91Ó°ÊÓ

For the laminar slug flow in a cylindrical tube, we consider the radial coordinate r. The governing equation of heat transfer in cylindrical coordinates is given by: \(\frac {1} { r } \frac {\partial} {\partial r } (r \frac {\partial T} {\partial r} ) = 0\)

There are two boundary conditions for this problem: 1. At r = 0 (axis of the tube), the temperature gradient is zero due to symmetry (dT/dr=0). 2. At r = R (tube wall), the temperature gradient is related to the uniform surface heat flux (q_s). The second boundary condition can be written as: \(-k \frac{\partial T}{\partial r} \Big|_{r=R} = q_s\), where k is the thermal conductivity of the fluid.

Integrate the governing equation for heat transfer with respect to r: \(\frac {d} {d r } ( r \frac {d T} {d r}) = C_{1}\) where \(C_{1}\) is the integration constant. Now, integrate the equation again with respect to r: \( r \frac {d T} {d r} = C_{1}r + C_{2}\), where \(C_{2}\) is the integration constant. Now, let's find the integration constants: 1. At r = 0, dT/dr = 0 (from the first boundary condition): \(0 = C_{1} \times 0 + C_{2}\), which implies \(C_{2} = 0\). 2. At r = R, we have the second boundary condition: \(-k \frac{dT}{dr}\Big|_{r=R} = q_s\) \(-k \frac{dT}{dr} \Big|_{r=R} = -k(C_{1}R)\) \(C_{1} = \frac{q_s}{k}\) Now, substitute the values of the constants back into the equation for the temperature gradient: \(r \frac{dT}{dr} = \frac{q_s}{k}r\) Integrate this equation again to obtain the temperature distribution T(r): \(T(r) = \frac{q_s}{2k}r^{2} + C_{3}\), where \(C_{3}\) is another integration constant.

The Nusselt number \(Nu_{D}\) is a dimensionless number that represents the ratio of convective to conductive heat transfer at the boundary. It can be defined as: \(Nu_{D} = \frac{hD}{k}\), where h is the heat transfer coefficient and D is the diameter of the tube. The heat transfer coefficient h can be expressed in terms of surface heat flux and temperature difference at the wall (using the second boundary condition): \(h = \frac{q_s}{T_{w} - T(r=R)}\), where \(T_{w}\) is the wall temperature. Now, we can substitute this expression for h into the Nusselt number equation: \(Nu_{D} = \frac{q_s D}{k (T_{w} - T(r=R))}\) Plug in the equation for temperature distribution \(T(r)\) at \(r=R\): \(Nu_{D} = \frac{q_s D}{k (T_{w} - (\frac{q_s}{2k}R^{2} + C_{3}))}\) Since the Nusselt number is a dimensionless quantity, the value of the integration constant C3 must be chosen in such a way that the dimension in the denominator is consistent. In this case, for fully developed slug flow, the temperature difference \(T_{w} - T(r=R)\) has the same dimension as \(\frac{q_s}{k}R^{2}\), so the integration constant should be zero (C3 = 0): \(Nu_{D} = \frac{q_s D}{k (T_{w} - \frac{q_s}{2k}R^{2})}\) Therefore, we have the Nusselt number \(Nu_{D}\) in terms of the surface heat flux \(q_s\) and the temperature difference at the wall.