The problem is the following:

*Processing Seafood*

Many foods, such as crabmeat, are sterilized by cooking. Harvested crabs are laden with bacteria, and the crabmeat must be steamed to reduce the bacteria population to an acceptable level. The longer the crabmeat is steamed, the lower the final bacteria count. But steaming forces moisture out of the meat, reducing the amount of crabmeat for sale. Excessive cooking also destroys taste and texture. The processor is therefore faced with a tradeoff when choosing an appropriate steaming time.

The basis for a choice of steaming time is the concept of "shelf life." After the steaming treatment is completed, the product is placed in a sterile package and refrigerated. Under refrigeration, the bacterial content in the meat slowly increases and eventually reaches a size where the crabmeat is no longer suitable for consumption. The time span during which packaged crabmeat is suitable for sale is called the shelflife of the product. We study the following problem: How long must the crabmeat be steamed to achieve a desired shelf life? The first step in modeling shelf life is to choose a model that describes the populati'bn dynamics of the bacteria. For simplicity, assume $\displaystyle \frac{dP}{dt} = k(T)P$ ( equation 1)

where $\displaystyle P(t)$ denotes the bacteria population at time t. In equation (1), $\displaystyle k(T) $represents the difference between birth and death rates per unit population per unit time. In this model k is not constant; it is a function of T, where T denotes the temperature of the crabmeat. [Note that $\displaystyle k(T)$ is ultimately a function of time, since the temperature T of the crabmeat varies with time in the steamer and in the refrigeration case.]

We need to choose a reasonabIe model for the bacteria growth rate, $\displaystyle k(T)$. We do so by reasoning as follows. At low temperatures (near freezing), the rate of growth of the bacteria population is slow; that's why we refrigerate foods. Mathematically, $\displaystyle k(t)$ is a relatively small positive quantity at those temperarures. As temperature increases, the bacterial growth rate, $\displaystyle k(T)$, first increases, with the most rapid rate of growth occurring near 90°F. Beyond this temperature, the growth rate begins to decrease. Beyond about 145°F, the death rate exceeds the birth rate and the bacteria population begins to decline. A simple model that captures this qualitative behavior is the quadratic function $\displaystyle k(T)=k_0+k_{1}(T-34)(140-T)$ ( equation 2)

where $\displaystyle k_0$ and $\displaystyle k_1$ are positive constants that are typically determined experimentally.

We also need a model that describes the thermal behavior of the crabmeat -how

the crabmeat temperature T varies in response to the temperature of the surroundings.

Assuming Newton's law of cooling, we have: $\displaystyle \frac{dT}{dt}= \eta[S(t)-T]$ ( equation 3)

In equation (3), $\displaystyle \eta$ is a positive constant and $\displaystyle S(t)$ is the temperature of the surroundings. Note that the surrounding temperature is not constant, since the crabmeat is initially in the steamer and then in the refrigeration case.

We now apply this model to a specific set of circumstances. Assume the following:(i) Initially the crabmeat is at room temperature (75°F) and contains about $\displaystyle 10^7$ bacteria per cubic centimeter.

(ii) The steam bath is maintained at a constant 2500°F temperature.

(iii) When the crabmeat is placed in the steam bath, it is observed that its temperature rises from 75°F to 200°F in 5 min.

(iv) When the crabmeat is kept at a constant 34°F temperature, the bacterial count in the crabmeat doubles in 60 hr.

(v) The bacterial count in the crabmeat begins to decline once the temperature exceeds 145°F, (see equation 2), $\displaystyle k(145)=0$

(vi) The bacterial count of $\displaystyle 10^5$ bacteria/cubic centimeter covers shelf life. Once this bacterial count is reached, the crabmean can no longer be offered for sale.

**Determine how long the crabmeat must be steamed to achieve a shelf life of 16 days**. Assume that the crabmeat goes directly from the 250°F steam bath to 34°F refrigeration case. Assume the 16-day shelf life requirement includes transit time to the point of sale, that is, assume that the measurement of shelf life begins the moment the crabmeat is removed from the steam bath.

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