Pressure variation with altitude in a static compressible fluid eg air：在一个静态的可压缩流体高度压力的变化如空气

1、Pressure variation with altitude in a static compressible fluid (e.g. air) with constant temperature gradient:_Basic equation:(1)This equation cannot be integrated since variation of with height is not known.Assume air is ideal:(2)From (1) & (2):(3)In the troposphere:(4)where = temperature lapse rat

2、e = 6.5 x 10-3 K/m = 6.5 K/kmfrom (3) & (4):(5) Variation of pressure with altitude in a static compressible fluid (e.g. air) at constant temperatureLower stratosphere ():Note: air is still treated as ideal.Temperature lapse rate in the atmosphere under polytropic conditionsStatic fluid:(1)Polytropi

3、c relationship:(n= polytropic index)(2)Assume air is ideal:(3)From (2):(4)From (3):(5)From (4) & (5):(6)From (1) & (6):Note: (i) for adiabatic conditions:n = k; k: adiabatic index (k = Cp / Cv)(ii) n = 1 in the lower stratosphere Variation of pressure with altitude in the atmosphere under polytropic

4、 conditionsBasic pressure-height relationship:(1)Assume air to be ideal:(2)Temperature variation with altitude:(3)(1) divided by (2):(4)with (3), (4) becomes(5)Note: the above is valid only in the troposphere. In the lower stratosphere, n = 1;the above will “blow up”Forces due to liquid pressure on

5、plane submerged surfaces:zFree surface of liquidAtmospheric pressureC(x,y)P(x,y)xLiquid density = yOyydAdFhhFRedge view of planedydAAAAAA: resultant force due to liquid pressureC: centriod of areaP: centre of pressure (i.e. point of application of FR)Note: P is always below C unless the surface is h

6、orizontal in which case P = CNeed:magnitude of FRdirection of FRline of action of FRNo shearing stresses in a static fluid; forces will be normal to surface independent of the orientation of the surface(the negative on right-hand side of equation indicates that the direction of is opposite to)Note:

7、positive direction of is the outward drawn normal to the areaResultant force:Pressure-height relationship in a static fluid:(h is positive downward from the free surface) h=ysindFhzAt the free surface i.e. at h = 0, p = patmalso, = const (liquid)Recall : first moment of area of surface about the x-a

8、xis using gauge pressures: (*)NOTE: (*) is based upon the assumption that pressure at the free surface of the liquid is atmospheric. More on this soon - imaginary free surface = const means that (*) is valid only for a single homogeneous liquidPoint of application of resultant force (centre of press

9、ure)Resultant force is equivalent to the individual forces if the moment of the resultant about any axis is the same as the sum of the moments of the individual forces about the same axisjiky-location of centre of pressureNOTE: is the second moment of area about the x-axisParallel-Axis Theorem: dAyB

10、BCyC: CentroiddAAMoment of inertia with respect to the AA axis moment of inertia first moment of area with respect to with respect to centroidal axisBB = 0 since BB = and where using gauge pressures, Similarly: where : product of inertiaUsing gauge pressures, NOTE: 1) and are referred to the origin

11、at the free surface 2) product of inertia may be positive or negative unlike Ix or Iy and Ixy = 0when one or both axes are axes of symmetryExampleFHingez5mPatm4mwater8mAn elliptical gate covers the end of a pipe 4m indiameter. If the gate is hinged at the top, what normal force, F, is required to op

12、en the gate when the water free surface is 8m above the top of the pipe and the pipe is open to atmosphere on the other side? Neglect the weight of the gate. Resultant hydrostatic force on gate Gauge pressures: mnnmy b3a512810Location of centre of pressure along the y-axisTaking moments about the hi

13、nge APPROACH #1Determination of the hydrostatic force on a curved submerged bodyConsider the following case:ydAydAzdAdAxxPatmzFig. 3.6 (Fox & McDonald)“Hydrostatic force acts, as in plane submerged surface, normal to surface. However, differentiation due to pressure on each element of the surface ac

14、ts in a different direction because of the surface curvature”Differential force: Resultant force: Similarly: angle between and respective unit vectors: angle between and respective unit vectorsIt is convenient to obtain the components of first and then can be expressed as the vector sum of the compo

15、nentsLine of action of each component of e.g. 2-D curved submerged surfacek32dAcos=-dAcos2=dAydAsin=dAcos3=dAzjdAdAdAzdAyFRyFRzzyzzzyz ALTERNATIVE APPROACHEnlarged view of dAdAh=dAcosdAv=dAsindA=wdsdxdFdFydFxdydsyxABdAp = yyConsider a 2-D curved surface AB as shown above. It is convenient to determi

16、ne the horizontal and vertical components of the hydrostatic force and to obtain the resultant vectorially from the components.-subscript v means projection of area to a vertical plane Note: (magnitude of Fx)wFx-subscript h means projection of area to a horizontal planedAydAxydAx Line of actionWi: contribution to the weight of