Interconnection Relationships. Derive Input/Output Models. School of Mechanical Engineering Purdue University

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Lorin Thompson
5 years ago
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hermal Systems Basic Modeling Elements esistance esistance Conduction Convection adiation Capacitance Interconnection elationships Energy Balance - st Law of hermodynamics Derive Input/Output Models

As a result, the dynamics of a thermal system has to be described by partial differential euations.

1 hermal Systems Basic Modeling Elements esistance esistance Conduction Convection adiation Capacitance Interconnection elationships Energy Balance - st Law of hermodynamics Derive Input/Output Models ME375 hermal Systems - Key Concepts : heat flow rate [J/sec W] ( ) : temperature [ o K] ] or [ o C] ( ) emperature in a body usually depends on spatial as well as temporal coordinates. As a result, the dynamics of a thermal system has to be described by partial differential euations. Moreover, nonlinearities are often essential in describing the heat h transfer by radiation and convection. However, very few nonlinear DEs have analytical (closed form) solutions. Usually, finite element methods (FEM) are used to numerically solve nonlinear DE problems. Our purpose is to try to use lumped model approimations ons of thermal systems to obtain linear ODEs that are capable of describing the dynamic response of thermal systems to a good first approimation. For many thermal system, an euilibrium condition eists that defines the nominal operating condition. In these cases, the deviation of the heat flow rate and temperature from their nominal values, and, are of interest. hus, we can define d the incremental heat flow rate ( ( t) ) and the incremental temperature ( () t ) to be: ( t ) ( t ) and ( t) ( t) ME375 hermal Systems - 2

2 Basic Modeling Elements hermal esistance E: wo bodies at temperatures and are Describes the heat transfer process separated by two elements with different through an element with the thermal resistance and 2. Heat flows characteristic that the heat flow rate through the two elements at a rate of.. Find across the element is proportional to the euivalent thermal resistance e and the temperature difference across the element, i.e. solve for the interface temperature between the two elements. + Δ Δ 2 2 or ( Δ 2) [ o K/W] 2 e ME375 hermal Systems - 3 Conduction Heat transfer through solid or continuous media via random molecular motion (diffusion). E: Calculate the euivalent thermal resistance of a wall with a window. Wall Window Area A W A G hickness d W d G α α W α G d Cross sectional area A α A d α A ( d 2) 2 α : thermal conductivity [W/m- o K] ME375 hermal Systems - 4 2

3 Convection Heat transfer between the interface of a solid material and a fluid material via bulk motion of the fluid. A : surface area [m 2 ] h : convective heat transfer coefficient [W/m 2 - o K] S : surface temperature [ o K] F : fluid temperature [ o K] F F F ha ( ) ha Δ S F ha h depends on surface geometry, fluid flow rate, temperature, flow direction,... ME375 hermal Systems - 5 adiation Heat transfer via electromagnetic waves. 4 4 σ F F A ( ) E V Surface Area A A : surface area [m 2 ] σ : Stefan-Boltzmann constant [W/m 2 - o K 4 ] F E : effective emissivity F V : view factor Nonlinear! Will not be considered in this course 2 Ecept for radiation, both conductive and convective heat transfer processes can be modeled as thermal resistances. In the previous discussions, the assumption is that the materials do not store thermal energy. In reality, materials do store a certain amount of thermal energy. Q: How would we model the process of storing thermal energy? ME375 hermal Systems - 6 3

4 Basic Modeling Elements hermal Capacitance If there is net heat flow into the he ability of a substance to hold or material, the temperature of the store heat is the heat capacity of the material and it behaves like a thermal material will change and the rate of temperature change is proportional to capacitance. Since the specific heat c the net heat flow rate SOE : can be interpreted as the heat storage capacity of the material per unit mass, c M d dt C SOE IN OU the total heat storage capability of a material is: c M We can define the thermal capacitance C c M c ρv + C IN C C OU Mass, M Volume, V Density, ρ IN OU C Note: he above relationship holds only if we assume that the temperature is uniform across the entire material. ME375 hermal Systems - 7 Interconnection Laws Energy Balance - st Law of hermodynamics Energy stored in the system is the sum of the net energy inflow, the energy generated within the system and the work done on the system: d dt W SOE IN OU + GENEAED' + WOK WIHIN DONE E: A material with a thermal capacitance C is surrounded by an insulation material with thermal resistance. Heat is added to the inner material at a rate of i (t). Find the system model, if the inner material temperature C is to be the output. C, C a i (t) ME375 hermal Systems - 8 4

5 In Class Eample E: he entium II processor under normal operation will generate heat at a rate of i (t). he processor itself has a specific heat of c. he cross sectional area of the chip is A with a thickness of d. he average density of the processor is ρ. o help dissipate the heat and reduce the processor temperature, a heat sink with the same cross sectional area and an average thickness of d S is added on top of the processor. he heat sink has a thermal conductivity ty of α S. o further improve heat dissipation, a fan is used to generate air flow on top of the heat sink, the effective convection coefficient is h A and the effective contact area between the heat sink and the air r flow is A S. he temperature inside the computer is maintained at A. Find the relationship between i (t)) and the temperature of the processor. A h A A Heat Sink d S c, ρ, d ME375 hermal Systems - 9 5

Thermal Systems. Basic Modeling Elements. Interconnection Relationships. Derive Input/Output Models. Resistance. Capacitance

Thermal Systems. Basic Modeling Elements. Interconnection Relationships. Derive Input/Output Models. Resistance. Capacitance hermal Systems Basic Modeling Elements Resistance Conduction Convection Radiation Capacitance Interconnection Relationships Energy Balance - 1st Law of hermodynamics Derive Input/Output Models ME375 hermal