Derivation Of Heat Equation For A One Dimensional Heat Flow

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:
Derivation Of Heat Equation For A One-Dimensional Heat Flow

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:

u_t = α²u_xx - βu + F(x,t)

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

Math

The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring
Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle
Derivation Of The Area Of An Ellipse
Derivation Of Volume Of A Cylinder
Derivation Of Volume Of A Sphere
Derivation Of Volume Of A Cone
Derivation Of Volume Of A Torus
Derivation Of Volume Of A Paraboloid
Volume Obtained By Revolving The Curve y = x² About The X Axis
Single Variable Functions
Absolute Value Functions
Conics
Real Numbers
Vector Spaces
Equation Of The Ascent Path Of An Airplane
Calculating Capacity Of A Video Adapter Board Memory
Probability Density Functions
Boolean Algebra - Logic Functions
Ordinary Differential Equations (ODEs)
Infinite Sequences And Series
Introduction To Group Theory
Advanced Calculus - Partial Derivatives
Advanced Calculus - General Charateristics Of Partial Differential Equations
Advanced Calculus - Jacobians
Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables
Advanced Calculus - Fourier Series
Advanced Calculus - Multiple Integrals
Production Schedule That Maximizes Profit Given Constraint Equation
Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation
Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions
Fourier Series
Derivation Of Heat Equation For A One-Dimensional Heat Flow
Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of matter and radiant energy. Matter is any kind of mass-energy that moves with velocities less than the velocity of light. Radiant energy is any kind of mass-energy that moves with the velocity of light.
Periodic Table
Composition And Structure Of Matter
How Matter Gets Composed
How Matter Gets Composed (2)
Molecular Structure Of Matter
Molecular Shapes: Bond Length, Bond Angle
Molecular Shapes: Valence Shell Electron Pair Repulsion
Molecular Shapes: Orbital Hybridization
Molecular Shapes: Sigma Bonds Pi Bonds
Molecular Shapes: Non ABn Molecules
Molecular Orbital Theory
More Pj Problem Strings

What is Time?
St Augustine On Time
Bergson On Time
Heidegger On Time
Kant On Time
Sagay On Time

What is Space?
Newton On Space
Space Governance
Leaders
Imperfect Leaders

Essence Of Mathematics
Toolness Of Mathematics
The Number Line
Variables
Equations
Functions

The Windflower Saga
Who Am I?
Primordial Equilibrium
Primordial Care
Force Of Being
Forgiveness

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