The ratio of thermal boundary layer thickness, 'xt', to hydrodynamic boundary layer thickness, 'x', for flow over a flat plate can be determined using the Prandtl number, NPr.

The Prandtl number is a dimensionless number that relates the momentum diffusivity (kinematic viscosity) to the thermal diffusivity of a fluid. It is defined as the ratio of the kinematic viscosity to the thermal diffusivity:

NPr = μ/α

where NPr is the Prandtl number, μ is the dynamic viscosity of the fluid, and α is the thermal diffusivity of the fluid.

The thermal boundary layer thickness, 'xt', is a measure of how the temperature of the fluid varies across the boundary layer. It represents the distance from the flat plate surface at which the temperature of the fluid reaches a certain percentage of the temperature difference between the plate surface and the free stream.

Similarly, the hydrodynamic boundary layer thickness, 'x', is a measure of how the velocity of the fluid varies across the boundary layer. It represents the distance from the flat plate surface at which the velocity of the fluid reaches a certain percentage of the free stream velocity.

The ratio of 'xt' to 'x' can be determined by considering the equations that govern the development of the thermal and hydrodynamic boundary layers. The equations for the thermal and hydrodynamic boundary layers are derived from the conservation equations for mass, momentum, and energy, and they include terms for diffusion and convection.

By comparing the equations for the thermal and hydrodynamic boundary layers, it can be shown that the ratio of 'xt' to 'x' is equal to the inverse of the Prandtl number raised to the power of one-third:

xt/x = NPr^(-1/3)

Therefore, the correct answer is option 'B', NPr^(1/3). This means that as the Prandtl number increases, the thermal boundary layer thickness becomes smaller relative to the hydrodynamic boundary layer thickness. Conversely, as the Prandtl number decreases, the thermal boundary layer thickness becomes larger relative to the hydrodynamic boundary layer thickness.