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Unformatted text preview: Non-isothermal Equations of change A methodology for solving energy transfer problems • Statement of problems 1.Define the domain boundary 2. Choose a coordinate system such that there are more vanishing terms (i.e. components of velocity, velocity gradient, temperature gradient, and forces). 3. State assumptions to simplify the problem (i.e. set zero more terms) 4. List heat sources/sinks and non-zero components of velocity, velocity/temperature gradients, forces, and works. 5. For each phase, apply directly the partial differential equations of continuity, motion, and energy (table lookup!). 6. Identify initial and boundary conditions, and then solve the velocity and temperature equations. 7. Determine other quantities related to velocity and temperature profiles. • Common boundary conditions Momentum balance: see previous lectures Energy Balance d. At interfaces the continuity of temperature and of the heat flux normal to the interface are required. e. Newton’s law of cooling is applicable at the fluid-solid interface f. Heat transfer by conduction is normally negligible in the directions of convective mass flow: g. Homogeneous and isotropic media / / i i k T x T x δ δ δ δ = ⇔ = Steady heat conduction with an electrical heating source 1. Assumptions: homogeneous and isotropic medium 2. Non-zero components: heat source S g , q r 3. Equation of energy Non-isothermal Equations of change 2 2 2 2 2 2 1 1 ˆ p T T T T C k T S k r S t r r r r z δ δ δ δ ρ ρ ρ δ δ δ θ δ ⎡ ⎤ ∂ ⎛ ⎞ = ∇ + = + + + ⎜ ⎟ ⎢ ⎥ ∂ ⎝ ⎠ ⎣ ⎦ 1 1 2 2 2 2 1 0 2 C =0 4 1 4 g g r r g g r R S r d d T d T k r S k C r dr dr dr q finite S r T C k S R r T T T T k R = = ⎛ ⎞ ⎛...
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- Fall '08
- Fluid Dynamics, dt, Incompressible Flow, non-zero components, Non-isothermal Equations