t∂ ρ∂ t∂ ρ∂ = ρ dt d

D. Keffer - ChE 240: Heat Transfer and Fluid Flowso the continuity equation becomes:−∂ρ 1= ∇ ⋅ρ v =∂ t2r2( r v ) 1 ∂( ρvsinθ) 1 ∂ρ ( )∂ρ∂ rr+r sinθSection 3.7 Now do the same analysis for a momentum balanceθv φ∂θ+r sinθE. Derive the general differential equation of change (momentum balance)accumulation = in - out + generation - consumptionDefine differential volume element as shown on page 167.Momentum is a vector. Since there is no intrinsic difference between x, y, and z coordinates, wecan derive the equation of change for the x-component of momentum and then make analogousstatements about the y and z components. Look only at x-component of momentum∂ ρacc = V∂ t( v ) ∂( ρv)x= ∆x∆y∆z∂ tMomentum can flow in and out by convection:x∂φin= inx+ iny+ inz= A y−zρvxvx | x + A x−zρvxvy | y + A x−y= ∆y∆zρvxvx|x+ ∆x∆zρvxvy|y+ ∆x∆yρvxvz|zρvxvz|zout= ∆y∆zρvx vx| x+ ∆x+ ∆x∆zρvxvy| y+∆y+ ∆x∆yρvxvz| z+∆zMomentum can flow in and out by molecular diffusion:in= inx+ iny+ inz= A y−zτxx| x + A x−zτyx| y + A x−y∆y∆zτxx|x+ ∆x∆zτyx|y+ ∆x∆yτzx|zρτzx|zout= ∆y∆zτxx | x+ ∆x+ ∆x∆zτyx| y+∆y+ ∆x∆yτzx| z+∆zMomentum can be generated in the differential volume by a body force like gravity:gen = Vρg= ∆x∆y∆zρx g x3-6