On the role of axial and transverse wall conduction in laminar counterflow parallel-plate heat exchangers with flat and corrugated plates (2016)

Counterflow parallel-plate heat exchangers are widely employed in the chemical, pharmaceutical, and food processing industries when a compact, flexible, liquid-to-liquid heat exchange system is required that does not entail high temperatures and pressures. Parallel-plate heat exchangers pack a very large surface area into a small volume and can be modified to suit different system requirements by simply changing the number and/or shape of the plates. In addition to being compact and flexible, they are relatively cheap and can be used with a wide range of fluids, including viscous and non-Newtonian liquids. However, analytical solutions to the equations governing the performance of parallel-plate heat exchangers are limited, and the design of these devices depends mainly on empirical correlations and numerical results. For the classical case of macro-heat exchangers axial wall conduction is usually neglected, but several studies have shown that this is not a good approximation in other cases, such as in polymer heat exchangers or in mini- and micro-devices. The main purpose of this thesis is to study the performance of laminar counterflow parallel-plate heat exchangers accounting for both axial and transverse wall conduction effects using a combined analytical and numerical approach. The analysis, carried out for constant property fluids, considers a fully developed laminar flow with moderately large Peclet numbers in the channels, so that longitudinal conduction can be neglected in the fluids in first approximation. The analytical solution for the temperature field is expressed in terms of eigenfunction expansions involving infinite sets of both positive and negative eigenvalues. Based on previous results, the eigenfunctions are expressed in terms of Whittaker functions, leading to an analytical eigencondition that provides the eigenvalues numerically. A suitably defined orthogonality condition enables the determination of the expansion coefficients through an infinite system of linear equations, which must be truncated to a sufficiently large number of terms in order to obtain a numerical solution. The exact solution for the temperature field provides analytical expressions for the interfacial and bulk temperatures, local heat transfer rates, overall heat-transfer coefficient, Nusselt numbers, and outlet bulk temperature of both fluids. The development of the analysis is divided in three stages: - First, a simplified mathematical model that neglects axial conduction in the wall but includes the transverse wall thermal resistance is developed. In addition to the exact analytical solution, an approximate solution that retains only the first two terms in the eigenfunction expansion is presented. The approximate two-term solution, which still incorporates the effect of higher order modes through apparent temperature offsets introduced at the inlet/outlet sections, provides an accurate representation of the temperature field away from the thermal entry regions, thereby enabling simplified expressions for the wall and fluid bulk temperatures, local heat transfer rate, Nusselt numbers, and overall heat transfer coefficient. The parameters involved in the model are the normalized Peclet number ratio, the normalized fluid-to-fluid and wall-to-fluid conductivity ratios (the inverse of the latter being the dimensionless wall thermal resistance), and the dimensionless heat exchanger length. The analysis shows that when axial wall conduction is absent increasing the wall thermal resistance (either by increasing its thickness or decreasing its thermal conductivity) always reduces the heat exchanger effectiveness and increases the Nusselt number of the fluid with the lower heat-capacity flow rate, bringing it closer to its theoretical value for a constant heat flux boundary condition. Asymptotic expressions for nearly balanced heat exchangers are also obtained, which describe the transition from exponential to linear behavior of the lowest order eigenmode as the heat exchanger approaches the balanced condition. - Second, an extension to the previous model is developed to include the effect of small scale wall corrugations using a multi-scale asymptotic approach. The amplitude and wavelength of the corrugations are assumed to be comparable to each other, but small compared to the channel half-widths. On the small (i.e., wall corrugation) scale, the shear-induced flow of two Newtonian fluids that exchange heat trough a thin corrugated wall is studied numerically, retaining both transverse and longitudinal conduction effects in the heat conducting wall. The analysis shows that far from the wall the effect of the corrugations is equivalent to that of a thickened virtual plane surface with partial slip along the corrugations and an apparent wall thermal resistance which has to be obtained numerically. The thermal performance of heat exchangers with flat and corrugated walls is then compared from the point of view of the first and second laws of thermodynamics. In the particular case of steady laminar flow considered here the results show that for high conductive (e.g., metallic) walls the addition of the corrugations always lowers the apparent wall conductivity (or increases the apparent wall thermal resistance), whereas for low conductive (e.g., polymeric) walls the addition of the corrugations may lead to a slightly larger apparent wall conductivity (or lower wall thermal resistance) under certain flow conditions, particularly for relatively thick walls and large corrugation amplitudes. - Finally, a general model that includes both axial and transverse wall conduction effects in laminar counterflow parallel-plate heat exchangers is presented. The model involves the dimensionless wall thickness as an additional parameter. Conditions for neglecting axial wall conduction are established starting from an upper bound for the ratio of the amount of heat conducted axially by the wall to that convected downstream by the fluids. A criterion for neglecting axial wall conduction is then proposed by requiring that this ratio be small compared to unity. A parametric study of the eigenvalue/eigenfunction spectrum and of the performance of heat exchangers with varying degrees of axial wall conduction is then presented. As main outcome, the analysis confirms that when axial wall conduction plays a key role the effectiveness of the heat exchanger varies non-monotonously with the wall thermal conductivity, so that an optimum wall conductivity exists that maximizes heat exchanger efficiency. The analysis of the thermally developed regime provides approximate analytical expressions for the optimum wall conductivity that agree well with the exact values obtained with the model, particularly for moderately long heat exchangers, although the optimum ceases to exist when the wall becomes moderately thick. Spatial distributions of the interfacial and bulk fluid temperatures are presented, together with the corresponding interfacial and wall axial heat fluxes, and Nusselt numbers of both fluids, aiming to illustrate the transition between the different operating regimes with/withouth significant effects of axial/transverse wall conduction.