Page 1 Module 2 : Convection Lecture 18 : Developed velocity and Developing temperature in Pipe flow with Constant Wall temperature Objectives In this class: The developing temperature profile for fully developed velocity profile and uniform circumferential heating with constant wall temperature is obtained. Analytical solutions cannot be obtained for the entire problem and numerical solutions are borrowed from text books where- ever required. Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-1 Now, look at the developing profile for a constant wall temperature. The variables are non-dimensionalized in a manner identical to that for the fully developed case. The only exception is the temperature. (18.1) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-2 The Governing Equation remains the same. (18.2) The Boundary conditions become (18.3) (18.4) (18.5) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-3 Again use separation of variables (18.6) The governing equ n (18.2) therefore becomes: (18.7) The â€˜Zâ€™ component of the equ n (18.7) gives: (18.8) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4 The â€˜Râ€™ component of the equ n (18.7) gives: (18.9) The associated boundary conditions are: Page 2 Module 2 : Convection Lecture 18 : Developed velocity and Developing temperature in Pipe flow with Constant Wall temperature Objectives In this class: The developing temperature profile for fully developed velocity profile and uniform circumferential heating with constant wall temperature is obtained. Analytical solutions cannot be obtained for the entire problem and numerical solutions are borrowed from text books where- ever required. Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-1 Now, look at the developing profile for a constant wall temperature. The variables are non-dimensionalized in a manner identical to that for the fully developed case. The only exception is the temperature. (18.1) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-2 The Governing Equation remains the same. (18.2) The Boundary conditions become (18.3) (18.4) (18.5) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-3 Again use separation of variables (18.6) The governing equ n (18.2) therefore becomes: (18.7) The â€˜Zâ€™ component of the equ n (18.7) gives: (18.8) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4 The â€˜Râ€™ component of the equ n (18.7) gives: (18.9) The associated boundary conditions are: (18.10) Unlike for the constant wall flux developing case, the boundary conditions provide a unique solution Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-5 Need to use numerical methods to obtain the solution. Some qualitative profiles are shown below: Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-6 Again the initial condition has to be satisfied and only a summation solution will be suitable (18.10) Imposing the initial condition in equ n (18.10) gives: (18.11) Equ n (17.28) continues to be valid since the differential equation has remained the same Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-7 Using equ n (17.28) in conjunction with equ n (18.11) the constants for the solution equ n (18.10) are evaluated as: (18.12) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-8 Again interest is in the wall quantities and since the temperature profile is known these are evaluated: Page 3 Module 2 : Convection Lecture 18 : Developed velocity and Developing temperature in Pipe flow with Constant Wall temperature Objectives In this class: The developing temperature profile for fully developed velocity profile and uniform circumferential heating with constant wall temperature is obtained. Analytical solutions cannot be obtained for the entire problem and numerical solutions are borrowed from text books where- ever required. Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-1 Now, look at the developing profile for a constant wall temperature. The variables are non-dimensionalized in a manner identical to that for the fully developed case. The only exception is the temperature. (18.1) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-2 The Governing Equation remains the same. (18.2) The Boundary conditions become (18.3) (18.4) (18.5) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-3 Again use separation of variables (18.6) The governing equ n (18.2) therefore becomes: (18.7) The â€˜Zâ€™ component of the equ n (18.7) gives: (18.8) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4 The â€˜Râ€™ component of the equ n (18.7) gives: (18.9) The associated boundary conditions are: (18.10) Unlike for the constant wall flux developing case, the boundary conditions provide a unique solution Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-5 Need to use numerical methods to obtain the solution. Some qualitative profiles are shown below: Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-6 Again the initial condition has to be satisfied and only a summation solution will be suitable (18.10) Imposing the initial condition in equ n (18.10) gives: (18.11) Equ n (17.28) continues to be valid since the differential equation has remained the same Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-7 Using equ n (17.28) in conjunction with equ n (18.11) the constants for the solution equ n (18.10) are evaluated as: (18.12) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-8 Again interest is in the wall quantities and since the temperature profile is known these are evaluated: (18.12a) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-9 First evaluate the blue term of equ n (18.12). Use the definition of bulk temperature Use manipulations similar to those used in the fully developed case: (18.13) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-10 Substitute the ? to obtain: (18.14) Use the original differential equ n (18.9), integrate it and use the boundary condition equ n (18.10) to get: (18.15) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-11 Substitute equ n (18.15) in equ n (18.14) to get: (18.16) The red term in equ n (18.12) is now evaluated: (18.17) Substitute equ n (18.16) and (18.17) in (18.12a) to get: Page 4 Module 2 : Convection Lecture 18 : Developed velocity and Developing temperature in Pipe flow with Constant Wall temperature Objectives In this class: The developing temperature profile for fully developed velocity profile and uniform circumferential heating with constant wall temperature is obtained. Analytical solutions cannot be obtained for the entire problem and numerical solutions are borrowed from text books where- ever required. Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-1 Now, look at the developing profile for a constant wall temperature. The variables are non-dimensionalized in a manner identical to that for the fully developed case. The only exception is the temperature. (18.1) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-2 The Governing Equation remains the same. (18.2) The Boundary conditions become (18.3) (18.4) (18.5) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-3 Again use separation of variables (18.6) The governing equ n (18.2) therefore becomes: (18.7) The â€˜Zâ€™ component of the equ n (18.7) gives: (18.8) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4 The â€˜Râ€™ component of the equ n (18.7) gives: (18.9) The associated boundary conditions are: (18.10) Unlike for the constant wall flux developing case, the boundary conditions provide a unique solution Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-5 Need to use numerical methods to obtain the solution. Some qualitative profiles are shown below: Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-6 Again the initial condition has to be satisfied and only a summation solution will be suitable (18.10) Imposing the initial condition in equ n (18.10) gives: (18.11) Equ n (17.28) continues to be valid since the differential equation has remained the same Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-7 Using equ n (17.28) in conjunction with equ n (18.11) the constants for the solution equ n (18.10) are evaluated as: (18.12) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-8 Again interest is in the wall quantities and since the temperature profile is known these are evaluated: (18.12a) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-9 First evaluate the blue term of equ n (18.12). Use the definition of bulk temperature Use manipulations similar to those used in the fully developed case: (18.13) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-10 Substitute the ? to obtain: (18.14) Use the original differential equ n (18.9), integrate it and use the boundary condition equ n (18.10) to get: (18.15) Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-11 Substitute equ n (18.15) in equ n (18.14) to get: (18.16) The red term in equ n (18.12) is now evaluated: (18.17) Substitute equ n (18.16) and (18.17) in (18.12a) to get: Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-12 For large values of â€˜zâ€™ only the first term is of importance and Nu = ß 1 = 3.66 which is the same as the fully developed situation. At large values of â€˜zâ€™ the influence of the developing region diminishes. The nondimensional length at which flow developes is approximately z/D/RePr ~ 0.1 Recap In this class: The developing temperature profile for fully developed velocity profile and uniform circumferential heating with constant wall temperature is obtained. Analytical solutions cannot be obtained for the entire problem and numerical solutions are borrowed from text books where- ever required. 18.18Read More

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