Computer Programs in Biomedicine 4 (1975) 214-218 © North-Holland Publishing Company

A M O D E L O F H E A T F L O W IN T H E S H E E P EXPOSED TO HIGH LEVELS OF SOLAR RADIATION

R.R. VERA, L.J. KOONG and J.G. MORRIS Department of Animal Science, University of California, Davis, Calif. 95616, USA The fleece is an important component in thermoregulation of sheep exposed to high levels of solar radiation. A model written in CSMP has been developed which represents the flow of energy between the sheep and its environment. This model is based on a set of differential equations which describe the flux of heat between the components of the system - fleece tip, skin, body and environment. It requires as input parameters location, date, time of day, temperature, relative humidity, cloud cover, wind movement, animal weight and linear measurements and fleece length. At each integration interval incoming solar radiation and its components, the heat arising from the animal's metabolism and the heat exchange by long-wave radiation, convection, conduction and evaporative coofing are computed. Temperatures at the fleece tip, skin and body core are monitored. Thermo-regulation of sheep

solar radiation

1. Introduction Sheep are the most populous and widely distributive domestic ruminants [16]. They are raised in environments which range from temperate to those which experience extremes of heat in summer and cold in winter. A computer model was developed to simulate the heat exchange of sheep with their environment, with emphasis on modelling the effects of high temperatures and levels o f solar radiation. The model has been used to identify areas in which further research is needed, for class demonstrations, and to describe the ecological niche o f sheep based on their physiological parameters and a description o f the environment.

CSMP

flected radiation from the ground, which is computed from solar and sky radiation and the albedo of the ground. The solar radiation intercepted by the sheep depends on the zenith angle o f the sun, the orientation o f the sheep in respect to the sun and the sheep's reflection coefficient [19]. All incoming radiant energy is summated in the fleece tip pool which also loses energy by long-wave radiation to the environment and by convection. Long-wave radiation is computed by the method proposed by Priestley [18], while the convection loss is based on the relationship between Nusselt and Reynolds numbers [12]. With the exception of wind speed, which is corrected for the animal's height above ground [11], all other environmental parameters are treated as inputs. 2.2. The animal

2. Program description 2.1. Simulation o f environmental conditions From the input parameters listed in fig. 1, the program computes the incoming direct solar, and sky or diffuse radiation incident to a horizontal surface. The computation follows the approach outlined by Klein [15]. As the sheep is represented by a horizontal cylinder above the ground surface, it also receives re-

The animal component o f the model was represented as a horizontal cylinder composed of three concentric compartments or pools - an outer fleece tip pool, a skin pool and a inner body core pool. The outer pool was represented by the fleece tip rather than the entire fleece as the tip temperature is a determinant of long-wave radiation and convective cooling loss and a large gradient exists between the fleece tip and the skin. Conduction down this gradient is a

R.R. Vera et al., Model of heat flow in sheep

215

FLOW DIAGRAM

SOLAR, SKY AND GROUNDRADIATION

CONVECTION (FTPCV)

LONG-WAVE RADIATION

(SWRFTP)

(FTPLWR)

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Skin temperature (SKITEM)

PERSPIRATION ~(

I BODYENERG d

Fleecetip (FTT)

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BODY HEAT POOL

Bodytemperature

(BHP)

ENERGY " INTAKE

(BCT)

~WARMING (COOLING) INSPIREDAIR RESPIRATORY EVAPORATION

Fig. 1. Flow diagram of the heat flow model o f a sheep in an open environment.

function of conductivity of the fleece, which was computed from data of Armstrong et al. [2] and varies with fleece length and air temperature. Changes in conductance of the skin due to vasomotor action were modelled in the absence of actual data for the sheep as being dependent on air temperature. Skin weight was taken as a function of body weight (W.N. Garrett, personal communication) and its specific heat was assumed to be 0.83 kcal C-1 kg -1 , in the absence of actual measurements reported in the literature [8]. In the sheep, sweating is a relatively minor source of heat loss from the skin, but the values reported in the literature are extremely variable. The program uses values derived from Brockway et al. [16] and Brook and Short [7]. In the model, the body pool exchanges heat with the skin pool and in addition, loses (or gains) heat through respiratory exchange. The latter is composed of two components: the evaporative loss, which de-

pends upon the air temperature and humidity, and direct exchange of heat by warming or cooling of the inspired ak. Ventilation rate was assumed to depend upon the body temperature, and the heat loss due to respiratory evaporation was computed from the formula given by N.A.S. [17]. An additional input to the body pool is the heat arising from metabolism and utilization of food. At maintenance in the ruminant, the heat production of the animal is equivalent to the uncorrected metabolizable energy of the food. This value is composed of three components: the maintenance energy expenditure of the animal, the heat arising from utilization of energy for maintenance (heat increment of maintenance), and the heat production from fermentation in the rumen [ 1]. For growing or lactating sheep additional components to describe inefficiencies of utilization of metabolizable energy for these processes (heat increment of growth or lactation)

216

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Fig. 2. Graphical representation of some of the outputs of a simulation: solar radiation, (upper curve); short-wave radiation entering fleece tip pool, (SWR); long-waveradiation leaving the fleece tip pool. (LWR) and heat loss from the body pool due to respiratory evaporation, (DEV). would have to be added to the model. The model has provision for increasing the metabolic rate of the sheep up to maximal or summit metabolism [3] when it is exposed to temperatures below the critical temperature [4]. At air temperatures greater than body temperature the Van 't Hoff effect occurs [13]. The specific heat of the body pool will depend on the percentage of fat in the body. Several runs of the model established 0.9 kcal C -1 kg -1 as an acceptable value for the body of an adult sheep.

3. Flow chart The flow diagram of the model is shown in fig. 1. The program is designed as a set of differential equations describing the rate of change, with respect to time, of the three pools: the fleece tip pool, the skin pool and the body heat pool. At each integration, the

magnitudes of the pools are calculated; and as each heat pool is the product of mass, specific heat and temperature, the temperature of each pool is also computed. The Runge-Kutta method variable step size was used for numerical integration. Initial conditions are given for the three pools and body energy.

4. Sample run Graphical results of several of the potential outputs of a simulation run of 24 hr are shown in figs. 2 and 3. The potential outputs are listed in table 2. In this simulation air temperature varied from 8 to 40°C, while wind speed and relative humidity were held constant. The behavior of the model compares very well with experimental observation [ 10].

R.R. Vera et al., Model o f heat flow in sheep

217

Table 1

Table 2

Inputs required

Outputs

(1)

Climatic and atmospheric parameters Air conductivity Air density Air viscosity Air temperature Aibedo of the animal Albedo of the ground Transmission coefficient of atmosphere Cloud cover Latitude Longitude Relative humidity Wind speed Hour

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

(2)

Animal parameters Weight Height Diameter Length Fleece length Energy intake

5. Hardware and software specifications

Solar, sky and ground radiation incident on the animaZ Net radiation Long-wave radiation emitted by the animal Convection Pool sizes and fluxes Mean fleece tip temperature Mean skin temperature Mean body temperature Loss of heat by perspiration Heat lost through respiratory evaporation Requirement of dietary energy to maintain energy equilibrium

The program was written in CSMP [14] and includes about 140 lines of formula, constants, etc. Core requirements vary with the type of simulation. The above sample run required, in a Borroughs 6700, 17 K bytes of core storage and its execution took 42 41

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Fig. 3. Changes in the temperature of the air; fleece tip, (FTT); skin, and body core, for the sheep (in the same simulation as presented in fig. 3).

218

R.R. Vera et al., Model o f heat flow in sheep

0.30 m i n . For class d e m o n s t r a t i o n s the program has b e e n stored o n disk and was utilized simultaneously from several T e x t r o n i x terminals.

6. Mode of availability Persons interested in obtaining a listing o f the program should contact the authors. The program has extensive d o c u m e n t a t i o n which explains the origin o f most o f the equations used.

References [1] A.R.C., The nutrient requirements of livestock, No. 2 Ruminants (Agricultural Research Council, London, 1965). [2] D.G. Armstrong et al., J. Agric. Sci., Cambridge 55 (1960) 395. [3] J.W. Bennett, Australian J. Agric. Res. 23 (1972) 1045. [4] K.L. Blaxter, The energy metabolism of ruminants (Hutchinson, London, 1962).

[5] K.L. Blaxter et al., J. Agric. Sci., Cambridge 52 (1959) 25. [6] J.M. Brockway et al., 1. Physiol., London 179 (1965) 554. [7] A.H. Brook and B.F. Short, Australian J. Agric. Res. 11 (1960) 557. [8] A.C. Burton and O.G. Edholm, Man in a cold environment (Hafner, New York, 1955). [9] J.L. Clapperton et al., J. Agric. Sci., Cambridge 64 (1965) 37. [10] E. Eyal, J. Agric. Sci., Cambridge 60 (1963) 159. [11 ] R.G. Feagle and J.A. Businger, An introduction to atmospheric physics (Academic Press, New York, 1963). [12] ta Fi~h~,~4~nand-D.A. Saunders, An introduction to heat transfer (Clarendon Press, Oxford, 1950). [13] N. McC. Graham et al., J. Agric. Sci., Cambridge 52 (1959) 13. [14] IBM, System/360 Continuous System Modelling Program, 5th ed. (IBM, New York. 1972). [ 15 ] W.H. KLein, J. Meteorology 5 (1948) 119. [ 16] G.R. Moule, Worm distribution of domestic animals, in: Adaptation of domestic animals, ed. E.S.E. Hafez (Lea and Febiger, Philadephia). [17] N.A.S., A guide to environmental research on animals (National Academy of Sciences, Washington, 1971) p. 335. [18] C.H.B. Priestley, Anstralian J. Agric. Res. 8 (1957) 271