AME 51 Principles of Combustion Lecture 7 Conservation equations Outline Conservation equations Mass Energy Chemical species Momentum 1
Conservation of mass Cubic control volume with sides dx, dy, dz u, v, w = velocity components in x, y and z directions v + v y dy 'dxdz udydz dy dx vdxdz u + u dx'dydz x Mass flow into left side mass flow out of right side m left = ua = udydz m right = u + u dxdydz x ' Net mass flow in x direction = sum of these terms m x = udydz u + u dx x ' dydz = u dxdydz x Conservation of mass Similarly for y and z directions m y = v y dxdydz; m = w z dxdydz z Rate of mass accumulation within control volume m t = V = dxdydz;v = volume t t Sum of all mass flows = rate of change of mass within control volume u dxdydz v w dxdydz dxdydz = x y z t dxdydz t = u + v + w ' x y z t + + u Note u = velocity vector = uîx + vîy + wîz î x,îy,îz = unit vectors in x, y, z directions = 0 4
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: de v in v = Q W + m out in hin + + gz in m out hout + + gz out dt E = energy within control volume = U + KE + PE as before Q, W = rates of heat work transfer in or out Watts Subscript in refers to conditions at inlets of mass, out to outlet s of mass m = mass flow rate in or out of the control volume h u + Pv = enthalpy Note h, u v are lower case, ie per unit mass; h = HM, u = UM, V = vm, etc; upper case means total for all the mass not per unit mass v = velocity, thus v is the KE term g = acceleration of gravity, z = elevation at inlet or outlet, thus gz is the PE term 5 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, eg via chemical reaction radiative transfer = q units power per unit volume Neglect potential gz and kinetic energy u for now Energy flow in from left side of CV E left = m left h + q left = uah ka T T = uhdydz dydz k x x ' Energy flow out from right side of CV E right = m right h + q right = u + u dx'dydz h + h x x dx T ' dydz k x + x k T, + dx' x - = uh + u h x dx + h u dx + u h x x x dx k T x + + x k T, 0 dx' 1 x - 4 dydx Can neglect higher order dx term 6
Conservation of energy Net energy flux E x in x direction = E left E right E x = u h x h u + x x k T, - + x ' dydzdx Similarly for y and z directions only y shown for brevity E y = v h y + h v y y k T, - + y ' dxdzdy Combining E x + E y E x + + E y = u h h v x y + x k T + x ' y k T y ' h u + v -+,+ x y '+ dxdzdy = { u 0 1h h 1 0 u + 1 0 k 1T }dxdzdy de CV dt term = m h P = h P V ' + V h -, = h 0 E CV t t t t P t + t P t + h t P t - P = h t P t + h t t P 0 t - 1V = h t + h t P 0 1 t dxdydz 1 V 7 Conservation of energy de CV dt = E x + E y + heat sourcessinks within CV h t + h t ' dxdydz = { u h h u + k T + q''' }dxdzdy + h t + u ' +, h t + u h - 0 1 k T = q''' First term = 0 mass conservation thus finally h t + u h ' k T = q''' 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: u dt dx k d T C P dx = q''' C P 8 4
Conservation of species Similar to energy conservation but Key property is mass fraction of species i Y i, not T Mass diffusion ρd instead of conduction units of D are m s Mass sourcesink due to chemical reaction = M i ω i units kgm s which leads to Y i t + u ' Y i DY i = M i i Special case: 1D, steady t = 0, constant ρd u dy i dx D d Y i dx = M i i Note if ρd = constant and ρd = kc P and there is only a single reactant with heating value Q R, then q = -Q R M i ω i and the equations for T and Y i are exactly the same kρc P D is dimensionless, called the Lewis number Le generally for gases D kρc P ν, where kρc P = α = thermal diffusivity, ν = kinematic viscosity viscous diffusivity 9 Conservation equations Combine energy and species equations Y i t + u Y ' i D Y i = M i i T t + u T ' k T C P + T t + u T ' + Y i t + u Y ' i Le = q''' = M i i Q R T = M i i Y i,, ; Y i = M i i ; Let T = T T, = T T,, Y = Y i C P Y i,, Q R C P T ad T, Y t + u Y ' Le t T + Y Add species energy equations for Le = 1: T + Y is constant, ie doesn t vary with reaction but If Le is not exactly 1, small deviations in Le thus T will have large impact on ω due to high activation energy Energy equation may have heat loss in q term, not present in species conservation equation Y + u T + Y = M i i Y i,, Y i,, ' T + Y = 0 10 5
Conservation equations - comments Outside of a thin reaction zone at x = 0 dt dx k d T uc P dx = 0;u = m = constant; nd order ODE A Boundary conditions upstream of reaction zone: x = 0,T = T ad ; x,t T Tx = T + T ad T e x ; k k = uc P S L C P Boundary conditions downstream of reaction zone: x = 0,T = T ad ; x,t T ad Tx = T ad = constant Temperature profile is exponential in this convectiondiffusion 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 δ 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 11 Schematic of deflagration from Lecture 1 Reaction zone 000K Product concentration Temperature Direction of propagation Speed relative to unburned gas = S L Reactant concentration 00K Distance from reaction zone Convection-diffusion zone S L = 0-6 mm 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 1 6
Conservation equations - comments In limit of infinitely thin reaction zone, T does not change but dtdx does; integrating across reaction zone u dt dx d T dx dx dx = 0 0 ' dt dt dx x= ' dt dt dx x= dx x=0 dx x=0 0 q''' dx C P ut ] 0 q''', = + k dx = 0 0 dt dx q''' Adx ka = Note also that from temperature profile: Tx = T + T ad T e x x 0 ' Tx = T ad = constant x 0 ' dt dt -, = T ad T + dx x= dx x=0 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 1 0 0 = q'''dv ka, = mq R + ka = -S L AC P T ad T - ka 0 q''' C P dx = T ad T - Conservation of momentum Apply conservation of momentum to our control volume results in Navier-Stokes equations: u t + u u = P + g + µ u or written out as individual components u u u + u + v t x y =-P x +g x +µ u x + v ' x momentum y v v v + u + v t x y =-P y +g y+µ u x + v ' y momentum y This is just Newton s nd Law, rate of change of momentum = dmudt = ΣForces Left side is just dmudt = mdudt + udmdt Right side is just ΣForces: pressure, gravity, viscosity 14 7