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Louisiana Tech University Ruston, LA Slide 1 Energy Balance Steven A. Jones BIEN 501 Wednesday, April 18, 2008

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Louisiana Tech University Ruston, LA Slide 2 Energy Balance Major Learning Objectives: 1.Provide the equations for the different modes of energy transfer. 2.Describe typical boundary conditions for heat transfer problems. 3.Derive complete solutions for heat tranfer problems without flow. 4.Derive complete solutions for specific heat transfer problems involving flow.

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Louisiana Tech University Ruston, LA Slide 3 Energy Balance Minor Learning Objectives: 1.Use Newton’s law of cooling. 2.Derive the complete solution for Couette flow of a compessible Newtonian fluid. 3.Describe the dimensional and non-dimensional parameters involved in heat transfer problems. 4.Distinguish between convection and conduction. 5.Obtain more facility with the separation of variables method for solving partial differential equations. 6.Examine the coupling between the energy equation and the momentum equation caused by viscous heating.

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Louisiana Tech University Ruston, LA Slide 4 Heat Conduction Equations The general equation for heat conduction is: Increase with time. Source of heat Difference between flux in and flux out.

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Louisiana Tech University Ruston, LA Slide 5 Heat Conduction Equations If thermal conductivity is constant: Increase with time. Source of heat Difference between flux in and flux out.

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Louisiana Tech University Ruston, LA Slide 6 Differential Form Describes how much a volume of material will increase in temperature with a given amount of heat input. I.e., if I add x number of Joules to a volume 1 cm 3, it will increase in temperature by T degrees. T Joules in

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Louisiana Tech University Ruston, LA Slide 7 Thermal Conductivity k – defines the rate at which heat “flows” through a material. Fourier’s law (A hot cup of coffee will become cold). Fourier’s law is strictly analogous to Fick’s law for diffusion. q is flux, i.e. the amount of heat passing through a surface per unit area. Gradient drives the flux.

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Louisiana Tech University Ruston, LA Slide 8 Heat Production, Example Consider the case of a resistor: V1V1 V2V2 The resistor dissipates power according to, (where I is the current) or, equivalently The volume of the resistor is. Therefore, at any spatial location within the resistor, it is generating: Joules/s/cm 3

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Louisiana Tech University Ruston, LA Slide 9 Boundary Conditions Constant temperature (T=T 0 ) Constant flux Heat transfer: More general: –Surface Condition or on the closed surface. –Initial condition within the volume.

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Louisiana Tech University Ruston, LA Slide 10 Semi-Infinite Slab (of marble) Flow of Heat T=T1T=T1 T=T 0 uniform How does temperature change with time? Initial Temperature Profile Boundary Conditions:

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Louisiana Tech University Ruston, LA Slide 11 Semi-Infinite Slab (of marble) Differential Equation (no source term): Boundary Conditions:

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Louisiana Tech University Ruston, LA Slide 12 Semi-Infinite Slab (of marble) For 1-dimensional geometry and constant k : This problem is mathematically identical to the fluid flow near an infinitely long plate that is suddenly set in motion.

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Louisiana Tech University Ruston, LA Slide 13 Semi-Infinite Slab (of marble) Dimensionless Temperature: Dimensionless temperature describes the difference between temperature and a reference temperature with respect to some fixed temperature difference, in this case T 1 – T 0.

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Louisiana Tech University Ruston, LA Slide 14 Semi-Infinite Slab (of marble) Equations in Terms of Dimensionless Temperature: Boundary Conditions: T * is valuable because it nondimensionalizes the equations and simplifies the boundary conditions.

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Louisiana Tech University Ruston, LA Slide 15 Semi-Infinite Slab (of marble) Assume a similarity solution, and define:. We make use of the following relationships:

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Louisiana Tech University Ruston, LA Slide 16 Semi-Infinite Slab (of marble) With these substitutions, the differential equation becomes: This can be divided by to yield: The combination can now be replaced with to give:.

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Louisiana Tech University Ruston, LA Slide 17 Semi-Infinite Slab (of marble) Instead of trying to solve directly for T *, try to solve for the first derivative:. The equation is rewritten as: which is separated as:. Thus:. So:

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Louisiana Tech University Ruston, LA Slide 18 Semi-Infinite Slab (of marble) Integrate:.. To obtain:. This integral cannot be evaluated in closed form by standard methods. However, it is tabulated in handbooks and it can be evaluated under standard software. The integral is called the Error function (because of it’s origins in probability theory, where Gaussian functions are important).

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Louisiana Tech University Ruston, LA Slide 19 Newton’s Law of Cooling Often we must evaluate the heat transfer in a body that is in contact with a fluid (e.g. heat dissipation from a jet engine, cooling of an engine by a radiator system, heat loss from a cannonball that is shot through the air). The boundary between the solid and fluid conforms neither to a constant temperature, nor to a constant flux. We make the assumption that the rate of heat loss per unit area is governed by:

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Louisiana Tech University Ruston, LA Slide 20 Newton’s Law of Cooling The heat transfer coefficent, h, is a function of the velocity of the cannonball. I.e. the higher the velocity, the more rapidly heat is extracted from the cannonball. One generally assumes that the heat transfer coefficient does not depend on temperature. However, in free convection problems, it can be a strong function of temperature because the velocity of the fluid depends on the fluid viscosity, which depends on temperature.

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Louisiana Tech University Ruston, LA Slide 21 Free Convection In free convection, the fluid moves as a result of heating of the fluid near the body in question. Fluid becomes less dense near the body. Bouyency causes it to move up, enhancing transfer of heat.

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Louisiana Tech University Ruston, LA Slide 22 Forced Convection In forced convection (vs. free convection), the velocity is better controlled because it does not depend strongly on the heat flow itself. A fan, for example, controls the velocity of the fluid.

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Louisiana Tech University Ruston, LA Slide 23 Convection vs. Conduction Convection enhances flow of heat by increasing the temperature difference across the boundary. Because “hot” fluid is removed from near the body, fluid near the body is colder, therefore the temperature gradient is higher and heat transfer is higher, by Fourier’s law. In other words, Fourier’s law still holds at the boundary. Small gradient, small heat transfer. Large gradient, large heat transfer.

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