# Calculation Methods - Pipes

It is not the intention in this section to provide detailed derivations or detailed explanations. You are referred to the many texts in the Reference Literature chapter to fill in the detail. Fluid Mechanics is based on the fundamental principles of conservation of mass, energy and momentum. When a fluid flows through a pipe a pressure loss is experienced. For all phase states the total pressure loss can be expressed mathematically as: ΔPtot = (ρ 1U1+ z1 ρ 1g + p1 + 0.5N1 ρ 1v1 2) - (ρ 2U2+ z2 ρ 2g + p2 + 0.5N2 ρ 2v2 2) Equation 1 FluidFlow works internally with total pressure losses, the calculation methods used within FluidFlow do not assume that energy is dissipated and so there is a need to keep track of total energy. In compressible, non isothermal liquid flow and two phase flow energy is transformed between the various forms of Equation 1, which means that a continuous energy balance is required throughout the system. Incompressible Newtonian Liquid For flow of an incompressible Newtonian liquid in a pipe, when the friction model is set to Moody (default setting), the friction loss is calculated according to the Darcy-Weisbach equation: (for liquid flow in a horizontal pipe, with no temperature change, and neglecting compressibility effects the friction loss and total pressure loss are equal). ΔPfric = 0.5fDLρv2/D If the friction model is set to Shell-MIT, friction loss is calculated according to the Shell-MIT equations. The solution approach uses a modified Reynolds Number which is a multiple of the normal Reynolds Number. Using the modified Reynolds Number, the friction factor is calculated depending on whether the flow is laminar or turbulent. Rm = Reynolds Number/7742. f = 0.00207 / Rm where the Reynolds Number is in the laminar region, i.e., lower than 2100. f = 0.0018 + 0.00662 * (1 / Rm)0.355 where the Reynolds Number is in the turbulent region, i.e., higher than 2100 Pm = 6.2191x1010 * f * Sg * Q2 / D5 (SI Units). where: f is the friction factor. Q is the flow rate (m3/h). Sg is the specific gravity of the fluid. D is the internal pipe diameter (mm). Compressible Fluid (Gas or Vapor) For flow of a compressible fluid (gas or vapor) in a pipe the Darcy-Weisbach equation is not applicable because both density and velocity change as the fluid flows down the pipe. For gas flow friction loss is never equal to total loss, because velocities and densities in and out of the pipe are always different. An approach often used in the literature is to assume ideal gas laws so that analytical equations can be derived. In many industrial processes significant departures from the ideal gas laws exist. Rather than make simplifying assumptions, FluidFlow uses a calculation procedure that solves the conservation equations together with an equation of state for small pipe increments. This means that we cannot provide an analytical expression for pipe friction loss. To help understand the solution process consider the following description: 1. A pipe increment is selected based on a small change in fluid static density ρ 1 and ρ 2. The length of this increment is not yet known. 2. The following upstream properties are calculated: Velocity, v1 = G/ρ 1 Static temperature, T1 = T0 1 - 0.5v1 2/Cp1 Static pressure P1 = P0 1 - 0.5ρ 1v1 2 Upstream stagnation enthalpy is obtained from the equation of state, knowing the upstream stagnation temperature, T0 and pressure P0. 3. Downstream stagnation enthalpy is calculated (any heat transfer is included at this stage). From the downstream stagnation enthalpy the downstream static temperature, T2 is calculated. 4. From the downstream static temperature, T2 and density, ρ 2 the equation of state is back solved to provide the downstream static pressure, P2. FluidFlow can use either Peng Robinson, Benedict Webb Rubin, or Lee Kesler equations of state. 5. An incremental form of the momentum equation (equation 1) is used to calculate the length of this pipe segment: ΔLsegment = (p1 - p2 - 2Dρ AvG2/(1/ρ 1 - 1/ρ 2)) / (fDG2) 6. Steps 1 to 5 are repeated until the end of the pipe is reached. The incremental length step size therefore shortens as the calculation moves down the pipe. For the last segment, which will never be the exact length required, we use interpolating functions based on results from previous segments. This basis of this method was originally developed for calculating high speed gas flows in pressure relief lines [6]. The Joule–Thomson Coefficient The Joule–Thomson effect describes the increase or decrease in the temperature of a gas when allowed to expand. The change in temperature (T) with a decrease of pressure (P) at constant enthalpy (H) is referred to as the Joule–Thomson coefficient (μJT). FluidFlow uses the Peng Robinson EOS to evaluate the Joule–Thomson coefficient which is expressed as: μJT = (∂T/∂P)H. The value of μJT is typically expressed in °C/bar and is dependent on the physical properties of the gas, as well as the temperature and pressure of the gas before expansion. Non-Newtonian or Non-Settling Slurry For flow a non-Newtonian or non-settling slurry in a pipe, the friction floss is calculated from the Darcy - Weisback equation: However, estimation of friction factor is not the same with how Newtonian flow systems are determined. Instead, it is based on the rheology data used to describe the viscosity as a function of shear rate. For flow of settling slurries in a pipe, the concept of additive pressure drop is applied. That is, the total friction loss for a slurry system is equal to the sum of the friction loss from the fluid alone and the additional pressure drop due to the presence of solid particles. This is represented by the following equation: Wherein the variable "phi" is determined using any of these empirical relationships: •Durand •Wasp •Wilson, Addie, Sellgren and Clift (WASC) •Sellgren, Wilson Four-Component Model •Liu Dezhong •Vsm •V50 •4CM Two Phase For flow of two phase mixtures in a pipe the friction loss can be calculated by a number of ways. The actual method used is set from the Calculation Options Dialog. The calculation of frictional pressure loss for two phase gas-liquid flow is complex. The coexistent flow of two phases complicates the theoretical and empirical approaches available, this means that a complete analytical solution is not possible. After 60 years of extensive research it is rare to find two correlations with exact predictions. In an effort to overcome these shortcomings FluidFlow provides six correlations that represent some of the most successful approaches to this complex problem. The pressure gradient (ΔP/L) for two phase flow is not constant but varies along the pipe as a function of temperature and pressure. This means that the pressure drop must be calculated by integrating the pressure gradient along the pipe. The calculation approach used by FluidFlow is similar to that used in compressible flow calculation. The following steps are made: 1. A pipe increment is selected based on a small change in pressure P1 and P2. The length of this increment is not yet known. 2. Determine the upstream temperature, pressure, quality and physical properties. Physical properties for each phase and the mixture physical properties are needed. 3. Make a flash calculation to determine the downstream quality. X2 = X1 + Cp(T1-Tb)/Hv 4. From the downstream properties determine the flow regime and then determine the incremental length of this segment. Determination of the incremental length depends on the calculation method used. The equations used are: Friedel: ΔLsegment = (P1 - P2)D5ρL/(CCfLΦm2) where CC is a constant ρL is the average liquid density over the increment fL is the liquid phase friction factor Φ is the Friedel correction factor. See [10] for further explanations. m is the mass flow Beggs Brill: ΔLsegment = (P1 - P2)D5ρmix/(CCfTPm2) where CC is a constant ρmix is the two phase mixture density over the increment. See [3] for further explanations. fTP is the two phase friction factor. See [3] for further explanations. m is the mass flow Drift Flux: ΔLsegment = (P1 - P2)D5ρmix/(CCfTPm2) where CC is a constant ρmix is the two phase mixture density over the increment. See [11] for further explanations. fTP is the two phase friction factor. See [11] for further explanations. m is the mass flow Muller Steinberg Heck: ΔLsegment = (P1 - P2) / (ΔPL+2(ΔPG-ΔPL)Xav(1-Xav)0.3333) + ΔPGXav 3) where ΔPG is the liquid phase pressure loss over the increment. See [12] for further explanations. ΔPL is the liquid phase pressure loss over the increment. See [12] for further explanations. Xav is the average vapor quality for the segment. For both Lockhart Martinelli and Chisholm Baroczy correlations the equations are identical to the Beggs Brill equation with the addition of a two phase multiplier. See [10] for further explanations. 5. Steps 1 to 4 are repeated until the end of the pipe is reached. The incremental length step size therefore shortens as the calculation moves down the pipe. For the last segment, which will never be the exact length required, we use interpolating functions based on results from previous segments. Pipe Heat Transfer For heat transfer through a pipe wall with or without insulation, FluidFlow uses an iterative solution to solve for an incremental pipe length: Q/L = Σ ΔT / (Ri + Rw + RLAG + Ro) where: Ri: is the inside film resistance and Ri = 1 / (Hi π Di) where Hi is the inside film coefficient of heat transfer.. Rw is the wall resistance and Rw = Xw / (Kw π Dw) where Xw is the pipe wall thickness, Kw is the thermal conductivity of the pipe wall and Dw is the log mean diameter of the pipe wall. RLAG: is the lagging/insulation resistance and RLAG = XL / (KL π DL) where XL is the insulation thickness, KL is the thermal conductivity of the insulation material and DL is the log mean diameter of the insulation. Ro: is the outside film resistance and Ro = 1 / (Ho π Do) where Ho is the outside film coefficient of heat transfer. The mechanism of heat transfer by radiation is superimposed on that of heat transfer by convection. FluidFlow therefore calculates the individual convection and radiation coefficients which are then summed to give a combined outside/surface film coefficient, i.e., Ho = hrad + hconv. Outside Film Radiation Heat Transfer The outside film radiation heat transfer coefficient is calculated from the equation: hrad = (σ)(e)(Tfluid)4 - (TAmbient)4) / (Tfluid - TAmbient). where: hrad is the radiation heat transfer coefficient (W/m2 K). σ is the Stefan-Boltzman constant (5.67 x 10-8 W/m2 K4). e is the surface emissivity. Tfluid is the temperature of the flowing fluid (K). TAmbient is the local ambient air temperature (K). Outside Film Convection Heat Transfer Coefficient The outside film convection heat transfer coefficient can be calculated from using either the equations based detailed in ASTM Standard C680 or by using the equations detailed in the VDI Wärmeatlas. You can choose your preferred method by selecting: Options | Calculation | Global Settings. ASTM Standard C680 When applying this method for above-ground pipelines, the outside film convection heat transfer coefficient is calculated as follows hconv = C (1/do) 0.2 x (1/Tavg) 0.181 x (Ts - TAmb) 0.266 x (1 + 1.277 (Wind)) 0.5. where: hconv is the convective heat transfer coefficient (W/m2 K). C is the constant depending of shape and heat flow direction (1.016 for horizontal pipes and 1.235 for vertical pipelines). do is the outer diameter (including insulation if present). Tavg is the average air film temperature, average of outside and ambient temperatures (oC). Ts is the outside surface temperature (oC). TAmb is the ambient temperature (oC). Wind is the wind speed (mph). VDI Wärmeatlas When applying this method for above-ground pipelines, the outside film convection heat transfer coefficient is calculated as follows: U = 1 / [Di/D*h in] + [Di*ln(Do/D)/2*k Pipe] + [Di*ln(Di/Do)/2*k Insulation] + [1/h air] where: Di is the outside diameter of the pipe insulation (if included in the design). D is the inside diameter of the pipe. hin is the inside surface heat transfer coefficient for the pipe. Do is the outside diameter of the pipe. kPipe is the thermal conductivity of the pipe. kInsulation is the thermal conductivity of the insulation. hair is the heat transfer coefficient of air at the outside surface of the pipe. Further information on the terms of the above equation is available in the VDI Wärmeatlas. Buried Pipe Heat Transfer FluidFlow splits each pipe into segment lengths and calculates the overall heat loss coefficient from the equation: Uoverall = 1 / Rt where Rt is the sum of the resistances to heat transfer which are: Rt = Ri + Rw + Rcoating + Rinsulation + Rbackfll + Rsoil. The resistance terms outlined in the above equation refer to the resistance of the pipe inside film, pipe wall, pipe coating, pipe insulation, backfill and soil respectively. Then for each segment: t2 = t1 - U A ΔT / (w Cp) where t1 is initial temperature, U is overall coefficient and is 1/Rt, A is heat transfer area, ΔT is temperature driving force (pipe outside wall temperature - ambient soil temperature), w is mass flow through the pipe and Cp is the fluid specific heat capacity.