CADFamily.com: FIDAP-Modeling Melting Apparent Specific Heat

Melting can be modeled by assuming that the change of state from solid to liquid takes place in a narrow range of temperature rather than at a fixed temperature. The latent heat of fusion is incorporated into the apparent specific heat by considering it as a continuous function of temperature. For example, say ice at -3⁰C is heated to water at 3⁰C. The break down of this process and the amount of heat required (per unit mass) for each step is:

1. Ice at -3⁰C is heated to ice at 0⁰C: (1)

2. Ice at 0⁰C changes to water at 0⁰C (phase change): where l is the latent heat of vaporization (2)

3. Water at 0⁰C is heated to water at 3⁰C: (3)

We approximate the heating process by introducing an apparent specific heat capacity value for the phase change:

(4)

Here c_p,max is the maximum value of c_pa, the apparent specific heat and T_max is the temperature at which this value is attained. It is equal to 0⁰C, the melting temperature of ice, in this case.

This is shown in Figure 1. The initial straight line portion of the curve parallel to the x-axis represents the part where ice is heated to a temperature of -1⁰C and the value on the y-axis gives the specific heat capacity for ice, . This is equal to.

The second and third part of the curve show that the specific heat increases to a very high value, c_pa,max and then falls downs. A linear dependence of specific heat on temperature has been assumed and is obtained by solving the following equations:

(5)

(6)

(7)

(6) & (7) are equations of the straight lines 2 and 3 respectively shown in Figure 1. In this case, T₁ and T₂ are assumed to be -1⁰C and 1⁰C respectively and the value of c_pa,max is found out by solving equations (5) through (7) knowing the values of c_pi (specific heat capacity of ice) and c_pw (specific heat capacity of water). Finally the dependence of c_pa1 and c_pa2 is obtained by substituting the value of c_pa,max in equations (6) and (7).

The last portion of the curve represented by 4 is the specific heat capacity of water which is assumed to be constant with a value of 4.18 kJ kg^-1 K^-1.

Figure 1: Apparent specific heat (kJ kg^{-1 0}C^-1)as a function of temperature (⁰C)

Modeling this in PreSTO is quite simple now. We will follow the same approach as we did for the Cryosurgery Tutorial. Specific heat is entered as a function of temperature by specifying a set of values of specific heat and temperature to represent the curve shown in Figure 1.

1) Open the Material Properties window: Define >> Material Properties >> Under Entity, select the entity for which you want to specify the properties of melting.

2) In the drop-down menu next to Specific Heat, select Temp. Curve. The Specific Heat Inputs window opens up. We will specify the specific heat of the material as a function of temperature to incorporate melting.

3) In the Number of Data text field, type the number of data points you want to specify. Let us specify 10 for this example. Remember here that the maximum number of data points that can be entered is 10 in PreSTO.

4) Click on Show. The specified number of Temperature- Specific Heat value text fields opens up.

5) Enter the values of temperature and specific heat as obtained from the equations earlier.

6) Click on Apply.

7) In the Material Properties window, click on Apply

8) Click on Close