OneDimensional SteadyState Conduction without Thermal Energy Generation Chapter

One-Dimensional, Steady-State Conduction without Thermal Energy Generation Chapter Three Sections 3. 1 through 3. 4

The Plane Wall • Consider a plane wall between two fluids of different temperature: • Heat Equation: (3. 1) • Implications: • Boundary Conditions: • Temperature Distribution for Constant This is the SS solution, so there is no change in time. But it is a second order equation in x. This means we need two boundary conditions to solve it. : But how to we get here? (3. 3)

Thermal Resistance Model

Thermal Resistance Model

Thermal Resistance Model

Thermal Resistance Model • Thermal Resistance Over a Unit Surface Area (Conduction and Convection): • Radiation Resistance (Total, and Over Area): (1. 9)

Plane Wall (cont. ) • Composite Wall with Negligible Contact Resistance: (3. 14) For the temperature distribution shown, k. A > k. B < k. C. • ΔT For Individual Sections: Since q through material is a constant, we can find the temperature change over any section by:

Plane Wall (cont. ) (3. 14) • Overall Heat Transfer Coefficient (U) : A modified form of Newton’s law of cooling to encompass multiple resistances to heat transfer. (3. 17) (3. 19)

Plane Wall (cont. ) • • Series – Parallel Composite Wall: Circuits based on assumption of isothermal surfaces normal to x direction or adiabatic surfaces parallel to x direction provide approximations for. Assuming isothermal surfaces perpendicular to x-direction. Assuming adiabatic surfaces parallel to x-direction. • Note departure from one-dimensional conditions for .

Tube Wall The Cylindrical Wall

Tube Wall • The Cylindrical Wall Heat Equation: (3. 28) • Temperature Distribution for Constant : (3. 31)

Tube Wall The Cylindrical Wall

Tube Wall (cont. ) • • Heat Flux and Heat Rate: [W/m 2] Heat flux (per unit area) [W/m] Heat loss over a unit length [W] Heat rate Conduction Resistance: (3. 33) Resistance over a unit length Why doesn’t a surface area appear in the expressions for thermal resistance?

Tube Wall (cont. ) • Composite Wall with Negligible Contact Resistance (3. 35) For the temperature distribution shown, k. A > k. B > k. C. but U itself is tied to specification of an interface. (3. 37)

Problem: Thermal Barrier Coating Problem 3. 23: Assessment of thermal barrier coating (TBC) for protection of turbine blades. Determine maximum blade temperature with and without TBC. SCHEMATIC:

Problem: Thermal Barrier Coating (cont. ) ANALYSIS: For a unit area, the total thermal resistance with the TBC is With a heat flux of the inner and outer surface temperatures of the Inconel are Inner surface temperature Outer surface temperature

Problem: Thermal Barrier Coating (cont. ) Without the TBC, Inner surface temperature Outer surface temperature

Problem: Heat Loss Through Window You have a double pane window that is 1 [m] x 1. 5 [m]. The glass is 4 [mm] thick with kglass = 0. 78 [W/m-K]. Between the panes is an air gap of 5 [mm], with kair = 0. 025 [W/m-K]. The outside air temperature is -20 °C with a convection coefficient of houtside= 20 [W/m 2 -K]. The inside air temperature is 20 °C with a convection coefficient of hinside= 40 [W/m 2 -K]. • • • Draw thermal resistance circuit for this system Calculate the heat rate through the window Calculate the change in temperature across each section