Chapter 2 Introduction to Conduction

Conduction Rate Equation Cartesian Cylindrical Spherical Isotherm: The direction of heat flow will always be normal to a surface of constant temperature.

Thermal Conductivity (W/mK) 300°K400°K500°K AgSilver AlAluminum AuGold CrChromium CuCopper MoMolybdenum NiNickel PbLead PtPlatinum SnTin TiTitanium WTungsten See Incropera Appendix A for full listing

Heat Diffusion Equation y x z q x+dx qxqx qzqz q z+dz q y+dy qyqy dz dy dx

Where Energy Balance

Recall so

From Fourier’s law Divide by

If k is constant or Whereis the thermal diffusivity which has units of m 2 s -1

Heat Diffusion Equation Another Approach y x z q x+Δx qxqx qzqz q z+Δz q y+Δy qyqy ΔzΔz ΔyΔy ΔxΔx

Energy Balance Recall So dividing the energy balance equation by and taking the limits to zero yields for the x direction

So again we find

Heat Diffusion Equation Cylindrical Coordinates Heat Diffusion Equation Spherical Coordinates

Boundary conditions for heat diffusion equation at surface x =0 unsteady State Constant Surface Temperature Constant finite surface heat flux Insulated Surface Convection x x x x

Temperature Distribution T1T1 T2T2 x1x1 x2x2 L One dimensional wall system No source or sink of energy Steady state Constant conductivity

integratingby separation of variables Integrating again by separation of variables

Finally we determine the constants using the boundary conditions and so and Therefore Applying Fourier’s law to determine the heat transfer rate

Temperature Distribution Similarly for a cylinder with no source or sink of energy, at steady state and with a constant conductivity Cold air Hot fluid L r2r2 r1r1 T2T2 T1T1

Integrating twice gives From the boundary conditions and we have Applying Fourier’s law to determine the heat transfer rate