AME 513 Principles of Combustion

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# AME 513 Principles of Combustion - PowerPoint PPT Presentation

AME 513 Principles of Combustion. Lecture 7 Conservation equations. Outline. Conservation equations Mass Energy Chemical species Momentum. Conservation of mass. Cubic control volume with sides dx, dy , dz u , v, w = velocity components in x, y and z directions

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### AME 513Principles of Combustion

Lecture 7

Conservation equations

Outline
• Conservation equations
• Mass
• Energy
• Chemical species
• Momentum

AME 513 - Fall 2012 - 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

AME 513 - Fall 2012 - 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

AME 513 - Fall 2012 - 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 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

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

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

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

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)

• 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

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

• 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

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

• 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

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