Numerical modelization by finite differences of a thermoelectric refrigeration device of double jump". Experimental validation.

Transcription

1 Numercal modelzaton by fnte dfferences of a thermoelectrc refrgeraton devce of double jump". Expermental valdaton. A. Rodríguez, J.G. Ván, D. Astran, Dpto. Ingenería Mecánca, Energétca y de Materales. Unversdad Públca de Navarra, Pamplona, Span Tel , e-mal: van@unavarra.es Abstract We have been developed a mathematcal model to study the operaton of a thermoelectrc refrgerator devce wth double jump. The computatonal model solves both thermoelectrcty and heat transfer equatons. The model calculates the value of the nternal temperature and the COP of the refrgerator. In an expermental phase a prototype was bult to adjust and valdate the computatonal model, and later to optmze the expermental applcaton. The model allows smulatng properly the system n both transent and steady states by usng the fnte dfferences numercal method. We have studed the nfluence of the dsspater of the Pelter module n both hot and cold sde, and we have made a comparson between a double jump and a sngle jump devce. Nomenclature A Area (m 2 ) C Thermal capacty (J K 1 ) COP Coeffcent of performance k Thermal conductvty (W m 1 K 1 ) l Length (m) Q Heat flux (W) q Rate of nternal generaton of heat (W m 3 ) R Thermal resstance (K W 1 ) T Temperature (ºC) T Temperature n next step (ºC) V Voltage (V) Greek letters T Temperature gap (K) τ Tme (s) Superscrpts, subscrpts dssp Dsspater Exp Expermental Node ns Insulaton J Joule effect j Node j Mod Model Introducton Most of the applcatons were Pelter modules are used are coolng applcatons. These devces allow warmng or keepng constant the temperature of objects where the varaton of the temperature s mportant. The thermoelectrc systems are used n applcatons where the power range s from mw to hundreds of W as shown n references [1] and [2]. Some examples are the mltary ndustry, aerospace applcatons, food coolng, laboratory equpment and medcal equpment. Some of these applcatons are consdered n references [3], [4] and [5]. The thermoelectrc technology offers more relablty reducng the polluton. The case of study s the calculaton of the dfferent temperatures as a functon of the tme n a thermoelectrc refrgerator. We wll use a confguraton n cascade of the coolng system, whch mght be formed by 3 Pelter modules. Two of these modules wll be n parallel and the other n seres. In the numercal model created the user mght fx the next varables: Room temperature Voltage suppled to the Pelter modules Tme step n the numercal method used. The smulaton tme duraton. Our model allows usng dfferent numercal methods to calculate the temperatures. The user wll receve the errors for each method, so they can be compared and check the convergence order, as well our study allows calculatng the transent state of a double jump refrgerator before reachng the steady state. Objectves The prncpal objectves are the followng: Development of a computatonal model to smulates the operaton of a double jump thermoelectrc refrgerator. Desgn and constructon of a prototype of a double jump thermoelectrc refrgerator Valdaton of the computatonal model developed prevously Study of the nfluence of the hot and cold sde dsspater n the nternal temperature of the refrgerator and the COP. Make a study of the double jump refrgerator and sngle jump refrgerator.

2 Computatonal model The basc prncple of the numercal approxmaton to the heat transfer problems s the substtuton of the dfferental equaton for the contnuous temperature n a heat conductor sold by a fnte dfference equaton that has to be fulfllng of each ponts of the sold. If a complex geometry s subdvded n a nodal net, t has to be fulfllng by the mplct formulaton the followng expresson n each node : ' ' t j t C ' + q = ( t t ) (1) R δτ j j The term q, rate of nternal generaton of heat by the node. Thus, the node future temperature s determned usng expresson (2): ' ' ' t j t q t = t + δ τ + j RjC C (2) Where the expresson of the thermal resstances between nodes and j are: lj R j = (3) ka j If al the resstances of the net are known and the capactes of all the cells nvolved, thus, wth a dstrbuton of the temperature at the startng tmng, equatons lke (2) can be wrtten for all the nodes. The result s that several equatons for future temperatures can be obtaned t of the nodal ponts as a functon of the future temperatures of the nodes around, once a tme step s chosen δτ. In order to obtan the thermoelectrc refrgerator COP there are several analytcal expressons as shown n reference [6], where s needed to know the temperatures of the Pelter module as boundary condtons. Mathematcal models (such as the developed n [7] or n ths paper) calculate the COP of the thermoelectrc refrgerator wthout knowng the temperatures of the module faces as an nput parameter. In Fg. 1 s shown the electrc thermal analogy of the problem to solve. The nodes of the analogy represent the elements of the thermoelectrc refrgerator. Fg. 1. double jump electrc thermal analogy for a double jump refrgerator. Each of the component temperature s assocated to a node, so n the analogy showed n Fg. 1, the temperature nodes are the followng: Node 1: Room temperature Node 2: Inner refrgerator temperature Node 3: Dsspater temperature of the Pelter module cold sde. Nodes 4, 9 y 12: Cold sde temperatures of the Pelter module Nodes 5, 10 y 13: Inner temperatures of the Pelter module Nodes 6, 11 y 14: Hot sde temperatures of the Pelter module Node 7: Heat-extender temperature placed next to Pelter module 1. Node 8: Heat-extender temperature placed next to Pelter module 2 and 3. Node 15: Dsspater temperature of the Pelter module hot sde. Node 16: Room temperature Wth the prevous dscretzaton the followng matrx system can be wrtten: ' δτ [ M ] [ T ] = [ T ] + Q C (4)

3 The terms Q 4, Q 9, Q 12 represent the heat fluxes due to the Pelter effect n the cold sde of the modules, faced to the nteror of the refrgerator. The Joule effect heat fluxes are Q 5, Q 10, Q 13 and the Pelter effects n the hot sde of the modules are Q 6, Q 11, Q 14. The methods useds to solve the problem have been Euler, Heun and Runge-Kutta (frst order methods) and Runge-Kutta 4 order method. Expermental assembly we have desgned and bult a double jump thermoelectrc coolng prototype. The refrgerator used to nstall the thermoelectrc devce has a nner volume of 55*10-3 m3. The thermoelectrc refrgerator system s formed by: A heat dsspater for the hot sde of the Pelter module placed n parallel (exteror to the refrgerator) A heat extender that connects the more nternal Pelter to the more external Pelter modules n parallel. A heat dsspater for the cold sde ntroduced where the heat have to be removed. A thrd Pelter module n contact wth the cold dsspater. A fan for the hot sde dsspater. As a scheme n Fg. 2 can be a secton of the refrgerator we have developed: Cold dsspater: 375 x 155 x 15 mm Fns: 400 x 50 x 1.5 mm Once the refrgeraton operaton was checked and n order to mprove ts features, a wnd tunnel was ncorporated n the hot sde dsspater wth a fan to make forced convecton on the fns of the dsspater. The fan s Axal, (dmensons 120 x 120 x 30 mm), wth a power consumpton of 5W. Model valdaton In order to valdate the mathematcal model, the voltage of the Pelter modules and the room temperature could be changed. In Table 1 all the confguratons that were tested are lsted for each test, n the valdaton process. Test 1 Test 2 Test 3 Test 4 Pelter voltage nº 1 5 V 6 V 8 V 6 V Pelter voltage nº 2 5 V 6 V 5 V 4 V Pelter voltage nº 3 5 V 6 V 5 V 4 V Fan voltage 12 V 12 V 12 V 12 V T Room 273 K 273 K 303 K 287 K Table 1. Tests made n order to valdate the thermoelectrc refrgerator model. The valdaton tests were made n a clmatc test room n order to work wth a room temperature fxed and controlled dependng on the condtons of each test. After makng the tests n the laboratory wth the prototype some smulatons were made usng the computatonal model. As shown n Fg. 3 the model s adjusted to the expermental values wth hgh accuracy for both transent and steady states. Fg. 2 Scheme of the coolng system. The dsspaters, both hot and cold sde were bult from an alumnum slab n the UPNA workshop. Fns were placed n the hot sde dsspater to ncrease the nterchange surface and thus obtan a better heat dsspaton. In the cold sde dsspater no fns were placed, n order to make the devce lghter and not to lose useful space n the nteror of the tank. The dmensons of the dsspaters and fns are: Hot dsspater: 400 x 155 x 15 mm Fg. 3. Comparson between prototype and model results. The greater dfference between the model and expermental temperatures s 1.5 ºC what makes accuracy very acceptable, as we can see n the Table 2.

4 dsspater has the same slope for all the room temperatures studed. Table 2. Comparson of the temperatures n the steady state. Results and dscusson One of the system varables wth hgher nfluence n the thermoelectrc refrgeraton s the thermal resstance of the dsspaters placed for the Pelter modules, as can be seen n references [8] and [9]. The model relates the thermal resstance of the dsspaters wth the smulated values of nternal temperature of the refrgerator what allow us to study the nfluence n the refrgerator operaton. Influence of the hot sde dsspater In ths pont the refrgerator operaton wll be study as a functon of the thermal resstance of the hot sde dsspater. The smulated values wth the matematcal model were calculated modfyng the datum of the thermal resstance of the hot sde dsspater and keepng constant all the varables of the system. The characterstcs of the smulaton are the followng: Suppled voltage of 4 V for all the Pelter modules Fan suppled wth 12 V Room temperature of 0 ºC, 10 ºC and 20 ºC. Wth these values a range of thermal resstance for the hot sde dsspater from 0.1 K/W to 1.0 K/W were made. The data of the temperature gap between the nteror of the refrgerator and the room temperature follow a lneal trend as shown n Fg. 4. The room temperature that makes the greater thermal gap was 20 ºC. Fg. 5. COP s as a functon of the thermal resstance of the hot sde dsspater. The smulatons allowed us to determne that a decrease of the 10% n the thermal resstance of the hot sde dsspater means an ncrease of the 3.4% for the COP value. Influence of the cold sde dsspater. The smulaton characterstcs for the suppled Pelter voltage and fan voltage, and the test condtons were the same than n the case of the hot sde dsspater. The thermal resstance value range was from 0.5 K/W to 1.5 K/W. Checkng the results from Fg. 6 t can be seen that the data present a trend smlar to the hot sde study. Fg. 6. Temperature gap between the nternal temperature and the room temperature as a functon of the thermal resstance for the cold sde dsspater. Fg.4. Thermal gap between the nteror temperature and the room temperature as a functon of the thermal resstance of the hot sde dsspater. As can be checked n Fg. 4, the greatest temperature gap s for a room temperature of 20 ºC, wth a thermal resstance for the hot sde dsspater of 0.1 K/W. In Fg. 5 are shown the results of the COP from the smulatons and It can be seen that the COP s curve as a functon of the hot sde The results of the COP of the thermoelectrc refrgerator are represented n Fg. 7. The greatest COP s obtaned for a room temperature of 20 ºC, wth the lowest thermal resstance of the cold sde 0.5K/W. The COP of a thermoelectrc devce s a functon of the room temperature where the Pelter modules are workng, beng necessary to use the most effcent modules for each room temperature.

5 Fg.7. COP s as a functon of the thermal resstance for the hot sde dsspater. A decrease of the 10% n the thermal resstance of the cold sde dsspater means an ncrease of the 2.2% n the COP value. The hot dsspater has a greater nfluence n the COP, as n a 10% of the decrease of the hot sde thermal resstance means an ncrease of the 3,4% whle n the cold sde makes a 2.2% ncrease. Comparson between the double jump and the sngle jump refrgerator. In order to compare the operaton between the double jump refrgerator (two temperature gaps due to two Pelter modules placed n seres thermally) and the sngle jump (a sngle temperature gap due to the Pelter modules), the gap between the nner temperature and external room temperature wll be analyzed usng the model. The suppled voltages wll be n a range from 1 to 12 V wth a varaton of 1 V between smulatons. The characterstcs of the smulatons are the followngs: Smulaton tme: 20 h. Room temperature: 10 ºC. Cold sde resstance: 1.03K/W Hot sde resstance: 0.22 K/W Double jump thermoelectrc refrgerator. For the double jump case, there are represented graphcally the values of the temperature gap between the room temperature and the nternal temperature, Fg. 8. Our model allows to calculate the voltage where the greatest temperature gap s placed. In ths case, the maxmum s produced whle feedng the Pelter modules wth 7V, gettng a temperature gap of 22.7 ºC. Fg. 8. Temperature gap between the nteror and the exteror as a functon of the voltage suppled to the Pelter modules. It s easy to check that f the objectve s to obtan the greatest temperature gap, the voltage needed s 7 V, but the COP values should be studed n order to check f that voltage produces a good COP value, or n the other hand, the power consumpton s so hgh that another confguraton wth lower power consumpton s recommended. The COP values are shown n Fg. 9. The best COP value s provded by a voltage of 7 V what concdes wth the best temperature gap between the nteror and the exteror. Fg. 9. COP s as a functon of the voltage for a double jump refrgerator. Sngle jump refrgerator. The sngle jump s smulated wth a 0V suppled voltage for Pelter module number 1 what makes a temperature gap between Pelter modules number 2 and 3. The optmum suppled voltage referred to the temperature gap and the COP value wll be studed.

6 Fg. 10. COP s as a functon of the voltage for the double jump refrgerator. In ths case the temperature gap s bgger as greater s the voltage of the Pelter modules, Fg 10. The slope of the curve decreases as the voltage ncreases. If ncreasng the voltage from 1 to 2 V makes a temperature gap of 25 ºC ncreasng the voltage from 11 to 12 V makes a temperature gap of 2 tenths of ºC. In Fg. 11 the COP of the sngle jump refrgerator s shown as a functon of the suppled voltage to the Pelter modules. The optmum value of COP s for a voltage of 2V for the Pelter modules, Fg. 11. In ths case the maxmum jump and the best value for the COP do not match at the same voltage as t happens for the double jump case. In ths stuaton t s needed to reach a compromse between the maxmum temperature gap and the best COP value. Fg. 11. COP as a functon of the voltage for the sngle jump refrgerator. Conclusons The conclusons due to ths work are: A computatonal model has been developed to smulate the steady and transent states for a thermoelectrc double jump refrgerator, usng the fnte dfferences method. Ths method smulates the temperatures of each part of the thermoelectrc devce (nodes). The accuracy of the model has been checked comparng expermental and smulated values wth errors lower than a 1ºC. The nfluence of the optmzaton of the heat dsspaters has been studed for the thermoelectrc refrgerators. From both dsspaters, the hot sde dsspater has more nfluence than the cold sde dsspater, as a decrease of the 10% n the thermal resstance n both devces makes an ncrease n the COP of the 3.4% and 2.2% respectvely. The COP for the double jump confguraton s a 25% greater than the sngle jump confguraton (when Pelter module number 1 s dsconnected). Our model calculates the voltage that provdes the maxmum value for the COP, 7V n case of the double jump confguraton and 2V for the sngle jump confguraton. References 1. Mn G. and Rowe D.M., Coolng performance of ntegrated thermoelectrc mcrocooler, Sold-State Electroncs 43 (1999) (5), pp Gordon J.M., et al., The electro-adsorpton chller: a mnaturzed coolng cycle wth applcatons to mcroelectroncs, Internatonal Journal of Refrgeraton 25 (2002) (8), pp Ván J.G., Astran D. and Domnguez M., Numercal modellng and desgn of a thermoelectrc dehumdfer, Appled Thermal Engneerng 22 (2002) (4), pp Ván J.G., Astran D. and Aguas J.J., Thermoelectrc equpment to keep laboratory test-tube at a controlled temperature, Journal of Thermoelectrcty (1999) (3), pp Ván J.G., Domínguez M., Astran D.,. Smulaton by electrc analogy of a thermoelectrc cheese dryer. Ffth european workshop on thermoelectrcs. Unversty of Pardubce (Czech Republc) Rowe D.M., CRC Handbook of Thermoelectrcs. ISBN , (1995). pp Astran D., Ván J.G. and Albzua J., Computatonal model for refrgerators based on Pelter effect applcaton, Appled Thermal Engneerng 25 (2005), pp Lau P. G.,. Rtzer T. M, Bust J. R., Thermodynamc optmzaton of heat/cold snks extenders n thermoelectrc coolng assembles, 13th Internatonal Conference on Thermoelectrcs, Kansas Cty, Mssour, D. Astran, J.G. Ván and M. Domínguez, Increase of COP n the thermoelectrc refrgeraton by the optmsaton of heat dsspaton, Appled Thermal Engneerng 23 (2003) (17), pp