Coupled thermal and concentration entry effects
From ThermalFluidsPedia
There are many transport phenomena problems in which heat and mass transfer simultaneously occur. In some cases, such as sublimation and vapor deposition, they are coupled. These problems are usually treated as a single phase. However, coupled heat and mass transfer should both be considered even though they are modeled as being single phase. In this article, coupled forced internal convection in a circular tube will be presented for both adiabatic and constant wall heat flux.
Sublimation inside an Adiabatic Tube
In addition to the external sublimation, internal sublimation is also very important. Sublimation inside an adiabatic and externally heated tube will be analyzed. The physical model of the problem under consideration is shown in figure to the right ^{[1]}. The inner surface of a circular tube with radius r_{o} is coated with a layer of sublimable material which will sublime when gas flows through the tube. The fullydeveloped gas enters the tube with a uniform inlet mass fraction of the sublimable substance, ω_{0}, and a uniform inlet temperature, T_{0}. Since the outer wall surface is adiabatic, the latent heat of sublimation is supplied by the gas flow inside the tube; this in turn causes the change in gas temperature inside the tube. It is assumed that the flow inside the tube is incompressible laminar flow with constant properties. In order to solve the problem analytically, the following assumptions are made:
1. The entrance mass fraction, ω_{0}, is assumed to be equal to the saturation mass fraction at the entry temperature, T_{0}.
2. The saturation mass fraction can be expressed as a linear function of the corresponding temperature.
3. The mass transfer rate is small enough that the transverse velocity components can be neglected.
The fully developed velocity profile in the tube is

where u_{m} is the mean velocity of the gas flow inside the tube. Neglecting axial conduction and diffusion, the energy and mass transfer equations are


where D is mass diffusivity. Equations (2) and (3) are subjected to the following boundary conditions:




Equation (7) implies that the latent heat of sublimation is supplied as the gas flows inside the tube. Another boundary condition at the tube wall is obtained by setting the mass fraction at the wall as the saturation mass fraction at the wall temperature ^{[2]}. According to the second assumption, the mass fraction and temperature at the inner wall have the following relationship:

where a and b are constants.
The following nondimensional variables are then introduced:

where T_{f} and ω_{f} are temperature and mass fraction of the sublimable substance, respectively, after heat and mass transfer are fully developed, and Le is Lewis number. Equations (2) – (8) then become






The heat and mass transfer eqs. (10) and (11) are independent, but their boundary conditions are coupled by eqs. (14) and (15). The solution of eqs. (10) and (11) can be obtained via separation of variables. It is assumed that the solution of θ can be expressed as a product of the function of η and a function of ξ, i.e.,
θ = Θ(η)Γ(ξ) 
Substituting eq. (16) into eq. (10), the energy equation becomes

where β is the eigenvalue for the energy equation. Equation (17) can be rewritten as two ordinary differential equations:
Γ' + β^{2}Γ = 0 

The solution of eq. (18) is

The boundary condition of eq. (19) at η = 0 is
Θ'(0) = 0 
The dimensionless temperature is then

Similarly, the dimensionless mass fraction is

where γ is the eigenvalue for the conservation of species equation, and Φ(η) satisfies

and the boundary condition of eq. (24) at η = 0 is
Φ'(0) = 0 
Substituting eqs. (22) – (23) into eqs. (14) – (15), one obtains
β = γ 

To solve eqs. (19) and (24) using the RungeKutta method it is necessary to specify two boundary conditions for each. However, there is only one boundary condition for each: eqs. (21) and (25), respectively. Since both eqs. (19) and (24) are homogeneous, one can assume that the other boundary conditions are Θ(0) = Φ(0) = 1 and the solve eqs. (19) and (24) numerically. It is necessary to point out that the eigenvalue, β, is still unknown at this point and must be obtained by eq. (27). There will be a series of β which satisfy eq. (27), and for each value of β_{n} there is one set of corresponding Θ_{n} and Φ_{n} functions .
If we use any one of the eigenvalues, β_{n}, and corresponding eigenfunctions, Θ_{n} and Φ_{n}, in eqs. (22) and (23), the solutions of eq. (10) and (11) become


which satisfy all boundary conditions except those at ξ = 0. In order to satisfy boundary conditions at ξ = 0, one can assume that the final solutions of eqs. (10) and (11) are


where G_{n} and H_{n} can be obtained by substituting eqs. (30) and (31) into eq. (12), i.e.,


Due to the orthogonal nature of the eigenfunctions Θ_{n} and Φ_{n}, expressions of G_{n} and H_{n} can be obtained by


The Nusselt number due to convection and the Sherwood number due to diffusion are


where T_{m} and ω_{m} are mean temperature and mean mass fraction in the tube.
The figure to right shows heat and mass transfer performance during sublimation inside an adiabatic tube. For all cases, both Nusselt and Sherwood numbers become constant when ξ is greater than a certain number, thus indicating that heat and mass transfer in the tube have become fully developed. The length of the entrance flow increases with an increasing Lewis number. While the fully developed Nusselt number increases with an increasing Lewis number, the Sherwood number decreases with an increasing Lewis number, because a larger Lewis number indicates larger thermal diffusivity or low mass diffusivity. The effect of (ah_{sv} / c_{p}) on the Nusselt and Sherwood numbers is relatively insignificant: both the Nusselt and Sherwood numbers increase with increasing (ah_{sv} / c_{p}) for Le < 1, but increasing(ah_{sv} / c_{p}) for Le > 1 results in decreasing Nusselt and Sherwood numbers.
Sublimation inside a Tube Subjected to External Heating
When the inner wall of a tube with a sublimablematerialcoated outer wall is heated by a uniform heat flux, q''(see figure to the right), the latent heat will be supplied by part of the heat flux at the wall. The remaining part of the heat flux will be used to heat the gas flowing through the tube. The problem can be described by eqs. (1) – (8), except that the boundary condition at the inner wall of the tube is replaced by

where the thermal resistance of the tube wall is neglected because the tube wall and the coated layer are very thin.
The governing equations for sublimation inside a tube heated by a uniform heat flux can be nondimensionalized by using the dimensionless variables defined in eq. (9), except the following:

where ω_{sat,0} is the saturation mass fraction corresponding to the inlet temperature T_{0}. The resulting dimensionless governing equations and boundary conditions are







where in eq. (43). The sublimation problem under consideration is not homogeneous, because eq. (45) is a nonhomogeneous boundary condition. The solution of the problem is consistent with its particular (fully developed) solution as well as the solution of the corresponding homogeneous problem ^{[3]}:
θ(ξ,η) = θ_{1}(ξ,η) + θ_{2}(ξ,η) 

While the fully developed solutions of temperature and mass fraction, θ_{1}(ξ,η) and , respectively, must satisfy eqs. (40) – (41) and (44) – (46), the corresponding homogeneous solutions of the temperature and mass fraction, θ_{2}(ξ,η) and , must satisfy eqs. (40), (41), (44), and (46), as well as the following conditions:



The fully developed profiles of the temperature and mass fraction are


The solution of the corresponding homogeneous problem can be obtained by separation of variables:


where


and β_{n} is the eigenvalue of the corresponding homogeneous problem. The Nusselt number based on the total heat flux at the external wall is

where θ_{w} and θ_{m} are dimensionless wall and mean temperatures, respectively.
The Nusselt number based on the convective heat transfer coefficient is

The Sherwood number is

When the heat and mass transfer are fully developed, eqs. (58) – (60) reduce to



The variations of the local Nusselt number based on total heat flux along the dimensionless location ξ are shown in the figure to the right. It is evident from part (a) of the figure that Nu increases significantly with increasing (ah_{sv} / c_{p}). The Lewis number has very little effect on Nu_{x} when (ah_{sv} / c_{p})= 0.1, but its effects become obvious in the region near the entrance when (ah_{sv} / c_{p}) = 1.0 and gradually diminishes in the region near the exit. has almost no influence on Nu in almost the entire region when (ah_{sv} / c_{p}) = 1.0, as seen in part (b) of the figure. When (ah_{sv} / c_{p})= 0.1, Nu_{x} increases slightly when ξ is small.
The variation of the local Nusselt number based on convective heat flux, Nu^{*}, is shown in part (a) of the figure to the right. Only a single curve is obtained, which implies that Nu^{*} remains unchanged when the mass transfer parameters are varied. The value of Nu^{*} is exactly the same as for the process without sublimation. Part (b) of the figure shows the Sherwood number for various parameters. It is evident that (ah_{sv} / c_{p}) and have no effect on Sh_{x}, and Le has an insignificant effect on Sh_{x} in the entry region.
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