## Reynolds Number Drag

Consider a road sign is placed in parallel with the direction of wind flowing 50 ft/s. If the sign’s Reynolds Number (Re) is 2 x 10^4 and its drag force is 0.1 lbf, what is the sign’s projected area in ft^2? Assume the density of air is 0.08 lb/ft^3.

Hint
The drag force on objects immersed in a large body of flowing fluid or objects moving through a stagnant fluid:
$$D=\frac{1}{2}\rho U^{2}C_{D}A$$$where $$C_D$$ is the drag coefficient, $$U$$ is the flowing fluid or moving object’s velocity, $$\rho$$ is the fluid density, and $$A$$ is the projected area of blunt objects with axes perpendicular to the flow. Hint 2 For flat plates placed parallel with the flow and have $$10^4 < Re<5\cdot10^5$$ : $$C_D=\frac{1.33}{\sqrt{Re}}$$$
where $$Re$$ is the Reynolds Number.
First, let’s find the road sign’s coefficient of drag. For flat plates placed parallel with the flow and have $$10^4 < Re<5\cdot10^5$$ :
$$C_D=\frac{1.33}{\sqrt{Re}}$$$where $$Re$$ is the Reynolds Number. $$C_D=\frac{1.33}{\sqrt{2\cdot 10^4}}=\frac{1.33}{141.42}=0.0094$$$
Next, the drag force on objects immersed in a large body of flowing fluid or objects moving through a stagnant fluid:
$$D_f=\frac{1}{2}\rho U^{2}C_{D}A$$$where $$C_D$$ is the drag coefficient, $$U$$ is the flowing fluid or moving object’s velocity, $$\rho$$ is the fluid density, and $$A$$ is the projected area of blunt objects with axes perpendicular to the flow. $$0.1lb_f=\frac{1}{2}(0.08\frac{lb}{ft^3})(50\frac{ft}{s})^{2}(0.0094)A$$$
Before solving for $$A$$ , recall that $$1\:lb_f=1\:lb \times gravity\approx 1\:lb \times 32.2 \frac{ft}{s^2}$$ . Thus,
$$0.1lb \times 32.2\frac{ft}{s^2}=\frac{1}{2}(0.08\frac{lb}{ft^3})(2,500\frac{ft^2}{s^2})(0.0094)A$$$$$3.22ft=(\frac{0.94}{ft})A$$$
$$A=\frac{3.22ft}{\frac{0.94}{ft}}=3.4\:ft^2$$$$$3.4\:ft^2$$$