流体力学与传热习题：problems and solutions forchapter1,2

1.3 A differential manometer as shown in Fig. is sometimes used to measure small pressure

difference. When the reading is zero, the levels in two reservoirs are equal. Assume that fluid B is

methane （甲烷）, that liquid C in the reservoirs is kerosene (specific gravity = 0.815), and that

liquid A in the U tube is water. The inside diameters of the reservoirs and U tube are 51mm and

6.5mm , respectively. If the reading of the manometer is145mm., what is the pressure difference

over the instrument In meters of water, (a) when the change in the level in the reservoirs is

neglected, (b) when the change in the levels in the reservoirs is taken into account? What is the

percent error in the answer to the part (a)?

Solution ：

p a =1000kg/m 3 p c =815kg/m 3 p b =0.77kg/m 3 D/d=8 R=0.145m

When the pressure difference between two reservoirs is increased, the volumetric changes in the reservoirs and U tubes

R d x D 2244ππ

= (1) so R D d x 2⎪⎭

⎫ ⎝⎛= (2) and hydrostatic equilibrium gives following relationship

g R g x p g R p A c c ρρρ++=+21 (3)

so

g R g x p p c A c )(21ρρρ-+=- (4)

substituting the equation (2) for x into equation (4) gives

g R g R D d p p c A c )(221ρρρ-+⎪⎭

⎫ ⎝⎛=- (5) （a ）when the change in the level in the reservoirs is neglected,

()Pa g R g R g R D d p p c A c A c 26381.98151000145.0)()(221=⨯-=-≈-+⎪⎭

⎫ ⎝⎛=-ρρρρρ

（b ）when the change in the levels in the reservoirs is taken into account

()Pa g R g R D d g R g R D d p p c A c c A c 8.28181.98151000145.081.9815145.0515.6)()(22221=⨯-+⨯⨯⨯⎪⎭

⎫ ⎝⎛=-+⎪⎭

⎫ ⎝⎛=-+⎪⎭

⎫ ⎝⎛=-ρρρρρρ error=％＝7.68

.2812638.281-

1.4 There are two U-tube manometers fixed on the fluid bed reactor, as shown in the figure. The readings of two U-tube manometers are R 1=400mm ，R 2=50mm, respectively. The indicating liquid is mercury. The top of the manometer is filled with the water to prevent from the mercury vapor diffusing into the air, and the height R 3=50mm. Try to calculate the pressure at point A and B .

Solution: There is a gaseous mixture in the U-tube manometer meter. The densities of fluids are denoted by Hg O H g ρρρ,,2, respectively. The pressure at point A is given by hydrostatic equilibrium

g R R g R g R p g Hg O H A )(32232+-+=ρρρ

g ρis small and negligible in comparison with Hg ρand ρH2O , equation above can be simplified

c A p p ≈=232gR gR Hg O H ρρ+

=1000×9.81×0.05+13600×9.81×0.05

=7161N/m²

1gR p p p Hg A D B ρ+=≈=7161+13600×9.81×0.4=60527N/m

Figure for problem 1.4