Tools › Fluids and Heat Transfer › Heat Exchangers › Heat Exchanger Sizing and Pressure Drop

Heat Exchanger Sizing and Pressure Drop

Thermal design of a two-stream heat exchanger by the LMTD method: the duty and the missing outlet temperatures from the energy balance, the log-mean temperature difference with the correction factor F of the arrangement (Bowman–Mueller–Nagle closed forms for 1-2 and 2-4 shell-and-tube, exact series for crossflow), the film coefficients on both sides (Dittus–Boelter, Gnielinski or Sieder–Tate in the tubes; Kern's method on the shell side), the overall coefficient clean and fouled, the area required against the area the geometry provides, and the pressure drop on each side against an allowable. Shell-and-tube exchangers are sized (shell diameter, tube count, baffles) or rated; double-pipe, gasketed-plate and air-cooled finned-tube options are included. Water and air properties come from the IAPWS-IF97 and NASA Glenn libraries; other fluids are entered.

  • Energy balance: the duty from the hot outlet, the cold outlet or an entered duty; the other outlet follows

  • LMTD and the correction factor F for 1-2, 2-4 and N-shell exchangers, parallel flow and the four crossflow cases

  • Tube-side and shell-side (Kern) film coefficients, wall and fouling resistances, U clean and fouled

  • Tube count by a geometric layout, shell diameter sized to the required area, pressure drops against allowables

  • Temperature-profile and F-factor charts, tube-layout sketch; double-pipe, plate and air-cooled options

Heat Exchanger Sizing

Unit System: /

Hot Stream (temperatures °F for USCS, °C for SI; viscosity in cP)










Cold Stream










Duty, Arrangement and Allowable Pressure Drops








Tubes (shared by the shell-and-tube, double-pipe and air-cooled options)










Shell-and-Tube Geometry (blank shell diameter, tube count or baffle spacing = sized by the tool)












Double Pipe (hairpins)




Gasketed Plate (chevron plates; constants illustrative, Martin 1996 cited)















Air-Cooled Finned Tube (hot stream in the tubes, air over annular fins)










The LMTD Design Method | Method page
×

Heat Exchanger Sizing and Pressure Drop

The LMTD Design Method

A heat exchanger is designed in three steps. The energy balance fixes the duty: Q = ṁh cp,h (Th,1 − Th,2) = ṁc cp,c (Tc,2 − Tc,1), so with both flows and three of the four terminal temperatures (or the duty itself) the fourth follows; the specific heats are taken at each stream's mean bulk temperature and the balance is iterated once or twice because the outlet moves the mean. The rate equation then gives the area, Q = U A F ΔTlm: the overall coefficient U, the log-mean temperature difference of the counterflow arrangement and the correction factor F of the arrangement actually built. Finally the geometry that provides that area is laid out and its pressure drops checked against what the pumps can afford. Sizing and rating are the same arithmetic run in opposite directions: in sizing the area is the unknown and the shell diameter, hairpins, plates or rows are chosen to supply it; in rating the geometry is given and the tool reports how much surface it has over (or under) the requirement.

LMTD and the correction factor F

For pure counterflow the mean driving force is the log mean of the two end differences, ΔTlm = (ΔT1 − ΔT2)/ln(ΔT1/ΔT2) with ΔT1 = Th,1 − Tc,2 and ΔT2 = Th,2 − Tc,1. Every other arrangement transfers less for the same terminal temperatures, and the deficit is written as a factor F ≤ 1 on the counterflow LMTD. Bowman, Mueller and Nagle (1940) collected the closed forms in terms of two numbers on the cold stream, P = (Tc,2 − Tc,1)/(Th,1 − Tc,1) and R = (Th,1 − Th,2)/(Tc,2 − Tc,1) = Cc/Ch. For one shell pass with two (or any even number of) tube passes,

F = √(R² + 1) ln[(1 − P)/(1 − RP)] / {(R − 1) ln[(2 − P(R + 1 − √(R² + 1)))/(2 − P(R + 1 + √(R² + 1)))]},

and the denominator's argument reaches zero at Pmax = 2/(1 + R + √(1 + R²)): no 1-2 exchanger, however large, can heat the cold stream beyond that fraction of the available difference. The 2-4 form is longer and is also closed. The tool evaluates both closed forms and, for every arrangement, the same quantity by a second route that needs no special formula: F = NTUcounterflow/NTUarrangement at the same P and R, where the NTU of the arrangement comes from its ε–NTU relation (the one the Effectiveness tool uses). The two routes agree to machine precision for the 1-2 and 2-4 shells, which is the check that the formulas are typed right. Crossflow with both streams unmixed has no closed form; the tool sums Mason's (1955) double series, which is exact, rather than the usual correlation that is good to about 1 %. An F below about 0.75 means the design sits near a temperature pinch, where a small change in a flow or an inlet moves the duty a lot; a second shell pass or true counterflow is the usual remedy, and the F–P chart on the results page shows how steeply the curve falls there.

Film coefficients and the overall coefficient

Inside the tubes the flow is fully developed turbulent pipe flow for all but the smallest exchangers, and the Nusselt number comes from Dittus–Boelter (0.023 Re0.8 Prn, the simplest and ±25 %), Gnielinski (the Petukhov friction form, ±10 % and valid down to Re 3,000) or Sieder–Tate (whose (μ/μw)0.14 factor matters for viscous liquids with a large wall-to-bulk temperature difference). The Convection Correlations library supplies them, with the laminar entry-length form and the 3.66 floor when a design drifts into laminar flow. On the shell side the flow crosses the bundle between baffles, leaks through the baffle clearances and bypasses the bundle round its edge; Kern's method (1950) folds all of that into one correlation on an equivalent diameter built from the tube pitch and OD and a crossflow area taken at the shell centre line, h De/k = 0.36 Re0.55 Pr1/3 (μ/μw)0.14 for Reynolds numbers from 2,000 to a million. It is a preliminary-design method: it tends to overstate the coefficient and the pressure drop for leaky bundles, and the Bell–Delaware method (Bell 1963), which corrects an ideal-bank coefficient for the baffle window, leakage, bypass and spacing effects separately, is the next step when the Kern design is to be refined. The overall coefficient referred to the outside tube area is the sum of five resistances, 1/Uo = 1/ho + Rf,o + do ln(do/di)/(2kw) + Rf,i do/di + do/(hi di); the table of resistances on the results page shows which one governs, which is where more velocity or more area pays. The fouling resistances are entered (the TEMA tables are the usual source) and the "clean" requirement is printed beside the fouled one so the cost of the fouling allowance is visible.

Tube count, shell diameter and baffles

The number of tubes a shell holds is a geometric question: the tube centres sit on a triangular or square lattice of pitch Pt, every tube must lie wholly inside the outer tube limit Dotl = Ds − clearance, and the pass partitions remove a lane of tubes for each division. The tool counts that lattice directly (and draws it), and prints beside it the common estimate Nt = 0.785 (CTP/CL) Ds²/(PR² do²), where CTP (0.93, 0.90, 0.85 for 1, 2 and 3-or-more passes) allows for the clearance and lanes and CL (1.0 square, 0.87 triangular) for the lattice density; tube-count tables in the handbooks are the same count tabulated for standard shells. In sizing, the required area divided by the area of one tube gives the tubes needed, the smallest shell whose lattice holds them is found, and the film coefficients are re-evaluated with the velocities that shell gives until the shell and U stop changing. Baffles are set at a spacing between 0.2 and 1.0 shell diameters (closer spacing raises the shell-side coefficient and the pressure drop together), and the number of baffles follows from the tube length. Kern's correlations were built for 25 % segmental cuts; the cut is recorded and drawn but does not enter his equations.

Pressure drops

On the tube side the loss is ordinary pipe friction over the total flow length, f (L np/di) ρv²/2 with the Colebrook friction factor, plus the turn-around losses in the headers, which Kern takes as four velocity heads per pass. On the shell side Kern writes ΔP = f Gs² Ds (Nb + 1)/(2 ρ De φs): one crossing of the bundle per baffle space, the friction factor read from his chart, for which the tool uses the published curve fit f = exp(0.576 − 0.19 ln Res). Both drops are compared with the allowables entered; a design that is short on area and long on pressure drop is told so, and the usual trades (more passes raise hi and ΔPt together, wider baffle spacing lowers ho and ΔPs together) can be tried from the form. Liquid tube velocities are kept between about 1 and 3 m/s (3 to 10 ft/s): below that fouling settles, above it erosion starts.

The other geometries

Double pipe: one tube inside a pipe, in hairpins; the annulus coefficient is taken on Kern's equivalent diameter (D2² − do²)/do and its friction on the hydraulic diameter D2 − do, with one velocity head per hairpin return. True counterflow, F = 1. Gasketed plate: chevron plates clamped in a frame, the two streams in alternate channels of gap b, hydraulic diameter 2b, single-pass counterflow; the Nusselt and friction constants depend on the chevron angle and the maker's plate, so they are entered (Martin's 1996 generalized correlation gives them as functions of the angle), the channel drop is 4f (Lp/Dh) ρv²/2 and the ports add 1.4 velocity heads. Air-cooled: the hot fluid in finned tubes, air drawn or blown across them; the air-side coefficient on the finned surface is entered (Briggs and Young's correlation or the maker's data), the annular-fin efficiency comes from the Bessel-function solution of the Fins library, and the overall coefficient is referred to the bare tube so the required area is a tube count and a number of rows; the bundle is crossflow with the air unmixed, which sets F.

What the tool does not do

No two-phase service (condensers and reboilers need their own film correlations and a zoned LMTD), no Bell–Delaware shell-side corrections, no vibration or thermal-expansion checks, no mechanical design of shell thickness, tubesheets or nozzles, and no cost. The property libraries cover liquid water and air; other fluids are entered at their mean bulk temperature. The results are a thermal design to Kern's level of accuracy, which is where a specification to a fabricator starts.



References

Bowman, R. A., Mueller, A. C., and Nagle, W. M. (1940). "Mean Temperature Difference in Design." Transactions of the ASME, Vol. 62, No. 4, pp. 283–294.

Kern, D. Q. (1950). Process Heat Transfer. McGraw-Hill, New York.

Mason, J. L. (1955). "Heat Transfer in Crossflow." In Proceedings of the Second U.S. National Congress of Applied Mechanics, American Society of Mechanical Engineers, New York, pp. 801–803.

U.S. Department of Energy (1992). DOE Fundamentals Handbook: Thermodynamics, Heat Transfer, and Fluid Flow, Volume 2 of 3. DOE-HDBK-1012/2-92, Washington, DC.

Segaser, C. L. (1952). Heat Exchanger Design Study for the ISHR. ORNL Central Files Number CF-52-10-195, Oak Ridge National Laboratory, Oak Ridge, TN.

Oak Ridge National Laboratory (1961). Heat Transfer and ΔP Design of MSRE Primary Heat Exchanger. ORNL Central Files Number CF-61-4-1, Oak Ridge National Laboratory, Oak Ridge, TN.

Dittus, F. W., and Boelter, L. M. K. (1930). "Heat Transfer in Automobile Radiators of the Tubular Type." University of California Publications in Engineering, Vol. 2, No. 13, pp. 443–461.

Sieder, E. N., and Tate, G. E. (1936). "Heat Transfer and Pressure Drop of Liquids in Tubes." Industrial and Engineering Chemistry, Vol. 28, No. 12, pp. 1429–1435.

Gnielinski, V. (1976). "New Equations for Heat and Mass Transfer in Turbulent Pipe and Channel Flow." International Chemical Engineering, Vol. 16, No. 2, pp. 359–368.

Colebrook, C. F. (1939). "Turbulent Flow in Pipes, with Particular Reference to the Transition Region Between the Smooth and Rough Pipe Laws." Journal of the Institution of Civil Engineers, Vol. 11, No. 4, pp. 133–156.

Taborek, J. (1983). "Shell-and-Tube Heat Exchangers: Single-Phase Flow." In Schlünder, E. U. (ed.), Heat Exchanger Design Handbook, Hemisphere Publishing, Washington, DC, Section 3.3.

Kakaç, S., and Liu, H. (2002). Heat Exchangers: Selection, Rating, and Thermal Design, 2nd ed. CRC Press, Boca Raton, FL.

Tubular Exchanger Manufacturers Association (2007). Standards of the Tubular Exchanger Manufacturers Association, 9th ed. TEMA, Tarrytown, NY.

Bell, K. J. (1963). Final Report of the Cooperative Research Program on Shell and Tube Heat Exchangers. Bulletin No. 5, University of Delaware Engineering Experiment Station, Newark, DE.

Martin, H. (1996). "A Theoretical Approach to Predict the Performance of Chevron-Type Plate Heat Exchangers." Chemical Engineering and Processing, Vol. 35, No. 4, pp. 301–310.

Shah, R. K., and Sekulić, D. P. (2003). Fundamentals of Heat Exchanger Design. John Wiley and Sons, Hoboken, NJ.

Briggs, D. E., and Young, E. H. (1963). "Convection Heat Transfer and Pressure Drop of Air Flowing Across Triangular Pitch Banks of Finned Tubes." Chemical Engineering Progress Symposium Series, Vol. 59, No. 41, pp. 1–10.

Robinson, K. K., and Briggs, D. E. (1966). "Pressure Drop of Air Flowing Across Triangular Pitch Banks of Finned Tubes." Chemical Engineering Progress Symposium Series, Vol. 62, No. 64, pp. 177–184.

Gardner, K. A. (1945). "Efficiency of Extended Surface." Transactions of the ASME, Vol. 67, No. 8, pp. 621–631.

Incropera, F. P., DeWitt, D. P., Bergman, T. L., and Lavine, A. S. (2007). Fundamentals of Heat and Mass Transfer, 6th ed. John Wiley and Sons, Hoboken, NJ.

International Association for the Properties of Water and Steam (2012). Revised Release on the IAPWS Industrial Formulation 1997 for the Thermodynamic Properties of Water and Steam. IAPWS R7-97(2012), Lucerne.

McBride, B. J., Zehe, M. J., and Gordon, S. (2002). NASA Glenn Coefficients for Calculating Thermodynamic Properties of Individual Species. NASA/TP-2002-211556, National Aeronautics and Space Administration, Cleveland, OH.

Worked Examples

Published examples this tool reproduces; each page lists the inputs to enter and the printed answers.

All worked examples

DISCLAIMER: The methods and calculation on aNOWysis.com are provided as means to educate the reader. There is no warranty, expessed or implied, as to the relevance or accuracy of the methods or calculations. All calculations should be checked by a qualified analyst for appropriateness and accuracy. We will not be held liable for any damages, injuries or deaths caused by using methods or calculations from this site.