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Main Question or Discussion Point
Hello, PF! Recently, while reading chapter 10 (microscopic energy balances) of the second edition of BSL, I found a minor discrepancy which is confusing me, especially when considering the mathematical analogies of heat and mass transfer.
In section 10.1, the authors introduce Newton's law of cooling as a boundary condition for a solidfluid interphase, stating that the heat flux normal to the surface is given by [itex]q=h(T_0T_b)[/itex]. Where h is the heat transfer coefficient, T_{0} is the temperature at the surface and T_{b} is the bulk temperature of the fluid. Then, in section 10.7, while analyzing a cooling fin (sketch attached) they use Newton's law as a heat loss term in the energy balance. In the model for the fin, heat is being conducted in the z direction, and the ends of the fin are insulated. Heat is lost from the surfaces at x = B and x = B. I suppose Newton's law can't be used as a boundary condition in this model because we are analyzing q_{z} and not q_{x}; and the authors never state Newton's law is restricted to be used only as a boundary condition. The microscopic energy balance for this model is
[tex]2BWq_z _z  2BWq_z _{z+\Delta z}  2W \Delta z h(TT_a) = 0[/tex]
Where T_{a} is the temperature of the surrounding air. To my understanding, if we were to consider also the heat conduction in the x direction, the energy balance would be
[tex]W \Delta x q_z _z  W \Delta x q_z _{z+\Delta z} + W \Delta z q_x _x  W \Delta z q_x _{x+\Delta x} = 0[/tex]
This time, Newton's law does not appear in the balance because it is a boundary condition for q_{x}, right?
Then, in chapter 18 (microscopic mass balances), section 18.1 we are introduced to two types of boundary conditions which are analogous to Newton's law of cooling. The mass flux normal to a surface in a solidfluid interphase, [itex]N_A = k_c (c_{A0}  c_{Ab})[/itex], where k_{c} is the mass transfer coefficient, c_{A0} is the concentration of A at the surface and c_{Ab} is the bulk concentration of A in the fluid stream. And the mass flux normal to a surface due to heterogeneous reaction, [itex]N_A = k_1'' c_{A0}[/itex], where k_{1}'' is the reaction rate and c_{A0} is the concentration of A at the surface where the reaction is occurring. This time, the authors explicitly state that these terms do not appear in mass balances, but rather used only as boundary conditions. My main concern is this, if we consider a system similar to the cooling fin, in which there is mass diffusion in the z direction, and mass transfer is occurring at surfaces at x = B and x = B, are we allowed to include the [itex]k_c (c_{A0}  c_{Ab})[/itex] term as a balance term and not as a boundary condition? And again, as done before, if we also consider diffusion in the x direction, then the interphase term is going to be used as a boundary condition for the mass flux in the x direction, and not in the balance, is that right?
I hope there's not trouble understanding what my concerns are, I tried to be as clear as possible. Thanks in advance for any input!
In section 10.1, the authors introduce Newton's law of cooling as a boundary condition for a solidfluid interphase, stating that the heat flux normal to the surface is given by [itex]q=h(T_0T_b)[/itex]. Where h is the heat transfer coefficient, T_{0} is the temperature at the surface and T_{b} is the bulk temperature of the fluid. Then, in section 10.7, while analyzing a cooling fin (sketch attached) they use Newton's law as a heat loss term in the energy balance. In the model for the fin, heat is being conducted in the z direction, and the ends of the fin are insulated. Heat is lost from the surfaces at x = B and x = B. I suppose Newton's law can't be used as a boundary condition in this model because we are analyzing q_{z} and not q_{x}; and the authors never state Newton's law is restricted to be used only as a boundary condition. The microscopic energy balance for this model is
[tex]2BWq_z _z  2BWq_z _{z+\Delta z}  2W \Delta z h(TT_a) = 0[/tex]
Where T_{a} is the temperature of the surrounding air. To my understanding, if we were to consider also the heat conduction in the x direction, the energy balance would be
[tex]W \Delta x q_z _z  W \Delta x q_z _{z+\Delta z} + W \Delta z q_x _x  W \Delta z q_x _{x+\Delta x} = 0[/tex]
This time, Newton's law does not appear in the balance because it is a boundary condition for q_{x}, right?
Then, in chapter 18 (microscopic mass balances), section 18.1 we are introduced to two types of boundary conditions which are analogous to Newton's law of cooling. The mass flux normal to a surface in a solidfluid interphase, [itex]N_A = k_c (c_{A0}  c_{Ab})[/itex], where k_{c} is the mass transfer coefficient, c_{A0} is the concentration of A at the surface and c_{Ab} is the bulk concentration of A in the fluid stream. And the mass flux normal to a surface due to heterogeneous reaction, [itex]N_A = k_1'' c_{A0}[/itex], where k_{1}'' is the reaction rate and c_{A0} is the concentration of A at the surface where the reaction is occurring. This time, the authors explicitly state that these terms do not appear in mass balances, but rather used only as boundary conditions. My main concern is this, if we consider a system similar to the cooling fin, in which there is mass diffusion in the z direction, and mass transfer is occurring at surfaces at x = B and x = B, are we allowed to include the [itex]k_c (c_{A0}  c_{Ab})[/itex] term as a balance term and not as a boundary condition? And again, as done before, if we also consider diffusion in the x direction, then the interphase term is going to be used as a boundary condition for the mass flux in the x direction, and not in the balance, is that right?
I hope there's not trouble understanding what my concerns are, I tried to be as clear as possible. Thanks in advance for any input!
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