# MHE: Heat Exchanger Effectiveness

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, THin and TCin and leave the exchanger at the same (unknown) temperature TM - which is somewhere in between THin and TCin.

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 TM. 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, THin and TCin. 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:

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

##### NOTE:

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!

##### OUTCOME:

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