Bird Metabolism During Flight: Evaluation of A Theory

A theory that accurately predicts the energetic cost of avian flight from a small number of easily measured parameters could be useful, for measurements of metabolic rate during flight under controlled conditions are difficult to make. Pennycuick (1969) has recently presented such a theory. However, the power requirements predicted by this theory for level flight at different speeds have not yet been compared with measured values. In this paper, I will compare the predictions of Pennycuick’s theory with the measured power requirements of budgerigar (Tucker, 1968) and the laughing gull (Tucker, 1972) in level flight at various speeds, and with some other data. Then I will add some new features to the theory and adjust its parameters so that the predictions fit the measured values more closely.

The International System of Units, based on the metre (m), kilogram (kg) and second (s), is used throughout this paper. In this system, weight, measured in newtons, is distinguished from mass, measured in kg. The relation between the two is W = mg where g = 9·81 m/s². The term ‘power’ describes energy transfer per unit time and is given in watts. One watt equals 0·860 kcal/h. Power input (P_i) is the rate at which free energy is released from substrates by oxidative metabolism, and the total power input is synonymous with one of the common definitions of metabolic rate. Power output (P₀) is the rate at which mechanical work is done by the system under consideration. For example, the power output of the flapping wings is the rate at which kinetic energy is added to the air. A speed of 1 m/s equals 3·60 km/h or 2·24 miles/h. The density of air in this study is 1·18 kg/m³, which describes air at sea level at a temperature of 23 °C and a relative humidity of 70% (Hodgman, 1959).

Accuracy is defined in terms of systematic error and imprecision as recommended by Eisenhart (1968) and Ku (1969).

Pennycuick’s theory fits the measured values for power input of the budgerigar and the gull at various speeds remarkably well, although it does not account for the rapid increase in the power input of the budgerigar at low speeds (Figs. 1, 2). At intermediate speeds the largest deviations from the theory are 10% for the budgerigar and 0% for the gull. The theory is less successful at predicting how the power input for flight of the gull changes with body mass (Fig. 3). Neither does it predict accurately the cost of transport for birds with weights much different from 3 N (0·3 kg) (Fig. 4). For the smallest (0·03 N, 0·003 kg) and largest (100 N, 10 kg) birds, the predicted costs of transport are low and high respectively by factors of more than 2. These discrepancies might be only partly due to failure of the theory, for the measurements are scattered and are from only a few, perhaps peculiar, species. At any rate, Pennycuick’s theory is a useful method of estimating metabolic rate during flight of the budgerigar and the gull at intermediate speeds.

The theory can be modified to fit the empirical curve for cost of transport more closely without much sacrifice of the agreement shown in Figs. 1 and 2. Although the modifications increase the complexity of the theory, they also introduce factors that, judging from aerodynamic and physiological data, should be accounted for in a theoretical treatment of flight energetics. In the following sections, I shall show the origin of the power terms in the theory, modify these terms and add some new ones, and adjust the parameters of the modified theory to an optimum fit with the empirical data.

Unless otherwise noted, force and velocity vectors in this paper are measured relative to a two-dimensional, orthogonal co-ordinate system on the bird’s body exclusive of the wings. Since flight is assumed to be horizontal, one axis of the co-ordinate system is vertical, and it, together with the other axis and the axis of the bird’s body, lies in a single plane. The bird’s body is taken as stationary relative to moving air, and air velocity (V) refers to a horizontal vector measured in the undisturbed air flow in front of the bird unless otherwise noted.

The aerodynamic quantities used are conventional and are described in a variety of textbooks such as Goldstein (1965), Prandtl & Tietjens (1957), and von Mises (1959).

The wings in level flight produce a mean aerodynamic force that balances two force vectors: weight, which is vertical, and body drag, which is horizontal. This mean force is conventionally resolved into the orthogonal components lift and thrust. The beating wings generate these forces by changing the momentum of the air in their vicinity. Thrust is generated as the wings accelerate air backwards, and lift is generated as air is accelerated downwards.

The motions of birds’ wings are similar to those of a pair of co-axial, counterrotating propellers or helicopter rotors (Fig. 5). During downstroke each wing is analogous to one blade of the pair of propellers or rotors, and during upstroke it is analogous to a blade of the other. One wing plays the role of first one propeller or rotor blade and then the counter-rotating one by twisting axially between upstroke and downstroke. Unlike the situation with propellers the magnitudes of the aerodynamic forces generated by the wings will differ during downstroke and upstroke.

Because flapping wings, unlike rotors, do not rotate through 360°, they also are similar to the fixed wings of conventional aircraft. The inner part of the flapping wing is primarily a lifting device, for it has a relatively small component of vertical motion and can produce a continuous upward force throughout the stroke cycle. The similarities between flapping wings, helicopter rotors and fixed wings are useful, because they allow the analysis of the energetic requirements of flapping flight to be made in terms of existing theories for helicopters (for example, see Shapiro, 1955) and fixed-wing aircraft. I shall now show how these theories are connected.

First, consider the power transferred to the air in the vicinity of a helicopter in vertical ascent or descent. It is assumed that: (1) the air is accelerated equally at all points on the disc in which the rotor rotates; (2) only axial, rather than rotational, kinetic energy is imparted to the air; and (3) there is no air friction. Thus, the air behaves as if it were accelerated uniformly at the disc in which the rotor rotates (the actuator disc, Fig. 6). Since the power output of the actuator disc to produce lift is the rate at which kinetic energy is added to the air,

The value of S can now be determined from Prandtl’s wing theory (Prandtl & Tietjens, 1957). This theory shows that the induced drag of a wing is least when the wing has an elliptical distribution of lift along its span, and in this case S is equal to the area of a disc with a diameter equal to the wing span (b). That is, the area S for a fixed wing with an elliptical lift distribution is the same as the area S_d of an actuator disc for a helicopter rotor of the same span. Thus, for both the helicopter in vertical flight, and the fixed-wing aircraft in horizontal flight, the mass of air that takes part in the change of momentum is that which flows through a great circle of a sphere of diameter b. Helicopter theory assumes that this great circle relation holds for vertical flight and for flight at all angles between vertical and horizontal (Shapiro, 1955). Because of the similarities of flapping wings to both rotors and fixed wings it is reasonable to assume that the great circle relation also holds for flapping flight, as Pennycuick has pointed out.

The preceding section has accounted for the induced power expended in accelerating an air mass vertically to produce lift, but what about the power that is expended in accelerating an air mass horizontally to produce thrust ? Thrust in level, unaccelerated flight overcomes the drag (parasite drag) of the body exclusive of the wings, and has the same magnitude as parasite drag but the opposite direction. The parasite power is simply the product of the parasite drag and the velocity of the bird’s body through the air.

The functional relation between parasite drag coefficient and (Re) has not been investigated for actual bird bodies. As an estimate, I shall assume that this relation has the same form as that for the drag coefficient of an infinitely thin flat plate oriented parallel to the direction of air flow. This assumption is accurate for streamlined bodies in wind tunnels since the drag of such bodies arises mainly from skin friction (Goldstein, 1965). The drag coefficient of a plate is proportional to (Re) raised to a power between −$12$ and −$15$⁠, depending on whether the boundary layer is laminar or turbulent, respectively (Goldstein, 1965). The boundary layer of a bird body might be laminar in some places and turbulent in others, depending on body size and the roughness of the feathers (Tucker, 1972). I shall assume that the parasite drag coefficient for bird bodies varies in proportion to (Re) raised to the power −$12$⁠, corresponding to a laminar boundary layer.

In the preceding treatment of aerodynamic forces generated by a change in momentum, I have assumed that the air is an ideal fluid and exerts no frictional forces or pressure drag on the rotor of wing that moves it. Actual air is viscous and will exert frictional forces and pressure drag. Profile drag comprises these forces, and the power required to overcome them is the profile power.

Profile power cannot be calculated accurately for birds because of uncertainties in the motions and aerodynamic characteristics of bird wings. Each region of the wings is exposed to an air velocity (relative to the wing region) which varies with time, and which at a given time is different for different regions of the wing (see Cone (1968) for a detailed description). In addition, the drag of each wing region depends on the air velocity relative to that region, the shape of the region and the angle of attack of the region relative to the air velocity. Profile power is the integral over space and time of the product of drag and velocity for each region of the wing.

Pennycuick assumes that the profile power is independent of flight speed and estimates it to be proportional to the minimum sum of induced and parasite power. He chooses a proportionality constant of 2 for his calculations and indicates how the results would differ if other constants were chosen. As a third modification to his theory, I shall assume that the profile power varies with flight speed and is proportional to the sum of parasite power and induced power at any given speed. The rationale for this assumption is that the wings expend power to overcome increased parasite and induced drag by increasing the momentum added to the air passing through the great circle previously described. This momentum increase can be accomplished either by moving the wings faster or by increasing their angles of attack. In either case profile drag and profile power will increase. There is no reason why the relation between profile power and the sum of induced and parasite power should be a proportional one, but without additional information this is the simplest assumption.

The power outputs of induced, parasite and profile power represent the rate at which work is done on the air surrounding the bird. The total power input to the bird must cover this work rate plus whatever losses occur in the power train between the point where energy is made available from fuel and the point where it is transferred to the air as work. For example, the mechanical work done by the flight muscles need not be transferred totally to the air. Some of this work increases the kinetic and potential energy of the wings and body during part of the wing-beat cycle and might be degraded to heat within the wings and body during another part of the wing-beat cycle (for details, see Cavagna, Saibene & Margaria, 1964; Weis-Fogh, 1972). An additional amount of work might be degraded to heat in overcoming viscosity and the friction of joints. I shall assume that the rate at which the wings do work on the air is 20% of the metabolic rate of the flight muscles. In vitro, vertebrate muscles convert up to 35% of their metabolic energy to mechanical work (Hill, 1939; Woledge, 1968), so I am assuming that less than half of the energy output of avian flight muscles is degraded to heat within the body.

Additional power is consumed for maintenance metabolism, circulation of the blood and ventilation of the respiratory system. I assume that the power for maintenance is the basal metabolic rate and will estimate the power outputs for circulation and respiration in the following sections. I also assume that the heat resulting from various losses within the body allows thermoregulation to be accomplished solely by the regulation of heat loss with no additional heat production.

Pennycuick assumes that the efficiency of the flight muscles is 20 % as I do, and that the total additional power expenditure is the basal metabolic rate. The addition of power terms for circulation and respiration is my fourth and final addition to his theory.

The power input (P_i,_r) for moving air through the respiratory system cannot be estimated as simply as that of the heart, for account must be taken of both velocity and pressure changes. The power input for ventilation in man has been analysed and various measurements and estimates made. For rates of oxygen consumption 10 –20 times the basal rate in man, the power input required for ventilation is estimated to be between 2 and 10% of the total power input (Otis, 1964). Accordingly, for birds in flight, I shall assume that 5 % of the total power input goes towards ventilation.

I assume that maintenance power is the basal metabolic rate (P_i,B) which can be measured directly or calculated from equations.

I measured the drags of bird bodies in a wind tunnel with a one-component straingauge flight balance similar to that shown in fig. 6.65 of Gorlin & Slezinger (1964). The wind tunnel (described in Tucker & Parrott, 1970) was run at an air speed (V₀) of 11·0 m/s, and the turbulence intensity was 0·7%. The strain gauges of the flight balance formed a four-arm bridge, and the degree of imbalance of the bridge was determined by integrating voltage over a period of 50 sec with a digital voltmeter. I calibrated the balance in its operating position by attaching weights to it with a thread that ran over a pulley. The relation between the force component applied parallel to the sensitive axis of the balance and the imbalance of the bridge was virtually linear with a bias of less than 1·5 × 10⁻³ N. The imprecision of the balance was less than a standard error of 0·3 × 10⁻³ N.

For measurements on birds, bodies of five birds (white-throated sparrow, Zonotrichia albicollis; budgerigar, Melopsittacus undulatus; starling, Sturnis vulgaris; laggar falcon, Falco jugger ; mallard, Anas platyrhynchos) were weighed, the wings were removed, and the bodies were frozen in a flight-like posture. A hole was drilled in the breast of each frozen body and a wooden plug was inserted and frozen into place. Then the plug was drilled and tapped to receive the rod from the flight balance. The body of the bird on the balance was oriented with respect to the air stream in the wind tunnel in what appeared to be a natural flight attitude. After each measurement, I calibrated the flight balance with the bird body in place.

Wing span varies during the stroke cycle, and I measured a maximum value when the wings were horizontal during the downstroke for budgerigars and gulls flying level in a wind tunnel. A remote-controlled camera placed directly behind the birds photographed the birds as they flew. Wing-span measurements made on the photographs and multiplied by a scale factor had a systematic error of less than 2 ×10⁻³ m and an imprecision estimated to be less than a standard error of 5·10⁻³ m.

I measured the maximum body widths and wing spans of the birds used for bodydrag measurements with the exception of the mallard and with the addition of the laughing gull. These measurements were made either on photographs taken in the wind tunnel as described above, or on dead birds with wings held at maximum span. Body width expressed as a percentage of total wing span varied from 9% (falcon) to 18% (sparrow), with a mean value of 13%.

I adjusted the parameters c and F in equation (49) until I obtained values for each that produced the closest simultaneous fit (as determined by eye) of this equation to three sets of measured data: the metabolic rates of the budgerigars and laughing gulls at different flight speeds between 6 and 13 m/s (equations (8) and (9), respectively) and the cost of transport for flying birds with weights between 0·03 and too N (3 × 10⁻³ and 10 kg, equation (11)).

The equivalent flat plate area of bird bodies varied according to the least-squares equation (Fig. 7).

The modified theory fits the empirical data most closely when c and F are assigned values of –0·5 and 1·8, respectively (Figs. 8, 9). The parameters used for the budgerigar and the gull (Table 2) in the modified theory were measured or calculated from data on other birds. The measurements show that both budgerigars and laughing gulls are atypical in that for their masses budgerigars have short wings and a small equivalent flat plate area, while laughing gulls have long wings.

The sensitivity of the modified theory to changes in various parameters depends on the values of the parameters. As an example, the percentage changes in power input that result from a 5 % increase of one parameter at a time are shown in Table 3 for a bird with a mass of 0·1 kg flying at 9·5 m/s.

The modified theory accounts for the variation in cost of transport with body mass more accurately than does Pennycuick’s theory. This improvement is largely due to allowing parasite and profile power to vary with Reynolds number. If these power terms are not allowed to vary with Reynolds number, the cost of transport of the modified theory varies with body mass in about the same way as predicted by Pennycuick’s theory (Fig. 10).

Neither Pennycuick’s theory nor the modified theory predict accurately the measured variation of power input with body mass in the flying gull (Fig. 11), although the modified theory offers a slight improvement.

The modified theory and Pennycuick’s theory are in fair agreement in their predictions of power inputs for medium-size birds, but Pennycuick’s theory yields lower values for small birds and higher values for large birds (Fig. 12).

Additional data on metabolic rates during flight have become available for bats (S. P. Thomas, unpublished) and for a crow (Bernstein, Thomas & Schmidt-Nielsen, 1973) since the greater part of this paper was written. Comparisons between measured values of metabolic rates at various speeds and predictions from both Pennycuick’s theory and the modified theory are shown in Table 2 and Fig. 13. The data for the crow has been fitted by linear least-squares with the equation

The modified theory fits the available data for metabolic rates in flight better than Pennycuick’s theory when metabolic rates are measured at the speeds where the ratio P_i/V is minimum (Table 2). The mean absolute value of the deviations of the predictions from the measurements is 8·3% (s.d. = 5·72) for the modified theory and 15·3% (s.d. = 8·60) for Pennycuick’s theory.

Equation (49) is tedious to solve without automatic computing equipment. Some of its solutions can be described approximately by means of equations that are quickly soluble with a slide rule. I have used least-squares fitting techniques to derive the following approximate equations.

This study was supported by grants GB 6160X and GB 29389 from the National Science Foundation.