Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

1

Effectiveness-NTU Method for Heat Exchangers

Chap. 16.2: NTU Method for Heat Exchangers

Maximum possible heat transfer:

-  Counter flow heat exchanger -  Infinitively long -  Maximum temperature difference for one fluid: ΔT = Th,i – Tc,i

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

2

Counter flow heat exchanger: Infinitively long

Maximum possible heat transfer:

hphcpc cmcm ,, ⋅<⋅ Case 1:

( )icihcpccpc TTcmTcmq ,,,max,max −⋅⋅=Δ⋅⋅=

hphcpc cmcm ,, ⋅>⋅ Case 2:

( )icihhphhph TTcmTcmq ,,,max,max −⋅⋅=Δ⋅⋅=

( ) ( )icihp TTcmq ,,max min −⋅⋅=

In other words:

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

3

Effectiveness of heat exchanger:

Effectiveness is the actual transferred heat divided by the maximum possible heat transfer.

( )( ) ( )

( )( ) ( )icihp

ohihhph

icihp

icoccpc

TTcmTTcm

TTcmTTcm

qq

,,

,,,

,,

,,,

max minmin −⋅⋅

−⋅⋅=

−⋅⋅

−⋅⋅==

ε

( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅=

p

p

cmcm

f

maxmin,NTUε

( )pcmAU⋅

⋅=

minNTU

On the other hand, effectiveness is a function of:

with Number of Transfer Units (NTU):

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

4

Effectiveness Relations:

hphcpc cmcm ,, ⋅>⋅ Parallel flow HX, Case 2:

( )( ) ( )

( )( ) icih

ohih

icihhph

ohihhph

icihp

ohihhph

TTTT

TTcmTTcm

TTcmTTcm

,,

,,

,,,

,,,

,,

,,,

min −

−=

−⋅⋅

−⋅⋅=

−⋅⋅

−⋅⋅=

ε

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅+

⋅⋅⋅−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

−

−

cpchphicih

ocoh

cmcmAU

TTTT

,,,,

,, 11ln

From LMTD method:

( )( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+⋅

⋅

⋅−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

−

−

p

p

picih

ocoh

cmcm

cmAU

TTTT

maxmin

1min

ln,,

,,

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

5

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+⋅−=

−

−

p

p

icih

ocoh

cmcm

TTTT

maxmin

1NTUexp,,

,,

Using the definition of NTU:

With the equation:

( )( ) ohih

icoc

p

p

TTTT

cmcm

,,

,,

maxmin

−

−=

⋅

⋅

This leads to:

( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+⋅−=

−

−

p

p

icih

ocoh

cmcm

TTTT

maxmin

11,,

,, ε

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

6

Resulting in:

( )( )

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+⋅−=⎟

⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+⋅−

p

p

p

p

cmcm

cmcm

maxmin

1NTUexpmaxmin

11 ε

And finally, a relation between NTU and effectiveness:

( )( )

( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+⋅−−

=

p

p

p

p

cmcm

cmcm

maxmin

1

maxmin

1NTUexp1

ε

hphcpc cmcm ,, ⋅<⋅ Parallel flow HX, Case 1:

→  Exactly the same result

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

7

Counter flow HX:

( )( )

( )( )

( )( ) ⎥

⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅−⋅−⋅

⋅

⋅−

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅−⋅−−

=

p

p

p

p

p

p

cmcm

cmcm

cmcm

maxmin

1NTUexpmaxmin

1

maxmin

1NTUexp1

ε

Effectiveness relations for other configurations:

Cross flow HX, unmixed:

( )( )

( )( ) ⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎥⎦

⎤

⎢⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅⋅−⋅⋅

⋅

⋅−= 1

maxmin

NTUexpNTUminmax

exp1 0.780.22

p

p

p

p

cmcm

cmcm

ε

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

8

NTU Relations:

( )( )

( )( )⎟

⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅

⋅+⋅−

−=

p

p

p

p

cmcm

cmcm

maxmin

1

maxmin

11ln

NTU

ε

Parallel flow HX:

Counter flow HX:

( )( )( )

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛−

⋅

⋅

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

⋅

⋅⋅−

=

1maxmin

1maxmin

1ln

NTU

p

p

p

p

cmcm

cmcm

εε

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

9

( ) ( )icihp TTcmqqqq

,,maxmax

min −⋅⋅⋅=⋅=→= εεε

Heat transfer by HX:

If the inlet temperatures, both flow rates and specific heat values, the surface area and the overall heat transfer are known, we can calculate the heat transfer between both fluids.

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

10

NTU Method for Heat Exchangers:

The NTU Method is used to evaluate heat exchangers for known inlet temperatures of the fluids and a known geometry of the heat exchanger (outlet temperatures unknown).

1) Calculate overall thermal resistance from known geometry. 2) Calculate NTU from overall thermal resistance. 3) Calculate effectiveness from NTU. 4) Calculate total heat transfer q from effectiveness and inlet temperatures. 5) Calculate outlet temperatures from known inlet temperatures and total heat transfer q.

Possible Procedure to Analyze Heat Exchanger:

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

11

Phase Change in Heat Exchanger:

Condensing vapor:

Phase change leads to constant temperature on hot side, flow direction of hot side is irrelevant.

Same result for: hphcpc cmcm ,, ⋅<<⋅

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

12

Evaporating liquid:

Phase change leads to constant temperature on cold side, flow direction of cold side is irrelevant.

Same result for: hphcpc cmcm ,, ⋅>>⋅

Phase Change in Heat Exchanger (2):

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

13

Same Fluid on Both Sides of Heat Exchanger:

( ) ( )pphphcpc cmcmcmcm ⋅=⋅→⋅=⋅ maxmin,,

( )[ ]2NTUexp15.0 ⋅−−⋅=ε ( )21ln5.0NTU ⋅−⋅−= ε

Parallel flow HX:

Counter flow HX:

NTU1NTU+

=εεε−

=1

NTU

Temperature difference ∆T is constant

Chap. 16.2: NTU Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

14