Material Balance Calculations • All material balance calculations are variations of a single theme: Given values of some input and output stream variables, derive and solve equations for the others • Solving the equations is a matter of simple algebra, however, you first need to: vderive the necessary equations from a description of the process and a collection of process data, and vdetermine what is known and what is required. • Developing a standard methodology to solve problems is the key to success! Some problems can be complex… The catalytic dehydrogenation of propane is carried out in a continuous packedbed reactor. One thousand kilograms per hour of pure propane is preheated to a temperature of 670°C before it passes into the reactor. The reactor effluent gas, which includes propane, propylene, methane, and hydrogen, is cooled from 800°C to 110°C and fed to an absorption tower, where the propane and propylene are dissolved in oil. The oil then goes to a stripping tower in which it is heated, releasing the dissolved gases; these gases are recompressed and sent to a distillation column in which the propane and propylene are separated. The propane stream is recycled back to join the feed to the reactor preheated. The product stream from the distillation column contains 98% propylene, and the recycle stream is 97% propane. The stripped oil is recycled to the absorption tower. To understand what is going on, it is necessary to draw a flowsheet to represent the process and material flows Problems involving Material Balances • Procedures will be outlined on singleunit processes (F&R 4.3) – No reaction (Consumption=Generation=0) – Continuous steadystate (Accumulation=0) • Develop good habits now!! Problems will get more complex as we extend the procedures to multipleunit processes (starting in Week 3) and processes with reaction (starting in Week 4 or 5) • Procedures are summarized in F&R Section 4.3 and include: – process diagram (4.3a) – selecting a basis of calculation (4.3b) – Setting up material balances (4.3c) – Performing a degree of freedom analysis (4.3d) Practice, Practice, Practice… • Read the textbook (Section 4.3)!! • Study and understand F&R examples (4.31 through 4.35) • CD with textbook: Interactive Tutorial #2 • Examples in next few lectures and Wk 2 tutorial • Posted assignment and solution from previous year • Assignment 2 (Material single unit processes without rxn) Flowcharts (F&R 4.3a) • A flowchart is a convenient way of organizing process information for subsequent calculations. • To obtain maximum benefit from the flowchart in material balance calculations, you must: 1. Write the values and units of all known stream variables at the locations of the streams on the chart. 2. Assign algebraic symbols to unknown stream variables and write these variable names and their associated units on the chart. Your flowsheet is an important part of the problem solution, and will be assigned marks for completeness Standard Notation The use of consistent notation is generally advantageous. In this course, the notation adopted in Felder and Rousseau will be followed. For example: m – mass & m – mass flow rate n – moles & n – molar flow rate V – volume (Volume is not conserved in a process!!) & V – volumetric flow rate x – component fractions (mass or mole) in liquid streams y – component fractions in gas streams Basis of Calculation (F&R 4.3b) • Basis of calculation – is an amount or flow rate of one of the process streams v If a stream amount or flow rate (an extensive variable) is given in the problem statement, use this as the basis of calculation v If no stream amounts or flow rates are known (i.e., only intensive variables), assume one, preferably a stream of known composition – if mass fractions are known, choose a total mass or mass flow rate of that stream (e.g., 100 kg or 100 kg/h) as a basis – if mole fractions are known, choose a total number of moles or a molar flow rate Methodology for Solving Material Balance Problems (F&R 4.3e) 1. Choose a basis of calculation 2. Draw and fully label a flowchart with all the known and unknown process variables as well as the basis of calculation. Be sure to include units!! 3. Express what the problem statement asks you to determine in terms of the variables labeled on your flowchart 4. State your assumptions (i.e., steadystate, ideal gas, etc.) 5. Determine the number of unknowns and the number of equations that can be written to relate them. That is, does the number of equations equal the number of unknowns? 6. Solve the equations 7. Check your solution – does it make sense? Calculate the quantities requested in the problem statement if not already calculated 8. Clearly present your solution with the proper units and the correct number of significant figures Degrees of Freedom Analysis (F&R 4.3d) • A degreeoffreedom analysis (DFA) is simply an accounting of the number of unknowns in a problem and the number of independent equations that can be written. The difference between the number of unknowns and the number of independent equations is the number of degreesoffreedom, DF or n , of the process. df unknowns - n ndf = n t independen equations • Possible outcomes of a DFA: – n = 0, there are n independent equations and n unknowns. The problem can be df solved. – n > 0, there are more unknowns that independent equations. The problem is df underspecified. n more independent equations or specifications are needed to df solve the problem. – n < 0, there are more independent equations than unknowns. The problem is df overspecified with redundant and possibly inconsistent relations. Sources of equations that relate unknown process variables include: 1. Material balances – for a nonreactive process, usually but not always, the maximum number of independent equations that can be written equals the number of chemical species in the process nd 2. Energy balances – 2 half of course 3. Process specifications – given in the problem statement 4. Physical properties and laws – e.g., density relation, gas law 5. Physical constraints – e.g., mass or mole fractions must add to 1 6. Stoichiometric relations – systems with reaction A set of equations is independent if you cannot derive one by adding and subtracting combinations of the others. Independent Equations è Linear Algebra Is this set of equations independent? 3×3 Matrix form: x + 2 y + z = 1 é1 2 1 ù é x ù é1 ù ú ê 2 1 -1ú ê y ú = ê 2 2 x + y - z = 2 ê úê ú ê ú ûë û ë ú ê0 1 2 ú ê z ú ê 5 ë û y + 2 z = 5 Gauss Matrices: ß Elimination An n x n matrix A has a rank r < n only if |A|=0 é1 0 0 ù é x ù é 6 ù ú ê0 1 0 ú ê y ú = ê -5 If |A|=0 then A is called a singular matrix. ê úê ú ê ú Otherwise, it is non singular. ê0 0 1 ú ê z ú ê 5 ú ë û ë û ë û An n x n matrix has rank r = n only if |A| ¹ 0 Rank = 3. No nonzero rows in reduced form The rank of matrix A is equal to the maximum number of linearly independent rows (or Solution: x= 6, y= –5, z = 5 columns) of A. Are these sets of equations independent? x + 2 y + z = 1 é1 2 1 ù é x ù é 1 ù (Eq. 2) 2 (Eq. 3) = 0 é1 2 1 ù é x ù é1 ù 2 y + 4 z = 10 ú ê0 2 4 ú ê y ú = ê10 ú ê0 0 0 ú ê y ú = ê0 y + 2 z = 5 ê úê ú ê ú ê0 1 2 ú ê z ú ê 5 ú ë ûë û ë û Þ ê úê ú ê ú ê0 1 2 ú ê z ú ê5 ë ûë û ë ú û x + 2 y + z = 1 é1 2 1 ù é x ù é1 (Eq. 1) + (Eq. 2) = (Eq. 3) é1 2 1 ù é x ù é1 ù ù 2 x + y - z = 2 ú ê 2 1 -1ú ê y ú = ê 2 ú ê3 3 0 ú ê y ú = ê3 x 3 + 3 y = 3 ê ë úê ú ê ú ê 3 3 0 ú ê z ú ê 3 ûë û ë ú û Þ ê úê ú ê ú ê3 3 0 ú ê z ú ê3 ûë û ë ú ë û Rank = 2 in both cases DF = 3 unknowns – 2 independent equations = 1 (underspecified) DF Analysis: For a system with N species, it is possible to formulate N+1 material balances. But only N of these (at most) will be independent! One thousand kg/h of an ethanol/methanol stream is to be separated in a distillation column. The feed has 40.0 wt% ethanol and the distillate has 90.0% methanol by wt. 80.0 wt%of the methanol is to be recovered as distillate. Determine the wt% methanol in the bottoms product. Distillate, D Feed, F Dist. Col. # of unknowns = Bottoms, B # of independent material balances = # of additional eqs = DF = There are two common situations where you will find fewer independent equations than species 1. Balance around a divider (splitter) – single input à two or more outputs of the same composition – x = x = x 1 2 3 – Only 1 independent equation – Splitters are used for: • Purge streams (reactor systems with recycle) • Total condensers at the top of distillation columns m kg/h 2 x kgA/kg 2 (1 x ) kgB/kg 2 m kg/h 1 DIVIDE x kgA/kg 1 m kg/h 3 (1 x ) kgB/kg 1 x kgA/kg 3 (1 x ) kgB/kg 3 # of species = 2 # of independent material balances = 1 2. If two species are in the same ratio to each other wherever they appear in a process and this ratio is incorporated in the flowsheet labeling – See F&R pg 127 – Classic example is air in nonreactive system (21 mol% O ; 79 mol% N ) 2 2 – E.g.; vaporization of liquid carbon tetrachloride into an air stream n mol O /s 3 2 n mol O /s 3.76 n mol N /s 3 2 1 2 3.76 n mol N /s n mol CCl / s 4 4(v) 1 2 n mol CCl / s 2 4(l) n mol CCl / s 5 4(l) # of species = 3 # of independent material balances = 2 – Better to treat air as a single species in this situation Recognize these common situations, and always check that your equations are independent!