As a result of radioactive decay, heat is generated uniformly throughout the interior of the earth at a rate of around 30 watts per cubic kilometer. (a watt is a rate of heat production.) the heat then flows to the earth's surface where it is lost to space. let f(x, y,z) denote the rate of flow of heat measured in watts per square kilometer. by definition, the flux of f across a surface is the quantity of heat flowing through the surface per unit of time.
(a) suppose that the actual heat generation is 33w/km^3. what is the value of div f?
(a) Suppose that the actual heat generation is 27W/km3 What is the value of div F? div F- Include units)
For this case the value for div F correspond to the generation of heat.
(b) Assume the heat flows outward symmetrically. Verify that where . Find a α, (Include units.)
For this case we can satisfy this condition:
And since we have the value for the we can find the value of like this:
(c) Let T (x,y,z) denote the temperature inside the earth. Heat flows according to the equation F= -k grad T where k is a constant. If T is in °C then k=27000 C/km. Assuming the earth is a sphere with radius 6400 km and surface temperature 20°C, what is the temperature at the center? 27,0 C/km.
For this case we have this:
And grad T represent the direction of the greatest decrease related to the temperature.
So we have this equation:
And we can solve for grad T like this:
Andif we integrate in order so remove the gradient on both sides we got:
For our case we have the following condition:
And we can solve for C like this:
So then our equation would be given by:
And for our case at the center we have that
And we got: