Log Mean Temperature Difference & Heat Exchanger Calculator
Flow Arrangement iHow the two fluids move relative to each other. Counterflow needs the least area. Every other arrangement uses a correction factor F on the counterflow LMTD.
Counterflow
Counterflow heat exchanger diagram showing the hot (red) and cold (blue) flow paths.
Terminal Temperatures & Sizing
Hot fluid
°F
°F
Cold fluid
°F
°F
Sizing (optional) iEnter any two. The third comes from Q = U × A × F × LMTD.
BTU/hr
BTU/hr·ft²·°F
ft²

Optional: enter any two of heat duty Q, overall coefficient U and area A to get the third.

Results
Effective LMTD for this arrangement—
Correction factor, F—1 for pure counterflow
Sizing (enter two of Q, U, A)—
Counterflow LMTD—
Parallel-flow LMTD—
P / R—P = (t2−t1)/(T1−t1), R = (T1−T2)/(t2−t1)
Effectiveness, ε—
Number of transfer units, NTU—UA / Cmin for this arrangement
Design Status
    Graphs

    The driving temperature difference changes exponentially along the exchanger. The LMTD is its true average. The arithmetic mean of the two end differences is always higher, so using it would undersize the exchanger.

    Assumes steady state, constant specific heats and overall coefficient U, no phase change across a temperature range, and no heat loss to the surroundings. Verify critical designs with a full thermal rating.

    F for crossflow with both fluids unmixed uses the standard approximate effectiveness correlation (within about 1%). All other arrangements use exact relations.

    Reference Guide: Log Mean Temperature Difference
    01

    What Is LMTD?

    LMTD stands for Log Mean Temperature Difference. It is the average temperature difference that drives heat from the hot fluid to the cold fluid in a heat exchanger.

    The temperature difference is not the same everywhere in an exchanger. It is large where the hot fluid meets cold fluid and small where the two have nearly matched. Heat transfer depends on that difference at every point, so the exchanger needs one number that represents the whole length. For steady flow with constant properties, that number is the logarithmic mean of the two end differences.

    With the LMTD known, the basic sizing equation is

    Heat exchanger rate equation\[ Q = U \, A \, F \, \Delta T_{lm,\,cf} \]

    where \(Q\) is the heat duty, \(U\) the overall heat transfer coefficient, \(A\) the heat transfer area, \(F\) the correction factor for the flow arrangement and \(\Delta T_{lm,\,cf}\) the counterflow LMTD.

    Counterflow double-pipe heat exchanger. The hot fluid flows left to right in the inner tube and the cold fluid flows right to left in the annulus around it.
    Counterflow, the Reference Case
    The two fluids move in opposite directions. For the same temperatures, counterflow needs the least area of any arrangement.
    Quick check: the LMTD always lies between the smaller end difference and the arithmetic mean of the two. For end differences of 140 °F and 80 °F, the LMTD is 107.2 °F and the arithmetic mean is 110 °F.
    02

    The LMTD Formula

    LMTD\[ \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln\!\left(\Delta T_1 / \Delta T_2\right)} \]
    Counterflow
    \[ \Delta T_1 = T_1 - t_2 \qquad \Delta T_2 = T_2 - t_1 \]
    Parallel Flow (Co-current)
    \[ \Delta T_1 = T_1 - t_1 \qquad \Delta T_2 = T_2 - t_2 \]

    \(T\) is the hot fluid and \(t\) the cold fluid. Subscript 1 is an inlet and 2 an outlet. It does not matter which end you call ΔT₁; the formula gives the same answer either way.

    When ΔT₁ = ΔT₂ the formula becomes 0/0. The limit is simply that common value, which happens in counterflow when both fluids have the same heat capacity rate.

    Where It Comes From

    Take a small slice of the exchanger with area \(dA\). The heat crossing it is \(dQ = U\,\Delta T\,dA\). The hot fluid loses that heat and the cold fluid gains it, so the local difference changes by

    \[ d(\Delta T) = -\,dQ \left(\frac{1}{C_h} \pm \frac{1}{C_c}\right) \]

    where \(C = \dot m\,c_p\) is each fluid's heat capacity rate (minus sign for counterflow, plus for parallel). Substituting \(dQ\) and integrating from one end to the other gives a logarithm:

    \[ \ln\frac{\Delta T_1}{\Delta T_2} = U A \left(\frac{1}{C_h} \pm \frac{1}{C_c}\right) \]

    The heat balance gives \(1/C_h \pm 1/C_c = (\Delta T_1 - \Delta T_2)/Q\). Combining the two results gives \(Q = U A\,\Delta T_{lm}\). The local difference falls exponentially along the length, which the Local ΔT graph above shows.

    03

    Heat Exchanger Flow Arrangements

    Counterflow heat exchanger with the hot fluid flowing right and the cold fluid flowing left.
    Counterflow
    Fluids flow in opposite directions. The cold outlet can rise above the hot outlet (a temperature cross). F = 1 by definition.
    Parallel-flow heat exchanger with both fluids flowing to the right.
    Parallel Flow (Co-current)
    Both fluids enter at the same end. The cold outlet can never exceed the hot outlet. Used when the wall must stay cool at the inlet or a product must not overheat.
    Shell and tube heat exchanger with one shell pass and two tube passes. The shell fluid weaves past baffles while the tube fluid goes out and back through a U-turn.
    Shell & Tube, 1 Shell Pass, 2 Tube Passes
    The most common industrial design. One tube pass runs counterflow to the shell fluid and the other runs parallel, so F < 1.
    Two shell and tube exchangers in series with the hot and cold fluids flowing through them in opposite order.
    Shells in Series (2 Shell Passes)
    Each added shell pass moves the exchanger closer to counterflow. Two shell passes can be two shells in series or one shell with a longitudinal baffle.
    Crossflow plate-fin core with the hot fluid flowing left to right and the cold fluid flowing top to bottom in separate finned channels.
    Crossflow, Both Fluids Unmixed
    Plate-fin cores, such as aircraft and automotive heat exchangers. Fins keep each stream in its own channels.
    Crossflow tube bank. The cold fluid flows inside the tubes while the hot fluid flows freely across them.
    Crossflow, One Fluid Mixed
    Unfinned tube banks and air coolers. The fluid outside the tubes can mix sideways; the fluid inside cannot.
    04

    The Correction Factor F

    Only pure counterflow and parallel flow have a simple LMTD. For every other arrangement the counterflow LMTD is multiplied by a correction factor \(F \le 1\). F depends on two temperature ratios:

    Temperature effectiveness and capacity ratio\[ P = \frac{t_2 - t_1}{T_1 - t_1} \qquad R = \frac{T_1 - T_2}{t_2 - t_1} = \frac{C_c}{C_h} \]

    P is how far the cold fluid is heated, as a fraction of the most it could be. R is the ratio of the two heat capacity rates.

    1 Shell Pass, Even Number of Tube Passes
    \[ F = \frac{\sqrt{R^2+1}}{R-1}\;\frac{\ln\!\left[\dfrac{1-P}{1-PR}\right]}{\ln\!\left[\dfrac{2-P\left(R+1-\sqrt{R^2+1}\right)}{2-P\left(R+1+\sqrt{R^2+1}\right)}\right]} \]

    This is Bowman's equation. For R = 1 use the limit \(F = \dfrac{\sqrt2\,P/(1-P)}{\ln\!\left[\dfrac{2-P(2-\sqrt2)}{2-P(2+\sqrt2)}\right]}\).

    How This Calculator Finds F

    For any arrangement, F is the ratio of the number of transfer units (NTU) that pure counterflow needs to reach the given P and R, to the NTU this arrangement needs:

    \[ F = \frac{NTU_{counterflow}(P,R)}{NTU_{arrangement}(P,R)} \]

    This one method gives F for any number of shell passes and for both kinds of crossflow. For one shell pass it reproduces Bowman's equation exactly.

    Design Rules of Thumb
    • Keep F ≥ 0.75 to 0.80. Below that the F curve is steep, so a small error in temperatures means a large error in area.
    • If F is too low, add a shell pass (or use shells in series) before adding area.
    • Each curve on the F chart ends at a maximum P. Past it, that arrangement cannot reach the outlet temperatures at any size.
    • If one fluid stays at constant temperature (condensing or boiling), F = 1 for every arrangement.
    05

    Effectiveness & NTU

    The effectiveness–NTU method describes the same physics as LMTD. It is more convenient when the outlet temperatures are unknown, for example when rating an existing exchanger.

    \[ \varepsilon = \frac{Q}{C_{min}\,(T_1 - t_1)} \qquad NTU = \frac{U A}{C_{min}} \qquad C_r = \frac{C_{min}}{C_{max}} \]

    When the outlet temperatures are known, use LMTD. When you know the size and need the outlets, use ε–NTU. This calculator reports both: ε comes straight from the temperatures, and NTU is solved for the chosen arrangement.

    Rating example: a counterflow exchanger with NTU = 2 and Cr = 0.5 has ε = (1 − e−1)/(1 − 0.5 e−1) = 0.775. It transfers 77.5% of the most heat possible, Q = 0.775 Cmin(T₁ − t₁), and both outlet temperatures follow from that Q.
    Arrangementε(NTU, Cr)
    Counterflow\( \dfrac{1-e^{-NTU(1-C_r)}}{1-C_r\,e^{-NTU(1-C_r)}} \)
    Parallel flow\( \dfrac{1-e^{-NTU(1+C_r)}}{1+C_r} \)
    1 shell pass\( 2\left[1+C_r+\sqrt{1+C_r^2}\,\dfrac{1+e^{-NTU\sqrt{1+C_r^2}}}{1-e^{-NTU\sqrt{1+C_r^2}}}\right]^{-1} \)
    Crossflow, Cmax mixed\( \dfrac{1}{C_r}\left(1-e^{-C_r\left(1-e^{-NTU}\right)}\right) \)
    Crossflow, Cmin mixed\( 1-e^{-\frac{1}{C_r}\left(1-e^{-C_r NTU}\right)} \)
    Crossflow, both unmixed\( 1-\exp\!\left[\dfrac{NTU^{0.22}}{C_r}\left(e^{-C_r NTU^{0.78}}-1\right)\right] \) (approx.)

    N shell passes in series use the 1-shell relation for each shell (NTU/N each) and combine them as counterflow stages.

    06

    Temperature Cross & Approach

    Approach is the smallest temperature difference anywhere in the exchanger. In counterflow it is the smaller of ΔT₁ and ΔT₂. Smaller approach means more heat recovered, but area grows quickly as the approach shrinks.

    A temperature cross means the cold outlet is hotter than the hot outlet (t₂ > T₂).

    • Counterflow handles a cross easily.
    • Parallel flow can never produce one. The calculator reports it as impossible.
    • A 1-2 shell and tube can only tolerate a small cross. Past that, F drops sharply and soon no area is enough. Add shells in series.
    Check before you size: both counterflow end differences must be positive. If the cold outlet would exceed the hot inlet, or the hot outlet would fall below the cold inlet, no exchanger of any size or arrangement can do the job.
    Use the right temperatures. LMTD uses the bulk fluid temperatures at the inlets and outlets, not wall temperatures. If a fluid changes phase partway through, split the exchanger into zones and use one LMTD per zone.
    07

    Worked Example: Sizing an Oil Cooler

    Hot oil is cooled from 250 °F to 150 °F by water heated from 70 °F to 110 °F in a 1-2 shell and tube exchanger. The duty is 1,000,000 BTU/hr and U is 75 BTU/hr·ft²·°F. (This is the calculator's Load Example.)

    1. Counterflow ends: ΔT₁ = 250 − 110 = 140 °F, ΔT₂ = 150 − 70 = 80 °F.
    2. \( \Delta T_{lm} = \dfrac{140 - 80}{\ln(140/80)} = 107.22\ \text{°F} \)
    3. P = (110 − 70)/(250 − 70) = 0.2222, R = (250 − 150)/(110 − 70) = 2.5
    4. Bowman's equation gives F = 0.9380, so the corrected LMTD = 0.9380 × 107.22 = 100.57 °F.
    5. \( A = \dfrac{Q}{U\,F\,\Delta T_{lm}} = \dfrac{1{,}000{,}000}{75 \times 100.57} = 132.6\ \text{ft}^2 \)
    Arrangement (same temperatures)FEffective LMTD
    Counterflow1.000107.22 °F
    Shell & tube, 2 shell passes0.985105.64 °F
    Crossflow, both unmixed0.951101.92 °F
    Shell & tube, 1 shell pass0.938100.57 °F
    Parallel flow0.86893.08 °F

    Same duty and U: parallel flow would need 1,000,000/(75 × 93.08) = 143.3 ft², about 15% more area than counterflow.

    08

    Typical Overall Heat Transfer Coefficients

    ServiceU, W/m²·KU, BTU/hr·ft²·°F
    Water to water850 – 1,700150 – 300
    Condensing steam to water1,000 – 3,500175 – 600
    Light organics to water350 – 90060 – 160
    Condensing refrigerant to water300 – 1,00050 – 175
    Water to oil100 – 35020 – 60
    Water to air, finned tubes (air-side area)25 – 504 – 9
    Gas to gas10 – 402 – 7

    Order-of-magnitude ranges for early sizing only. Real values depend on velocities, viscosity, fouling, geometry and which area U is based on. 1 BTU/hr·ft²·°F = 5.678 W/m²·K.

    09

    Common Mistakes

    • Pairing the wrong ends. Counterflow pairs hot inlet with cold outlet. Parallel flow pairs the two inlets.
    • Using the arithmetic mean. It is always larger than the LMTD, so it undersizes the exchanger. The error is about 4% when one end difference is twice the other, and grows quickly beyond that.
    • Forgetting F. A 1-2 shell and tube with the counterflow LMTD and no F can be undersized by 10–25%.
    • Mixing units. A temperature difference of 1 °C equals 1 K, and 1 °F equals 1 °R. Keep U, A and Q in one consistent system.
    • One LMTD across a phase change. A condenser that also subcools has two zones with different temperature profiles. Size each zone separately.
    10

    Frequently Asked Questions

    What is the full form of LMTD?
    Log Mean Temperature Difference: the logarithmic average of the hot-to-cold temperature difference at the two ends of a heat exchanger.
    What is the LMTD formula?
    LMTD = (ΔT₁ − ΔT₂) / ln(ΔT₁/ΔT₂). For counterflow, ΔT₁ = T₁ − t₂ and ΔT₂ = T₂ − t₁. For parallel flow, ΔT₁ = T₁ − t₁ and ΔT₂ = T₂ − t₂.
    Why is it a log mean and not an average?
    The local temperature difference falls exponentially along the exchanger, and the true average of an exponential curve is the logarithmic mean. The arithmetic mean always comes out higher.
    Can the LMTD be negative or zero?
    No. Both end differences must be positive for heat to flow from hot to cold everywhere. A zero or negative end difference means the temperatures you entered are impossible for that arrangement.
    Do I use °C or K, °F or °R?
    Either. Only differences enter the formula, and a difference of 1 °C equals 1 K (1 °F equals 1 °R). The calculator gives the same LMTD in both.
    What does F = 0.7 mean?
    The arrangement transfers heat like a counterflow exchanger with only 70% of the driving force, so it needs about 1/0.7 = 1.43 times the area. More importantly, it is on the steep part of its curve. Add shell passes instead.
    What LMTD do I use for a condenser or boiler?
    If one side stays at a constant temperature, the flow pattern no longer matters: F = 1, and the counterflow and parallel LMTDs are equal.

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