Why does the heat equation use second derivatives instead of first derivatives?

[Tekneek]

In terms of Heat equation it is d²T/dx₂ + q_gen = q_stored assuming steady state, heat flows in one direction.

But I have seen examples where energy is balanced in terms of q_conv + q^cond + q_gen= q_stored

Why is the first equation just not the first derivative of Temperature with respected to x?

Why does one equation have second derivative while the other has just one (for example: q_cond = -k(dT/dx). For instance, when u calculate heat generated wouldn't the answer be different? I am confused.

 

[Chestermiller]

In terms of Heat equation it is d²T/dx₂ + q_gen = q_stored assuming steady state, heat flows in one direction.

But I have seen examples where energy is balanced in terms of q_conv + q^cond + q_gen= q_stored

Why is the first equation just not the first derivative of Temperature with respected to x?

Why does one equation have second derivative while the other has just one (for example: q_cond = -k(dT/dx). For instance, when u calculate heat generated wouldn't the answer be different? I am confused.

In the first equation your wrote, the term involving the second derivative of T represents an energy balance on a segment of the rod between x and x + dx. So you have to subtract the heat conducted in at x (involving a first derivative of T) from the heat conducted out at x + dx (also involving a first derivative of T) to get the net heat accumulated between x and x + dx. This leads to the second derivative.

Chet