By now we are all familiar with how the classroom temperature increases between

Question

By now we are all familiar with how the classroom temperature increases between the time we start the lecture and when we end. We would like to have a simple model to describe how temperature changes with time due to the metabolic heat generated by us. Consider the walls, including windows, to be perfectly insulating, i.e., no heat flow. The room is well-mixed (i.e. no spatial gradient in the room and the room is at one temperature). Cold air comes in and leaves at the room temperature. 1) Perform an energy balance for the room in terms of incoming colder air temperature Tin, air flow rate, number of students N, metabolic heat generation per person Q, air properties and room dimensions from which you can calculate the room temperature, T. 2) Solve for room temperature as function of time. 3) What is the room temperature at the end of a 45 minute lecture? 4) Comment on how accurate your prediction is and discuss the major assumptions made here that would influence your prediction. The door dimensions are 1m × 2.5m, the dimensions of the room are 5m × 3m × 3m, and the average air velocity through the doors is 0.1 m/s. The metabolic heat generation per person is 60 W and the total number of persons is 80. The initial room temperature and incoming air temperature are both 25oC. The properties of air at 25oC are density of 1.1769 kg/m3 and specific heat of 1006 J/kg.K. Ignore temperature variation in the properties as the incoming air heats up in the room.

Explanation / Answer

1)    Incoming Cold Air =    Qair * Cp * Tin

      Outgoing Warm Air =    Qair * Cp * Troom

      Heat generation = N * Q

      Energy Balance    ( Troom is the room temperature, Vroom is the volume of room)

     Qair * Cp * Tin + N * Q - Qair * Cp * Troom = rho * Cp * DTroom / dt * Vroom

2)    DTroom / dt     +   Qair * Troom / rho * Vroom    = Qair * Tin / rho * Vroom + N * Q / (rho * Cp * Vroom)

       Troom   = (Qair*Tin/ rho*Vroom + N*Q / (rho*Cp*Vroom)) / (Qair / rho*Vroom) - exp(t) / (Qair / rho * Vroom)

4) Assumption : Constant specific heat capacities

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