A system is being designed to remove heat from a server rack in a data center. The figure below provides a schematic of this system. A pump circulates water through a heat sink in the rack to remove 5000 W of heat. To keep the electronics below the temperature at which damage may occur, the water leaving the server rack will be limited to 80 °C. That temperature will result in the most optimal system, so you can assume that it is the temperature of the water leaving the server rack. This water is then circulated to a heat exchanger that can be modeled as a counter flow model with a stream of air to remove the heat. Air with an inlet temperature of 20 C at a mass flow rate of 0.1 kg/s is blown through this heat exchanger. Your task is to determine the size of the air-to-water heat exchanger (i.e., its area) and the mass flow rate of water to minimize the life cycle cost of the system. The cost, which is dominated by the first cost of the heat exchanger and the energy to run the pump, has been estimated to be: Cost = 45*A + 0.01*mw² Where A: area of heat exchanger mw: mass flow rate of water Other information: The U value of the heat exchanger is 40 W/(m²*K) (a) Provide the equations describing the optimization problem, including the objective function, the design variables, and any constraint equations. (b) Convert the formulation into an unconstrained optimization in a single variable (i.e., use the equations in part (a) to eliminate all but one unknown variable) (c) Use a Fibonacci search to determine the optimum settings of your design variables using the equation developed in part (b). The mass flow rate of water is expected to be between 1 kg/s and 12 kg/s, and you would like to know the optimum setting to within 0.5 kg/s.

Given:To find:The heat balance equation is:Here, is the specific heat capacity of water.Tout is the outlet temperature of water.Tin is the inlet temperature of water.Formulate the unconstrained optimization in a single variable:We can eliminate the design variable A from the objective function by using the heat balance equation to express it in terms of mw. Substituting the heat balance equation into the objective function, we get:Now, the objective function only depends on the single design variable mw.import numpy as np def fibonacci_search(f, lb, ub, tol): """ Fibonacci search to find the minimum of a function. Args: f: Function to be minimized. lb: Lower bound of the search interval. ub: Upper bound of the search interval. tol: Convergence tolerance. Returns: x: Optimum setting of the design variable. fmin: Minimum value of the objective function. """ # Generate a Fibonacci sequence. f_seq = [0, 1] while f_seq[-1] < ub - lb: f_seq.append(f_seq[-1] + f_seq[-2]) # Initialize the search interval. x_l = lb x_r = ub # Start the Fibonacci search. while x_r - x_l > tol: # Calculate the Fibonacci numbers. f_l = f_seq[-2] f_r = f_seq[-1] # Evaluate the objective function at the Fibonacci points. f_l_val = f(x_l) f_r_val = f(x_r) # Move the search interval.