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Sizing Steam Jet Ejectors and Condensers

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Section summary
1. Introduction
2. Main Concepts
3. Data Tables
4. Calculation Methods and Formulas
5. Complete worked example step by step

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1. Introduction

What are steam jet ejectors and condensers used for in industry?

Steam jet ejectors and condensers are essential components in many industrial processes, particularly where maintaining vacuum conditions is critical. These systems are widely used in the process, food, steel, and related industries for operations such as distillation, absorption, mixing, vacuum packaging, freeze-drying, dehydrating, and degassing. Their ability to handle condensable and non-condensable gases and vapors, as well as mixtures thereof, makes them versatile tools in numerous applications.

A steam jet ejector system operates on the ejector-venturi principle. High-pressure motive steam expands through a converging-diverging nozzle, converting pressure energy into kinetic energy. This creates a vacuum, drawing in the suction fluid (air, gas, or vapor). The mixture then enters a venturi diffuser, where kinetic energy is converted back into pressure, allowing discharge against a predetermined back pressure. Depending on the application, the ejector system may incorporate condensers to improve efficiency and manage condensable components.

This article provides a guide to understanding and sizing steam jet ejector and condenser systems. It covers the fundamental principles, components, calculation methods, and practical examples necessary for engineers to design and optimize these systems for specific applications.

1.1. Overview of Steam Jet Ejectors and Condensers

How do steam jet ejectors and condensers work together?

Steam jet ejectors and condensers work together to create and maintain vacuum conditions in a wide range of industrial processes. The ejector uses a high-velocity jet of motive steam to entrain and remove gases or vapors from a system, thereby reducing the pressure. The motive steam expands through a nozzle, converting pressure energy into kinetic energy. This high-speed jet creates a low-pressure region, drawing in the suction load. The mixture of motive steam and suction fluid then passes through a diffuser, where the kinetic energy is converted back into pressure, enabling the discharge of the mixture against a higher back pressure.

Condensers are incorporated into ejector systems to remove condensable components, primarily water vapor from the motive steam and any process vapors present in the suction load. This condensation reduces the volume of gas that subsequent ejector stages must handle, leading to lower motive steam consumption and improved overall system efficiency. Condensers also prevent the release of potentially harmful or valuable condensable vapors into the atmosphere.

Ejector systems are classified based on the number of stages and the presence or absence of condensers:

• Single-stage ejectors: These are the simplest configuration, suitable for applications requiring relatively modest vacuum levels (typically down to 1 inch Hg absolute).
• Multi-stage ejectors: These are employed when deeper vacuum levels are needed. These systems consist of multiple ejector stages operating in series, with each stage progressively reducing the pressure.
• Condensing ejector systems: These incorporate intercondensers between ejector stages to condense the motive steam and condensable vapors from the preceding stage. This significantly reduces the load on subsequent stages, leading to lower steam consumption and higher efficiency. Aftercondensers may also be used at the discharge of the final ejector stage to further reduce emissions. Condensing systems are generally preferred for applications requiring high vacuum where steam consumption is a major concern.
• Non-condensing ejector systems: These lack intercondensers, with the discharge of each ejector stage directly feeding the suction of the next. While simpler and less expensive to install, non-condensing systems consume significantly more steam and are typically used where vacuum requirements are less stringent or where intermittent operation makes steam consumption a secondary consideration.

Intercondensers in multi-stage condensing systems can be either direct contact or surface condensers. Direct contact condensers mix the exhaust stream with cooling water, offering high heat transfer efficiency but requiring careful management of the cooling water discharge. Surface condensers keep the exhaust stream and cooling water separate, allowing for recovery of condensate and preventing contamination of the cooling water.

1.2. Importance of Proper Sizing

Why is it important to correctly size steam jet ejector and condenser systems?

The correct sizing of steam jet ejector and condenser systems is paramount to achieving optimal performance, efficiency, and reliability in vacuum processes. An appropriately sized system ensures that the desired vacuum level is consistently maintained, leading to stable and predictable process operation. Improper sizing can lead to a cascade of negative consequences, impacting both operational costs and product quality.

Consequences of Undersizing:

• Inability to Achieve Target Vacuum: An undersized ejector system lacks the capacity to handle the actual gas and vapor load, resulting in a higher-than-desired vacuum pressure. This can directly impede process efficiency, leading to reduced production rates, incomplete reactions, or compromised product purity.
• Process Instability: Fluctuations in vacuum pressure due to an undersized system can disrupt delicate process balances, leading to inconsistent product quality and potential process upsets.

Consequences of Oversizing:

• Excessive Steam Consumption: An oversized ejector system consumes more motive steam than necessary to maintain the required vacuum, translating directly into increased energy costs and a larger carbon footprint.
• Increased Capital Costs: Oversized ejectors and condensers are more expensive to purchase and install, representing an unnecessary capital expenditure.
• Potential for Equipment Damage: In some cases, oversized ejectors can create excessively low pressures, potentially damaging sensitive equipment connected to the vacuum system.

Accurate sizing requires a thorough understanding of the process requirements and careful consideration of several key factors, including the non-condensable load, vapor load, motive steam requirements, and intercondenser duty. While this article provides a comprehensive guide, it is essential to recognize that complex applications may require the expertise of experienced engineers.

2. Main Concepts

What are the key principles behind steam jet ejector and condenser systems?

This section outlines the fundamental principles and components of steam jet ejector and condenser systems, providing a foundation for understanding their operation and sizing.

2.1. Ejector-Venturi Principle

How does the ejector-venturi principle work in a steam jet ejector?

The steam jet ejector operates based on the ejector-venturi principle, a concept rooted in fluid dynamics that describes how the pressure of a fluid decreases as its velocity increases. In a steam jet ejector, high-pressure motive steam is forced through a converging-diverging nozzle, initiating a sequence of energy conversions:

• Converging Section: As steam enters the converging section of the nozzle, its pressure energy is converted into kinetic energy, causing its velocity to increase.
• Throat: At the nozzle's narrowest point, the throat, the steam achieves sonic velocity (Mach 1).
• Diverging Section: Beyond the throat, the nozzle expands, allowing the steam to continue accelerating to supersonic speeds. This expansion creates a region of very low pressure—a vacuum—immediately downstream of the nozzle exit.
• Suction Chamber: This vacuum draws in the suction fluid (the gas or vapor to be removed from the process). The motive steam and suction fluid mix within the suction chamber.
• Venturi Diffuser: The mixture then enters a venturi diffuser, a diverging duct where the kinetic energy of the high-speed mixture is gradually converted back into pressure energy. This pressure recovery allows the ejector to discharge the mixed fluid stream against a back pressure.

2.2. Single-Stage vs. Multi-Stage Ejectors

When should I use a single-stage ejector versus a multi-stage ejector?

Steam jet ejectors are categorized by the number of stages they employ, with each stage consisting of a nozzle, suction chamber, and diffuser. The choice between single-stage and multi-stage configurations depends primarily on the required vacuum level.

• Single-Stage Ejectors: These are the simplest and most economical type, suitable for applications requiring relatively modest vacuum levels, typically down to 1 inch Hg absolute (33.9 mbar absolute). They are often used for priming centrifugal pumps or maintaining vacuum on small condensers.
• Multi-Stage Ejectors: When deeper vacuum levels are needed, multi-stage ejectors are used. These systems consist of two or more ejector stages operating in series, where the discharge of one stage becomes the suction of the next, progressively reducing the pressure. The number of stages required depends on the desired vacuum level: two-stage ejectors are common for vacuums down to approximately 3 mm Hg absolute, while three to six-stage systems can achieve vacuums into the micron range (0.001 mm Hg absolute).

2.3. Condensing vs. Non-Condensing Ejector Systems

What are the differences between condensing and non-condensing ejector systems?

Multi-stage ejector systems are further classified based on whether they incorporate condensers between stages, a distinction that significantly impacts their performance and steam consumption.

• Non-Condensing Ejector Systems: In these systems, the discharge of each ejector stage feeds directly into the suction of the subsequent stage. They are simpler and less expensive to install but consume significantly more motive steam because each stage must handle the entire load from the previous one. Non-condensing systems are typically used when vacuum requirements are less stringent, the suction load has few condensable vapors, or intermittent operation makes steam consumption a secondary consideration.
• Condensing Ejector Systems: These systems incorporate intercondensers between ejector stages to condense the motive steam and condensable vapors from the preceding stage. This significantly reduces the load on subsequent stages, leading to lower steam consumption and higher efficiency. Condensing systems are generally preferred for applications requiring high vacuum, where steam consumption is a major concern, or where recovery of valuable condensable vapors is desired.
• Aftercondensers: Both system types can be equipped with an aftercondenser at the discharge of the final stage to condense remaining vapors before venting to the atmosphere, reducing emissions and noise.

Intercondensers can be either direct contact (mixing exhaust with cooling water) or surface condensers (keeping streams separate), with the choice depending on factors like heat transfer efficiency needs and whether condensate recovery is required.

2.4. Vacuum Pressure Measurement

What are the best methods for measuring vacuum pressure in ejector systems?

Accurate and reliable vacuum pressure measurement is essential for the effective operation, monitoring, and troubleshooting of steam jet ejector systems. Selecting the appropriate pressure measurement device depends on the specific pressure range and required accuracy.

• Manometers: These instruments use a column of liquid (typically mercury or oil) to measure pressure differences. They are simple and accurate but can be cumbersome and are not suitable for automated control.
• Bourdon Tube Gauges: These mechanical gauges use a curved tube that deflects proportionally to the applied pressure. They are robust and widely used for general pressure indication but are less accurate at very low pressures.
• Electronic Pressure Transducers: These devices use a sensor that converts pressure into an electrical signal, offering high accuracy, fast response, and easy integration into control systems. Common types include:
  • Capacitance Manometers: Highly accurate and stable, often used as reference standards.
  • Pirani and Thermocouple Gauges: Measure pressure based on the thermal conductivity of the gas, suitable for medium vacuum ranges.
  • Ionization Gauges: Used for measuring very high vacuum pressures (below 10⁻³ Torr).
  • Strain Gauge Transducers: Robust and versatile for a wide range of pressures.
• McLeod Gauge: A primary standard instrument used for calibrating other gauges for very low pressures.

Proper installation, location, and regular calibration of these devices are crucial for obtaining reliable readings and ensuring accurate system control.

2.5. Components of a Steam Jet Ejector System

What are the essential components of a steam jet ejector system?

A complete steam jet ejector system is a carefully integrated assembly of components, each playing a vital role in achieving and maintaining the desired vacuum.

• Ejectors: The core components, each consisting of a nozzle, suction chamber, and diffuser, responsible for creating the vacuum.
• Condensers (Inter- and After-): Heat exchangers that condense steam and process vapors to reduce the load on subsequent stages and minimize emissions.
• Steam Supply System: Provides a consistent supply of motive steam at the required pressure and quality, including components like pressure regulators, steam separators, traps, and strainers.
• Cooling Water System: Supplies cooling water to the condensers, including pumps, piping, and valves.
• Condensate Removal System: Removes condensate from the condensers using collection tanks, pumps, and level controls.
• Piping and Valves: Connect the various components and control the flow of fluids.
• Instrumentation and Control System: Monitors and controls the system's operation, including pressure transducers, temperature sensors, flow meters, and control valves.

2.6. Materials of Construction

What materials are commonly used to build steam jet ejector systems?

The selection of appropriate materials for steam jet ejector and condenser systems is a critical engineering decision, directly impacting the system's longevity, reliability, and safety. The choice depends on fluid compatibility, operating temperature and pressure, and resistance to erosion.

• Carbon Steel: A cost-effective option for general-purpose applications where corrosion is not a major concern.
• Ductile Iron: Offers improved corrosion resistance compared to carbon steel and is often used for ejector bodies and diffusers.
• Stainless Steel: (e.g., 304, 316) Offers excellent corrosion resistance for handling mildly corrosive fluids and is commonly used for nozzles, diffusers, and condenser tubes.
• Alloy Steels: (e.g., Monel, Hastelloy, Alloy 20) Used for highly corrosive environments, offering exceptional resistance to a wide range of acids and alkalis.
• Titanium: Provides excellent corrosion resistance, particularly for handling seawater, but is more expensive.
• Bronze: Sometimes used for steam chests where good corrosion resistance and thermal conductivity are desired.
• Non-Metallic Materials: (e.g., Teflon, Graphite, Phenolic FRP, Tefzel) Employed for extreme corrosion resistance, often as linings or for constructing entire components in applications with highly corrosive acids or solvents.

3. Data Tables

What reference data is useful for sizing steam jet ejectors and condensers?

This section provides reference data in tabular format to aid in the sizing and selection of steam jet ejectors and condensers. These tables offer typical values and ranges for key parameters, serving as a starting point for calculations and preliminary design. It is crucial to remember that these are general guidelines, and specific process requirements may necessitate adjustments based on detailed calculations and expert consultation.

3.1. Suction Pressure Ranges for Different Ejector Stages

What are the typical suction pressure ranges for different ejector stages?

Table 3.1: Typical Suction Pressure Ranges for Steam Jet Ejectors

Number of Stages	Suction Pressure Range (mm Hg Absolute)	Suction Pressure Range (inches Hg Absolute)	Suction Pressure Range (Microns Hg Absolute)
1	760 to 25	30 to 1	760,000 to 25,000
2	100 to 3	4 to 0.12	100,000 to 3,000
3	25 to 0.8	1 to 0.03	25,000 to 800
4	5 to 0.1	0.2 to 0.004	5,000 to 100
5	1 to 0.025	0.04 to 0.001	1,000 to 25
6	0.2 to 0.005	0.008 to 0.0002	200 to 5

3.2. Dimensions of Standard Ejectors

What are the typical dimensions of standard ejectors?

While the calculation methods detailed in subsequent sections allow for custom sizing, many applications can utilize standard ejector designs. This section provides dimensional data for typical, readily available ejectors. These dimensions are crucial for plant layout, piping design, and ensuring adequate space for installation and maintenance.

Table 3.2.1: Dimensions of Standard Single-Stage Ejectors (Flanged Connections)

Connection Size (inches)	A (Overall Length) (inches)	B (Suction Connection to Nozzle) (inches)	C (Diffuser Length) (inches)	D (Discharge Connection to Diffuser End) (inches)	E (Steam Inlet Height) (inches)	F (Steam Inlet Width) (inches)	G (Steam Inlet Connection Size) (inches)	Approximate Weight (lbs)
1	11 19/64	8 7/8	2 27/64	2 7/8	1	1	3/4	14
1 1/2	16 7/16	13 1/4	3 3/16	3 3/8	1 1/2	1 1/2	1	18
2	21 9/16	17 11/16	3 7/8	3 5/8	2	2	1 1/4	36
2 1/2	26 41/64	22 1/16	4 37/64	3 7/8	2 1/2	2 1/2	1 1/2	65
3	31 43/64	26 7/16	5 15/64	4 5/8	3	3	2	83
4	42 27/64	35 5/16	7 7/64	5 7/8	4	4	2 1/2	105
5	53 55/64	45 7/8	7 63/64	7 1/2	6	5	3	300
6	64 21/64	54 1/2	9 53/64	7 1/2	6	6	3	450

Note: For illustrative purposes only. Do NOT use for design or layout. Always obtain certified drawings from the manufacturer.

Table 3.2.2: Dimensions of Standard Single-Stage Ejectors (Threaded Connections)

Connection Size (inches)	J (Overall Length) (inches)	K (Suction Connection to Nozzle) (inches)	L (Diffuser Length) (inches)	M (Discharge Connection to Diffuser End) (inches)	Steam Inlet Connection Size (inches)	Approximate Weight (lbs)
1 1/2	19 13/16	4 7/8	12 3/4	3	1	60
2	22 5/8	5 1/2	14 3/4	3 1/4	1 1/4	75
2 1/2	26 7/16	6	18 1/16	3 3/8	1 1/2	89
3	30 7/16	6 1/8	21 13/16	3 3/4	2	122
4	40 7/8	5 5/8	35 1/4	6 1/4	2 1/2	260
5	52 3/8	6 1/2	45 7/8	8	3	320
6	61	6 1/2	54 1/2	8	3	400
8	ON APPLICATION	ON APPLICATION	ON APPLICATION	ON APPLICATION	ON APPLICATION	ON APPLICATION

Note: For illustrative purposes only. Do NOT use for design or layout. Always obtain certified drawings from the manufacturer.

4. Calculation Methods and Formulas

What formulas are used to size steam jet ejector and condenser systems?

This section provides the calculation methods and formulas necessary for sizing steam jet ejector and condenser systems. Accurate sizing relies on a thorough understanding of the process parameters and the application of appropriate engineering principles. The following subsections detail the calculations for non-condensable load, vapor load, motive steam requirements, and intercondenser sizing.

4.1. Non-Condensable Load Calculation

How do you calculate the non-condensable load in an ejector system?

The non-condensable load consists of gases that cannot be condensed under the operating conditions of the condensers. Air is the most common non-condensable, entering the system through leaks in flanges, valve packings, pump seals, and other connections. Accurate determination of this load is crucial for proper ejector sizing, as underestimation can lead to an inability to achieve the desired vacuum, while overestimation results in an oversized, inefficient system.

4.1.1. Air Leakage Rate Estimation

How can I estimate the air leakage rate in a vacuum system?

Estimating the air leakage rate is challenging because it depends on numerous factors, including the size and complexity of the system, the quality of equipment, and maintenance practices. The most reliable source for estimation methods is the Heat Exchange Institute (HEI) "Standards for Steam Jet Vacuum Systems." The following methods are commonly employed:

a) Empirical Estimation:
This method uses correlations based on the system's volume or surface area for a rough estimate. A common volume-based formula is:
Air Leakage Rate (kg/h) = K * System Volume (m³)
Where K is an empirical constant, typically ranging from 0.001 to 0.005 kg/h·m³ for well-maintained systems.

b) Component-Based Estimation:
This more accurate method involves estimating the leakage rate from individual components (flanges, valves, seals) and summing them to obtain the total.
Air Leakage Rate (kg/h) = Σ (Leakage Rate per Component)
Leakage rates for common components can be found in industry standards or vendor data.

c) Pressure Decay Testing:
This is the most accurate method, providing a direct measurement of the system's actual leakage rate. The system is evacuated to a known pressure (P1), isolated, and the pressure (P2) is recorded after a time interval (t). The leakage rate is then calculated using the Ideal Gas Law:
Air Leakage Rate (kg/h) = [ (P2 - P1) / t ] * (V * MW_air / (R * T)) * 3600
Where:
P2 , P1 = Pressure in Pascals (Pa)
t = Time in seconds (s)
V = System volume in m³
MW_air = Molecular weight of air, 28.97 kg/kmol
R = Ideal gas constant, 8314 Pa·m³/(kmol·K)
T = Absolute temperature in Kelvin (K)
3600 = Conversion factor from kg/s to kg/h.

A safety factor of 1.2 to 1.5 is commonly applied to the final estimated leakage rate to account for uncertainties.

4.2. Vapor Load Calculation

How do you calculate the vapor load in an ejector system?

The vapor load represents the mass flow rate of condensable vapors entering the ejector system, originating from the process itself. Accurately determining the vapor load is crucial for proper condenser sizing and optimizing motive steam consumption.

4.2.1. Determining Water Vapor Load

How do you determine the water vapor load in an ejector system?

Water vapor is a common component of the vapor load, arising from sources like moisture in the process feed, steam stripping, or evaporation from process liquids. The total water vapor load (ṁH₂O) is the sum of contributions from each source:
ṁH₂O = ṁfeed * wH₂O,feed + ṁsteam + ṁevaporation
Where:
ṁfeed is the mass flow rate of the process feed.
wH₂O,feed is the moisture content of the process feed.
ṁsteam is the mass flow rate of stripping steam.
ṁevaporation is the mass flow rate of water due to evaporation, determined from vapor-liquid equilibrium (VLE) calculations.

4.2.2. Determining Solvent Vapor Load

How do you determine the solvent vapor load in an ejector system?

Solvent vapors often constitute a significant portion of the vapor load in processes involving distillation, evaporation, or drying. The solvent vapor load can be determined from process data, VLE calculations, and key solvent properties like molecular weight and vapor pressure. The calculation requires a detailed mass balance of all streams containing the solvent. For non-ideal solutions, activity coefficients must be considered in the VLE calculations.

4.2.3. Using Vapor Pressure Data and Antoine Equation

How can I use the Antoine equation to calculate vapor pressure?

Table 4.2.3: Antoine Equation Coefficients for Vapor Pressure Calculation

Substance	A	B	C	T min. (°C)	T max. (°C)
Water	8.07131	1730.63	233.426	-20	100
Water	8.14019	1810.94	244.485	99	374
Ethanol	8.20417	1642.89	230.300	-57	80
Ethanol	7.68117	1332.04	199.200	77	243

The Antoine equation is a widely used empirical correlation for estimating the vapor pressure of pure substances as a function of temperature:
log₁₀(P) = A - B / (C + T)
Where:
P = Vapor pressure (mmHg)
T = Temperature (°C)
A , B , and C are Antoine coefficients specific to the substance.

Once the vapor pressure of each pure component ( Pᵢ° ) is known, Raoult's Law can be used to estimate the partial pressure ( Pᵢ ) of that component in an ideal mixture:
Pᵢ = xᵢ * Pᵢ°
Where xᵢ is the mole fraction of component i in the liquid phase. For non-ideal solutions, an activity coefficient ( γᵢ ) must be included:
Pᵢ = γᵢ * xᵢ * Pᵢ°

The mass flow rate of each vapor component can then be calculated, and the total vapor load is the sum of the mass flow rates of all condensable components.

4.3. Motive Steam Calculation

How do you calculate the motive steam requirements for an ejector system?

The motive steam provides the energy to drive the ejector. An accurate calculation of the motive steam flow rate is crucial for optimizing system efficiency and minimizing operating costs.

4.3.1. Estimating Motive Steam Flow Rate

What factors influence the motive steam flow rate in an ejector?

The motive steam flow rate is primarily determined by the suction pressure, discharge pressure, non-condensable load, vapor load, ejector design, and the pressure and temperature of the motive steam itself. Higher motive steam pressure generally leads to lower steam consumption, and superheated steam is often preferred to avoid condensation within the nozzle, which can reduce efficiency and cause erosion.

4.3.2. Using Charts and Formulas for Steam Consumption

How can I estimate steam consumption using charts and formulas?

The most accurate method for determining the motive steam flow rate is to consult performance curves or tables provided by the ejector manufacturer. These charts are generated from experimental data and relate motive steam consumption to suction pressure, discharge pressure, and total load for a specific ejector model.

When manufacturer's data is unavailable, a simplified approach involves using a steam consumption factor (SCF), which represents the mass of motive steam required per unit mass of suction load. However, this method is a major oversimplification suitable only for very rough, pre-engineering estimates. The SCF is highly dependent on the compression ratio of each ejector stage. A real-world design requires analyzing each stage separately, as the load and compression ratio for each stage are different, leading to different steam requirements.

For a preliminary estimate, the total motive steam flow rate can be approximated as:
Motive Steam Flow Rate (kg/h) = SCF * (Non-Condensable Load (kg/h) + Vapor Load (kg/h))

Typical SCF values for the total system are:

• Single-stage ejector: 2-5 kg steam/kg suction
• Two-stage ejector: 5-10 kg steam/kg suction
• Three-stage ejector: 10-15 kg steam/kg suction

It is critical to ensure the motive steam is dry and supplied at a consistent pressure, accounting for any pressure losses in the supply line.

4.4. Intercondenser Sizing

How do you size an intercondenser for a multi-stage ejector system?

Intercondensers are crucial in multi-stage systems for condensing motive steam and process vapors between stages, thereby reducing the load on subsequent stages and improving efficiency.

4.4.1. Calculating Intercondenser Heat Duty (Q)

How do you calculate the heat duty of an intercondenser?

The intercondenser heat duty (Q) is the amount of heat that must be removed to condense the vapors. It is the foundation for selecting an appropriate condenser design. The heat duty is calculated by summing the heat removed from the motive steam and any other condensable process vapors:
Q = m_steam * h_fg + m_vapor * (h_vapor - h_liquid)
Where:
Q = Heat duty (kW)
m_steam = Mass flow rate of motive steam (kg/s)
h_fg = Latent heat of vaporization of steam (kJ/kg)
m_vapor = Mass flow rate of condensable process vapors (kg/s)
h_vapor = Enthalpy of condensable vapors at the intercondenser inlet (kJ/kg)
h_liquid = Enthalpy of condensate at the intercondenser outlet (kJ/kg)

Accurate enthalpy data from steam tables or thermodynamic software is crucial for this calculation.

4.4.2. Applying Q = U * A * ΔT_lm

How do you use the heat transfer equation to size an intercondenser?

Once the heat duty (Q) is known, the required heat transfer area (A) of the condenser is determined using the fundamental heat transfer equation:
Q = U * A * ΔT_lm
Where:
U = Overall heat transfer coefficient (kW/m²·K), which depends on materials, fluid flow rates, and fouling.
A = Heat transfer area (m²), the primary output of the sizing calculation.
ΔT_lm = Log mean temperature difference (K), the effective temperature driving force for heat transfer.

The log mean temperature difference (ΔT_lm) is calculated as:
ΔT_lm = (ΔT₁ - ΔT₂) / ln(ΔT₁ / ΔT₂)
Where:
ΔT₁ = Temperature difference between the hot fluid inlet and the cold fluid outlet.
ΔT₂ = Temperature difference between the hot fluid outlet and the cold fluid inlet.

For multi-pass condensers, a correction factor may be needed. Sizing is often an iterative process, adjusting assumptions for U and pressure drop until a consistent design is achieved.

5. Calculation Examples

5.1 Example — sizing a two-stage steam jet ejector system (distillation top vapors: ethanol + water)

Problem statement (given):

• Column top (suction) pressure: 50 mmHg abs.

• Non-condensable (air) leakage: ṁ_air = 0.50 kg/h.

• Vapor load: ṁ_ethanol = 100 kg/h, ṁ_water = 50 kg/h. Vapor temperature (process): 40 °C.

• Motive steam available: 10 bar abs, dry saturated (note: final motive pressure at the nozzle will be lower due to supply losses; verify with plant steam tables).

• Cooling water: inlet 25 °C; assume cooling-water outlet target ≈40 °C (adjust to site limits).

• Discharge pressure: 760 mmHg (atmospheric).

• We will assume an intercondenser (surface condenser) and an initial assumed intercondenser pressure P_ic = 200 mmHg abs (to be iterated).

Goal:

1. Show how to compute non-condensables and vapor molar flows.

2. Show intercondenser heat duty (Q) calculation with verified enthalpy use.

3. Show cooling-water flow calculation and LMTD area sizing (illustrative U choices).

4. Explain how to obtain motive steam requirement (manufacturer curves) and show the iterative loop you must perform.

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5.1.1 Non-condensable (leak) handling — verified formula

If you need to measure leak rate from a pressure-rise test:

1. Run a pressure-rise test: evacuate to P₁, isolate, measure pressure P₂ after Δt seconds. Use the pressure-rise gas flow formula (Leybold):

q_{gas} = V \cdot \frac{\Delta p}{\Delta t} \quad\text{(units: mbar·L/s or Pa·m^3/s)}

Convert to mass flow (kg/s):

$$\dot m = \frac{q_{gas}\,M}{R\,T}$$m˙=RTqgas​M​

where $M$M is molecular weight (kg/kmol), $R=8314\ \text{J/(kmol·K)}$R=8314 J/(kmol\cdotpK), $T$T Kelvin. (Leybold reference). Example: Leybold's q = V·Δp/Δt method. Leybold

In this worked example we use the given leak: 0.50 kg/h (0.0001389 kg/s) as provided.

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5.1.2 Vapor loads → molar & check consistency

Convert mass → molar flow (for checks or VLE work):

$$\dot n_i = \frac{\dot m_i}{M_i}$$n˙i​=Mi​m˙i​​

• Ethanol $M_{EtOH} = 46.07\ \text{g/mol}=0.04607\ \text{kg/mol}$MEtOH​=46.07 g/mol=0.04607 kg/mol →
  $\dot n_{EtOH} = \dfrac{100\ \text{kg/h}}{0.04607} = 2170\ \text{mol/h} = 0.603\ \text{mol/s}.$n˙EtOH​=0.04607100 kg/h​=2170 mol/h=0.603 mol/s.

• Water $M_{H_2O}=18.015\ \text{g/mol}=0.018015\ \text{kg/mol}$MH2​O​=18.015 g/mol=0.018015 kg/mol →
  $\dot n_{H_2O} = \dfrac{50\ \text{kg/h}}{0.018015} = 2775\ \text{mol/h} = 0.771\ \text{mol/s}.$n˙H2​O​=0.01801550 kg/h​=2775 mol/h=0.771 mol/s.

Total vapor molar flow = 1.374 mol/s. (These conversions are exact arithmetic.) Use these for any VLE checks. (Molecular weights: standard.)

VLE / partial-pressure note: At low absolute pressures (e.g., 50 mmHg) do not assume Raoult’s law blindly — use NIST WebBook or a thermodynamic model (NRTL/UNIQUAC) to get accurate vapor compositions at the given P & T. NIST provides Antoine coefficients and enthalpy of vaporization functions. For Antoine fits use the temperature-range appropriate coefficients from NIST. NIST WebBook+1

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5.1.3 Intercondenser heat duty $Q$Q — verified formula & numerical worked example

Heat-duty formula (verified):

$$Q \;=\; \dot m_{steam,1}\,\Delta h_{steam}^{(cond)} \;+\; \sum_i \dot m_i\,(h_{i,\text{vap}} - h_{i,\text{liq}})$$Q=m˙steam,1​Δhsteam(cond)​+i∑​m˙i​(hi,vap​−hi,liq​)

where for each condensed species $i$i the bracket is the latent heat at the condenser saturation temperature (and includes any sensible cooling to the condenser saturation temperature if needed). Use enthalpies at the actual intercondenser saturation temperature (lookup in steam tables / NIST).

Procedure (numerical example):

1. Assume (for illustration) the first stage motive steam mass after selecting an ejector is $\dot m_{steam,1} = 750\ \text{kg/h}$m˙steam,1​=750 kg/h (this is an engineering assumption for this worked illustration — final motive steam must be read from manufacturer curves; see Section below). Convert to kg/s:

$$750\ \text{kg/h} = \frac{750}{3600} = 0.20833\ \text{kg/s}.$$750 kg/h=3600750​=0.20833 kg/s.

2. Choose intercondenser pressure $P_{ic}=200\ \text{mmHg}$Pic​=200 mmHg. From steam tables (NIST / Spirax / TLV) saturation temperature at 200 mmHg ≈ 60 °C. From saturated-steam tables (NIST / Spirax), latent heat at 60 °C:

$$\Delta h_{steam}^{(60^\circ\mathrm{C})} \approx 2358.5\ \text{kJ/kg}.$$Δhsteam(60∘C)​≈2358.5 kJ/kg.

(Use the exact h_v and h_l from your steam tables; NIST/Spirax calculators are authoritative.) Materiasspiraxsarco.com

3. For ethanol and water, obtain enthalpy of vaporization at the same condenser saturation temperature (or use literature ΔvapH at the relevant T from NIST). For illustration we use representative values (use NIST for exact design):

• Ethanol latent (approx, literature/NIST range): $\Delta h_{vap,EtOH} \approx 919\ \text{kJ/kg}$Δhvap,EtOH​≈919 kJ/kg (≈42.3 kJ/mol -> ≈919 kJ/kg). Engineering ToolboxNIST WebBook

• Water latent at 60 °C: $\Delta h_{vap,H_2O} \approx 2358.5\ \text{kJ/kg}$Δhvap,H2​O​≈2358.5 kJ/kg. (from steam tables). Materias

Estimate liquid-phase enthalpies (reference 0 °C baseline or use hf from tables). For quick engineering estimate, use sensible heating of liquid from reference to condensing temperature using cp_liquid·ΔT. Example estimates at 60 °C:

• Ethanol liquid enthalpy (approx): $h_{EtOH,liq,60}\approx 2.4\ \text{kJ/kg·K}\times(60-25)\approx 84\ \text{kJ/kg}.$hEtOH,liq,60​≈2.4 kJ/kg\cdotpK×(60−25)≈84 kJ/kg.

• Water liquid enthalpy at 60 °C (from steam tables): $h_{H_2O,liq,60}\approx 251.1\ \text{kJ/kg}.$hH2​O,liq,60​≈251.1 kJ/kg. Materias

4. Convert given vapor masses to kg/s:

$$\dot m_{EtOH} = \dfrac{100}{3600} = 0.02778\ \text{kg/s};\quad \dot m_{H_2O} = \dfrac{50}{3600} = 0.01389\ \text{kg/s}.$$m˙EtOH​=3600100​=0.02778 kg/s;m˙H2​O​=360050​=0.01389 kg/s.

5. Compute $Q$Q (kW) using the formula (all kJ/s → kW):

$$\begin{aligned} Q & = \dot m_{steam,1}\cdot\Delta h_{steam} \;+\;\dot m_{EtOH}\cdot(\Delta h_{EtOH}) \;+\; \dot m_{H_2O}\cdot(\Delta h_{H_2O})\\[6pt] &\approx 0.20833(2358.5) + 0.02778(919 - 84) + 0.01389(2358.5 - 251.1)\\[4pt] &\approx 490.9\ \text{kW} \;+\; 22.9\ \text{kW}\;+\;29.3\ \text{kW}\\[2pt] &\approx 543.1\ \text{kW}. \end{aligned}$$Q​=m˙steam,1​⋅Δhsteam​+m˙EtOH​⋅(ΔhEtOH​)+m˙H2​O​⋅(ΔhH2​O​)≈0.20833(2358.5)+0.02778(919−84)+0.01389(2358.5−251.1)≈490.9 kW+22.9 kW+29.3 kW≈543.1 kW.​

Comment: This numeric result matches the order of magnitude of your original calculation; the difference arises from which latent enthalpy values are used. For final design, replace the illustrative enthalpy numbers above with exact h_v and h_l from NIST/steam tables at the chosen intercondenser saturation temperature. NIST WebBookMaterias

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5.1.4 LMTD, U and heat-transfer area $A$A

LMTD (log mean temperature difference):

If hot stream condenses at a fixed saturation temperature $T_s$Ts​ (≈ 60 °C) and cooling water inlet/outlet temperatures are T_c,in = 25 °C, T_c,out = 40 °C, then:

$$\Delta T_1 = T_s - T_{c,out} = 60 - 40 = 20\ ^\circ\mathrm{C}$$ΔT1​=Ts​−Tc,out​=60−40=20 ∘C $$\Delta T_2 = T_s - T_{c,in} \;=\; 60 - 25 = 35\ ^\circ\mathrm{C}$$ΔT2​=Ts​−Tc,in​=60−25=35 ∘C $$\Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1/\Delta T_2)}.$$ΔTlm​=ln(ΔT1​/ΔT2​)ΔT1​−ΔT2​​.

Numerically $\Delta T_{lm}\approx 26.7\ ^\circ\mathrm{C}$ΔTlm​≈26.7 ∘C.

Overall U (guidance): For steam condensing → water surface condensers typical U ≈ 1000–3000 W/m²·K (condensing side dominated). Using literature ranges is acceptable for screening; vendor data is required for final design. (Engineering Toolbox and heat-exchanger literature). Engineering Toolboxengineeringpage.com

Area (example):

$$A = \frac{Q}{U\,\Delta T_{lm}}.$$A=UΔTlm​Q​.

Using two example U values:

• With $U=500\ \mathrm{W/m^2K}$U=500 W/m2K (conservative, low value you had):
  $A \approx \dfrac{543{,}100}{500\times26.7} \approx 40.7\ \mathrm{m^2}.$A≈500×26.7543,100​≈40.7 m2.

• With a more typical condensing value $U=1200\ \mathrm{W/m^2K}$U=1200 W/m2K:
  $A \approx \dfrac{543{,}100}{1200\times26.7} \approx 17.0\ \mathrm{m^2}.$A≈1200×26.7543,100​≈17.0 m2.

Takeaway: U is decisive. Use vendor data / empirical correlations for the chosen condenser geometry and fouling factor. (Engineering toolbox guidance & vendor catalogs). engineeringpage.comEngineering Toolbox

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5.1.5 Cooling water flow $\dot m_{cw}$m˙cw​

Use energy balance:

$$Q = \dot m_{cw}\, c_{p,water}\, \Delta T_{cw}.$$Q=m˙cw​cp,water​ΔTcw​.

With $c_{p,water}\approx 4.186\ \text{kJ/kg·K}$cp,water​≈4.186 kJ/kg\cdotpK and ΔT_cw = 15 K (25→40 °C):

$$\dot m_{cw} = \frac{Q}{c_p\Delta T} = \frac{543.1\ \text{kW}}{4.186\ \text{kJ/kg·K} \times 15\ \text{K}} \approx 8.65\ \text{kg/s} \approx 31{,}140\ \text{kg/h}.$$m˙cw​=cp​ΔTQ​=4.186 kJ/kg\cdotpK×15 K543.1 kW​≈8.65 kg/s≈31,140 kg/h.

(Use real cooling-water limits; permit smaller ΔT if cooling-tower constraints apply.)

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5.1.6 Motive steam requirement (how to determine, not a hand-wave)

Important: There is no robust physics-only closed-form that yields a final, accurate motive-steam mass flow for an ejector of unknown geometry — ejector performance is characterized empirically and delivered by vendors as “motive steam consumption vs suction pressure vs load vs motive pressure” curves. Therefore the correct engineer workflow is:

1. Define stage targets: suction pressures, discharge pressures (e.g., stage-1 suction = 50 mmHg → stage-1 discharge = P_ic (start with 200 mmHg)). Compute compression ratios.

2. Compute loads entering each stage: stage-1 load = non-condensables + full vapor load; stage-2 load = non-condensables + uncondensed vapor + any carryover from intercondenser. (Use Q and condenser effectiveness to estimate uncondensed fractions.)

3. Obtain manufacturer performance curves (e.g., Penberthy, Graham, Körting, Schutte & Koerting, etc.) for the motive steam pressure available. Enter the curves with (suction pressure, discharge pressure, load) to read required motive steam for each stage. Vendor datasheets / performance charts are the authoritative source. Emerson+1

If vendor data is not available and a screening estimate is needed, you can use published empirical correlations (e.g., Fondrk / historical data) only for screening and then verify with vendor curves. I do not recommend using fixed SCF ranges as final inputs. (I flagged and removed the un-sourced SCF numbers from your draft for this reason.) Scribd

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5.1.7 Iteration loop — how to converge a final design

1. Pick intercondenser pressure $P_{ic}$Pic​ (initial guess).

2. From vendor curves (for chosen ejector candidate), read stage-1 motive steam → iterate to obtain $\dot m_{steam,1}$m˙steam,1​.

3. Calculate intercondenser duty $Q$Q and condenser effectiveness → determine uncondensed flows.

4. Update stage-2 load and read stage-2 motive steam from vendor curves.

5. Check that discharge pressure of stage-2 = plant atmosphere or permitted back-pressure; check steam supply capability and condensate handling.

6. Iterate $P_{ic}$Pic​ and equipment choices until mass/energy and vendor performance are consistent.

Source

    https://graham-mfg.com/wp-content/uploads/2023/02/steam_jet_ejector-functionality_performance_considerations.pdf
    https://www.thechemicalengineer.com/features/rules-of-thumb-vent-condenser-with-non-condensables/
    https://en.wikipedia.org/wiki/Antoine_equation