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