Heat Conduction in a cooling fin

ADVANCED TRANSPORT PROCESSES /

TRANSPORT PHENOMENACCB/CBB 30335.Energy TransportLesson 20. Heat Conduction through a CylinderLesson 21. Heat Conduction Through a laminated tube2At the end of the lesson the student should be able to 20. Solve problems concerning heat transfer through a cylinderwhen the wall temperatures inside and outside are givenwhen the bulk temperatures of the fluid inside and outside the cylinder are given21. Solve problems about heat transfer through composite cylinders

Lesson outcomes3Conduction through a cylinder when wall temperatures are given : The internal and external wall temperatures of the cylinder are T0 and T1 , respectively, and T0 > T1 . Determine the temperature distribution, the heat flux and the heat flow. Heat Conduction through a cylinder: Case 1

34Assumptionnon-flow system, v =0No energy generation( production)No work done on the system by external force

SolutionB.C.1 T=T0 at r = r0

B.C.1 T=T1 at r = r1 Heat Conduction through a Slab: Case 1Boundary ConditionsEnergy Transport Mechanism5Shell Energy BalanceTqr|rqr|r+rrr

(1)(2)Rate of Energy out by molec. transportRate of Energy in by molec. transport-= 0Dividing both sides by 2Lr

6

Taking the limit as r is close to zero(3)Integrating (3)Applying Fouriers Law of heat conduction qr = -k dT/drRearranging and integrating (5)(4)(5)

(6) Heat Conduction through a cylinder: Case 17Similarly with T = T0 at r = r0 and T = T1 at r = r1Dividing (6) by (7) we get the temperature distribution(7)(8)(9)

Temperature distributionRearranging (7) and using it with (4)

Heat Conduction through a cylinder: Case 18Rearranging (9)The heat flow is given by(10)(11) Heat Flux and Heat Flow

Heat flux distribution

Heat flow9 Heat Conduction through a cylinder: Case 2

Conduction through a cylinder when fluid bulk temperatures are given : The bulk temperature of the fluid inside and outside the tube are Ta and Tb, respectively, and Ta > Tb . Determine the heat flux at the inside surface and the heat flow.910Assumptionnon-flow system, v =0No energy generation( production)No work done on the system by external force

Solution Heat Conduction through a Slab: Case 1Boundary ConditionsEnergy Transport Mechanism

Newtons Law of cooling11Shell Energy BalanceTqr|rqr|r+rrr

(1)(2)Rate of Energy out by molec. transportRate of Energy in by molec. transport-= 0Dividing both sides by 2Lr

12

Taking the limit as r is close to zero(3)Integrating (3)Applying Fouriers Law of heat conduction qr = -k dT/drRearranging and integrating at r0 and r1(4)(5)

(6) Heat Conduction through a cylinder: Case 213Using Newtons Law of cooling

Adding (6) , (7) and (8) together(7)(8)

(9)(10)Rearranging (9) we getHeat flux at r0 Heat Conduction through a cylinder: Case 2

14The heat flow at the inner surface is given by(11)Heat Flow

Heat flow

Using (10) in (11) we get the heat flow(12)The heat flow in terms of the overall heat transfer coefficient U0From (12) and (13) (13)(14)15Conduction through composite cylinder:- The composite cylinders shown in the Figure below is exposed to a fluid at bulk temperatures of Ta and Tb where Ta > Tb . Determine the heat flux and the heat flow at the surface for steady state condition. Heat Conduction through a Composite Cylinder

1516Assumptionnon-flow system, v =0No energy generation( production)No work done on the system by external force

Solution Heat Conduction through a Slab: Case 1Boundary ConditionsEnergy Transport Mechanism

Newtons Law of cooling17Shell Energy BalanceTqr|rqr|r+rrr

(1)(2)Rate of Energy out by molec. transportRate of Energy in by molec. transport-= 0Dividing both sides by 2Lr

18

Taking the limit as r is close to zero(3)Integrating (3)Applying Fouriers Law of heat conduction(4)(5a)(5b)

(5c) Heat Conduction through a Comp. Cylinder19Integrating (5a), (5b) and (5c) assuming constant heat capacitiesUsing Newtons law of cooling at the two fluid-solid interfaces

(6a)

(6b)

(6b)

(7)(8) Heat Conduction through a Comp. Cylinder20Adding (6a), (6b) , (6c), (7) and (8) together

(9)(10)Rearranging (9) we get the heat flux at the inner surface Heat Conduction through a Comp. Cylinder21The heat flow at the inner surface is given by(11)Heat Flow

Using (10) in (11) we get the heat flow(12)The heat flow in terms of the overall heat transfer coefficient U0From (12) and (13) (13)(14)

22In general for a system with n annular layers we get the definition for overall heat transfer coefficient for composite cylinderOverall Heat Transfer Coefficient for Comp. Cylinder

(16)2319. Solve problems concerning heat transfer through a cylinderwhen the wall temperatures inside and outside are givenwhen the bulk temperatures of the fluid inside and outside the cylinder are given20. Solve problems about heat transfer through composite cylinders

Lesson outcomes