answer:To optimize the yield of the reaction between sulfuric acid (H2SO4) and sodium hydroxide (NaOH) for the production of sodium sulfate (Na2SO4) and water (H2O), given a fixed amount of reactants and reaction conditions, you can follow these steps: 1. Determine the stoichiometry of the reaction: The balanced chemical equation for this reaction is: H2SO4 + 2 NaOH → Na2SO4 + 2 H2O This means that one mole of sulfuric acid reacts with two moles of sodium hydroxide to produce one mole of sodium sulfate and two moles of water. 2. Calculate the limiting reactant: Determine the amount (in moles) of each reactant present, and compare their ratios to the stoichiometry of the reaction. The reactant that is present in the lowest ratio relative to its stoichiometric requirement is the limiting reactant. This reactant will be completely consumed during the reaction and will determine the maximum yield of the product. 3. Optimize reaction conditions: Ensure that the reaction conditions, such as temperature, pressure, and concentration, are optimal for the reaction to proceed efficiently. For this reaction, the conditions are generally ambient temperature and pressure, and the reactants should be mixed in their stoichiometric ratio. 4. Minimize side reactions: If there are any side reactions that could consume the reactants or produce unwanted byproducts, take steps to minimize their occurrence. For the reaction between sulfuric acid and sodium hydroxide, side reactions are generally not a concern. 5. Monitor reaction progress: Monitor the reaction to ensure that it is proceeding as expected. This can be done using various analytical techniques, such as titration or spectroscopy, to determine the concentration of reactants and products over time. 6. Maximize product recovery: After the reaction is complete, take steps to efficiently separate and purify the product (sodium sulfate) from the reaction mixture. This may involve techniques such as filtration, crystallization, or evaporation. By following these steps, you can optimize the yield of sodium sulfate from the reaction between sulfuric acid and sodium hydroxide, given a fixed amount of reactants and reaction conditions.,To optimize the yield of the reaction between sulfuric acid and sodium hydroxide for the production of sodium sulfate, given a fixed amount of reactants and reaction conditions, you can follow these steps: 1. Ensure an appropriate stoichiometric ratio: Maintaining the correct stoichiometric ratio between sulfuric acid (H2SO4) and sodium hydroxide (NaOH) is crucial to maximize the yield. The balanced chemical equation for this reaction is: H2SO4 (aq) + 2 NaOH (aq) → Na2SO4 (aq) + 2 H2O (aq) 2. Always use concentrated H2SO4 and NaOH solutions: Concentrated solutions provide a higher concentration of reactants, increasing the probability of successful collisions and a higher overall reaction rate. 3. Control the temperature: Conduct the reaction at an optimal temperature to increase the reaction rate without causing any side reactions. For this reaction, the optimal temperature is typically around 20-30°C. 4. Maintain a slow and steady addition rate: Adding the reactants slowly and steadily ensures a more controlled reaction, allowing for better control over the stoichiometric ratio and minimizing any side reactions. 5. Use a suitable catalyst: Adding a catalyst can help to increase the reaction rate and optimize yield. However, it's essential to ensure that the catalyst does not adversely affect your reaction. 6. Properly monitor and control the pH: The reaction between H2SO4 and NaOH is a pH-sensitive reaction. By monitoring and controlling the pH, you can ensure more efficient and complete reactions, maximizing the yield of Na2SO4. 7. Avoid side reactions: The addition of a buffer solution or a buffering couple can help to minimize any side reactions that may negatively impact your yield. 8. Separate the product: After the reaction is complete, separate the product, sodium sulfate, from the unreacted H2SO4 and NaOH using techniques such as filtration or crystallization. By following these steps, you can optimize the yield of the reaction between sulfuric acid and sodium hydroxide for the production of sodium sulfate, given a fixed amount of reactants and reaction conditions., To optimize the yield of the reaction between sulfuric acid (H2SO4) and sodium hydroxide (NaOH) for the production of sodium sulfate (Na2SO4), follow these steps: 1. Balance the chemical equation: Make sure that the chemical reaction is balanced. For this reaction, the balanced equation is: 2H2SO4 + 2NaOH → 2Na2SO4 + 2H2O 2. Determine the limiting reactant: Identify which reactant is present in a lesser amount, as this will limit the overall yield of the reaction. To do this, calculate the stoichiometric amounts of both reactants based on the balanced equation and compare them with the given amounts. 3. Maximize reaction volume: To ensure complete reaction between the reactants, provide sufficient volume for them to mix thoroughly. In an industrial setting, this might mean using larger reaction vessels or multiple reaction vessels in parallel. 4. Control reaction temperature: Maintain the reaction temperature within the optimal range. For the reaction between sulfuric acid and sodium hydroxide, the optimal temperature range is around 100-150°C (212-302°F). At lower temperatures, the reaction rate will be slower, while at higher temperatures, the risk of byproduct formation increases. 5. Ensure good mixing: To promote rapid reaction kinetics and increase the likelihood of reaching equilibrium, mix the reactants well. In a laboratory setting, this might mean stirring the reaction mixture mechanically. In an industrial setting, this could mean using impellers, baffles, or other mixing equipment. 6. Monitor reaction progress: Periodically sample the reaction mixture and analyze the concentrations of reactants and products. Based on this information, you can adjust reaction conditions as necessary to maximize product yield. 7. Remove heat of reaction: Both sulfuric acid and sodium hydroxide are exothermic reactants, meaning they release heat when they react. To minimize byproduct formation, manage the heat of reaction by adding cooling equipment, such as jacketed reactors or external cooling coils, to maintain an optimal reaction temperature. 8. Remove byproducts: If byproducts form, consider installing separation equipment to remove them from the reaction mixture. Removing byproducts can help shift the equilibrium toward the desired product and increase overall yield. 9. Recycle unreacted reactants: If any reactants remain unreacted after the reaction is complete, consider recycling them back into the reaction mixture to maximize reactant usage and reduce waste. 10. Optimize equipment design: In an industrial setting, optimize the design of reaction vessels and associated equipment to maximize efficiency and minimize waste. This might involve optimizing flow rates, pressure drops, heat transfer coefficients, or other engineering parameters.

answer:To design a distillation column to separate a mixture of benzene and toluene, we need to determine the reflux ratio and the number of theoretical trays required to meet the purity specification. We will use the Fenske-Underwood-Gilliland (FUG) method to estimate these parameters. 1. Determine the relative volatility (α) of benzene to toluene at 1 atm: At 1 atm, the boiling points of benzene and toluene are 80.1°C and 110.6°C, respectively. Using the Clausius-Clapeyron equation, we can estimate the vapor pressures of benzene (Pb) and toluene (Pt) at their boiling points. Then, we can calculate the relative volatility as α = Pb/Pt. For this problem, we will assume α ≈ 2.3. 2. Calculate the minimum reflux ratio (Rmin) using the Underwood equation: Rmin = (α - 1) / (α - xF), where xF is the mole fraction of benzene in the feed. Rmin = (2.3 - 1) / (2.3 - 0.4) ≈ 0.76 3. Choose an operating reflux ratio (R) above the minimum reflux ratio: A common practice is to use 1.2 times the minimum reflux ratio to ensure adequate separation. R = 1.2 * Rmin ≈ 1.2 * 0.76 ≈ 0.91 4. Calculate the number of theoretical trays (N) using the Fenske equation: N = ln(xD * (1 - xB) * (1 - xF)) / (xB * (1 - xD) * xF) / ln(α), where xD is the mole fraction of benzene in the distillate and xB is the mole fraction of benzene in the bottoms. For this problem, xD = 0.9 (90% purity of benzene in the distillate) and xB = 0.1 (assuming 10% benzene in the bottoms to ensure adequate separation). N ≈ ln(0.9 * 0.9 * 0.6) / (0.1 * 0.1 * 0.4) / ln(2.3) ≈ 7.5 Since the number of theoretical trays must be an integer, we will round up to 8 trays to ensure adequate separation. 5. Design the distillation column: The distillation column should be designed with 8 theoretical trays and an operating reflux ratio of 0.91. The column should be able to process a feed rate of 500 kg/h of the benzene-toluene mixture at 1 atm and constant temperature., To design a distillation column, we need to use some assumptions and simplifications. I will use the Fenske-Underwood-Gilliland method for this design. The following assumptions are made: 1. Constant molar overflow 2. Equimolar overflow 3. Ideal liquid and vapor phases 4. Steady-state operation 5. Complete separation in each tray 6. Constant relative volatility between benzene and toluene The relative volatility of benzene to toluene can be calculated using the following formula: α = (Kd / Kl) = (Pv_benzene / Py_benzene) / (Pv_toluene / Py_toluene) where Kd and Kl are the distribution coefficients of benzene and toluene, Pv_benzene and Pv_toluene are the vapor pressures at the column temperature, and Py_benzene and Py_toluene are the partial pressures of benzene and toluene. At 1 atm and at the boiling point of the mixture, the vapor pressures are approximately equal to the operating pressure. Thus, α ≈ (Pv_benzene / Py_benzene) ≈ (1 / 0.4) ≈ 2.5. Now, we can calculate the minimum number of trays required using the Fenske equation: Nmin = (ln(xDR / xD)(xR / xF)) / (ln α) where Nmin is the minimum number of trays, xDR, xD, xR, and xF are the mole fractions of benzene in the distillate, reflux, bottoms, and feed, respectively. For the given purity specification, we want xDR ≥ 0.9 and xF = 0.4. Since the bottoms stream should be pure toluene, xR = 0. To find xR, we can use a mass balance on benzene: F * xF = D * xDR + B * xR where F is the feed flow rate (500 kg/h), D is the distillate flow rate, and B is the bottoms flow rate. Since the column operates at a constant temperature, the total flow rate out of the column is equal to the feed flow rate: F = D + B Solving for D and xR, we get: D = F / (1 + (xR / xDR)) xR = (xF - xDR * D / F) / (B / F) Since xR = 0, we can calculate the bottoms-to-feed ratio (B / F): B / F = (xF - xDR * D / F) / xR = (0.6 - 0.9 * D / F) / 0 D / F = 0.6 / (1 - xDR) ≈ 1 B / F ≈ (0.6 - 0.9 * 0.6) / 0.9 ≈ 0.22 Now, we can calculate the minimum number of trays: Nmin = (ln(0.9 / 0.22)(0 / 0.4)) / (ln 2.5) ≈ 5.06 Using the Underwood method, we can calculate the minimum reflux ratio: Rmin = 1 / ((α - 1) * xDR - α * xR) Since xR = 0, we get: Rmin = 1 / ((2.5 - 1) * 0.9) ≈ 0.77 A safety factor can be added to the reflux ratio to ensure better separation: Reflux ratio = Rmin * SF Let's choose a safety factor of 1.5: Reflux ratio = 0.77 * 1.5 ≈ 1.15 Finally, using the Gilliland correlation, we can calculate the number of actual trays needed based on the reflux ratio (we'll use an efficiency of 70%): N = Nmin / 1 - exp(-(Nmin + 1) * (1 - q) * (α - 1) / (R + 1 + (α - 1)))^0.5 where N is the number of actual trays, and q is the liquid flow ratio (q = L / D, where L is the liquid flow rate). Since the column operates at a constant temperature, q ≈ 1. N ≈ 5.06 / 1 - exp(-(5.06 + 1) * (1 - 1) * (2.5 - 1) / (1.15 + 1 + (2.5 - 1)))^0.5 ≈ 7.88 Therefore, to separate the mixture of benzene and toluene with the given purity specification, a distillation column with at least 8 theoretical trays and a reflux ratio of approximately 1.15 is required. The actual number of trays will be higher due to the efficiency of the trays.,To design the distillation column for this separation, we need to determine the necessary reflux ratio (R) and the number of theoretical trays (N) required. We can use the VLE (Vapor-Liquid Equilibrium) data and an equilibrium stage model for this purpose. 1. Determine the boiling points of benzene and toluene at 1 atm: Benzene boiling point: 80.1 C Toluene boiling point: 111 C 2. Determine the relative volatility (α) of benzene and toluene (α = k1 / k2): Assuming the boiling points and VLE data remain unchanged, we can estimate the relative volatility. In reality, it would be necessary to use VLE data for accurate calculations. Let's assume α = 1.2 (a reasonable value for this separation). 3. Determine the reflux ratio (R) from the given feed composition and desired distillate purity: R = (L/D) = (1 - xF)/(α * xF) where xF is the mole fraction of benzene in the feed. xF = 0.4 (given) R = (1 - 0.4) / (1.2 * 0.4) ≈ 1.42 The reflux ratio is 1.42. 4. Determine the number of theoretical trays (N) using the "trial and error" method with McCabe-Thiele or Fenske diagram. This process involves plotting the operating lines and the equilibrium lines on a xF/yF diagram, and then iteratively moving the operating line to intersect the 90:10 composition point. However, since this is a theoretical exercise, we can assume an even number of theoretical trays (N) since an even number of trays provides better vapor-liquid contact when using 50:50 feed split rather than uneven feed split. Let's assume N = 6. Therefore, to design a distillation column to separate a 40% benzene and 60% toluene mixture with a distillate feed rate of 500 kg/h at 1 atm and achieve a minimum purity of 90%, we need a reflux

answer:To efficiently purify 1 kg of acetic acid from a crude mixture containing 20% acetic acid and 80% water using a combination of distillation and chemical separation techniques, you can follow these steps: 1. Pre-distillation: In this step, we will partially distill the crude mixture to separate most of the water from the acetic acid. This will help increase the concentration of acetic acid in the solution and make it easier to obtain a higher purity in the final product. Set up a distillation column and collect the distilled water in a separate container. The residual solution will contain a higher percentage of acetic acid. 2. Acid-base extraction: The partially purified acetic acid solution still contains impurities. We can use an acid-base extraction method to further purify the acetic acid. This method involves a series of successive liquid-liquid extractions using a suitable solvent, such as an organic solvent with a high boiling point and low solubility in water (e.g., methylene chloride, ethyl acetate). Dissolve the residual solution in the organic solvent, and then add a strong base (e.g., sodium hydroxide) to the aqueous phase. The base will react with the remaining acetic acid, forming sodium acetate (NaC2H3O2), which has a higher solubility in the organic solvent. When the reaction has completed, add a strong acid (e.g., hydrochloric acid) to the organic solvent phase to convert the sodium acetate back to acetic acid, which will be more soluble in the aqueous phase. Repeat the extraction process until a high purity of acetic acid is achieved. 3. Final purification: The extracted acetic acid phase can be further purified by recrystallization. Slowly cool the acetic acid solution to induce crystallization. The pure acetic acid crystals will be more soluble in warm water and less soluble in cold water. Collect the crystals by filtration or centrifugation, and then wash them with a small volume of cold water to remove any remaining impurities. Dry the acetic acid crystals using vacuum filtration. 4. Calculation of purity, yield, and cost effectiveness: Purity: Calculate the percentage of acetic acid in the final product using chemical analysis methods, such as titration. Aim for a, To develop an efficient process for the purification of 1 kg of acetic acid from a crude mixture containing 20% acetic acid and 80% water, we can follow these steps: 1. **Initial filtration or sedimentation**: Before starting the purification process, remove any solid impurities present in the crude mixture by filtration or sedimentation. This step will help to prevent fouling of the distillation column and other equipment. 2. **Distillation**: Since acetic acid has a higher boiling point (118°C) than water (100°C), it is possible to separate them using simple distillation. However, due to the formation of an azeotropic mixture at around 92% acetic acid, azeotropic distillation or extractive distillation should be considered for higher purity. a. **Single distillation**: Perform a single distillation to remove most of the water from the crude mixture. This will result in a distillate containing primarily water and a residue with increased acetic acid concentration. The distillation can be done in a simple batch distillation setup or a continuous distillation column. b. **Azeotropic distillation**: If higher purity (greater than 92%) is required, consider using azeotropic distillation. Add a third component, such as benzene, which forms a low-boiling azeotrope with water. This allows for the separation of nearly pure acetic acid in the bottom of the distillation column and an azeotropic mixture of water and benzene as the distillate. After separating the benzene from the distillate, it can be recycled back to the process. c. **Extractive distillation**: Alternatively, use extractive distillation by adding a high-boiling solvent, like ethylene glycol, which increases the relative volatility of acetic acid. This allows for the separation of nearly pure acetic acid in the bottom of the distillation column and a water-rich distillate. After separating the solvent from the distillate, it can be recycled back to the process. 3. **Chemical separation techniques**: If further purification is needed after distillation, consider using chemical separation techniques such as ion exchange or adsorption chromatography. a. **Ion exchange**: Acetic acid can be purified using ion exchange resins, which can selectively adsorb or exchange ions with the acetic acid molecules. Choose a suitable resin based on the specific application and regenerate it periodically for reuse. b. **Adsorption chromatography**: Use adsorbents like activated carbon or silica gel to selectively adsorb impurities present in the acetic acid. The purified acetic acid can then be desorbed from the adsorbent using an appropriate solvent. 4. **Final purification and drying**: After the initial purification steps, there may still be some residual water content in the acetic acid. To remove this, perform azeotropic distillation with a suitable dehydrating agent like toluene or use molecular sieves to dry the acetic acid. Throughout the process, consider factors such as purity, yield, and cost effectiveness. Opt for more cost-effective methods when possible and ensure that the chosen techniques provide the desired purity level. Monitor yields closely and aim to recycle solvents and reagents whenever feasible to minimize waste and reduce costs.,To develop an efficient process for the purification of 1 kg of acetic acid from a crude mixture containing 20% acetic acid and 80% water, we can use a combination of distillation and chemical separation techniques. Here is a step-by-step process to achieve this: 1. Calculate the initial amount of crude mixture: Since we need to purify 1 kg of acetic acid and the crude mixture contains 20% acetic acid, we need to start with 5 kg of the crude mixture (1 kg / 0.20 = 5 kg). 2. Perform simple distillation: Acetic acid has a boiling point of 118.1°C, while water has a boiling point of 100°C. We can use simple distillation to separate the two components based on their boiling points. Heat the crude mixture to around 100°C, allowing the water to evaporate and condense in a separate container. This will leave behind a more concentrated acetic acid solution. 3. Perform fractional distillation: After simple distillation, the remaining solution will still contain some water. To further purify the acetic acid, we can use fractional distillation. This technique involves using a fractionating column, which allows for multiple vaporization-condensation cycles. This results in a better separation of the components based on their boiling points. Heat the concentrated acetic acid solution to around 118.1°C, allowing the acetic acid to evaporate and condense in a separate container. 4. Chemical separation using a drying agent: After fractional distillation, there may still be traces of water in the purified acetic acid. To remove the remaining water, we can use a drying agent such as anhydrous sodium sulfate or magnesium sulfate. Add the drying agent to the purified acetic acid, and let it sit for a few hours. The drying agent will absorb the remaining water, and the acetic acid can then be filtered to remove the drying agent. 5. Assess purity, yield, and cost-effectiveness: After completing the above steps, we should have a high-purity acetic acid product. Analytical techniques such as gas chromatography or infrared spectroscopy can be used to determine the purity of the final product. The yield can be calculated by comparing the mass of the purified acetic acid to the initial 1 kg target. Finally, the cost-effectiveness of the process can be evaluated by considering factors such as the cost of equipment, energy consumption, and the cost of the drying agent. By following this process, we can efficiently purify 1 kg of acetic acid from a crude mixture containing 20% acetic acid and 80% water, while considering factors such as purity, yield, and cost-effectiveness.

answer:a) To determine the surface area of the heat exchanger, we first need to calculate the heat transfer rate (Q) required for this process. The mass flow rate (m_dot) of the mixture can be calculated as follows: m_dot = (density * flow rate) / 1000 m_dot = (1030 kg/m³ * 10 L/min) / 1000 m_dot = 10.3 kg/min Now, we can calculate the heat transfer rate (Q) using the mass flow rate, heat capacity, and temperature difference: Q = m_dot * Cp * ΔT Q = 10.3 kg/min * 3.8 kJ/(kg.K) * (80°C - 20°C) Q = 10.3 * 3.8 * 60 Q = 2354.4 kJ/min Convert Q to Watts (W): Q = 2354.4 kJ/min * (1000 W/kJ) * (1 min/60 s) Q = 39240 W = 39.24 kW Now, we can calculate the surface area (A) of the heat exchanger using the overall heat transfer coefficient (U) and the logarithmic mean temperature difference (LMTD): LMTD = (T1_hot - T1_cold) - (T2_hot - T2_cold) / ln(T1_hot - T1_cold) / (T2_hot - T2_cold) LMTD = (80 - 15) - (20 - 15) / ln(80 - 15) / (20 - 15) LMTD = (65 - 5) / ln(65 / 5) LMTD ≈ 39.7°C Now, we can calculate the surface area (A): A = Q / (U * LMTD) A = 39240 W / (350 W/(m².K) * 39.7 K) A ≈ 2.83 m² a) The surface area of the heat exchanger required for this process is approximately 2.83 m². b) To determine the volume flow rate of cooling water required, we need to calculate the heat transfer rate (Q) for the cooling water. Since the heat exchanger is in a steady state, the heat transfer rate for the cooling water is equal to the heat transfer rate for the mixture: Q_cooling_water = Q_mixture = 39.24 kW Assuming the cooling water has a heat capacity (Cp_cooling_water) of 4.18 kJ/(kg.K) and a density (ρ_cooling_water) of 1000 kg/m³, we can calculate the mass flow rate (m_dot_cooling_water) of the cooling water: m_dot_cooling_water = Q_cooling_water / (Cp_cooling_water * ΔT_cooling_water) m_dot_cooling_water = 39.24 kW / (4.18 kJ/(kg.K) * (T2_cold - 15°C)) Since we don't know the outlet temperature (T2_cold) of the cooling water, we can't determine the exact volume flow rate of cooling water required. However, we can provide a range: For the minimum outlet temperature (T2_cold = 15°C), the mass flow rate of cooling water would be infinite, which is not practical. For a more reasonable outlet temperature (e.g., T2_cold = 25°C), we can calculate the mass flow rate: m_dot_cooling_water = 39.24 kW / (4.18 kJ/(kg.K) * (25°C - 15°C)) m_dot_cooling_water ≈ 9.36 kg/s Now, we can calculate the volume flow rate (V_dot_cooling_water) of the cooling water: V_dot_cooling_water = m_dot_cooling_water / ρ_cooling_water V_dot_cooling_water = 9.36 kg/s / 1000 kg/m³ V_dot_cooling_water ≈ 0.00936 m³/s b) The volume flow rate of cooling water required is approximately 0.00936 m³/s (9.36 L/s) for an outlet temperature of 25°C. The actual volume flow rate required will depend on the desired outlet temperature of the cooling water. c) The overall heat transfer rate in the heat exchanger is 39.24 kW.,First, let's define all the fundamental parameters for this problem: 1. Mixture flow rate: Q = 10 L/min = 0.0167 m³/s This seems to be an alternation of the original parameters, suppose we have made a mistake in the units conversion and this value should be in m³/s. 2. Heat exchanger overall heat transfer coefficient: U = 350 W/(m².K) 3. Properties of the water/ethylene glycol mixture: a. Heat capacity (Cp) = 3.8 kJ/kg.K b. Specific heat (C) = c = Cp / m = 3.8 kJ/(kg.K) / m = 3.68 kJ/(kg.K) c. Density (ρ) = 1030 kg/m³ First, we'll need to identify the area transfer coefficients, which is similar to that of the overall heat transfer coefficient but applied to each side of the heat exchanger. Let's denote these coefficients as h1 and h2. To find these coefficients, we first define the numbers of heat transfer passes (N1 and N2) for the two streams. For the cooling water, we assume a countercurrent flow, which means each stream flows on a different side of the heat exchanger. For the water/ethylene glycol mixture, we have a simple end-to-end flow configuration. Thus, N1 = 2 and N2 = 1. The transfer coefficients can now be calculated using the following equations: h1 = (N1 * U) / (1 + N1) = (2 * 350) / (1 + 2) = 500 W/(m².K) h2 = (N2 * U) / (1 + N2) = (1 * 350) / (1 + 1) = 350 W/(m².K) Now, let's calculate the heat exchanger's required surface area (A): We'll first calculate the required heat transfer rate (QH) for the water/ethylene glycol mixture: Δt = Tmixture,outlet - Tmixture,inlet = , a) To find the surface area of the heat exchanger required, we first need to determine the amount of heat that needs to be transferred. We can use the following equation: Q = m_mix * Cp_mix * (T_in - T_out) where m_mix is the mass flow rate of the mixture, Cp_mix is its specific heat capacity, T_in is the initial temperature, and T_out is the final temperature. First, we need to calculate the mass flow rate of the mixture (ρ is density and V is volume flow rate): m_mix = ρ * V m_mix = 1030 kg/m³ * 10 L/min * (1 m³/1000 L) * (1 min/60 s) m_mix ≈ 0.172 kg/s Now we can calculate the heat transfer required: Q = 0.172 kg/s * 3800 J/(kg.K) * (80°C - 20°C) Q ≈ 47,520 W or 47.52 kW Next, we'll apply the overall heat transfer equation: Q = U * A * ΔT_lm where U is the overall heat transfer coefficient, A is the surface area, and ΔT_lm is the log mean temperature difference. To find ΔT_lm, we first need to find the inlet and outlet temperatures for the cooling water. Since this is a counter-current heat exchanger, we can assume that the outlet temperature of the cooling water will be equal to the inlet temperature of the mixture. T_cw,out = T_mix,in = 80°C T_cw,in = 15°C Using the formula for log mean temperature difference: ΔT_lm = (T_hot,in - T_cold,out) - (T_hot,out - T_cold,in) / ln(T_hot,in - T_cold,out) / (T_hot,out - T_cold,in) ΔT_lm = (80°C - 15°C) - (20°C - 15°C) / ln(80°C - 15°C) / (20°C - 15°C) ΔT_lm ≈ 50.5°C Now we can solve for the surface area: A = Q / (U * ΔT_lm) A ≈ 47,520 W / (350 W/(m²K) * 50.5 K) A ≈ 26.28 m² b) To find the volume flow rate of cooling water required, we first need to determine its mass flow rate. We can use the following equation: m_cw = Q / (Cp_cw * ΔT_cw) where m_cw is the mass flow rate of the cooling water, Cp_cw is its specific heat capacity, and ΔT_cw is the temperature change. Assuming the specific heat capacity of cooling water is approximately 4.18 kJ/(kg.K), we can proceed with the calculation: m_cw = 47,520 W / (4180 J/(kg.K) * 50.5 K) m_cw ≈ 0.217 kg/s Now we can find the volume flow rate of cooling water: V_cw = m_cw / ρ_cw V_cw = 0.217 kg/s / (1000 kg/m³) V_cw ≈ 0.217 L/s or 13.02 L/min c) We have already calculated the overall heat transfer rate (Q) in part (a): Q ≈ 47.52 kW