A 0.2 m \times 0.2 m vertical plate has a surface temperature that is maintained at 40^{\circ} C . Air at 20^{\circ} C is flowing in parallel over the plate with a velocity of 0.4 m / s. Determine the Nusselt number for both assisting flow and opposing flow (Fig. 9-38).
Question : A 0.2 m × 0.2 m vertical plate has a surface temperature tha...
SOLUTION Air is flowing over a vertical plate. Nusselt numbers for both assisting and opposing flows are to be determined.
Assumptions 1 Steady operating conditions exist. 2 Properties are constant. 3 The surface temperature is constant. 4 Air is an ideal gas. 5 Heat transfer by radiation is negligible.
Properties The properties of air (1 atm ) at 30^{\circ} C \text { are }
k=0.02588 W / m \cdot Kv=1.608 \times 10^{-5} m ^{2} / s
and \operatorname{Pr}=0.7282 Table (A-15).
Also, \beta=1 / T_{f}= 0.0033 K ^{-1}
Analysis The Reynolds and Grashof numbers are
\operatorname{Re}=\frac{V L}{\nu}=\frac{(0.4 m / s )(0.2 m )}{1.608 \times 10^{-5} m ^{2} / s }=4975Gr _{L}=\frac{g \beta\left(T_{s}-T_{\infty}\right) L^{3}}{v^{2}}=\frac{\left(9.81 m / s ^{2}\right)\left(0.0033 K ^{-1}\right)(40-20) K (0.2 m )^{3}}{\left(1.608 \times 10^{-5}\right)^{2} m ^{4} / s ^{2}}
=2.003 \times 10^{7}
Hence,
\frac{ Gr _{L}}{ Re ^{2}}=\frac{2.003 \times 10^{7}}{(4975)^{2}}=0.809Noting that Gr _{L} / Re ^{2} \approx 1, both natural convection and forced convection are significant, and we have mixed flow. The Nusselt numbers for the natural and forced convection cases are determined from relevant relations to be
Nu _{\text {natural }}=0.59 Ra _{L}^{1 / 4}=0.59\left(2.003 \times 10^{7} \times 0.7282\right)^{1 / 4}=36.46Nu _{\text {forced }}=0.664 Re ^{0.5} Pr ^{13}=0.664(4975)^{0.5}(0.7282)^{1 / 3}=42.14
Finally, the combined Nusselt numbers for the cases of assisting flow (flowing upward) and opposing flow (flowing downward) become
\text { Assistingflow: } Nu _{\text {combined }}=\left( Nu _{\text {forced }}^{3}+ Nu _{\text {natural }}^{3}\right)^{1 / 3}=\left(42.14^{3}+36.46^{3}\right)^{1 / 3}=49.8
\text { Opposingflow: } Nu _{\text {combined }}=\left( Nu _{\text {forced }}^{3}- Nu _{\text {natural }}^{3}\right)^{1 / 3}=\left(42.14^{3}-36.46^{3}\right)^{1 / 3}=29.8
Discussion Note that the Nusselt number for the assisting flow is about 67 percent higher than that for the opposing flow. Therefore, natural convection must be taken into consideration when it is significant.
