01 The rate of heat gained by the ball due to convection is Rate of heat gained

01 the rate of heat gained by the ball due to

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Chapter 10.01 The rate of heat gained by the ball due to convection is Rate of heat gained due to convection a hA , (3) where h = the convective cooling coefficient, K m 2 / W . A = surface area of ball, 2 m a ambient temperature of the hot water, K As you can see we have the expression for the rate at which heat is gained (not the heat gained), so we rewrite the heat energy balance as Rate at which heat is gained - Rate at which heat is lost =Rate at which heat is stored (4) This gives us dt d mC hA a (5) Equation (5) is a first order ordinary differential equation that when solved with the initial condition 0 ) 0 ( , would give us the temperature of the spherical ball as a function of time. However, we made a large assumption in deriving Equation (5) - we assumed that the system is lumped. What does a lumped system mean? It implies that the internal conduction in the sphere is large enough that the temperature throughout the ball is uniform. This allows us to make the assumption that the temperature is only a function of time and not of the location in the spherical ball. The system being considered lumped for this case depends on: material of the ball, geometry, and heat exchange factor (convection coefficient) of the ball with its surroundings. What happens if the system cannot be treated as a lumped system? In that case, the temperature of the ball will now be a function not only of time, but also the location.
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  • Spring '16
  • Partial differential equation, Autar Kaw, Sri Harsha Garapati

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