Steady-periodic heating in parallel-plate microchannel flow with participating walls

Steady-state Green’s function solution for moving media with axial conduction

“…Here, S(x, y, z) is the volumetric heat source, C = ρc p is the thermal capacitance, k is the thermal conductivity, and u is the velocity of the moving body. The term ω 2 T is often known as the fin effect and it also emerges during steady-periodic heating as described in Cole [23]. Functional dependences are indicated by C = C(y, z), k = k(y, z), and u = u(y, z) while each remains independent of the axial coordinate x.…”

“…Then, the substitution of the Green's function from Equations (18a) and (18b) into Equation (17) produces the temperature solution as (19) Furthermore, let T w (x) = T i when x < 0 and T w (x) = T w , as a constant, when x ≥ 0 and then, the above equation, when x < 0, reduces to (20) while when x > 0, the solution has the form (21) To compare Equations (20) and (21)(20) and (21) with the existing solutions given in [18], the contour integrals can be converted to volume integrals. To perform this task, one can substitute f m (y, z)exp (−λ m x) from Equation (9) into Equation (3) with ω 2 = 0 and S(x, y, z) = 0 to get the relation (22) The integration of Equation (22) over the cross-sectional area A and after using the divergence theorem provides the relation (23) for positive and negative values of λ m . To verify the accuracy of this solution obtained by the application of the Greens function, it is possible to acquire a solution for this special case by an existing procedure presented in [9,10,18].…”

The effect of axial conduction on heat transfer in a liquid microchannel flow

“…The outside wall of the flow channel is heated, and the flow between parallel plates is fully-developed laminar. The plate spacing is L and the wall thickness is W. The theoretical discussion given below is similar that for the steady-periodic theory developed previously [16], except here the frequency of heating is zero. The temperature satisfies the following equations:…”

Microchannel Heat Transfer with Slip Flow and Wall Effects

“…The plate spacing is L and the wall thickness is L 0 . The theoretical discussion given next is similar to that developed previously [16], so only a brief outline is given. The discussion is limited to steady-periodic heating and steady periodic temperature.…”

“…This approach requires use of a series involving the hypergeometric function with challenging series convergence behavior [16]. In contrast, the layered approach given here involves a closed-form GF in each layer combined with a simple matrix solution.…”