Methods of material balance and heat balance

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Section 3: Methods of Material Balance and Heat Balance. Since material balance and heat balance form the basis of chemical engineering design, and serve as important tools for inspection, evaluation, and quota analysis in production, they also constitute the foundation for chemical engineering calculations in this course. Therefore, it is necessary to understand the methods of material and heat balance before moving on to the subsequent chapters. I. Specific methods for material balance For continuous and stable processes, the equation for material balance is: ΣMλ=ΣM_out, or In = Out; this is an algebraic equation. For unstable processes, the equation for material balance is: ΣMλ=ΣM_out + M_accumulated, or In = Out + Accumulated; this is also an algebraic equation. Within the scope of this book, the vast majority of cases involve continuous and stable processes, so emphasis will be placed on discussing these situations. Under normal circumstances, the steps for material balance are as follows: First, identify the object of balance; depending on the requirements of the problem, this can be the total amount of material, a certain component, or a specific element ； Secondly, determine the scope of accounting; as required by the question, it can be a system, a workshop, or a certain piece of equipment ； A certain part of the device, etc…… ； Finally, determine the basis for measurement; as required by the problem, it can be unit mass, unit time, etc. After the above three are determined, specific material balance equations (one or more) are formulated based on the material balance equations, and then the equations or system of equations is solved. Here are some examples: Example 1-1: As shown in Figure 1-2A, an aqueous KNO3 solution with a concentration of 20% (by mass, the same hereafter) is fed into an evaporator at a flow rate of 1000 kg/hr. At a certain temperature, some of the water is evaporated, resulting in an aqueous KNO3 solution with a concentration of 50%. This solution is then fed into a crystallizer where it is cooled to cause KNO3 crystals containing 4% moisture to precipitate, which are subsequently removed continuously. The saturated KNO3 mother liquor with a concentration of 37.5% is returned to the evaporator for recycling; this is a continuous and stable process. Determine: 1. The amount of crystalline product P, and the amount of water evaporated W ； 2. Recirculated mother liquor volume R, concentration amount S. Solution to Figure 1-2A: 1. Substance under consideration: total amount of KNO3. Scope of consideration: the area enclosed by a closed dashed line as shown in the figure. Basis for calculation: per unit time – per hour. Equation: Any substance that enters the scope of consideration by passing through the closed dashed line is considered an input ； Any material that exits the calculated range by passing through the closed dashed line is considered to be outgoing ； The total mass ΣMλ = ΣM_out; therefore, F = W + P (1). Similarly, for KNO3, 0.2F = 0×W + P(1–4%) (2). By substituting the values from equations (1) and (2) into the relevant data, we obtain a system of equations: 1000 = W + P, 0.2×1000 = P(1–4%). Solving this system yields: P = 208.3 kg/hr, W = 791.7 kg/hr. 2. The baseline of the object under consideration remains unchanged, while the scope of calculation is altered, as shown in Figure 1-2B. Total material S = R + P. KNO3: 0.5 × S = 0.375 × R + P(1–4%). By substituting these two equations and solving them together, we obtain: R = 766.6 kg/hr and S = 947.9 kg/hr. These are the values sought. Figure 1-2B It should be noted that it is important to determine the appropriate range for calculation; if this range is not properly defined, it will be impossible to find a solution. Generally, it is necessary to ensure that the known conditions and the quantities to be found fall within the scope of the accounting, so that they can be included in the accounting equation. This skill can be gradually mastered through future study and practice. II. Specific methods for heat balance calculation: Similar to mass balance calculation, for heat balance calculation, since the object of balance has already been determined, it is only necessary to define the scope of the balance and the basis for calculation. However, the basis for heat balance has its own characteristics, which include a quantity basis, a temperature basis, and a phase state basis. The quantity benchmark refers to which quantity is chosen as a basis for measuring heat. For continuous operations, the feed rate per unit time is generally used as a reference; whereas the determination of temperature and phase status allows one to obtain the relevant thermodynamic data from appropriate handbooks for use in calculations. An example is given below: Example 1-2 In a heat exchanger, air at 298 K is heated using saturated steam at a pressure of 136 kPa. The air flow rate is 1 kg/s, while the steam flow rate is 0.01 kg/s; the condensed water is discharged at a saturation temperature of 381 K. If the average specific heat of air is taken as 1.005 kJ/kg·K, calculate the air outlet temperature (ignoring heat loss). Solution to Figures 1-3: The process is as shown in Figure 1-3. Quantity basis: Flow rate of steam and air per second. Temperature basis: The temperature of the inlet air is taken as 298 K; hence its enthalpy is zero, i.e., H_cold air = 0. Phase basis: For saturated steam at a pressure of 136 kPa, its enthalpy H_steam at 281 K is 2690 KJ/kg. Meanwhile, the enthalpy of condensed water at 281 K is 452.9 KJ/kg. Scope of calculation: The area enclosed by the dashed line. For this continuous and steady-state process, ΣQ_in = ΣQ_out. Where: ΣQ_in = ΣQ_steam + ΣQ_cold air = 0.001 × H_steam + 1 × H_cold air; ΣQ_out = ΣQ_hot air + ΣQ_condensed water = 1 × 1.005 (T – 298) + 0.001 × H_cold air. T: Temperature of the hot air. By substituting the values, T = 320.3 K. Section 4: Chemical Process Flow Diagrams and Units In chemical engineering problems, complex production processes are often encountered. For example, in the production of soda ash via the ammonia-alkali process, it starts with the ammoniation and carbonation of saturated brine, followed by processes such as filtration, calcination, and washing; the filtrate then undergoes ammonia evaporation for desorption and is reused in a cycle. When describing such a complex process, it is necessary to use simple methods to organize the available technical information and list the known and unknown conditions; the best way to do this is to represent the process as a flowchart. Flowcharts used in the chemical industry generally include process flowcharts that show the flow of products, and construction flowcharts that are actually used in plant construction. The latter can be further subdivided into piping diagrams, instrument control diagrams, electrical wiring diagrams, and utility process diagrams, depending on the construction requirements. A process flow diagram (also known as a production flow diagram or industrial process diagram) shows the manufacturing steps taken from raw materials to the final product. It depicts the equipment used in each step in geometric form at appropriate scales, and connects these devices based on their relative positions and relationships. Such a diagram that illustrates the entire production process is called a production process flow diagram, or simply a process flow diagram. The production process flow reflects the actual conditions of the factory or workshop, that is, it shows all the main equipment designed, along with the calculated material balance and heat balance, on the flow diagram. However, the production processes in textbooks are mostly schematic processes for illustrative purposes. It is a schematic diagram of the production process, which merely provides a qualitative depiction of the main pathways in the chemical processes and equipment involved in transforming raw materials into finished products. The equipment is shown in its approximate geometric shape; even a block diagram can be used. The relative positions of the equipment do not need to be accurate. Using block diagrams for various calculations is simple, clear, and convenient. As has been used many times in the earlier sections of this chapter. The main equipment shown in the process flow diagram includes reactors, towers, heat exchangers, heating furnaces, filters, centrifuges, dryers, compressors, pumps, and all other vessels used in unit operations. The geometric shapes of these devices are recognized in the chemical industry as standard symbols for major equipment, and will be introduced gradually in subsequent courses. An example of a process flow diagram can be found in Figure 9-25 on page 280 of the textbook. Regarding the system of units, this course uses the International System of Units, namely the SI system. When other units of measurement appear in this book, their conversion factors to the SI system will be provided. If encountered in example problems or *problems, it should be converted to the SI system. Otherwise, the calculation results will differ significantly from the accurate values due to inconsistent units. Your current location: \Chapter 1 Introduction\Section 3 Methods of Material and Heat Balance\Thermochemical data are fundamental values related to the heat effects of physical and chemical changes; they typically include heat capacity, heat of phase transition, heat of chemical reaction, and heat of mixing. These data constitute a basic set of information used in the study of chemical engineering thermodynamics, and they find wide application in thermodynamic analyses and calculations as well as in engineering design. Heat capacity refers to the amount of heat required to raise the temperature of a certain amount of homogeneous material by 1 K, in the absence of phase changes and chemical reactions. If it is 1 mol of a substance, the heat required is the molar heat capacity. The molar heat capacity Cp under isobaric conditions is called the molar heat capacity at constant pressure. The molar heat capacity Cv under isochoric conditions is called the constant-volume molar heat capacity. The relationship between the molar heat capacity at constant pressure and temperature is usually expressed as a polynomial. The coefficients in the polynomials are related to material properties, phase states, and temperature ranges, and are published in relevant handbooks. Phase change heat refers to the heat absorbed or released by a substance during a phase change. The main types include: ① Vaporization heat, which is the phase change heat that occurs when a substance changes from a liquid state to a gas state ； ②Enthalpy of fusion, the phase transition heat when changing from the solid phase to the liquid phase ； ③Sublimation heat is the phase transition heat when changing directly from the solid phase to the gas phase. In the chemical industry, the enthalpy of vaporization is the most commonly used. There are two common methods for determining the phase transition heat: ① direct calorimetry. For example, the energy required for a certain amount of liquid to evaporate is measured under isobaric conditions (pressure equal to the saturated vapor pressure), thereby calculating the enthalpy of vaporization. ②First, measure the saturated vapor pressures at different temperatures, and then calculate the enthalpy of vaporization using the Clausius-Clapeyron equation (see vapor pressure equation). The phase transition heat data for many substances are published in relevant handbooks, and can also be calculated using empirical formulas. Among them, the data on vaporization heat are relatively complete, and the accuracy of the empirical formulas is also high. The Riedel equation is commonly used to calculate the enthalpy of vaporization: △Hv＝1.093RTb(lnPc-1)/(0.930-Tbr), where Tbr＝Tb/Tc. Here, △Hv represents the enthalpy of vaporization at the normal boiling point ； R is the molar gas constant ； Tb and Tc are the normal boiling point temperature and critical temperature, respectively; Pc is the critical pressure (kg/cm2). The prediction error of this formula is no more than 2%. The Watson equation is also commonly used to calculate the enthalpy of vaporization: 0.38, where subscripts 1 and 2 denote two different temperatures respectively. This formula is used to calculate the enthalpy of vaporization at another temperature from that at one temperature. The melting heat depends not only on temperature but also on the crystal form of the solid phase; therefore, there is no universal calculation method. For single-atom substances, it can be estimated using the following formula: Hm = RTm, where Hm is the heat of fusion ； Tm is the normal melting point ； R is the molar gas constant. The sublimation heat can generally be considered as the sum of the melting heat and the vaporization heat. The heat of chemical reaction, abbreviated as reaction heat, is the heat released or absorbed in a chemical reaction at constant temperature. In principle, the reaction heat can be determined by two experimental methods: ① Direct measurement using a calorimeter, for example by carrying out the reaction in an adiabatic closed container, and the reaction heat can be calculated through energy balance; ② First, determining the reaction equilibrium constants at different temperatures, and then using thermodynamic equations that relate reaction heat, reaction equilibrium constants, and temperature to calculate the reaction heat. For chemical reactions whose reaction heat or equilibrium constants are difficult to control and determine, it is possible to calculate them indirectly using the enthalpy of formation (the change in enthalpy when 1 mole of a certain compound is synthesized from the most stable elements at a constant temperature) or the enthalpy of combustion (the change in enthalpy when 1 mole of a certain substance burns completely), based on Hess’s law proposed by G.I. Hess in 1840 (the heat effect of a chemical reaction or physical change is independent of the path taken). The specific calculation formula is as follows: or, where △Hr°, △Hfi°, and △Hci° represent the reaction heat, the enthalpy of formation of substance i, and the enthalpy of combustion of substance i, respectively ； The superscript “°” indicates that all substances involved in the reaction are in their thermochemical standard states (gases are ideal gases at a pressure of 101.325 kPa, while liquids and solids are pure liquids and pure solids at a pressure of 101.325 kPa). vi is the stoichiometric coefficient of substance i in the reaction equation; it is negative for reactants and positive for products (see stoichiometry). Data on the heat of combustion and formation of many substances are published in relevant handbooks. Heat of mixing: The heat absorbed or released when different substances are mixed under constant temperature and pressure. No heat of mixing is generated when ideal gases are mixed. The heat of mixing for low-pressure gases is very small and can be ignored. Some liquids with similar structures (such as toluene and benzene) also have a low heat of mixing. For high-pressure gases and liquids with significant structural differences, the heat of mixing must be taken into account. The mixing process is sometimes similar to a reaction process, and it is difficult to draw a strict distinction between the two; for example, the heat of mixing of an aqueous hydrochloric acid solution and an aqueous sodium hydroxide solution is actually the heat of the reaction in which hydrogen ions and hydroxide ions combine to form water. The heat of mixing when gases, liquids, and solids dissolve in a solvent to form solutions is commonly referred to as the heat of solution. There are mainly four types of heat of solution data: ① Integrated heat of solution, which is the amount of heat absorbed or released when 1 mole of solute dissolves in a certain amount of solvent to form a solution of a specific concentration ； ②Differential heat of solution refers to the heat effect produced when 1 mole of solute is dissolved in a large amount of solution at a constant concentration, without causing any change in the concentration of the solution ； ③Integrated dilution heat is the heat effect when a solution containing 1 mole of solute is diluted from one concentration to another ； ④Differential dilution heat refers to the heat effect produced by adding 1 mole of solvent in a large amount of solution at a constant concentration, without changing the concentration of the solution. Thermochemical data are fundamental values related to the heat effects of physical and chemical changes; they typically include heat capacity, heat of phase transition, heat of chemical reaction, and heat of mixing. These data constitute a key aspect of chemical engineering thermodynamics and are widely used in thermodynamic analyses and calculations as well as in engineering design. Heat capacity refers to the amount of heat required to raise the temperature of a certain amount of homogeneous material by 1 K, in the absence of phase changes and chemical reactions. If it is 1 mol of a substance, the heat required is the molar heat capacity. The molar heat capacity Cp under isobaric conditions is called the molar heat capacity at constant pressure. The molar heat capacity Cv under isochoric conditions is called the constant-volume molar heat capacity. The relationship between the molar heat capacity at constant pressure and temperature is usually expressed as a polynomial. The coefficients in the polynomials are related to material properties, phase states, and temperature ranges, and are published in relevant handbooks. Phase change heat refers to the heat absorbed or released by a substance during a phase change. The main types include: ① Vaporization heat, which is the phase change heat that occurs when a substance changes from a liquid state to a gas state ； ②Enthalpy of fusion, the phase transition heat when changing from the solid phase to the liquid phase ； ③Sublimation heat is the phase transition heat when changing directly from the solid phase to the gas phase. In the chemical industry, the enthalpy of vaporization is the most commonly used. There are two common methods for determining the phase transition heat: ① direct calorimetry. For example, the energy required for a certain amount of liquid to evaporate is measured under isobaric conditions (pressure equal to the saturated vapor pressure), thereby calculating the enthalpy of vaporization. ②First, measure the saturated vapor pressures at different temperatures, and then calculate the enthalpy of vaporization using the Clausius-Clapeyron equation (see vapor pressure equation). The phase transition heat data for many substances are published in relevant handbooks, and can also be calculated using empirical formulas. Among them, the data on vaporization heat are relatively complete, and the accuracy of the empirical formulas is also high. The Riedel equation is commonly used to calculate the enthalpy of vaporization: △Hv＝1.093RTb(lnPc-1)/(0.930-Tbr), where Tbr＝Tb/Tc. Here, △Hv represents the enthalpy of vaporization at the normal boiling point ； R is the molar gas constant ； Tb and Tc are the normal boiling point temperature and critical temperature, respectively; Pc is the critical pressure (kg/cm2). The prediction error of this formula is no more than 2%. The Watson equation is also commonly used to calculate the enthalpy of vaporization: 0.38, where subscripts 1 and 2 denote two different temperatures respectively. This formula is used to calculate the enthalpy of vaporization at another temperature from that at one temperature. The melting heat depends not only on temperature but also on the crystal form of the solid phase; therefore, there is no universal calculation method. For single-atom substances, it can be estimated using the following formula: Hm = RTm, where Hm is the heat of fusion ； Tm is the normal melting point ； R is the molar gas constant. The sublimation heat can generally be considered as the sum of the melting heat and the vaporization heat. The heat of chemical reaction, abbreviated as reaction heat, is the heat released or absorbed in a chemical reaction at constant temperature. In principle, the reaction heat can be determined by two experimental methods: ① Direct measurement using a calorimeter, for example by carrying out the reaction in an adiabatic closed container, and the reaction heat can be calculated through energy balance; ② First, determining the reaction equilibrium constants at different temperatures, and then using thermodynamic equations that relate reaction heat, reaction equilibrium constants, and temperature to calculate the reaction heat. For chemical reactions whose reaction heat or equilibrium constants are difficult to control and determine, it is possible to calculate them indirectly using the enthalpy of formation (the change in enthalpy when 1 mole of a certain compound is synthesized from the most stable elements at a constant temperature) or the enthalpy of combustion (the change in enthalpy when 1 mole of a certain substance burns completely), based on Hess’s law proposed by G.I. Hess in 1840 (the heat effect of a chemical reaction or physical change is independent of the path taken). The specific calculation formula is as follows: or, where △Hr°, △Hfi°, and △Hci° represent the reaction heat, the enthalpy of formation of substance i, and the enthalpy of combustion of substance i, respectively ； The superscript “°” indicates that all substances involved in the reaction are in their thermochemical standard states (gases are ideal gases at a pressure of 101.325 kPa, while liquids and solids are pure liquids and pure solids at a pressure of 101.325 kPa). vi is the stoichiometric coefficient of substance i in the reaction equation; it is negative for reactants and positive for products (see stoichiometry). Data on the heat of combustion and formation of many substances are published in relevant handbooks. Heat of mixing: The heat absorbed or released when different substances are mixed under constant temperature and pressure. No heat of mixing is generated when ideal gases are mixed. The heat of mixing for low-pressure gases is very small and can be ignored. Some liquids with similar structures (such as toluene and benzene) also have a low heat of mixing. For high-pressure gases and liquids with significant structural differences, the heat of mixing must be taken into account. The mixing process is sometimes similar to a reaction process, and it is difficult to draw a strict distinction between the two; for example, the heat of mixing of an aqueous hydrochloric acid solution and an aqueous sodium hydroxide solution is actually the heat of the reaction in which hydrogen ions and hydroxide ions combine to form water. The heat of mixing when gases, liquids, and solids dissolve in a solvent to form solutions is commonly referred to as the heat of solution. There are mainly four types of heat of solution data: ① Integrated heat of solution, which is the amount of heat absorbed or released when 1 mole of solute dissolves in a certain amount of solvent to form a solution of a specific concentration ； ②Differential heat of solution refers to the heat effect produced when 1 mole of solute is dissolved in a large amount of solution at a constant concentration, without causing any change in the concentration of the solution ； ③Integrated dilution heat is the heat effect when a solution containing 1 mole of solute is diluted from one concentration to another ； ④Differential dilution heat refers to the heat effect produced by adding 1 mole of solvent in a large amount of solution at a constant concentration, without changing the concentration of the solution. The temperature boundary layer, also known as the thermal boundary layer, is a thin layer near a wall surface where a temperature gradient is formed due to heating or cooling as the fluid flows past it; it is the region where the thermal resistance to convective heat transfer lies. Outside this region, both the temperature gradient and thermal resistance can be ignored. Therefore, research on convective heat transfer is limited to the temperature boundary layer range. The concept of a temperature boundary layer is an extension of the concept of a flow boundary layer to non-isothermal flow conditions. By utilizing the properties of the temperature boundary layer to simplify the energy equation, and following the calculation methods for flow boundary layers, it is possible to compute convective heat transfer, determine the temperature distribution, and obtain the heat transfer coefficient. Formation of the temperature boundary layer on the wall: When a fluid flows over a solid surface with a different temperature from its own, different temperature distributions occur, depending on whether the fluid is heated or cooled at the wall surface (temperature distribution near the wall in Figure 1). The change in fluid temperature can, in theory, extend to infinity, but the changes primarily occur in a thin layer near the wall surface. Generally, the thickness δt of the temperature boundary layer is defined as the distance from the wall surface to the point where the temperature difference between the fluid and the wall surface reaches 99% of the temperature difference between the fluid itself and the wall surface. That is, the temperature at the outer boundary of the temperature boundary layer must satisfy the following equation: (T – Tw) = 0.99(Tf – Tw), where Tf and Tw represent the temperatures of the fluid bulk and the wall surface, respectively ； T is the temperature at the outer boundary of the temperature boundary layer. The thickness of the temperature boundary layer increases continuously in the flow direction. The thinner δt, the greater the temperature gradient within the layer. During convective heat transfer, the temperature field in the fluid is divided into two regions: one is the temperature boundary layer where a temperature gradient exists, and heat conduction plays a role there ； The other is the outer main flow region where the temperature gradient can be ignored; in this region, convective heat transfer plays the dominant role, and the effect of heat conduction can be disregarded. The temperature boundary layer on the wall can form simultaneously with the flow boundary layer, or the flow boundary layer can form first, with the temperature boundary layer forming subsequently (Figure 2: Flow boundary layer and temperature boundary layer). The ratio of the thicknesses of the two boundary layers is related to the Prandtl number Pr. When two boundary layers begin to form simultaneously, their approximate relationship is given by: where δt and δ represent the thicknesses of the temperature boundary layer and the flow boundary layer, respectively. When Pr=1, the thicknesses of the two are equal. Calculation of the temperature boundary layer on a flat surface. The heat transferred through thermal conduction within the temperature boundary layer is given by: where q represents the heat flux ； λ is the thermal conductivity of the fluid, and ∂T/∂y is the temperature gradient in the direction of the heat flow ； The subscript w denotes the wall surface. By combining Newton’s law of cooling (see convective heat transfer), the formula for calculating the heat transfer coefficient can be derived: if the temperature distribution can be determined and the wall temperature gradient is known, then the heat transfer coefficient can be calculated. For laminar flow in the boundary layer over a flat wall, the thickness of the temperature boundary layer can be calculated as: where x is the distance from the end of the flat wall ； Re is the Reynolds number. The temperature distribution at this time can be approximated as: where y is the vertical distance from the wall. The formula for calculating the average heat transfer coefficient am is: where L is the length of the flat wall.