AME 513 Principles of Combustion Lecture 7 Conservation equations

2 AME Fall Lecture 7 - Conservation equations Outline  Conservation equations  Mass  Energy  Chemical species  Momentum

3 AME Fall Lecture 7 - Conservation equations Conservation of mass  Cubic control volume with sides dx, dy, dz  u, v, w = velocity components in x, y and z directions  Mass flow into left side & mass flow out of right side  Net mass flow in x direction = sum of these 2 terms

4 AME Fall Lecture 7 - Conservation equations Conservation of mass  Similarly for y and z directions  Rate of mass accumulation within control volume  Sum of all mass flows = rate of change of mass within control volume

5 AME Fall Lecture 7 - Conservation equations Conservation of energy – control volume  1st Law of Thermodynamics for a control volume, a fixed volume in space that may have mass flowing in or out (opposite of control mass, which has fixed mass but possibly changing volume):  E = energy within control volume = U + KE + PE as before  = rates of heat & work transfer in or out (Watts)  Subscript “in” refers to conditions at inlet(s) of mass, “out” to outlet(s) of mass  = mass flow rate in or out of the control volume  h  u + Pv = enthalpy  Note h, u & v are lower case, i.e. per unit mass; h = H/M, u = U/M, V = v/M, etc.; upper case means total for all the mass (not per unit mass)  v = velocity, thus v 2 /2 is the KE term  g = acceleration of gravity, z = elevation at inlet or outlet, thus gz is the PE term

6 AME Fall Lecture 7 - Conservation equations Conservation of energy  Same cubic control volume with sides dx, dy, dz  Several forms of energy flow  Convection  Conduction  Sources and sinks within control volume, e.g. via chemical reaction & radiative transfer = q’’’ (units power per unit volume)  Neglect potential (gz) and kinetic energy (u 2 /2) for now  Energy flow in from left side of CV  Energy flow out from right side of CV  Can neglect higher order (dx) 2 term

7 AME Fall Lecture 7 - Conservation equations Conservation of energy  Net energy flux (E x ) in x direction = E left – E right  Similarly for y and z directions (only y shown for brevity)  Combining E x + E y  dE CV /dt term

8 AME Fall Lecture 7 - Conservation equations Conservation of energy  dE CV /dt = E x + E y + heat sources/sinks within CV  First term = 0 (mass conservation!) thus (finally!)  Combined effects of unsteadiness, convection, conduction and enthalpy sources  Special case: 1D, steady (∂/∂t = 0), constant C P (thus ∂h/∂T = C P ∂T/∂t) & constant k:

9 AME Fall Lecture 7 - Conservation equations Conservation of species  Similar to energy conservation but  Key property is mass fraction of species i (Y i ), not T  Mass diffusion  D instead of conduction – units of D are m 2 /s  Mass source/sink due to chemical reaction = M i  i (units kg/m 3 s) which leads to which leads to  Special case: 1D, steady (∂/∂t = 0), constant  D  Note if  D = constant and  D = k/C P and there is only a single reactant with heating value Q R, then q’’’ = -Q R M i  i and the equations for T and Y i are exactly the same!  k/  C P D is dimensionless, called the Lewis number (Le) – generally for gases D ≈ k/  C P ≈, where k/  C P =  = thermal diffusivity, = kinematic viscosity (“viscous diffusivity”)

10 AME Fall Lecture 7 - Conservation equations Conservation equations  Combine energy and species equations  is constant, i.e. doesn’t vary with reaction but  If Le is not exactly 1, small deviations in Le (thus T) will have large impact on  due to high activation energy  Energy equation may have heat loss in q’’’ term, not present in species conservation equation

11 AME Fall Lecture 7 - Conservation equations Conservation equations - comments  Outside of a thin reaction zone at x = 0  Temperature profile is exponential in this convection- diffusion zone (x ≥ 0); constant downstream (x ≤ 0)  u = -S L (S L > 0) at x = +∞ (flow in from right to left); in premixed flames, S L is called the burning velocity   has units of length: flame thickness in premixed flames  Within reaction zone – temperature does not increase despite heat release – temperature acts to change slope of temperature profile, not temperature itself

12 AME Fall Lecture 7 - Conservation equations Schematic of deflagration (from Lecture 1)   Temperature increases in convection-diffusion zone or preheat zone ahead of reaction zone, even though no heat release occurs there, due to balance between convection & diffusion   Temperature constant downstream (if adiabatic)   Reactant concentration decreases in convection-diffusion zone, even though no chemical reaction occurs there, for the same reason

13 AME Fall Lecture 7 - Conservation equations Conservation equations - comments  In limit of infinitely thin reaction zone, T does not change but dT/dx does; integrating across reaction zone  Note also that from temperature profile:  Thus, change in slope of temperature profile is a measure of the total amount of reaction – but only when the reaction zone is thin enough that convection term can be neglected compared to diffusion term

14 AME Fall Lecture 7 - Conservation equations Conservation of momentum  Apply conservation of momentum to our control volume results in Navier-Stokes equations: or written out as individual components  This is just Newton’s 2 nd Law, rate of change of momentum = d(mu)/dt =  (Forces)  Left side is just d(mu)/dt = m(du/dt) + u(dm/dt)  Right side is just  (Forces): pressure, gravity, viscosity