One way of measuring a heat exchanger's performance is to
calculate its effectiveness. The heat exchanger effectiveness,
$\epsilon$ (note that this is *not* the same notation as
the book), is defined as the ratio of the actual heat transfer to
the heat transfer attainable in an infinitely long counterflow
exchanger. We choose an infinitely long exchanger since that will
yield the *maximum* heat transfer that can take place (i.e.,
eventually, enough heat will be transferred so that the driving
force will *disappear* - the streams will reach the same
temperature). To understand the rationale behind the choice of a
*counterflow* exchanger (in this definition), it is
instructive to look at both cases: an infinitely long parallel
exchanger and one an infinitely long counterflow exchanger.

In the parallel flow case, the two fluids enter the exchanger
with (typically known) inlet temperatures,
*T*_{Hin} and
*T*_{Cin} and leave the
exchanger at the same (unknown) temperature
*T*_{M} - which is somewhere in between
*T*_{Hin} and
*T*_{Cin}.

Now, in order to figure out the heat transfer (so that we might
plug it into the denominator of our effectiveness definition) one
needs to determine *T*_{M}. This can be
calculated by equating the two heat duties, $\displaystyle{q =
\dot m_Cc_C(T_M-T_{C_{in}}) = \dot m_Hc_H(T_{H_{in}}-T_M)}$, to
give $\displaystyle{T_M = \frac{\dot m_Hc_HT_{H_{in}}+\dot
m_Cc_CT_{C_{in}}}{\dot m_Cc_C+\dot m_Hc_H}}$ (i.e., the weighted
average of the temperatures). Plugging this solution into either of
the above equations then tells us our heat duty.

$\displaystyle{q = \dot m_Cc_C(\frac{\dot m_Hc_HT_{H_{in}}+\dot m_Cc_CT_{C_{in}}}{\dot m_Cc_C+\dot m_Hc_H} - T_{C_{in}}) = \dot m_Hc_H(T_{H_{in}}-\frac{\dot m_Hc_HT_{H_{in}}+\dot m_Cc_CT_{C_{in}}}{\dot m_Cc_C+\dot m_Hc_H}) }$

An alternative is to operate the exchanger in counterflow. In
this case, again, we will assume that we know the inlet
temperatures, *T*_{Hin} and
*T*_{Cin}. Also, as before
there will (eventually) be no driving force (in an infinitely long
exchanger). There are two ways in which this can happen: the hot
stream can leave at the same temperature as the cold stream inlet,
or the cold stream can leave at the same temperature as the hot
stream inlet (clearly, if *both* of these happened at the
same time the streams would have to be the same temperature
*everywhere*! (this happens when the fluids have the
*exact same* mass flow rate times heat capacity!)).

Which of the two cases actually happens in a given scenario is
determined by which of the capacity coefficients is larger, $\dot
m_Cc_C$ or $\dot m_Hc_H$ . If $\dot m_Cc_C \gt \dot m_Hc_H$
then the hot stream will leave at the same temperature as the cold
stream (and vice versa). It is easier to understand why if you
consider the case where the fluid properties of the hot and cold
streams are the *same* (except for temperature, of course),
but the *flow rate* is vastly different (so that the
capacity coefficients differ). Consider cold water rushing
exceedingly quickly through the exchanger. It will change in
temperature only a small amount since it will only be in the
exchanger for a short time. In contrast if hot water is barely
trickling through the exchanger it will have a *long* time
to cool down and will eventually reach the temperature of the cold
stream (in this example $\displaystyle{\dot m_C \gg \dot
m_Hc_H}$). In any event, in order to determine the heat transfer
(so that we might plug it into the denominator of our effectiveness
definition) all we then need to know is whether $\dot m_Cc_C$ or
$\dot m_Hc_H$ is larger since $q = (\dot mc)_{min}
(T_{H_{in}}-T_{C_{in}})$ in either case (where $(\dot mc)_{min}$
is the *smaller* one).

So, by defining the effectiveness in this way, we have a much simpler expression for our effectiveness:

$\displaystyle{\epsilon = \frac{\dot m_Cc_C(T_{C_{out}}-T_{C_{in}})}{(\dot mc)_{min}(T_{H_{in}}-T_{C_{in}})} = \frac{\dot m_Hc_H(T_{H_{in}}-T_{H_{out}})}{(\dot mc)_{min}(T_{H_{in}}-T_{C_{in}})}}$

In one of these two equations the $\dot mc$ will cancel since either the cold or the hot must be the minimum fluid!

If, for any given exchanger, the effectiveness is known we can then write our heat duty equation as:

$\displaystyle{q = \epsilon(\dot mc)_{min}(T_{H_{in}}-T_{C_{in}})}$

*without knowing the exit temperatures beforehand!*

Calculate the heat flow in an exchanger from the exchanger's effectiveness