Does an object storing more internal energy emit more thermal radiation? There are two objects of the same material, similar geometry but different dimensions and same temperature. The larger object is supposed to contain more quantity of heat because it has larger volume.
Now I know that the quantity of emitted thermal radiation is directly proportional to temperature, but is it also regarding to internal energy? 
How do I know which object emits more thermal radiaion? (without measurement)
 A: If you just mean the blackbody radiation, then that the radiated power both depends on the object's surface area, $A$, and its temperature to the forth power, $T^4$, according to $P_{\rm rad} = \epsilon\sigma A T^4$, where $\epsilon$ is the object's emissivity (material dependent) and $\sigma=5.67\times10^{-8}{\rm\,W\,m^{-2}\,K^{-4}}$ is the Stefan-Boltzmann constant. Hence it's the surface area which determines the radiation.
Since two objects can have the same volume but vastly different surface areas (think of a sheet of paper which has a large surface area but a very small volume), the thermal radiation is not dependent on the volume. However if you have two objects of the same shape, e.g. two cubes or two spheres, then the surface area scales with the volume as $A\propto V^{2/3}$, which would then also affect the thermal radiation, but only for two object of the same shape.
To answer your final question: If you have two objects at the same temperature and material, then the one with the larger surface area will radiate more energy.
A: @Andreas Sundstrom is correct that the equation for radiation does not depend on the volume (mass) of the radiating object. But that is because the equation assumes the temperature of the emitting body is constant. 
As the body radiates heat its internal energy is reduced thus reducing the temperature of the body and the rate of radiant heating. How quickly this occurs depends, of course, on the mass of the body and its specific heat and the variables in the radiant heat transfer equation. I should think the temperature of a sheet of paper would would drop quickly due to its low mass (volume).
Bottom line, to the extent that the temperature of the radiating body depends on its internal energy and the rate of radiant heat transfer depends on temperature, the rate of radiant heat transfer depends on the volume of the emitting object.
Hope this helps
