Lecture 21ChE 333 1

Effectiveness Concept for Heat Exchangers

The Design Equation for a Heat Exchanger

QH = UA

∆T2 – ∆T1

ln ∆T2

∆T1

= UA∆T lm

A typical problem in the analysis of a heat exchanger is the Performancecalculation. That is, we are asked , given inlet conditions to evaluate howthe exchanger performs, that is what are the outlet temperatures. With theequation given above, the solution may be reached only by trial-and-error.

EffectivenessAn alternate approach lies in the notion of exchanger effectiveness, E.

E = actual heat transfermaximum possible heat transfer

Overall Energy BalanceThe actual heat transfer is given by the energy balance

(wCp ∆T)hot = (wCp∆T)cold

The maximum possible temperature rise is the difference between thetemperatures of the two entering streams (Thin - Tcin). Which fluidundergoes the maximum temperature rise ? Of course, it is the one withthe least heat capacitance

(wCp)min (∆T)max = (wCp)max (∆T)min

It follows then thatQmax = (wCp)min (Thin - Tcin)max

Lecture 21ChE 333 2

Definitions of Effectiveness

For the Double-Pipe Heat Exchanger there are four possible cases:

Co-Current Counter-CurrentHot Fluid minimum Case 1 Case 3Cold Fluid minimum Case 2 Case 4

Case 1- Co-Current flow, hot fluid minimal

E =

wCp hTh1 – Th2

wCp hTh1 – Tc1

=Th1 – Th2

Th1 – Tc1

Case 2- Co-Current flow, cold fluid minimal

E =

wCp cTc2 – Tc1

wCp cTh1 – Tc1

=Tc2 – Tc1

Th1 – Tc1

Case 3- Counter-Current flow, hot fluid minimal

E =

wCp hTh1 – Th2

wCp hTh1 – Tc2

=Th1 – Th2

Th1 – Tc2

Case 4- Counter-Current flow, cold fluid minimal

E =

wCp cTc1 – Tc2

wCp cTh1 – Tc2

=Tc1 – Tc2

Th1 – Tc2

Lecture 21ChE 333 3

Number of Transfer Units

Recall the definition of the ratio of thermal capacitances

R =

WCp c

WCp h

= Cc

Ch

= Th1 – Th2

Tc2 – Tc1

Also we can reexamine the Design Equation and rewrite it in the followingform:

lnTh2 – Tc2

Th1 – Tc1

= – UACc

1 + R

or

Th2 – Tc2

Th1 – Tc1

= e– UAC c

1 + R

We need to express this temperature ratio in terms of the effectiveness, E.

A good deal of algebra leads for Case 1 to

E = 1 + e– UA

C c1 + R

1 + R

For case 2 the equation is the relation is very similar and indeed would bethe same if R were replaced by Rmin = Cmin/Cmax.

E = 1 – e– UA

Cmin1 + R min

1 + Rmin

Lecture 21ChE 333 4

For case 3 and case 4, the equation can be expressed as a single relation.

E = 1 – e– UA

C min1 – Rmin

1 + Rmine– UACmin

1 – Rmin

We can define a dimensionless group as the Number of Transfer Units(NTU)

NTU = UACmin

This whole concept can be extended to all kinds of exchangerconfigurations, e.g.,.shell and tube with n tube passes and one shell pass; across-flow exchanger.

Cross-flow Heat Exchangers

Types - MixedUn-Mixed

Example Automobile radiator

Cross-flowMixed/Unmixed Exchangerunmixed stream- minimal stream

E = R 1 – exp – R 1 – e–NTU

mixed stream- minimal stream

E = 1 – exp – R 1 – e –RNTU

Lecture 21ChE 333 5

Unmixed/Unmixed ExchangerCross-flow

E = 1 – exp R NTU 0.22 exp –R NTU 0.78 – 1