Outlines 1. Interpolation: Basic Theory 2. Methods of interpolation 3. Interpolation in Excel 4. Interpolation in chem.eng
Interpolation (basic theory) Interpolation function a function that passes exactly through a set of data points. Interpolating functions to interpolate values in tables find functional values at other values of the independent variable, e.g. sin(0.63253)
Interpolation methods Linear Interpolation Linear interpolation is obtained by passing a straight line between2 data points the exact function for which values are known only at a discrete set of data points the interpolated approximation to the data points(also referred to as interpolation points or nodes)
Interpolation methods Linear Interpolation In tabular form:
Interpolation methods Linear Interpolation If is a linear function then To pass through points and
Interpolation methods Linear Interpolation Advantages: Easy to use (homework, exams) Exact for linear functions Disadvantages: Not very accurate for nonlinear functions
Interpolation methods Linear Interpolation Sin( 0.63 ) = ??
Interpolation methods Error in Linear Interpolation Error is defined as:
Interpolation methods Polynomial Interpolation Two steps: obtain polynomial coefficients evaluate the value of the polynomial at the desired location (x i ) For N data points, there is a unique polynomial (usually of order n-1) that goes through each point This is an interpolating polynomial, because it goes exactly through each data point
Interpolation methods Polynomial Interpolation First step: obtain polynomial coefficients For general n, the goal is to determine the coefficients a 1, a 2,…,a n so that for all i = 1:n P n-1 (x) = a 1 + a 2 x + a 3 x 2 +…+ a n x n-1 Writing these equations in matrix-vector form, we obtain Vandermonde matrix
Interpolation methods Polynomial Interpolation Second step: evaluate the value of the polynomial at the desired location (xi) P n-1 (xi) = a 1 + a 2 xi + a 3 xi 2 +…+ a n xi n-1 Sin( 0.63 ) = ??
Interpolation in excel Lookup Tables Often, we need to retrieve data that is stored in a table For example, consider these tables:
Interpolation in excel LOOKUP(Lookup_value,Lookup_vector,Result_vector) search down the lookup vector find either a matching value, or a value greater than the lookup value, whichever occurs first go to the same position in the result vector, and extract the value there
Interpolation in excel Metal Modulus of Elasticity, psi Density, lb/in 3 Yield Strength, psi Aluminum 2014-T6 10,600,0000.10160,000 Aluminum 6061-T6 10,000,0000.09837,000 Stainless Steel 304 28,000,0000.28430,000 Structural Steel A36 29,000,0000.28436,000 Since the properties to be retrieved are in columns, the table is called a vertical lookup table The vertical lookup table command in Excel is: VLOOKUP(lookup value, table range, column number, true/false)
Interpolation in excel VLOOKUP(Lookup_value,Table_array,col_index_num,Range_lookup) search down left column of table array to find match Which column number for extraction of result? 1, 2, 3 or 4? result optional argument: if FALSE, requires an exact match
Interpolation in excel Examples Using VLOOKUP to find the freezing point of a 33% solution The formula =VLOOKUP(F3,$A$3:$D$54,2,0)
Interpolation in excel Other ways to Lookup Tables Match(Lookup_value, lookup_array, match type) Returns the relative position in the array that matches the lookup value. Index(range, position) Returns the value from the range in the position specified
Interpolation in excel Example =MAX($B$5:$B$l6)83119
Interpolation in excel Example use the MATCH function to find the position of the maximum value in the range MATCH(lookup_value,lookup_array,match_type_num) =MATCH(83119,$B$5:$B$16,0)
Interpolation in excel Example use the INDEX function to return the value in the same position in the array of months: =INDEX( $A$5:$A$16,4) =INDEX( $A$5:$A$16, MATCH( MAX( $B$5: $B$16), $B$5:$B$16,0))
Interpolation in excel Lookup value for two-way tables
Interpolation in excel Lookup value for two-way tables =VLOOKUP(Temp,Table, MATCH( Percent, P-Row, 1 )+ 1,1) =VLOOKUP( M2, $A$4:$1$32, MATCH( N2, $B$3:$K$3,1 )+1, 1)
Interpolation in excel A Note About Lookup Tables Lookup tables will not interpolate values! When looking up a numerical value, if an exact match is not found (and the “TRUE” option allows for an approximate match), then the value searched for is rounded down to the next tabulated value. We will illustrate this in the following example
Interpolation in excel Example The population data of a town is given in the table We want to use a lookup table to report the population for any year entered
Interpolation in excel Example Note that using the VLOOKUP function (with the TRUE/FALSE option left off or set to TRUE) returns the population for the next year in the table lower than the input value:
Interpolation in excel Linear interpolation in excel For each population interval, construct an IF statement to see if the input year falls within the interval range… If the input year is within the interval, then the interpolation formula should be used… If the input year is not within the interval, then a space is entered into the cell to make it blank.
Interpolation in excel Linear interpolation in excel IF statements add a great deal of flexibility to Excel spreadsheets – allow control over how calculations are performed Lookup tables are convenient for accessing data that is stored in table form. Important to remember that lookup tables do not interpolate between values Linear interpolation formulas can be added to tables in order to provide a better estimate of values between tabulated points
Interpolation in excel Application of VLOOKUP: Linear interpolation from a table of physical properties For any %HNO 3 from 0 to 90, we want an automatic estimate of the heat capacity
Interpolation in excel Application of VLOOKUP: Linear interpolation from a table of physical properties Create index columns to the right and left of the table
Interpolation in excel Application of VLOOKUP: Linear interpolation from a table of physical properties Enter VLOOKUP functions based on “NitricTable2” and using the low and high row indices to extract the bracketing values of %HNO 3 and Heat Capacity.
Interpolation in excel LowPct HighPct LowHC HighHC linear approximation between to locations in the table input %HNO 3 HeatCap estimate
Interpolation in excel Automatic linear interpolation formula and results
Interpolation in excel Change values of the input to get different results [including bad results when input outside of acceptable range] the goodthe badand the ugly!
Interpolation in excel Linear Interpolation using TREND function TREND can be used to perform linear interpolation between two adjacent data points. TREND( known_y's, known_x's, new_x 's, const) =TREND( B20: B21, A20:A21, F18,1)
Interpolation in excel Linear Interpolation in a Two-way Table Find the viscosity value in the table for x = 76°F, y = 56.3 wt% ethylene glycol
Interpolation in excel Linear Interpolation in a Two-way Table First step, interpolate the value at y = 56.3 Second step, interpolate the value at x = 76
Extrapolation Definition The process of taking data values at points x 1,..., x n, and approximating a value outside the range of the given points. The process of estimating and forecasting Sin (2.6) = ??
Extrapolation Methods Linear extrapolation If the two data points nearest the point to be extrapolated are and, linear extrapolation gives the function: Polynomial extrapolation
Extrapolation Methods Linear extrapolation Given the following data which is known to be linear, extrapolate the y value when x = 2.3. (0.3 0.80), (0.7, 1.3), (1.2, 2.0), (1.8, 2.7)
Interpolation in Chem. Eng Example - Vapor Pressure of Water Determine vapor pressure of water at: 25 °C 92 °C 105 °C
Interpolation in Chem. Eng Example - “Real” Gases p- pressure z- compressibility factor R- gas constant T- temperature - Molar volume Find z at T = 725K and p = 8bar.