## 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})/(Bc/100)`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