Post on 23-Feb-2016
description
AME 513
Principles of Combustion
Lecture 7Conservation equations
2AME 513 - Fall 2012 - Lecture 7 - Conservation equations
Outline Conservation equations
Mass Energy Chemical species Momentum
3AME 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
4AME 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
5AME 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
6AME 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
7AME 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
8AME 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:
9AME 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)
which leads to
Special case: 1D, steady (∂/∂t = 0), constant rD
Note if rD = constant and rD = k/CP and there is only a single reactant with heating value QR, then q’’’ = -QRMiwi and 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”)
10AME 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 w due to high activation energy Energy equation may have heat loss in q’’’ term, not present in
species conservation equation
11AME 513 - Fall 2012 - 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 = -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
12AME 513 - Fall 2012 - 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
13AME 513 - Fall 2012 - 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
14AME 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