Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

1

Log Mean Temperature Difference for Heat Exchangers

Chap. 17.1: LMTD Method for Heat Exchangers

( )ohihhph TTcmq ,,, −⋅⋅=

( )icoccpc TTcmq ,,, −⋅⋅=

ohihh mmm ,, ==

ocicc mmm ,, ==

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

2

hhph dTcmdq ⋅⋅−= ,

ccpc dTcmdq ⋅⋅= ,( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

⋅+

⋅⋅−=−=−

cpchphchch cmcm

dqTTddTdT,,

11

( )ch TTdAUdq −⋅⋅=

On the other hand, the heat transfer between both sides can be calculated as:

Chap. 17.1: LMTD Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

3

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅+

⋅⋅−⋅⋅−=−

cpchphchch cmcmTTdAUTTd

,,

11

( ) ( )( )

dAcmcm

UTTTTd

TTd

cpchphch

ch ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅+

⋅⋅−=

−

−=

Δ

Δ

,,

11

Combining the heat transfer equation and the energy balance:

After integration:

( ) ( )[ ]ocohicihcpchph

TTTTqAU

cmcmAU

TT

,,,,,,1

2 11ln −−−⋅⋅

−=⎟⎟⎠

⎞⎜⎜⎝

⎛

⋅+

⋅⋅⋅−=⎟⎟

⎠

⎞⎜⎜⎝

⎛

Δ

Δ

( )12ln TTTTAUq io

ΔΔ

Δ−Δ⋅⋅= 1: x = 0

2: x = L

Chap. 17.1: LMTD Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

4

( ) ( )12

12

12 lnln TTTTAU

TTTTAUq io

ΔΔ

Δ−Δ⋅⋅=

ΔΔ

Δ−Δ⋅⋅=

1: x = 0, inlet for cold and hot 2: x = L, outlet for cold and hot

LMTD Method for Parallel Flow Heat Exchangers

icihi TTTT ,,1 −=Δ=Δ

ocoho TTTT ,,2 −=Δ=Δ

Chap. 17.1: LMTD Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

5

( )12

12

ln TTTTAUqΔΔ

Δ−Δ⋅⋅=

1: x = 0, hot inlet and cold outlet 2: x = L, cold inlet and hot outlet

LMTD Method for Counter Flow Heat Exchangers

ocihch TTTTT ,,1,1,1 −=−=Δ

icohch TTTTT ,,2,2,2 −=−=Δ

Chap. 17.1: LMTD Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

6

crosslmTFAUq ,Δ⋅⋅⋅=

LMTD Method for Cross Flow Heat Exchangers

icih

icoc

TTTT

P,,

,,

−

−=

icoc

ohih

TTTT

R,,

,,

−

−=

( ) ( )( ) ( )[ ]icohocih

icohocihcrosslm TTTT

TTTTT

,,,,

,,,,, ln −−

−−−=Δ

Chap. 17.1: LMTD Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

7

convhotcondwallconvcoldtot RRRR ,,, ++=

Overall Heat Transfer:

Cold fluid: Rcold,conv

Wall: Rwall,cond

Hot fluid: Rhot,conv Thot flow

Thot wall

Tcold wall

Tcold flow

hothotcwallcoldcoldtot AhAk

tAh

R⋅

+⋅

+⋅

=11

( )flowcoldflowhotwettottot

wet TTAUqR

AU ,,1

−⋅⋅=→=⋅

Chap. 17.1: LMTD Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

8

Log Mean Temperature Difference Method:

The LMTD Method is used to design heat exchangers for known inlet and outlet temperatures of the fluids and a known geometry of the heat exchanger.

1) Determine known or specified inlet and outlet temperatures. 2) Calculate total heat transfer from inlet and outlet temperatures and fluid properties. 3) Calculate LMTD using formula for the given heat exchanger configuration (parallel, counter, cross flow). 4) Calculate overall thermal resistance using q and LMTD. 5) Calculate geometry from overall thermal resistance and heat transfer coefficients.

Possible Procedure to Design Heat Exchanger:

Chap. 17.1: LMTD Method for Heat Exchangers

Prof. Nico Hotz

ME 150 – Heat and Mass Transfer

9