Tag: heat exchanger

Heat Exchanger Rating (Bell-Delaware Method)

Heat Exchanger Rating (Bell-Delaware Method)

Stream Analysis Heat Exchanger

In Bell Delaware method, the fluid flow in the shell is divided into a number of individual streams. Each of these streams introduces a correction factor which is used to correct heat transfer coefficient and pressure drop across the shell. This article gives step-by-step guidance on doing heat exchanger rating analysis based on Bell-Delware method.

Shell Side Heat Transfer Coefficient, hs

Cross flow area, Sm is the minimum flow area in one baffle space at the center of the tube bundle. It is calculated by following equation:

Sm = B[(Ds - DOTL) + (DOTL - Do)(PT - Do)/PT,eff ]

where, PT is tube pitch, B is central baffle spacing, DOTL is outer tube limit diameter, Ds is shell diameter and Do is tube outside diameter.

PT,eff = PT for 30° and 90° layouts
PT,eff = 0.707*PT for 45° layout

Shell side cross flow mass velocity, GS is defined as:

GS = mS/Sm

where, mS is shell side mass flow rate. Shell side Reynolds number ReS is then calculated from

ReS = Do.GS / μS

where, μS is the shell side fluid dynamic viscosity at average bulk temperature.

Shell side Prandtl number PrS is calculated as following :

PrS = CP,SS / kS

where, CP,S is the shell side fluid specific heat and kS is the shell side fluid thermal conductivity.

Colburn j-factor for an ideal tube bank is defined as:
Colburn j-factor
where a1, a2, a3 and a4 are correlation constants listed below.
Correlation constants j factor
The ideal tube bank based coefficient is calculated from –
ideal heat transfer coefficient
where, μS,W is shell side fluid viscosity at wall temperature.

Correction factor for Baffle Window Flow, JC

The factor JC accounts for heat transfer in the baffle windows. It has a value of 1.0 for exchanger with no tubes in the windows.

JC = 0.55 + 0.72FC
FC = 1 - 2FW
FW = (θCTL - Sin(θCTL))/2π
θCTL = 2cos-1(Ds(1 - 2*Bc/100)/DCTL)

where, Bc is segemental baffle cut in %.

Correction factor for Baffle Leakage, JL

The correction factor JL considers the effects of the tube-to-baffle and shell-to-baffle leakage streams on heat transfer.

JL = 0.44(1-rS) + (1-0.44(1-rS))exp(-2.2rL)
rS = Ssb /(Ssb + Stb)
rL = (Ssb + Stb)/ Sm
Ssb = Ds*DSB(π - 0.5θDS)
Stb = (π/4)((Do+LTB)2 - Do2)Nt(1-FW)
θDS = 2cos-1(1 - 2Bc/100)

where, Nt is number of tubes, DSB is diametral clearance between shell & baffle and LTB is diametral clearance between tube and baffle.

Correction factor for Bundle Bypass effects, JB

Bundle bypass correction factor JB accounts for the bundle bypass stream flowing in the gap between the outermost tubes and the shell. The number of effective rows crossed in one cross flow section, Ntcc between the baffle tips is provided by following equation.

Ntcc = (Ds/Pp)(1 - 2Bc/100)
Pp = PT 30.5/2 for 30° layout
Pp = PT / 20.5 for 45° layout
Pp = PT for 90° layout

Ratio of sealing strips to tube rows rss is provided by

rss = Nss/ Ntcc

where Nss is number of sealing strips (pairs) in one baffle.

The bundle bypass flow area, Sb is defind as

Sb = B(Ds - DOTL - Do/2)

where, B is central baffle spacing. Correction factor JB is then calculated as following –

JB = exp(-Cj(Sb / Sm)(1 - (2rss)1/3)) for rss < 0.5
JB = 1 for rss >= 0.5
Cj = 1.35 for ReS < 100
Cj = 1.25 for ReS >= 100

Correction factor for adverse temperature gradient, JR

The factor JR accounts for the decrease in the heat transfer coefficient with downstream distance in laminar flow.

Ntcw = (0.8/Pp)(Ds(Bc/100) - (Ds-(DOTL-Do))/2 )
NB = 1 + (int)(L - 2Ls - LBIn - LBOut)/(Bc/100)
NC = (Ntcw + Ntcc)(1 + NB)
JRL = (10/NC)0.18
JR = 1, ReS > 100
JR = JRL + (20-ReS)(JRL - 1)/80, ReS <= 100, ReS > 20
JR = JRL, ReS <= 20

where, L is tube length, Ls is tubesheet thickness, LBIn is inlet baffle spacing and LBOut is outlet baffle spacing.

Correction factor for unequal baffle spacing, JS

n1 = 0.6, ReS >= 100
n1 = 1/3, ReS < 100
JS = ((NB-1)+(LBIn/B)1-n1 + (LBOut/B)1-n1)/((NB-1)+(LBIn/B) + (LBOut/B))

Shell side heat transfer coefficient is calculated as

hs = hIdeal(JC.JL.JB.JS.JR)

Shell Side Pressure Drop, ΔPs

Friction factor for ideal tube bank is calculated as following –
friction factor for ideal tube bank
where b1, b2, b3 and b4 are correlational constants listed below.
friction factor constants
Pressure drop for an ideal tube bank is calculated from

ΔPIdeal = 2f(GS²/ρS)(μSS,W)0.14 Ntcc

Correction factor for Baffle Leakage, RL

RL = exp(-1.33(1+rS)rLp)
p = 0.8 - 0.15(1+rS)

Pressure drop for window section, ΔPW

Following terms are calculated as –

SWG = (Ds²/8)(θDS - Sin(θDS))
SWT = Nt.FW(πDo²/4)
GW = mS/(Sm.SW)0.5
DW = 4.SW /(π.Do.Nt.FW + θDS.Ds)

Pressure drop for laminar and turbulent flow is calculated.

ΔPW, Turb = NB.RL(2+0.6*Ntcw).GW²/(2.ρS)

pressure drop for laminar flow

ΔPW = ΔPW, Turb , ReS >= 100
ΔPW = ΔPW,Laminar , ReS < 100

Correction factor for Bundle Bypass effect, RB

RB = exp(-Cr(Sb / Sm)(1 - (2rss)1/3)) for rss < 0.5
RB = 1 for rss >= 0.5
Cr = 4.5 for ReS < 100
Cr = 3.7 for ReS >= 100

Correction factor for unequal baffle spacing, RS

n = 0.2, ReS >= 100
n = 1.0, ReS < 100
RS = 0.5((B/LBIn)2-n + (B/LBOut)2-n)

Pressure drop in Central Baffle spaces, ΔPC is defined as –

ΔPC = (NB - 1)ΔPIdeal.RL.RB

Pressure drop in entrance & exit baffle spaces, ΔPE is calculated as –

ΔPE = ΔPIdeal(1 + Ntcw/Ntcc).RB.RS

Shell side pressure drop is calculated as following –


Tube Side Heat Transfer Coefficient, ht

Reynold’s number and Prandtl number are calculated as following –

ReT = Di.v.ρtt
PrT = Cp,tt/kt

where, Di is tube inside diameter, v is velocity, ρt is density, μt is viscosity, kt is thermal conductivity and Cp,t is specific heat for fluid on tube side.

For laminar flow, ReT < 2300, Sieder and Tate correlation is used for Nusselt’s nubmer.

Nu = 1.86(ReT.PrT.Di/Leff)1/3
Leff = L - 2*Ls

For turbulent flow, ReT > 10,000, following equation developed by Petukhov-Kirillov can be used.

Nu = (f/2)ReT.PrT/(1.07+12.7(f/2)0.5(PrT2/3-1))
f = (1.58 ln(ReT) - 3.28)-2

For transient flow, Nusselt number can be interpolated from Nu Laminar & Nu Turbulent.

Heat transfer coefficient is calculated as following –

ht = Nu.(kt/Di)(μtt, w)0.14

Tube Side Pressure Drop, ΔPt

Tube side pressure drop is calculated by following equation –

ΔPt = (4.f.Leff.Np/Di + 4.Np)ρt.v²/2

where, Np is number of tube passes.

Overall Heat Transfer Coefficient, U

Resistance due to tube wall is calculated as following

Rtube = Do/(2.ln(Do/Di).ktube)

where, ktube is thermal conductivity of tube material. Overall clean heat transfer coefficient, UClean is calculated as per below equation

UClean = 1/(hS + Do/(Di.ht) + Rtube)

Overall dirty heat transfer coefficient, UDirty is calculated as per below expression

UDirty = 1/(1/UClean + fshell + ftube)

where, fshell & ftube are fouling factors for shell and tube side.

Heat transfer coefficient required, URequired is calculated as following

URequired = Q /(A x LMTDcorrected)

where, Q is heat duty, A is heat transfer area and LMTDcorrected is corrected logarithmic mean temperature difference.

Over Surface, % = (UClean/URequired - 1)*100
Over Design,  % = (UDirty/URequired - 1)*100

Web based calculation available at CheCalc.com


Spreadsheet for Heat Exchanger Rating based on Bell-Delaware Method


LMTD Correction Factor Charts

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)


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.


Spreadsheet for LMTD Correction factor charts.


  • 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

Double Pipe Heat Exchanger Design

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

Determine Heat Load
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)

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

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

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

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)

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

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)

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