Cooking Potato

Novi Wijaya (A0068355X)Yong Wai Fen (A0068349R)

Mathematical Modelling and Simulation for the Cooking of Potato

By early 1990s, the petroleum and petrochemical industries have had computer

simulators for their processes for years[1]. This means that the processes happening in

the industries are well-understood. The same cannot be said for the food industries; a

major difficulty with food systems is the lack of mathematical models to describe

mass and energy balances in the unit operations.

There had been many papers published in the field of food processing, particularly the

potato processing. Potato may have caused many interests because it is globally

consumed, cheap and easily available in various sizes. Potato have a water content of

about 80 wt%, a starch content of 16 wt% and cellulosic material content of 4 wt% [2].

Starch is a long chain of sugar molecules called a polysaccharide[3] and it composes of

two polymeric units, i.e. amylase (a linear polymer) and amylopectin (a highly-

branched polymer). The size of each amylopectin group is about 9 to 10 nm. The

amylose fraction exists naturally and is randomly interspersed among the amylopectin

molecules in both the amorphous and crystalline regions of starch. Thus, small

amylose molecules located at the periphery are free to leach out of the granule and

undergo gelatinization. Two obvious changes occur in potato tissue during cooking:

the starch is gelatinized which occurs at 65 0C and, owing to a marked decrease in cell

adhesion, the tissue is softened so that it can be readily mashed. These two alterations

are those prerequisite to the conversion of raw tissue into a product having the

physical qualities associated with cooked potato[4].

The cooking of potato involves heat transfer and change in its material content due to

gelatinization. In this assignment we will model the heat transfer and the

gelatinization in a potato, and simulate the cooking of a potato based on the model.

To be able to derive a first principle mathematical model, we made several

assumptions. The assumptions made are:

1 Kozempel, M., Craig, J. C., Jr., Sullivan, J. F. and Damert, W. Computer simulation of potato processing. Biotechnology Progress, 4, 63-67 (1988).2 Chen, X.D. Cooking Potatoes Experimentation and Mathematical Modeling. 2002.3 P. Barham. The science of cooking. 2001.4 Personius, C. J. and P. F. Sharp, 1939. Simulation, by chemical agents, of cooking of potato tissue, Food Res. 4. 469.


1. The potatoes have spherical shapes.

2. The potatoes are homogenous throughout.

3. Cooking corresponds to the gelatinization.

4. The gelatinization moves on a uniform front.

5. Little swelling occurs during the gelatinization process and we therefore

assumes a constant radius while cooking.

6. The boundary between gelatinized to ungelatinized starch is relatively small

compare to the size of the potato.

7. We conclude that the potato is cooked based on the ‘watermark’ dividing the

cooked and uncooked regions.

8. At temperature above 65oC, the gelatinization occurs at its maximum rate.

9. Since the gelatinization is very fast, the heat transfer becomes the limiting

step. Hence the rate of cooking is determined by the rate at which heat arrives

at the cooking interface.

10. Constant driving force for heat transfer.

11. All the heat conducted through the shell to the interface is consumed by the

interface cooking reaction.

For this simulation we will use a laptop computer with MATLAB, to solve the partial

differential equations in our model.

First Principle Model

The equation of energy in terms of q, for spherical systems is:


For one-dimensional, conductive heat transfer, however, the equation above can be

simplified into:

ρ C p∂ T∂ t

= 1

r2

∂∂ r

(r2qr )

Fourier’s law of heat conduction for one dimensional heat transfer in a spherical

coordinate:

qr=k∂ T∂ r

The equation of heat transfer can be written as:

ρ C p∂ T∂ t

= 1

r2

∂∂ r (r2 k

∂ T∂ r )

ρ C p∂ T∂ t

=k∂2 T∂ r2 + 2k

r∂T∂ r

The properties for potato is taken from Food Properties Handbook, and5, and is shown

in Table 1 below.

TABLE 1Properties of Potatoes

cp (J/kg.K) : 3500

k (W/m.K) : 0.624 + 1.19x10-3T (oC)

ρstarch (kg/m3) : 1500

It is also stated in literature that a potato consists of 20% starch and 80% water.

At the surface, convective heat transfer takes place:

k∂ T∂r

=h∗(T s−T ∞ )

Chen6 observed that the convection is strong at the boundary, hence the boundary

temperature can be assumed to be constant, i.e.:

T s=T ∞ at r=ro

The change in temperature will stop at the center, hence:

5 Rahman, S., Food Properties Handbook, CRC Press, Boca Raton, FL (1995).6 Chen, X. D. Cooking potatoes: Experimentation and mathematical modeling. Chemical Engineering Education, Winter 2002.


∂ T∂r

=0 at r=0

The initial condition for the potato is:

T (r )=T 0at t=0

We would assume that the initial temperature of the potato is 20oC.

According to Chen, the reacting interface temperature is 65oC. So when the

temperature of the potato reaches 65oC, the gelatinization occurs so fast that the rate

of heat transfer becomes the limiting rate.

Then we can simulate the heat profile with different cooking time.

We use pdepe in MATLAB to solve our partial differential equation for T.

Experiment


We next design an experiment that can be use to validate our model.

The objectives of the experiment are:

To quantify the gelatinization of a potato visually by cooking potato samples at

the same temperature for different intervals of time. The gelatinization front can

be distinguished as a watermark between cooked and uncooked potato (Figure 1).

However, this watermark may be hard to distinguish, so we propose the use of

iodine to mark the barrier between the gelatinized and the ungelatinized potato.

Iodine only stained starch purple, so only the ungelatinized potato will be colored

purple by the iodine.

We predicted that the gelatinization front will move closer to the center of the

potato when time increasing.

Figure 1 Watermark between the cooked and uncooked regions of the potato after being heated in water for few minutes

Materials used are:

Spherical potatoes with a radius of 25 mm

Iodine

Distilled water, to avoid mineral interaction with the potato

Latex gloves

Vernier calipers

Experiment procedures:


1. Fill an aluminium tank with 50 % of distilled water. Attach heating element at

both sides of the tank. Place small bucket into the tank and fill with distilled

water until the water level in the bucket is greater than the water level in the tank.

Turn on the heater and heat until 65oC (gelatinization start).

2. Place the potato samples into the bucket.

3. Remove potato samples at every interval of 10 minutes.

4. Slice the potato samples through the circumference.

5. Wear latex gloves and wet end of index finger with iodine. Rub the iodine onto

the potato cross-sections.

6. Place the potato for 3 minutes and rinse the potato with water.

7. Take pictures of the potato cross-section and compare the distance of the iodine

barrier mark between the different interval times. It will look like the illustration

in Figure 2.

Figure 2 Illustration of the partially cooked potato

Simulation


We will do several simulations to observe how our model behaves.

Several variations we will use are:

- The length of cooking time

- The radius of potato

- The temperature of water used to cook the potato

Several things that we will observe:

- The temperature profile of the potato

- The radius of uncooked potato

- The cooking time

Temperature plot for a potato with a radius of 2.5 cm and cooking time 15 minutes

(900 seconds)

Figure 3 is the contour plot of cooking time, potato radius, and potato temperature.

00.005

0.010.015

0.020.025

0

500

1000280

300

320

340

360

380

Distance from center (m)

The Cooking of Potato

Time (s)

Tem

pera

ture

(K

)

Figure 3 The contour plot of a simulation of the cooking of a potato

Various length of cooking time


Figure 4 shows how temperature changes throughout the potato, with different

cooking time.

0 0.005 0.01 0.015 0.02 0.025290

300

310

320

330

340

350

360

370

380

Distance from center(m)

Tem

pera

ture

(K

)Temperature profile at different time

t=0.2 mint=3 mint=5 mint=6 mint=10 mint=15 min

Figure 4 Temperature-distance profile at different cooking times (potato radius=2,5 cm and the initial temperature=20oC)

Because we assume that gelatinization occurs at 65oC, then we can conclude that a

potato is cooked once it reaches 65oC (or 338.15 K). Hence, from the data we

obtained from MATLAB, we can plot a graph of uncooked radius versus time (Figure

5).

0 1 2 3 4 5 6 7 8 9 100

0.005

0.01

0.015

0.02

0.025

0.03

Time (min)

Unco

oked

radi

us (m

)

Figure 5 Uncooked radius of the potato versus cooking time

Various radius of potato


Figure 6 shows how temperature changes with time, in potatoes of different sizes.

0 500 1000 1500 2000 2500290

300

310

320

330

340

350

360

370

380

Time (s)

Tem

pera

ture

(K

)Temperature profile at different radius

r=2.5 cmr=3 cm

r=3.5 cm

r=4 cm

r=4.5 cmr=5 cm

Figure 6 Temperature-time profile at different potato radius (cooking time=35 minutes and the initial temperature=20oC)

From the data we obtained from MATLAB, we can plot a graph of cooking time

versus the radius of potato (Figure 7).

2 4 6 8 10 12 14 160

1

2

3

4

5

6

Cooking time (min)

Radi

us o

f pot

ato

(cm

)

Figure 7 How the radius of potato influences cooking time

The temperature of the water used to cook potato


We used potatoes of radius 2.5 cm and cooking time is 10 minutes.

The water used to cook potato is set to be 60oC, 70oC, 80oC, 90oC and 100oC. The

temperature profile of the potato for each water temperature can be seen in Figure 8

below.

0 0.005 0.01 0.015 0.02 0.025310

320

330

340

350

360

370

380

Distance from center(m)

Tem

pera

ture

(K

)

Temperature profile at different time

T=100 C

T=90 C

T=80 CT=70 C

T=60 C

Figure 8 Temperature-distance profile at different cooking water temperature (potato radius=2,5 cm and the initial temperature=20oC)

From the simulations results, we can conclude several things:

1. To cook a potato with a radius of 2.5 cm in boiling water, we need about 9

minutes.

2. The time needed to cook a potato relates to the size of the potato; an increase in

the size of potato will increase the time needed to cook the potato, roughly

linearly.

3. To cook a potato with a radius of 2.5 cm in 10 minutes, the water temperature

must be 90oC or higher.

(Note: the first two simulations are written in pdepotato.m, while the last simulation

is written as pdepotatoinwater.m)


Conclusion/Insight of the Problems

We tried to derive a first principle mathematical model of potato cooking process. We

found the thermal properties of potato (i.e. thermal conductivity and the specific heat

capacity) and the physical properties of potato (composition and density) from papers

and textbooks. Then we determine that the gelatinization temperature is 65oC, based

on literature. From there we can calculate the cooking time of a potato, also the radius

of gelatinization as a function of time.

We assume that the gelatinization front moved through potato in a uniform way. The

longer the cooking time, the faster the gelatinization occurs and the farther the

gelatinization front move from the boundary towards the center of the potato. This

logic agrees with our model and simulations. So we can say that to some extent, our

model successfully represents the physical phenomenon that happens inside a potato

when it is cooked.

In our mathematical model, there is no free convection term; free convection term at

high temperature can greatly influence the conduction of heat into a submerged

sphere. Convection within a fluid increases at high temperature. This proportionally

increases heat transfer in the potato.

Our mathematical model can simulate the heat transfer due to convection. However,

to get a more accurate representation of the phenomenon of starch gelatinization

inside a potato, other variables need to be taken into account, such as water absorption

and the hardness of the potato. The change of the hardness should be studied as well,

as this is the result of starch gelatinization in excess water. In addition, our experiment

procedure should able to define a good visual guide for the amount of starch

gelatinization that occurs. One of them is, in order to have a better accuracy, the

amount of the iodine used should be quantified.

In our project, we use pdepe in MATLAB to solve our mathematical model. Other

than pdepe, we can also use the long and gruesome finite differential method such as

Euler method. In the finite differential method, the sample is subdivided into elements

of arbitrary size and shape. The temperature field at a given time in each element is

expressed as a function of the temperatures at a small number of nodes at the corners

and edges of the element. Spatial discretization of the heat transfer equation will give


a system of ordinary differential equations. From this system of ordinary differential

equations, we can calculate the unknown temperatures at the nodes. A simple

representation can be seen at Figure 9

Figure 9 Finite element grid of the potato sample to calculate starch gelatinization process.

Cooking Potato

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