Maximum rate of heat transfer in Effectiveness NTU method for a heat exchanger

Question

The effectiveness NTU method is used for determining the heat transfer rate for a heat exchanger, with known inlet temperatures and surface area.

It makes use of effectiveness which is defined as

$$\epsilon = \frac{Q_{actual}}{Q_{max}}$$

What the term in the denominator truly represents?

I read that maximum rate of heat transfer is obtained in a counter flow heat exchanger with given inlet temperatures and heat capacity rates. Why is this so?

Answer 1

$q_{max }$ is the maximum heat that could be transferred between the fluids per unit time, in the ideal case that the temperature difference between the input and output of a flow is equal to the maximum possible (see below).

---

$q_{max }$ is calculated as the product of :

• the maximum temperature difference i.e. the hot inlet $T_{h,i}$ minus the the cold inlet temperature $T_{c,i}$ $$\Delta T_{max} = T_{h,i}- T_{c,i}$$
• the minimum product of mass rate and heat capacity
  $$C_{min}= \min\left(\dot{m}_h\cdot c_{p,h}, \dot{m}_c\cdot c_{p,c}\right)$$

so $q_{max }$ is given by: $$q_{max } = C_{min}\Delta T_{max}$$

The reason that is done, because in the best case scenario, the maximum heat is transferred when the hot reaches the inlet cool temperature (and vice versa). Beyond that point there is no exchange if there is no temperature difference.

The minimum $\dot{m}\cdot c_{p}$ is selected because depending on the properties, -in the generic case - only one of the flows will reach the limiting temperature (be that cold or hot). That will be depended on the mass flow, and on the material heat capacity properties (i.e. how much energy is required to increase by 1 deg the temperature of 1 kg).