Newton's And Fourier's Cooling Laws Applied To Heat Flow Boundary Conditions

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Newton's And Fourier's Cooling Laws Applied To Heat Flow Boundary Conditions

Many important physical phenomena can be modeled as problems of systems of *partial differential equations* (PDEs) or ordinary differential equations (ODEs). Usually, the mathematical expressions of the *initial conditions* (IC) and *boundary conditions* associated with a particular problem are stated with the PDEs or ODEs. The PDE, BC and IC, together constitute an *Initial-Boundary-Value-Problem* (IBVP).

Consider the laterally insulated one-dimensional copper rod with length L (figure 14.12(a)), the ends of which are enclosed in containers of liquids at temperatures described by the functions *g _{1}(t)* and

Use Newton's and Fourier's cooling laws to express the boundary conditions at the ends of the rod in mathematical terms.

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S_{7}P_{2}A_{21} (Physical Property)
**The math**:

Pj Problem of Interest is of type *identity*. Problems of mathematical modelind are *identity* problems. The primary interes is to identitify the mathematical structure of the physical problem being modeled. It is in this sense that the Pj Problem of Interest is of type *identity*.

**Newton's Law Of Cooling**:
**Outward flux of heat (at x= 0) = h[u(0,t) - g _{1}(t)]**-------(1)

Where,

Outward flux of heat across a boundary is proportional to the inward normal derivative across the boundary (figure 14.12(b)):

where k = thermal conductivity of rod.

Equating equations (1) and (3):

k∂u(0,t)/∂x = h[u(0,t) - g

So,

Equating equations (2) and (4):

k∂u(L,t)/∂x = h[u(L,t) - g

So,

Equations (5) and (6) express the boundary conditions in mathematical terms.

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.

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Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

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