IN the last fifteen years, variational inequalities have gainedimportance in analysis, both from the theoretical and the practical pointsof view. At the beginning of this work we emphasize the formulation ofvarious types of problems in a variational setting. We deal with variousmathematical aspects of many of these special variational formulationsand with the numerical analysis for some of these problems. To complete4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.4-1 ConclusionsFrom the experimental results presented and discussed in the previous chapter, the following conclusions may be made :1- The heat transfer characteristics of tubes in fixed volume , bothfor staggered and in-line arrangements are better than those forsingle tube at the same conditions .2- Staggered arrangements of tubes in fixed volume gives better heat transfer coefficient compared with in line one and at the same conditions .3- The optimal tube spacing for different arrangements at is Sh /d and S/d = 1 at S/d = 1 , for both in-line and staggered arrangements.4- A correlation for in-line and staggered arrangements of the present data are developed. This correlation satisfactorily expresses the influences of Re, Pr, and cylinder spacing on the Nusselt Number Nu . The correlation can be used to predict the heat transfer coefficient with reasonable accuracy not exceeding ± 30% of measured values.4-2 Recommendations For Future WorkFrom the experimental study of heat transfer coefficient of single smooth tube, in-line and staggered arrangements of smooth tubes in a fixed volume , we recommend the following research points for the future work as follows:1- Heat transfer coefficient of the different arrangements with anther Sh/d and S/d for both in-line and staggered arrangements in the different fixed volumes.2- Heat transfer coefficient of enhanced tubes surface at the same conditions for both in-line and staggered arrangements in the same fixed volume.3- Optimum design of tube surface enhancement for both in-line and staggered arrangements .4- Study the effect of using hot and moist air on the heat transfer coefficient of the same different arrangements ( in-line and staggered) in the same fixed values.