Proving Heat exchanger equations

[metalscot]

For a double-pipe heat exchanger operating with countercurrent flow, the operating equation is:

S = (1 - exp X)/ (R - exp X) X = ((U*A)/(Ma*Ca)) (1-R)

where R and S are defined by:

R = (Ma*Ca)/( Mb*Cb) = (Tb2 - Tb1)/(Ta1 - Ta2)

S = (Ta1 - Ta2)/(Ta1- Tb1)

(i) Show how there is a potential problem using the above equation if the value of R is equal to 1.

The potential problem with the equation for S can be solved using an appropriate substitution. For small values, the exponential can be written as: exp(cy)≈ 1+ cy

(ii) Use the substitution

y=1-R

to show that as the value of R approaches 1, the exponential disappears and the operating equation reduces to a simplified function of (UA/MC).


Attemped solution:

(i) For R to = 1

R = (Ma*Ca)/( Mb*Cb) = (Tb2 - Tb1)/(Ta1 - Ta2) = 1

This would mean that (Tb2 - Tb1) and (Ta1 - Ta2) were the same, say Tb2=30 Tb1=10 Ta1=40 and Ta2=20

For the value of S however because Ta1 is used twice, S would equal 2/3

But when R= 1 in the operating equation S must also equal 1.

(ii) exp(cy)≈ 1+ cy

y=1-R, as R approaches 1 y approaches 0.
when y becomes 0 cy becomes 0 meaning that exp(cy)≈1

when exp (cy) is 1, X=0 in the operating equation when X = 0, (UA/MaCa) or (1-R) must equal zero. From the previous equation we know that it is 1-R that equals 0 leaving us with the simplified (UA/MaCa)

Can someone please tell me if my methods and answers are correct?
Thank you in advance :)

 

[anathema]

Your methods and answers are mostly correct. However, there are a few minor mistakes and clarifications that can be made.

(i) When R = 1, it does not necessarily mean that (Tb2 - Tb1) and (Ta1 - Ta2) are the same. It just means that the ratio of (Ma*Ca)/(Mb*Cb) is equal to 1. This could happen if (Tb2 - Tb1) and (Ta1 - Ta2) are both equal to 10, or if they are both equal to 100, for example. So, the specific values of Tb2, Tb1, Ta1, and Ta2 are not necessarily important in this case. The important thing to note is that when R = 1, the ratio between the temperatures (Tb2 - Tb1) and (Ta1 - Ta2) is equal to 1, which means they are the same.

(ii) The substitution y = 1 - R is correct, but the explanation is a bit unclear. Instead of saying "when y becomes 0, cy becomes 0," you can say "when R approaches 1, y approaches 0, which means that the exponential term becomes 1, and X becomes 0." This is because when R = 1, y = 0, so the exponential term becomes exp(0) = 1, and X becomes 0.

Also, it should be noted that when R = 1, the operating equation becomes S = 0/0, which is an indeterminate form. This means that the equation cannot be solved as is, and further analysis or substitutions may be necessary to find a solution.

Overall, your understanding and approach to the problem seem to be correct. Just be careful with the notation and make sure to clearly explain your steps and reasoning.