What the calculation gives you
A heat exchanger moves heat from a hot stream to a cold one through a wall. Sizing it means finding the surface area that transfers the required duty with the temperatures you have, or, for an exchanger that exists, finding the duty it can deliver. The log mean temperature difference (LMTD) method answers the first question directly: duty, temperature difference and overall coefficient in, area out.
The duty
Q = ṁ cp (Tin − Tout)
ṁ mass flow of one stream · cp its specific heat · the same Q written for the other stream gives its outlet temperature
An energy balance on either stream fixes the duty; the balance on the other stream then fixes the one temperature you do not know. For a phase change, ṁ times the latent heat replaces the sensible term.
The temperature difference
The driving force varies along the exchanger as the streams approach each other. For pure counterflow or parallel flow its correct average is the log mean:
LMTD = (ΔT1 − ΔT2) / ln(ΔT1 / ΔT2)
ΔT1, ΔT2 the hot-to-cold differences at the two ends of the exchanger
Counterflow always gives the larger LMTD for the same four temperatures, and it is the only arrangement in which the cold outlet can rise above the hot outlet. For shell-and-tube and crossflow arrangements the counterflow LMTD is multiplied by a correction factor F (from the standard charts, usually 0.8 to 1); an arrangement with F under 0.75 is a poor one and should be changed, not corrected.
The overall coefficient
1 / (U A) = 1 / (hi Ai) + Rwall + 1 / (ho Ao) + Rf
hi, ho film coefficients on the two sides · Rwall the conduction resistance of the tube, ln(ro/ri) / (2π k L) · Rf fouling allowance
The resistances add in series, so the smallest h controls U. Each film coefficient comes from a Nusselt number correlation for its geometry and flow, h = Nu k / D: Dittus–Boelter (Nu = 0.023 Re0.8 Pr0.4) inside tubes in turbulent flow, Churchill–Bernstein or Zukauskas for flow across tubes, and coil-specific correlations for helical tubes, where the secondary flow raises h above the straight-tube value. A copper or thin steel wall adds little; a glycol or oil film on the outside, or a gas on either side, adds a lot. Typical overall values: 800 to 1500 W/m2K water to water, 50 to 250 W/m2K with a gas on one side.
The area
A = Q / (U · F · LMTD)
With the area, the number and length of tubes follow from the chosen tube size, and the pressure drop on each side must then be checked: a compact exchanger with a high velocity has a high U and a pressure drop that may be more than the pump can give. When instead the exchanger is fixed and the outlet temperatures are unknown, the effectiveness–NTU method avoids iterating: NTU = U A / Cmin, and the effectiveness follows from NTU and the ratio of the heat capacity rates for each flow arrangement.
Worked example
The worksheet below sizes a double-helix coil exchanger that recovers heat from a sewage stream at 12 °C into a 60% glycol solution at 4 °C, inside a 6 in pipe shell 2 m tall. From the chosen tube size, coil gap and height it builds the length and the inside and outside surface areas of the two helices, then computes the velocity, Reynolds number and friction factor on the sewage side with the coil correction to the friction factor, the head loss through the bends and the straight runs, the Prandtl and Nusselt numbers on each side, the film coefficients, the overall resistance, and finally the heat the exchanger can recover. The sliders set the flow, the coil gap and the glycol velocity; the pipe size and series or parallel flow are selected from lists.
Common mistakes
- Arithmetic mean instead of log mean. They agree within 5% only when the two end differences are within a factor of about 1.7 of each other. With a close approach at one end the arithmetic mean overstates the driving force badly.
- Counterflow LMTD for a multi-pass exchanger. Apply the F factor; a 1-2 shell-and-tube with a temperature cross may have F below 0.8.
- No fouling allowance. Clean U is the day-one value. Sewage, cooling tower water and process streams foul; standard fouling resistances add 20 to 50% to the area.
- Properties at the wrong temperature. Viscosity drives Re and Pr; evaluate it at the mean film temperature of each stream. Glycol at 4 °C is ten times more viscous than water.
- Area on the wrong side. U is defined with respect to one surface, usually the outside. Use the same one throughout.
- Pressure drop checked last, or not at all. A coil with many turns can have a head loss that halves the flow the pump delivers, which then changes h.
Questions
- LMTD or effectiveness–NTU?
- LMTD when you know all four temperatures and want the area. NTU when you know the exchanger and want the outlet temperatures. They are the same physics; one just avoids iteration in each case.
- How much margin should I add?
- Beyond the fouling allowance, 10 to 20% on the area is usual for uncertainty in U. Oversizing has a cost in pressure drop and, for a controlled exchanger, in controllability.
- Why is the glycol side the bottleneck?
- Its film coefficient is low: high viscosity means a low Reynolds number, often laminar, and a low thermal conductivity. Raising its velocity, or putting it inside the tube rather than outside, is where the design effort pays.
