Given:
Diameter of cylindrical pin fin, d = 0.6 cm
Length of cylindrical pin fin, L = 3 cm
Heat loss from tip, Q = 0
Efficiency of the fin, η = 0.7

To find:
Effectiveness of the fin

Solution:

1. Formula for efficiency of a fin:

η = (q / A) * (1 / (T_b - T_a))

where,
q = heat transfer rate from fin
A = surface area of fin
T_b = base temperature (temperature at the base of the fin)
T_a = ambient temperature (temperature of the surrounding medium)

2. Formula for effectiveness of a fin:

ε = (T_b - T_a) / (T_b - T_a)_max

where,
(T_b - T_a)_max = (q / h_p * A_c)

h_p = heat transfer coefficient at the fin surface
A_c = cross-sectional area of the fin

3. Expressions for surface area and cross-sectional area of the cylindrical pin fin:

Surface area, A = π * d * L

Cross-sectional area, A_c = π * d^2 / 4

4. Heat transfer coefficient at the fin surface:

h_p = k * (Nu - 1) * (d / L)

where,
k = thermal conductivity of the fin material
Nu = Nusselt number

5. Expression for Nusselt number for a cylindrical pin fin:

Nu = (h_p * d) / k = (2 + (d / L) * (Nu_D - 2)) * (k * L / d)

where,
Nu_D = Nusselt number for a cylinder of diameter d in cross-flow

6. Nusselt number for a cylinder of diameter d in cross-flow:

Nu_D = 0.3 * (Re_D)^0.5 * (Pr / (1 + (0.4 / Pr)^(2/3))^0.25) * (1 + (d / L)^(2/3))^0.5

where,
Re_D = Reynolds number based on diameter, d
Pr = Prandtl number

7. Expressions for Reynolds number and Prandtl number:

Re_D = (ρ * u * d) / μ

Pr = μ * c_p / k

where,
ρ = density of the fluid
u = velocity of the fluid
μ = dynamic viscosity of the fluid
c_p = specific heat of the fluid at constant pressure
k = thermal conductivity of the fluid

8. For air at 25°C, properties are:

ρ = 1.18 kg/m^3
μ = 1.84 × 10^-5 Pa·s
c_p = 1.005 kJ/kg·K
k = 0.0263 W/m·K
Pr = 0.71

9. Calculation:

Using the above expressions and properties, we can calculate the efficiency, η, of the fin:

η = 0.7 = (q / A) * (1 / (T_b - T_a))

q / A = 0.7 * (T_b - T_a)

q = (0.7 * (T_b