EXAMLE HERMAL DAMING wrk in air sealed utlet A BIYLE UM WIH HE OULE EALED When the pistn is depressed, a fixed mass f air is cmpressed mechanical wrk is dne he mechanical wrk dne n the air is cnerted t heat the air temperature rises A temperature difference between the air and its surrundings induces heat flw entrpy is prduced he riginal wrk dne is nt recered when the pistn is withdrawn t the riginal pistn aailable energy is lst Md im Dyn yst hermal damping example page
MODEL HI YEM GOAL the simplest mdel that can describe thermal damping (the lss f aailable energy) ELEMEN WO KEY HENOMENA wrk-t-heat transductin a tw prt capacitr represents therm-mechanical transductin entrpy prductin a tw prt resistr represents heat transfer and entrpy prductin BOUNDARY ONDIION Fr simplicity assume a flw surce n the (fluid-)mechanical side a cnstant temperature heat sink n the thermal side Md im Dyn yst hermal damping example page
A BOND GRAH I A HOWN Q(t) f d/dt (fluid) mechanical dmain gas d gas /dt thermal dmain 0 R e d /dt AUAL ANALYI he integral causal frm fr the tw-prt capacitr (pressure and temperature utputs) is cnsistent with the bundary cnditins and with the preferred causal frm fr the resistr Md im Dyn yst hermal damping example page 3
ONIUIE EQUAION Assume air is an ideal gas and use the cnstitutie equatins deried abe R c exp R c + exp Assume Furier s law describes the heat transfer prcess Q ka l ( - ) Md im Dyn yst hermal damping example page 4
ANALYI Fr simplicity, linearize the capacitr equatins abut a nminal perating pint defined by and ( ) c R ( ) + c R Inerse capacitance + c R equality f the ff-diagnal terms (the crssed partial deriaties) is established using mr Linearized cnstitutie equatins ( ) + c R where -, -, - (, ), - (, ) Md im Dyn yst hermal damping example page 5
NEWORK REREENAION he linearized mdel may be represented using the fllwing bnd graph - F 0 / his representatin shws that / / in the isthermal case ( 0) the fluid capacitance is fluid in the cnstant-lume case ( 0) the thermal capacitance is thermal the strength f therm-fluid cupling is his uses the cnentin that the transfrmer cefficient is fr the flw equatin with utput flw n the utput pwer bnd and hence thugh causal cnsideratins may require the inerse equatins Md im Dyn yst hermal damping example page 6
ALERNAIELY It may be useful t express the parameters in term f easily-measured reference ariables and as fllws - F 0 /mr / /mr his representatin shws that the strength f the cupling is mr prprtinal t the (nminal) gas lume inersely prprtinal t the mass f gas and mr mr Md im Dyn yst hermal damping example page 7
REIOR EQUAION he tw-prt resistr cnstitutie equatins are l ka Q l ka Q Linearize the resistr cnstitutie equatins abut a nminal perating pint defined by and l ka his is in cnductance frm, Ge f Nte that this cnductance matrix is singular 0 G this is because bth entrpy flws are assciated with the same heat flw Md im Dyn yst hermal damping example page 8
LINEARIZE ABOU ZERO HEA FLOW If the tw nminal perating temperatures are equal,, the linearized cnstitutie equatins are l ka his simple frm can be represented by an equally simple bnd graph R l/ka his fllws the usual cnentin f writing the resistr parameter in resistance frm Md im Dyn yst hermal damping example page 9
AEMBLE HE IEE LINEARIZED BOND GRAH f - gas gas gas F 0 / e / / R l/ka Nte the sign change n the capacitr thermal prt (t aid a superfluus 0-junctin) ausal assignment indicates a first-rder system ime-cnstant is determined by thermal (cnductin) resistance and thermal capacitance Gas pressure is determined by fluid capacitance and (reflected) thermal capacitance and resistance Md im Dyn yst hermal damping example page 0
INLUDE ION INERIA BOND GRAH I F m pistn /A pistn gas F 0 / gas - ut_f_gas e ambient / / R l/ka ausal analysis indicates a third-rder system capable f resnant scillatin In this mdel the nly damping is in the thermal dmain heat transfer, entrpy flw Md im Dyn yst hermal damping example page
UMMARIZING HE GA ORE ENERGY It als acts as a transducer because there are tw ways t stre r retriee this energy tw interactin prts energy can be added r remed as wrk r heat he energy-string transducer behair is mdeled as a tw-prt capacitr just like the energy-string transducers we examined earlier Md im Dyn yst hermal damping example page
IF OWER FLOW IA HE HERMAL OR, AAILABLE ENERGY I REDUED the system als behaes as a dissipatr he dissipatie behair is due t heat transfer Gas temperature change due t cmpressin and expansin des nt dissipate aailable energy If the walls were perfectly insulated, n aailable energy wuld be lst, but then, n heat wuld flw either Withut perfect insulatin temperature gradients induce heat flw Heat flw results in entrpy generatin Entrpy generatin means a lss f aailable energy HE EOND LAW Md im Dyn yst hermal damping example page 3
DIUION ALL MODEL ARE FALE It is essential t understand what errrs ur mdels make, and when the errrs shuld nt be ignred It is cmmnly assumed that mdeling errrs becme significant at higher frequencies nt s! mpressin and expansin f gases is cmmn in mechanical systems Hydraulic systems typically include accumulatrs (t preent er-pressure during flw transients) he mst cmmn design uses a cmpressible gas mpressin and expansin f the gas can dissipate (aailable) energy Md im Dyn yst hermal damping example page 4
his dissipatin requires heat flw, but heat flw takes time Fr sufficiently rapid cmpressin and expansin, little r n heat will flw, and little r n dissipatin will ccur he simplest mdel f a gas-charged accumulatr may justifiably ignre thermal damping hat is an eminently reasnable mdeling decisin but that mdel will be in errr at lw frequencies nt high frequencies HI I A GENERAL HARAERII OF HENOMENA DUE O HERMODYNAMI IRREERIBILIIE Md im Dyn yst hermal damping example page 5