Conservation of Mass

In the Reynolds transport theorem for conservation of mass, let B = m, i.e. B_sys = m_sys = mass of the system,

For our system, we know that

So, Reynold's Transport Theorem (R.T.T.) becomes

• Let's analyze the last term in the equation:
• n = outward normal unit vector (perpendicular to surface area element dA)
• V = velocity vector, which can be in any arbitrary direction
  
  
• Recall, from definition of dot product:
• Outlets: V is outward (mass leaving C.V.)
• Inlets: V is inward (mass entering C.V.)
  
• Now, it turns out that
   = mass flow rate outward through a surface
  so, if we integrate over the entire surface,
  
  and the conservation of mass equation becomes:
  
  *this is the most useful form.
  
• e.g. Given: A rigid tank of volume V with p = p₀ at t = 0. Air is pumped in at constant mass flow rate isothermally.
  
  Find: p in tank as a function of time
  Solution: first, draw a C.V. inside the entire tank.
  
  
  Now use our conservation of mass equation.
  
  assume density is equal everywhere in the tank, and only varies with time.  There are no outlets and the only term remaining is the mass term at the inlet:
  
  This is a differential equation we must solve by separating the variables and integrating from t=0 to some arbitrary t.
  
  Finally, use the ideal gas law to get the pressure. Thus,
  
  (NOTE: pressure varies linearly with t)
  
  Special Case: STEADY FLOW
  For a steady flow, nothing is a function of time, the d/dt term in the conservation of mass equation goes away.
  
  in other words, "What goes in must come out"
  If more is going in than out, mass will be accumulating with time inside the control volume, and it would not be steady state!! ( and you need to include the unsteady volume term)
  
  

• Since the density is constant,
  
  take density outside of the integral
  
  
  So the integral term is equal to Q, the volume flow rate
  i.e. where mass flow rate = density X volume flow rate
  
  So the conservation of mass equation becomes:
  
  "volume flow rate it" = "volume flow rate out"
  

• Another simplification: 1-D inlets & outlets (for incompressible flow)
  

  Consider an outlet:
  

  
  Where:
  • 1-D outlet means velocity is parallel to n, the unit normal vector.
  • V = constant across the outlet.
  • Density = constant across the outlet
    
  
  
  thus, Q_out = V A. Therefore, at an inlet
  and
  
  we usually say:
  
  

  ….since the negative sign is accounted for in the conservation of mass equation,
  

  i.e.
  
  

• Average Velocity
  

  If an inlet or exit is not 1-D, we can still use but V_AV must be the average velocity
  

  • e.g.: At an outlet, for incompressible flow ( density = constant)
    
    Let us define:
    
    

    but for most problems V is parallel to n……
    
    

    An equivalent 1-D outlet will have the same mass flow rate as the actual outlet.
    

• EXAMPLE 3.14 in text:
  

  A container of water,
  

  
  

  Given: V_AV, Q₃, D₁, D₂ and h equal constants.
  V₁ = 3 m/s; Q₃ = 0.01 m³/s; h = constant; D₁ = 0.05m; D₂= 0.07m
  

  Find: Average exit velocity V₂
  

  Solution: Use conservation of mass. First draw the C.V. shown.
  

  Since it is steady and incompressible,

  
  
  Using the above equations, solve for V₂: