Overall effectiveness of 3 HX's in series

[rikhaz]

Dear all,

I have a question about three heat exchangers in series. For each individual heat exchanger the effectiveness is known, and I have to show that the overall effectiveness is given by:

eff = [ eff1 + eff2 + eff3 - eff1*eff2*eff3*C + (C+1)*(eff1*eff2*eff3*(C + 1) - eff1*eff2 - eff2*eff3 - eff3*eff1 ) ] / [ 1 - C*(eff1*eff2 + eff2*eff3 + eff3*eff1) + C*(C+1)*eff1*eff2*eff3 ]

In which eff1, eff2 and eff3 are the effectiveness of each individual heat exchanger and C is equal to Cmin / Cmax, the ratio of the smaller capacity on one side to the larger capacity on the other side in the heat exchanger (equal for each individual heat exchanger).

Can anyone help me with this?

Kind regards,

Rik

 

[KataKoniK]

Dear Rik,

Thank you for your question about the overall effectiveness of three heat exchangers in series. To begin, let's define the terms used in the equation you provided. The effectiveness (eff) of a heat exchanger is a measure of how well it transfers heat from one fluid to another. It is expressed as the ratio of the actual heat transfer rate to the maximum possible heat transfer rate. In your case, eff1, eff2, and eff3 are the effectiveness of each individual heat exchanger.

Now, let's break down the equation to understand how it is derived. The first term in the numerator, eff1 + eff2 + eff3, represents the sum of the individual effectiveness of each heat exchanger. Next, we have the term - eff1*eff2*eff3*C. This term takes into account the decrease in overall effectiveness due to the unequal capacities of the heat exchangers. As C is the ratio of the smaller capacity to the larger capacity, multiplying it with the product of the individual effectiveness values will account for the decrease in overall effectiveness.

Moving on to the next term, (C+1)*(eff1*eff2*eff3*(C + 1) - eff1*eff2 - eff2*eff3 - eff3*eff1 ), we can see that it is a combination of the individual effectiveness values and the capacity ratio. This term is derived from the fact that the overall effectiveness is also affected by the interaction between the three heat exchangers. This term takes into account the positive and negative effects of this interaction on the overall effectiveness.

Finally, in the denominator, we have the term 1 - C*(eff1*eff2 + eff2*eff3 + eff3*eff1) + C*(C+1)*eff1*eff2*eff3. This term is derived from the fact that the overall effectiveness is also affected by the cumulative effect of the individual heat exchangers. It takes into account the positive and negative effects of this cumulative effect on the overall effectiveness.

I hope this explanation helps you understand the equation better. If you have any further questions, please don't hesitate to ask.