The present work investigates the energy equation of a general fluid, Newtonian or non-Newtonian, with variable thermal conductivity in turbulent flow. The usual energy equation, without the dissipation terms, is taken into account with the fluctuating terms in the temperature as well as in the thermal conductivity. The energy equation is written for the average temperature, for the fluctuating temperature one as well as for the square of the fluctuating temperature. Besides the usual Reynolds stresses, a new term appears, which is the product of the fluctuation of the thermal conductivity and the gradient of the temperature fluctuation. This new term is interpreted and introduced in the energy equation where the variable is the square of the temperature fluctuation where new terms appear. A possible physical interpretation is given to the different terms. Assuming a polynomial relation between thermal conductivity and temperature it is then possible to write an expression for the average and the fluctuating thermal conductivity. The expressions are then simplified on the basis of physical and mathematical considerations. Specifically, the heat flux due to the fluctuating thermal conductivity is then expressed as the product of the derivative of the thermal conductivity with the mean temperature to the gradient of the square of the temperature fluctuation. Further considerations allow to write a new energy equation of the average temperature which include the new term. The solution of this energy equation is possible with the coupled solution of the equation for the square of the fluctuating temperature. The introduction of this new term in the energy equation can be of some importance in problems related to liquid metals flowing in turbulent flow and/or in very low temperature applications where the thermal conductivity becomes very high.

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