Question

For flow in a pipe of uniform diameter D the loss in hydraulic head, h, is given by: L u2 D 2 where L is the length of the pipe, u is the average flow velocity, g is the acceleration due to gravity and f is a constant. During reconfiguration of a plant it is estimated that the length of the pipe could be reduced by 15% and its diameter increased by 6%. Calculate the maximum percentage by which The acceleration due to gravity on Mars is 3.71 m s2 compared with 9.81 m s2 on Earth. If the plant were designed to function on Mars rather than on Earth, by what percentage would the loss in hydraulic head change? Both parts must be attempted using a partial differentiation methood [9 marks]

Answer 1

z t(x,y) If Concept: Sy Small chany es in x andy and are Sx ove rall chonge in z, 22.Sy Approximate + 9and f is constant. t deniva h've 2 h = henu 2 Dg will be 2ero, Thug (Lu01) h= f 29 head, overai change in h Sf + CLR C aand both cne constants as Now iength is Yeducud b 15 y. SL= - 0.15 SD= O.O6 inneased by diumeter fu2 4e 2 Dg (5.)) 4e 2 Dg L L (20) (Su) e 2 D9 ALu SD) 202g RITLU 2 2 D'g Thus, - 4LU2 + 2fL0 (o.06D) 2D'g 2 Dg MulHplying cand dinding by U. 2 Dg ,2 - O 15 + 2 Su Sh = J O.Cn 2D9

l0ss o1 head. without ince asing $2 S h fLu2 [2 Su- 0.21 2Dg O Su J 2 21/ Thus, percentage vdocity S change in 2 (0 5

emaing so me Su= SL SD U, L, D Part (b), 9.8) on easth 3.1 mars 9 on -6-1 3 1-9.81 - 6.1 m/s2 deccas ing by NOw h.189) flu2 Sh. = (59) 2 D ALU2 2 Dg Sh 1 (1.9-)- 9.81 S C6218 Sh 0-6218h 62-180/0 in corases by Thus hydrautic head