OBJECTIVE Natural Convetion Experiment Measurements from a Vertial Surfae 1. To demonstrate te basi priniples of natural onvetion eat transfer inluding determination of te onvetive eat transfer oeffiient. 2. To demonstrate te boundary layer arater of external natural onvetion. BACKGROUND Sine te density of most fluids vary wit temperature, temperature gradients witin a fluid medium will give rise to density gradients. If tese density gradients are su tat te fluid is in an unstable situation, eavy fluid on top of ligt fluid, te fluid will begin to move. Tis motion is termed natural onvetion. Newton's law of ooling governs tis pysial proess wi states tat te eat transfer from te surfae is diretly proportional to te temperature differene between te surfae and te fluid far away from te surfae or ( T - T ) q& (1) surfae fluid Introduing te onvetive eat transfer oeffiient as te onstant of proportionality, we ave ( T - T ) q& = (2) surfae fluid It is lear tat te major obstale in utilizing Eq. (2) for onvetive eat transfer alulations is te evaluation of te onvetive eat transfer oeffiient. Tere are tree standard metods used to evaluate. Te first involves a matematial solution to te onservation equations in differential form. For problems were tese equations are too ompliated to be solved analytially, we an employ te seond metod, a omputational solution. Finally for problems tat are so ompliated tat we annot even write te appropriate onservation equations, we must go into te laboratory and make measurements in employing an experimental solution. Before we ontemplate employing one of tese solution metods, it is useful to use our intuition to figure out upon wat te onvetive eat transfer oeffiient depends. Our intuition tells us tat te tree major fators assoiated wit te alulation of sould be (i) fluid meanis 1
(ii) (iii) fluid transport properties geometry Starting wit te fluid meanis, we reognize tere are a variety of ways to araterize te flow. We an onsider wat is driving te flow and lassify it as fored or natural onvetion. Next, we onsider ow many boundaries te fluid flow interats wit and lassify it as external or internal flow. Reall tat te differene between internal flows and external flows is often one of perspetive. A tird way to araterize te flow is by te presene or absene of turbulene. An important onsideration in te andling of onvetive eat transfer oeffiients is te notion of dynami similarity. It is found tat ertain systems in fluid meanis or eat transfer are found to ave similar beaviors even toug te pysial situations may be quite different. Reall te fluid meanis of flow in a pipe. Wat we are able to do is to take data as sown in Fig. 1 for different fluids and pipe diameters and by appropriately saling ollapse tese urves into one urve. Figure 1. Dynami Similarity for Pipe Flow Pressure Drop Frition Fator Veloity Reynolds Number In onvetive eat transfer we may apply dynamis saling to make a parallel transformation. 2
Figure 2. Dynami Similarity for Convetive Heat Transfer Convetive Heat Transfer Coeffiient Nusslet Number Temperature Differene Rayleig Number We ave defined a dimensionless onvetive eat transfer oeffiient alled te Nusselt number as Nu = L k (3) were : onvetive eat transfer oeffiient L: arateristi lengt k: termal ondutivity of te fluid. Te arateristi lengt is osen as te system lengt tat most affets te fluid flow. For flow along a flat plate our arateristi lengt is te plate lengt, L, and we write L Nu = D (4) k Te Rayleig number is indiative of te buoyany fore tat is driving te flow and is given by gβts T Ra = να f L 3 (5) 3
were g: aeleration due to gravity β: fluid termal expansion oeffiient T s : surfae temperature T f : fluid temperature L: arateristi lengt ν: fluid kinemati visosity α: fluid termal diffusivity. Anoter important feature introdued by te fluid meanis is te loal nature of te onvetive eat transfer oeffiient. If as te fluid flows over different regions of te surfae, te fluid meanis ange, ten te onvetive eat transfer oeffiient will ange. For natural onvetion over a vertial flat plate we will ave te boundary layer flow sown in Fig. 3. Figure 3. Flat Plate Boundary Layer Development δ x r g As te boundary layer tikness grows, te fluid meanis ange signifiantly, so tat te onvetive eat transfer oeffiient will vary along te lengt of plate and we will ave a loal onvetive eat transfer oeffiient, (x). Ten we may also define loal Nusselt and Rayleig numbers as Nu x = (x) x k (6) 4
gβts (x) T Ra = να f x 3 (7) Toug loal eat transfer onditions an be extremely important, an average eat transfer oeffiient over te entire surfae lengt is often desirable. By definition we ave 1 = L l, avg 0 (x) dx (8) wit an average Nusselt Number given as Nu L,avg =,avg k L (9) Te dimensionless parameter wi is used to represent te affet of fluid properties is te Prandtl number ν Pr = (10) α Te influene of geometry may be seen in a ouple of ways. First, for tose onfigurations tat ave two lengt dimensions, su as a ylinder, we introdue a dimensionless geometri parameter D Χ = (11) L Te seond way in wi we see geometrial influenes is troug te funtional form of te Nusselt number orrelation. In general we may write Nu = fn(ra,pr,x) (12) For simple situations tese may often be written as power law relationsips n Pr m Nu = a Ra (13) were te onstants a, m, and n will ange for different geometries. 5
In tis experiment te student will develop te relationsip among te Nusselt number and oter dimensionless parameters for natural onvetion from a vertial flat surfae. Te students will determine loal eat transfer oeffiients wi are indiative of a boundary layer penomena. Tese loal measurements will ten be averaged and ompared to publised orrelations. Natural onvetion eat transfer from a vertial surfae will be investigated using an eletrially eated flat plate. A semati of te apparatus is sown in Figure 4. Figure 4. Semati of Experimental Apparatus Rotary Seletion Swit Power Supply Flat Plate Voltage DMM Current DMM Test Setion T/C DMM Assuming te eating is uniform, te eat flux at any loation on te plate is given by, V I q& = (14) 2 w L were V and I are te eletri voltage and urrent supplied to te plate and w and L are te widt and lengt of te eated surfae. Temperature measurements are made along te lengt of te plate wit termoouple embedded beneat te eated surfae. 6
Te loal eat transfer oeffiient (te eat transfer oeffiient at a ertain distane from te leading edge) is ten alulated as q& (x) = T (x) - T s f (15) Sine in most engineering appliations te interest would be in an average eat transfer oeffiient for te entire surfae, we use te matematial definition 1 L = (x) dx (16) L 0 Now substituting and reognizing tat te termoouple is designed to read te temperature differene diretly, we ave q& L dx = L T(x) 0 (17) We now apply a trapezoidal rule integration to obtain q& N-1 1 1 = 0.5 + L i=1 Ti Ti-1 ( x - x ) i i-1 (18) were N is te number of temperature data points along te plate and i=0 orresponds to te point at te leading edge. Te average Nusslet number an ten alulated. PROCEDURE 1. Cek tat te power supply is unplugged and tat te transformer is set at zero. 2. Plug in te power supply and turn on te transformer. 5. Set te transformer at a power level of about 5% and allow te system to stabilize. You will want to ave measurements for at least five power settings. Sine natural onvetion is a relatively poor eat transfer proess you will want to monitor te maximum surfae temperature to prevent burnout of te system. It is reommended tat you do not exeed 25% power level on te transformer. 6. Read and reord te voltage measurements assoiated wit te eletrial voltage and urrent supplied for eating and te termoouples. 7
7. Inrease te power level, allow te system to stabilize, and reord appropriate measurements. Repeat te experiment as needed 8. Turn off te transformer and unplug te power supply. DATA ANALYSIS 1. Ea ase sould be reorded on te Exel spread seets provided in te lab. 2. From te experimentally determined values of te loal eat transfer oeffiient alulate te average eat transfer oeffiient and surfae temperature for ea power setting. 3. Plot te loal eat transfer oeffiient versus distane from te leading edge. 4. Calulate te average Nusselt number and Rayleig number for ea power setting. 5. Plot te Nusselt number versus Rayleig number for ea ase along wit te publised orrelations. 6. Estimate te unertainty error in your experimentally determined Nusselt number and Rayleig number. 7. Provide one sample and alulation of your data proessing. SUGGESTIONS FOR DISCUSSION 1. How does te experimental data ompare to te publised orrelations? 2. Wat are some possible errors in te experiment? 3. Wat an you tell about te struture of te boundary layer from te loal eat transfer oeffiient? 4. Can you identify a transition from laminar to turbulent flow? 8