1. CONVECTIVE HEAT TRANSFER Mohammad Goharkhah Department of Mechanical Engineering, Sahand Unversity of Technology, Tabriz, Iran

2. LAMINAR BOUNDARY LAYER FLOW CHAPTER 3- PART4

3. Approximate Solutions- Integral Method

4. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Introduction when an exact solution is not available or can not be easily obtained. when the form of the exact solution is not convenient to use. Examples include solutions that are too complex, implicit or require numerical integration. When we use approximate solutions?  The integral approach to solving the boundary layer equations is an important piece of analysis developed by Prandtl’s disciples Pohlhausen (doctoral student) and von K´arm´an (postdoc) in the first decades of this century.  In the integral method, we look at the definitions of τ and h and recognize that what we need is not a complete solution for the velocity u(x,y) and temperature T(x,y) near the wall, but only the gradients ∂(u,T)/∂y evaluated at y = 0. Because the y > 0 variation of u and T is not the most relevant to evaluating τ and h, we have the opportunity to simplify the boundary layer Equations by eliminating y as a variable. Integral Method

5. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Introduction  The integral method is used extensively in fluid flow, heat transfer and mass transfer.  Because of the mathematical simplifications associated with this method, it can deal with such complicating factors as turbulent flow, temperature dependent properties and non-linearity. Applications

6. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Differential vs. Integral Formulation A differential element measuring dx ×dy is selected. The three basic laws are formulated for this element. The resulting equations thus apply to any point in the region and the Solutions to these equations satisfy the basic laws exactly. A differential element measuring dx×δ is selected. (infinitesimal in x but finite in y). The three basic laws are formulated for this element. The resulting equations satisfy the basic laws for an entire cross section δ and not at every point. Solutions to this type of formulation are approximate in the sense that they do not satisfy the basic laws at every point. A key simplification in integral method is a reduction in the number of independent variables. For example, for two-dimensional problems, instead of solving a PDE, one solves an ODE in integral formulation.

7. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Procedure (1) Integral formulation of the basic laws (principles of conservation of mass, momentum and energy.) (2) Assumed approximate velocity and temperature profiles which satisfy known boundary conditions. An assumed profile can take on different forms. However, a polynomial is usually used in Cartesian coordinates. An assumed profile is expressed in terms of a single unknown parameter or variable which must be determined. (3) Determination of the unknown parameter or variable. Substituting the assumed velocity profile into the integral form of conservation of momentum and solving the resulting equation gives the unknown parameter. Similarly, substituting the assumed velocity and temperature profiles into the integral form of conservation of energy yields an equation whose solution gives the unknown parameter in the temperature profile.

8. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Accuracy of the Integral Method Since an assumed profile is not unique (several forms are possible), the accuracy of integral solutions depends on the form of the assumed profile. In general, errors involved in this method are acceptable in typical engineering applications. The accuracy is not very sensitive to the form of an assumed profile. While there are general guidelines for improving the accuracy, no procedure is available for identifying assumed profiles that will result in the most accurate solutions. An assumed profile which satisfies conditions at a boundary yields more accurate information at that boundary than elsewhere.

9. Integral Formulation of the Basic Laws

10. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of Conservation of Mass boundary layer flow over a curved porous surface P is wall porosity

11. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of Conservation of Momentum

12. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of Conservation of Momentum

13. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of Conservation of Momentum (1) Fluid entering the element through the porous surface has no axial velocity. Therefore it has no x-momentum. (2) There is no shearing force on the slanted surface since the velocity gradient at the edge of the boundary layer vanishes, i.e. Əu(x, δ ) / Əy ~0. (3) The equation applies to laminar as well as turbulent flow. (4) Since the porous surface is curved, the external flow velocity and pressure vary along the surface. (5) The effect of gravity is neglected. (6) Although u is a function of x and y, once the integrals are evaluated one obtains a first order ordinary differential equation with x as the independent variable.

14. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Conservation of Momentum- Special Cases Case 1: Incompressible fluid Boundary layer approximation: The x-momentum equation for boundary layer flow At the edge of the boundary layer

15. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Conservation of Momentum- Special Cases Case 2: Incompressible fluid and impermeable flat plate. At the edge of boundary layer flow the fluid is assumed inviscid

16. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of Conservation of Energy

17. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of Conservation of Energy Mass entering the element from the external flow is at the free stream temperature Heat conduction at the porous surface is determined using Fourier’s law Energy convected with fluid flow within the boundary layer

18. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of Conservation of Energy Special Case: Constant properties and impermeable flat plate Setting P = 1 and assuming constant density and specific heat

19. Integral Formulation of the Basic Laws- The Second Approach

20. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of the Basic Laws- The Second Approach The Integration formulations can be obtained by integrating the boundary layer equations term by term from y = 0 to y = Y, where Y > max(δ,δT) is situated in the free stream.

21. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of the Basic Laws- The Second Approach Integrating above equations from y = 0 to y = Y and using Leibnitz’s integral formula yields Because the free stream is uniform, we note that (∂/∂y) Y = 0, u Y = U∞, and T Y = T∞

22. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Integral Formulation of the Basic Laws- The Second Approach We evaluate v Y by performing the same integral on the continuity equation The wall is impermeable, v 0 = 0

23. Integral Formulation of the Basic Laws- Examples

24. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 1- Uniform Flow over a Semi-Infinite Plate- Flow Field Solution Blasius laminar flow problem Assume a velocity profile For laminar flow over a flat plate, a polynomial is a reasonable representation of the velocity profile The coefficients are determined using the known exact and approximate boundary conditions on the velocity

25. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 1- Uniform Flow over a Semi-Infinite Plate- Flow Field Solution Thus the assumed velocity is expressed in terms of the unknown variable δ(x). This variable is determined using the integral form of the momentum equation (1) The second and third conditions are approximate since the edge of the boundary layer is not uniquely defined. (2) Condition (4) is obtained by setting y= 0 in the x- component of the Navier equations of motion.

26. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Compare above results with:

27. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Important Notes (1) The integral and Blasius solutions for δ(x) and C f have the same form. (2) The constant 5.2 in Blasius solution for δ(x) differs by 10.8% from the corresponding integral solution of 4.64. However, it must be kept in mind that the constant in Blasius solution for δ (x) is not unique. It depends on how δ(x) is defined. (3) The error in C f is 2.7%. (4) Predicting C f accurately is more important than predicting δ(x).

28. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Boundary layer flow over a semi infinite plate- Temperature Distribution  A leading section of the plate of length x o is insulated and the remaining part is at uniform temperature.  Of interest is the determination of the thermal boundary layer thickness, local heat transfer coefficient, and Nusselt number.  Since the velocity field is independent of temperature, the integral solution for the velocity and boundary layer thickness obtained previously is applicable to this case.

29. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Boundary layer flow over a semi infinite plate- Temperature Distribution For laminar flow over a flat plate a polynomial is a reasonable representation for the temperature profile.  Note that the second and third conditions are approximate since the edge of the thermal boundary layer is not uniquely defined.  The fourth condition is obtained by setting y=0 in the energy equation Assume a temperature profile

30. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Boundary layer flow over a semi infinite plate- Temperature Distribution Eliminating δ(x) in the above gives a first order ordinary differential equation for δ t (x).

31. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING For Prandtl numbers greater than unity the thermal boundary layer is smaller than the viscous boundary layer

32. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Boundary layer flow over a semi infinite plate- Temperature Distribution Nusselt Number

33. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Boundary layer flow over a semi infinite plate- Temperature Distribution Special Case: Plate with no Insulated Section  The solution to this case is obtained by setting x o =0 in the more general case of a plate with a leading insulated section.  Thermal boundary layer thickness, heat transfer coefficient, and Nusselt number are obtained by settig x 0 =0

34. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Comparison with Pohlhausen’s results For the limiting case of Pr = 1, δ t / δ= 1. This has an error of 2.5%. We examine next the accuracy of the local Nusselt number For Pr>10 Pohlhausen’s solution is error 2.4%. Setting Pr=1 in integral solution

35. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Laminar Boundary Layer Flow over a Flat Plate: Uniform Surface Temperature Linear velocity and temperature profiles Momentum Equation

36. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Laminar Boundary Layer Flow over a Flat Plate: Uniform Surface Temperature Energy Equation

37. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 2- Laminar Boundary Layer Flow over a Flat Plate: Uniform Surface Temperature

38. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Effect of Shape Function Let us assume that the shape of the longitudinal velocity profile is described by m is an unspecified shape function that varies from 0 to 1, and n = y/δ

39. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Effect of Shape Function The resulting expressions for local boundary layer thickness and skin friction coefficient are

40. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Effect of Shape Function Heat transfer coefficient information is extracted in a similar fashion with dT∞/dx = 0. Thus, we assume the temperature profile shapes (a function of Prandtl number) Based On the assumptions δT < δ (high-Pr fluids), the integral energy equation reduces to

41. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Effect of Shape Function Assuming the simplest temperature profile, m = p: which is numerically identical to the scaling law for Pr>>1 fluids

42. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Effect of Shape Function In the case of liquid metals Δ>> 1, we obtain

43. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Effect of Shape Function

44. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 3- Uniform Surface Flux A flat plate with an insulated leading section of length x o. The plate is heated with uniform flux q’’ along its surface We consider steady state, laminar, two-dimensional flow with constant properties. We wish to determine surface temperature distribution and the local Nusselt number For constant properties, the velocity distribution is independent of temperature. Assume a third degree polynomial for the temperature profile

45. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 3- Uniform Surface Flux

46. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 3- Uniform Surface Flux For the special case of a plate with no insulated section, setting x 0 =0 This result is in good agreement with the more accurate differential formulation solution

47. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 4- Laminar Boundary Layer Flow over a Flat Plate: Variable Surface Temperature

48. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 4- Laminar Boundary Layer Flow over a Flat Plate: Variable Surface Temperature

49. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 4- Laminar Boundary Layer Flow over a Flat Plate: Variable Surface Temperature

50. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE The integral solution for heat transfer with an unheated starting length is the building block for the construction of heat transfer results for more complicated situations. Consider, for example, heat transfer from the heated spot x 1 <x< x 2, The wall temperature upstream and downstream from the heated spot is equal to the constant free-stream value, T ∞, while the spot temperature is T 0 Since the integral energy equation is linear in temperature, the thermal boundary layer generated by the T 0 spot can be reconstructed as the superposition of two thermal boundary layers.

51. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING The first thermal boundary layer, δ T,1 is the fingerprint of wall heating (T∞+ ΔT) downstream from x= x 1. The second thermal boundary layer is the result of wall cooling (T∞−ΔT) downstream from x = x 2. The superposition of the two thermal layers constitutes the thermal boundary layer due to spot heating. Of interest is the heat flux q’’ from the wall to the fluid.

52. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE To calculate q’’, we identify three distinct wall regions: 1.0 < x < x1, the unheated started length, where q’’ = 0 because the wall is in thermal equilibrium with the free stream 2- x1 < x < x2, the heated spot, where : 3- x > x2, the trailing section, where q is the superposition of two effects. Note that since x2 > x1, the heat flux q in region 3 is negative. This means that in the trailing section, the wall reabsorbs part of the heat released earlier in region 2.

53. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE The heat flux from the wall to the fluid, downstream from N step changes T i in wall temperature, is given by where x i is the longitudinal position of each temperature step change Ti If the wall temperature varies smoothly, T 0 (x), above formula is replaced by its integral limit (the limit of infinitesimally small steps): The factor 0.332, which appears on the right-hand side of the equations, was borrowed from the similarity solution. The actual factor generated by the integral solution with cubic profile is 0.331.

54. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE The method of superposition can be used to develop heat transfer solutions for the boundary layer for an arbitrary wall temperature variation because of the linearity of the energy differential equation of the boundary layer. Let ɵ(ξ, x, y) be a solution to energy for constant-property, constant free- stream velocity flow along a flat plate for the step-function boundary condition t 0 = t ∞ for x ξ. Then ɵ will be defined such that Then a solution to energy equation for any arbitrary variation in surface temperature t 0, but with free-stream temperature constant, is

55. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE The heat flux from the wall surface is determined from

56. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE The previously derived step-function solution for the laminar boundary layer can be written as: As an example of the method, consider a plate with a step in wall temperature at the leading edge, followed by a linear wall- temperature variation. Above equation then provides the method for the calculation of heat-transfer rates from a flat plate with a laminar boundary layer and any axial wall temperature distribution. It is necessary to insert only the desired dt 0 / d ξ as a function of ξ in the integral and any abrupt changes in wall temperature in the summation.

57. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE There is only one step in wall temperature, at ξ = 0, so there is only one term in the summation. Thus

58. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE The following change of variable transforms the integral into the form of the beta function. The integral is then readily evaluated by use of beta function tables

59. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE

60. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING 5- CONSTANT FREE-STREAM VELOCITY FLOW ALONG A SEMI-INFINITE PLATE WITH ARBITRARILY SPECIFIED SURFACE TEMPERATURE

61. CONVECTIVE HEAT TRANSFER- CHAPTER3 By: M. Goharkhah SAHAND UNIVERSITY OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING Questions?