Question

In the energy balance formulation of the finite difference method, it is recommended that all heat transfer at the boundaries of the volume element be assumed to be into the volume element even for steady heat conduction. Is this a valid recommendation even though it seems to violate the conservation of energy principle?

Short answer

Short Answer: Assuming all heat transfer at the boundaries of the volume element to be into the volume element, even for steady heat conduction, is a valid recommendation as it does not violate the conservation of energy principle. While it may appear to contradict conservation of energy at first glance, it simplifies the problem and allows for a tractable solution to solving heat conduction problems by accounting for both the heat entering and leaving the volume element. In steady heat conduction, the net heat transfer is zero, and this assumption helps establish energy balances for each volume element in the finite difference method.

Step-by-step solution

Understand the conservation of energy principle

The conservation of energy principle states that energy cannot be created or destroyed, but can only be converted from one form to another. In the context of steady heat conduction, the rate at which energy enters a volume element is equal to the rate at which it is conducted away from it. Therefore, if the energy balance is maintained, it means that the conservation of energy principle is not violated.

Examine the energy balance formulation of the finite difference method

The finite difference method is an approach to solve heat conduction problems by discretizing the domain and breaking it into smaller volume elements. The energy balance formulation considers the energy entering and leaving these volume elements, and it assumes that all heat transfer occurs at the boundaries of the volume elements.

Analyze the recommendation

The recommendation is to assume that all heat transfer at the boundaries of the volume element is into the volume element, even for steady heat conduction. While this assumption seems to violate the conservation of energy principle at first glance, it is actually a reasonable simplification and approximation method to make the heat conduction problem more tractable.

Demonstrate why the recommendation is valid

In steady-state heat conduction, the temperature within each volume element remains constant over time and the net heat transfer into the volume element is zero. However, at the boundaries, there still exists heat flow from one part to another, and this heat flow must be accounted for to quantitatively analyze the heat conduction problem. By assuming that all the heat transfer is into the volume element, we are essentially accounting for both the heat entering and leaving the element. For steady heat conduction, the amount of heat that enters the volume element is equal to the amount of heat that leaves it. In that sense, this assumption does not violate the conservation of energy principle as the net heat transfer is zero. With this assumption, we can easily establish energy balances for each volume element, which enables us to solve for temperature distributions and heat transfer rates across the domain, making it a valid recommendation.

Conclusion

The recommendation to assume all heat transfer at the boundaries of the volume element be into the volume element, even for steady heat conduction, is a valid one, as it does not violate the conservation of energy principle. Instead, it offers a simplification that helps us solve complex steady heat conduction problems more easily.