Point heat steady versus transient

The three-dimensional delta function, $\delta^{(3)}(\vec{x})$, is a mathematical tool defined by its integral property, where integrating over a volume $V$ yields a result of 1 only if the origin ($\vec{x}=0$) is contained within $V$, and 0 otherwise. This function is utilized in heat conduction to describe the concentration of an extensive quantity, such as a heat source, precisely located at a single point $\vec{x}_0$, where the source term $\kappa$ is given by $\kappa(\vec{x})=K \delta^{(3)}\left(\vec{x}-\vec{x}_0\right)$ for a source releasing heat at rate $K$. Although using the delta function is typically a simplification, it yields reasonably good results when the source size is indistinguishable from a point relative to measurement precision. Furthermore, the simulation of these point heat sources reveals two fundamentally different physical behaviors: if the source is continuous (steady state), the system achieves equilibrium and the temperature distribution near the source is classically inversely proportional; however, if the source is an instantaneous pulse (unsteady state), the entire system remains transient as the heat spreads outwards as a thermal wave, causing the temperature at every point to rise, peak, and subsequently decay back to the ambient zero level as the heat fully dissipates.

📎IllustraDemo

A derivative illustration based on our specific text and creative direction

🏗️Structural clarification of Condensed Notes

block-beta
columns 5
CC["Criss-Cross"]:5

%% Condensed Notes

CN["Condensed Notes"]:5
RF["Relevant File"]:5
NV["Narrated Video"]:4 VO["Voice-over"] 
PA("Plotting & Analysis")AA("Animation & Analysis")KT("Summary & Interpretation") ID("Illustration & Demo") PO("Polyptych")

%% Proof and Derivation

PD["Proof and Derivation"]:5
AF("Derivation Sheet"):5
NV2["Narrated Video"]:4 VO2["Voice-over"]
PA2("Plotting & Analysis")AA2("Animation & Analysis")KT2("Summary & Interpretation") ID2("Illustration & Demo") PO2("Polyptych")

classDef color_1 fill:#8e562f,stroke:#8e562f,color:#fff
class CC color_1

classDef color_2 fill:#14626e,stroke:#14626e,color:#fff
class CN color_2
class RF color_2

classDef color_3 fill:#1e81b0,stroke:#1e81b0,color:#fff
class NV color_3
class PA color_3
class AA color_3
class KT color_3
class ID color_3

classDef color_4 fill:#47a291,stroke:#47a291,color:#fff
class VO color_4
class PO color_4

%% Proof and Derivation

classDef color_5 fill:#307834,stroke:#307834,color:#307834
class PD color_5
class AF color_5

classDef color_6 fill:#38b01e,stroke:#38b01e,color:#38b01e
class NV2 color_6
class PA2 color_6
class AA2 color_6
class KT2 color_6
class ID2 color_6

classDef color_7 fill:#47a291,stroke:#47a291,color:#47a291
class VO2 color_7
class PO2 color_7

🗒️Downloadable Files - Recursive updates

‣

Copyright Notice

All content and images on this page are the property of Sayako Dean, unless otherwise stated. They are protected by United States and international copyright laws. Any unauthorized use, reproduction, or distribution is strictly prohibited. For permission requests, please contact [email protected]

</aside>