent amount of heat transfer from the increased fin surface area. The above solution is valid for the case that the cross-sectional area is uniform. If the cross-sectional area is not uniform [e.g., Fig. 1(c)], the second term in eq. (7) is retained and the solution becomes more complicated. Figure 3 shows the physical model of heat transfer in an annular fin with uniform thickness. The perimeter and the cross-sectional area of the annular fin are

P = 2πr
A = 2πrt
Figure 3: Heat transfer in an annular fin with uniform thickness

The energy equation (7) becomes

$\frac{{{d^2}\vartheta }}{{d{r^2}}} - \frac{1}{r}\frac{{d\vartheta }}{{dr}} - {m^2}\vartheta = 0,{\rm{ }}{r_1} < r < {r_2} \qquad \qquad(27)$

where

$m = \frac{{2h}}{{kt}} \qquad \qquad(28)$

The boundary conditions for eq. (27) are

$\theta = {\vartheta _0},{\rm{ }}r = {r_1} \qquad \qquad(29)$
$\frac{{d\vartheta }}{{dr}} = 0,{\rm{ }}r = {r_2} \qquad \qquad(30)$

Equation (68) is a modified Bessel’s equation of zero order and has the following general solution

$\vartheta (r) = {C_1}{I_0}(mr) + {C_2}{K_0}(mr) \qquad \qquad(31)$

where I0 and K0 are zero order modified Bessel function of the first and second kinds, respectively (See Appendix G). Substituting eq. (31) into eqs. (29) and (30), the two integral constants in eq. (31) can be determined and the temperature distribution in the annular fin becomes

$\theta (r) = \frac{{{K_1}(m{r_2}){I_0}(mr) + {I_1}(m{r_2}){K_0}(mr)}}{{{K_1}(m{r_2}){I_0}(m{r_1}) + {I_1}(m{r_2}){K_0}(m{r_1})}} \qquad \qquad(32)$

The heat transfer rate can be determined using Fourier’s law at the base (r = r1), i.e.,

${q_f} = - k2\pi {r_1}t{\left. {\frac{{d\vartheta }}{{dx}}} \right|_{r = {r_1}}} \qquad \qquad(33)$

Substituting eq. (32) into eq. (33) yields

${q_f} = k2\pi {r_1}t{\vartheta _0}m\frac{{{I_1}(m{r_2}){K_1}(m{r_1}) - {K_1}(m{r_2}){I_1}(m{r_1})}}{{{I_1}(m{r_2}){K_0}(m{r_1}) + {K_1}(m{r_2}){I_0}(m{r_1})}} \qquad \qquad(34)$

If the entire fin surface is at the base temperature, the heat transfer amount is

${q_{f,\max }} = 2\pi (r_2^2 - r_1^2)h{\vartheta _0} \qquad \qquad(35)$

The fin efficiency for a straight annular fin is

${\eta _f} = \frac{{{q_f}}}{{{q_{f,\max }}}} = \frac{{2{r_1}}}{{m(r_2^2 - r_1^2)}}\frac{{{I_1}(m{r_2}){K_1}(m{r_1}) - {K_1}(m{r_2}){I_1}(m{r_1})}}{{{I_1}(m{r_2}){K_0}(m{r_1}) + {K_1}(m{r_2}){I_0}(m{r_1})}} \qquad \qquad(36)$

The fin efficiencies for other configurations can be found in Incropera et al. (2006).