t∂ ρ∂ t∂ ρ∂ = ρ dt d

D. Keffer - ChE 240: Heat Transfer and Fluid Flowin= inx+ iny+ inz= Ay−zρvx | x + Ax−zρvy | y + Ax−y= ∆y∆zρvx|x+ ∆x∆zρvy|y+ ∆x∆yρvz|zρvz|zout= ∆y∆zρvx | x+ ∆x+ ∆x∆zρvy| y+∆y+ ∆x∆yρvz | z+∆zPut these five terms in mass balance:∆−∂ρx∆y∆z= ( ∆y∆zρvx | x + ∆x∆zρvy | y + ∆x∆yρvz | z )∂ t( ∆y∆zρv| + ∆x∆zρv| + ∆x∆yv | )xDivide by differential volume:x+ ∆xy y+∆yρzz+∆z∂ρ ⎛ρv=⎜x |∂ t ⎝ ∆x⎛ρv⎜x | x−⎝ ∆xx+ ∆x++ρvρv∆yyy||y∆y+y+∆yρvz |∆z+zρv⎞⎟⎠| z∆zz+ ∆z⎞⎟⎠Rearrange into a form recognizable as the definition of a derivative:−∂ρ=∂ tρvx|x+∆x−∆xρvx|x+ρvy|y+∆y−∆yρvy|y+ρvz|z+∆z −∆zρvz|zTake limits as differential elements approach 0 and apply the definition of the derivative:−∂ρ=∂ t( v ) ∂ρ ( v y ) ∂( ρv)∂ ρ∂ xx+∂ y+∂ zz= ∇ ⋅ρvNow consider the law for the derivative( v ) ∂( v ) ∂( ρ)∂ ρ∂ xx= ρx∂ x+vx∂ xand the same for y and z−Dρ⎛∂v= ρ⎜Dt ⎝ ∂ xx+∂ vy∂ y+∂ v∂ zz⎞⎟⎠3-4