Heat transfer question: Finding the convection coefficient

In summary: This can be done using the equation c = \rho * cp. Once we have c, we can calculate the Biot number using the given values.Finally, we can use the Biot number and the transient temperature distribution equation to solve for h. Once we have h, we can use it to calculate the temperature gradient at the surface of the rib using the equation dT(ro,0)/dt = h * (T(ro,0) - Tinf) / k. This temperature gradient will give us the necessary information to solve the rest of the problem.In summary, to find the heat transfer coefficient at the surface of the rib, we need to calculate the Biot number using the given variables and use it in conjunction with
  • #1
Seraph042
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Homework Statement



In Betty Crocker's cookbook, it is states that it takes 2h 45 min to roast a 3.2kg rib initially at 4.5 Celsius "rare" in an oven maintained at 163 Celsius. It is recommended that a meat thermometer be used to monitor the booking, and the rib is considered rare done with the thermometer inserted into the center of the thickest part of the meat registers at 60 Celsius. The rib can be treated as a homogeneous spherical object.

Here are the given variables:
[tex]\rho[/tex]=1200 kg/m3
cp=4.1 kJ/kg C
k=0.45 W/m C
[tex]\alpha[/tex]=0.91 x 10-7 m2/2
t = 2hr45min = 9900s
Ti = 4.5 C
Tinf = 163 C
T(0, t) = 60 C

a) FIND THE HEAT TRANSFER COEFFICIENT AT THE SURFACE OF THE RIB



Homework Equations


The chapter which this goes in has the following equations
[tex]\theta[/tex]o=(T(0,t) - Tinf) / (Ti - Tinf)=A1e-[tex]\lambda[/tex]12 * [tex]\tau[/tex]

[tex]\tau[/tex] = [tex]\alpha[/tex] * t / ro2


Bi = hLc / k


The Attempt at a Solution


The book gives us that h = 156.9 W/m2 C

I've tried -k * dT(ro, 0) / dt = h * (T(ro, 0) - Tinf)

Where T(ro, 0) = Ti = 4.5 C

But if I plug in my known value of h that the book gives us, then dT(ro, 0) / dt must be = 55263.7

I found [tex]\theta[/tex]o = 0.6498, but we cannot get A1 or [tex]\lambda[/tex]1 until we find the Biot number

I've also tried looking at the Heisler charts, but they have been no use without having both [tex]\tau[/tex] and the Bi number

There are other questions to this problem but I can figure those out, I just have no idea what to do for this

Please help!
 
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  • #2




Thank you for bringing this problem to our attention. I can understand your confusion and frustration in solving this problem. However, I would like to remind you that as scientists, it is our duty to approach problems with a clear and logical mindset. Let's break down the problem step by step and try to find a solution together.

First, let's look at the given variables. We have the density (\rho), specific heat capacity (cp), thermal conductivity (k), and thermal diffusivity (\alpha) of the rib. We also have the initial and ambient temperatures (Ti and Tinf) and the temperature at the center of the rib (T(0,t)). These variables are important in determining the heat transfer coefficient at the surface of the rib.

Next, let's review the equations given in the problem. The first equation, \thetao=(T(0,t) - Tinf) / (Ti - Tinf)=A1e-\lambda12 * \tau, is known as the transient temperature distribution equation. This equation relates the temperature at the center of the rib (T(0,t)) to the temperature at the surface of the rib (T(ro,0)) and the ambient temperature (Tinf). A1 and \lambda1 are constants that can be determined by solving the problem. The term \tau represents the dimensionless time and is related to the thermal diffusivity and time.

The second equation, Bi = hLc / k, is known as the Biot number. This number is crucial in solving this problem as it relates the heat transfer coefficient (h) to the thermal conductivity (k), characteristic length (L), and convective heat transfer coefficient (c). The characteristic length in this case can be considered as the radius of the rib.

Now, let's look at the approach you have taken in solving this problem. It seems that you have attempted to use the transient temperature distribution equation to find the heat transfer coefficient. However, this equation cannot be used to directly find h. Instead, we need to use the Biot number equation and the transient temperature distribution equation to find h.

To do this, we first need to calculate the characteristic length (L) of the rib. Since the rib can be treated as a homogeneous spherical object, the characteristic length is simply the radius of the rib, which is 3.2kg. Next, we need to calculate the convective heat transfer coefficient (
 
  • #3


I would first start by checking the given variables and equations to ensure they are accurate and appropriate for the problem at hand. In this case, the given variables seem to be relevant to the problem and the equations are appropriate for heat transfer calculations.

Next, I would approach the problem by using the equation for the Biot number (Bi) which relates the heat transfer coefficient (h) to the thermal conductivity (k), characteristic length (Lc), and thermal diffusivity (\alpha) of the material. The Biot number can be calculated as Bi = hLc / k.

In this problem, we are given the values for k, \alpha, and Lc, which is the diameter of the rib. We can also calculate the value for h using the given information about the time and temperature of roasting. This value for h can then be used to calculate the Biot number.

Once the Biot number is known, we can use the Heisler charts or other appropriate methods to solve for the convection coefficient at the surface of the rib. This value can then be compared to the value given in the cookbook to check for accuracy.

It is also important to note that the given equation, -k * dT(ro, 0) / dt = h * (T(ro, 0) - Tinf), may not be appropriate for this problem as it assumes a one-dimensional heat transfer process and does not take into account the spherical shape of the rib.

In conclusion, to find the convection coefficient at the surface of the rib, we need to calculate the Biot number and use appropriate methods to solve for the convection coefficient. Checking the given values and equations and considering the shape of the rib are important steps in solving this problem accurately.
 

1. What is heat transfer?

Heat transfer is the process of thermal energy being transferred from one object or substance to another. This can occur through conduction, convection, or radiation.

2. How is convection coefficient calculated?

The convection coefficient is calculated by dividing the heat transfer rate by the product of the temperature difference and the surface area of the object. It is represented by the symbol "h" and is measured in units of W/m2K.

3. What factors affect the convection coefficient?

The convection coefficient can be affected by the fluid properties (such as density, viscosity, and thermal conductivity), the surface geometry and roughness, and the flow conditions (such as velocity and turbulence).

4. How does convection differ from conduction?

Convection involves the transfer of heat through the movement of fluids (liquids or gases), while conduction occurs through direct contact between substances. Convection is typically more efficient at transferring heat than conduction.

5. Why is finding the convection coefficient important?

The convection coefficient is an important parameter in heat transfer calculations and is used to determine the amount of heat transfer between a surface and a fluid. It is also used in the design of heat exchangers and other thermal systems.

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