Logarithmic Mean Temperature Difference

Many solutions for the same problem

Introduction

After the 2024 International Meeting of Slide Rule Collectors, FanHsiu from Taiwan contacted me with a question about the calculation of Logarithmic Mean Temperature Differences. He had found three slide rules for this application that each worked in a different way: the Hemmi 257L, the Tad Products Corporation LMTD and the Perrygraf Chemical Mixing Calculator. He could explain the mathematics of the first two slide rules, but the third one remained obscure.

LMTD

The Logarithmic Mean Temperature Difference.[1] (LMTD) is used in determining the heat transfer in flow systems, such as a double pipe counter-flow heat exchanger. The LMTD is the logarithmic average of the temperature difference ΔT between the hot and cold feeds at each end of the two pipes a and b. For a given heat exchanger, the larger the LMTD, the more heat is transferred. The LMTD is defined as:

LMTD = (ΔTa − ΔTb) / ln(ΔTa / ΔTb)

for ΔTa ≠ ΔTb and LMTD = ΔTa for ΔTa = ΔTb.

This can also be read as:

LMTD = (ΔTa − ΔTb) / (ln ΔTa – ln ΔTb)

For example, for a heat exchange system with ΔTa = 90°C, ΔTb = 30°C: LMTD = (90 − 30) / ln(90/30) ≈ 54.61°C

Hemmi 257L

The slide of the Hemmi 257L Chemical slide rule (Figure 1) has an LM scale which consists of two parts: an a/b scale labeled in red and an lm(a:b) scale labeled in black. The LM scale is used in combination with a red and black mmHg/atm scale on the body. Note that pressures do not appear in the LMTD calculation, the pressure scale is just reused for this application. This slide rule, dating from 1971 to ca. 1975, has been described before by Richard Smith Hughes.[3]
The earlier Hemmi 257 chemical slide rule does not have the LM scale.

Fig. 1. Back of Hemmi 257L (courtesy of sliderulemuseum.com[2])

To calculate the LMTD for ΔTa = a = 90°C, ΔTb = b = 30°C (Figure 2)

1. Set LM a/b scale left index opposite 30 on the mmHg scale.

2. Move cursor to 90 on the mmHg scale and read a:b = 3 on the LM a/b scale.

3. Keep the slide still, move cursor to 3 on the lm(a:b) scale and read ~54.5°C on the atm scale.

FanHsiu explained the Hemmi 257L as follows:

LM a/b, mmHg, atm are all logarithmic scales.

LMTD = (a − b)/ln(a/b) = b×(a/b − 1)/ln(a/b)

Define lm(a:b) = (a/b − 1)/ln(a/b).
This makes lm(a:b) only a function of the ratio a:b.

LMTD = b×lm(a:b)

In the example lm(a:b) = lm(90:30)

lm(3) = (3/1 − 1)/ln(3) = 2/ln3 ≈ 1.82

LMTD = 30×1.82 ≈ 54.5°C

Note that when you set the left index on the LM a/b scale opposite b = ΔTb on the mmHg scale,
you also set the left index of the lm(a:b) scale opposite b = ΔTb on the atm scale.

Figure 2. Calculation of LTMD(90°C, 30°C) on a Hemmi 257L

Tad Products Corporation LMTD

The Tad Products Corporation LMTD (Figure 3) is a circular slide rule from FanHsiu’s collection.

The rim of the disc contains an outer half-scale ΔTmax, and an inner half-scale “Logarithmic Mean Temperature Difference.”
The inner rotating disc has two half-scales, ΔTmin and ΔTmax − ΔTmin, and a window labeled “Read KEY NO.” and “Set KEY NO.”

Figure 3. Tad Products Corporation LMTD

To calculate LMTD for ΔTa = ΔTmax = 90°C and ΔTb = ΔTmin = 30°C (Figure 4):

1. Set 30 on ΔTmin opposite 90 on ΔTmax

2. Read KEY=3 at upper arrow in window

3. Set 3 at lower arrow.

4. Calculate ΔTmax − ΔTmin = 90 – 30 = 60 in your head.

5. Opposite 60 on the ΔTmax − ΔTmin scale, read ~54°C on the LMTD scale.

One can see that the Tad device is much less convenient than the Hemmi 257L since you have to set the scales twice, and perform a mental calculation.

The Tad's scales are quite straightforward.
ΔTmax and ΔTmin are logarithmic scales, and the upper KEY is simply ΔTmax/ΔTmin.

The lower KEY is ln(ΔTmax/ΔTmin) so that

LMTD = (ΔTmax − ΔTmin)/ln(ΔTmax/ΔTmin)

= (90 − 30)/ln(90/30) = 60/ln3 ≈ 54°C

Tad Products Corp., Beverly, MA, was in business from 1963 to 1990.[5]

Figure 4. Tad LMTD step 1,2 (left) and 3,5 (right)

Perrygraf Chemical Mixing Calculator

Figure 5. Perrygraf Chemical Mixing Calculator (courtesy of sliderulemuseum.com[4])

Following the instructions on the back of the slide chart for ΔTa = 90°C, ΔTb = 30°C (Figure 6) we would follow these steps:

Align 90 on the LM1 scale with 90 on LM2 scale.

Opposite 30 on the LM2 scale, read the log mean value ~54.6 on the LM1 scale.

So you only move the slide one time and there is no need for a “KEY”.

It is not immediately clear how this slide chart works mathematically. The LM1 and LM2 scales are not linear or logarithmic.
The LM1 is a twice enlarged version of the LM2 scale (Figure 7.)

Figure 6. LMTD(90°C, 30°C) on a Perrygraf

Figure 7. Comparing the Perrygraf scale LM1 with the twice-enlarged LM2

How are these scales constructed? Let’s measure tick marks on the LM1 scale in “pixels” and assume a function f(x) which converts an LM1 ΔT‑value to “pixels”, so f(x)/2 converts LM2 values to “pixels”, then, for the described operation to work, function f should comply to:

f(a) − f((a − b)/ln(a/b)) = (f(a) − f(b))/2 for a,b ≥ 1

(f(a) + f(b))/2 = f((a − b)/ln(a/b))
i.e.
(f(a) + f(b))/2 = f(LMTD)

The function f plays the same role as the generator of a quasi-arithmetic mean.[6]
For some well-known means f corresponds to:

· arithmetic (x1+x2+…+xn)/n: f(x) = x

· geometric n√(x1×x2×…×xn): f(x) = log(x)

· harmonic n/(1/x1+1/x2+...1/xn) : f(x) = 1/x

· power ((x1p ×x2p×…×xnp)/n)1/p : f(x) = xp

· log semiring, defined as log(exp(x1)+exp(x2)+…+exp(xn)) – log(n): f(x) = exp(x)

LMTD does not appear in this list. The French Wikipedia page on quasi-arithmetic means[6] explicitly states that the logarithmic mean is not a quasi-arithmetic mean. The logarithmic mean used here has only two arguments (n=2) but there exists a more complicated generalized logarithmic mean for n>2. The logarithmic mean is always greater than or equal to the geometric mean and less than or equal to the arithmetic mean.

We still need to explain the Perrygraf LM scales.

After measuring 36 tick marks ranging from 1 to 100 on the LM1 scale, converting them to a linear scale between 0 and 1, and fitting a large variety of functions, the best fit was a simple power function f(x) = A×xB − C, with B = 0.3134.
Note that A and C drop out of the calculation for LMTD, so the LMDT is approximated as a power means. A more rigorous minimization[7] of the RMS error between (f(a) + f(b))/2 and f((a − b)/ln(a/b)) for 121 (a,b) pairs with a,b between 1 and 100 results in a slightly higher B = 0.3165. The value of B is not 1/3, or 1/e or 1/π, so what could it really be?

A nomogram

A literature search revealed a 1954 article by Hermann Wundt,[8] describing an LMTD nomogram designed by W. Mehner.[9]
This nomogram consists of three parallel lines and is probably the same as the one shown in a late 1960’s Walter Roller catalog[10] (Figure 8). The lines of the nomogram all have the same scales. The LMTD (Δtm) is read from the middle scale, which has equal distances to the outer scales. So Δtm is the average of f(Δt1) and f(Δt2), as on the Perrygraf.

The scales of the Walter Roller nomogram do not perfectly match the photograph of Perrygraf scale LM1 (Figure 9). This might be due to some distortion caused by the cursor.

Fgure 8. Nomogram in Walter Roller Catalog[10]

Figure 9. Comparing the Mehner nomogram Δt scale (Walter Roller[10]) with the Perrygraf LM1 scale

Wundt found for f :

This function has to be calculated by numerical integration since there is no closed form. After calculating the integrals for Wundt’s f using Python,[11] with a typical numerical error less than 10−8, it turned out that scales based on Wundt’s function performed worse than the f(x) = x0.3134 from the fit on the measured Perrygraf scales or f(x) = x0.3165 from the 121 pair minimization (Table 1).

In 1996 P. Kahlig and J. Matkowski[12] showed that there are no nontrivial differentiable and strictly monotonous solutions for f. They cite two consecutive papers from the 1950’s, by Miktós Hosszu[13] and E. Stiefel.[14]
The first paper is a mathematical proof that the only differentiable solution to Wundt’s equation is the trivial function f = constant. The second paper uses “nomogram-theory” to prove that no parallel-lines nomogram exists for this problem and suggests a nomogram with curved lines in the ξ,η plane:

ΔTa : ξ = 1/ln(ΔTa) and η = ΔTa/ln(ΔTa)

ΔTb : ξ = 1/ln(ΔTb) and η = ΔTb/ln(ΔTb)

LMTD: ξ = 0 and η = LMTD

Although mathematically correct, this nomogram is useless because the ΔTa and ΔTb curves coincide and have large nearly straight sections, making it hard to correctly set the isopleth (Figure 10).

Figure 10. LMTD(18,8) on Stiefel’s nomogram

Another nomogram

A more useful nomogram with curved lines was published in 1941 in a publication by Bauer[15] (Figure 11).
Values of Δ' = ΔTa, Δ'' = ΔTb and ΔTm are all on the same horizontal, just as they appear in the Mehner nomogram. The ΔTm scale is linear.

Bauer’s nomogram can be constructed by taking a fixed starting point ΔTb, at a certain distance d from the central line, choosing a ΔTa between 0.2 and 12, calculating the corresponding ΔTm, marking the intersection of the line (d,ΔTb)-(0,ΔTm) with the horizontal ΔTa and repeating this for another ΔTa. The ΔTb curve is just a mirror of the ΔTa curve. However, this procedure gives a slightly different curve if one of the previously found points is used as a fixed starting point for ΔTb, so this nomogram is not completely mathematically correct.

Figure 11. LMTD(18,8) on Bauer’s nomogram

Conclusion

Four practical methods for calculating Logarithmic Mean Temperature Differences have been found. Two of them have been explained mathematically: the methods used by the Hemmi 257L and the Tad Products Corporation LMTD. The mathematical foundation of the method shared by the Perrygraf Chemical Mixing Calculator and the Mehner nomogram remains unknown, as does Bauer’s nomogram.

Acknowledgements

Many thanks to FanHsiu for kicking off this article by explaining the Hemmi and Tad slide rules and challenging me to explain the Perrygraf.

Notes

Rajendra Bhatia, "The Logarithmic Mean", Resonance, June 2008, pages 583–594.
Hemmi 257L, sliderulemuseum.com . Manual available at https://sliderules.nl/media/slide_instructions/Hemmi-257L_Chemical.pdf
Richard Smith Hughes, "A Layman’s Guide to Chemical Slide Rules", Journal of the Oughtred Society, 22:1, 2013, pages 2–5. Richard used T1 and T2 in his description, which should read ΔT1 and ΔT2. The numerator of the LMTD formula contains a difference of temperature differences, and the denominator a ratio of temperature differences. The slide rule can be used for various temperature scales (°C, K, °F), as long as you use only one temperature scale in a calculation.
Perrygraf Chemical Mixing Calculator, sliderulemuseum.com
Tad Products, sliderulemuseum.com
Wikipedia: Quasi-arithmetic mean; Moyenne quasi-arithmétique, (accessed Nov. 9, 2024).
Python function scipy.optimize.minimize_scalar version 1.4.1
Hermann Wundt, "Über eine Funktionalgleichung aus der Wärmeleitung", Zeitschrift für angewandte Mathematik und Physik, 5, 1954, page 172–175. Animation at https://www.rekeninstrumenten.nl/MIR/supplement/wundtLMTD.htm
W. Mehner, Mittlere logarithmische Temperaturdifferenz für Gleich- und Gegenstrom-Wärmetauscher, BWK [Brennstoff, Wärme, Kraft] Arbeitsblatt 12, 1951.
Snippet at https://books.google.nl/books?id=k2l5Wuce2qkC&q=Mehner
Walter Roller, Kondensatoren Katalog Nr. 11, ca. 1969, page 7.
Python function scipy.integrate.quad version 1.4.1
P. Kahlig, J. Matkowski, "Functional equations involving the logarithmic mean", Z. angew. Math. Mech. 76:7, 1996, page 385–390.
Miktós Hosszu, "Eine Bemerkung zur Mitteilung von H. Wundt: «Über eine Funktionalgleichung aus der Wärmeleitung»", Zeitschrift für angewandte Mathematik und Physik, 6, 1955, page 143–144.
E. Stiefel, "Bemerkung zur obigen Mitteilung von Herrn M. Hosszu", Zeitschrift für angewandte Mathematik und Physik, 6, 1955, page 144–145.
Animation at https://www.rekeninstrumenten.nl/MIR/supplement/nomogramLMTD.htm
G. Bauer, W. Brose, Der Schiffsmaschinenbau. Anhang zu Band 3, Walter de Gruyter GmbH & Co KG, 2019 (original: 1941) page 534.
Animation at https://www.rekeninstrumenten.nl/MIR/supplement/bauerLMTD.htm

An earlier version of this paper appeared in the Journal of the Oughtred Society, 34:1, 2025, page 57-62

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