hermal Systems Basic Modeling Elements Resistance Conduction Convection Radiation Capacitance Interconnection Relationships Energy Balance - 1st Law of hermodynamics Derive Input/Output Models ME375 hermal Systems - 1
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 transfer by radiation and convection. However, very few nonlinear PDEs have analytical (closed form) solutions. Usually, finite element methods (FEM) are used to numerically solve nonlinear PDE problems. Our purpose is to try to use lumped model approximations of thermal systems to obtain linear ODEs that are capable of describing the dynamic response of thermal systems to a good first approximation. For many thermal system, an euilibrium condition exists 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 the incremental heat flow rate ( t ( )) and theincremental temperature ( t () ) to be: t ( ) t () and t () t () ME375 hermal Systems - 2
Basic Modeling Elements hermal Resistance Describes the heat transfer process through an element with the characteristic that the heat flow rate across the element is proportional to the temperature difference across the element, i.e. Ex: wo bodies at temperatures 1 and 2 are separated by two elements with different thermal resistance R 1 and R 2. Heat flows through the two elements at a rate of. Find the euivalent thermal resistance R e and solve for the interface temperature between the two elements. R = 1 2 R + 1 2 R 12 1 2 R or 1 ( R 1 R 1 2) [ o K/W] 1 R 1 R 2 2 1 2 R e ME375 hermal Systems - 3
hree ypes of Heat ransfer Conduction Heat transfer through solid or continuous media via random molecular motion (diffusion). 1 2 Ex: Calculate the euivalent thermal resistance of a wall with a window. Wall Window Area A W A G hickness d W d G W G 1 d Cross sectional area A 2 x A d A ( d R 1 2) 12 thermal conductivity [W/m- o K] ME375 hermal Systems - 4
hree ypes of Heat ransfer 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 x x ha( ) ha S F 1 R ha h depends on surface geometry, fluid flow rate, temperature, flow direction,... ME375 hermal Systems - 5
hree ypes of Heat ransfer Radiation Heat transfer via electromagnetic waves. 2 1 4 4 F F A( ) E V Surface Area A A : surface area [m 2 ] : 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 1 2 Except 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
Basic Modeling Elements hermal Capacitance he ability of a substance to hold or store heat is the heat capacity of the material and it behaves like a thermal capacitance. Since the specific heat c P can be interpreted as the heat storage capacity of the material per unit mass, the total heat storage capability of a material is: c M P + C If there is net heat flow into the material, the temperature of the material will change and the rate of temperature change is proportional to the net heat flow rate SORE : c M d dt P C SORE IN OU We can define the thermal capacitance C c M c V P P IN C C OU Mass, M Volume, V Density, 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 - 1st 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 SORE IN OU GENERAED' WORK WIHIN DONE Ex: A material with a thermal capacitance C is surrounded by an insulation material with thermal resistance R. 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 R i (t) ME375 hermal Systems - 8
In Class Example Ex: he Pentium II processor under normal operation will generate heat at a rate of i (t). he processor itself has a specific heat of c P. he cross sectional area of the chip is A P with a thickness of d P. he average density of the processor is P. o help dissipate the heat and reduce the processor temperature P, 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 of 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 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 P. A h A A P Heat Sink d S c P, P, P d P ME375 hermal Systems - 9