Developed Velocity & Developing Temperature in Pipe Flow with Constant Wall Temperature - Convection Mechanical Engineering Notes | EduRev

Mechanical Engineering : Developed Velocity & Developing Temperature in Pipe Flow with Constant Wall Temperature - Convection Mechanical Engineering Notes | EduRev

 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 ‘Z’ component of the equ
n
 (18.7) gives:
 (18.8)
Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4
The ‘R’ 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 ‘Z’ component of the equ
n
 (18.7) gives:
 (18.8)
Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4
The ‘R’ 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 ‘Z’ component of the equ
n
 (18.7) gives:
 (18.8)
Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4
The ‘R’ 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 ‘Z’ component of the equ
n
 (18.7) gives:
 (18.8)
Pipe - Fully Dev. Vel. Profile, Developing Temp. Profile Constant Wall Temp.-4
The ‘R’ 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 ‘z’ 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 ‘z’ 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.
18.18
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