ThePhysicsofbakinggoodPizza

AndreyVarlamov1,AndreasGlatz2,3,SergioGrasso41CNR-SPIN,VialedelPolitecnico1,I-00133,Rome,Italy

2MaterialsScienceDivision,ArgonneNationalLaboratory,9700SouthCassAvenue,Argonne,Illinois60439,USA3DepartmentofPhysics,NorthernIllinoisUniversity,DeKalb,Illinois60115,USA

4FoodAnthropologist,Rome,Italy

Abstract:Physicalprinciplesareinvolvedinalmostanyaspectofcooking.Hereweanalyzethespecificprocess of baking pizzas, deriving in simple terms thebaking times for twodifferent situations: Forabrickoveninapizzeriaandamodernmetallicovenathome.Ourstudyisbasedonbasicthermodynamicprinciplesrelevanttothecookingprocessandisaccessibletoundergraduatestudents.Westartwithahistoricaloverviewofthedevelopmentandartofpizzabaking,illustratetheunderlyingphysicsbysomesimplecommonexamples,andthenapplythemindetailtotheexampleofbakingpizza.Historicalbackground:Thehistoryofpizzaiscrowdedwithtales,legendsandanecdotes.Notwithstandingthat,Italiansaretrustedastheinventorsofthishumble,deliciousand“universal”flatbread.The precursors of pizza are theNeolithicunleavened flatbreads baked on fire-heatedrocks and made from coarse grains flour (spelt, barley, emmer), developedautonomouslyinseveralareasfromChinatotheAmericas.The Italian word “pizza” first appeared on a Latin parchment (Codex DiplomaticusCajtanus) reportinga listofdonationsduebyadomain tenant to thebishopofGaeta(Naples).Thedocumentdated997ADfixesasupplyofduodecimpizze("twelvepizzas")everyChristmasDayandEasterSunday.Etymologistsdebateondifferentoriginsoftheword“pizza”:fromtheByzantineGreekπίτα=bread,cake,pie,pitta(attestedin1108);fromGreekπηκτή=congealed;fromLatinpicta=painted,decorated,orpinsere=flatten,enlarge.Sincethe8thcenturyBC,theGreekcoloniesinsouthernItaly(Napoliitselfwasfoundedaround 600 B.C. as a Greek town), produced the Plakuntos, a flat baked grain-basedsourdough covered with oil, garlic, onion, herbs, occasionally minced meat or smallfishes,almostsimilar toactualTurkishpide.Platomentions“cakes”madefrombarleyflour,kneadedandcookedwitholivesandcheese.Greeks familiarized the whole Mediterranean with two Egyptian procedures: theleavening and kneading of the dough to get a more digestible bread, and the use ofcupola-roof-ovens instead of open fires. Greek bakers who became popular in Romesince the 4th century BC converted flavored and garnishedPlakuntos into the romanPlacenta=flat-bread.TheRomanpoetVirgilwrote:

Theirhomelyfaredispatch’d,thehungryband Invadetheirtrenchersnext,andsoondevour,Tomendthescantymeal,theircakesofflour.Ascaniusthisobserv’d,andsmilingsaid: “See,wedevourtheplatesonwhichwefed.”

No doubt that more than twenty centuries ago Greeks and Romans

createdtheprototypesofpizza,butitwastheNeapolitans,“inventors”of this inexpensive food that could be consumed quickly, who wereresponsible for the addition of the ingredients universally associatedwithpizzatoday—tomatoandmozzarellacheese(Fig.2).Neapolitans

Figure1.BustofVergilfromtheTombofVergilinNaples,Italy

become familiar with the “exotic” tomato plant after Columbus but the fruit wasbelieved to be poisonous. Barely and hazardously eaten by peasants, tomatoes firstappeared in the recipebook “IlCuocoGalante” (TheGallantCook)written in1819bychefVincenzoCorrado.Thefirstmentioningof“mozzarella”canbefoundintherecipe-book“Opera”(1570)byBartolomeoScappi.Uptopresenttime,genuinemozzarellaisstillproducedfromfatbuffalomilkinthevicinityofNaples(Fig.3).Itisaverydelicateproduct, not only in taste, but also to store – mozzarella does not handle lowtemperatureswell – if kept ina refrigerator, itbecomes “rubbery”. It shouldonlybestoredinitsownwheyatroomtemperatureforafewdays.

Documents demonstrate that until the18thcentury, theNeapolitanpizzawasasimple dish of pasta baked or fried,flavored with lard, pecorino-cheese,olives, salt or small fishes calledcecinielli. During the 19th century up totwo hundred “pizzaioli” (pizza-makers)crowdedthestreetsofNaplessellingfora nickel baked or fried pizzas dressed

withtomatosauceandbasilleaves.In1889,afewyearsaftertheunificationofItaly,thepizzaioloRaffaeleEspositodecidedtopay

homage to the ItalianQueenbyaddingmozzarella to the traditional tomatoandbasilpie.Thecombinationofred,white,andgreensymbolizedthecolorsof the Italian flagandthetri-colorpizza,stillknownasPizzaMargherita,transformedthemodestpietoasuccessthatthemodestpizzaiolocouldneverhaveimagined.InItalytoday,pizzaismadeindifferentregionalstyles.Neapolitansarefamousfortheirround-shaped pizzas with a high crusty edge or cornicione. Beyond the mentionedMargherita the standard isPizzaNapoletana (protected by the European Union as aGuaranteed Traditional Speciality), dressedwithtomato,mozzarellaandanchovies.Thesimplest version is calledMarinara, simplytoppedwith tomato, garlic, oregano andoil.ElaborateandopulentistheCalzoneacircleof pizza dough folded into a half-moon

stuffed with ricotta, mozzarella, salami orprosciutto.Romans,devoted to crunchy flatbreads, addmorewater to the flour (up to 70%) and add olive-oilor lard to the dough, so theirpizzas can be stretched to the thickness of canvas without losing its toothsomechewiness;condimentsarethesamefortheNapoletanabutthepizzaservedatthetableof a romanpizzeria iswithout cornicione. Romanbakeries and groceries usually sellpizzabiancabyweight,a“white”,rectangular,crustypizzatoppedonlywithoilandsalt.Sardenaira is a flatbread with tomatoes, olives & anchovies typical of Liguria butoriginated in the nearby Provence where it is called pissaladière (from the anchovypastecalledpissalat=saltfish),TypicaloftheAbruzzi’straditionisthePizzadiSfrigolimadebydeeplykneadinglard,flour,andsalt,thenincorporatinglittlepigbits(sfrigole)beforebaking,whileinApuliathe beloved recipe isPizzaPugliese, thin andcoveredwith tomato sauce and a lotofstewed onions (anchovies and olives are optional). Another Apulian specialty is the

Figure3.BuffaloranchnearCapaccio(Campaniaregion),Italy

Figure2.MozzarelladiBufalaandcherrytomatoes.

Panzerotto, a littlepizzapocket served tomark thebeginningofCarnival.PanzerottodiffersfromtheNeapolitancalzoneinbothsizeandmethodofcooking(it’sdeep-fried,notbaked).Theclassicfillingistomatosauceandfreshmozzarella,butmanyvariationsexist. Calabrians - devoted to sharp and spicy flavors - enrich their pizzas with hotsalami (soppressata) or ‘nduja, a spreadable and fiery blend of lard and chili peppersprobablyintroducedintoCalabriabytheSpanish.

The Sicilian version of pizza is called sfinciuni (from thelatin spongia=sponge), a rectangular thick and cushionedflatbreadgenerouslydressedwitholiveoil,onions,sheep'scheeseandsun-driedtomatoes.Scaccia-aspecialtyoftheRagusaprovince inSicily– isa thinlyrolleddoughspreadwithtomatosauceandcheeseandthenfoldeduponitselftoresembleastrudel;thelong,rectangularpizzasarethen

sliced,revealingthelayersofcrust,sauce,andcheese.Sardinian panada is a nourishing pizza shield filled with

eggplant,lambandtomatoes–orinaseafoodversion,stuffedwithfishorbutterylocaleel. Essentially an “open sandwich”,Neapolitan pizza disembarked toUS in the late 19thand early 20th centurieswhen Italian immigrants, as didmillions of Europeans,werecoming toNewYork,Trenton,NewHaven,Boston,ChicagoandSt. Louis.Flavorsandaromas of humble pizzas sold on the streets by Neapolitan pizzaioli to the countryfellows began to intrigue the Americans. Neapolitan immigrant Antonio Pero beganmakingpizza forLombardi’sgrocerystore,whichstillexists inNY’sLittle Italyand in1905 Mr. Gennaro, Lombardi’s owner, was licensed to open the first Pizzeria inManhattanonSpringSteet.DuetothewideinfluenceofItalianimmigrantsinAmericanculture, the U.S. has developed regional forms of pizza, some bearing only a casualresemblance to the Italian original. Chicago has its own style of a deep-dish pizza.Detroitalsohasitsuniquetwice-bakedstyle,withcheeseallthewaytotheedgeofthecrust, andNewYorkCity's thin crustpizzasarewell-known. St. Louis,Missouriusesthin crustsand rectangular slices in its localpizzas,whileNewHaven-stylepizza is athin crust variety that does not include cheese unless the customer asks for it as anadditionaltopping.Maintext:Beingcurious,theauthorsbeganlookingintothesecretsofmakingpizza.Rulenumberone,asItalianstoldthem,wastoalwayslookforapizzeriawithawoodburningoven(notwithanelectricalone).Goodpizzeriasareproudoftheir“forno”(“oven”inItalian),inwhich you can seewith your own eyes thewhole processof baking. Thepizzaioloformsadoughdisc,coversitwithtopping,placesthefreshpizzaontopofawoodenoraluminumspade,andfinally transfers it intotheoven. Acoupleofminutes later it issittinginfrontofyou,coveredwithmouth-wateringbubblesofcheese,encouragingyoutoconsumeitandwashitdownwithapitcherofgoodbeer.TheauthorsreceivedusefuladvicefromafriendlypizzaiolowhowasworkinginalocalRoman pizzeria, frequently visited by them when they lived in that neighborhood:“Alwayscomeforapizzaeitherbefore8p.m.orafter10p.m.,whenthepizzeriaishalf-empty.”Theadvicewasalsoconfirmedbyoneofthepizzeria’sfrequentvisitors:abig

Figure4.Pizzamarinara

greycat.Whenthepizzeriawasfull,thecatwouldleave,anddidnotshowanyinterestinwhatwasonthepatron’splates.

The reason for thisadvicewasverysimple–oven capacity. As thepizzaiolo explained, 325- 330°C1is the optimaltemperature for Romanpizza baked in a woodburningovenwithafire-brick bottom. In thiscase,a thinRomanpizzawill be done in 2

minutes. Thus, even putting two pizzas into the oven, the pizzaiolo can serve 50-60clientswithinanhour. Duringpeakhours,aboutonehundredcustomersfrequentthepizzeriaandat least tenclientsarewaiting fora take-outpizza.Tomeet thedemand,thepizzaioloincreasesthetemperatureintheovenupto390°C,andpizzas“flyout”ofthe oven every 50 seconds (hence, each one requires a “baking time” of around 1½minutes).However, theirquality isnot thesame: thebottomandthecrustarea little“overdone”(slightlyblack),andthetomatoesarealittleundercooked.Since it isnotalwayseasyto findapizzeriawithabrickoven, letus takea lookwhatadvantagesithascomparedtoanelectricovenandwhetherthereisawaytoimprovethelattertoproduceadecentpizza.

Toillustratethephysicalprinciplesinvolvedinbakingpizzas,letusconsideracommonexampleofhowheatis transferred. Imagine when you were a child andhad a fever, but no thermometer at hand. Yourmother would put her hand on your forehead andquicklysay:“youhaveahightemperature,noschoolfor you tomorrow”. To investigate this processscientifically, we start with simplifying the problem.Let us imagine that your mom is touching yourforeheadwithherownforeheadratherthanherhand.In that case, if the temperature of your foreheadwould have been 38°С, and yourmother’s 36°С, it isclear by the symmetry of the problem that the

temperatureattheinterface(T0)betweenthetwoforeheadswillbe37°С,andthatyourmotherwouldfeeltheflowofheatcomingfromyourforehead(theactualtemperaturedistributionintimeisshowninFig.6).

1 Obviously, the temperature depends on theway the dough was prepared and stored. The pizzaioloAntonio prepares the dough well in advance – 24 hours before baking the pizza. After mixing allingredientsand kneading itwell, he leaves the dough “to rest” for several hours,and thencuts it intopiecesandformsball-shapedportions.ForNaples-stylepizzatheportionis180-250grams,forRoman-style less. Theseportionsareusedforasinglepizza.Thenheputsthedoughballs intowoodenboxes,wherethedoughwillberisefor4-6hours.Afterthatitisreadytobebakedorplacedinarefrigeratorforlateruse.

Figure5.TwomodernpizzaiolosinRomeinfrontofabrickpizzaovenandpizzamagherita

Figure6.Temperatureprofilewithinmother’sandchild’shead-0.1s,1s,10s,and60saftertheymadecontact.

Nowletusassumethatyourmother’sheadismadeof steel, and her temperature is the same – 36°С.Intuitively, it is clear the temperature at theinterfacewilldecrease, letus say, to36.3°С.This isrelated to the fact that the steel will draw off theheatfromtheinterfaceregioneffectivelytoitsbulk,since its heat conductivity is large. It is also clearthat this removal becomes more efficient, when asmaller amount of heat needs to be drawn awayfrom the interface region (i.e., the heat removalincreases when the specific heat capacity of the“mother’shead”materialdecreases,anillustrationis

showninFig.7).Let us now analyze the process of pizza baking more scientifically. We start byremindingthereaderofthemainconceptsofheattransfer[1]. Whenwespeakabout“heat”,weusuallyhaveinmindtheenergyofasystem(likethemother’shead,theoven,or the pizza itself) associatedwith the chaoticmotionof atoms,molecules and otherparticlesitiscomposedof.Weinheritedthisconceptofheatfromthephysicsofapastera. Physicists say that heat is not a function of the state of a system, its amountdepends on theway the system achieved this state. Likework, heat is not a type ofenergy, but rather a value convenient to use to describe energy transfer [2,3]. Theamountofheat,necessarytoraisethetemperatureofamassunitofthematerialbyonedegree,iscalledaspecificheatcapacityofthematerial:

𝑐 = ∆$%∆&

(1)

HereMisthemassofthesystemand∆𝑄isthequantityofaheatrequiredforheatingthesystembyatemperature∆𝑇.FromthisexpressionitisclearthattheheatcapacityismeasuredinJ´kg-1´K-1inSIunits.

In thecaseofa thermalcontactbetweenthetwosystemswith different temperatures, the heat will go from thewarmersystemtothecoolerone.Theheatfluxdensity𝑞istheamountofheat∆𝑄that flows throughaunitareaperunittimeinthedirectionoftemperaturechange:

𝑞 = ∆$*∆+ (2)

In the simplest case of a homogenous non-uniformlyheatedsystem,usingEq.(1),onefinds

𝑞 = ,%∆&*∆+

= 𝑐𝜌 (∆/)1

∆+2∆&∆/3 = −𝜅 6&

6/, (3)

where𝜌is the mass density2.Assuming that Δx is small, we identified the value inparentheses as the derivative of the temperature by the coordinate x and took intoaccount the fact that the temperature decreases in x-direction (see Fig. 8). In thegeneralcase,qisthevectorandthederivativeinEq.(3)isreplacedbythegradient𝛁𝑇,whichdescribestherateoftemperaturechangeinspace.Thecoefficient𝜅inEq.(3)is2Onecanalsousethisformulatoeasilyfindtheheatlossthroughthewallsofoneshouseduringacoldwinter.Inthisstationarysituation,thetemperaturedistributiondoesnotchangewithtime.

Figure7.ThesameasFig.6,butwithacoolersteel“head”.

Figure8.Heatflowinasmallcylinderfromhot(T0+DT)tocold(T0).Notice,thetemperaturedecreasesfromlefttoright!

thethermalconductivity,whichdescribestheabilityofamaterialtotransferheatwhena temperature gradient is applied3. Eq. (3) expresses mathematically the so-calledFourier'slaw,whichisvalidwhenthetemperaturevariationissmall.Next, letusanalyzehowa “temperature front”penetratesamedium from its surface,when a heat flow is supplied to it (see Fig. 8). Assume that during time t thetemperatureinthesmallcylinderoftheheight𝐿(𝑡)andcross-section𝑆haschangedby∆𝑇.4LetusgetbacktoEq.(3)andrewriteitbyreplacing𝛥𝑥by𝐿(𝑡):

,?@(+)∆&+

= 𝜅 ∆&@(+)

. (4)

SolvingEq.(4)withrespecttothelength𝐿(𝑡)onefinds:

𝐿(𝑡)~CD+,?= √𝜒𝑡, (5)

i.е.,thetemperaturefrontentersthemediumbythesquare-rootlawoftime.Thetimeafter which the temperature at depth L will reach a value close to the one of theinterface depends on the values𝜅, с, and𝜌. The parameter𝜒 = 𝜅/𝑐𝜌is called thethermal diffusivity or coefficient of temperature conductivity and the heating time ofthewholevolumecanbeexpressedinitsterms:𝜏~𝐿I/𝜒.Of course,our consideration of theheat penetration problem into amedium is just asimple evaluation of the value𝐿(𝑡). A more precise approach requires solution ofdifferentialequations.Yet,thefinalresultconfirmsourconclusion(5),justcorrectedbyanumericalfactor:

𝐿(𝑡) = √𝜋𝜒𝑡. (6)Now that we know how heat transfer works, let us get back to the problem ofcalculatingthetemperatureattheinterfacebetweentwosemi-spaces:ontheleftwithparameters𝜅K,с1,𝜌KandtemperatureT1at−∞,andontherightwithparameters𝜅I,с2,𝜌IandtemperatureT2at+∞.Letusdenotethetemperatureat theboundarylayerasT0.Theequationoftheenergybalance,i.е.,therequirementoftheequalityoftheheatflowing from thewarm, right semi-space through the interface to the cold, left semi-space,canbewrittenintheform

𝑞 = 𝜅K&NO&PQRSN+

= 𝜅I&PO&1QRS1+

. (7)

Herewesimplifiedtheproblemassumingthatall temperaturechangeshappenat thecorrespondingtemperaturedependentlength(6).Solvingthisequationwithrespectto𝑇Tonefindsthat

𝑇T = &NUV1N&1KUV1N

, (8)where

𝜈IK =D1DNCSNS1= C

D1с1?1DNсN?N

. (9)

3Thedefinitionofthethermalconductivity𝜅 = 𝑐𝜌 (∆/)1

∆+usedinEq.(3)requiresaclarification:Whileour

simplifiedderivationsuggestsageometrydependence,weemphasizethatinrealityitisdeterminedonlybymicroscopicpropertiesofthematerial.4Thisprocessisnotstationaryanymoreandtheflowqnotconstant,sincetheheatwillbepartiallyusedfortheheatingofthecylindermaterial.Therefore,unlikeinthestationaryprocess,therateoftemperaturechangedT/dxinthemediumisafunctionofspaceandtime.

Onenotices, that timedoesnotenter inexpression(8)(i.е., the interfacetemperatureremains constant in the processof the heat transfer, see Figs. 6,7&9). In the case ofidenticalmediawithdifferent temperaturesonecaneasily find:𝑇T =

&NU&1I. This is the

quantitativeproofoftheintuitiveresponseweprovidedinthebeginningofthearticlefor the temperature 37°С of the interface between themother’s hand and the child’sforehead. If the mother’s hand would be made of steel,𝜈IK ≫ 1 and𝑇T ≈ 𝑇I , itstemperature would remain almost unchanged after contact with the hot forehead,meaningthatshewouldnotbeabletonoticethechild’sfever.Finally,wearereadytodiscusstheadvantagesofthebrickoven.Letusstartfromthecalculationofthetemperatureattheinterfacebetweenthepizzaplacedintothebrickoven and its heated baking surface. All necessary parameters are shown in the tablebelow:

Property

Material

Heat capacity с [J/(kg´K)]

Thermal conductivity k [W/(m´K)]

Mass density r [kg/m3]

Temperature conductivity c [m2/s]

𝝂𝟐𝟏5

dough6 2-2.5´103 0.5 0.6-0.8´103 2.5-4.2´10-7 1 food grade steel (Х18Н10Т)

4.96´102 18 7.9´103 4.5´10-6 0.1

fire brick 8.8´102 0.86 2.5´103 4.0´10-7 0.65 Water (@25C) 4.2´103 0.58 1.0´103 1.4´10-7 0.2

Assuming the initial temperature of the pizza dough as𝑇T6` =20°С, and the temperature inside theoven, аsourpizzaiolo claimed, being about𝑇Ka` =330°С, we find forthetemperatureattheboundarylayerbetweentheovensurfaceandpizzabottom

𝑇Ta` = 330 СT + 0.65 ⋅ 20 СT

1.65 ≈ 208 СT .

As we know from the words of the same pizzaiolo, apizza is perfectly baked in two minutes under theseconditions.Letusnowrepeatourcalculations for theelectricovenwith its baking surface made of steel. For an electricoven the ratio will be𝜈i` = 0.1, and if heated to thesametemperatureof330°С,thetemperatureatthebottomofthepizzawillbeequalto

330 СT + 0.1 ⋅ 20 СT

1.1 ≈ 300 СT .

Thatistoomuch!Thepizzawill justturnintocoal!Thisinterfacetemperatureisevenmuchhigher than inNaples’ pizzerias,whereoven temperatures between 400-450°Carecommon.

5 Fordough,steel,andbrick,material“2”isdough.Forwater,material“1”issteel.6Thematerialparametersfordoughshouldbeconsideredasanestimate.Itisclearthattheexactnumbersstronglydependonthetypeofflourandthefermentation/risingtimeofthedough(inthelatterprocess,thedoughisenrichedbygases,whichchangeitsdensity).

Figure9.Temperatureprofileinabrickovenwithpizzaatdifferenttimes.At60sthetopsurfaceofthepizzareaches100°C(redcircle).Hereweonlytakethermaldiffusionintoaccount.Evaporationandradiationareneglected.

Table1.Physicalpropertiesofdifferentmaterials,includingheatcapacity,thermalconductivity,densityandtemperatureconductivity

Well, letusformulatetheproblemdifferently.Letusassumethatgenerationsofpizzamakers,whowereusingwoodenspadetotransferpizzasintotheoven,areright: thetemperatureatthe(Roman)pizza’sbottomshouldbeabout210°С.Whatwouldbethenecessarytemperatureforanelectricovenwithsteelsurface?TheanswerfollowsfromtheEq.(8)withcoefficient𝜈i` = 0.1andsolvedwithrespectto𝑇Ki`whenthetemperatureatthebottomofthepizzaisthesameasinthewoodoven:𝑇Ti` = 𝑇Ta` . The result of this exercise shows that the electric oven should bemuchcolderthanthebrickone:𝑇Ki` ≈ 230 СT . Itseemsthat ifone isable to forget the flavorofburningwood,sizzlingdryairin thebrick oven and the other natural features, the problemwould be solved – let us setelectric stove controls to 230°C and in a couple ofminuteswe can take an excellentpizzaoutoftheoven.But:Isitthateasy?In order to answer this question, we first need to consider the second importantmechanism of a heat transfer: thermal radiation [4]. Its intensity, the amount ofradiationenergyarrivingeachsecondto1сm2ofsurfaceintheoven,isdeterminedbytheStefan-Boltzmannlaw:

𝐼 = 𝜎𝑇l, (10)where𝜎 = 5.67 ∙ 10Oo𝑊/(𝑚I𝐾l)istheso-calledStefan-Boltzmannconstant.A typical brick oven has a double-crownvault filled with sand, which is kept atalmost constant temperature. Its walls aswell as the bottom part, are also heated to𝑇Ka` = 330°C = 603K, meaning that thecomplete volume of the oven is “filled” byinfrared radiation.With a temperature thathigh,thisradiationbecomessignificant:thepizzahereiscontinuously“irradiated”fromboth sides by a “flow” of infrared radiationoftheintensity

𝐼a` = 𝜎(𝑇Ka`)l = 5.67 ∙ 10Oo(603)l= 7.5𝑘𝑊/𝑚I,

i.е., eachsecondanamountofenergyclosetothe0.75Jarrivesat1сm2ofpizza.7Here one should notice, that, in its turn, the pizza also irradiates out a “flow” of theintensity𝐼wxyyz = 𝜎(𝑇wxyyz)l.Sincethemajorpartofthebakingtimeisrequiredfortheevaporationofwatercontainedinthedoughandtoppings,wecanassume𝑇wxyyz = 𝑇{ =100℃ = 373𝐾,which results in a radiation intensity of𝐼wxyyz = 𝜎(𝑇{)l = 1.1𝑘𝑊/𝑚I,i.e.,15%oftheobtainedradiation,thepizza“returns”totheoven.Forthemuchlessheatedelectricoven,thecorrespondingamountofenergy,incidentto1сm2ofpizzasurface,ismorethantwiceless:

𝐼i` = 𝜎(𝑇Ki`)l = 5.67 ∙ 10Oo(503)l𝑊

𝑚I = 3.6𝑘𝑊/𝑚I,

7Hereweassumethatthepizzabehavesasablackbody.Inrealityitisslightlyreflective,reducingtheamountofheatitabsorbs.

Figure10.Heattransfermechanismsinthepizzaoven.

whilethereturnedradiationisthesame:1.1𝑘𝑊/𝑚I.Now it is a time to evaluatewhat amount ofheat 1 cm2of pizza receives per secondthroughitsbottom.Bythedefinitionitisdeterminedbytheheatflow(3)andinordertogetitsnumericvaluewewillevaluatethetemperaturegradientattheovensurfaceinthesamewayaswasalreadydoneinEq.(7):

𝑞(𝑡) = 𝜅𝑇K` − 𝑇T√𝜋𝜒𝑡

,

where𝑇K`is the temperature of the oven. One can see, that, contrary to the Stefan-Boltzmann radiation, the heat flux arriving into the pizza by means of a heatconductancedependsontime.Correspondingly,theamountofheattransferredto1cm2ofpizzainthiswayfromtheovenduringtimetisdeterminedby

𝑄(𝜏) = } 𝑞(𝑡)𝑑𝑡 =�

T2𝜅(𝑇K` − 𝑇T)�

𝜏𝜋𝜒.

Therefore,thetotalamountofheat,arrivingat1cm2ofpizzaduringtime𝜏,is

𝑄+`+(𝜏) = 𝜎[(𝑇K`)l − (𝑇{)l]𝜏 + 2𝜅(𝑇K` − 𝑇T)C�RS. (11)

Thisvalueisusedtoheat1cm2ofpizzafromthedoughtemperature𝑇I6 =20°Сtotheboilingtemperatureofwater𝑇{ =100°С:

𝑄�iz+ = 𝑐6 𝜌6 𝑑�𝑇{ − 𝑇I6�. Yet, this is not all. During the process of baking the perfect pizza we apparentlyevaporatewater fromthedough, tomatoes,cheese,andother ingredients.Weneedtotake the requiredenergy for this intoaccountaswell. If oneassumes that thewatermassfractionαevaporatesfromthedoughandalltoppingonegets:

𝑄{`x� = αL𝜌az+i�𝑑. Heredisthethicknessofthepizza,whichweassumetobed=0.5cm,while𝐿 = 2264.76J ∙ 𝑔OKisthelatentheatofevaporationforwater.Collectingboththesecontributionsinone,wecanwrite

𝑄+`+ = 𝑄�iz+ + 𝑄{`x� = 𝑐6`𝜌6`𝑑�𝑇{ − 𝑇I6� + αL𝜌az+i�𝑑. (12)EquatingEqs.(11)and(12)onefindsthefinalequationdeterminingthe“bakingtime”ofpizza:

𝜎[(𝑇K`)l − (𝑇{)l]𝜏 + 2𝜅(𝑇K` − 𝑇T)C�RS= 𝑐6`𝜌6`𝑑�𝑇{ − 𝑇I6� + αL𝜌az+i�𝑑. (13)

In order toobtain a realistic answer for the baking time, it is important to know theamountofwater,which isevaporatedduringthebakingprocess. A typicalrecipe forpizzaMargaritacallsfor240gofdoughand90goftoppings(consistingoftomatoesandmozzarella). Thedoughcontainsaboutone-thirdofwaterandthetoppings80%(therest is mostly fat from the cheese). Together with a weight loss of 30g, a goodassumptionisa20%lossofwater,i.e.α=0.2.Usingthiswiththevaluesofspecificheatcapacityanddensityfordoughfromthetableabove,onefindsthat𝑄+`+ = (70 + 226)𝐽/𝑐𝑚I ,which gives for the baking time in thewood oven𝜏a` ≈ 125𝑠. For the electricovenananalogouscalculationresultsinanalmost50%longertime𝜏i` ≈ 170𝑠.Weseethat we have succeeded to reproduce the value disclosed to us by our pizzaiolo: 2minutes for baking in a wood oven. The result of an attempt to bake a pizza in theelectricovenwillbethementionedunbalancedproduct.

UsingEq.(8)onecaneasilyfindthatthetemperatureattheinterfacebetweenthepizzaand oven surface reaches 240°С, when the temperature in wood fired brick ovenincreases to 390°С. Replacing correspondingly𝑇Tin Eq. (13) one can find the bakingtime under these extreme conditions to be approximately 82 seconds, hence theproductivityoftheovenincreasesbyalmost50%!A final “trick” disclosed to us is the importance for pizzas with water-rich toppings(eggplants,tomatoesslices,orothervegetables).Inthiscase,theexpertfirstbakesthepizza in theregularwayontheovensurface,butwhenthepizza’sbottomisreadyheliftsitwiththewooden/aluminumspadeandholdsitelevatedfromthebakingsurfaceforanotherhalfminuteormoreinordertoexposethepizzatojustheatirradiation.Inthiswaytheyavoidburningthedoughandgetwellcookedtoppings.Certainly,asisroutinelydoneinphysics,inordertogettothecoreofthephenomenon,we examined only the simplest model here (in particular, we ignored the thirdmechanismoftheheattransfer:convection,whichwecanassumetohaveonlyasmalleffect.SeeFig.~10),whichcapturestheessentialphysicalprocesses.As a final note, we remark that it is difficult to build a classic brick oven, andmanycustomers do not appreciate the difference between an excellent and decent pizza.These are the reasons why engineers invent all sorts of innovations: for example,inserting a ceramic bottom made of special ceramics to imitate the bottom of brickovens inamodernprofessional electricoven.Tobakeapizzaevenly, rotatingbakingsurfaces are available – convection ovens emulate the gas flows in brick ovens, andmanyotherthings.But,thedryheatandthesmellofwoodintraditionalfirebrickovensremaintheidealwaytobaketheperfectpizza.AcknowledgementsWewanttoexpressourgratitudetotheRomanpizzaiolosAntonioandVincenzoforliftingtheveil to some secrets of the pizza baking art. Special thanks goes toMr. ZhengZhou, a youngstudent of a high school in Shanghai, attending the lectures of one of the authors (AV) formakingseveralvaluablecomments,whichhelpedustoimprovethefinaltextofthisarticle.References:[1] Theodore L. Bergman, Adrienne S. Lavine, Frank P. Incropera, David P. DeWitt,

IntroductiontoHeatTransfer,6thEdition,Wiley(June13,2011)[2] Richard P. Feynman, Robert B. Leighton, Matthew Sands, The Feynman Lectures on

Physics,Vol.I:MainlyMechanics,Radiation,andHeat,TheNewMillenniumEdition,BasicBooks(October4,2011)

[3] FrederickReif,FundamentalsofStatisticalandThermalPhysics,WavelandPrInc(2008)[4] L. D. Landau, E.M. Lifshitz, StatisticalPhysics.Vol.5, 3rd ed., Butterworth-Heinemann

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