### Tag: lmtd

LMTD Correction Factor Charts

## LMTD Correction Factor Charts

Heat transfer rate in the exchanger is represented by

``q = U * A * F * LMTD``

here F (< 1) is interpreted as a geometric correction factor, that when applied to the LMTD (Log Mean Temperature Difference) of a counter flow heat exchanger, provides the effective temperature difference of the heat exchanger under consideration.

It is a measure of the heat exchanger’s departure from the ideal behavior of a counter flow heat exchanger having the same terminal temperatures. The F-LMTD method is widely used in heat exchanger analysis, particularly for heat exchanger selection, (sizing problems) when as a result of the process requirements the temperatures are known and the size of the heat exchanger is required.

Log Mean Temperature Difference is defined as

``LMTD = (ΔT1 - ΔT2) / ln(ΔT1 / ΔT2)``

where,

````ΔT1 = T1 - t2`
`ΔT1 = T2 - t1````

T1, T2 are inlet and outlet temperature of Fluid 1;  t1, t2 are inlet and outlet temperature of Fluid 2.

Log Mean Temperature Difference Correction Factor F is dependent on temperature effectiveness P and heat capacity rate ratio R for a given flow arrangement. Temperature effectiveness P is different for each fluid of a two fluid exchanger.

For fluid 1, it is defined as the ratio of the temperature range of fluid 1 to the inlet temperature difference.

``P1 = ( T2 - T1 ) / ( t1 - T1 )``

Heat Capacity Ratio R is defined as

``R1 = ( t1 - t2 ) / ( T2 - T1 )``

### N (Shell) – 2M (Tube) Pass Tema E

Following general equation is used for shell and tube heat exchanger having N shell passes and 2M tube passes per shell.

````S = (R1² + 1)0.5 / (R1 - 1)`
`W = [(1 - P1.R1)/(1 - P1)]1/N`
`F = S.ln(W)/ ln[( 1 + W - S + S.W) /( 1 + W + S - S.W)]````

For limiting case of R1 = 1,

````W' = (N - N.P1)/( N - N.P1 + P1 )`
`F = 20.5 [(1 - W')/ W' ]/ln[( W'/(1-W') + 1/20.5)/( W'/(1-W') - 1/20.5)]````

For plotting the correction factor charts P1 values are listed from 0.01 to 1 with increment of 0.01 and then F values are calculated for each P1 and R1 based on above equations.

For following type of exchangers F values depend on NTU along with P1 and R1, where NTU is defined as number of transfer units. Range of NTU values are listed from 0.01 to 32 and F value is calculated for each value. LMTD Correction factor, F is determined as following –

``F = [ 1/(NTU1.(1 - R1))].ln[(1 - R1.P1)/(1 - P1)]  ---- (1)``

For limiting case of R1 = 1,

``F = P1 / [NTU1 . (1 - P1)] ---- (2)``

### Cross Flow Fluid 1 Unmixed Tema X

Following relation is used to calculate P1 using NTU.

````K = 1 - exp(-NTU1)`
`P1 = [1 - exp(-K.R1)]/ R1````

F factor is calculated as per equations (1) & (2).

### Cross Flow Fluid 2 Unmixed Tema X

Following relation is used to calculate P1 using NTU.

````K = 1 - exp(-R1.NTU1)`
`P1 = 1 - exp(-K/R1)````

F factor is calculated as per equations (1) & (2).

### Cross Flow Both Fluid Mixed Tema X

Following relation is used to calculate P1 using NTU.

````K1 = 1 - exp(-NTU1)`
`K2 = 1 - exp(-R1.NTU1)`
`P1 = 1/[1/K1 + R1/K2 - 1/NTU1]````

F factor is calculated as per equations (1) & (2).

In a similar way, LMTD correction charts can be prepared for different type of geometries based on relation between NTU, P & R.