MECH 6120: Combustion
Coupling Kinetics with Thermodynamics
At this point in the course, we have covered the topics of thermo
dynamics and chemical kinetics – which are both fairly distinct topics
(that is, one is not derived from the other). Our next item of business
will be to couple these two topics together to predict the performance
of reactors.
In a sense, what we will do here is analogous to one of the first topics
covered in heat transfer, e.g., the ‘lumped capacity’ solution for transient
cooling of an object. To review, say we have a control mass subject to
transient heat transfer. The first law, written on a rate basis, would be
dU
dt
=
˙
Q
(1)
Simply put: the rate at which the internal energy changes is equal to
the rate of heat transfer to/from the system (no work here). The above
statement is fundamentally sound and complete – no approximations.
We can, however, develop a more descriptive prediction of the system
by making some assumptions, in particular
1. The temperature of the system, at any point in time, is nearly
uniform so that
dU
=
mdu
=
mc
v
dT
=
ρV c
v
dT
.
2. The rate of heat transfer is given by the convective rate law:
˙
Q
=

hA
(
T
s

T
∞
), where
T
s
is the surface temperature,
h
is the heat
transfer coefficient, and
A
the surface area. Since we assume the
temperature of the system is uniform, then
T
s
=
T
.
Putting these two assumptions together gives the familiar first–order
cooling law,
ρc
v
V
dT
dt
=

hA
(
T

T
∞
)
which we can solve easily for the temperature as a function of time.
What was the rationale for assuming that the temperature was uni
form throughout the system?
You might recall that this assumption
implies that the resistance to heat transfer
within
the system is sig
nificantly smaller than the resistance to heat transfer
from
the system.
Indeed, we could define the ratio of these resistances in a quantity known
as the Biot number, and our solution would be valid providing that the
Biot number was small.
We want to do something similar here – except expanded to include
chemical reactions within the system. Kinetics has provided us with rate
laws for chemical reactions, analogous to how Newton’s law of cooling
(convection) gave us a rate law for heat transfer.
And, as was done
above, we want to make a similar assumption regarding the state of
the system as it evolves in time. In particular, we will assume that the
properties of the system (
T
,
P
, mass fractions of chemical species) are
uniform
throughout the system at any point in time.
The validity of
this assumption follows a criterion similar to that for the simple heat
transfer problem, in that it assumes that transport (heat, mass, momen
tum)
within
the system is significantly faster than transport to/from the
system. This assumption is known as the well–stirred (or well–mixed)
approximation.
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 Summer '09
 Thermodynamics, Heat, Combustion, Chemical reaction, Yf

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