Adiabatic wall temperature is the temperature acquired by a wall in liquid or gas flow if the condition of thermal insulation is observed on it: (∂T/∂n)_{w} = 0 or
= 0. It is denoted as either T_{r} or T_{eq}, and is sometimes called an *equilibrium temperature* and, in aerodynamics, a *recovery temperature*. A distinction between the adiabatic wall temperature and a characteristic flow temperature may depend on the dissipative heat release in the boundary layer, on the existence of a different nature, in the flow of internal heat sources and on the thermal effect of other bodies (walls). In this case, if there is heat transfer between the wall and the flow, i.e., at
= 0, the temperature field in the fluid can be represented as a superposition of the temperature field at
= 0 on the natural field, which would be produced by the wall in the absence of disturbing factors, i.e., internal heat release or the effect of other walls. This superposition makes it possible to represent the law of heat transfer between the flow and the wall as:

where T_{r} is the adiabatic wall temperature for given conditions of the problem and α_{0} the heat transfer coefficient under undisturbed conditions.

The concept of adiabatic wall temperature is used in the field of high velocity aerodynamics. The temperature profile in the boundary layer of a high-velocity gas flow over an adiabatic surface is displayed in Figure 1 (Curve 1).

Curve 2 in Figure 1 shows a typical distribution of temperature for the case where heat is being added through the surface and Curves 3 and 4 are typical of cases where heat is being removed from the fluid via the surface. It is obvious that T_{r} > T_{∞}, i.e., an adiabatic wall is heated in relation to the thermodynamic temperature of the flow or, in other words, aerodynamic heating of the surface immersed in the flow occurs. The adiabatic wall temperature is determined by the relation:

where T_{∞} is the thermodynamic temperature, u_{∞} the velocity of the external flow and c_{p} the isobaric heat capacity of fluid. For a compressible gas, the equation is:

where γ is the isoentropic exponent, Ma_{∞} is the Mach number in the external flow and r is the *recovery coefficient*. r is not a constant but depends, in particular, on the character of the flow on the surface, the flow regime, and the thermal properties of the medium. For some simple cases, its value can be estimated as follows: At the front stagnation point of bodies in the flow, r = 1; in a laminar boundary layer on a plane plate,
for Prandtl numbers 0.5 < Pr <10; in a turbulent boundary layer on a plate,
for Prandtl numbers close to 1. These estimates, with small corrections, also hold for gases flowing in pipes. For supersonic flows when variations of the thermal properties of gas become significant, the above relations hold for the enthalpy field

In estimating r, the Prandtl number is chosen at the reference *Eckert enthalpy* h_{*} = h_{∞}0.5(h_{w} − h_{∞}) + 0.22(h_{r} − h_{∞}) .

#### REFERENCES

Dorrance, W. H. (1962) Viscous Hypersonic Flow. Theory of Reacting and Hypersonic Boundary Layers, McGraw-Hill, New York.

Echert, E. R. G. and Drake, R. M. (1972) Analysis of Heat and Mass Transfer, McGraw-Hill, New York.

#### References

- Dorrance, W. H. (1962) Viscous Hypersonic Flow. Theory of Reacting and Hypersonic Boundary Layers, McGraw-Hill, New York.
- Echert, E. R. G. and Drake, R. M. (1972) Analysis of Heat and Mass Transfer, McGraw-Hill, New York.

Heat & Mass Transfer, and Fluids Engineering