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• Defining LMTD as

• The total heat transfer rate for all single-pass flow arrangements (PF, CF, evaporator, condenser)

• In case of CF HEX with Ch=Cc, DTlm is indeterminate

• Therefore from L’Hospital: DTlm= DT1= DT2

1 2

1 2ln( )lm

T TT

T T

lmQ AU T

1 2 2 1 1 2( ) ( ) and h h c cT T T T T T

• In case of PF HEX, the same definition of LMTD (previous slide) is valid. But obviously

• LMTD represents the maximum temperature potential for heat transfer that can only be obtained in a counter flow heat exchanger. Therefore, the surface area required to obtain a prescribed heat transfer rate Q is smaller for a counter flow arrangement than that for a parallel flow arrangement, assuming that U is the same for both cases.

• Also note that Tc2 can exceed Th2 for counter flow but not for parallel flow.

• Example 2.4 in book

)(T and )( 222111 chch TTTTT

Multipass and Crossflow HX

• LMTD defined for single pass (PF or CF) HEX is not valid for multipass and crossflow HEX. This time by defining a mean temperature difference DTm

• We can determine DTm in terms of LMTD of CF HEX, and two quantities P and R.

mQ AU T

2 1 1 2

,2 1 1 2

( ) ( )

ln ( ) ( )h c h c

lm cfh c h c

T T T TT

T T T T

2 1

1 1 max

c c c

h c

T T TP

T T T

1 2

2 1

c h h

h c c

C T TR

C T T

Meanings of Tlm,cf P and R

• DTlm,cf is the log-mean temperature difference for a counter flow arrangement with the same fluid inlet and outlet temperatures.

• P is a measure of the ratio of the heat actually transferred to the cold fluid to the heat which would be transferred if the same fluid were to be raised to the hot fluid inlet temperature; therefore is the temperature effectiveness of the HEX on the cold fluid side.

• R is the ratio of the mcp value of the cold fluid to that of the hot fluid and it is called the heat capacity rate ratio.

Correction factor F

• With the definition of the correction factor F heat transfer rate for a multipass or crossflow HEX can be given as

• F is a nondimensional term which depends on the temperature effectiveness P, the heat capacity rate ratio R, and the flow arrangement.

,lm cfQ UAF T

Charts for correction factors

• Depending on the flow arrangement there are series of charts to obtain F using calculated values of P and R.

• Such figures are:In our textbook: Figures 2.7 through 2.14.In Incropera and DeWitt: Figures 11.10

through 11.13.• It is immaterial whether the cold fluid flows through the

shell or inside the tubes• If temperature change of one fluid is negligible, F = 1• Some examples of charts are...

• LMTD correction factor F for a shell-and-tube heat exchanger – one shell pass and two or multiple of two tube passes

• Example 2.5 in book now, also study examples 2.6-2.8

Effectiveness-Number of Transfer Units (e-NTU) for HEX Analysis

• When inlet and exit temperatures are unknown, a trial and error procedure may be needed. Instead, the method of number of transfer units (NTU) based on HEX effectiveness may be used.

• The e - NTU method is based on the fact that the inlet or exit temperature differences of a heat exchanger are a function of UA/Cc and Cc/Ch.

• The HEX heat transfer equations may be written in dimensionless form resulting in some dimensionless groups.

Dimensionless groups

1. Heat capacity rate ratio: , C* 1

2. HEX heat transfer effectiveness:

e is the ratio of the actual heat transfer rate in a HEX to the thermodynamically limited maximum possible heat transfer rate if an infinite heat transfer area were available in a counter flow HEX.

min

max

CC

C

max

Q

Q

• The actual heat transfer is obtained either by the energy given off by the hot fluid or the energy received by the cold fluid

If Ch > Cc, then (Th1-Th2) < (Tc2-Tc1)If Ch < Cc, then (Th1-Th2) > (Tc2-Tc1)

• The fluid that might undergo the maximum temperature difference is the fluid having the minimum heat capacity rate Cmin.

1 2 2 1( ) ( ) ( ) ( )p h h h p c c cQ mc T T mc T T

• Maximum heat transfer:

or

• Therefore, HEX effectiveness can be written as

• The above equation is valid for all heat exchanger flow arrangements. The value of e ranges between 0 and 1.

• For a given e and Qmax, the actual HT rate isQ = e (mcp)min(Th1-Tc1)

max 1 1( ) ( ) if p c h c c hQ mc T T C C

max 1 1( ) ( ) if p h h c h cQ mc T T C C

1 2 2 1

min 1 1 min 1 1

( ) ( )

( ) ( )h h h c c c

h c h c

C T T C T T

C T T C T T

3. Number of Transfer Units:

The third dimensionless number NTU shows the nondimensional heat transfer size of the HEX

min min

1NTU

A

AUUdA

C C

Single pass heat exchanger Cc>Ch

•Ch=Cmin and Cc=Cmax

•We had obtained

•By using definition of NTU

where the + is for counter flow and the – is for parallel flow

2 1 1 2

1 1( )exph c h c

c h

T T T T UAC C

min2 1 1 2

max

( ) exp NTU 1-h c h c

CT T T T

C

• Using

and

Th2 and Tc2 can be eliminated to obtain for CF

• If Cc < Ch, the result will be the same

1 2 2 1( ) ( ) ( ) ( )p h h h p c c cQ mc T T mc T T

1 2 2 1

min 1 1 min 1 1

( ) ( )

( ) ( )h h h c c c

h c h c

C T T C T T

C T T C T T

min max

min max min max

1 exp NTU(1-

1 ( )exp NTU(1-

C C

C C C C )

• For Cmin/Cmax=1, this result is indeterminate, but by applying L’Hospital’s rule, the following result is obtained for counter flow

• and for parallel flow

NTU

1 NTU

2NTU1(1 )

2e

• For Cmin/Cmax=0, both CF and PF HXs

• Similar expressions for all flow arrangements including multipass and crossflow can be obtained following the same approach. Some of those results are presented as tables like the following. Also there are charts like Fig 2.15 of the text book

NTU1 e

e-NTU Expressions (Table 2.2 of the book, more detail in book)

Type of HEX e(NTU,C*) NTU(e,C*)

Counterflow

Parallel Flow

Cross flow, Cmin mixed and Cmax unmixed

Cross flow, Cmax mixed and Cmin unmixed

1 to 2 shell-and-tube HEX

NTU1exp1

NTU1exp1

CC

C

1

1ln

1

1NTU

C

C

NTU1exp11

1

C

C

C

C NTUexp1exp1

NTUexp1exp11

CC

2/121NTUexp1

2/121NTUexp1

2/1211

2

C

C

CC

C

C11ln

1

1NTU

1ln1ln1

NTU CC

CC

1ln1

1-lnNTU

2/1

2112

2/12112

ln2/12

1

1NTU

CC

CC

C

LMTD and e-NTU relations

LMTD e-NTU

F = f (P, R, flow arrangement) e = f (NTU, C*, flow arrangement)

cflmTUAFQ , )11(min cThTCQ

21ln

21, TT

TTcflmTLMTD

11min

12

11min

21

cThTC

cTcTcC

cThTC

hThThC

122,211 cThTTcThTT

12

21,

11

12

cTcT

hThTR

cThT

cTcTP

max

min

max

min

pcm

pcm

C

CC

AUdACC

UA

min

1

min

NTU

Example 2.9

A two-pass tube, baffled single-pass shell, shell-and-tube HEX is used as an oil cooler. Cooling water flows through the tubes at 20oC at a flow rate of 4.082 kg/s. Engine oil enters the shell side at a flow rate of 10 kg/s. The inlet and outlet temperatures of oil are 90oC and 60oC, respectively. Determine the surface are of the HEX using both the LMTD and e-NTU methods, if the overall heattransfer coefficient based on the outside tube area is 262 W/m2K. The specific heats of water and oil are 4179 J/kgK and 2118 J/kgK, respectively.

Heat Exchanger Sizing

If inlet temperatures, one of the outlet temperatures and mass flow rates are known, we can use LMTD method for sizing problem:

1. Calculate Q and the unknown temperature2. Calculate LMTD and obtain F if necessary3. Calculate U4. Determine A from A=Q/(UFDTlm,cf)

Heat Exchanger RatingFor an available heat exchanger (size, mass flow rates, inlet temperatures and materials are known) using e-NTU method we can rate the heat exchanger:

1. Calculate C*=Cmin/Cmax and NTU=UA/Cmin

2. Determine e from appropriate charts or e-NTU equations

3. Calculate Q=e Cmin(Th1-Tc1) 4. Calculate outlet temperatures

Sizing Using e-NTU method

1. Calculate e using Cmin, Cmax and temperatures

2. Calculate C*=Cmin/Cmax

3. Calculate U4. Determine NTU from charts or equations5. When NTU is known calculate heat transfer area

from A=(CminNTU)/U

Rating using LMTD

• LMTD method can also be used for rating but for the unknown temperature trial and error approach is required.

• In general LMTD is more appropriate for sizing and e-NTU is more appropriate for rating problems.

• Variable U may complicate the calculations, next figure shows some cases where U is variable. Variation dependent on flow Reynolds number, HT surface geometry, fluid physical properties