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In effectiveness NTU method for heat exchangers, we have effectiveness of the heat exchanger defined as

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

Where $$Q_{max} $$ is the maximum rate of heat transfer in a counter flow heat exchanger with some prescribed inlet temperature and heat capacity rates.

$$\text{when}\begin{cases} C_H< C_C, & \dot{Q}_{max} = C_H(T_{h,i}- T_{c,i})\\ C_C< C_H, & \dot{Q}_{max} = C_C(T_{h,i}- T_{c,i})\\ \end{cases} $$

What will be the maximum rate of heat transfer that will be taken when heat exchanger is balanced i.e. $$C_h = C_c $$

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  • $\begingroup$ O dear. When C_h = C_c then C_h*(T_hi-T_ci)=C_c(T_hi-T_ci) .... $\endgroup$
    – mart
    Aug 19 at 6:53
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It depends on the length of the heat exchanger.

generally, in the steady state there will be an steady temperature difference across the length of the heat exchanger. The difference will be smaller for higher lengths.

If the length is long enough, then eventually:

  • the out temperature of the hot $t_{h,o}$ will be equal to the inlet of the cool $t_{c,i}$
  • the out temperature of the cold $t_{c,o}$ will be equal to the inlet of the hot $t_{h,i}$

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Figure : counter flow heat exchanger with 6000 m ( 3kg/s mass rate of water, cool:400K, hot:800, initial condition 300K )

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Figure : counter flow heat exchanger with 60 m ( 3kg/s mass rate of water, cool:400K, hot:800, initial condition 300K )

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  • $\begingroup$ If we consider long enough length case then in a counter flow heat exchanger with heat capacity rates equal, the temperature profiles become linear and parallel. So, if at any length Th,o=Tc,i and Tc,o=Th,i then wouldnt the temperature profiles of the hot and cold fluids coincide? in which case the heat transfer will stop? $\endgroup$ Aug 19 at 5:29
  • $\begingroup$ @HarshitRajput heat transfer doesn't stop, you just can't get any more out of the system because heat only flows from hot to cold. $\endgroup$
    – Tiger Guy
    Aug 19 at 13:01
  • $\begingroup$ I have edited in the question what I want to say, that as we keep on increasing the length the temperature profiles of the hot and cold fluids will approach each other and then finally coincide $\endgroup$ Aug 20 at 12:16
  • $\begingroup$ That's what I've been trying to say all along, that if the length is loooong enough and the flow characteristics are equal then the temperatures will eventually be reversed. $\endgroup$
    – NMech
    Aug 20 at 12:37
  • $\begingroup$ Oh okay so the temperatures profiles will coincide right at long enough length? $\endgroup$ Aug 20 at 12:43
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When the two heat capacities are the same, then Hot out = cold in, the same as the CH < CC. You can't get more heat out of the hot side than the delta between the hot entering temp and the cold entering temp, so this max occurs for all conditions where the cold side heat capacity is equal to or greater than the hot side.

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