MATH-UA 251 Intro to Math Modeling Final Project A Model of Ice Melting Yue Sun May 11, 2017 Abstract This project constructs a model for one-dimensional ice melting process, and numerically simulates the melting time of certain length of ice slab under different melting temperatures. The model involves solving the heat equation with a moving boundary condition, hence we study the Stefan problem, i.e. the moving boundary condition problem. This project solves both analytically and numerically for this one-dimensional heat equation with moving bound- ary condition. 1 Introduction The incentive of this project is to calculate the melting time of a given amount of ice, for example, how long it would take for all the ice to be melted on Gould Plaza after it snows in New York City. However, it would require a construction of a three-dimensional model, which would be too complicated for this project. Here we simplify the project onto a one-dimensional semi-infinite plane; instead of ice cube, we consider ice slab in this project. Since the melting process involves phase change, it is natural to consider heat equation for this model. As mentioned in lectures that heat equation is a form of partial differential equation, we would need the initial conditions and the boundary conditions to find a solution. The key thing to notice is that the boundary condition, h(t), is not fixed, but dependent with time. This particular moving boundary condition problem, also called the Stefan problem, has been studied by Josef Stefan. The classical Stefan problem aims to describe the temperature distribution in a homoge- neous medium undergoing a phase change, for example ice passing to water: this is accomplished by solving the heat equation imposing the initial temperature distribution on the whole medium, and a particular boundary condition, the Stefan condition, on the evolving boundary between its two phases. In this paper, we examine a model of ice melting through Stefan problem. This problem is estab- lished in Section 2, analytically solved in Section 4 and numerically solved in Section 5. In Section 3 we introduce the dimensionless parameter, the Stefan number. 2 The Mathematical Model of Melting Processes Before we derive a mathematical model for the melting process, we first discuss the physical properties of phase change, and the assumptions needed to simplify this model in Section 2.1. Then we derive the heat equation, the basis of this model in Section 2.2. In Section 2.3, we explain and derive the Stefan condition. 1 2.1 Physical Properties and Assumptions For this project, we restrict ourselves to the ice melting process only, i.e. one-phase, hence we consider a material in a liquid and solid phase separated by an interface. For simplification, we assume that the liquid phase density ρL and the solid phase density ρS to be constant, i.e. ρL = ρS = ρ. What’s more, we assume a constant atmosphere temperature u0, a constant melting temperature uB and latent heat L. For each phase, we have constant thermal conductivities kL, kS, kL 6= kS and constant specific heats cL, cS, cL 6= cS. Also, we do not consider other physical effects other than isotropic heat convection in this model. We restrict our domain onto the one- dimensional semi-infinite case, i.e. Ω = [0; +1), and without any heat sink or source. We denote the interface separating the water and ice as h(t), 0 ≤ h(t) ≤ l, and we assume it to be without surface tension and of zero thickness. 2.2 The Heat Equation We have proved in lectures that, for heat conduction in an infinite rod, we can write the heat equation, i.e. one dimensional diffusion as @u @u = α : @t @x2 Now we derive the one-dimensional heat equation in this project. The melting of ice involves phase change from solid state to liquid state. As a result, molecular movement is involved in the heat conduction, which is represented by temperature. When a change of phase takes place, a latent heat L is either absorbed or released, while the temperature of the material changing its phase remains constant. The fundamental physical concept of the heat equation is the heat flux. In the references of this project, there are detailed derivation of the heat equation. Thus we leave out the proof and only state the heat equation. @u(x; t) k @2u(x; t) = (2.1) @t cρ @x2 Common boundary condition for the case Ω = [0; +1) are an imposed temperature, u(0; t) = u0(t) and lim u(x; t) = u1(t) (2.2) x!1 2.3 The Stefan Problem For our melting process, we divide the domain Ω into a solid phase and a liquid phase, separated by an interface h(t), as illustrated in Figure 1. Figure 1: Liquid and solid phases with interface h(t) and imposed boundary temperatures 2 By studying the Stefan problem, we would derive a Stefan condition, i.e. moving boundary con- dition, which is crucial in solving a Stefan problem. Here we review the derivation of the Stefan condition based on the references, but we leave out the detailed proof. The key to derive a boundary condition is to write out the total enthalpy referred to the melting temperature uB, and here we denote the cross section of Ω as A, h Z h(t) Z l i E(t) = A ρcL(u(x; t) − uB) + ρLdx + ρcS(u(x; t) − uB) + ρLdx (2.3) 0 h(t) with L as the latent heat of the material. We apply Leibniz’s integral, the heat equation to (2.3) imposed with u(h(t); t) = uB, and we get 1 dE @u(h(t); t) @u(0; t) dh(t) @u(l; t) @u(h(t); t) = k − k + ρL + k − k (2.4) A dt L @x L @x dt S @x S @x With rearrangement of heat fluxes and simplification, we have the one-dimensional Stefan condition dh(t) @u(h(t); t) @u(h(t); t) ρL = k − k (2.5) dt S @x L @x dh(t) which indicates that the velocity of the interface dt of the interface h(t) is proportional to the jump of the heat flux across the interface. This condition allows us to construct a mathematical model for a phase change process under the assumptions and simplifications in Section 2.1. Since here we only consider one-phase, i.e. the ice slab is initially solid at melting temperature uB, we have dh(t) @u(h(t); t) ρL = −k (2.6) dt L @x where kL = k denotes the thermal conductivity of the liquid phase. 2.4 Equations Sum-up Define u(x; t) as the temperature distribution, h(t) as the moving boundary. • Heat equation: @u(x; t) k @2u(x; t) = ; 0 ≤ x ≤ h(t); 0 ≤ h(t) ≤ l; t > 0 (2.7) @t cρ @x2 where c is the specific heat of the liquid phase, ρ is the density of the liquid, and k is the thermal conductivity of the liquid phase. • Stefan condition: dh(t) k @u(x; t) = − ; h(0) = 0; t > 0 (2.8) dt Lρ @x x=h(t) where L is the latent heat of melting. • Initial boundary condition: u(0; t) = u0 u(h(t); t) = uB u(0; 0) = uB where u0 is atmosphere temperature, and uB is the temperature at the interface, i.e. the melting temperature. 3 3 Nondimensionalization To nondimensionalize the heat equation and the Stefan condition derived in Section 2, we define x k u h Xe = ; τ = 2 t; Ue = ; eh = l ρcl u0 l This nondimensionalization would allow to set the problem for a finite unit ice slab 0 ≤ Xe ≤ 1, as illustrated in Figure 2. Figure 2: Liquid and solid phases with interface eh(τ) and imposed boundary temperatures in dimensionless form We can rewrite the equations into their nondimensionalized form, and we will be using this form for following calculation. • Heat equation: @u(X;e τ) @2u(X;e τ) e = e ; 0 ≤ Xe ≤ eh(τ); 0 ≤ eh(τ) ≤ 1; τ > 0 (3.1) @τ @Xe 2 • Stefan condition: dh(τ) @u(X; τ) e e e = −Ste ; eh(0) = 0 (3.2) dτ @Xe Xe=eh(τ) c∆u where Ste = L , and ∆u as the temperature difference between two phases. • Initial boundary condition: ue(0; τ) = ue0 ue(eh(τ); τ) = ueB ue(0; 0) = ueB The key thing to notice is that, we have derived a dimensionless parameter, i.e. the Stefan number, in this nondimensionalization: c∆u Ste = L which is crucial in analyzing Stefan problem. 4 Analytical Solution of One Dimensional Heat Equation After deriving the equations for this model, it is natural to find an analytical solution to it. Here, we solve for (3.1) and (3.2). However, since we have not systematically learned how to solve for 4 partial differential equation with moving boundary conditions, we only introduce the solutions learned from the references in this project. For easy understanding, let us denote ue(eh(τ); τ) = u(h(t); t) If the initial boundary position and the initial temperature distribution are h(0) = h0 (4.1) 8 erf( px ) <u − ∆u 2 t0 ; 0 ≤ x ≤ h u(x; 0) = 0 erf(λ) 0 (4.2) :uB; x > h0 where 2 Z x erf(x) = p exp(−s2)ds; (4.3) π 0 λ is the solution of the equation c∆u Ste λexp(λ2)erf(λ) = p = p (4.4) L π π 2 2 and t0 = h0 =(4λ ), then the solution of the Stefan problem is: p h(t) = 2λ t + t0 (4.5) 8 erf( px ) <u − ∆u 2 t+t0 ; 0 ≤ x ≤ h(t) u(x; t) = 0 erf(λ) (4.6) :uB; x > h(t) The key thing to notice for this analytical solution is that,p given the initial boundary position and initial temperature distribution, we wouldp find h(t) / t.