τ = RC = [K1 + K2 EMBED Equation.3 avgε + K3 / (V0 + VT/2)]C (6) where τ is the time constant (s), R is the respiratory resistance (N s/m5), V0 is the initial lung volume (m3), which uses average flow rate over the waveshape and average lung volume during the waveshape.
Initial lung volume was assumed to be FRC = 0.0025 m3.
Where a dual sign appears, the top sign corresponds to inhalation and the bottom sign to exhalation.
Average flows, inhalation times and exhalation times are given in Table 2 for exercise conditions of rest, light, moderate, heavy and very heavy. 4.comPARING WORK RATES Results from this comparison appear in Tables 3 and 4.
As expected, the rectangular waveform gave the lowest work rates for all four exercise levels.
This is the result of the lowest maximum flow rate of any waveshape.
Expressions appearing in Table 1 clearly show the penalty paid for higher maximum flow rates, with EMBED Equation.3 and EMBED Equation.3 terms appearing often. Using the rectangular waveform as a base, the sinusoidal inspiratory waveform costs from 9% more at light exercise to 16% more at very heavy exercise.
There is not a great penalty paid for breathing with a trapezoidal waveshape: the trapezoid costs 3% more at light exercise, increasing to 7% more at very heavy exercise. The trends for the inspiratory exponential waveshape costs are opposite from the others.
The truncated exponential costs 30% more at light exercise, decreasing to 9% more at very heavy exercise.
The inspiratory hybrid exponential costs 29% more at light exercise, decreasing to 12% more at very heavy exercise.
These trends are the result of shortening the inhalation time as exercise proceeds.
Calculated time constant remains at about 0.39 s and, with shorter waveform duration, the exponential waveshapes more closely approach the trapezoidal waveshape as exercise intensity increases.
Because of this, maximum flow rates relative to the trapezoidal and sinusoidal waveshapes decrease with exercise for the exponential waveshapes. Exhalation work rates were generally lower than inhalation work rates as a result of exhalation times longer than inhalation times.
As exhalation time shortens, exhalation work rates overtake inhalation work rates.
This is because the EMBED Equation.3 R(3) component depends inversely on lung volume, which is higher during inspiration than expiration as long as initial lung volume is fixed at FRC. All elastic work terms were calculated to be identical for all waveshapes.
This is a consequence of setting all average flow rates equal, since all elastic work rate terms became: EMBED Equation.3 R(4) = EMBED Equation.3 (7) Johnson  gives the value of physical work for circulation and respiration for rest as 2 N m/s and for light activity as 6 N m/s.
He also estimates resting cardiac power output at about 1.8 N m/s.
Values appearing in Table 4 are thus very reasonable in magnitude. Efficiency of the respiratory muscles has been estimated at 7–11% .
Total physiological work demands on the body are thus about 10 times the amounts appearing in Table 3.
The energy cost of very light work has been estimated at 183 N m/s .
The sinusoidal waveshape, assuming passive exhalation, gives an energy cost of about 4.68 N m/s, or 3% of the body’s energy expenditure.
The energy cost of heavy work is about 707 N m/s.
The values from Table 3, assuming active inhalation and exhalation, both using trapezoidal waveform, give 118.9 N m/s energy cost of respiration, or about 17% of the total.
This number is expected to be about half that percentage.
One difference between these results and expectations is that, for all conditions except resting, lung volume does not begin at FRC.
Since one of the largest contributors to the total work rate, in most cases, is elastic work ( EMBED Equation.3 (4)), a change in beginning position of the lung can have a significant effect on total energy cost.
For instance, elastic work for an expiratory trapezoidal waveform beginning at FRC during heavy work is 5.83 N m/s, whereas for the same exhalation waveform beginning at FRC + VT elastic work is 0.442 N m/s (Table 5).
A trapezoidal inhalation beginning at FRC followed by a trapezoidal exhalation beginning at FRC + VT expends about 65 N m/s physiological work, or 9% of the total cost of heavy exercise. — 14.
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