Numercal ransent Heat Conducton Experment OBJECIVE 1. o demonstrate the basc prncples of conducton heat transfer.. o show how the thermal conductvty of a sold can be measured. 3. o demonstrate the use of fnte dfference to solve transent heat conducton problems. BACKGROUND he prmary law that descrbes conducton heat transfer s Fourer's Law of Heat Conducton. Fourer's Law s based upon the observaton that the conductve heat flux s drectly proportonal to the negatve of the temperature gradent or q & x - x (1) Introducng the thermal conductvty as the constant of proportonalty, we then have the equaton q & x = - k x () In ths experment we wll utlze Fourer's Law to study the problem of transent, one dmensonal heat conducton n a cylnder and to use the law n determnng the thermal conductvty of a sold. A cylndrcal element whch s embedded wth several thermocouples s heated at one end by an electrc hot plate and cooled at the other end by flowng water. he sde of the cylnder s very well nsulated so that the heat conducton s assumed to be one dmensonal. A schematc of the apparatus s shown below.
Fgure 1. Expermental Apparatus for Cylndrcal Element Water Flow hermocuples Hot Plate We consder the transent problem n whch the cylnder begns at some constant, unform temperature and then suddenly the hot plate s turned on so that a heat flux s mposed at the lower boundary. Our descrbng equaton for conservaton of energy balances the nternal energy change wth the axal heat conducton. Hence, n dfferental form we wrte = α t z (3) where α s the thermal dffusvty of the cylnder materal. One way to solve ths equaton s to dscretze the space doman, wrte t n fnte dfference form, and solve the subsequent system of algebrac equatons for the dscrtzed temperatures. If we use a second order correct approxmaton, the fnte dfference form of Eq. (3) becomes - t (j-1) = α (?) +1 + (?) ( z) - (?) -1 (4) he subscrpt on the represents the space (or z) dscetzaton whle the superscrpt represents the tme dscetzaton. Note that n Eq. (4) we have not specfed the tme dscretzaton (j or j-1) for the spatal dervatve. here are two choces for the tme dscretzaton for the temperatures n the spatal dervatve. hey could be evaluated at the prevous tme step, j-1, or at the current tme step, j. he algorthm s called explct f the temperatures are evaluated at the prevous tme step, and clearly ths makes the algebrac system very easy to solve snce Eq. (4) can be solved drectly for
wth no couplng to the other spatal nodes (+1 and -1). However, the explct approach does not always lead to a stable soluton (can you say blow-up) and n fact stablty s only guaranteed when α t ( z) < 0.5 Snce for most materals α s of the order of 10-5, ths stablty crtera often leads to unacceptable tme steps or spatal grds. he mplct algorthm, when the spatal dervatve s evaluated at the current tme step, does not have ths stablty problem, but does requre smultaneous soluton of the spatal node equatons. Fortunately, Mcrosoft Excel s a powerful spreadsheet tool that can carry out these smultaneous calculatons. Hence, for a spatal doman wth N spatal nodes, we would have N smultaneous equatons of the form - t (j-1) = α +1 + ( z) - -1 (5) to be solved for the 's. In steady state the axal temperature profle should be lnear whch confrms Fourer's Law. he steady state heat transfer s determned by measurng the mass flow rate and temperature change of a coolant stream whch passes over one end of the element, or q& = m& c p ( - ) out n coolant (6) hen the thermal conductvty can be calculated by q/a & cross-secton,element k = slope of vs z graph (7) PROCEDURE #1 (For sectons dong heatng up experment) 1. Makng sure that all coolant sample valves are closed turn on the water supply to the apparatus table.
. Open the valves to provde coolng to the hot plate assembles. Valves should be turned so as to pont at ether coolant n or coolant out. 3. Check to make sure all panel swtches and the hot plate swtches are n the off poston. Plug n the cable and turn on the power to the apparatus table. 4. urn the panel swtch for the hot plates (3 UNI 4) to the on poston. 5. Record the temperatures for the ten thermocouples for Unt 4. hese wll serve as your ntal temperatures for the transent conducton process. 6. Immedately set the hot plate swtch for Unt 4 to approxmately 50 C and start the stop watch. 7. At ten mnute ntervals record the temperatures for the ten thermocouples for Unt 4. Also record the tme requred to record these temperatures. After one hour the system should have reached steady state, whch can be confrmed by the lnearty of the temperature data wth poston or lttle change n the slope calculaton. 8. At steady state the energy delvered to the coolng water wll be determned. Place the empty beaker on the scale and zero the scale. Usng the coolant sample valve allow water to flow nto the beaker, measurng the tme wth a stop watch. When the beaker s nearly full turn the coolant sample valve to off. Record the tme. Place the flled beaker on the scale and record the mass. 9. At steady state record thermocouple readngs from channels 4&5 on unt 5. PROCEDURE # (For sectons dong coolng down experment) 1. If your secton s runnng a coolng down transent then the system s already at the heated steady state. At steady state the energy delvered to the coolng water wll be determned. Place the empty beaker on the scale and zero the scale. Usng the coolant sample valve allow water to flow nto the beaker, measurng the tme wth a stop watch. When the beaker s nearly full turn the coolant sample valve to off. Record the tme. Place the flled beaker on the scale and record the mass.. At steady state record thermocouple readngs from channels 4&5 on unt 5. 3. Record the temperatures for the ten thermocouples for Unt 4. hese wll serve as your ntal temperatures for the transent conducton process as well as the steady state temperatures from whch to determne the thermal conductvty. 4. Immedately turn the hot plate swtch for Unt 4 off and start the stop watch.
5. At ten mnute ntervals record the temperatures for the ten thermocouples for Unt 4. Also record the tme requred to record these temperatures. After one hour the system should have completely cooled down, whch can be confrmed by a constant temperature along the element. 6. urn the panel swtch for the hot plates (3 UNI 4) to the off poston. 7. Check to make sure all panel swtches and the hot plate swtches are n the off poston. Unplug n the cable and turn off the power to the apparatus table. 8. Close the valves to provde coolng to the hot plate assembles. 9. urn off the water supply to the apparatus table. Your data wll be entered on an Excel spreadsheet smlar to that shown below. ME 41 One Dmensonal, ransent Heat Conducton Experment Materal hermal Dffusvty : 1.17E-04 m^/s /C Dstance: 0.046 m emperature emperature Locaton Expermental Numercal Locaton Expermental Numercal 1 1 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 me = 0 sec. me = 600 sec. d/dx= d/dx= Steady State Calculaton of hermal Conductvty Coolant Mass: gm Coolant n: C me: sec Coolant out: C Mass Flow: kg/sec Coolant avg: C Water Cp: J/kg K Heat Flow: W hermal Conductvty: W/m K Whte cells ndcate data nput by the student. Lghtly shaded cells have calculaton equatons suppled.
DAA ANALYSIS 1. Graph both the expermental temperature and the numercal temperature versus poston at three dfferent tmes (early tme, moderate tme, and steady state). Show a sample calculaton for the heat suppled at steady state by the hot plate to the element. 3. Show a sample calculaton for the thermal conductvty of the element. SUGGESIONS FOR DISCUSSION 1. How do the numercal and expermental temperatures compare? Explan any dfferences or trends n the comparson.. What sort of shape does the temperature dstrbuton have at early tmes? How does ths compare wth what s predcted by analytcal solutons? 3. Based on the thermal conductvty, what would you suppose the element to be made of? 4. How could you check the one-dmensonalty of the experment?
able 1 Dameter (n.) Length (n.) 11 Poston of frst thermocouple from hot plate (n.) 1 Dstance between thermocouples (n.) Cylndrcal Element Unt 4 3 8 15 16 31 3 Unt 5 hermocouple Channels Coolant n: Channel 5 Coolant out: Channel 4