The image illustrates a cooling system for a server rack using a closed-loop configuration. It consists of three main components:1. **Server Rack**: A rectangular box labeled "Server Rack," indicating where the heat is generated.2. **Radiator**: This component is shown to the right of the server rack. It resembles a metal structure with fins, designed for heat dissipation. The radiator releases the heat transferred from the server rack.3. **Pump**: Located below the radiator and server rack, this device circulates the coolant throughout the loop, ensuring the continuous transfer of heat away from the server rack.**Flow of Coolant**:- From the server rack, cooled fluid moves to the radiator, as indicated by the blue arrows pointing to the right.- After the fluid dissipates heat in the radiator, it moves downward and enters the pump.- The pump circulates the fluid back to the server rack, completing the cycle, as shown by the blue arrows pointing left.This configuration maintains an efficient cooling process by circulating coolant between the server rack and the radiator. A system is being designed to remove heat from a server rack in a data center. The schematic provided shows the system's setup. A pump circulates water through a heat sink in the rack, removing 5000 W of heat. To prevent electronic damage, the water leaving the server rack is kept at a max temperature of 80 °C, which is considered optimal for the system. This water then circulates to a heat exchanger, modeled as a counterflow heat exchanger, where air is used to remove the heat. Air at 20 °C with a mass flow rate of 0.1 kg/s is blown through this exchanger.The goal is to determine the optimal size for the air-to-water heat exchanger and the water's mass flow rate to minimize the system's life cycle cost. This cost, primarily from the heat exchanger's initial cost and pump energy, is given by:\[ \text{Cost} = 45*A + 0.01*m_w^2 \]Where:- \( A \): area of the heat exchanger- \( m_w \): mass flow rate of waterAdditional info:- The heat exchanger's U value is 40 W/(m²*K)Tasks:(a) Define the optimization problem with objective function, design variables, and constraints.(b) Reformulate it into a single-variable optimization problem.(c) Utilize a Fibonacci search to find optimal settings within a water mass flow rate range of 1 kg/s to 12 kg/s, accurate to 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.