High-Temperature Flows: Basic Examples

Overview

The high temperature effects on thermodynamic properties were discussed at length in Chapter 16 and in this chapter, the principles of high-temperature flow will be presented and.

Chapter Roadmap

Sections

Introduction to Local Thermodynamic and Chemical Equilibrium

The concepts thermodynamic and chemical equilibrium are introduced and defined. The difference between local and complete equilibrium is discussed in some detail.

Equilibrium Normal Shock Flows

Will we be able to use the relations obtained in Chapter 3 to analyze a normal shock when the shock is strong enough for high-temperature effects to be important? The governing equations derived in Chapter 3 are valid for all gases but in the examples in the same chapter, calorically perfect gas was assumed. For the high-temperature case we will have to use other gas relations.

Equilibrium Quasi-One-Dimensional Nozzle Flows

In this section the relations derived in Chapter 5 are revisited. The quasi-one-dimensional relations are relevant to study at elevated temperatures since one of the main applications studied in Chapter 5 was the convergent divergent nozzle, which if used as the propelling unit of a space launcher will operate at really high temperatures. Again, the governing equations derived in Chapter 5 can be used for the analysis since no assumptions regarding gas properties were made in the derivation. However, as in the normal shock case, the gas model has to be adjusted to account for the high-temperature effects.

It is shown that equilibrium chemically reacting flow through the nozzle is isentropic (unless normal shocks appear inside the nozzle of course).

Frozen and Equilibrium Flows: Specific Heats

Equilibrium flow implies infinite chemical and vibrational rates. In cases where this is not true, other gas models will have to be used. This section presents a few examples where this is the case.

Equilibrium Speed of Sound

The speed of sound is a thermodynamic property that is very important for the analysis of compressible flows. For calorically perfect gases the speed of sound can be calculated as \(a=\sqrt{\gamma RT}\) but for equilibrium gases it is not as straight forward.

Study Guide

The questions below are intended as a "study guide" and may be helpful when reading the text book.

  1. Assume that you would try to predict the stagnation temperature for a reentry vehicle approaching earth at a Mach number of 32.5 using a model based on a calorically perfect gas assumption. At this Mach number there will be a strong shock in front of the vehicle and with the calorically perfect gas assumption you will severely overestimate the temperature ratio over the shock - explain why.
  2. Define thermodynamic and chemical equilibrium.
  3. Explain the concepts local thermodynamic equilibrium and global thermodynamic equilibrium.
  4. Explain the concepts local chemical equilibrium and global chemical equilibrium.
  5. What gas model is applicable in each of the cases listed below?
    • local thermodynamic equilibrium and local chemical equilibrium
    • local thermodynamic equilibrium and chemical non-equilibrium
    • local thermodynamic equilibrium and frozen composition
    • thermodynamic non-equilibrium and frozen composition
  6. If the flow is both chemically and vibrationally frozen we will be able to treat the flow in the same way as if it would have been calorically perfect - explain why.
  7. Is equilibrium chemically reacting flow through a convergent-divergent nozzle without the presence of normal shocks isentropic?
  8. Can we still use the area-velocity relation for convergent-divergent nozzle flows derived in Chapter 5 at elevated temperatures?
  9. Consider the normal shock relations derived in Chapter 3. What are the differences when analyzing a normal shock strong enough for high-temperature effects to be important?
  10. High temperature effects in compressible flows are found when analyzing for example very strong shocks or nozzle flows with extremely high total pressure and total enthalpy. What is the root cause of these effects and what do we mean by equilibrium gas? What kind of thermodynamic relations are needed to compute the flow of equilibrium gas?
  11. Using equilibrium gas assumption in the analysis of chemically reacting nozzle flow will lead to higher exhaust temperatures than if calorically perfect gas assumption is used for the same analysis. Explain why.