(a) The thermal circuit diagram is given in Figure (b). The rate of surfaceconvection heat transfer from the bare and finned areas of the base is given by Q_{ku}=\left\langle Q_{ku}\right\rangle _{s-\infty }=(A_{b}+A_{f}\eta _{f})\left\langle Nu\right\rangle _{w}\frac{k_{f}}{w}(T_{s,0}-T_{f,\infty }) . Here we use the circular fins and replace \left\langle Nu\right\rangle _{w}k_{f} /w with \left\langle Nu\right\rangle _{D}k_{f} /D,
i.e.,
−Q_{s} = Q_{ku} = [(A − N_{f} A_{k}) + N_{f} A_{ku,f} η_{f} ]\left\langle Nu\right\rangle _{D}\frac{k_{f}}{D} (T_{s,0}-T_{f,\infty })
(b) To determine the Nusselt number \left\langle Nu\right\rangle _{D}, the correlation we use (an approximation) is from Table, i.e.,
\left\langle Nu\right\rangle _{D}=a_{1}Re_{D}^{a_{2}}Pr^{1/3} Table
where, as defined in \left\langle Q_{ku}\right\rangle _{L( or D)}=\frac{T_{s}-T_{f,\infty }}{\left\langle R_{ku}\right\rangle_{L(or D)} } =A_{ku}\frac{k_{f}}{L(or D)}\left\langle Nu\right\rangle _{L(or D)}(T_{s}-T_{f,\infty }), Re_{L(or D)}=\frac{u_{f,\infty }L(or D)}{v_{f}} ,
Re_{D}=\frac{u_{f,\infty }}{v_{f}}
The thermophysical properties for air are obtained from Table at T = 300 K, i.e.,
ν_{f} = 15.66 × 10^{−6} m^{2}/s Table
k_{f} = 0.0267 W/m-K Table
Pr = 0.69 Table
Also from Table , we have for aluminum (pure) at T = 300 K
k_{s} = 237 W/m-K Table
Then
Re_{D}=\frac{1(m/s) × 7 × 10^{-3}(m)}{15.66 × 10^{−6} (m^{2}/s)} =447.0
From Table
a_{1} = 0.683 , a_{2} = 0.466 Table
\left\langle Nu\right\rangle _{D} = 0.683(447.0)^{0.466}(0.69)^{1/3} = 10.37.
(c) The fin efficiency ηf is given by \eta _{f}=\frac{tanh(mL_{c})}{mL_{c}} ,0\leq \eta _{f}\leq 1 fin efficiency
=\frac{tanh[(R_{k,s}/\left\langle R_{ku}\right\rangle_{w} )^{1/2}]}{(R_{k,s}/\left\langle R_{ku}\right\rangle_{w} )^{1/2}} =\frac{tanh(Bi_{w}^{1/2})}{Bi_{w}^{1/2}} as
\eta _{f}=\frac{tanh(mL_{c})}{mL_{c}}
where L_{c} and m are given by L_{c}=L+\frac{D}{4} and m=\left(\frac{P_{ku,f}\left\langle Nu\right\rangle _{w}k_{f}}{A_{k}k_{s}w} \right) ^{1/2}=\left(\frac{P_{ku,f}\left\langle Nu\right\rangle _{w}\frac{k_{f}}{w}\frac{L_{c}^{2}}{L_{c}^{2}} }{A_{k}k_{s}}\right) ^{1/2}=\left(\frac{P_{ku,f}L_{c}}{A_{k}}\frac{\left\langle Nu\right\rangle _{w}\frac{k_{f}}{w}}{k_{s}/L_{c}} \right)^{1/2} \frac{1}{L_{c}} =\left(\frac{R_{k,s}}{\left\langle R_{ku}\right\rangle_{w} } \right) ^{1/2}\frac{1}{L_{c}} \equiv Bi_{w}^{1/2}\frac{1}{L_{c}} , i.e.,
L_{c}=L+\frac{D}{4}
m=\left(\frac{P_{ku,f}\left\langle Nu\right\rangle _{D}k_{f}}{A_{k}k_{s}D} \right) ^{1/2}
where P_{ku,f} = πD, and A_{k} = πD^{2}/4. Then
m=\left[\frac{\pi D\left\langle Nu\right\rangle_{D}k_{f} }{(\pi D^{2}/4)k_{s}D} \right] ^{1/2}
or
m=\left(\frac{4 \left\langle Nu\right\rangle_{D}k_{f} }{D^{2}k_{s}} \right) ^{1/2}=\left(\frac{4 \left\langle Nu\right\rangle_{D}}{D^{2}}\frac{k_{f}}{k_{s}} \right) ^{1/2}
=\left[\frac{4 × 10.37}{(7 × 10^{−3} )^{2} (m^{2} )} \frac{0.0267(W/m-K)}{237(W/m-K)} \right] ^{1/2}= 9.766 1/m.
Then
mL_{c} = 9.766(1/m) × [7 × 10^{−2} (m) + (7 × 10^{−3} /4)(m)] = 0.7006.
From Table, tanh(0.7006) = 0.6044, and for \eta _{f} we have
\eta _{f}=\frac{tanh(ml_{c})}{mL_{c}} =\frac{0.6044}{0.7006} = 0.8627 fin efficiency.
(d) The relation for Q_{ku} was listed earlier, and we have
A_{ku,f} = P_{ku,f} L_{c} = πD(L + D/4)
Q_{ku} = [(a × a − N_{f} πD^{2} /4) + N_{f} πD(L + D/4) × \eta _{f} ]\left\langle Nu\right\rangle _{D}\frac{k_{f}}{D} (T_{s,0} − T_{f,∞})
Q_{ku} = [(0.1)^{2} (m)^{2} − 10^{2} × π × (7 × 10^{−3})^{2} (m)^{2} /4 + 10^{2} × π × 7 × 10^{−3} (m)
× (7 × 10^{−2} + 7 × 10^{−3} /4)(m) × 0.8627] × 10.37
× \frac{0.0267(W/m-K)}{7 × 10^{−3}(m)} × (80 − 20)(^{\circ }C)
= [0.01(m)^{2} − 3.847 × 10^{−3} (m)^{2} + 0.1361(m)^{2} ] × 39.55(W/m-K) × 60(^{\circ } C)
= 345.2 W.
