CONSERVATION OF MASS Control Volumes By: Bashir Momodu

INTRODUCTION The concept of defining a volume of fluid in a flow field surrounded by an imaginary enclosing surface is extremely useful in fluid mechanics. Such a volume is called a control volume and its surrounding, a controlled surface.

Mass Conservation Control volumes and surfaces can be useful in the application of the laws of conservation of mass, momentum, and energy to the fluid flow.

DEFINITION The conservation of mass in a fluid flow requires the accumulation of mass inside a control volume is accounted for by the net flow of mass across the control surface because mass can neither be destroyed or created within the control volume.

Real Life Comparison “Mass conservation can be compared to a bank account. The increase in value of the principal is equal to the deposit plus interest payments minus the withdrawals plus service charges. Flow into the control volume is equivalent to a credit, while outflow is a debit to the account.”

Fluid System and Control Volume The concept of a free body diagram as used in statics of rigid bodies and in fluid statics is usually inadequate for the analysis of moving fluid. The concept of fluid system and a control volume is preferred for these type of analyses. A fluid system refers to specific mass of fluid within the boundaries defined by the closed surface. The shape of such systems and boundaries may change with time. As a fluid moves and deforms, so does the system containing the fluid. However, in contrast control volume refers to a fixed position in space which does not change shape or move.

Control Volume for Steady Flow Diagrams

Diagram Discussion Considering the linear momentum of the fluid system and control volume defined within the stream tubes of figure 6.1, the fixed control volume lies between sections 1 & 2. The moving fluid system consists of the fluid mass contained at time (t) in the control volume. At a short time interval, Δt, we assumes that the fluid moves a short distance Δs1in section 1 and Δs2 at section 2. d(mV)s = d(mV)cv + d(mV)out cv - d(mV)in cv dt dt dt dt -subscript s denotes the moving fluid system -subscript cv denotes the fixed control volume

Unsteady Flow The following equation is the momentum equation for unsteady flow: Σ F = d(mV)cv + d(mV)out cv - d(mV)in cv dt dt dt On the right side of this equation, the force time represents the rate of change or the accumulation of momentum within the control volume. The second and third terms respectively represent the rate at which momentum enters and leaves the control volume. The above equation states that the result of a force acting on a fluid mass is equal to the rate of change of momentum of the fluid mass.

Steady Flow In steady flow cases, the condition between the control volume does not change, so d(mV)cv/dt = 0 and the momentum equation reduces to ΣF = d(mV)out cv - d(mV)in cv dt dt For steady flow, the net force on the fluid mass is equal to the net rate of outflow of momentum across the control surface. If it is specified that the velocity is constant where it cuts across the control surface, and assuming there is steady flow: d(mV)1 = dm1*V1=m1V1=ρ1Q1V1 dt dt and the same relationships hold for section 2. Therefore for steady flow: ΣF = m2V2 -m1V1= ρ2Q2V2 - ρ1Q1V1 The direction of the sum of F must be the same as that of the velocity change, delta V. Note that the ΣF represents the sum of all forces acting on the fluid mass and the control volume, including gravity forces, shared forces, and the pressure forces.

Discussion Momentum principle is particularly important in flow problems when forces need to be determined. Such forces are encountered whenever the velocity of a stream changes in magnitude or direction. By the law of action and reaction, the fluid exerts an equal and opposite force on the body producing the change. Momentum principle is derived from Newton’s Second Law of Motion. It does not matter if the flow is compressible or incompressible, real (friction) or ideal (frictionless), steady or unsteady and the equation is not limited to flow along a stream line.

Discussion, con’t. In applying the energy equation to real fluid, the energy loss must be computed. This is not a problem with momentum analysis, since it can be expressed from Newton’s Second Law. This law states the sum of the forces (F) on a body of fluid or system(S) is equal to the rate of change of linear momentum (mV) of that body or system. All the previously developed momentum equations in this write-up were all derived from Newton’s Second Law: