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Unformatted text preview: nd integral ∙ These are all now integrals over the same volume, except for the viscous term ∙ Where the over‐bar on the viscous term indicates that it is per unit volume ∙ This is the x‐component of the momentum equation in differential form we also have y and z components ∙ ∙ If you put the three component equations back together you can get a vector form Energy Equation Conservation of energy This is a statement of the first law of thermodynamics The change in energy in a system de equals the heat added to the system from the surroundings plus the work done on the system. Heat and work are both forms of energy We will divide this by time so we end up constructing a rate equation If we take some small control volume to be our system and consider the energy within it, the energy takes the form of internal energy and kinetic energy 1 2 We can divide this by m and get 2 The rate of energy carried by fluid entering and leaving the volume V across its boundary S is then ∙ 2 If we consider the change in energy of the fluid already in the volume V we have 2 So the rate of change in energy contained within V is the sum of these 2 2 ∙ This is the left side of our energy equation Now let us consider the volumetric heating of the fluid. This can be from absorption of radiation, or from nuclear decay, or from chemical reaction. We assign a function wh...
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