Encyclopaedia Index

### SPECIFIC HEATs at constant pressures, CP1 and CP2

CP1 and CP2 are used in PHOENICS in two ways, namely:
1. for deducing the material temperature from its enthalpy, when the latter is the solved-for variable; and
2. for deducing the influence of heat sources on the material temperature, as it flows through space and time, when the temperature is the solved-for variable.

The details of the first use may be inspected in the open-source Fortran sub-routine GXTEMPR.HTM, where the formula used depends upon the value ascribed to TMP1 or TMP2, etc. If an In-Form statement for TMP1 exists in the Q1 file however, this will over-write whatever GXTEMPR computes.

The second use is expressed internally within the EARTH solver by coding which implies that the enthalpy is equal to the specific heat times the absolute temperature. This "effective" specific heat, Cp_eff, is defined as the enthalpy of the material at the prevailing temperature minus the enthalpy of the same material when the temperature is zero on the currently-used temperature scale.

Thus:

Cp_eff = (H - H_0) / Tabs
where h_0 is the temperature at absolute zero

This differs from the conventional specific heat at constant pressure, Cp, which is defined as the rate of change of enthalpy with temperature; but it has the same dimensions, and is of the same order of magnitude.

The following diagram explains the relation between the quantities.

```
|                   .*    ---------- H
|                 . *
enthalpy           .  * |        The *** curve represents the
|             .  * ^ |        enthalpy-versus-temperature
|           .  *  ^  |        relationship.
|         .  *   ^
|       .  *    ^    T        The ^^^ line represents the
|     .  *     ^              tangent at the working point
|   .  *                      H~T, and has the slope Cp.
| . *
.*    -------- H_0            The ... line has the slope
|                             Cp_eff used by PHOENICS
|
*----------------------------------------------------------
0      Tabs, Absolute Temperature
```

The following remarks may be helpful:

1. The reason for using Cp_eff rather than Cp is computational and internal-coding economy: for the above equation makes if easy to deduce temperature from enthalpy or enthalpy from temperature.

2. The temperature which is deduced from enthalpy (because the latter is the solved-for quantity) is conventionally given the name TMP1 or TMP2, according to phase.

3. The temperature which is solved for directly is conventionally given the name TEM1 or TEM2, according to phase. Then the enthalpies are the derived quantities.

4. Temperature can be measured on any scale chosen by the user; but the absolute-temperature Kelvin scale and the Celsius scale (the zero of which is 273 on the Kelvin scale) are the most common.

5. The value of H_0 depends of course on which temperature scale is in use, as does the value of Cp_eff.

6. Cp_eff and Cp are identical when the H-versus-T is linear, as is often very close to the truth.

7. They are also nearly equal when, though the H-versus-T relation is not linear, the working point is close to the zero location of the temperature scale.

8. When the material in question does not engage in chemical reaction, it is permissible to set H_0 to zero.

9. H_0 is deduced from the setting of TMP1 or TMP2. It can also be set by ascribing values to the STOREd 3D variables H0_1 and H0_2 in GROUND or via In-Form.