**3. Numerical modeling**

A numerical model using Ansys Fluent is developed to study the stratification and self-pressurization phenomenon in a cryogenic storage tank. Smooth wall condition is considered and VOF (Volume of Fluid) transient method is used for the simulation. The governing equations are:

Conservation of mass:

$$\frac{\partial \rho}{\partial t} + \nabla . \left( \rho \vec{V} \right) = \mathbf{0} \tag{1}$$

Conservation of momentum:

$$\frac{\partial(\rho \overrightarrow{V})}{\partial t} + \nabla.\left(\rho \overrightarrow{V} \overrightarrow{V}\right) = -\nabla P - \rho \rho \overrightarrow{\mathbf{g}} \left(T - T\_0\right) + \nabla.\left[\mu\_{\text{eff}} \left(\nabla \overrightarrow{V} + \nabla \overrightarrow{V}^T\right)\right] + \overrightarrow{F} \tag{2}$$

Conservation of energy:

$$\frac{\partial(\rho CT)}{\partial t} + \nabla.\left(\overrightarrow{V}(\rho CT + P)\right) = \nabla.\left(\mathbf{K}\_{\text{eff}} \nabla T\right) + \mathbf{S}\_h\tag{3}$$

The location of the interface is identified by solving the continuity equation for the volume fraction of the second phase.

$$\frac{\partial}{\partial t}(a\_v \rho\_v) + \nabla. \left(a\_v \rho\_v \overrightarrow{V}\right) = \dot{m} \tag{4}$$

Where m represents the phase change mass at the interface due to evaporation or condensation. The density, viscosity and thermal conductivity is defined in terms of volume fractions.

$$\text{Density}, \rho = a\_l \rho\_l + a\_v \rho\_v \tag{5}$$

$$\text{Viscosity}, \mu = a\_l \mu\_l + a\_v \mu\_v \tag{6}$$

$$\text{Thermal conductivity}, k = a\_l k\_l + a\_v k\_{v.} \tag{7}$$

The assumptions used are,


Boundary conditions: On the side walls,

$$-K\frac{\partial T}{\partial n} = q\_w \tag{8}$$

At the bottom and top surfaces, the walls are adiabatic,

$$\frac{\partial T}{\partial n} = \mathbf{0} \tag{9}$$

Where n is the direction normal to the wall

### **3.1 Numerical implementation**

A cylindrical tank with 0.5 m diameter and 1 m height is used for the studies. The wall thickness is considered as 0.003 m and 2-D geometry is considered. Commercial CFD package Ansys 15 is used for solving the conservation equations. Axis-symmetric condition is selected because of the nature of physics, geometry and boundary conditions. The Rayleigh number corresponds to all operating condition is above the critical value and flow is always turbulent. So, k-ϵ turbulence model with enhanced wall function approach is applied. A constant heat flux of 10 W/m2 is applied on the left sidewall. The pressure–velocity coupling algorithm selected is SIMPLEC (Semi-Implicit Method for Pressure-Linked Equation- Consistent). The converged solution is easy to achieve by using this method than a SIMPLE algorithm. The body forced weighted average scheme is used for solving the momentum equation. For tracking the liquid-vapor interface, the Geometric Reconstruction Scheme is applied. Since the problem is transient, a time step of 0.001 s is selected so that the Courant number is less than 0.1.

### **3.2 Grid independency**

Three grid systems with mesh numbers 15288, 22893 and 30671 are used to test the resolution. **Table 1** shows the variation of pressure inside the tank with mesh number for the time period of 100 s. The self-pressurization for the two grids (22893 and 30671) is almost the same as the maximum pressure difference is less than 165 kPa. For the mesh number of 30671, the solution is time-consuming. The grids with 22893 quadrilateral grid elements are used with successively increasing mesh numbers towards the wall is selected for the present work.


#### **Table 1.** *Effect of mesh size on self-pressurization.*

*Effect of Roughness Elements on the Evolution of Thermal Stratification in a Cryogenic… DOI: http://dx.doi.org/10.5772/intechopen.98404*

**Figure 2.** *Validation of the numerical model with experimental result.*

#### **3.3 Validation of the model**

The numerical model developed has to be validated with the experimental results to prove the validity of the model in self-pressurization studies. The numerical model is validated with experiments conducted by Vishnu et al. [21]. The experiment set up consists of cylindrical test tank with a diameter of 0.11 m and height of 1.4 m. The liquid nitrogen cryogen was used as the model propellant and fill height was 0.7 m. The overall thermal conductivity of the tank wall was calculated as 0.01957 W/m.K for a tank thickness of 0.15378 m. Applying tank wall temperature of 300 K gives equivalent heat flux as that of experimental condition. The initial liquid temperature selected is 79.1 K which is equal to the initial interface temperature corresponds to the experimental condition and initial pressure corresponds to the saturation value of interface temperature. The initial temperature is assumed to be same throughout vapor and liquid domain. **Figure 2** compare the predicted pressure evolution against the experimental data. The maximum under-prediction of tank pressure by the numerical model compared to the experimental pressure data is not more than 2.6 percent. The tank numerical pressurization rate is in good agreement with the experiment, and hence self-pressurization is satisfactorily predicted.

## **4. Effect of obstruction elements**

This section discusses about the impact of stratification due to the presence of roughness elements on the propellant tank wall. The nature of flow through a tank wall with roughness vary significantly to that of a smooth wall tank. The presence of roughness elements affects the formation of velocity and thermal boundary layer. The natural convection flow over a roughness element is more analogous to flow behavior over large scale obstruction elements such as forward-facing steps, backward facing steps, ribs and fences [20]. Based on the geometry, tank surface with obstruction elements can be classified as grids and ribs.

#### **4.1 Comparison between flow over a smooth and rough wall**

While comparing the performance of tanks with and without obstruction elements, proper scaling of tank geometry should be considered. Thermal

stratification of a smooth wall with same liquid filling height as run length can be compared with that of a rough wall tank, but the volume of liquid and heated surface area will not be identical. Similarly, the heat flux applied could be matched but the filling height or run length would not be identical. In this analysis, the liquid filling height and tank radius considered are similar between smooth and rough wall tanks. Both tanks are having a radius of 0.25 m and filling height 0.5 m which is filled with liquid hydrogen. An axial heat flux of 10 W/m<sup>2</sup> is applied on the side wall. There are 25 number of obstruction elements with 0.178 cm height and 0.076 cm thickness. The details are shown in **Table 2**. The total run length along the obstruction surface is 108.9 cm which is 8.9% more than that of smooth wall tank. The smooth wall tank has a volume of 196349.54 cm3 whereas rough wall tank has 196286.87 cm<sup>3</sup> which is 0.0319% lesser than smooth wall. The total heated surface area of rough wall tank is 17105.008 cm2 which is 8.90% more than smooth wall tank and hence the volume to surface area ratio becomes 8.189% lesser for rough wall tank. The tank geometry and obstruction element layout is shown in **Figure 3**.
