When a single-phase fluid flows in a hydraulic system which generally consists of elements such as constant-cross-section conduits; throttles (valves, diaphragms, filters, etc.); variable-cross-section channels, e.g., nozzles and reducers; bends, tanks, basins, heat exchangers, and so on, the fluid pressure changes both as a result of conversion of the kinetic energy to potential energy and vica versa and an irreversible conversion of part of the mechanical energy of the fluid to heat due to viscous friction forces in the flow. In actual fact, static pressure is most often constant over the flow cross section and changes only longitudinally, while the fluid flow can be considered as one-dimensional and characterized in each reference cross section by the mean parameters, i.e., the mean mass velocity , where is the mass flow rate of the fluid, S the flow cross-section area, and by the bulk velocity , where ρ(p, hB) is the bulk density of the fluid determined depending on the local static pressure p and the bulk enthalpy . Along with the fluid static pressure, i.e., the pressure which an instrument moving together with the flow would indicate, hydromechanics often deals with the total pressure p0 ( ) that is a sum of static pressure p, dynamic pressure , and hydrostatic pressure ρBgz , where g is the gravitational acceleration, z the height of the flow axis above the ground reckoned from some fixed zero mark. The pressure difference in the initial (1) and end (2) reference flow cross sections is referred to as the pressure drop or hydraulic resistance and denoted as Δp = p1 − p2 or Δp0 = p01 − p02. Frequently Δp ≠ Δp0 , therefore, one should mention without fail which pressures we are having to do with. In pipes and channels the length derivative dp/dx is called the pressure gradient and dp0/dx the hydraulic slope. Determination of Δp and Δp0 under given flow conditions is the most important problem of hydrodynamic analysis. Its results are used to find the parameters of a pump, a fan, or a compressor responsible for fluid motion or the draught needed in circuits with natural circulation.

Pressure drops are determined using the equations of continuity, energy, and fluid motion. We consider some of the solutions that are valid for one-dimensional or quasi-one-dimensional flows with a constant flow rate. The continuity equation for this flow has the form: ρBuBS = const or in a differential form

(1) Using the energy equation for the stream

(2) (3) where is the rate of heat input per unit time into a flow of a fluid flowing at kg/s, u the specific internal energy, v = 1/ρ the specific volume, WT the useful work, WF = work againt friction forces in fluid fully converted to friction heat , yields, for a flow doing no useful work, dWT = 0, the relationships

(a) for the flow of incompressible fluid for which v is assumed constant and dv to be zero. The flows of dropping liquids, such as water and petroleum products, and gases at Δv/v < 5% (if T const at Dp/p < 5%) are close to this case. The heat put into or removed at d ≠ 0 is mainly expended for changing the internal energy of fluid and does not affect the mechanical energy balance (3). Integrating (3) at p dV = 0 and averaging the integral quantities over the flow yields

(4) Equation (4) is said to be the Bernoulli equation. The value > 0 due to the work of friction in the flow is known as pressure loss or friction loss (hydraulic resistance). The first term on the right-hand side of Eq. 4 is the dynamic pressure difference, the second term is the hydrostatic pressure difference. The coefficient , where u is the local longitudinal component of the fluid velocity, takes into account the actual non-uniformity of the velocity profile and, consequently, that of kinetic energy distribution over the channel cross section and is referred to as the Coriolis coefficient. For fully developed turbulent flows in pipes α = 1.06-1.07, for fully developed laminar flows α = 2.0 in a circular pipe and α = 1.54 in plane and annular pipes for D2/D1 → 1. Equation (4) readily yields Δp0 = Δpf in uniform (u2 = u1, α1 = α2) or homogeneous (α1 = α2 = 1) flows of incompressible fluid, i.e., the total pressure is reduced by the value of hydraulic losses.

(b) for flows of compressible fluids: v = v(P, T) (gases at Δv/ v > 5%)

(5) where is the effective density of fluid depending on the character of thermodynamic process, . An exact solution (5) can be obtained only for some particular cases. Thus, for an isothermal gas flow obeying the Clapeyron equation of state

(6') where ε = p2/p1 is the backpressure ratio. When the gas flow is polytropic,

(6'') where n is the polytropic exponent. Forthe adiabatic process n g = cp/cv, where γ is the isentropic exponent (see Adiabatic conditions.) As follows from Eqs. (4) and (5), in cases where the hydrostatic pressure difference and friction loss are low in relation to the dynamic head the static pressure drops, Δp > 0, as a result of fluid acceleration, (uB2 > uB1); in the case of stagnation (uB2 < uB1), the static pressure increases (is recovered) Δp < 0.

Conversion of the pressure energy to kinetic energy of the flow and vica versa is performed using channels (nozzles) of variable cross section (Figure 1) such as contractions (duB > 0). These channels are as a rule short and therefore, the hydrostatic pressure difference can be neglected and the flow can be considered adiabatic. Solving Eqs. 4 and 5 together with equation of continuity (1) for these conditions, we derive the relations Figure 1. Flow through nozzles.

(a) for incompressible fluid

(7) in particular cases, for instance, when the fluid outflows from the reservoir (uB1/uB2; and S1/S2 → 0) or if a deep stagnation occurs in a diffuser such that uB1/uB2 and S1/S2 → 0, Eq. (7) can be simplified. This equation is used to calculate the Venturi tubes (Fig. 1) employed for measuring the flow rate,

(b) in the case of compressible gas (p/ργ = const) for a stream tube without friction the differential equations

(8) are valid, where Ma = u/usound is the Mach number, is the local velocity of sound. In the subsonic flow (Ma < 1) the flow velocity increases (du > 0) at dS < 0, i.e., in contracting channels, while flow stagnation occurs (du < 0) in expanding channels (dS > 0). In the supersonic flow (Ma > 0) the ratio du/dS changes sign. Therefore, in order to initiate supersonic flow a Laval nozzle is used that is a combination of the subsonic and supersonic contractors (Figure 1). Equation (8) implies the relations

(9) (10) where Ma1 = u/usound is the Mach number in entrance section 1. Acceleration of the subsonic flow in a smoothly shaped contracting nozzle is possible up to

(11) At ε = ε* the local sound velocity is achieved in the exit section and the critical outflow regime sets in under which the maximum possible gas flow rate is reached that is retained even at ε < ε*.

Determination of pressure losses Δpf in many cases is a most important element in hydrodynamic analysis (see Hydraulic Resistance).

Hydraulic loss in liquid filtration through a bulk porous layer can be determined by the formula

(12) where is the mass flow rate per unit area of the layer, Π the layer porosity, H the layer height, is the arbitrary diameter of the medium pores (α1 is the viscous coefficient of resistance in the linear filtration region) (see also Porous Medium). Using Figure 2, the ξp values can be found as a function of the Reynolds number Rep = , α1 = 10.5 ad−2 P −4.4, d is the mean diameter of particles. The approximate a and C values in Figure 2 for bulk layers are presented in TABLE1. Figure 2. Resistance factor for flow in a porous medium.

Table 1. The pressure drop can be also determined using the equation of fluid motion (the momentum equations). This equation in the differential form is known as the Navier-Stokes equation and gained currency in theoretical solution to fluid dynamics problems. This approach is advantageous in analyzing pressure losses of fluids with variable density which flow in the channels with heat input and removal when there occurs an interconversion of thermal and mechanical energy which hinders analyzing Eqs. (2) and (3).

An integral equation of motion for fluid flow in the channel with impermeable walls depicted in Figure 3 is of the form

(13) where P1 and P2 are the total pressure forces at the inlet and outlet of the channel as shown in Figure 3. The vector polygon of forces corresponding to Eq. (13) is also shown in Figure 3. If the pressure can be assumed constant over the flow section, then P = pS. The coefficient β in Eq. 13 is referred to as the momentum coefficient or the Boussinesq coefficient and allows for a nonuniform distribution of momentum over the flow section with ingomogeneous distribution of fluid velocity and density. For fully developed turbulent flows in pipes β = 1.02-1.04 and for stabilized laminar flow in circular pipes β = 1.33, in plane and annular pipes at D2/ D1 → 1, β = 1.2. The force R, i.e., the reaction of the channel walls, is a resultant of forces acting on the lateral surface of the flow, including the friction force on the wall. The same force but opposite in direction acts from the side of fluid on the channel section under consideration. Figure 3.

In the case of a straight uniform section tube often encountered in practice Eq. (13) takes the form

(14) Using equation of continuity (1) for S = const yields

(15) where is the result of acceleration of the flow, the friction drag (τw is the local tangential stress on the wall averaged over perimeter Π), is the hydrostatic pressure difference. Here φ is the angle formed by the vectors of velocity and gravitational acceleration, the fluid density averaged over cross section, i.e., commonly and β = 1. In the case of constant β and ρ

(16) i.e., the equation of motion coincides with the Bernoulli equation (5) and, hence, Δpf = Δpτ. Δp, as well as Δpf, in tubes and in channels are determined by the Darcy-Weisbach formula

(17) (18) where and f are the average and the local friction factors respectively. We note that Δpf, that is an energy characteristic of friction, and the dynamic characteristic Δpτ with interconversion of kinetic, potential, and thermal energies occurring in the flow have, generally, different values in the same way as the coefficients λ and f do. The components Δpw and Δpg can be expressed in Eq. (15) by relations similar to (17) and (18), fw and fg are respectively the coefficients of internal and hydrostatic resistance. In case fi, ρb, and ub substantially vary over the tube length the differences Δpi are determined by integration of equations of the type of Eq. (18). In the case of flow in tubes of a fluid with variable physical properties Δpτ and f are experimentally determined often using a simplified version of Eq. (14) in which it is assumed that β2 = β1 = const = 1, βρ = 1,

(14') This equation complies with a one-dimensional, or homogeneous, flow model, and the , , f0, etc. values are said to be one-dimensional. Actual values may differ from them (this can occur, e.g., in transcritical states of fluid), therefore for, in engineering calculations one should use an equation of motion which was applied to obtain f, fw, fg. As an example we refer to Eq. (14) for a flow of heated gas in a tube derived by Guggenheim. Writing it in a differential form and multiplying its both sides by p, we have

(19') Since for gas p/ρ = RT, then, discarding the intermediate manipulations, we obtain

(20') for f = const. Under flow regimes far from critical (Ma << 1) the values of T1,T2, and the pipe-averaged temperature can be found from the thermal balance and the last term in brackets can be neglected in the first approximation.

Under the conditions when Eq. (16) is valid, f can be determined by the Poiseuille, Blasius, and Colebrook-White formulas (see Hydraulic resistance). In the case of fluid heating or cooling when its viscosity and density cannot be assumed constant over the cross section, f can be evaluated by formulas for liquids for which only viscosity variation is essential

(21') When the flow is laminar, the mean length coefficient is determined over the length l, by the Poiseuille formula at the Reynolds number Re0 at the pipe inlet, where at Pe dh/l < 1500, C = 2.3, m = – 0.3, at Pe dh/1 > 1500, C = 0.535, m = – 0.1. The values of η0 are taken for the pipe inlet, ηw for the wall temperature. If the flow is turbulent, η0 = ηb, , and n = 0.33 for heating and n = 0.24 for cooling of fluid. For gases

(22') If the gas in the laminar flow is heated, n = –1, if cooled, n = 0. For fluid heated in the transcritical region of states n = 0.4.

In analyzing pressure loss in channels with permeable walls, e.g., heaters and manifolds, thermal pipes in evaporation and condensation zones, we use an equation of variable mass flow which for incompressible fluid takes the form

(23') Here θ is the projection of the velocity vector of separated or added fluid masses in the direction of the main flow, and β the Boussinesq coefficient which plays an active role in this equation.