In heat exchanger applications, it is often necessary to analyze the way in which heat conduction in solid components interacts with the convection processes in the fluid medium, with the objective of optimization of design or performance. The context of this analysis may be the study of extended surface design (fins) or the design of laminar flow heat exchangers, where heat conduction along separating walls may have a significant effect on heat exchanger performance.

A simple method of dealing with coupled conduction/convection heat transfer may be exemplified by the analysis of heat dissipation from a plain fin, such as the one illustrated in Figure 1.

In a surface extension such as a fin, there exists an internal distribution of energy transfer by conduction which is dependent on convective dissipation around the fin boundary. The function of the fin is to enhance heat transfer beyond that possible with a plain surface, and the most effective fin would be one in which the entire surface is at the same temperature as the base T_{b}. (See Augmentation of Heat Transfer.) To determine heat dissipation rate from a hot fin, it is necessary to know the temperature distribution along the fin. This may be predicted from the principle of conservation of energy applied to a representative differential section of the fin with width dx. The analysis is easier if simplifying assumptions are made: that conduction is quasi-one-dimensional in the x-direction (reasonable for thin fins with high conductivity); that the system has reached a steady-state; that λ and α are constant; and that there is no internal energy conversion and radiation effects are negligible. The energy balance on the differential element is thus:

where the convective losses from the perimeter of the element [surface area P.dx, where P = 2(w + t)] to the ambient fluid at T_{a} are denoted by
. These losses may be calculated, using Newton's Law of Cooling, as
, where T is the temperature of the fin material at position x. A Taylor Series expansion may be used to approximate
:

and Fourier's Law may be used to give , where A (= w.t) is the cross-sectional area of the fin, thus enabling the energy balance on the element to be written as:

Defining m^{2} =
and introducing a new dependent variable θ = (T – T_{a}) this energy balance can be written as:

which is a linear, homogeneous, second-order differential equation with constant coefficients having a general solution of the form:

The constants C_{1} and C_{2} are evaluated using boundary conditions which proscribe the fin temperature distribution. Four possible tip boundary conditions are illustrated schematically in Figure 2.

1) The temperature at the base of the fin (x = 0) is T_{b}, i.e.,

2) There are a number of possible conditions which may exist at the tip of the fin. The simplest condition is that which would occur at the end of a very long (L → ∞) fin when the tip would approach the ambient fluid temperature, i.e., θ(L) = 0 as indicated in Figure 2a. The tip may be insulated, in which case, the tip temperature gradient is zero, Figure 2b. Alternatively, the tip could be maintained at some prescribed temperature T_{L}, Figure 2c. However, a more likely condition is that the tip temperature is unknown but it can be assumed that energy arriving at the tip by conduction is lost by convection through the tip end area, where a heat balance would give

Using this latter tip condition gives:

The particular solution for fin temperature for these boundary conditions can then be obtained as:

The form of this temperature distribution is shown in Figure 2d, which illustrates the way in which convective heat loss from the fin, including the tip, affects conduction rate along the fin. The amount of heat dissipated by the fin may be calculated by noting that the conduction rate into the fin through the base must equate to convective loss from the fin surfaces in contact with the fluid, i.e., heat dissipation rate = = . Differentiating the expression for the fin temperature distribution with respect to x, and then setting x = 0 gives:

The effectiveness of finning is usually assessed by a *fin efficiency factor*, defined as the ratio of the fin assembly heat transfer rate (as exemplified by the above expression) to the heat transfer rate of the finless system, in this case αA (T_{b} — T_{a}), which should generally be greater than 2. A comprehensive review of the coupling effects occurring in extended surface systems is given by Kem and Krauss (1972).

In the above example of heat conduction in a fin interacting with convection from the fin surfaces, it has been assumed that the heat transfer coefficient was both constant and known *a priori* so that temperature distribution in the solid was easily obtained with this single piece of information about thermal interaction with the fluid. Problems in which the thermal boundary condition at a fluid boundary is not known *a priori* are called conjugate problems and require simultaneous solution of temperature fields in both the solid and adjacent convecting fluid. Such problems are of particular interest in the design of compact heat exchangers where the flow is *laminar* since it is then possible that *conduction along the walls of the exchanger* can compromise the exchanger performance.

In the design of heat exchangers, it is usual to assume that the film coefficients have constant values estimated from the appropriate correlations and sizing is then usually satisfactory for turbulent flow conditions. For laminar flow conditions, thermal inlet lengths can be of the same order of magnitude as flow path length, and there can therefore exist continuous variation in film coefficient within the exchanger. Since the two streams are thermally-coupled by the wall, the boundary conditions for both streams cannot be defined *a priori* and axial conduction in either or both the wall and the fluid may also be significant. The body of knowledge on conjugated heat transfer problems has grown since the early discussion of Shah and London (1978), and Pagliarini and Barozzi (1991) have given a recent literature survey. Figure 3 illustrates conjugation in a simple and idealized counterflow heat exchanger system in which two flow passages are separated by an insulated wall containing a conducting window through which hot and cold fluids thermally interact, and along which energy may be transferred by conduction.

**Figure 3. Model of an idealized counterflow heat exchanger. From Demko, J. A. and Chow, L. C., AIAA J., 22(5), 705, (1984). Copyright © 1984 AIAA. Reprinted with permission.)**

A simplfied hydrodynamic system serves to illustrate the effects of conjugation. Both flows are uniform in velocity and temperature as they approach the conducting window, which is sharply defined by the adjacent insulated walls. The heat capacity flow rates (C = ) of the hot and cold streams are identical. The plate may be one of many making up an idealized plate heat exchanger, and the flow centerlines may be considered as axes of symmetry across which no energy is transferred. The inlet fluid temperatures are known and uniform, but the convective heat transfer coefficients at the solid/fluid interfaces are not known, nor is the plate temperature distribution here assumed to be one-dimensional, i.e., the window is long and thin. The temperature fields in both fluids and the solid must be obtained simultaneously with no prior knowledge of the temperature or heat flux at the two solid/fluid interfaces.

Three energy balances need to be solved simultaneously; one for each of the flow streams and one for the intervening wall. The dimensionless energy balances for each convective flow are both of the form:

Assuming that the wall is relatively thin and that the conductivities of the hot and cold fluids are equal, the energy balance for the solid is:

where λ_{s} is the thermal conductivity of the window material and λ_{f} is the thermal conductivity of both fluids. Referring to Figure 3, the following dimensionless variables are employed in the above energy balances: streamwise distance, x = X/D; cross-stream distance, y = Y_{h}/D or Y_{c}/D; streamwise velocity component, u = u_{1}/ U_{1} or u_{2}/U_{2}; cross-stream velocity component, v = v_{1}/U_{1} or v_{2}/U_{2}; hot stream temperature at (x,y); θ_{h} = 2(T_{h} – T_{av})/(T_{1} – T_{2}); cold stream temperature at (x,y), θ_{c} = 2(T_{c} – T_{av})/(T_{1} – T_{2}); window temperature at (x,y), θ_{s} = 2(T_{s} – T_{av})/(T_{1} – T_{2}, where T_{av} = (T_{1} – T_{2})/2. The Peclet Number, Pe(=RePr), and the conductivity ratio, Λ = (d/D)(λ_{s}/λ_{f}), emerge as important parameters. Figures 4a and 4b show the effect of A on the plate and hot bulk temperature distributions for Pe = 10.

**Figure 4. Effect of conjugation temperature distributions in a simple counterflow heat transfer device. From Demko, J. A. and Chow, L. C., AIAA J., 22(5), 705, (1984). Copyright © 1984 A1AA. Reprinted with permission.**

For Peclet Numbers less than 50, axial fluid conduction can be significant compared with axial bulk convective transport so that hot fluid may lose energy and cold fluid may gain energy prior to their entry into the conducting window area and heat transfer across the flow to and from the wall will be higher than at high Pe values. The wall conduction parameter determines, in part, how much of the energy from the hot fluid will be conducted axially along the wall. The effects of wall conduction are more significant at low Pe. For laminar flow, heat conducted along the plate may be of the same magnitude as the energy transferred to the cold fluid.

Numerical solution of the above system of equations is possible [Demko and Chow (1984)] and the influence of Pe and on heat exchanger effectiveness e is shown in Figure 5, using the usual definition of effectiveness (see Heat Exchangers).

where C represents the heat capacity of the stream, , and the subscripts min and h signify the stream with lower and higher capacities, respectively.

**Figure 5. Influence of conjugation on heat exchanger effectiveness. From Demko, J. A. and Chow, L. C., AIAA J., 22(5), 705, (1984). Copyright © 1984 AIAA. Reprinted with permission.)**

The effectiveness ε is the ratio of the total amount of heat actually transferred in a heat exchanger to the maximum possible amount of heat exchange between the flow streams at their respective inlet temperatures. For Λ = 0 (the nonconjugated case), the extrapolations to the left ordinate (at — ∞) show that ε is not monotonic with Pe; but for all Pe, ε decreases as Λ (or wall conduction) increases, the rate of decrease depending on Pe. As Pe → 0, fluid conduction dominates the entire process and the temperature of both the hot and cold fluids approach each other and that of the wall; consequently ε → 0.5. As Pe → ∞, axial bulk convection dominates and the hot and cold fluid temperatures will not change much so that ε → 0 and the effect of wall conduction is small.

In general, axial wall conduction lowers the bulk temperature of the hot fluid near the exchanger entrance but results in higher exit temperatures, with the net result that effectiveness is decreased.

#### REFERENCES

Demko, J. A. and Chow, L. C. (1984) *Heat Transfer Between Counterflowing Fluids Separated by a Heat Conducting Wall*, AIAA Jnl., 22(5).

Kern, D. Q. and Kraus, A. D. (1972) *Extended Surface Heat Transfer*, McGraw-Hill, New York.

Pagliarini, G. and Barozzi, G. S. (1991) Thermal Coupling in Laminar Flow Double-Pipe Heat Exchangers, *J Hear Transfer*, 113,526-534.

Shah, R. K. and London, A. L. (1978) *Laminar Flow Forced Convection in Ducts*, Academic Press, New York.

#### References

- Demko, J. A. and Chow, L. C. (1984)
*Heat Transfer Between Counterflowing Fluids Separated by a Heat Conducting Wall*, AIAA Jnl., 22(5). - Kern, D. Q. and Kraus, A. D. (1972)
*Extended Surface Heat Transfer*, McGraw-Hill, New York. - Pagliarini, G. and Barozzi, G. S. (1991) Thermal Coupling in Laminar Flow Double-Pipe Heat Exchangers,
*J Hear Transfer*, 113,526-534. DOI: 10.1115/1.2910595 - Shah, R. K. and London, A. L. (1978)
*Laminar Flow Forced Convection in Ducts*, Academic Press, New York.