- #1

HumanistEngineer

- 18

- 2

I am working on modelling of a heat storage tank. More specifically, I need to find out the transient temperature variation through the tank height.

My question is about the plume entrainment considering the case that hot water is injected from the bottom part of a heat storage tank. This bottom part is initially with cold water mass. So I need to model the plume entrainment (the hot water naturally rises to a higher level above the cold region due to its low density).

I could find a solution procedure (print-screen below) but I don't know how to replace these derivatives (Eq. 6, 7 and 8) with the finite difference quotients/approximations. Would you please help me solving these equations or any other approach is welcome?

The reference (of the print-screen) can be reached from https://www.researchgate.net/profile/Hossein_Khorasanizadeh2/publication/258446557_Effect_of_an_Incoming_Jet_on_Thermal_Stratification_of_Hot_Water_Tanks/links/547df0570cf285d6caa99543/Effect-of-an-Incoming-Jet-on-Thermal-Stratification-of-Hot-Water-Tanks.pdf?_sg%5B0%5D=hrAxHaPXcYNiXYkj__x1IRAO7It9HuvSQOeprFhP9HMIMF6I4eEuZLvZtxO7urTn-266pLke9XoMC8HEAzyUew.rymjI4Md_XKMYh0NtvMmDYvN2XC71vY32wxogZL727v6TR31DGT3A9uAFHRyRW8KN3jxRsRlrQLf9hOkRtPoKA&_sg%5B1%5D=zOlVzI6fVeWfpofXWMIhT1GkYWS8bwV8OE8DoWc1X8TlD7MniMslQ5-vBJRxC9rwIb1YmMGEo6LUnWbef7qfTvVZLTraCIOXWO4k4VzoFq9X.rymjI4Md_XKMYh0NtvMmDYvN2XC71vY32wxogZL727v6TR31DGT3A9uAFHRyRW8KN3jxRsRlrQLf9hOkRtPoKA&_iepl=.

My question is about the plume entrainment considering the case that hot water is injected from the bottom part of a heat storage tank. This bottom part is initially with cold water mass. So I need to model the plume entrainment (the hot water naturally rises to a higher level above the cold region due to its low density).

I could find a solution procedure (print-screen below) but I don't know how to replace these derivatives (Eq. 6, 7 and 8) with the finite difference quotients/approximations. Would you please help me solving these equations or any other approach is welcome?

The reference (of the print-screen) can be reached from https://www.researchgate.net/profile/Hossein_Khorasanizadeh2/publication/258446557_Effect_of_an_Incoming_Jet_on_Thermal_Stratification_of_Hot_Water_Tanks/links/547df0570cf285d6caa99543/Effect-of-an-Incoming-Jet-on-Thermal-Stratification-of-Hot-Water-Tanks.pdf?_sg%5B0%5D=hrAxHaPXcYNiXYkj__x1IRAO7It9HuvSQOeprFhP9HMIMF6I4eEuZLvZtxO7urTn-266pLke9XoMC8HEAzyUew.rymjI4Md_XKMYh0NtvMmDYvN2XC71vY32wxogZL727v6TR31DGT3A9uAFHRyRW8KN3jxRsRlrQLf9hOkRtPoKA&_sg%5B1%5D=zOlVzI6fVeWfpofXWMIhT1GkYWS8bwV8OE8DoWc1X8TlD7MniMslQ5-vBJRxC9rwIb1YmMGEo6LUnWbef7qfTvVZLTraCIOXWO4k4VzoFq9X.rymjI4Md_XKMYh0NtvMmDYvN2XC71vY32wxogZL727v6TR31DGT3A9uAFHRyRW8KN3jxRsRlrQLf9hOkRtPoKA&_iepl=.