Dimensional Analysis for Cooking a Turkey 

This worksheet is used to investigate the accuracy of the traditional rule of thumb that turkey should be cooked in a 350?F oven for 20 min/lb. 

Dimensional and Dimensionless Analysis 

In the SI system, there are 7 fundamental dimensional units: mass, length, time, current, temperature, amount of substance, and luminous intensity.  For our example, we will only be concerned with mass, length, and time, and these will represent our fundamental dimensional units. 

 

 

 

 

 

The remainder of our descriptive quantities in our model have dimensions that are products of powers of our fundamental dimensional units. 

 

time	length	density	oven temperature	thermal conductivity
(1.1)	(1.2)	(1.3)	(1.4)	(1.5)

(Here temperature is defined as energy per volume, and thermal conductivity is energy times the length divided by the product area, the time, and the temperature) 

 

Using T as the secondary quantity 

Since there are three fundamental dimensional units, we will have three primary quantities which can be chosen arbitrarily from our four variables.  We will choose l, ρ, and κ as our primary variables, giving us three linearly independent vectors with respect our fundamental dimensional units , , and : 

 

 

We will now create our primary quantity D, such that [D]=[t], where , where are linearly independent primary quantities.  To do this, we need to solve .  First, we will construct the augmented matrix: 

(1.1.1)

 

 

 

(1.1.2)

 

Solving the above matrix yields: 

 

 

(1.1.3)

 

 

From this, we can solve for our variables α, β, and γ: 

 

(1.1.4)

 

 

Therefore, , which has the same dimensions of .  Next, we will calculate based on our secondary quantity , such that [D]=[T].  To do this, we solve .  First, we construct the augmented matrix: 

 

 

 

(1.1.5)

 

Solving the matrix yields: 

 

(1.1.6)

 

From this, we can solve for our variables α, β, and γ: 

 

(1.1.7)

 

 

Therefore, , which has the same dimensions of  . 

 

To create our dimensionless quantities and , we can simply divide our primary and secondary quantities by our newly created quantities and .  This give us:
 

Applying the Buckingham Pi Theorem, we obtain the dependency , with an unknown function F.  In order to gain further information on F, we will need to invoke experimental data.  Before doing that, we will simplify our equation by making a few assumptions.  Since we are testing the validity of the rule of thumb, we will first assume that we'll keep the oven temperature constant at 350 ?F.  Also, it would be reasonable to assume that the thermal conductivity and density of turkeys does not vary much from one bird to the next.  The final assumption we'll make is that the mass of a turkey (M) is proportional to the volume of the turkey.  Since volume is in and mass is in , we can make the substitution .  Making the substitution will give us .  This relationship is rather useless, since all it shows is that is a function of .  Looking at the equation, we know that is dimensionless, and temperature, density and thermal conductivity are kept fixed.  Only the variable remains, and we have to proceed by testing simple hypotheses for . 

 

The simplest hypotheses is that is constant.  If this is the case, than cooking time is proportional to as opposed to being a linear function as stated by the rule of thumb.  However, is not known to be constant.  If, for example where C is some constant, than the resulting dependence on would in fact be linear. 

 

Using κ as the secondary quantity 

Repeating a similar process as above, we will choose l, ρ, and T as our primary variables: 

 

 

We will now solve .  First, we will construct the augmented matrix: 

(1.2.1)

 

 

 

(1.2.2)

 

Solving the above matrix yields: 

 

 

(1.2.3)

 

 

From this, we can solve for our variables α, β, and γ: 

 

(1.2.4)

 

 

Therefore, , which has the same dimensions of .  Next, we will calculate based on our secondary quantity , such that [D]=[κ].  To do this, we solve .  First, we construct the augmented matrix: 

 

 

 

(1.2.5)

 

Solving the matrix yields: 

 

(1.2.6)

 

From this, we can solve for our variables α, β, and γ: 

 

(1.2.7)

 

 

Therefore, , which has the same dimensions of  . 

 

To create our dimensionless quantities and , we can simply divide our primary and secondary quantities by our newly created quantities and .  This give us:
 

Applying the Buckingham Pi Theorem, we obtain the dependency .  Making the same assumptions about temperature, thermal conductivity and mass, we will get .  

 

Using mass instead of density and length 

A third option is to use T, κ, and mass M.  This will give us: 

 

We will now solve .  First, we will construct the augmented matrix: 

(1.3.1)

 

 

 

(1.3.2)

 

Solving the above matrix yields: 

 

 

(1.3.3)

 

 

 

From this, we can solve for our variables α, β, and γ: 

 

(1.3.4)

 

 

 

Therefore, .  Since there is no secondary quantity, we can simply create our dimensionless quantity as .  Making the same assumptions about temperature, thermal conductivity and mass, we will get .  

 

 

 

Evaluating the Functions 

We now have three functions to describe the cooking time of a turkey in addition to the rule of thumb.  In order to determine which option yields the best results, we will need to analyze some data.  The following chart has actual cooking times for various sizes of turkeys: 

 

weight (lbs)	5	6	10	12	15	20	25	30
time (hrs)	2	2.5	3.4	3.5	4.5	5.4	5.8	7

 

Because we do not have any information regarding the functions , , and , we will just test the models under the assumptions that these functions are constant.  We will now plot the dependence of against the weight where p=2/3, 2/5, 1/3 respectively, along with a line of best fit. 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

As we can see, all three can be represented by a linear function, but only p=2/3 is essentially constant.  Therefore the cooking time would be best represented by the function  .  Simplifying this will give us , where is some constant 

 

A New Rule 

The constant can easily be determined from the line of best fit = .  The slope is 0.7185 hrs/lbs ^(2/3), or roughly 45 min/lbs^(2/3).  Therefore, our new and old rules of thumb are: 

 

 

(3.1)

 

 

(3.2)

 

 

where x is the weight in pounds, and time is in minutes.  Plotting these equations against the cooking times provided give us: 

 

 

 

 

From this plot, we can clearly see that our new function is much closer to the experimental cooking times.  Even when a fifteen minute error is allowed, the old rule is only accurate when the weight of the turkey is between 9-14 pounds, whereas the new rule is accurate for the whole range of weights. 

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