The concept of heat transfer coefficients is a construction relating the heat flux to a temperature difference,usually the wall temperature (Ts) and the mixing-cup (or free-stream) fluid temperature (Tb), i.e. q1=h*(Ts-Tb). This is known as Newtons law of cooling. On the other hand, the local heat flux is also determined by Fourier's law, stating that the heat flux is proportional to the fluid thermal conductivity (k) and the temperature gradient at the surface (dT/dy|s), i.e. q2=-k*dT/dy. It is required that q1=q2. If, for instance you have a situation of fluid heating (i.e. Ts>Tb and dT/dy|s

Given that heat flux, h, is  defined to be in the direction hot to cold and the gradient of temperature, T, is in the direction of cold to hot, the usual relation is h=-k grad(T) where k > 0. So it follows that heat transfer coefficient, k

The concept of heat transfer coefficients is a construction relating the heat flux to a temperature difference,usually the wall temperature (Ts) and the mixing-cup (or free-stream) fluid temperature (Tb), i.e. q1=h*(Ts-Tb). This is known as Newtons law of cooling. On the other hand, the local heat flux is also determined by Fourier's law, stating that the heat flux is proportional to the fluid thermal conductivity (k) and the temperature gradient at the surface (dT/dy|s), i.e. q2=-k*dT/dy. It is required that q1=q2. If, for instance you have a situation of fluid heating (i.e. Ts>Tb and dT/dy|s

In general, the heat transfer coefficient for a steady setup should be strictly positive, as Peter Bollada points out. Apart from unsteady examples as the one given by Erling Naess, a negative heat transfer coefficient may just be the result of a heat flux modelled solely using Newton's or Fourier's law in a situation where these are not the only, or not the dominant mechanisms, e.g. when the dufour effect (heat flux due to a concentration gradient) or the thermoelectric effect (heat flux due to electric potential gradients) are existing and significant.

@ Peter, thanks for the update. What Erling mentioned is exactly what I encountered.

@ Erling, thanks for the update. 

What you specified is exactly what I encountered. So, you mean there is no physical significance of negative heat transfer coefficient. But still it is confusing.

@ Christian, thanks for the update

A negative heat transfer coefficient just implies that the heat flux is directed opposite to the temperature difference (Ts-Tb). You may also encounter an infinite heat transfer coefficient even though the heat flux is finite (meaning that the overall temperature difference is zero). The choice of Tb being either mixing-cupor free-stream temperature is made forconvenienceand may, in some instances,cause funny behavior of the associated heat transfer coefficient.

@ Erling, thanks.

h=q/(Ts-Tb)

If Ts-Tb value is negative, then h value will be negative.

It means Heat is from surroundings to Surface .

it implies the rate of change of temperature has an in flexion point. It implies an periodical variation ( irregular or regular) of temperature ( the sigmoidal shape). It happens when the coolant mixes with gas.