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Proving Heat exchanger equations

  1. Nov 11, 2011 #1
    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 :)
     
  2. jcsd
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