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.