  In this paper we study quantum radiation from an evaporating spherically
symmetric non-singular black hole. We used a modified Hayward metric for a
description of a non-singular black hole. We assume that the mass parameter of
this metric depends on the advanced time, and choose this dependence so that it
properly reproduces both black hole formation and its subsequent evaporation.
We consider a quantum massless scalar field propagating in this geometry and
use 2D approximation for the calculation of the quantum average of the
stress-energy tensor. For the calculation of this quantity it is sufficient to
find a map between the Killing times $u_+$ and $u_-$ at the future and past
null infinities, established by the propagation of the radial null rays. In
this formalism the quantum energy flux at the future null infinity can be
expressed in terms of the function $u_+(u_-)$ and its derivatives. We developed
a special formalism, which allows one to reduce the problem of the calculation
of the quantum energy flux and other observables to a solution of a simple set
of ordinary differential equations. We used this approach to study quantum
effects in two cases: i) with the trivial, $\alpha=1$ and ii) the non-trivial,
$\alpha\ne1$, redshift function. In both cases there exists an outburst of the
quantum energy radiation from the inner domain of the black hole, close to the
inner part of its apparent horizon. For $\alpha=1$ this outburst is
exponentially large. It is a direct consequence of the so-called mass inflation
effect. We showed, that this severe problem can be solved by a proper choice of
the redshift function. However, even in this case the emitted energy can be
much larger than the initial mass of the evaporating black hole. This means
that for a construction of a self-consistent model of a non-singular
evaporating black hole the back-reaction effects are highly important.
