A Method for Efficient Evaluation of Temperature using the NASA Polynomials

Modeling variable specific heat is a crucial aspect of aerospace engineering, playing a significant role in designing and analyzing high-enthalpy internal and external flows. Moreover, variable specific heat effects are becoming progressively more important with the increasing interest in hypersonic flight, where accurate temperature prediction is critical to determine chemical reaction rates and heat flux. Similarly, the influence of variable specific heat is significant in combustion, where temperature predictions are necessary to determine chemical reaction rates and other quantities of interest. The NASA polynomials are frequently employed in computational simulations to model these variable specific heat gases, which are a function of temperature only. One challenge when using NASA polynomials is determining the temperature T given the static enthalpy h calculated from the energy balance. As a result, a non-linear root-finding problem for T given h arises. Solving this non-linear root-finding problem can be computationally expensive and unstable, especially for gradient methods due to the high-order terms. Therefore, an approximate lower-order polynomial has particular advantages over iteration. Specifically, a known closed-form expression avoids iteration and provides an unconditionally stable evaluation of the roots. The roots may then be examined to identify the most appropriate choice from the available roots, i.e., the root must be real, positive, and contained in the appropriate temperature range of the polynomial used. Roots are assessed for suitability, i.e., real, positive, and within the polynomial's valid temperature range. This note briefly reviews NASA polynomials. A projection-based technique defines an approximate polynomial to the original NASA polynomials. This approximate polynomial determines the temperature T given the enthalpy h when utilizing a thermally perfect ideal gas model.