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. 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

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 = R 1 + R 2. If T L is the temperature at the left, and T R is the temperature at the right, the heat transfer rate is given by

Wall Surfaces with Convection Boundary conditions: R conv,1 R cond R conv,2 T1T1 T2T2

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 = Q 1 + Q 2  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 Homogeneous and constant property material

At any radial location the surface are for heat conduction in a solid cylinder is: At any radial location the surface are for heat conduction in a solid sphere is: The GDE for cylinder:

The GDE for sphere: General Solution for Cylinder: General Solution for Sphere:

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: h,T  TsTs riri roro

Electrical analogy: As the outside radius, r o, increases, then in the denominator, the first term increases but the second term decreases. Thus, there must be a critical radius, r c, that will allow maximum rate of heat transfer, Q The critical radius, r c, can be obtained by differentiating and setting the resulting equation equal to zero.

T i,T b, k, L, r o, r i are constant terms, therefore: When outside radius becomes equal to critical radius, or r o = r c, 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 C to C. Insulation calculations should aim to keep the outside temperature of the insulation around 60 0 C. 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 R conv,1 R pipe R conv,2 T1T1 T2T2 R insulation R Al

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:

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.

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.