Equations of heat distribution within the body

A vector integral equation describing heat distribution within the body has been derived. The factors considered are heat conduction, forced convection via the circulatory system, environmental exchange, metabolic heat production, and change in heat content. The vector partial differential equation and alternative forms incorporating boundary conditions were also developed. A difference equation based on a first-order approximation to the fundamental equations was derived to form the basis of a model for heat distribution within the body. It has been shown that factors involving conduction and convection must be considered independently unless the temperature of the blood flowing from a region of the body is equal to the average temperature of the tissue in that region. If this relation between tissue and blood temperature does exist, only a single temperature from each eleeent is needed to describe the heat distribution. In this latter case, models which ascribe all heat transfer to “equivalent” conduction or to convection can give valid predictions.     

Theoretical and Experimental Studies of Energy Exchange from Jackrabbit Ears and Cylindrically Shaped Appendages

Convection properties of jackrabbit ears were examined in a wind tunnel and in the field in an attempt to study the possible thermal role of the large ears. This work was part of a study on energy exchange of appendages. Cylindrical copper models of various shapes, aluminum castings of domestic and jackrabbit ears, and an amputated jackrabbit ear were studied in a wind tunnel (a) to define the range for convective heat loss for appendages of various shapes, and (b) to study the effect on convection of model shape and orientation to the wind. Shape, i.e. length and closure, proved important. Orientation to the wind produced no consistent or significant variation in the convection coefficient. The convection coefficients from the ear castings fell within the range generated from the cylindrical models. The convection coefficients for the amputated rabbit ear fell partially within the range. Net thermal radiation loss at midday from the jackrabbit ears was found to be small. Convection from the ears, however, could account for the loss of over 100% of the animal's metabolic heat at an air temperature of 30°C. If air temperature exceeds body temperature, the animal must either store heat or resort to the evaporation of water.     

Boundary layer properties of highly dissected leaves: an investigation using an electrochemical fluid tunnel

Abstract. A method for modelling heat and mass transfer by diffusion-controlled electrode reactions in a fluid tunnel is described. In this procedure, a nickelplated leaf functions as a test electrode, and the convective transfer of ions to the leaf cathode in an electrolyte-filled flow tunnel is measured as a function of flow rate. The method permits the simulation of water vapour and heat transfer, and in particular, the determination of boundary layer conductances, by analogy with observed ion transfer. The approach is applicable to many problems in modelling heat and mass transfer between leaves and their surroundings, and is especially useful in examining the properties of leaves in which surface characteristics or overall shape are complex. Using this method, the properties of the highly dissected leaves of Achillea lanulosa with regard to forced convection were investigated. The leaves showed high transfer conductances, indicating that the effective unit of heat transfer was probably the individual leaf subelements. Conductances tended to be greater and effective characteristic dimensions smaller for the larger, more open leaves of a lower altitude population in contrast with leaves from high altitude plants. While the results provide insight into the properties of these complex leaf shapes, difficulties in interpreting the findings are discussed, and a number of exploratory approaches are suggested for data analysis and interpretation.     

Experimental study on convective heat transfer coefficient of the human body

1. 1. The convective heat transfer coefficient of the human body is essential to predict convective heat loss from the body.

2. 2. The object of this paper is to calculate the convective heat transfer coefficient of the human body using heat flow meters and to estimate the thermally equivalent sphere and cylinder to the human body.

3. 3. The experimental formulae of the convective heat transfer coefficient for the whole body were obtained by regression analysis for natural, forced and mixed convection.

4. 4. Diameters of the thermally equivalent sphere and cylinder of the human body were calculated as 12.9 and 12.2 cm, respectively.

Estimating metabolic heat loss in birds and mammals by combining infrared thermography with biophysical modelling

Infrared thermography (IRT) is a technique that determines surface temperature based on physical laws of radiative transfer. Thermal imaging cameras have been used since the 1960s to determine the surface temperature patterns of a wide range of birds and mammals and how species regulate their surface temperature in response to different environmental conditions. As a large proportion of metabolic energy is transferred from the body to the environment as heat, biophysical models have been formulated to determine metabolic heat loss. These models are based on heat transfer equations for radiation, convection, conduction and evaporation and therefore surface temperature recorded by IRT can be used to calculate heat loss from different body regions. This approach has successfully demonstrated that in birds and mammals heat loss is regulated from poorly insulated regions of the body which are seen to be thermal windows for the dissipation of body heat. Rather than absolute measurement of metabolic heat loss, IRT and biophysical models have been most useful in estimating the relative heat loss from different body regions. Further calibration studies will improve the accuracy of models but the strength of this approach is that it is a non-invasive method of measuring the relative energy cost of an animal in response to different environments, behaviours and physiological states. It is likely that the increasing availability and portability of thermal imaging systems will lead to many new insights into the thermal physiology of endotherms.     

A metabolic derivation of tritium transfer coefficients in animal products

Tritium is a potentially important environmental contaminant originating from the nuclear industry, and its behaviour in the environment is controlled by that of hydrogen. Animal food products represent a potentially important source of tritium in the human diet and a number of transfer coefficient values for tritium transfer to a limited number of animal products are available. In this paper we present an approach for the derivation of tritium transfer coefficients which is based on the metabolism of hydrogen in animals. The derived transfer coefficients separately account for transfer to and from free (i.e. water) and organically bound tritium. A novel aspect of the approach is that tritium transfer can be predicted for any animal product for which the required metabolic input parameters are available. The predicted transfer coefficients are compared to available independent data. Agreement is good (R ²=0.97) with the exception of the transfer coefficient for transfer from tritiated water to organically bound tritium in ruminants. This may be attributable to the particular characteristics of ruminant digestion. We show that tritium transfer coefficients will vary in response to the metabolic status of an animal (e.g. stage of lactation, diet digestibility etc.) and that the use of a single transfer coefficient from diet to animal product is inappropriate. It is possible to derive concentration ratio values from the estimated transfer coefficients which relate the concentration of tritiated water and organically bound tritium in an animal product to their respective concentrations in the animals diet. These concentration ratios are shown to be less subject to metabolic variation and may be more useful radioecological parameters than transfer coefficients. For tritiated water the concentration ratio shows little variation between animal products ranging from 0.59 to 0.82. In the case of organically bound tritium the concentration ratios vary between animal products from 0.15 (goat milk) to 0.67 (eggs). Received: 28 May 2001 / Accepted: 20 August 2001     

Computer-based analysis of steady-state and transient heat transfer of small-sized leaves by free and mixed convection

In order to study convective heat transfer of small leaves, the steady‐state and transient heat flux of small leaf‐shaped model structures (area of one side = 1730 mm²) were studied under zero and low (= 100 mm s^?1) wind velocities by using a computer simulation method. The results show that: (1) distinct temperature gradients of several degrees develop over the surface of the model objects during free and mixed convection; and (2) the shape of the objects and onset of low wind velocities has a considerable effect on the resulting temperature pattern and on the time constant τ. Small leaves can thus show a temperature distribution which is far from uniform under zero and low wind conditions. The approach leads, however, to higher leaf temperatures than would be attained by ‘real’ leaves under identical conditions, because heat transfer by transpiration is neglected. The results demonstrate the fundamental importance of a completely controlled environment when measuring heat dissipation by free convection. As slight air breezes alter the temperature of leaves significantly, the existence of purely free convection appears to be questionable in the case of outdoor conditions. Contrary to the prognoses yielded by standard approximations, no quantitative effect of buoyancy on heat transfer under the considered conditions could be detected for small‐sized leaf shapes.     

Respiratory heat loss in the sheep: a comprehensive model

A model is presented for the respiratory heat loss in sheep, considering both the sensible heat lost by convection ( C(R)) and the latent heat eliminated by evaporation ( E(R)). A practical method is described for the estimation of the tidal volume as a function of the respiratory rate. Equations for C(R) and E(R) are developed and the relative importance of both heat transfer mechanisms is discussed. At air temperatures up to 30 degrees C sheep have the least respiratory heat loss at air vapour pressures above 1.6 kPa. At an ambient temperature of 40 degrees C respiratory loss of sensible heat can be nil; for higher temperatures the transfer by convection is negative and thus heat is gained. Convection is a mechanism of minor importance for the respiratory heat transfer in sheep at environmental temperatures above 30 degrees C. These observations show the importance of respiratory latent heat loss for thermoregulation of sheep in hot climates.     

哺乳动物毛被传热性能及其影响因素

毛被能够加强或减弱动物向周围环境的热量散失,毛被的形态结构和颜色是传热性能的决定因素,其传热过程往往是传导、对流和辐射3个过程的耦合。以往研究发现环境因子中,风可增加机体向环境中的散热速率,且散失量与风速正相关,且动物通过调节在风场中的姿态来适应不同风向。动物体与环境间的温差是影响散热速率的另一因素,不同环境中的动物通过改变毛被结构来适应温差变化。毛被含水率上升会引起导热和蒸发冷却作用加强,动物通过行为或毛被结构变化来调节毛被含水率。毛色决定毛被吸收和反射热辐射的能力。毛被传热性能直接把动物的生理特点与环境因子关联起来,这对揭示动物的适应、进化都具有重要意义。同时提出,毛被结构和传热性能的研究还有助于仿生学意义的挖掘。因此,今后应重点在毛被结构和物理性能、研究技术与方法以及毛被生物学和仿生学意义等方面开展研究。     

A diffusion model for drying of a heat sensitive solid under multiple heat input modes

To obtain optimal drying kinetics as well as quality of the dried product in a batch dryer, the energy required may be supplied by combining different modes of heat transfer. In this work, using potato slice as a model heat sensitive drying object, experimental studies were conducted using a batch heat pump dryer designed to permit simultaneous application of conduction and radiation heat. Four heat input schemes were compared: pure convection, radiation-coupled convection, conduction-coupled convection and radiation-conduction-coupled convection. A two-dimensional drying model was developed assuming the drying rate to be controlled by liquid water diffusion. Both drying rates and temperatures within the slab during drying under all these four heat input schemes showed good accord with measurements. Radiation-coupled convection is the recommended heat transfer scheme from the viewpoint of high drying rate and low energy consumption.     

An Analytical Model of the Counter-Current Heat Exchange Phenomena

An analytical model for the counter-current heat exchange mechanism in animals has been formulated and a solution has been obtained. The nondimensional parameters that govern the mechanism have been determined in terms of the properties of the animal. The normalized temperatures are functions of normalized distance and, in general, three nondimensional heat transfer conductances. Graphical results are presented for two representative physiological systems. These results allow a delineation of those situations in which counter-current heat transfer is important, and also a quantitative prediction of the heat transfer and temperature distributions. The theory is compared to the available experimental results.     

A model of heat flow in the sheep exposed to high levels of solar radiation.

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 cooling are computed. Temperatures at the fleece tip, skin and body core are monitored.     

A generic geometric transformation that unifies a wide range of natural and abstract shapes

To study forms in plants and other living organisms, several mathematical tools are available, most of which are general tools that do not take into account valuable biological information. In this report I present a new geometrical approach for modeling and understanding various abstract, natural, and man-made shapes. Starting from the concept of the circle, I show that a large variety of shapes can be described by a single and simple geometrical equation, the Superformula. Modification of the parameters permits the generation of various natural polygons. For example, applying the equation to logarithmic or trigonometric functions modifies the metrics of these functions and all associated graphs. As a unifying framework, all these shapes are proven to be circles in their internal metrics, and the Superformula provides the precise mathematical relation between Euclidean measurements and the internal non-Euclidean metrics of shapes. Looking beyond Euclidean circles and Pythagorean measures reveals a novel and powerful way to study natural forms and phenomena.     

Cutaneous heat flux models do not reliably predict metabolic rates of marine mammals

Heat flux models have been used to predict metabolic rates of marine mammals, generally by estimating conductive heat transfer through their blubber layer. Recently, Kvadsheim et al. (1997) found that such models tend to overestimate metabolic rates, and that such errors probably result from the asymmetrical distribution of blubber. This problem may be avoided if reliable estimates of heat flux through the skin of the animals are obtained by using models that combine calculations of conductive heat flux through the skin and fur, and convective heat flux from the surface of the animal to the environment. We evaluated this approach based on simultaneous measurements of metabolic rates and of input parameters necessary for heat flux calculations, as obtained from four harp seals (Phoca groenlandica) resting in cold water. Heat flux estimates were made using two free convection models (double-flat-plate and cylindrical geometry) and one forced convection model (single-flat-plate geometry). We found that heat flux estimates generally underestimated metabolic rates, on average by 26-58%, and that small variations in input parameters caused large variations in these estimates. We conclude that cutaneous heat flux models are too inaccurate and sensitive to small errors in input parameters to provide reliable estimates of metabolic rates of marine mammals.     

Air movement and heat loss from sheep. I. Boundary layer insulation of a model sheep, with and without fleece

A model sheep, made from metal cylinders and hemispheres, was heated electrically. Heat loss by forced convection in a wind tunnel was analysed in terms of the dependence of the Nusselt number (Nu) on Reynolds number (Re). For a bare trunk Nu = 0.095 Re0.684, but with fleece covering the trunk to a depth of 3.5 cm, Nu = 0.0112 Re0.875 when the mean radiative temperature of the the coat was taken as the surface temperature. Heat transfer by convection from the whole body, including legs, was described by Nu = 0.029 Re0.80. However, a bulk Nesselt number should not be used to estimate heat loss from a live sheep in a hot environment if the windspeed is below about 4 m s-1 because the relation between mean surface temperature, Nusselt number and convective heat flux is not unique.     

Natural convection from leaves at realistic Grashof numbers

Abstract. The boundary layer resistance of model leaves was measured in still air, at a range of leaf-to-air temperature differences. The results were compared to those calculated from standard formulae for natural convection. The agreement between observed and calculated was only satisfactory when Grashof numbers exceeded about 105. At the lower Grashof numbers, which often prevail in nature, the observed rates of heat transfer considerably exceeded those calculated.     

RADIATION AND CONVECTION IN CONIFERS

The transfer of energy to and from a conifer branch involves solar radiation, thermal radiation from the ground, atmosphere, and surroundings, thermal emission by the branch, and free convection in still air and forced convection in wind. It is necessary to know the actual surface area of the branch, the effective area for absorbing sunlight, the effective area for absorbing long wave thermal radiation and for emission, and the free and forced convection coefficients. These parameters are determined using silver castings of blue spruce and white fir branches suspended in an evacuated radiation chamber and in a wind tunnel. The actual surface area of a branch is determined by means of an electrolytic technique. Numerical examples are given for energy transfer in a natural environment for conifers and comparison is made to a broad deciduous type of leaf. The role of transpiration in the energy transfer process is discussed.     

Triangular node for Transmission-Line Modeling (TLM) applied to bio-heat transfer

Transmission-Line Modeling (TLM) is a numerical method used to solve complex and time-domain bio-heat transfer problems. In TLM, rectangles are used to discretize two-dimensional problems. The drawback in using rectangular shapes is that instead of refining only the domain of interest, a large additional domain will also be refined in the x and y axes, which results in increased computational time and memory space. In this paper, we developed a triangular node for TLM applied to bio-heat transfer that does not have the drawback associated with the rectangular nodes. The model includes heat source, blood perfusion (advection), boundary conditions and initial conditions. The boundary conditions could be adiabatic, temperature, heat flux, or convection. A matrix equation for TLM, which simplifies the solution of time-domain problems or solves steady-state problems, was also developed. The predicted results were compared against results obtained from the solution of a simplified two-dimensional problem, and they agreed within 1% for a mesh length of triangular faces of 59 µm±9 µm (mean±standard deviation) and a time step of 1 ms.     

Heat loss from giant extinct reptiles

There is disagreement in the literature about the relative rates of heat loss from a large animal surrounded by either air or water. Here, it is shown that, in most circumstances, the rate at which heat is lost by a large body is significantly greater when it is immersed in water than when it is surrounded by air, assuming that the two fluids are at the same temperature. The only circumstance when this may not apply is when comparing air with fresh water when both are at a temperature somewhere between 0 degrees C and 6 degrees C, the animal is still and water or air currents are negligible. Under these conditions, free convection in water is weak or non-existent, and so the combined effect of conduction and free convection in air becomes comparable to or even greater than that of conduction alone in water. However, in these circumstances, radiation is the dominant mode of heat loss to both media, and so heat losses are approximately the same in both air and water.     

Heat and mass exchange processes between the surface of the human body and ambient air at various altitudes

 The rates of convection and evaporation at the interface between the human body and the surrounding air are expressed by the parameters convective heat transfer coefficient h _c, in W m^–2°C^–1 and evaporative heat transfer coefficient h _e, W m^–2 hPa^–1. These parameters are determined by heat transfer equations, which also depend on the velocity of the airstream around the body, that is still air (free convection) and moving air (forced convection). The altitude dependence of the parameters is represented as an exponential function of the atmospheric pressure p: h _c∼p ⁿ and h _e∼p ^1–n, where n is the exponent in the heat transfer equation. The numerical values of n are related to airspeed: n=0.5 for free convection, n=0.618 when airspeed is below 2.0 ms^–1 and n=0.805 when airspeed is above 2.0 ms^–1. This study considers the coefficients h _c and h _e with respect to the similarity of the two processes, convection and evaporation. A framework to explain the basis of established relationships is proposed. It is shown that the thickness of the boundary layer over the body surface increases with altitude. As a medium of the transfer processes, the boundary layer is assumed to be a layer of still air with fixed insulation which causes a reduction in the intensity of heat and mass flux propagating from the human body surface to its surroundings. The degree of reduction is more significant at a higher altitude because of the greater thickness of the boundary layer there. The rate of convective and evaporative heat losses from the human body surface at various altitudes in otherwise identical conditions depends on the following factors: (1) during convection – the thickness of the boundary layer, plus the decrease in air density, (2) during evaporation (mass transfer) – the thickness of the boundary layer, plus the increase with altitude in the diffusion coefficient of water vapour in the air. The warming rate of the air volume due to convection and evaporation is also considered. Expressions for the calculation of altitude dependences h _c (p) and h _e (p) are suggested. Received: 23 June 1998 / Accepted: 10 February 1999