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.
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.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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use LMTD for part a and b
Transcribed Image Text: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.
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