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 optimum values of Q and delta T, we need to formulate an optimization problem. Let's define our objective function and constraints.
Objective: Minimize the operating cost.
Constraints:
1. Drying time should be equal to or less than 60 days.
2. Q and delta T should be greater than zero.
Let's proceed with solving this optimization problem using geometric programming.
Step 1: Define the objective function.
We need to minimize the operating cost, which includes heating cost and blower cost.
Operating Cost (OC) = Heating Cost + Blower Cost
OC = 0.002Q(delta T) + 2.6 * 10^(-9)Q^3
Step 2: Formulate the constraints.
- The drying time should be equal to or less than 60 days.
(80 * 10^6) / (Q^2 * delta T) <= 60
- Ensure Q and delta T are greater than zero.
Q > 0
delta T > 0
Step 3: Convert the problem into a standard geometric programming form.
To convert the non-linear constraints into geometric form, we take the logarithm of both sides.
log((80 * 10^6) / (Q^2 * delta T)) <= log(60)
log(80 * 10^6) - 2 * log(Q) - log(delta T) <= log(60)
Step 4: Solve the geometric programming problem.
You can use optimization software or programming language libraries that support geometric programming to solve this problem. These tools use numerical algorithms to find the optimal solution.
The solution will provide the minimum operating cost and the optimum values of Q and delta T that will achieve the desired drying time of 60 days.