A hot-air balloon with a diameter of 15 ft, height of 13 ft, weight of 200 lb, and cargo (4 ft × 4 ft × 4 ft) weighing 300 lb ascends at a constant velocity of 25 ft/sec in the atmosphere to a height of 10,000 ft above sea level (standard atmosphere at 10,000 ft above sea level, ρ = 0.0017555 slugs

Question

A hot-air balloon with a diameter of 15 ft, height of 13 ft, weight of 200 lb, and cargo (4 ft × 4 ft × 4 ft) weighing 300 lb ascends at a constant velocity of 25 ft/sec in the atmosphere to a height of 10,000 ft above sea level (standard atmosphere at 10,000 ft above sea level, [latex] ho = 0.0017555 slugs/ft^{3} [/latex], [latex]\mu = 0.35343 imes 10^{-6} lb-sec/ft^{2} [/latex], [latex]\gamma = 0.056424 lb/ft^{3} [/latex]), as illustrated in Figure EP 10.26. Assume the drag coefficient for the hot-air balloon, [latex] C_{D} [/latex] = 1.3. (a) Determine the lift force. (b) Determine the lift coefficient.

Accepted Answer

(a) In order to determine the lift force, Equation 10.28 [latex] \sum\limits_{z}{F_{z}} = F_{L} + F_{B} - W - F_{D} = ma_{z} = 0 [/latex] (Newton’s second law of motion) is applied in the vertical direction. The frontal area is used to compute the drag force for the hot-air balloon, and the drag force is determined by applying Equation 10.12 [latex] F_{D} = C_{D} \frac{1}{2} ho v^{2}A [/latex] as follows:

[latex] \sum\limits_{z}{F_{z}} = F_{L} + F_{B} - W - F_{D} = ma_{z} = 0 [/latex]

[latex] D_{b}: = 15 ft [/latex] [latex] A_{frontb}: = \frac{\pi . D_{b}^{2}}{4} = 176.715 ft^{2} [/latex] [latex] h_{b}: = 13 ft [/latex] [latex] C_{Db}: = 1.3 [/latex]

[latex] V: = 25 \frac{ft}{sec} [/latex] [latex] b_{c}: = 4 ft [/latex] [latex] d_{c}: = 4 ft [/latex] [latex] h_{c}: = 4 ft [/latex] [latex] \gamma _{air}: = 0.056424 \frac{lb}{ft^{3}} [/latex]

[latex] V_{bc}: = \frac{\pi .D_{b}^{3}}{12} + \frac{1}{3} \left[ \frac{\pi .D_{b}^{2} (h_{b} + h_{c})}{4} ight] - \frac{1}{3} \left(\frac{\pi . b_{c}^{2} h_{c}}{4} ight) + b_{c}. d_{c} . h_{c} = 1.932 imes 10^{3} ft^{3} [/latex]

[latex] W_{b}: = 200 lb [/latex] [latex] W_{c}: = 300 lb [/latex] [latex] W_{bc}: = W_{b} + W_{c} = 500 lb [/latex]

Guess value: [latex] F_{B}: = 100 lb [/latex] [latex] F_{D}: = 100 lb [/latex] [latex] F_{L}: = 100 lb [/latex]

Given

[latex] F_{L} + F_{B} - W_{bc} - F_{D} = 0 [/latex] [latex] F_{B} = \gamma _{air} . V_{bc}[/latex] [latex] F_{D} = C_{Db} \frac{1}{2} ho _{air} .V^{2} . A_{frontb} [/latex]
[latex] \left ( \begin{matrix} F_{B} \ F_{D} \ F_{L} \end{matrix} ight ) : =Find ( F_{B},F_{D}, F_{L}) [/latex]

[latex] F_{B} = 109.022 lb [/latex] [latex] F_{D} = 170.631 lb [/latex] [latex] F_{L} = 561.609 lb [/latex]

(b) The planform area of the hot-air balloon is used to model the lift force for the hotair balloon, and the lift coefficient is determined by applying the lift force equation, Equation 10.21 [latex] F_{L} = C_{L} \frac{1}{2} ho.V^{2} . A[/latex], as follows:

[latex] slug: = 1 lb \frac{sec^{2}}{ft} [/latex] [latex] ho _{air} : = 0.0017555 \frac{slug}{ft^{3}} [/latex] [latex]\mu _{air}: = 0.35343 imes 10^{-6} lb \frac{sec}{ft^{2}} [/latex]

[latex] Aplanc: = \frac{\pi . D_{b}^{2}}{4} = 176.715 ft^{2} [/latex]

Guess value: [latex] R : = 10000 [/latex] [latex] C_{L} : = 1.0 [/latex]

[latex] F_{L} = C_{L} \frac{1}{2} ho _{air} .V^{2} . Aplanc [/latex] [latex] R = \frac{ ho _{air} .V .D_{b}}{\mu _{air}} [/latex]
[latex] \left ( \begin{matrix} R \ C_{L} \end{matrix} ight ) : = Find (R, C_{L}) [/latex]
[latex] R = 1.863 imes 10^{6}[/latex] [latex] C_{L} = 5.793 [/latex]

Since R = 1.863 [latex] imes 10^{6} [/latex] , this confirms the assumption that R > 10,000 (see Table 10.6). Furthermore, it is important to note that because such simplistic assumptions (for instance, the planform area of the hot-air balloon in the lift equation) have been made in the estimation and modeling of the lift force, [latex] F_{L} [/latex], the estimated value of the lift coefficient, [latex] C_{L} [/latex] is a rough estimate at best.

The Drag Coefficient, [latex] C_{D} [/latex] for Three-Dimensional Bodies at Given Orientation to the Direction of Flow, v for R > 10,000
			[latex] C_{D} [/latex] = 1.3		ν (m/s)	[latex] C_{D} [/latex]
					10	1.0 - 1.4
					20	0.8 - 1.2
					30	0.5 - 0.9
					40	0.3 - 0.7
Parachute, A = [latex] \pi D^{2}/4 [/latex]				Tree, A= frontal area
			[latex] C_{D} [/latex] = 1.0 - 1.4		Position	[latex] C_{D} [/latex] A ([latex] ft^{2} [/latex])
					Standing	9
					Sitting	6
					Crouching	2.5
High-rise buildings, A = frontal area				Average person
	Position	A ([latex] ft^{2} [/latex])	[latex] C_{D} [/latex]			[latex] C_{D} [/latex] = 0.4
	Upright	5.5	1.1
	Racing	3.9	0.88
	Drafting	3.9	0.5
	Streamlined	5.0	0.12
Bikes				Large birds, A = frontal area
			Without deflector [latex] C_{D} [/latex] = 0.96			Minivan: [latex] C_{D} [/latex] = 0.4
			With deflector: [latex] C_{D} [/latex] = 0.76			Passengercar or sports car: [latex] C_{D} [/latex] = 0.3
Tractor-trailer truck, A= frontal area				Automotive, A = frontal area

Question 10.26: A hot-air balloon with a diameter of 15 ft, height of 13 ft,...

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