Question
Download Solution PDF\(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} = 0\) is an equation for: [where T = Temperature]
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFExplanation:
Generalised 3D conduction equation is given by Fourier equation which is:
\(\frac{{{\partial ^2}T}}{{\partial {x^2}}} + \frac{{{\partial ^2}T}}{{\partial {y^2}}} + \frac{{{\partial ^2}T}}{{\partial {z^2}}} + \frac{{\dot q}}{k} = \frac{1}{\alpha }\left( {\frac{{\partial T}}{{\partial \tau }}} \right)\)
3D, steady, state heat equation without heat generation and with constant thermal conductivity
\(\frac{{{\partial ^2}T}}{{\partial {x^2}}} + \frac{{{\partial ^2}T}}{{\partial {y^2}}} + \frac{{{\partial ^2}T}}{{\partial {z^2}}} = 0\)
\({\nabla ^2}T = 0\) ⇒ Laplace equation
Points to remember
\({\nabla ^2}T + \frac{q}{k} = 0 \Rightarrow\) Poisson equation
\({{\rm{\Delta }}^2}T = \frac{1}{\alpha }\frac{{\partial T}}{{\partial t}}\) ⇒ Fourier equation
Last updated on Jul 15, 2025
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