The document discusses the governing equations for reacting flows, including conservation of mass, momentum, molecular species, and energy. It outlines the continuity, momentum, species transport, and energy equations. The species transport equation accounts for convection, diffusion, and chemical reaction sources. The energy equation considers changes in enthalpy due to convection, diffusion, pressure work, and radiation. Simplifications are discussed under certain assumptions, such as a single diffusion coefficient and negligible pressure work/radiation terms, in which case enthalpy behaves as a passive scalar. Other relationships presented include the equation of state and definitions of specific heat capacity and density.

1. Reactive Flows
Dr. Mohammad Jadidi
(Ph.D. in Mechanical Engineering)
Combustion modeling in HRSG boiler

2. Presented by: Mohammad Jadidi 2
Equations governing reacting flowsReactive Flows
 Conservation equations
 Continuity equation (conservation of mass)
 Transport of momentum
 Transport of molecular species
 Transport of Energy
 Equation of State
 Turbulence Transport
Current lecture
See next lecture
 Favre averaging (density weighted averaging)

3. Presented by: Mohammad Jadidi 3
Governing equations for combusting flows (1)Reactive Flows
Continuity
It should be noted that density in combusting flows is a
variable, and depends on pressure, temperature and
species concentration.
Momentum equations
𝜏𝑖𝑗 is the viscous stress tensor and 𝐹𝑖 is
the body force (which includes gravity)

4. Presented by: Mohammad Jadidi 4
Governing equations for combusting flows (2)Reactive Flows
Transport equations for species (k)
1- Rate of change
of mass of
species k
2- Net rate of
decrease of mass
of species k due
to convection
3-Net rate of
increase of mass of
species k due to
diffusion
4-Net rate of increase
of mass of species k
due to sources
(Reaction:
(It is also shown by 𝑅 𝑘)
Needs modeling

5. Presented by: Mohammad Jadidi 5
Governing equations for combusting flows (3)Reactive Flows
Transport equations for species (k)
It is common practice to assume a single diffusion coefficient for all species
Whilst the single D assumption is not always accurate (and may in fact be quite
inaccurate) it is very attractive, since it enables far-reaching simplification of
combustion calculations.

6. Presented by: Mohammad Jadidi 6
Governing equations for combusting flows (4)Reactive Flows
Energy equation
Rate of
change of
enthalpy
Net rate of
decrease of
enthalpy due
to convection
Net rate of increase of
enthalpy due to diffusion
a along gradients of
enthalpy
Net rate of increase of enthalpy
due to mass diffusion a along
gradients of species concentration
Net rate of increase
of enthalpy due to
pressure work
Net rate of increase
of enthalpy due to
radiative heat
transfer
 ℎ is the mixture enthalpy per unit mass
 ℎ 𝑘 is the specific enthalpy of species 𝑘, and the summation is carried out over all 𝑁
species considered in the chosen reaction mechanism

7. species Schmidt number (𝑆𝑐 𝑘 )
Presented by: Mohammad Jadidi 7
Governing equations for combusting flows (5)Reactive Flows
Prandtl number (𝜎ℎ )
Lewis number (𝐿𝑒 𝑘 )
Energy equation

8. Presented by: Mohammad Jadidi 8
Reactive Flows
If a single diffusion coefficient is used( i.e. 𝐷 𝑘= 𝐷 for 𝑘 = 1, 2, … 𝑁)
If Lewis number is unity ( 𝐿𝑒 𝑘 = 1)
For low-speed flows ( Τ𝜕𝑝 𝜕𝑡 = 0)
If the radiation source term is also small ( 𝑆𝑟𝑎𝑑~0)
The enthalpy is a conserved or passive scalar
Why is it
important?
Energy equation
Governing equations for combusting flows (6)

9. Presented by: Mohammad Jadidi 9
Governing equations for combusting flows (7)Reactive Flows
Some combustion models do not require
a transport equation for enthalpy: for example,
in the laminar flamelet model the temperature
is obtained from the laminar flamelet library
curves. Other combustion submodels,
however, require the solution of the transport
equation for enthalpy.

10. The temperature can be calculated from
the enthalpy by means of
where ∆ℎ 𝑓𝑘 is the enthalpy of formation.
Presented by: Mohammad Jadidi 10
Governing equations for combusting flows (8)Reactive Flows Other relationships
The total of mass fractions of fuel, oxidant
and inert species is equal to 1
The average value of the specific heat
𝑐 𝑝 is defined as follows
local density of the mixture
where 𝑀𝑊𝑘is the molecular weight of species k and
𝑅 𝑢 is the universal gas constant (8.314 kJ/kmol.K).

11. 11
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Reactive Flows
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Dr. Mohammad Jadidi
(Ph.D. in Mechanical Engineering)
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