Heat and mass transfer -- Boundary conditions & balance terms

[MexChemE]

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 solid-fluid interphase, stating that the heat flux normal to the surface is given by $q=h(T_0-T_b)$. Where h is the heat transfer coefficient, T₀ 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
$$2BWq_z |_z - 2BWq_z |_{z+\Delta z} - 2W \Delta z h(T-T_a) = 0$$
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
$$W \Delta x q_z |_z - W \Delta x q_z |_{z+\Delta z} + W \Delta z q_x |_x - W \Delta z q_x |_{x+\Delta x} = 0$$
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 solid-fluid interphase, $N_A = k_c (c_{A0} - c_{Ab})$, 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, $N_A = k_1'' c_{A0}$, where k₁'' 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 $k_c (c_{A0} - c_{Ab})$ 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!

 

[Chestermiller]

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 solid-fluid interphase, stating that the heat flux normal to the surface is given by $q=h(T_0-T_b)$. Where h is the heat transfer coefficient, T₀ 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
$$2BWq_z |_z - 2BWq_z |_{z+\Delta z} - 2W \Delta z h(T-T_a) = 0$$
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
$$W \Delta x q_z |_z - W \Delta x q_z |_{z+\Delta z} + W \Delta z q_x |_x - W \Delta z q_x |_{x+\Delta x} = 0$$
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 solid-fluid interphase, $N_A = k_c (c_{A0} - c_{Ab})$, 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, $N_A = k_1'' c_{A0}$, where k₁'' 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 $k_c (c_{A0} - c_{Ab})$ 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 the fin analysis, they are assuming that, within the fin, the temperature gradient in the x direction is negligible, so that the temperature is uniform in the x direction at each z location. This is a very good approximation. If you use the Newton's law equation to estimate the temperature gradient in the x direction (say by assuming that the temperature in the x direction is a parabola), you will find that it is much lower at all locations than the temperature gradient in the z direction. The first equation you wrote is a combination of the heat conduction equation and the Newton boundary condition.

Chet

 

[MexChemE]

This is a very good approximation.

Last night, while reflecting on the subject, I got to the conclusion that a shell balance is an empirical model we develop to model a real system. There's no fundamental law that says what can be put into a balance and what not. Even most constitutive relations are empirical and not fundamental. We get to decide what and what not to include in a balance, depending on the desired complexity and accuracy.

So for example, in the cooling fin model, it is up to the analyst to decide whether to develop the model taking into account conduction in the x and z directions, and solve a partial differential equation; or sacrifice some accuracy (which is actually not that much in the case of the fin) and consider conduction in the z direction only, adding a heat loss term in the form of Newton's law of cooling, and solve an ordinary differential equation.

 

[Chestermiller]

I have something to add regarding your example. The analyst has more than an "accuracy choice" to make. With one method, he can obtain his result analytically with very little effort in about 15 minutes. With the other method, he can spend lots of effort to solve a partial differential equation (probably numerically), and have to expend about a couple of days getting his results. (The cost of this in man-hours might be a couple of thousand dollars). The logical extension of this type of thinking is he can decide to get a very accurate answer by solving the problem using Molecular Dynamics (solving Newton's 2nd law for every molecule comprising the cooling fin), and wait several thousand years for the computer to deliver his answer. Then he can apply this result thousands of year in the future to design his finned tube heat exchanger that needs to be designed right now.

Chet

 

[MexChemE]

The logical extension of this type of thinking is he can decide to get a very accurate answer by solving the problem using Molecular Dynamics (solving Newton's 2nd law for every molecule comprising the cooling fin), and wait several thousand years for the computer to deliver his answer. Then he can apply this result thousands of year in the future to design his finned tube heat exchanger that needs to be designed right now.

I definitely agree with you, and I liked your hyperbole. I wasn't trying to imply that the most accurate model, and consequently hardest and most expensive to solve, is the best model. One has to find the adequate balance between accuracy and solving time/cost, especially in an industrial setting. As I said, it depends on the needs and resources of the modeller, IMHO.