the seasonal operating cost in dollars per square meter of grain bed for such a dryer consists of the cost of heating the air.
heating cost=0,002Q(delta T)
and blower operating cost
Blower cost=2.6^(10^-9)Q^3
where Q= air quantity deliverd through the bed during season, m3/m2 bed area.
(delta T)= rise in temperature through heater.C
the values of Q and delta T also influence the time required ofor adequate drying of the grain according to the equation
Drying time = (80*10^6)/((Q^2) * delta T) days
using the geometric-programing method of constrained optimization .
-compute the minimum operating cost and optimum value of Q and delta T that will achive adequate drying in 60 days.
To find the minimum operating cost and the optimum values of Q and delta T that achieve adequate drying in 60 days, we can use the geometric programming method of constrained optimization.
1. Define the objective function:
The objective is to minimize the operating cost, which consists of the heating cost and the blower operating cost. We can represent the objective function as follows:
Minimum Cost = Cost of Heating the Air + Blower Operating Cost
2. Determine the constraints:
The constraint is that the drying time should be equal to or less than 60 days. So, the equation for drying time can be rewritten as:
Drying time = 80 * 10^6 / (Q^2 * delta T) <= 60
3. Convert the problem into geometric programming form:
To convert the problem into a geometric programming form, we need to transform the objective function and the constraint.
a. Objective function transformation:
As the heating cost is given by 0.002Q(delta T) and the blower operating cost is given by 2.6^(10^-9)Q^3, we can rewrite the objective function as:
Minimum Cost = 0.002(Q * delta T) + 2.6^(10^-9)Q^3
b. Constraint transformation:
Using the constraint from step 2, we can rewrite it as:
80 * 10^6 / (Q^2 * delta T) - 60 <= 0
4. Solve the geometric programming problem:
Using a suitable solver or optimization software, input the objective function and the constraint in the geometric programming form, and solve the problem. The solver will find the optimum values of Q and delta T that minimize the operating cost while satisfying the drying time constraint.
5. Interpret the results:
Once the solver provides the solution, the optimum value of Q and delta T can be used to achieve adequate drying in 60 days while minimizing the operating cost.
Note: Since the specific values of the constants in the problem statement are not provided, you would need to substitute the appropriate values for Q, delta T, heating cost, and blower operating cost in the equations and constraints while solving the problem.