Question

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.

Answer 1

a)
`div F = 27 (W)/(km^3)`

b)
`\alpha=(27 W/km^3)/(3)= 9(W)/(Km^3)`

c)
`T(0,0,0)=-(10)/(2(27000)) (0) +7605.185=7605.185`

Explanation:

(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.

`div F = 27 (W)/(km^3)`

(b) Assume the heat flows outward symmetrically. Verify that
`F= \alpha r` where
`r=xi +yj+zk`. Find a α, (Include units.)

For this case we can satisfy this condition:

`div[\alpha (xi +yj +z k)]]=\alpha(1+1+1)=3\alpha`

And since we have the value for the
`div F` we can find the value of
`\alpha` like this:

`\alpha=(27 W/km^3)/(3)= 9(W)/(Km^3)`

(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:

`F =-k grad T`

And grad T represent the direction of the greatest decrease related to the temperature.

So we have this equation:

`10(xi +yj+zk)=-27000 grad T`

And we can solve for grad T like this:

`grad T = -(10)/((27000)) (xi+yj+zk)`

Andif we integrate in order so remove the gradient on both sides we got:

`T=-(10)/(2(27000)) (x^2 +y^2 +z^2) +C`

For our case we have the following condition:

`x^2 +y^2 +z^2 = 6400 , T=20 C`

`T=-(1)/(54000) (6400^2)+C =20`

And we can solve for C like this:

`C= 20+(6400^2)/(5400)= 7605.185`

So then our equation would be given by:

`T=-(10)/(2(27000)) (x^2 +y^2 +z^2) +7605.185`

And for our case at the center we have that
`x^2+ y^2+ z^2 =0`

And we got:

`T(0,0,0)=-(10)/(2(27000)) (0) +7605.185=7605.185`