Heat Conduction in a cooling fin

ADVANCED TRANSPORT PROCESSES /

TRANSPORT PHENOMENACCB/CBB 30335.Energy TransportLesson 23. Energy Transport With Energy Dissipation2At the end of the lesson the student should be able to 23. Solve problems concerning heat transfer through a cylinder with viscous energy dissipation Lesson outcomesCourse Outcomes

Semester May 2013CLO1Explain the theoretical aspect of momentum, mass and energy transportCLO2Apply mathematical and numerical methodology in analyzing momentum transfer problemCLO3Apply mathematical and numerical methodology in analyzing heat transfer problemCLO4Apply mathematical and numerical methodology in analyzing mass transfer problemCLO5Analyze and solve transport phenomena using Computational Fluid Dynamics (CFD) tools.34Heat Conduction with Viscous Energy Dissipation:- Consider an incompressible Newtonian fluid between two coaxial cylinders Shown in Figure 10.4-1. The surface of the inner and outer cylinders are maintained at T=T0 and T=Tb, respectively .

Determine the temperature distribution. If T0=Tb what will be the temperature distribution, the radius at which there will be maximum temperature and the maximum temperature.

Heat Conduction with Viscous Dissipation

Stationary surfaceCurvature of the bounding surface neglectedTop surface moved with velocity vb=R45 Heat Conduction with a Heat SourceNote that the problem involves both momentum and energy transport. The momentum transport will give us the velocity distribution and the energy transport will lead us to Temperature distribution.Momentum transportA case of two parallel plates where the top one is moving at a velocity vb=R and the bottom one is stationary.6SolutionSchematic diagram for simplified model

AssumptionsIt is a flow system Laminar flow, vz=f(x) vx=vy=0- There is energy generation due to viscous dissipation Heat Conduction with a Heat Source70

vx=0

Momentum Transport MechanismEnergy Transport Mechanism

(1)(3)(2)(4) Heat Conduction with a Heat Source8

vy=0vx=000

Therefore using (5) in (3), the combined energy flux can be described as (5)(6)Shell Energy Balancex

xx+x Heat Conduction with a Heat Source9Shell Energy Balance Equation

(8)(7)(9)(10)Using (6) in (10)

(11) Heat Conduction with a Heat Source10Applying Fouriers Law of Heat conduction and Newtons Law of viscosity in (10)

(12)Note that from the assumptions vz in (12) is a function of x and the velocity distribution should be found from momentum balance to integrate (12) Momentum BalanceSimplifying the equation of motion for the given problem

0000(13)11

Integrating (14)(14)(15)Using Newtons Law of viscosity and rearranging(16)

(17)Integrating (16) Heat Conduction with a Heat Source12

(19)(20)Using (18) and B.C.2. at x=b vz=vb in (17)Using (19) and (18) in (17)

(18)Using B.C.1. at x=0 vz=0Taking the derivative of (20) with respect to x

(21) Heat Conduction with a Heat Source13

Using (20) and (21) in (12)(22)

Rearranging (22)Integrating (23) with the boundary conditions as follows:(23)(24a)at any x the temperature is T and at x=0 T=T0

Rearranging we get Heat Conduction with a Heat Source14

(25)Rearranging (24b) and using it in (24a) to eliminate c1x/k and rearranging Where the dimensionless Brinkman number (Br) is defined as

(24b)Similarly at x=b T=Tb and at x=0 T=T0 Heat Conduction with a Heat Source15Solve the problem if T0=Tb

(26)When T0=Tb (24b) can be rearranged to Using (26) in (24a) and rearranging the temperature distribution is obtained as

(27) Heat Conduction with a Heat Source16Lesson 23. Solving problems concerning heat transfer through a cylinder with viscous energy dissipation Lesson outcomes