(Refer to Example Problem 10.17.) Air at standard atmosphere at an altitude of 25,000 ft above sea level flows at a velocity of 800 ft/sec over a prototype 1500-ft-long square shaped cylinder with a side of 6ft, as illustrated in Figure EP 11.4. A smaller model of the larger prototype is designed

Question

(Refer to Example Problem 10.17.) Air at standard atmosphere at an altitude of 25,000 ft above sea level flows at a velocity of 800 ft/sec over a prototype 1500-ft-long square shaped cylinder with a side of 6ft, as illustrated in Figure EP 11.4. A smaller model of the larger prototype is designed in order to study the flow characteristics of turbulent flow over the blunt body. The model fluid is carbon dioxide at [latex] 68^{\circ} F [/latex], and the model scale, λ is 0.20. (a) Determine the drag force on the prototype square cylinder. (b) Determine the velocity of flow of the carbon dioxide over the model square cylinder and the required pressure of the carbon dioxide in order to achieve dynamic similarity between the model and the prototype. (c) Determine the drag force on the model square cylinder in order to achieve dynamic similarity between the model and the prototype.

Accepted Answer

(a) In order to determine the drag force on the prototype square cylinder, the drag force equation, Equation 11.91 [latex] F_{D} = C_{D} \frac{1}{2} ho.v^{2} . A [/latex], is applied, where the frontal area of the square cylinder is used to compute the drag force. Because the Reynolds number, R > 10,000 (turbulent flow) over a blunt body is assumed, the drag coefficient, [latex] C_{D} [/latex] becomes independent of R, as illustrated in Figure 10.12 for two-dimensional bodies. However, Table 10.4 for two-dimensional bodies presents an easier-to-read magnitude for the drag coefficient, [latex] C_{D} [/latex] for the square cylinder, assuming turbulent flow. Furthermore, Figure 10.19 illustrates that for blunt (square-shaped) two-dimensional bodies, the drag coefficient, [latex] C_{D} [/latex] is independent of M for M ≤ 0.5, while the drag coefficient, [latex] C_{D} [/latex] is dependent on M for M>0.5. The fluid properties for air are given in Table A.1 in Appendix A. The R and [latex] M = C^{0.5}[/latex] are computed as follows:

[latex] slug: = 1 lb \frac{sec^{2}}{ft} [/latex] [latex] ho _{p} : =0.0010663 \frac{slug}{ft^{3}} [/latex] [latex] \mu _{p} : = 0.32166 imes 10^{-6} lb \frac{sec}{ft^{2}} [/latex]

[latex] C_{p}: = 1016.11 \frac{ft}{sec} [/latex] [latex] E_{vp}: = c_{P}^{2} ho _{P} = 1.101 imes 10^{3} \frac{lb}{ft^{2}} [/latex]

[latex] D_{P}: = 6 ft [/latex] [latex] L_{p}: = 1500 ft[/latex] [latex] V_{P}: = 800 \frac{ft}{sec} [/latex]

[latex] R_{p}: = \frac{ ho _{p} .V_{p} .D_{p}}{\mu _{p}} = 1.591 imes 10^{7} [/latex] [latex] C_{p}: = \frac{ ho _{p} .V_{p}^{2} }{E_{vp}} = 0.62 [/latex] [latex] M_{P}: = \sqrt{C_{p}} = 0.787 [/latex]

Thus, since R >10,000, flow is turbulent and thus the drag coefficient, [latex] C_{D} [/latex] is independent of R, and since M> 0.5, the drag coefficient,[latex] C_{D} [/latex] is dependent on M; thus, Figure 10.19 is used to determine the drag coefficient, [latex] C_{D} [/latex] as follows:

[latex] C_{Dp}: = 2.1 [/latex] [latex] A_{p}: = L_{p} . D_{p} = 9 imes 10^{3} ft^{2} [/latex]
[latex] F_{Dp}: = C_{Dp} \frac{1}{2} ho _{p}.V^{2}_{p} . A_{p} = 6.449 imes 10^{6} lb [/latex]

(b) To determine the velocity of flow of the carbon dioxide over the model square cylinder and the required pressure of the carbon dioxide in order to achieve dynamic similarity between the model and the prototype, for turbulent flow over a blunt body, the geometry, [latex] L_{i}/L [/latex] must remain a constant between the model and prototype as follows:

[latex] \left(\frac{L_{i}}{L} ight)_{p} = \left(\frac{L_{i}}{L} ight)_{m} [/latex]

Furthermore, in order to determine the geometry L and D of the model square cylinder, the model scale, λ (inverse of the length ratio) is applied as follows:

[latex] \lambda : = 0.20 [/latex]

Guess value: [latex] D_{m}: = 1 ft [/latex] [latex] L_{m}: = 1 ft[/latex]

Given

[latex] \lambda = \frac{D_{m}}{D_{p}} [/latex] [latex] \lambda = \frac{L_{m}}{L_{p}} [/latex]
[latex] \left ( \begin{matrix} D_{m} \ L_{m} \end{matrix} ight ) : = Find ( D_{m},L_{m}) = \left ( \begin{matrix} 1.2 \ 300 \end{matrix} ight ) ft [/latex]

And, the geometry is modeled as follows:

[latex] \frac{L_{p}}{D_{p}} = 250 [/latex] [latex] \frac{L_{m}}{D_{m}} = 250 [/latex]

And, although the relative roughness, ɛ/L should remain a constant between the model and prototype as follows:

[latex] \left(\frac{\varepsilon }{L} ight)_{p} = \left(\frac{\varepsilon }{L} ight)_{m} [/latex]

Figure 10.18is for a sphere of diameter, D and not for a square shaped cylinder. As such, the dependence of the drag coefficient, [latex] C_{D} [/latex] on the relative roughness, ɛ/L will not be modeled in the example problem herein. Furthermore, the C (or M) must remain a constant between the model and prototype as follows:

[latex]\underbrace{\left[\left(\frac{ ho v^{2}}{E_{v}} ight)_{p} ight] }_{C_{p}} = \underbrace{\left[\left(\frac{ ho v^{2}}{E_{v}} ight)_{m} ight] }_{C_{m}} [/latex]

The fluid properties (gas constant and specific heat ratio, which are in dependent of the gas pressure and are only a function of the temperature of the gas) for carbon dioxide are given in TableA.5 in Appendix A. Furthermore, [latex] E_{v} [/latex] for the car bon dioxide is determined by assuming isentropic conditions for the compression and expansion of the gas; thus, [latex] E_{v} [/latex] = kp, where the pressure is determined by applying the ideal gas law, p = ρRT (see Chapter 1). The sonic velocity for the carbon dioxide, [latex] c = \sqrt{E_{v}/ ho } = \sqrt{kRT} [/latex] , which is independent of the gas pressure and is only a function of the temperature of the gas.

[latex] T_{m}: = 68^{\circ } F = 527 imes 67^{\circ } R [/latex] [latex] Rgasc_{m}: = 1123 \frac{ft^{2}}{sec^{2} {^{\circ}}R} [/latex] [latex] k_{m}: = 1.28 [/latex]

Guess value: [latex]V_{m}: = 0.1 \frac{ft}{sec} [/latex] [latex]P_{m}: = 1 \frac{lb}{ft^{2}} [/latex] [latex]E_{vm}: = 1 \frac{lb}{ft^{2}} [/latex]

[latex] ho _{m} : =0.00354 \frac{slug}{ft^{3}} [/latex] [latex] c_{m}: = 1000 \frac{ft}{sec} [/latex] [latex] c_{m}: = 0.5 [/latex]

Given

[latex] C_{m} = \frac{ ho _{m} .V_{m}^{2} }{E_{vm}} [/latex] [latex] P_{m} = ho _{m} . Rgasc_{m} . T_{m} [/latex] [latex]E_{vm}= k_{m}.P_{m}[/latex]

[latex] C_{m} = C_{p} [/latex] [latex] c_{m} = \sqrt{\frac{E_{vm}}{ ho _{m}} } [/latex] [latex] c_{m} = \sqrt{k_{m}.Rgasc_{m}. T_{m}} [/latex]
[latex]\left ( \begin{matrix} V_{m} \ P_{m} \ E_{Vm} \ ho _{m} \ c_{m} \ C_{m} \end{matrix} ight ) : = Find (V_{m}, P_{m}, E_{Vm}, ho _{m}, c_{m}, C_{m}) [/latex]

[latex]V_{m} = 685.686 \frac{ft}{s} [/latex] [latex]P_{m} = 79.432 \frac{lb}{ft^{2}} [/latex] [latex]E_{vm} = 101.673 \frac{lb}{ft^{2}} [/latex]

[latex] ho _{m} = 1.34 imes 10^{-4} \frac{slug}{ft^{3}} [/latex] [latex] c_{m} = 870.916 \frac{ft}{s} [/latex] [latex] C_{m} = 0.62 [/latex]

(c) To determine the drag force on the model square cylinder in order to achieve dynamic similarity between the model and the prototype for turbulent flow over a blunt body, the drag coefficient, [latex] C_{D} [/latex] must remain a constant between the model and the prototype (which is a direct result of maintaining a constant C, and a constant [latex] L_{i}/L [/latex] between the model and the prototype) as follows:

[latex] \underbrace{\left[\frac{F_{D}}{\frac{1}{2} ho v^{2}A} ight]_{p} }_{c_{D_{p}}} = \underbrace{\left[\frac{F_{D}}{\frac{1}{2} ho v^{2}A} ight]_{m} }_{c_{D_{m}}} [/latex]

Furthermore, the frontal area of the square cylinder is used to compute the drag force as follows:

[latex] A_{m}: = L_{m} . D_{m} = 360 ft^{2} [/latex]

Guess value: [latex] F_{Dm}: = 1 lb [/latex] [latex] C_{Dm}: = 1 [/latex]

Given

[latex] C_{Dm} = \frac{ F_{Dm}}{\frac{1}{2} ho _{m} . v_{m}^{2}.A_{m}} [/latex]

[latex] C_{Dm} = C_{Dp} [/latex]
[latex] \left ( \begin{matrix} F_{Dm} \ C_{Dm} \end{matrix} ight ) : = Find (F_{Dm}, C_{Dm}) [/latex]

[latex] F_{Dm} = 2.382 imes 10^{4} lb [/latex] [latex] C_{Dm} = 2.1 [/latex]

Therefore, although the similarity requirements regarding the independent π term, [latex] L_{i}/L [/latex] and the independent π term, C (“elastic model”) are theoretically satisfied ([latex] C_{p} [/latex]= [latex] C_{m} [/latex] = 0.62), the dependent π term (i.e., the drag coefficient, [latex] C_{D} [/latex]) will actually/practically remain a constant between the model and its prototype ([latex] C_{Dp} [/latex] = [latex] C_{Dm} [/latex] = 2.1) only if it is practical to maintain/attain the model velocity, drag force, fluid, scale, and cost. Furthermore, because the drag coefficient, [latex] C_{D} [/latex] is independent of R for blunt bodies, R does not need to remain a constant between the model and the prototype as follows:

[latex] \mu _{m} : = 0.310 imes 10^{-6} lb \frac{sec}{ft^{2}} [/latex]

[latex] R_{m}: = \frac{ ho _{m} .V_{m} .D_{m}}{\mu _{m}} = 3.558 imes 10^{5} [/latex] [latex] R_{p}= 1.591 imes 10^{7} [/latex]

One may ask: Is the model speed too high to be able to maintain for the model? If it is too high, then this is a “distorted model” that needs to be adjusted in order to achieve a “true model” or as close to it as possible. The answer is no, the speed of carbon dioxide under pressure is not too high for the model.

Table 10.4
The Drag Coefficient, [latex] C_{D} [/latex] for Two-Dimensional Bodies at Various Orientations to the Direction of Flow, v for R > 10,000
	Sharp corners:				[latex] C_{D} [/latex]
	L/D	[latex] C_{D} [/latex]		L/D	Laminar	Turbulent
	0.0 *	1.9		1	1.2	0.30
	0.1	1.9		2	0.60	0.20
	0.4	2.3		4	0.35	0.15
	0.5	2.5		8	0.25	0.10
	0.7	2.7
	1.0	2.2
	1.2	2.1
	2.0	1.7
	2.5	1.4
	3.0	1.3
	*Corresponds to thin plate
Rectangular rod			Elliptical rod
	Round front edge:			Sharp corners: [latex] C_{D} [/latex] = 2.2
	L/D	[latex] C_{D} [/latex]
	0.5	1.16
	1.0	0.90		Round corners (r/D = 0.2): [latex] C_{D} [/latex] = 1.2
	2.0	0.70
	4.0	0.68
	6.0	0.64
Rectangular rod			Square rod

Semicircular rod	Semicircular shell		Equilateral triangular rod		Hexagonal rod

Table A.1
Physical Properties for the International Civil Aviation Organization (ICAO) Standard Atmosphere as a Function of Elevation above Sea Level
Elevation above Sea Level ft	Temperature (θ) [latex]^{\circ } F[/latex]	Absolute Pressure (p) psia	Density [latex]\left( ho ight) [/latex] [latex]slug/ft^{3} [/latex]	Specific Weight [latex]\left(\gamma ight) [/latex] [latex]lb/ft^{3} [/latex]	Absolute (Dynamic) Viscosity [latex]\left(\mu ight) [/latex] [latex]10^{-6} lb - sec/ft^{2} [/latex]	Kinematic Viscosity (ν) [latex]10^{-3} ft^{2}/sec [/latex]	Speed of Sound (c) ft/sec	Acceleration due to Gravity (g) [latex]ft/sec^{2} [/latex]
0	59.000	14.69590	0.002376800	0.0764720	0.37372	0.15724	1116.45	32.174
5000	41.173	12.22830	0.002048100	0.0658640	0.36366	0.17756	1097.08	32.158
10,000	23.355	10.10830	0.001755500	0.0564240	0.35343	0.20133	1077.40	32.142
15,000	5.545	8.29700	0.001496100	0.0480680	0.34302	0.22928	1057.35	32.129
20,000	-12.255	6.75880	0.001267200	0.0406940	0.33244	0.26234	1036.94	32.113
25,000	-30.048	5.46070	0.001066300	0.0342240	0.32166	0.30167	1016.11	32.097
30,000	-47.832	4.37260	0.000890650	0.0285730	0.31069	0.34884	994.85	32.081
35,000	-65.607	3.46760	0.000738190	0.0236720	0.29952	0.40575	973.13	32.068
40,000	-69.700	2.73000	0.000587260	0.0188230	0.29691	0.50559	968.08	32.052
45,000	-69.700	2.14890	0.000462270	0.0148090	0.29691	0.64230	968.08	32.036
50,000	-69.700	1.69170	0.000363910	0.0116520	0.29691	0.81589	968.08	32.020
60,000	-69.700	1.04880	0.000225610	0.0072175	0.29691	1.31600	968.08	31.991
70,000	-67.425	0.65087	0.000139200	0.0044485	0.29836	2.14340	970.90	31.958
80,000	-61.976	0.40632	0.000085707	0.0027366	0.30182	3.52150	977.62	31.930
90,000	-56.535	0.25540	0.000053145	0.0016950	0.30525	5.74360	984.28	31.897
100,000	-51.099	0.16160	0.000033182	0.0010575	0.30865	9.30180	990.91	31.868
Elevation above Sea Level Km	Temperature (θ) [latex]^{\circ } C[/latex]	Absolute Pressure (p) kPa abs	Density [latex]\left( ho ight) [/latex] [latex]kg/m^{3} [/latex]	Specific Weight [latex]\left(\gamma ight) [/latex] [latex]N/m^{3} [/latex]	Absolute (Dynamic) Viscosity [latex]\left(\mu ight) [/latex] [latex]10^{-6} N - sec/m^{2} [/latex]	Kinematic Viscosity (ν) [latex]10^{-6} m^{2}/sec [/latex]	Speed of Sound (c) m/sec	Acceleration due to Gravity (g) [latex]m/sec^{2} [/latex]
0	15.000	101.325	1.22500	12.0131	17.894	14.607	340.294	9.80665
1	8.501	89.876	1.11170	10.8987	17.579	15.813	336.430	9.80360
2	2.004	79.501	1.00660	9.8652	17.260	17.147	332.530	9.80050
3	-4.500	70.121	0.90925	8.9083	16.938	18.628	328.580	9.79740
4	-10.984	61.660	0.81935	8.0250	16.612	20.275	324.590	9.79430
5	-17.474	54.048	0.73643	7.2105	16.282	22.110	320.550	9.79120
6	-23.963	47.217	0.66011	6.4613	15.949	24.161	316.450	9.78820
8	-36.935	35.651	0.52579	5.1433	15.271	29.044	308.110	9.78200
10	-49.898	26.499	0.41351	4.0424	14.577	35.251	299.530	9.77590
12	-56.500	19.399	0.31194	3.0476	14.216	45.574	295.070	9.76970
14	-56.500	14.170	0.22786	2.2247	14.216	62.391	295.070	9.76360
16	-56.500	10.352	0.16647	1.6243	14.216	85.397	295.070	9.75750
18	-56.500	7.565	0.12165	1.1862	14.216	116.860	295.070	9.75130
20	-56.500	5.529	0.08891	0.8664	14.216	159.890	295.070	9.74520
25	-51.598	2.549	0.04008	0.3900	14.484	361.350	298.390	9.73000
30	-46.641	1.197	0.01841	0.1788	14.753	801.340	301.710	9.71470

Table A.5
Physical Properties for Some Common Gases at Standard Sea-Level Atmospheric Pressure at Room Temperature [latex](68^{\circ } F or 20^{\circ }C )[/latex]
Gas at [latex]68^{\circ }[/latex]F	Chemical Formula	Molar Mass (m) slug=slug- mol	Density (ρ) [latex]slug/ft^{3} [/latex]	Absolute (Dynamic) Viscosity (μ) [latex]10^{-6} Ib-sec/ft^{2} [/latex]	Gas Constant (R) [latex]ft-Ib/(slug-^{\circ }R )=ft^{2}/(sec^{2} -^{\circ }R )[/latex]	Specific Heat		Specific Heat Ratio, [latex]K=C_{ ho }/C_{\upsilon }[/latex]
						[latex]C_{ ho }[/latex]	[latex]C_{\upsilon } [/latex]
						[latex]ft-Ib/(slug-^{\circ }R )=ft^{2}/(sec^{2} -^{\circ }R )[/latex]
Air		28.960	0.002310	0.376	1715	6000	4285	1.40
Carbon dioxide	[latex]CO_{2} [/latex]	44.010	0.003540	0.310	1123	5132	4009	1.28
Carbon monoxide	CO	28.010	0.002260	0.380	1778	6218	4440	1.40
Helium	He	4.003	0.000323	0.411	12,420	31,230	18,810	1.66
Hydrogen	[latex]H_{2}[/latex]	2.016	0.000162	0.189	24,680	86,390	61,710	1.40
Methane	[latex]CH_{2}[/latex]	16.040	0.001290	0.280	3100	13,400	10,300	1.30
Nitrogen	[latex]N_{2}[/latex]	28.020	0.002260	0.368	1773	6210	4437	1.40
Oxygen	[latex]O_{2}[/latex]	32.000	0.002580	0.418	1554	5437	3883	1.40
Water vapor	[latex]H_{2}O[/latex]	18.020	0.001450	0.212	2760	11,110	8350	1.33
at [latex]20^{\circ } C[/latex]		kg/kg-mol	[latex]kg/m^{3} [/latex]	[latex]10^{-6} N-sec/m^{2} [/latex]	[latex]N-m/(kg-^{\circ}K )=m^{2} /(sec^{2}-^{\circ}K )[/latex]	[latex]N-m/(kg-^{\circ}K )=m^{2} /(sec^{2}-^{\circ}K )[/latex]
Air		28.960	1.2050	18.0	287	1003	716	1.40
Carbon dioxide	[latex]CO_{2} [/latex]	44.010	1.8400	14.8	188	858	670	1.28
Carbon monoxide	CO	28.010	1.1600	18.2	297	1040	743	1.40
Helium	He	4.003	0.1660	19.7	2077	5220	3143	1.66
Hydrogen	[latex]H_{2}[/latex]	2.016	0.0839	9.0	4120	14,450	10,330	1.40
Methane	[latex]CH_{2}[/latex]	16.040	0.6680	13.4	520	2250	1730	1.30
Nitrogen	[latex]N_{2}[/latex]	28.020	1.1600	17.6	297	1040	743	1.40
Oxygen	[latex]O_{2}[/latex]	32.000	1.3300	20.0	260	909	649	1.40
Water vapor	[latex]H_{2}O[/latex]	18.020	0.7470	10.1	462	1862	1400	1.33

Question 11.4: (Refer to Example Problem 10.17.) Air at standard atmosphere...

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