# 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?