# ODE, step change, Heat transfer

• Dynamo
In summary, the author discusses how to solve an equation for delta T (change in temperature) due to a step change in source temperature. He then shows how to do this using the second equation. However, he can't seem to get it right himself, and would appreciate help.
Dynamo
Hi all.

I am doing some work with temperature equations. I have a book that gives an equation and then manipulates it. However, I can not follow what the author does, so can anyone help:

He starts with:

dT/dt = Q/MC - (T-O)/RMC

(in the following I use the text "delta" to represent lower case delta)

He then says he solves this for delta T (change in temperature), due to a step change in source temperature (delta O).

The equation then becomes:

delta T(t) = delta O (1- e^(-t/RMC))

I use this 2nd equation in my work, but just can't follow how the author jumps from the 1st eqn to this one.

I 'solve' the original ODE - but it comes no where near what he gets.

Any help would be great.

This is what I get:

For:

$$\frac{dT}{dt}=\frac{Q}{MC}-\frac{(T-O)}{RMC}$$

I get:

$$T(t)=RQ+O-Ke^{\frac{-t}{RMC}}$$

with K the integration factor.

Then the differential of T with respect to O is:

$$dT=\frac{dT}{dO}\Delta O$$

$$dT=\Delta O$$

See, not happening for me. Perhaps someone can help us.

Thanks for having a look.

Thats pretty much what I get..

From: $$\frac{dT}{dt} = \frac{Q}{MC} - \frac{1}{RMC} (T- \theta)$$

I then diff with respect to theta, and get what you get,

I can't understand how he, and I quote...

"This equation can be solved for the change in temperature $$\delta T$$ due to a change in the temperature of the medium $$\delta\theta$$.

The result for a unit step change in $$\theta$$ is:

$$\delta T(t) = \delta \theta (1-e^{\frac{-t}{RMC}})$$"

I just can't get this (although I know that the first equation is correct, and so is the final equation - the 2nd equaion I have used a lot in my work, and it is correct).

Anyone else got any ideas?

(PS just realized the form can use latex).

## What is an ODE?

An ODE (Ordinary Differential Equation) is a mathematical equation that describes the rate at which one or more variables change with respect to another variable, typically time. It is used to model various physical phenomena in fields such as physics, engineering, and chemistry.

## What is a step change?

A step change is a sudden and instantaneous change in the value of a variable. In the context of ODEs, it represents a sudden change in the rate at which a variable is changing, often due to a sudden external influence or input.

## How does heat transfer relate to ODEs?

Heat transfer is a process by which thermal energy is transferred from one object or medium to another. ODEs are often used to model heat transfer because they describe the rate of change of temperature with respect to time, which is a key factor in heat transfer.

## What are some common methods for solving ODEs?

There are several methods for solving ODEs, including analytical methods such as separation of variables and integrating factors, and numerical methods such as Euler's method and Runge-Kutta methods. The choice of method depends on the complexity of the ODE and the desired level of accuracy.

## How is heat transfer affected by step changes?

Step changes can have a significant impact on heat transfer. For example, a sudden increase in temperature can cause a rapid transfer of heat between two objects, while a sudden change in the surface area of an object can affect the rate of heat transfer through convection. ODEs can be used to model these changes and their effects on heat transfer.

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