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A differential equation is a mathematical equation that relates a function to its derivatives. In the context of heat transfer, it helps us understand how the temperature of an object changes over time. For this dishwasher scenario, the plate's temperature changes over time due to heat exchange with its surroundings. To derive the differential equation for our plate, we start by figuring out how much heat is being transferred. The heat transfer from the plate is captured by the equation:\[ Q = h A_s (T_p - T_x) \]Here, \( Q \) is the heat transferred, \( h \) is the heat transfer coefficient, \( A_s \) is the plate's surface area, \( T_p \) is the plate's temperature, and \( T_x \) is the temperature of the surrounding air.The rate of change of heat in terms of temperature for the plate is given by:\[ \frac{dQ}{dt} = M c \frac{dT_p}{dt} \]Where \( M \) is the plate's mass, \( c \) is its specific heat, and \( \frac{dT_p}{dt} \) is the rate of temperature change with time.By combining these equations, we obtain the differential equation describing how the plate's temperature changes:\[ \frac{dT_p}{dt} = \frac{h A_s}{M c} (T_p - T_x) \]This equation will help us predict the temperature timeline of the plate during its drying cycle.

The temperature-time history of an object tells us how its temperature changes over a period. For our dishwasher plate, this means tracking its temperature as it dries. Initially, the plate has a temperature of \( 65^{\circ} \mathrm{C} \), and the environment (saturated air) is at \( 55^{\circ} \mathrm{C} \). As time progresses, heat is exchanged between the plate and the surrounding environment, resulting in a changing temperature on the plate. The rate of this change is captured by the differential equation we previously derived:\[ \frac{dT_p}{dt} = \frac{h A_s}{M c} (T_p - T_x) \]This equation is crucial because it tells us how the rate of temperature change (\( \frac{dT_p}{dt} \)) evolves over time as a response to the heat exchange with the surroundings. Understanding the temperature-time history can help in predicting how fast the plate will cool down and when it might reach a steady state, making it crucial for designing efficient drying processes in applications like dishwashers.

The heat transfer coefficient \( h \) is a parameter that quantifies how easily heat is transferred between a solid surface and a fluid in contact with it. In this case, it tells us how effectively heat passes from the plate to the air and vice versa.In our scenario, we're given that the heat transfer coefficient is \( 3.5 \, \mathrm{W/m^2 \, K} \). This means for every square meter of surface area, and for every degree of temperature difference between the plate and the air, 3.5 watts of heat can be transferred.This coefficient is crucial in our differential equation for calculating how quickly the plate's temperature changes:\[ \frac{dT_p}{dt} = \frac{h A_s}{M c} (T_p - T_x) \]A higher heat transfer coefficient indicates a more efficient heat exchange, resulting in a faster change in temperature over time. It is influenced by factors like the properties of the plate material, the fluid's motion, and the characteristics of the surface involved.

Specific heat \( c \) is a property that describes how much heat energy is needed to change the temperature of a unit mass of a substance by one degree Celsius. In simple terms, it tells us how resistant a material is to changing its temperature.For the plate in our exercise, specific heat helps in defining how quickly it will heat up or cool down in response to the heat being transferred.The rate of temperature change of the plate, as expressed in the differential equation, depends inversely on specific heat:\[ \frac{dT_p}{dt} = \frac{h A_s}{M c} (T_p - T_x) \]This means if the plate has a high specific heat, it would require more heat to change its temperature, hence slowing the rate of temperature change. The specific heat helps in determining the thermal inertia of the plate.When designing systems that involve heating or cooling, knowing the specific heat is essential for predicting how an object's temperature will respond to various amounts of heat being lost or gained.