 ## Engineering Problem Statement:

The dimension of room model is 573×387×254 cm. The rotating speed of fan is 250 rpm. The ceiling fan has three blades.The purpose of the simulation is to analyse the flow in the room.

Multiple Reference Frame:

A Moving Reference Frame (MRF) is a relatively simple, robust, and efficient steady-state, Computational Fluid Dynamics (CFD) modeling technique to simulate rotating machinery. For example, the ceiling fan on a room can be modeled with MRFs.

The MRF model accounts for complete blade design and fan details to simulate complex turbo machinery. It is a steady state approximation where the fluid zone in the fan region is modeled as a rotating frame of reference and the surrounding zones are modeled in a stationary frame.  Contrary to the Body Force Model, the MRF model includes the geometry of the fan blades. The fan blades are modeled stationary but since the fluid domains surrounding them are in a rotating frame, the pressure jump and the swirl components are given by the presence of blades as wall.

An MRF assumes that an assigned volume has a constant speed of rotation and the non-wall boundaries are surfaces of revolution. In the case of the ceiling fan the volume between the fan are designated as MRFs, assigned rotational speeds, and embedded within a multi-volume flow domain. MRF is also known as the 'frozen rotor approach'.

In case of ceiling fan Flow field around a ceiling fan depends on variety of factors; distance between the rotor and the ceiling, solidity of the rotors, the number of blades, and speed of the rotor and blade profiles.

Relating rotating frames to stationary frames:

The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. It begins with the relation between a particle's coordinates in a rotating frame and its coordinates in an inertial (stationary) frame. Then, by taking time derivatives, formulas are derived that relate the velocity of the particle as seen in the two frames, and the acceleration relative to each frame. Using these accelerations, the fictitious forces are identified by comparing Newton's second law as formulated in the two different frames.

Relation between positions in the two frames:

To derive these fictitious forces, it's helpful to be able to convert between the coordinates ( x',y',z') of the rotating reference frame and the coordinates ( x, y, z ) of an inertial reference frame with the same origin. If the rotation is about the z axis with an angular velocity \Omega and the two reference frames coincide at time t=0, the transformation from rotating coordinates to inertial coordinates can be written Whereas the reverse transformation is This result can be obtained from a rotation matrix. Introduce the unit representing standard unit basis vectors in the rotating frame. The time-derivatives of these unit vectors are found next. Suppose the frames are aligned at t = 0 and the z-axis is the axis of rotation. Then for a counterclockwise rotation through angle: Where the (x, y) components are expressed in the stationary frame. Likewise, Thus the time derivative of these vectors, which rotate without changing magnitude, is Where This result is the same as found using a vector cross product with the rotation vector Ω pointed along the z-axis of rotation Ω= (0, 0, Ω), namely Time derivatives in the two frames:

Introduce the unit vectors representing standard unit basis vectors in the rotating frame. As they rotate they will remain normalized. If we let them rotate at the speed of about an axis then each unit vector of the rotating coordinate system abides by the following equation: and we want to examine its first derivative we have (using the product rule of differentiation): Where is the rate of change of f as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as: This result is also known as the Transport Theorem in analytical dynamics and is also sometimes referred to as the Basic Kinematic Equation.

Relation between velocities in the two frames:

A velocity of an object is the time-derivative of the object's position, or The time derivative of a position in a rotating reference frame has two components, one from the explicit time dependence due to motion of the particle itself, and another from the frame's own rotation. Applying the result of the previous subsection to the displacement , the velocities in the two reference frames are related by the equation Where subscript i means the inertial frame of reference, and r means the rotating frame of reference.

Model: The model of room with ceiling fan is created in Design modeler. The model is created in three parts Stator, Rotor or Ceiling fan. Boolean Subtraction is use to the fan is modeled by a rotating reference frame. The blades' surface is set to rotate at the rotation rate of the rotating reference frame and the hub surface of the fan is fixed in relation to the absolute reference frameto properly reflect the real fan operation.

Meshed Model:  Solution Setup

Models: Enable the standard k-epsilon turbulence model.

The K-epsilon model is one of the most common turbulence models, although it just doesn't perform well in cases of large adverse pressure gradients It is a two equation model that means, it includes two extra transport equations to represent the turbulent properties of the flow. This allows a two equation model to account for history effects like convection and diffusion of turbulent energy

Cell Zone Conditions:Rotor

• Enable Frame Motion.
• In Reference Frame Retain the Rotation-Axis Origin default setting of (0, 0). This is the center of curvature for the circular boundaries of the rotating zone.
• Provide 250rpm for Speed in the Rotational Velocity group box.

Results:

There are various factors affecting the flow pattern  of the ceiling fan such as presence of solid boundaries like ceiling and walls and geometrical parameters of the fan blades are some of them. There are range of possible variations available for these geometrical  design parameters at the manufacturing stage.  Discussion:

A ceiling fan pushes air downward, (or sucks it upward depending on the rotation direction), creating flow streamlines in the shape of a torus (the size of the room). When fan moves, air flow passes over & under the surface of a fan blade. The air flow at the tip of the blade create vortices and make the flow turbulent. Dynamic pressure is created due to this vortex generation. This dynamic pressure is lowest closest to the axis of the fan, and increases as we move away from it, in accordance with Bernoulli’s principle.

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