THE MELTING OF A HAILSTONE: ENERGY, HEAT

AND MASS TRANSFER EFFECTS

Yi Ying ChinAkinola Oyedele Emilio Ramirez

M 475 SPRING 2014 F INAL PROJECT

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

MotivationSource: National Oceanic & Atmospheric Administration

(NOAA) Photo Library

Problem

MATHEMATICAL MODEL Energy Conservation used to calculate rate of energy

change:

where = density (constant)

= energy (enthalpy)

= heat flux

Heat Flux utilized Fourier’s Law:

k = thermal conductivity

MATHEMATICAL MODEL Stefan problem for modeling phase change (melting)

𝑡

𝑥

𝐿 𝑆

𝑥=Χ (𝑡)

𝓁

𝑇 𝑡=𝛼𝐿𝑇 𝑥𝑥 𝑓𝑜𝑟 0<𝑥<Χ (𝑡 ) , 𝑡>0

𝑇 ( Χ (𝑡 ) , 𝑡 )=𝑇𝑚

𝜌ℒ Χ ′ (𝑡)=−𝑘𝐿𝑇 𝑥+𝑘𝑆𝑇 𝑥¿𝑥=Χ (𝑡 )

𝑇 (𝑥 ,0 )=𝑇 𝑖𝑛𝑖<𝑇𝑚

𝑇 (0 ,𝑡 )=𝑇 𝐿(¿𝑇𝑚)

Initial Condition:

Boundary Conditions:

Stefan Condition:

𝑇 𝑡=𝛼𝑆𝑇 𝑥𝑥 𝑓𝑜𝑟 Χ (𝑡)<𝑥<ℓ ,𝑡>0

−𝑘𝑆𝑇 𝑥(ℓ ,𝑡 )=0

Interface Condition:

Thermal diffusivit

y,

MATHEMATICAL MODEL Enthalpy formulation:

𝐸=𝜌𝑒

𝑇

𝐿

𝑆𝑇𝑚

𝜌ℒ Jump in energy at

𝐸 (𝑇 )={ 𝜌𝑐𝑝𝑆 [𝑇 −𝑇𝑚 ] ,𝑇 <𝑇𝑚

[0 ,𝜌ℒ ] ,𝑇=𝑇𝑚

𝜌ℒ+𝜌𝑐𝑝𝐿 [𝑇 −𝑇𝑚 ] ,𝑇>𝑇𝑚

}

MATHEMATICAL MODEL Liquid Fraction: Equation of state

𝜆={ 0 ,𝐸≤0 (𝑠𝑜𝑙𝑖𝑑 )𝐸𝜌ℒ

,0<𝐸<𝜌ℒ

1 ,𝐸≥𝜌ℒ ( 𝒍𝒊𝒒𝒖𝒊𝒅 )

(𝒎𝒖𝒔𝒉𝒚 )}a b

METHODS

1. Time stepping through the time explicit scheme for time steps

2. At each time step:

Compute fluxes, at

Update from energy conservation law

From Equation of state find: liquid fraction, new temperature,

*The CFL condition to minimize growth of errors:

Forward Euler (Explicit) Time Discretization

𝛥𝑡≤ ∆𝑟2

¿¿

𝜕𝜕𝑡

(𝜌𝑒)+𝛻 ∙�⃑�=0

𝐸 (𝑇 )={ 𝜌𝑐𝑝𝑆 [𝑇 −𝑇𝑚 ] ,𝑇 <𝑇𝑚

[0 ,𝜌ℒ ] ,𝑇=𝑇𝑚

𝜌ℒ+𝜌𝑐𝑝𝐿 [𝑇 −𝑇𝑚 ] ,𝑇>𝑇𝑚

}𝐸𝑖

𝑛+1=𝐸𝑖𝑛+∆ 𝑡∆V i ((𝐴𝑞)

𝑖−12

𝑛 −(𝐴𝑞)𝑖+12

𝑛 )

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?

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)

METHODS: GEOMETRY DISCRETIZATION

𝒓 𝒊−𝟏𝟐

𝒓 𝒊

𝒓 𝒊+𝟏𝟐

×× ××𝑖+12𝑖−

12

𝑟 𝑖1+121−

12𝑟1 𝑀+

12

𝑀−12𝑟𝑀

a b

𝐸𝑖𝐸𝑀𝐸1

𝐸0 𝐸𝑀+1𝑑𝑟

Area,

Volume, a b

METHODS: DISCRETE MODEL

Initial Condition:

Boundary Conditions

𝐴𝑡 𝑟=0 :𝑞 (0 , 𝑡 )=0⇒𝑇0𝑛=𝑇 1

𝑛

𝑞𝑖 − 12

𝑛 =𝑇 𝑖−1−𝑇𝑚

𝑅 𝑖−1

+𝑇𝑚−𝑇 𝑖

𝑅 𝑖

,𝑖=2 ,…,𝑀

𝑅𝑖=∆𝑟 [ 1− 𝜆𝑖

𝑘𝑆 +𝜆𝑖

𝑘𝐿 ]

Area,

Volume,

(solid)

𝐴𝑡 𝑟=ℓ :𝑇 (ℓ , 𝑡)=𝑇 ∞

APPROXIMATE SOLUTIONApproximate 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:

The local heat flux from the sphere to the surrounding is

Initial condition:

Boundary condition:

VERIFICATION1. Numerical heat conduction spherical model verified using

approximate solution

2. Recktenwald (2006) utilized an infinite series solution:

With positive roots: 𝐹𝑜=𝛼𝑡ℛ2≫1

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 ∞

RESULTS: TEMPERATURE PROFILE AT VARIOUS POSITIONS ALONG THE RADIUS

400 sec 1200 sec2000 sec2800 sec3200 sec4000 sec

r(cm)

= 40°C

= -20°C

r = 3 cm

RESULTS: PROFILE AT SPATIAL NODES

1 M/2 M

RESULTS: NUMERICAL MODEL OUTPUTS

6 nodes

Outer

node

Middle

node

Inner nodetime(sec)

RESULTS: PROFILE AT SPATIAL NODES

Inner node Middle node Outer node

32 nodes

Outer

node

Middle

node

Inner node

RESULTS: THE MELT FRONT

Liquid

Solid

DISCUSSION/CONCLUSIONSIntuitive 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

REFERENCES1) 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.