# Heat Exchanger Rating (Bell-Delaware Method)

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, h_{s}

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

`S`_{m} = B[(Ds - D_{OTL}) + (D_{OTL} - Do)(P_{T} - Do)/P_{T,eff} ]

where, P_{T} is tube pitch, B is central baffle spacing, D_{OTL} is outer tube limit diameter, Ds is shell diameter and Do is tube outside diameter.

`P`

_{T,eff}= P_{T}for 30° and 90° layouts`P`

_{T,eff}= 0.707*P_{T}for 45° layout

Shell side cross flow mass velocity, G_{S} is defined as:

`G`_{S} = m_{S}/S_{m}

where, m_{S} is shell side mass flow rate. Shell side Reynolds number Re_{S} is then calculated from

`Re`_{S} = Do.G_{S} / μ_{S}

where, μ_{S} is the shell side fluid dynamic viscosity at average bulk temperature.

Shell side Prandtl number Pr_{S} is calculated as following :

`Pr`_{S} = C_{P,S}.μ_{S} / k_{S}

where, C_{P,S} is the shell side fluid specific heat and k_{S} is the shell side fluid thermal conductivity.

Colburn j-factor for an ideal tube bank is defined as:

where a1, a2, a3 and a4 are correlation constants listed below.

The ideal tube bank based coefficient is calculated from –

where, μ_{S,W} is shell side fluid viscosity at wall temperature.

#### Correction factor for Baffle Window Flow, J_{C}

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

`J`

_{C}= 0.55 + 0.72F_{C}`F`

_{C}= 1 - 2F_{W}`F`

_{W}= (θ_{CTL}- Sin(θ_{CTL}))/2π`θ`

_{CTL}= 2cos^{-1}(Ds(1 - 2*Bc/100)/D_{CTL})`D`

_{CTL}= D_{OTL}- Do

where, Bc is segemental baffle cut in %.

#### Correction factor for Baffle Leakage, J_{L}

The correction factor J_{L} considers the effects of the tube-to-baffle and shell-to-baffle leakage streams on heat transfer.

`J`

_{L}= 0.44(1-r_{S}) + (1-0.44(1-r_{S}))exp(-2.2r_{L})`r`

_{S}= S_{sb}/(S_{sb}+ S_{tb})`r`

_{L}= (S_{sb}+ S_{tb})/ S_{m}`S`

_{sb}= Ds*D_{SB}(π - 0.5θ_{DS})`S`

_{tb}= (π/4)((Do+L_{TB})^{2}- Do^{2})Nt(1-F_{W})`θ`

_{DS}= 2cos^{-1}(1 - 2Bc/100)

where, Nt is number of tubes, D_{SB} is diametral clearance between shell & baffle and L_{TB} is diametral clearance between tube and baffle.

#### Correction factor for Bundle Bypass effects, J_{B}

Bundle bypass correction factor J_{B} 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, N_{tcc} between the baffle tips is provided by following equation.

`N`

_{tcc}= (Ds/P_{p})(1 - 2Bc/100)`P`

_{p}= P_{T}3^{0.5}/2 for 30° layout`P`

_{p}= P_{T}/ 2^{0.5}for 45° layout`P`

_{p}= P_{T}for 90° layout

Ratio of sealing strips to tube rows r_{ss} is provided by

`r`_{ss} = N_{ss}/ N_{tcc}

where N_{ss} is number of sealing strips (pairs) in one baffle.

The bundle bypass flow area, S_{b} is defind as

`S`_{b} = B(Ds - D_{OTL} - Do/2)

where, B is central baffle spacing. Correction factor J_{B} is then calculated as following –

`J`

_{B}= exp(-Cj(S_{b}/ S_{m})(1 - (2r_{ss})^{1/3})) for r_{ss}< 0.5`J`

_{B}= 1 for r_{ss}>= 0.5`Cj = 1.35 for Re`

_{S}< 100`Cj = 1.25 for Re`

_{S}>= 100

#### Correction factor for adverse temperature gradient, J_{R}

The factor J_{R} accounts for the decrease in the heat transfer coefficient with downstream distance in laminar flow.

`N`

_{tcw}= (0.8/P_{p})(Ds(Bc/100) - (Ds-(D_{OTL}-Do))/2 )`N`

_{B}= 1 + (int)(L - 2Ls - LB_{In}- LB_{Out})/B`N`

_{C}= (N_{tcw}+ N_{tcc})(1 + N_{B})`J`

_{RL}= (10/N_{C})^{0.18}`J`

_{R}= 1, Re_{S}> 100`J`

_{R}= J_{RL}+ (20-Re_{S})(J_{RL}- 1)/80, Re_{S}<= 100, Re_{S}> 20`J`

_{R}= J_{RL}, Re_{S}<= 20

where, L is tube length, Ls is tubesheet thickness, LB_{In} is inlet baffle spacing and LB_{Out} is outlet baffle spacing.

#### Correction factor for unequal baffle spacing, J_{S}

`n1 = 0.6, Re`

_{S}>= 100`n1 = 1/3, Re`

_{S}< 100`J`

_{S}= ((N_{B}-1)+(LB_{In}/B)^{1-n1}+ (LB_{Out}/B)^{1-n1})/((N_{B}-1)+(LB_{In}/B) + (LB_{Out}/B))

Shell side heat transfer coefficient is calculated as

`h`_{s} = h_{Ideal}(J_{C}.J_{L}.J_{B}.J_{S}.J_{R})

### Shell Side Pressure Drop, ΔP_{s}

Friction factor for ideal tube bank is calculated as following –

where b1, b2, b3 and b4 are correlational constants listed below.

Pressure drop for an ideal tube bank is calculated from

`ΔP`_{Ideal} = 2f(G_{S}²/ρ_{S})(μ_{S}/μ_{S,W})^{0.14} N_{tcc}

#### Correction factor for Baffle Leakage, R_{L}

`R`

_{L}= exp(-1.33(1+r_{S})r_{L}^{p})`p = 0.8 - 0.15(1+r`

_{S})

#### Pressure drop for window section, ΔP_{W}

Following terms are calculated as –

`S`

_{WG}= (Ds²/8)(θ_{DS}- Sin(θ_{DS}))`S`

_{WT}= Nt.F_{W}(πDo²/4)`S`

_{W}= S_{WG}- S_{WT}`G`

_{W}= m_{S}/(S_{m}.S_{W})^{0.5}`D`

_{W}= 4.S_{W}/(π.Do.Nt.F_{W}+ θ_{DS}.Ds)

Pressure drop for laminar and turbulent flow is calculated.

`ΔP`_{W, Turb} = N_{B}.R_{L}(2+0.6*N_{tcw}).G_{W}²/(2.ρ_{S})

`ΔP`

_{W}= ΔP_{W, Turb}, Re_{S}>= 100`ΔP`

_{W}= ΔP_{W,Laminar}, Re_{S}< 100

#### Correction factor for Bundle Bypass effect, R_{B}

`R`

_{B}= exp(-Cr(S_{b}/ S_{m})(1 - (2r_{ss})^{1/3})) for r_{ss}< 0.5`R`

_{B}= 1 for r_{ss}>= 0.5`Cr = 4.5 for Re`

_{S}< 100`Cr = 3.7 for Re`

_{S}>= 100

#### Correction factor for unequal baffle spacing, R_{S}

`n = 0.2, Re`

_{S}>= 100`n = 1.0, Re`

_{S}< 100`R`

_{S}= 0.5((B/LB_{In})^{2-n}+ (B/LB_{Out})^{2-n})

Pressure drop in Central Baffle spaces, ΔP_{C} is defined as –

`ΔP`_{C} = (N_{B} - 1)ΔP_{Ideal}.R_{L}.R_{B}

Pressure drop in entrance & exit baffle spaces, ΔP_{E} is calculated as –

`ΔP`_{E} = ΔP_{Ideal}(1 + N_{tcw}/N_{tcc}).R_{B}.R_{S}

Shell side pressure drop is calculated as following –

`ΔP`_{S} = ΔP_{W} + ΔP_{C} + ΔP_{E}

### Tube Side Heat Transfer Coefficient, h_{t}

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

`Re`

_{T}= Di.v.ρ_{t}/μ_{t}`Pr`

_{T}= C_{p,t}.μ_{t}/k_{t}

where, Di is tube inside diameter, v is velocity, ρ_{t} is density, μ_{t} is viscosity, k_{t} is thermal conductivity and C_{p,t} is specific heat for fluid on tube side.

For laminar flow, Re_{T} < 2300, Sieder and Tate correlation is used for Nusselt’s nubmer.

`Nu = 1.86(Re`

_{T}.Pr_{T}.Di/Leff)^{1/3}`Leff = L - 2*Ls`

For turbulent flow, Re_{T} > 10,000, following equation developed by Petukhov-Kirillov can be used.

`Nu = (f/2)Re`

_{T}.Pr_{T}/(1.07+12.7(f/2)^{0.5}(Pr_{T}^{2/3}-1))`f = (1.58 ln(Re`

_{T}) - 3.28)^{-2}

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

Heat transfer coefficient is calculated as following –

`h`_{t} = Nu.(k_{t}/Di)(μ_{t}/μ_{t, w})^{0.14}

### Tube Side Pressure Drop, ΔP_{t}

Tube side pressure drop is calculated by following equation –

`ΔP`_{t} = (4.f.Leff.Np/Di + 4.Np)ρ_{t}.v²/2

where, Np is number of tube passes.

### Overall Heat Transfer Coefficient, U

Resistance due to tube wall is calculated as following

`R`_{tube} = Do/(2.ln(Do/Di).k_{tube})

where, k_{tube} is thermal conductivity of tube material. Overall clean heat transfer coefficient, U_{Clean} is calculated as per below equation

`U`_{Clean} = 1/(h_{S} + Do/(Di.h_{t}) + R_{tube})

Overall dirty heat transfer coefficient, U_{Dirty} is calculated as per below expression

`U`_{Dirty} = 1/(1/U_{Clean} + f_{shell} + f_{tube})

where, f_{shell} & f_{tube} are fouling factors for shell and tube side.

Heat transfer coefficient required, U_{Required} is calculated as following

`U`_{Required} = Q /(A x LMTD_{corrected})

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

`Over Surface, % = (U`

_{Clean}/U_{Required}- 1)*100`Over Design, % = (U`

_{Dirty}/U_{Required}- 1)*100

Web based calculation available at CheCalc.com

### Spreadsheet

Spreadsheet for Heat Exchanger Rating based on Bell-Delaware Method

### References

- Chapter 4, Design Fundamentals of Shell-And-Tube Heat Exchanger
- Process Heat Transfer – Principles and Applications, 2007 – Robert W. Serth
- Chemical Process Computations, 1985 – Raghu Raman

## 18 Replies to “Heat Exchanger Rating (Bell-Delaware Method)”

Realy usefull and grat job, thanks so much

There is some criterion for the dynamic pressure(DensityxVelocity^2) in the nozzles in existing equipment.

Excellent Spreadsheet, It can be increased with Triffin tubes.

Very interesting. A real great job. I have one curiosity. How to compute the effect of baffle orientation? I mean, those correlations are for vertical baffle cuts. I’ve checked in HTRI, and if you use horizontal the baffle cuts, the Reynolds number will increase by about 12%, and shell heat transfer coefficient will increase by about 17%. Do you have the correlations for that correction? I know horizontal baffle cuts are unusual due to accumulation of solids, but I know one case of heat exchanger with this configuration.

There is a discrepancy between the Excel spreadsheet calculation and the online website calculation. The shell-side h seems to be incorrect in the Excel spreadsheet.

Please disregard. Just found my typo. Sorry about that.

Nevermind again. There is definitely an error with the shell-side HT coefficient calculation in the Excel spreadsheet.

JL was off — found discrepancy in G-K61: Diametral Shell-Baffle Clearance calculation. One website, it is an input, not a calculation – regardless, these numbers are what caused the biggest discrepancy. I am still looking for one more discrepancy that relates to Js.

Hello, great job!

But I have one question, my appologies if I’m wrong but shouldn’t this equation:

NB = 1 + (int)(L – 2Ls – LBIn – LBOut)/(Bc/100)

be this:

NB = 1 + (int)(L – 2Ls – LBIn – LBOut)/(B) ?

It makes more sense.

Also you have it in spreadsheet as NB = 1 + (int)(L – 2Ls – LBIn – LBOut)/(B/100), but dividing B by 100 doesn’t make any sense. I believe you corrected Bc to B and then forgot to delete the 100?

Thanks for finding it, I have corrected the equation.

Is the equation for number of baffles “NB” correct?

“Bc” in equation[NB = 1 + (int)(L – 2LS – LBin – LBout ) / (Bc/100)] is baffle cut %. I think that the center baffle spacing “LBc” is correct. Actually, “LBc” is using in spreadsheet. But the calculation for “NB” in this spreadsheet may be not correct,too. I think it is not “(Bc/100)” but “(LBc) in equation and it is not “LBc/100” but “LBc/1000”.

Thanks for finding it, I have corrected it in the spreadsheet.

Thank you for your reply. I undestand that abbrevation “B” means the central baffle spacing in this text. But the abbrevation of the central baffle spacing is “LBc” in spreadsheet. I think this is slightly confusing.

One more comment. Is the equation for “Overall H.T.C Dirty (UDirty)” in spread sheet correct? Equation is [= 1/(1/Uclean + “Tube Side Fouling Factor” + “Shell Side Fouling Factor”)] I think there is no correction of tube diameter for the tube side fouling factor and it is not “Tube Side Fouling Factor” but “(Tube Side Fouling Factor) x (Do / Di)”.

Anyway your information in this site was very useful for me. Thank you very much.

I agree with you. To be consistent, it should be“(Tube Side Fouling Factor) x (Do / Di)”.

I have one more question. Generally, the diamentral shell-baffle clearance DSB means the difference of shell inside diameter and baffle outside diameter. So half of DSB as the clearance of shell and baffle should be used for calculation of the shell to baffle leakage area SSB. If DSB is the diamental clearance, the equation for SSB may be not [Ds x DSB x ( PI() – 0.5 x θDS)] but [Ds x DSB / 2 x ( PI() – 0.5 x θDS)] . Do you think which is correct?

Hi! What are typical values for Dotl?

Thank you!