Thermal shock arises when a solid at uniform temperature T_{0} is suddenly brought in contact with a fluid at a different temperature T_{f}. If the solid is small, it may be assumed that its temperature will remain uniform throughout the solid although it will change with time. It can be shown that this variation is given by the equation,

where T is the temperature of the solid at time t, A and V are the surface area and volume of the solid, respectively, ρ is the density, α is the surface heat transfer coefficient and c the specific heat capacity of the solid. The above equation may be expressed in terms of the Biot and the Fourier Numbers by noting that V/A is a linear dimension which can be denoted by L,

where λ is the thermal conductivity of the solid.

When the internal resistance of the solid is not negligible, the temperature varies with position as well as with time and may be represented by

Solutions for simple geometries may be found in Heisler (1947) and Schneider (1963). In the case of one-dimensional heat flow an analytical solution is possible but more general cases require numerical analysis (see also Conduction.)

In the presence of temperature differences within the solid, thermal stresses will arise, increasing in magnitude as the temperature gradient increases. Plastic deformation or fracture may result. Consider the simple example, illustrated in Figure 1, of a system consisting of three bars fixed to rigid cross members at the ends. If the central bar is plunged into a large liquid bath of temperature Tf, and it is assumed that the internal resistance of the bars is negligible, the temperature history is given by Equation (1) and the thermal stress is

where β is the coefficient of thermal expansion and E is Young's modulus.

The thermal stress is thus seen to depend on the Biot and Fourier numbers. A similar situation arises when the solid is in the form of, say, a thick slab, in which case the surface temperature rises rapidly to approach that of the fluid while the rest of the body lags behind. The thermal stresses resulting will be compressive at the surface and tensile in the interior of the body.

The magnitude and spacial distribution of the thermal stresses depend on the temperature field which, in turn, depends on the Biot and Fourier numbers. As a general rule, the greater the conductivity of the solid, the smaller the temperature gradient and hence the thermal stresses. Materials with poor thermal conductivity, such as ceramics, are prone to fracture under thermal shock while good conductors, such as metals, can withstand more severe shocks without any significant ill effects.

See also Johns (1965) for the calculation of thermal stresses.

#### REFERENCES

Heider, M. P. (1947) Temperature charts for induction and constant temperature heating, *Trans. Am. Soc. Mech. Eng.*, 69, 227–36.

Schneider, P. J. (1963) *Temperature Response Charts.*

Johns, D. J. (1965) *Thermal Stress Analyses*, Pergamon.