Steam

Equations

SC-GCM-118 CM Issue 4 © Copyright 2007 Spirax-Sarco Limited

Block 16 Equations

Module 16.1

Module 16.1 Equations

The Steam and Condensate Loop

16.1.1

Equations

Block 16 Equations

Module 16.1

Equations Block 1: Introduction Equation number

Equation

There are no equations in Block 1

Block 2: Steam engineering principles and heat transfer Equation number

2.1.1

Equation

Thermodynamic temperature

Density of a material

2.1.2

Where: r = Density (kg /m³) m = Mass (kg) V = Volume (m³) vg = Specific volume (m³ /kg) Specific gravity of a material

2.1.3 Energy transfer equation

2.1.4

Where: Q = Quantity of energy (kJ) m = Mass of the substance (kg) cp = Specific heat capacity of the substance (kJ /kg °C ) DT= Temperature rise of the substance (°C) Change in entropy

2.1.5 Change in specific entropy 2.1.6 Total enthalpy of saturated steam 2.2.1

16.1.2

Where: hg = Total enthalpy of saturated steam (Total heat) (kJ/kg) hf = Liquid enthalpy (Sensible heat) (kJ /kg) hfg = Enthalpy of evaporation (Latent heat) (kJ /kg) The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Enthalpy of evaporation of wet steam 2.2.2

Where: hfg = Enthalpy of evaporation (Latent heat) (kJ /kg) c = Dryness fraction Total enthalpy of wet steam

2.2.3

Where: hf = Liquid enthalpy (Sensible heat) (kJ /kg) hfg = Enthalpy of evaporation (Latent heat) (kJ /kg) c = Dryness fraction Specific volume of wet steam

2.2.4

Where: vg = Specific volume of dry steam at same pressure c = Dryness fraction Flash steam produced from hot water and condensate

2.2.5

Where: P1 = Initial pressure P2 = Final pressure hf = Liquid enthalpy (kJ /kg) hfg = Enthalpy of evaporation (kJ /kg) Carnot efficiency

2.3.1

Where: Ti = Temperature at turbine inlet (K) Te = Temperature at turbine exhaust (K) Rankine efficiency

2.3.2

Where: Hi = Heat at turbine inlet (kJ /kg) He = Heat at turbine exhaust (kJ /kg) he = Sensible heat in condensate (kJ /kg)

The Steam and Condensate Loop

16.1.3

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Dalton’s law of partial pressures 2.4.1

Heat transfer by conduction through a layer (Fourier’s law)

2.5.1

Where: Q = Heat transferred per unit time (W) k = Thermal conductivity of the material (W/m K or W/m°C) A = Heat transfer area (m²) DT = Temperature difference across the material (K or °C) ƒ = Material thickness (m) Heat transfer by convection (Newton’s law of cooling)

2.5.2

Where: Q = Heat transferred per unit time (W) h = Convective heat transfer coefficient of the process (W/m² °C) A = Heat transfer area of the surface (m²) DT = Temperature difference between the surface and the bulk fluid (K or °C) General heat transfer

2.5.3

Where: Q = Heat transferred per unit time (W) U = Overall heat transfer coefficient (W/m² °C) A = Heat transfer area (m²) DT = Temperature difference between the primary and secondary fluid (K or °C) Note: Q will be a mean heat transfer rate (QM) if DT is a mean temperature difference (DTLM or DTAM). Arithmetic mean temperature difference (AMTD or DTAM)

2.5.4

16.1.4

Where: Ts = Steam temperature (°C) T1 = Secondary fluid in temperature (°C) T2 = Secondary fluid out temperature (°C)

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Log mean temperature difference (LMTD or DTLM)

2.5.5

Where: Ts = Steam temperature (°C) T1 = Secondary fluid in temperature (°C) T2 = Secondary fluid out temperature (°C) ln = A mathematical function known as ‘natural logarithm’ Rate of heat transfer across a barrier knowing the thickness and conductivity

2.5.6

Where: Q = Heat transferred per unit time (W ) A

= Heat transfer area (m²)

DT = Temperature difference across the barrier (°C) ƒ/ = Barrier thickness / material thermal conductivity k Rate of heat transfer across a barrier knowing thermal resistance

2.5.7

Where: Q = Heat transferred per unit time (W ) A = Heat transfer area (m²) DT = Temperature difference across the barrier (°C) R = Thermal resistance of the barrier (m2 °C / W) Resistivity from conductivity

2.5.8

Where: r = Thermal resistivity (m°C / W) k = Thermal conductivity (W / m°C) Thermal transmittance (heat transfer coefficient) from thermal resistance

2.5.9

Where: U = Thermal transmittance of the barrier (W / m2 °C) R = Thermal resistance of the barrier (m2 °C / W)

The Steam and Condensate Loop

16.1.5

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Thermal transmittance (U) from the individual thermal resistances

2.5.10

Where: R1 = Resistance of the air film R2 = Resistance of the condensate film R3 = Resistance of the scale film on the steam side R4 = Resistance of the of the metal wall R5 = Resistance of the scale film on the water side R6 = Resistance of the product film Thermal transmittance (U) from the individual thicknesses and conductivities

2.5.11

Energy requirement for a non-flow application (e.g. batch or tank)

2.6.1

Where: Q = Mean heat transfer rate (kW (kJ /s)) m = Mass of the fluid (kg) c p = Specific heat capacity of the fluid (kJ /kg °C) DT = Increase in fluid temperature (°C) t = Time for the heating process (seconds) Quantity of heat transferred by condensing steam

2.6.2

16.1.6

Where: Q = Quantity of heat (kJ) ms = Mass of steam (kg) hfg = Specific enthalpy of evaporation of steam (kJ /kg)

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Heat transfer of condensing steam 2.6.3

Where: Q = Mean heat transfer rate (kW or kJ /s) ms = Mean steam consumption (kg /s) hfg = Specific enthalpy of evaporation of steam (kJ /kg) Energy balance between steam and secondary fluid of a non-flow process

2.6.4

Where: ms = Mean steam consumption rate (kg /s) hfg = Specific enthalpy of evaporation of steam (kJ /kg) Q = Mean heat transfer rate (kW (kJ /s)) m = Mass of the secondary fluid (kg) c p = Specific heat capacity of the secondary fluid (kJ /kg °C) DT = Temperature rise of the secondary fluid (°C) t = Time for the heating process (seconds) Energy requirement for a flow-type application (e.g heat exchanger)

2.6.5

Where: Q = Mean heat transfer rate (kW) m = Mean secondary fluid flowrate (kg /s) c p = Specific heat capacity of the secondary fluid (kJ/kg K) or (kJ/kg °C) DT= Temperature rise of the secondary fluid (K or °C) Energy balance between steam and fluid of a flow-type application

2.6.6

Where: ms = Mean steam consumption rate (kg /s) hfg = Specific enthalpy of evaporation of steam (kJ /kg) Q = Mean heat transfer rate (kW (kJ /s)) m = Mass flowrate of the secondary fluid (kg /s) c p = Specific heat capacity of the secondary fluid (kJ /kg °C) DT = Temperature rise of the secondary fluid (°C) Mean steam consumption of a flow type application

2.6.7

Where: ms = Mean steam consumption rate (kg /s) m = Mass flowrate of the secondary fluid (kg /s) c p = Specific heat capacity of the secondary fluid (kJ /kg °C) DT = Temperature rise of the secondary fluid (°C) hfg = Specific enthalpy of evaporation of steam (kJ /kg)

The Steam and Condensate Loop

16.1.7

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Mean steam consumption of a flow type application

2.6.8

Where: ms = Mean steam consumption rate (kg /s) Q = Mean heat transfer rate (kW) hfg = Specific enthalpy of evaporation of steam (kJ /kg) To determine the required steam flowrate from a kW rating

2.8.1 To determine the steam flowrate for a steam injection process

2.11.1

Where: ms = Mean steam flowrate (kg /s) Q = Mean heat transfer rate (kW) hg = Specific total enthalpy of the steam upstream of the control valve (kJ /kg) T = Final temperature of the water c p = Specific heat capacity of the water (kJ /kg °C) Steam consumption to provide tank heat losses

2.11.2

Where: ms = Mean steam flowrate to provide the heat losses from the tank (kg /s) Q = Q(sides) + Q(surface) (kW) 2256.7 = Enthalpy of evaporation at atmospheric pressure (kJ / kg) Mass and heat balance for steam injection into a tank

2.11.3

16.1.8

Where: m = Initial mass of water in the tank (kg) ms = The mass of steam to be injected (kg) h1 = The heat in the water at the initial temperature (kJ /kg) h2 = The heat in the water at the final temperature (kJ /kg) hg = The total enthalpy of the steam upstream of the control valve (kJ /kg)

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Steam consumption by injection into a tank

2.11.4

Where: ms = The mass of steam to be injected (kg) m = Initial mass of water in the tank (kg) h1 = The heat in the water at the initial temperature (kJ /kg) h2 = The heat in the water at the final temperature (kJ /kg) hg = The total enthalpy of the steam upstream of the control valve (kJ /kg) Steam start-up load to bring steam pipework to operating temperature

2.12.1

Where: ms = Mean rate of condensation of steam (kg / h) W = Total weight of pipe plus flanges and fittings (kg) Ts = Steam temperature (°C) Tamb = Ambient temperature (°C) c p = Specific heat of pipe material (kJ / kg °C) hfg = Enthalpy of evaporation at operating pressure (kJ / kg) t = Time for warming up (minutes) Steam running load to keep a steam main at operating temperature

2.12.2

Where: ms = Rate of condensation (kg /h) Q = Heat emission rate (W/m) L = Effective length of pipe allowing for flanges and fittings (m) f = Insulation factor (dimensionless) hfg = Enthalpy of evaporation at operating pressure (kJ / kg) Note: The constant 3.6 gives the answer in kg / h Steam condensing rate for air heating equipment

2.12.3

Where: ms = Rate of steam condensation (kg /h) V = Volumetric flowrate of air being heated (m³/s) DT = Air temperature rise (°C) c p = Specific heat of air at constant pressure (kJ / m³ °C) hfg = Enthalpy of evaporation of steam in the coils (kJ / kg) Note: The constant 3 600 gives the solution in kg / h

The Steam and Condensate Loop

16.1.9

Equations

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Module 16.1

Equation

Steam condensing rate for horizontal pipes in still air

2.12.4

Where: ms = Rate of steam condensation (kg / h) Q = Heat emission from pipe (W/m) L = Effective length of pipes (m) hfg = Enthalpy of evaporation at the working pressure (kJ / kg) Note: The constant 3.6 gives the answer in kg / h Mean steam flowrate to a storage calorifier

2.13.1

Where: ms = Mean rate of condensation (kg / h) m = Mass of water heated (kg) c p = Specific heat of water (kJ / kg °C) DT = Change in temperature of water (°C) hfg = Enthalpy of evaporation of steam (kJ / kg) t = Recovery time to heat the water (hours) Steam consumption of drying cylinders

2.14.1

Where: ms = Mass flowrate of steam (kg / h) W w = Throughput of wet material (kg / h) W d = Throughput of dry material (kg / h) T2 = Temperature of material leaving the machine (°C) T1 = Temperature of material entering the machine (°C) hfg = Enthalpy of evaporation of steam in cylinders (kJ / kg) The kinetic energy in steam

2.16.1

16.1.10

Where: E = Kinetic energy (kJ) m = Mass of the fluid (kg) u = Velocity of the fluid (m /s) g = Acceleration due to gravity (9.806 65 m /s²) J = Joule’s mechanical equivalent of heat (101.972 m kg /kJ)

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Velocity of steam passing through an orifice in terms of kinetic energy

2.16.2

Where: u = Velocity of the fluid (m /s) E = Kinetic energy (kJ) g = Acceleration due to gravity (9.806 65 m /s²) J = Joule’s mechanical equivalent of heat (101.972 m kg /kJ) m = Mass of the fluid (kg) Velocity of steam passing through an orifice in terms of heat drop

2.16.3

Where: u = Veolocity of the fluid (m/s) h = Heat drop per unit mass (kJ/kg) Mass flow of steam through an orifice

2.16.4 Velocity of steam passing through an orifice in terms of heat drop

2.16.5

Where: u = Velocity of the fluid in m/s h = Heat drop in J/kg 2 = Constant of proportionality incorporating the gravitational constant ‘g’.

The Steam and Condensate Loop

16.1.11

Equations

Block 16 Equations

Module 16.1

Block 3: The boiler house Equation number

Equation

Stress in a boiler shell resulting from boiler pressure

3.2.1

Where: s = Hoop stress (N /m²) P = Boiler pressure (N /m² = bar x 105) D = Diameter of cylinder (m) ƒ = Plate thickness (m) Relating boiler pressure to heat transfer rate

3.2.2

Where: P = Boiler pressure (N /m² = bar x 105) Q = Heat transfer rate (kW) To determine the evaporation factor of a boiler from its ‘From & At’ rating

3.5.1

Where: A = Specific enthalpy of evaporation at atmospheric pressure. B = Specific enthalpy of steam at operating pressure. C = Specific enthalpy of water at feedwater temperature. To determine the actual evaporation rate of a boiler from its kW rating and the energy required to be added to the feedwater to make steam

3.5.2 Where: m = Steam output (kg/h) Q = Boiler rating (kW) To determine boiler horse power from heat transfer area 3.5.3

Where: BoHP = Boiler horsepower Calculating boiler efficiency

3.6.1

16.1.12

The Steam and Condensate Loop

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Block 16 Equations

Equation number

Module 16.1

Equation

To determine the TDS of a sample by the density method 3.12.1 To determine the TDS of a sample by the conductivity method 3.12.2 To correct the conductivity of a sample at a temperature from 25°C

3.12.3

Where: sT = Conductivity at temperature T (µS / cm) s25 = Conductivity at 25°C (µS / cm) a = Temperature coefficient, per °C (Typically 0.02 / °C or 2%°C) T = Temperature (°C) The electrical resistance of a conductivity probe

3.12.4

Where: R = Resistance (Ohm) K = Cell constant (cm-1) s = Conductivity (S / cm) To determine the blowdown rate of a boiler

3.12.5

Where: F = Feedwater TDS (ppm). S = Steam generation rate (kg / h). B = Required boiler water TDS (ppm). Ohm’s Law

3.16.1

Where: I = Current (amperes) V = Voltage (volts) R = Resistance (ohms) Capacitance Law

3.16.2

Where: C = Capacitance (farad) K = Dielectric constant (non-dimensional) A = Area (m²) D = Distance between plates (m)

The Steam and Condensate Loop

16.1.13

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Steam injection required to power a steam deaerator

3.21.1

Where: ms = Mass of steam to be injected (kg / h) m = Maximum boiler output at the initial feedwater temperature (kg / h) h1 = Enthalpy of water at the initial temperature (kJ / kg) h2 = Enthalpy of water at the required temperature (kJ / kg) hg = Enthalpy of steam supplying the control valve (kJ / kg) Sizing a control valve for saturated steam

3.21.2

Where: ms = Steam mass flowrate (kg /h) K v = Valve coefficient required P1 = Pressure upstream of the control valve (bar a) P2 = Pressure downstream of the control valve (bar a)





Sizing a control valve for liquid

3.21.3

Where: V = Volumetric flowrate (m3 /h) K v = Valve coefficient required DP= Pressure drop across the valve (bar) G = Relative density of fluid (water = 1) Steam storage capacity of an accumulator

3.22.1

16.1.14

The Steam and Condensate Loop

Equations

Block 16 Equations

Module 16.1

Block 4: Flowmetering Equation number

Equation

To determine the Absolute or Dynamic viscosity of a fluid by dropping a sphere through a fluid

4.1.1

Where: µ = Absolute (or dynamic) viscosity (Pa s) D r = Difference in density between the sphere and the liquid (kg / m3) g = Acceleration due to gravity (9.81 m / s 2) r = Radius of sphere (m) u

=

To determine the Kinematic viscosity of a fluid

4.1.2

Where: v = Kinematic viscosity (centistokes) µ = Dynamic viscosity (Pa s) r = Density (kg / m3) To determine the Reynolds number of a fluid in a circular pipe

4.1.3

Where: R e = Reynolds number (dimensionless) r = Density (kg /m3) u = Mean velocity in the pipe (m /s) D = Internal pipe diameter (m) µ = Dynamic viscosity (Pa s) To determine volumetric flowrate from velocity

4.1.4

Where: qv = Volume flow (m3/s) A = Cross sectional area of the pipe (m2) u = Velocity (m / s) To determine mass flowrate from volumetric flowrate

4.1.5

Where: qm = Mass flow (kg / s) qv = Volume flow (m3/s) v g = Specific volume (m3/ kg)

The Steam and Condensate Loop

16.1.15

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

To determine mass flowrate from velocity

4.1.6

Where: qm = Mass flow (kg / s) A = Cross sectional area of the pipe (m2) u = Velocity (m /s) v g = Specific volume (m3/ kg) To determine the turndown ratio of a steam flowmeter

4.2.1 Bernoulli’s Equation for a liquid

4.2.2

Where: P1 and P2 u1 and u2 h1 and h2 r g

= = = = =

Pressure at points within a system (Pa) Velocities at corresponding points within a system (m /s) Relative vertical heights within a system (m) Density (kg / m3) Gravitational constant (9.81 m /s²)

Bernoulli’s Equation multiplied throughout by ‘r g’

4.2.3

Where: P1 and P2 u1 and u2 h1 and h2 r g

= = = = =

Pressure at points within a system (Pa) Velocities at corresponding points within a system (m /s) Relative vertical heights within a system (m) Density (kg / m3) Gravitational constant (9.81 m /s²)

Bernoulli’s Equation with constant potential energy terms 4.2.4

Where: P1 and P2 = Pressure at points within a system (Pa) u1 and u2 = Velocities at corresponding points within a system (m /s) r = Density (kg / m3) Bernoulli’s Equation with constant potential energy terms and frictional losses

4.2.5

16.1.16

Where: P1 and P2 u1 and u2 r hf

= = = =

Pressure at points within a system (Pa) Velocities at corresponding points within a system (m /s) Density (kg / m3) Friction loss (Pa)

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

The pressure drop for a liquid equals the friction loss 4.2.6

Where: P1 = Upstream pressure (Pa) P2 = Downstream pressure (Pa) hf = Friction loss (Pa) Potential energy

4.2.7

Where: m = Mass of all the molecules directly between and including molecule 1 and molecule 2. g = Gravitational constant (9.81 m/s2) h = Cumulative height of molecules above the hole Kinetic energy

4.2.8

Where: m = Mass of the object (kg) u = Velocity of the object at any point (m/s) Potential energy at Kinetic energy at = the start of process the end of process

4.2.9

Where: m = Mass of the object (kg) g = Gravitational constant (9.81 m/s2) h = Height of the object above a reference point (m) Velocity of liquid through an orifice

4.2.10

Where: u = Velocity (m / s) g = Gravitational constant (9.81 m/s2) h = Pressure head (m) Volumetric flowrate of liquid through an orifice

4.2.11

Where: qv = Volumetric flowrate (m3/s) C = Coefficient of discharge (dimensionless) A = Area of orifice (m2) g = Gravitational constant (9.8 m/s2) h = Differential pressure (m)

The Steam and Condensate Loop

16.1.17

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Volumetric flowrate of liquid is proportional to the square root of pressure drop 4.2.12

Where: qv = Volumetric flowrate (m3 / s) Dp= Pressure drop (m) The liquid velocity measured by a Pitot tube

4.2.13

Where: u1 = The fluid velocity in the pipe DP= Static pressure - dynamic pressure r = The fluid density To determine the b ratio for an orifice plate

4.3.1 Note: Diameters must be in the same unit of measurement To determine the vortex shedding frequency around a bluff body

4.3.2

Where: f = Shedding frequency (Hz) Sr = Strouhal number (dimensionless) u = Mean pipe flow velocity (m/s) d = Bluff body diameter (m) The volumetric flowrate from the shedding frequency

4.3.3

16.1.18

Where: qv = Volumetric flowrate (m3/s) A = Cross sectional area of the orifice (m2) f = Shedding frequency (Hz) k = A constant for all fluids for a given design of meter

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Percentage error when using a velocity sensing meter which is not pressure compensated 4.4.1

Where: e = Flow error expressed as a percentage of the actual flow Specified r = Density of steam at the specified steam line pressure Actual r = Density of steam at the actual line pressure Percentage error when using a pressure difference meter which is not pressure compensate

4.4.2

Where: e = Percentage flow error Actual r = Density of steam at actual pressure (kg /m3) Specified r = Density of steam at specified pressure (kg /m3) To determine the density of steam with known dryness fraction

4.4.3

Where: r = Density of steam with dryness fraction c n g = Specific volume of dry steam c = Dryness fraction Approximation of relationship between indicated and actual flowrate with a deviation in dryness fraction

4.4.4

Actual value of superheated steam flowing through a flowmeter calibrated for saturated steam 4.4.5

Block 5: Basic control theory Equation number

Equation

There are no equations in Block 5

The Steam and Condensate Loop

16.1.19

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Block 16 Equations

Module 16.1

Block 6: Control hardware: Electric/pneumatic actuation Equation number

Equation

Closing force to close a valve (A x DP) + Friction allowance = F 6.1.1

Where: A = Valve seating area (m2) DP = Differential pressure (kPa) F = Closing force required (kN) Calculate valve Kv for liquids

6.3.1

Where: K v = Flow of liquid that will create a pressure drop of 1 bar (m³/ h bar) V = Flowrate (m³/h) G = Relative density /specific gravity of the liquid (dimensionless). DP = Pressure drop across the valve (bar) Volumetric flow of water through a valve

6.3.2

Where: V = Flowrate (m³ /h) K v = Flow of liquid to create a pressure drop of 1 bar (m³ /h bar) DP = Pressure drop across a valve (bar) The flow of liquid through a constant bore pipe relative to pressure loss

6.3.3

Where: V1 = Flowrate at pressure loss P1 V2 = Flowrate at pressure loss P2 Valve authority

6.3.4

16.1.20

Where: N = DP 1 = DP 2 = DP1 + DP2 =

Valve authority Pressure drop across a fully open control valve Pressure drop across the remainder of the circuit Pressure drop across the whole circuit

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Critical pressure ratio for dry steam and gases 6.4.1

Where: g = Isentropic exponent of the steam or gas Speed of sound in steam

6.4.2

Where: C = Speed of sound in steam (m / s) 31.6 = Constant of proportionality g = Steam isentropic exponent (1.135 : saturated, 1.3 : superheated) R = 0.461 5 the gas constant for steam (kJ / kg) T = Absolute steam temperature (K) Steam flow through a valve under critical flow conditions

6.4.3

Where: ms = Mass flow through a valve (kg/h) K v = Valve capacity (m3/h bar) P1 = Upstream pressure (bar a) Volumetric flow through an equal pecentage valve

6.5.1

Where: V = Volumetric flow through the valve at lift H. x = (ln t) H Note: ‘In’ is a mathematical function known as ‘natural logarithm’. t = Valve rangeability (ratio of the maximum to minimum controllable flowrate, typically 50 for a globe type control valve) H = Valve lift (0 = closed, 1 = fully open) V max = Maximum volumetric flow through the valve Required capacity of a water control valve

6.5.2

Where: Kvr = The actual valve capacity required by the installation (m³/h bar) V = Flowrate through the valve (m3/h) DP = The differential pressure across the valve (bar)

The Steam and Condensate Loop

16.1.21

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Percentage lift of an equal percentage valve in terms of relative flow

6.5.3

Where: H% = In = V = t = V max =

Percentage lift Natural logarithm Flow through the valve at lift ‘H’ (m3/h) Valve rangeability Maximum flow through the valve at full lift (m3/h)

Percentage lift of an equal percentage valve in terms of relative Kv

6.5.4

Where: H% = Percentage lift In = Natural logarithm K vr = Required capacity at lift ‘H’ (m3/h bar) t = Valve rangeability Kvs = Valve capacity full open (m3/h bar) The required capacity for a steam valve under sub-sonic flow

6.5.5

Where: K vr = Required capacity at lift ‘H’ (m3/h bar) ms = Steam mass flowrate (kg/h) P1 = Upstream pressure (bar a) P2 = Downstream pressure (bar a) x = (P1 - P2) / P1

Block 7: Control hardware: Self-acting actuation Equation number

Equation

Stem force required to close a control valve

7.1.1

16.1.22

Where: d = Diameter of valve orifice (mm) DP = Differential pressure (bar)

The Steam and Condensate Loop

Equations

Block 16 Equations

Module 16.1

Block 8: Control applications Equation number

Equation

There are no equations in Block 8

Block 9: Safety valves Equation number

Equation

Flow area of a safety valve 9.1.1

Where: d = The area of the inlet port at its narrowest point Curtain area of a safety valve

9.1.2

Where: d1 = Minimum area of opening between the valve and seat L = Maximum lift from seat to valve Required opening force for a safety valve with the spring housing vented via the discharge vent pipe

9.2.1

Where: PV = Fluid inlet pressure AN = Nozzle area FS = Spring force PB = Backpressure Required opening force for a safety valve with the spring housing vented to atmosphere

9.2.2

Where: PV = Fluid inlet pressure AN = Nozzle area FS = Spring force PB = Backpressure A D = Disc area Required opening force for a safety valve with the spring housing vented via the discharge vent pipe and taking into effect the build-up backpressure

9.2.3

Where: PS = Set pressure of safety valves AN = Nozzle area FS = Spring force PB = Backpressure PO = Overpressure

The Steam and Condensate Loop

16.1.23

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Required opening force for a balanced safety valve 9.2.4

Where: PV = Fluid inlet pressure AN = Nozzle area FS = Spring force Cold differential pressure

9.3.1

Where: CDSP = Cold differential set pressure RISP = Required installed set pressure CBP = Constant backpressure Coefficient of discharge

9.4.1

Where: K d = Coefficient of discharge Critical pressure ratio

 9.4.2



Where: PB = Critical backpressure (bar a) P1 = Actual relieving pressure (bar a) k = Isentropic coefficient of the gas or vapour at the relieving conditions AD-Merkblatt valves - Minimum flow area for steam

9.4.3

Where: AO = Minimum cross sectional flow area (mm2) c = Pressure medium coefficient m = Mass flow to be discharged (kg / h) a W = Outflow coefficient PR = Absolute relieving pressure (bar a) AD-Merkblatt valves - Minimum flow area for dry gases and air

9.4.4

16.1.24

Where: AO = Minimum cross sectional flow area (mm2) m = Mass flow to be discharged (kg / h) Y = Outflow function a W = Outflow coefficient PR = Absolute relieving pressure (bar a) T = Inlet temperature (K) M = Molar mass (kg / kmol) Z = Compressibility factor The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

AD-Merkblatt valves - Minimum flow area for liquids

9.4.5

Where: AO = Minimum cross sectional flow area (mm2) m = Mass flow to be discharged (kg / h) a W = Outflow coefficient r = Density (kg / m3) DP = PR - PB PR = Absolute relieving pressure (bar a) PB = Absolute backpressure (bar a) Compressibility factor for compressible steam and dry gases

9.4.6

Where: Z = Compressibility factor PR = Safety valve relieving pressure (bar a) n = Specific volume of the gas at the actual relieving pressure and temperature (m3 / kg) M = Molar mass (kg / kmol) R u = Universal gas constant (8 314 Nm / kmol K) T = Actual relieving temperature (K) Proportion of vapour in two phase discharge

9.4.7

Where: n = The proportion of discharge fluid which is vapour hf1 = Enthalpy of liquid before the valve (kJ / kg) hf2 = Enthalpy of liquid after the valve (kJ / kg) hfg2 = Enthalpy of evaporation after the valve (kJ / kg) ASME (API RP 520) valves - Minimum flow area for steam

9.4.8

Where: AO = Required effective discharge area (in2) m = Required mass flow through the valve (lb / h) PR = Upstream relieving pressure (psi a) K d = Effective coefficient of discharge K SH = Superheat correction factor ASME (API RP 520) valves - Minimum flow area for dry gases and air

9.4.9

Where: AO = Required effective discharge area (in2) V = Required volume flow through the valve (ft3 / min) T = Relieving temperature (°R) Z = Compressibility factor G = Specific gravity of the air or gas Cg = Nozzle gas constant K d = Effective coefficient of discharge PR = Upstream relieving pressure (bar a) K B = Backpressure correction factor for gas and vapour

The Steam and Condensate Loop

16.1.25

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

ASME (API RP 520) valves - Minimum flow area for liquids

9.4.10

Where: AO = Required effective discharge area (in2) V1 = Required volume flow through the valve (U.S. gal / min) K d = Effective coefficient of discharge (specified by the manufacturer) K µ = Viscosity factor K W = Backpressure correction factor for liquids (bellows balanced valves only) G = Specific gravity (ratio of molar mass of the fluid to the molar mass of air PR = Upstream relieving pressure (psi a) PB = Absolute backpressure (psi a) ASME (API RP 520) valves - Nozzle gas constant

9.4.11





Where: C g = Nozzle gas constant k = Isentropic coefficient of the gas or vapour at the relieving conditions ASME (API RP 520) valves - Backpressure correction factor

9.4.12

Where: K B = Backpressure correction factor C1 = Capacity of valve with backpressure applied C2 = Capacity of valve when discharging to atmosphere ASME (API RP 520) valves - Bellows balanced valves

9.4.13

Where: PB = Backpressure (psi g) PS = Set pressure (psi g) ASME (API RP 520) valves - Conventional valves

9.4.14

Where: PB = Backpressure (psi g) PR = Relieving pressure (psi g) ASME (API RP 520) valves - Reynolds number: Metric units

9.4.15

16.1.26

Where: R e = Reynolds number m = Mass flow to be discharged (kg / h) µ = Dynamic viscosity (Imperial – cP, Metric – Pa s) AO = Discharge area (Imperial – in2, Metric – mm2) The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

ASME (API RP 520) valves - Reynolds number: Imperial units

9.4.16

Where: Re = Reynolds number V = Volume flow to be discharged (U.S. gal / min) m = Mass flow to be discharged (kg / h) µ = Dynamic viscosity (Imperial – cP, Metric – Pa s) AO = Discharge area (Imperial – in2, Metric – mm2) BS 6759 valves - Minimum orifice area for steam

9.4.17

Where: AO = Flow area (mm2) m = Mass flow to be discharged (kg / h) PR = Absolute relieving pressure (bar a) K dr = Derated coefficient of discharge K SH = Superheat correction factor BS 6759 valves - Minimum orifice area for air

9.4.18

Where: AO = Flow area (mm2) V = Volumetric flow to be discharged (l / s) PR = Absolute relieving pressure (bar a) K dr = Derated coefficient of discharge T = Inlet temperature (K) BS 6759 valves - Minimum orifice area for dry gases

9.4.19

Where: AO = Flow area (mm2) m = Mass flow to be discharged (kg / h) PR = Absolute relieving pressure (bar a) Cg = Nozzle gas constant K dr = Derated coefficient of discharge Z = Compressibility factor T = Inlet temperature (K) M = Molar mass (molecular weight) (kg / kmol)

The Steam and Condensate Loop

16.1.27

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

BS 6759 valves - Minimum orifice area for liquids

9.4.20

Where: AO = Flow area (mm2) m = Mass flow to be discharged (kg / h) K dr = Derated coefficient of discharge K µ = Viscosity correction factor r = Density (kg / m3) DP = PR - PB PR = Absolute relieving pressure (bar a) PB = Absolute backpressure (bar a) BS 6759 valves - Minimum orifice area for hot water

9.4.21

Where: AO = Flow area (mm2) Q = Hot water heating capacity (kW) PR = Absolute relieving pressure (bar a) K dr = Derated coefficient of discharge BS 6759 valves - Nozzle gas constant

9.4.22





Where: k = Isentropic coefficient of gas or vapour EN ISO 4126 valves - Minimum orifice area for steam, air and dry gas at critical flow

9.4.23

Where: A = Flow area (not curtain area) (mm2) m = Mass flowrate (kg / h) C = Function of the isentropic exponent K dr = Certified derated coefficient of discharge P o = Relieving pressure (bar a) ng = Specific volume at relieving pressure and temperature (m³/kg) EN ISO 4126 valves - Minimum orifice area for wet steam at critical flow

9.4.24

16.1.28

Where: A = Flow area (not curtain area) (mm2) m = Mass flowrate (kg / h) C = Function of the isentropic exponent K dr = Certified derated coefficient of discharge P o = Relieving pressure (bar a) ng = Specific volume at relieving pressure and temperature (m³/kg) x = Dryness fraction of wet steam

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

EN ISO 4126 valves - Minimum orifice area for air and dry gas at sub-critical flow

9.4.25

Where: A = Flow area (not curtain area) (mm2) m = Mass flowrate (kg / h) C = Function of the isentropic exponent K dr = Certified derated coefficient of discharge K b = Theoretical correction factor for sub-critical flow P o = Relieving pressure (bar a) ng = Specific volume at relieving pressure and temperature (m³/kg) EN ISO 4126 valves - Minimum orifice area for liquids

9.4.26

Where: A = Flow area (not curtain area) (mm2) m = Mass flowrate (kg / h) K dr = Certified derated coefficient of discharge K v = Viscosity correction factor P o = Relieving pressure (bar a) Pb = Backpressure (bar a) ng = Specific volume at relieving pressure and temperature (m³/kg) Safety valve vent pipe diameter

9.5.1

Where: d = Pipe diameter (mm) Le = Equivalent length of pipe (m) m = Discharge capacity (kg / h) P = Safety valve set pressure (bar g) x 0.1 vg = Specific volume of steam at the pressure (P) (m3 / kg) Reaction force at the end of a safety valve vent pipe

9.5.2

Where: F = Reaction force at the point of discharge to atmosphere (newtons) m = Discharge mass flowrate (kg / s) k = Isentropic coefficient of the fluid T = Fluid temperature (K) M = Molar mass of the fluid (kg / kmol) A = Area of the outlet at the point of discharge (mm2) P = Static pressure at the outlet at the point of discharge (bar g)

The Steam and Condensate Loop

16.1.29

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Sound power level at the safety valve outlet

9.5.3

Where: LP = Sound power level in dB (A) m = Mass flow (kg / h) u = Speed of sound in an ideal gas (m / s), k = Isentropic coefficient of the gas R u = Universal gas constant (8 314 J / kmol K) T = Absolute gas temperature at the safety valve outlet (K) M = Molar mass (kg / kmol) Sound pressure level at the safety valve outlet

9.5.4

Where: L = Sound pressure level in dB (A) LP = Sound power level in dB (A) R = Distance from the source (m) Sound power level from EN ISO 4126

9.5.5

Block 10: Equation number

Where: LP = Sound power level in dB (A) DA = Discharge pipe bore (mm) = Specific volume at relieving pressure and temperature (m3 / kg) u = Velocity of fluid in the outlet pipe (m / s)

Steam distribution Equation

The ‘SI based’ D’Arcy equation for determining pressure drop due to frictional resistance

10.2.1

16.1.30

Where: hf = Head loss to friction (m) f = Friction factor (dimensionless) L = Length (m) u = Flow velocity (m /s) g = Gravitational constant (9.81 m /s²) D = Pipe diameter (m)

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

The ‘Imperial based’ D’Arcy equation for determining pressure drop due to frictional resistance

10.2.2

Where: hf = Head loss to friction (m) f = Friction factor (dimensionless) L = Length (m) u = Flow velocity (m /s) g = Gravitational constant (9.81 m /s²) D = Pipe diameter (m) Friction factor for turbulent fluids (Colebrook-White formula)

10.2.3

Where: f = Friction factor (Relates to the SI Moody chart) kS = Absolute pipe roughness (m) D = Pipe bore (m) Re = Reynolds number (dimensionless) ‘SI based’ friction factors - f

10.2.4

Where: f = Friction factor Re = Reynolds number ‘Imperial based’ friction factors - f

10.2.5

Where: f = Friction factor Re = Reynolds number Reynolds number

10.2.6

Where: Re = Reynolds number r = Density of water (kg /m3) u = Velocity of water (m /s) D = Pipe diameter (m) m = Dynamic viscosity of water (kg /m s) Relative pipe roughness

10.2.7



Where: ks = Pipe roughness (m) D = Pipe bore (m)

The Steam and Condensate Loop

16.1.31

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Pressure factor

10.2.8

Where: F = Pressure factor P1 = Factor based on the inlet pressure P2 = Factor based on the pressure at a distance of L metres of pipe L = Equivalent length of pipe (m) Pressure drop formula 1

10.2.9

Where: P1 = Upstream pressure (bar a) P2 = Downstream pressure (bar a) L = Length of pipe (m) m = Mass flowrate (kg /h) D = Pipe diameter (mm) Pressure drop formula 2 (Maximum pipe length: 200 metres)

10.2.10

Where: DP = Pressure drop (bar) L = Length of pipe (m) vg = Specific volume of steam (m³ /kg) m = Mass flowrate (kg /h) D = Pipe diameter (mm) Thermal expansion of pipe

10.4.1

Block 11: Equation number

Where: L = Length of pipe between anchors (m) DT = Temperature difference between ambient temperature and operating temperatures (°C) a = Expansion coefficient (mm /m °C x 10-3)

Steam trapping Equation

There are no equations in Block 11

16.1.32

The Steam and Condensate Loop

Equations

Block 16 Equations

Block 12:

Module 16.1

Pipeline ancillaries

Equation number

Equation

Pressure drop across a valve in a liquid system

12.2.1

Where: DP = Pressure drop across the valve (bar) G = Specific gravity of the liquid (non-dimensional) V = Flowrate of liquid (m³ / h) Kv = Valve flow coefficient (m³/h bar) Equivalent water flowrate through a check valve

12.3.1

Where: Vw = Equivalent water volume flowrate (m³ / h) r = Density of the liquid (kg / m³) V = Volume flowrate of liquid (m³ / h) Converting water mass flow to volumetric flow

12.3.2

Where: V = Volume flowrate (m³ / h) m = Mass flowrate (kg / h) ng = Specific volume (m³ / kg) Largest particle size through a strainer screen

12.4.1

Where: a = Mesh hole length b = Mesh hole width c = Particle size Pressure drop across a steam valve

12.4.2

Where: DP = Pressure drop across the valve (bar) ms = Steam flowrate (kg / h) Kv = Valve flow coefficient (m3 / h bar) P1 = Upstream pressure (bar a)

The Steam and Condensate Loop

16.1.33

Equations

Block 16 Equations

Block 13: Equation number

Module 16.1

Condensate removal Equation

Calculating the heating area of a heat exchanger

13.2.1

Where: A = Area of heating surface (m²) Q = Mean heat transfer rate (W) U = Heat transfer coefficient (W / m² °C) DT M = Mean temperature difference. The heat exchanger temperature design constant

13.2.2

Where: TDC = Temperature design constant of the heat exchanger = Steam temperature (°C) Ts T1 = Secondary fluid inlet temperature (°C) T2 = Secondary fluid outlet temperature (°C) The steam temperature at any load

13.2.3

Where: TDC = Temperature design constant of the heat exchanger Ts = Steam temperature (°C) T1 = Secondary fluid inlet temperature (°C) T2 = Secondary fluid outlet temperature (°C) The secondary fluid inlet temperature at any load

13.2.4

Where: TDC = Temperature design constant of the heat exchanger Ts = Steam temperature (°C) T1 = Secondary fluid inlet temperature (°C) T2 = Secondary fluid outlet temperature (°C) The secondary fluid outlet temperature at any load

13.2.5

16.1.34

Where: TDC = Temperature design constant of the heat exchanger Ts = Steam temperature (°C) T1 = Secondary fluid inlet temperature (°C) T2 = Secondary fluid outlet temperature (°C)

The Steam and Condensate Loop

Equations

Block 16 Equations

Equation number

Module 16.1

Equation

Mean differential temperature between the primary and secondary fluids

13.3.1

Where: DT M = Mean temperature difference. Note: DTM may be either DTLM (LMTD) or DTAM (AMTD) Q = Mean heat transfer rate (W) U = Heat transfer coefficient (W / m² °C) A = Heating area (m²) The secondary inlet temperature at any load

13.4.1

Where: Tx = The secondary inlet temperature at any load factor ‘x’ (°C) T1 = The secondary inlet temperature at full-load (°C) T2 = The secondary outlet temperature at full-load (°C) x = The load factor The stall load for a constant flow secondary

13.5.1

Where: A = The steam temperature in the steam space at full-load (°C) B = The secondary fluid outlet temperature (°C) D = The backpressure equivalent saturated steam temperature (°C) Calculating the stall load with a variable flow secondary

13.6.1

Where: A = Steam temperature at full-load (°C) B = Secondary fluid outlet temperature at full-load (°C) C = Secondary fluid inlet temperature at full-load (°C) D = Equivalent backpressure steam temperature (°C)

The Steam and Condensate Loop

16.1.35

Equations

Block 16 Equations

Block 14:

Module 16.1

Condensate recovery

Equation number

Equation

Cost of fuel saved by returning condensate

14.1.1

Where: X = Expected improvement in condensate return expressed as a percentage between 1 and 100 A = Cost of fuel to provide 1 GJ of energy B = Energy required per kilogram of make-up water to reach condensate temperature (kJ/kg). C = Average boiler evaporation rate (kg / h) D = Operational hours per year (h / year) E = Boiler efficiency (%) Cost of water saved by returning condensate

14.1.2 Cost of effluent saved by returning condensate 14.1.3

Calculating pump delivery head

14.4.1

Where: h d = Total delivery head hs = Pressure required to raise the water to the desired level (static head) hf = Pressure required to move the water through the pipes (friction head) h p = Pressure in the condensate system Calculate condensate velocity in a pipe

14.4.2

Block 15: Equation number

Desuperheating Equation

Calculate cooling water flowrate for a desuperheater

15.1.1

16.1.36

Where: mcw = Mass flowrate of cooling water (kg / h) ms = Mass flowrate of superheated steam (kg / h) hs = Enthalpy at superheat condition (kJ / kg) h d = Enthalpy at desuperheated condition (kJ / kg) hcw = Enthalpy of cooling water at inlet connection (kJ / kg)

The Steam and Condensate Loop