# Transient Energy Balance for Heating a Tank of Water

• hinjab
In summary, the conversation is about calculating the energy and rate of heat transfer required to heat a tank of water to a specific temperature, as well as the time it would take to reach that temperature using a transient energy balance and a closed form solution. The convective heat transfer coefficient and surface area are given, and the ambient air temperature is constant. The conversation also touches on solving a linear first order ordinary differential equation with constant coefficients.
hinjab

## Homework Statement

In an industrial process, a tank containing 200 liters of water must be heated from a temperature of 293 K to 372 K at a constant pressure. There is a negligible change in the volume of the water. The water is stirred during this process to maintain a uniformly distributed temperature. During the process, the convective heat transfer coefficient between the water and the ambient air is 8.5 W/(m2K) and the surface area equals 2.0 m2. The ambient air remains at a constant temperature of 293 K. Neglect the work associated with stirring the tank.

I was first told to find the energy required to heat the water to its final temperature assuming the tank is perfectly insulated and does not convectively lose energy. I got 66107 kJ and it is correct.

I think had to calculate the rate of heat transfer at the final temperature from the water, which is its steady state temperature. I got Qdot = 1343 W which is correct.

I now have to calculate the time to heat the water to within 0.985 of its final temperature assuming that the heat transfer calculated in part (b) is the heat input to the device. For this problem you need to use the transient form of the energy balance. Plot the temperature time history of the water. While this question can be answered using a numerical method, you can investigate it using a differential equation and a closed form solution.

## Homework Equations

I know that my transient energy balance is power in = Time derivative of the internal energy change + convective heat loss.

## The Attempt at a Solution

power in (1364) = mcdT/dt + hA ((T - Tair), T = Twater which is what I need to solve for. I know that my solution will be something like T = Cexp (-t) but I'm having such a hard time getting to this conclusion.

Hello Hinjab,

Since the ambient air temperature is constant, you can add (or subtract) its time derivative to your equation:$$\dot Q = mc {d\over dt}(T-T_{air}) + hA(T-T_{air} )$$Does that make it easier for you ?

i see

Last edited by a moderator:
hinjab said:
Hi - kind of but not really, can you go over what my next step would be ?
Are you saying that you don't know how to solve a linear first order ordinary differential equation with constant coefficients?

Chestermiller said:
Are you saying that you don't know how to solve a linear first order ordinary differential equation with constant coefficients?
sorry read it wrong- got it thanks

## 1. How does the transient energy balance equation for heating a tank of water differ from the steady-state equation?

The transient energy balance equation for heating a tank of water takes into account the change in temperature over time, while the steady-state equation assumes a constant temperature. This means that the transient equation includes a term for the rate of change of temperature, while the steady-state equation does not.

## 2. What factors affect the transient energy balance for heating a tank of water?

The factors that affect the transient energy balance for heating a tank of water include the initial temperature of the water, the temperature of the heating source, the surface area of the tank, the thermal conductivity of the tank material, and the rate of heat transfer.

## 3. How does the size of the tank affect the transient energy balance?

The size of the tank affects the transient energy balance by changing the surface area-to-volume ratio. A larger tank will have a smaller surface area-to-volume ratio, meaning that it will take longer to heat up and will experience less heat loss to the surrounding environment.

## 4. Can the transient energy balance equation be applied to other heating systems besides tanks of water?

Yes, the transient energy balance equation can be applied to any system that involves heating a material or substance over time. This includes heating systems for buildings, industrial processes, and even food preparation.

## 5. How can the transient energy balance equation be used in practical applications?

The transient energy balance equation can be used to determine the time it will take to heat a tank of water to a desired temperature, the amount of heat transfer needed to heat the water, and the efficiency of the heating system. It can also be used to optimize heating processes and improve energy efficiency.

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