- #1

wmrunner24

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## Homework Statement

Sketched below is a solid, truncated cone, with a side profile of y=Cx. Based on the geometry, the area (in m

^{2}) of the left (truncated, x = 1m) and the right face of the cone is 4∏

and 36∏, respectively. The temperature of the left face is T

_{1}=50°C and T

_{2}=30°C. Assuming that the system is one-dimensional, steady state, and has constant thermal conductivity k=1/∏, answer the following questions:

1) What is the actual value of "C"? What is the location of the right face?

2) Show that the temperature profile in the cone can be described as T(x)=a(1/x)+b. Determine the values of a and b.

3) Determine the expression of heat flux q"(x) and the heat transfer rate rate q(x), as a function of x.

## Homework Equations

Heat Equation:

[itex]\frac{∂}{∂x}[/itex](k[itex]\frac{∂T}{∂x}[/itex])+[itex]\frac{∂}{∂y}[/itex](k[itex]\frac{∂T}{∂y}[/itex])+[itex]\frac{∂}{∂z}[/itex](k[itex]\frac{∂T}{∂z}[/itex])+q = ρc

_{p}[itex]\frac{∂T}{∂t}[/itex]

## The Attempt at a Solution

The main issue I'm having is with Part 2.

Under the assumptions of constant k, no internal heat generation, and steady-state one-dimensional heat flow along the axial direction x, the heat equation simplifies to:

[itex]\frac{∂^2T}{∂x^2}[/itex] = 0

There are two boundary conditions prescribed:

T(1m) = 50°C and T(3m) = 30°C (3m was obtained as the x-position of the right face of the cone during Part 1)

My issue begins after integrating the relevant form of the heat equation twice. Once integrated, the temperature will fit the form:

T(x) = ax + b

Which is not the form outlined in the statement of the problem. Why does this happen? Is it an error of my own or in the formulation of the problem statement?

Any help anyone can provide will be greatly appreciated.