2 Principles of Heat Transfer and Thermodynamics N 2. and H 2. r fr. 1.4 kj/kgk for carbohydrates. 1.6 kj/kgk for proteins. c p. 1.

Transcription

1 2 Principles f Heat ransfer and herdynaics 17 2 Principles f Heat ransfer and herdynaics In this Chapter thse principles which are iprtant fr essential technical calculatins in dairy practice have been extracted fr the extensive literature n heat transfer and therdynaics. Particular prbles in heat transfer are dealt with in each sectin f this bk as they arise. Fr re details the relevant literature shuld be cnsulted. 2.1 General Cncepts eperature is a easure f the theral state f a bdy and is easured in degrees Celsius C and, as abslute teperature, in Kelvin K. A teperature difference can be expressed an C r as K, as the difference ϑ in C is the sae as in K. he teperature f elting ice at pressure f N/ Pa is ϑ = 0 C r = 273K (re accurately: = K). At the sae pressure water bils at ϑ = 100 C r = 373K (see als the water vapur table in Sectin 23). he heat Q, which is a quantity equivalent t wrk r energy, is expressed in J (Jule) = N. (Newtn x eter) = kg 2 /s 2 and the heat flw rate Q, & which is equivalent t pwer r t energy flw rate, is expressed in W (Watt) = J/s = N. /s = kg. 2 /s 3. he heat flw rate per unit area 1 2 is &q = Q/A & expressed as W/ 2. Cnversin factrs fr r int frerly used units are given in sectin 23. he energy state f a aterial is given by the enthalpy h (frerly als called heat cntent). Its diensin per unit weight in kg is J/kg = N. /kg = 2 /s 2. Values are ften given in kj/kg. he specific heat capacity f a aterial in defined as the aunt f heat which is necessary t raise the teperature f 1 kg f the aterial by 1 C r 1K withut changing the state f the atter. Its diensins are therefre (J/kg. K) r, ften (kj/kg K). he specific heat f gases generally increases with increasing teperature, that f liquids ay increase r decrease depending n the teperature. hese differences are very sall, hwever, ver the teperature ranges which are f interest in fd prcessing, s that it is justified t use average values fr specific heats. When cnsidering the specific heat f gases a distinctin has t be ade between heat transfer at cnstant pressure c p and at cnstant vlue c v. c p is generally used in the prcesses which are dealt with here. he specific heats c p f air, O 2, N 2 and H 2 are 1.4 ties greater than c v, and c p f water vapur is 1.33 ties greater than c v. he differences is due t the wrk f expansin. Enthalpy is defined as: h = 0 J/kg fr water at 0 C At a given teperature ϑ, h beces: h = c. p ϑ fr gases h = c. ϑ fr liquids and slids. It shuld be nted that, in case f hygrscpic dry prducts with a very lw isture cntent, the heat f absrptin f absrbed liquid has t be taken int accunt. If there is a change in phase, r fr, the latent heat f fusin r slidificatin, r r, the latent heat f evapratin r cndensatin, has t be taken int accunt. Its diensin is J/kg. he enthalpy f ice at a teperature belw 0 C is: h = cl ϑ fr rfr + cice ϑ ϑ fr d i (2.1) he enthalpy f superheated stea is: h = cl ϑev + r+ cpv ϑ ϑev b g (2.2) Where ϑ ev biling teperature at the prevailing vapur pressure ϑ final teperature ϑ fr teperature f fusin r slidificatin respectively c 1 specific heat f the liquid c ice specific heat f the slidified liquid c pv specific heat f the vapur r latent heat f the evapratin at the biling teperature r fr heat f fusin r slidificatin at ϑ fr he rder f agnitude f a nuber f c values: c w 4.2 kj/kgk fr water c ice 2.0 kj/kgk fr ice c c 1.4 kj/kgk fr carbhydrates c p 1.6 kj/kgk fr prteins c f 1.7 kj/kgk fr fats c a 0.8 kj/kgk fr ash (inerals) he specific heat f cplex prducts such as fd can be calculated fr the ass fractin f the individual cnstituents. i c c = i i 1 ges (2.3)

2 18 2 Principles f Heat ransfer and herdynaics 2.2 heral Expansin Expansin f Slids and Liquids Materials generally expand when heated. heir linear expansin l is prprtinal t the initial length l 0 and t the increase in teperature : l = βl l0 ϑ (2.4) where β L is the cefficient f linear expansin which dentes the change in length per unit length and degree abslute teperature (Kelvin) (able 2.1). In lng pipe lines cpensatrs have be fitted between fixed pints t accdate the expansin thrugh heat. Fig. 2.1 shws a U shaped cpensatr ften used in industry t cpensate the expansin due t teperature changes. he cefficient f expansin fr an area is abut twice the linear expansin cefficient. βa 2 βl (2.5) he cubic theral expansin is given by the Equatin belw: ab Cefficients f linear expansin aterial steel, cncrete aluiniu (99.5%) cpper stainless steel ice rigid PVC plyethylen glass cefficient f linear expansin β L /K /K /K /K /K /K /K /K Fig U shaped cpensatr V 3 βl V ϑ = βv V ϑ (2.6) When the teperature is raised by ϑ the riginal vlue V 0 is increased by V. he cefficient f cubic expansin is abut three ties linear expansin cefficient (β v 3. β L ). Fr instance β v fr ils / 3 K ethanl / 3 K tluene / 3 K ercury / 3 K he cefficient f expansin β v is ften nly cnstant ver a certain teperature range. eperature has a great influence n water, as shwn by the nnlinear change in specific vlue with teperature (Fig. 2.2). Water is especially analus because it even cntracts between the teperature f 0 and 4 C and has its highest density at 4 C. he largest increase in vlue n freezing is ntewrthy. Fig. 2.2 als shws that the relatinship between the specific vlue and teperature is linear fr ice and butterfat and nn-linear fr skied ilk. he reciprcal f the specific vlue v in 3 /kg is the density ρ in kg/ 3. ρ = v 1 (2.7) Expansin f Gases fixing pint he cefficient f expansin is alst the sae fr all gases. hey behave like ideal gases if the utual interactin between the gas lecules due t attractive frces is sall and if the vlue f the le- I F A? EB E? L L K E A! C " & $ " ' & ' $ ice water butterfat skied ilk " " & J A F A H = J K H A E + Fig he specific vlues f se prducts as a functin f the teperature & % % & '! ' ' ' $ ' "! ' $ ' A I EJ ρ OE C! "

3 2 Principles f Heat ransfer and herdynaics 19 cules is sall cpared t the ttal vlue. Fr each degree rise in teperature at a cnstant pressure a gas expands by 1/ f its vlue V 0 which it wuld ccupy at 0 C r K. he cefficient f cubical expansin f the gas at 0 C is therefre abut v / K β = = = (2.8) If p = cnst. the increase in vlue is V = βv V ϑ (2.9) r 1 b g (2.10) V V = V r V = V (2.11) where V vlue f the gas at V 0 vlue f the gas at 0 If the vlue f the gas is kept cnstant (V = cnst.) the sae relatinship is btained fr the increase in pressure with increasing teperature. p = p (2.12) Where p is the pressure f the gas at teperature is the pressure at teperature 0 When is cnstant the law f Byle-Maritte states: p11 V = p2v2 =... = pv = cnst., (2.13) i.e. at cnstant teperature the vlue f a gas is inversely prprtinal t its pressure. Fig. 2.3 illustrates the increase in vlue with increasing teperature when = cnst. V' = V (2.14) When = cnst. p V = p V' (2.15) Substituting V fr Equatin 2.14 yields Gay-Lussac s law: p V = p V (2.16) Using the general gas law p V = R R M p V i = = (2.17) the density f a gas M p ρ = = V R an its specific vlue V v = 1 R ρ = = M p can be btained, where ass f the gas M relative lecular ass R i special gas cnstant R universal gas cnstant (2.18) (2.19) 2.3 Balances fr the Deterinatin f the State f a Mixture Mass balance fr a discntinuus syste p V 0 V ' V = cnst. = cnst. = cnst. 0 Fig Changes in pressure and vlue f gases with teperature + = t t Mass balance fr a cntinuus syste (2.20) & & 1A + & & & 2A (2.21) & + & = & 1A + & 2A &

4 20 2 Principles f Heat ransfer and herdynaics he su f the ttal ass flw entering the syste equals the su f the ttal ass flws leaving it; cnsidering changes in the syste: & E = & A & (2.22) Heat balance fr a discntinuus syste: h h h + h = ( h) t ( h) t Heat balance fr a cntinuus syste: & & h h (2.23) & 1A h1a & h & 2A h (2.24) 2A + ( ) In the absence f heat lsses the su f the heat entering the syste is equal t the su f the heat leaving it; cnsidering changes in the syste: ( h ) = ( h ) ( h ) & & & (2.25) E A Fr this the teperature btained n ixing ay be fund. & c ϑ + & c ϑ+... = = & 1A c1a + & 2A c2a+... ϑ b g (2.26) 2.4 Heat ransfer In any fd prcessing peratins heat exchange between tw bdies r tw streas f aterial is f great iprtance. A knwledge f the rate and extent f heat transfer is necessary if sterilisatin, freezing, evapratin, drying etc. are t be carried ut crrectly. When cnsidering these prcesses a distinctin has t be ade between steady and unsteady heat exchange. he frer is a lcally develped teperature pattern which, nce established, reains the sae. It is fund in alst all cntinuus prcesses such as cntinuus flw heating, cling and evapratin. Unsteady heat exchange cnditins ccur during heating and cling where teperatures change with tie, as in the sterilisatin f gds in cans, the cling f yghurt cups, etc. hree ain types f heat transissin are radiatin, cnductin and cnvectin. Special electrical ethds, such as heating by icrwaves, in which the bdy t be heated acts as dielectric in an high frequency alternating field, are n re than entined here because they have prved unecnical in industrial use and have therefre gained little iprtance heral Radiatin A war bdy eits electragnetic waves f a wavelength f > hese are absrbed t a varying degree by a cler bdy. Accrding t Stefan-Bltzann, the rate f heat flw fr a bdy is: & (2.27) 4 Q= A σb Where σ b is the theral radiatin cnstant (Stefan- Bltzann cnst.). A perfect black bdy has the axiu pssible value f this cnstant. 8 W J σb = 5,67 10 = K s K (2.28) All bdies et with in practice radiate less energy at any given teperature than a perfect black bdy. heir theral radiatin cnstant is less than σ b. he reductin is expressed by the eissin degree ε = σ/σ b. he fllwing is a rugh guide t the value f σ (in 10-8 W/( 2 K 4 )) σ = 0.1 t 0.2 fr highly plished etal surfaces = 0.2 t 0.4 fr att etal surfaces = 0.5 t 3 fr re r less strngly xidised etals = 4 t 5.5 fr all liquids, plastics, paints, building aterials, ice, snw, fds, nn-etallic packaging aterials bdy 1 Q Rad bdy 2 Fig Exchange f radiatin between tw bdies

5 2 Principles f Heat ransfer and herdynaics 21 Fr purpses f calculatin the assuptin is ade that the eissin capacity is equal t the absrptin capacity. When radiatin in exchanged between tw bdies (Fig. 2.3) the fllwing Equatin then applies: &Q 12 = A1 σ Φ 12 (2.29) where σ 12 is the theral radiatin cnstant, cnsidering the exchange between the bdies 1 and 2: 1 σ12 = = 1 A σ1 A 2 σ2 σ b σb = (2.30) 1 A ε A ε he rati A 1 /A 2 is ften very sall r the theral radiatin cnstant f bdy 2 is clse t that f a black bdy. σ 12 then beces apprxiately equal t σ 1. he extent f the exchange f radiatin will be reduced if the radiatin eitted fr surface A 1 ipinges nly partly n surface A 2, as in the case f ppsite walls. he rati f the absrbed radiatin n the ttal ne appears in Equatin (2.29) as the factr Φ 12. Φ 12 is called the irradiatin nuber and its value is fund purely by geetrical eans. he influence f radiatin in industrial practice is nt t be underestiated. he cntributin ( t heat lsses) f radiatin fr the equipent (dryers, evapratrs) r the absrptin f the heat fr the envirnent by cld strage tanks can be quite iprtant Heat Cnductin Every aterial, whether slid, liquid r gas, cnducts heat when there is a teperature gradient dϑ dt. he carriers f the energy are lecules r electrns which transit the energy thrugh cllisin between free lecules, r, if bund, thrugh vibratin r ther types f tin. Furier s law f heat cnductin: &Q A k d ϑ = (2.31) dx k is the theral cnductivity f a aterial expressed in W /( K) = J /(s K). In the steady state the rate f heat flw thrugh the crss sectin A f a wall f thickness s (Fig. 2.5) is: Fig Heat cnductin thrugh a wall b g (2.32) &Q= A k ϑ1 ϑ2 s Values fr the heat cnductivity f a nuber f ilk prducts as a functin f the teperature are given in Fig hey are based n a large nuber f data fund in the literature but ainly n values fr a suary by KOSAROPOULOS [1971]. Mre data are fund in Sectin 23. In the case f several layers jined in series (Fig. 2.7) the rate f heat flw is btained by a siple derivatin: heat cnductivity k in W/( K) ϑ ϑ 1 0 k 0 s range f butter Q ϑ 2 range f pure butterfat surface f wall A teperature in C Fig he effect f teperature n the heat cnductivity f ilk prducts