Temperature Control Mechanism by Butterfly Wings by Milad Arkian

Temperature Control Mechanism echanism by Butterfly utterfly Wings

3rd Year Mechanical Engineering - Final Report Author: Milad Arkian Project Supervisor: upervisor: Professor Hector Iacovides

Acknowledgements

“Αιέν αριστεύειν” “Forever improving”

My sincerest and deepest gratitude is due to my personal tutor Professor H. Iacovides, without whom this report would have lost its heart and its eyes. His guidance and direction has been essential and have increased my desire and passion for the world of science. 2

Contents Abstract……………………………………………………………………………………………….

Glossary……………………………………………………………………………………………….

Introduction…………………………………………………………………………………………

Butterfly Anatomy……………………………………………………………………………….

2.1

General body……..………………………………………………………………………..........

2.2

Wings………………………………………………………………………………………………....

2.3

Head & Thorax……………………………………………………………………………..……..

2.4

Proboscis……………………………………………………………………………………………..

Literature Survey………………………………………………………………………………….

3.1

Butterfly Heat Transfer Models …………………………………………………………..

3.2

Behavioural Habits & Body Traits in Effecting Thermoregulation ………..

3.2.1

Posture…………………………………………………………………………………………………

3.2.2

Shivering…………………………………………….………………………………………………..

3.2.3

Wing Level..…………………………………………………………………………………………

3.2.4

Abdominal Pumping…………………………………………………………………………….

3.2.5

Tilting………………………………………………………………………………………………….

3.2.6

Fur Thickness……………………………………………………………………………………….

3.2.7

Aposematic Colours……………………………………………………………………….…...

3.2.8

Wind Shielding………………………………………….………………………………………...

Conclusions of the Literature Survey………………………………..………………….

Heat Balance………………………………………………………………………………………..

General Analysis and Key Assumptions……………………………..…………………

3.3 4 4.1 4.2 4.3 4.4

Wing Angle: 90 ……………………………..……………..……………………………..…….. o

Wing Angle: 45 ……………………………..……………..……………………………..…….. o

Wing Angles: 46-89 ……………………………..……………..…………………………….. o

34 39 40

4.5

Wing Angles: 10-44 ……………………………..……………..……………………………..

4.6

Results & Conclusions……………………………..……………………………..……………

4.7

Summary of Conclusions………………………………………………………………………

Future Work…………………………………………………………………………………………

Solidworks Simulations……………………………..……………..………………………….

Appendix………………………………………………………………………………….…………..

5 5.1 6

6.1

Databank……………………………………………………………………………………………..

6.2

Nomenclature………………………………………………………………………………………

References……………………………………………………………………………………………

Gantt Chart…………………………………………………………………………………………..

List of Figures Figure No

Caption

An annotated diagram of the general body parts of the butterfly species……..

Annotated butterfly showing difference between fore-wing and hindwing………………………………………………………………………………………………………………...

Difference between the ventral and dorsal side of the wings ………………………..

Flight stroke positions…….…………………….…………………….………………………………….

Wing scale structure…………………….…………………….…………………….…………………….

Annotated picture showing the upper body parts of the butterfly………………….

Uncurled proboscis…………………….…………………….…………………….………………………

Proboscis…………………….…………………….…………………….……………………………………..

Representation of the yaw angle superimposed over the butterfly body……….

Page No

Nusselt number vs. Reynolds number for butterflies at a yaw angle of 45 . The dashed lines represent individual butterflies with the solid line the model cylinder at a yaw angle of y = 90o……………………………………………………………………. Effects of fur thickness on flight time and solar absorptivity: Each line represents a different value for butterfly fur thickness (mm). The research sites are Montrose (elevational height=1.5km) and Skyland (elevational height= 2.8km) …………………….…………………….…………………….……………………..…….

Butterfly body (depicted by the cylinder) and wings (symmetrical lines) are shown in relation to the orientation angle Ψ (normal to the thorax) and wing angle θ (angle between the wings the orientation angle).………………………………

Basking postures: pictorial illustrations of the lateral, dorsal and reflectance basking postures used by butterflies to regulate their temperature.…………………….…………………….…………………….………………………………..

Reflectance basking: The black basal absorption areas are responsible for taking in solar radiation and increasing the body temperature through heat conduction. The hatched distal region of the wings was not seen to effect body temperature. The white medial regions reflect solar radiation from the wings onto the thorax or abdomen. …………………….………………………………………………………………………………………………...

Melanisation in Pierid butterfly wings. Where + indicates an increase in temperature when melanisation occurs. O corresponds to no effect and – as a

decrease in temperature. …………………….…………………….……………………………….

Relationship between body and ambient temperature of perched male black swallowtails in the field. Solid lines indicate points where body temperatures equal ambient. Dotted lines represent pattern of thoracic temperature. Black spots represent thoracic temperature; white spots are the abdominal temperatures. …………………….…………………….…………………………………………………..

A graph comparing the various postures taken up by the Swallowtail butterfly for given ambient temperatures and levels of solar radiation………………………..

Close-up of butterfly fur…………………….…………………….…………………….………........

19 & 20

Aposematic colours of the unpalatable Birdwing butterfly.…………………………….

Aposematic colours of the white Pierid butterflies…………………….…………………..

Wind shielding of the thorax by the abdomen…………………….………………………….

Proportion of solar radiation striking the butterfly body…………………………………

Butterfly body, wings and interaction with incoming solar radiation………………

Forced convection along the butterfly body…………………………………………………….

View factor to the surroundings ~ 0.5……………………………………………………………..

Proportion of body acquiring incident radiation………………………………………………

View factor to the surroundings for wing angles of 46-89 …………………………….. o

View factor to the surroundings for wing angles of 10-44 ………………………………

Solidworks model of the butterfly (view 1)……………………………………………………..

Solidworks model of the butterfly (view 2)……………………………………………………..

Applying ambient heat conditions on meshed butterfly body…………………………

Table 1

Body Dimensions……………………………………………………………………………………………..

Table 2

Equilibrium temperature and proportions of solar radiation from sun and wings

31 42

Table 3

Proportions of key experiment parameters

Body temperatures of Swallowtails in field studies (oC). Mean ± sd above, range below.…………………….…………………….…………………….…………………….…………

Various parameters in relation to different wing and abdominal positions Mean ± sd. (Sample size) ………………….…………………….…………………….……………….

Critical thoracic temperatures for various activities of black Swallowtails in the flight cage. Mean (N = sample size) above, range below (oC)……………………

Identification, sex, means and standard deviations (SD) of body mass m [mg], wing length R [mm], wing loading pw [N m-2], thoracic temperature Tth [oC], ambient temperature Ta [oC], thoracic excess ΔT=Tth-Ta [oC] and solar irradiance I [W m-2] for two species of Danaine butterfly………………………………..

Solar absorptivity, thoracic fur thickness and thoracic diameter of four butterfly species in central Colorado…………………………………………………………......

Cumulative daily flight activity time (KFAT) in hours for the three sites of different elevational heights…………………….…………………….………………………………

A2 A3 A4

A5 A6

Sensitivity analysis of the energy balance model, where Td is body temperature excess (labelled as Tex in the nomenclature), Tex = Tb-Ta. This graph relates how each parameter may affect the butterfly’s body temperature…………………….…………………….…………………….………………………………... Table of results for experiments carried out at high elevations……………………….

49 49

Abstract The aim of this project is to investigate the biomimetic process by which butterflies regulate and maintain their body temperatures by modelling and understanding the heat energy balance equations for different wing postures. The Swallowtail species of butterfly was taken for the body dimensions. Three main postures are responsible for the basking practice butterflies use to regulate temperature behaviourally: dorsal, lateral and reflectance basking. Thermoregulation in the wings occurs through a combined effect of colour pigmentation and skeletal structure supplemented with behavioural heat regulation habits. Wing angles between 10-44o provided the largest equilibrium body temperature (76.2oC) and the radiation reflected onto the body is an order of magnitude above wing angles of 45 and above. Using a higher wing angle has an advantage of reducing the heat loss via radiation and convection and may be adopted at low wind speeds where the butterfly is at a favourable body temperature. The equilibrium body temperature is approximately constant for wing angles above 45o which indicates that the reduction in convective and radiative heat loss is balanced by a lower intensity of heat uptake. Radiation in-take is highly dependent on wing melanisation and heat loss can be reduced by behavioural postures such as tilting the body away from the solar rays or having thick fur to prevent convective heat loss.

Glossary Abdominal Pumping: Contraction of the abdominal muscles that results in the expansion of the air sacs. This forces greater active ventilation, as opposed to passive ventilation that occurs by normal breathing. Aposematic coloration: In biology, the technical name for warning coloration markings that make a dangerous, poisonous, or foul-tasting animal particularly conspicuous and recognizable to a predator. Examples include the yellow and black stripes of bees and wasps, and the bright red or yellow colours of many poisonous frogs and snake, ref Cooling curve: A curve obtained by plotting time against temperature for a solid-liquid mixture cooling under constant conditions. DFW, Dorsal Fore Wing: top side (posterior) of the butterfly wings located at the larger fore wings. DHW, Dorsal Hind Wing: top side of the butterfly wings located at the smaller hind wings. Diffuse Radiation: radiation that has been scattered by atmospheric constituents (e.g. clouds, particulates, aerosols). Delineate: To represent pictorially. Dimorphism: Are the systematic differences acquired in form that occurs due to a difference in gender amongst the same species. Common examples include colour, size or the absence of certain body organs such as antlers or tusks. Direct Radiation: Portion of radiation emitted by a radiation source which reaches the observed receiving point via the shortest distance, possibly weakened by existing shielding walls. The direct radiation is distinguished from scattered (diffuse) radiation which may reach the receiving point indirectly due to scattering on other media. Electromagnetic spectrum: The complete range of frequencies of electromagnetic waves including, in order of lowest to highest: radio, infrared, visible light, ultraviolet, X-ray, and gamma ray waves. Emissivity: defined as the ratio of the energies emitted radiated by the material and by a black body at the same temperatures. Heat Flux: Heat flux is the rate of heat energy transfer through a given surface. Hemolymph: The circulatory fluid found in invertebrates. It is a freely flowing fluid that moves on an open plane around the invertebrate’s body. Hydrophobic: A substance/molecule/object that repels water or is incapable of dissolving in water. Irradiance: Irradiance is the term for used in radiometry for the power of electromagnetic radiation at a surface, per unit area. Irradiance is used when the electromagnetic radiation is incident on the surface. The SI units for all of these quantities are watts per square metre (W·m−2). Melanisation: Melanin is a substance known to darken the appearance of the object it is concentrated on. Melanisation is the process by which butterflies have darker pigments on their bodies due to a local concentration of melanin. Mesothorax: The middle of three segments of the thorax on an insect’s body. The mesothorax houses the second pair of legs. 8

Monochromatic: Pertaining to radiation composed of only one wavelength. Monochromatic Absorptivity: Defined as the ratio of the absorbed radiation at a specific wavelength and temperature to the absorbed radiation by a black body at the same wavelength and temperature. Perching: The butterfly rests or perches at a position or spot for roosting. Photoperiod: The duration of the organism’s daily exposure to light, especially in regards to the effect of its growth and development. Quiescent: Still, inactive or at rest. Radiance: Radiance or spectral radiance are radiometric measures that describe the amount of light that passes through or is emitted from a particular area, and falls within a given solid angle in a specified direction. They are used to characterize both emission from diffuse sources and reflection from diffuse surfaces. The SI unit of radiance is watts per steradian per square metre (W·sr-1·m-2). Irradiance: Total amount of radiative flux incident upon a point on a surface from all directions above the surface hemisphere. Roosting: The butterfly settles down for rest or sleep. Specular radiation: The incident radiation rays are reflected according to the law of reflection. The law of reflection states that should a construction line normal to the flat reflective surface, the incident and reflected rays will exhibit equal angles. Solar spectrum: The spectrum of the sun's electromagnetic radiation extending over the whole electromagnetic spectrum. Thermocouple: A junction between two different metals that produces a voltage based on temperature difference. VFW, Ventral Fore Wing: underneath surface (anterior) of the butterfly wing located at the larger fore wings VHW, Ventral Hind Wing: underneath surface of the butterfly wing located at the smaller hind wings. Yaw Angle: The angle between a butterfly’s longitudinal body axis and its line of travel, as seen from above. Zenith angle: The angle at the earth's surface measured between the Sun and an observer's or an object’s zenith (a point directly above the observed object.

1 Introduction The study of biomimicry has exposed many solutions to human related problems. For example, spiderâ&#x20AC;&#x2122;s silk is known to have five times the tensile strength of steel for a given diameter (Heimbuch, 2010). Furthermore the sonar system that bats use to navigate blindly around caves is now being mimicked for submarine/air craft radar systems. As butterflies have to survive through the daily challenges of varying temperature conditions, there is sufficient purpose to research and understand the structure of the wings and their heat regulation behaviour. Butterflies are known to live and survive under delicate environmental conditions. They are biologically cold blooded and some form of basking (reclining under solar radiation to increase body temperature) is required in order to raise their body temperatures for flight. This basking and its link with thermoregulatory practices of the butterflies will be under investigation. An attempt to accurately represent the physical heat transfer mechanism between the butterfly and its surroundings will be made. This will include analysing the proportions of incoming solar radiation to out-going heat loss via convection and radiation and obtaining the equilibrium body temperature for the butterfly. The main parameter used to quantify thermoregulation is wing angle and four bands of wing angles (10-44o, 45o, 46-89o and 90o) will be used to better understand how the butterfly uses solar radiation to increase its body temperature. In providing a heat balance for the butterfly, crucial body dimensions will be taken from the literature survey and any assumptions made will be backed by either previous scientific researchers or used with supporting reasoning. The databank in the appendix will provide a useful source of experimental values for the key parameters in the heat balance equations. The final section of the research consists of a complimentary conclusion section that will attempt to explain the significance of the results and provide insight into the benefits of thermoregulation for the butterflies. The industrial relevance of this project lies with better understanding how a natural phenomenon such as butterfly thermoregulation may benefit current research on solar panel efficiency. Other potential engineering benefits include using the wing structure to model Nano-scale computer chips to better dissipate heat. Although not considered in this study, the way in which light interacts with the wings is becoming increasingly important and may benefit areas of optical research.

2 Butterfly Anatomy Prior to proceeding with the investigation inve it is useful to understand some of the organs of the butterfly species that may relate to its method of temperature regulation.

Figure 1, An annotated diagram of the general body parts of the butterfly species, species (Wilson, 2010)

2.1 General body The butterfly anatomy can be broadly labelled la into three main sections: the head, thorax and abdomen (figure 1).. 6 legs and 4 wings are attached to the the thorax, with the wing appearance (colour, shape) being the mode of identifying and naming a butterfly. Butterflies are cold blooded and instead of having a method of internal heat production they rely on external heat sources to generate the temperatures required for flight and other energy intensive activities. activities A butterfly’s skeleton is hard case on the outside outside and on the inside there is only blood nerves and organs (Thinkquest, n.d).. Butterflies have an open blood circulation i.e. they have no veins and the whole of the inside of the body is covered or flooded with blood (Thinkquest, n.d). In vertebrate species the blood circulatory system is closed. The blood flow provides two functions: gas exchange and nutrient/waste exchange. The heart pumps blood to the tissues allowing the cells to exchange material with the blood stream. stream In invertebrates or insects the gas and nutrient/waste exchange are exclusive. Gas is exchanged with the surroundings via the trachea or ‘windpipe’ opening directly into the air. The process of gas exchange is by simple diffusion through the trachea branches. Butterflies and other invertebrates have a different liquid for circulatory purposes, called hemolymph that carries nutrients and waste. Due to the open circulation system in butterflies the 11

hemolymph isnâ&#x20AC;&#x2122;t constrained to arteries and veins. A dorsal tube tube (rudimentary butterfly heart) pumps hemolymph over all its organs, circulating freely throughout the abdomen. The hemolymph is collected back into the heart via simple diffusion.

Fore-wings

Hind-wings

Figure 2, Annotated butterfly showing difference between fore-wing wing and hind-wing hind (Wong, n.d)

Dorsal, top side of the wings

Ventral, underside of the wings

Figure 3,, Difference between the ventral and dorsal side of the wings (Anonymous 1, 2010)

Flight stroke begins, wings held together Completed stroke Mid-stroke

Figure 4, Flight stroke positions (Smetacek, 2000)

Figure 5, Wing scale structure (Horton, 2010)

2.2 Wings Butterflies have 4 wings (2 fore-wings fore and 2 hind-wings), with the fore-wings being the top, larger wings and the hind-wings wings the smaller, small lower set (figure 2). The fore-wingss and hind-wings are symmetrical. Butterfly wing ing colour is based on the reflective tendency of each wing scale and the wavelength of light that is not absorbed by the wings is reflected and this gives the iridescently powerful vibrancy of the wings. The entire body of the butterfly (including the wings) is covered with a hydrophobic waxx layer to protect the species from water related damage. (Thinkquest, n.d). Butterfly flight occurs (figure 4)) by the beating of the wings from 5-10o (above above their thorax) and o swing through an arch of almost 180 at which the stroke is completed. The structure of the butterfly wing consists of thousands of microscopic scales split into two to three layers (Horton, 2010).. Each of these scales is further split into multiple layers separated by air (Horton, 2010). These multiple scale layers provide numerous numerous instances of constructive interference. interference In constructive interference two waves meet with the resulting wave being the sum of the preceding amplitudes. Consequently,, when light beams interact and reflect off these layers, the intense butterfly wing colours are produced (Horton, 2010). A simplified overview of the butterfly wing w structure is demonstrated in figure 5. 5

Head

Compound eyes

Proboscis Thorax Figure 6,, Annotated picture showing the upper body parts of the butterfly (Anonymous 3, n.d). n.d)

2.3 Head & Thorax The thorax horax is the middle of the three main parts of a butterflyâ&#x20AC;&#x2122;s body, body, between the head and the abdomen.. The thorax houses all the legs and wings of the butterfly therefore it is the crucial organ of the body when it comes to thermoregulation. Should the thorax be at a temperature below or above that of the acceptable flight range the butterfly would suffer from from reduced flight capabilities. capabilities This could affect territorial defence behaviour, behavi mating and escape from a predator.

Uncurled proboscis

Proboscis

Figure 7, uncurled proboscis, (Knew, 2008)

Figure 8,, Proboscis, (Valentino, 2006)

2.4 Proboscis The butterfly proboscis (figuress 7 & 8) or â&#x20AC;&#x2DC;tongueâ&#x20AC;&#x2122; providess the vehicle through which butterflies feed (predominately nectar) but also sweet fruit occasionally. After a butterfly lands on a source of food there is a reaction that causes the proboscis to uncurl and extend to the source of the food. This high surface area curl allows the t butterfly to keep its long proboscis compact until required for use.. The proboscis is shaped like a straw when uncurled and provides the vehicle through which the butterflies suck up nectar or other viable food products such as water or tree sap.

3 Literature Survey The aim of the literature survey is to compile scientific research previously written on similar aspects of this study. As the heat balance is at the heart of this study the first section (3.1) of the literature search will rely on obtaining derivations of the heat balance with accompanying source details. The heat balances required are the steady state and transient derivations including specifics of convection and radiation heat loss. Section 3.2 will solely focus on behavioural habits or body traits that effect thermoregulation of the butterfly such as tilting, fur thickness or wind shielding. Where there are differences in notation between the different research papers, a master label has been used in the nomenclature for ease of use. For example Td as well as Tex have been used in the research papers for body temperature excess (Tb - Ta). Here the label Tex has been chosen as the principal identity for the expression of body temperature excess. Similarly where there is a difference in units between researchers, SI units have been used as the universal set of units. Any data from previous papers will be converted into SI units for calculations and data handling during the simulation of the heat balance. All of the terms are tabulated under Nomenclature in the Appendices.

3.1 Butterfly Heat Transfer Models Research paper reference: (Kingsolver, 1982) The purpose of the literature search is to identify research produced by scientists that conform to similar outcomes required in this report. Additional information in each journal provides a useful databank from which pools of data may be pulled for the heat balance. One particular paper (Kingsolver, 1982) provided a strong background model for the heat balance. Kingsolver provides a derivation of the heat balance equation (steady and transient) whilst providing additional commentary on the effects of yaw angle and other thermoregulatory parameters in regards to the heat balance. The foremost purpose of Kingsolverâ&#x20AC;&#x2122;s (1982) thesis is to determine the convective heat transfer hf for real and model butterflies. A significant section of the research is carried out on a set of model and real butterflies in an open circuit wind tunnel under Reynolds numbers Re of around 0 to 3,000. A graph (figure 10) of the Nusselt number Nu (Nu = hfDeff/k) against Re is made, with 0, 45 and 90 degrees of yaw angle (rotation about the vertical axis (figure 9)). These yaw angles will compare the real and model butterflies and whether their orientation to the wind makes a change to the convective heat coefficient. The Nusselt number is the ratio of convective to conductive heat transfer normal to the boundary surface of a body. Further tests from the author attempted to explore the effects of fur on the coefficient of convective heat transfer. The tests concluded that fur acts as an insulation layer to reduce convective heat loss.

y â&#x20AC;&#x201C; Yaw angle: rotation about the vertical a axis

Figure 9, Representation of the yaw y angle superimposed over the butterfly body (Hicker, n.d) The yaw angle (figure 9) is the angle between a butterflyâ&#x20AC;&#x2122;s longitudinal body axis and its line of travel, as seen from above the butterfly. butterfly The mathematical derivation (Kingsolver 1982) is described below: Reynolds Re number was defined as

(5-1)

Where

(5-2)

V, the volume and L, the longitudinal length of the butterfly model were measured experimentally. Deff is the characteristic dimension of the butterfly model (taken to be the maximum width of the mesothorax including the fur). The thorax has three sections, the mesothorax being bein the middle segment. To study the forced convective heat transfer transfe the author defines its equation as:

(5-3)

Kingsolver used a combination of steady state and transient methods for estimation of heat transfer coefficients. For the transient model, the butterfly model is heated and time constant Ď&#x201E; estimated from the resulting cooling curve:

# $ % ln $ &

(5-4)

The total heat transfer coefficient hT is then calculated from:

' 16

()* &

(5-5)

Where the area of the model: A=Ď&#x20AC;DeffL. The forced convective coefficient, correction factors for radiation and conduction heat transfer are required. When free convection is negligible the forced convective coefficient may be written as:

' +

(5-6)

The radiation correction factor hR may be estimated by: 0 ,-. /+ 0 1 + /+

(5-7)

The correction factor for conductive heat transfer of thermocouple wires and support structures:

23 4 . /5 4 1 4 . /5 1

(5-8)

For the transient derivation, the criterion used for experimental conditions (where free convection is negligible) is stated below:

76 8 0.1

(5-9)

Gr is the Grashof number = (gÎ˛(Tb-Ta)D3eff/Î˝2), a dimensionless measure of the free convective heat transfer. The ratio (Archimedes number Ar) indicates the relative magnitude of free vs. forced convection. In the steady-state analysis, the butterfly model is heated internally with a resistance wire. The power input to the heater and steady-state model temperature and ambient air temperature are being measured here. From the steady-state energy balance, the total heat transfer is estimated to be: 0 < = ,- 0 =

23 4 4 4

(5-10)

Kingsolver writes that the free convective heat transfer coefficient is a function of the temperature difference Î&#x201D;T between the model and the air, whereas the forced convection coefficient hf is not. For the state-state method, a plot of hf against Î&#x201D;T at low values of Re confirmed that free convection was negligible for the experimental conditions used. The verification of the experimental procedure was based on a standard cylinder at a reference wind velocity, where the results from the authorâ&#x20AC;&#x2122;s experiment agreed within Âą10% for all data.

Figure 10: Nusselt number Nu as a function of Reynolds number Re and yaw angle y for one species of Colias Butterfly (Kingsolver 1982). Figure 10 shows that the Nusselt number is essentially independent of the yaw angle for a given Reynolds number (common Reynolds numbers experienced by the butterflies in external fields are 25-1200). For a given yaw angle, as the Reynolds number increases so does the Nusselt number. number Yaw o o angle y at 45 gave the largest increase of Nu with 90 giving the lowest and intermediate angles such as 30o to 60o fitting in in-between. between. Further work by the author based on fur and non-fur non models show that fur has a distinctive effect on the heat transfer process, especially as an insulation barrier against convective heat loss. In one experiment on regional height experiments, Kingsolver (1988) chose three three sites in Colorado, Colorado USA are used to compare the relative differences that an elevational gradient will bring to the rate of flight activity. They are Montrose (height h=1.5km), Skyland h=2.8km and Mesa Seco h=3.3-3.6km. h=3.3 At each site, flight activity is given to be a function of VHW absorptivity and fur thickness. An example of the relevant data collected is shown in figure 11 below.

Figure 11, Effects of fur thickness on flight time and solar absorptivity: Each line represents a different value for butterfly fur thickness (mm). The research sites are Montrose (elevational ( height=1.5km) and Skyland (elevational height= 2.8km) (Kingsolver 1988). The various line numbers for each respective site are different fur thicknesses ranging from 0 to 1.5mm, which are the possible useful ranges of fur thickness for thermal regulation. There are clear indications that for a given % solar absorptivity and fur thickness there is a much longer flight time for the butterflies habituating at lower altitudes. Conclusions • This thesis provided a strong background on the steady state state and transient heat balance models taking into any considerations that could have affected the results. • Fur helps in insulating the butterfly’s body from convective heat loss. • The Nusselt number is independent of the yaw angle towards the wind direction. • For a given solar absorptivity and fur thickness, butterflies exhibit longer flying time at lower altitudes. Research paper reference: (Kingsolver, 1983): A second noticeable report (Kingsolver, 1983) deals with elevational effects on flight activity times for butterflies. This is split into three separate regions of low, mid and high elevational regions measured from the ground. These The regions are represented by Montrose at height h=1.5km, h=1.5km Skyland at h=2.8km and Mesa Seco at h=3.3-3.6km.. The author touches on the diverse meteorological conditions placed on the respective butterfly populations living in the low to high regions. Some of the conditions include wind velocity, butterfly wing absorptivity, fur thickness and cloud coverage. c

To develop a model of the heat transfer processes Kingsolver begins by stating a general set of conditions, widely applicable for the Colias butterflies. Firstly their body temperatures are assumed to be isothermal and the ideal position of rest is at the top of a vegetational layer. For these set of conditions the steady-state energy balance is:

>? ># = >

(5-11)

Where Qs is total solar radiative heat flux, Qt is thermal radiative heat flux and Qc is convective heat flux. Furthermore Kingsolver defines an equation for a resting butterfly and the corresponding solar radiative energy flux:

>? >?. <@ = >?. < = >?.@

A ?. <@ B?. <@ = A ?.@ B?.@ = A6H ?.@ B?.##I cos .G1

(5-12)

(5-13)

Qs.dir, Qs.dif and Qs.ref are the direct, diffuse and reflected solar radiative heat fluxes respectively (equation 2.3). The direct heat flux is the solar radiation, emitted onto the butterflyâ&#x20AC;&#x2122;s body and similarly the diffuse heat transfer is the proportion of heat transferred onto the body under cloudy conditions. The reflected heat flux is the solar radiation reflected off the wings and onto the body with the units being [W/m2s] for each three. Hs.dir, Hs.dif and Hs.ttl are the direct, diffuse and total solar radiative horizontal flux densities or irradiance [W/m2]. As.dir, As.ref and As.ttl are the corresponding direct, reflected and total heat transfer surface areas. Îą is the solar absorptivity, rg is the substrate (ground/vegetation) solar reflectivity and z is the zenith angle (defined under glossary). For basking Colias butterflies orientated perpendicularly to the solar beams Kingsolver has assumed that As.dir = As.ref = 0.5As.ttl. This assumption is made because at any one time the proportion of solar radiation reaching the butterfly body is 0.5 as the other half of the butterfly body will be shaded. Values of the total solar horizontal flux density were measured in the field. For sunny conditions the relative proportion of direct to diffuse sunlight was taken to be a function of the elevation, location, date and time of day. For z < 80o, Hs.dir and Hs.ttl are given to be nearly constant (0.92). In cloudy conditions the solar radiation is taken to be completely diffuse. Substrate reflectivity rg is assumed to be 0.3 (a typical value for grassland vegetation). The thermal radiative flux is given to by: 0 0 0 ># 0.5 # ,- ?K = 0.5 # ,- H0

(5-14)

At is the thermal radiative heat transfer surface area, Tg is ground surface temperature and Tsky is the equivalent black body sky temperature. Thermal emissivity , is proposed to be 1 (in the thermal infra-red spectrum at about 5 Âľm. As the angle of view of a butterfly is close to the normal, it is appropriate to give a value of 1 for the emissivity. If the emissivity was given a value less than 1, the temperature and temperature differences would have been underestimated (Clark et al, 1973). Moreover the equivalent black body temperature is estimated from the Brunt equation (Sutton, 1965): 20

?K .( = Lâ&#x2C6;&#x161; 1- 0

(5.15)

Where m and n are constants, e is vapour pressure in the lower levels of the atmosphere, - is the Stefanâ&#x20AC;&#x2122;s constant and T is the absolute temperature. The convective heat flux is given by:

> # . 1

(5-16)

For high wind speed and low intensity radiation conditions there is negligible free convection therefore from one of his previous thesisâ&#x20AC;&#x2122;, Kingsolver uses the relationship between the Reynolds and Nusselt numbers for a bare cylinder (similar to butterfly models without fur) in eqn (5-17):

NO 0.6 / Conclusions â&#x20AC;˘ Body emissivity of the butterfly is approximately 1.

â&#x20AC;˘

Free convection is negligible at high wind speeds and low solar intensity.

(5-17)

3.2 Behavioural Habits & Body Traits in Effecting Thermoregulation 3.2.1 Posture Research paper reference: (Kingsolver, 1988) This section of the literature search deals with the behavioural basking used by butterflies, b to either increase or to reduce body temperature depending on their required choice.

Figure 12,, Butterfly body (depicted by the cylinder) and corresponding orientation angle Ψ (normal to the thorax) and wing angle θ (Kingsolver, 1988).. Kingsolver (1988) outlines three three main postures as the typical butterfly postures for thermoregulation; they are lateral, dorsal and reflectance (figure 13). Lateral ateral basking is when the wings are closed over the body and orientated perpendicularly towards the sun’s solar beam. This posture is mainly used to avoid temperature increases in the body. In dorsal basking the butterfly opens its wings normal to the solar rays (θ=90o) thereby directly heating the thorax and abdomen. Finally, in reflectance basking the butterfly will open its wings at an angle, reflecting reflect solar rays off the reflective region of its wings onto the thorax and abdomen (figure 15).

Figure 13,, Basking postures: lateral, dorsal and reflectance basking (Kingsolver, 1988). 1988) The basal regions of the butterfly terfly wing due to their melanisation (influenced by photoperiod during the larval stage) are responsible for the majority of the conductive heat absorption by the wings. Melanisation is the process by which butterflies have darker pigments on their wings and bodies due to a local concentration of substance known as melanin. For or this reason the Kingsolver (1988) asserts that butterflies living in cooler habitats tend to be darker in colour, aiding the harsher heat regulation conditions as opposed to butterflies living in more favourable mild climates. Furthermore Kingsolver (1988) argues that there are primarily four parameters that help to quantify the link between butterfly thermoregulation characteristics characteristics and flight activity time: body b size (thoracic diameter), thermoregulatory posture (figure ( 13), ), solar absorptivity and fur thickness.

Figure 14,, Reflectance basking: The black basal absorption areas are responsible for heat conduction. The hatched distal region has little or no effect. The white medial regions reflect solar radiation from the wings onto the thorax or abdomen (Kingsolver 1988).

The author asserts that there is a link between wing melanisation (defined above) and body temperature based on figure 14. 14 He states that the greater the melanisation % on the wings the larger the wing angle required to maximise body temperature. Figure 14 only applies to one set of butterfly species and each species have their own proportions of basal, medial and distal wing regions.

Figure 15, Melanisation in Pierid butterfly wings. Where + indicates an increase in temperature where melanisation occurs. O corresponds to no effect and â&#x20AC;&#x201C; as a decrease in temperature. Left wing is the dorsal side, right wing is ventral (Kingsolver 1988). As shown in figure 15 the author uses a functional map to show the effects of melanisation in Pierid butterflies. There is a striking difference between the dorsal and ventral areas of the wing. The sides of the wing with the greater melanisation will have darker patterns patterns (i.e. larger absorptivity coefficients). The only region of the wings where this may aid heat gain is in the near wing basal region where small amounts of heat are conducted to the butterflyâ&#x20AC;&#x2122;s body. The medial and distal regions are lighter in colour (coefficient of reflection Ď is higher) and this is more beneficial for the butterfly in reflecting solar radiation onto the body.

3.2.2 Shivering Research paper reference: (Rawlins 1980) Shivering occurs when the muscles contract allowing for a quick increase in body temperature. temperature A very expensive process in terms of energy dissipation, it is only used in situations of absolute need, i.e. avoiding predators by sufficiently raising their body temperature for required flight. Rawlins also considers minimum inimum and maximum temperature ranges that the butterflies can can withstand without procuring fatal injuries (Rawlins, 1980). 1980) Rawlins also asserts that shivering may be used by the butterflies to improve basking sites under low solar radiation, selecting suitable roosting sites in the evening and to regain a roosting spot after being dislodged. It may also be a good spontaneous tactic to avoid predation (a fatal attack on the butterfly from a predator) when ambient temperatures are below b low those required for flight.

3.2.3 Wing Level Research paper reference: (Rawlins 1980)

Figure 16,, Relationship between betwe body and ambient temperature of perched male black swallowtails. Solid line indicates indicate a region where the body temperature is equal to the ambient temperature (i.e. Tb=Ta).. Dotted line represents the general pattern of spots (black being thoracic temperatures and white ambient temperatures) (Rawlins, 1980). Based on figure 16, for a given ambient temperature, temperature thoracic and abdominal temperatures are higher than the ambient temperature. In terms of thermoregulatory ulatory practise Rawlins (1980) states that for low ambient temperature conditions the butterflies would usually raise rais their abdomen above the wings, exposing it to direct solar radiation and raising the abdominal temperature Tab. Conversely onversely for hotter weather conditions, conditions butterflies often level their abdomens abdomen to wing height or just below thereby shading them from direct sunlight.

3.2.4 Abdominal pumping

Figure 17:: A graph comparing the various postures taken up by the Swallowtail butterfly for given ambient temperatures and levels of solar radiation (Rawlins 1980). 1980) Research paper reference: (Rawlins 1980) The author Rawlins (1980) in his research paper states that at thoracic temperatures temperature Tth of 37.1 Âą 1.5 o sd ( C), the he butterfly began struggling, struggling pumping its abdomen. Abdominal pumping is the contraction of the abdominal muscles that results in the expansion of the air sacs. This occurs mainly when insects are active and require cooling through greater respiratory exchange. During abdominal pumping there iss a decrease between the thoracic and a abdominal temperatures.. This is due to a zero net increase in thoracic temperature, temperat where the abdominal temperaturess are seen to increase in temperature, indicating heat transfer from the butterflyâ&#x20AC;&#x2122;s thorax into the abdomen. This practise is mainly evident when the thorax is exposed to solar radiation and the abdomen is shaded. The transfer of heat from the thorax to the abdomen reduces the likelihood of thorax over-heating. Moreover Rawlins (1980) says that conditions under which h heat exchange is carried out between the thorax and abdomen may depend solely on whether the butterfly exhibits abdominal pumping. During abdomen-shade shade posture, pumping occurs and heating exchange via the hemolymph ymph is maximised. In cases where the thorax is overheated and the abdomen shaded, a useful cooling procedure (heat transferred from thorax to abdomen) occurs to reduce stress from excessive temperatures.

3.2.5 Tilting Research paper reference: (Shelly and Ludwig, 1985): A report by Shelly & Ludwig (1985) dealt with better understanding the behaviour of Calisto butterflies under a forest location as opposed to the more common open land habitats from previous reporters. Tilting behaviour was analysed and found to elevate the rate of heat he intake in the thorax and so reduce the time required for thoracic temperature Tth to rise. Titling behaviour occurs in lateral baskers (figure 13), 13), where the butterfly positions its body for the most effective angle for increased rate of heat intake from solar radiation.. It is especially used for short duration basking where it is a useful way of heating the body more quickly. quickly

Figure 18,, Close-up of butterfly fur, (Anonymous 2, n.d) An additional chapter of the literature survey has been assigned to general features that the butterflies have,, that aids thermoregulation. thermoregulation. These include fur thickness, aposematic colouring of the wings and other heat regulation techniques. These do not directly link in with the proposed heat balance alance that is to be carried out but provide useful background information that that may be called upon when required.

3.2.6 Fur Thickness Research paper reference: (Kingsolver and Watt, 1984): The fundamental reason for this paper by Kingsolver & Watt (1984) lies with two important constraints that can be used as effective parameters in varying optimal conditions for maximum flight activity. Namely fur thickness and solar absorptivity and they are tested with three different habitats of butterflies ranging from fr low to mid and high elevations (Montrose Montrose h=1.5km, Skyland height=2.8km and Mesa Seco h=3.3-3.6km. h=3.3 The main article of importance sprung from a better understanding of the effects of pubescence (fur) on butterfly thermoregulation. According to Watt & Kingsolver 1984,, fur decreases the butterfly bodyâ&#x20AC;&#x2122;s sensitivity to temperature changes. This is advantageous for high elevation butterflies butt (that have more fur) controlling their body temperatures at higher wind speeds. Conversely butterflies dwelling in lower elevations have less fur but this allows them to fly for extended periods pe when there is little wind

3.2.7 Aposematic Colours

Figures 19 & 20,, Aposematic colours of the unpalatable Birdwing butterfly. Left (Wong, 2010), right (Mun, 2010) Research paper reference: (Dudley 1991): The author Dudley (1991) attempts to link relations between palatability (figures 19 & 20) 20 in butterflies and the difference between the thoracic and ambient temperature labelled as thoracic excess. Palatable butterflies according to Dudley fly in arbitrary flight patterns at the cost of expensive energy consumption expended due to the increased wing beat count and metabolism rates. In contrast unpalatable Danaine butterflies fly more slowly and soar for longer as they have less need of avoiding predators and as there is a connection between etween predation rates and flying fly speeds. This thesis didnâ&#x20AC;&#x2122;t bring to light direct heat transfer information for the Danaine butterfly species but rather highlighted a reason for elevated or reduced body temperatures for butterflies depending on their predatory desirability.

Figure 21,, Aposematic colours of the white Pierid butterflies,, (Jack, 2010) Research paper reference: (Kingsolver, Kingsolver, 1987) In another paper on palatability, Kingsolver (1987) writes about a special al white pigment on the Pierid butterfly wings that deter predators from attacking the butterflies as itâ&#x20AC;&#x2122;s a sign of un-palatability. un 28

Kingsolver justifies a hypothesis that the white wing wing pigment represents aposematic coloration (figures 19 & 20) that may ward of predators. Aposematic colours specifically warn off predators from poisonous, dangerous or bas tasting animals, i.e. bright yellow colour of golden poison frog. Furthermore from his previous studies Kingsolver relates a second useful function for the white wing pigment-aa reflective colour to aid reflectance basking in the Pierid species. Therefore this white pigment assists predation avoidance as well as aiding thermoregulation thermoregulation through basking practise. He also argues that melanisation nisation of the wings can increase the rate of thermoregulation during basking due to increased solar absorption tion.

3.2.8 Wind Shielding

The abdomen acts as a â&#x20AC;&#x2DC;shieldâ&#x20AC;&#x2122; and prevents convective cooling of the thorax from the wind

Wind direction irection Figure 22,, Wind shielding of the thorax by the abdomen, (Toogood, n.d) Research paper reference: (Polcyn & Chappell, 1986) This paper throws light on thorax temperatures temperature when light and wind are applied across a butterflyâ&#x20AC;&#x2122;s body at different angles. The researchers Polcyn & Chappell attempt different combinations of wind/lightt angles and wind velocities, citing the different temperature variations that each parameter has on the butterfly utterfly thorax. They also assert that closing the wings actually increases Tth (due to a reduction in convective cooling) by maximising the increasess in temperature due to wind shielding of the thorax by the abdomen. Wind shielding occurs when the butterfly flies/lies flies stationary against gainst the wind direction and the abdomen acts as a buffer for the thorax from direct convective cooling from the wind (figure 24).

3.3 Conclusions of the Literature Survey

∼

∼ ∼ ∼ ∼ ∼ ∼ ∼

Conductive heat gain occurs strictly at the basal region of the butterfly wing. For heat conduction to occur the butterfly exhibits the lateral basking posture and this allows the solar radiation from the sun to directly heat the ventral basal region of the wings. The heated basal region of the wings then proceeds to conductively transport heat into the thorax. This process is enhanced by the local dark colouration that grants greater thermal absorptivity of solar radiation. The heat conduction process is aided by the close proximity of the basal region of the wings to the thorax and abdomen. The rate of the conduction is however very slow as compared to the convection and radiation. The butterfly angles its body towards the solar rays thereby heightening its chances of heat gain in a behavioural posture called tilting. Fur helps to reduce convective heat loss and is more apparent on butterflies existing in colder climates. At high wind speeds and low solar radiation free convection is negligible. Changing the wing angles can either help to increase body temperature by reflecting solar radiation onto the body: reflectance basking, or alternatively cool the butterfly down by angling wings perpendicularly to the sun’s rays: lateral basking. When the butterfly is flying against the wind direction its abdomen is responsible for shielding the thorax from convective heat loss. This reduces the chances of the thorax temperature dropping below an acceptable flight range. Melanisation of the wings is one of the most valuable traits of their wings providing a much increased rate of solar radiation intake by wing reflection, especially in colder climates where it is more present in the wings. Under high temperatures the butterflies often shade their abdomens below their wings, allowing the excess heat from the thorax to conductively transfer into the abdomen.

4 Heat Balance 4.1 General Analysis and Key Assumptions The purpose of the heat balance is to identify an equilibrium temperature that a butterfly species may require for controlled body temperatures. In order to carry out an effective heat balance, precise information on the butterfly’s body dimensions is required. Regular body dimensions where taken from the Swallowtail species and are tabulated in table 1 below, where average values are detailed in parenthesis. The butterfly body area is modelled as a cylinder for simplification.

Body part(s)

Body dimensions [mm], [mm2]

Body length L (top of head to abdomen end) Thorax width Cylinder radius (thorax width/2) Area of cylinder (2πr2+2πrL) Total wing area (both wings) Wing span Wing thickness ~ thickness of paper (Kingsolver & Koehl, 1985)

25-32 (28) 3.5 1.75 308 [mm2] 6643 [mm2] 161.5 0.2-0.4 (0.3)

Table 1, Body dimensions The thorax is the most crucial organ of the butterfly body in relation to its ability to regulate temperature effectively. For this reason, the base of the energy balance model is set from the thorax, whether the butterfly is heating up or cooling down. Initially, conduction was deemed to be a part of effective heat transfer in the butterfly body yet after careful consideration of the literature survey it became more feasible to abandon it. Conduction occurs only in the basal region of the wings and there it is also very gentle, therefore negligible overall. For the purposes of analysing the effects of wing angle on the heat transfer rates, four different groups of wing angles were chosen: 10-44o, 45o, 46-89o, 90o. 10o is the estimated physical minimum wing angle that the butterfly can achieve owing to its body radius (Kingsolver 1985). 45o is a special case where reflection off incident radiation will mirror perfectly horizontally and either reflect off the opposing wing and out into the surroundings or be intercepted by the butterfly’s body. Between the wing angles of 46-89o it is expected that the area of the wing that is used for heating the body reduces gradually with increasing wing angle up until 90o where it would have no effect on heating the body. 90o is also another special case where the wings are essentially exclusive of providing reflection radiation and consequently heat gain to the body. At 10o it is expected that the highest concentration of radiation is reflected off the wings and onto the body due to the steep gradient of the wings and the proportion of reflective wing length. The solar irradiance was chosen to be 579 W/m2, a value consistent with typical ambient conditions in the Colorado region in the USA, where the majority of the most progressive research on butterflies has been carried out. This value assumes constant direct radiation as opposed to a more genuine variation in a nominal day where periods of direct and diffuse radiation occur, owing to interspersed cloud coverage. Diffuse radiation was primarily left out due to the lack of tangible ambient data. The value of the body absorptivity is 0.95 (Berthier, 2005), with the solar absorptivity 0.54 (Kingsolver, 1983). This value of the solar absorptivity is consistent for weather conditions of 31

direct sunlight, in the summer months of June, where ambient temperatures may vary between 2040 oC in Colorado. The solar rays are assumed to be radiating from directly above (Î¨=0o) the butterflyâ&#x20AC;&#x2122;s body as shown by figure 23 below:

Figure 23, Proportion of solar radiation striking the butterfly body The reflection coefficient of solar wings interacting with the wings is taken to be 1 from a similar investigation carried out by Kingsolver (1985). All other parameters such as the wind speed and body radius were taken for the most general cases of butterfly body dimensions and ambient conditions, with the intention of drawing an overall understanding of the critical processes that are occurring. An outlined picture of the butterfly wing angle and solar radiation orientation are shown in figure 24. The heat balance is to be carried out without the addition of the fur thickness, with the principal reason being that the effects of the fur become more pronounced for butterflies in flight, whereas this study is strictly for stationary butterflies, adding unnecessary complexity to the calculations. The fur thickness also becomes more important for the butterflies in colder conditions (Ta<20o) whereas the ambient temperatures taken here are in the summer months. As stated above the ambient temperature is generally above 20o in the day (between 20o-40o) when most measurements were carried out in the literature survey in Colorado. The temperature excess Tex= Tb-Ta was required in order to obtain a prospective value of the Grashof number and subsequently the Nusselt and coefficient of convective heat transfer in free convection. A value of 6oC was chosen and this is based on research carried out (in the literature survey) on temperature differences between the surface of the body and ambient temperatures, a reasonable average being 6oC for the stated environmental conditions.

Figure 24, Butterfly body, wings and interaction with incoming solar radiation The nominal wind speed taken to calculate the Reynolds number and consequently the forced convection is 3.14 m/s. This was taken from samples of data from ambient conditions of the most typical weather conditions that the butterfly species may face. At this wind speed the ratio of free to forced convection is approximately 0.84 at a wing angle of 90o. At 2.15 m/s the ratio of free/forced convection is 1 and any further decreases in wind speed would represent a majority of free over forced convection. Hence a wind speed of 2.15 m/s is the balancing tip between forced and free convection. Increasing the wind speed from 3.14 m/s decreases the body equilibrium temperature of the butterfly Tb. For the previous calculations the temperature excess Tex=Tb-Ta was kept at 6oC in order to better understand the effect of the other parameters in changing the heat transfer equations. Increasing Tex by a constant amount does not affect the forced convection but rather decreases the free convection. This is due to the fact that the film temperature rises and the inverse occurs with coefficient of volumetric expansion β, thereby reducing the Nusselt number and subsequently coefficient of convective heat transfer h. In short, the key ambient conditions and constants for this study are under ambient conditions based on Colorado, USA, in the summer month of June, under direct sunlight. A summary of the parameters are shown below: • • • • • • 33

Steady state heat balance (independent of time measurement). Butterfly body is approximated as a cylinder. Butterfly body will be assumed to be above the wings as opposed to being level with them. Butterfly is upright (i.e. no tilting is assumed for the heat balance). Orientation angle of butterfly body to the sun Ψ: 0o. Ambient temperature: 20oC.

• • • • • • • •

• • •

Solar absorptivity: 0.54 (Kingsolver, 1983). Body emissivity: 1 (Clark et al, 1973). Wind speed: 3.14 m/s. Solar irradiance: 579 W/m2. Temperature excess Tex=60 (required for Grashof number in free convection). Wing angle bands: 10-44o, 45o, 46-89o, 90o. Yaw angle (direction of wing in relation to longitudinal length of butterfly body): 0o. The butterfly wing is taken to be totally reflective, i.e. (ρ=1) for all parts of the wing. This assumption allows for a more general analysis to take place on the wing geometry with the wing melanisation also being particularly complicated to examine, mainly because of the uniqueness of each wing pattern for each species. Wind shielding by the abdomen has not been accounted for as it is assumed that the wind direction is towards the head of the butterfly and along the body longitudinally. Radiation emitted by the butterfly that reflects off the ground and back onto the body has not been considered to exclusively emphasise the effects of the wing angle(s) to the sun. Radiation heat transfer will be based on ambient sky temperature and the ground temperature will be disregarded (eqn: 5-14).

4.2 Wing angle: 90o Incident Radiation The incident solar radiation is a function of the solar irradiance, absorptivity and size of the body being radiated to. When the wings are held at 90o to the vertical there is no radiation from the wings onto the body and the butterfly body receives all radiation directly. The equation for the heat transfer onto the butterfly body is:

>< < # A / 7 /

(6-1)

Convection model Convection is split into its dual constituents of forced and free convection, and each will be derived separately. Forced Convection Reference for derivation of equations: (Cotton, 2010)

Figure 25, Forced convection along the butterfly body 34

A research paper deemed paper to be a suitable approximation in place of the thermal conductivity of butterfly wings. It is anticipated that other factors such as the specific heat capacity and the butterfly body width were considered in choosing paper as a good estimate for thermal conductivity. With this assumption it is then possible to estimate the butterfly body as a flat plate (as the wings are very thing when facing the wind axially). The wind direction is taken to be facing the axial direction of the butterfly body as shown above in figure 25. From figure 25 there is an indication of a growing boundary layer from the leading edge (or butterfly head) in the x direction. The Nusselt number as well as the coefficient of heat transfer also varies along the length of the body in the flow direction. The local Nusselt or Reynolds numbers may be obtained along the length of the body, yet it is more appropriate to acquire the average values across the whole body. The Reynolds numbers for typical wind speeds experienced by the butterfly are Re < 5x105, i.e in the laminar region of flow. The equation for the average Nusselt number is:

SSSSSR NO

SSS R 2

(6-2)

SSSV the average coefficient of heat transfer along the flat Where SSSSS NuV is the average Nusselt number, h SSSV may be further described by its own length. L is the body length and k is the thermal conductivity. h definition as:

1 R SSS R X Y [Y Z

(6-3)

In the laminar region of flow, the flat plate surface temperature Tw is constant and as the Nusselt number is a function of the Reynolds and Prandtl numbers, the forced convection average Nusselt is equal to equation (6-4) for a horizontal flat plate:

SSSSSS NOY

SSS R \ 0.332 / Y / _6 ` 2

(6-4)

Where The Reynolds and Prandtl numbers are defined as:

Y _6

$ \ a

)* b 2

(6-5)

(6-6)

Hence:

SSS Y / \ 0.332 / $ _6 ` 2 a / \

SSS Y $ / \ c 0.332 / _6 ` 2 a

SSS R _6 ` 1 R c 0.332 / $ X \ [\ 2 Z a

SSS R _6 ` 1 0.332 / $ d2\ / eRZ 2 a

SSS R $ _6 ` c 2 / 0.332 / 2 a

SSS R $ 0.664 / _6 ` 2 a

SSSSSR 0.664 / R _6 ` 2NOR NO

(6-7)

(6-8)

(6-9)

(6-10)

(6-11)

(6-12)

(6-13)

The average value of the Nusselt number is thus double the local Nusselt number along the flat plate in the laminar flow region. The average Nusselt number is then used to obtain the coefficient of convective heat transfer and subsequently the forced heat transfer rate:

NO / 2

> @ @ . 1

(6-14)

(6-15)

Free Convection An assumption is made here that the butterfly is stationary in a quiescent manner and thus can be modelled as a horizontal cylinder. The derivation of the heat transfer is begun with the coefficient of volumetric expansion below:

1 [ [.hL 1 [.hL 1 1 g i [ * [ [ *

(6-16)

Where Tf is the film temperature (mean of the surface and free-stream temperatures). The Grashof number is represented by:

jf. 1 ` a

(6-17)

Here D is the diameter of the cylinder and a is the kinematic viscosity of air taken from steam tables. In order to obtain the Nusselt number for free convection, the product of the Grashof and Prandtl numbers is required for a horizontal cylinder.

76_6

(6-18)

Once the Nusselt is obtained, the coefficient of convective transfer can be calculated from the equation below:

NO / 2

(6-19)

A value of the free convective heat transfer is then obtained:

> @ @ . 1

(6-20)

The total combined heat transfer from forced and free convection is:

># # I # # I . 1

(6-21)

# # I @ = @

(6-22)

Where htotal is defined by:

Radiation Model One assumption made in the radiation model is that the emitted radiation onto the surface of the wings and body has 0 transmissivity, and all radiation is either due to absorption (direct radiation onto the body) or reflected radiation (off the wings and deflected onto the body).

For the first case of radiative heat transfer where the wings do not provide any heat gain benefits, the wing angle are set to 90o from the vertical. The equation for the radiative heat transfer is:

>@ -, k 0 0

(6-23)

Where - is the Stefan-Boltmann constant, , the emissivity off the butterfly body and f is the view factor to the surroundings. The view factor (0.5) was calculated based on the direct radiation to the ambient surroundings as well as radiation reflected onto the surrounding off the wings. As shown by figure 26 below, the highlighted blue portion of the emitted radiation returns to the body leaving 50% radiating to the surroundings.

Figure 26, View factor to the surroundings ~ 0.5

Overall Heat Balance The overall heat balance states that the incident solar radiation must balance the heat lost by the butterfly at the equilibrium body temperature:

A7 f -, . 0 0 1 = # # I . 1

(6-24)

The value of the equilibrium temperature may be obtained from an in-built feature (goal seek) in Microsoft Excel that equates both sides of the equation to 0 and thereby obtains Tb.

4.3 Wing angle: 45o For the special case of the wings being angled at 45o to the vertical there will be an evident change in the equilibrium body temperature Tb. It is predicted that this value will drop below the value obtained when the wings were completely horizontal. The aided benefit of angled wings will aid reflective basking and so reduce the equilibrium temperature.

Incident Radiation Here the incident radiation is split into two entities, direct and reflected solar radiation. The direct solar radiation will be exactly equal to case 1 (wing angle=90o) as the direction of the solar radiation hasnâ&#x20AC;&#x2122;t changed and neither has the body area. As before the direct heat transfer is represented by:

> <@ # A / 7 /

(6-25)

The reflected radiation will supplement the heat transfer to the butterfly body and a larger proportion (579 W/m2) of the incident radiation will be used by the butterfly. Lr is the length section of the wings that will reflect solar radiation onto the butterflyâ&#x20AC;&#x2122;s body and for this case can be approximated to be the body radius of the butterfly (Kingsolver 1985). The radiation area is the multiplication of the reflective wing length by the wing width (approximated to be the body length). The total estimated wing area of reflection is stated below and is demonstrated in figure 27:

kh )%lm nlLj o6 o @ /

(6-26)

Therefore the reflective heat transfer is:

>@ I # p / 7 / / @

Figure 27, Proportion of body acquiring incident radiation

(6-27)

The total heat transfer emitted onto the body of the butterfly is a summation of the direct and reflected radiation as stated in equation (6-28) below:

># # I > <@ # = >@ I #

(6-28)

Convection Model Forced Convection Forced convection for angled wings is the same as the horizontal wing case because the heat transfer equations are exempt of geometric considerations. Free Convection Free convection can be split into two sections; from the upper/dorsal surface of the wings and from the down/ventral surface of the wings. For the upside surface of the wings most of the parameters required for the Grashof number originate from the film temperature (previously stated as being the mean of the free stream and surface temperature of the body). For the ventral side of the wings there are slightly different derivations for the Nusselt number owing to the geometry positioning of the lower wing face.

4.4 Wing angles: 46o-89o In deciding which wing angles to use between 46o and 89o for the analysis, it became clear that the average value of the range of angles would give a realistic indication of the thermoregulatory behaviour. For example the reflective wing length Lr was calculated based on the angle of the wings and the reflective area of the wings was obtained by multiplying Lr by the body length. The accuracy of this method of obtaining the capacitive area of the wings for reflection depends on the degree of melanisation of the wings as well as the region of the wings that are melanised (distal, medial or basal (figure 13)). The view factor to the surroundings was calculated by taking a line perpendicular to each wing angle to the outer tangent of the butterflyâ&#x20AC;&#x2122;s body. This configuration for the view factor owes to the fact that the reflective coefficient of the wings Ď is 1. A diagram of the geometric relationships is shown in figures 28 & 29 and mathematically stated as being:

ml n ko)%q6 %q % rO66qOL[lLjr

180 = .90 u1 / 2 360

(6-29)

Figure 28, View factor to the surroundings for wing angles of 46-89o

4.5 Wing angles: 10o-44o

Figure 29, View factor to the surroundings for wing angles of 10-44o

4.6 Results & Conclusions Wing Angle(s) [o] 10-44 (avg) 45 46-89 (avg) 90

Tb [oC]

QIR [W]

Qwings [W]

Qtotal [W]

76.169 26.131 26.976 26.753

0.0481 0.0481 0.0481 0.0481

0.818 0.0284 0.0134 0

0.866 0.0765 0.0616 0.0481

Table 2, Equilibrium temperature and proportions of solar radiation from sun and wings As shown by table 2 above, widening the wing angle decreases the total heat transfer to the butterfly body. The heat transfer to the body is constant as the angle of incident radiation was taken to be 0o or directly above the butterfly’s head and the wings do not obstruct direct radiation. The maximum heat transfer reflected onto the body from the wings occurs at 10o and gradually reduces until zero effect at 90o. Table 2 shows final equilibrium temperatures obtained for each band of wing angle(s). As previously stated, at 90o the wings provide no heat gain to the butterfly body via the wings and overall there is minimum heat transferred to the body. ΔT is approximately 55oC at wing angles of 10-44o, with a high equilibrium temperature of 76.2oC due to the long reflective wing length apparent for this band of angles. This high value indicates that wing angles lower than 45o can only be displayed by the butterfly for a finite amount of time. As time was not reckoned into the study it is possible to deduce that any amount of time between instant up until the fatal body temperature (~50oC) may be possible. Assuming it does take longer than an instant transition from the body temperature to change to 76.2oC, it is possible for the butterfly to apply this angled posture and swiftly change to wing angles of 45o and above when a suitable body temperature has been acquired. This provides several advantages, particular in emergency situations such as avoiding predation where body temperature can be raised above the flight minimum in a very short amount of time.

Wing Angle(s) [o] 10-44 (avg) 45 46-89 (avg) 90

Hfree [W/m2K]

Hforced [W/m2K]

Htotal [W/m2K]

Qtotal [W]

Qconv [W]

Qrad [W]

Tb [oC]

32.644

11.000

43.644

0.866

0.101

0.00930

76.169

26.023 14.084

11.000 11.000

36.253 25.084

0.0765 0.0616

0.0670 0.0463

0.00815 0.00688

26.131 26.976

9.201

11.000

20.201

0.0481

0.0373

0.00543

26.753

Table 3, Proportions of key experiment parameters Table 3 shows that the forced convection was constant throughout the different wing angles, mainly due to the wind direction that is travelling in the longitudinal direction of the butterfly’s body, and therefore in exclusion of geometric wing changes. Any changes in Qconv would have therefore come about from free convection. Qconv and Qrad gradually increase proportionally with decreasing wing angle but at wing angles between 10-44o there is a sharp rise in Qtotal. There are two reasons for this; 42

firstly that the wing is considered completely reflective (ρ=1) along its entire length and secondly due to the large increase in the reflective wing length (figure 29). Kingsolver (1985) states that any reflective index less than 1 would see a decrease in the intensity (by a factor of ρ2) of the radiation reflecting off each wing until it is intercepted by the body. This is the main reason for the large increase of incoming radiation at wing angles less than 45o. It is also remarkable that between 45-90o the equilibrium body temperature is fairly constant, indicating that although decreasing the wing angle causes a rise in the incoming heat to the body, there is a complimentary rise in heat loss via convection and radiation. This special occurrence is only applicable for the conditions stated but may differ significantly when for example radiation is diffuse, and other parameters are accounted for, such as wing melanisation, fur thickness and tilting etc.

4.7 Summary of Conclusions

∼

∼ ∼

∼

The greatest heat gain was produced at lower wing angles (10-44o). At these wing angles however, the equilibrium body temperature is fatal and so it is suggested that the butterfly would only display these wing angles for a finite or very short amount of time (requires experimental confirmation). An increasing wing angle lessens the heat loss via free convection, and perhaps would be used in situations of low wind speed to conserve a favourable body temperature. At the equilibrium temperature of approximately 26o, the butterfly species are comfortable in that they have the capacity to fly without requiring any heat gain or heat loss through thermoregulation. Therefore between 45-90o, it is possible to say that the most stable body temperatures are acquired for the stated conditions. The wing span as well as the reflective wing length can greatly increase the amount of radiation received by the butterfly. This is particularly emphasised when the butterfly species has a lighter coloured wing colour (higher coefficient of reflection ρ).

5 Future Works It is desired that a better understanding of the structural aspects of the butterfly wing are understood. This will include a more in depth study of the Nano-scale scans of the wing. Moreover a detailed investigation will occur on the full effects of the wing structure and its direct effects on thermoregulation. An extended understanding of the radiation model of the heat transfer equations. This includes a thorough search on the emissive and absorptive properties of the wings. A better appreciation of the wings absorptivity to transmissivity and reflectivity is required. A question of considerable interest is whether the wings are more efficient than solar panels in absorbing solar radiation. Does the material/powder coating of the wings enhance/decrease heat transfer rates? Transient heat balance including time. Comparing various body radius sizes and how each respective length can be advantages/disadvantages for the given ambient conditions. Use accurate CAD model of butterfly to simulate equilibrium body temperatures during flight. Consider how the uptake of solar radiation may be different for tilted body posture, especially as wing patterns are different for the ventral side of the wings as opposed to the dorsal. Produce extended studies in ambient conditions of less than 20oC. Current wind speed (3.14 m/s) is for average ground heights in Colorado USA, yet it may become more remarkable to study how at higher altitudes where the wind speeds would be expected to be much higher. This would change the proportion of forced/free convection and help in understanding how the butterfly mitigates the effects of convective heat loss. Fur has been reckoned to be an effective aid in battling the effects of convective heat loss (literature survey) and yet was left out of the main study due to the lack of accurate data on temperature changes in the body. It is desired that a greater understanding of the relationship of the fur in regulating temperature is required particularly between warm and cooler climates.

5.1 Solidworks Simulation

Figure 30, Solidworks model of the butterfly (view 1) Figure 30 & 31 show a model of the butterfly made on CAD software (Solidworks), (Solidworks which can be used to provide a more accurate study of the equilibrium body temperature with closer contour detail being considered. Meshing the butterfly model (figure 31) allows for a more accurate analysis of the areas which are more susceptible to heat he gain or heat loss. Solidworks has an in--built simulation feature for thermal analysis and it allows the three main modes of heat transfer to be evaluated as well as setting other parameters such as ambient temperatures, view factors and emissivities.

Figure 30, Solidworks model of the butterfly (view 2) 45

Figure 31, Applying mesh to butterfly body & hind-wings

6 Appendix 6.1 DataBank

Figure A1, Body temperatures of Swallowtails in field studies (oC). Mean Âą sd above, range below, below (Rawlins, 1980).

Figure A2, Various parameters in relation to different wing and abdominal positions posi Mean Âą sd. (Sample size), (Rawlins, 1980).

Figure A3, Critical thoracic temperatures for various activities of black Swallowtails Swallowtail in the flight o cage. Mean (N = sample size) above, range below ( C), (Rawlins 1980).

Figure A4, Identification, sex, means and standard deviations (SD) of body mass m [mg], wing length R [mm], wing loading pw [N m-2], thoracic temperature Tth [oC], ], ambient temperature Ta [oC], thoracic excess Î&#x201D;T=Tth-Ta [oC] and solar irradiance I [W m-2] for two species of Danaine butterfly, (Dudley, 1991).

Figure A5, Solar absorptivity, thoracic fur thickness and thoracic diameter of four butterfly species in central Colorado, (Kingsolver, 1983).

Figure A6, Cumulative daily flight activity time (KFAT) in hours for the three sites of o different elevational heights, (Kingsolver 1983).

Figure A7, Sensitivity analysis of the energy balance model, where Td is body temperature excess (labelled as Tex in the nomenclature), nomenclature) Tex = Tb-Ta. This graph relates how each parameter may affect the butterflyâ&#x20AC;&#x2122;s body temperature, (Kingsolver, 1983).

Figure A8, Table of results for experiments carried out at high elevations, elevations, (Kingsolver & Watt, 1984).

6.2 Nomenclature Symbol

Quantity

Units

A Ar

Effective area of butterfly model Archimedes number: 2 Ar = Gr/Re Convective heat transfer surface area Direct solar radiative heat transfer surface area Reflected solar radiative heat transfer surface area Total solar radiative heat transfer surface area Thermal radiative heat transfer surface area Cross-sectional area of wire Specific heat of butterfly model Characteristic dimension of the butterfly model (maximum width of mesothorax including the fur): 1/2 Deff = (4V/πL) Vapour pressure Gravitational constant Solar radiation intensity 3 2 Grashof number = (gβ(Tb-Ta)Deff/ν ) Boundary layer conductance Conductive heat transfer coefficient Convective heat transfer coefficient Fur layer conductance Radiative heat transfer coefficient Total heat transfer coefficient Direct solar radiative horizontal flux densities Diffuse solar radiative horizontal flux densities Coefficient of free convective heat transfer Coefficient of forced convective heat transfer Total coefficient of convective transfer (free+ forced) Total solar radiative horizontal flux densities Solar Irradiance: amount of solar power received over a certain area Thermal conductivity of air Thermal conductivity of fur Thermal conductivity of wire Length of butterfly model Length of wire Mass of model Nusselt number: Nu = hfDeff/k Wing loading: the weight of the butterfly divided by its wing area Heat transfer rate by forced convection Rate of internal heat input Incident radiation heat transfer Convective heat flux

m ---

Ac As,dir As.ref As.ttl At Aw cp Deff e g Gd Gr hb hc hf hfur hr ht Hs.dir Hs.dif Hfree Hforced Htotal Hs.ttl I k ke kW L Lw m Nu pw v conv v in QIR Qconv 50

m 2 m 2 m 2 m 2 m 2 m -1 -1 J kg K m

Pascals -2 ms -2 Wm ---2 -1 Wm K -2 -1 Wm K -2 -1 Wm K -2 -1 Wm K -2 -1 Wm K -2 -1 Wm K -2 Wm -2 Wm -2 -1 Wm K -2 -1 Wm K -2 -1 Wm K -2 Wm -2 Wm -1

-1

Wm K -1 -1 Wm K -1 -1 Wm K m m kg ---2

Nm W W W W

Qrad Qs Qs.dif Qs.dir Qs.ref Qt Qwings rg ri R Re t T Ta Tab Tba Tb Tex To Tsky Tt Tth Tw T∞ U V y z α β w ε σ τ ν Ψ

Radiation heat transfer Total Solar radiative heat flux Diffuse solar radiative heat flux Direct solar radiative heat flux Reflected solar radiative heat flux Thermal radiative heat flux Wing reflected heat transfer Ground solar reflectivity Body radius Wing length Reynolds number: Re = UDeff/ν Time Absolute temperature Ambient temperature Abdominal temperature Basking temperature = steady state thoracic temperature of butterfly in basking posture with the wings held at a perpendicular angle to the sun’s beams. Body temperature Temperature excess, which equates to the body temperature minus the ambient temperature: Tex = (Tb - Ta) Initial body temperature Black body sky temperature Body temperature at time t Thoracic temperature Wire temperature Final body temperature Wind velocity Volume of body Yaw angle Zenith angle Solar absorptivity of wing Thermal expansion coefficient Thoracic fur thickness Thermal emissivity of model (=1) Stefan-Boltzmann constant Transient time constant Kinematic viscosity of air Orientation angle

W W W W W W W --m m --s K K K K

K K

K K K K K K -1 ms 3 m Degrees Degrees ---1 K m ---2 -1 Wm K s 2 -1 m s o

7 References Anonymous 1, 2010 ‘Painted Lady Butterfly’ Anonymous 2, 2010 ‘High Mag Butterfly Focus Stack Tutorial’ http://www.fredmiranda.com/forum/topic/936420/0#8841929 Anonymous 3, n.d, ‘no title’, Berthier, S., 2005 ‘Thermoregulation and spectral selectivity of the tropical butterfly Prepona meander: a remarkable example of temperature auto-regulation’ Clark, A. J., Cena, K., Mills, J. N., 1973 ‘Radiative Temperatures of Butterfly Wings’ Dudley, R., 1991, ’Thermoregulation in unpalatable Danaine Butterflies’ Heimbuch, J., 2010, ‘Researchers create artificial spider’s silk spinner’ Heinrich B., 1986 ‘Thermoregulation and Flight Activity Satyrine’ Hicker R., n.d, ‘Pink Cattlefish Butterfly’, http://www.hickerphoto.com/pink-cattleheart-butterfly10295-pictures.htm, Horton. J., 2008 ‘Where do butterflies get their striking colors?’, Available at: http://animals.howstuffworks.com/insects/butterfly-colors.htm Kingsolver J., 1983 ‘Thermoregulation and Flight in Colias Butterflies: Elevational Patterns and Mechanistic Limitations’ Kingsolver, J., 1985, Thermoregulatory significance of wing melanisation is Pieris butterflies: physics, posture, and pattern Kingsolver J., 1987, ’Predation, Thermoregulation, and Wing Colour in Pierid Butterflies’ Kingsolver J., 1988, ‘Thermoregulation, Flight, and the Evolution of Wing Pattern in Pierid Butterflies: The Topography of Adaptive Landscapes’ Kingsolver, K., Koehl M., 1985 ‘Aerodynamics, thermoregulation, and the evolution of insect wings: differential scaling and evolutionary change’ Evolution,39(3), pp. 488-504 Kingsolver, J., Moffat, R., 1982 ‘Thermoregulation and the Determinants of Heat Transfer in Colias Butterflies’ Kingsolver, J., Watt, W., 1984, ‘Mechanistic Constraints and Optimality Models: Thermoregulatory Strategies in Colias Butterflies’ Miakar, P., n.d, ‘Boquet’, Available at: http://pixdaus.com/single.php?id=268335&f=rs Mun B, 2010 ‘Butterfly of the Month’, Available at: http://www.butterflycircle.com/?m=201001&paged=2 Polcyn, D., Chappell, M., 1986 ‘Analysis of Heat Transfer in Vanessa Butterflies: Effects of Wing Position and Orientation to Wind and Light’ Rawlins J., 1980 ‘Thermoregulation by the Black Swallowtail Butterfly, PapilioPolyxenes’ Shelly, T., Ludwig D., 1985, ‘Thermoregulatory Behaviour of the Butterfly Calisto nubile in a Puerto Rican Forest’ 52

Smetacek, P., 2000, ‘The Study of Butterflies’, Resonance-The study of Indian Butterflies’, No.6, pp 814 Sutton, O. G., 1965 ‘Biographical Memoirs of Fellows of the Royal Society’, Vol. 11, pages 41-52 Thinkquest, n.d, Available at: http://library.thinkquest.org/C002251/cgibin/default.cgi?language=english&chapter=3&section=2&mode=chapter&outputmode=0&navmenu =0&javascript=0 Toogood, P., n.d, ‘Mission Beach High Resolution Image Gallery’, Available at: http://www.missionbeach.me/cairns-birdwing.jpg Valentino, J. A., 2006, ‘Butterfly portrait’, Available at: http://www.pbase.com/alvalentino/image/59094508 White, F., n.d, ‘Heat Transfer’ Wong A., n.d, ‘Great Mormon’, Available at: http://butterflycircle.blogspot.com/2009/08/lifehistory-of-great-mormon.html Wong A., 2010 ‘Butterfly of the Month’, Available at: http://www.butterflycircle.com/?m=201001&paged=2, 2010

8 Gantt Chart

Thermoregulation By Butterfly Wings Project Author Milad Arkian

Research and acquire relevant thesis information Matlab programming practise Background reading on butterfly biology

02-May-11

25-Apr-11

18-Apr-11

11-Apr-11

04-Apr-11

28-Mar-11

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31-Jan-11

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20-Dec-10

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13-Dec-10

31/01/11 31/01/11 31/01/11 18/04/11 04/04/11 25/04/11 25/04/11 25/04/11

Final Report work Research the gaps in knowledge from the initial report carry out training of simulation software for the modelling of the heat equations Obtain a better understanding of the butterfly wing structure Refine asbtract, introduction and the literary survey Develop links between biology of the butterfly and possible engineering applications Refine reference list, nomenclature Prepare final report structure Finish final report

06-Dec-10

74 42 32 74

29-Nov-10

04/03/11 31/01/11 04/03/11 04/03/11

22-Nov-10

20/12/10 20/12/10 31/01/11 20/12/10

Poster work Research graphics tools and software that may be required for the poser Research colour designs and effective use of space on the poster organise material that will be used on the poster from the report Design final poster

15-Nov-10

81 81 63 52 35 11 7

08-Nov-10

17/12/10 17/12/10 29/11/10 18/11/10 29/11/10 17/12/10 09/12/08

01-Nov-10

27/09/10 27/09/10 27/09/10 27/09/10 25/10/10 06/12/10 02/12/08

Initial Report work Begin write up and organisation of the initial report, including abstract and introduction Select research papers that will provide the main backbone of the project Build up data bank of useful information Research key features of the butterflies anatomy that relates to its thermoregulation Derive the general heat balance equations Sum up the report with a conclusion Check report, organise and label figures, tables, glossary and nomenclature

25-Oct-10

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18-Oct-10

03/05/11 03/05/11 03/05/11

11-Oct-10

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27-Sep-10

Start

Pre term work

Week Starting

Main Tasks

17/06/2010 Duration (Days)

Start Date: