The Pandolf equation: what a backpack adds to walking energy

The Pandolf equation predicts the metabolic rate, in watts, for walking with a load on a grade. Multiply the rate by the time in seconds and divide by 4,184 J per kilocalorie to get the energy for the walk. The equation was published in 1977 for standing or walking very slowly, and the energy tool on this site accepts speeds from 0.5 to 6 km/h. For a 75 kg walker with 12 kg, at 4 km/h on a 3% grade, the rate is 379.6 W.

The equation in four terms

M = 1.5W  +  2.0(W + L)(L ÷ W)²  +  η(W + L)(1.5V² + 0.35VG)

W  body mass (kg)            L  load mass (kg)
V  walking speed (m/s)       G  grade (percent slope)
η  terrain factor (no unit)

Wikipedia’s summary gives the terrain factor η as 1.0 for a blacktop road or treadmill, 1.1 for a dirt road, 1.2 for light brush, 1.5 for heavy brush and 1.8 for swampy bog. The energy tool offers the first four.

The first term, 1.5W, is the cost of standing with no load. The second is the extra cost of the load, and it grows with the square of the load ratio, L ÷ W. The third is the walking term. It scales with total mass, W + L, and rises with the square of speed and with the product of speed and grade. Each term can be checked separately, which is useful when a number looks wrong.

Where a 12 kg load adds its energy

Take a 75 kg walker at 4 km/h, which is V = 4 ÷ 3.6 = 1.111 m/s, on a 3% grade over firm ground. Compare no load with a 12 kg load.

standing     1.5 × 75                        = 112.5 W
load term    2.0 × 87 × (12 ÷ 75)²           = 2.0 × 87 × 0.0256 = 4.45 W
walking      1.0 × 87 × 3.0185               = 262.6 W
M            112.5 + 4.45 + 262.6            = 379.6 W

Most of the extra cost comes from the walking term, because that term scales with total mass. The load term is small at this ratio. It grows faster than the load itself, because it depends on the square of the load ratio.

Pack weight at fixed walking settings

The table keeps the walker, speed, grade and terrain fixed and changes only the pack. The energy column is the rate for one hour divided by 4,184 J per kcal. The band column applies the tool’s ±25% sensitivity band, which shows how far the answer might move with the inputs. It is not a confidence interval.

The 20 kg row is 26.7% of body mass (20 ÷ 75), within the tool’s limit of 50%. Moving from 12 kg to 20 kg adds 412.8 − 379.6 = 33.2 W, which is about 4 W for each extra kilogram. The early kilograms are not free, but the walking term makes the effect of grade larger than the effect of load in this range.

Why grade outweighs a few kilograms

The same 12 kg pack on a 10% grade gives 616.4 W, or 530 kcal in one hour. Against 379.6 W at 3%, the grade adds 616.4 − 379.6 = 236.8 W. That is about 5.8 times the 40.7 W the pack itself adds at 3%. Walking with no pack at 10% gives 543.1 W, which is already more than the 12 kg pack at 3%.

Terrain has a smaller but real effect. The same 12 kg on light brush, with η = 1.2 and 3% grade, gives 432.1 W, which is 432.1 − 379.6 = 52.5 W above firm ground. The Loaded miles tool with the grade set to 10% shows the steeper case directly.

Checking the model against measurements

A 2017 study by Drain and colleagues tested the equation for contemporary military load carriage. Sixteen male participants completed ten trials, at five walking speeds from 2.5 to 6.5 km/h, with two external loads of 22.7 kg and 38.4 kg. The abstract reports a mean bias of 124.9 W, with 95% limits of agreement from −48.7 W to 298.5 W. It also reports that the equation systematically under-predicted metabolic rate across all ten speed and load combinations.

Errors ranged from 12% to 33%. They were smallest at the moderate speeds of 4.5 to 5.5 km/h, at 12% to 17%, and larger at slower and faster speeds, at 21% to 33%. The loads in that study are heavier than the pack loads in the table above, and the authors advise applying the equation to military load carriage with caution. The abstract gives no correction factor, so this guide does not apply one.

What the energy tool leaves out

The tool accepts uphill grades from 0 to 15%. Downhill walking is outside the tool and outside the equation as this site uses it. The model has no terms for heat, altitude, fitness or cold, and it describes the energy of the walk, not a full day’s intake. Its output is gross walking energy, so it does not show how much food the walk requires once the rest of the day is counted.

Using the pack total in the energy tool

The energy tool takes the load the walker actually carries, so the input should be the total pack weight, including consumables, rather than base weight alone. The base weight guide works through a ledger that reaches 10.33 kg with food, water and fuel.

Walking energy is one part of the daily total. The food weight guide turns a calorie target into grams of food, and the target itself is the reader’s own. Route timing, which sets how long the walk lasts, is covered in Naismith’s rule, and the effect of grade on speed is in Tobler’s hiking function.