Answer:
Given that
D= 4 mm
K = 160 W/m-K
h=h = 220 W/m²-K
ηf = 0.65
We know that
[tex]m=\sqrt{\dfrac{hP}{KA}}[/tex]
For circular fin
[tex]m=\sqrt{\dfrac{4h}{KD}}[/tex]
[tex]m=\sqrt{\dfrac{4\times 220}{160\times 0.004}}[/tex]
m = 37.08
[tex]\eta_f=\dfrac{tanhmL}{mL}[/tex]
[tex]0.65=\dfrac{tanh37.08L}{37.08L}[/tex]
By solving above equation we get
L= 36.18 mm
The effectiveness for circular fin given as
[tex]\varepsilon =\dfrac{2\ tanhmL}{\sqrt{\dfrac{hD}{K}}}[/tex]
[tex]\varepsilon =\dfrac{2\ tanh(37.08\times 0.03618)}{\sqrt{\dfrac{220\times 0.004}{160}}}[/tex]
ε = 23.52
This question involves the concepts of the efficiency and the effectiveness of the circular fin.
(a) Fin length is found to be "35.2 mm".
(b) Fin effectiveness is found to be "23.52".
(a)
For circular fin:
[tex]m=\sqrt{\frac{4h}{KD}}[/tex]
where,
m = effective mass = ?
h = convective coefficient = 220 W/m².K
K = Thermal Conductivity = 160 W/m.K
D = diameter = 4 mm = 0.004 m
Therefore,
[tex]m=\sqrt{\frac{(4)(220)}{(160)(0.004)}}\\\\m =37.1[/tex]
Now, we will use the formula for efficiency of the circular fin to find out the corrected length of the fin ([tex]L_c[/tex]):
[tex]efficiency=\frac{tanh\ mL_c}{mL_c}\\\\0.65 = \frac{tanh\ (37.1)L_c}{(37.1)L_c}[/tex]
using the iterative method to solve this equation, we get the following answer:
Lc = 0.0362 m = 36.2 mm
Now, the actual length of the fin (L) can be found using the following formula:
[tex]L=L_c-\frac{D}{4}\\\\L=36.2\ mm-\frac{4\ mm}{4}\\\\[/tex]
L = 35.2 mm
(b)
Now, the formula for the fin effectiveness of the circular fin is given as follows:
[tex]effectiveness = \frac{2tanh\ mL_c}{\sqrt{\frac{hD}{K}}}\\\\effectiveness = \frac{2tanh\ (37.1)(0.0362)}{\sqrt{\frac{(220)(0.004)}{160}}}\\\\[/tex]
Effectiveness = 23.52
Learn more about fin effectiveness here:
https://brainly.com/question/15841378?referrer=searchResults
The attached picture shows the different configurations of fins and their formulae.
