The Diet Problem

This Excel spreadsheet will help you find the cheapest combination of foods that fulfills a set of desired nutritional requirements.  This is known as The Diet Problem.  It's a classic example of linear programming and originates from the work of George Stigler, an economist who studied this problem in the 1940s. It was only in the 1950s with the development of the simplex algorithm by Dantzig that fast and accurate solutions became available.

Click this picture for an overview of the spreadsheet.

There are three primary steps in using this worksheet.
  1. Specify the upper and lower bounds on your desired nutritional requirements.  You can set minimum and maximum values for the total calories, fat, saturated fat, trans fat, cholesterol, sodium, carbohydrates, fibre, sugar, protein, vitamin A, vitamin C, calcium and iron.
  2. Specify a possible range of foods, their nutritional data and the cost per portion.  I've included a range of foods, but if you want to modify this list then has a database of foods and their nutritional qualities.
  3. Use Excel's Solver to minimize the total cost (number of portions x cost per portion) while maintaining the nutritional requirements by varying the portions of each food (follow the steps in the picture above).

You may need to enable the Solver add-in via File>Options>Add-Ins.  I've saved my Solver model settings in the range A33:A57 in case you want to return to my defaults.

Of course, if you're seriously thinking about using this spreadsheet to optimize your diet, then I seriously suggest you reconsider - life is too short.

In fact, read George Dantzig's 1990 article The Diet Problem, in which he describes his attempt to follow a diet derived from this method.  He nearly overdosed on Bovril Bouillion cubes closely followed by two pounds of bran.  He ended up following his wife's advice and lost over 20 pounds. 

Calculating Two-Phase Pressure Drop with the Lockhart-Martinelli Correlation

The Lockhart and Martinelli correlation is commonly used to calculate the pressure-drop of two-phase flow in pipe. It's simpler than other methods, and is particularly suitable for low liquid and gas flows in small diameter pipes.

This Excel spreadsheet helps you calculate the two-phase pressure drop in a pipe.  It has several stages
  • First the pressure drops of the individual phases are calculated assuming that each phase is alone in the pipe (i.e. separated flow). 
  • Then, the Lockhart-Martinelli parameter is calculated. 
  •   Finally, the Chisholm (1967) correlation is used to calculate the total pressure drop.

    Balancing a Pump Curve against a System Curve

    This article will demonstrate how you can balance a pump curve against a system curve to calculate liquid velocity with Excel.

    First, we'll develop the equations that determine the liquid velocity in a simple pump and pipe system.  Then we'll discuss how these equations be solved using Excel's Goal Seek feature. Finally, we'll show Visual Basic code that can be used to automate Goal Seek so that any parameter change will automatically calculate the new liquid velocity.

    The spreadsheet can be downloaded here, but read the rest of this article if you'd like to understand the theory.

    Pump and Pipe System
    Consider a centrifugal pump receiving liquid from a reservoir and forcing liquid through a pipe to a reservoir.

    First consider the pump.  Its flowrate-head curve is can be described by a polynomial derived from empirical data, where a, b and c are best-fit coefficients, and Q is the volumetric flowrate

    Equation 1
    But the volumetric flowrate is

    Equation 2
    where A is the cross-sectional area of the pipe and V is the liquid velocity through the pipe.  Substituting Equation 2 into Equation 1 to eliminate Q gives

    Equation 3
    This equation now describes the head produced by the pump as a function of the liquid velocity through the pipe.

    Now consider the pipe.  Frictional head loss through the pipe can be described by the Bernoulli equation and written as

    Equation 4
    We'll call Equation 4 the System Curve. f is the friction factor, given by the Haaland Equation.  

    Equation 5
     where Re is the Reynolds Number.

    Equation 6
    The Haaland equation is only valid in turbulent flow, i.e. if the Reynolds Number is over 2500. 

    For our pump and pipe system, the pump head is equal to the head loss in the pipe.  Hence
    Equation 7
    We can now use Excel to find the liquid velocity that satisfies Equation 7 (effectively determining the intersection between the pump curve and the system curve).

    Excel Implementation
    The Excel spreadsheet uses this cell coloring convention.

    Step 1.  First define the parameters and calculate the cross-sectional area of the pipe.

    Step 2. Now define the coefficients of the pump curve

    Step 3. Set up the calculations required by Goal Seek

    Step 4. Go to Data > What-If Analysis > Goal Seek.  Make the changes such that we find the liquid velocity that makes difference between pump curve and the system curve equal to zero.

    You should now have the correct value of the liquid velocity.

    Ensure that the Reynolds number is greater than 2500 so that our assumption of turbulent flow (and hence the use of the Haaland equation) is verified.

    Visual Basic Macro to Automate Goal Seek
    If you're really keen, you can use Visual Basic to automate Goal Seek.

    Private Sub Worksheet_Change(ByVal Target As Range)
    Dim bSuccess As Boolean
        On Error Resume Next
        bSuccess = Range("C23").GoalSeek(0, Range("c18"))
        On Error GoTo 0
        If Not bSuccess Then
            MsgBox "Goal Seek Failed"
        End If
    End Sub

    Whenever any value in the worksheet is changed, the Worksheet_Change() event is initiated .  The VB code then asks GoalSeek() to find the liquid velocity ("C18") that makes the difference between the pump and system curve ("C23") equal to zero.

    Preliminary Heat Exchanger Design

    This article will help you understand how you can estimate the initial design characteristics of a heat exchanger.  An Excel spreadsheet using the equations developed in this article is also provided. The equations are derived from a simple heat balance, and a few other elementary relationships.

    These equations act as initial estimates, and the results will need to be refined by more sophisticated calculations.  If you just want the spreadsheet then click here, but read the rest of the article if you want to understand the theory.

    Consider a heat exchanger operating in countercurrent flow.

    Assume that we know the
    • desired input and output temperatures of all streams, 
    • specific heat capacities, 
    • the overall heat transfer coefficient, 
    • and the mass flowrate of the cold stream.  
    We will now calculate
    • the flowrate of the hot stream, 
    • the heat transfer rate 
    • and the heat transfer area.
    A heat balance on the cold and hot streams gives

    Qc gives the overall heat transfer rate.

    But Qh = Qc.  Equating both equations and rearranging for the mass flowrate of the hot liquid stream gives

    The log mean temperature difference is

    The overall heat transfer rate Qc can be defined in terms of the log mean temperature difference

    The final two equations can be easily rearranged to give the overall heat transfer area A in terms of the heat transfer rate Qc, the heat transfer area A and the log mean temperature difference.

    Excel Implementation
    Implementing these basic heat transfer equations in Excel is easy, and no special explanation is required.

    Download Excel Spreadsheet for Preliminary Heat Exchanger Design

    Related article: Modeling the Temperature Dynamics of a Cross-Flow Heat Exchanger

    The Three Reservoir Problem

    This article discusses how you can solve the Three Reservoir Problem with Excel. First, we develop the governing equations by applying Bernoulli's Equation and the Continuity Equation. We then explore how these equations can be solved in Excel.

    If you just want the tutorial spreadsheet, click here, but I encourage you to read the rest of the article so you understand how the spreadsheet was developed.  Read on for the Three Reservoir Problem solution.

    Three reservoirs at different elevations are connected by a pipe network.  The common junction of the piping network is subject to an external demand Qj of 0.01 m3/s.  We will develop the theory required to calculate the flowrates in each pipe (Q1, Q2 and Q3), the head at the junction (Hj) and determine whether liquid is flowing into or out of each reservoir

    Assuming that the liquid level in each reservoir is constant and the surface is open to atmosphere, the Bernoulli Equation for Reservoir i (where i=1, 2 and 3) is

    Equation 1
    where zi is the elevation, fi is the friction factor, Li and Di are the length and diameter of the pipe connecting the reservoir to the junction, Vi is the liquid velocity and g is the gravitational constant.

    But the volumetric flowrate Qi and the cross sectional area Ai of the pipe are

    Equation 2
    Equation 3

    Substituting Equations 2 and 3 into Equation 1 to eliminate Vi gives

    Equation 4
    To determine whether liquid is flowing into or out of a reservoir, we need to preserve the sign on the Qi^2 term by writing Equation 4 thus

    Equation 5
    If Qi is positive, liquid is flowing out of the reservoir, and if Qi is negative, liquid is flowing into the reservoir.

    We only need a few more relationships to completely specify the system.  The friction factor fi is given by the Haaland approximation to the Colebrook-White Equation,

    where Rei is the Reynolds Number,

    Additionally, the sum of the flowrates from each reservoir is equal to the external demand

    Excel Implementation
    Moving all terms in Equation 5 to the right-hand side gives

    Equation 6
    However, if we don't know the exact values of the flowrates in each pipeline (Qi) or the head at the junction (Hj) then we can define an error for each pipe.

    Equation 7
    We'll use Excel's Solver add-in to find the values of Q1, Q2, Q3 and Hj that minimize the total error...

    Equation 8
    ...while keeping the total flowrate at the junction equal to the external demand.

    Equation 9
    Step1. Specify fixed parameters (such as densities, viscosities, reservoir heights, pipe diameters and roughnesses etc)

    Step 2. Set initial guess values for the flowrates in each pipe


    Step 3. Specify calculated values

    Step 4. Specify an initial guess value for the head at the junction, and the sum of all flowrates in each pipe (as given by Equation 9).  The External Demand will act as the constraint for Excel's Solver

    Step 5.  Specify the errors for each pipeline (as given by Equation 7), and the total error (as given by Equation 8). 

    We can now use Excel's Solver Add-in to find the flowrates (Q1, Q2 and Q3) and head at the junction (Hj) that minimize the total error (as set in Step 5) subject to the flowrate constraint (as set in Step 4).

    Step 6.  Initiate Excel's Solver menu (if you haven't already, you'll need to load it in the File > Options > Add-ins menu)

    Step 7. Make the appropriate changes in the Solver window such that you minimise the total error by varying the flowrates and the junction head while maintaining the external demand at a set value (for this example, I've set the external demand to 0.01 m3/s).  Additionally, set the solving method to GRG Nonlinear.

    Step 8.  Click Solve.  After dismissing the following window, you'll find that the flowrates in each pipeline, and the junction head have changed.  Bear in mind that positive flowrates indicate flow out of a reservoir, while negative flowrates indicate liquid flow into a reservoir.

    Step 9. We're not finished yet! Check that the Total Error specified in Step 5 is a very small number, and the External Demand (in Step 5) is equal to the value specified in Step 7.

    Dynamic Model of a Cross-Flow Heat Exchanger

    This article will develop a dynamic model of a cross-flow heat exchanger from first principles, and then discretize the governing partial differential equation with finite difference approximations.  It will then demonstrate how this equation can be implemented in Excel (or indeed any other math tool)

    If you just want the Excel implementation, then click here, but I encourage you to read the rest of the article so you understand how the spreadsheet is implemented.

    First Principles Modeling
    Consider liquid flowing (at mass flowrate F) through a length Δx of pipe (diameter D), subject to cooling by cross-flow air (at temperature Ta and heat transfer coefficient U)

    A heat balance over time Δt gives the following.

    Dividing by Δx and Δt and simplifying gives

    As Δx and Δt tend to zero, we get the following parabolic partial differential equation

    Equation 1

    Finite Difference Approximation
    A forward difference approximation for the first of temperature with respect to time is

    Equation 2
    A backward difference approximation for the first of temperature with respect to space is

    Equation 4
    Substituting Equation 2 and 3 into Equation 1, and rearranging gives

    Equation 4

    We only need to know the temperature of the bar at time t (on the RHS of Equation 4) to calculate the temperature at time t + Δt (on the LHS of Equation 4).

    Implementating in Excel
    This is how Equation 4 will be implemented in Excel

    We will now discuss the individual steps in detail.

    Step 1  - Specify your parameters, including your chosen time and space step.  I've named the cells in Column C with the names in Column E.  I'll use named values when entering Equation 4.

    Step 2 - Create a column and row containing your space and time steps

    Step 3 - Fill in your initial conditions at time t = 0 (this will be the inlet liquid temperature as specified in the parameters).

    Step 4 - Insert your boundary conditions at distance x = 0 (this will be the inlet liquid temperature - the same as the initial condition).

    Step 5 - Implement Equation 4 into the first empty cell (at t = Δt and x = Δx)

    Step 5 - Copy this formula to all other times and positions.  For my implementation, I go up to t = 1 and x = 0.4.

    The techniques I've demonstrated above can be applied to many other challenges in science, engineering and math.  If you have any requests, then let me know.