AME 513 Principles of Combustion

1. AME 513Principles of Combustion Lecture 7 Conservation equations

2. Outline • Conservation equations • Mass • Energy • Chemical species • Momentum AME 513 - Fall 2012 - Lecture 7 - Conservation equations

3. 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 AME 513 - Fall 2012 - Lecture 7 - Conservation equations

4. 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 AME 513 - Fall 2012 - Lecture 7 - Conservation equations

5. 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 v2/2 is the KE term • g = acceleration of gravity, z = elevation at inlet or outlet, thus gz is the PE term AME 513 - Fall 2012 - Lecture 7 - Conservation equations

6. 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 (u2/2) 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 AME 513 - Fall 2012 - Lecture 7 - Conservation equations

7. Conservation of energy • Net energy flux (Ex) in x direction = Eleft – Eright • Similarly for y and z directions (only y shown for brevity) • Combining Ex + Ey • dECV/dt term AME 513 - Fall 2012 - Lecture 7 - Conservation equations

8. Conservation of energy • dECV/dt= Ex+ Ey+ 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 CP (thus ∂h/∂T = CP∂T/∂t) & constant k: AME 513 - Fall 2012 - Lecture 7 - Conservation equations

9. Conservation of species • Similar to energy conservation but • Key property is mass fraction of species i (Yi), not T • Mass diffusion rD instead of conduction – units of D are m2/s • Mass source/sink due to chemical reaction = Miwi (units kg/m3s) which leads to • Special case: 1D, steady (∂/∂t = 0), constant rD • Note if rD = constant andrD= k/CPandthere is only a single reactant with heating value QR, then q’’’ = -QRMiwiand the equations for T and Yi are exactly the same! • k/rCPD is dimensionless, called the Lewis number (Le) – generally for gases D ≈ k/rCP ≈n, where k/rCP= a= thermal diffusivity, n = kinematic viscosity (“viscous diffusivity”) AME 513 - Fall 2012 - Lecture 7 - Conservation equations

10. 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 wdue to high activation energy • Energy equation may have heat loss in q’’’ term, not present in species conservation equation AME 513 - Fall 2012 - Lecture 7 - Conservation equations

11. 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 = -SL (SL > 0) at x = +∞ (flow in from right to left); in premixed flames, SL is called the burning velocity • d 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 AME 513 - Fall 2012 - Lecture 7 - Conservation equations

12. Schematic of deflagration (from Lecture 1) • Temperature increases in convection-diffusion zone or preheat zoneahead 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 AME 513 - Fall 2012 - Lecture 7 - Conservation equations

13. 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 AME 513 - Fall 2012 - Lecture 7 - Conservation equations

14. 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 2nd Law, rate of change of momentum = d(mu)/dt = S(Forces) • Left side is just d(mu)/dt = m(du/dt) + u(dm/dt) • Right side is just S(Forces): pressure, gravity, viscosity AME 513 - Fall 2012 - Lecture 7 - Conservation equations