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Typical Solutions in Transport Phenomena

Typical Solutions in Transport Phenomena. Multi-dimensional Transport. For Nonisothermal Systems. Chap.12 Temperature Distributions with More Than One Independent Variable.

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Typical Solutions in Transport Phenomena

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  1. Typical Solutions in Transport Phenomena Multi-dimensional Transport For Nonisothermal Systems

  2. Chap.12 Temperature Distributions with More Than One Independent Variable In this chapter, the application of the equations of change for nonisothermal systems with heat transport was illustrated by some typical examples. Two methods will be discussed in this lesson: Asymptotic solution. Series solution by the method of Sturm-Liouville eigenvalue problems.

  3. Solution by the method of Sturm-Liouville eigenvalue theorem The Sturm-Liouville theorem says: (1) For an ODE in the following form, (a.1) if k(x) , q(x) and p(x) are positive and k(x) ,k’(x) ,q(x) and p(x) are continuous in [a, b], there must be numberless eigenvalues

  4. Solution by the method of Sturm-Liouville eigenvalue problems. As  takes one of the eigenvalues n, Eq.(a.1) must have a nontrivial solution fn(x) to satisfy the B.C.s, which is called the eigenfunction corresponding ton. (2) The eigenfunctions are weightedly orthogonal to each other in [a , b] (a.2)

  5. Solution by the method of Sturm-Liouville eigenvalue problems. (3) Any function, which has piecewise continuous first and second derivatives in[a , b] and satisfies the B.C.s, can be expanded as an absolutely and uniformly convergent series of the eigenfunctions (a.3) in which (a.4)

  6. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (1) Description of the problem: A Newtonian fluid flows through a long, straight circular tube. At a position far from the entrance, an electrical heating coil is set at the wall. It is required to find the analytical solution of the temperature distribution along the length and radius of the tube.

  7. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (2) 1. Physical Model: Constant physical properties; Steady developed laminar flow; Steady thermal state; Constant heat flux at wall; Negligible axial conduction; Circumferentially uniform; Negligible viscous dissipation; No internal heat source.

  8. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (3) 2. Mathematical Model: 1) Choose a cylindrical coordinate system as z-axis along the centerline of tube, and set z=0 at the starting point of heating . 2) Simplifications: (1) From P.M. (1) & (2),

  9. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (4) (2) From P.M. (3) (3) From P.M. (5) (4) From P.M. (6) (5) From P.M. (7) (6) From P.M. (8) (7) From P.M. (1)

  10. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (5) 3) Simplify the equation of energy in cylindrical coordinates, Eq.(B.9-2) It reduces to

  11. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (6) With the above, simplification (1) and P.M. (4), we get the M.M. as (10.8-12)

  12. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (7) 3. Solution to M.M. 1) Make M.M. dimensionless Choosing R as the characteristic length is a natural choice which results and

  13. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (8) The B.C.2 can be rearranged as Let The B.C.1 and B.C.2 become The B.C.1 can be homogenized by letting

  14. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (9) Then and The dimensionless number can be combined into * make the equation simpler.

  15. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (10) Eq.(10.8-12) becomes (10.8-19)

  16. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (11) 2) Asymptotic solution for large distance Eq.(10.8-19) has a nonhomogeneous B.C. at =1, so the method of separation of variables cannot be applied to it directly. Because of neglecting the axial heat conduction, it is rational to suppose that the temperature profile in the region far from the start point of heating should be fully developed, that is, the shape of temperature profile keeps unchanged as the temperature rises proportionally to z due to the constant heat flux at wall.

  17. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (12) This supposition can be expressed mathematically as (10.8-23) Inserting it into Eq.(10.8-19) gives (10.8-26)

  18. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (13) The general solution to Eq. (10.8-26) is easy to find by integrating it twice, (10.8-23) By B.C.3, C1=0. By B.C.2, C0= 4. Then This solution does not satisfy B.C.1 so C2 cannot be determined by B.C.1.

  19. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (14) In order to determine C2 , we make an overall heat balance from z = 0 toz = z. Or Then (10.8-31)

  20. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (15) 3) Complete solution The asymptotic solution satisfy both Eq.(10.8-19) andB.C.2 at =1, so it can be utilized to homogenize the B.C.2 of Eq.(10.8-19). Let (12.2-4) Inserting it into Eq.(10.8-19), we obtain the M.M. about d as

  21. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (16) (b.1) The operator on the left side involves in only and that on the right side only . So, we can try the method of separation of variables.

  22. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (17) Let (12.2-8) Inserting it into Eq. (b.1), we have Dividing the both sides by , we get (b.2) It can be seen that the left side is a function of  and the right side one of . The only way for the identity being true is that the both sides are equal to a constant (known as the separation constant).

  23. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (18) That is It can be separated into two ODEs. (12.2-9*) (12.2-10*) The general solution of Eq. (12.2-9*) is Since  should be finite as → , there must beC 0, denoted as C = -c2.

  24. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (19) Then, (b.3) (12.2-10) Let =  2 and  = c2/4, Eq. (12.2-10) becomes (b.4) It is a Sturm-Liouville problem.

  25. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (20) Then according to the Sturm-Liouville theorem, there must be numberless eigenvalues k and eigenfunctions kfor this equation and any solution to it can always be expressed as a series of the eigenfunctions Thus the solution to Eq.(b.1) can be expressed as

  26. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (21) The coefficients Bk can be determined by B.C.1of Eq.(b.1): The eigenvalues k and eigenfunctions kthemselves should be found by solving Eq.(b.4) with its B.C.s. R. Siegel has done it for k up to 7.

  27. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (22) 4. Analysis of the results The complete solution Siegel’s results are

  28. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (23) It can be seen that exp(-ck2 ) converges rapidly with k increasing provided is not very small. For example, at =0.05, It means that the error of taking d =d1is less than 2% as  0.05.

  29. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (24) 2) The asymptotic solution for large  At =0.1, It means that the error of taking = is less than 2% as  0.1.

  30. §12.2-1 Laminar Tube Flow with Constant Heat Flux at The Wall (25) The physical significance of  0.1 can be seen as follows, At Re=2000, For air, Pr = 0.7, z/R=140; For water, Pr=2.2 (20C), z/R=440 .

  31. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(1) At very small value of , the terms of higher degree in the complete solution should be included, which makes the solution complicated and inconvenient to use. For sake of this, an asymptotic solution for the entrance region is derived, which is based on the finite depth of heat penetration at small .

  32. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(2) 1) Physical simplifications employed (1) Linearization in geometry A flat wall is used to replace the cylindrical wall. (2)Semi-infinite space approximation The outer boundary of fluid is extended to infinite far from the wall. (3) Linearization of velocity profile The velocity is expanded in Taylor’s series at wall and only the linear part is retained.

  33. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(3) 2) Simplified M.M. (1) Choosea Cartesian coordinate system with y representing the distance from wall. (2) The expression for velocity is (3) The equation of energy reduces to (12.2-13)

  34. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(4) Or Differentiating its both sides with respect to y, we have With We get (12.2-14)

  35. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(5) To make M.M. dimensionless, let (12.2-15) We get (12.2-16) With

  36. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(6) 3) Solution by combination of variables (1) To get the combined variable Choosing a set of characteristic quantities and taking Eq.(12.2-16) may be rewritten as The similar solution to it is

  37. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(7) The following two ways are equivalent to each other for finding the value of * : To change {*,*} with {0 ,0}keeping constant To change {0 ,0}with {* ,*} keeping constant So we might as well take Then we get

  38. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(8) Let Inserting it into Eq.(12.2-16), we have or Taking we get an ODE (12.2-20)

  39. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(9) (2) To solve the ODE Eq.(12.2-20) may be rearranged as Integration of this equation gives Or Then by B.C.1,

  40. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(10) By B.C.2, Therefore, in which (x)is the Gamma function, defined as

  41. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(11) (3) To find the temperature distribution Or

  42. §12.2-2 Laminar Tube Flow with Constant Heat Flux at The WallAsymptotic solution for the entrance region(12) With original variables, we have in which (x, y)is the incomplete Gamma function defined as

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