Multiphysics Simulation of Infrared Signature of an Ice Cube

Transcription

1 Muliphysics Simulaion of Infrared Signaure of an Ice Cube Infrared (IR) refers o a band of elecromagneic waves beween he 1 mm (frequency of 300 GHz) o 0.7 μm wavelenghs (frequency of 430 THz) and phoon energy from 1.24 mev o 1.7 ev as shown in he elecromagneic specrum in Figure 1. Mos of he hermal radiaion from objecs around 273K is in he range of IR. Thermal radiaion is generaed due o he ineraomic moion of he paricles. I happens in any maer above absolue zero (zero degrees Kelvin). Thermal radiaion is emied regardless of he physical sae (solid, liquid and gas) of he maer [1]. The surface emperaure can be deermined using he Sefan-Bolzmann Law [2] based on he amoun of hermal energy emied from he objec s surface as given in Equaion (1). q = ε σ A (T s 4 T 4 ) (1) Where q is hea ransfer per uni ime (W), ε is emissiviy in comparison o black body (dimensionless), σ is Sefan-Bolzmann consan (W/(m 2.K 4 )), A is area of emiing surface (m 2 ), T s is surface emperaure (K) and T is he room (surrounding) emperaure (K). The radiaive emissiviy of an objec varies wih he wavelengh. Figure 2 shows how he emissiviy of pure ice made from disilled waer varies wih wavelengh [3]. I shows ha he value of ice emissiviy varies from o in he range of 4μm o 13μm wavelenghs which means ha ice has high radiaive emiance in he hermal and he far IR ranges.

2

3 IR deecion devices such as IR cameras scan over a range of wavelenghs and average over he resuls o calculae he IR signaure [4]. The oupu of an IR device is a emperaure profile superimposed over geomerical feaures as shown in Figure 3. In his work, hree differen mehodologies are used o solve he 3D ransien hea equaion o simulae he infrared signaure of an ice cube. These mehodologies are he finie difference mehod (FDM), he specral mehod, and he ANSYS Muliphysics sofware. The size of he cube is 15cm x 15cm x 15cm. A cubical geomery was chosen o simplify he problem. Also, he hermal signaure is symmeric in a cubical geomery, which allows for easy modelling using simulaion echniques. The resuls show he variaion in emperaure when an ice cube iniially a a consan emperaure of -28 C is lef in room emperaure condiions (25 C) o warm up. The emperaure profiles on he surface of he cube are compared in he ime and space domains. The underlying physics of hea ransfer hrough conducion in a solid medium can be solved mahemaically using he hea equaion [5] as given in Equaion (2). ρc T = q + T (k x x ) (2) Where ρ is densiy of he medium (kg/m 3 ), c is specific hea capaciy (J/(kg K)), q is he volumeric energy generaion erm (W/m 3 ), k is coefficien of hermal conduciviy (W/ (m.k)), T is emperaure (K), x refers o spaial posiion (m) and is he ime (s). The exended form of he above equaion in hree spaial dimensions wih no energy

4 generaion erm [6] is given in Equaion (3). T = α T ( 2 x T y T z2) (2) Where x, y and z refer o spaial posiions (m) in hree dimensions and α is he hermal diffusiviy erm (m 2 /s) as given in (4). α = k ρc (3) To solve Equaion (3), he boundary, and he iniial condiions are required. The convecive boundary condiions [7] are applied on each exernal surface of he cubical geomery as given in Equaion (5). k T s x = h(t T s ) (4) Where T s is he surface emperaure (K), T is he surrounding emperaure (K) and h is convecive hea ransfer coefficien (W/(m 2.K)). This boundary condiion represens he exisence of convecion heaing or cooling a a surface and is obained hrough he energy balance a he surface. The iniial condiion, in his case, is he uniform emperaure hroughou he cube ha is se o -28 C (245K). This emperaure varies as he ice cube sars o warm up. Densiy(ρ), specific hea capaciy (c) and coefficien of hermal conducion (k) vary wih emperaure [8] as shown in Figure 4. The convecive hea ransfer coefficien may also vary beween 4-10 W/(m 2.K) depending on he surrounding condiions. The range of ineres in his problem is from -28 C o 0 C. A 0 C ice will sar phase change from solid o liquid waer wih no variaion in emperaure. Phase change is no simulaed in his sudy. Physical and hermal properies of ice beween -28 C o 0 C are given in Table 1. As found, he sandard deviaion is less han 5% of he average values hence hese values can be considered as consans for seing up he simulaions. The values of he consans se are given in Table 2. The following assumpions are considered in his sudy, Energy ransfer from ice cube hrough he mode of radiaion is minimal. Energy ransfer from he ice cube o he surrounding is only hrough naural convecion. Variaion in physical and hermal properies wih emperaure are no significan. Values of se consans are given in Table 2.

5 Densiy of Ice (ρ) kg/m 3 Temperaure ( C) Value Average value Sandard Deviaion (% of Average) 1.37 (0.15 %) Specific Hea Capaciy of Ice (c) J/kg/K Temperaure ( C) Value Average value Sandard Deviaion (% of Average) 60.9 (3.1 %) Coefficien of Thermal Conducion of Ice (k) W/(m.K) Temperaure ( C) Value Average value 2.35 Sandard Deviaion (% of Average) (4.4 %)

6 Consan Value Unis Radiaive Emissiviy of Ice (ε) Dimensionless Sefan-Bolzmann consan (σ) x 10-8 W/(m 2.K 4 ) Surrounding Temperaure (T ) 25 (298) C (K) Densiy of Ice (ρ) a 919 kg/m 3 Specific Hea Capaciy of Ice (c) a 1970 J/kg K Coefficien of Thermal Conducion of Ice (k) a 2.35 W/(m.K) Thermal Diffusiviy (α) x 10-6 m 2 /s Convecive Hea Transfer Coefficien of surrounding (h) b 5.00 W/(m 2.K) a Average value from Table 1. b Depends on surrounding condiions

7 Finie difference mehod (FDM) is a numerical mehod for solving differenial equaions such as he hea equaion given in Equaion (3). This mehod approximaes he differenials wih differences by discreizing he dependen variables (emperaure) in he independen variable domains (space and ime) [9]. Each discreized value of he dependen variable is referred o as a nodal value. In his case, hea equaion given in Equaion (3) is discreized using FDM forward-ime cenral-space (FTCS). The discreized equaion is given in Equaion (6). T +1 i,j,k = T i,j,k + α (T i+1,j,k 2T i,j,k + T i 1,j,k ( x) 2 +α (T i,j+1,k +α (T i,j,k+1 2T i,j,k ( y) 2 2T i,j,k ( z) 2 ) + T i,j 1,k ) + T i,j,k 1 ) (5) Where superscrip and subscrip i,j,k refer o ime and posiion for a value of nodal emperaure respecively. is a imesep size (s) and x, y, z are he differences in he spaial posiion of emperaure nodes. The convecive boundary condiion is also discreised using FDM and only applied o he ouer surfaces as given in Equaion (7). k (T i+1,j,k T i,j,k ) = h(t x T i,j,k ) (6) I is vial for he sabiliy and accuracy of FDM o choose he correc ime sep value. In his work, Couran Friedrichs Lewy (CFL) condiion [9, 10] is used o decide he ime sep size. CFL condiion for he hea equaion is given in Equaion (8). 2α ( x) 2 (7) Equaions (6) and (7) are solved and pos-processed in MATLAB. Resuls are discussed in secions 5 and 6. The specral mehod is also a numerical mehod for solving differenial equaions such as hea equaion given in Equaion (3). This mehod assumes he soluion can be wrien as a sum of cerain basic funcions such as Fourier series or as, in his case, polynomials. The specral mehod gives a lower numerical error for mahemaically coninuous soluions in comparison o oher numerical mehods [11]. In his mehod, space dimensions are needed o be discreized using Chebyshev poins, which gives an uneven spaial grading and is beer for fiing polynomials. An example of 2D Chebyshev-Lobao poins grid [12] is shown in Figure 5. This mehod can be exended o muli-dimensions such as hea equaion as given in Equaion (3) by adding he derivaives of polynomials. Time sepping is essenial for sabiliy and accuracy of he numerical resuls. In his case, Runge-Kua ime sepping mehod (RK4) is used. Equaion (3) is solved using he specral mehod in MATLAB. Resuls are discussed in secions 5 and 6.

8 ANSYS Muliphysics sofware offers o simulae various physical phenomena [13]. The mehod of soluion is based on finie elemen mehod (FEM) [14]. In his work, ANSYS Muliphysics hermal module is used o solve hermal signaure of an ice cube. To do so a cubic geomery is buil in ANSYS Muliphysics Graphics User Inerface (GUI) and meshed using hermal mass Solid Brick 8 noded 278 elemens [15]. This mesh is esed for space and ime sensiiviy. The mesh buil in ANSYS Muliphysics is shown in Figure 6.

9 Temperaure ( C) Temperaure ( C) Temperaure ( C) Temperaure ( C) The iniial condiion is consan emperaure hroughou he mesh. The convecive boundary condiion is applied on hree boundary surfaces. Symmery (zero hea flux) is applied o he res of he hree boundary surfaces o reduce he run ime of simulaion. A program chosen algorihm is used o conrol he ime sepping. Resuls are discussed in secions 5 and 6. This secion discusses he resuls obained by FDM, specral mehod and ANSYS Muliphysics simulaions. Surface emperaure conour plos obained hrough FDM and Specral mehods a various ime inervals are given in Figure 7. I is noable ha he emperaure profile is esablishing a 100s. The difference beween max and min values of emperaure is around 3 C. Around 500s, he emperaure profile is more esablished, and he difference beween max and min values has risen o around 5 C. A he 1500s, he emperaure profile is fully developed, and he gap beween he maximum and minimum values of emperaure is abou 6 C. A 4500s, he paern appears o be similar as noed earlier however he difference beween max and min has sared o drop (abou 4.5 C). This difference falls slighly as surface emperaure reaches close o meling

10 emperaure (0 C) and emperaure hroughou he cube ends o sabilize o a consan value. Since simulaion does no ake ino accoun he phase change condiion, herefore, resuls afer 0 C are no valid. From all above plos, i can also be observed ha he max value of emperaure is a he corners, and he minimum value is in he surface cenre. Surface emperaure conour plos a various ime inervals obained hrough hermal simulaion in ANSYS Muliphysics are given in Figure 8. All indicaed values are in C. ANSYS Muliphysics also demonsraed same behaviour as earlier discussed in FDM resuls. Variaion in he emperaure of a corner of he cube is ploed agains ime for all hree mehods for a comparison as given in Figure 9. The resul indicaes he oal ime aken by an ice cube o reach meling emperaure, which is found o be approximaely 5500s. The comparison also shows close agreemen beween he hree mehods.

11 Three differen mehods, namely finie difference mehod (FDM), specral mehod and ANSYS Muliphysics sofware, are used o simulae he hermal image of an ice cube, when warming from -28 C o reach meling poin under room emperaure condiions. Resuls revealed ha ice cube of dimensions (15 X 15 X 15 cm) akes approximaely 5500s o reach meling emperaure. During his ime emperaure profile develops wihin he ice cube wih a emperaure difference of 5-6 C. These resuls are imporan o undersanding he hermal behaviour of ice. Fuure work is proposed o capure he hermal image of an ice cube hrough a hermal imaging device (e.g. IR camera) and compare wih given simulaion mehods. This work will also help o build he physical relaion beween hermal imaging devices and underlying physics of hea ransfer. The work repored in his paper is funded by he Norges forskningsråd, projec no /160 in collaboraion wih Faroe Peroleum. We would also like o acknowledge he suppor given by Prof. James Mercer a he UiT The Arcic Universiy of Norway.

12 [1] Howell, J.R., P. Menguc, and R. Siegel, Thermal Radiaion Hea Transfer, 5h Ediion. 2010: Taylor & Francis. [2] G.F.S, The consan σ of he Sefan-Bolzmann law. Journal of he Franklin Insiue, (1): p. 64. [3] Wan, Z., MODIS (Moderae Resoluion Imaging Specromeer) UCSB Emissiviy Library. 1999: Universiy of California, Sana Barbara. [4] Rogalski, A., Infrared Deecors, Second Ediion. 2010: CRC Press. [5] Cannon, J.R., The One-Dimensional Hea Equaion. 1984: Cambridge Universiy Press. [6] Widder, D.V., The Hea Equaion. 1976: Elsevier Science. [7] Moran, M.J., Inroducion o hermal sysems engineering: hermodynamics, fluid mechanics, and hea ransfer. 2003: Wiley. [8] Perenko, V.F. and R.W. Whiworh, Physics of Ice. 2002: OUP Oxford. [9] Paankar, S., Numerical Hea Transfer and Fluid Flow. 1980: Taylor & Francis. [10] Couran, R., K. Friedrichs, and H. Lewy, Über die pariellen Differenzengleichungen der mahemaischen Physik. Mahemaische Annalen, (1): p [11] Trefehen, L.N., Specral Mehods in MATLAB. 2000: Sociey for Indusrial and Applied Mahemaics. [12] Canuo, C., e al., Specral Mehods: Fundamenals in Single Domains. 2007: Springer- Verlag. [13] ANSYS, Academic Research. release [14] Kim, M., Finie Elemen Mehods wih Programming and Ansys. 2013: LULU Press. [15] ANSYS, Academic Research, Theory Reference, in Mechanical APDL Guide. release 14.0.