2.9 - Fr{_5.;‘1y6 (0‘7/174'0‘.’ (f EX’...

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Unformatted text preview: Fr{_5.;‘1y6 (0‘7/174'0‘.’ (f EX’ *Cw—uxfie/ 4 lfiéJpn/‘g 7&4// “~77 Mij/x3,4/C Ihrfi¢¢ 7‘!~;.‘04—, W4,V( /r.q,(_; 4 Mcy— W Aptf/Z; ' IC/Lu'tx/ /€4m¢; 77'! [V‘r7(a(/ “40% ’57]? 77c 3 ' 776 744,: Jh/fiflc o‘F 7’7( ‘f/e-/ d’ i 724 (X%Wq,/ #/6.'/ «(fl/{rah 1,? 47! mm #:3, L @‘P6 3 (71“‘07’4'V74 flVfJ'J'erJ a: (1.47b) si‘tive in the .sistent, and Is force per nodynamic me way as ner as Eqs. >e found in newtonian ' examples resistance nin g, fluid strong, as g case of a as to flow. ivior after eopectic / _Common fluids \ - lOlli’OPIC Time tin rate; yield may also be nonlinear. An example of a yielding fluid is toothpaste, which will not flow out of the tube until a finite stress is applied by squeezing. A further complication of nonnewtonian behavior is the transient effect shown in Fig. 1.6b. Some fluids require a gradually increasing shear stress to maintain a constant strain rate and are called rheopectic. The opposite case of a fluid which thins out with time and requires decreasing stress is termed thixotropic. We shall neglect nonnewtonian effects in this book; see Ref. 5 for further study. Surface Tension A liquid, being unable to expand freely, will form an interface with a second liquid or gas. The physical chemistry of such interfacial surfaces is quite complex, and whole textbooks are devoted to this specialty [l3]. Molecules deep within the liquid repel each other because of their close packing. Molecules at the surface are less dense and attract each other. Since half of their neighbors are missing, the mechanical effect is that the surface is in tension. We can account adequately for surface effects in fluid mechanics with the concept of surface tension. If a cut of length dL is made in an interfacial surface, equal and opposite forces of magnitude T dL are exposed normal to the cut and parallel to the surface, where T is called the coefficient of surface tension. The dimensions of T are {F /L}, with SI units of newtons per meter and BG units of pounds force per foot. An alternate concept is to open up the cut to an area dA; this requires work to be done of amount T (M. Thus the coefficient T can also be regarded as the surface energy per unit area of the interface, in newton-meters per square meter or foot- pounds force per square foot. The trvo most common interfaces are water-air and mercury-air. For a clean surface at 20°C = 68°F, the measured surface tension is Y _ 0.0050 lbf/ft = 0.073 N/m air-water (I 48) — 0.033 lbf/ft = 0.48 N/m air-mercury ' These are design values and can change considerably if the surface contains con- taminants like detergents or slicks. Generally Y decreases with liquid temperature and is zero at the critical point. If the interface is curved, a mechanical balance shows that there is a pressure difference across the interface, the pressure being higher on the concave side. This is illustrated in F ig. 1.7. In Fig. 1.7a, the pressure increase in the interior of a liquid cylinder is balanced by two surface-tension forces 2RL Ap = ZTL T or Ap = R (1.49) We are not considering the weight of the liquid in this calculation. In Fig. 1.7b, the pressure increase in the interior of a spherical droplet balances a ring of surface- VISCOSITY AND OTHER SECONDARY PROPERTIES/ 1.7 .: m Mr...” .nl 2RL Ap 1R2 Ap (a) (b) Ap (M .m. l e A (C) ‘n .- Fig. 1.7 Pressure change across a curved interface due to surfatx tension: (a) interior of a liquid cylinder; (b) interior of a spherical droplet; (c) general curved interface. tension force 11R2 Ap = 21rRY ' 2r or Ap = i— (1.50) We can use this result to predict the pressure increase inside a soap bubble, which has two interfaces with air, an inner and outer surface of nearly the same radius R 4T Apbubble z 2 Apdrnplet = 17 32 INTRODUCTION 1R2 4p ZIRT (b) :e tension: (a) interior of curved interface. (1.50) a soap bubble, which rly the same radius R (1.51) Figure 1.7:: shows the general case of an arbitrarily curved interface whose princi- pal radii of curvature are R , and R,. A force balance normal to the surface will show that the pressure increase on the concave side is Ap=T(R;" +R;‘) (1.52) Equations (1.49) to (1.51) can all be derived from this general relation; e.g.. in Eq. (1.49) R, = R and R2 a co. A second important surface effect is the contact angle 9 which appears when a liquid interface intersects with a solid surface, as in Fig. 1.8. The force balance would then involve both T and 0. If the contact angle is less than 90°. the liquid is said to wet the solid; if 0 > 90°, the liquid is termed nonwetting. For example. water wets soap but does not wet wax. Water is extremely wetting to a clean glass surface, with 9 z 0°. Like T, the contact angle 0 is sensitive to the actual physi- cochemical conditions of the solid-liquid interface. For a clean mercury-air-glass interface, 0 = 130°. Example 1.13 illustrates how surface tension causes a fluid interface to rise or fall in a capillary tube. EXAMPLE 1.13 Derive an expression for the change in height h in a circular tube of a liquid with surface tension T and contact angle 0, as in Fig. £1.13. solution The vertical component of the ring surface-tension force at the interface in the tube must balance the weight of the column of fluid of height h 21tRT cos 0 = pgnR 2h Solving for h, we have the desired result = 2]” cos 0 h 09R Ans. Thus the capillary height decreases inversely with tube radius R and is positive if 0 < 90° (wetting liquid) and negative (capillary depression) if 9 > 90". Gas Nonwetting fl Solid Fig. 1.8 Contact-angle effects at a liquid-gas-solid interface. If 0 < 90°, the liquid “wets” the solid; if 9 > 90°, the liquid is nonwetting. vrscosnv AND OTHER SECONDARY PROPERTIES/1.7 33 Fig. 121.13 Suppose that R = 1 mm. Then the capillary rise for a water-air-glass interface, 0 2 0°, T = 0.073 N/m, and p -= 1000 kg/m3 is _ 2(0.073 N/m)(cos 0°) . 2 —. - h _ 1000 kg/m (9.81 m/s )(0'001 m) 0.015 (N 5 )/kg — 0.015 m — 1.5 cm For a mercury-air-glass interface, with 9 = 130°, T = 0.48 N/m, and p = 13,600 kg/m’, the capillary rise is h _ 2(0.48)(cos 130°) ‘ 131300931 )(0.001) = ‘0'46 cm When a small-diameter tube is used to make pressure measurements (Chap. '2), these capillary effects must be corrected for. Vapor Pressure Vapor pressure is the pressure at which a liquid boils and is in equilibrium with its own vapor. For example, the vapor pressure of water at 68°F is 49 lbf/ft’, while that of mercury is only 0.0035 lbf/ftz. If the liquid pressure is greater than the vapor pressure, the only exchange between liquid and vapor is evaporation at the interface. If, however, the liquid pressure falls below the vapor pressure, vapor bubbles begin to appear in the liquid. If water is heated to 212°F, its vapor pressure rises to 2116 lbf/ft2 and thus water at normal atmospheric pressure will boil. If the liquid pressure is dropped below the vapor pressure due to a flow phenomenon, we call the process cavitation. As we shall see in Chap. 2, if water is accelerated from rest to about 50 ft/s, its pressure drops by about 15 lbf/inz, or 1 atm. This can cause cavitation. Figure 1.9 shows cavitation occurring in the low- pressure region associated with the tip vortices shed by a marine propeller. The dimensionless parameter describing flow-induced boiling is the cavitation number Ca = Fifi—{E (1.53) 34 INTRODUCTION ...
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This note was uploaded on 04/07/2008 for the course AE 2020 taught by Professor Ruffin during the Summer '07 term at Georgia Tech.

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2.9 - Fr{_5.;‘1y6 (0‘7/174'0‘.’ (f EX’...

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