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Computational Fluid Mechanics and Heat Transfer Third Edition_5

Tham khảo tài liệu 'computational fluid mechanics and heat transfer third edition_5', kỹ thuật - công nghệ, điện - điện tử phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 4.2 The general solution 145 Figure 4.2 One-dimensional heat conduction in a ring. T T t only If T is spatially uniform it can still vary with time. In such cases V2T q - 1 k a st -0 and dT dt becomes an ordinary derivative. Then since a - k pc d - pc 4.2 This result is consistent with the lumped-capacity solution described in Section 1.3. If the Biot number is low and internal resistance is unimportant the convective removal of heat from the boundary of a body can be prorated over the volume of the body and interpreted as effective - -h T d y Tr- W m3 4.3 volume and the heat diffusion equation for this case eqn. 4.2 becomes -- hA T - Too 4 4 dt pcV T 1f The general solution in this situation was given in eqn. 1.21 . A particular solution was also written in eqn. 1.22 . 146 Analysis of heat conduction and some steady one-dimensional problems 4.2 Separation of variables A general solution of multidimensional problems Suppose that the physical situation permits us to throw out all but one of the spatial derivatives in a heat diffusion equation. Suppose for example that we wish to predict the transient cooling in a slab as a function of the location within it. If there is no heat generation the heat diffusion equation is d2T 1 dT dx2 a dt 4.5 A common trick is to ask Can we find a solution in the form of a product of functions of t and x T T t X x To find the answer we substitute this in eqn. 4.5 and get X T 1 T X a 4.6 where each prime denotes one differentiation of a function with respect to its argument. Thus T dT dt and X d2X dx2. Rearranging eqn. 4.6 we get X 1 T_ X a T 4.7a This is an interesting result in that the left-hand side depends only upon x and the right-hand side depends only upon t. Thus we set both sides equal to the same constant which we call -À2 instead of say À for reasons that will be clear in a moment X 1 7 -Ả2 a constant X a T 4.7b It follows that the differential eqn. 4.7a can be resolved into two ordinary differential equations X -Á2X and T