Heat Exchanger Sizing and Pressure Drop: Method

The LMTD Design Method

A heat exchanger is designed in three steps. The energy balance fixes the duty: Q = ṁ_h c_p,h (T_h,1 − T_h,2) = ṁ_c c_p,c (T_c,2 − T_c,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 ΔT_lm: 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, ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) with ΔT₁ = T_h,1 − T_c,2 and ΔT₂ = T_h,2 − T_c,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 = (T_c,2 − T_c,1)/(T_h,1 − T_c,1) and R = (T_h,1 − T_h,2)/(T_c,2 − T_c,1) = C_c/C_h. 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 P_max = 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 = NTU_counterflow/NTU_arrangement 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 Re^0.8 Prⁿ, the simplest and ±25 %), Gnielinski (the Petukhov friction form, ±10 % and valid down to Re 3,000) or Sieder–Tate (whose (μ/μ_w)^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 D_e/k = 0.36 Re^0.55 Pr^1/3 (μ/μ_w)^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/U_o = 1/h_o + R_f,o + d_o ln(d_o/d_i)/(2k_w) + R_f,i d_o/d_i + d_o/(h_i d_i); 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 P_t, every tube must lie wholly inside the outer tube limit D_otl = D_s − 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 N_t = 0.785 (CTP/CL) D_s²/(PR² d_o²), 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 n_p/d_i) 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 G_s² D_s (N_b + 1)/(2 ρ D_e φ_s): 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 Re_s). 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 h_i and ΔP_t together, wider baffle spacing lowers h_o and ΔP_s 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 − d_o²)/d_o and its friction on the hydraulic diameter D₂ − d_o, 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 (L_p/D_h) 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.