Calculation of evaporation length of a liquid bridge flowing between inclined hot tubes

The bridge consists of liquid held by surface tension forces between two inclined tubes in an LNG heat exchanger. The shape of the bridge is calculated by the hydrostatic equation, which is reduced to a nonlinear integral equation and resolved by the Newton method. The velocity and temperature fields in the bridge are described by the Navier-Stokes and energy equations, respectively. They are reduced to the boundary integral equations and calculated by the method of boundary elements. Heat transfer coefficient is calculated for evaporating bridge and the length of total bridge evaporation is estimated.


Bridge shape and velocity of liquid in flowing down bridge
The cooling mixture flows down the spiral tubes of the LNG heat exchanger first as a continuous film of liquid, but since the film evaporates, at the bottom of the column the film disappears and the liquid starts to flow in the form of bridges sandwiched between the tubes. Due to the inclination of the tubes to the horizon at an angle J , the liquid in the bridge flows along the axis of the tubes. The paper considers heat transfer coefficient in such flow and length of bridge flow zone. Various bridge shapes and liquid velocities in flowing down bridges were calculated in our paper [1]. Here we shortly describe the algorithm of these calculations.
Let us introduce the dimensionless values for the bridge problem: the coordinates x and y, the radius of tubes r, the free surface coordinate G , and the gap between the tubes d: is the characteristic Laplace scale (V is the surface tension). It is convenient to set the contact points by angles D and E (as it is shown in Fig.1   In the limited formulation and without heat exchange analysis, similar bridge flows have been considered in [2,3]. From hydrostatics equation after integration we obtain the differential equation describing the free surface of liquid: where all coordinates of the contact points are determined by formulas (2). Equation (4) connects two angular coordinates of contact points D and E with the parameters given in the problem: contact angle T , inter-tube distance d, and tube radius r . Choosing some angle D and solving equation (4) by the Newton method, we obtain the second angle E as a function of D and above-listed parameters. Using the calculated value E D , we calculate the free surface by equation (3). In Fig.2 where * is the bridge contour; coordinates [ r (observation points) and coordinates x r (integration points) are laying on the contour.
From the boundary conditions for velocity w and its normal derivative, we obtain the boundary conditions on the surfaces of tubes and on the free surface of the liquid ( 6 ), respectively, C w w w w . These quantities are substituted into integral equation (5). Unknown values of normal derivative / v n w w on the surfaces of tubes and function v itself on the free surface of the liquid ( 6 ) are determined from the system of linear equations resolved by boundary element method (BEM). Equation (5) is the basic boundary integral equation in the Boundary Element Method [4].

Temperature field in the bridge
Bridge evaporation is a weak process. It will be shown later that the length of total bridge evaporation is very large. Therefore we can ignore the convection term in the equation for temperature field, and describe temperature T only by heat conductivity effects via the Laplace equation where the observation ( [ r ) and integration points ( x r ) are both placed on the bridge contour * .
Boundary conditions in the dimensionless form are as follows: a) on the free boundary of the bridge 1 t ; b) on the symmetry axis 0 where the parameter of conjugate heat transfer is  Boundary integral equations (5) and (7) can be resolved by the Boundary Element Method [4], which is briefly described below. The boundary integral equations contain the functions and their normal derivatives, which are partially specified on the contour * via boundary conditions a), b), c). These known quantities are substituted into integral equations (5), (7), and unknown values of normal derivatives and functions are determined from the system of linear equations. We obtain this system of equations by dividing contour * into N small linear segments and integrating integrals (5), (7) over them. In our calculations, N was up to N=1000. During integration, the unknown values are considered as constant along the elements. Integrals along the elements of function G and its normal derivative can be calculated analytically. The resulting system of equations is solved by the Gauss method. BEM belongs to a class of direct (not iterative) methods of solution to the elliptic problems of a potential theory. In Fig. 3  T . Note the very large values of the dimensionless temperature gradient at contact points A and D. Therefore we use very small steps along the contour and it leads to a large number of control points on the contour N~1000. But the built BEM-based algorithm is so effective that calculation of solution for the velocity and temperature fields on the contour takes only a few seconds. Therefore we can calculate the effect of all parameters in detail.

The bridge evaporation length and Nusselt criterion
While moving along z axis, the liquid bridge is evaporating permanently. The balance equation for the heat of liquid mass, evaporated from the free surface, supplied through a wall, for length dz takes the , where Q is a mass flow rate of liquid in the bridge, F is a heat flux through the bridge (per a unit of a tube length), ∆h is the specific heat of liquid evaporation (difference between vapor and liquid enthalpies). The heat flux F is calculated by the integral over wetted surfaces of tubes and contains two parts In the dimensionless form, we obtain where J is the angle of tube inclination relative to the horizon; Prandtl, Kutateladze, and Kapitza criteria are also introduced as: ln 1 0.08 10 In Fig. 4,b the Nusselt criterion calculated according to this simple model is compared with directly calculated values of Nu for parameters r= 2.5 and θ =1 0 , 20 0 . One can conclude, that this formula can catch the main features of the conjugate heat transfer. The large deviation is seen only at very small contact angles (θ =1 0 ) for tubes made of aluminum with very high parameter C>2000.

Results of calculations
For cooling ethane-methane mixture with a composition of 50%+50%, the physical properties at low cryogenic temperatures (-150 0 ...-60 0 C) can be estimated according to [5]   The tubes in the LNG heat exchanger according to [6,7]  . According to [6] this temperature difference is about C and the angle of inclination of tubes relative to horizon is J =7 0 . The criteria in formula (8) have the following values: Pr=3.3; Ku=27; Ka 1/2 =4.5×10 5 .
According to paper [6] in the LNG column, the aluminum tubes with a tube diameter of 9.5 mm are used. The dimensionless radius is / 2.5 r R / . In Fig. 5  Therefore, the domain in the LNG column, occupied by bridge flow in the case of stainless-steel tubing is about 3...4.5 m, which seems relatively large. It is not very good due to the lower total heat transfer coefficient in the bridge flow (in comparison with film flow, [6]).

Conclusion
The nonlinear equation for the shape of the bridge is solved by Newton's method, and free surface coordinate is calculated for fixed parameters of the lower contact point, tube radius, tube spacing, and contact angle. The velocity and temperature fields are described in the laminar flow regime by elliptical equations (Navier-Stokes, Laplace), which are reduced to boundary integral equations and calculated by the BEM. It is assumed that the liquid evaporates from the free surface of the bridge into the surrounding vapor. At the surface of the bridge, the temperature is assumed to be equal to the temperature of saturated steam. Heat transfer from hot flow in tubes through the bridge is calculated taking into account the thermal resistance of the tube wall. In the vicinity of contact points, heat flux density becomes very large especially for the case of aluminum tubes with high thermal conductivity. Heat transfer coefficients referring to the wetted surface turn out to be rather high: 1000...4000 (W/m K). However, since the area wetted by the bridge is not large, the heat transfer coefficient referred to the whole surface of the tube is much smaller, 300...800 (W/m K). The estimate for the heat transfer coefficient of the LNG column made in paper [6] is 3...5 times higher than our values. The reason for this discrepancy is that the flow regime in the paper [6] was not laminar bridge flow, but turbulent film flow.
The formula for the bridge evaporation length follows from the balance equation of heat flux and flow rate of evaporating liquid. The evaporation length turns out to be large for cases with large tube spacing (d>1). For stainless steel tubes, this evaporation length is 1.5...2.5 times longer than for aluminum tubes, which is due to the lower thermal conductivity of steel in comparison to aluminum. According to the column's height, the size of the area occupied by bridge flow is obtained by multiplication of bridge evaporation length by the sine of slope angle, which in our case (7 Deg.) gives a multiplier of 0.12. We have obtained the estimate of the height of bridge mode area in the column of 3...6 m (for steel tubes), which is quite significant taking into account the total length of the column of 15...20 m [6].