Numerical solution to the moving boundary problem for transient

The best way to implement a numerical solution of a moving boundary problem is the enthalpy method (Voller, et al., 1980). In the enthalpy method, the enthalpy of each node is used to determine its temperature using enthalpy-temperature relations. The heat transfer between nodes is calculated based on the conduction equation (Eqn. 7.1) and the temperature of the nodes. Thus, the conservation of energy will be applied directly rather than indirectly as in the finite difference method (Kakac, et al., 1993).

The enthalpy method will be discussed in the form of an algorithm. Referring to Figure 39, the entire domain (rpi<r<ro) is discretised into cylindrical shell elements of thickness Δr. The position of the element is denoted by variable j, and the radial position (rj) is taken in the centre of the element, with its boundaries at

rj+ Δr/2 and rj-Δr/2 respectively. Δr j j+1 j-1 rj ro rpi

Step 1 – Calculate position, volume and mass of each node

The first step is to calculate the position, volume and mass constants for each element, which are stored in vectors r(j), V(j) and M(j) respectively. The density of the element is taken as that of the solid AlSi12 at the melting point. The r(j), V(j) and M(j) is calculated by equations 7.10, 7.11 and 7.12 respectively.

( ) ( ) (7.10)

( ) (( ( ) ) ( ( ) ) ) (7.11)

( ) ( ) (7.12)

Step 2 – Calculate the thermal resistance between each node

In section 7.1 it has been established that the heat transfer through the entire problem is conductive. The steady state analytical solution for equation 7.1 can be implemented using the resistance method (Kakac, et al., 1993). In this sense, the heat transfer between elements is treated as quasi-steady within each time step. The thermal resistances between the nodes (Figure 40) are stored in vector R(j), and are calculated using equation 7.13.

j j+1

Figure 40 - Resistances between elements

( ) (

( ) ( ) )

(7.13)

The boundary conditions of the problem can also be represented using the resistance model for heat transfer. Generally the heat transfer at the inner pipe at rpi depends on the heat transfer mechanism on that boundary, and similarly on the

outside, ro, either way, the heat transfer coefficient of this boundary condition can be calculated with the appropriate heat transfer correlation which will be discussed along with the analysis. Convective boundary conditions are generally represented by equation 7.14 where h represents the heat transfer coefficient.

_{( )} (7.14)

The initial conditions can be initialized by using the temperature vector, T(j), which is simply the temperature of each node, or by using the internal energy vector, E(j). The internal energy vector can be used to describe an initial condition with a solid-liquid interface somewhere within the domain to represent a partial charge or discharge.

If the initial temperature, Ti, is given, then:

( ) (7.15)

If Ti is above or below the melting point, the internal energy is calculated as:

( ) ( ) ∫ ( ) if (7.16) ( ) ( ) (∫ ( ) ∫ ( ) ) if (7.17)

The initial enthalpy matrix H(j), is calculated by

( ) ( ) ( ) (7.18)

Step 4 – Calculate the temperature of each node

Essential to the enthalpy method is the temperature-enthalpy (T-H) relationships. The enthalpy-temperature relationships can be derived using a DSC machine, yielding complex T-H curves. Fortunately, eutectic materials such as AlSi12, have a very sharp melting temperature, which results in a fairly simple, linear T-H relations (Voller, et al., 1992). The T-H diagram for a eutectic of pure substance is shown in Figure 41.

Tm T (°C) H (kJ /kg) λ CpL Cps H3 H₂

Figure 41 - T-H diagram for a eutectic or pure metal

The T-H relations for the eutectic system are defined in equations 7.19 through 7.24. (7.19) ∫ ( ) (7.20) (7.21) ( ) ( ) if ( ) (7.22) ( ) if ( ) (7.23) ( ) ( ( ) ) if ( ) (7.24)

Because the whole system is essentially on the melting point, the specific heat will be taken as close to the melting temperature a possible and will be treated as a constant.

Step 5 – Re-evaluate resistances for new temperatures

If the thermal conductivity of the PCM varies significantly within the temperature range of the model, it might be necessary to re-evaluate the thermal resistances at the newly calculated temperatures.

Step 6 – Calculate the heat transfer between elements

The heat transfer between the nodes is calculated using the resistance model. The heat transfer between the elements is calculated using equation 7.25.

̇( ) ( ) ( )

( ) (7.25)

Step 7 – Calculate new internal energy and enthalpy for each node

The internal energy of each node is calculated using an energy balance on the node. The internal energy of each node is calculated using equation 7.26.

( ) ( ) ( ̇( ) ̇( )) (7.26) Step 8 – Repeat steps 5 to 7 for each time step until maximum time steps is complete.

The entire algorithm is presented in Figure 42.

Step 9 – Print results

Because equation 7.26 implies the explicit method, stability needs to be checked using equation 7.27.

(7.27)

Problem description: rpi, ri, ro, L, Δr, Δt,

Ti, hpi, ho , Cp, k, ρ

Step 1 – Calculate position, volume and mass of each node

5.7,5.8,5.9

Step 2 – Calculate the thermal resistance between each node

5.10

Step 3 – Define boundary and initial conditions

5.11,5.12,5.13,5.14,5.15

Step 4 – Calculate the temperature of each node

5.16,5.17,5.18,5.19,5.20,5.21 Stable?

(5.24) Yes

Step 5 – Re-evaluate resistances for new temperatures

5.10

Step 6 – Calculate the heat transfer between elements

5.22

Step 7 – Calculate new internal energy and enthalpy for each node

5.23 t+Δt

No

Is t>tmax

Print results Yes

E H T M R r V Q