Derivation Of Heat Equation For A One Dimensional Heat Flow
TECTechnics Classroom   TECTechnics Overview

Derivation Of Heat Equation For A One Dimensional Heat Flow

The flow of heat is a consequence of temperature gradient. Consider the one-dimensional rod of length L in figure 8.105. The following assumptions apply to the rod:
(1) The rod is made of a single homogeneous conducting material
(2) The rod is laterally insulated, that is, heat flows only in the x-direction.
(3) The rod is thin, that is, the temperature at all points of a cross section is constant.
(4) The principle of the conservation of energy can be applied to the heat flow in the rod.

(a) Derive the heat equation for a one-dimensional heat flow.
(b) How does the heat equation change if the rod is not laterally insulated, the surrounding is kept at zero, and the heat flow in and out across the lateral boundary, is at a rate proportional to the temperature gradient between the temperature u(x,t) in the rod and its surrounding.

The strings:

S7P4A41 (Heat Flow - Linear Motion)

The math:

Pj Problem of Interest is of type motion because it is the flow of the heat that is of primary interest.

(a) Derivation Of Heat Equation:

F(x,t) = (1/cρ)f(x,t) = Heat source density.

(b) When rod is not laterally insulated under the given conditions, heat equation becomes:

ut = α2uxx - βu + F(x,t)

Where, u = u(x,t)
ut = ∂u/∂t;
ux = ∂u/∂x;
uxx = ∂2u/∂x2
β = the rate constant for the lateral heat flow.(β > 0).

Blessed are they that have not seen, and yet have believed. John 20:29

TECTechnic Logo, Kimberlee J. Benart | © 2000-2021 | All rights reserved | Founder and Site Programmer, Peter O. Sagay.