Heat Transfer Equation: Balance of Energy Explained

In summary, the conversation discusses two different equations for the heat equation, one involving a second derivative and the other involving a first derivative. The second derivative is necessary to account for the energy balance on a segment of the rod, while the first derivative only considers the heat conducted at a specific point. This can lead to different calculations for the heat generated.
  • #1
Tekneek
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In terms of Heat equation it is d2T/dx2 + qgen = qstored assuming steady state, heat flows in one direction.

But I have seen examples where energy is balanced in terms of qconv + qcond + qgen= qstored

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: qcond = -k(dT/dx). For instance, when u calculate heat generated wouldn't the answer be different? I am confused.
 
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  • #2
Tekneek said:
In terms of Heat equation it is d2T/dx2 + qgen = qstored assuming steady state, heat flows in one direction.

But I have seen examples where energy is balanced in terms of qconv + qcond + qgen= qstored

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: qcond = -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
 

Related to Heat Transfer Equation: Balance of Energy Explained

1. What is the heat transfer equation?

The heat transfer equation is a fundamental equation used in thermodynamics to describe the transfer of thermal energy between a system and its surroundings. It is based on the principle of conservation of energy and states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.

2. How is the heat transfer equation derived?

The heat transfer equation is derived from the first law of thermodynamics, which states that energy cannot be created or destroyed, only transferred or converted from one form to another. By applying this law to a closed system, we can derive the heat transfer equation by considering the different forms of energy (internal, kinetic, potential) and their changes.

3. What are the different modes of heat transfer?

The three main modes of heat transfer are conduction, convection, and radiation. Conduction is the transfer of heat through a solid or stationary fluid, convection is the transfer of heat through the movement of fluids, and radiation is the transfer of heat through electromagnetic waves.

4. How is the heat transfer equation used in real-world applications?

The heat transfer equation is used in various engineering and scientific fields, such as in the design of heating and cooling systems, thermal management of electronics, and analysis of heat exchangers. It is also used in the study of thermodynamics and heat transfer processes in nature, such as the Earth's climate system.

5. What factors affect the rate of heat transfer?

The rate of heat transfer is affected by several factors, including the temperature difference between the system and its surroundings, the thermal conductivity of the material, the surface area for heat transfer, and the presence of insulating materials. Other factors such as fluid flow, surface roughness, and heat transfer coefficients also play a role in determining the rate of heat transfer.

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