Problem 4.3. Problem 4.4

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Marilyn Waters
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1 Problem 4.3 Problem 4.4 Problem 4.5 Problem 4.6 Problem 4.7 This is forced convection flow over a streamlined body. Viscous (velocity) boundary layer approximations can be made if the Reynolds number Re x > 100. Thermal (temperature) boundary layer approximations can be made if the Peclet number Pe x = Re x Pr > 100. The Reynolds number decreases as the distance along the plate is decreased. The surface is streamlined. The fluid is water. Inertia and viscous effects can be estimated using scaling. If a viscous term is small compared to inertia, it can be neglected. Properties should be evaluated at the film temperature T ( T T ) / 2. The surface is streamlined. The fluid is water. Convection and conduction effects can be estimated using scaling. If a conduction term is small compared to convection, it can be neglected. The scale for t / L depends on whether t or t. Properties should be evaluated at the film temperature T ( T T ) / 2. The fluid is air. Dissipation and conduction can be estimated using scaling. Dissipation is negligible if the Eckert number is small compared to unity. The surface is streamlined. The fluid is air. f f s s

2 Problem 4.9 Problem 4.11 Problem 4.12 This is a forced convection problem over a flat plate. At the edge of the thermal boundary layer, the axial velocity is u V. Blasius solution gives the distribution of the velocity components u(x,y) and v(x,y). Scaling gives an estimate of v(x,y). This is a laminar boundary layer flow problem. Blasius solution gives the velocity distribution for the flow over a semi-infinite flat plate. (iii) A solution for the boundary layer thickness depends on how the thickness is defined. Since the flow within the boundary layer is two-dimensional the vertical velocity component does not vanish. Thus stream lines are not parallel. Blasius solution is valid for laminar boundary layer flow over a semi-infinite plate. The transition Reynolds number from laminar to turbulent flow is Boundary layer approximations are valid if the Reynolds number is greater than 100. Problem 4.13 Problem 4.14 Problem 4.16 This is an external flow problem over a flat plate. Blasius s solution for the velocity distribution and wall shearing stress is assumed to be applicable. Of interest is the value of the local stress at the leading edge of the plate. This is an external flow problem over a flat plate. The force needed to hold the plate in place is equal to the total shearing force by the fluid on the plate. Integration of wall shear over the surface gives the total shearing force. Blasius s solution for the velocity distribution and wall shearing stress is assumed to be applicable.

3 Problem 4.18 Problem 4.19 Problem 4.20 Of interest is the region where the upstream fluid reaches the leading edge of the plate. The fluid is heated by the plate. Heat from the plate is conducted through the fluid in all directions. Pohlhausen s solution assumes that heat is not conducted upstream from the plate and therefore fluid temperature at the leading edge is the same as upstream temperature. This is a forced convection problem over a flat plate. At the edge of the thermal boundary layer, fluid temperature is T T. Pohlhausen s solution gives the temperature distribution in the boundary layer. The thermal boundary layer thickness t increases with distance from the leading edge. t depends on the Prandtl number. Pohlhausen s solution for the temperature distribution and heat transfer coefficient is assumed to be applicable. Problem 4.22` Of interest is the value of the local heat flux at the leading edge of the plate. Knowing the local transfer coefficient and using Newton s law, gives the heat flux Pohlhausen s solution for the temperature distribution is assumed to be applicable. Of interest is the value of the normal temperature gradient at the surface. This is an external forced convection problem over two flat plates. Both plates have the same surface area. For flow over a flat plate, the heat transfer coefficient h decreases with distance from the leading edge. Since the length in the flow direction is not the same for the two plates, the average heat transfer coefficient is not the same. It follows that the total heat transfer rate is not the same.

4 Problem 4.23 Problem 4.24 Problem 4.25 The flow over a flat plate is laminar if the Reynolds number is less than Pohlhausen s solution for the temperature distribution and heat transfer coefficient is assumed to be applicable. Of interest is the value of the heat transfer rate from a section of the plate at a specified location and of a given width. Newton s law of cooling gives the heat transfer rate. Of interest is the variation of the local heat transfer coefficient with free stream velocity and distance from the leading edge. Pohlhausen's solution applies to this problem. This is an external flow problem. At the edge of the thermal boundary layer, stream temperature. That is, T T and T ( T T ) /( T ) 1. * y t, fluid temperature approaches free s T s According to Pohlhausen's solution, Fig. 4.6, the thermal boundary layer thickness depends on the Prandtl number, free stream velocity V, kinematic viscosity and location x. Problem 4.26 Problem 4.27 The Reynolds number and Peclet number should be checked to determine if the flow is laminar and if boundary layer approximations are valid. Pohlhausen's solution is applicable if 100 < Re x < and Pex = Re x Pr > 100. Thermal boundary layer thickness and heat transfer coefficient vary along the plate. Newton s law of cooling gives local heat flux. (vi) The fluid is water. This is an external forced convection problem over a flat plate.

5 Problem 4.28 Problem 4.29 Problem 4.30 Increasing the free stream velocity, increases the average heat transfer coefficient. This in turn causes surface temperature to drop. Based on this observation, it is possible that the proposed plan will meet design specification. Since the Reynolds number at the downstream end of the package is less than 500,000, it follows that the flow is laminar throughout. Increasing the free stream velocity by a factor of 3, increases the Reynolds number by a factor of 3 to 330,000. At this Reynolds number the flow is still laminar. The power supplied to the package is dissipated into heat and transferred to the surroundings from the surface. Pohlhausen's solution can be applied to this problem. The ambient fluid is unknown. Convection heat transfer from a surface can be determined using Newton s law of cooling. The local heat transfer coefficient changes along the plate. The total heat transfer rate can be determined using the average heat transfer coefficient. For laminar flow, Pohlhausen's solution gives the heat transfer coefficient. For two in-line fins heat transfer from the down stream fin is influenced by the upstream fin. The further the two fins are apart the less the interference will be. Pohlhausen s solution for the temperature distribution and heat transfer coefficient is assumed to be applicable. Knowing the heat transfer coefficient, the local Nusselt number can be determined. the Newton s law of cooling gives the heat transfer rate. Pohlhausen s solution gives the thermal boundary layer thickness. Convection heat transfer from a surface can be determined using Newton s law of cooling.

6 Problem 4.31 Problem 4.32 Problem 4.33 The local heat transfer coefficient changes along the plate. For each triangle the area changes with distance along the plate. The total heat transfer rate can be determined by integration along the length of each triangle. Pohlhausen's solution may be applicable to this problem. Heat transfer rate can be determined using Newton s law of cooling. The local heat transfer coefficient changes along the plate. The area changes with distance along the plate. The total heat transfer rate can be determined by integration along the length of the triangle. Pohlhausen's solution may be applicable to this problem. Heat transfer rate can be determined using Newton s law of cooling. The local heat transfer coefficient changes along the plate. The area changes with distance along the plate. The total heat transfer rate can be determined by integration along over the area of the semi-circle. Pohlhausen's solution gives the heat transfer coefficient. Heat transfer rate can be determined using Newton s law of cooling. The local heat transfer coefficient changes along the plate. The area changes with distance along the plate. The total heat transfer rate can be determined by integration along the length of the triangle. Pohlhausen's solution may be applicable to this problem.

7 Problem 4.34 Problem 4.36 This problem involves determining the heat transfer rate from a circle tangent to the leading edge of a plate Heat transfer rate can be determined using Newton s law of cooling. The local heat transfer coefficient changes along the plate. The area changes with distance along the plate. The total heat transfer rate can be determined by integration along the length of the triangle. Pohlhausen's solution may be applicable to this problem. The flow field for this boundary layer problem is simplified by assuming that the axial velocity is uniform throughout the thermal boundary layer. Since velocity distribution affects temperature distribution, the solution for the local Nusselt number can be expected to differ from Pohlhausen s solution. The Nusselt number depends on the temperature gradient at the surface. Problem 4.37 Problem 4.38 The flow field for this boundary layer problem is simplified by assuming that the axial velocity varies linearly in the y-direction. Since velocity distribution affects temperature distribution, the solution for the local Nusselt number can be expected to differ from Pohlhausen s solution. The Nusselt number depends on the temperature gradient at the surface. The flow and temperature fields for this boundary layer problem are simplified by assuming that the axial velocity and temperature do not vary in the x-direction. The heat transfer coefficient depends on the temperature gradient at the surface. Temperature distribution depends on the flow field. The effect of wall suction must be taken into consideration.

8 Problem 4.39 Problem 4.40 Problem 4.41 Problem 4.42 This is a forced convection flow over a plate with variable surface temperature. The local heat flux is determined by Newton s law of cooling. The local heat transfer coefficient and surface temperature vary with distance along the plate. The variation of surface temperature and heat transfer coefficient must be such that Newton s law gives uniform heat flux. The local heat transfer coefficient is obtained from the local Nusselt number. This is a forced convection flow over a plate with variable surface temperature. The Reynolds number should be computed to determine if the flow is laminar or turbulent. The local heat transfer coefficient and surface temperature vary with distance along the plate. The local heat transfer coefficient is obtained from the solution to the local Nusselt number. The determination of the Nusselt number requires determining the temperature gradient at the surface. This is a forced convection flow over a plate with variable surface temperature. The Reynolds number should be computed to determine if the flow is laminar or turbulent. Newton s law of cooling gives the heat transfer rate from the plate. The local heat transfer coefficient and surface temperature vary with distance along the plate. Thus determining the total heat transfer rate requires integration of Newton s law along the plate. The local heat transfer coefficient is obtained from the local Nusselt number. This is an external forced convection problem of flow over a flat plate Convection heat transfer from a surface can be determined using Newton s law of cooling. The local heat transfer coefficient and surface temperature vary along the plate. For each triangle the area varies with distance along the plate.

9 Problem 4.43 Problem 4.44 Problem 4.45 Problem 4.46 The total heat transfer rate can be determined by integration along the length of each triangle. This is a forced convection boundary layer flow over a wedge. Wedge surface is maintained at uniform temperature. The flow is laminar. The fluid is air. Similarity solution for the local Nusselt number is presented in Section The Nusselt number depends on the Reynolds number and the dimensionless temperature gradient at the surface d ( 0) / d. (vii) Surface temperature gradient depends on wedge angle. This is a forced convection boundary layer flow over a wedge. Wedge surface is maintained at uniform temperature. The flow is laminar. The average Nusselt number depends on the average heat transfer coefficient.. Similarity solution for the local heat transfer coefficient is presented in Section Newton s law of cooling gives the heat transfer rate from a surface. Total heat transfer from a surface depends on the average heat transfer coefficient h. Both flat plate and wedge are maintained at uniform surface temperature. Pohlhausen s solution gives h for a flat plate. Similarity solution for the local heat transfer coefficient for a wedge is presented in Section The flow field for this boundary layer problem is simplified by assuming that axial velocity within the thermal boundary layer is the same as that of the external flow. Since velocity distribution affects temperature distribution, the solution for the local Nusselt number differs from the exact case of Section The local Nusselt number depends the local heat transfer coefficient which depends on the temperature gradient at the surface.

10 Problem 4.47 The flow field for this boundary layer problem is simplified by assuming that axial velocity within the thermal boundary layer varies linearly with the normal distance. Since velocity distribution affects temperature distribution, the solution for the local Nusselt number differs from the exact case of Section The local Nusselt number depends on the local heat transfer coefficient which depends on the temperature gradient at the surface.

CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW

CHAPTER 4 BOUNDARY LAYER FLOW APPLICATION TO EXTERNAL FLOW 4.1 Introduction Boundary layer concept (Prandtl 1904): Eliminate selected terms in the governing equations Two key questions (1) What are the