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The Heat Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi An Easy Solution to Industrial Heat Transfer.

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Presentation on theme: "The Heat Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi An Easy Solution to Industrial Heat Transfer."— Presentation transcript:

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2 The Heat Conduction Equation P M V Subbarao Associate Professor Mechanical Engineering Department IIT Delhi An Easy Solution to Industrial Heat Transfer Problems…

3 The Heat Equation Incorporation of the constitutive equation into the energy equation above yields: Dividing both sides by r Cp and introducing the thermal diffusivity of the material given by

4 For constant thermal properties and no heat generation. This is often called the heat equation.

5 General conduction equation based on Cartesian Coordinates

6 For an isotropic and homogeneous material:

7 General conduction equation based on Polar Cylindrical Coordinates

8 Thermal Conductivity of Brick Masonry Walls

9 Thermally Heterogeneous Materials

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11 Satellite Imaging : Remote Sensing

12 Ultra-sound Imaging of Brain

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15 Steady-State One-Dimensional Conduction Assume a homogeneous medium with invariant thermal conductivity ( k = constant) : For one-dimensional steady state conduction with no energy generation, the heat equation reduces to : One dimensional Transient conduction with heat generation.

16 Steady-State One-Dimensional Conduction For one-dimensional heat conduction in a variable area geometry. We can devise a basic description of the process. The first law in control volume form (steady flow energy equation) with no shaft work and no mass flow reduces to the statement that  Q = 0  for all surfaces. From Fourier law of conduction, the heat transfer rate in at the left (at x) is:

17 Taylor’s Theory of Continuum For a function converging & well behaving… For a pure steady state conduction:

18 Substitute Fourier’s law of conduction:

19 If k is constant (i.e. if the material is homogeneous and properties of the medium are independent of temperature), this reduces to Pure radial conduction through A Sphere.

20 Surface area of a sphere at r

21 Heat transfer through a plane slab

22 Isothermal Wall Surfaces

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24 Wall Surfaces with Convection Boundary conditions:

25 Wall with isothermal Surface and Convection Wall Boundary conditions:

26 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 Q is 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

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29 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 R = R 1 + R 2. If T L is the temperature at the left, and T R is the temperature at the right, the heat transfer rate is given by

30 Wall Surfaces with Convection Boundary conditions: R conv,1 R cond R conv,2 T1T1 T2T2

31 R conv,1 R cond R conv,2 T1T1 T2T2


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