_{1}

^{*}

Laminar flow and heat transfer in different protruding-edged plate systems are modelled and analyzed in the present work. These include the Parallel Flow (PF) and the Counter Flow (CF) protruding-edgedplate exchangers as well as those systems being subjected to Constant Wall Temperature (CWT) and Uniform Heat Flux (UHF) conditions. These systems are subjected to normal free stream having both power-law velocity profile and same average velocity. The continuity, momentum and energy equations are transformed to either similarity or nonsimilar equations and then solved by using well validated finite difference methods. Accurate correlations for various flow and heat transfer parameters are obtained. It is found that there are specific power-law indices that maximize the heat transfer in both PF and CF systems. The maximum reported enhancement ratios are 1.075 and 1.109 for the PF and CF systems, respectively, at
*Pr* = 100. These ratios are 1.076 and 1.023 for CWT and UHF conditions, respectively, at
*Pr* = 128. Per same friction force, the CF system is preferable over the PF system only when the power-law indices are smaller than zero. Finally, this work demonstrates that by appropriately distributing the free stream velocity, the heat transfer from a plate can be increased up to 10% fold.

Conversion and utilization of energy often involve heat transfer process. This process is encountered in many engineering applications. These applications include steam generation and condensation in power plants; sensible heating and cooling of viscous fluids as in thermal processing of pharmaceutical, agricultural and hygiene products; evaporation and condensation of refrigerants in refrigeration and air-conditioning systems; cooling of engine and turbomachinery systems; and cooling of electrical appliances and electronic devices. It is well known that improving heat transfer over that in the typical practice results in significant increases in both the thermal efficiency and the economics of the plant operation. Improving heat transfer is a terminology that is frequently referred to it in the literature as heat transfer enhancement or augmentation [

Heat transfer enhancement mechanisms basically reduce the thermal resistance in a conventional thermal system by promoting higher convective heat transfer coefficient that can be accompanied with surface area increase. Consequently, the size of a thermal system can be reduced, or the heat duty of an existing thermal system can be increased, or the pumping power requirements can be reduced [

When a normal free stream strikes a plate having a protruding edge at its inlet, stagnation flow occurs along the plate length which has its stagnation line coinciding with the plate inlet edge. This flow is characterized by having an increasing axial velocity in the vicinity of the plate from zero at the inlet to maximum at the exit [

In the next section, the geometries of various analyzed systems composed of plates with protruding edges are explained. These systems include the Parallel Flow (PF) and the Counter Flow (CF) protruding-edged plate exchangers. These systems are exposed to normal free stream having both power-law velocity profile and same average velocity. The continuity, axial momentum and energy equations of the fluids adjacent to the plate are transformed to either similarity and nonsimilarity equations. Also, various similarity equations are obtained for protruding-edged plates subjected to either constant wall temperature or uniform heat flux conditions. The governing equations are solved numerically and are validated against well-established special cases. Different accurate correlations for flow and heat transfer parameters are obtained. An extensive parametric study has been conducted in order explore the influence of power-law index, Prandtl numbers and relative Reynolds numbers on Nusselt numbers and different heat transfer enhancement indicators.

The proposed two types of protruding-edged plate heat exchangers are shown in

are at the plate entrances and they ban fluid flows in the opposite directions. In the CF system, the plate entrance of one face is opposing the entrance of the other face. Both entrances contain side protrusions so as to force the induced hot and cold fluid stagnation flows to have counter current directions as shown in

Consider that the normal streams approaching the faces of the protruding-edged plate have the following velocity profile along the face length

where

where

where

where

Define the following independent and dependent variables:

Equations 4(a), 4(b) are transformed to the given similarity equations when Equations (8) and (9) are used:

The transformed boundary conditions are equal to:

The average skin friction coefficient denoted by

If

then, the energy equations of the hot and cold fluids are given by (Bejan, 2013):

where

where

Define the enhancement ratio

Invoking the similarity variables given by Equations (8) and (9), Equations (15) and (16) for the PF system reduce to the following similarity equations and boundary conditions:

The dimensionless heat transfer rate per unit width denoted by

Invoking the following nonsimilarity variables:

to Equations (14) and (15) for the CF system where

The average skin friction coefficient

When

For this case, the local Nusselt number is defined as:

where

When the plate is generating uniform heat flux

This is in order to reduce the energy equation given by Equation 15(b) to a similarity equation. This similarity equation is given by:

The boundary conditions for this case are given by:

For this case, the local Nusselt number is defined as:

The average Nusselt number relationship is given by:

In terms of average convection heat transfer coefficients, the energy balance given by Equation (17) can be reduced to one equation given by:

The definition of average Nusselt number can be used to show that

Equation (10) was discretized using three points center differencing after substituting

where

Also, Equations (19), (20), (29) and (34) were discretized using three points center differencing quotients and the resulted tri-diagonal system of algebraic equations have been solved using the Thomas algorithm without iterations. The left side of Equations 21(d) and 26(d) were discretized using two points difference quotients.

Under assumed plate temperatures, the solutions of the discretized forms of Equations (24) and (25) were obtained using the Thomas algorithm [

The marching procedure used for solving Equations (24) and (25) were repeated by letting the assumed plate temperatures

When

Correlation for transformed axial velocity and the average skin friction coefficient

where

The coefficients

m | Condition | ||
---|---|---|---|

−1 | CWT | 0.664 | 0.66412 (0.0181%) |

−1 | UHF | 0.458 | 0.45897 (0.211%) |

0 | CWT | 0.495 | 0.49587 (0.175%) |

i | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

b_{i}_{,1} | 1.23348 | −0.160569 | 0.129839 | −2.10319 × 10^{−}^{3} | 1.28401 × 10^{−}^{2} |

b_{i}_{,2} | 1.41036 | −2.80954 × 10^{−}^{2} | 0.243632 | 8.37931 × 10^{−}^{4} | 2.98980 × 10^{−}^{2} |

b_{i}_{,3} | −5.34578 × 10^{−}^{2} | 0.139322 | 0.131941 | −3.06229 × 10^{−}^{3} | 2.03408 × 10^{−}^{2} |

b_{i}_{,4} | −0.357418 | 1.25455 × 10^{−}^{2} | −5.29649 × 10^{−}^{3} | −2.78334 × 10^{−}^{3} | −2.15775 × 10^{−}^{7} |

b_{i}_{,5} | −5.53362 × 10^{−}^{2} | −1.37009 × 10^{−}^{2} | −2.11865 × 10^{−}^{2} | 1.60357 × 10^{−}^{3} | −3.02098 × 10^{−}^{3} |

b_{i}_{,6} | 0.67520 | 0.768542 | 0.670481 | −4.18399 × 10^{−}^{2} | −0.294328 |

b_{i}_{,7} | −0.250973 | −9.8122 × 10^{−}^{2} | −0.244653 | −0.213869 | −1.70334 × 10^{−}^{3} |

b_{i}_{,8} | −0.1949221 | −6.34351 × 10^{−}^{2} | −2.71828 × 10^{−}^{2} | 3.91614 × 10^{−}^{2} | −1.94741 × 10^{−}^{3} |

b_{i}_{,9} | −6.87427 × 10^{−}^{3} | 4.37413 × 10^{−}^{4} | 1.46370 × 10^{−}^{3} | −5.01454 × 10^{−}^{3} | 2.67294 × 10^{−}^{4} |

b_{i}_{,10} | 9.26946 × 10^{−}^{5} | −4.44071 × 10^{−}^{6} | −6.14264 × 10^{−}^{5} | 2.58304 × 10^{−}^{4} | −1.45535 × 10^{−}^{5} |

i | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|

b_{i}_{,1} | 0.281228 | 0.199668 | 6.23405 × 10^{−}^{2} | 4.14688 × 10^{−}^{3} | 1.26094 × 10^{−}^{2} |

b_{i}_{,2} | 0.536739 | 0.394869 | 0.155267 | 5.53394 × 10^{−}^{3} | 2.95620 × 10^{−}^{2} |

b_{i}_{,3} | 0.106061 | 0.196443 | 0.119232 | −4.20443 × 10^{−}^{3} | 1.98301 × 10^{−}^{2} |

b_{i}_{,4} | −5.88725 × 10^{−}^{2} | −3.17318 × 10^{−}^{2} | 6.82691 × 10^{−}^{3} | −3.37330 × 10^{−}^{3} | −5.98251 × 10^{−}^{4} |

b_{i}_{,5} | −8.59925 × 10^{−}^{3} | −1.78048 × 10^{−}^{2} | −1.95314 × 10^{−}^{2} | 1.56172 × 10^{−}^{3} | −3.25532 × 10^{−}^{3} |

b_{i}_{,6} | 0.625636 | 0.703080 | 0.673154 | −8.60474 × 10^{−}^{2} | −0.309082 |

b_{i}_{,7} | −8.96274 × 10^{−}^{2} | −0.101322 | −0.242239 | −0.18175 | −6.22788 × 10^{−}^{3} |

b_{i}_{,8} | −5.10059 × 10^{−}^{2} | −5.28074 × 10^{−}^{2} | −2.59353 × 10^{−}^{2} | 3.11246 × 10^{−}^{2} | −9.47830 × 10^{−}^{4} |

b_{i}_{,9} | −1.63201 × 10^{−}^{3} | 5.28279 × 10^{−}^{5} | 1.42072 × 10^{−}^{3} | −3.97059 × 10^{−}^{3} | 1.28336 × 10^{−}^{4} |

b_{i}_{,10} | 1.80196 × 10^{−}^{5} | 2.89120 × 10^{−}^{6} | −6.01733 × 10^{−}^{5} | 2.02796 × 10^{−}^{4} | −6.65839 × 10^{−}^{6} |

Correlation (46) has maximum relative error less than 0.026% when

Correlation for the transformed boundary layer thickness

The edge of the transformed boundary layer

Correlation (47) has maximum relative error less than 0.031% when

Correlations for average Nusselt number for CWT and UHF conditions

The average Nusselt number for CWT and UHF conditions can be shown to be correlated to

where

The coefficients

relative error in computing

condition when

i | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

d_{i}_{,1} | 0.237089 | 0.701739 | 9.62241 × 10^{−}^{2} | 9.97888 × 10^{−}^{4} | 0.777659 |

d_{i}_{,2} | 0.262489 | 0.7666 | 0.105549 | 1.09794 × 10^{−}^{3} | 0.883683 |

d_{i}_{,3} | 5.1566 × 10^{−}^{2} | 0.144598 | 1.97739 × 10^{−}^{2} | 2.05205 × 10^{−}^{4} | 0.111896 |

d_{i}_{,4} | 1.03331 × 10^{−}^{3} | 2.75865 × 10^{−}^{3} | 3.7421 × 10^{−}^{4} | 3.87755 × 10^{−}^{6} | −7.7778 × 10^{−}^{2} |

d_{i}_{,5} | 1.3379 | 1.30361 | 1.30382 | 1.30552 | 1.15327 |

d_{i}_{,6} | 0.451707 | 0.425895 | 0.423815 | 0.42372 | 0.152245 |

d_{i}_{,7} | 3.1727 × 10^{−}^{2} | 2.8567 × 10^{−}^{2} | 3.81949 × 10^{−}^{2} | 2.81163 × 10^{−}^{2} | −0.102728 |

i | 6 | 7 |
---|---|---|

d_{i}_{,1} | 4.42076 × 10^{−}^{2} | 1.59237 × 10^{−}^{4} |

d_{i}_{,2} | 4.93237 × 10^{−}^{2} | 1.67905 × 10^{−}^{4} |

d_{i}_{,3} | 6.00407 × 10^{−}^{3} | 9.89797 × 10^{−}^{6} |

d_{i}_{,4} | −4.27051 × 10^{−}^{3} | −1.88004 × 10^{−}^{5} |

d_{i}_{,5} | 1.13872 | 1.08047 |

d_{i}_{,6} | 0.146785 | 7.30291 × 10^{−}^{2} |

d_{i}_{,7} | −0.100184 | −0.123036 |

i | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|

d_{i}_{,1} | 0.237089 | 0.70174 | 9.62241 × 10^{−}^{2} | 9.97887 × 10^{−}^{4} | 0.777658 |

d_{i}_{,2} | 0.208973 | 0.619674 | 8.5159 × 10^{−}^{2} | 8.84996 × 10^{−}^{4} | 1.11645 |

d_{i}_{,3} | 8.74216 × 10^{−}^{3} | 2.31176 × 10^{−}^{2} | 3.18616 × 10^{−}^{3} | 3.35662 × 10^{−}^{5} | 0.417543 |

d_{i}_{,4} | −1.08765 × 10^{−}^{4} | −2.50112 × 10^{−}^{4} | −3.53111 × 10^{−}^{5} | −3.82297 × 10^{−}^{7} | 4.47731 × 10^{−}^{2} |

d_{i}_{,5} | 1.42369 | 1.3712 | 1.3635 | 1.36176 | 1.46612 |

d_{i}_{,6} | 0.561028 | 0.510083 | 0.503006 | 0.501575 | 0.575081 |

d_{i}_{,7} | 5.27399 × 10^{−}^{2} | 4.44569 × 10^{−}^{2} | 4.32554 × 10^{−}^{2} | 4.31231 × 10^{−}^{2} | 6.7219 × 10^{−}^{2} |

i | 6 | 7 |
---|---|---|

d_{i}_{,1} | 4.42076 × 10^{−}^{2} | 1.59239 × 10^{−}^{4} |

d_{i}_{,2} | 6.30676 × 10^{−}^{2} | 1.46677 × 10^{−}^{4} |

d_{i}_{,3} | 2.3008 × 10^{−}^{2} | −1.05239 × 10^{−}^{5} |

d_{i}_{,4} | 2.34532 × 10^{−}^{3} | −4.9774 × 10^{−}^{6} |

d_{i}_{,5} | 1.468 | 0.968052 |

d_{i}_{,6} | 0.572198 | −3.28294 × 10^{−}^{2} |

d_{i}_{,7} | 6.598905 × 10^{−}^{2} | −4.01647 × 10^{−}^{2} |

Correlations for maximum average Nusselt numbers and critical power-law indices

The maximum average Nusselt numbers for CWT and UHF conditions can be shown to be correlated to

These correlations have maximum relative error of 0.202% and 0.233% for the CWT and UHF conditions, respectively, when

to the following correlations:

These correlations have maximum relative error of 0.0355% and 0.0309% for the CWT and UHF conditions, respectively, when

Correlations for exit Nusselt number and critical power law index for UHF condition

The maximum Nusselt number at the plate exit for the UHF condition can be shown to be correlated to

This correlation has maximum relative error of 0.583% when

This correlation has maximum relative error of 0.631% when

In

The heat transfer rate per same friction force is proportional to

corresponding to the CWT condition while it is vice versa for the critical power-law index plots. It is shown in

It is shown in

Laminar flow and heat transfer in various protruding-edged plate systems are modeled and investigated in the present work. These systems include the Parallel Flow and the Counter Flow protruding-edged plate exchangers as well as those systems being subjected to CWT and UHF conditions. These systems are exposed to normal free stream having both power-law velocity profile and same average velocity. The continuity, axial momentum and energy equations are transformed to similarity equations for CWT and UHF conditions as well as for the Parallel Flow system while they are transformed to non-similarity equations for the Counter Flow system. These equations are solved by using an accurate finite difference method. Excellent agreement is obtained between the numerical results and reported solutions of well-established special cases. Accurate correlations for different flow and heat transfer parameters are generated by using modern regression tools. It is found that there are always local maximum values for Nusselt numbers for both CWT and UHF conditions at specific power-law indices. Also, it is found that there are specific power-law indices that can maximize the heat transfer rate in the Parallel and Counter Flow systems. The maximum enhancement ratios for the Parallel and Counter Flow systems that are identified in this work are 1.075 and 1.109, respectively, which occur at Pr = 100. These ratios are 1.076 and 1.023 for CWT and UHF conditions, respectively, at Pr = 128. Per same friction force, the counter flow system is found to be preferable over the Parallel Flow system only when the power-law indices are smaller than zero. Finally, this work paves a way for new passive heat transfer enhancement method that can enhance heat transfer from a plate by a magnitude of 10% fold which is by appropriately distributing the free stream velocity.

Abdul Rahim A.Khaled, (2015) Modelling and Theoretical Analysis of Laminar Flow and Heat Transfer in Various Protruding-Edged Plate Systems. Journal of Electronics Cooling and Thermal Control,05,45-65. doi: 10.4236/jectc.2015.53004