I&D Chapter 6 Supplemental Material

I&D Chapter 6 Supplemental Material - c06_supl.qxd...

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W-21 6S.1 Derivation of the Convection Transfer Equations In Chapter 2 we considered a stationary substance in which heat is transferred by conduction and developed means for determining the temperature distribution within the substance. We did so by applying conservation of energy to a differential control volume (Figure 2.11) and deriving a differential equation that was termed the heat equation . For a prescribed geometry and boundary conditions, the equation may be solved to determine the corresponding temperature distribution. If the substance is not stationary, conditions become more complex. For exam- ple, if conservation of energy is applied to a differential control volume in a moving ±uid, the effects of ±uid motion ( advection ) on energy transfer across the surfaces of the control volume must be considered, along with those of conduction. The resulting differential equation, which provides the basis for predicting the tempera- ture distribution, now requires knowledge of the velocity ²eld. This ²eld must, in turn, be determined by solving additional differential equations derived by applying conservation of mass and Newton’s second law of motion to a differential control volume. In this supplemental material we consider conditions involving ±ow of a vis- cous Fuid in which there is concurrent heat and mass transfer . Our objective is to develop differential equations that may be used to predict velocity, temperature, and species concentration ²elds within the ±uid, and we do so by applying Newton’s second law of motion and conservation of mass, energy, and species to a differential control volume. To simplify this development, we restrict our attention to steady, two-dimensional Fow in the x and y directions of a Cartesian coordinate system. A unit depth may therefore be assigned to the z direction, thereby providing a differ- ential control volume of extent ( ). 6S.1.1 Conservation of Mass One conservation law that is pertinent to the ±ow of a viscous ±uid is that matter may neither be created nor destroyed. Stated in the context of the differential con- trol volume of Figure 6S.1, this law requires that, for steady ±ow, the net rate at which mass enters the control volume (in±ow 2 out±ow) must equal zero . Mass enters and leaves the control volume exclusively through gross ±uid motion. Trans- port due to such motion is often referred to as advection . If one corner of the control volume is located at ( x, y ), the rate at which mass enters the control volume through the surface perpendicular to x may be expressed as , where is the total mass density ( ) and u is the x component of the mass average velocity . The control volume is of unit depth in the z direction. Since and u may vary with x , the r r 5 r A 1 r B r ( r u ) dy dx z dy z 1
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W-22 6S.1 j Derivation of the Convection Transfer Equations rate at which mass leaves the surface at x + dx may be expressed by a Taylor series expansion of the form Using a similar result for the y direction, the conservation of mass requirement becomes
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I&D Chapter 6 Supplemental Material - c06_supl.qxd...

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