MODUL I adiation DITION HT TNSF Definition adiation, energy transfer across a system oundary due to a ΔT, y the mechanism of photon emission or electromagnetic wave emission. Because the mechanism of transmission is photon emission, unlike conduction and convection, there need e no intermediate matter to enale transmission. The significance of this is that radiation will e the only mechanism for heat transfer whenever a vacuum is present. lectromagnetic Phenomena. We are well acuainted with a wide range of electromagnetic phenomena in modern life. These phenomena are sometimes thought of as wave phenomena and are, conseuently, often descried in terms of electromagnetic wave length, λ. xamples are given in terms of the wave distriution shown elow:
X ays UV μm Visile Light ht, 0.-0.7 Thermal adiation 0-5 0-0 - 0-0 - 0-0 0 0 0 0 0 5 Wavelength, λ, μm Microwaves One aspect of electromagnetic radiation is that the related topics are more closely associated with optics and electronics than with those normally found in mechanical engineering courses. Nevertheless, these are widely encountered topics and the student is familiar with them through every day life experiences. From a viewpoint of previously studied topics students, particularly those with a ackground in mechanical or chemical engineering, will find the suject of adiation Heat Transfer a little unusual. The physics ackground differs fundamentally from that found in the areas of continuum mechanics. Much of the related material is found in courses more closely identified with uantum physics or electrical engineering, i.e. Fields and Waves. t this point, it is important for us to recognize that since the suject arises from a different area of physics, it will e important that we study these concepts with extra care.
Stefan-Boltzman Law Both Stefan and Boltzman were physicists; any student taking a course in uantum physics will ecome well acuainted with Boltzman s work as he made a numer of important contriutions to the field. Both were contemporaries of instein so we see that the suject is of fairly recent vintage. (ecall that the asic euation for convection heat transfer is attriuted to Newton.) σt as where: missive Power, the gross energy emitted from an ideal surface per unit area, time. σ Stefan Boltzman constant, 5.670-8 W/m K T as solute temperature of the emitting surface, K. Take particular note of the fact that asolute temperatures are used in adiation. It is suggested, as a matter of good practice, to convert all temperatures to the asolute scale as an initial step in all radiation prolems. You will notice that the euation does not include any heat flux term,. Instead we have a term the emissive power. The relationship etween these terms is as follows. Consider two infinite plane surfaces, oth facing one another. Both surfaces are ideal surfaces. One surface is found to e at temperature, T, the other at temperature, T. Since oth temperatures are at temperatures aove asolute zero, oth will radiate energy as descried y the Stefan-Boltzman law. The heat flux will e the net radiant flow as given y: " - σt - σt Plank s Law While the Stefan-Boltzman law is useful for studying overall energy emissions, it does not allow us to treat those interactions, which deal specifically with wavelength, λ. This prolem was overcome y another of the modern physicists, Max Plank, who developed a relationship for waveased emissions.
λ ƒ(λ) We plot a suitale functional relationship elow: λ, W/m μm rea Wavelength, λ, μm We haven t yet defined the Monochromatic missive Power, λ. n implicit definition is provided y the following euation: dλ λ We may view this euation graphically as follows: 0 λ, W/m μm rea Wavelength, λ, μm definition of monochromatic missive Power would e otained y differentiating the integral euation: λ d dλ
The actual form of Plank s law is: λ 5 λ C [ ] e C λt where: C πhc o.70 8 Wμm /m C hc o /k.90 μmk Where: h, c o, k are all parameters from uantum physics. We need not worry aout their precise definition here. This euation may e solved at any T, λ to give the value of the monochromatic emissivity at that condition. lternatively, the function may e sustituted into the integral d 0 λ λ to find the missive power for any temperature. While performing this integral y hand is difficult, students may readily evaluate the integral through one of several computer programs, i.e. MathCad, Maple, Mathmatica, etc. d 0 λ σ λ mission Over Specific Wave Length Bands Consider the prolem of designing a tanning machine. s a part of the machine, we will need to design a very powerful incandescent light source. We may wish to know how much energy is eing emitted over the ultraviolet and (0 - to 0. μm), known to e particularly dangerous. T ( ) 0. μm 0. 000 0. 0. 00 dλ With a computer availale, evaluation of this integral is rather trivial. lternatively, the text ooks provide a tale of integrals. The format used is as follows: μm λ
( 0. 00 0. ) m μm λ dλ dλ 0. μ 0. μ 0. 00 0 0 λ 0 m dλ λ dλ λ 0. 000 0. 0 μm dλ λ F( 0 0. ) F( 0 0. 000) dλ λ eferring to such tales, we see the last two functions listed in the second column. In the first column is a parameter, λt. This is found y taking the product of the asolute temperature of the emitting surface, T, and the upper limit wave length, λ. In our example, suppose that the incandescent ul is designed to operate at a temperature of 000K. eading from the tale: λ., μm T, K λt, μmk F(0 λ) 0.000 000 0. 0 0. 000 600 0.0000 F(0. 0.000μm) F(0 0.μm)- F(0 0.000μm) 0.0000 This is the fraction of the total energy emitted which falls within the I and. To find the asolute energy emitted multiply this value times the total energy emitted: I F(0. 0.000μm)σT 0.00005.670-8 000.7 W/m Solar adiation The magnitude of the energy leaving the Sun varies with time and is closely associated with such factors as solar flares and sunspots. Nevertheless, we often choose to work with an average value. The energy leaving the sun is emitted outward in all directions so that at any particular distance from the Sun we may imagine the energy eing dispersed over an imaginary spherical area. Because this area increases with the distance suared, the solar flux also decreases with the distance suared. t the average distance etween arth and Sun this heat flux is 5 W/m, so that the average heat flux on any oject in arth orit is found as: G s,o S f cos θ c Where S Solar Constant, 5 W/m c
f correction factor for eccentricity in arth Orit, (0.97<f<.0) θ ngle of surface from normal to Sun. Because of reflection and asorption in the arth s atmosphere, this numer is significantly reduced at ground level. Nevertheless, this value gives us some opportunity to estimate the potential for using solar energy, such as in photovoltaic cells. Some Definitions In the previous section we introduced the Stefan-Boltzman uation to descrie radiation from an ideal surface. σ T as This euation provides a method of determining the total energy leaving a surface, ut gives no indication of the direction in which it travels. s we continue our study, we will want to e ale to calculate how heat is distriuted among various ojects. For this purpose, we will introduce the radiation intensity, I, defined as the energy emitted per unit area, per unit time, per unit solid angle. Before writing an euation for this new property, we will need to define some of the terms we will e using. ngles and rc Length We are well accustomed to thinking of an angle as a two dimensional oject. It may e used to find an arc length: α L L r α Solid ngle
We generalize the idea of an angle and an arc length to three dimensions and define a solid r dω angle, Ω, which like the standard angle has no dimensions. The solid angle, when multiplied y r the radius suared will have dimensions of length suared, or area, and will have the magnitude of the encompassed area. Projected rea The area, d, as seen from the prospective of a viewer, situated at an angle θ from the normal to the surface, will appear somewhat smaller, as cos θ d. This smaller area is termed the projected area. θ d d cos θ projected cos θ normal Intensity The ideal intensity, I, may now e defined as the energy emitted from an ideal ody, per unit projected area, per unit time, per unit solid angle. I d cosθ d dω
Spherical Geometry Since any surface will emit radiation outward in all directions aove the surface, the spherical coordinate system provides a convenient tool for analysis. The three asic coordinates shown are, φ, and θ, sin θ representing the radial, azimuthal and zenith directions. In general d will correspond to the emitting surface or the source. The surface d will correspond to the receiving surface or the target. Note that the area proscried on the hemisphere, d, may e written as: d d θ Δφ φ or, more simply as: d [( sin θ ) dϕ] [ dθ ] d sinθ dϕ dθ ] ecalling the definition of the solid angle, we find that: d dω dω sin θ dθ dφ eal Surfaces Thus far we have spoken of ideal surfaces, i.e. those that emit energy according to the Stefan-Boltzman law: σ T as
eal surfaces have emissive powers,, which are somewhat less than that otained theoretically y Boltzman. To account for this reduction, we introduce the emissivity, ε. ε so that the emissive power from any real surface is given y: ε σ T as eceiving Properties Targets receive radiation in one of three ways; they asorption, reflection or transmission. To account for these characteristics, we introduce three additional properties: sorptivity, α, the fraction of incident radiation asored. eflected adiation sored adiation Transmitted adiation eflectivity, ρ, the fraction of incident radiation reflected. Transmissivity, τ, the fraction of incident radiation transmitted. We see, from Conservation of nergy, that: α + ρ + τ In this course, we will deal with only opaue surfaces, τ 0, so that: α + ρ Opaue Surfaces Incident adiation, G
elationship Between sorptivity,α, and missivity,ε Consider two flat, infinite planes, surface and surface B, oth emitting radiation toward one another. Surface B is assumed to e an ideal emitter, i.e. ε B.0. Surface will emit radiation according to the Stefan-Boltzman law as: Surface, T Surface B, TB and will receive radiation as: ε σ T G α σ T B The net heat flow from surface will e: ε σ T - α σ T B Now suppose that the two surfaces are at exactly the same temperature. The heat flow must e zero according to the nd law. If follows then that: α ε Because of this close relation etween emissivity, ε, and asorptivity, α, only one property is normally measured and this value may e used alternatively for either property. Let s not lose sight of the fact that, as thermodynamic properties of the material, α and ε may depend on temperature. In general, this will e the case as radiative properties will depend on wavelength, λ. The wave length of radiation will, in turn, depend on the temperature of the source of radiation. The emissivity, ε, of surface will depend on the material of which surface is composed, i.e. aluminum, rass, steel, etc. and on the temperature of surface.
The asorptivity, α, of surface will depend on the material of which surface is composed, i.e. aluminum, rass, steel, etc. and on the temperature of surface B. In the design of solar collectors, engineers have long sought a material which would asor all solar radiation, (α, T sun ~ 5600K) ut would not re-radiate energy as it came to temperature (ε <<, T collector ~ 00K). NS developed an anodized chrome, commonly called lack chrome as a result of this research. Black Surfaces Within the visual and of radiation, any material, which asors all visile light, appears as lack. xtending this concept to the much roader thermal and, we speak of surfaces with α as also eing lack or thermally lack. It follows that for such a surface, ε and the surface will ehave as an ideal emitter. The terms ideal surface and lack surface are used interchangealy. Lamert s Cosine Law: surface is said to oey Lamert s cosine law if the intensity, I, is uniform in all directions. This is an idealization of real surfaces as seen y the emissivity at different zenith angles: 0...6.8.0 Dependence of missivity on Zenith ngle, Typical Metal. 0...6.8.0 Dependence of missivity on Zenith ngle, Typical Non-Metal.
The sketches shown are intended to show is that metals typically have a very low emissivity, ε, which also remain nearly constant, expect at very high zenith angles, θ. Conversely, non-metals will have a relatively high emissivity, ε, except at very high zenith angles. Treating the emissivity as a constant over all angles is generally a good approximation and greatly simplifies engineering calculations. elationship Between missive Power and Intensity By definition of the two terms, emissive power for an ideal surface,, and intensity for an ideal surface, I. I hemisphere cos θ dω eplacing the solid angle y its euivalent in spherical angles: π 0 π 0 I cosθ sin θ dθ dϕ Integrate once, holding I constant: π I π 0 cosθ sin θ dθ Integrate a second time. (Note that the derivative of sin θ is cos θ dθ.) π sin θ π I π I 0 π I
adiation xchange During the previous lecture we introduced the intensity, I, to descrie radiation within a particular solid angle. d I cosθ d dω This will now e used to determine the fraction of radiation leaving a given surface and striking a second surface. earranging the aove euation to express the heat radiated: d I cosθ d dω Next we will project the receiving surface onto the hemisphere surrounding the source. First find the projected area of surface d, d cos θ. (θ is the angle etween the normal to surface and the position vector,.) Then find the solid angle, Ω, which encompasses this area. d Sustituting into the heat flow euation aove: d cos θ d I cosθ d cosθ d To otain the entire heat transferred from a finite area, d, to a finite area, d, we integrate over oth surfaces: I cos θ d cosθ d To express the total energy emitted from surface, we recall the relation etween emissive power,, and intensity, I.
View Factors-Integral Method emitted π I Define the view factor, F -, as the fraction of energy emitted from surface, which directly strikes surface. F emitted I cos θ d π I cosθ d after algeraic simplification this ecomes: F cosθ cosθ d d π xample Consider a diffuse circular disk of diameter D and area j and a plane diffuse surface of area i << j. The surfaces are parallel, and i is located at a distance L from the center of j. Otain an expression for the view factor F ij. dr d j j θ i D θ j L d i d i
The view factor may e otained from: cos cos d d F π θ θ is a differential area Since d i cos cos d F π θ θ Sustituting for the cosines and the differential area: ( ) dr r L F π π fter simplifying: dr r L F Let ρ L + r. Then ρ dρ r dr. d L F ρ ρ ρ fter integrating, 0 D L L L F + ρ ρ
Sustituting the upper & lower limits F L L + D L D 0 D L + D This is ut one example of how the view factor may e evaluated using the integral method. The approach used here is conceptually uite straight forward; evaluating the integrals and algeraically simplifying the resulting euations can e uite lengthy. nclosures In order that we might apply conservation of energy to the radiation process, we must account for all energy leaving a surface. We imagine that the surrounding surfaces act as an enclosure aout the heat source which receive all emitted energy. Should there e an opening in this enclosure through which energy might e lost, we place an imaginary surface across this opening to intercept this portion of the emitted energy. For an N surfaced enclosure, we can then see that: N j F i, j This relationship is known as the Conservation ule. xample: Consider the previous prolem of a small disk radiating to a larger disk placed directly aove at a distance L. The view factor was shown to e given y the relationship: D F From our conservation rule we have: L + D Here, in order to provide an enclosure, we will define an imaginary surface, a truncated cone intersecting circles and.
N j F i, j F, + F, + F, Since surface is not convex F, 0. Then: F D L + D eciprocity We may write the view factor from surface i to surface j as: cos θi cosθ j di i Fi j j i π d j Similarly, etween surfaces j and i: cosθ j cosθi dj j Fj i j i π d i Comparing the integrals we see that they are identical so that: i F i j j F j i This relationship is known as eciprocity.
xample: Consider two concentric spheres shown to right. ll radiation leaving the outside of surface will strike surface. Part of the radiant energy leaving the inside surface of oject will strike surface, part will return to surface. To find the fraction of energy leaving surface which strikes surface, we apply reciprocity: the F, F, F, F, D D ssociative ule Consider the set of surfaces shown to the right: Clearly, from conservation of energy, the fraction of energy leaving surface i and striking the comined surface j+k will eual the fraction of energy emitted from i and striking j plus the fraction leaving surface i and striking k. k j i F i ( j+ k ) Fi j + F i k This relationship is known as the ssociative ule. adiosity We have developed the concept of intensity, I, which let to the concept of the view factor. We have discussed various methods of finding view factors. There remains one additional concept to introduce efore we can consider the solution of radiation prolems. adiosity,, is defined as the total energy leaving a surface per unit area and per unit time. This may initially sound much like the definition of emissive power, ut the sketch elow will help to clarify the concept.
ε + ρ G ε ρ G G Net xchange Between Surfaces Consider the two surfaces shown. adiation will travel from surface i to surface j and will also travel from j to i. i j i i F i j j likewise, j i j j F j j i The net heat transfer is then: j i (net) i i F i j - j j F From reciprocity we note that F F so that j j j i (net) F - i i i j j F i i j i F i j ( i j ) Net nergy Leaving a Surface The net energy leaving a surface will e the difference etween the energy leaving a surface and the energy received y a surface: ε ρ G G [ε α G] Comine this relationship with the definition of adiosity to eliminate G. ε + ρ G G [ - ε ]/ρ
{ε α [ - ε ]/ρ} ssume opaue surfaces so that α + ρ ρ α, and sustitute for ρ. {ε α [ - ε ]/( α)} Put the euation over a common denominator: ( ) + α α ε α ε α α ε α If we assume that α ε then the euation reduces to: ( ) ε ε ε ε ε lectrical nalogy for adiation We may develop an electrical analogy for radiation, similar to that produced for conduction. The two analogies should not e mixed: they have different dimensions on the potential differences, resistance and current flows. uivalent uivalent Potential Current esistance Difference I ΔV Ohms Law Net nergy ε ε Leaving Surface - Net xchange F Between Surfaces i j
xample: Consider a grate fed oiler. Coal is fed at the ottom, moves across the grate as it urns and radiates to the walls and top of the furnace. The walls are cooled y flowing water through tues placed inside of the walls. Saturated water is introduced at the ottom of the walls and leaves at the top at a uality of aout 70%. fter the vapour is separated from the water, it is circulated through the superheat tues at the top of the oiler. Since the steam is undergoing a sensile heat addition, its temperature will rise. It is common practice to sudivide the super-heater tues into sections, each having nearly uniform temperature. In our case we will use only one superheat section using an average temperature for the entire region. SH section Surface Superheat Tues Coal Chute Surface Water Walls Surface Burning Coal nergy will leave the coal ed, Surface, as descried y the euation for the net energy leaving a surface. We draw the euivalent electrical network as seen to the right: ε ε σ T The heat leaving from the surface of the coal may proceed to either the water walls or to the super-heater section. That part of the circuit is represented y a potential difference etween adiosity: It should e noted that surfaces and F F
will also radiate to one another. F F F F It remains to evaluate the net heat flow leaving (entering) nodes and. ε ε F F ε ε ε ε lternate Procedure for Developing Networks Count the numer of surfaces. ( surface must e at a uniform temperature and have uniform properties, i.e. ε, α, ρ.) Draw a radiosity node for each surface. Connect the adiosity nodes using view factor resistances, / i F i j. Connect each adiosity node to a grounded attery, through a surface resistance, [ ε ε ]. This procedure should lead to exactly the same circuit as we otain previously.
Simplifications to the lectrical Network Insulated surfaces. In steady state heat transfer, a surface cannot receive net energy if it is insulated. Because the energy cannot e ε stored y a surface in ε steady state, all energy must e re-radiated ack into the enclosure. Insulated surfaces are often termed as reradiating surfaces. lectrically cannot flow through a attery if it is not grounded. Surface is not grounded so that the attery and surface resistance serve no purpose and are removed from the drawing. Black surfaces: lack, or ideal surface, will have no surface resistance: ε 0 ε In this case the nodal adiosity and emissive power will e eual. This result gives some insight into the physical meaning of a lack surface. Ideal surfaces radiate at the maximum possile level. Nonlack surfaces will have a reduced potential, somewhat like a attery with a corroded terminal. They therefore have a reduced potential to cause heat/current flow. Large surfaces: Surfaces having a large surface area will ehave as lack surfaces, irrespective of the actual surface properties:
ε ε ε ε 0 Physically, this corresponds to the characteristic of large surfaces that as they reflect energy, there is very little chance that energy will strike the smaller surfaces; most of the energy is reflected ack to another part of the same large surface. fter several partial asorptions most of the energy received is asored. Solution of nalogous lectrical Circuits. Large nclosures Consider the case of an oject,, placed inside a large enclosure,. The system will consist of two ojects, so we proceed to construct a circuit with two radiosity nodes. /( F ) Now we ground oth adiosity nodes through a surface resistance. (-ε)/(ε ) /( F ) (-ε )/(ε ) σt σt Since is large, 0. The view factor, F
(-ε)/(ε ) /( F ) σt σt Sum the series resistances: Ohm s law: Series (-ε )/(ε ) + / /(ε ) i ΔV/ or y analogy: Δ / Series ε σ(t T ) You may recall this result from Thermo I, where it was introduced to solve this type of radiation prolem. Networks with Multiple Potentials Systems with or more grounded potentials will reuire a slightly different solution, ut one which students have previously encountered in the Circuits course. The procedure will e to apply Kirchoff s law to each of the adiosity junctions. F
i i 0 In this example there are three junctions, so we will otain three euations. This will allow us to solve for three unknowns. adiation prolems will generally e presented on one of two ways: o The surface net heat flow is given and the surface temperature is to e found. o The surface temperature is given and the net heat flow is to e found. eturning for a moment to the coal grate furnace, let us assume that we know (a) the total heat eing produced y the coal ed, () the temperatures of the water walls and (c) the temperature of the super heater sections. pply Kirchoff s law aout node, for the coal ed: Similarly, for node : + + + + + + + + (Note how node, with a specified heat input, is handled differently than node, with a specified temperature. nd for node : + + + + 0 0 0
The three euations must e solved simultaneously. Since they are each linear in, matrix methods may e used: The matrix may e solved for the individual adiosity. Once these are known, we return to the electrical analogy to find the temperature of surface, and the heat flows to surfaces and. Surface : Find the coal ed temperature, given the heat flow: 0.5 + σ σ T T Surface : Find the water wall heat input, given the water wall temperature: T σ Surface : (Similar to surface ) Find the water wall heat input, given the water wall temperature: T σ