Effect of co-ordinate system on thermal gradient calculations.xls

KNOWN: Plane wall with prescribed thermal conductivity, thickness and surface temperature.

FIND: Heat flux q'x, and temperature gradient, for the three different coordinate systems.

ASSUMPTIONS: 1) steady-state conditions.

2) One-dimensional heat flow.

3) No internal heat generation.

4) Constant properties.

Calculation Reference

Fundamentals of Heat and Mass Transfer - Frank P. Incropera

The choice of coordinate system can affect the direction and sign of thermal gradient calculations. Let's consider the three different coordinate systems commonly used in heat transfer analysis: Cartesian, cylindrical, and spherical.

1. Cartesian Coordinate System: In the Cartesian coordinate system, the thermal gradient is calculated as the change in temperature with respect to the spatial coordinate x. The heat flux, q'x, is given by Fourier's Law:

q'x = -k * dT/dx

Where k is the thermal conductivity, dT/dx is the temperature gradient in the x-direction.

2. Cylindrical Coordinate System: In the cylindrical coordinate system, the thermal gradient is calculated as the change in temperature with respect to the radial coordinate r. The heat flux, q'r, is given by Fourier's Law:

q'r = -k * (1/r) * dT/dr

Where k is the thermal conductivity, dT/dr is the temperature gradient in the r-direction, and the term (1/r) accounts for the geometry of the cylindrical coordinate system.

3. Spherical Coordinate System: In the spherical coordinate system, the thermal gradient is calculated as the change in temperature with respect to the radial coordinate r. The heat flux, q'r, is given by Fourier's Law:

q'r = -k * (1/r^2) * dT/dr

Where k is the thermal conductivity, dT/dr is the temperature gradient in the r-direction, and the term (1/r^2) accounts for the geometry of the spherical coordinate system.

It is important to note that the signs and specific form of the temperature gradient and heat flux equations depend on the coordinate system being used. The choice of coordinate system should align with the geometry and symmetry of the problem at hand to accurately calculate the thermal gradients and heat fluxes.