大连理工大学化工传递过程课后题参考答案Reference answers of homework 1.pdf

1 reference answers of homework 1 1 what are the orders of the following quantities 2 write out the following summations 4 evaluate 5 v erify gbfauwvdcbawuvwv 14 3 1 2 k k 2 3 2 2 2 1 3 1 2 aaaa k k 333323321331322322221221311321121111 3 1 3 1 babababababababababa jk kjjk 22 3 2 2 2 1 2 3 1 2 aaaa j j 22 3 2 2 2 1 4 3 2 3 2 2 4 2 2 3 2 1 2 2 2 1 4 1 3 1 3 1 22 222aaaaaaaaaaaaaa jk kj 3 a vector has components v5 2 1 zyx vvv a vector has components 1 1 3 zyx www w evaluate 4 wv1 1 v 0 2 0 1 1 zyx wv 7163 30 v zy w 1 1 321 33132 222 3 2 2 2 1 2 3 2 2 2 1 2 332211 2 1221 2 3113 2 2332 2 332211 22 wvwwwvvv wvwvwvwvwvwvwvwvwv wvwvwvwvwvwvwv ji i ijk jk k http hdhe tk index html 2 evaluate 7 evaluate 8 a field is said to be irrotational if which of the following fields are irrotational b c 9 evaluate and for the four fields in exercise 8 0 b 0 0 000 00 000 b 6 the components of a symmetric tensor are 3 xx 2 xy 1 xz 2 yx 2 yy 1 yz 1 zx 1 zy 4 zz the components of a vector are 5 x v3 y v2 z v zyx v 101423 177 vv v zyx v 101423 41 464 6915 101525 vv zyx 23 1 0 1221 3 0 21 i i i zyxv 0 v v v 000 000 00b 000 00 00 b b 000 00 00 b b http hdhe tk index html 3 2 1 please obtain and in any orthogonal curvilinear coordinates and their special forms in the three kinds of coordinates rectangular cylindrical and spherical coordinates v v v 1 1 11 11 321 3 231 2 132 1321 2 vhh q vhh q vhh qhhh hh v q h h v hh v q v hhh v hh v q v h q h hq h hhq v h v hh v qh v j i j i k j j i i j i j q h ji j ij ij i i i i i ij q h ji j ij i i i i q h ik j ik ijk ijq h ji j i i i i ijk k j k k ij j i j i i i i i i i ijij jqij i i q v ij i j jj i i i i z v y v x v v z y x z v r v rv rrr v z v r v r v v z r rzr 1 v r v r vr rr v r sin 1 sin sin 1 1 2 2 3 2 12 21 2 1 21 12 1 2 3 13 31 3 1 31 13 1 13 3 23 2 3 2 2 32 3 2 31122 21 23311 31 12233 32 2121 2121 211211 1 1 1 11 11 11 1 1 122113312332 211332 q h hh v q v hq h hh v q v hq h hh v q v hq h hh v q v h v q h hhq v h v q h hhq v h vh q vh qhh vh q vh qhh vh q vh qhh v q h hhq v h v q h hhq v h v q h hhq v h v q h hhq v h v q h hq h hhq v h v qhq v h v qh v kijki i ji j ijk i i i kijk ikl ji j kijk ijk i i i likl ikl ki j kijk ijk i j j k k i i ijk ij j kijk ijk i j j k k k ij i j i ij i i j kijk ijk i j i ij i i j i jij i i ji i i i ji ji ki ki kii iii z x y y zx x y z y v x v x v z v z v y v v http hdhe tk index html 4 z rzr r z r v r v r v r v z v z v r v v sin sin cos sin r v r v r v r v r v r v v rr v r v v rr r j i j i ij i i i i j j j j ij i i i i k k ik i i j i j i ij i i j ij i ji j k j k k ij i j i ij i i j ij i ji v q h hh v q h q v h hh v q h hh v q h hhq v h v q h hq h hhq v h v jji kji kji 1 1 1 1 1 1 1 1 1 2121 3311 3121321 3 23 23 13 1 3 3 3 3 3 3 2 2 23 3 3 3 1 1 13 2 2 2 3 3 32 2 32 32 12 1 2 2 2 2 2 2 1 1 12 1 13 3 31 1 12 2 21 1 31 31 21 21 1 q h hh v q h hh v q v h v q h q v h hh v q h q v h hh v q h q v h hhq h hh v q h hh v q v h v q h q v h hh v q h q v h hh v q h q v h hhq h hh v q h hh v q v h v z v z v z v y v y v y v x v x v x v v z y x z y x z y x z v z v z v r v r v r v r v r v r v r v r v v zr zr zr cossin sin 1 cos sin 1 sin sin 1 1 1 1 vv v r v v r v v r v r v v r v v r r v r v r v v r r r r r http hdhe tk index html 5 2 if is the instantaneous position vector for a particle show that the velocity and acceleration of the particle are given by in cylindrical coordinate the dots indicate time derivatives of the coordinates solution 3 suppose we have a velocity profile please find the track line and the stream line which pass when t 0 solution 1 stream line r zrr dt rd v zr zrrrr dt vd a zr 2 2 zr zrr trr tzz f i t cos sin 1 r r t v 2 1 m 2 22 1 11 v dqh v dqh sin cos 1 dr t rdr cr t ln sin 1 1 zrr z d rr dt d zz dt d rr dt zd dt rd dt rd v zr z r r z z r r zr zrrrr zrrrrr dt zd dt rd dt rd dt vd a zr zrr zr 2 2 2 z zz yz xz y zyyyxy x zx yx xx zyxzyxzyx zrz i i iz i rr i i i i rrr i i ir i rqhrqhrrqh 11 11 111 sin cos1 sin cos121 sin cos11 sin cos121 1 sin cos121 rr i i i i rr i i i i rrrr i i ir i rrrqh rrrrqh rrrqh http hdhe tk index html 6 c 1 so the stream line is ln sin1 r 2 track line from r t dt dr 1 1 22 2 1 2 1 cttr c1 1 2 12 2 ttr from cos sin dt rd cos sin 1 dt dt 2 ln ln sin 1ln ct c2 1 1arcsin t 4 homework for you suppose there is a shear stress on the gas liquid interface is a known value please find the velocity profile for the cases when gas flows in cocurrent or countercurrent with liquid respectively solution boundary conditions b c 气体向上流动时取 号 向下流动时取 号 将微分方程积分后 21 2 2 cos cxcx g vz 可以定出 1 2 2 2 cos c g c 当气体向下流动时 1g c xxg v g z 11 2 cos 2 2 当气体向上流动时 1g c xxg v g z 11 2 cos 2 2 0 x z g dx dv g 0cos 2 2 g dx vd z 0 g x z dx dv 0 x z v http hdhe tk index html 7 5 please find 1 the relationship between rr z s dr dv and 2 why re 16 f try to prove it 3 how to calibrate a flow meter with the relationship between q and solution 1 r dz dp dr dv d rr z 2 1 rr z s dr dv r dz dpd 2 1 2 from re 1682 2 aa s rvv f 3 从 可知 两者为线性关系 不同流量下测定压力降 即可获得流量 值 6 problems for you to consider as your homework 1 try to find a method to make the inner pipe without shear stress acting on it 2 how to make the shear stresses acting on inner and outer cylinders be equal 如果使内筒以速度v0沿轴向运动 则微分方程 的解成为 ln ln 4 4 1 2 1 2 2 1 2 20 2 2 2 r r r r rr dz dp v rr dz dp v d d z 相应地 ln 2 4 1 2 2 1 2 20 max r r rr dz dp v r d 如果令rmax r1 可求出v0 使内筒不受力 此外 由内外筒所受力的表达式 dz dpd dz dpd 1 4 1 22 r r r dz dp v d z 2 2 1 as vf 2 0 2 8 1 2 1 r dz dp drvr r v d z r a dz dpr vrq d a 8 4 2 1 dr dv r dr d rdz dp zd http hdhe tk index html 8 1 2 1 2 max 2 1 1 1 r rr dz dpd rr rzsr 2 2 max 2 2 2 1 2 2 r rr dz dpd rr rzsr 如果令两者相等 可求出相应的v0 7 1 try to find the velocity profile of fluid within two parallel plates after one plate suddenly moves 2 what is the final velocity profile after time approaches infinity can you prove your answer solution 1 微分方程的解仍为 边界条件成为 由边界条件可以定出 c2 1 t b x dxe c 4 0 1 2 1 速度分布为 4 4 0 0 0 0 2 2 t b erf erf dxe dx