Convective boundary condition

The heat transfer model, energy and material balance equations plus boundary condition and initial conditions are shown in Figure 4. The energy balance partial differential equation (PDE) (Equation 10) assumes two dimensional axial conduction. Figure 5 illustrates the rectangular cross-section of the composite part. Convective boundary conditions are implemented at the interface between the walls and the polymer matrix. 

Figure 3. Finite element simulation of plane Couette flow with thermal dissipation and conductive heat transfer. (f) — fixed temperature condition (c) — convective boundary condition.

When Bi zero dimensional or lumped model [2, 11], On the other hand, if Bi 1, the fluid can be considered isothermal and Ts = Too, which changes the convection boundary condition to a thermal equilibrium condition. 

Figure 4.2 Infinitely long two-dimensional fin with convective boundary conditions.

Consider the infinitely long two-dimensional fin with convective boundary conditions, as shown in Fig. 4.2. The temperature of the surrounding fluid is T and the heat transfer coefficient is h. The x-axis is taken to be the axis of symmetry. The governing equations of the problem may be stated as below. 

When the solid is exposed to some convection boundary condition, the temperatures at the surface must be computed differently from the method given above. Consider the boundary shown in Fig. 3-7. The energy balance on node (m, n) is... 

An equation of this type must be written for each node along the surface shown in Fig. 3-7. So when a convection boundary condition is present, an equation like (3-25) is used at the boundary and an equation like (3-24) is used for the interior points. 

Fig. 3-7 Nomenclature for nodal equation with convective boundary condition.

Derive an equation equivalent to Eq. (3-25) for a one-dimensional convection boundary condition. 

Rework Prob. 3-50 with the surface absorbing a constant heat flux of 300 W/m2 instead of the convection boundary condition. The bottom surface still remains at 200°C. 

Sunderland, J. E., and K. R. Johnson Shape Factors for Heat Conduction through Bodies with Isothermal or Convective Boundary Conditions, Trans. ASHAE, vol. 70, pp. 237-241, 1964. 

Fig. 4-5 Temperature distribution in the semi-infinite solid with convection boundary condition.

A semi-infinite aluminum cylinder 5 cm in diameter is initially at a uniform temperature of 200°C. It is suddenly subjected to a convection boundary condition at 70°C with h = 525 W/m2 °C. Calculate the temperatures at the axis and surface of the cylinder 10 cm from the end 1 min after exposure to the environment. 

The difference equations given above are useful for determining the internal temperature in a solid as a function of space and time. At the boundary of the solid, a convection resistance to heat flow is usually involved, so that the above relations no longer apply. In general, each convection boundary condition must be handled separately, depending on the particular geometric shape under consideration. The case of the flat wall will be considered as an example. 

Fig. 4-20 Nomenclature for numerical solution of unsteady-state conduction problem with convection boundary condition.

The plane wall shown has internal heat generation of 50 MW/tn and thermal properties of k = 19 W/m °C, p = 7800 kg/mJ, and c = 460 J/kg °C. It is initially at a uniform temperature of 100°C and is suddenly subjected to the heat generation and the convective boundary conditions indicated in the figure. Calculate the temperature distribution after several time increments. 

When a convection boundary condition is involved, the construction at the boundary must be modified. Rewriting Eq. (4-39), we have... 

Fig. 4 24 Graphical technique of representing convection boundary condition with the Schmidt plot.

Fig- 4-25 Schmidt plot for four time increments, including convection boundary condition. 

A node like that shown in Table 3-2d has both x and y increments equal to 1.0 cm. The convection boundary condition is at 50 C and h = 60 W/m2 - °C. The solid material is stainless steel (18% Cr, 8% Ni). Using the thermal resistance and capacitance formulation for a transient analysis write the nodal equation for this node and determine the maximum allowable time increment. 

The analytical solutions obviously afford the user greater accuracy than the charts, but they may be more cumbersome to handle. The method selected for a problem solution may depend on uncertainties in the convection boundary conditions. In some cases a combination may be used perhaps the analytical method to find 80 and the graphs to evaluate 8100. This might avoid the need for evaluating Bessel functions. 

Convection is probably the most common boundary condition encountered in practice since most heat transfer surfaces are exposed to an environment at a specified temperature. The convection boundary condition is based on a surface energy balance expiressed as... 

For one-dimensional heat transfer in the. v direction in a plate of thickness the convection boundary conditions on both surfaces can be expressed as... 

Convection boundary conditions on the two surfaces of a plane wall. 

Discussion Note that the temperature of the inner surface of the base plate Is 13°C higher than the temperature of the outer surface when steady operating conditions are reached. Also note that this heat transfer analysis enables us to calculate the temperatures of surfaces that we cannot even reach. This example demonstrates how the heat flux and convection boundary conditions are applied to heat transfer problems. 

Consider a long pipe of inner radius r, outer radius r2> and thermal conductivity k. The outer surface of the pipe is subjected to convection to a medium at T. with a heat transfer coefficient of h, but the direction of heat transfer is not known. Express the convection boundary condition on the outer surface of the pipe. 

A plane wall of lliickness L is subjected to convection at both surfaces with ambient temperature and heat transfer coefficient at inner surface, and corresponding and values at the outer surface. Taking the positive direction of x to be from the inner surface to the outer surface, the correct expression for the convection boundary condition is... 

The application of either the explicit or the implicit formulation to each of the A/ - 1 interior nodes gives M — I equations. The remaining two equations are obtained by applying the same method to the two boundary nodes unless, of course, the boundary temperatures are specified as constants (invariant with time). For example, the formulation of the convection boundary condition at the left boundary (node 0) for the explicit case can be e.xpressed as (Fig. 5-41)... 

Blackwell, B. F. (1990) Temperature Profile in Semi-Infinite Body with Exponential Sources and Convective Boundary Condition, Journal of Heat Transfer, Vol. 112, pp. 567-571. 

F.G. Elmor, R.O.C. Guedes, and F.N. Scofano, Improved Lumped Solution for Conjugate Heat Transfer In Channel Flow with Convective Boundary Condition, Int. J. Heat Technology, pp. 78-88 (2005). 

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