The Melting of an hailstone: Energy, Heat and Mass Transfer Effects
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The document presents a mathematical model for simulating the melting of a hailstone. It uses energy conservation principles and Fourier's law of heat transfer to model the heat flux and phase change during melting. A finite volume method is used to numerically solve the partial differential equations describing the transient heat transfer. The model was verified against an approximate analytical solution and showed good agreement. Results demonstrate that longer times are required to melt the center of the hailstone compared to the outer parts.
The Melting of an hailstone: Energy, Heat and Mass Transfer Effects
- 1. THE MELTING OF A HAILSTONE: ENERGY, HEAT AND MASS TRANSFER EFFECTS Yi Ying Chin Akinola Oyedele Emilio Ramirez M 475 SPRING 2014 FINAL PROJECT
- 2. INTRODUCTION Hail is defined as precipitation in the form of small balls or irregular lumps of ice and compact snow, each of which is called a hailstone. Hail is one of the most significant thunderstorm hazards to aviation. Hailstones accumulating on the ground can be hazardous to landing aircraft. How much time and energy are required to melt hailstones? Acquire a fundamental understanding of melting hailstones Create a mathematical model that simulates the melting of a hailstone to make better predictions of the rate at which hailstones melt, based on conditions it is exposed to. Goal Motivation Source: National Oceanic & Atmospheric Administration (NOAA) Photo Library Problem
- 3. MATHEMATICAL MODEL Energy Conservation used to calculate rate of energy change: 𝜕 𝜕𝑡 (𝜌𝑒) + 𝛻 ∙ 𝑞 = 0 where 𝜌 = density (constant) 𝑒 = energy (enthalpy) 𝑞 = heat flux Heat Flux utilized Fourier’s Law: 𝑞 = −𝑘𝛻𝑇 k = thermal conductivity
- 4. MATHEMATICAL MODEL Stefan problem for modeling phase change (melting) 𝑡 𝑥 𝐿 𝑆 𝑥 = Χ(𝑡) 𝓁 𝑇𝑡 = 𝛼 𝐿 𝑇𝑥𝑥 𝑓𝑜𝑟 0 < 𝑥 < Χ 𝑡 , 𝑡 > 0 𝑇 Χ 𝑡 , 𝑡 = 𝑇 𝑚 𝜌ℒΧ′ (𝑡) = −𝑘 𝐿 𝑇𝑥 + 𝑘 𝑆 𝑇𝑥| 𝑥=Χ(𝑡) 𝑇 𝑥, 0 = 𝑇𝑖𝑛𝑖 < 𝑇 𝑚 𝑇 0, 𝑡 = 𝑇𝐿(> 𝑇 𝑚) Initial Condition: Boundary Conditions: Stefan Condition: 𝑇𝑡 = 𝛼 𝑆 𝑇𝑥𝑥 𝑓𝑜𝑟 Χ(𝑡) < 𝑥 < 𝓁, 𝑡 > 0 −𝑘 𝑆 𝑇𝑥(𝓁, 𝑡) = 0 Interface Condition: Thermal diffusivity, 𝛼 = 𝑘 𝜌𝑐 𝑝
- 5. MATHEMATICAL MODEL Enthalpy formulation: 𝐸 = 𝜌𝑒 𝑇 𝐿 𝑆 𝑇 𝑚 𝜌ℒ Jump in energy at 𝑇 𝑚 𝐸 𝑇 = 𝜌𝑐 𝑝 𝑆 𝑇 − 𝑇 𝑚 , 𝑇 < 𝑇 𝑚 0, 𝜌ℒ , 𝑇 = 𝑇 𝑚 𝜌ℒ + 𝜌𝑐 𝑝 𝐿 𝑇 − 𝑇 𝑚 , 𝑇 > 𝑇 𝑚
- 6. MATHEMATICAL MODEL Liquid Fraction: Equation of state 𝜆 = 0 , 𝐸 ≤ 0 𝑠𝑜𝑙𝑖𝑑 𝐸 𝜌ℒ , 0 < 𝐸 < 𝜌ℒ 1 , 𝐸 ≥ 𝜌ℒ 𝒍𝒊𝒒𝒖𝒊𝒅 (𝒎𝒖𝒔𝒉𝒚) a b
- 7. METHODS 1. Time stepping through the time explicit scheme for 𝑛 time steps 𝑌𝑛+1 = 𝑌𝑛 + ∆𝑡 ∙ 𝑓 𝑡 𝑛, 𝑌𝑛 , 𝑛 = 0, 1, … , 𝑁 2. At each time step: Compute fluxes, 𝑞 at 𝑇𝑖 Update 𝐸𝑖 from energy conservation law From Equation of state find: liquid fraction, 𝜆𝑖 new temperature, 𝑇𝑖 *The CFL condition to minimize growth of errors: Forward Euler (Explicit) Time Discretization 𝛥𝑡 ≤ ∆ 𝑟2 )4(max 𝛼 𝜕 𝜕𝑡 (𝜌𝑒) + 𝛻 ∙ 𝑞 = 0 𝐸 𝑇 = 𝜌𝑐 𝑝 𝑆 𝑇 − 𝑇 𝑚 , 𝑇 < 𝑇 𝑚 0, 𝜌ℒ , 𝑇 = 𝑇 𝑚 𝜌ℒ + 𝜌𝑐 𝑝 𝐿 𝑇 − 𝑇 𝑚 , 𝑇 > 𝑇 𝑚 𝐸𝑖 𝑛+1 = 𝐸𝑖 𝑛 + ∆𝑡 ∆Vi (𝐴𝑞) 𝑖− 1 2 𝑛 −(𝐴𝑞) 𝑖+ 1 2 𝑛
- 8. METHODS Easy and clear to understand One step, explicit scheme can be easily checked by hand The stability requirement may not impose undue restrictions in situations where the time-step must be small for physical reasons Explicit schemes may turn out to be as efficient as implicit schemes Why use forward Euler Scheme?
- 9. METHODS ⇨ 1. Numerical spherical model used forward Euler enthalpy method 2. Discretization of the system PDE solved numerically from control volume to control volume 3. Applies to all types of PDEs in general 4. Exact conservation and stable with CFL stability condition 5. Volume tracking scheme as opposed to a front tracking scheme does not require the melting front to be resolved Finite Volume Method (FVM)
- 10. METHODS: GEOMETRY DISCRETIZATION × ×× 𝑖 + 1 2𝑖 − 1 2 𝑟𝑖 1 + 1 21 − 1 2 𝑟1 𝑀 + 1 2 𝑀 − 1 2 𝑟 𝑀 a b 𝐸𝑖 𝐸 𝑀𝐸1 𝐸0 𝐸 𝑀+1 𝑑𝑟 Area, 𝐴𝑖−1 2 = 4𝜋𝑟𝑖−1 2 2 Volume, ∆𝑉𝑖= 4 3 𝜋(𝑟𝑖+1 2 3 − 𝑟𝑖−1 2 3 )a b
- 11. METHODS: DISCRETE MODEL 𝐸𝑖 𝑛+1 = 𝐸𝑖 𝑛 + ∆𝑡 ∆Vi (𝐴𝑞) 𝑖− 1 2 𝑛 −(𝐴𝑞) 𝑖+ 1 2 𝑛 , 𝑖 = 1, … , 𝑀 Initial Condition: Boundary Conditions 𝐴𝑡 𝑟 = 0: 𝑞 0, 𝑡 = 0 ⇒ 𝑇0 𝑛 = 𝑇1 𝑛 𝑞 𝑖− 1 2 𝑛 = 𝑇𝑖−1 − 𝑇 𝑚 𝑅𝑖−1 + 𝑇 𝑚 − 𝑇𝑖 𝑅𝑖 , 𝑖 = 2, … , 𝑀 𝑅𝑖 = ∆𝑟 1 − 𝜆𝑖 𝑘 𝑆 + 𝜆𝑖 𝑘 𝐿 Area, 𝐴𝑖−1 2 = 4𝜋𝑟𝑖−1 2 2 Volume, ∆𝑉𝑖= 4 3 𝜋(𝑟𝑖+1 2 3 − 𝑟𝑖−1 2 3 ) 𝑇 𝑟, 0 = 𝑇𝑖𝑛𝑖 < 𝑇 𝑚 (solid) 𝐴𝑡 𝑟 = 𝓁: 𝑇(𝓁, 𝑡) = 𝑇∞ ⇒ 𝑇 𝑀+1 𝑛 = 𝑇∞
- 12. APPROXIMATE SOLUTION Approximate solution for transient, 1-D heat conduction. The external surface of the sphere exchanges heat by convection The temperature field is governed by the heat equation in spherical coordinates: 𝜕𝑇 𝜕𝑡 = 𝛼 𝑟2 𝜕 𝜕𝑟 1 𝑟2 𝜕𝑇 𝜕𝑟 , (𝑟, 𝑡) ∈ Ω The local heat flux from the sphere to the surrounding is 𝑞 = ℎ(𝑇𝑠 − 𝑇∞) Initial condition: 𝑇 𝑟, 0 = 𝑇𝑖 Boundary condition: 𝑘 𝜕𝑇 𝜕𝑟 | 𝑟=𝑟0 = ℎ(𝑇∞ − 𝑇𝑠)
- 13. VERIFICATION 1. Numerical heat conduction spherical model verified using approximate solution 2. Recktenwald (2006) utilized an infinite series solution: With positive roots: 𝐹𝑜 = 𝛼𝑡 ℛ2 ≫ 1
- 14. RESULTS: VERIFICATION OF NUMERICAL MODEL Good agreement between numerical & approximate solution Surface of sphere Center of sphere Surface – approximate solution Center – approximate solution Surface – numerical model Center – numerical model 𝑇∞
- 15. RESULTS: TEMPERATURE PROFILE AT VARIOUS POSITIONS ALONG THE RADIUS 400 sec 1200 sec 2000 sec 2800 sec 3200 sec 4000 sec r(cm) 𝑇∞ = 40°C 𝑇𝑖𝑛𝑖 = -20°C r = 3 cm
- 16. RESULTS: PROFILE AT SPATIAL NODES 1 M/2 M
- 17. RESULTS: NUMERICAL MODEL OUTPUTS 6 nodes Outer node Middle node Inner node time(sec)
- 18. RESULTS: PROFILE AT SPATIAL NODES Inner node Middle node Outer node 32 nodes Outer node Middle node Inner node
- 19. RESULTS: THE MELT FRONT Liquid Solid
- 20. DISCUSSION/CONCLUSIONS Intuitive result: A relatively longer time is required to melt the center of a hailstone as compared to the outer parts Slightly counter-intuitive result Affirms the challenges in melting hailstones Case studied here is a simple case of a spherical hailstone in 1D Possible expansion into a model that better resembles reality (variable density & heat capacity) Study projected to provide insight into how to utilize the results in melting hailstones effectively and efficiently
- 21. REFERENCES 1) Alexiades, V. and Solomon, A.D., 1993. Mathematical Modeling of Melting and Freezing Processes, Hemisphere Publishing Corporation, USA. 2) Peiro, J. and Sherwin, S. “Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations”. Department of Aeronautics, Imperial College, London, UK. 2005. 3) Recktenwald, G, Transient, 2006. One-Dimensional Heat Conduction in a Convectively Cooled Sphere.