One Dimensional Steady Heat Conduction problems

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Simple ideas for complex Problems…

Electrical Circuit Theory of Heat Transfer

• Thermal Resistance• A resistance can be defined as the ratio of a

driving potential to a corresponding transfer rate.

i

VR

Analogy:

Electrical resistance is to conduction of electricity as thermal resistance is to conduction of heat.

The analog of Q is current, and the analog of the temperature difference, T1 - T2, is voltage difference.

From this perspective the slab is a pure resistance to heat transfer and we can define

thR

TQ

The composite Wall

• The concept of a thermal resistance circuit allows ready analysis of problems such as a composite slab (composite planar heat transfer surface).

• In the composite slab, the heat flux is constant with x.

• The resistances are in series and sum to R = R1 + R2.

• If TL is the temperature at the left, and TR is the temperature at the right, the heat transfer rate is given by

Wall Surfaces with Convection

2112

2

0 CxCTCdx

dT

dx

TdA

Boundary conditions:

110

)0(

TThdx

dTk

x

22 )(

TLThdx

dTk

Lx

Rconv,1 Rcond Rconv,2

T1 T2

Heat transfer for a wall with dissimilar materials

• For this situation, the total heat flux Q is made up of the heat flux in the two parallel paths:

• Q = Q1+ Q2

with the total resistance given by:

Composite Walls

• The overall thermal resistance is given by

Desert Housing & Composite Walls

One-dimensional Steady Conduction in Radial Systems

0

drdrdT

kAd

Homogeneous and constant property material

0

drdrdT

Ad

At any radial location the surface are for heat conductionin a solid cylinder is:

rlAcylinder 2

At any radial location the surface are for heat conductionin a solid sphere is:

24 rAsphere

The GDE for cylinder:

0

drdrdTrd

The GDE for sphere:

0

2

drdrdT

rd

General Solution for Cylinder:

21 ln CrCrT

General Solution for Sphere:

r

CCrT 1

2

Boundary Conditions

• No solution exists when r = 0. • Totally solid cylinder or Sphere have no physical relevance!

• Dirichlet Boundary Conditions: The boundary conditions in any heat transfer simulation are expressed in terms of the temperature at the boundary.

• Neumann Boundary Conditions: The boundary conditions in any heat transfer simulation are expressed in terms of the temperature gradient at the boundary.

• Mixed Boundary Conditions: A mixed boundary condition gives information about both the values of a temperature and the values of its derivative on the boundary of the domain.

• Mixed boundary conditions are a combination of Dirichlet boundary conditions and Neumann boundary conditions.

• If A, is increased, Q will increase. • When insulation is added to a pipe, the outside

surface area of the pipe will increase. • This would indicate an increased rate of heat

transfer

• The insulation material has a low thermal conductivity, it reduces the conductive heat transfer lowers the temperature difference between the outer surface temperature of the insulation and the surrounding bulk fluid temperature.

• This contradiction indicates that there must be a critical thickness of insulation.

• The thickness of insulation must be greater than the critical thickness, so that the rate of heat loss is reduced as desired.

Mean Critical Thickness of Insulation

Heat loss from a pipe:

TThAQ s

h,T

Ts

ri

ro

Electrical analogy:totalR

TerheattransfofRate

ooi

o

i

Lhrrr

Lk

TTQ

21

ln2

1

As the outside radius, ro, increases, then in the denominator, the first term increases but the second term decreases.

Thus, there must be a critical radius, rc , that will allow maximum rate of heat transfer, Q

The critical radius, rc, can be obtained by differentiating and setting the resulting equation equal to zero.

Ti,Tb, k, L, ro, ri are constant terms, therefore:

01

2

ooo rh

k

r

When outside radius becomes equal to critical radius, or ro = rc, we get,

Safety of Insulation

• Pipes that are readily accessible by workers are subject to safety constraints. 

• The recommended safe "touch" temperature range is from 54.4 0C to 65.5 0C.  

• Insulation calculations should aim to keep the outside temperature of the insulation around 60 0C. 

• An additional tool employed to help meet this goal is aluminum covering wrapped around the outside of the insulation.  

• Aluminum's thermal conductivity of 209 W/m K does not offer much resistance to heat transfer, but it does act as another resistance while also holding the insulation in place. 

• Typical thickness of aluminum used for this purpose ranges from 0.2 mm to 0.4 mm. 

• The addition of aluminum adds another resistance term, when calculating the total heat loss:

Structure of Hot Fluid Piping

Rconv,1 Rpipe Rconv,2

T1 T2

Rinsulation RAl

• However, when considering safety, engineers need a quick way to calculate the surface temperature that will come into contact with the workers. 

• This can be done with equations or the use of charts. 

• We start by looking at diagram:

At steady state, the heat transfer rate will be the same for each layer:

Alinsulationpipe R

TT

R

TT

R

TTQ 433221

Solving the three expressions for the temperature difference yields:

Each term in the denominator of above Equation is referred to as the “Thermal resistance" of each layer. 

totalAlinsulationpipe R

TT

R

TT

R

TT

R

TTQ 41433221

Design Procedure

• Use the economic thickness of your insulation as a basis for your calculation.

• After all, if the most affordable layer of insulation is safe, that's the one you'd want to use. 

• Since the heat loss is constant for each layer, calculate Q from the bare pipe.

• Then solve T4 (surface temperature).   • If the economic thickness results in too high a surface temperature,

repeat the calculation by increasing the insulation thickness by 12 mm each time until a safe touch temperature is reached.

• Using heat balance equations is certainly a valid means of estimating surface temperatures, but it may not always be the fastest. 

• Charts are available that utilize a characteristic called "equivalent thickness" to simplify the heat balance equations. 

• This correlation also uses the surface resistance of the outer covering of the pipe.