Question

The partial differential equation for two-dimenisional steady-state conduction assuming no volumetric thermal energy generation and constant propertiesis ∂²T/∂x²+∂²T/∂y² =0 The following temperature distribution is valid under the stated conductions. T=ax+by

a. True.
b. Fatse.
c. It is not possible to determine whether the profile is valid.

Answer 1

The temperature distribution T = ax + by satisfies the partial differential equation for two-dimensional steady-state conduction without thermal energy generation (∂²T/∂x² + ∂²T/∂y² = 0), therefore it is valid.

Step-by-step explanation:

The partial differential equation for two-dimensional steady-state conduction, assuming no volumetric thermal energy generation and constant properties, is given by ∂²T/∂x² + ∂²T/∂y² = 0. To verify if the given temperature distribution T = ax + by is valid under the stated conditions, we would need to insert it into the equation and check for equality:

• Calculating the second partial derivative with respect to x, we get ∂²T/∂x² = ∂²(ax)/∂x² = 0, since a is a constant.
• Calculating the second partial derivative with respect to y, we get ∂²T/∂y² = ∂²(by)/∂y² = 0, since b is a constant.
• Adding these results, we find 0 + 0 = 0, which satisfies the original differential equation.

So, the answer is option (a), as T = ax + by is a valid temperature distribution under the conditions described.