Cost optimization of a conical electric heater

Electric heaters are used extensively in many industrial applications including air conditioning, water heating, and drying of crop and other materials. There are several interacting parameters that affect heater performance and contribute to its operating and manufacturing costs. The design parameters of interest included coil length, coil diameter, helix diameter, coil pitch, number of turns, coil emissivity, heater wall emissivity, applied voltage, air flow rate, air temperature at the heater outlet, insulation thickness, and the heater dimensions. An objective function for the heater cost was defined in terms of the total annual costs for energy usage and manufacturing. The manufacturing cost included the material cost of the heater, the cost of coating the heater coil, and the cost of insulating the heater. A detailed model describing the performance of a conical heater was utilized to evaluate and minimize the objective function. The optimization problem was completely defined by introducing fourteen inequality constraints, three equality constraints, and ten design variables. The univariate search was used to reduce the number of inequality constraints. The direct search and variable metric methods were used to optimize the objective function at different values of the applied power and air temperature at the heater outlet, leading to the same results. The optimization results indicate that the optimum heater design was found at the minimum values of the coil diameter, conical angle, coil emissivity, and the maximum value of coil temperature. The maximum operating temperature of the optimum heater design (a heater with a silver coated coil) is substantially lower than the maximum normal operating temperature so that longer life for the heating element is expected. The effect of varying the applied power on the objective function showed that as the applied power increases the objective function may increase or decrease, depending on the combined effect of increasing the coil length and heater efficiency and decreasing the insulation thickness. The effect of varying the air temperature at the heater outlet on the objective function indicated that as the air temperature increases the objective function increases.