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CHAPTER 5

CHAPTER 5. APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD. 5.1 Introduction. Why approximate solution?. When exact solution is:. Not Available. Complex. Requires numerical integration. Implicit. Approximate solution by integral method:.

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CHAPTER 5

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  1. CHAPTER 5 APPROXIMATE SOLUTIONS: THE INTEGRAL METHOD 5.1 Introduction • Why approximate solution? • When exact solution is: • Not Available • Complex

  2. Requires numerical integration • Implicit • Approximate solution by integral method: • Advantages: simple, can deal with • complicating factors • Used in fluid flow, heat transfer, • mass transfer

  3. 5.2 Differential vs. Integral Formulation • Result: Solutions are exact

  4. Result: Solutions are approximate

  5. 5.3 Integral Method Approximation: Mathematical Simplification • Number of independent variables are reduced • Reduction in order of differential equation 5.4 Procedure • Integral formulation of the basic laws • Conservation of mass • Conservation of momentum • Conservation of energy

  6. (2) Assumed velocity and temperature profiles • Satisfy boundary conditions • Several possibilities • Examples: Polynomial, linear, exponential • Assumed profile has an unknown parameter or variable (3) Determination of the unknown parameter or variable • Conservation of momentum gives the unknown variable in the assumed velocity

  7. Conservation of energy gives the unknown variable in the assumed temperature 5.5 Accuracy of the Integral Method • Accuracy depends on assumed profiles • Accuracy is not sensitive to form of profile • Optimum profile: unknown

  8. : • Conservation of mass for element 5.6 Integral Formulation of the Basic Laws 5.6.1 Conservation of Mass • Boundary layer flow • Porous curved wall

  9. (a)

  10. = mass from the external flow = mass through porous wall = mass from boundary layer rate entering element at x (b) dmo= ρvo Pdx (c) One-dimensional mass flow rate: Apply (b) to porous side, assume that injected is the same as external fluid

  11. = density (d) P = porosity Applying (b) to dy Integrating (c) and (d) into (a)

  12. (5.1) Apply momentum theorem to the element (a) = External x-forces on element 5.6.2 Conservation of Momentum

  13. = x-momentum of the fluid entering = x-momentum of the fluid leaving

  14. Fig. 4 and (a):

  15. = pressure = velocity at the edge of the boundary layer (b) = wall stress (c) and

  16. ( ) ¶ dp u x , 0 ( ) - d - m - = 1 P ¶ dx y d d ( x ) ( x ) d d ( ) ( ) ò ò 2 r - r - r u dy V x u dy V x P v ¥ ¥ o dx dx 0 0 • x-momentum of (b) and (c) into (a) • Shear force on slanted surface? • (5.2) applies to laminar and turbulent flow

  17. Curved surface: and (4.12) • (5.2) is the integral formulation of conservation of momentum and mass • (5.2) is a first order O.D.E. with x as the independent variable Special Cases: (i) Case 1: Incompressible fluid For boundary layer flow

  18. (4.5) Apply (4.5) at where (5.3) (5.3) into (5.2), constant (5.4)

  19. (d) (e) (5.5) (ii) Case 2: Incompressible fluid and impermeable flat plate Flat plate, (5.3) Impermeable plate (d) and (e) into (5.4)

  20. 5.6.3 Conservation of Energy Neglect: • Changes in kinetic and potential energy (2) Dissipation (3) Axial conduction

  21. Conservation of energy element

  22. (a) (b) Energy with mass Rearranging Fourier’s law

  23. Use (5.1) for (c) (d) (e) Energy of injected mass Energy convected with fluid

  24. (5.6) (b)-(e) into (a) Note • (5.6) is integral formulation of conservation • energy • (5.6) is a first order O.D.E. with x as the • independent variable

  25. (5.7) Special Case: Constant properties and impermeable flat plate Simplify (5.6)

  26. 5.7.1 Flow Field Solution: Uniform Flow over a Semi-Infinite Plate 5.7 Integral Solutions • Integral solution to Blasius problem • Integral formulation of momentum (5.5)

  27. (5.5) • Assumed velocity profile (5.8) (a) Example, a third degree polynomial

  28. Boundary conditions give (1) (2) (3) (4) (2) B.C. (4): set in the x-component of the Navier equations, (2.10x) Note: • B.C. (2) and (3) are approximate. Why? (3) 4 B.C. and (a) give:

  29. , , (5.9) (b) (5.9) into (5.5), evaluate integrals

  30. (b) is first order O.D.E. in Integrate, use B.C. (5.10) Evaluate integrals

  31. (5.10) into (5.9) gives Use (4.36) and (4.37a) Friction coefficient : Use (5.10) to eliminate (5.11)

  32. , Blasius solution (4.46) , (4.48) Blasius solution (2) Error in is 10.8% Accuracy: Compare (5.10) and (5.11) with Blasius solution Note: • Both solutions have same form

  33. (3) Error in is 2.7% (4) Accuracy of is more important than • Insulated section 5.7.2 Temperature Solution and Nusselt Number: Flow over a Semi-Infinite Plate (i) Temperature Distribution • Flat plate

  34. Surface at • Determine: , , • Must determine: and (2) (5.7) • Laminar, steady, two-dimensional, constant properties boundary layer flow Integral formulation of conservation of energy

  35. Integral Solution to and : (5.9) (5.10) (5.12) Assumed temperature

  36. (a) Boundary conditions give (1) (2) (3) (4) Let

  37. (2) B.C. (4): set in the x-component of the energy equation (2.19) , Note • B.C. (2) and (3) are approximate. Why? (3) Four B.C. and (a) give:

  38. (5.13) (5.14) • (5.14) is simplified for (5.15) , for Therefore (5.9) and (5.13) into (5.7) and evaluating the integral

  39. (b) Use (5.10) for (c) Last two term in (5.14): Simplify (5.14)

  40. (d) Solve (d) for . Let (e) (f) (c) into (b) (e) into (d)

  41. (g) C = constant. Use boundary condition on (h) (i) Separate variables and integrate Apply (h) to (g)

  42. (5.16) Use (c) to eliminate in (5.16) (5.17a) (5.17b) (i) into (h) or

  43. is the local Reynolds (5.18) (j) (k) (ii) Nusselt Number Local Nusselt number: h is given by

  44. (5.19) Use (5.17a) to eliminate in (5.19) (5.20) (5.21) Use temperature solution (5.13) in (k) Substitute into (j)

  45. in above solution set (5.22) (5.23) (iii) Special Case: Plate with no Insulated Section

  46. (5.24) (5.25) Set in (5.22) (1) for . , (2) Compare with Pohlhausen’s solution. For (4.72c) Accuracy of integral solution: Error is 2.5% Error is 2.4%

  47. (a) (1) (2) f , (b) Example 5.1: Laminar Boundary Layer Flow over a Flat Plate: Uniform Surface Temperature Use a linear profiles. Velocity: Boundary conditions

  48. (5.26) (c) (1) , (2) (d) Apply integral formulation of momentum, (5.5). Result: Temperature profile: Boundary conditions

  49. (5.27) Special case: Apply integral formulation of energy, (5.7). Result: Comments • Linear profiles give less accurate results • than polynomials. (ii) More accurate prediction of Nusselt number than viscous boundary layer thickness.

  50. Insulated section of length • Plate is heated with uniform flux • Determine and 5.7.3 Uniform Surface Flux • Flat plate

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