Question

Question (1)

A- Write full Navier Stokes equation and energy equation in cylindrical coordinates. From this derive equation for asymmetrical coordinates. Explain reasons for each assumption that you make while deriving the equation.

B- Develop a relation for velocity and temperature distribution in a circular pipe which is experiencing a constant pressure gradient and a constant wall temperature. Explain each boundary condition

Answer 1

In many engineering problems, approximate solutions concerning the overall properties of a fluid system can be obtained by application of the conservation equations of mass, momentum and energy written in integral form, for a conveniently selected control volume. This approach necessitates in general introduction of simplifying assumptions, regarding in particular the spatial distributions of the different variables (e.g., uniform velocity at the open boundaries) and the neglect of terms that are anticipated to give a relatively small contribution to the overall balances. This integral approach is however not suitable when one is interested in computing local properties of the flow (e.g., distributions of velocity v¯(¯x, t), density ρ(¯x, t), pressure p(¯x, t), etc). For that purpose, the conservation equations in integral form

Internal and external flows
Depending on how the fluid and the solid boundaries contact each other, the flow may be classified as internal flow or external flow. In internal flows, the fluid moves between solid boundaries. As is the case when fluid flows in a pipe or a duct. In external flows, however, the fluid is flowing over an external solid surface, the example may be sited is the flow of fluid over a sphere as shown in
Let us now consider the example of fluid flowing over a horizontal flat plate as shown in The velocity of the fluid is V Infinity before it encounters the plate. As the fluid touches the plate, the velocity of the fluid layer just adjacent to the plate surface becomes zero due to the no slip boundary condition. This layer of fluid tries to drag the next fluid layer above it and reduces its velocity. As the fluid proceeds along the length of the plate (in x-direction), each layer starts to drag adjacent fluid layer but the effect of drag reduces as we go further away from the plate in y-direction. Finally, at some distance from the plate this drag effect disappears or becomes insignificant. This region where the velocity is changing or where the velocity gradients exists, is called the boundary layer region. The region beyond boundary layer where the velocity gradients are insignificant is called the potential flow region.
As depicted in , the boundary layer keeps growing along the x-direction, and may be referred to as the developing flow region. In internal flows (e.g. fluid flow through a pipe), the boundary layers finally merge after flow over a distance as shown in below.
Developing flow and fully developed flow region
The region after the point at which the layers merge is called the fully developed flow region and before this it is called the developing flow region. In fact, fully developed flow is another important assumption which is taken for finding solution for varity of fluid flow problem. In the fully developed flow region (as shown in the velocity vz is a function of r direction only. However, the developing flow region, velocity vz is also changing in the zdirection.
Main axioms of transport phenomena
The basic equations of transport phenomena are derived based on following five axioms.
Mass is conserved, which leads to the equation of continuity. Momentum is conserved, which leads to the equation of motion. Moment of momentum is conserved leads to an important result that the 2nd order stress tensor τ~ is symmetric. Energy is conserved, which leads to equation of thermal energy. Mass of component i in a multi-component system is conserved, which leads to the convective diffusion equation.
The solution of equations, resulting from axiom 2, 4 and 5 leads to the solution of velocity, temperature and concentration profiles. Ones these profiles are known, all other important information needed can be determined. We first take the axiom -1. Other axioms will be taken up one by one letter on.
There are three types of control volumes (CV) which may be chosen for deriving the equations based these axioms.
Rectangular shaped control volume fixed in space In this case, the control volume is rectangular volume element and is fixed in space. This method is the easiest to understand but requires more number of steps. Irregular shaped control volume element fixed in space In this case, the control volume can be of any shape, but it is again fixed in space. This method is somewhat more difficult than the previous method as it requires little better understanding vector analysis and surface and volume integrals. Material volume approach In this case, the control volume can be of any shape but moves with the velocity of the flowing fluid. This method is most difficult in terms of mathematics, but requires least number of steps for deriving the equations.
All three approaches when applied to above axiom, lead to the same equations. In this web course, we follow the first approach. Other approaches may be found elsewhere.
Axioms-1
Mass is conserved
Consider a fluid of density ρ flowing with velocity as shown in Fig. 8.4. Here, ρ and are functions of space (x,y,z) and time (t). For conversion of mass, the rate of mass entering and leaving from the control volume (net rate of inflow) has to be evaluated and this should be equal to the rate of accumulation of mass in the control volume (CV). Thus, conservation of mass may be written in words as given below
Fixed rectangular volume element through which fluid is flowing
The equation is then divided by the volume of the CV and converted into a partial differential equation by taking the limit as all dimensions go to zero. This limit effectively means that CV collapses to a point, thereby making the equation valid at every point in the system.
Let m and m+Δm be the mass of the control volume at time t and t+Δt respectively. Then, the rate of accumulation,
In order to evaluate the rate of inflow of mass into the control volume, we need to inspect how mass enters the control volume. Since the fluid velocity has three components vx, vy and vz, we need to identify the components which cause the inflow or the outflow at each of the six faces of the rectangular CV. For example, it is the component vx which forces the fluid to flow in the x direction, and thus it makes the fluid enter or exit through the faces having area ΔyΔz at x = x and x = x+Δx respectively. The component vy forces the fluid in y direction, and thus it makes the fluid enter or exit through the faces having area ΔxΔz at y = y and y = y+Δy respectively. Similarly, the component vz forces the fluid to flow in z direction, and thus it makes fluid enter or exit through the faces having area ΔxΔy at z = z and z = z+Δ z respectively.
The rate mass entering in x direction through the surface ΔyΔz is (ρvxΔyΔz|x), the rate of mass entering in y direction through the surface ΔxΔz is (ρvyΔxΔzy) and the rate of mass entering from z direction through the surface ΔxΔy is (ρvzΔxΔy|z). In a similar manner, expressions for the rate of mass leaving from the control volume may be written.
Thus, the conservation of mass leads to the following expressions
Dividing the by the volume ΔxΔyΔz, we obtain
Note that each term in has the unit of mass per unit volume per unit time. Now, taking the limits Δx→0, Δy→0 and Δz→0, we get
and using the definition of derivative, we finally obtain
Equation (8.4) is applicable to each point of the fluid. Rearranging the terms, we get the equation of continuity, may be written as given below.
We need not to derive the equation of continuity again and again in other coordinate system (that is, spherical or cylindrical). The idea is to rewrite Equation (8.5) in vector and tensor form. Once it is written in this form, the same equation may be applied to other coordinate system as well. Thus, the Equation may be rewritten in vector and tensor form as shown below.
Vector and tensor analysis of cylindrical and spherical coordinate systems is not done here, and can be looked up elsewhere. Thus, the final expressions in cylindrical and spherical coordinates are given as below.
The second term in Equation may be broken into two parts as shown below. Partial derivative present in the Equation (8.6) can be converted into substantial derivative using vector and tensor identities.
In some cases, the fluid may be incompressible, i.e. density ρ is a constant with time as well as space coordinates. For example, water may be assumed as an incompressible fluid under isothermal conditions. In fact, all liquids may be assumed as incompressible fluids under isothermal conditions. For this special case, the equation of continuity may be further simplified as shown below

B) When a viscous fluid flows along a fixed impermeable wall, or past the rigid surface of an immersed body, an essential condition is that the velocity at any point on the wall or other fixed surface is zero. The extent to which this condition modifies the general character of the flow depends upon the value of the viscosity. If the body is of streamlined shape and if the viscosity is small without being negligible, the modifying effect appears to be confined within narrow regions adjacent to the solid surfaces; these are called boundary layers. Within such layers the fluid velocity changes rapidly from zero to its main-stream value, and this may imply a steep gradient of shearing stress; as a consequence, not all the viscous terms in the equation of motion will be negligible, even though the viscosity, which they contain as a factor, is itself very small. A more precise criterion for the existence of a well-defined laminar boundary layer is that the Reynolds number should be large, though not so large as to imply a breakdown of the laminar flow. F luid flow in circular and noncircular pipes is commonly encountered in practice. The hot and cold water that we use in our homes is pumped through pipes. Water in a city is distributed by extensive piping networks. Oil and natural gas are transported hundreds of miles by large pipelines. Blood is carried throughout our bodies by arteries and veins. The cooling water in an engine is transported by hoses to the pipes in the radiator where it is cooled as it flows. Thermal energy in a hydronic space heating system is transferred to the circulating water in the boiler, and then it is transported to the desired locations through pipes. Fluid flow is classified as external and internal, depending on whether the fluid is forced to flow over a surface or in a conduit. Internal and external flows exhibit very different characteristics. In this chapter we consider internal flow where the conduit is completely filled with the fluid, and flow is driven primarily by a pressure difference. This should not be confused with open-channel flow where the conduit is partially filled by the fluid and thus the flow is partially bounded by solid surfaces, as in an irrigation ditch, and flow is driven by gravity alone. We start this chapter with a general physical description of internal flow and the velocity boundary layer. We continue with a discussion of the dimensionless Reynolds number and its physical significance. We then discuss the characteristics of flow inside pipes and introduce the pressure drop correlations associated with it for both laminar and turbulent flows. Then we present the minor losses and determine the pressure drop and pumping power requirements for real-world piping systems. Finally, we present an overview of flow measurement devices.

Transfer of thermal energy by the movement of fluid and, as a consequence, such transfer is dependent on the nature of the flow. Heat transfer by convection may occur in a moving fluid from one region to another or to a solid surface, which can be in the form of a duct, in which the fluid flows or over which the fluid flows. Convective heat transfer may take place in boundary layers, that is, to or from the flow over a surface in the form of a boundary layer, and within ducts where the flow may be boundary-layer-like or fully-developed. It may also occur in flows which are more complicated, such as those which are separated, for example, in the aft region of a cylinder in cross-flow or in the vicinity of a backward-facing step. The flow may give rise to convective heat transfer where it is driven by a pump and is referred to as forced convection, or arise as a consequence of temperature gradients and buoyancy, referred to as natural or free convection. Examples are given later in this Section and are shown in to facilitate introduction to terminology and concepts.

4

Velocity and temperature profiles in boundary-layer and separated flows.

The boundary layer on the flat surface of Figure 1 has the usual variation of velocity from zero on the surface to a maximum in the free-stream. In this case, the surface is assumed to be at a higher temperature than the free-stream and the finite gradient at the wall confirms the heat transfer from the surface to the flow. It is also possible to have zero temperature gradient at the wall so there is no heat transfer to or from the surface but heat transfer within the flow. If the flow is laminar, heat transfer from the surface is given by the Fourier flux law, that is:

where q represents the rate of heat transfer per unit surface area, λ is the thermal conductivity, T is the temperature and y is distance measured from the surface. The same expression applies to any region of the flow and also in the case of the adiabatic wall where zero temperature gradient implies zero heat transfer. It should be noted that the surface can be horizontal as shown, with air flow driven by a fan or a liquid flow by a pump, and that it can equally be vertical, with buoyancy providing the driving force for the flow. In the latter case, the free-stream velocity would be zero so that the corresponding profile would have zero values at the wall and far from the wall.

The backward-facing step of results in a more complicated flow and several boundary layers can be identified within the flow as a consequence of separation and reattachment. The details of flows of this type are not well-understood so it is difficult to identify the characteristics of the boundary layers and it can be imagined that the shapes of the velocity and temperature profiles— and therefore of the local heat transfer within the fluid and to the wall—will vary considerably from one location to another. It is known, for example, that the rate of heat transfer can become high at the location of reattachment of the upstream flow on to the surface of the step, as is also the case at the leading edge of a cylinder in cross-flow, but the detailed mechanisms remain incompletely understood and research continues.

It is well known that even comparatively simple geometrical configurations, such as those of can give rise to heat transfer rates which vary considerably depending on the nature of the flow and of the surface. With laminar flows, heat transfer to or from the wall varies with distance from the leading edge of a boundary layer. Turbulent flows can give rise to heat transfer rates which are much larger than those of laminar flows, and are caused by the manner in which the turbulent fluctuations increase mixing; they also affect the heat transfer to and from the surface, especially where the free-stream fluid is able to penetrate to the wall even for short periods of time. The nature of the surface, for example the degree or type of roughness, usually affects heat transfer to or from it, and in some circumstances to a large extent. It is convenient, therefore, to represent the heat transfer at the wall by the expression

where again represents the rate of heat transfer from the wall, this time over unit area of surface; the temperature difference refers to that between the wall and the free stream; and α is the “heat transfer coefficient” which is a characteristic of the flow and of the surface. The two temperatures can vary with x-distance and it can be difficult to identify a free-stream temperature in some complex flows. Typical values of α are shown in Table 1, from which it can be seen that increases in velocity generally result in increases in heat transfer coefficient, so that α is smallest in natural convection and increases to 100 and more on flat surfaces with air velocities greater than around 50 m/s. The heat transfer coefficient is considerably greater with liquid flows and greater again with two-phase flows.