## 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.

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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.

Spreadsheet for LMTD Correction factor charts.

### References

• Journal of Heat Transfer, Vol. 125, June 2003 – A General Expression for the Determination of the Log Mean Temperature Correction Factor for Shell and Tube Heat Exchangers – Ahmad Fakheri
• Handbook of Heat Transfer 3rd Edition – Chapter 17 – Heat Exchangers by R.K. Shah and D.P. Sekulic

## Double Pipe Heat Exchanger Design

This article shows how to do design for Double Pipe Heat Exchanger and estimate length of double pipe required.

Obtain flowrate (W ), inlet, outlet temperatures and fouling factor for both hot and cold stream. Calculate physical properties like density (ρ), viscosity (μ), specific heat (Cp) and thermal conductivity (k) at mean temperature. Determine heat load by energy balances on two streams.

````Q = mH.CpH(THot In - THot Out)`
`  = mC.CpC(tCold Out - tCold In)````

where,
mH , mC: Mass flow rate of Hot and Cold Stream
CpH , CpC: Specific Heat of Hot and Cold Stream
THot In , THot Out: Inlet and outlet temperature of Hot Stream
tCold In , tCold Out: Inlet and outlet temperature of Cold Stream

Calculate Logarithmic Mean Temperature Difference (LMTD)

``LMTD = (ΔT1 - ΔT2)/ln( ΔT1 / ΔT2)``

For Counter-current flow

````ΔT1 = THot In - tCold Out`
`ΔT2 = THot Out - tCold In````

For Co-current flow

````ΔT1 = THot In - tCold In`
`ΔT2 = THot Out - tCold Out````

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Calculate Film Coefficient
Allocate hot and cold streams either in inner tube or annular space. General criteria for fluid placement in inner tube is corrosive fluid, cooling water, fouling fluid, hotter fluid and higher pressure stream. Calculate equivalent diameter (De) and flow area (Af) for both streams.

Inner Tube

````De = Di`
`Af = π Di²/4````

Annular Space

````De = D1 - Do`
`Af = π (D1² - Do²)/4````

where,
Di : Inside Pipe Inner Diameter
Do : Inside Pipe Outer Diameter
D1 : Outside Pipe Inner Diameter

Calculate velocity (V), Reynolds No. (Re) and Prandtl No. (Pr) number for each stream.

````V = W / ( ρ Af )`
`Re = De V ρ / μ`
`Pr = Cp μ / k````

For first iteration a Length of double pipe exchanger is assumed and heat transfer coefficient is calculated. Viscosity correction factor (μ / μw)0.14 due to wall temperature is considered 1.

For Laminar Flow (Re <= 2300), Seider Tate equation is used.

``Nu = 1.86 (Re.Pr.De/L )1/3(μ/ μw)0.14``

For Transient & Turbulent Flow (Re > 2300), Petukhov and Kirillov equation modified by Gnielinski can be used.

````Nu = (f/8)(Re - 1000)Pr(1 + De/L)2/3/[1 + 12.7(f/8)0.5(Pr2/3 - 1)]*(μ/μw)0.14`
`f  = (0.782* ln(Re) - 1.51)-2````

where,
L : Length of Double Pipe Exchanger
μw : Viscosity of fluid at wall temperature
Nu : Nusselts Number (h.De / k)

Estimate Wall Temperature

Wall temperature is calculated as following.

``TW = (hitAve + hoTAveDo/Di)/(hi + hoDo/Di)``

where,
hi : Film coefficient Inner pipe
ho : Film coefficient for Annular pipe
tAve : Mean temperature for Inner pipe fluid stream
TAve : Mean temperature for Annular fluid stream

Viscosity is calculated for both streams at wall temperature and heat transfer coefficient is multiplied by viscosity correction factor.

Overall Heat Transfer Coefficient
Overall heat transfer coefficient (U) is calculated as following.

``1/U = Do/hi.Di + Do.ln(Do/Di)/2kt + 1/ho+ Ri.Do/Di + Ro``

where,
Ri : Fouling factor Inner pipe
Ro : Fouling factor for Annular pipe
kt : Thermal conductivity of tube material

Calculate Area and length of double pipe exchanger as following.

````Area = Q / (U * LMTD )`
`L = Area / π * Do````

Compare this length with the assumed length, if considerable difference is there use this length and repeat above steps, till there is no change in length calculated.

Number of hair pin required is estimated as following.

``N Hairpin = L / ( 2 * Length Hairpin )``

Calculate Pressure Drop
Pressure drop in straight section of pipe is calculated as following.

``ΔPS = = f.L.G²/(7.5x1012.De.SG.(μ/ μw)0.14)``

where,
ΔP : Pressure Drop in PSI
SG : Specific Gravity of fluid
G : Mass Flux ( W / Af ) in lb/h.ft²

For Laminar flow in inner pipe, friction factor can be computed as following.

``f = 64/Re``

For Laminar flow in annular pipe.

````f = (64 / Re) * [ (1 - κ²) / ( 1 + κ² + (1 - κ²) / ln κ) ]`
`κ = Do / D1````

For turbulent flow in both pipe and annular pipe

``f = 0.3673 * Re -0.2314``

Pressure Drop due to Direction Changes

For Laminar Flow

``ΔPR = 2.0x10-13. (2NHairpin - 1 ).G²/SG``

For Turbulent Flow

``ΔPR = 1.6x10-13. (2NHairpin - 1 ).G²/SG``

Total Pressure Drop

``ΔPTotal = ΔPS + ΔPR``

Spreadsheet for Double Pipe Exchanger Design