Solutions of the Conduction Equation

1. Solutions of the Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi An Idea Generates More Mathematics…. Mathematics Generate Mode Ideas…..

2. The Conduction Equation Incorporation of the constitutive equation into the energy equation above yields: Dividing both sides by rCpand introducing the thermal diffusivity of the material given by

3. Thermal Diffusivity • Thermal diffusivity includes the effects of properties like mass density, thermal conductivity and specific heat capacity. • Thermal diffusivity, which is involved in all unsteady heat-conduction problems, is a property of the solid object. • The time rate of change of temperature depends on its numerical value. • The physical significance of thermal diffusivity is associated with the diffusion of heat into the medium during changes of temperature with time. • The higher thermal diffusivity coefficient signifies the faster penetration of the heat into the medium and the less time required to remove the heat from the solid.

4. This is often called the heat equation. For a homogeneous material:

5. This is a general form of heat conduction equation. Valid for all geometries. Selection of geometry depends on nature of application.

6. General conduction equation based on Cartesian Coordinates

7. For an isotropic and homogeneous material:

8. General conduction equation based on Polar Cylindrical Coordinates

9. General conduction equation based on Polar Spherical Coordinates Y X

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11. Thermally Heterogeneous Materials

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17. One Dimensional Heat Conduction problems P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi Simple ideas for complex Problems…

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19. Steady-State One-Dimensional Conduction • For conduction through a large wall the heat equation reduces to: • Assume a homogeneous medium with invariant thermal conductivity ( k = constant) : One dimensional Transient conduction with heat generation.

20. Steady Heat transfer through a plane slab No heat generation

21. Isothermal Wall Surfaces Apply boundary conditions to solve for constants: T(0)=Ts1; T(L)=Ts2 The resulting temperature distribution is: and varies linearly with x.

22. Applying Fourier’s law: heat transfer rate: heat flux: Therefore, both the heat transfer rate and heat flux are independent of x.

23. Wall Surfaces with Convection Boundary conditions:

24. Wall with isothermal Surface and Convection Wall Boundary conditions:

25. Electrical Circuit Theory of Heat Transfer • Thermal Resistance • A resistance can be defined as the ratio of a driving potential to a corresponding transfer rate. Analogy: Electrical resistance is to conduction of electricity as thermal resistance is to conduction of heat. The analog of Qis current, and the analog of the temperature difference, T1 - T2, is voltage difference. From this perspective the slab is a pure resistance to heat transfer and we can define

28. The composite Wall • The concept of a thermal resistance circuit allows ready analysis of problems such as a composite slab (composite planar heat transfer surface). • In the composite slab, the heat flux is constant with x. • The resistances are in series and sum to Rth = Rth1 + Rth2. • If TLis the temperature at the left, and TRis the temperature at the right, the heat transfer rate is given by

29. T1 T2 Rconv,1 Rcond Rconv,2 Wall Surfaces with Convection Boundary conditions:

30. Heat transfer for a wall with dissimilar materials • For this situation, the total heat flux Q is made up of the heat flux in the two parallel paths: • Q = Q1+ Q2 • with the total resistance given by:

31. Composite Walls • The overall thermal resistance is given by

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