Back in 2008, Panofsky gave an empirical formula, $T=11.5W2/3$, for turkey baking time, T, in hours vs turkey weight, W, in pounds, the so-called Panofsky formula or Panofsky constant. Compared to the previously existed recipes that are based on the simple linear relationship between turkey weight and baking time, the Panofsky formula provides a more accurate estimate for the baking time. For instance, a general guideline of 13–20 min/lb was widely recommended in all previous turkey baking recipes. In this work, we conduct a comprehensive study of the turkey baking process that leads to a mathematical derivation of the Panofsky formula under some approximations. We also generalize the Panofsky formula to define a general Panofsky formula, $T=1PW2/3$, where P is defined as the Panofsky constant. Under spherical approximations, we then apply an accurate physical solution of the heat transfer equation and use the rigorous solution with numerical methods to study the generalized Panofsky formula and the Panofsky constant. We found that the generalized Panofsky formula can be perfectly applied to all turkey baking scenarios for baking time calculations. Furthermore, we did a careful analysis of the Panofsky constant, which equals 1.5 in the original Panofsky formula. The dependency of the new Panofsky constant on thermal properties of the turkeys and other initial parameters of baking, e.g., initial and final center temperature of the turkeys, oven temperature, thermal conductivity, specific heat, and turkey’s density, was carefully analyzed and mapped out. The Panofsky constant, P, could vary from 1.1 to 1.9 depending on these thermal parameters.

To most physicists, it is an essential belief that all natural phenomena can be understood and interpreted in both qualitative and quantitative ways according to general physics principles.1–8 A great example the authors have been looking carefully into is the culinary arts.

Human beings have been cooking with a long history.1–4 Cooking provides such a rich set of opportunities and examples to learn and study science, in particular, physics.5,6 It is an ideal and easy-to-access tool for learning and studying physics, including performing both theoretical and experimental research studies. It is easy to perform even in a regular home setting. Understanding the science in cooking, supported with interesting home-kitchen laboratory experiments, generates enthusiasm and provides strong motivation for anyone who is interested in physics and science. Many factors could affect the result of making a successful dish. Those factors include but are not limited to cooking time, cooking temperature, heating method, and size of the food. Each recipe is an experiment; deviating from tradition may result in culinary disaster. An intrepid cook is fundamentally a scientist who needs to explore a wide range of variables through careful experimentation by systematically varying one parameter of the recipe at a time and observing the outcome. Understanding of the cooking process of food is crucial in understanding the underlying physical and chemical phenomena, which is key to optimizing the culinary quality in terms of process, texture, and flavor optimization, as well as safety of meat and other types of foods’ consumption.7,9

In the past, many research studies have been performed, both experimentally and theoretically, and many publications’ references have highlighted the importance of physical modeling.8–21

Varlamov et al.15,16 utilized a simplified heat transfer model to study pizza baking,15 boiling, steaming, and rinsing Chinese foods16 and derived the quadratic power relationship with the food dimension. Olszewski8 used a rigorous heat transfer equation to solve the short cylinder to study the baking time of cake and its dependence on the dimensions of the cake. Nelson et al.11 studied steak cooking with a two-dimensional mathematical model. Unsworth and Durate,17 Buay et al.,18 Roura et al.,19 and Williams20 all have studied boiled eggs based on solving the thermal diffusion equations. Chang et al. modeled heat transfer during roasting of unstuffed turkeys.21

A great example of applying physical thinking to culinary arts is the story about how SLAC former director, Professor WKH “Pief” Panofsky, developed the formula for baking of turkeys.22

People have been baking turkeys for a long time.22–32 In all previously existed recipes, from generations to generations, a “15–30 min/lb” recipe was widely adopted.22–32 It says that the baking time of a turkey is linearly proportional to the weight of the turkey. There are many variations of this linear recipe, which we can easily find in every turkey cook book or via online searching. A revised recipe may call that the time per pound rate varies according to the range of the turkey weight. As the turkey becomes larger and much heavier, for instance, over 20 lb, then the minutes per pound rate should go down slightly.22–32

In 2008,22 according to the original note from Nicholas Panofsky, “There was a point in time when my grandfather [SLAC Director Emeritus] WKH “Pief” Panofsky was not satisfied with the cooking times for turkeys of “30 min/lb.” This is of course reasonable because the time for which a turkey should be cooked is not a linear equation. “So Pief Panofsky derived an equation based on the ratio between the surface area and the mass of a turkey. He determined that the cooking time for a stuffed turkey in a 325 °F oven is $T=11.5W2/3$, where T is the cooking time in hours and W is the weight of the turkey in pounds. The constant 1.5 was determined empirically.”22

The so-called Panofsky formula, $T=11.5W2/3$, provides a more accurate estimate for the baking time compared to the previously existed recipes with the simple linear relationship. The power of 2/3 dependency on the turkey weight is relatively easy to understand. The power of 1/3 of the weight essentially converts the weight into a characteristic dimension of the turkey, such as the radius or diameter or length of the turkey, and we also know that, in principle, the cooking time has a power of 2 dependent on the linear dimension of the food piece, that is, doubling the size of the food would get a cooking time increased by four times.

However, the exact physics beyond the above brief description could not be easily found or available online or in the textbooks or scientific literature. The questions we have been asking are the following: (1) How sound is the Panofsky formula from a full and accurate physical consideration? (2) What kinds of assumptions and approximations have been applied in order to get to use the formula? (3) How could we understand the physics behind the constant 1.5? (4) How does this constant depend on the physical and thermal parameters of the turkey cooking process?

The motivation of this research is to use accurate physical modeling along with some approximations to deal with the cooking of turkeys in order to make the complicated real-life problem truly understandable. We try to simplify the physics but meanwhile do not lose the essentials, as evident from how close the calculated cooking time is related to the real-life data. We try to find a unified framework to treat the cooking in order to provide us a clear overall picture on how to calculate and address the cooking time and recipe, which depends primarily on the size of the food, heating profile (i.e., temperature), and food thermal properties.8–21

In this paper, we conduct a comprehensive study of the turkey baking process that leads to a mathematical derivation of the Panofsky formula with some approximations. We also generalize the Panofsky formula to define a general Panofsky formula, $T=1PW2/3$, where P is defined as the Panofsky constant. Adopting the spherical approximations, we derive an accurate physical solution of the heat transfer equation. We then use the rigorous solution with numerical methods to study the generalized Panofsky formula and the Panofsky constant. We found that the generalized Panofsky formula can be perfectly applied to all turkey baking scenarios for baking time descriptions. Furthermore, we did a careful analysis of the Panofsky constant, which equals 1.5 in the standard Panofsky formula. The dependency of the new Panofsky constant on thermal properties of the turkeys and other initial parameters of baking, e.g., initial and final center temperature of the turkeys, oven temperature, thermal conductivity, specific heat, and turkey’s density, was carefully analyzed and mapped out. The Panofsky constant, P, could vary from 1.1 to 1.9 depending on these thermal parameters.

The outline of the article is as follows: First, we apply the spherical approximation and then derive the heat transfer partial differential equation and finally find the solution of the heat transfer equations. Then, we mathematically derive the Panofsky formula under some approximations. We then consider all the other shapes and irregularities with a defined shape factor. Next, after a careful survey of the thermal properties and parameters of turkeys, we carefully study the generalized Panofsky formula’s soundness and the dependency of the Panofsky constant on all the key physical and thermal parameters. In Sec. V, we touch upon the issues of weight loss and so forth.

According to Fourier’s law,33–38 the heat flux q (in W/m2), as shown schematically in Fig. 1, is proportional to the temperature gradient, i.e., q = −k$⋅dTdx$ for one-dimensional systems. For the three-dimensional system, $q⃗$ = −k · $∇$T, where $q⃗$ is a vector, $∇$ is the gradient, and k is the thermal conductivity in W/(cm K),

$Qnet=A⋅qx+δx−qx=−kA⋅∂T∂xx+δx−∂T∂xx=−kA⋅∂T∂xx+δx−∂T∂xxdx⋅dx$
$=−kA⋅∂2T∂x2⋅dx.$
$−Qnet=dUdt=ρcA⋅dT−Trefdt⋅dx=ρcA⋅dTdt⋅dx.$

This leads to the one-dimensional heat diffusion equation

$∂2T∂x2=ρck∂T∂t=1α∂T∂t,$

where T is a function of location and time, i.e., T(x, t), and $α$ is the thermal diffusivity with the unit of m2/s. $α$ can be described as k/$ρ$c, with $ρ$ representing the density (kg/m3) and c representing the specific heat [J/(kg K)].

FIG. 1.

Schematic representation of the one-dimensional heat flow and transfer.

FIG. 1.

Schematic representation of the one-dimensional heat flow and transfer.

Close modal

In three-dimensions, the heat transfer equation becomes

$∇⋅∇T=1α∂T∂t,$

where

$∇⋅∇T=∂2T∂x2+∂2T∂y2+∂2T∂z2forCartesiancoordinates=1r2⁡sin⁡θsin⁡θ∂∂rr2∂T∂r+∂∂θsin⁡θ∂T∂θ+1sin⁡θ∂2T∂φ2forsphericalcoordinates=1r∂∂rr∂T∂r+1r2∂2T∂θ2+∂2T∂z2forcylindricalcoordinates.$

For a sphere with azimuthal symmetry, as shown schematically in Fig. 2, during the heat transfer,2–18,18–25 we have $∂T∂θ=0$ and $∂2T∂φ2=0$, and the heat transfer equation becomes

$1r2∂∂rr2∂T∂r=1α∂T∂t.$

If we apply V = r · T to the above equation, we get

$∂2V∂r2=1α∂V∂t.$

We can decouple V(r, t) into

$V(r,t)=R(r)⋅T(t).$

We get

$∂V∂t=R(r)⋅∂T∂t=R(r)⋅T′(t)$

and

$∂2V∂r2=T(t)⋅R″(r).$

Then, we have

$T(t)⋅R″(t)=1α⋅R(r)⋅T′(t).$

It can be rearranged into

$R″(r)R(r)=1α⋅T′(t)T(t).$

Since the left side is only r-dependent and the right side is only t-dependent and they are equal to each other, they must be neither r- or t-dependent. So, we have

$R″(r)R(r)=1α⋅T′(t)T(t)=−λ.$

Then, we have

$R″+λR=0$

and

$T′+λαT=0.$

From the above equation, we have

$dTdt=−λαT,$
$dTT=−λα⋅dt,$
$∫0tdTT=−λα⋅∫0tdt,$
$lnT(t)−lnT(0)=−λαt,$
$T(t)=e−λαt⋅T(0).$

For

$R″+λR=0,$
$d2R(r)dr2=−λ⋅R(r),$
$R(r)=A⁡cosλ⋅r+B⁡sinλ⋅r.$

Now, we have

$V(r,t)=∑λA⁡cosλ⋅r+B⁡sinλ⋅r⋅e−λαt,$
$T(r,t)=∑λA⁡cosλ⋅r+B⁡sinλ⋅r⋅e−λαtr.$

For cooking (heating) food starting with low temperature T0 and surrounded at a high temperature (bath temperature) Th, we have the following boundary conditions:

$For0≤r≤R,T(r,o)=T0,whereRistheradiusofthesphere,$
$T(≥R,t)=Th.$

We have

$A=0andλ=nπR2,wheren=1,2,3,….$

We then have

$T(r,t)=Th−2RTh−T0π⋅r∑n=1∞×−1n+1nsinnπrR⋅e−αn2π2t/R2for(0≤r≤R).$

We define

$τ=R2π2⋅αasthetimeconstant.$

Thus, we have

$T(r,t)=Th−2RTh−T0π⋅r∑n=1∞−1n+1nsinnπrR⋅e−t/τ.$

With r = 0, we can then get the temperature at the center of the sphere,

$Tc=Th−2Th−T0∑n=1∞−1n+1⋅e−t/τ.$

We can spell out the equation with some of the initial (and deciding) terms,

$Tc=Th−2Th−T0e−t/τ−e−4t/τ+e−9t/τ−e−16t/τ+e−25t/τ−e−36t/τ+e−49t/τ−⋯,$
(1)

where $τ=R2π2⋅α$ and $α=kρc$.

FIG. 2.

The spherical approximation is used in our calculation, and R is the radius of the sphere. Th is the temperature of the environment which the sphere is put into, and T0 is the initial temperature of the sphere prior to heating.

FIG. 2.

The spherical approximation is used in our calculation, and R is the radius of the sphere. Th is the temperature of the environment which the sphere is put into, and T0 is the initial temperature of the sphere prior to heating.

Close modal

When we carefully look at Eq. (1), the value of the second term and also all the later terms in the parentheses goes down very quickly compared to the first term. The other higher order terms have even smaller contribution that can be ignored without causing any meaningful error in the calculations. In all our numerical calculations discussed later (i.e., Figs. 38), in order to build our confidence, we intentionally compared the calculation results with only the first term, the initial two terms, the initial three terms, the initial four terms, and the initial ten terms.

FIG. 3.

Calculated cooking time for turkeys, in hours, as a function of weights, in pounds. The dots are the calculated cooking time with the temperature at the center of the turkey reaching 85 °C. The dotted curves are the fitting curves with the fitting parameters listed on the figure. The blue, gray, and orange dots are the calculated cooking time with different Th and T0: for the blue dots, Th = 163 °C, T0 = 20 °C; for the gray dots, Th = 163 °C, T0 = 10 °C; and for the orange dots, Th = 177 °C, T0 = 20 °C.

FIG. 3.

Calculated cooking time for turkeys, in hours, as a function of weights, in pounds. The dots are the calculated cooking time with the temperature at the center of the turkey reaching 85 °C. The dotted curves are the fitting curves with the fitting parameters listed on the figure. The blue, gray, and orange dots are the calculated cooking time with different Th and T0: for the blue dots, Th = 163 °C, T0 = 20 °C; for the gray dots, Th = 163 °C, T0 = 10 °C; and for the orange dots, Th = 177 °C, T0 = 20 °C.

Close modal
FIG. 4.

Dependency of the calculated Panofsky constant vs the initial temperature. With all the other parameters from the baseline, the Panofsky constant can approach 1.5 with the initial temperature around −5 °C.

FIG. 4.

Dependency of the calculated Panofsky constant vs the initial temperature. With all the other parameters from the baseline, the Panofsky constant can approach 1.5 with the initial temperature around −5 °C.

Close modal
FIG. 5.

The calculated Panofsky constant as a function of the thermal conductivity. A perfect linear relationship between the Panofsky constant and the thermal conductivity is demonstrated.

FIG. 5.

The calculated Panofsky constant as a function of the thermal conductivity. A perfect linear relationship between the Panofsky constant and the thermal conductivity is demonstrated.

Close modal
FIG. 6.

The calculated Panofsky constant as a function of the specific heat. It shows that the Panofsky constant is inversely proportional to the specific heat.

FIG. 6.

The calculated Panofsky constant as a function of the specific heat. It shows that the Panofsky constant is inversely proportional to the specific heat.

Close modal
FIG. 7.

The calculated Panofsky constant as a function of the density.

FIG. 7.

The calculated Panofsky constant as a function of the density.

Close modal
FIG. 8.

(a) The calculated cooking time as a function of the weight. (b) The calculated average cooking time per pound. (c) The calculated instant rate of cooking time per pound, that is, the slope of the curve in (a).

FIG. 8.

(a) The calculated cooking time as a function of the weight. (b) The calculated average cooking time per pound. (c) The calculated instant rate of cooking time per pound, that is, the slope of the curve in (a).

Close modal

Based on our numerical calculation, in the case that we are confident that the first term dominates the result, we can ignore all the higher order terms other than the first term in the parentheses, and we get

$Tc=Th−2Th−T0e−t/τfromEq.(1).$

We can re-arrange the equation into

$t=τ⋅ln2Th−ToTh−Tc;$

since $τ=R2π2⋅α$ or $τ=L2π2⋅α$ and $α=kρc$, we have

$t=R2π2⋅αln2Th−ToTh−Tcforspheres.$
(2)

Thus, the new approximated equation with only the first term considered gives a square dependence on the critical dimension, i.e., the radius R for the sphere.

Now, let us convert Eq. (2) into a baking time dependency on the weight of the turkey, which is essentially the Panofsky formula or an approximation.

Since the weight of the turkey is W = ρ · V, where V is the volume $V=43πR3$, we have $W=43πρR3$. Thus, we have

$t=3W4πρ2/3π2⋅αln2Th−ToTh−Tc.$
$So,wehavet=34πρ2/3π2⋅αln2Th−ToTh−Tc⋅W2/3,$

where

$α=kρc.$

We define the Panofsky constant, P, as

$t(hour)=1P⋅W2/3,Winpound,$
$P=3600⋅π2⋅α38.82⋅πρ2/3⁡ln2Th−ToTh−Tc.$
(3)

In the Panofsky constant formula (3), both ρ and α are in SI units.

With Eq. (3) and the standard39,40 density around 1050 kg/m3, the specific heat c around 3530 J/(kg K), the thermal conductivity K around 0.5 W/(m K),39,40 the oven temperature at 163 °C (325 °F), the initial turkey temperature at 20 °C, and the center temperature of the spherical turkey at 85 °C (185 °F), our calculated P is 1.16, which is much smaller than 1.5, which Dr. Panofsky suggested in 2008 through his famous formula.

We do realize in the process of deriving Eq. (3) that all the terms except the first term in Eq. (1) are neglected. In Sec. IV, we will try to calculate the Panofsky constant and demonstrate the feasibility of the Panofsky formula through a rigorous numerical calculation with higher terms considered.

Since the turkey is never a spherical shape, this section introduces a shape factor18 to discuss the possible deviation caused by our approximation with the shape.

We will use the other geometry shapes such as cube and short cylinder to provide some insights.

Because the received heat is proportional to the surface area and the received energy per volume is inversely proportional to the total volume, the inverse of the baking time (or heating time) is proportional to the surface area and inversely proportional to the volume of the piece. We then have

$1bakingtime∼A(totalsurfacearea)V(totalvolume)=S(shapefactor).$

For a sphere, we have

$S=4πR243πR3=3R,andwegetR=3S.$

For a cube with a side length of a, we have

$S=6a2a3=6a=312a.$

If the cube has the same volume as the sphere, $43πR3=a3$, we have $12a≅0.8R$. That is, from thermal transfer (heating) perspective, a cube with the same volume as a sphere is equivalent to a sphere with 0.8 (80%) of the diameter, i.e., a ≅ 0.8 · (2R).

For a short cylinder (such as a cake) with a radius r and height h,

$S=2πr2+2πrhπr2h=4πr2πr3(forh∼r)=4r=30.75r.$

With the volume of the short cylinder the same as the sphere, $43πR3=πr2h=πr3forh∼r,wegetr≅1.1R$, so we have 0.75r = 0.75 · 1.1R ≈ 0.83R.

That is, from the thermal transfer (heating) perspective, a short cylinder (radius and height similar) with the same volume as a sphere is equivalent to a sphere with 0.83 (83%) of the radius.

From another perspective, the above analysis also proves that the sphere is the most efficient geometry in heat transfer. In other words, with the same weight and density, the shape deviation from a perfect sphere would result in a slower rise (longer heating time) in the center temperature, i.e., a less efficient heating process. We will come back to this discussion further in Sec. IV A when we discuss the generalized Panofsky formula.

In our work, a through literature survey of the thermal properties of turkeys was conducted. However, their thermal parameters vary hugely in some of the reported research studies, as listed in Table I.

TABLE I.

Turkey’s thermal properties, i.e., density, thermal conductivity, and specific heat, are listed. The data are based on available sources.

ParameterUnitValueCondition or trendReferences
Density kg/m3 1050  26 and 27
1070 20 °C 46
∼1000 80 °C 46
Thermal conductivity W/(m K) 0.5  39 and 40
0.464  47
0.3 20 °C 46
0.5 60–80 °C 46
Specific heat J/(kg K) 3530  39 and 40
2850 30 °C 46
3380 80 °C 46
3450 20 °C 48
ParameterUnitValueCondition or trendReferences
Density kg/m3 1050  26 and 27
1070 20 °C 46
∼1000 80 °C 46
Thermal conductivity W/(m K) 0.5  39 and 40
0.464  47
0.3 20 °C 46
0.5 60–80 °C 46
Specific heat J/(kg K) 3530  39 and 40
2850 30 °C 46
3380 80 °C 46
3450 20 °C 48

According to Refs. 39–51, which summarize all the essential thermal parameters for all the foods, turkey has a reported density around 1050 kg/m3, the specific heat c around 3530 J/(kg K), and the thermal conductivity K around 0.5 W/(m K).39,40

According to Ref. 46, with temperature between 20 and 60 °C, the thermal conductivity K increases linearly from 0.3 to 0.5 W/(m K). Between 60 and 80 °C, it remained constant. From 20 to 40 °C, the density of the turkey decreases slightly as a function of temperature. A transition phase was observed from 40 to 60 °C, which was followed by a decrease from 60 to 80 °C. There was a decrease of about 50 kg/m3 between the density of a raw product at room temperature (at maximum 1070 kg/m3) and the product heated to 80 °C (at minimum 970 kg/m3) due to the gelation or setting of the structure.

For the specific heat, the heat capacity increased linearly from 2850 to 3380 J/(kg K) for temperature from 30 to 80 °C. The heat capacity includes the enthalpy change as a result of the chemical reactions from the cooking process.

In Ref. 47, the thermal conductivity K is 0.464 W/(m K) for turkeys.

In Ref. 48, the specific heat c is 3450 J/(kg K) at 20 °C.

The temperature dependence of specify heat and thermal conductivity is partially due to the change of composition in the turkey in the heating process.

In this section, we have done the numerical calculations based on the rigorous equation (1). As the baseline, we adopt the thermal parameters from Refs. 39 and 40, that is, turkey has a density around 1050 kg/m3, the specific heat c around 3530 J/(kg K), and the thermal conductivity K around 0.5 W/(m K).39,40 We also use the baseline with the oven temperature set at constant 163 °C (325 °F) and initial turkey temperature at 20 °C.

A typical turkey weighs between 8 lb (3.63 kg) and 25 lb (11.3 kg). We approximate the turkey with a spherical shape. Based on the density and weight, we can calculate the effective radius of the turkey based on the weight. We have radius at 115 mm for 6.8 kg, 127 mm for 9.1 kg (20 lb), and 137 mm for 11.3 kg.

We calculate the cooking time based on reaching a center temperature of 85 °C (185 °F). Table II shows the calculated results.

TABLE II.

Turkeys with various representative weights, effective radius calculated based on weights and density, and the calculated cooking time for the turkeys to reach the temperature of 85 °C (185 °F) at the center of the turkey.

Weight (kg)Effective radius (mm)Time taken to reach 85 °C at the center
3.63 (8 lb) 93.8 141 min (2.35 h)
4.54 (10 lb) 101 163 min (2.72 h)
6.8 (15 lb) 115 212 min (3.53 h)
9.07 (20 lb) 127 258 min (4.31 h)
11.33 (25 lb) 137 302 min (5.03 h)
Weight (kg)Effective radius (mm)Time taken to reach 85 °C at the center
3.63 (8 lb) 93.8 141 min (2.35 h)
4.54 (10 lb) 101 163 min (2.72 h)
6.8 (15 lb) 115 212 min (3.53 h)
9.07 (20 lb) 127 258 min (4.31 h)
11.33 (25 lb) 137 302 min (5.03 h)

We plot out the calculated time (in minutes) with the turkey’s weight (in pounds). It is displayed in blue dots in Fig. 3.

In Fig. 3, the dots are the calculated data. The dotted line is a power function fitting to the calculated data. It shows that the cooking time has a 0.667 (2/3) power relationship with the weight of the turkey, with a fitting equation $T(hours)=0.5862⋅W2/3=11.71W2/3$ (W is the weight in pounds), with the fitting R2 = 1.0.

Our fitting formula of the turkey baking time (in hours) in relation to the weight of the turkey (in pounds)

$T(hours)=11.71W2/3$
(4)

is very close to the Panofsky formula, which is $T(hours)=11.5W2/3$.22

The constant of 1/1.5 in the Panofsky equation or formula was determined empirically.22

However, our fitting formula is fitted to all theoretical calculations based on the approximation to treat turkeys as a spherical body from a simple heat transfer perspective. It is amazing that we could fit all the strictly calculated data so neatly with this simple formula but generalized the so-called Panofsky constant.

The above calculated times match well with the normal recommended turkey baking time (Table III) from, for example, http://www.csgnetwork.com/turkeycookingtimecalc.html. Our calculated times also match well with our own turkey baking times conducted in our own experiments.

TABLE III.

The cooking time vs weight for various sizes of turkeys suggested by www.csgnetwork.com.

Turkey weight (pound)Cooking time in a 325 °F oven (hour)Internal temperature (°F)
6–8 2.25–2.75 185
8–12 2.75–3.25 185
12–16 3.25–4.25 185
16–20 4.25–4.75 185
20–24 4.75–5.25 185
Turkey weight (pound)Cooking time in a 325 °F oven (hour)Internal temperature (°F)
6–8 2.25–2.75 185
8–12 2.75–3.25 185
12–16 3.25–4.25 185
16–20 4.25–4.75 185
20–24 4.75–5.25 185

Our rigorous calculation based on the baseline thermal parameters gives a Panofsky constant at 1.71, which is closer but larger than the 1.5.

In the following study, we generalize the Panofsky formula into

$T=1PW2/3,$
(5)

and we define P as the so-called Panofsky constant. For the standard Panofsky formula, the Panofsky constant is 1.5. For our above baseline case, P has a value of 1.71. In the following, we conduct a careful study of the Panofsky constant and its dependency on other parameters.

In Sec. II D, we indicate that the sphere is the most efficient geometry in heat transfer. With the same weight and density, the shape deviation from a perfect sphere would result in a slower rise (longer heating time) in the center temperature, i.e., a less efficient heating process. It means that the Panofsky constant, P, will be an upper limit when we approximately consider the turkey with the same weight as a perfect spherical shape. After considering the real shape, which is deviated from the sphere, we should re-write the Panofsky formula into

$T=1PrealW2/3=1∈⋅PsW2/3,$
(6)

where Preal is the Panofsky constant for real turkeys, Ps is the Panofsky constant with an ideal spherical shape, and ∈ is the shape correction factor. Since the sphere is the most efficient shape in heating, ∈ is always equal to or less than 1.0.

#### 1. The initial temperature of the turkey

The initial temperature of the turkey at 20 °C is our baseline. Obviously, if this temperature is different, then the calculated Panofsky parameter would change. The gray dots in Fig. 3 are for an initial temperature of 10 °C. It gives a Panofsky constant P = 1.62. Figure 4 shows the dependency of the calculated Panofsky constant vs the initial temperature.

#### 2. The oven temperature

The orange dots in Fig. 3 are for the oven temperature at 177 °C (350 °F). The calculated Panofsky constant is P = 1.82 with all the other parameters as used in the baseline case.

#### 3. The impact of thermal conductivity

We find that the Panofsky constant is very sensitive to the value of the thermal conductivity. With the thermal conductivity varying from 0.4 to 0.5 W/(m K), the Panofsky constant increases from 1.36 to 1.71, as shown in Fig. 5. All the other parameters are the same as the baseline cases. From the fitting in Fig. 5, it shows that the Panofsky constant is in a linear relationship with the thermal conductivity, K. This result agrees well with Eq. (3), which is derived with an approximation from Eq. (1).

#### 4. The impact of specific heat

We find that the Panofsky constant also is very sensitive to the value of the specific heat. With the specific heat varying from 2800 to 3980 J/(kg K), the Panofsky constant decreases from 1.93 to 1.36, as shown in Fig. 6. In Fig. 6, all the other parameters are the same as in the baseline cases except that the thermal conductivity of 0.45 W/(m K) rather than 0.5 W/(m K) is used. From the fitting in Fig. 6, it shows that the Panofsky constant is in an inverse relationship with the specific heat c. This result also agrees well with Eq. (3) that is derived with an approximation from Eq. (1).

#### 5. The impact of density

The dependency of the Panofsky constant on the density is more complicated than what it seems to be. From Eq. (3), it seems that we should find a power of (−1/3) dependency between the Panofsky constant K and the density ρ. Our calculation results shown in Fig. 7 show that the Panofsky constant decreases from 1.56 to 1.51 with the increase in the density from 1000 to 1100 kg/m3. However, the dependency is not in a power of (−1/3) relation based on the fitting in Fig. 7. The reason for that is that the change in the density also changes the weight if the dimension of the turkey does not change. Or if the weight is kept the same, then the dimension would change.

#### 6. The rate of cooking time per pound

All the turkey baking recipes prior to the Panofsky formula relied on a simple rate number, which is the cooking time (in minutes) per pound of the turkey. In the following, we look at this rate in more detail based on our numerical calculation.

Figure 8 shows a set of calculated results with the following parameters: 1050 kg/m3 for density, 3600 J/(kg K) for the specific heat, 0.45 W/(m K) for the thermal conductivity, 163 °C (325 °F) for the oven temperature, 20 °C for the initial turkey temperature, and 85 °C for ending center temperature. Figure 8(a) shows the calculated cooking time as a function of the weight, which is in a perfect Panofsky formula. The fitting gives a Panofsky constant of 1.5. Figure 8(b) is the calculated average cooking time per pound. It shows that the average cooking time per pound is at 25 min/lb for a 4-lb turkey and then goes down to around 15 min/lb for a 20-lb turkey and then further goes down to about 13 min/lb for a 30-lb turkey. Those values are in line with all the numbers recommended in the existing recipes. Figure 8(c) is the calculated instant rate of cooking time per pound, that is, the slope of the curve in Fig. 8(a). It says that for a turkey of 5 lb, its cooking time increases about 14.5 min for a 1-lb increase in weight. This rate is about 10 min/lb for a 20-lb turkey.

Since all foods contain a significant amount of water and some fat, naturally, during a high temperature process, the cooking process leads to cooking loss in weight and food shrinkage as a result.

Depending on food types and cooking methods, the cooking process resulted weight loss or food shrinkage happen at different rates. An oven-baked turkey also shrinks and loses much less than an openly roasted Peking duck. For a turkey baked by a typical closed oven method, the weight loss is not significant.52 Thus, we did not consider the impact of weight loss and shrinkage issue.

Back in 2008, Panofsky gave an empirical formula, $T=11.5W2/3$, for turkey baking time T in hours vs turkey weight W in pounds, the so-called Panofsky formula or Panofsky constant. Compared to the previously existed recipes that are based on the simple linear relationship between turkey weight and baking time, the Panofsky formula provides a more accurate estimate for the baking time. For instance, a general guideline of 13–20 min/lb was widely recommended in all previous turkey baking recipes. In this work, we conduct a comprehensive study of the turkey baking process that leads to a mathematical derivation of the Panofsky formula under some approximations. We also generalize the Panofsky formula to define a general Panofsky formula, $T=1PW2/3$, where P is defined as the Panofsky constant. Under the spherical approximations, we then apply an accurate physical solution of the heat transfer equation and use the rigorous solution with numerical methods to study the generalized Panofsky formula and the Panofsky constant. We found that the generalized Panofsky formula can be perfectly applied to all turkey baking scenarios for baking time calculations. Furthermore, we did a careful analysis of the Panofsky constant, which equals 1.5 in the original Panofsky formula. The dependency of the new Panofsky constant on thermal properties of the turkeys and other initial parameters of baking, e.g., initial and final center temperature of the turkeys, oven temperature, thermal conductivity, specific heat, and turkey’s density, was carefully analyzed and mapped out. The Panofsky constant, P, could vary from 1.1 to 1.9 depending on these thermal parameters.

L. R. Wang and Y. J. Jin would like to acknowledge both The Pennington School and Phillips Academy, Andover, for their support throughout this study.

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
M.
Nathan
,
C.
Young
, and
M.
Bilet
,
Modernist Cuisine: The Art and Science of Cooking
, 1st ed. (
The Cooking Lab
,
2011
), ISBN: 978-0-9827610-0-7, Vol. 1.
2.
J.
Potter
,
Cooking for Geeks: Real Science, Great Hacks, and Good Food
(
O'Reilly Media
,
2010
), ISBN-13: 978-1491928059.
3.
J. J.
Provost
,
K. L.
Colabroy
,
B. S.
Kelly
, and
M. A.
Wallert
,
The Science of Cooking: Understanding the Biology and Chemistry Behind Food and Cooking
, 1st ed. (
Wiley
,
2016
), ISBN-13: 978-1118674208.
4.
E.
Yin-Fei Lo
,
Mastering the Art of Chinese Cooking
(
Chronicle Books
,
2009
).
5.
P.
Barham
, “
Physics in the kitchen
,”
Flavour
2
,
5
(
2013
).
6.
A. C.
Rowat
,
N. N.
Sinha
et al, “
The kitchen as a physics classroom
,”
Phys. Educ.
49
(
5
),
512
(
2014
).
7.
J. M.
Aguilera
, “
Relating food engineering to cooking and gastronomy
,” in
Comprehensive Reviews in Food Science and Food Safety 1021
(
Wiley
,
2018
), Vol. 17.
8.
E. A.
Olszewski
, “
From baking a cake to solving the diffusion equation
,”
Am. J. Phys.
74
,
502
(
2006
).
9.
P. B.
Pathare
and
A. P.
Roskilly
, “
Quality and energy evaluation in meat cooking
,”
Food Eng. Rev.
8
,
435
447
(
2016
).
10.
N.
Huda
,
A. A.
Putra
, and
R.
, “
Proximate and physicochemical properties of Peking and Muscovy duck breasts and thighs for further processing
,”
J. Food Agric. Environ.
9
(
1
),
82
88
(
2011
).
11.
H.
Nelson
,
S.
Deyo
,
S.
Granzier-Nakajima
,
P.
Puente
,
K.
Tully
, and
J.
Webb
, “
A mathematical model for meat cooking
,”
Eur. Phys. J. Plus
135
,
322
(
2020
).
12.
N.
Huda
and
R.
, “
Proximate and physicochemical properties of Peking and Muscovy duck breasts and thighs for further processing
,”
J. Food Agric. Environ.
9
(
1
),
82
88
(
2011
).
13.
S.
Pappas
, How to cook the perfect steak (with science), https://www.livescience.com/44852-cook-perfect-steak-science.html.
15.
A.
Varlamov
et al, “
The physics of baking good pizza
,”
Phys. Educ.
53
,
065011
(
2018
).
16.
A.
Varlamov
,
Z.
Zheng
, and
Y.
Chen
, “
Boiling, steaming or rinsing? (the physics of Chinese cuisine)
,”
Phys. Econ.
1
,
55
68
(
2018
).
17.
J.
Unsworth
and
F. J.
Duarte
, “
Heat diffusion in a solid sphere and Fourier theory: An elementary practical example
,”
Am. J. Phys.
47
,
981
(
1979
).
18.
D.
Buay
,
S. K.
Foong
,
D.
Kiang
,
L.
Kuppan
, and
V. H.
Liew
, “
How long does it take to boil an egg? Revisited
,”
Eur. J. Phys.
27
,
119
131
(
2006
).
19.
P.
Roura
,
J.
Fort
, and
J.
Saurina
, “
How long does it take to boil an egg? A simple approach to the energy transfer equation
,”
Eur. J. Phys.
21
,
95
100
(
2000
).
20.
C. D. H.
Williams
,
The Science of Boiling an Egg
(
University of Exeter
,
2002
).
21.
H. C.
Chang
,
J. A.
Carpenter
, and
R. T.
Toledo
, “
Modeling heat transfer during oven roasting of unstuffed turkeys
,”
J. Food Sci.
63
(
2
),
257
(
2008
).
22.
See https://today.slac.stanford.edu/a/2008/11-26.htm for The Panofsky Turkey Constant.
23.

https://www.allrecipes.com/article/turkey-cooking-time-guide/ [20 min/lb general rule, 350 °F (175 °C) oven, 10–18 lb for 3–3.5 h, 18–22 lb for 3.5–4 h, 22–24 lb for 4–4.5 h, and 24–29 lb for 4.5–5 h for unstuffed turkeys, and add 0.75–1.25 for stuffed turkeys. Interior temperature is 75 °C].

24.

https://www.southernliving.com/thanksgiving/turkey-cooking-time-per-pound (325 °F for oven, 8–12 lb, 2.75–3 h, 12–14 lb, 3–3-3.75 h, 14–18 lb, 3.75–4.25 h, 18–20 lb, 4.25–4.5 h, and 20–24 lb, 4.5–5 h for unstuffed turkeys).

25.

https://www.delish.com/holiday-recipes/thanksgiving/a44899/how-long-cook-turkey/ (325 °F oven, 8 lb, 2.75 h, 10 lb, 2.9 h, 12 lb, 3 h, 14 lb, 3.75 h, 16 lb, 4 h, 18 lb, 4.25 h, 20 lb, 4.5 h, 22 lb, 4.75 h, and 24 lb, 5 h for unstuffed turkeys. For stuffed one, add more time).

26.

https://www.huffpost.com/entry/how-long-to-cook-a-turkey-per-pound_n_56566cc5e4b072e9d1c1c469 (USDA minimal 165 °F for interior, 325 °F oven, 8–12 lb, 2.75–3 h, 12–14 lb, 3–3.75 h, 14–18 lb, 3.75–4.25 h, 18–20 lb, 4.25–4.5 h, and 20–24 lb, 4.5–5 h for unstuffed turkeys).

27.

https://www.epicurious.com/holidays-events/the-easiest-way-to-cook-turkey-article (12–14 lb, 350 °F for 2.75–3 h; 325 °F for 3–3.75 h; 15–16 lb, 350 °F for 3.5–3.75 h, 325 °F for 3.75–4 h; 18–20 lb, 350 °F for 4–4.5 h, 325 °F for 4.25–4.5 h; 21–22 lb, 350 °F for 4.5–4.75 h, 325 °F for 4.75–5 h; 24 lb, 350 °F for 4.75–5 h, and 325 °F for 5–5.25 h).

28.

https://www.bhg.com/recipes/how-to/handling-meat/how-long-to-cook-a-turkey/ (325 °F. 8–12 lb, 2.75–3 h; 12–14 lb, 3–3.75 h; 14–18 lb, 3.75–4.25 h; 18–20 lb, 4.25–4.5 h; and 20–24 lb, 4.5–5 h).

29.

https://dinersjournal.blogs.nytimes.com/2011/11/24/how-long-should-i-cook-my-turkey/ (325 °F oven, unstuffed, 15 min/lb. 350 °F oven, 13 min/lb).

30.

https://parade.com/236361/smccook/how-long-to-cook-the-perfect-turkey-and-how-to-know-when-a-turkey-is-done/ (325 °F oven, unstuffed, 8–12 lb, 2.75–3 h; 12–14 lb, 3–3.75 h; 14–18 lb, 3.75–4.25 h; 18–20 lb, 4.25–4.5 h; and 20–24 lb, 4.5–5 h).

33.
Y. A.
Cengel
,
R. H.
Turner
, and
J. M.
Cimbala
,
Fundamentals of Thermal-Fluid Sciences
, 3rd ed. (
McGraw-Hill
,
2008
).
34.
J. C.
Polking
, “
An introduction to partial differential equations in the undergraduate curriculum
,” LECTURE 12 Heat Transfer in the Ball, Lecture note (Harvey Mudd College,
2021
).
36.
M. J.
Hancock
, Solutions to problems for 2D & 3D heat and wave equations, linear partial differential equations, https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/assignments/probpde3dsolns.pdf.
37.
T.
Jan
and
P.
Ocłoń
, “
Transient heat conduction in sphere
,” in
Encyclopedia of Thermal Stresses
(
Springer
, Dordrecht,
2014
), pp.
6186
6198
.
39.
ASHRAE
,
2006 ASHRAE Handbook: Refrigeration
, SI ed. (International System of Units) (
American Society of Heating
,
2006
), ISBN 13: 9781931862875.
40.

Composition data from USDA (1996). Initial freezing point data from Table I in Chap. 30 of the 1993 ASHRAE Handbook—Fundamentals. Specific heats calculated from equations in this chapter. Latent heat of fusion obtained by multiplying water content expressed in decimal form by 334 kJ/kg, the heat of fusion of water (Table I in Chap. 30 of the 1993 ASHRAE Handbook—Fundamentals).

41.
M. K.
Krokida
,
P. A.
Michailidis
,
Z. B.
Maroulis
, and
G. D.
Saravacos
, “
Literature data of thermal conductivity of foodstuffs
,”
Int. J. Food Prop.
5
(
1
),
63
111
(
2002
).
42.
W. E. L.
Spiess
,
E.
Walz
,
P.
,
M.
Morley
,
I. A.
van Haneghem
, and
D. R.
Salmon
, “
Thermal conductivity of food materials at elevated temperatures
,”
High Temp. - High Pressures
33
(
6
),
693
697
(
2001
).
43.
B. R.
Becker
and
B. A.
Fricke
, “
Thermal properties of foods
,” in
Encyclopedia of Food Sciences and Nutrition
, 2nd ed. (
Elsevier Science
,
2003
).
44.
Z.
Berk
, “
Physical properties of food materials
,” in
Food Process Engineering and Technology
, Food Science and Technology (
Elsevier Science
,
2018
), Chap. 1.
45.
P. J.
Fellows
,
Food Processing Technology: Principles and Practice
, 3rd ed., A Volume in Woodhead Publishing Series in Food Science, Technology and Nutrition (
,
2009
).
46.
M.
Marcotte
,
A.
Taherian
, and
Y.
Karimi
, “
Thermophysical properties of processed meat and poultry products
,”
J. Food Eng.
88
,
315
(
2008
).
47.
S.
Tavman
,
S.
Kumcuoglu
, and
V.
Gaukel
, “
Apparent specific heat capacity of chilled and frozen meat products
,”
Int. J. Food Prop.
10
(
1
),
103
112
(
2007
).
48.
B.
Karunakar
,
S. K.
Mishra
, and
S.
, “
Specific heat and thermal conductivity of shrimp meat
,”
J. Food Eng.
37
,
345
351
(
1998
).
49.
N. N.
Mohsenin
,
Thermal Properties of Foods and Agricultural Materials
(
Gordon and Breach Science Publishers
,
New York
,
1980
).
50.
M. E.
Shmalko
,
R. O.
Morawicki
, and
L. A.
Ramallo
, “
Simultaneous determination of specific heat and thermal conductivity using the finite-difference method
,”
J. Food Eng.
31
,
531
540
(
1996
).
51.
V. E.
Sweat
, “
Thermal properties of foods
,” in
Engineering Properties of Foods
, edited by
M. A.
Rao
,
S. S. H.
Rizvi
, and
M.
Dekker
(
CRC Press
,
Basel, NY
,
1995
), pp.
99
138
.
52.
USDA table of cooking yields for meat and poultry, 2012, http://www.ars.usda.gov/nutrientdata.