Heat Transfer, Convective and Conductive Rates

[chriskay301]

Homework Statement

A 1.0 cm diameter steel rod with k = 20W/(m-K) is 20 cm long. It has one end maintained at 35°C and the other at 100°C. It is exposed to convection heat transfer with h=65W/(m^2-K) and an ambient air stream at 20°C. a) Sketch the distribution of temp within the rod. b) Determine the rate of convective heat transfer from the rod. c) What are the conductive heat transfer rates at each end of the rod?

Homework Equations

Rate of Convection = (h)(Area)(T-T∞)
Rate of Conduction = -k(Area)(dT/dx)

The Attempt at a Solution

I'm not really sure how to attempt this problem. I know that using an energy balance E(in)=E(out) I could solve for a diff. eq. for the temperature anywhere in the rod. But I'm not sure how that would be useful. For conduction, if its at the end of the rod how is there a change in x direction? And for convection, I'm not sure what temperature I would use for T to find the rate since the entire rod is going to be at different temperatures.

 

[Chestermiller]

Hi chriskay301. Welcome to physics forums!

You are going to be assuming that the temperature within the rod is a function only of x, where x is the distance from the 35C end. The idea is to first find this temperature profile. Do a heat balance on the section of rod between x and x + dx. In terms of the cross sectional area A and the thermal conductivity k, what is the rate of heat flow by conduction into this control volume at x? What is the rate of heat flow by conduction out of this control volume at x + dx? What is the rate of heat flow by convection out of the control volume (in terms of the perimeter P= πD)? Combine these into your differential heat balance. You should have a second order ODE that you can solve subject to the temperature boundary conditions at the two ends. Solve this set of equations for the temperature as a function of x. Now you have what you need to answer the rest of the questions.

Chet

 

[chriskay301]

Okay, so I got a temperature distribution equation in respect to my left and right boundary conditions. I'm still not entirely sure what to do from here.

I know I can use this equation to get the temperature of any x, but how does that help me for convection or conduction? Do I just use the center of the rod for the convection temperature? Do I use a point such as x=.1 and use that to determine the conduction at the left end?

Still a bit confused!

 

[Chestermiller]

Okay, so I got a temperature distribution equation in respect to my left and right boundary conditions. I'm still not entirely sure what to do from here.

I know I can use this equation to get the temperature of any x, but how does that help me for convection or conduction? Do I just use the center of the rod for the convection temperature? Do I use a point such as x=.1 and use that to determine the conduction at the left end?

Still a bit confused!

The rate of convective heat transfer from the rod is equal to the net rate of conductive heat transfer into the rod at its two ends. These are determined by the temperature gradient at the ends times the thermal conductivity times the cross sectional area. You can also get the rate of convective heat transfer another way, by integrating the local rate of convective heat transfer with respect to x over the surface of the rod. Both these methods will give you the same answer.
For part (c), you already calculated the heat transfer rates at the ends in part (b).