- Email: [email protected]
Comments on “Dynamic response of a packed bed thermal storage system”
314
Letters to the Editor
C o m m e n t s o n " D y n a m i c r e s p o n s e of a p a c k e d b e d t h e r m a l s t o r a g e s y s t e m " Dear Sir, The authors have proposed a one-dimensional model of the transient thermal response of a packed bed which includes axial thermal dispersion and intraparticle conduction effects. The predictions of this model are compared to measurements made in this laboratory and reported by Vanden Broek and Clark[I] of the mixed mean exit fluid temperature from a bed of randomly packed spheres of uniform diameter. The authors attribute the discrepancies between their model and the measured values to energy losses from the bed. This statement is at least partly true, as observed steady-state losses were equivalent to - 1.5~ drop in mixed mean exit fluid temperature. However, the author's statements, "Vanden Broek and Clark[I] reported radial temperature variations.., which can be attributed to heat losses. The temperature variations are not due to velocity non-uniformities.", require closer evaluation. The temperature variations in [1] for a transient generated by an increase in the inlet fluid temperature show solid temperatures at the wall higher than those at the center of the bed; such a distribution cannot arise from thermal losses but rather, as recent analysis has demonstrated[2], is caused be velocity non-uniformities due to the wall effect[3,4]. This effect is caused by an increased void fraction at the wall and the geometry of the bounding surface. The author's statement that uniform velocities were measured is in error. Higher velocities were measured in the near-wall region ([1], Appendix II). Recently a two-dimensional model of flow and heat transfer in packed beds was developed[2] which demonstrates the effect of non-uniform velocity on tern-
perature distribution, as well as the effect of wall heat capacity and energy losses, as was suggested by Saez and McCoy (Solar Energy 29, 201 (1982)). We feel that the discrepancies in the author's model are due to the failure to include velocity non-uniformities, wall heat capacity and energy losses, as well as to the overestimation of axial dispersion. The two-dimensional effects may be eyen stronger in a commercial sized rock bed. as opposed to a bed of uniform spheres[2].
Solar Energy Laboratory Department of Afechanical Engineering and Applied Mechanics Unirersity o,f Michigan Ann Arbor, MI 48109 U.S.A.
DONALD E. BEASLEY JOHN A. CLARK
REs I. C. Vanden Broek and J. A. Clark, Von Karman Institute for Fluid Dynamics. Lecture Series Vol. I (1979). 2. J. A. Clark and D. E. Beasley, Progress in Solar Energy 525 (1982). 3. R. Newell and N. Standish, Metallurgical Trans. 4, 1851 (1973). 4. H. S. Mickley, K. A. Smith and E. I. Korchak, Chem. Engr. ScL 20, 237 (1965).
R e s p o n s e to c o m m e n t s b y Drs. D. E. Beasley a n d J. A. C l a r k Dear Sir, We join Drs. Beasly and Clark in emphasizing the importance of non-uniform velocity profiles in the experiments of Vanden Broek and Clark[l]. The discrepancies between our theory[2] and the experimental data[I] are likely due to the effect of these velocity non-uniformities, since one of the basic assumptions in our model is that the velocity is constant with respect to radial position in the bed. The wall effect is important when the ratio of column to particle diameter is less than 20, as shown by Cohen and Metzner[3]. This ratio was 14.4 for the experiments performed by Vanden Broek and Clark[ll. For pebble bed thermal storage systems the column to particle diameter ratio would probably be greater than 30, for which wall effects are negligible [3]. Even with the neglect of wall effects the absolute deviations between our model and the experimental values were generally less than I0 percent, indicating a good agreement for many practical purposes. The model includes axial dispersion and intraparticle temperature profiles, which are commonly neglec-
ted, but has the advantage of simplicity in the final analytical solution. The effect of heat losses can be included in the model without complicating the solution, as shown by Saez and McCoy[4]. A thorough evaluation of this more general model is currently underway.
Departmetzt of Chemical Engineering Unirersity of Califim6a Daris, CA 95616 U.S.A.
A.E. SAEZ B.J. McCoy
REFERENCES L C. Vanden Broek and J. A. Clark, Von Karman Institute for Fluid Dynamics, Lecture Series Vol. I (1979). 2. A. E. Saez and B. J. McCoy, Solar Energy 29, 201 0982). 3. Y. Cohen and A. B. Metzner, A.I.Ch.E.J. 27, 705 (1981). 4. A. E. Saez and B. J. McCoy, Int. J. Heat and Mass Transfer (1983).
Letters to the Editor
C o m m e n t s o n " D y n a m i c r e s p o n s e of a p a c k e d b e d t h e r m a l s t o r a g e s y s t e m " Dear Sir, The authors have proposed a one-dimensional model of the transient thermal response of a packed bed which includes axial thermal dispersion and intraparticle conduction effects. The predictions of this model are compared to measurements made in this laboratory and reported by Vanden Broek and Clark[I] of the mixed mean exit fluid temperature from a bed of randomly packed spheres of uniform diameter. The authors attribute the discrepancies between their model and the measured values to energy losses from the bed. This statement is at least partly true, as observed steady-state losses were equivalent to - 1.5~ drop in mixed mean exit fluid temperature. However, the author's statements, "Vanden Broek and Clark[I] reported radial temperature variations.., which can be attributed to heat losses. The temperature variations are not due to velocity non-uniformities.", require closer evaluation. The temperature variations in [1] for a transient generated by an increase in the inlet fluid temperature show solid temperatures at the wall higher than those at the center of the bed; such a distribution cannot arise from thermal losses but rather, as recent analysis has demonstrated[2], is caused be velocity non-uniformities due to the wall effect[3,4]. This effect is caused by an increased void fraction at the wall and the geometry of the bounding surface. The author's statement that uniform velocities were measured is in error. Higher velocities were measured in the near-wall region ([1], Appendix II). Recently a two-dimensional model of flow and heat transfer in packed beds was developed[2] which demonstrates the effect of non-uniform velocity on tern-
perature distribution, as well as the effect of wall heat capacity and energy losses, as was suggested by Saez and McCoy (Solar Energy 29, 201 (1982)). We feel that the discrepancies in the author's model are due to the failure to include velocity non-uniformities, wall heat capacity and energy losses, as well as to the overestimation of axial dispersion. The two-dimensional effects may be eyen stronger in a commercial sized rock bed. as opposed to a bed of uniform spheres[2].
Solar Energy Laboratory Department of Afechanical Engineering and Applied Mechanics Unirersity o,f Michigan Ann Arbor, MI 48109 U.S.A.
DONALD E. BEASLEY JOHN A. CLARK
REs I. C. Vanden Broek and J. A. Clark, Von Karman Institute for Fluid Dynamics. Lecture Series Vol. I (1979). 2. J. A. Clark and D. E. Beasley, Progress in Solar Energy 525 (1982). 3. R. Newell and N. Standish, Metallurgical Trans. 4, 1851 (1973). 4. H. S. Mickley, K. A. Smith and E. I. Korchak, Chem. Engr. ScL 20, 237 (1965).
R e s p o n s e to c o m m e n t s b y Drs. D. E. Beasley a n d J. A. C l a r k Dear Sir, We join Drs. Beasly and Clark in emphasizing the importance of non-uniform velocity profiles in the experiments of Vanden Broek and Clark[l]. The discrepancies between our theory[2] and the experimental data[I] are likely due to the effect of these velocity non-uniformities, since one of the basic assumptions in our model is that the velocity is constant with respect to radial position in the bed. The wall effect is important when the ratio of column to particle diameter is less than 20, as shown by Cohen and Metzner[3]. This ratio was 14.4 for the experiments performed by Vanden Broek and Clark[ll. For pebble bed thermal storage systems the column to particle diameter ratio would probably be greater than 30, for which wall effects are negligible [3]. Even with the neglect of wall effects the absolute deviations between our model and the experimental values were generally less than I0 percent, indicating a good agreement for many practical purposes. The model includes axial dispersion and intraparticle temperature profiles, which are commonly neglec-
ted, but has the advantage of simplicity in the final analytical solution. The effect of heat losses can be included in the model without complicating the solution, as shown by Saez and McCoy[4]. A thorough evaluation of this more general model is currently underway.
Departmetzt of Chemical Engineering Unirersity of Califim6a Daris, CA 95616 U.S.A.
A.E. SAEZ B.J. McCoy
REFERENCES L C. Vanden Broek and J. A. Clark, Von Karman Institute for Fluid Dynamics, Lecture Series Vol. I (1979). 2. A. E. Saez and B. J. McCoy, Solar Energy 29, 201 0982). 3. Y. Cohen and A. B. Metzner, A.I.Ch.E.J. 27, 705 (1981). 4. A. E. Saez and B. J. McCoy, Int. J. Heat and Mass Transfer (1983).