One Dimensional Steady Heat Conduction problems P M

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, T 1 - T 2, 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 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 Boundary conditions: T 1 T 2 Rconv, 1 Rcond Rconv, 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.

Mean Critical Thickness of Insulation Heat loss from a pipe: h, T • 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 ri Ts ro • 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.

Electrical analogy: 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: 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 0 C to 65. 5 0 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 T 1 T 2 Rconv, 1 Rpipe Rinsulation RAl Rconv, 2

• 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 T 4 (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.