Example 14.4-1. Estimation of the Solar Constant

The solar constant is the radiant heat flux entering the earth's atmosphere from the sun.

Figure 3: Sun and the Earth as Black Bodies (taken from BSL, p.444)

> restart;

> dQ[12]:=sigma*T[1]^4*cos(theta[1])*dA[1]*dA[2]*cos(theta[2])/(Pi*r[12]^2); Eqn. 14.4-5 and in the discussion

> dSC:=dQ[12]/cos(theta[2])/dA[2];

> SC:=int(dSC/dA[1], A[1]); need to remove dA1 for Maple

> A[1]:=Pi*d[1]^2/(4*cos(theta[1])); cos(theta1) can be included because the integral(cos(theta))dA is the area of the sun, which is nearly pi*D1^2/4. For simplicity, we simply remove it after the integral has been calculated.

> SC;

> sigma:=q[b1]/T[1]^4;

> SC;

>

>

Diameter of the sun:

> d[1]:=8.60e5*miles;

Distance between dA1 and dA2:

> r[12]:=9.29e7*miles;

Heat flux:

> q[b1]:=2.0e7*Btu/hr/ft^2;

> SC;

Which is reasonably close to the book's answer.