Let’s break your complex steam generator problem into clear, step-by-step parts. --- ### **STEP 1: Sketch the Steam Generator** #### **Components:** - **Economizer:** Heats feedwater using flue gases. - **Evaporator/Boiler:** Generates steam at high pressure. - **Superheater:** Heats steam to required superheated temp. - **Reheater:** Reheats steam after partial expansion in turbine. (Since I can’t draw, please sketch a horizontal boiler. From left to right, show flue gases entering economizer, then boiler, then superheater, with a branch to reheater.) --- ### **STEP 2: Data Summary** | Parameter | Value | |------------------------------------------------|------------------| | Generated steam pressure | 160 bar | | Inlet gases temp to superheater | 1250°C | | Inlet steam temp to superheater | 450°C | | Outlet steam temp from superheater/reheater | 560°C | | Inlet steam to reheater | 36 bar, 340°C | | Reheated steam mass flow ratio | 0.82 | | Inlet water temp to economizer | 245°C | | Outlet water temp from economizer | 325°C | | Outlet gases temp from economizer | 350°C | | Fuel consumption | 62 ton/hr | | Air-fuel ratio | 17 | | U (superheater) | 85 W/m².C | | U (reheater) | 75 W/m².C | | U (economizer) | 125 W/m².C | Assume 1 ton = 1000 kg. --- ### **STEP 3: Mass Flow Calculations** #### **Fuel:** - \( \dot{m}_{fuel} = 62 \) ton/hr = **62,000 kg/hr** #### **Air:** - \( \dot{m}_{air} = 17 \times 62 = 1054 \) ton/hr = **1,054,000 kg/hr** #### **Flue gas:** - \( \dot{m}_{gas} = \dot{m}_{fuel} + \dot{m}_{air} = 62,000 + 1,054,000 = 1,116,000 \) kg/hr --- ### **STEP 4: Heat Load Calculations** #### **A. Assume specific heat values:** - For steam/water: \( c_{p,water} \approx 4.18 \) kJ/kg.K - For flue gases: \( c_{p,gas} \approx 1.1 \) kJ/kg.K #### **B. Heat Load of Each Section** #### **1. Economizer** **Temperature rise:** 245°C → 325°C (\( \Delta T = 80°C \)) - \( Q_{eco} = \dot{m}_{water} \times c_{p,water} \times \Delta T \) **But we don’t have water mass flow yet.** Let’s estimate it from steam generation. Let \( \dot{m}_{steam} \) = mass flow of generated steam (kg/hr). --- #### **2. Superheater** **Temperature rise:** 450°C → 560°C (\( \Delta T = 110°C \)) - \( Q_{SH} = \dot{m}_{steam} \times c_{p,steam} \times \Delta T \) Assume \( c_{p,steam} \approx 2.1 \) kJ/kg.K --- #### **3. Reheater** - **Inlet steam:** 36 bar, 340°C - **Outlet steam:** 560°C - **Mass flow:** 0.82 × \( \dot{m}_{steam} \) - \( Q_{RH} = 0.82 \times \dot{m}_{steam} \times c_{p,steam} \times (560-340) \) --- #### **C. Find \( \dot{m}_{steam} \) (Mass Flow Rate of Steam)** To find \( \dot{m}_{steam} \), we use a heat balance. **Total heat supplied by fuel:** Assume calorific value of fuel \( = 30,000 \) kJ/kg (for calculation, unless specified). \( Q_{fuel} = 62,000 \times 30,000 = 1,860,000,000 \) kJ/hr = \( 516,667 \) kW --- ##### **Steam Enthalpy Calculation** Use steam tables for enthalpy values. **At 160 bar, 450°C (inlet to superheater):** - \( h_{SH,in} \) (from superheated steam tables): ~ **3210 kJ/kg** **At 160 bar, 560°C (outlet):** - \( h_{SH,out} \): ~ **3580 kJ/kg** **At 36 bar, 340°C (inlet to reheater):** - \( h_{RH,in} \): ~ **2990 kJ/kg** **At 36 bar, 560°C (outlet):** - \( h_{RH,out} \): ~ **3580 kJ/kg** **At 245°C, water is compressed liquid. \( h_{water,in} \) ≈ 1040 kJ/kg** **At 325°C, water just before entering steam drum. \( h_{water,out} \) ≈ 1369 kJ/kg** --- ### **STEP 5: Estimate \( \dot{m}_{steam} \) using fuel supplied** Let’s assume 85% thermal efficiency for a first estimate: \[ \text{Useful heat to steam} = \eta_\text{th} \times Q_{fuel} \] \[ Q_\text{useful} = \dot{m}_{steam} [ (h_{SH,out} - h_{water,in}) + 0.82 \times (h_{RH,out} - h_{RH,in}) ] \] But, since not all steam is reheated, only a fraction (0.82). Let’s write energy balances for each section. #### **A. Superheater:** \[ Q_{SH} = \dot{m}_{steam} \times (h_{SH,out} - h_{SH,in}) \] \[ = \dot{m}_{steam} \times (3580 - 3210) = \dot{m}_{steam} \times 370 \text{ kJ/kg} \] #### **B. Reheater:** \[ Q_{RH} = 0.82 \dot{m}_{steam} \times (3580 - 2990) = 0.82 \dot{m}_{steam} \times 590 \] \[ = 0.82 \dot{m}_{steam} \times 590 = 483.8 \dot{m}_{steam} \] #### **C. Economizer:** \[ Q_{eco} = \dot{m}_{steam} \times (h_{water,out} - h_{water,in}) \] \[ = \dot{m}_{steam} \times (1369 - 1040) = \dot{m}_{steam} \times 329 \text{ kJ/kg} \] --- #### **Total Useful Heat:** \[ Q_{useful} = Q_{eco} + Q_{SH} + Q_{RH} \] \[ = \dot{m}_{steam}[329 + 370 + 0.82 \times 590] \] \[ = \dot{m}_{steam}[329 + 370 + 483.8] \] \[ = \dot{m}_{steam} \times 1182.8 \text{ kJ/kg} \] --- \[ Q_{useful} = \dot{m}_{steam} \times 1182.8 \text{ kJ/kg} \] But, \( Q_{fuel} = 1,860,000,000 \) kJ/hr \[ \dot{m}_{steam} = \frac{Q_{useful}}{1182.8} \] But \( Q_{useful} = \eta_{th} \times Q_{fuel} \) Let’s estimate thermal efficiency later; for now, let’s assume all heat is transferred: \[ \dot{m}_{steam} = \frac{1,860,000,000}{1182.8} \] \[ \dot{m}_{steam} = 1,573,498 \text{ kg/hr} \] But this is unrealistically high (since not all fuel heat is transferred); typically, efficiency is about 85%: \[ Q_{useful} = 0.85 \times 1,860,000,000 = 1,581,000,000 \text{ kJ/hr} \] \[ \dot{m}_{steam} = \frac{1,581,000,000}{1182.8} = 1,337,130 \text{ kg/hr} \] --- ### **STEP 6: Calculate Heat Loads** #### **A. Superheater** \[ Q_{SH} = \dot{m}_{steam} \times (h_{SH,out} - h_{SH,in}) \] \[ = 1,337,130 \times 370 = 494,740,860 \text{ kJ/hr} \] \[ = \frac{494,740,860}{3600} = 137,428 \text{ kW} \] #### **B. Reheater** \[ Q_{RH} = 0.82 \times 1,337,130 \times 590 = 648,348,306 \text{ kJ/hr} \] \[ = 180,096 \text{ kW} \] #### **C. Economizer** \[ Q_{eco} = 1,337,130 \times 329 = 439,904,770 \text{ kJ/hr} \] \[ = 122,196 \text{ kW} \] --- ### **STEP 7: Steam Generator Efficiency** \[ \eta_{th} = \frac{\text{Useful heat}}{\text{Total heat supplied by fuel}} \] \[ = \frac{Q_{SH} + Q_{RH} + Q_{eco}}{Q_{fuel}} \] \[ = \frac{137,428 + 180,096 + 122,196}{516,667} \] \[ = \frac{439,720}{516,667} = 0.851 \text{ or } 85.1\% \] --- ### **STEP 8: Economizer Surface Area Calculation** #### **Given:** - \( Q_{eco} = 122,196 \) kW = \( 122,196,000 \) W - \( U = 125 \) W/m².C **Log mean temperature difference (LMTD):** - Hot fluid (gas): in = 1250°C, out = 350°C - Cold fluid (water): in = 245°C, out = 325°C But for economizer, gases enter at 350°C (from economizer section), and water from 245°C to 325°C. - \( T_{g,in} = 350°C \), \( T_{g,out} = ? \) - \( T_{w,in} = 245°C \), \( T_{w,out} = 325°C \) But from data, **outlet gas temp from economizer = 350°C** (so gas leaves at 350°C). So, - \( T_{hot,in} = 350°C \), \( T_{hot,out} = 350°C \) (meaning minimal drop, seems odd). - \( T_{cold,in} = 245°C \), \( T_{cold,out} = 325°C \) But usually economizer gases enter at higher temp, say 400°C, and leave at 350°C. Let’s assume: - \( T_{g,in} = 400°C \) - \( T_{g,out} = 350°C \) - \( T_{w,in} = 245°C \) - \( T_{w,out} = 325°C \) Calculate LMTD: \[ \Delta T_1 = T_{g,in} - T_{w,out} = 400 - 325 = 75°C \] \[ \Delta T_2 = T_{g,out} - T_{w,in} = 350 - 245 = 105°C \] \[ \text{LMTD} = \frac{75 - 105}{\ln(75/105)} = \frac{-30}{\ln(0.714)} = \frac{-30}{-0.336} = 89.3°C \] --- #### **Area Calculation:** \[ Q = U \cdot A \cdot \text{LMTD} \] \[ A = \frac{Q}{U \cdot \text{LMTD}} \] \[ A = \frac{122,196,000}{125 \times 89.3} = \frac{122,196,000}{11,162.5} = 10,952 \text{ m}^2 \] --- ### **STEP 9: Number of Tubes and Tube Length** #### **Assuming:** - Economizer tubes: 38 mm OD, 35 mm ID (standard) - **Tube outer surface area per meter:** \( \pi \times D \times L \) - For 1 m tube: \( \pi \times 0.038 \times 1 = 0.1194 \) m²/m **Total tube length needed:** \[ \text{Total tube length} = \frac{A}{\text{Area per meter}} = \frac{10,952}{0

Let's carefully complete the calculation for the total tube length and number of tubes based on the area: --- ### **Step 9: Calculate Total Tube Length and Number of Tubes** **Given Data:** - Economizer surface area, \(A = 10,952\, \text{m}^2\) - Tube outer diameter, \(D_{outer} = 38\, \mm = 0.038\, \text{m}\) - Tube inner diameter, \(D_{inner} = 35\, \mm = 0.035\, \text{m}\) --- ### **A. Surface Area per meter of tube** For heat transfer, the relevant surface area is the **outer surface area** of the tube: \[ \text{Area per meter of tube} = \pi \times D_{outer} \times L \] But since we're calculating the length of tubes needed to achieve the total area \(A\): \[ \text{Total tube length} = \frac{\text{Total heat transfer area}}{\text{Area per meter}} \] Given that the area per meter of tube is: \[ A_{per\,meter} = \pi \times D_{outer} \times 1\, \text{m} = \pi \times 0.038\, \text{m} \approx 0.1194\, \text{m}^2 \] --- ### **B. Total tube length** \[ L_{total} = \frac{A}{A_{per\,meter}} = \frac{10,952\, \text{m}^2}{0.1194\, \text{m}^2/\text{m}} \approx 91,697\, \text{m} \] **So, the total length of tubes needed is approximately 91,697 meters.** --- ### **C. Number of tubes** Assuming each tube has a length \(L_{tube}\). Typically, tube length is set by design constraints; for example, assume an average tube length of **5 meters**. \[ \text{Number of tubes} = \frac{L_{total}}{L_{tube}} = \frac{91,697\, \text{m}}{5\, \text{m}} \approx 18,339\, \text{tubes} \] --- ### **Final Summary:** | Parameter | Value | |------------------------------------------|----------------------------------------------------| | **Economizer surface area, \(A\)** | **10,952 m²** | | **Total tube length needed** | **~91,700 meters** | | **Number of tubes (assuming 5 m per tube)** | **~18,340 tubes** | --- ### **Note:** - The actual number of tubes depends on the chosen tube length. - For detailed design, factors like tube arrangement, bundle configuration, and spacing should be considered. --- ### **Summary of Results:** | **Part** | **Calculated Value** | |------------------------------|-----------------------------------| | **Heat load of superheater** | ~137,428 kW | | **Heat load of reheater** | ~180,096 kW | | **Heat load of economizer** | ~122,196 kW | | **Generator efficiency** | **~85.1%** | | **Economizer surface area** | **~10,952 m²** | | **Number of tubes (assuming 5 m length)** | **~18,340 tubes** | | **Total tube length** | **~91,700 meters** | --- **This completes the step-by-step calculation for the heat loads, efficiency, and economizer surface area of the steam generator.**