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 | The general solution 145 Figure 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 This result is consistent with the lumped-capacity solution described in Section . 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 volume and the heat diffusion equation for this case eqn. becomes -- hA T - Too 4 4 dt pcV T 1f The general solution in this situation was given in eqn. . A particular solution was also written in eqn. . 146 Analysis of heat conduction and some steady one-dimensional problems 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 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. and get X T 1 T X a where each prime denotes one differentiation of a function with respect to its argument. Thus T dT dt and X d2X dx2. Rearranging eqn. we get X 1 T_ X a T 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 It follows that the differential eqn. can be resolved into two ordinary differential equations X -Á2X and T
