In many engineering situations, means are often sought to improve heat dissipation from a surface to its surroundings. The Newton-Rikhman relation Q = hA (t – ta) reveals that the convective heat flow can be enhanced by increasing the film coefficient h, the surface area A and the temperature difference (f – ta).
The convective coefficient is a function of the geometry, fluid properties and the flow rate. Control of h through these parameters helps to obtain its optimum value. With regard to the effect of temperature excess (t – ta), difficulties are encountered when the ambient temperature ta is too high particularly in hot weather conditions.
The surface area exposed to the surroundings is frequently increased by the attachment of protrusions to the surfaces, and the arrangement provides a means by which heat transfer rate can be substantially improved. The protrusions are called fins or spines, and these extensions can take a variety of forms.
A straight fin is an extended surface attached to a plane wall- the cross-sectional area of the fin may be uniform or it may vary with distance from the wall. Annular fins are attached circumferentially to a cylindrical surface and their cross-sectional area varies with radius from the centre line of the cylinder.
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However, both the straight and annular fins are of rectangular cross-section, whose area can be expressed as a product of the fin thickness δ and width b for straight fins or the circumference 2πr for annular fins. In contrast, a pin fin or a spine is an extended surface of circular cross-section which may be uniform or non-uniform. Thus a spine represents a thin cylindrical or conical rod protruding from a wall.
Common applications of finned surfaces are:
i. Economisers for steam Dower plants.
ii. Convectors for steam and hot water heating of systems.
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iii. Air cooled cylinders of aircraft engines, I.C. engines and air compressors.
iv. Electrical transformers and motors.
v. Cooling coils and condenser coils in refrigerators and air conditioners.
vi. Electronic equipments.
Steady Flow of Heat along a Rod:
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Consider a straight rectangular fin or a pin fin (spine) protruding from a wall surface. The characteristic dimensions of the fin are its length I, constant cross-sectional area Ac and the circumferential parameter P. Thus for a rectangular fin-
Ac = b δ; P = 2 (b + δ)
and for the spine-
Ac = π/4 d2; p = πd
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The temperature at the base of the fin is t0 and the temperature of the ambient fluid into which the rod extends is considered to be constant at temperature ta. The base temperature t0 is highest and the temperature along the fin length goes on diminishing.
Analysis of heat flow form the finned surface is made with the following assumptions:
i. Thickness of the fin is small compared with the length and width; temperature gradients over the cross-section are neglected and heat conduction treated one dimensional
ii. Homogeneous and isotropic fin material; the thermal conductivity k of the fin material is constant
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iii. Uniform heat transfer coefficient h over the entire fin surface
iv. No heat generation within the fin itself
v. Joint between the fin and the heated wall offers no bond resistance; temperature at root or base of the fin is uniform and equal to temperature t0 of the wall
vi. Negligible radiation exchange with the surroundings; radiation effects, if any, are considered as included in the convection coefficient h
vii. Steady state heat dissipation
Heat from the heated wall is conducted through the fin and convected from the sides of the fin to the surroundings. Let attention be focused on an infinitesimal element of the fin; the element has thickness δx and is located at a distance x from base wall.
(i) Heat conducted into the element at plane x-
(ii) Heat conducted out of the element at plane (x + δx)-
(iii) Heat convected out of the element between the planes x and (x + δx)-
Here temperature t of the fin has been presumed to be uniform and non-variant for the infinitesimal element.
A heat balance on the element gives:
Equation 5.4 is further simplified by transforming the dependent variable by defining the temperature excess ϴ as,
ϴ(x) = t(x) – ta
Since the ambient temperature ta is constant, we get by differentiation
Equations 5.4 and 5.5 provide a general form of the energy equation for one-dimensional heat dissipation from an extended surface. For a given fin, the parameter m is constant provided the convective film coefficient h is constant over the entire surface and the thermal conductivity k is constant within the considered temperature range. Then the general solution of this linear, homogeneous second order differential equation is of the form
The constant q and C2 are to be determined with the aid of relevant boundary conditions.
Example:
Three rods, one made of silver (k = 420 W/m-deg), second made of aluminium (k = 210 W/m-deg) and the third made of wrought iron (k = 70 W/m-deg) are coated with a uniform layer of wax all around. The rods are placed vertically in a boiling water bath with 250 mm length of each rod projecting outside. If all rods are 15 mm diameter, 300 mm length and have identical surface coefficient 12.5 W/m2-deg, work out the ratio of lengths upto which wax will melt on each rod.
Solution:
Let l1, l2, and l3 be the lengths upto which wax will melt on each rod. Then
Heat Dissipation from an Infinitely Long Fin:
The relevant boundary conditions are:
(i) Temperature at the base of fin equals the temperature of the surface to which the fin is attached.
t = t0 at x = 0
In terms of excess temperature-
t – ta = t0 – ta
Or ϴ = ϴ0 at x = 0
(ii) Temperature at the end of an infinitely long fin equals that of the surroundings.
t = ta at x = ∞
ϴ = 0 at x = ∞
Substitution of these boundary conditions in equation 5.6 is given:
ϴ = C1 emx + C2 e-mx …….(5.6)
We get,
C1 + C2 = ϴ0 … (a)
C1 em¥ + C2 e-m(∞) = 0 … (b)
Since the term C2 e-m(∞) is zero, the equality is valid only if C1 = 0. Then it follows from relation (a) that C2 = ϴ0. Substituting these values of constants C1 and C2 in equation 5.6, one obtains the following expression for temperature distribution along the length of the fin-
ϴ = ϴ0 e-mx; (t – ta) = (t0 – ta) e-mx … (5.7)
Fig. 5.4 shows the dependence of dimensionless temperature (t – ta)/(t0 – ta) along the length of fin for different values of parameter m (m1 < m2 < w3). The plot indicates that the dimensionless temperature falls more with increase in factor m. With the fin length extending to infinity, x → ∞ all the curves approach (t – ta)/(t0 – ta) = 0 asymptotically.
The most important design variable for a fin is the amount of heat that it can remove from the heated wall and dissipate it to the surroundings.
An estimate of the heat flow rate can be made by writing the Fourier rate equation corresponding to root section of the fin-
Alternatively the heat flow rate can be worked out by integrating the expression for convective heat transport from the infinitesimal element of the fin surface to the surroundings.
which is the same as evaluated above (Equation 5.8)
Equations 5.7 and 5.8 are reasonable approximations of temperature distribution and heat flow rate in a finite fin if its length is very large compared to its cross-sectional area.
The temperature distribution (Equation 5.7) would suggest that the temperature drops towards the tip of the fin. Hence area near the fin tip is not utilized to the extent as the lateral area near the base. Obviously an increase in the fin length beyond a certain point does not pay much regarding an increase in the heat dissipation. A tapered fin is then considered to be a better design as it has more lateral area near the base where the difference in temperature is high.
Heat flow rates through solids can be compared by having an arrangement consisting essentially of a box to which rods of different materials are attached (Ingen-Hausz experiment). The rods are of same length and area of cross-section (same size and shape); their outer surfaces are electroplated with the same material and are equally polished. This is to ensure that for each rod, the surface heat transfer will be same. The procedure would involve coating the rods with wax and filling the box with be same. Heat flow from the box along the rod would melt the wax for a distance which would depend upon the rod material.
Let-
ϴO = excess of temperature of the hot bath above the ambient temperature
ϴm = excess of temperature of melting point of wax above the ambient temperature
l1, l2, l3 … = lengths upto which wax melts.
Then for different rods (treating each as fin of infinite length),
Thus, the thermal conductivity of the material of the rod is directly proportional to the square of the length upto which the wax melts on the rod.
Heat Dissipation from a Fin Insulated at the Tip:
The fin is of any finite length with the end insulated and so no heat is transferred from the tip.
Therefore, the relevant boundary conditions are:
Substituting these values of constant C1 and C2 in equation 5.6, i.e –
ϴ = C1 emx + C2 e-mx …….(5.6)
One obtains the following expression for temperature distribution along the length of fin-
Heat Dissipation from a Fin Losing Heat at the Tip:
The fin is of any finite length with the tip exposed for heat dissipation.
The relevant boundary conditions are:
(i) ϴ = ϴ0 at x = 0
(ii) The fin is losing heat at the tip, i.e., the heat conducted to the fin at x = l equals the heat convected from the end to the surroundings-
At the tip of fin, the cross-sectional area for heat conduction Ac equals the surface area Ag from which the convective heat transport occurs. Thus-
Substituting these values of constants C1 and C2 in equation. 5.6, one obtains the following expression for temperature distribution along the length of the fin.
Utility of a Fin in Dissipating Heat:
The utility of a fin in dissipating a given quantity of heat is generally assessed on the basis of the following parameters:
1. Efficiency of fin relates the performance of an actual fin to that of an ideal or fully effective fin. A fin will be most effective, i.e., it would dissipate heat at maximum rate if the entire fin surface area is maintained at the base temperature.
ηf = actual heat transfer rate from the fin/heat that would be dissipated if the whole surface of the fin were maintained at the base temperature
Thus for a fin insulated at tip-
The parameter PI represents the total surface area exposed for convective heat flow. Upon simplification,
An estimate of the fin efficiency can thus be made by substituting the value of m in the above relation. An insight into this expression shows that,
(i) For a very long fin-
tanh ml/ml → 1/large number
Obviously the fin efficiency drops with an increase in its length.
(ii) For small values of ml, the fin efficiency increases. When the length is reduced to zero, then,
tanh ml/ml → ml/ml = 1
Thus the fin efficiency reaches its maximum value of 100% for a trivial value of l = 0, i.e., no fin at all. Naturally maximization of fin performance with respect to its length does not constitute the design criterion for a fin. The efficiency of fin, however, forms a criterion for judging the relative merits of fins of different geometries or materials.
2. Effectiveness of fin (ϵf) represents the ratio of the fin heat transfer rate (heat dissipation with a fin) to the heat transfer rate that would exist without a fin.
Fig. 5.8 shows the base heat transfer surface before and after the fin has been attached.
The heat transfer through the root area Ac before fin attachment is:
Qfin = h Ac (t0 – ta)
After the attachment of an infinitely long fin, the heat transfer rate through the root area becomes:
In case of a straight rectangular fin of thickness δ and width b,
